Displacement theory and multiscale numerical

Displacement theory and multiscale numerical modeling of three-phase flow in porous media by Ruben Juanes

Engineering (University of La Coru˜ na, Spain) 1997 M.S. (University of California at Berkeley) 1999

A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Engineering — Civil and Environmental Engineering in the GRADUATE DIVISION of the UNIVERSITY OF CALIFORNIA, BERKELEY

Committee in charge: Tadeusz W. Patzek, Chair Francisco Armero Keith Miller Robert L. Taylor Spring 2003

The dissertation of Ruben Juanes is approved:

Chair

Date

Date

Date

Date

University of California, Berkeley

Spring 2003

Displacement theory and multiscale numerical modeling of three-phase flow in porous media

Copyright 2003 by Ruben Juanes

1 Abstract

Displacement theory and multiscale numerical modeling of three-phase flow in porous media by Ruben Juanes Doctor of Philosophy in Engineering — Civil and Environmental Engineering University of California, Berkeley Tadeusz W. Patzek, Chair

Three-phase flows in porous media occur in applications of high socioeconomic impact, such as enhanced oil recovery, and environmental remediation of the vadose zone. This investigation addresses some of the unresolved issues in the mathematical and numerical modeling of such flows. The traditional macroscopic description of three-phase flow relies on a multiphase extension of Darcy’s equation. When capillarity effects are neglected, the mathematical model leads to a 2 × 2 system of conservation laws —the saturation equations— whose character depends exclusively on the relative permeabilities. It is well known that widely-used relative permeability models lead to regions in the saturation space where the system is elliptic, rather than hyperbolic —the so-called elliptic regions. It was concluded in some investigations that elliptic regions are unavoidable when sufficiently general relative permeability functions are employed. In this dissertation, we show that this conclusion is not quite correct. We argue that elliptic regions are the artifacts of an incomplete mathematical

2 model, and that they are not physically plausible. The key element of our analysis is to understand relative permeabilities as functionals of the various fluid/rock descriptors, and not as fixed functions of saturations alone (or even saturation history). We derive conditions that the relative permeabilities must satisfy, so that the system is everywhere strictly hyperbolic. These conditions depend, in an essential way, on the fluid viscosity ratios and the gravity number. They are supported by the physics of multiphase displacements, and are also in good agreement with experimental data. After observing that an appropriate choice of the relative permeabilities leads to a strictly hyperbolic system, we derive the general analytical solution to the Riemann problem of three-phase flow. We present, for the first time, the complete catalogue of solutions that may arise, and conclude that the wave structure is restricted to only 9 solution types. In the second part of this dissertation, we develop stabilized finite element methods for the simulation of miscible, two-phase, and three-phase flows. The key idea of the formulation is a multiscale decomposition into resolved grid scales and unresolved subgrid scales. Incorporating the effect of the subgrid scales onto the coarse scale problem results in a method with enhanced stability, and not overly diffusive. The multiscale formalism, which is now dominant in fluid mechanics, is adopted and extended here for the simulation of three-phase flows. We illustrate the performance of the method with representative examples, which demonstrate its great potential for the numerical solution of complex multiphase compositional flows.

Tadeusz W. Patzek Dissertation Committee Chair

i

To my parents, Ovidio and Mar´ıa

ii

Contents List of Figures

vi

List of Tables

xi

1 Introduction 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 The need for revisiting the mathematical description 1.1.2 The need for consistent relative permeabilities . . . . 1.1.3 The need for a general analytical solution . . . . . . . 1.1.4 The need for stabilized numerical methods . . . . . . 1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Review of displacement theory . . . . . . . . . . . . . . . . . 1.3.1 Displacement theory . . . . . . . . . . . . . . . . . . 1.3.2 Relative permeability experiments . . . . . . . . . . . 1.3.3 Relative permeability models . . . . . . . . . . . . . 1.3.4 Limitations and inconsistencies of current models . . 1.4 Review of stabilized and multiscale methods . . . . . . . . . 1.4.1 Alternative approaches to advection-dominated flows 1.4.2 Stabilized finite element methods . . . . . . . . . . . 1.4.3 The multiscale approach . . . . . . . . . . . . . . . . 1.5 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 3 9 11 12 13 15 15 19 22 24 30 32 36 42 45

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Displacement theory

2 Mathematical formulation of three-phase flow 2.1 Introduction . . . . . . . . . . . . . . . . . . . . 2.2 Continuum equations . . . . . . . . . . . . . . . 2.3 Relative permeabilities and capillary pressures . 2.4 Equations in dimensionless form . . . . . . . . . 2.5 Flow regions and reduced saturations . . . . . .

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iii 2.6

Special cases . . . . . . . . . . . . . 2.6.1 Negligible gravity effects . . 2.6.2 Negligible capillarity effects 2.6.3 Two-phase flow . . . . . . . 2.6.4 Miscible flow . . . . . . . .

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3 Relative permeabilities for strictly hyperbolic models 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Mathematical model . . . . . . . . . . . . . . . . . . . 3.2.1 System of governing equations . . . . . . . . . . 3.2.2 Character of the system of equations . . . . . . 3.3 Relative permeabilities for strict hyperbolicity . . . . . 3.3.1 Loss of strict hyperbolicity in traditional models 3.3.2 Conditions for strict hyperbolicity . . . . . . . . 3.3.3 A simple model . . . . . . . . . . . . . . . . . . 3.4 Validation with experimental data . . . . . . . . . . . . 3.4.1 Description of the “endpoint-slope” analysis . . 3.4.2 Two-phase flow experiments . . . . . . . . . . . 3.4.3 Three-phase flow experiments . . . . . . . . . . 3.5 Concluding remarks . . . . . . . . . . . . . . . . . . . .

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69 69 70 70 71 74 75 83 94 100 101 102 105 110

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112 112 113 114 116 117 118 121 124 126 127 132 133 134 144

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146 146 147 148 148

4 Strictly hyperbolic models of co-current flow with gravity 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Character of the equations . . . . . . . . . . . . . . . 4.1.2 Effects of gravity . . . . . . . . . . . . . . . . . . . . 4.1.3 Chapter outline . . . . . . . . . . . . . . . . . . . . . 4.2 Mathematical model . . . . . . . . . . . . . . . . . . . . . . 4.3 Conditions for a strictly hyperbolic system . . . . . . . . . . 4.3.1 Conceptual picture of three-phase displacements . . . 4.3.2 Conditions for co-current flow . . . . . . . . . . . . . 4.3.3 Conditions for strict hyperbolicity . . . . . . . . . . . 4.3.4 Discussion of conditions . . . . . . . . . . . . . . . . 4.4 A simple model . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Conditions for co-current flow . . . . . . . . . . . . . 4.4.2 Conditions for strict hyperbolicity . . . . . . . . . . . 4.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . 5 Analytical solution to the Riemann problem 5.1 Introduction . . . . . . . . . . . . . . . . . . . 5.2 Governing equations . . . . . . . . . . . . . . 5.2.1 Two-phase flow . . . . . . . . . . . . . 5.2.2 Three-phase flow . . . . . . . . . . . .

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iv 5.3

5.4

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5.6

II

Mathematical structure of the equations . . . 5.3.1 Two-phase flow . . . . . . . . . . . . . 5.3.2 Three-phase flow . . . . . . . . . . . . Solution to the Riemann problem . . . . . . . 5.4.1 Introduction . . . . . . . . . . . . . . . 5.4.2 Riemann problem for two-phase flow . 5.4.3 Riemann problem for three-phase flow Application example: water-gas injection . . . 5.5.1 Description of the problem . . . . . . . 5.5.2 Exact solution . . . . . . . . . . . . . . 5.5.3 Approximate solution . . . . . . . . . . 5.5.4 Discussion . . . . . . . . . . . . . . . . Concluding remarks . . . . . . . . . . . . . . .

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Multiscale numerical modeling

149 150 152 157 157 160 166 181 181 182 183 185 189

191

6 Multiscale finite elements for miscible flow and two-phase 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Numerical formulation . . . . . . . . . . . . . . . . . . . . . 6.2.1 Initial and boundary value problem . . . . . . . . . . 6.2.2 Weak form . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Classical Galerkin method . . . . . . . . . . . . . . . 6.2.4 Multiscale approach . . . . . . . . . . . . . . . . . . 6.2.5 Shock-capturing techniques . . . . . . . . . . . . . . 6.3 Representative numerical simulations . . . . . . . . . . . . . 6.3.1 One-dimensional miscible flow . . . . . . . . . . . . . 6.3.2 Two-dimensional miscible flow . . . . . . . . . . . . . 6.3.3 One-dimensional immiscible flow . . . . . . . . . . . 6.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . .

flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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192 192 195 195 196 197 198 206 207 208 218 223 231

7 Multiscale finite elements for three-phase flow 7.1 Introduction . . . . . . . . . . . . . . . . . . . . 7.2 Multiscale numerical formulation . . . . . . . . 7.2.1 Initial and boundary value problem . . . 7.2.2 Weak form . . . . . . . . . . . . . . . . . 7.2.3 Classical Galerkin method . . . . . . . . 7.2.4 Multiple-scale approach . . . . . . . . . 7.2.5 Matrix of stabilizing coefficients . . . . . 7.2.6 Shock-capturing techniques . . . . . . . 7.3 Representative numerical simulations . . . . . . 7.3.1 Oil filtration in relatively dry soil . . . .

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7.3.2 Water-gas injection in a reservoir . . . . . . . . . . . . . . . . 273 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

8 Closure 283 8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 8.2 Future extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 Nomenclature

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A Algorithms for the analytical solution of the Riemann problem A.1 Solution algorithms for the wave curves of three-phase flow . . . . . A.1.1 Rarefaction curves . . . . . . . . . . . . . . . . . . . . . . . A.1.2 Shock curves . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.3 Rarefaction-shock curves . . . . . . . . . . . . . . . . . . . . A.2 Solution algorithms for selected solution types of three-phase flow . A.2.1 S1 S2 solution . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.2 R1 R2 solution . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.3 R1 S1 R2 S2 solution . . . . . . . . . . . . . . . . . . . . . . . B Implementation of the multiscale finite B.1 The variational multiscale formulation B.2 Galerkin contribution . . . . . . . . . . B.2.1 Finite element residual . . . . . B.2.2 Finite element tangent . . . . . B.3 Subgrid-scale contribution . . . . . . . B.3.1 Finite element residual . . . . . B.3.2 Finite element tangent . . . . .

element method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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[The table of contents] may come at the beginning or at the end. But if it has to be in the beginning, it should truly be there. In some Anglo-Saxon books it appears after the preface, and often after the preface, the introduction to the first edition and the introduction to the second edition. Nonsense. To do something this stupid, one may equally well place it in the middle of the book. — UMBERTO ECO, Come si fa una tesi di laurea (1977)

vi

List of Figures 1.1 1.2 1.3 1.4 2.1 2.2

Steps in the development of a numerical model. . . . . . . . . . . . . 2 Conceptual model of three-phase flow displacement of Kyte et al. [1956]. 17 Elliptic region inside the saturation triangle for the Stone I model, as presented by Bell et al. [1986]. . . . . . . . . . . . . . . . . . . . . . . 25 Evolving solution of a mixed elliptic/hyperbolic system for left and right states inside the elliptic region, as presented by Bell et al. [1986]. 31 Saturation triangle and ternary diagram. . . . . . . . . . . . . . . . . Schematic of the map between the space of actual saturations, and the space of reduced saturations. . . . . . . . . . . . . . . . . . . . . . . .

Schematic representation of fast eigenvectors for the models analyzed by Shearer [1988] and Holden [1990a]. . . . . . . . . . . . . . . . . . . 3.2 Eigenvectors along the edges of the saturation triangle for models with an umbilic point on the WG edge. . . . . . . . . . . . . . . . . . . . . 3.3 Eigenvectors along the edges of the saturation triangle for the type of models we propose. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Oil isoperms for the simple model given by Equation (3.49). . . . . . 3.5 Check of strict hyperbolicity on edges of the saturation triangle. . . . 3.6 Markedly different qualitative behavior of the slope of the relative permeability of a phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Relative permeability curves of water and gas for the two-phase drainage experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Behavior of the relative permeability as a function of its own saturation in the neighborhood of the “endpoint saturation”, for the two-phase drainage experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Relative permeability curves of water and gas for the two-phase imbibition experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Behavior of the relative permeability as a function of its own saturation in the neighborhood of the “endpoint saturation”, for the two-phase imbibition experiment. . . . . . . . . . . . . . . . . . . . . . . . . . .

60 61

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vii 3.11 Saturation path for the drainage-dominated three-phase relative permeability experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12 Behavior of the relative permeability as a function of its own saturation in the neighborhood of the “endpoint saturation”, for the three-phase drainage-dominated experiment. . . . . . . . . . . . . . . . . . . . . . 3.13 Saturation path for the imbibition-dominated three-phase relative permeability experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.14 Behavior of the relative permeability as a function of its own saturation in the neighborhood of the “endpoint saturation”, for the three-phase imbibition-dominated experiment. . . . . . . . . . . . . . . . . . . . . 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9

Schematic representation of fast eigenvectors for the models analyzed by Shearer [1988] and Holden [1990a]. . . . . . . . . . . . . . . . . . . Schematic representation of eigenvectors along the edges of the saturation triangle for the type of models we propose. . . . . . . . . . . . Schematic of the profiles of water and oil relative mobilities along the OW edge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Admissible values of the endpoint-slope βg of the gas relative permeability as a function of the gravity number. . . . . . . . . . . . . . . . Relative permeability of gas as a function of its own saturation for two different values of the endpoint-slope: βg = 0 and βg = 0.1. . . . . . . Check of strict hyperbolicity on edges of the saturation triangle. . . . Contour plot of the discriminant δ := (ν2 − ν1 )2 , evaluated on the saturation triangle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fast characteristic paths evaluated on the small region indicated in Figure 4.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Idealized behavior of the relative permeability functions for an oil-water system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Idealized behavior of the flux function in two-phase flow, for different values of the viscosity ratio. . . . . . . . . . . . . . . . . . . . . . . . Convexity regions of the flux function in two-phase flow. . . . . . . . The two-phase Buckley-Leverett flow problem develops a nonphysical triple-valued solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . Isoperms of all three phases. . . . . . . . . . . . . . . . . . . . . . . . Contour plots of the fractional flow functions of all three phases. . . . Example of a single rarefaction solution to the Riemann problem of two-phase flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic of a shock. . . . . . . . . . . . . . . . . . . . . . . . . . . . Example of a single shock solution to the Riemann problem of twophase flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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110 122 123 130 138 139 140 141 142 150 151 152 153 158 158 161 163 163

viii 5.10 Example of a composite rarefaction-shock solution to the Riemann problem of two-phase flow. . . . . . . . . . . . . . . . . . . . . . . . . 165 5.11 Algorithm for obtaining the wave structure for two-phase flow. . . . . 166 5.12 Schematic representation of the generic solution to the Riemann problem of three-phase flow. . . . . . . . . . . . . . . . . . . . . . . . . . 167 5.13 Integral curves for the relative mobilities (5.18)–(5.20). . . . . . . . . 170 5.14 Plot of the Hugoniot loci of both characteristic families, for the relative mobilities (5.18)–(5.20). . . . . . . . . . . . . . . . . . . . . . . . . . 172 5.15 Contour plots of eigenvalues and the inflection loci of both characteristic families, for the relative mobilities (5.18)–(5.20). . . . . . . . . . 174 5.16 Rarefaction-shock curves of the 1-family, for the relative mobilities (5.18)– (5.20). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 5.17 Schematic tree with all possible combinations of solutions to the Riemann problem of three-phase flow. . . . . . . . . . . . . . . . . . . . 177 5.18 Examples of the saturation paths for all 9 solution types. . . . . . . . 179 5.18 Examples of the saturation paths for all 9 solution types (continued). 180 5.19 Sketch of the injection problem. . . . . . . . . . . . . . . . . . . . . . 182 5.20 Saturation path of the exact solution to the water-gas injection problem.183 5.21 Saturation profiles of the exact solution to the water-gas injection problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 5.22 Saturation path of the approximate solution to the water-gas injection problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 5.23 Saturation profiles of the approximate solution to the water-gas injection problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 5.24 Comparison of the dimensionless oil production rate predicted by the exact solution and the approximate solution. . . . . . . . . . . . . . . 188 5.25 Comparison of the dimensionless cumulative oil production predicted by the exact solution and the approximate solution. . . . . . . . . . . 188 6.1 6.2 6.3 6.4 6.5 6.6 6.7

ASGS and Galerkin solutions for the one-dimensional miscible flow problem with zero distributed sources. . . . . . . . . . . . . . . . . . Amount of shock-capturing diffusion introduced by the “canonical” formulation and the proposed formulation. . . . . . . . . . . . . . . . Detail of the shock and boundary layer for the one-dimensional miscible flow problem with zero distributed sources. . . . . . . . . . . . . . . . Convergence of the ASGS method with shock-capturing for the onedimensional miscible flow problem with zero distributed sources. . . . ASGS and Galerkin solutions for the one-dimensional miscible flow problem with production. . . . . . . . . . . . . . . . . . . . . . . . . ASGS and Galerkin solutions for the one-dimensional miscible flow problem with reaction and production. . . . . . . . . . . . . . . . . . Schematic of the two-dimensional miscible flow problem. . . . . . . .

211 212 212 213 216 219 220

ix 6.8 6.9 6.10 6.11 6.12 6.13 6.14 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10

7.11

7.12

7.13 7.14 7.15

Galerkin and ASGS solutions for transient and steady-state conditions for the two-dimensional miscible flow problem. . . . . . . . . . . . . . Contour plots at steady-state for the two-dimensional miscible flow problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fractional flow function f , and capillary diffusion function D used in the immiscible flow simulations. . . . . . . . . . . . . . . . . . . . . . ASGS and Galerkin solutions to the one-dimensional immiscible flow problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Amount of shock-capturing diffusion introduced by the “canonical” formulation and the proposed formulation. . . . . . . . . . . . . . . . Detail of the shock and boundary layer for the one-dimensional immiscible flow problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evolution of the Newton iterative scheme for the one-dimensional immiscible flow problem. . . . . . . . . . . . . . . . . . . . . . . . . . .

221 222 225 227 228 229 230

Comparison of alternative definitions of the diffusion correction factor. 252 Sketch of the oil filtration problem. . . . . . . . . . . . . . . . . . . . 262 Saturation path of the exact solution to the oil filtration problem. . . 263 Saturation profiles of the exact solution to the oil filtration problem. . 264 Saturation profiles of the standard Galerkin solution to the oil filtration problem on a fine mesh of 4000 elements. . . . . . . . . . . . . . . . . 266 Saturation path for the oil filtration problem obtained by the standard Galerkin method on a fine mesh of 4000 elements. . . . . . . . . . . . 266 Saturation profiles of the standard Galerkin solution to the oil filtration problem on a coarse mesh of 40 elements. . . . . . . . . . . . . . . . . 268 Saturation profiles of the ASGS solution (τ formulation given by Hughes and Mallet [1986a]) to the oil filtration problem on the coarse mesh. . 269 Saturation profiles of the ASGS solution (τ formulation given by Codina [2000]) to the oil filtration problem on the coarse mesh. . . . . . 269 Saturation profiles of the ASGS solution to the oil filtration problem. Formulation of τ given by Hughes and Mallet [1986a]. Shock-capturing diffusion in “canonical form”. . . . . . . . . . . . . . . . . . . . . . . 270 Saturation profiles of the ASGS solution to the oil filtration problem. Formulation of τ given by Hughes and Mallet [1986a]. Shock-capturing diffusion in “quadratic form”. . . . . . . . . . . . . . . . . . . . . . . 270 Saturation profiles of the ASGS solution to the oil filtration problem. Formulation of τ given by Hughes and Mallet [1986a]. Proposed formulation of shock-capturing diffusion. . . . . . . . . . . . . . . . . . . 271 Profiles of shock capturing diffusion introduced by the “canonical form”.272 Profiles of shock capturing diffusion introduced by the “quadratic form”.272 Profiles of shock capturing diffusion introduced by the proposed formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

x 7.16 Sketch of the water-gas injection problem. . . . . . . . . . . . . . . . 274 7.17 Saturation path of the exact solution to the water-gas injection problem.275 7.18 Saturation profiles of the exact solution to the water-gas injection problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 7.19 Saturation profiles of the standard Galerkin solution to the water-gas injection problem on a fine mesh of 4000 elements. . . . . . . . . . . . 277 7.20 Saturation path for the water-gas injection problem, obtained by the standard Galerkin method on a fine mesh of 4000 elements. . . . . . . 277 7.21 Saturation profiles of the standard Galerkin solution to the water-gas injection problem on a coarse mesh of 40 elements. . . . . . . . . . . 278 7.22 Saturation profiles of the ASGS solution (τ formulation given by Hughes and Mallet [1986a]) to the water-gas injection problem on the coarse mesh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 7.23 Saturation profiles of the ASGS solution to the water-gas injection problem. Formulation of τ given by Hughes and Mallet [1986a]. Proposed formulation of shock-capturing diffusion. . . . . . . . . . . . . . 280 7.24 Profiles of shock capturing diffusion introduced by the proposed formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 A.1 Newton algorithm for obtaining the 1-shock curve. . . . . . . . . . . . A.2 Predictor-corrector algorithm for obtaining the rarefaction-shock curve of the 1-characteristic family. . . . . . . . . . . . . . . . . . . . . . . . A.3 Schematic of the kth iteration of the predictor-corrector algorithm for obtaining a rarefaction-shock of the 1-characteristic family. . . . . . . A.4 Schematic of the S1 S2 solution path in the ternary diagram. . . . . . A.5 Profiles of wave speeds and saturations for the S1 S2 solution. . . . . . A.6 Performance of the Newton iterative scheme for the S1 S2 solution. . . A.7 Predictor-corrector algorithm for obtaining the R1 R2 solution. . . . . A.8 Schematic diagram of the kth iteration of the predictor-corrector procedure for a R1 R2 intersection. . . . . . . . . . . . . . . . . . . . . . A.9 Schematic of the R1 R2 solution path in the ternary diagram. . . . . . A.10 Profiles of wave speeds and saturations for the R1 R2 solution. . . . . A.11 Performance of the predictor-corrector iterative scheme for the R1 R2 solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.12 Predictor-corrector algorithm for obtaining the R1 S1 R2 S2 solution. . A.13 R1 S1 R2 S2 solution path in the ternary diagram. . . . . . . . . . . . . A.14 Wave speeds and saturations for the R1 S1 R2 S2 solution. . . . . . . .

348 350 350 354 354 355 356 357 358 358 359 360 362 362

xi

List of Tables 3.1 3.2 3.3 3.4 3.5 3.6 3.7 4.1 4.2 4.3 6.1

Summary of conditions along the OW edge. . . . . . . . . . . . . . . . 90 Summary of conditions along the OG edge. . . . . . . . . . . . . . . . 91 Summary of conditions along the WG edge. . . . . . . . . . . . . . . . 93 Parameters of the power-law fitting for the two-phase drainage experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Parameters of the power-law fitting for the two-phase imbibition experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Parameters of the power-law fitting for the three-phase drainage-dominated experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Parameters of the power-law fitting for the three-phase imbibitiondominated experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Summary of conditions for strict hyperbolicity along the edges saturation triangle, in terms of the fractional flow functions. . Summary of conditions for strict hyperbolicity along the edges saturation triangle, in terms of the fluid relative mobilities. . . Summary of conditions for strict hyperbolicity at the vertices saturation triangle, in terms of the fluid relative mobilities. . .

of the . . . . 127 of the . . . . 128 of the . . . . 129

Expressions of the characteristic times for advection, diffusion and reaction processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

xii

Acknowledgments Many people have contributed, directly or indirectly, to the completion of this dissertation. My gratitude towards them will not be justly reflected in this note. I am very grateful to my advisor, Tadeusz W. Patzek, for his guidance and support. Working with him has been stimulating and rewarding in every possible aspect. He has been able to establish a flexible and creative research environment, which has fostered my own academic maturity. I want to thank Francisco Armero for his help and advice during the development of this thesis, as well as for the courses he imparted while I was at Berkeley. He has been a most inspiring example, research and otherwise. I want to express my esteem and admiration for Robert L. Taylor, whose finite element code FEAP has been an invaluable tool at several stages of this dissertation. I am also thankful to Keith Miller, for reviewing the thesis and offering helpful comments. Special thanks are due to Dmitriy Silin from the Lawrence Berkeley Lab, for following my work with interest, carefully reviewing many pages of manuscripts, and providing constructive criticism. I have greatly profited from his involvement in my research. I am also indebted to Martin Blunt from Imperial College, for his comments, our vigorous discussion on elliptic regions, and for providing an electronic version of the relative permeability data in Chapter 3. I gratefully acknowledge the time and attention dispensed by Ramón Codina from Universitat Politècnica de Catalunya, during his sabbatical year at Berkeley and, later, for many insightful comments and suggestions on my work on stabilized finite elements. I also want to thank my fellow students in the research group, Ahmed Al-Futaisi, Pascual Benito, Elizabeth Duff, Guodong Jin, Matthew Small, and Yasir Zargar, for their companionship and good humor in many sleepless nights at the Lab.

xiii My experience at Berkeley would not have been nearly as enjoyable without so many good friends. I especially thank Alberto Figueroa for his long friendship, Ignacio Romero for many illuminating discussions, Julio Garc´ıa for our lively afternoon coffee breaks, and Alex Lago in representation of the entire Spanish community gathered around the Iberia-Berkeley student association. I do not want to forget my faculty and colleagues from the University of La Coru˜ na. I am particularly grateful to Javier Samper, who guided me during my early research, and encouraged me to continue my education in the United States. I am thankful to Jorge Molinero for his friendship, his collaboration across the Atlantic, and for all the good times. I also want to thank Manuel Casteleiro, Ferm´ın Navarrina, and Ignasi Colominas, who have helped me in so many different ways. I am much obliged to the Barrié de la Maza Foundation, whose fellowship provided funding during my first two years at Berkeley. I have received additional funding through the Jane Lewis Fellowship, the Repsol-YPF Fellowship, the Forum Filatélico Fellowship, and the UC Oil Consortium. Their financial support is gratefully acknowledged. I am immensely grateful to Laura, my partner, for all these years together. She has been a constant source of inspiration and kind support. Finally, I want to express my everlasting gratitude to my parents, Ovidio and Mar´ıa, and to my sister, Sara. My love and admiration for them is present in each and every page of this dissertation.

“I thank you, Samana, for watching out over my sleep.” — HERMANN HESSE, Siddhartha (1922)

1

Chapter 1 Introduction This dissertation is concerned with three-phase flow in porous media, that is, simultaneous filtration of three immiscible fluids through permeable rocks and soils. Such flows are often encountered in groundwater hydrology, oil reservoir engineering, and other branches of engineering. In this context, mathematical and numerical models are required to perform quantitative predictions of oil and gas recovery in reservoirs, and contamination events by nonaqueous phase liquids in the shallow subsurface. The steps involved in the development of a numerical model are summarized in Figure 1.1. There are many difficulties associated with the correct conceptualization and mathematical formulation of the problem, due to scarcity of data and complexity of the physical processes governing flow of several phases through a porous medium. In addition, the nature of the mathematical problem also poses nonstandard challenges in obtaining numerical solutions. The dissertation aims to improve our current understanding of three-phase flow systems, with an emphasis on the mathematical and numerical formulation of these processes. It will become apparent that the physics of the problem has profound

2

Chapter 1. Introduction Observable reality

-

Conceptual model

identification of processes

- Mathematical

model

mathematical formulation

-

Numerical model

space and time discretization

Figure 1.1. Steps in the development of a numerical model. implications, and poses essential requirements, on the structure of the mathematical equations. This fact is recognized in this work, and used to elaborate answers to some of the unresolved mathematical and numerical issues. In the first section of this chapter we motivate the problem, emphasizing its huge practical application, and highlighting the need for significant advances in our description of three-phase porous media flow. The specific objectives of the dissertation, listed in Section 1.2, try to address some of these issues. In Sections 1.3 and 1.4 we give a review of the technical literature on the subject. In the final section of this introductory chapter we discuss the scope, and give a brief overview, of this dissertation.

1.1

Motivation

The importance of two-phase flow in porous media has long been recognized in many fields, like soil physics [Richards, 1931], groundwater hydrology [Bear, 1972; Philip, 1957, 1969], and petroleum engineering [Buckley and Leverett, 1942; Muskat and Meres, 1936]. Although investigated from the very onset [Leverett and Lewis, 1941; Muskat, 1949], the need for quantitative predictions involving flow of three fluid phases is more recent. However, there is now little doubt that a good description of three-phase flow is essential in practical applications like enhanced oil recovery [Lake, 1989] and environmental remediation of the unsaturated zone [Hunt et al., 1988a].

Chapter 1. Introduction

3

Flow of three immiscible fluids —denoted hereafter as water, oil, and gas— occurs in a variety of flow situations in the subsurface, including: 1. Primary oil production below bubble point and with movable water [Allen III, 1985; Coats, 1982; Muskat, 1949]. 2. Gas injection into hydrocarbon reservoirs [Muskat, 1949]. 3. Water flooding in the presence of free gas [Holmgren and Morse, 1951; Kyte et al., 1956; Willhite, 1986]. 4. Water alternating gas (WAG) injection [Christensen et al., 2001; Guzmán et al., 1994; Larsen and Skauge, 1999]. 5. Steam floods [Falta et al., 1992a,b; Hunt et al., 1988a,b; Mandl and Volek, 1969; Shutler, 1969]. 6. Immiscible CO2 floods [Holm, 1976; Pope, 1980; SPE, 1999] and geological CO2 sequestration [Pruess et al., 2002; Pruess and Garc´ıa, 2002; Tsang et al., 2002]. 7. Pollutant migration of nonaqueous phase liquids in the shallow subsurface, and associated remediation techniques [Abriola and Pinder, 1985a,b; Adenekan et al., 1993; Falta et al., 1992a,b; Forsyth and Shao, 1991; Lenhard et al., 1995; Sleep and Sykes, 1993a,b; White et al., 1995]. This wide range of applications —and the extent of their socioeconomic impact— makes three-phase flow in porous media a particularly relevant research topic.

1.1.1

The need for revisiting the mathematical description

Mathematical modeling of multiphase flow in porous media is, to a large extent, still an open issue [Miller et al., 1998]. The main difficulty stems from its inherent


4

multiscale character. In fact, we regard multiphase flow not only as a multiscale problem, where the parameters and the variables of interest are scale-dependent, but also a multiphysics problem —different processes dominate at different scales. The microscale is controlled by capillary forces, whereas viscous and gravity forces usually dominate at the macroscale. This complex behavior should be contrasted with the development of simplistic mathematical models, which are extensions of models proven successful for single-phase flow. This is particularly true for three-phase flow, which has been traditionally modeled using a direct application of two-phase flow formulations [Aziz and Settari, 1979; Chavent and Jaffré, 1986; Muskat, 1949; Peaceman, 1977] (see also Miller et al. [1998] and the references therein). As it turns out, three-phase flow is less forgiving than two-phase flow, and exposes the physical and mathematical inconsistencies of the classical mathematical formulation. In our opinion, the use of the simplistic classical approach has been favored by two factors: first, the limited understanding of the physics of flow of several phases in a porous medium; and second, the challenge of posing the mathematical problem in a tractable form that allows the development of predictive tools. Classical macroscopic descriptions of multiphase flow rely on the continuum approximation: the porous medium is regarded as a superposition of several continua, each filling the entire medium [Bear, 1972; Coussy, 1995]. The key ingredients of such formulations are mass conservation equations, and a straightforward multiphase extension of Darcy’s equation [Muskat, 1949].1 Darcy’s equation for single-phase flow is an approximate form of the fluid momentum balance in creeping flow through porous media. Underlying the extension of the motion equation of a single fluid to the simultaneous flow of two or more fluids is the concept of relative permeability [Bear, 1

In this work, the so-called Darcy’s law is always referred to as Darcy’s equation or Darcy’s “law” in the context of multiphase flows. It is not a physical law but, rather, an empirical relation.


5

1972; Muskat, 1949]. Relative permeabilities account for the reduction in the flow of each phase due to the mutual interaction of the different flowing phases. Applicability of the traditional formulation just described relies heavily on the following: 1. Appropriateness of the continuum approximation. 2. Validity of Darcy’s equation and its extension to multiphase flow. 3. Proper model for the relative permeabilities. Each of these items will be now discussed. 1.1.1.1

Appropriateness of the continuum approximation

Because the multiphase system is regarded as a superposition of several continua, the interest is necessarily constrained to gross —macroscopic— quantities. This macroscopic viewpoint is certainly the most adequate for practical purposes. The usefulness of this approach should be ultimately measured in terms of how well it compares with experiments, and how accurate and reliable are the predictions of the model. In the context of continuum theories, flow is governed by conservation laws, which take the form of field equations or jump conditions, depending on whether smooth or discontinuous solutions are relevant [Truesdell and Noll, 1965]. The crux of the continuum formulation lies on the specification of constitutive equations, which provide closure to the mathematical problem. In the context of isothermal multiphase flow, constitutive equations relate phase pressures and fluid fluxes. The common multiphase extension of Darcy’s equation is one example. Although plagued with difficulties, other options are possible. One such alternative is to derive the macroscopic constitutive equations from the microscopic

6


equations through averaging techniques [Bachmat and Bear, 1986; Bear and Bachmat, 1986; Gray, 1999; Gray and Hassanizadeh, 1998; Hassanizadeh and Gray, 1979a,b, 1980, 1990, 1993; Marle, 1982; Whitaker, 1986a,b], or statistical mechanics [Sposito, 1978a,b; Sposito and Chu, 1981]. 1.1.1.2

Validity of Darcy’s “law”

Darcy’s law is an empirical relation that replaces —at the macroscopic level— the equations of momentum conservation. For single-phase flow, Darcy’s law states that the volumetric fluid flux q is a linear function of the piezometric gradient ∇h [Bear, 1972]: k q = − ρg∇h, µ

(1.1)

where k is the absolute permeability tensor of the medium, and µ is the dynamic viscosity of the fluid. The piezometric head h is given by h=

p + z, ρg

(1.2)

where p is the fluid pressure, ρ is the fluid density, g is the gravitational acceleration, and z is the elevation. The pressure and gravity components of the hydraulic gradient are indistinguishable from the point of view of fluid flow: all that matters is the sum of the two. Equation (1.1) is a postulate, supported for single-phase flow by experimental evidence and by volume averaging as a first-order approximation [Hassanizadeh, 1986; Whitaker, 1986a]. A major assumption was introduced by Muskat [1949, Chap. 7], who extended Darcy’s equation to model multiphase flow. The constitutive relation for the volumetric fluid flux q α of a given phase α takes the form: qα = −

kkrα ∇(pα + ρα gz), µα

(1.3)


7

where krα is the relative permeability to the α-phase. In the expression above, the driving force is a combination of viscous, capillary and gravity forces, and the relative permeability is assumed to be independent of what process or processes dominate.2 The multiphase extension of Darcy’s equation given by Equation (1.3) may be described as a quasi-linear relation, because the fluid flux depends linearly on the “driving force” —which includes viscous, capillary, and gravity forces— and all the nonlinearity is agglutinated in the relative permeabilities. Several experimental studies have (partially) confirmed the applicability of this working assumption for unconsolidated sands [Leverett, 1939; Leverett and Lewis, 1941; Muskat and Meres, 1936; Wyckoff and Botset, 1936] and some consolidated rocks [Botset, 1940; Geffens et al., 1951; Osoba et al., 1951; Richardson et al., 1952; Sandberg et al., 1958], and verified that relative permeability is independent of fluid viscosities and total fluid rate. However, while Darcy’s law for single-phase flow has been obtained from fundamental principles after volume averaging and linearization [Hassanizadeh, 1986; Whitaker, 1986a], its extension to multiphase systems is not rigorous [Hassanizadeh and Gray, 1993], therefore limiting its potential applicability. The physical limitations of Darcy’s “law” are easily understood from the multiscale/multiphysics character of simultaneous flow of several fluid phases through permeable media [Juanes and Patzek, 2002e,f]. The overall displacement of one fluid by another is a consequence of the pore-scale mechanisms that actually take place in the porous medium. Indeed, the behavior of the transition zones in multiphase flows 2

Muskat [1949] was aware of this assumption: “. . . while the Darcy law representation . . . is a natural generalization to multiphase-flow systems of the original homogeneous-fluid ‘law of force’ . . . , its quantitative validity is not so well established as in the latter case. Such validity implies that the permeability functions ko , kg , and kw are determined only by the saturation distribution but are independent of the fluid viscosity and pressure gradient. With respect to the former . . . no significant effect of the fluid viscosity was found. . . . As to the dependence of the permeabilities on the pressure gradient, the situation is rather less satisfactory.”


8

may vary wildly (see, e.g., Akin et al. [2000]), and may depend on the wettability properties of the fluids, the fluid viscosity ratios, the fluid density ratios, the displacement process —drainage or imbibition— and the displacement history that determined the pore-scale configuration of the fluids [Avraam and Payatakes, 1995a,b; Lenormand, 1986; Lenormand et al., 1988]. To account properly for the physics of multiphase flow, it is likely that one has to resort to a multiscale formulation. The development of such a formulation is an open issue and, although several approaches have been proposed [Avraam and Payatakes, 1999; Barenblatt et al., 1990; Gray and Hassanizadeh, 1998; Hassanizadeh and Gray, 1990; Silin and Patzek, 2002; Valavanides et al., 1998], they are immature and have to be explored fully. However, there is still a great interest in Darcy-like formulations, because they are almost universally used in hydrogeology and petroleum engineering, and because alternative approaches have not yet been reduced to a tractable form. Following the program initiated in Juanes and Patzek [2002e,g], we adopt the multiphase extension of Darcy’s equation as a working assumption, rather than a physical law, and concentrate on the development of improved relative permeabilities, which result in a mathematical model that adheres to the essential physics of multiphase flow. 1.1.1.3

Relative permeabilities as key flow descriptors

Darcy-type models do not reflect many of the multiple-scale considerations described above. The only way in which they can capture at least a shadow of the behavior of the actual multiphase displacement is through the relative permeability functions, because these are the only “degrees of freedom” of the formulation. In this framework, success of the formulation depends heavily on the use of “correct” relative permeabilities. Traditionally, they are taken as functions of current fluid saturations


9

alone. This is a very strong assumption, which does not account for: (1) hysteretic effects [Lenhard and Parker, 1987; Parker and Lenhard, 1987], which include the past saturation history into the formulation; (2) nonequilibrium effects [Barenblatt et al., 1990; Hassanizadeh and Gray, 1990; Silin and Patzek, 2002], which introduce the concept of a relaxation time for pore-scale rearrangement of fluid saturations; and (3) the flow regime, determined by the ratios of viscous, capillary, and gravity forces [Lenormand, 1986; Lenormand et al., 1988], which influences the pore-scale mechanisms of fluid displacement. Therefore, relative permeabilities cannot be understood as fixed functions of saturation, or even saturation history. They depend intrinsically on the flow regime and properly should be called functionals rather than functions. This is certainly not an ideal description, as the very presence of functionals is a reflection of having an incomplete formulation [Barenblatt, 1993; Hassanizadeh and Gray, 1993]. In summary, we regard the relative permeabilities as nothing else than functionals used in the constitutive model, which may —and in fact should — be influenced by the fluid viscosity ratios and the gravity number. As shown in Juanes and Patzek [2002e,f,g], it is precisely the influence of viscosity and gravity on the relative permeabilities that allows one to remove some of the mathematical inconsistencies of the classical formulation of three-phase flow.

1.1.2

The need for consistent relative permeabilities

Given the state of affairs discussed in Section 1.1.1, it is not surprising that the traditional mathematical formulation of three-phase flow —using classical expressions for the relative permeabilities— may be inconsistent, and yield nonphysical predictions. In Section 1.3.4, and in Chapters 3 and 4, we describe in detail the source, character, and implications, of the mathematical singularities that appear in the three-phase


10

flow equations. Here we give a brief summary of this discussion. In the context of multiphase displacements, capillarity effects lead to a nonlinear diffusion term in the macroscopic continuum conservation equations (see Chapter 2). Here, the role of capillarity is to smear the moving fronts that arise from the displacement of one fluid by another.3 As capillarity effects vanish, one expects that the solution to the macroscopic equations will develop sharp features, such as shocks and boundary layers. In order to isolate as much as possible the dependence on the relative permeabilities, we focus our attention on the case of small capillarity effects —vanishing diffusion. Under these conditions, the mathematical model leads to a 2 × 2 system of first-order partial differential equations —the saturation equations. It was long believed that, when capillarity is ignored, this system of equations would be strictly hyperbolic for any relative permeability functions. This is far from being the case and, in fact, most relative permeability models used today give rise to systems which are not strictly hyperbolic for the entire range of admissible saturations [Bell et al., 1986; Fayers, 1987; Hicks Jr. and Grader, 1996; Holden, 1990a; Shearer, 1988; Shearer and Trangenstein, 1989]. Loss of strict hyperbolicity typically occurs at bounded regions of the saturation triangle —the so-called elliptic regions— where the system is elliptic in character. We find this behavior disturbing for many reasons. In our opinion, elliptic regions are artifacts of an incorrect mathematical model, and their presence contradicts the expected physical behavior of a three-phase displacement.4 Arguments supporting this view are given in Juanes and Patzek [2002e,f,g], and elaborated further from a more physical perspective in Juanes and Patzek [2003a] 3

The term smeared front is referred to as viscous profile in the field of fluid mechanics. We have avoided this terminology because the shape of fronts in multiphase displacements is governed by capillary forces, not viscous forces! 4 We agree completely with Shearer and Trangenstein [1989, p. 523], when they say that “We have no reason to believe that the elliptic regions are physical; rather, we believe that they are an unintended consequence of the forms of the three-phase flow models.”


11

and Juanes et al. [2003]. Essentially, our thesis is that elliptic regions should not be justified simply because they appear as a consequence of using Muskat’s extension of Darcy’s equation, and common relative permeability functions. Therefore, we propose that if appropriate relative permeabilities are employed, nonphysical behavior of the solution is avoided. Juanes and Patzek [2002e,f] show that it is indeed possible to impose conditions on the relative permeability functions so that the system of saturation equations —with and without the gravity term— is strictly hyperbolic inside the entire saturation triangle and, moreover, that the required conditions are in agreement with pore-scale physics and experimental data.

1.1.3

The need for a general analytical solution

Today, most experimental investigations for the determination of relative permeabilities are based on high-rate unsteady displacements. The unsteady method is often the preferred experimental technique because it is much faster than the steadystate method, and because only one phase is injected. Many experiments include not only production data, but also in-situ saturation measurements, and the common procedure for the interpretation of these experiments is through some kind of history matching with a numerical simulator (see Section 1.3.2.3 below). However, interpretation of relative permeability functions would always be faster and more reliable if the forward problem is solved analytically. One of the features of high-rate experiments is that the effects of capillarity are minimized, which results in displacement processes where sharp fronts can be approximated macroscopically by traveling discontinuities. Juanes and Patzek [2002a,g] show that, under capillarity-free conditions, it is possible to obtain a general analytical so-


12

lution reproducing laboratory conditions. The value of an analytical solution is not restricted to allowing faster interpretation of laboratory experiments. It is extremely useful on many counts, such as: 1. To reveal the structure of the solution, thus giving insight into more efficient recovery —or sequestration— of fluids in the subsurface. 2. As a building block for certain numerical methods, such as the Godunov method [Godunov, 1959; LeVeque, 1992]. 3. For its implementation in streamline/streamtube simulators [Batycky et al., 1997; Blunt et al., 1996; King and Datta-Gupta, 1998; Thiele et al., 1995a,b]. 4. To develop improved relative permeability models [Chavent et al., 1999]. 5. For its use as a benchmark solution to validate numerical methods. Since a complete, and general, analytical solution to one-dimensional three-phase flow was not available, such solution constitutes by itself a fundamental advance in the area of displacement theory of multiphase flow.

1.1.4

The need for stabilized numerical methods

Despite the usefulness of analytical solutions, in all real-life cases the mathematical problem needs to be solved numerically. Some of the factors ruling out analytical solutions in real problems are the presence of heterogeneities and fractures, complex geometries, and time-dependent boundary conditions. Development of novel numerical methods for the full equations of multiphase compositional flow in multidimensions must necessarily start from simplified models in


13

one space dimension. The equations studied in this dissertation are a simplified version of the general multiphase, compositional model. They display, however, some of the essential features that pose difficulties in obtaining satisfactory numerical solutions. In particular, the problem is extremely nonlinear, almost hyperbolic for the case of vanishing capillarity, and the solution naturally develops sharp features like shocks and boundary layers. It is well known that, for the type of problems described above, classical numerical methods either lack stability and produce globally oscillatory solutions, or accuracy —solutions are overly diffusive. An answer to this dichotomy was provided with the introduction of stabilized methods, devised with the intention of obtaining stable solutions which retain high-order accuracy (see, e.g., Franca (ed.) [1998] and the references therein). Stabilized numerical methods are of great interest in reservoir simulation because they allow dealing with very large element Peclet numbers.5 This feature permits reducing drastically the number of gridblocks that would otherwise be required for stability of the computed solution. In Section 1.4 we give an overview of alternative approaches to deal with advection-dominated flows, and concentrate on the development of stabilized finite element methods, and their recent re-interpretation from the point of view of multiscale phenomena.

1.2

Objectives

This dissertation aims at improving the current mathematical description of threephase flow in porous media, and its numerical solution. In doing so, we attempt to 5

The Peclet number is a dimensionless quantity that measures the importance of advection relative to diffusion. Large Peclet numbers imply that the problem is advection-dominated.


14

provide an answer to some of the issues raised in the previous section. We summarize the specific objectives of the dissertation in the following items: 1. Analysis of the mathematical character of the three-phase flow equations. The goal is to show that key mathematical properties of the governing equations are linked to, and understood from, the physics of three-phase flow. In particular, the mathematical character of the capillarity-free equations should be compatible with the notion of fluid displacements that the equations are supposed to describe. 2. Derivation of appropriate conditions on the relative permeabilities. The existing literature on the subject first assumes a particular behavior of the relative permeabilities and then infers the mathematical character of the system of equations. We adopt the opposite viewpoint. We first impose that the mathematical character of the equations be such that it preserves physical behavior of the solution, and then derive conditions on the relative permeabilities so that this requirement is satisfied. The conditions derived in this manner should then be justified on physical grounds, and compared against experimental data. 3. Extension of the Buckley-Leverett theory to three-phase flow. The solution to the Riemann problem6 of two-phase flow with negligible capillarity was presented in Buckley and Leverett [1942]. One of the objectives of this dissertation is to derive a general solution to the Riemann problem of three-phase flow for any initial and injected conditions. 4. Development and application of stabilized finite element methods. The final 6

The Riemann problem refers to a mathematical problem defined by an evolution equation in an unbounded one-dimensional domain, with initial constant data separated by a single discontinuity (see Chapter 5).


15

goal of this work is to formulate novel techniques for the numerical solution of the challenging problem of multiphase flow with vanishing capillarity effects. The proposed formulation is a stabilized finite element method derived from the notion of multiscale phenomena [Hughes, 1995], where the stabilizing terms arise naturally in a variational multiscale method [Hughes et al., 1998]. By performing a multiple-scale decomposition of the solution, we acknowledge the fact that some of features of the solution cannot be captured with any grid. This idea, which is now dominant in computational fluid mechanics, is adopted here for the simulation of multiphase, porous media flow. The major benefit of this numerical formulation is that the oscillatory behavior of the classical Galerkin method is drastically reduced. This is achieved without compromising the computational cost of the method, or the accuracy of the solution.

1.3 1.3.1

Review of displacement theory Displacement theory

By displacement theory we refer to the mathematical description of one-dimensional multiphase displacements under the conditions of small capillarity effects. If the capillarity term is indeed neglected, and dropped from the formulation, the solution is allowed to be discontinuous. Otherwise, the solution is continuous, and capillarity will affect only the local structure of shocks, but not the global structure of the solution. 1.3.1.1

Two-phase flow

Displacement theory in multiphase systems was first described by Buckley and Leverett [1942] using a fractional flow formulation. They developed a mathematical


16

solution to the problem of one-dimensional displacement of oil by water or gas ignoring the effects of gravity and capillarity. Their solution consisted of a shock and a rarefaction —“initial” and “subordinate” phases of displacement, respectively— thus avoiding a triple-valued solution. A quantitative analysis of the effects of fluid viscosity, gravity and capillarity in two-phase immiscible flow was presented in Terwilliger et al. [1951], Rapoport and Leas [1953], Kyte and Rapoport [1958], Fayers and Sheldon [1959], and Cardwell Jr. [1959]. Welge [1952] used the self-similarity of the Buckley-Leverett solution to the “saturation” equation to obtain relative permeability ratios from two-phase displacement experiments. This technique was extended by Johnson, Bossler and Naumann [1959], who described a method —now known as the JBN method— for calculating individual relative permeabilities, based on incorporating the “pressure” equation in the ´ analysis. A similar procedure had been published in the Russian literature by Efros [1956]. Graphical constructions for the JBN method were later presented by Jones and Roszelle [1978]. Today, relative permeability experiments are interpreted by history matching the production and/or saturation data, usually accounting for gravity and capillarity. These methods assume a certain functional form of the relative permeability and capillary pressure curves, and optimize parameters to minimize the discrepancy between predictions and measurements. The forward model may use an analytical [Helset et al., 1998] or a numerical solution [Kulkarni et al., 1998; Mejia et al., 1995; Nævdal et al., 2000].


17

Figure 1.2. Conceptual model of three-phase flow displacement of Kyte et al. [1956], as a sequence of two successive two-phase flow displacements. The conceptual model is correct only under very restrictive conditions. 1.3.1.2

Three-phase flow

The presence of free gas was found, very early on, to affect significantly oil recovery by water flooding (see Holmgren and Morse [1951] and the references therein). The first conceptual model of three-phase flow displacement was presented by Kyte et al. [1956], who assumed that the process consisted in two successive two-phase displacement events (see Figure 1.2). This conceptual model, which is correct only under very restrictive initial and injected conditions, has been very influential and widely used in the analysis of water floods [Pope, 1980; Willhite, 1986]. Extensions of the fractional flow approach —Buckley-Leverett model— to threephase flow first appeared in the Russian literature [Filinov, 1967; Stklyanin, 1960]. A detailed mathematical formulation is given by Peaceman [1977], and Aziz and Settari [1979], where the hyperbolic form of the three-phase flow equations is presented, and the “pressure” and “saturation” equations are derived. Application of the fractional flow theory to multiphase/multicomponent flow was presented by Pope [1980] and Helfferich [1981], exploiting the analogy with multicomponent chromatog-


18

raphy [Helfferich and Klein, 1970]. Their discussion of three-phase flow is restricted, however, to linear relative permeability functions. Extensions of the theory, confined to the case where each relative permeability is a function of its own phase saturation, and for particular initial and injected states, have been presented by Shalimov [1972], and Falls and Schulte [1992a,b]. Similar extensions have been proposed for steamdrive processes (see, e.g., Bruining and van Duijn [2000] and the references therein). Marchesin and Plohr [2001] present a survey of recent mathematical theory of immiscible three-phase flow, which we review critically in Section 1.3.4. Of particular relevance in relation with the scope of this dissertation are the developments of Guzmán [1995], and Guzmán and Fayers [1997a,b]. They constructed analytical solutions to the Riemann problem of three-phase flow, with and without gravity, and using different classical relative permeability models. The limitations of their work come in two flavors: 1. Although singularities like elliptic regions are found for the relative permeability models they use, their impact is disregarded, claiming that “The elliptic regions are generally so small that posing initial states inside these regions has negligible practical interest.” [Guzmán and Fayers, 1997b, p. 310] 2. Only generic guidelines for the construction of solutions is given, arguing that “A complete and detailed guide to determine solutions would be very extensive and impractical due to the large number of possible types of solutions.” [Guzmán, 1995, p. 165] As shown in Juanes and Patzek [2002e,f], the first limitation may be overcome — rather than dismissed— by imposing appropriate conditions on the relative permeability functions, so that the equations are everywhere strictly hyperbolic. Exploiting


19

this fact, Juanes and Patzek [2002a,g] present the complete analytical solution to the Riemann problem, concluding that only 9 combinations of rarefactions, shocks and rarefaction-shocks are possible. The first attempt to extend the Welge/JBN method [Johnson et al., 1959; Welge, 1952] for the interpretation of three-phase displacement experiments is that of Sarem [1966]. His analysis assumed that the relative permeability of each phase is a function of the saturation of that phase. An extension of the method, which overcomes this limitation, was proposed independently by Virnovskii [1984], and Grader and O’Meara Jr. [1988]. However, their analysis is valid only after breakthrough of the injected phase and, therefore, incomplete. The existing mathematical theory has been used in the forward simulation of three-phase displacement experiments [Hicks Jr. and Grader, 1996; Sahni et al., 1996; Siddiqui et al., 1996]. Because a general analytical framework has been lacking, all optimization methods of three-phase flow functions [Chavent et al., 1999; Mejia et al., 1996; Nordtvedt et al., 1997] still rely on a numerical solution of the forward problem.

1.3.2

Relative permeability experiments

It is recognized in this dissertation that relative permeabilities must ultimately be determined experimentally. Here we summarize laboratory experiments reported in the literature for the determination of relative permeabilities. Reviews of three-phase experiments are given by Saraf et al. [1982], Honarpour et al. [1986], Baker [1988], and Blunt [2000].

Chapter 1. Introduction 1.3.2.1

20

Early experiments of two-phase flow

In his pioneering work, Richards [1931, p. 323] observed that “the essential difference between flow through a porous medium which is saturated and flow through a medium which is unsaturated is that under this latter condition the pressure is determined by capillary forces and the conductivity depends on the moisture content of the medium.” In petroleum engineering, the experimental study of Wyckoff and Botset [1936] on the flow of gas and liquid through a horizontal core of unconsolidated sand, allowed them to obtain relative permeability-saturation curves, and was the basis for the theoretical description of two-phase flow of Muskat and Meres [1936] and Muskat et al. [1937]. A descriptive account of early experiments of two-phase flow [Botset, 1940; Dunlap, 1938; Leverett, 1939; Muskat et al., 1937; Wyckoff and Botset, 1936] is given by Muskat [1949]. The main observation is that relative permeability curves depend mostly on the pore-size distribution and cementation of the porous medium, and are almost insensitive to viscosities and interfacial tension of the fluids, and the total flow rate. It was also recognized early on [Geffens et al., 1951; Levine, 1954; Muskat, 1949; Osoba et al., 1951] that relative permeabilities are not single-valued functions of saturation, but depend also on saturation history. 1.3.2.2

Steady-state experiments of three-phase flow

The first experimental study of three-phase flow (steady-state experiments on unconsolidated sands) was conducted as early as in 1941 by Leverett and Lewis [1941]. Because of the inherent laboratory difficulties, the number of experimental studies of three-phase relative permeabilities conducted since then is relatively small [Baker, 1993; Caudle et al., 1951; Corey et al., 1956; Jerauld, 1997b; Oak, 1990, 1991; Oak et al., 1990; Reid, 1956; Saraf et al., 1982; Saraf and Fatt, 1967; Schneider and Owens,


21

1970; Snell, 1962]. Except for the more recent experiments reported by Oak [1991] and Jerauld [1997b], all other investigations are for water-wet rocks. Saraf et al. [1982] and Oak et al. [1990] provide a summary of the saturation paths —sequence of steady states— and the shape of the isoperms, obtained in each investigation. A common observation is that the relative permeability to the wetting phase is essentially a function of the wetting phase saturation, and almost identical to that of two-phase flow. On the other hand, relative permeabilities of the nonwetting and intermediatewetting phases vary with the fluid saturations of all three phases, the intermediatephase isoperms deviating the most from straight lines. Interpretation of steady-state relative permeability experiments relies on minimizing end-effects to obtain uniform saturations, and applying the multiphase extension of Darcy’s law directly. 1.3.2.3

Unsteady experiments of three-phase flow

The importance of connate water and initial gas saturation on dynamic displacement of oil was confirmed as early as in the 1940’s [Dickey and Bossler, 1944; Holmgren, 1949; Holmgren and Morse, 1951; Welge, 1949]. The lack of quantitative theory of displacement experiments delayed reports of unsteady relative permeabilities until the work of Sarem [1966], Donaldson and Dean [1966], Saraf et al. [1982], and van Spronsen [1982]. In fact, it was not until the 1980’s that the equivalent of the Welge [1952] construction of the two-phase Buckley-Leverett theory was successfully extended to three-phase flow [Grader and O’Meara Jr., 1988] (see Section 1.3.1). Nowadays, most relative permeability studies are conducted using displacement experiments, which normally include in-situ saturation measurements [DiCarlo et al., 2000a,b; Eleri et al., 1995; Goodyear and Jones, 1995; Hicks Jr. and Grader, 1996;


22

Naylor et al., 1995; Nordtvedt et al., 1997; Sahni et al., 1998; Siddiqui et al., 1996; Skauge et al., 1994].

1.3.3

Relative permeability models

We give in this section a thorough account of existing relative permeability models, which we organize in the following categories: 1. Capillary models. The first three-phase relative permeability model should be attributed to Rose [1949]. He used an analogy of the Kozeny-Carman equation [Carman, 1937; Kozeny, 1927] for multiphase flow, and the capillarity theory of Leverett [1941], to obtain wetting-phase relative permeabilities. Extension to the nonwetting and intermediate-wetting phases was done using heuristic arguments. Similar expressions were obtained by Purcell [1949] using the analogy of a bundle of capillary tubes, and Rapoport and Leas [1951] using thermodynamic arguments. Later modifications of the Rose-Purcell model to account for tortuosity of the porous medium include those of Fatt and Dykstra [1951] and Burdine [1953]. Corey, Rathjens, Henderson and Wyllie [1956] and Brooks and Corey [1966] used the model of Burdine [1953] with particularly simple capillary pressure curves.7 Models of non-wetting and intermediate-wetting phase trapping were proposed by Naar and Henderson [1961], Naar and Wygal [1961] and Land [1968] for imbibition relative permeabilities. A capillary model of relative permeabilities similar to that of Rose-Purcell was derived by Mualem [1976] for two phase flow, and later used by Lenhard and Parker [Lenhard and 7

Contrary to the wide-spread terminology of “Corey-type models” as models in which the relative permeability of each phase depends solely on the saturation of that phase, Corey et al. [1956] proposed a model of oil relative permeability which depends on all fluid saturations, so that the oil isoperms are not straight lines (see Equation (1) and Figure 4 of Corey et al. [1956]).


23

Oostrom, 1998; Lenhard and Parker, 1987; Lenhard et al., 1989; Parker and Lenhard, 1987; Parker et al., 1987] in a sophisticated three-phase hysteretic model of multiphase flow constitutive relations. 2. Probability models. These models estimate three-phase oil relative permeability from two-phase data, relying on the assumption that “each fluid establishes its own tortuous paths, which form very stable channels” [Bear, 1972, p. 457]. Probably, the most popular three-phase relative permeability models in petroleum engineering are the Stone I and Stone II models [Stone, 1970, 1973], normalized by Aziz and Settari [1979, pp. 33–37]. Different interpolation schemes for the residual oil saturation, which is a free parameter in the Stone I model, are given in Fayers and Matthews [1984] and Fayers [1987]. Alemán and Slattery [1988], and Robinson and Slattery [1994] have proposed a model similar to Stone I. 3. Power-law models. These purely empirical models have been used both in twophase [Irmay, 1954] and three-phase flow [Delshad and Pope, 1989]. 4. Saturation-weighted models. The first saturation-weighted interpolation of twophase relative permeabilities was proposed by Baker [1988]. Improved saturationweighted models, which account for trapping, are presented in Jerauld [1997a] and Blunt [2000], among others. 5. Pore-network and process-based models. The quasi-static pore-network models of multiphase flow allow one to predict relative permeabilities in two-phase [AlFutaisi and Patzek, 2002; Blunt, 2001; Blunt and King, 1991; Øren et al., 1998; Patzek, 2001], and three-phase flow [Al-Futaisi and Patzek, 2003; Fenwick and Blunt, 1998a,b; Lerdahl and Øren, 2000; Piri and Blunt, 2002; van Dijke et al., 2002], if topology and geometry of the pore space are known [Øren and Bakke,


24

2002]. Recently, van Dijke, McDougall and Sorbie [2001] and van Dijke, Sorbie and McDougall [2001] proposed three-phase relative permeability models with “correct physics” limited to a bundle of capillaries.

1.3.4

Limitations and inconsistencies of current models

The fractional flow approach for immiscible incompressible three-phase flow leads to a 2×2 system of conservation laws —the “saturation” equations— and a “pressure” equation whose solution is trivial in the one-dimensional case [Chavent and Jaffré, 1986; Peaceman, 1977]. Capillarity effects enter the formulation as a diffusion-like term. In the absence of capillary forces, it seems natural to think that the system of partial differential equations should be strictly hyperbolic for any relative permeability functions. The theory of strictly hyperbolic systems was put together by Lax [1957], and extended by Liu [1974, 1975] to systems with fields which are neither genuinely nonlinear nor linearly degenerate (for scalar conservation laws, the theory of Liu [1974] reduces to the case of nonconvex flux functions studied by Oleinik [1957]). Some important remarks about the structure of the solution are given in a recent paper by Ancona and Marson [2001]. An introduction to the theory of conservation laws is given in the books by LeVeque [1992], Smoller [1994], and Dafermos [2000]. Charny [1963] is one of the first to study the mathematical character of the threephase flow equations. He mentions that one could envision such a situation when the system of equations would be of mixed elliptic/hyperbolic type, leading to a Tricomi problem [Tricomi, 1961]. However, he concludes that, in physically realistic flows in porous media, the system of equations is hyperbolic. This work did not permeate to the Western literature, where three-phase flow through porous media caught the attention of the mathematically-oriented community ever since Bell, Trangenstein


25

Figure 1.3. Elliptic region inside the saturation triangle for the Stone I model, as presented by Bell et al. [1986]. The parameters of the two-phase relative permeability curves were selected to magnify the extent of the non-hyperbolic region. and Shubin [1986] showed that, for certain relative permeability functions, the system is not necessarily hyperbolic. In particular, they observed that Stone I relative permeabilities give rise to regions inside the saturation triangle where the system of equations was elliptic in character —the so-called elliptic regions— (see Figure 1.3). We summarize the conclusions of recent research on this topic in the following points: 1. The analysis of Shearer [1988], Shearer and Trangenstein [1989], and Holden [1990a,b], suggests that elliptic regions are unavoidable in three-phase flow models with “general” relative permeabilities, that is, models where all relative permeabilities are allowed to be functions of both the water and gas saturations. 2. It was concluded in Marchesin and Medeiros [1989], and Trangenstein [1989], that the only relative permeability models which do not produce elliptic regions are those where the relative permeability of each phase depends solely on the saturation of that phase —“Corey-type” models. These models display,


26

however, isolated umbilic points, where the system is not strictly hyperbolic. 3. It has been shown that the effect of gravity is not always stabilizing, so that the size and strength of the elliptic regions may increase [Guzmán and Fayers, 1997a; Hicks Jr. and Grader, 1996; Jackson and Blunt, 2002; Shearer and Trangenstein, 1989]. 4. Therefore, all commonly used models of relative permeabilities produce one of the following mathematical singularities: (a) Models with umbilic points —nonstrictly hyperbolic systems. Umbilic points act as “repellers” for classical waves [Guzmán, 1995; Guzmán and Fayers, 1997b; Marchesin and Plohr, 2001] and, as a result, solutions to the nonstrictly hyperbolic system require nonclassical waves —termed transitional waves [Isaacson et al., 1989]. The most salient features of the solution are that: (1) it is sensitive to the particular form of the diffusion term due to capillary effects [Isaacson et al., 1992]; (2) the saturation path may be the same for wildly different initial and injected states [Falls and Schulte, 1992a,b; Guzmán, 1995]; and (3) in WAG displacement, the intermediate oil bank may be split in two [Marchesin and Plohr, 2001]. A classification of nonstrictly hyperbolic systems with quadratic flux functions is given by Schaeffer and Shearer [1987a,b], where cases I and II in their classification are relevant to three-phase flow. Solutions of the Riemann problem have been studied by Isaacson et al. [1989, 1990, 1992], de Souza [1992, 1995], Marchesin et al. [1997], and Schecter et al. [1996], among others. (b) Models with elliptic regions —mixed elliptic/hyperbolic systems. From a purely mathematical viewpoint, mixed elliptic/hyperbolic systems are par-


27

ticularly challenging: the solution structure in the neighborhood of the elliptic regions is extremely complicated [Holden, 1987; Holden and Holden, 1989; Holden et al., 1990], and the correct entropy condition is not yet known, which leads to nonunique solutions [Azevedo and Marchesin, 1990, 1995]. Loss of strict hyperbolicity in the theory of one-dimensional incompressible three-phase flow has been identified by many researchers [Azevedo and Marchesin, 1990; Bell et al., 1986; Chavent and Jaffré, 1986; de Souza, 1992, 1995; Falls and Schulte, 1992a,b; Fayers, 1987; Guzmán, 1995; Hicks Jr. and Grader, 1996; Holden, 1990a,b; Isaacson et al., 1992; Jackson and Blunt, 2002; Marchesin and Medeiros, 1989; Marchesin and Plohr, 2001; Marchesin et al., 1997; Schaeffer and Shearer, 1987a,b; Schecter et al., 1996; Shalimov, 1972; Shearer, 1988; Shearer and Trangenstein, 1989; Trangenstein, 1989]. Some defend the implications of such mathematical singularities [de Souza, 1995; Falls and Schulte, 1992a,b; Isaacson et al., 1992; Jackson and Blunt, 2002; Marchesin and Plohr, 2001; Schaeffer and Shearer, 1987a; Shalimov, 1972], whereas others do not attach any physical significance to them [Azevedo and Marchesin, 1990; Balbinski et al., 1999; Chavent and Jaffré, 1986; Chavent et al., 1999; Fayers, 1987; Guzmán, 1995; Marchesin et al., 1997; Miller et al., 1998; Sahni et al., 1996; Schecter et al., 1996; Shearer and Trangenstein, 1989; Trangenstein, 1989]. Jackson and Blunt [2002] took a unique step in justifying elliptic regions as physically plausible, by using a serial model of capillary bundles to demonstrate that elliptic regions exist in a simplistic but physically realizable porous medium. From our discussion in Section 1.1.2, it is difficult to justify the physical meaning of mathematical singularities like elliptic regions and umbilic points. Recently, Juanes


28

and Patzek have revisited the problem of change of type in three-phase flow models, and offered an new perspective on this topic. We summarize the main findings in four points, which address each of the items above. 1. Juanes and Patzek [2002e,g] identify that the presence of elliptic regions is a direct consequence of the behavior assumed for the two-phase relative permeability functions at the edges of the saturation triangle. More precisely, Shearer [1988] and Holden [1990a] assume, without further discussion, that all relative permeability functions have zero derivative at the endpoint saturation. This assumption is critically reviewed in Juanes and Patzek [2002e], where a novel approach is presented: to honor the physics of multiphase displacements, strict hyperbolicity of the equations is enforced, and appropriate conditions on the relative permeabilities are derived. It is also shown that these conditions are justified from pore-scale considerations, and supported by experimental data. 2. The fact that the only relative permeability models known not to produce elliptic regions are “Corey-type” models is not appealing. In these models, the relative permeability of each phase is allowed to depend only on the saturation of that phase. This overly simplistic behavior is not supported by experimental observations (see, e.g., Oak et al. [1990]) and pore-scale models [Al-Futaisi and Patzek, 2003; van Dijke et al., 2002]. 3. When Marchesin and Medeiros [1989] and Trangenstein [1989] determine that “Corey-type” models are the only relative permeability models that do not produce elliptic regions, they do so by neglecting a priori any potential dependence of the relative permeabilities on the gravity number, and assuming from the very onset that relative permeabilities are fixed functions of fluid saturations. The


29

same hypothesis is made in all other investigations that study the effects of gravity on the character of the equations [Guzmán, 1995; Guzmán and Fayers, 1997a; Hicks Jr. and Grader, 1996; Jackson and Blunt, 2002; Shearer and Trangenstein, 1989]. In view of our comments in Section 1.1.1, this assumption is not realistic. By understanding the relative permeability as functionals that depend on the gravity number and the fluid viscosity ratio, Juanes and Patzek [2002f] extended the analysis in Juanes and Patzek [2002e], and demonstrated that it is possible to remove elliptic behavior also when gravity is included in the formulation. 4. Regarding the physical interpretation of elliptic regions, there seem to be many reasons why mixed elliptic/hyperbolic character does not conform to the expected behavior of multiphase displacements. For example, the principle of causality 8 may be violated, implying that boundary conditions may need to be imposed at later times to control the solution at earlier times [Fayers, 1987]. The analysis of Bell et al. [1986] shows that the solution is unstable inside the elliptic regions. One of the consequences is that, for arbitrarily close left and right states inside the elliptic region, the solution develops wildly oscillatory waves (see Figure 1.4), which are never observed in experiments. Moreover, the wave pattern is unstable with respect to the initial states. Juanes and Patzek [2002e] and Juanes et al. [2003] give additional physical arguments against change of type in the governing equations. In relation to the justification of elliptic regions as physically plausible, given by Jackson and Blunt [2002], it is shown in Juanes et al. [2003] that a three-phase displacement process in a sequence of 8

“The causality of natural processes may be interpreted as implying that the conditions in a body at time t are determined by the past history of the body, and that no aspect of its future behavior need to be known in order to determine all of them.” [Truesdell and Noll, 1965, p. 56]


30

bundles cannot be described in the form proposed by the authors. Therefore, the relative permeabilities used in Jackson and Blunt [2002] should be understood as another empirical model, and the presence of elliptic regions in their model does not imply that they exist in reality.

1.4

Review of stabilized and multiscale methods

In Chapter 2 we adopt the fractional flow formalism [Chavent and Jaffré, 1986; Chen and Ewing, 1997] in deriving the governing equations of three-phase flow. Under the assumption of incompressibility of the rock and the fluids, the pressure equation is of elliptic type. If compressibility effects are taken into account, then it is a (usually stiff) parabolic equation. The current trend for its numerical solution is the use of mixed or hybrid finite element methods [Brezzi and Fortin, 1994], since they provide a better resolution of the total velocity than standard approximations [Durlofsky, 1994; Ewing, 1996]. Stabilized finite element methods, which are the subject of our review in Section 1.4.2, have recently been proposed in this context [Masud and Hughes, 2002]. Because the solution to the pressure equation is trivial in the one-dimensional case studied here, we focus on the numerical solution of the system of saturation equations. As mentioned in Section 1.1.4, we are interested in the case of small capillarity, for which the system of equations becomes almost hyperbolic (see the discussion in Section 1.3.4). As a result, the mathematical problem is in the realm of advectiondominated flows. The search for a successful numerical strategy is complicated by the fact that the equations are highly nonlinear.


31

Figure 1.4. Evolving solution of a mixed elliptic/hyperbolic system for left and right states inside the elliptic region, as presented by Bell et al. [1986]. Water and gas saturation profiles are plotted at four different times. A nonphysical wave structure emerges from arbitrarily close initial and injected states.


1.4.1

32

Alternative approaches to advection-dominated flows

An enormous body of literature has emerged to address the fundamental problem of advection-dominated flows. It is not the purpose of this section to give a complete account, and a historical perspective, of the development of such methods. We limit our review to some of the most relevant contributions in the area. The origin of most numerical methods for advection-dominated flows —modeled by nonlinear systems of equations— starts from the numerical solution of the model advection-diffusion equation, which is a linear scalar equation. Ewing and Wang [2001] organize these methods in two categories: (1) Eulerian methods, and (2) characteristics methods. We shall add a third category, encompassing (3) moveable-node methods. 1.4.1.1

Eulerian methods

Eulerian methods use a fixed grid to discretize the equations in space, typically using finite difference, finite element, or finite volume techniques [Thomée, 2001]. Regarding the time discretization, there are essentially two possibilities: (a) The method of lines. The space discretization is performed first to obtain a system of ordinary differential equations. This system is then solved by a time stepping technique like finite differences, backward difference formulae, or Runge-Kutta methods [Thomée, 1997]. (b) A space-time discretization. The equations are discretized simultaneously in space and time, generally using a single time slab. Marching in time is accomplished by weakly enforcing continuity of the solution between time steps. Recent methods of solution falling in the category of Eulerian methods include:


33

1. Total variation diminishing (TVD) methods. These methods require that the total variation of the numerical solution decreases as it evolves with time [LeVeque, 1992]. This property is achieved through the use of flux limiters [Harten, 1983; Roe, 1985; Sweby, 1984] or slope limiters [Colella, 1985; Goodman and LeVeque, 1988; van Leer, 1973, 1974, 1977a,b, 1979] 2. Essentially nonoscillatory (ENO) schemes. These are high-order methods which also limit the total variation of the solution, by locally adapting the stencil of the interpolation scheme [Harten, 1987; Harten et al., 1987, 1986; Shu, 1997]. 3. Discontinuous Galerkin (DG) methods. These are finite element methods that employ completely discontinuous piecewise polynomials for both the trial and test function spaces [Johnson et al., 1984; Johnson and Pitkäranta, 1986]. They have been recently combined with slope limiters and Runge-Kutta time stepping techniques [Cockburn et al., 1990, 1989; Cockburn and Shu, 1989, 1998]. 4. Stabilized finite elements. Because these are the methods of choice in this dissertation, they are described more fully in Section 1.4.2. 1.4.1.2

Characteristics methods

The distinctive feature of all characteristic methods is that the time integration of the equations is performed along the characteristics of the solution. Among them, we find: 1. Particle tracking. In this method, the domain is divided into a number of cells, and a number of particles is placed in each cell. The governing equation is then used to move the particles from cell to cell [Farmer, 1980; Garder et al., 1964].


34

2. Eulerian-Lagrangian methods. One such method is the one developed by Neuman [1981, 1984], where a combination of forward and backward tracking of the characteristics is used. 3. Modified method of characteristics (MMOC). This method employs backward tracking of the characteristics, and standard finite element test functions [Douglas Jr. and Russell, 1982; Pironneau, 1982; Thomaidis et al., 1988]. 4. Eulerian-Lagragian localized adjoint methods (ELLAM). This formalism was introduced in Celia et al. [1990], and analyzed in Herrera et al. [1993], for the solution of the one-dimensional advection-diffusion equation. It is based on a space-time formulation of the problem, where a characteristic split is used to define the test functions. This formulation has been studied in a large number of publications (see, e.g., Binning and Celia [1996]; Dahle et al. [1995]; Wang, Dahle, Ewing, Espedal, Sharpley and Man [1999]; Wang, Ewing, Qin, Lyons, Al-Lawatia and Man [1999]). 5. Characteristic mixed finite element method (CMFEM). This method shares features with ELLAM, but uses a mixed method by introducing a new variable for the fluid flux [Arbogast et al., 1992; Arbogast and Wheeler, 1995; Yang, 1992]. The main underlying idea of characteristics methods is to split the advective and diffusive terms, adopting a Lagrangian viewpoint for the former. The splitting procedure symmetrizes and stabilizes the equations, allowing for larger time steps [Ewing and Wang, 2001]. These methods, however, rely on a good approximation of the characteristics, which requires a fine mesh for highly nonlinear flows, or flows in heterogeneous media, thus restricting severely the range of element Peclet numbers the method can handle.


35

Moveable-node methods

The moving finite element (MFE) method, originally introduced by Miller [Miller, 1981; Miller and Miller, 1981], is a finite element method for transient problems, in which the nodal locations evolve along with the nodal values. These methods have proved very successful for the numerical solution of problems that develop large gradients and shocks with complex structure. Their success stems from automatically concentrating the nodes, and aligning the mesh, along the sharp features of the solution. The method, as originally devised by Miller, has been studied extensively [Baines, 1991, 1994; Baines and Wathen, 1986, 1988; Gelinas et al., 1981; Johnson et al., 1988; Wathen, 1984; Wathen and Baines, 1985]. It was soon recognized that robust techniques for regularization of the equations were needed [Miller, 1983]. This observation lead to the gradient-weighted moving finite element (GWMFE) method, described fully in Miller [1997], and Carlson and Miller [1998a,b]. A comparison of the MFE and GWMFE methods with other moving-grid methods for one-dimensional problems, is given in Furzeland et al. [1990], and Zegeling and Blom [1992]. An area in which moving finite element methods have traditionally had less success is in modeling steady-state, or near steady-state, advection-dominated problems. The reason is that the nodes continue to move with the characteristic speed, and pile up at the outflow boundary of the domain. This issue has recently been addressed — and corrected— in Miller and Baines [2001], with the introduction of a least squares version of the method (LSMFE). Despite their success in several areas of computational mechanics, moveable-node methods have restricted applicability to subsurface flows. The reason is twofold: 1. In these methods, the node locations are unknowns of the problem, therefore


36

introducing a (substantial) additional computational cost. 2. More importantly, real-life problems of flow in porous and fractured media are highly heterogeneous. The mesh used for the numerical solution should honor the heterogeneity of the medium. This requirement seems to be in direct confrontation with the philosophy of moveable-node methods, in which the grid nodes are allowed to move freely.

1.4.2

Stabilized finite element methods

Quoting from the preface to the special issue on stabilized finite element methods in Computer Methods in Applied Mechanics and Engineering [Franca (ed.), 1998], “Stabilized finite elements are constructed by modifying the variational form of a particular problem, such that enhanced numerical stability is achieved without compromising consistency.” Here, we give a succinct review of published work on stabilized finite element methods. We admittedly concentrate on the fluid mechanics literature, and do not include relevant —and sometimes related— developments in the area of solid mechanics, such as: 1. Assumed-strain or enhanced mixed finite element methods [Armero, 2000; Glaser and Armero, 1997; Kasper and Taylor, 2000a,b; Simo and Armero, 1992; Simo et al., 1993; Taylor, 2000]. 2. Localization and damage in models of continuum mechanics [Armero, 1999a,b, 2001; Armero and Callari, 1999; Armero and Garikipati, 1996]. The presentation of stabilized methods given here is strongly influenced by the paper by Codina [1998]. Stabilized methods can deal with, at least, two types of instabilities:


37

1. The so-called velocity-pressure instability arising in mixed formulations [Brezzi and Fortin, 1994], where more than one approximation space is used. 2. Instabilities due to a dominant advection term in advection-diffusion equations [Brooks and Hughes, 1982]. The role of stabilized formulations for instabilities of the first type above was elucidated by Franca and Hughes [1988], where they identified two classes of mixed methods: (1) those satisfying the Babuska-Brezzi (BB) condition [Babuska, 1971; Brezzi, 1974], and (2) those circumventing the BB condition. It is precisely the methods that circumvent the BB condition that fall in category of stabilized methods. We are interested, however, in the stabilization of advection, and we shall restrict the scope of this review to stabilized methods developed for that purpose. 1.4.2.1

Stabilized variational formulations: SUPG, GLS, and beyond

The origin of stabilized finite element methods —at least in the form in which they are used today— can be traced back to the upwinding technique of Hughes and Brooks [1979], where the artificial diffusion introduced by the method acts only in the direction of streamlines. The streamline diffusion method was then incorporated in the context of a variational formulation, leading to the streamline upwind Petrov-Galerkin (SUPG) method [Brooks and Hughes, 1982; Hughes and Brooks, 1982]. The analysis of this method —under the name of streamline diffusion (SD)— was initiated in Johnson and Nävert [1982] for the linear advection-diffusion equation, and continued with extensions to time-dependent problems in Johnson et al. [1984], and to nonlinear hyperbolic conservation laws in Johnson and Szepessy [1987] (see also Johnson [1987]). The SUPG formulation was then extended to systems of equations in Hughes and Mallet [1986a], and analyzed in Hughes et al. [1987].


38

Numerical investigation of the Stokes problem [Hughes and Franca, 1987; Hughes, Franca and Balestra, 1986] originated the idea of the Galerkin least-squares (GLS) method, introduced in Hughes et al. [1989], and extended to transient problems in Shakib and Hughes [1991] using a space-time formulation based on a time discontinuous Galerkin strategy. A slightly, but essentially, different stabilized method was introduced by Douglas Jr. and Wang [1989] for the Stokes problem. The difference with respect to the GLS method of Hughes and Franca [1987] was only in the sign of the viscous operator applied to the test function. This apparently small change results in improved stability properties. The method of Douglas Jr. and Wang [1989] was generalized by Franca et al. [1992], and applied to the advection-diffusion equation [Franca et al., 1992], and the incompressible Navier-Stokes equations [Franca and Frey, 1992]. This formulation was later understood as a particular instance of a class of stabilization techniques emanating within the framework of multiscale phenomena (see Section 1.4.3). 1.4.2.2

Design of the matrix of stabilizing coefficients

Common to the developments of most —if not all— of the methods mentioned above, is the need for a good design of the matrix of stabilizing coefficients τ , which invariably appears in the stabilizing terms added to the variational formulation. This fact has been sharply summarized in Shakib et al. [1991, p. 151]: “The structure of τ is the crux of this method. Although the construction of the least-squares operator is simple and straightforward, the construction of τ is not.” Here we review briefly several options that have been considered in the literature. An optimal expression of the intrinsic time τ was known, in the context of upwinding techniques with linear elements, for the scalar, steady-state, one-dimensional,


39

advection-diffusion equation [Heinrich and Zienkiewicz, 1979]. This expression, which leads to nodally-exact numerical solutions, was adopted in the framework of the SUPG method in Hughes and Brooks [1982], and Brooks and Hughes [1982]. Similar developments for quadratic elements were proposed by Christie and Mitchell [1978] and Heinrich [1980], and revisited in the context of stabilized methods in Shakib and Hughes [1991], Franca et al. [1992], and Codina et al. [1992]. Extensions in the formulation of τ for the scalar advection-diffusion equation with a reaction term were proposed by Tezduyar and Park [1986], Harari and Hughes [1994], Idelsohn et al. [1996], Codina [1998], and Franca and Valentin [2001]. All these formulations were reviewed, and their performance compared in both the exponential and propagating regimes of the equation, by Hauke and Garc´ıa-Olivares [2001] and Hauke [2002]. It was soon realized that extension of the τ formulation for the scalar equation to systems of advection-diffusion-reaction equations was not trivial. Effort in this direction was initiated in Hughes and Mallet [1986a]. The basis for the proposed extension is to diagonalize the system of equations, by solving an eigenvalue problem, and transform the matrix of stabilizing coefficients accordingly. A different formulation of the matrix of stabilizing coefficients was presented in Shakib et al. [1991] in the context of the space-time Galerkin least-squares (GLS) method, and it involves calculation of the square-root inverse of a matrix. This formulation was later justified in Codina and Blasco [2002] using a completely different argument. Yet another design of the matrix of intrinsic time scales was proposed in Codina [2000] for systems of advection-diffusion-reaction equations in multidimensions. It is similar to the one proposed in Shakib et al. [1991], but slightly simpler, and involves the calculation of a matrix inverse. A radically different approach for the evaluation of τ was presented


40

in Tezduyar and Osawa [2001], where it is computed from the element-level matrices and vectors. Several of these formulations are used and contrasted in Chapter 7, for the simulation of three-phase flow in porous media. It is important to note that these “generic” formulations of τ will not work for all problems of interest. In Codina [2000, 2001, 2002], examples are given of problems that can be recast in the form of advection-diffusion-reaction systems, for which ad hoc expressions of τ must be used. The choice should be ultimately dictated, in any case, by convergence analysis of the method. 1.4.2.3

Shock-capturing techniques

Even when stabilized finite element methods are successful at attaining global stability of the numerical solution, they do not rule out the possibility of local overshoots and undershoots in the neighborhood of sharp gradients. The reason for this behavior is that these methods do not emanate from a monotonicity argument [LeVeque, 1992]. It is well known that a linear monotonicity-preserving method is at most first order accurate [Godunov, 1959; LeVeque, 1992]. Therefore, to obtain a high-order method that precludes spurious oscillations, one must necessarily resort to nonlinear methods, that is, a numerical scheme that will be nonlinear even if the equation being solved is linear. The underlying idea of all shock-capturing techniques is to increase the numerical dissipation in the regions where the solution is not smooth. One of the earliest attempts in this direction was presented in the seminal paper by von Neumann and Richtmyer [1950]. For a review of many other strategies in the context of conservation laws and finite difference methods, see LeVeque [1992]. In the framework of finite element variational formulations, most discontinuitycapturing formulations can be expressed as an extra diffusion term [Codina, 1993],


41

which should satisfy the following design conditions [Hughes and Mallet, 1986b; Shakib et al., 1991]: (1) consistency, which implies that the operator has to be proportional to the residual of the Euler-Lagrange equations; (2) enhanced stability, by providing extra control over the gradient of the numerical solution; and (3) accuracy, by vanishing quickly in the regions where the solution is smooth. The literature on discontinuity-capturing formulations is large [Codina, 1993; Dutra do Carmo and Galeão, 1991; Galeão and Dutra do Carmo, 1988; Hughes and Mallet, 1986b; Hughes, Mallet and Mizukami, 1986; Johnson, 1987, 1992; Johnson and Szepessy, 1987; Johnson et al., 1990; Shakib et al., 1991; Tezduyar and Park, 1986]. In Chapter 7, some of these formulations are adapted, and a novel expression for the shock-capturing diffusion, first proposed in Juanes and Patzek [2002b, 2003b], is compared with previous formulations. 1.4.2.4

Time integration schemes

It was also realized that transient advection-dominated problems suffer from stabilization issues that are distinct from those associated with stationary problems. The literature on the topic of time integration for stable solutions is so vast that we simply refer to the specialized articles by Shakib and Hughes [1991], Shakib et al. [1991], Donea and Quartapelle [1992], and Simo and Armero [1994], and to the classical books by Hughes [1987], Johnson [1987], Thomée [1997], and Zienkiewicz and Taylor [2000].


1.4.3

The multiscale approach

1.4.3.1

Multiscale formulations in fluid mechanics

42

The paper by Hughes [1995] set a landmark in the theory of stabilized methods, and opened new vistas for their development. A multiple-scale formalism was introduced, which allowed to derive variational formulations of problems encompassing multiscale phenomena. The key idea is to decompose the variable of interest (and the test function) into resolved coarse scales, and unresolved subgrid scales. Such decomposition allows to split the weak form of the problem into two subproblems: one for the coarse scales and one for the subgrid scales. After a locality assumption, the subgrid scales are solved for analytically, and their (nonlocal) effect is incorporated into the coarse scales. This procedure yields an enriched problem for the resolvable scales, which is then solved numerically. The formalism described above allows —among other things— a physical interpretation of stabilized methods, as techniques which aim to capture subgrid variability in the solution. In particular, it provides a natural derivation of the method of Douglas Jr. and Wang [1989], as generalized by Franca et al. [1992], which is known as algebraic subgrid scale (ASGS) method thereafter. Moreover, it provides a long sought explanation to the intrinsic time τ , a formula for which now emanates from the identification of stabilized methods with a class of subgrid-scale models. The multiscale formalism of Hughes [1995] was extended and given a formal variational framework in Hughes et al. [1998], where the term variational multiscale method was coined. One of the most important extensions was consideration of distributional effects (Dirac’s layers), which give rise to inter-element jump terms in the formulation. Proper evaluation of these terms, required for consistency of the numerical method, pose additional difficulties which are not yet fully resolved. An approach suggested in Hughes et al. [1998] is the use of edge functions,


43

together with a localization assumption to avoid global coupling of the subgrid-scale equations. A different way to palliate the weak consistency associated with neglecting these boundary terms was proposed in Jansen et al. [1999], and Whiting and Jansen [2001], based on either a global or a local reconstruction of second-derivative terms (the diffusive fluxes). The multiscale method of Hughes [1995] and Hughes et al. [1998] is closely related to other numerical methods of interest for advection-dominated flows, such as: 1. Bubble functions [Baiocchi et al., 1993; Brezzi et al., 1992; Franca and Farhat, 1994, 1995] and residual-free bubbles [Brezzi et al., 1997, 1998; Franca and Russo, 1996, 2000]. 2. Subgrid stabilization of Galerkin approximations [Guermond, 1999, 2001]. 3. The nonlinear Galerkin method [Ammi and Marion, 1994; Burie and Marion, 1997]. The multiscale decomposition originally proposed in Hughes [1995] and Hughes et al. [1998] has been extended to other linear [Codina, 1998, 2000; Hauke, 2002; Hauke and Garc´ıa-Olivares, 2001; Jansen et al., 1999; Masud and Hughes, 2002; Oberai and Pinsky, 1998, 2000] and nonlinear problems [Codina, 2001, 2002; Garikipati and Hughes, 1998, 2000; Hughes et al., 2000; Hughes, Mazzei and Oberai, 2001; Hughes, Oberai and Mazzei, 2001]. The common approach to deal with nonlinear problems is to linearize the equations upfront, using either a Picard or a Newton strategy [Codina, 2001, 2002]. The publications referenced above are just a small sample of a giant effort towards the development of improved multiscale and stabilized methods, which is now a recurrent theme in conferences on computational mechanics [Masud et al., 2001, 2002, 2003].


44

Multiscale formulations in flow in porous media

Multiscale-stabilized methods, which are now dominant in computational fluid dynamics, were introduced in the context of subsurface flow for linear transport and nonlinear two-phase flow problems in Juanes and Patzek [2002b,c]. Additional physical interpretation of the performance of the method was given in Juanes and Patzek [2002d]. These results were then extended in Juanes and Patzek [2003b] to the much more complicated problem of three-phase flow, which is governed by a nonlinear system of conservation laws. The distinctive features of the proposed formulation is that nonlinearity of the equations is retained at the time of invoking the multiscale split, and that a pointwise reconstruction of the multiscale variable is avoided. In addition, several expressions of the matrix of intrinsic time scales are compared, and a novel subscale-driven shock-capturing technique is proposed. It is important to realize that the formulation presented in the aforementioned publications is very different from other methods that account for multiple-scale phenomena, such as the multiscale finite element method [Hou and Wu, 1997], the subgrid upscaling technique [Arbogast, 2000, 2002], and the mortar upscaling method [Peszynska et al., 2002], where the main objective is to incorporate the small-scale heterogeneity. On the other hand, the recent paper by Masud and Hughes [2002] applies the original variational multiscale formulation of Hughes [1995] and Hughes et al. [1998] to porous media flows. It is restricted, however, to the linear scalar equation describing steady-state, single-phase, Darcy flow, and the objective is to remove velocity-pressure instabilities, rather than instabilities arising from the nearly hyperbolic character of the equations.


1.5

45

Overview

This dissertation is organized in two parts. Part I, consisting of Chapters 2 through 5, deals with the physical and mathematical aspects of displacement theory. Part II, which comprises Chapters 6 and 7, is devoted to multiscale numerical methods of solution. In Chapter 2 we review the mathematical formulation of one-dimensional, immiscible, three-phase flow to be used in later chapters. The general formulation is particularized to several special cases of interest. The mathematical character of the equations, and its implications on the relative permeability functions, are studied in Chapter 3. The problem of mixed elliptic/hyperbolic behavior is addressed. We identify conditions that must be satisfied by the relative permeability functions, so that the system of equations is strictly hyperbolic everywhere in the saturation triangle. In Chapter 4, the analysis of the previous chapter is extended to the case when a gravity term is included in the formulation. We argue that, because a Darcytype formulation is necessarily incomplete, relative permeabilities must depend on the gravity number. This observation allows us to remove the nonphysical elliptic behavior. The complete and general analytical solution to the Riemann problem of threephase flow is given in Chapter 5. The wave structure of the solution is discussed in detail, and motivated from the well-known two-phase flow case. By means of a representative example of water/gas injection in a hydrocarbon reservoir, we show that assuming linear saturation paths —a common assumption in petroleum engineering— may lead to very inaccurate predictions. In Chapter 6, a multiscale formulation is proposed for efficient numerical solution


46

of the two-phase flow equation with vanishing capillarity effects. The developments are first presented in the framework of miscible displacements, which are modeled by a linear advection-diffusion equation. Several representative simulations are given, which illustrate the performance of the proposed formulation. The extension of the variational multiscale formulation to three-phase flow is addressed in Chapter 7. The process is governed by a nonlinear system of conservation laws which, in the case of interest, is almost hyperbolic. The derivation of the methodology is given in full detail. Special attention is given to the choice of the matrix of stabilizing coefficients, and a novel discontinuity-capturing technique is proposed and compared with existing formulations. The methodology is applied to the simulation of two problems of great practical interest: oil filtration in the vadose zone, and water-gas injection in a hydrocarbon reservoir. The numerical simulations show the potential and applicability of the formulation for solving three-phase porous media flow on very coarse grids. Chapter 8 concludes this dissertation by summarizing the main results, and suggesting directions for further research. Efficient algorithms for the evaluation of the analytical solution to the three-phase flow Riemann problem are given in Appendix A. In Appendix B, we describe the implementation of the variational multiscale formulation in a finite element code.


47 It is as ridiculous to deride continuum physics because it is not obtained from nuclear physics as it would be to reproach it with lack of foundation in the Bible. — CLIFFORD TRUESSDELL and WALTER NOLL, The Non-Linear Field Theories of Mechanics (1965)

“Simulators solve equations. We must ask, first, Are we solving the right equations, and second, Are the equations correctly solved?” — ROBERT ROSNER, University of Chicago Magazine (December 2002)

48

Part I Displacement theory

49

Chapter 2 Mathematical formulation of three-phase flow 2.1

Introduction

In this chapter we derive the mathematical formulation of three-phase flow. The mathematical model is based on the general considerations of Chapter 1. We adopt a continuum approach, where three-phase flow is modeled through conservation equations, together with certain constitutive relations. Classical formulations rely on a multiphase flow extension of Darcy’s equation, originally proposed by Muskat [1949]. We shall make use of this extension —which is not rigorously derived from first principles [Hassanizadeh and Gray, 1993]— as a modeling assumption, rather than a physical law. By invoking the fractional flow formalism [Chavent and Jaffré, 1986], the problem may be written in terms of a pressure equation and a system of saturation equations. For the purpose of obtaining a mathematical model that is analytically tractable, we assume that relative permeabilities and capillary pressures are functions

Chapter 2. Mathematical formulation of three-phase flow

50

of saturations alone, and that all other factors are accounted for in the particular expression of the functions used for each specific problem. The governing equations are also expressed in dimensionless form. This form of the equations is convenient both for an analytical and a numerical analysis, and elucidates the scaling of the different terms present. In the last section of this chapter we particularize the general three-phase flow equations to several cases of interest, including: (1) negligible gravity effects, (2) negligible capillarity effects, and (3) two-phase flow.

2.2

Continuum equations

Here we present a standard derivation of the equations of three-phase flow in porous media under the following assumptions: 1. One-dimensional flow, 2. Zero distributed sources and sinks, 3. Immiscible fluids, 4. Incompressible fluids, 5. Rigid homogeneous porous medium, 6. Multiphase flow extension of Darcy’s equation, Similar derivations of the governing equations may be found in Aziz and Settari [1979], Peaceman [1977], Chavent and Jaffré [1986], Lake [1989], and Guzmán [1995], among others.


51

The equation of mass conservation of a chemical component i in a one-dimensional medium without distributed sources (assumptions 1 and 2) is: ∂t mi + ∂x Fi = 0,

0 < x < L,

t > 0,

(2.1)

where mi is the mass of the i-component per unit bulk volume, Fi is the mass flux of component i, ∂t and ∂x denote the partial derivatives with respect to time and space, respectively, and L is the length of the domain. By virtue of assumption 3, one can identify any chemical component i with the generic fluid phase α. We shall consider three fluid phases: water (w), oil (o) and gas (g). The mass of the α-phase per unit bulk volume is: mα = ρα Sα φ,

(2.2)

where ρα is the density of the α-phase, Sα is the saturation of the α-phase, and φ is the porosity. In view of assumptions 4 and 5, the phase densities and the porosity are taken as constants. We write the mass flux of the α-phase in the form: Fα = ρα vα φ,

(2.3)

where vα is the velocity of the α-phase. We use Muskat’s extension of Darcy’s equation [Muskat, 1949] to model the fluid velocities (assumption 6): vα = −

k krα (∂x pα + ρα g∂x z), φ µα

(2.4)

where k is the absolute permeability, krα , µα , and pα are the relative permeability, the dynamic viscosity, and the pressure of the α-phase, respectively, g is the gravitational acceleration, and z is the elevation. To simplify notation, we define gx := g∂x z.

(2.5)

For convenience, we also define the relative mobility of the α-phase, λα :=

krα . µα

(2.6)


52

We define the capillary pressures Pcα in a standard way, as differences of the phase pressures pα and a reference pressure p. For the purpose of this derivation, we take the oil pressure as the reference pressure, p := po ,

(2.7)

Pcw := pw − p,

(2.8)

Pcg := pg − p.

(2.9)

and define:

We note in passing that the water capillary pressure is usually defined with opposite sign (see, e.g., Aziz and Settari [1979]). Definition (2.8) has been used here to preserve symmetry of the formulation. Substituting Equations (2.2) and (2.3) into Equation (2.1), the mass conservation equations for the α-phase reads: ∂t Sα + ∂x vα = 0,

α = w, o, g.

(2.10)

Equations (2.10) are subject to the additional constraint that the fluids fill up the entire pore space: X

α=w,o,g

Sα ≡ 1.

(2.11)

The essence of the fractional flow approach [Aziz and Settari, 1979; Chavent and Jaffré, 1986; Peaceman, 1977] is to combine Equations (2.10) with the algebraic constraint (2.11) to obtain a “pressure equation” of elliptic type, and a system of “saturation equations” of parabolic type. Summation of the mass conservation equations and use of the saturation constraint yields the pressure equation: ∂x vT = 0,

(2.12)


53

where we have defined the total velocity vT as vT :=

X

vα .

(2.13)

α=w,o,g

Substituting the expression of the fluid velocities (2.4) and the capillary pressures (2.8)– (2.9) in Equation (2.13): vT = −

¢ k¡ λT ∂x p + (ρw λw + ρo λo + ρg λg )gx + λw ∂x Pcw + λg ∂x Pcg φ

(2.14)

or, rearranging,

¢ k k¡ − λT ∂x p = vT + (ρw λw + ρo λo + ρg λg )gx + λw ∂x Pcw + λg ∂x Pcg , φ φ

(2.15)

where

λT :=

X

λα

(2.16)

α=w,o,g

is the total mobility. Substituting Equation (2.15) in Equations (2.4), and rearranging, we obtain: · µ ¶ ¸ k λw λg vw = v T fw − λw 1 − ∂x Pcw − λw ∂x Pcg , φ λT λT · µ ¶ ¸ k λg λw λg 1 − vg = v T fg − ∂x Pcg − λg ∂x Pcw . φ λT λT

(2.17)

where fw , fg are the fractional flow functions —hyperbolic part of the flux— defined as: µ ¶ ¡ ¢ k λw 1− gx (ρw − ρo )λo + (ρw − ρg )λg , fw := λT vT φ ¶ µ ¢ ¡ λg k fg := 1+ gx (ρo − ρg )λo + (ρw − ρg )λw . λT vT φ

(2.18)

Using Equations (2.12) and (2.17) in (2.10), we obtain the system of saturation equations: · µ µ ¶ ¶¸ k λg λw ∂t S w + v T ∂x f w − ∂ x ∂x Pcw − λw ∂x Pcg λw 1 − = 0, φ λT λT ¶ · µ µ ¶¸ λw k λg −λg ∂x Pcw + λg 1 − ∂t S g + v T ∂x f g − ∂ x ∂x Pcg = 0. φ λT λT

(2.19) (2.20)


54

Remarks 2.1. 1. The mathematical problem is complete up to imposition of initial and boundary conditions, and definition of the relative permeability and capillary pressure functions. 2. The pressure equation (2.12) and the system of saturation equations (2.19)– (2.20) are coupled through the capillary pressures. 3. This coupling is of little consequence in one-dimensional models, because the solution to the pressure equation is trivial in this case. The total velocity is at most a function of time and, therefore, dependent only on the boundary conditions. 4. In the multidimensional case, the pressure and saturation equations may be decoupled by introducing a global pressure [Chavent and Jaffré, 1986]. 5. By virtue of the pressure equation (2.12), the total velocity is at most a function of time and depends only on the boundary conditions. When gravity effects are included in the formulation, we shall further assume that this quantity is constant, so that the fractional flow functions do not have an explicit time dependence.

2.3

Relative permeabilities and capillary pressures

As we discussed in Chapter 1, relative permeabilities are the key descriptors of classical Darcy-type formulations of multiphase flow through porous media. Strictly speaking, relative permeabilities should depend not only on the fluid saturations, but also on the saturation history, wettability characteristics, gravity effects, and


55

fluid viscosities [Juanes and Patzek, 2002e,f]. Thus, they should be properly called functionals, rather than functions. For the purpose of the mathematical formulation presented here, however, we shall understand the relative permeabilities as functions of fluid saturations alone. We assume that the dependency on other factors is accounted for in the expressions of the functions used. In particular, we shall use relative permeabilities that satisfy Stone’s assumptions [Stone, 1970, 1973], that is, water and gas relative permeabilities depend only on the water and gas saturations, respectively, and oil relative permeability depends of both: krw = krw (Sw ), kro = kro (Sw , Sg ),

(2.21)

krg = krg (Sg ). Similar considerations apply to the capillary pressures. Summarizing our discussion of the previous chapter, capillarity effects lead, in the context of multiphase displacements, to a nonlinear diffusion term. The physical consequence of this diffusion term is to smear the moving fronts —shocks— that arise in the displacement process. The detailed structure of these shocks —and consequently the capillary pressures— should depend on several factors, including wettability properties, viscosity ratios, displacement process (drainage or imbibition) and pore-scale fluid configuration [Lenormand, 1986]. For the purpose of the mathematical formulation, however, we shall use Leverett’s assumption [Aziz and Settari, 1979; Leverett and Lewis, 1941] that the water and gas capillary pressures depend only on the water and gas saturations, respectively: Pcw = Pcw (Sw ), Pcg = Pcg (Sg ).

(2.22)


2.4

56

Equations in dimensionless form

It is convenient, both for a mathematical and a numerical analysis, to write the governing equations (2.19)–(2.20) in dimensionless form. We define the dimensionless space and time coordinates: x , LZ 1 t vT (τ ) dτ, tD := L 0

xD :=

(2.23) (2.24)

respectively. When gravity is included in the formulation, we further assume that the total velocity vT is constant, so that the fractional flow functions do not depend explicitly on time. In this case, the dimensionless time takes the simple form: tD =

vT t . L

(2.25)

We define the gravity number and the density ratio as follows: (ρo − ρg )k gx , µo vT φ ρw − ρ g ρD := . ρo − ρ g

ND :=

(2.26) (2.27)

We define also the water and gas capillary numbers: ∗ (k/φ)Pcw , vT µw L ∗ (k/φ)Pcg Cg := , vT µg L

Cw :=

(2.28) (2.29)

∗ ∗ where Pcw and Pcg are characteristic values of the water and gas capillary pressures,

e.g., ∗ Pcw ∗ Pcg

:= :=

Z

Z

1 0

|Pcw (s)| ds,

(2.30)

|Pcg (s)| ds.

(2.31)

1 0


57

Dimensionless water and gas capillary pressures are defined as follows: Pcw , ∗ Pcw Pcg := ∗ . Pcg

D Pcw := D Pcg

(2.32) (2.33)

Using the definitions above, and the fact that the total velocity vT is at most a function of time, the system of saturation equations may be written as: " # µ ¶ D D λw ∂Pcw ∂fw ∂ λg ∂Pcg ∂Sw C w µw λw 1 − = 0, + − − C g µg λw ∂tD ∂xD ∂xD λT ∂xD λT ∂xD " # µ ¶ D D ∂fg ∂ ∂Sg λg ∂Pcg λw ∂Pcw + − + C g µg λg 1 − −Cw µw λg = 0. ∂tD ∂xD ∂xD λT ∂xD λT ∂xD

(2.34) (2.35)

Using the Leverett assumption (2.22) for the capillary pressures, and defining ²w := Cw µw ,

²g := Cg µg ,

(2.36)

we write the system in the equivalent form: " # µ ¶ D D λw dPcw ∂fw ∂ ∂Sw λg dPcg ∂Sg ∂Sw ²w λ w 1 − = 0, (2.37) + − − ² g λw ∂tD ∂xD ∂xD λT dSw ∂xD λT dSg ∂xD " # µ ¶ D D ∂Sg ∂Sw ∂Sg λw dPcw λg dPcg ∂fg ∂ −²w λg = 0. (2.38) + − + ² g λg 1 − ∂tD ∂xD ∂xD λT dSw ∂xD λT dSg ∂xD Using Equations (2.26)–(2.27), the water and gas fractional flow functions take the following expressions: · ¶¸ µ λw µo fw := 1 − ND (ρD − 1)kro + ρD krg , λT µg · ¶¸ µ λg µo 1 + ND ρD krw + kro , fg := λT µw

(2.39)

λw [1 − M ((ρD − 1)λo + ρD λg )] , λT λg [1 + M (ρD λw + λo )] , fg := λT

(2.40)

or, alternatively, fw :=


58

where M := ND µo .

(2.41)

Understanding the independent variables x and t as their dimensionless counterparts in Equations (2.23) and (2.24), we write the saturation equations in their final form as a nonlinear system of conservation laws: ∂t u + ∂x f − ∂x (D∂x u) = 0,

(2.42)

where 



 Sw  u :=   , Sg    fw  f :=   , fg     ´ D ³ D λg dPcg dPcw λw −²g λw λT dSg  Dww Dwg  ²w λw 1 − λT dSw D :=  ³ ´ D, = D dPcg λg cw 1 − −²w λg λλwT dP ² λ Dgw Dgg g g dSw λT dSg

(2.43)

(2.44)

(2.45)

are the vector of unknown saturations, the fractional flow vector and the capillary diffusion tensor, respectively. Remarks 2.2. 1. The fractional flow and the diffusion tensor are (nonlinear) functions of the unknown saturations, i.e., f = f (u),

D = D(u).

(2.46)

2. The solution vector u is restricted to lie in the saturation triangle: T := {(Sw , Sg ) : Sw ≥ 0,

Sg ≥ 0,

Sw + Sg ≤ 1}.

(2.47)


59

The saturation triangle is usually represented as a ternary diagram, in which the pair (Sw , Sg ) corresponds to the triple (Sw , Sg , So ), where So ≡ 1 − Sw − Sg (see Figure 2.1). Under a (linear) change of variables, the water and gas saturations may be understood as normalized saturations, rather than actual saturations. This renormalization is the subject of the next section. 3. The character of the system (2.42) depends on the eigenvalues and eigenvectors of the Jacobian matrix f 0 . In Chapter 3 [see also Juanes and Patzek, 2002e] we argue that this matrix must have real and distinct eigenvalues for the solution to be physically plausible, and we derive conditions on the relative permeability functions so that this requirement is satisfied. Here, we further assume that the capillary diffusion tensor is positive semi-definite. Under these conditions, the system of equations is parabolic, and strictly hyperbolic in the limit of vanishing diffusion [Zauderer, 1983].

2.5

Flow regions and reduced saturations

Experimental evidence suggests that there is a threshold saturation for each phase, below which that phase is immobile. This fact has been observed since the earliest two-phase and three-phase flow experiments [Botset, 1940; Leverett and Lewis, 1941; Muskat et al., 1937; Wyckoff and Botset, 1936]. As a result, three-phase flow takes place only for phase saturations in a region inside the saturation triangle. The nature of these threshold saturations depends on the wettability of the fluids, and on the displacement process [Geffens et al., 1951]. For the most wetting phase, the term “connate” —or “irreducible”— saturation would be appropriate both in drainage and imbibition. For the most nonwetting phase, the term “critical” saturation would


60

1 0.8

Sg 0.6 0.4 0.2 0

0

0.2

Sw

G 0

PSfrag replacements

1

0.8

0.6

0.4

1

0.2

0.8

0.4

0.6

0.6

0.4

0.8

0.2

1

0.8

0.6

0.4

0.2

0

1

O

0

W

Figure 2.1. Saturation triangle (top) and ternary diagram (bottom). be applicable in drainage, and “trapped” —or “residual”— saturation in imbibition. For the purpose of the description herein we lump the terminology above in the term “immobile” saturation, and assume that appropriate values are used for each fluid and for the particular process —saturation path— of interest. In principle, these threshold or endpoint saturations need not be constant, and the immobile saturation of each phase may vary with the saturations of the other two phases. This is a well-known behavior for the oil phase [Land, 1968], and several correlations for the “residual” oil saturation have been proposed [Fayers, 1987; Fayers and Matthews, 1984]. The relative permeability of a phase is zero if that phase is immobile, and it is positive otherwise. For each phase, there is a line separating the region where the phase is mobile from the region where it is immobile. These lines

61


0.2

Immobile water 0.2

˜ u

0.2

Immobile gas

(a) Space of actual saturations

0 1

0.8

0.6

0.2

0

1

0.8

0.6

0

W oilO Immobile

1

0.4

0.4

0.8

mo

0.4

il

Im

0.6

ϕ−1

eo bil

0.2

0.8

0.6

mo

Im

0.8

bil ew a te

r

0.6

u

1

0.4

0.4 0.6

0

0

0 0.2

0.8

Immobile gas

1

O

G

PSfrag replacements

1

0.4

G

PSfrag replacements

W

(b) Space of reduced saturations

Figure 2.2. Schematic of the map between (a) the space of actual saturations, and (b) the space of reduced saturations. The lines defining immobile phases subdivide the ternary diagram into regions of three-phase (center), two-phase (along the edges) and one-phase flow (near the vertices). The three-phase flow region (shaded area) is mapped onto the entire ternary diagram. define regions of one-phase, two-phase and three-phase flow in the ternary diagram. The three-phase flow region in the space of actual saturations u = (Sw , Sg )t can be ˜= mapped onto the entire ternary diagram of reduced —normalized — saturations u (S˜w , S˜g )t , as shown in Figure 2.2: ϕ : R 2 → R2 , ˜ 7→ u = ϕ(˜ u u).

(2.48) (2.49)

We assume that this map is C 1 invertible and orientation-preserving, so that by simple change of variables we can study three-phase flow in terms of reduced saturations: ∂t [ϕ(˜ u)] + ∂x [f (ϕ(˜ u))] − ∂x (D(ϕ(˜ u))∂x [ϕ(˜ u)]) = 0.

(2.50)

Equation (2.50) can be reduced to its canonical form (see, e.g., Dafermos [2000]) if the regions of mobile and immobile phases are separated by straight lines. In this


62

case the map between actual and reduced saturations is linear, ˜, u = ϕ(˜ u) = u0 + ϕ0 u

(2.51)

where ϕ0 ≡ Du˜ ϕ is the Jacobian matrix of the mapping, which is constant for a linear map. The system of conservation laws (2.50) is expressed as ¡ ¢ ˜ xu ˜ + ∂x f˜ − ∂x D∂ ˜ = 0, ∂t u

(2.52)

˜ are related to the original ones where the newly defined flux f˜ and diffusion tensor D by the following expressions: f˜ (˜ u) := (ϕ0 )−1 f (ϕ(˜ u)),

(2.53)

˜ u) := (ϕ0 )−1 D(ϕ(˜ D(˜ u)) ϕ0 .

(2.54)

Relative permeabilities and, consequently, fractional flows, are expressed in Equation (2.52) as functions of reduced saturations. By definition, they take a zero value along one of the edges of the ternary diagram, and are positive everywhere else. To simplify notation we shall drop the tildes from Equation (2.52) and write ∂t u + ∂x f − ∂x (D∂x u) = 0,

(2.55)

but still refer to the system in terms of reduced saturations. Remark 2.3. If the lines defining immobile regions are straight lines parallel to the edges of the ternary diagram, the tree-phase flow region is an equilateral triangle. In this particular case, reduced saturations take the following simple expression: S˜α :=

Sα − Sαi , 1 − (Swi + Soi + Sgi )

α = w, o, g,

where Sαi is the immobile saturation —now constant— of the α-phase.

(2.56)


2.6

63

Special cases

Equation (2.55) is the general equation describing one-dimensional three-phase flow with a Darcy formulation. It accounts for viscous, gravity, and capillary forces. It takes the general form of a nonlinear 2×2 system of conservation laws of advectiondiffusion type. These equations are now restricted to special cases, which will be relevant later on.

2.6.1

Negligible gravity effects

When flow is horizontal, or when gravity effects are negligible, we may use a zero value of the gravity number (2.26). The system of equations is still given by (2.55), but with simplified expressions of the fractional flow functions,     λw  fw   λT  f =   =  , λg fg λT

and the capillary diffusion tensor,     D D dPcg dPcw −²g λw fg dSg  Dww Dwg  ²w λw (1 − fw ) dSw D= . = D D dPcg cw −²w λg fw dP ² λ (1 − f ) Dgw Dgg g g g dSg dSw

2.6.2

(2.57)

(2.58)

Negligible capillarity effects

From the definition of the capillary numbers (2.28)–(2.29) it is apparent that, in a Darcy formulation of multiphase flow, capillarity scales with the inverse of the domain size. If capillarity effects are negligible for a particular situation, the capillary diffusion term is dropped from the formulation. The system of governing equations is, in this case, ∂t u + ∂x f = 0.

(2.59)


64

The flux function may or may not include the gravity term, and is given by Equations (2.44) or (2.57), respectively. It is essential to note that, because the term with the highest derivative is dropped from the formulation, the system of equations involves first-order derivatives only. This issue will be discussed at length in the following chapters, where we focus precisely on the capillarity-free system.

2.6.3

Two-phase flow

The equations governing two-phase flow through porous media under the same assumptions of Section 2.2 may be obtained analogously [Bear, 1972; Chavent and Jaffré, 1986; Peaceman, 1977]. Following the derivation in Juanes and Patzek [2002b], we define the capillary pressure of a water/oil immiscible system as: Pc := p − pw ,

(2.60)

where the reference pressure p is taken simply as the oil-phase pressure. Taking the same steps as in the three-phase case, we obtain the pressure equation: ∂x vT = 0,

(2.61)

which dictates that the total velocity, k vT := vw + vo = − [λT ∂x p − (λw ρw + λo ρo )gx − (λo − λw )∂x Pc ] , φ is at most a function of time. The saturation equation takes the form: µ ¶ k λw λo ∂t Sw + vT ∂x fw − ∂x −2 ∂x Pc = 0, φ λT where the water fractional flow fw is given by: · ¸ λw k fw = 1− gx (ρw − ρo )λo λT vT φ

(2.62)

(2.63)

(2.64)


65

The equation above may be written in dimensionless form. To this end, we define the dimensionless parameters: (ρw − ρo )k gx µo vT φ (k/φ)2Pc∗ Cw := vT µo L

ND :=

(gravity number),

(2.65)

(capillary number).

(2.66)

As in the three-phase case, we also define a characteristic value of the capillary pressure Pc∗ , defined as in Equation (2.30), and a dimensionless capillary pressure PcD , as in Equation (2.32). Understanding x and t as dimensionless space and time coordinates —Equations (2.23) and (2.24)— and assuming that the relative permeabilities and the capillary pressure are unique functions of the water saturation (see Section 2.3), the saturation equation of two-phase flow reads: ∂t Sw + ∂x fw − ∂x (Dw ∂x Sw ) = 0,

(2.67)

where the fractional flow fw and the capillary diffusion Dw are λw [1 − ND µo λo ] , λT µ ¶ λw λo dPcD Dw = C w µ o − . λT dSw fw =

2.6.4

(2.68) (2.69)

Miscible flow

Even though it is not a particular case of three-phase flow, we shall develop here the governing equations for miscible flow or tracer transport. The reason is that the traditional mathematical formulation of these processes results in a linear equation, which will be used in Chapter 6 to motivate the numerical method used for modeling two-phase and three-phase flow. Anticipating these developments, we


66

derive here the equations in several space dimensions, considering a reactive tracer —which undergoes radioactive decay— and allowing for a distributed source term. For a complete derivation, see Bear [1972] or Chavent and Jaffré [1986]. We consider two substances which are perfectly miscible, and assume that one is present in very small proportions. To fix ideas, we may think of a fluid mixture consisting of water and a radioactive tracer. A mass conservation statement for each species reads: Water: ∂t mw + ∇ · F w = Qw ,

(2.70)

Tracer:

(2.71)

∂t ms + ∇ · F s = Qs − σms ,

where mw (resp. ms ) is the mass of water (resp. tracer) per unit volume of porous medium, F w (resp. F s ) is the water (resp. tracer) mass flux, Qw (resp. Qs ) is the water (resp. tracer) distributed source term, and σ is the decay constant of the radioactive tracer. Let c be the mass fraction of tracer in the mixture (c ¿ 1), we express: mw = ρφ(1 − c),

(2.72)

ms = ρφc,

(2.73)

where ρ is the density of the mixture, and φ is the porosity of the medium. Dependence of the fluid density on the mass fraction is neglected. Similarly, we write the source term for each component as: Qw = ρφqT (1 − c∗ ),

(2.74)

Qs = ρφqT c∗ ,

(2.75)

where qT is the total volumetric source of fluid, and c∗ is the tracer mass fraction of the source. In the case of a negative source (qT < 0), the produced fluid has a tracer mass fraction equal to that of the reservoir, i.e., c∗ = c.


67

The total fluid mass flux (water and tracer) is F T := F w + F s = ρφv T .

(2.76)

The total fluid velocity v T is given by Darcy’s law for a single-phase system (excluding gravity for expositional simplicity): vT = −

k/µ ∇p, φ

(2.77)

where k is the absolute permeability tensor, µ is the fluid dynamic viscosity, and p is the fluid pressure. We use an advection-diffusion formulation for the mass flux of the tracer: F s = ρφ (v T c − Dh ∇c) ,

(2.78)

where Dh is the diffusion tensor, which includes the effects of molecular diffusion and hydrodynamic dispersion [Bear, 1972]. We assume it is independent of the mass fraction c. Assuming incompressibility of the fluid and the medium, summation of equations (2.70) and (2.71) yields the continuity, or “pressure” equation: ∇ · v T = qT − σc,

(2.79)

where the total velocity is given by Equation (2.77). This is a linear elliptic equation, to be solved for the pressure p and the total velocity v T . Substituting Equation (2.78) in (2.71) we obtain the mass fraction, or “concentration” equation: ∂t c + ∇ · (v T c − Dh ∇c) = qT c∗ − σc.

(2.80)

This is an equation of advection-diffusion-reaction type, to be solved for the mass fraction c. Equation (2.80) is linear and, assuming that c ¿ 1, it is almost decoupled from Equation (2.79). Equation (2.80) may be written in dimensionless form, through


68

a similar procedure to that of Section 2.4. Understanding all independent variables as dimensionless, and denoting the unknown as u ≡ c, the final equation governing tracer transport takes the following form: ∂t u + ∇ · (au − D∇u) = q − σu.

(2.81)

Someone told me that each equation I included in the book would halve the sales. I therefore resolved not to have any equations at all. In the end, however, I did put in one equation, Einstein’s famous equation, E = mc2 . I hope that this will not scare off half of my potential readers. — STEPHEN HAWKINGS, A Brief History of Time (1988)

69

Chapter 3 Relative permeabilities for strictly hyperbolic models 3.1

Introduction

We study one-dimensional horizontal flow through porous media of three immiscible, incompressible fluids. The mathematical formulation makes use of the commonlyused extension of Darcy’s equation. As shown in Chapter 2, this model leads to a 2 × 2 system of saturation equations. It was long believed that, when capillarity is ignored, this system of equations would be strictly hyperbolic for any relative permeability functions. This is far from being the case and, in fact, most relative permeability models used today give rise to systems which are not strictly hyperbolic for the entire range of admissible saturations [Bell et al., 1986; Fayers, 1987; Hicks Jr. and Grader, 1996; Holden, 1990a; Shearer, 1988; Shearer and Trangenstein, 1989]. Loss of strict hyperbolicity typically occurs at bounded regions of the saturation triangle —the so-called elliptic regions— where the system is elliptic in character. We find

Chapter 3. Relative permeabilities for strictly hyperbolic models

70

this behavior disturbing for many reasons, and are of the opinion that elliptic regions are the artifacts of an incorrect mathematical model. The objective of this chapter is to show that it is possible to choose relative permeability functions that preserve strict hyperbolicity of the three-phase flow equations, even if the usual extension of Darcy’s equation is employed and the relative permeabilities take a very simple form. The chapter is organized as follows. In Section 3.2 we summarize the mathematical model of three-phase flow, and introduce the classification of the system of governing equations. In Section 3.3 we derive necessary conditions that the relative permeability functions must satisfy for the system of equations to be strictly hyperbolic for all admissible saturation states. We show, in Section 3.4, that the essential condition that needs to be imposed agrees well with experimental data. Finally, in Section 3.5 we give some concluding remarks.

3.2

Mathematical model

In this section we revisit the traditional mathematical formulation of three-phase flow of immiscible incompressible fluids, derived in Chapter 2.

3.2.1

System of governing equations

We study one-dimensional three-phase flow under the assumptions of Section 2.2. Furthermore, we do not include gravitational and capillary forces in the analysis. The mathematical problem is described by a 2 × 2 system of first-order partial differential equations, known as the saturation equations. When written in vector notation, the system takes the form: ∂t u + ∂x f = 0,

(3.1)

Chapter 3. Relative permeabilities for strictly hyperbolic models where

     u  S w  u= =  v Sg

71

(3.2)

is the vector of unknown saturations, and       λw  f   fw   λ  f = = = T λg fg g λT

(3.3)

is the vector of fractional flow functions. We understand that the equations are in dimensionless form (see Section 2.4) and that fluid saturations have been re-normalized following the procedure in Section 2.5.

3.2.2

Character of the system of equations

For the purpose of the classification of the system (3.1), we write it in quasilinear form: ∂t u + A(u)∂x u = 0, where



(3.4) 

f,u (u) f,v (u) A(u) := f 0 (u) ≡ Du f ≡   g,u (u) g,v (u)

(3.5)

is the Jacobian matrix of the system at point u. Subscripts after a comma denote differentiation (e.g., f,u ≡ ∂u f ). The classification of the 2 × 2 system of first-order partial differential equations (3.4) is based on the properties of the characteristic curves [Smoller, 1994; Zauderer, 1983]. Definition 3.1. A characteristic curve of (3.4) is a curve on the (x, t)-plane, along which ∂t u and ∂x u cannot be specified (if indeed they can be determined at all) if the initial data are prescribed along that same curve.


72

From the definition above, it follows [Dafermos, 2000; Zauderer, 1983] that a characteristic curve of the system (3.4) associated with a classical —smooth— solution u(x, t) is a function x = x(t), which is an integral curve of the ordinary differential equation dx = ν(u(x, t)), dt

(3.6)

where ν(u) is an eigenvalue of the Jacobian matrix A(u). As a result, the classification of the system (3.4) reduces to analyzing the behavior of the eigenvalue problem Ar = νr,

(3.7)

where the Jacobian matrix A, the eigenvalue ν, and the right eigenvector r, are evaluated at a point u. For the eigenvalue problem (3.7) with a 2 × 2 real matrix, it is well-known [Coddington and Levinson, 1955] that there exists a real nonsingular matrix T such that, after the change of variables z = T r, the equivalent eigenvalue problem

has a real coefficient matrix

¡

¢ T AT −1 z = νz

(3.8)

J := T AT −1 ,

(3.9)

which has one of the following canonical forms:   λ 0  1.   , λ 6= µ, 0 µ 



α −β  2.  , β α 



λ 0  3.  , 0 λ

β 6= 0,

Chapter 3. Relative permeabilities for strictly hyperbolic models 

73



λ 1  4.  . 0 λ

These four canonical forms provide the basis for the classification of the system of firstorder partial differential equations. Following the terminology in Zauderer [1983], we denote the system whose Jacobian matrix has the canonical form of cases 1 through 4 above, respectively: 1. Strictly hyperbolic. The eigenvalue problem has two real, distinct eigenvalues. The Jacobian matrix is diagonalizable and there are two real and linearly independent eigenvectors [Axler, 1996]. Therefore, the system has two distinct families of characteristic curves, which carry waves traveling at different characteristic speeds. 2. Elliptic. The eigenvalues are complex conjugates, and there are no real characteristic curves that may act as carriers of possible discontinuities in the solution [Dafermos, 2000; Zauderer, 1983]. 3. Nonstrictly hyperbolic. In this case, there is a double real eigenvalue, and the Jacobian matrix is diagonalizable. Every direction is characteristic, so one can pick any two linearly independent vectors as eigenvectors. The system is hyperbolic —real eigenvalues and linearly independent eigenvectors— but not strictly hyperbolic, which requires that the eigenvalues be distinct. 4. Parabolic. The system has a real, double eigenvalue, and the Jacobian matrix is defective, that is, non-diagonalizable. There is only one eigenvector and, therefore, there is only one real characteristic direction. This completes the classification of the system at any given state u. It is important to note that this classification is restricted to 2×2 real systems of first-order equations.


74

As we shall see, the character of the system may in principle be different in different regions of the phase space, i.e., the saturation triangle. Remarks 3.2. 1. The eigenvalues νi , i = 1, 2 of the original Jacobian matrix (3.5) are given by · ¸ q 1 f,u + g,v ∓ (f,u − g,v )2 + 4f,v g,u . (3.10) ν1,2 = 2 The physical interpretation of the eigenvalues —when they are real— is the

characteristic speeds at which waves describing changes in saturation propagate through the domain. In the strictly hyperbolic case, there exist two distinct waves which travel at different characteristic speeds. It is common to use the terms slow wave and fast wave for the waves associated with the smaller and larger eigenvalue, respectively. 2. The right eigenvectors r i = [riu , riv ]t , i = 1, 2, which correspond to eigenvalues νi , i = 1, 2, respectively, are calculated by the following expressions: ν1 − f,u g,u r1v = = , r1u f,v ν1 − g,v f,v ν2 − g,v r2u = = . r2v ν2 − f,u g,u

(3.11) (3.12)

When they are real, the right eigenvectors correspond to the directions (in the phase space) of admissible changes in saturation. In the strictly hyperbolic case, the changes in saturation associated with the slow (resp. fast) wave have a direction dictated by r 1 (resp. r 2 ) and propagate with velocity ν1 (resp. ν2 ).

3.3

Relative permeabilities for strict hyperbolicity

In this section we investigate the character of the system of equations (3.1) describing one-dimensional three-phase flow of immiscible incompressible fluids when


75

gravity and capillary forces are negligible. The objectives are to understand better the interplay between relative permeabilities and the nature of the system, and to derive conditions on phase mobilities —and ultimately on relative permeabilities— so as to preserve strict hyperbolicity.

3.3.1

Loss of strict hyperbolicity in traditional models

It was believed that the system (3.1) was strictly hyperbolic for any relative permeability functions. In this case, the theory of Lax [1957] as extended by Liu [1974, 1975] would apply. In the seminal work of Charny [1963], it is concluded that, for physically realistic flows, the system of equations should be hyperbolic. It was recognized, however, that certain relative permeability functions might lead to a system of mixed elliptic/hyperbolic type. Bell, Trangenstein and Shubin [1986] showed that, indeed, the system is not necessarily hyperbolic for all relative permeability functions. In particular, they observed that Stone I [Stone, 1970] relative permeabilities may give rise to elliptic regions inside the saturation triangle. Elliptic regions are portions of the saturation triangle where the eigenvalues are complex, so the system is locally elliptic rather than hyperbolic. In subsequent publications [Fayers, 1987; Hicks Jr. and Grader, 1996; Holden, 1990a; Shearer, 1988; Shearer and Trangenstein, 1989], it was shown that occurrence of elliptic regions is the rule rather than exception for the most common relative permeability models. Loss of strict hyperbolicity of three-phase flow models was analyzed by Shearer [1988] and Holden [1990a]. They also used reduced saturations, and therefore limited their analysis to the three-phase flow region. From the point of view of studying the character of the system, this assumption is not particularly restrictive, for it can be shown that elliptic regions cannot exist in the one-phase and two-phase flow


76

regions [Falls and Schulte, 1992b]. The analysis of Shearer [1988] and Holden [1990a] starts by assuming the behavior of relative permeabilities in two-phase flow. Relative permeabilities —and, therefore, relative mobilities— of both phases, say water and gas, are taken as functions of the reduced water saturation: λw = λw (u),

λg = λg (u).

(3.13)

These functions are assumed to have a zero value and a zero derivative at their endpoint saturations, λw (0) = λ0w (0) = 0,

λg (1) = λ0g (1) = 0.

(3.14)

This behavior, which is taken for granted without further discussion, is then used as a guidance to impose conditions on the three-phase relative permeabilities along the edges of the saturation triangle. In three-phase flow, relative permeabilities are assumed to be functions of the water and gas reduced saturations: λw = λw (u, v),

λg = λg (u, v),

λo = λo (u, v).

(3.15)

Shearer [1988] imposes two types of conditions on the edges: 1. Consistency conditions (termed “(B-L) conditions”), which reduce three-phase relative mobilities to the assumed two-phase flow behavior —Equation (3.14)— when one of the phases is not mobile. For example, on the edge of zero reduced gas saturation (v = 0, 0 < u < 1), it is required: λw (0, 0) = λw,u (0, 0) = 0, λg (u, 0) ≡ 0, λo (1, 0) = λo,u (1, 0) = 0.

(3.16)


77

Similar conditions hold on the other two edges. 2. A first interaction condition (termed “(I1) condition”), that limits the effect of the immobile phase on the flow, compared with that of the other two phases. For instance, on the edge v = 0, 0 < u < 1: λg,v
0,

(3.18)

λg,u (λw + λo ) − λg (λw,u + λo,u ) > 0. Holden [1990a] imposes very similar conditions. The following is required on the edges:


78

1. The value of the relative mobility of a phase is zero along the edge of zero reduced saturation of that phase, i.e., λw = 0 on u = 0, 0 < v < 1, λg = 0 on v = 0, 0 < u < 1,

(3.19)

λo = 0 on v = 1 − u, 0 < u < 1. 2. The derivative of the relative mobility of a phase along the normal direction to the edge of zero reduced saturation is also zero: λw,u = 0 on u = 0, 0 < v < 1, λg,v = 0 on v = 0, 0 < u < 1,

(3.20)

−λo,u − λo,v = 0 on v = 1 − u, 0 < u < 1. We note that the conditions above on the normal derivatives imply that the “(I1) interaction condition” of Shearer [1988] is immediately satisfied and, as a result, the eigenvector of the fast family is parallel to the edges of the saturation triangle. The properties near the corners are introduced next: 3. Holden [1990a] considered three possible types of behavior near the vertices of the saturation triangle, based on the sign of the off-diagonal terms of the Jacobian matrix, f,v , g,u : (A1) Both are positive: f,v > 0, g,u > 0. (A2) Have different sign: f,v g,u < 0. (A3) Both are negative: f,v < 0, g,u < 0.


79

Using a wettability argument, it is suggested that condition (A1) is the most reasonable in all three corners. This condition implies that the “(I2) interaction condition” of Shearer [1988] is automatically satisfied. We can summarize the conditions imposed by Shearer [1988] and Holden [1990a] as follows (see Figure 3.1): 1. The right eigenvector associated with the fast characteristic family, r 2 , is parallel to the edges of the triangle of reduced saturations. 2. The fast eigenvector r 2 points into the triangle, for saturation states near the vertices. The assumed behavior at the edges and corners of the saturation triangle has a profound impact on the character of the system. The first consequence is that each vertex of the saturation triangle is an umbilic point, i.e., eigenvalues are equal and the system is not strictly hyperbolic at those points. The second consequence is that, in general, an elliptic region must exist inside the saturation triangle. This general result may be proved using ideas of projective geometry [Holden, 1990a; Schaeffer and Shearer, 1987a; Shearer, 1988]. Naturally, the question of whether elliptic regions in the saturation space are physically-plausible arises. A full discussion on this topic is given in Juanes et al. [2003]. However, we briefly point out some of the reasons why local elliptic behavior seems to be an undesirable artifact of the mathematical model, rather than a necessary consequence dictated by physics. 1. The first remark that one should bear in mind is that Equation (3.1) is a system of first-order equations in space-time coordinates. Thus, the physical meaning of a system with mixed elliptic/hyperbolic behavior is very different from that


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0.8

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0.6

r2 0.4

r2 0.2

0

1

O

0.2

0

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Figure 3.1. Schematic representation of the direction of fast eigenvectors r 2 along the edges of the saturation triangle for the models analyzed by Shearer [1988] and Holden [1990a]. For models of this type, vertices are umbilic points, and there must be an elliptic region inside the saturation triangle, usually very close to the oil-water edge. when the independent variables are two space coordinates, such as in steady transonic flow [Courant and Friedrichs, 1948; Keyfitz, 1990]. In the former case, “initial data” should be imposed in such a way that the principle of causality1 is not violated [Fayers, 1987]. 2. Saturation states inside the elliptic region give rise to linearly ill-posed problems. More precisely, a bounded solution to the linearized Cauchy problem 2 does not exist when arbitrarily close —but not equal— asymptotic left and right states are inside the elliptic region. This fact is in frontal disagreement with the notion of three-phase flow displacement, where one expects a bounded and 1

“The causality of natural processes may be interpreted as implying that the conditions in a body at time t are determined by the past history of the body, and that no aspect of its future behavior need to be known in order to determine all of them.” [Truesdell and Noll, 1965, p. 56] 2 The Cauchy problem refers to an initial value problem on an unbounded domain with general initial conditions

81


monotonic transition between the right (initial) state and the left (injected) state. However, the question of whether ill-posedness remains when nonlinear effects are accounted for is still unclear [Azevedo and Marchesin, 1990; Holden et al., 1990]. 3. In connection with the loss of well-posedness of the problem, appropriate entropy conditions have not yet been found so as to allow both existence and uniqueness of solutions [Azevedo and Marchesin, 1990, 1995].

Although a

few qualitative properties are known [Holden, 1987; Holden and Holden, 1989; Holden et al., 1990], a complete theory of mixed elliptic/hyperbolic systems does not yet exist. 4. More specifically to models of three-phase flow in porous media, it has been found [Hicks Jr. and Grader, 1996] that different models matching experimental data equally well, produce elliptic regions in opposite corners of the saturation triangle. This result suggests a nonphysical arbitrariness to the location of elliptic behavior in phase space for traditional models of three-phase flow. 5. If capillarity is introduced in the formulation and a traveling wave solution is sought for the Riemann problem 3 , the critical points of the associated 2 × 2 dynamical system are spiral points [Coddington and Levinson, 1955]. If a traveling wave solution exists, it will necessarily present a spiral-like behavior near the critical points, which translates into oscillatory —nonmonotonic— saturation profiles. Validity of this type of solution is questionable on several counts: (1) as the capillarity effects are taken to zero, oscillations collapse into a singular 3

The Riemann problem is a particular case of the Cauchy problem —initial value problem on an unbounded domain— where the initial data are piecewise constant states separated by a single discontinuity


82

shock, of dubious physical interpretation; (2) introducing “sufficient” capillarity will not cure the problem of oscillatory behavior, as the spiral-like orbit will persist asymptotically. 6. Numerical simulations seem to corroborate, at least in first instance, the nonphysical behavior of solutions inside the elliptic region [Bell et al., 1986; Jackson and Blunt, 2002]. For arbitrarily close left and right states inside the elliptic region, the solution develops wildly oscillatory waves, which are never observed in experiments. Moreover, the wave pattern is unstable with respect to the initial states [Bell et al., 1986]. The only relative permeability models which do not produce elliptic regions (under the assumed behavior at the edges and corners) are those where the relative permeability of a phase depends solely on the saturation of that phase [Marchesin and Medeiros, 1989; Trangenstein, 1989]. This simplistic behavior is not supported by experimental results [see, e.g., Oak et al., 1990] and pore-scale models [Al-Futaisi and Patzek, 2003; van Dijke et al., 2002]. For such relative permeability models, the elliptic region, where eigenvalues are complex conjugates, shrinks to an isolated umbilic point —a saturation state with a real double eigenvalue— which cannot be removed by further approximation of the relative permeabilities. As shown by Holden [1990a], it may be possible to reduce the elliptic region to an isolated umbilic point for more general relative permeability models. However, this requires a continuous deformation of the relative permeability functions, which is physically unappealing. Umbilic points also act as “repellers” for classical waves [Guzmán, 1995; Guzmán and Fayers, 1997b; Marchesin and Plohr, 2001] and, as a result, solutions to the nonstrictly hyperbolic system require nonclassical waves, termed transitional waves [Isaacson et al., 1989]. The most salient features of these solutions are that: (1) they are sensitive


83

to the particular form of the diffusion term due to capillary effects [Isaacson et al., 1992]; (2) the saturation path may be the same for wildly different initial and injected states [Falls and Schulte, 1992a,b; Guzmán and Fayers, 1997a,b]; and (3) in WAG displacement, the intermediate oil bank may be split in two [Marchesin and Plohr, 2001]. From the observations above, it is difficult to justify the physical relevance of mathematical singularities like elliptic regions and umbilic points [Chavent and Jaffré, 1986; Chavent et al., 1999; Fayers, 1987; Miller et al., 1998; Sahni et al., 1996; Shearer and Trangenstein, 1989; Trangenstein, 1989]. In fact, the presence of elliptic regions in models of three-phase flow has not yet been successfully justified on physical grounds. The first attempt in this direction we are aware of is the recent paper by Jackson and Blunt [2002], which we review critically in Chapter 4 (see also Juanes and Patzek [2002f], and Juanes et al. [2003]). We are of the opinion that these singularities are mere artifacts of an incorrect mathematical model. As discussed in Chapter 1, inappropriateness of the formulation may have several sources. In the context of this dissertation, where the traditional multiphase flow extension of Darcy’s equation is used, the element of the formulation that first needs to be revisited is the relative permeability model and, in particular, the assumed behavior at the edges of the saturation triangle. In fact, it is widely recognized that the slope of experimental relative permeabilities near the endpoints is often ill-defined [Fayers, 1987].

3.3.2

Conditions for strict hyperbolicity

The generic approach in the existing literature can be summarized as follows: a certain behavior of the relative permeabilities is assumed, and loss of strict hyperbolicity inside the saturation triangle is inferred. We adopt the opposite viewpoint:


84

we enforce that the system be strictly hyperbolic, and investigate the conditions on relative permeabilities as functions of saturation such that strict hyperbolicity is preserved. In doing so we keep, however, the restriction that eigenvectors should not rotate along edges of the ternary diagram. What is more, we require that one of the eigenvectors is parallel to any given edge. The reason for this restriction is to preserve the property that the edges of the ternary diagram are invariant lines for the system of equations [Shearer, 1988], i.e., if one phase is immobile everywhere at the initial time, that phase remains immobile, because its reduced saturation is zero everywhere at all times. This is a very simplistic condition, which in fact does not have to hold if curved lines defining immobile saturations are used, or if hysteretic effects are considered. If the latter are accounted for, the boundary of the three-phase flow region changes with the saturation path of interest. It is easy to check that the requirement of having one eigenvector parallel to each edge, precludes the possibility of having a strictly hyperbolic system everywhere along the edges of the ternary diagram. There are two different ways in which strict hyperbolicity may fail on the boundary of the saturation triangle: 1. The system is strictly hyperbolic at all three vertices. For vertices to be strictly hyperbolic, eigenvectors lying on each of the two edges must be of different family (e.g., at the O corner, r 1 is parallel to the OW edge and r 2 is parallel to the OG edge). But then, there must exist at least one edge that has a parallel eigenvector of the fast family near one vertex, and of the slow family near the other vertex. Inevitably, an umbilic point —where characteristic speeds of the slow and fast characteristic families coincide— must exist somewhere on this edge, because eigenvectors are not allowed to rotate along the edge (Figure 3.2).


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0

1

O

r2

Umbilic point

0

W

Figure 3.2. Schematic representation of the direction of fast (r 2 ) and slow (r 1 ) eigenvectors along the edges of the saturation triangle for models with strictly hyperbolic vertices. An umbilic point must exist somewhere on the WG edge, where eigenvalues of the slow and fast characteristic families coincide. 2. At least one of the vertices is an umbilic point. As we show below, it is possible to have a model that will be nonstrictly hyperbolic at the G vertex and strictly hyperbolic everywhere else (Figure 3.3). Having the considerations above in mind, the key observation is that, whenever gas is present as a continuous phase, the mobility of gas is much higher than that of the other two fluids, water and oil. To honor this physical behavior, we associate fast characteristic paths with displacements involving changes in gas saturation, even in the region of small gas saturation. The immediate consequence is that the eigenvector associated with the fast family of characteristics (r 2 ) is transversal —and not parallel— to the oil-water edge of the ternary diagram (Figure 3.3). As we shall see, this conceptual picture permits that the system will be strictly hyperbolic everywhere inside the saturation triangle. The G vertex, corresponding to 100% reduced gas saturation, remains an umbilic point because fast paths corresponding to the OG


86

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Figure 3.3. Schematic representation of the direction of fast (r 2 ) and slow (r 1 ) eigenvectors along the edges of the saturation triangle for the type of models we propose. The system is strictly hyperbolic everywhere inside the saturation triangle, and the only umbilic point is located at the G vertex, where the fast paths corresponding to the OG and WG edges coalesce. and WG edges coalesce. This umbilic point could be further removed if one allows for fast eigenvectors to rotate along the OG and WG edges. This was not done here to prevent saturation paths from falling outside the three-phase flow region. Let us recapitulate the conceptual picture expressed in Figure 3.3: 1. Along the oil-water (OW) edge, the eigenvector associated with the slow characteristic family (r 1 ) is parallel to the edge. The system is strictly hyperbolic everywhere along the edge, including the O and W vertices. 2. Along the oil-gas (OG) and water-gas (WG) edges, the eigenvector associated with the fast characteristic family (r 2 ) is parallel to these edges. The system is strictly hyperbolic everywhere along the edges except at the G vertex, which is an umbilic point. Below we present a systematic study of the general conditions that ensure strict


87

hyperbolicity of the system. On each edge we identify two types of conditions. Condition I enforces that eigenvectors of the appropriate family are parallel to the edge. Condition II guarantees strict hyperbolicity of the system along the edge. The latter condition is further specialized to both vertices of each edge, which provides additional insight into the behavior of the relative permeabilities. The analytical developments are expressed most effectively in terms of water and gas fractional flows (f and g, respectively) and their derivatives with respect to water and gas saturations (u and v, respectively). We then translate these requirements into conditions that the relative permeabilities must satisfy. We emphasize that relative permeabilities should be understood as functionals of the various system descriptors. For the simplified case studied here, where capillarity and gravity effects are not included, only the fluid saturations and the fluid viscosities enter the formulation. Therefore, we express the relative permeabilities as functions of saturations alone, but it is implicitly understood that the functional form that is actually used may depend on the displacement process —drainage or imbibition— and on the viscosity ratios: λα = λα (u, v; viscosity ratios, displacement process), 3.3.2.1

α = w, o, g.

(3.21)

Analysis along the OW edge.

This edge corresponds to the line of zero reduced gas saturation, v = 0. The mathematical condition for the slow eigenvector to be parallel to the OW edge (r 1 = [1, 0]t ) is: g,u = 0.

(3.22)

When expressed in terms of mobilities, Condition I above reads: λg,u = 0,

(3.23)

88


that is, the derivative of the gas relative mobility with respect to water saturation is zero. This condition is immediately satisfied for any model, as the gas mobility is identically zero along this edge. For the eigenvector r 1 to be parallel to the edge, it is also necessary that the denominator in Equation (3.11) is different from zero: ν1 − g,v 6= 0.

(3.24)

Moreover, introducing Equation (3.22) into the expression of the eigenvalues, Equation (3.10), one obtains: ν1,2 =

   f

,u

(3.25)

  g,v

Condition (3.24), together with the condition for strict hyperbolicity, ν 1 < ν2 , implies that How := g,v − f,u > 0,

(3.26)

λw . λT

(3.27)

which is equivalent to λg,v > λw,u − λT,u

Condition II above is the fundamental requirement for strict hyperbolicity of the system of equations of three-phase flow. When this condition is evaluated at the vertices of the OW edge, one obtains: λg,v > λw,u

at the O vertex,

(3.28)

λg,v > −λo,u

at the W vertex,

(3.29)

where the inequalities above are strict. In particular, Equations (3.28)–(3.29) require that the gas relative permeability does not have zero-derivative at its endpoint saturation. A summary of the conditions at the OW edge is given in Table 3.1.


89

Remarks 3.3. 1. The requirement of a nonzero endpoint slope of the gas relative permeability is a necessary condition for strict hyperbolicity, which is violated by the models of all previous studies on this subject. 2. This behavior of gas relative permeability is in good agreement with experimental observations of two-phase [Botset, 1940; Geffens et al., 1951; Muskat et al., 1937; Osoba et al., 1951; Richardson et al., 1952; Wyckoff and Botset, 1936] and three-phase flow [Leverett and Lewis, 1941; Oak, 1990, 1991; Oak et al., 1990; Saraf et al., 1982; Schneider and Owens, 1970], both in drainage and imbibition. We demonstrate this agreement in Section 3.4. 3. A finite positive slope for the gas relative permeability can also be justified from the point of view of pore-scale processes. Gas is the most nonwetting fluid, so gas flow takes place through the middle region of the pores (bulk flow). In a drainage process, gas flow starts with the first percolating cluster. In an imbibition process, gas flow ceases with the last trapped cluster. In both cases, the transition between zero flow and nonzero flow is rather abrupt, thus justifying a positive slope at the endpoint of the relative permeability curve. 4. In contrast, near their endpoint saturations, the most wetting and intermediate wetting fluids flow through a continuous network of films [DiCarlo et al., 2000a] (corner flow and/or film flow). The effective cross-sectional area of this network varies depending on the local level of capillary pressure. Since the fluid conductance is proportional to the cross-sectional area, it seems plausible that that the relative permeability will approach zero as a quadratic or higher-order power function of saturation and, thus, with zero slope.


90

Table 3.1. Summary of conditions along the OW edge. Condition

3.3.2.2

Frac. flows

Mobilities

I

g,u = 0

⇔

λg,u = 0

II

g,v − f,u > 0

⇔

λg,v > λw,u − λT,u λλwT

II at O

λg,v > λw,u

II at W

λg,v > −λo,u

Analysis along the OG edge.

This edge corresponds to the line of zero reduced water saturation, u = 0. A necessary condition for the fast eigenvector to be parallel to the OG edge (r 2 = [0, 1]t ) is: f,v = 0.

(3.30)

In terms of mobilities, Condition I reads: λw,v = 0,

(3.31)

that is, the derivative of the water relative mobility with respect to gas saturation is zero. This condition is immediately satisfied because water mobility is identically zero along this edge. We also require that the denominator of Equation (3.12) is nonzero: ν2 − f,u 6= 0.

(3.32)

By virtue of Equation (3.30), the eigenvalues take the following expressions along the OG edge: ν1,2 =

   f,u   g,v

(3.33)


91

Table 3.2. Summary of conditions along the OG edge. Condition

Frac. flows

Mobilities

I

f,v = 0

⇔

λw,v = 0

II

g,v − f,u > 0

⇔

λg,v > λw,u + λT,v λλTg

II at O

λg,v > λw,u

II at G

λw,u + λo,v = 0

We impose that the system is strictly hyperbolic everywhere along the OG edge, excluding the G vertex. The condition of strict hyperbolicity, ν1 < ν2 , implies that (Condition II): Hog := g,v − f,u > 0,

(3.34)

λg . λT

(3.35)

or, equivalently: λg,v > λw,u + λT,v

When we specialize Condition II at the O vertex, we obtain: λg,v > λw,u .

(3.36)

Equation (3.36) requires again that the gas relative permeability has a positive slope at its endpoint saturation. The G vertex is assumed to be an umbilic point, where the slow and fast characteristic speeds coincide, that is, ν1 = ν2 . When expressed in terms of relative mobilities, the condition reads: λw,u + λo,v = 0. The conditions at the OG edge are summarized in Table 3.2.

(3.37)

Chapter 3. Relative permeabilities for strictly hyperbolic models 3.3.2.3

92

Analysis along the WG edge.

This edge corresponds to the line of zero reduced oil saturation, v = 1 − u. The analysis at the WG edge is complicated by the fact that it is a tilted line in the (u, v)-plane. The fast eigenvector will be parallel to the WG edge (r 2 = [−1, 1]t ) if: ν2 − f,u = −f,v .

(3.38)

Substituting the expression for the eigenvalues —Equation (3.10)— into the condition above, one arrives at: f,v + g,v = f,u + g,u .

(3.39)

In terms of mobilities, because the oil mobility is zero along the WG edge, Condition I reads: λo,v = λo,u ,

(3.40)

that is, the derivatives of the oil relative mobility with respect to gas and water saturations are equal. As for the other two edges, this condition is identically satisfied by all models. Using Equation (3.39), the eigenvalues take the following expressions along the WG edge: ν1 = g,v + f,v

(3.41)

ν2 = g,v − g,u

(3.42)

The system is strictly hyperbolic everywhere along the WG edge (ν1 < ν2 ), excluding the G vertex, if: Hwg := −g,u − f,v > 0,

(3.43)

λw λg (λg,v − λg,u ) + (λw,u − λw,v ) > −λo,u . λT λT

(3.44)

or, equivalently:


93

Table 3.3. Summary of conditions along the WG edge. Condition

Frac. flows

I

f,v + g,v = f,u + g,u

II

−g,u − f,v > 0

Mobilities ⇔ λo,v = λo,u ⇔

λw (λg,v λT

− λg,u )

+ λλTg (λw,u − λw,v ) > −λo,u II at W

λg,v > −λo,u

II at G

λw,u + λo,v = 0

The strict inequality at the W vertex reads: λg,v − λg,u > −λo,u ,

(3.45)

and the equality at the G vertex (umbilic point) imposes that λw,u − λw,v = −λo,u .

(3.46)

Substituting Condition I along all three edges —Equations (3.23), (3.31) and (3.40)— into (3.45)–(3.46) above, we obtain the conditions at the vertices in their final form: at the W vertex,

(3.290 )

λw,u + λo,v = 0 at the G vertex.

(3.370 )

λg,v > −λo,u

In Table 3.3 we summarize the conditions at the WG edge. Remark 3.4. The conditions expressed in Tables 3.1–3.3 are necessary conditions for strict hyperbolicity of the system of equations everywhere in the saturation triangle (with the exception of the G vertex, which is an umbilic point). They are not sufficient.


3.3.3

94

A simple model

Our interest here reduces to presenting a simple model that satisfies the conditions above. A common practice in petroleum engineering [Stone, 1970, 1973] is to assume that relative permeabilities of the most and least wetting fluids (usually water and gas) depend only on their own saturation, whereas the relative permeability of the intermediate wetting fluid (usually oil) depends on all saturations. Although we do not defend this assumption in general, here we show that it is possible to obtain models which are strictly hyperbolic everywhere in the three-phase flow region. We take, for example: λw = (1/µw )u2 , ¡ ¢ λg = (1/µg ) βg v + (1 − βg )v 2 ,

(3.47) βg > 0

λo = (1/µo )(1 − u − v)(1 − u)(1 − v).

(3.48) (3.49)

The most important feature of the model is the positive derivative of the gas relative permeability function as it approaches zero. For the particular function used here, oil isoperms are slightly convex (see Figure 3.4). It is immediate to check that the relative mobilities (3.47)–(3.49) satisfy Condition I on all three edges. Whether Condition II is satisfied will depend, in general, on the values of the fluid viscosities and the endpoint-slope of the gas relative permeability. 3.3.3.1

Analysis along the OW edge

We need to study admissible values of the endpoint slope of the gas relative permeability, βg , such that Equation (3.26) is satisfied for any given fluid viscosities. The


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Figure 3.4. Oil isoperms for the simple model given by Equation (3.49). For this particular case, oil isoperms are slightly convex towards the O corner. derivatives of the fractional flow functions along the OW edge (v = 0) are as follows: f,u =

1 u2 µw

g,v =

+

(1−u)2 µo

1 u2 µw

+

(1−u)2 µo

·2 ·

u(1 − u) , µo u2 + µw (1 − u)2

βg . µg

(3.50) (3.51)

Then, the condition for strict hyperbolicity along the OW edge reads: How

· ¸ βg 1 = g,v − f,u = − F (u) > 0, D(u) µg

(3.52)

where D(u) :=

1 u2 µw

> 0 ∀u ∈ [0, 1],

+

(1−u)2 µo

u2

u(1 − u) . + µw (1 − u)2

(3.53)

and F (u) := 2

µo

(3.54)

Chapter 3. Relative permeabilities for strictly hyperbolic models Defining M := max

0

µg , µ ¯

µ ¯=

√

µ o µw .

(3.56)

Despite the fact that Equation (3.56) is restricted to the simple model considered here, it is illuminating with regard to the required behavior for the relative permeability of the most nonwetting phase. Equation (3.56) expresses that there is a lower bound in the endpoint slope of the nonwetting phase relative permeability, if the threephase flow model is to be strictly hyperbolic. This threshold is proportional to the ratio between the viscosity of the nonwetting phase and the average viscosity of the other two phases. This is perfectly consistent with the expected behavior in real displacements: 1. If the viscosity of the most nonwetting phase is very small compared to that of the other phases, one expects fingering —unstable displacement— when the nonwetting phase is injected. This results in an early breaktrough with small abrupt changes between having no flow and some flow at the outlet. This implies a small value of the endpoint slope of the relative permeability, βg . 2. On the other hand, if the viscosity of the nonwetting phase is comparable with that of the other phases in the medium, the displacement of that phase will be stable, and will cause that the transition between zero flow and nonzero flow at the outlet will be more drastic, resulting in a larger value of βg .

97

Chapter 3. Relative permeabilities for strictly hyperbolic models 3.3.3.2

Analysis along the OG edge

We now study if the relative permeability model satisfies Condition II along the OG edge of the saturation triangle (u = 0). Along this edge, the derivatives of the fractional flow functions involved in Equation (3.34) are: f,u = 0, g,v =

(3.57)

1 1−v ³ µg µo βg v+(1−βg )v2 + µg

(1−v)2 µo

´2 · (βg + (2 − βg )v).

(3.58)

For strict hyperbolicity along the OG edge (excluding the G vertex), we require: Hog = g,v − f,u =

1−v 1 ³ µg µo βg v+(1−βg )v2 + µg

(1−v)2 µo

´2 · (βg + (2 − βg )v) > 0,

(3.59)

which is always satisfied for all v ∈ [0, 1), as long as the endpoint slope βg > 0. At the G vertex (u = 0, v = 1), we obtain Hog = 0, so this point is an umbilic point, as required. 3.3.3.3

Analysis along the WG edge

We perform now the same analysis on the edge of zero reduced oil saturation, u = 1 − v. The expressions of the fractional flow derivatives in Equation (3.43) are: f,v = ³

g,u = ³

−1

βg v+(1−βg )v 2 µg

+

−1

βg v+(1−βg )v 2 µg

+

·

¸ βg + 2(1 − βg )v v(1 − v) (1 − v)2 + , · ´2 · µg µo µw (1−v)2 µw

¸ 2(1 − v) v(1 − v) βg v + (1 − βg )v 2 − · . ´2 · µw µo µg (1−v)2 µw

·

(3.60)

(3.61)

The condition for strict hyperbolicity along this edge (not including the G vertex) is Hwg

¸ · 1 Cwg (v) Cog (v) Cwo (v) = −g,u − f,v = > 0, + + E(v) µw µg µo µg µw µo

(3.62)

98

Chapter 3. Relative permeabilities for strictly hyperbolic models where E(v) =

µ

βg v + (1 − βg )v 2 (1 − v)2 + µg µw

¶2

,

(3.63)

Cwg (v) = (2 − (2 − βg )(1 − v))(1 − v),

(3.64)

Cog (v) = −(1 − (1 − βg )(1 − v))v 2 (1 − v),

(3.65)

Cow (v) = −v(1 − v)3 .

(3.66)

At the G vertex, corresponding to v = 1, it is clear that Hwg = 0. On the other hand, it is not easy to infer the conditions on the fluid viscosities and the endpoint slope β g such that the strict inequality (3.62) is satisfied on the entire edge. It is possible, however, to identify the conditions for strict hyperbolicity along this edge near the G vertex. Let u = ε, v = 1−ε with ε → 0, that is, a state on the WG edge near the G corner. The first-order Taylor expansion of Hwg about ε = 0 is · ¸ 2 1 Hwg = µg ε + O(ε2 ). − µw µo

(3.67)

Therefore, for Hwg > 0 in the neighborhood of the G corner, we obtain the condition µw < 2µo .

(3.68)

This imposes an additional restriction —one that is not obvious to anticipate— on the values of the fluid viscosities, if the relative permeability model (3.47)–(3.49) is to result in a strictly hyperbolic system. 3.3.3.4

Summary of conditions for strict hyperbolicity of the model

We have arrived at the conclusion that the very simple relative permeability model (3.47)–(3.49) may yield strictly hyperbolic behavior everywhere in the saturation triangle, except at the vertex of 100% reduced gas saturation, which is an


99

How (u), Hog (v), Hwg (u)

8

PSfrag replacements

7

OW

6

OG

5 4

WG

3 2 1 0

0

0.2

0.4

u, v

0.6

0.8

1

Figure 3.5. Check of strict hyperbolicity on edges of the saturation triangle (Condition II), which requires that all three functions How (u), Hog (v), and Hwg (u) are positive everywhere. umbilic point. The only two conditions that the parameters of the model need to satisfy are: βg > √

µg , µ o µw

(3.560 ) (3.680 )

µw < 2µo .

For illustrative purposes, we take reasonable values of the viscosities (what really matters is the viscosity ratios): µw = 0.875,

µg = 0.03,

µo = 2 cp,

(3.69)

and a small value of the endpoint slope: βg = 0.1. These values of the parameters satisfy the two conditions above. In Figure 3.5 we represent graphically the functions How (u) along OW, Hog (v) along OG, and Hwg (u) along WG. Inequalities (3.26), (3.34), and (3.43) are satisfied, and the system is strictly hyperbolic, if all three curves are positive everywhere. The curves for the OG edge and the WG edge reach a zero value for v = 1 and u = 0, respectively, so that the G vertex is an umbilic point.


3.4

100

Validation with experimental data

In Section 3.3 we derived the necessary conditions that must be satisfied by the relative permeability functions, if the system of equations describing three-phase flow is to be strictly hyperbolic everywhere inside the saturation triangle. The essential requirement for strict hyperbolicity turns out to be that, at the edge of zero reduced gas saturation, the relative permeability of gas (the most nonwetting phase) must have a positive derivative with respect to its own saturation —Equation (3.27). In this section we verify how realistic this condition is, by means of comparison with experimental data. To this end, we use Oak’s steady-state experiments [Oak, 1990; Oak et al., 1990], which are arguably the most extensive and reliable data set available. The fact that we use steady-state relative permeability data in a dynamic fluid displacement model should not invalidate the conclusions, because it has been reported that the relative permeabilities measured with steady and unsteady methods ´ are similar (see, e.g., Efros [1956]; Johnson et al. [1959]; Osoba et al. [1951]; Welge [1952]). The data set consists of over 1800 two-phase and three-phase relative permeability measurements, obtained using a fully automated steady-state method. Three fired Berea sandstone cores were employed, with absolute permeabilities of 200 md (Sample 6), 800 md (Sample 14), and 1000 md (Sample 13). Water, oil, and gas viscosities were 1.06, 1.77, and 0.0187 cp, respectively. The study includes over 30 combinations of rock and fluid systems and saturation histories. A complete description of the experimental apparatus and procedure is given in the original references Oak [1990], and Oak et al. [1990].

101

Chapter 3. Relative permeabilities for strictly hyperbolic models 6

6

krα

krα

-

Sα (a) Zero-slope behavior

-

Sα (b) Positive-slope behavior

Figure 3.6. Markedly different qualitative behavior of the slope of the relative permeability of a phase, in the region near the “immobile” saturation of that phase: (a) zero-slope, and (b) positive slope.

3.4.1

Description of the “endpoint-slope” analysis

We are interested in the qualitative behavior of the relative permeability of each phase in the region of low reduced saturation of that phase. More precisely, we want to determine whether the relative permeability of a phase, when expressed as a function of its own saturation only, takes off with a zero or a positive slope. This fundamentally different behavior is shown schematically in Figure 3.6. The steps involved in our “endpoint-slope” analysis of Oak’s relative permeability data are described below: 1. Select an experiment. An experiment consists of several —sometimes dozens— relative permeability measurements. 2. Regardless of the type of experiment —two-phase, or three-phase; drainage or imbibition— tabulate the relative permeability of a phase against its own saturation. 3. For each phase α, identify a maximum saturation Sα,max that defines the range


102

of saturations to be used in the analysis, and use only the data points for saturations Sα < Sα,max . This range should be small enough to be considered close to the immobile saturation, but should have a sufficient number of data points to indicate a trend. 4. For each phase α, fit the power-law expression krα = Cα (Sα − Sαi )mα ,

Sα < Sα,max ,

(3.70)

using a least squares procedure [Coleman et al., 2000], with the following constraints: Cα > 0,

Sαi ≥ 0,

mα ≥ 1.

(3.71)

Of the three parameters to be optimized, the most relevant for our discussion is the exponent mα . A value of 1 or close to 1 is indicative of a linear behavior of the relative permeability and, thus, a positive slope at the endpoint saturation. On the other hand, an exponent larger than 2 suggests that the relative permeability will approach a zero value (at the endpoint saturation) with zero slope. Here we present the results of the analysis of four representative experiments, to show how well the condition of positive endpoint slope for the gas relative permeability is satisfied by actual data.

3.4.2

Two-phase flow experiments

The selected data come from two consecutive experiments, performed on a fired Berea sandstone core of about 1000 md (Sample 13), using water and gas as the wetting and the nonwetting phase, respectively.


103

0

Relative Permeability

10

−1

10

PSfrag replacements

−2

10

−3

10

Gas Water

−4

10

0

0.2

0.4

0.6

0.8

1

Water Saturation

Figure 3.7. Relative permeability curves of water and gas for the two-phase drainage experiment (Sample 13, Experiment 16a of Oak’s dataset). The solid square mark (¥) indicates the initial saturation state of the core. 3.4.2.1

Primary drainage experiment

The first experiment —Oak’s Sample 13, Experiment 16a— corresponds to a drainage process, where gas is injected into an initially water-filled core. This is done through a sequence of steady states: water and gas are injected at constant rates and, when steady state is achieved, the relative permeabilities and average saturations are measured; then, the ratio of gas/water flow rates is increased, the system reaches a new steady state, and the process continues. The relative permeability curves for this experiment are plotted in semi-log scale in Figure 3.7. The results of the power-law fitting of the data from the two-phase drainage experiment are presented in Table 3.4. The most important observation is the essential difference in the value of the exponent mα for the wetting phase (mw ≈ 8) and the nonwetting phase (mg ≈ 1.1). The actual fit of the relative permeability data of water and gas is shown graphically in Figure 3.8. From this figure it is apparent that the water relative permeability reaches a value of zero with a zero value of the slope, whereas the slope of the gas relative permeability curve is finite and positive.

104


Table 3.4. Parameters of the power-law fitting for the two-phase drainage experiment. Note the major difference in the value of the exponent mα for water and gas. Water

Gas

Sα,max

0.7000

0.2500

Cα

2.6362

0.1902

Sαi

0.0000

0.0404

mα

8.0282

1.1377

0.16

0.035

Gas Relative Perm

PSfrag replacements

Water Relative Perm

0.14

PSfrag replacements

0.12 0.1

Water Saturation

0.08 0.06

Gas Saturation

0.04 0.02 0

Gas Relative Perm

Water Relative Perm 0

0.2

0.4

0.6

0.03

0.025 0.02

0.015 0.01

0.005

0.8

Water Saturation (a) Water relative permeability

0

0

0.05

0.1

0.15

0.2

0.25

Gas Saturation (b) Gas relative permeability

Figure 3.8. Behavior of the relative permeability as a function of its own saturation in the neighborhood of the “endpoint saturation”, for the two-phase drainage experiment: (a) the water relative permeability curve shows a high-order behavior and, therefore, a zero slope near the connate water saturation; (b) on the other hand, the gas relative permeability curve displays an almost-linear relation against gas saturation, which can be assimilated to a nonzero slope near the critical gas saturation. This experimental observation is in agreement with the key condition (3.27) proposed in this investigation. 3.4.2.2

Secondary imbibition experiment

The data analyzed here —Oak’s Sample 13, Experiment 16b— correspond to the imbibition process following the primary drainage experiment described in the


105

0

Relative Permeability

10

−1

10

PSfrag replacements

−2

10

−3

10

Gas Water

−4

10

0

0.2

0.4

0.6

0.8

1

Water Saturation

Figure 3.9. Relative permeability curves of water and gas for the two-phase imbibition experiment (Sample 13, Experiment 16b of Oak’s dataset). The solid square mark (¥) indicates the saturation state of the core after primary drainage. previous paragraph. As before, the experiment involves a sequence of steady states, except that now, when a steady state is achieved, the ratio of gas/water flow rates is decreased. The relative permeability curves obtained in this way are plotted in semi-log scale in Figure 3.9. The results of the power-law fitting to the data are presented in Table 3.5. Once again, the values of the exponent mα for the wetting phase (mw ≈ 4) and the nonwetting phase (mg = 1) are fundamentally different. The actual fit to the relative permeability data of water and gas is shown graphically in Figure 3.10, and the same comments as in the drainage experiment follow: a zero slope of the water relative permeability, and a positive slope of the gas relative permeability near their respective immobile saturations.

3.4.3

Three-phase flow experiments

We repeat the same analysis for experiments involving simultaneous flow of three fluids: water, oil, and gas. Water is the most wetting phase, and gas is the least

106

Chapter 3. Relative permeabilities for strictly hyperbolic models Table 3.5. experiment.

Parameters of the power-law fitting for the two-phase imbibition Water

Gas

Sα,max

0.6700

0.4500

Cα

3.9379

1.2051

Sαi

0.2785

0.3212

mα

3.9030

1.0000 0.16 0.14

0.1

PSfrag replacements

0.08

Water Saturation

0.06

Gas Saturation

0.12 0.1

0.08 0.06

0.04

0.04

0.02

Gas Relative Perm

Gas Relative Perm

PSfrag replacements

Water Relative Perm

0.12

Water Relative Perm 0 0.2

0.3

0.4

0.5

0.6

0.7


0.02 0 0.3

0.35

0.4

0.45

0.5


Figure 3.10. Behavior of the relative permeability as a function of its own saturation in the neighborhood of the “endpoint saturation”, for the two-phase imbibition experiment. Remarks in Figure 3.8 apply. wetting. We present the results of two experiments: one simulating gas injection, and the other characteristic of water flooding. 3.4.3.1

Drainage-dominated experiment

The first of the three-phase flow experiments consists in a sequence of steady states of increasing average gas saturation. The jump in saturations from one steady-state to the next is achieved by appropriately modifying the flow rates at which each fluid is injected into the core. In particular, the ratio of gas/water flow rates is increased,


107

G 0

1

0.2

0.8

0.4

0.6

0.6

PSfrag replacements

0.4

0.8

0.2

0 1

0.8

0.6

0.4

0.2

0

1

O

W

Figure 3.11. Saturation path for the drainage-dominated three-phase relative permeability experiment (Sample 6, Experiment 15a of Oak’s dataset). The square mark (¤) indicates the saturation state of the core at the beginning of the experiment (initially water-filled). while the water/oil ratio is held constant. The resulting saturation path for this experiment is shown on a ternary diagram in Figure 3.11. In Table 3.6 we present the parameters of the power-law fit for the water and gas phases. In accordance with the two-phase flow results, we observe a high value of the exponent (mw ≈ 3) for water, and a value close to one (mg ≈ 1.2) for gas. The obvious interpretation is that the water relative permeability curve reaches zero with a zero-value of the slope, whereas the slope is positive for the gas relative permeability (see Figure 3.12). 3.4.3.2

Imbibition-dominated experiment

Our last example is a three-phase flow experiment, in which fluids are injected in such a way that steady states of increasing water saturation are obtained. This sequence is achieved by decreasing the ratio of gas/water flow rates, while keeping

108


Table 3.6. Parameters of the power-law fitting for the three-phase drainagedominated experiment. Water

Gas

Sα,max

0.4840

0.4000

Cα

0.1065

0.7162

Sαi

0.2092

0.0941

mα

2.8623

1.2552

−3

x 10

0.2

2.5

PSfrag replacements

PSfrag replacements

2

1.5

Gas Saturation

Water Saturation

1

0.15

0.1

0.05

0.5

Gas Relative Perm

Gas Relative Perm

Water Relative Perm

3

Water Relative Perm 0 0.2

0.25

0.3

0.35

0.4

0.45

0.5


0

0

0.1

0.2

0.3

0.4


Figure 3.12. Behavior of the relative permeability as a function of its own saturation in the neighborhood of the “endpoint saturation”, for the three-phase drainagedominated experiment. Resemblance of actual data to the conceptual picture of Figure 3.6 is apparent. constant the water/oil ratio. In Figure 3.13, we depict the saturation path for this experiment, whose starting point is the saturation state obtained at the end of drainage-dominated process analyzed before. Table 3.7 has the numeric values of the power-law fit, and Figure 3.14 shows the experimental data and the fitted relative permeability curves. The same qualitative behavior as that of the previous examples is observed (mw ≈ 3, mg = 1).


109

G 0

1

0.2

0.8

0.4

0.6

0.6

PSfrag replacements

0.4

0.8

0.2

0 1

0.8

0.6

0.4

0.2

0

1

O

W

Figure 3.13. Saturation path for the imbibition-dominated three-phase relative permeability experiment (Sample 6, Experiment 15b of Oak’s dataset). The square mark (¤) indicates the saturation state of the core at the end of the drainage-dominated experiment (gas injection). Table 3.7. Parameters of the power-law fitting for the three-phase imbibitiondominated experiment. Water

Gas

Sα,max

0.5000

0.6000

Cα

0.4737

2.1985

Sαi

0.2569

0.3530

mα

3.2804

1.0000

Remark 3.5. The particular examples presented herein are representative of more than one hundred experiments in Oak’s dataset. Experimental data seems to corroborate the fundamental requirement for strict hyperbolicity of the model, that is, a positive slope of the relative permeability of the most nonwetting phase near its immobile saturation.

110

Chapter 3. Relative permeabilities for strictly hyperbolic models −3

x 10

0.7

4

PSfrag replacements

PSfrag replacements

3

Water Saturation

0.3 0.2

1

Water Relative Perm Gas Relative Perm

0.5 0.4

2

Gas Saturation

Gas Relative Perm

0.6

Water Relative Perm

5

0 0.25

0.3

0.35

0.4

0.45

0.5


0.1 0 0.3

0.4

0.5

0.6


Figure 3.14. Behavior of the relative permeability as a function of its own saturation in the neighborhood of the “endpoint saturation”, for the three-phase imbibitiondominated experiment.

3.5

Concluding remarks

Traditional formulations of three-phase flow in porous media employ the usual extension of Darcy’s equation to model fluid fluxes. Within this framework, it was believed that elliptic regions were unavoidable when generic relative permeability functions were used in models of one-dimensional immiscible incompressible threephase flow. This conclusion was inferred after a particular behavior of the relative permeabilities along the edges of the saturation triangle was assumed. In this chapter, we show it is possible to identify conditions which the relative permeability functions must satisfy for the system of equations to be strictly hyperbolic everywhere in the saturation triangle. By means of a specific example, we suggest how strict hyperbolicity may be invoked to impose constraints on the parameters of the relative permeability model. It turns out that the fundamental requirement is a finite positive slope of the gas relative permeability at the saturation where gas becomes mobile. This condition is consistent with a pore-scale description of multiphase flow


111

and, as shown here, is also supported by experimental relative permeability data. This important result is restricted to the case when gravitational effects are not accounted for, which is sensible only when the gravity number is small, or when flow is horizontal. It is possible, however, to extend this analysis to the case when flow is not horizontal and gravity is included, by allowing that relative permeabilities may vary with the gravity number [Juanes and Patzek, 2002f]. This dependence is physically reasonable, and can be justified in terms of the stability of the displacement of one fluid by another. This is discussed in the following chapter.

The ellipsis, or speech by half-words [is the peculiar talent] of ministers and politicians. — ALEXANDER POPE, The Art of Sinking in Poetry (1727)

112

Chapter 4 Strictly hyperbolic models of co-current flow with gravity 4.1

Introduction

Traditional formulations of multiphase flow model macroscopic fluid fluxes with a straightforward extension —first proposed by Muskat [1949]— of Darcy’s equation for single-phase flow. As we discussed in Chapter 1, Darcy-type models do not reflect the multiscale character of multiphase flow. The only way in which they can capture at least a shadow of the behavior of the actual displacement is through the relative permeability functions, because these are the only “degrees of freedom” of the formulation. Therefore, relative permeabilities cannot be understood as fixed functions of saturation, or even saturation history. They depend intrinsically on the flow regime and properly should be called functionals rather than functions. In this context, we regard the relative permeability as nothing else than a functional used in the constitutive model, which may —and in fact should — be influenced by the fluid viscosity

Chapter 4. Strictly hyperbolic models of co-current flow with gravity

113

ratios and the gravity number. It is precisely the influence of viscosity and gravity on the relative permeabilities that allows one to remove some of the mathematical inconsistencies of the classical formulation of three-phase flow.

4.1.1

Character of the equations

It was conjectured in Charny [1963], and shown in Bell et al. [1986], that certain relative permeabilities would yield a system of equations that is not strictly hyperbolic for all saturation states. Indeed, regions in the saturation triangle where the system is locally elliptic —the so-called elliptic regions— are present for most relative permeability functions used today [Bell et al., 1986; Fayers, 1987; Hicks Jr. and Grader, 1996; Holden, 1990a; Shearer, 1988; Shearer and Trangenstein, 1989]. The only models that do not display elliptic regions are those where the relative permeability of each phase is a function of its own saturation only [Marchesin and Medeiros, 1989; Trangenstein, 1989]. In this case, elliptic regions shrink to isolated umbilic points, where eigenvalues of different families are equal. The analysis of Shearer [1988] and Holden [1990a] suggests that elliptic regions are an unavoidable consequence of three-phase flow models. This important result has had two consequences: 1. Further investigation on the theory of mixed elliptic/hyperbolic systems (see, e.g., the monographs Keyfitz and Shearer [1990], and Lindquist [1989] and the references therein). 2. A widespread controversy about whether elliptic regions are physical [Jackson and Blunt, 2002]. As for (1) above, although a few qualitative aspects are known [Holden, 1987; Holden


114

and Holden, 1989; Holden et al., 1990], a complete mathematical theory of mixed elliptic/hyperbolic systems is still lacking. In particular, correct entropy conditions —needed to remove nonuniqueness of the solutions— are not yet known [Azevedo and Marchesin, 1990, 1995]. As for (2), we think that the presence of elliptic regions should not be justified simply because they appear in a Darcy-type formulation of three-phase flow with relative permeability that are fixed functions of saturations. In particular, the analysis in Shearer [1988] and Holden [1990a] assumes a particular behavior of the relative permeabilities on the edges of the saturation triangle. In the previous chapter (see also Juanes and Patzek [2002e]), we argued that the system describing three-phase displacements should be strictly hyperbolic, and we identified conditions on the relative permeabilities so that this essential feature is preserved. The generic conditions that need to be imposed on the relative permeability functions seem to be in agreement with experimental observations and pore-scale physics. We now extend the previous analysis to incorporate gravity effects.

4.1.2

Effects of gravity

When gravity is included in the analysis, it was shown that a large class of models of three-phase relative permeabilities may yield elliptic regions [Trangenstein, 1989]. The effect of gravity on elliptic regions has also been illustrated elsewhere [Guzmán and Fayers, 1997a; Hicks Jr. and Grader, 1996; Jackson and Blunt, 2002; Shearer and Trangenstein, 1989]. In all these investigations, the usual extension of Darcy’s equation to multiphase flow is adopted without further discussion. In addition, relative permeabilities are taken as fixed functions of saturations, independent of the flow regime. Ellipticity of the equations arises because the “driving force” is modified by including a dominant gravity term, while the relative permeability functions are kept


115

unchanged. In Juanes and Patzek [2002e] we conjectured that relative permeabilities would vary in such a way that the system of equations —including the gravity term— would remain everywhere strictly hyperbolic. In this chapter we conclude that, under certain physical conditions, our conjecture was correct. An essential feature of our study is that we restrict the analysis to displacement processes, where the role of capillarity effects is simply to smear out the moving fronts. Physically, we are limited to systems with co-current flow for all saturation states. In situations with mixed co-current and counter-current flow, the effect of capillarity cannot be neglected. In those situations, the capillarity-free model is no longer valid. To avoid misunderstanding, we want to distinguish our approach from that of other investigations, such as those of Trangenstein [1989], and Jackson and Blunt [2002]. Trangenstein [1989] neglects any potential dependence of the relative permeabilities on the flow regime, and he states this assumption explicitly: “. . . I will assume that the relative permeabilities are only allowed to depend on the phase saturations. Thus, given the relative permeability functions, it will be fair game to vary the viscosities, mass densities, total fluid velocity and reservoir dip angle in order to obtain complex characteristic speeds.” [Trangenstein, 1989, p. 149] In view of our observations in Chapter 1, this assumption is not realistic, because the presence of gravity modifies pore-scale displacements which, in turn, yield different macroscopic relative permeabilities. Similarly, Jackson and Blunt [2002] assume that relative permeabilities are fixed functions of fluid saturations alone, and evaluate the character of the system for large values of the gravity number. Their values are well outside the range yielding cocurrent flow. Our analysis does not apply to the mixed co-current/counter-current


116

conditions they encounter. In fact, and this is partly what they observe, we think that in such cases capillarity cannot be dropped from the formulation. Jackson and Blunt [2002] take a unique step in justifying elliptic regions as physically plausible. They use a serial model of capillary bundles to demonstrate that elliptic regions exist in a simplistic but physically realizable porous medium. However, a three-phase displacement process in a sequence of bundles cannot be described in the form proposed by the authors [Juanes et al., 2003]. The constraints imposed by the fractional flow formalism and by the proposed communication between bundles reduce their model to a single bundle of capillary tubes. The latter model is inappropriate in the context of a displacement process described by the volume-averaged mass and momentum balances dominated by viscous and gravity forces. Therefore, the relative permeabilities used in Jackson and Blunt [2002] should be understood as another empirical model. The presence of elliptic regions in their model does not imply that they exist in reality.

4.1.3

Chapter outline

In this chapter we derive conditions that the relative permeabilities must satisfy for the system of saturation equations to be strictly hyperbolic, when gravity is included in the formulation. The crucial point is to acknowledge that, because a Darcy-like formulation is incomplete, the relative permeability functions may depend on the fluid viscosity ratios and on the gravity number. It is reasonable to think that the effect of gravity and viscosity ratios will be most pronounced in the relative permeability to gas, which is usually the least wetting, the least viscous, and the least dense fluid phase. By means of an illustrative example, we show that this dependence may not be strong, but sufficient to remove elliptic behavior. The required conditions


117

for strict hyperbolicity, which are derived from strictly mathematical arguments, are then analyzed from a physical viewpoint. It is found that they are consistent with, and may justified from, the observed behavior of fluid displacements that take place in porous media. An outline of this chapter is as follows. In Section 4.2, we present the governing equations of one-dimensional, immiscible three-phase flow with gravity, derived in Chapter 2. We express the system of equations in dimensionless form and discuss its mathematical character. In Section 4.3, we derive conditions for strict hyperbolicity of the system, and show they are closely-related to the conditions for co-current flow. These conditions are then examined from a physical viewpoint. In Section 4.4, we give an example of how to accommodate the abstract conditions for strict hyperbolicity for a particularly simple relative permeability model, and illustrate the effects of removing elliptic behavior. Finally, in Section 4.5, we summarize the main results of this chapter.

4.2

Mathematical model

In this section, we give a brief account of the mathematical model describing onedimensional three-phase flow with gravity under negligible capillarity effects. The mathematical formulation was presented in detail in Chapter 2. The 2 × 2 system of governing equations is expressed in its final form:

where

∂t u + ∂x f = 0,

(4.1)

     u  S w  u= =  Sg v

(4.2)

Chapter 4. Strictly hyperbolic models of co-current flow with gravity is the vector of unknown saturations, and       λw [1 − M ((ρD − 1)λo + ρD λg )]  f   fw    u =   =   =  λT λ  g g fg [1 + M (ρD λw + λo )] λT

118

(4.3)

is the vector of fractional flow functions, where (see Chapter 2) (ρo − ρg )k gx , vT φ ρw − ρ g ρD := . ρo − ρ g M :=

We shall also make use of the the alternative form of the flux functions: · µ ¶¸ λw µo fw := 1 − ND (ρD − 1)kro + ρD krg , λT µg µ · ¶¸ µo λg 1 + ND ρD krw + kro , fg := λT µw

(4.4) (4.5)

(4.6)

where ND is the gravity number, ND =

M . µo

(4.7)

We understand throughout that the system (4.1) is in dimensionless form, using the definitions in Section 2.4 and that fluid saturations have been re-normalized, as explained in Section 2.5. The classification of this system depending on its mathematical character is given in Section 3.2.2.

4.3

Conditions for a strictly hyperbolic system

It is by now a well-known fact that the system of equations (4.1) describing onedimensional three-phase flow may exhibit mixed hyperbolic/elliptic character for most relative permeability functions used today [Bell et al., 1986; Falls and Schulte, 1992a,b; Fayers, 1987; Guzmán and Fayers, 1997a,b; Hicks Jr. and Grader, 1996; Holden,


119

1990a; Miller et al., 1998; Shearer, 1988; Shearer and Trangenstein, 1989]. Strict hyperbolicity of the equations is typically lost at: 1. Elliptic regions: bounded regions of the saturation triangle, where the eigenvalues are complex conjugates, and the system is locally elliptic. 2. Umbilic points: saturation states for which there is a double eigenvalue, and the system is not strictly hyperbolic. The existing literature seems to suggest that the presence of elliptic regions and umbilic points inside the saturation triangle is an unavoidable feature of three-phase flow models in porous media. The approach taken in previous investigations [Holden, 1990a; Shearer, 1988] is to assume a particular behavior of the relative permeabilities and, from this assumed behavior, infer that the system of equations cannot be strictly hyperbolic everywhere in the saturation triangle. One of the key assumptions is the “zero-derivative” condition, which states that the relative permeability of a phase has a zero normal derivative at the edge of zero saturation of that phase. This condition implies, for example, that the derivative of the gas relative mobility with respect to gas saturation is zero along the oil-water edge of the saturation triangle: λg,v = 0 on v = 0, 0 < u < 1.

(4.8)

Under this assumed behavior along the edges of the saturation triangle, it can be shown that an elliptic region must be present inside the ternary diagram [Holden, 1990a; Schaeffer and Shearer, 1987a; Shearer, 1988]. When gravity is included in the formulation, it was shown that a large class of models of three-phase relative permeabilities may yield elliptic regions [Trangenstein, 1989] (see also Shearer and Trangenstein [1989], Hicks Jr. and Grader [1996], Guzmán and Fayers [1997a], Guzmán and Fayers [1997b], and Jackson and Blunt [2002]).


120

Common to all these investigations is the use of Muskat’s multiphase flow extension of Darcy’s equation, and relative permeabilities which are taken as fixed functions of saturations independent of the flow regime. Under these assumptions, the analysis in Trangenstein [1989] shows that for any relative permeability model where one or more relative permeabilities are functions of two saturations, the system of saturation equations will have elliptic regions for some combination of viscosities, densities and gravity numbers. We find the mixed elliptic/hyperbolic behavior disturbing for many reasons, and are of the opinion that elliptic regions are artifacts of an incorrect mathematical model. In the previous chapter, we have argued that strict hyperbolicity is an essential property of a displacement process, which should be preserved by our three-phase flow models. From an analysis of the character of the system of equations governing three-phase immiscible incompressible flow we conclude that —when gravity effects are not included— it is possible to choose relative permeability functions so that this system is strictly hyperbolic everywhere in the saturation triangle. Moreover, the generic conditions that need to be imposed on the relative permeability functions to preserve strict hyperbolicity seem to be in agreement with experimental observations and pore-scale physics. The question remains: is it possible to derive similar conditions when gravity is included, so that the system of saturation equations is strictly hyperbolic? The analysis of Trangenstein [1989] suggests that the answer is negative. His analysis is limited, however, by the fact that fixed relative permeability functions are used. Ellipticity of the equations arises because the “driving force” in the multiphase extension of Darcy’s equation is modified by including a dominant gravity term, while the relative permeability functions are kept unchanged. In our opinion, using fixed relative perme-


121

ability functions for all gravity numbers is not realistic. Dominance of gravity forces modifies the displacements of one fluid by another which, in turn, may yield different macroscopic relative permeabilities. Thus, in the analysis below, we adopt the opposite viewpoint to that of previous investigations: we do not rule out the possibility that the macroscopic relative permeabilities may depend on the gravity number. In doing so, we acknowledge that the common extension of Darcy’s equation does not capture all the physics of multiphase flow.

4.3.1

Conceptual picture of three-phase displacements

The behavior of relative permeabilities, assumed in traditional models of threephase flow, limits the influence of a phase on the flow of the other two phases near the edge where that phase is immobile [Holden, 1990a; Shearer, 1988]. This behavior is expressed effectively through the eigenvectors of the Jacobian matrix of the system of governing equations along the edges of the ternary diagram (see Figure 4.1, or Figure 3.1 in Chapter 3): 1. The right eigenvector associated with the fast characteristic family, r 2 , is parallel to the edges of the triangle of reduced saturations. 2. The fast eigenvector r 2 points into the triangle, for saturation states near the vertices. The most relevant impact of this conceptual picture on the mathematical character of the system is that, in general, an elliptic region must exist inside the saturation triangle. This general result can be proved using ideas of projective geometry [Holden, 1990a; Schaeffer and Shearer, 1987a; Shearer, 1988].


122

G 0

1

0.2

0.8

r2

r2

0.4

PSfrag replacements

0.6

0.6

r2

0.4

Elliptic region

0.8

r2

1

0.8

0.6

r2 0.4

r2 0.2

0

1

O

0.2

0

W

Figure 4.1. Schematic representation of the direction of fast eigenvectors r 2 along the edges of the saturation triangle for the models analyzed by Shearer [1988] and Holden [1990a]. For models of this type, vertices are umbilic points, and there must be an elliptic region inside the saturation triangle. Our conceptual picture of three-phase displacements differs from the traditional one. The difference, although subtle, is essential. The key observation is that, whenever gas is present as a continuous phase, the mobility of gas is much higher than that of the other two fluids, water and oil. This implies that the wave associated with changes in gas saturation is in fact the fast wave, even in the neighborhood of the edge of zero reduced gas saturation. The proposed behavior requires that the eigenvector of the fast characteristic family (r 2 ) is transversal —and not parallel— to the oil-water edge of the ternary diagram (see Figure 4.2, or Figure 3.3 in Chapter 3). In Juanes and Patzek [2002e] we showed that this conceptual picture permits that the system of equations be strictly hyperbolic everywhere inside the saturation triangle. The only saturation state for which strict hyperbolicity is lost is the vertex of 100% reduced gas saturation, due to the additional requirement that the eigenvectors are not allowed to rotate along the edges of the ternary diagram.


123

G 0

1

Umbilic point

0.2

PSfrag replacements

0.8

r2 0.4

r2 0.6

0.6

0.4

r2 0.8

r2

1

0.8

0.6

r1 0.4

r1 0.2

0

1

O

0.2

0

W

Figure 4.2. Schematic representation of the direction of fast (r 2 ) and slow (r 1 ) eigenvectors along the edges of the saturation triangle for the type of models we propose. The system is strictly hyperbolic everywhere inside the saturation triangle, and the only umbilic point is located at the G vertex, where the fast paths corresponding to the OG and WG edges coalesce. The picture of slow and fast characteristic paths along the edges of the saturation triangle shown in Figure 4.2 was developed for the case where gravity effects were not included in the formulation [Juanes and Patzek, 2002e]. We preserve this conceptual picture when gravity effects are taken into account. This is sensible, however, only if the three-phase flow is a displacement process. Therefore, we limit our description of three-phase flow with gravity to situations of co-current flow, that is, when all three phases have fluid velocities that contribute positively to the total flow. In fact, the situation changes dramatically if flow of one of the fluids is counter-current for certain saturation states. It is obvious that flow cannot be counter-current in the entire saturation triangle and, therefore, mixed co-current and counter-current flow will take place for certain saturation paths. In this case, capillarity effects cannot be neglected and the three-phase displacement model analyzed here is no longer valid.


124

In summary, we restrict our attention to displacement processes, characterized by co-current flow. In the following sections we derive mathematical conditions for co-current flow and strict hyperbolicity of the system.

4.3.2

Conditions for co-current flow

The conditions for co-current flow can be expressed succinctly as follows: f ≥ 0,

(4.9)

g ≥ 0,

(4.10)

1 − f − g ≥ 0.

(4.11)

When written in terms of the relative mobilities, Equations (4.3), the equations above take the following form: 1 − M (ρD λg + (ρD − 1)λo ) > 0,

(4.90 )

1 + M (ρD λw + λo ) > 0,

(4.100 )

1 − M (λg − (ρD − 1)λw ) > 0.

(4.110 )

Conditions (4.90 )–(4.110 ) must hold for any saturation state in the ternary diagram. It is possible, however, to ascertain which saturation states impose the most restrictive constraints on the gravity number. Condition (4.90 ) of positive water fractional flow will impose the strictest restriction on the gravity number wherever the factor Ξw := (ρD − 1)λo + ρD λg

(4.12)

is largest. The oil and gas relative mobilities will increase as the water saturation decreases. Therefore, it will be sufficient to analyze the factor Ξw on the OG edge,


125

where the reduced water saturation is zero. Moreover, the functions λo and λg are convex along the OG edge, so Ξw will take a maximum value at one of the vertices, corresponding to either 100% reduced oil saturation (the O vertex) or 100% reduced gas saturation (the G vertex). In all practical cases, the gas viscosity is much smaller than the oil viscosity (µg ¿ µo ) and, as a result, Ξw will be largest at the G vertex. Evaluating condition (4.90 ) at this saturation state, one obtains: 1 , ρD λmax g

(4.13)

1 µg . max µo ρD krg

(4.14)

M< or, equivalently, ND
0

g,v − f,u > 0

−g,u − f,v > 0

Along each of the three edges, two types of conditions need to be imposed: • Condition I enforces that eigenvectors of the appropriate family are parallel to the edge. • Condition II enforces strict hyperbolicity along the edge. We shall not repeat the entire analysis along each of the edges of the saturation triangle, and we refer the reader to Chapter 3, and to Juanes and Patzek [2002e], for the mathematical considerations that lead to the conditions shown in Table 4.1. When these conditions are expressed in terms of fluid relative mobilities through Equations (4.3), we obtain restrictions that the relative mobilities must satisfy for the system to be strictly hyperbolic. The conditions along the edges are summarized in Table 4.2, and the conditions at the vertices of the saturation triangle are given in Table 4.3. The analysis is analogous to that carried out for the case without gravity, the only difference being that the water and gas fractional flows —f and g, respectively— have more complicated expressions due to the gravity term.

4.3.4

Discussion of conditions

Several aspects of the conditions for strict hyperbolicity derived above deserve further discussion and interpretation.


128

Table 4.2. Summary of conditions for strict hyperbolicity along the edges of the saturation triangle, in terms of the fluid relative mobilities. Condition I OW edge:

λg,u = 0

OG edge:

λw,v = 0

WG edge:

λo,v = λo,u

Condition II OW edge:

(λw + λo )[1 + M (ρD λw + λo )]λg,v > λo [1 − M (ρD − 1)λo ]λw,u − λw [1 + M (ρD − 1)λw ]λo,u

OG edge:

(λg + λo )[1 − M (ρD λg + (ρD − 1)λo )]λw,u < λo [1 + M λo ]λg,v − λg [1 − M λg ]λo,v

WG edge:

(λg + λw )[1 − M (λg − (ρD − 1)λw )](−λo,u ) < λw [1 + M ρD λw ](λg,v − λg,u ) + λg [1 − M ρD λg ](λw,u − λw,v )

1. If the gravity number is zero, the conditions reduce to those obtained in Chapter 3, where gravity was not included in the analysis. 2. It is interesting to see how the conditions for co-current flow relate to the conditions for strict hyperbolicity. When specialized to each of the three edges (that is, λg = 0 along the OW edge, λw = 0 along the OG edge, and λo = 0 along the WG edge), relations (4.90 )–(4.110 ) imply that all the terms in square brackets in Table 4.2 are strictly positive. Similarly, if the co-current flow conditions are specialized at the vertices of the saturation triangle, it is immediate to see that both the numerator and denominator of the fractions in Table 4.3 are also


129

Table 4.3. Summary of conditions for strict hyperbolicity at the vertices of the saturation triangle, in terms of the fluid relative mobilities. Condition I O vertex:

λg,u = 0;

λw,v = 0

W vertex:

λg,u = 0;

λo,v = λo,u

G vertex:

λw,v = 0;

λo,v = λo,u

Condition II O vertex:

λg,v >

1 − M (ρD − 1)λo λw,u 1 + M λo

W vertex:

λg,v >

1 + M (ρD − 1)λw (−λo,u ) 1 + M ρ D λw

G vertex:

λw,u =

1 − M λg (−λo,u ) 1 − M ρ D λg

strictly positive. These restrictions, which arise physically from limiting our study to co-current flow, are crucial to obtaining a strictly hyperbolic model. Indeed, only in co-current flow it is possible to satisfy the conditions for strict hyperbolicity with finite derivatives of the relative mobilities. 3. A positive derivative of the gas relative mobility with respect to gas saturation along the OW edge is the fundamental requirement for strict hyperbolicity of the system. It means that the gas relative permeability curve —as a function of its own saturation— must have a positive slope at its endpoint saturation. This essential condition was first identified in Juanes and Patzek [2002e], where it was justified in terms of pore-scale fluid displacements, and shown to be in good qualitative agreement with experimental data. 4. The condition of a positive endpoint-slope of the gas relative permeability is


130

6

λw

λo λ O

u

-

W

Figure 4.3. Schematic of the profiles of water and oil relative mobilities along the OW edge. The value u corresponds to the reduced water saturation for which oil and water relative mobilities are equal: λw = λo = λ. generalized in this chapter to account for gravity effects. To understand better the influence of gravity, we study this condition at a saturation state (u, 0) on the OW edge such that the water and the oil relative mobilities are equal (see Figure 4.3), that is, λw (u, 0) = λo (u, 0) = λ.

(4.23)

The condition for strict hyperbolicity at this saturation state reduces to: Mλ 1 [ρD λw,u − λo,u ]. λg,v > λmin g,v = [λw,u − λo,u ] − 2 1 + M (ρD + 1)λ

(4.24)

Since ρD > 0, it is evident that the terms enclosed in brackets in expression (4.24) are always positive. Because the density and viscosity of gas are much lower than those of water and oil, it is reasonable to think that gravity will influence the relative mobility of gas more than the relative mobilities of oil and water. Thus, we may assume that λw,u and λo,u do not vary greatly with the gravity number. Under this assumption, we find that the minimum endpoint-slope of the gas relative mobility, λmin g,v , shows the following trend with the gravity number: ¯ ¯ ¯ min ¯ min ¯ ¯ ≥ λ ≥ λ λmin g,v M >0 . g,v M =0 g,v M 0), the gas displacement will tend to be less stable than in the case of horizontal flow [Lake, 1989; Yortsos et al., 2001]. At any given cross-section of the porous medium, the transition between zero gas flow and nonzero gas flow will be less abrupt. This gentler transition will translate into a smaller endpoint-slope of the gas relative permeability. On the other hand, if flow is in the direction of the gravity force (M < 0), the displacement will be stabler than for horizontal flow. In this case, an abrupt transition from zero to nonzero gas flow is expected, resulting in a larger endpoint-slope of the gas relative permeability function. (b) Decreasing gas saturation. We associate this type of displacement with imbibition, where fluids more wetting than gas —water and oil— are injected into the porous medium. There is an essential difference with respect to a drainage process: because gas is the least wetting fluid, there are now mechanisms for trapping it. Therefore, the behavior of the relative permeability curve in the neighborhood of zero mobile gas saturation will be determined by the last trapped cluster. In the case of flow upwards


132

(M > 0), gravity effects will enhance the stability of gas displacement by water and oil [Lake, 1989; Zhang et al., 2000], resulting in a smaller fraction of “disconnected” gas clusters and a more gradual and efficient sweeping of the gas phase. This continuous dependence of the gas flow on the amount of gas present will translate into a smaller endpoint-slope of the gas relative permeability. In contrast, when flow is in the direction of gravity (M < 0), the displacement process is less stable. It is more likely that water and oil will take the most favorable paths, leaving behind large amounts of gas. These paths may eventually merge and form disconnected, trapped gas clusters. At any given cross-section of the flow domain, the transition between nonzero and zero gas flow may occur for very small changes in gas saturation. It seems natural to reproduce this behavior through a larger endpoint-slope of the gas relative permeability. 6. It would be interesting to validate, at least qualitatively, the conditions obtained in this chapter with experiments. However, this was not pursued here.

4.4

A simple model

The purpose of this section is to demonstrate it is indeed possible to devise relative permeability models that satisfy the conditions for co-current flow and strict hyperbolicity derived in Section 4.3. As a result, the system of equations describing three-phase co-current flow with gravity will be strictly hyperbolic everywhere inside the saturation triangle. To illustrate our analysis, we use the following model of


133

relative mobilities: λw = (1/µw )u2 , ¡ ¢ λg = (1/µg ) βg v + (1 − βg )v 2 ,

λo = (1/µo )(1 − u − v)(1 − u)(1 − v).

(4.26) (4.27) (4.28)

This model belongs to a widely used class of models [Stone, 1970, 1973], where the water and gas relative permeabilities depend solely on their own saturations, whereas the oil relative permeability depends on two saturations. The most important features of this model are: 1. The only relative permeability function whose form is allowed to change with the viscosity ratio and the gravity number is the gas relative permeability. This is accomplished here in the simplest possible way, through a single parameter: the endpoint-slope βg . 2. In view of the conditions for co-current flow and strict hyperbolicity derived in the previous section, we anticipate that the endpoint-slope of the gas relative permeability function will be positive, that is, βg > 0.

4.4.1

Conditions for co-current flow

It is immediate to specialize condition (4.22) on the allowable range of the gravity number ND to the simple model studied here. Because the relative permeability max functions are assumed to be normalized, we take krα = 1, α = w, o, g. Defining the

viscosity ratios: ± µw D := µw µo , ± µgD := µg µo ,

(4.29) (4.30)


134

we express the conditions for co-current flow as follows: ¾ ¾ ½ g ½ w µD g µD . ;µ max − ; −1 < ND < min ρD ρD D

4.4.2

(4.31)

Conditions for strict hyperbolicity

It can be readily checked that the relative permeability model (4.26)–(4.28) automatically satisfies Condition I (see Table 4.1 and Table 4.2), which imposes that eigenvectors do not rotate along the edges of the saturation triangle. Condition II for strict hyperbolicity along the edges of the ternary diagram (see Table 4.1) reads: How := g,v − f,u > 0

along the OW edge,

(4.32)

Hog := g,v − f,u > 0

along the OG edge,

(4.33)

Hwg := −g,u − f,v > 0 along the WG edge.

(4.34)

When gravity is not included in the analysis, it is possible to obtain closed-form expressions on the parameter βg , so that these conditions are satisfied [Juanes and Patzek, 2002e]. This is not viable in the present case, because gravity effects yield much more complicated expressions. 4.4.2.1

Analysis along the OW edge

Specializing the relative permeabilities to the OW edge, i.e., v = 0, 0 ≤ u ≤ 1, substituting the appropriate expressions into condition (4.32) (see also Table 4.2),


135

and after some algebraic manipulations, we obtain: How

√ p w ¤ µw ¢£ ρD 2 2 2 D βg + µD (1 − u) 1 + ND ( w u + (1 − u) ) ∝ u/ µD µgD ¤ £ − 2 1 − ND (ρD − 1)(1 − u)2 (1 − u)2 u £ ρD − 1 2 ¤ 2 u u (1 − u) > 0. − 2 1 + ND µw D ¡

2

p

µw D

(4.35)

The expression above translates into a condition on the endpoint-slope βg of the gas relative permeability as a function of the dimensionless parameters: βg > where

βgmin

© ª µgD := √ w max F (u; ND , µw D , ρD ) , µD 0≤u≤1

¤ £ ¤ 1 − ND (ρD − 1)(1 − u)2 (1 − u)2 u + 1 + ND ρDµw−1 u2 u2 (1 − u) D ¡ √ w √ w ¢£ ¤ F := 2 . u2 / µD + µD (1 − u)2 1 + ND ( µρDw u2 + (1 − u)2 ) £

(4.36)

(4.37)

D

It is worth noting that the restrictions on the gravity number due to the co-current flow conditions imply that all terms in square brackets in Equation (4.37) are positive. 4.4.2.2

Analysis along the OG edge

Specializing the relative mobilities at the OG edge, u = 0, 0 < v < 1, condition (4.33) reads: ¶ µ ¤ £ ND 2 Hog ∝ 2 1 − g v (1 − v) − 2 1 + ND (1 − v)2 (1 − v)2 v µD µ ¶ £ ¤ ND 2 2 2 − 2 g v(1 − v) + 1 + ND (1 − v) (1 − v) (1 − 2v) βg > 0. µD

(4.38)

Although difficult to derive analytically, it can be checked that condition (4.38) is always satisfied as long as the conditions of co-current flow are met. The analytical calculations may be carried out in the neighborhood of the G vertex. If we let v = 1−ε, the first-order Taylor expansion of Hog about ε = 0 is: Hog ∝ 2ε + O(ε2 ) > 0.

(4.39)


136

At ε = 0, Hog = 0, so that the G vertex is an umbilic point, as required. For all states on the OG edge near the G vertex, Hog > 0, and the system is strictly hyperbolic. 4.4.2.3

Analysis along the WG edge

We study the strict hyperbolicity condition along the WG edge following the same procedure. As was the case for the analysis on the OG edge, the condition Hwg > 0 is difficult to verify analytically. It was found, however, that if it is satisfied in the neighborhood of the G vertex, it is also satisfied everywhere along the edge. Therefore, we analyze this condition for a saturation state (u, v) = (ε, 1 − ε). The first-order Taylor expansion about ε = 0 reads: ¡ ¢ g w 2 Hwg ∝ (2 − µw D ) − (2ρD − µD )ND /µD ε + O(ε ) > 0.

(4.40)

µw D < min{2ρD ; 2},

(4.41)

For condition (4.40) to be satisfied, we require that:

ND
βgmin

0.1 0.08

βg

PSfrag replacements

0.06 0.04

βg < βgmin

0.02 0 −0.4

−0.3

−0.2

−0.1

0

ND Figure 4.4. Admissible values of the endpoint-slope βg of the gas relative permeability as a function of the gravity number ND , for the gravity and viscosity ratios in Equation (4.43). The shaded area (βg > βgmin ) represents the set of values of the parameter βg for which the system is strictly hyperbolic everywhere in the saturation triangle. and analyze two different values of βg , one violating condition (4.36), and the other one satisfying it: βgell = 0,

βghyp = 0.1.

(4.48)

The relative permeability of gas as a function of its own reduced saturation, given by Equation (4.27), is shown in Figure 4.5 for the two values of the endpoint-slope β g above. Remark 4.3. Both curves in Figure 4.5 are very close to one another and, would, in principle, match relative permeability data equally well (or equally badly). However, the implications of using one endpoint-slope or the other are far-reaching: one will yield a system of equations with mixed elliptic/hyperbolic behavior, while the other will produce a system that is everywhere strictly hyperbolic.


139

1 0.8 0.6

krg

PSfrag replacements

βg = 0.1

0.4

βg = 0

0.2 0 0

0.5

v

1

Figure 4.5. Relative permeability of gas as a function of its own saturation for two different values of the endpoint-slope: βg = 0 and βg = 0.1. In Figure 4.6 we plot the functions How (u) along OW, Hog (v) along OG, and Hwg (u) along WG, for each of the two values of βg . The curves along the OG edge and the WG edge reach a zero value for v = 1 and u = 0, respectively, so that the G vertex is an umbilic point. Inequalities (4.32)–(4.34) are satisfied —and the system is strictly hyperbolic— if all three curves are positive everywhere. As expected, this condition is violated when the value βg = 0 is used —Figure 4.6(a)— and it is satisfied when βg = 0.1 is employed —Figure 4.6(b). For each of these two cases we search for elliptic regions in the saturation triangle. An elliptic region is a set of points in the saturation space where the eigenvalues of the Jacobian matrix of the system of governing equations are complex conjugates. Thus, the (complex) eigenvalues at any point in the elliptic region take the form: ν1 = 0

λo = (1/µo )(1 − u − v)(1 − u)(1 − v).

(5.19) (5.20)

The most important feature of the model is the positive derivative of the gas relative permeability function as it approaches zero. The model considers that the relative permeabilities of the most and least wetting fluids (usually water and gas) depend only on their own saturation, whereas the relative permeability of the intermediate wetting fluid (usually oil) depends on all saturations. This is a common assumption in hydrogeology [Parker et al., 1987]) and petroleum engineering [Stone, 1970, 1973]. Although we do not defend this assumption in general, it can be shown [Juanes and Patzek, 2002e] that the relative mobilities (5.18)–(5.20) yield a system which is strictly hyperbolic everywhere in the saturation triangle, as long as the following conditions are satisfied: βg > √

µg , µ o µw

(5.21)

µw < 2µo .

(5.22)

For illustrative purposes, we take reasonable values of the viscosities: µw = 0.875,

µg = 0.03,

µo = 2 cp,

(5.23)

and a small value of the endpoint slope: βg = 0.1. These values of the parameters satisfy the two conditions (5.21)–(5.22) above. Relative permeabilities for all three

Chapter 5. Analytical solution to the Riemann problem

157

phases are shown in Figure 5.5. We use this model to carry out all our sample calculations. 5.3.2.3

Fractional flow functions

From the definition (5.6), water and gas fractional flow functions are, respectively (see Figure 5.6), λw (u) , λT (u, v) λg (v) g(u, v) = . λT (u, v)

f (u, v) =

5.4 5.4.1

(5.24) (5.25)

Solution to the Riemann problem Introduction

The Riemann problem consists in finding a (usually weak) solution to the system of conservation laws: ∂t u + ∂x f = 0, with initial condition u(x, 0) =

−∞ < x < ∞, t > 0,    u

l

  u r

if x < 0,

(5.26)

(5.27)

if x > 0.

For three-phase flow, the system of governing equations is the 2 × 2 system (5.3), whereas for two-phase flow it reduces to the scalar equation (5.1). Unrealistic as it may seem —unbounded domain, and piecewise constant initial data with a single discontinuity— the solution to the Riemann problem is extremely valuable for practical applications. Many laboratory experiments reproduce in fact the conditions of

PSfrag replacements krg = 0.6

PSfrag replacements

PSfrag replacements krg = 0.6

krg = 0.8

krg = 0.8

= 0.2 0.6 kkrw ro =

0.2

1

kro = 0.2 (b) Gas relative permeability

kro = 0.4

0

W

1

0

krg fo = =0.8 0.9O W

0.8

0.6

0.4

0.2

0

1

0.8

0.6

0.4

0.2

0

0.4

0.4

1

kro = 0.2 (a) Water relative permeability kro = 0.2 kro = 0.4

0.6

krg fo = =0.6 0.1

1

1

0

0.8

0.8

0.2

0.6 = 0.6 krg fo = = 0.1 rw krg = foW = 0.8 0.9O

1

0.6

0.2

kfrg =0.1 0.6 o = kfrg =0.9 0.8 O o =

0.8

0.8

fg = = 0.2 0.1 krg krw = 0.2 0.4 fg = = 0.4 0.9 krg krw = 0.4

0.6

0.6

kfrg =0.9 0.4 g =

0.4

0.6

0.8

G

fw = =0.4 0.1 krg = k0.8 rw krg = 0.6 fw = =0.6 0.9 krw krg = 0.4 kro = 0.2 0.6 frgg = = 0.1 kkrg 0.2 =0.2 kro = 0.4 0.4 fg = =0.4 0.9 krg kro = 0.6

0.2

fw = = 0.6 0.9 krw

0.8

0.4

kfrg =0.1 0.2 g =

0.2

0.2

kfrw =0.9 0.6 w =

1

0

fw = = 0.4 0.1 krw

1

0

0

kfrw =0.1 0.4 w =

krw = 0.2

G

0.8

G

(c) Oil relative permeability

kro = 0.4

Figure 5.5. Isoperms of all three phases for the relative mobilities given by Equations (5.18)–(5.20).

0.2

0.2

0 1

0.8

O

0.6

W

0.4

fo = 0.1

0.4

0

0

0.8

0.6

0.4

0.2

0

1

0.8

0.6

0.4

0.2

fo = 0.9

1

(b) Gas fractional flow

0.6

0.8

0

fg = 0.9

1

(a) Water fractional flow

0

fo W = 0.9O

0

0.4

fo = 0.1

1

fo = 0.9O

fg = 0.1

0.8

0.8

0.2

fo = 0.1

fg = 0.1

0.8

0.4

0.4

fw = 0.1

0.6

0.6

0.6

0.6

fg = 0.9

fg = 0.9

1

0.2

0.6

fw = 0.9

0.8

0.4

fw = 0.9

G

fw = 0.1

1

0.2

0.2

fw = 0.9

0.8

0.4

fg = 0.1

0

0

fw = 0.1

1

kro = 0.6

G

0.2

kro = 0.6

G

1

kro = 0.6


kkro =0.6 0.2 rw=

0.6

kro = 0.4

0.4

kro = 0.4

0.2

kro = 0.2

0

kro = 0.2

W

(c) Oil fractional flow

158

Figure 5.6. Contour plots of the fractional flow functions of all three phases, for the relative mobilities given by Equations (5.18)–(5.20), with viscosities (5.23).


159

the Riemann problem: the medium has initially homogeneous saturations, and the proportion of injected fluids is held constant during the experiment. The solution to the Riemann problem gives also information about the structure of the system of equations, and can be used as the building block for obtaining solutions to problems with more complex initial conditions (as in the Godunov method [Godunov, 1959; LeVeque, 1992]). The system (5.26) is invariant under uniform stretching of coordinates (x, t) 7→ (cx, ct) and, therefore, admits self-similar solutions. The property of self-similarity has been termed “stretching principle” [Welge, 1952] or “coherence condition” [Helfferich, 1981; Pope, 1980] in the petroleum engineering literature. It means that the solution at different times “can be obtained from one another by a similarity transformation.” [Barenblatt, 1996] We seek a solution of the form u(x, t) = U (ζ),

(5.28)

where, in our case, the similarity variable ζ is simply ζ :=

x . t

(5.29)

It can be shown (see, e.g., Dafermos [2000]) that self-similar solutions (5.28) of the Riemann problem (5.26)–(5.27) are composites of constant states, shocks joining constant states, and rarefaction waves connecting constant states or contact discontinuities. More precisely, since for a strictly hyperbolic system waves of different families are strictly separated, any self-similar solution to the Riemann problem for a n × n system comprises n + 1 constant states: W

W

Wn−1

W

n 1 2 u n = ur . ul = u0 −→ u1 −→ · · · −→ un−1 −→

(5.30)


160

States ui−1 and ui are joined by a wave of the i-family (Wi ) which, in general, may consist of i-rarefactions, i-shocks and/or i-contact discontinuities. Next, the admissible wave structure for two-phase and three-phase flow is described, and the complete set of solutions to the Riemann problem is given.

5.4.2

Riemann problem for two-phase flow

The solution to the two-phase displacement problem was presented originally by Buckley and Leverett [1942]. Many of the features in the displacement theory of three-phase flow are natural extensions of concepts already present in the twophase flow case. Some concepts are more easily understood in the latter case, which involves a scalar equation rather than a system. 5.4.2.1

Wave structure

In the two-phase flow case, the Riemann problem involves the scalar equation (5.1). Therefore, the left and right states —ul and ur , respectively— are joined by a single wave: W

ul −→ ur .

(5.31)

When the flux function f of a scalar conservation law is convex, the wave appearing in the solution is either a shock or a rarefaction. When f is not convex, as in the present case (see Figure 5.3), the characteristic wave may involve both [LeVeque, 1992]. We now study in more detail the structure of the wave connecting the two constant states ul (left) and ur (right). There are three possible wave types, namely: a single rarefaction, a single shock, and a composite rarefaction-shock.

PSfrag replacements Chapter 5. Analytical solution to the Riemann problem

ul

(a)

1

1

(b)

(c)

u

ur

161

0

1

f

0

0

0

2.5

f0

0

2.5

x

1

t

(d)

0

0

2.5

x Figure 5.7. Example of a single rarefaction solution to the Riemann problem of twophase flow. The characteristic speed f 0 increases monotonically from the left state to the right state, and the solution is smooth and single-valued. Single rarefaction (R). A rarefaction (a term coined in the context of gas dynamics) is a smooth solution U (ζ), with U (ζl ) = ul and U (ζr ) = ur . For a smooth, continuous and differentiable solution, substitution of Equations (5.28)–(5.29) into Equation (5.1), yields: f 0 (U )U 0 = ζU 0 .

(5.32)

If the solution is not a constant function, that is, if U 0 6= 0, it must satisfy: ζ = f 0 (U ).

(5.33)

Clearly, the solution is admissible only if the characteristic speed f 0 increases monotonically from the left state to the right state. Otherwise, characteristics intersect on the x-t plane, and the solution is not single-valued. For the characteristic speed to increase monotonically, both left and right states must lie on the same convexity


162

region, that is, ul > u r ≥ u 0 ,

or ul < ur ≤ u0 .

(5.34)

An example of a rarefaction wave solution is shown in Figure 5.7 for the case ul > ur ≥ u0 , using the fractional flow function (5.10) with µ ˜ = 0.5. The figure includes a plot of: (a) the flux function indicating the left and right states; (b) the flux derivative; (c) the solution profile at t = 1; and (d) the characteristics on the x-t plane. The solution is constant along characteristics, and characteristics spread from the origin in a rarefaction fan. Single shock (S). A shock is a traveling discontinuity. Discontinuities are allowed in the context of weak solutions, and they develop whenever the characteristic speed f 0 at the left state is larger than that at the right state. Otherwise, characteristics would intersect and the solution would be multiple-valued. If a shock connects two states, u− = U (ζ− ) and u+ = U (ζ+ ), as shown schematically in Figure 5.8, the speed of propagation σ is determined by the Rankine-Hugoniot condition: f (u+ ) − f (u− ) = σ · (u+ − u− ).

(5.35)

This condition is, in essence, a mass conservation statement when the solution is discontinuous. In the scalar case, it is equivalent to the equal-area rule [Buckley and Leverett, 1942; LeVeque, 1992]. For the solution to be admissible, the shock must satisfy the Lax entropy condition for genuine shocks [Lax, 1957; LeVeque, 1992; Smoller, 1994]: f 0 (u− ) > σ > f 0 (u+ ).

(5.36)

In Figure 5.9 we show the case of a single shock between two constant states, including: (a) the flux function indicating the left and right states; (b) the flux derivative and the shock speed; (c) the solution profile at t = 1; and (d) the characteristics

163

Chapter 5. Analytical solution to the Riemann problem u− - σ

u+

PSfrag replacements

ζ Figure 5.8. Schematic of a shock.

(a)

1

1

(b) u

ul

(c)

σ ur 0

1

f

0 0

σ

0

2.5

f0

0

2.5

x

1

t

(d)

0

0

2.5

x Figure 5.9. Example of a single shock solution to the Riemann problem of two-phase flow. The speed of propagation of the discontinuity is determined by the RankineHugoniot condition, and it is readily checked that characteristics go into the shock. on the x-t plane. The solution satisfies the Lax entropy condition, which implies that characteristics go into the shock. Composite rarefaction-shock (RS). It might not be possible to connect left and right states with a simple wave. In some cases, a composite wave consisting of a rarefaction and a shock is required. The left and right states must lie on different


164

convexity regions so that the characteristic speed is not monotonic, that is, ul > u 0 > u r ,

or ul < u0 < ur .

(5.37)

The solution involves at most one rarefaction and one shock, because the fractional flow function has one inflection point only. Moreover, since the inflection point corresponds to a maximum value of the derivative, the rarefaction fan of a composite wave is always slower than the shock [Ancona and Marson, 2001]. The rarefaction and the shock are connected at some intermediate point u∗ , called the post-shock value. This is the value of u at which the left characteristic speed (rarefaction fan) coincides with the speed of the right discontinuity (shock): f 0 (u∗ ) = σ∗ :=

f (u∗ ) − f (ur ) . u∗ − u r

(5.38)

Equation (5.38) is a nonlinear equation, which can be solved, for example, by Newton iteration. The wave joining the left and right constant states is, thus, a rarefaction-shock (W ≡ RS), which we express schematically as follows: R

S

ul −→ u∗ −→ ur .

(5.39)

The appropriate criterion for ascertaining admissible shocks when the flux function is not convex is the Oleinik entropy condition [Dafermos, 2000; Oleinik, 1957; Smoller, 1994], which states that any discontinuity joining the states u− and u+ must satisfy: f (u) − f (u− ) f (u+ ) − f (u− ) f (u+ ) − f (u) , ≥ ≥ u − u− u+ − u − u+ − u

(5.40)

for all u between u− and u+ . This entropy criterion is equivalent to the concave hull construction [LeVeque, 1992]. In the particular case of two-phase flow, where the flux function has only one change in convexity, associated with a maximum value of the

PSfrag replacements

165


ul

(a)

1

1

(b) u

u∗

(c)

ur 0

1

f

0 0

f0

0

σ∗ 2.5

0

2.5

x

1

t

(d)

0

0

2.5

x Figure 5.10. Example of a composite rarefaction-shock solution to the Riemann problem of two-phase flow. Left and right states lie necessarily on opposite sides of the inflection point and the rarefaction is always behind the shock. The speed of propagation of the discontinuity is determined by the concave hull construction. derivative, condition (5.40) reduces to checking that the characteristic speed f 0 must increase from left to right along the rarefaction fan, and that the shock satisfies: f 0 (u∗ ) ≥ σ∗ > f 0 (ur ).

(5.41)

A solution which involves a composite rarefaction-shock wave is shown in Figure 5.10, computed for the fractional flow function (5.10) with µ ˜ = 0.5, as before. 5.4.2.2

Complete set of solutions

The solution to the Riemann problem of two-phase flow consists in a single wave that joins the left and right constant states. As shown above, this wave may only be a rarefaction, a shock, or a rarefaction-shock. Therefore, only three types of solutions are possible. In Figure 5.11, we present an algorithm which summarizes the process


166

• Given left and right states: ul , ur • Trial shock speed: σ trial =

f (ul ) − f (ur ) ul − u r

IF f 0 (ul ) > σ trial > f 0 (ur ) THEN • S: Single shock with speed σ = σ trial ELSE IF f 0 (ul ) < f 0 (ur ) & f 00 (ul )f 00 (ur ) > 0 THEN • R: Single rarefaction ELSE • RS: Composite rarefaction-shock · Post-shock value u∗ such that f 0 (u∗ ) = σ∗ · Shock speed: σ∗ =

f (u∗ ) − f (ur ) u∗ − u r

END END Figure 5.11. Algorithm for obtaining the wave structure for two-phase flow. of obtaining the wave structure for two-phase flow.

5.4.3

Riemann problem for three-phase flow

The system of conservation laws (5.3) describing three-phase flow is a 2×2 system, which is strictly hyperbolic for all saturation paths of interest [Juanes and Patzek, 2002e]. This implies that there are two separated waves connecting three constant

167

Chapter 5. Analytical solution to the Riemann problem G

PSfrag replacements

0

1

ul 0 .2

¼

0.8

W1

0 .4

um 0.6

W2 À

ur

ul 0 .6 0.8

W1

W2

0.2

0 1

0.8

0.6

0.4

ur 0.2

0

1

O

ζ

0.4

um

W

Figure 5.12. Schematic representation of the generic solution to the Riemann problem of three-phase flow. The solution comprises two distinct waves —slow and fast waves— connecting three constant states, and each wave might involve traveling discontinuities. On the left plot we show a possible configuration of the wave curves in the saturation space. On the right plot we display the corresponding saturation profile for one of the components (gas saturation, say) against the similarity variable ζ = x/t. states: ul (left), um (middle), and ur (right). Therefore, the solution to the Riemann problem for three-phase flow reduces to finding the intermediate constant state u m as the intersection of an admissible 1-wave W1 (slow wave) and an admissible 2-wave W2 (fast wave) on the saturation triangle (see Figure 5.12): W

W

1 2 ul −→ um −→ ur .

(5.42)

The theory of strictly hyperbolic systems was compiled by Lax [1957]. The solution to the Riemann problem was restricted to systems whose characteristic fields are either genuinely nonlinear or linearly degenerate. The notion of genuine nonlinearity, which is made precise below, is central to solving the Riemann problem of three-phase flow. Basically, genuine nonlinearity of a characteristic field is the natural extension of


168

convexity of the flux function for scalar equations. As we shall see, this property does not hold for the three-phase flow system, much in the same way as the flux function in two-phase flow is not convex. The theory of Lax was extended by Liu [1974, 1975] to find Riemann solutions for systems with nongenuinely nonlinear fields. This theory is used here to describe the admissible wave structure, and the complete set of solutions to the Riemann problem. 5.4.3.1

Wave structure

We now describe the structure of the waves in the Riemann solution. From the theory of strictly hyperbolic conservation laws (see, e.g., Dafermos [2000]), a wave of the i-family consists of i-rarefactions, i-shocks and/or i-contact discontinuities. This is discussed next. Integral curves and rarefactions. Rarefactions are smooth waves joining constant states or contact discontinuities. If the solution is smooth, using Equations (5.28)– (5.29) in Equation (5.26), a self-similar solution of the Riemann problem satisfies the system of ordinary differential equations A(U )U 0 = ζU 0 ,

(5.43)

where A(U ) is the Jacobian matrix of the system, Equation (5.14). This is an eigenvalue problem, where the similarity variable ζ = x/t is an eigenvalue, and U 0 is a right eigenvector. Because the system is strictly hyperbolic (see Section 5.3.2.1), there exist two distinct eigenvalues νi , and two linearly independent eigenvectors r i , corresponding to the two different characteristic families i = 1, 2. This leads to the following

169


Definition 5.3. An i-rarefaction is a smooth function U i (ζ) satisfying Equation (5.43), where the parameter ζ is not arbitrary, but the i-eigenvalue of the Jacobian matrix of the system: ζ = νi (U i (ζ)).

(5.44)

It follows that an i-rarefaction curve (in phase space) must lie on an integral curve of the i-family, that is, a curve whose tangent at any point U is in the direction of the i-eigenvector r i (U ) at that point. The two families of integral curves, usually termed as slow and fast paths, are depicted in Figure 5.13 for the relative mobilities (5.18)– (5.20). A rarefaction curve U i (ζ) will provide an admissible single-valued solution only if the similarity variable parameter ζ = νi increases monotonically along the curve from the left state to the right state. Rarefaction curves can be calculated by simple numerical integration with a Runge-Kutta algorithm, as explained in Section A.1.1 of Appendix A. Hugoniot loci and shocks. Any propagating discontinuity connecting two states u− = U (ζ− ) and u+ = U (ζ+ ), must satisfy an integral conservation equation for each variable, known as the Rankine-Hugoniot jump condition [LeVeque, 1992]: f (u+ ) − f (u− ) = σ(u+ ; u− ) · (u+ − u− ),

(5.45)

where σ(u+ ; u− ) is the speed of propagation of the discontinuity. For a fixed state u− , one can determine the set of states u+ which can be connected to u− such that Equation (5.45) is satisfied. There are two families of solutions, one for each characteristic family, which form two curves passing through the reference state u− : H1 (u− ) and H2 (u− ) (see Figure 5.14). The set of points on each of these curves is called the

170

Chapter 5. Analytical solution to the Riemann problem G 0

1

0 .2

2-family 0.8

0 .4

PSfrag replacements

1-family

0.6

0 .6

0.4

0.8

0.2

0 1

0.8

0.6

0.4

0.2

0

1

O

W

Figure 5.13. Integral curves for the relative mobilities (5.18)–(5.20). Integral curves of the 1- and 2-family are usually termed slow and fast paths, respectively. Hugoniot locus. It is easy to show [LeVeque, 1992] that the Hugoniot curves are tangent to the corresponding eigenvectors at the reference point u− . Moreover, since the system is strictly hyperbolic, Hugoniot loci do not have detached branches and are transversal to each other [Dafermos, 2000]. Not every discontinuity satisfying the Rankine-Hugoniot condition is a valid shock. For a genuine shock of the i-family (an i-shock) to be physically admissible, it must satisfy the Lax entropy condition [Dafermos, 2000; Lax, 1957; LeVeque, 1992; Smoller, 1994]: νi (u− ) > σi (u+ ; u− ) > νi (u+ ),

(5.46)

where u− and u+ are the values at the left and at the right of the discontinuity, respectively. Condition (5.46) implies that characteristics of the i-family go into the shock. Definition 5.4. A shock curve of the i-family passing through point u− , denoted


171

as Si (u− ), corresponds to a subset of the Hugoniot locus Hi (u− ), for which the entropy condition (5.46) is satisfied. An algorithmic procedure for the calculation of the Hugoniot loci, based on a Newton iterative scheme, is detailed in Section A.1.2 of Appendix A. Inflection loci and rarefaction-shocks. The notion of genuine nonlinearity is crucial to the wave structure arising in multiphase flow. Definition 5.5. The i-field is said to be genuinely nonlinear if the i-eigenvalue ν i varies monotonically along integral curves of the i-family. This is expressed mathematically as: ∇νi (U ) · r i (U ) 6= 0 for all U ,

(5.47)

where ∇νi (U ) := [∂νi /∂u, ∂νi /∂v]t is the gradient of νi (U ). This condition is equivalent to that of convexity, f 00 (u) 6= 0 ∀u, for scalar conservation laws. Definition 5.6. The i-field is said to be linearly degenerate if νi is constant along integral curves of the i-family, that is, ∇νi (U ) · r i (U ) ≡ 0 for all U .

(5.48)

Of course, the value of νi (U ) may vary from one integral curve to the next. As it turns out, the characteristic fields of the system describing three-phase flow are neither genuinely nonlinear nor linearly degenerate: eigenvalues attain local maxima along integral curves. This observation leads to the following Definition 5.7. The inflection locus Vi for the i-characteristic field as the set of

172


G 0

1

0 .2 0 .4

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0.2

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(a) Reference state u− = (0.1, 0.5)

G 0

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1

O

W

(b) Reference state u− = (0.4, 0.3)

Figure 5.14. Plot of the Hugoniot loci of both characteristic families, H 1 (u− ) and H2 (u− ), for the relative mobilities (5.18)–(5.20) and two different reference states u− (4).


173

points U such that ∇νi (U ) · r i (U ) = 0,

(5.49)

that is, the locations at which νi attain either a maximum or a minimum value when moving along integral curves of the i-family. In Figure 5.15, we show contour plots of eigenvalues and the inflection loci for both characteristic families. We note that in all realistic models of multiphase flow, the inflection locus corresponds to maxima of eigenvalues. This is consistent with the well-known behavior of the flux function for the two-phase flow case, where the fractional flow function is S-shaped, and the inflection point corresponds to the maximum value of the derivative (see Figure 5.3). For a strictly hyperbolic system whose characteristic fields are genuinely nonlinear, any wave connecting two constant states ul and ur can only be a rarefaction or a genuine shock, and any discontinuity must satisfy the Lax entropy condition (5.46). If the characteristic fields are nongenuinely nonlinear, each wave might consist in a combination of rarefactions and discontinuities [Liu, 1974, 1975]. In our case, since the inflection locus for each field is a single connected curve satisfying certain orthogonality conditions with respect to integral curves, the composite wave has at most one rarefaction and one discontinuity. Moreover, because the inflection loci correspond to local maxima of eigenvalues along integral curves, the rarefaction is always slower than the shock [Ancona and Marson, 2001]. Definition 5.8. More precisely, a rarefaction-shock curve of the i-family connecting the left and right states ul and ur , respectively, is a curve on the phase plane consisting of an i-rarefaction curve emanating from ul , connected to an i-shock curve at some intermediate point u∗ , which ends at the right state.

174


G 0

1

Inflection Locus V1

0.02 0 .2

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0

O

0.2

1

0.5 0.02 2 0.05 5 0.1

0.6

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W

0.2 (a) 1-family

0.5

G 0

1

0.2

Inflection Locus V2

0. 0

2

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Inflection Locus V1

W

(b) 2-family

Figure 5.15. Contour plots of eigenvalues νi and the inflection loci Vi of both characteristic families, for the relative mobilities (5.18)–(5.20). Inflection loci correspond to local maxima of eigenvalues when moving along integral curves.


175

We denote this rarefaction-shock curve as Ri Si (ul , ur ) and, unlike rarefaction curves or shock curves alone, is defined through both endpoints. The intermediate state u∗ is the post-shock state, at which the following property holds: νi (u∗ ) = σi (ur ; u∗ ).

(5.50)

A necessary condition for a Ri Si (ul , ur ) wave is that the left and right states lie on opposite sides with respect to the inflection locus Vi . This rules out the possibility of two such states being connected by a rarefaction wave, since the characteristic speed would not be monotonically increasing and, as a result, the solution would not be single-valued. Figure 5.16 shows two rarefaction-shock curves for the first characteristic family, corresponding to the same left state but two different right states. Note that the post-shock value u∗ , at which the R1 and S1 curves are connected, is different for each case. This connection is always very smooth. In fact, it can be shown [Lax, 1957] that both curves are connected with second order tangency (same slope and curvature). If the i-characteristic field is nongenuinely nonlinear, any discontinuity in the iwave, connecting two states u− (left) and u+ (right), must satisfy the Liu entropy condition [Liu, 1974, 1975], which states that σi (u+ ; u− ) ≤ σi (u; u− ),

(5.51)

for all states u ∈ Si (u− ) between u− and u+ . This condition generalizes the Oleinik entropy condition (5.40) for scalar equations to systems of conservation laws. In the particular case of three-phase flow, it is possible to arrive at a simpler condition. Because inflection loci are single connected curves, which correspond to maxima of eigenvalues, it can be shown [Ancona and Marson, 2001] that condition (5.51) is

176


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1

0 .2

0.8

V1

0 .4

0.6

ul 0 .6

ur u∗

0.4

R1

0.8

S1

0.2

0 1

0.8

0.6

0.4

0

0.2

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O

W

(a) Right state with ur ≈ 0.17

PSfrag replacements

G 0

1

0.2

0.8

V1

0.4

0.6

ul 0.6

u∗

ur

0.4

R1

0.8

S1

0.2

0 1

0.8

0.6

0.4

0.2

0

1

O

W

(b) Right state with ur ≈ 0.02

Figure 5.16. Rarefaction-shock curves of the 1-family with the same left state ul = (0.5, 0.5) and two different right states, for the relative mobilities (5.18)–(5.20). The dash-dotted curve is the inflection locus of the 1-family. Note that the post-shock state u∗ is different for each case.


177

ur um

S R RS

S

ul

R

S R RS

S1 S2 S1 R2 S1 R2 S2 R1 S2 R1 R2 R1 R2 S2

RS S R

R1 S1 S2 R1 S1 R2

RS

R1 S1 R2 S2

Figure 5.17. Schematic tree with all possible combinations of solutions to the Riemann problem of three-phase flow. equivalent to νi (u− ) ≥ σi (u+ ; u− ) > νi (u+ ).

(5.52)

Algorithms for calculating rarefaction-shock curves, based on a predictor-corrector strategy that achieves quadratic convergence, are given in Section A.1.3 in Appendix A. 5.4.3.2

Complete set of solutions

Based on the analysis of the wave structure in Section 5.4.3.1, a wave of the i-family connecting two constant states may only be one of the following: an i-rarefaction (Ri ), an i-shock (Si ), or an i-rarefaction-shock (Ri Si ). Since the full solution to the Riemann problem is a sequence of two waves, W1 and W2 , there are only 9 possible combinations of solutions. A schematic tree with all possible solution types is shown in Figure 5.17. In Figure 5.18 we present the saturation paths in the ternary diagram for all 9 solution types. These are: (a) S1 S2 : both waves are genuine shocks and, therefore, the solution comprises three constant states separated by two discontinuities.


178

(b) S1 R2 : the solution consists of a 1-shock and a 2-rarefaction. (c) S1 R2 S2 : the solution comprises a genuine 1-shock through the left state and a composite 2-rarefaction-shock through the right state. (d) R1 S2 : the left state and the right state are joined by a 1-rarefaction followed by a 2-shock. (e) R1 R2 : both waves are rarefactions, so the solution is continuous everywhere. (f) R1 R2 S2 : a 1-rarefaction from the left state is followed by a composite 2rarefaction-shock to the right state. (g) R1 S1 S2 : the slow wave emanating from the left state is a composite rarefactionshock, which is followed by a genuine 2-shock to the right state. (h) R1 S1 R2 : the left state is joined to the intermediate constant state by a composite rarefaction-shock, and the right state is reached along a 2-rarefaction. (i) R1 S1 R2 S2 : both waves are rarefaction-shocks. All cases discussed above give the complete set of solutions to the Riemann problem of three-phase flow, when the following physical properties are satisfied: (1) the system is strictly hyperbolic; and (2) the inflection loci are single connected curves, transversal to the integral curves, and correspond to maxima of the eigenvalues. Efficient algorithms for the complete calculation of the solution are given in Appendix A. They are based on a predictor-corrector strategy coupled with a full Newton iteration, which achieves quadratic convergence in all cases.

S1

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R1

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0

W

0

Figure 5.18. Examples of the saturation paths for all 9 solution types. These 9 cases constitute the complete set of solutions to the Riemann problem of three-phase flow.

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Chapter 5. Analytical solution to the Riemann problem 179

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O ur

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G 1

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(i) R1 S1 R2 S2

ul

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Figure 5.18. Examples of the saturation paths for all 9 solution types (continued).

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(g) R1 S1 S2

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Chapter 5. Analytical solution to the Riemann problem 180


5.5

181

Application example: water-gas injection

In this section we describe in some detail a synthetic example, where we apply the analytical solution of three-phase flow presented in Section 5.4. The objective is twofold: 1. Illustrate the applicability and potential of the theory to develop exact solutions for problems of great practical interest. 2. Show that the wave structure arising in three-phase flow displacements should not be approximated by a sequence of two consecutive two-phase flow displacements.

5.5.1

Description of the problem

The problem involves simultaneous injection of water and gas into a core that is initially filled with oil and water, as shown in Figure 5.19. Initially, the core has constant reduced —normalized— saturations of 0.95 oil saturation, and 0.05 water saturation. Gas and water are injected in such proportion that the reduced water and gas saturations at the inlet are 0.5 and 0.5, respectively. The injected reduced saturations are assumed to be constant throughout the experiment. This physical problem is modeled mathematically as a Riemann problem, where two initially constant states are separated by a single discontinuity. Here, the left (injected) saturation state is ul = (0.5, 0.5), and the right (initial) saturation state is ur = (0.05, 0). We use the relative mobilities in Equations (5.18)–(5.20) with β g = 0.1, and the fluid viscosities in (5.23).

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Chapter 5. Analytical solution to the Riemann problem Initial saturations

Injected saturations Sw = 0.5

Sw = 0.05

So = 0

So = 0.95

Sg = 0.5

Sg = 0

Figure 5.19. Sketch of the injection problem. Water and gas are injected into a core initially filled with oil and water.

5.5.2

Exact solution

The exact solution turns out to be of type R1 S1 R2 S2 , that is, both waves are rarefaction-shocks: W1 ≡ R1 S1 , and W2 ≡ R2 S2 . The variables that need to be determined to characterize fully the solution are: the intermediate constant state u m , the shock speeds σ1 and σ2 , and the post-shock states u∗1 and u∗2 of each wave. The constant state um corresponds to the intersection of the two wave curves, while the post-shock states are the points where the rarefaction curve and the shock curve of the same family are joined. Schematically, this can be represented as follows: R

S

R

S

2 1 2 1 ur . um −→ u∗2 −→ ul −→ u∗1 −→

(5.53)

A detailed algorithm to obtain this solution is given in Section A.2.3 in Appendix A. In Figure 5.20 we show the saturation path of the exact solution. The corresponding saturation profiles are plotted in Figure 5.21 against the similarity variable ζ = x/t. Of course, the right state coincides with the initial saturations (95% oil and 5% water) and the left state is given by the injected saturations (50% water and 50% gas). Because the characteristic speeds of the slow and fast waves are very different, the entire saturation profile shown on the right plot 5.21(b) does not allow to visualize the structure of the slow wave (1-wave). A detail of the 1-wave is shown on the left plot 5.21(a). It is apparent that the 1-wave involves changes in the saturation

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R1

R2 0.2

0

1

O

u∗1

um

0.8

u∗2

0.6

W

Figure 5.20. Saturation path of the exact solution to the water-gas injection problem. Both waves are rarefaction-shocks (R1 S1 R2 S2 solution). Dash-dotted curves represent the inflection loci. of all three fluids.

5.5.3

Approximate solution

Classical analytical solutions of three-phase flow are usually restricted to the case when saturation paths are straight lines [Helfferich, 1981; Lake, 1989; Pope, 1980; Willhite, 1986]. This approximation dates back to the early conceptual model of a water flood in the presence of gas by Kyte et al. [1956]. In general, straight saturation paths arise only when the relative permeability of each phase is assumed to be a linear function of its own saturation. The physical motivation for assuming saturation paths that are straight lines parallel to the edges of the ternary diagram is to split the actual three-phase flow displacement into a sequence of two successive two-phase displacements. In the context of the water-gas injection problem described above, this approximation is equivalent

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Chapter 5. Analytical solution to the Riemann problem zoom

R1 1

1

gas

S1

gas

saturation

σ1

oil

R2 S2

R1

oil

S2

R2

S1

water

σ2 0 0

ζ = x/t

σ1

(a) Detail of the 1-wave

0.1

0 0

ζ = x/t

σ2

10

(b) Entire saturation profile

Figure 5.21. Saturation profiles of the exact solution to the water-gas injection problem. Saturations of each phase are plotted against the similarity variable ζ = x/t. The right plot (b) shows the entire saturation profile. Because of the very different characteristic speeds of the slow and fast waves, we show a detail of the 1-wave on the left plot (a). to assuming that the fast wave is a displacement of oil exclusively by gas, and that the slow wave is a displacement of oil exclusively by water. Therefore, it is assumed that the water saturation is constant along the fast wave, and the gas saturation is constant along the slow wave. The immediate benefit of this simplification is that the solution may be computed using the theory of two-phase Buckley-Leverett flow. Here we evaluate the accuracy of this simplifying assumption. In Figure 5.22 we show the saturation paths that result from the assumption described above. The intermediate constant state um is obvious to calculate as the intersection of the two wave paths. Each wave is then resolved using the catalogue of two-phase flow solutions in Section 5.4.2. It turns out that the slow wave is a 1-shock (W1 ≡ S1 ), and the fast wave is a 2-rarefaction-shock (W2 ≡ R2 S2 ). Thus, the wave structure is different from that of the exact solution. In Figure 5.23 we plot the saturation profiles of the approximate solution against the similarity variable ζ. The

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ur

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O

0

1

u∗1

0.4

R2

0.2

0.8

u∗2

ul

W

Figure 5.22. Saturation path of the approximate solution to the water-gas injection problem, which assumes that saturation paths are straight lines, parallel to the edges of the ternary diagram. The slow wave is a 1-shock, and the fast wave is a 2rarefaction-shock (S1 R2 S2 solution). Each individual wave is fully determined using the theory of two-phase displacements. right figure 5.23(b) shows the entire saturation profile, and the left figure 5.23(a) a detail of the 1-wave. It is evident that, while the qualitative behavior of the fast wave is similar to that of the exact solution (see Figure 5.21(b)), the structure of the slow wave is very different (compare with Figure 5.21(a)).

5.5.4

Discussion

To better evaluate the accuracy of the straight-line approximation of the saturation paths, we compare the oil production of the exact and the approximate solutions. This is done by taking a fixed length L of the core, and calculating the amount of oil displaced at the outlet at any given time. Using the definitions of dimensionless space and time coordinates in Chapter 2 —Equations (2.23) and (2.24)— and the similarity variable in Equation (5.29), we plot the results against the dimensionless

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Chapter 5. Analytical solution to the Riemann problem zoom

R1 1

R1 1 gas

gas

S1

saturation

σ1

oil

R2

S1

water

S2

S2

R2

oil

σ2 0 0

ζ = x/t

σ1

0.1

0 0

ζ = x/t

σ2

10

(b) Entire saturation profile

(a) Detail of the 1-wave

Figure 5.23. Saturation profiles of the approximate solution to the water-gas injection problem, which assumes that the three-phase flow displacement is split into a sequence of two successive two-phase displacements. Saturations of each phase are plotted against the similarity variable ζ = x/t. The right plot (b) shows the entire saturation profile. The left plot (a) shows a detail of the 1-wave. time τ := 1/ζ = t/x.

(5.54)

The variables of interest are: 1. Oil production rate. This quantity is proportional to the oil fractional flow f oil at the outlet face, that is: foil (τ ) = 1 − f (τ ) − g(τ ).

(5.55)

2. Cumulative oil production. This quantity is proportional to the fraction Q oil of the original oil in place that has been swept through the outlet face, that is: Qoil (τ ) =

Z

τ

foil (η) dη. 0

(5.56)


187

In Figure 5.24 we plot the dimensionless oil production rate foil defined in Equation (5.55), as predicted by the exact solution and the approximate solution. The approximate solution agrees well with the exact solution at early times (roughly, for τ < 3). The reason for this good agreement is that, at early times, only the fast wave has reached the outlet face, and the exact saturation path of the fast wave may be approximated accurately by a straight line of constant water saturation (compare Figure 5.22 with 5.20, and Figure 5.23(b) with 5.21(b)). However, the approximate solution deviates very significantly from the exact solution for times τ > 3, because both the saturation and the speed of propagation of the oil bank are predicted incorrectly. A faulty behavior of the approximate solution is better visualized by plotting the dimensionless cumulative oil production Qoil defined in Equation (5.56), which is simply the area under the curve in Figure 5.24. This quantity is shown in Figure 5.25 for both the exact and the approximate solutions. The curve given by the exact solution tends asymptotically to a value of Qoil,max = 0.95, which is precisely the initial reduced oil saturation. This is required for mass conservation. In contrast, the curve predicted by the approximate solution reaches a plateau of Qoil,max ≈ 0.67 at time τ ≈ 15. This behavior illustrates that the approximate solution is not mass conservative. The results presented in this section motivate the following remarks. Remarks 5.9. 1. Rarefaction-shocks waves, common in two-phase displacements, appear also in realistic scenarios of three-phase flow. 2. In the realm of Buckley-Leverett models of three-phase flow, individual waves involve simultaneous three-phase displacements.

188


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τ = 1/ζ = t/x Figure 5.24. Comparison of the dimensionless oil production rate predicted by the exact solution and the approximate solution. The approximate solution deviates very significantly from the exact solution because both the saturation and the speed of propagation of the oil bank are incorrect.

1

Exact Approx

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τ = 1/ζ = t/x Figure 5.25. Comparison of the dimensionless cumulative oil production predicted by the exact solution and the approximate solution. The exact solution tends asymptotically to a value of 0.95, consistent with the original reduced oil saturation. The approximate solution does not, which indicates it does not satisfy the overall mass balance.


189

3. In general, the saturation paths induced by the exact characteristic waves are not straight lines on the ternary diagram. Saturation paths are straight lines only for linear relative permeability functions, or for very particular initial and injected saturation states. 4. Splitting a three-phase displacement into a sequence of two successive two-phase displacements, for which saturation paths are straight lines parallel to the edges of the ternary diagram, is an assumption that may not be appropriate. 5. In particular, it was shown by means of a representative example that the approximate solution calculated under this assumption does not satisfy overall mass balance, and leads to very inaccurate predictions of oil recovery.

5.6

Concluding remarks

The key result of this chapter is the complete general analytical solution to the Riemann problem of three-phase flow of immiscible, incompressible fluids, when gravity and capillarity are not included in the formulation. The solution comprises two waves, a (slow) 1-wave and a (fast) 2-wave, separated by an intermediate constant saturation state. Each of these two waves may only be a rarefaction, a shock, or a rarefaction-shock. Thus, there can only be 9 possible combinations of admissible waves. All these combinations, which constitute the complete set of solutions to the Riemann problem, are discussed in this chapter. We demonstrate that, in general, a three-phase flow displacement should not be approximated by a sequence of two successive two-phase displacements. Such approximation does not satisfy an overall mass balance, and may lead to very inaccurate predictions of oil recovery.


190

“We Europeans, we cannot. We have as little time as our great and finely articulated continent has space, we must be as economical of the one as of the other, we must husband them, Engineer.” — THOMAS MANN, The Magic Mountain (1924)

191

Part II Multiscale numerical modeling

192

Chapter 6 Multiscale finite elements for miscible flow and two-phase flow 6.1

Introduction

The main difficulty when solving flow and transport in fractured porous media stems from the fact that, very often, these processes are not dominated by diffusion. This makes the mathematical problem almost hyperbolic, which naturally develops sharp features in the solution. Classical numerical methods produce a solution that either lacks stability, resulting in nonphysical oscillations, or accuracy, by showing excessive numerical diffusion. A vast literature, succinctly reviewed in Section 1.4.1 of Chapter 1, has emerged to provide solutions to this fundamental problem. Despite the attention advectiondominated flow has received from the scientific community in the past decades, there is still a need for new numerical techniques. Modern characteristics methods like the Eulerian-Lagrangian Localized Adjoint Method (ELLAM) [Celia et al., 1990]

Chapter 6. Multiscale finite elements for miscible flow and two-phase flow

193

require a fine grid to track accurately the characteristics in a highly nonlinear problem, and state-of-the-art stabilized methods like Streamline-upwind/Petrov-Galerkin (SUPG) [Brooks and Hughes, 1982] or Galerkin Least Squares (GLS) [Hughes et al., 1989], are not as effective in the presence of reaction and production terms [Codina, 1998]. Recently, stabilized finite element methods have been re-interpreted from the point of view of multiscale phenomena [Hughes, 1995], where the stabilizing terms arise naturally in a variational multiscale method [Hughes et al., 1998]. The idea of a multiple-scale decomposition of the solution, which is now dominant in fluid mechanics, is adopted here for the simulation of subsurface flow and transport. Development of novel numerical methods for the complete equations of multiphase compositional flow in multidimensions must necessarily start from simplified models in one space dimension. These reduced model problems should display, however, the key features which pose difficulties in obtaining satisfactory numerical solutions such as, for instance, wild nonlinearity, shocks or near-shocks, boundary layers and degenerate diffusion. Here we study the physical processes of miscible flow —or tracer transport— and two-phase flow. In both cases, the mathematical formulation is given by a scalar equation: a linear advection-diffusion-reaction equation, and a nonlinear conservation law of advection-diffusion type, respectively (see Chapter 2). It is important to note that the source of the diffusion term is different in each case. In the tracer transport model it combines the effect of physical diffusion and hydrodynamic dispersion, while in the two-phase flow model it is related to capillary pressure effects. The key point of the proposed formulation is a multiscale decomposition of the variable of interest into resolved grid scales and unresolved subgrid scales, which acknowledges the fact that the fine-scale structure of the solution cannot be captured by


194

any mesh. However, the influence of the subgrid scales on the resolvable scales is not negligible. By accounting for the subgrid scales, the oscillatory behavior of classical Galerkin is drastically reduced and confined to a small neighborhood containing the sharp features, while the solution is high-order accurate where the solution is smooth. This ensures that the numerical solution is not globally deteriorated. The method does not emanate from a monotonicity argument and, therefore, it does not rule out small overshoots and undershoots near the sharp layers. These localized wiggles may be further reduced or completely eliminated by using a discontinuity-capturing technique. In this chapter we present a novel shock-capturing mechanism, and we compare it with previous formulations. One of the original contributions of the numerical formulation presented here is the particular implementation of the multiscale formalism for nonlinear problems. We perform the multiscale decomposition directly on the weak form of the equations, prior to any linearization. This approach is different from the most common one, which relies on linearizing the equations first, and then resorting to a multiscale decomposition [Codina, 2001, 2002]. The derivation of the numerical formulation presented in this chapter is very brief. Our intention is to introduce the main ideas, and test the performance of the method. A complete derivation and discussion is given in Chapter 7, for the much more complicated problem of three-phase flow. An outline of this chapter is as follows. The mathematical and numerical formulations are described in Section 6.2, within the unified framework of conservation laws. Under certain simplifying assumptions, miscible flow takes the form of a linear advection-diffusion equation, while immiscible flow leads to the classical BuckleyLeverett equation. Several numerical simulations for both miscible and immiscible flow are presented in Section 6.3. In Section 6.4 we gather the main conclusions.

195


6.2 6.2.1

Numerical formulation Initial and boundary value problem

In Chapter 2 we derived the governing equations of miscible flow and two-phase —immiscible— flow. Here we model miscible and immiscible flow in porous media in the unified framework of scalar conservation laws of the form: ∂t u + ∇ · F = q − σu,

x ∈ Ω, t ∈ (0, T ],

(6.1)

where u is the conserved quantity, F is the total flux of that quantity, q is the rate of production (per unit volume), σ is the decay constant, Ω is the spatial domain, and (0, T ] is the time interval of interest. It is understood that the problem has been re-scaled and that all variables are dimensionless. The total flux has the form: F = f (u) − D(u)∇u,

(6.2)

where f is the hyperbolic part of the flux and D is the diffusion tensor. In principle, both are allowed to be nonlinear functions of the unknown u. Let ∂Ω be the boundary of the domain, Γu ⊂ ∂Ω, and Γn = ∂Ω \ Γu . We shall consider the following boundary conditions (Dirichlet and Neumann, respectively): u = u¯ on Γu , F · n = F¯

on Γn ,

(6.3) (6.4)

where n is the outward unit normal to the boundary. The initial conditions are u(x, t = 0) = u0 (x),

¯ x ∈ Ω.

(6.5)


196

For the tracer transport model, governed by a linear equation, we introduce the following equivalent notation: ∂t u + Lu = q,

x ∈ Ω, t ∈ (0, T ],

(6.6)

where Lu is the linear advection-diffusion-reaction operator in conservation form, Lu := ∇ · (au − D∇u) + σu,

(6.7)

and the advective velocity a, the diffusion tensor D, and the decay constant σ are independent of u. The boundary and initial conditions are given by Equations (6.3)– (6.5), as before.

6.2.2

Weak form

The weak form of the mathematical problem relaxes the regularity requirements of the solution u. It is obtained by multiplying the differential equation by a smooth function v which vanishes on Γu , integrating over the entire domain Ω, and applying Green’s formula to the flux term, to get the integral equation: Z Z Z Z Z ∂t u v dΩ − F · ∇v dΩ + σu v dΩ = qv dΩ − Ω

Ω

Ω

Ω

F¯ v dΓ.

(6.8)

Γn

The relation above needs to be satisfied at each fixed time t for all functions v belonging to some appropriate space of functions. We introduce the functional spaces: V := {v ∈ W : v = u¯ on Γu },

(6.9)

V0 := {v ∈ W : v = 0 on Γu },

(6.10)

which are, respectively, the space of trial functions where the solution is sought, and the space of test functions employed to test the weak form (6.8). The choice of the Sobolev space W depends on the form of the diffusion tensor and, for the purpose

197


of this presentation, it is sufficient to understand it as comprising smooth-enough functions. It is well known (see, e.g., Friedman [1969]) that, in the linear case, it is sufficient to have W ≡ H 1 (Ω), that is, the space of all square-integrable functions in Ω, whose first derivatives are also square-integrable. The weak form of problem (6.1)– (6.5) is then stated succinctly as follows: find u ∈ V for each fixed t ∈ (0, T ], such that (∂t u, v) + a(u, v; u) = l(v) ∀v ∈ V0 ,

(6.11)

u(x, t = 0) = u0 (x), where (∂t u, v) :=

Z

∂t u v dΩ, Z Z Z a(u, v; w) := − f (w) · ∇v dΩ + D(w)∇u · ∇v dΩ + σu v dΩ, Ω Ω Ω Z Z F¯ v dΓ. l(v) := qv dΩ −

(6.12)

Ω

Ω

(6.13) (6.14)

Γn

The weak form of the linear problem given by (6.6)–(6.7) with boundary and initial conditions (6.3)–(6.5) is to find u ∈ V for each fixed t ∈ (0, T ], such that (∂t u, v) + a(u, v) = l(v) ∀v ∈ V0 ,

(6.15)

u(x, t = 0) = u0 (x). The only difference with respect to Equation (6.11) is that a(u, v) ≡ a(u, v; u) is now a bilinear form.

6.2.3

Classical Galerkin method

With the notation above, it is straightforward to introduce the standard Galerkin approximation. Rather than considering the infinite-dimensional spaces V and V 0 ,


198

one employs the finite-dimensional subspaces Vh ⊂ V,

Vh,0 ⊂ V0 .

(6.16)

These are conforming spaces of functions with compact support, defined on a finite element mesh of the domain Ω. The standard Galerkin approximation of (6.11) is simply to find uh ∈ Vh for each fixed t, such that (∂t uh , vh ) + a(uh , vh ; uh ) = l(vh ) ∀vh ∈ Vh,0 ,

(6.17)

and uh (x, t = 0) is the projection of the initial function u0 (x) onto space Vh . This constitutes a system of nonlinear ordinary differential equations. The fully discrete system is obtained by further discretizing in time [Thomée, 1997]. Remark 6.1. The trial functions uh and the test functions vh , which are usually piecewise polynomials, can only capture variability at a scale larger than the mesh resolution. All subgrid variability, that is, all the features at a scale smaller than the element size, is automatically neglected. The well-known fact that the standard Galerkin method lacks stability for advection-dominated problems can be understood in this context. If the subscales are not captured adequately (or if they are completely ignored, as in the classical Galerkin method), their effects can propagate to larger scales, and deteriorate the coarse-scale calculations. In Section 6.3 we show examples of this behavior.

6.2.4

Multiscale approach

The fundamental principle of the multiscale approach is to acknowledge the presence of fine scales, which cannot be captured by the mesh. This is particularly important for advection-dominated problems, where the solution develops sharp fea-

199


tures that would require an impractical grid resolution. The formulation is based on a multiple-scale decomposition of the variable of interest u ∈ V [Hughes, 1995]: u = uh + u˜,

(6.18)

where uh is the part that can be resolved by the grid, and u˜ the unresolved part. This decomposition is unique if we can express the original functional space V as the direct sum of two spaces: ˜ V = Vh ⊕ V,

(6.19)

where Vh is the space of resolved scales and V˜ is the space of subgrid scales. The space V˜ is an infinite-dimensional space that completes Vh in V. This space is generally unknown, and it is the role of the subgrid model to provide a successful approximation to it. 6.2.4.1

The linear problem

For the linear advection-diffusion problem, the multiscale decomposition allows one to split the original problem into two. To this end, we invoke a multiscale decomposition of the trial function u and the test function v: ˜ u = uh + u˜ ∈ V = Vh ⊕ V,

(6.20)

˜ v = vh + v˜ ∈ V0 = Vh,0 ⊕ V,

(6.21)

and exploit the linearity of all the terms in Equation (6.11) with respect to v. We obtain one equation for the grid scales, (∂t (uh + u˜), vh ) + a(uh + u˜, vh ) = l(vh ) ∀vh ∈ Vh ,

(6.22)

and one for the subscales, ˜ (∂t (uh + u˜), v˜) + a(uh + u˜, v˜) = l(˜ v ) ∀˜ v ∈ V.

(6.23)

200


The former is a finite-dimensional problem, whereas the latter is infinite-dimensional. We now assume that the subscales are quasi-static [Codina, 2002], that is, ∂t u˜ ≈ 0.

(6.24)

Integrating by parts on each element and making use of the linearity of a(·, ·) and continuity of diffusive fluxes of the exact solution u = uh + u˜ across interelement boundaries [Codina, 2000], we get the following equation for the subscales: XZ e

Ωe

v˜ L˜ u dΩ =

XZ e

Ωe

˜ v˜ Ruh dΩ ∀˜ v ∈ V,

(6.25)

where Ruh := q − ∂t uh − Luh

(6.26)

is the grid scale residual. At this point, there are several options to solve the subgrid problem (6.25), most of which resort to some kind of localization assumption [Arbogast, 2002; Brezzi et al., 1997; Hughes, 1995]. In this investigation, the subscale problem is modeled —rather than solved— using an algebraic subgrid-scale (ASGS) approximation [Codina, 2000; Hughes, 1995]: u˜ ≈ τ Ruh ,

(6.27)

where the algebraic operator τ is called intrinsic time —or relaxation time. The expression of τ is one of the most difficult issues when devising stabilized methods. It should depend on the parameters of the problem, and on the actual discretization. From a numerical standpoint, a proper formulation of the intrinsic time should enhanced stability of the coarse-scale calculations without degrading the order of accuracy of the method. Here we use the following expression, which has proven useful in the context of linear systems of advection-diffusion-reaction equations [Codina,

201

Chapter 6. Multiscale finite elements for miscible flow and two-phase flow 2000]: τ=

µ

|a| kDk + c3 σ c1 2 + c 2 h h

¶−1

,

(6.28)

where h is a characteristic length of the element under consideration, and c1 = 4, c2 = 2, c3 = 1 for linear elements [Codina, 1998, 2000]. Equation (6.28) used to model the relaxation time can also be justified from heuristic physical considerations (see Section 6.3.1) and from an asymptotic Fourier analysis [Codina and Blasco, 2002]. This completes the description of the subgrid scales. After integration by parts on each element of the term a(˜ u, vh ) in Equation (6.22), the equation for the grid scales reads (∂t uh , vh ) + a(uh , vh ) +

X ·Z e

∗

Ωe

u˜ L vh dΩ +

Z

∗

Γe

¸

u˜ b vh dΓ = l(vh ) ∀vh ∈ Vh,0 , (6.29)

where L∗ is the adjoint of the differential operator L defined in Equation (6.7), L∗ v := −a · ∇v − ∇ · (D∇v) + σv,

(6.30)

and b∗ v is the associated boundary operator, b∗ v := (D∇v) · n.

(6.31)

Remarks 6.2. 1. The final equation for the resolved scales, Equation (6.29), includes the usual Galerkin terms (see Equation (6.17)) and some additional volume and boundary integrals evaluated element by element. These stabilizing terms, which are similar to those of other stabilized formulations [Codina, 1998], arise naturally in the multiscale approach [Hughes, 1995; Hughes et al., 1998].


202

2. The contribution from the boundary integrals in (6.29) has been neglected in the calculations. This is sensible only when the process is not dominated by diffusion. 3. The subscales u˜ are modeled analytically and eliminated from the global problem. With the algebraic approximation used here, they are proportional to the grid scale residual (see Equation (6.27)). The method is residual-based and, therefore, automatically consistent. 4. The new term in the grid-scale equation (6.29) is similar to that of other stabilized formulations, the only difference being the form of the operator multiplying the subscales [Codina, 1998]: ASGS: SUPG: GLS:

Z

ZΩ

ZΩ

e

u˜ L∗ vh dΩ,

L∗ v := −a · ∇v − ∇ · (D∇v) + σv,

u˜ (−Ladv vh ) dΩ, e

u˜ (−Lvh ) dΩ,

Ωe

−Ladv v := −a · ∇v,

−Lv := −∇ · (av) + ∇ · (D∇v) − σv.

(6.32) (6.33) (6.34)

The multiscale approach has several advantages over other stabilized formulations: (1) the stabilizing term arises naturally; (2) it is not restricted to a particular subgrid model; and (3) the ASGS formulation is endowed with better stability properties than SUPG and GLS for problems with reaction and source terms [Codina, 1998, 2000; Douglas Jr. and Wang, 1989; Franca et al., 1992]. 6.2.4.2

The nonlinear problem

Extension of the multiscale approach to the nonlinear problem given by Equations (6.1)–(6.5) is not straightforward, mainly because the form a(u, v; w) in Equa-


203

tion (6.11) is not linear in w. One of the key features of our approach is that nonlinearity of the equations is retained at the time of invoking the multiscale split. This distinguishes our method from the most common approach, which relies on linearizing the conservation equations upfront [Codina, 2001, 2002]. Linearizing the equations is not desirable when the solution is not smooth, since the conservation properties of the method might be compromised [LeVeque, 1992]. Other approaches which are related to ours have been presented in a different context by Garikipati and Hughes [1998] for the simulation of strain localization, Hughes et al. [2000] for the simulation of turbulent flows, and Ainsworth and Oden [2000] to obtain a posteriori error estimators for nonlinear elliptic problems. Specifically, we propose an incremental formulation and a multiple scale decomposition of the increment: (k)

(k)

(k)

u(k+1) ≈ uh + δu(k) = uh + δuh + δ˜ u(k) ,

(6.35)

where the index (k) refers to the iteration level. In this context, the incremental subscale δ˜ u(k) may be understood as a perturbation, that is, (k+1)

u(k+1) ≈ uh

+ δ˜ u(k) .

(6.36)

In what follows, we shall omit superscripts referring to the iteration level and simply write: u ≈ uh + δ˜ u.

(6.37)

Since the form a(u, v; w) is linear with respect to the second argument —the test function— the multiscale approach for problem (6.11) leads to a grid scale problem: (∂t (uh + δ˜ u), vh ) + a(uh + δ˜ u, vh ; uh + δ˜ u) = l(vh ) ∀vh ∈ Vh,0 ,

(6.38)

and a subscale problem: ˜ (∂t (uh + δ˜ u), v˜) + a(uh + δ˜ u, v˜; uh + δ˜ u) = l(˜ v ) ∀˜ v ∈ V.

(6.39)


204

Expanding the constitutive relations f (u) and D(u) to first order about an approximate coarse-scale solution uh , integrating by parts on each element some of the terms, assuming continuous interelement fluxes for the solution u, and considering quasistatic subscales as before, Equation (6.39) takes the form XZ e

Ωe

=

£ ¤ v˜ ∇ · ((f 0 (uh ) − D0 (uh )∇uh )δ˜ u − D(uh )∇δ˜ u) + σδ˜ u dΩ XZ e

Ωe

£ ¤ ˜ (6.40) v˜ q − ∂t uh − ∇ · (f (uh ) − D(uh )∇uh ) − σuh dΩ ∀˜ v ∈ V.

The variable a(uh ) := f 0 (uh ) − D0 (uh )∇uh

(6.41)

plays the role of an advective velocity. Defining the linearized advection-diffusionreaction operator, Luh v := ∇ · (a(uh )v − D(uh )∇v) + σv,

(6.42)

the expression above is written succinctly as XZ e

Ωe

u dΩ = v˜ Luh δ˜

XZ e

Ωe

˜ v˜ R(uh ) dΩ ∀˜ v ∈ V,

(6.43)

where R(uh ) is the grid-scale residual, R(uh ) := q − ∂t uh − ∇ · (f (uh ) − D(uh )∇uh ) − σuh .

(6.44)

Using an algebraic approximation to the subscales, the equation for the subgrid scales is, finally, δ˜ u ≈ τuh R(uh ),

(6.45)

where τuh —the intrinsic time— is now a nonlinear function of the grid scale solution uh : τuh =

µ

|a(uh )| kD(uh )k + c2 + c3 σ c1 2 h h

¶−1

.

(6.46)

205


Integrating by parts some of the terms in Equation (6.38) and rearranging conveniently, the coarse-scale equation reads (∂t uh , vh ) + a(uh , vh ; uh ) +

X ·Z e

Ωe

δ˜ u L∗uh vh

dΩ +

Z

Γe

δ˜ u b∗uh vh

dΓ

¸

= l(vh ) ∀vh ∈ Vh,0 , (6.47) where L∗uh v := −a(uh ) · ∇v − ∇ · (D(uh )∇v) + σv,

(6.48)

is the adjoint of the linearized operator Luh , and b∗uh v := (D(uh )∇v) · n

(6.49)

is the associated boundary operator. Equations (6.45) and (6.47) define the algebraic subgrid scale finite element method for a nonlinear advection-diffusion equation. Remarks 6.3. 1. Equations (6.45) and (6.47) are analogous to equations (6.27) and (6.29) for the linear case. In both cases, the formulation is residual-based and, thus, automatically consistent. In the nonlinear case, both coarse-scale and subgridscale equations need to be solved using an iterative procedure, e.g., Newton’s method. 2. Application of the multiple scale framework to nonlinear problems was investigated for one-dimensional strain localization in Garikipati and Hughes [1998]. In this reference, the incremental solution δu = δuh + δ˜ u is reconstructed after each iteration.


6.2.5

206


We further improve the method by incorporating a shock-capturing technique, based on increasing the amount of numerical dissipation in the neighborhood of strong gradients in the solution. By doing this, we avoid oscillations where sharp features occur, while keeping the method high-order accurate in smooth regions. The discontinuity-capturing scheme consists in adding the following integral to the lefthand side of Equation (6.47): XZ e

Ωe

Dsc (uh )∇uh · ∇vh dΩ,

(6.50)

where Dsc is a nonlinear diffusion coefficient. This leads necessarily to a nonlinear method, even if the underlying equation is linear. The success of this technique depends on the design of the numerical diffusion. The “canonical” form of the artificial diffusion is [Codina, 1993]: Dsc = αsc h

|R(uh )| |∇uh |

(6.51)

where µ

1 αsc = max 0, Csc − Pe

¶

,

(6.52)

P e being the element Peclet number and Csc a constant coefficient. In the advective limit, αsc approaches a constant value and, for steady-state conditions and linear elements, R(uh ) = −a · ∇uh . For one-dimensional problems this gives the same artificial diffusion everywhere, even though numerical dissipation is required only in the vicinity of sharp gradients. In this investigation, we propose a novel dimensionallyconsistent, “subscale-driven” artificial diffusion, given by Dsc = Csc

|δ˜ u| h|a(uh )|, Usc

(6.53)


207

where Usc is a characteristic value of the solution near the shock, and Csc is a constant coefficient. The added numerical diffusion will be significant only where the absolute value of the subgrid scales and the advective velocity are important.

6.3

Representative numerical simulations

In this section we present some numerical simulations of miscible and immiscible flow. We concentrate, for the most part, on the one-dimensional problem. For each of the test cases presented here, we compare the standard Galerkin solution with the algebraic subgrid scale (ASGS) solution. This comparison may seem a little unfair, as the test cases involve advection- and reaction-dominated flows, for which the classical Galerkin method is known to have unstable behavior. The motivation is to show the stabilizing effect of the new terms in the ASGS formulation, which arise from consideration of the subgrid scales. It is interesting to note that: 1. The ASGS method is in fact a Galerkin method, because the coarse-scale trial and test functions belong to the same finite element space. The difference with respect to the classical Galerkin method is that the subgrid scales are modeled separately. 2. The computational cost of the ASGS method is essentially the same as that of the standard Galerkin method, as the former involves the calculation of just a few additional integrals, which are evaluated elementwise. For some test cases, we also show the effect of shock-capturing on the numerical solution, and we compare the performance of the “canonical” form and the “subscaledriven” form of the discontinuity-capturing dissipation. Convergence behavior of Newton iteration is also discussed.

208


6.3.1

One-dimensional miscible flow

We consider one-dimensional flow of a tracer that is perfectly miscible with water. The governing equation of tracer transport is a transient linear advection-diffusionreaction equation with production —Equation (6.6)— which, for the one-dimensional case, takes the form: ∂t u + a∂x u − D∂xx u + σu = q,

x ∈ (0, 1), t ∈ (0, T ].

(6.54)

We investigate three test cases, each one with particular values of the advective velocity a, the diffusion coefficient D, the decay constant σ, and the production term q: 1. Dominant advection with zero production: a = 1, D = 10−4 , σ = 0, q = 0. 2. Dominant advection with production: a = 1, D = 10−4 , σ = 0, q = 1. 3. Advection and reaction with production: a = 1, D = 10−4 , σ = 4, q = 1. It is illustrative to compute the intrinsic times for each of the physical processes involved, taking as a reference length one-half of the element size. The expressions for the characteristic times of advection (τa ), diffusion (τd ), and reaction (τr ), are given in Table 6.1. The intrinsic times are related to the Courant, Peclet and Damkholer numbers: 1 δt a δt = , h 2 τa ah τd Peclet #: P e := =2 , D τa 2 τd σh =4 . Damkholer #: Da := D τr Courant #:

Co :=

(6.55) (6.56) (6.57)

The Courant number indicates whether the time discretization is fine enough to capture the advective flux from element to element. It is usually restricted to be less


209

Table 6.1. Expressions of the characteristic times for advection, diffusion and reaction processes, taking as a reference length one-half of the element size. Diffusion τd =

h2 4D

Advection τa =

Reaction

h 2a

τr =

1 σ

than 1.0 for stability or accuracy requirements. The Peclet and Damkholer numbers measure, respectively, the preeminence of advection and reaction with respect to diffusion. Values of these dimensionless numbers much greater than 1.0 imply that the problem is not dominated by diffusion, suggesting that the solution might present sharp features. The expression of the intrinsic time used in the subgrid model (6.27) is the harmonic mean of the characteristic time of each process, that is, ¢−1 ¡ = τ = τd−1 + τa−1 + τr−1

µ

D a 4 2 +2 +σ h h

¶−1

.

(6.58)

The expression provides a physical interpretation of the intrinsic time (6.28) used in this investigation. 6.3.1.1

Test Case 1: Dominant advection with zero production

The governing equation is (σ = 0, q = 0 in Equation (6.54)): ∂t u + a∂x u − D∂xx u = 0.

(6.59)

We solve the problem on a unit segment with Dirichlet boundary conditions: u(0, t) = 1,

u(1, t) = 0,

(6.60)

and zero initial data: u(x, 0) = 0.

(6.61)


210

For early times, that is, before breakthrough of the inlet boundary conditions, the analytical solution is given by [Lapidus and Amundson, 1952]: µ · ¶ µ ¶¸ ³ ax ´ x + at x − at 1 exp erfc √ + erfc √ , u(x, t) = 2 D 4Dt 4Dt

(6.62)

where erfc(·) is the complementary error function [Abramowitz and Stegun, 1972]. Because the value of the diffusion coefficient is so small, the solution is essentially a sharp front, moving with speed a = 1. For long simulation times, the injected boundary conditions reach the outlet face of the domain, and the solution reaches a steady-state, given by: u(x) = C + (1 − C) exp

³ ax ´ D

,

(6.63)

where C=

exp (aL/D) . exp (aL/D) − 1

(6.64)

with L = 1. For the advection-dominated case of interest, the expression above takes the approximate form: u(x) = 1 −

exp (ax/D) , exp (aL/D)

(6.65)

which is a boundary layer of width of the order of D/a. Solution obtained by the classical Galerkin method and the subgrid scale approach (ASGS), with and without shock-capturing, are compared against the analytical solution for a very coarse and a finer discretizations. The coarse discretization employs Ne = 20 elements, and a time step of δt = 0.02. The corresponding Peclet and Courant numbers are P e = 500, and Co = 0.4, respectively. The finer discretization uses Ne = 100 elements, and a time step of δt = 0.004, corresponding to P e = 100, and Co = 0.4. Linear elements and a Crank-Nicolson time-stepping scheme are used in both cases.

211

Chapter 6. Multiscale finite elements for miscible flow and two-phase flow 2 1.5

2

Galerkin ASGS Analytical

1.5

t = 0.2

t = 0.5

u

1

u

1 t=2

t = 0.2

0.5

0.5

0

0

−0.5 0


0.2

0.4

x

0.6

0.8

(a) Coarse grid. Galerkin vs. ASGS

1

−0.5 0

0.2

0.4

t = 0.5

x

0.6

t=2

0.8

1

(b) Fine grid. Galerkin vs. ASGS

Figure 6.1. One-dimensional miscible flow with zero distributed sources. ASGS and Galerkin solutions at three different times, for a very coarse (left) and a finer discretization (right). The solution displays a moving front during the transient phase, and develops a very sharp boundary layer when stationary conditions are reached. All methods predict the speed of the front accurately, even with the coarse mesh. The ASGS method is able to reduce the oscillations near the front compared with the classical Galerkin method, as shown in Figure 6.1 for observation times t = 0.2 and t = 0.5. However, the most relevant difference arises for long simulation times, when the solution is steady state (shown is the solution at t = 2). The classical Galerkin solution shows spurious oscillations in the entire domain, whereas the ASGS solution has no oscillations whatsoever and is extremely accurate: the boundary layer is captured within a single element. Localized wiggles of the ASGS solution during transient conditions are further reduced by using a shock-capturing technique, as described in the previous section. Two different formulations are used for the discontinuity-capturing numerical dissipation: the “canonical” form (CASC) given by Equation (6.51), and the “subscale-driven”


0.06

212

SGSC (Csc=5) CASC (C =1) sc

t=2

Dsc

0.04

PSfrag replacements

0.02

0 0

t = 0.5

0.2

0.4

x

0.6

1

0.8

Figure 6.2. Amount of shock-capturing diffusion introduced by the “canonical” formulation and the proposed formulation, shown at two different times.

1.4 1.2

SGSC (C =5) sc CASC (Csc=1) Analytical

u

1 0.8

t = 0.5

0.6

t=2

0.4 0.2 0 −0.2 0.4

0.5

x

0.6

0.9

0.95

1

x

Figure 6.3. Detail of the shock (left) and boundary layer (right) for the onedimensional miscible flow problem with zero distributed sources. Comparison of the ASGS solution with the “canonical” and the proposed forms of shock-capturing diffusion.


log10 residual norm

0

213

CASC SGSC

−4

−8

−12 0

1

2

3

4

5

6

Newton iteration

Figure 6.4. Convergence of the ASGS method with shock-capturing for onedimensional miscible flow problem with zero distributed sources. Convergence of the “classical” and the proposed shock-capturing schemes are compared at t = 0.5. form (SGSC) of Equation (6.53). In Figure 6.2 we plot the profile of numerical diffusion added by each of these formulations for the coarse grid at two different times: t = 0.5 (transient conditions) and t = 2 (stationary conditions). The “canonical” form of shock-capturing diffusion displays an erratic oscillatory behavior for transient conditions, with significant diffusion everywhere and not just in the vicinity of the front. When steady state is met, the artificial diffusion added by this model is constant over the entire domain, which is in disagreement with the very concept of discontinuity capturing. On the other hand, the novel formulation of nonlinear dissipation proportional to the absolute value of the subscales yields the expected behavior: the profile of artificial diffusion has a maximum value near the location of the layer, and decays rapidly away from it. A detail of the numerical solutions obtained by the ASGS method with shock-capturing are shown in Figure 6.3 for the fine grid. Both discontinuity-capturing schemes are effective at reducing localized oscillations near the layers. When the shock-capturing technique is used, the method is nonlinear even if the underlying equation is linear, because the amount of dissipation is a nonlinear function


214

of the solution. A Newton iterative scheme was used to achieve convergence at every time step. In Figure 6.4 we show the evolution of the L2 -norm of the residual for a typical time step (in this case t = 0.5). The ASGS method with “subscale-driven” shock-capturing converges quadratically for all time steps. Convergence when the “canonical” form of discontinuity-capturing is used is always slower and often not monotonic. This is not surprising given the erratic behavior of the numerical diffusion introduced by the method (see Figure 6.2). 6.3.1.2

Test Case 2: Dominant advection with production

The governing equation is (σ = 0 in Equation (6.54)): ∂t u + a∂x u − D∂xx u = q.

(6.66)

The problem is solved on the unit segment with zero initial data, u(x, 0) = 0, and homogeneous Dirichlet boundary conditions, u(0, t) = 0,

u(1, t) = 0.

(6.67)

At early times, the behavior of the solution is as follows: 1. Away from both boundaries. Because the initial concentration is uniform, and so is the source term, the concentration gradients will be zero as long as the region is not affected by the boundaries. Since ∂x u ≈ 0, the equation reduces to: ∂t u ≈ q.

(6.68)

Therefore, the solution consists in a plateau, rising at a rate of q = 1 concentration units per unit time.


215

2. Near the left (inlet) boundary. Physically, what happens is that water with zero concentration from the inlet boundary “washes” water with tracer inside the domain. A steady tracer profile is established near the inlet, accommodating the effects of advection and distributed sources. Since the concentration profile is steady (∂t u ≈ 0) and diffusion is negligible (D ≈ 0), the approximate governing equation for this conditions is: a∂x u ≈ q.

(6.69)

The solution near the left boundary is, thus, a ramp of slope q/a = 1. 3. Near the right (outlet) boundary. In the neighborhood of this boundary, diffusion cannot be ignored. A boundary layer develops to connect the solution far away from the boundary (given by the rising plateau) to the Dirichlet boundary condition. The width of this layer is of the order of D/a = 10−4 . For long simulation times, steady-state conditions are reached when the effect of the inlet boundary is felt at the outlet. The solution then consists of a ramp of slope q/a = 1 and a sharp boundary layer whose width is of the order of D/a = 10−4 . The problem is solved numerically on a very coarse grid of Ne = 40 linear elements (the corresponding element Peclet number is P e = 250). A Crank-Nicolson time stepping is used, with a constant time step of δt = 0.01, so that the Courant number is Co = 0.4. Solution by the classical Galerkin method and the algebraic subgrid scale (ASGS) method are shown in Figure 6.5 at three different times: t = 0.2, t = 0.5, and t = 2, for which a steady-state solution has been reached. At time t = 0.2, the standard Galerkin solution is wildly oscillatory. Oscillations are more pronounced near the outlet face, but significant in more than half of the computational domain. For later times, the solution is globally polluted with nonphysical oscillations. The

PSfrag replacements Chapter 6. Multiscale finite elements for miscible flow and two-phase flow 1

Galerkin ASGS

216

t=2

0.8 0.6

u

t = 0.5

0.4 t = 0.2 0.2 0

0

0.2

0.4

x

0.6

0.8

1

Figure 6.5. ASGS and Galerkin solutions for the one-dimensional miscible flow problem with production, at three different times. oscillatory behavior arises because the method lacks stability: the boundary layer cannot be resolved with the discretization used (the boundary layer width is two orders of magnitude smaller than the element size), and this loss of accuracy at the subgrid scale “propagates” to degrade the coarse-scale calculations. On the other hand, the solution obtained by the ASGS method is perfectly nonoscillatory. The calculated concentration profiles reproduce the transient behavior described above: a ramp near the inlet boundary, a rising plateau in the center region, and a sharp layer at the outlet boundary. The slope of the ramp and the rate of increase of the plateau concentration agree with the predicted values. Moreover, the boundary layer is reproduced in the best possible way given the actual discretization: it is captured with just one element, and without a single overshoot.

217

Chapter 6. Multiscale finite elements for miscible flow and two-phase flow 6.3.1.3

Test Case 3: Advection and reaction with production

The governing equation is the full Equation (6.54), with zero initial and boundary conditions. We can identify the following features in the solution: 1. Away from both boundaries. Using the same arguments as before, this is a region of uniform concentration u∗ , described by the initial value problem: du∗ + σu∗ = q, dt

u∗ (t = 0) = 0.

(6.70)

The solution is: u∗ =

q (1 − exp(−σt)) , σ

(6.71)

so the rising plateau tends asymptotically to a steady value of q/σ = 0.25. 2. Near the left (inlet) boundary. It is difficult to obtain an analytical description of the solution during the transient phase. However, when steady-state conditions are reached (∂t u = 0) and neglecting the effects of diffusion, the governing equation reduces to: a

du + σu = q, dx

u(x = 0) = 0.

(6.72)

The solution to the problem above is the concentration profile: u=

³ σ ´´ q³ 1 − exp − x , σ a

(6.73)

The width of this profile can be estimated by the value of x such that u ≈ 0.95umax , which gives a width of approximately 0.75. 3. Near the right (outlet) boundary. The solution consists of a boundary layer with the same characteristics as those of the previous test cases.


218

We solve the problem numerically on a coarse grid of 40 linear elements. Marching in time is performed with a Crank-Nicolson method and a constant time step of δt = 0.01. The characteristic dimensionless parameters for this simulation are: P e = 250, Co = 0.4, and Da = 25. In Figure 6.6 we show the results obtained with the classical Galerkin method and the ASGS approach, at three different times: t = 0.2, t = 0.4, and t = 2 (steady-state). The standard Galerkin solution reproduces the rising plateau, and it captures the structure of the solution at the left boundary. We recall that, for the parameters used in this simulation, the concentration profile at the inlet has a width of the order of 2 or 3 elements. However, the classical Galerkin method displays nonphysical oscillations at the right boundary, which propagate well into the domain. This nonlocal oscillatory behavior denotes the lack of stability of the method, and its inability to “damp out” subgrid effects appropriately. By contrast, the ASGS solution is accurate and stable: it captures sharply all the features of the solution, and does not present spurious oscillations. We remark that the ASGS formulation introduces just a marginal additional computational cost with respect to the classical Galerkin method.

6.3.2

Two-dimensional miscible flow

In this section, we apply the multiscale method to the solution of a two-dimensional tracer transport problem of a conservative tracer (σ = 0) and without source terms (q = 0). The governing equation is the scalar linear advection-diffusion equation in 2D: ∂t u + ∇ · (au − D∇u) = 0.

(6.74)

Chapter 6. Multiscale finite elements for miscible flow and two-phase flow 0.4

Galerkin ASGS

0.3

u

219

t=2 t = 0.4

0.2

t = 0.2 0.1

0 0

0.2

0.4

x

0.6

0.8

1

Figure 6.6. ASGS and Galerkin solutions for the one-dimensional miscible flow problem with reaction and production, at three different times. We take an isotropic diffusion coefficient, D = D1, where 1 is the 2 × 2 identity tensor. We solve this equation on the unit square Ω = [0, 1]×[0, 1], with homogeneous Dirichlet boundary conditions on the bottom and right edges (u = 0 on x = 1, y = 0), and unit Dirichlet boundary conditions on the top and left edges (u = 1 on x = 0, y = 1). We take a uniform velocity field of unit magnitude, tilted 30◦ with respect to the x-axis, a = (cos 30◦ , sin 30◦ ), so that it is not aligned with the mesh. We consider a very small diffusion coefficient D = 10−6 . A schematic of this test case is shown in Figure 6.7. During the transient phase, the solution displays a front moving from left to right, an internal layer forming a 30◦ -angle with the bottom boundary, and a boundary layer at the top edge. Once the moving front reaches the right boundary, stationary conditions are established. The sharp features of the steady-state solution include an internal layer, which develops from the lower-left corner and is tilted 30 ◦ , and a boundary layer at the upper part of the right boundary, required to satisfy the

Chapter 6. Multiscale finite elements for miscible flow and two-phase flow 6

220

¸ 9

u=1 >

a

1

θ = 30◦ u=0 ?

:

° ¾

1

-

Figure 6.7. Schematic of the two-dimensional miscible flow problem. homogeneous boundary condition at the right edge. We solve the problem using a uniform grid of 20 × 20 bilinear quadrilaterals, and backward Euler time-stepping with δt = 0.01. The element Peclet number is 50,000. Such coarse mesh is clearly insufficient to resolve the abrupt discontinuities of the solution, but the objective is to analyze the stability properties of the numerical method, that is, whether the inaccuracies will propagate to produce a globally oscillatory solution. Solution obtained by the classical Galerkin method is completely oscillatory both for transient and stationary conditions (see Figure 6.8(a) and Figure 6.8(c)). This behavior is in agreement with the well known lack of stability of the Galerkin method for advection-dominated problems. In contrast, the subgrid-scale formulation (ASGS method) produces an effectively stabilized solution. It captures rather sharply the advancing front during the transient phase and the internal and boundary layers which develop at steady state (see Figure 6.8(b) and Figure 6.8(d)). It is important to note that the ASGS method only requires computation of some additional integrals, evaluated element-by-element, so the computational cost is roughly

PSfrag replacements PSfrag replacements Chapter 6. Multiscale finite elements for miscible flow and two-phase flow

2.5

2 1.5 2.5

2

31

1.5 −100 1

−100 0.5

3

1000.5 0 200 1

221

100

PSfrag replacements

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0 200 1 1

1 0.5

300

300

0.5 0 0

400

0.5 0 0

400

(a) Galerkin solution, t = 0.5

0.5

(b) ASGS solution, t = 0.5

1.5 2 2.5

2 1.5 2.5

400 300

3

31

200

−100 0.5

100 0

100 0 200 1

−100 1

1

1 0.5

0.5 0 0

(c) Galerkin solution, steady-state

300 400

0.5

0.5 0 0

(d) ASGS solution, steady-state

Figure 6.8. Galerkin and ASGS solutions for transient and steady-state conditions for the two-dimensional miscible flow problem.

Chapter 6. Multiscale finite elements for miscible flow and two-phase flow PSfrag replacements PSfrag replacements

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

PSfrag replacements

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0.4

0.6

0.8

1

0 0

(a) ASGS solution, 20 × 20 elmts.

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.2

0.4

0.6

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(c) ASGS solution, 80 × 80 elmts.

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(b) ASGS solution, 40 × 40 elmts.

1

0 0

222

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0 0

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0.4

0.6

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(d) ASGS solution, 125 × 125 elmts.

Figure 6.9. Contour plots at steady-state for the two-dimensional miscible flow problem, on 20 × 20, 40 × 40, 80 × 80, and 125 × 125 grids.


223

the same as that of Galerkin method. Local overshoots in the vicinity of discontinuities, which are still present in the stabilized solution, could be further reduced or completely removed using a shock-capturing technique [Codina, 1993], as in the one-dimensional case. This possibility was not explored here. In Figure 6.9, we show contour plots of the ASGS solution at steady state for increasingly refined grids of 20×20, 40×40, 80×80, and 125×125 elements. Contours correspond to the values u = 0.1, 0.2, . . . , 0.9. This figure effectively shows how the ASGS solution converges as the mesh is refined.

6.3.3

One-dimensional immiscible flow

We present in this section the numerical simulation of one-dimensional immiscible flow of two phases, which we shall denote ‘water’ and ‘oil’. As discussed in Chapter 2, the mathematical problem is defined by a nonlinear scalar conservation law of advection-diffusion type —the saturation equation— which we recall here: ∂t u + ∂x (f (u) − D(u)∂x u) = 0,

x ∈ (0, 1), t ∈ (0, T ].

(6.75)

We assume that this equation has been re-scaled, so that all the variables are dimensionless. In Equation (6.75) above, u is the water saturation, f is the water fractional flow, and D is the capillary diffusion coefficient. Neglecting gravitational effects, functions f and D take the following expressions (see Equations (2.68)–(2.69) in Chapter 2): λw , λw + λ o µ ¶ dPcD λw λo − . D(u) = Cw µo λw + λ o dSw f (u) =

(6.76) (6.77)

The fractional flow function f is typically S-shaped and, thus, nonconvex. We


224

shall consider the following simple model [Dahle et al., 1995; LeVeque, 1992]: f (u) =

u2 , u2 + µ ˜(1 − u)2

(6.78)

where µ ˜ is the water-oil viscosity ratio, taken here as 1. This model is a consequence of considering quadratic relative permeabilities, and is used for illustrative purposes in Chapter 5 also. A graphical representation of the fractional flow function is given in Figure 6.10(a). The diffusion coefficient D, which arises from capillarity effects, is typically degenerate at the endpoint saturations, that is, it vanishes for u = 0 and u = 1, and is positive otherwise [Chavent and Jaffré, 1986]. To mimic this behavior, we choose the following expression [Dahle et al., 1995] (see Figure 6.10(b)): D(u) = ²u(1 − u).

(6.79)

We solve Equation (6.75) on the unit segment Ω = [0, 1] with Dirichlet boundary conditions, u(0, t) = 1,

u(1, t) = 0,

(6.80)

and zero initial conditions, u(x, 0) = 0.

(6.81)

These conditions are representative of a linear water flood in a hydrocarbon reservoir [Buckley and Leverett, 1942; Rapoport and Leas, 1953]. We take a value of ² = 10−4 for the constant in the degenerate diffusion coefficient. We use such a small value of this parameter to minimize the effects of capillary pressure, and solve the near-hyperbolic problem. The diffusion-free problem, or Buckley-Leverett problem [Buckley and Leverett, 1942], admits a straightforward analytical solution, discussed in Chapter 5. During

225

Chapter 6. Multiscale finite elements for miscible flow and two-phase flow 1


f (u)

D(u)/²

PSfrag replacements

f (u)

D(u)/²

0.25

0 0

u (a) Fractional flow function

1

0 0

u

1

(b) Capillary diffusion function

Figure 6.10. Fractional flow function f (left), and capillary diffusion function D (right), used in the immiscible flow simulations. The function f is typically S-shaped and, thus, nonconvex. The function D typically vanishes at the endpoint saturations, and is positive elsewhere. the transient phase, before breakthrough of the injected fluid, the solution consists of a rarefaction fan and a shock. Both the shock speed and the post-shock value are constant, and easily computable from the flux function [LeVeque, 1992]. As a result, the solution “stretches” with time in a self-similar fashion. For long simulation times, after breakthrough, the system reaches a quasi-steady state. Dirichlet boundary conditions are particularly challenging, because they force a very fast initial transient at the inlet, and a sharp boundary layer at the outlet after breakthrough. This problem “exhibits several difficult features beyond the usual ones of advection-dominated flow: degenerate diffusion, sharpening near-shock solutions, and capillary outflow boundary layers” [Dahle et al., 1995, p. 247]. Numerical solutions to the Buckley-Leverett problem include the early works of Todd et al. [1972], Settari and Aziz [1975], Aziz and Settari [1979] and, more recently, Dahle et al. [1995], and Binning and Celia [1999].


226

Since we use a very small value of the parameter ², capillary diffusion effects do not greatly influence the global structure of the solution. However, capillarity is not negligible in the neighborhood of sharp features, because of the extremely high saturation gradients. In particular, the width of the traveling shock (before breakthrough) and the boundary layer at the outlet face (after breakthrough) are of the order of ² = 10−4 . Of course, resolving the fine-scale structure of the solution would require elements smaller than this length. This is not feasible in practical problems, and the goal is to obtain an accurate numerical solution on a coarse grid, which preserves the global structure of the exact solution. The following numerical methods were tested in this investigation: 1. Galerkin: standard Galerkin approximation —Equation (6.17). 2. ASGS: multiscale formulation, with an algebraic approximation to the subgridscales —Equation (6.47). 3. CASC: ASGS method with the “canonical” expression for the shock-capturing dissipation —Equation (6.51). 4. SGSC: ASGS method with the proposed “subgrid-scale” shock-capturing dissipation —Equation (6.53). The expression of the relaxation time used in the ASGS formulation is: τuh =

¡

τd−1

+

¢−1 τa−1

=

µ

a(uh ) D(uh ) 4 2 +2 h h

¶−1

,

(6.82)

where the “advective velocity” a(uh ) comes from the proposed linearization of the problem, and is given by: a(uh ) = f 0 (uh ) + D0 (uh )∂x uh .

(6.83)


1

1

t = 1.0

0.8

t = 1.0

0.8 t = 0.4

t = 0.4

u

0.6

u

0.6 1.2

227

1.2

0.4 0.2 0 −0.2 0

0.4 0.2


0.2

0 0.4

x

0.6

0.8

1

(a) Coarse grid. Galerkin vs. ASGS

ASGS Analytical

−0.2 0

0.2

0.4

x

0.6

0.8

1

(b) Fine grid. Galerkin vs. ASGS

Figure 6.11. ASGS and Galerkin solutions to the one-dimensional immiscible flow problem at two different times, for a very coarse (left) and a fine discretization (right). Results for a very coarse grid (Ne = 20, δt = 0.01), and a finer grid (Ne = 500, δt = 0.0005), are provided. These two discretizations correspond to element Peclet numbers of P e ≈ 2, 500 and P e ≈ 100, respectively.1 In all cases we used a backward Euler time-stepping scheme. In Figure 6.11(a), the numerical solutions obtained by the standard Galerkin method and the ASGS method on the coarse grid are compared with the analytical solution of the hyperbolic problem. Both numerical solutions predict correctly the shock location.2 The classical Galerkin method produces a big overshoot and a nonphysical saturation plateau upstream of the front during the transient state (shown is a snapshot of the solution at t = 0.4). More noticeably, it gives a completely oscillatory solution after breakthrough (the results are displayed at t = 1). On the other 1

This range of Peclet numbers should be compared with that of simulations using characteristics methods [Binning and Celia, 1999; Dahle et al., 1995], where the highest Peclet number considered is about 2. 2 Correct prediction of the shock location and convergence to the physically correct solution for the Galerkin method is in contrast to the findings of Todd et al. [1972] and Aziz and Settari [1979], where midpoint weighting (equivalent to classical Galerkin) is shown to converge to a nonphysical solution. The reason for this discrepancy [LeVeque, 1992] is that the conservation form of the “saturation” equation is used here, whereas the aforementioned references discretize a nonconservation form.

Chapter 6. Multiscale finite elements for miscible flow and two-phase flow 0.05 0.04

228

CASC (Csc=1) SGSC (Csc=2)

Dsc

t = 1.0 0.03 0.02

t = 0.4

0.01 0 0

0.2

0.4

x

0.6

0.8

1

Figure 6.12. Amount of shock-capturing diffusion introduced by the “canonical” formulation and the proposed formulation, shown at two different times. hand, the ASGS solution is not globally polluted with oscillations, and preserves a sharp definition of the saturation front and the boundary layer. The ASGS solution is remarkably accurate wherever the actual solution is smooth —along the rarefaction fan and the constant initial state— even though an extremely coarse mesh of just 20 linear elements was used. The oscillatory behavior of the numerical solution is confined to a single undershoot at the downstream end of the traveling shock, and a single overshoot at the boundary layer for the long-time solution. These observations are further confirmed by the simulations on a finer grid of 500 linear elements, shown in Figure 6.11(b). In this case, the standard Galerkin method did not converge at all for t = 1 and, thus, is not shown in the figure. The most important feature of the ASGS solution is that the advancing front (before breakthrough) and the boundary layer (after breakthrough) are captured sharply, avoiding the excessive smearing of traditional upwind formulations. The localized wiggles that remain in the solution can be successfully removed by using a shockcapturing technique, which introduces numerical dissipation only in the neighborhood of discontinuities.


1 0.8

ASGS CASC (Csc=1) SGSC (C =2)

229

t = 1.0

sc

Analytical

t = 0.4

u

0.6 0.4 0.2 0 −0.2 0.47

x

0.5

0.985

x

1

Figure 6.13. Detail of the shock (left) and boundary layer (right) for the onedimensional immiscible flow problem. Comparison of the ASGS solution with the “canonical” and the proposed forms of shock-capturing diffusion. We analyze the effect of using a shock-capturing scheme in conjunction with the ASGS method, and compare the performance of the “canonical” form (CASC) and the “subgrid-scale” form (SGSC). In Figure 6.12 we plot the profiles of artificial diffusion added by each method on the coarse grid at two different times. For both methods, the amount of numerical dissipation is significant only in the neighborhood of layers, but it decays more rapidly for the SGSC formulation. In Figure 6.13 we show magnified plots of all three numerical solutions (ASGS, CASC and SGSC) computed on the fine grid at times t = 0.4 and t = 1. Both shock-capturing schemes are effective at reducing or eliminating local overshooting, the novel SGSC formulation being (slightly) less diffusive. A Newton scheme was used in all cases to solve the system of nonlinear algebraic equations at each time step. In Figure 6.14 we show the evolution of the L2 -norm of the residual for two typical time steps (t = 0.4 for transient conditions, and t = 1.0 for quasi-steady conditions) of the coarse grid simulations. The main observation is


230

0 log10 residual norm

t = 0.4

t = 1.0

−4 −8 −12 0

Galerkin ASGS CASC SGSC

4 2 6 Newton iteration

0

4 2 6 Newton iteration

Figure 6.14. Evolution of the Newton iterative scheme for the one-dimensional immiscible flow problem. Comparison of the convergence of the standard Galerkin and ASGS methods (with and without shock-capturing) on the coarse grid, for two typical time steps (t = 0.4 and t = 1). Convergence is monotonic and quadratic in both cases. that convergence is monotonic and asymptotically quadratic for all methods (classical Galerkin and ASGS, with and without shock-capturing). The additional stabilizing term of the ASGS formulation does not degrade convergence of the Newton iterative scheme. On the contrary, convergence of the ASGS method on the finer grid is also quadratic at all times, whereas the standard Galerkin method fails to converge shortly after breakthrough, due to unbounded growth of spurious oscillations. The ASGS method with “classical” and “subscale-driven” shock-capturing diffusion — CASC and SGSC methods, respectively— take one or two more iterations during the transient period, while the situation is reversed for steady state.


6.4

231

Concluding remarks

We have presented a formalism for the numerical solution of nonlinear conservation laws, which is based on a multiscale decomposition of the variable of interest, and applied it to the problems of miscible and immiscible two-phase flow in porous media. The key idea is to acknowledge that the fine-scale structure of the solution cannot be captured by any grid, and to incorporate the net effect of the subgrid scales onto the scales resolved by the mesh [Hughes, 1995; Hughes et al., 1998]. An algebraic approximation of the subscales [Codina, 2000] is used to model subgrid variability. The key parameter of the formulation is the intrinsic time τ , which is calculated as the harmonic mean of the characteristic times of diffusion, advection and reaction at the length scale of half the element size. This multiscale approach leads to a stabilized finite element method with excellent properties. The numerical simulations clearly show the ability of the method to simulate accurately miscible and immiscible flows on very coarse grids, even when the solution develops sharp features. Furthermore, the proposed methodology involves only a marginal additional computational cost with respect to the standard Galerkin method. In the following chapter, the multiscale formalism is extended to systems of equations, and applied to the simulation of one-dimensional three-phase flow.

The system of Planets is stable, the system of Comets is unstable. — HENRY MOSELEY, Astronomy (1839)

232

Chapter 7 Multiscale finite elements for three-phase flow 7.1

Introduction

A multiscale formulation for the numerical solution of one-dimensional threephase flow through porous media is presented in this chapter. The mathematical model, described in Chapter 2, is based on the multiphase extension of Darcy’s equation and the fractional flow formalism. We are particularly interested in the case of vanishing capillarity effects. Even though the equations studied here are a simplified version of the full nonisothermal, multiphase, compositional model in several dimensions, they display some of the essential features that pose numerical difficulties. In particular, the problem is extremely nonlinear, almost hyperbolic for the case of interest —vanishing capillarity— and the solution naturally develops shocks and boundary layers. In an attempt to obtain stable solutions which retain high-order accuracy, the

Chapter 7. Multiscale finite elements for three-phase flow

233

equations are solved here using the stabilized finite element method of [Hughes, 1995], where the stabilizing terms arise naturally in a variational multiscale method [Hughes et al., 1998]. The specific contributions of this chapter may be succinctly summarized as follows: 1. The multiscale formalism is applied to the equations governing one-dimensional three-phase flow through porous media. Previous work on miscible and immisicible two-phase flow [Juanes and Patzek, 2002b,d, see also Chapter 6] — described by scalar equations— is extended here to nonlinear systems of conservation laws. 2. Nonlinearity of the equations is retained at the time of invoking the multiscale split. Proper linearization of the stabilizing terms is introduced after the multiple-scale decomposition into resolved and unresolved scales. Furthermore, the multiple-scale solution is not reconstructed from point values of coarse and subgrid scales. 3. Several definitions of the key parameter of the formulation —the matrix of stabilizing coefficients— are tested and compared [Codina, 2000; Hughes and Mallet, 1986a; Shakib et al., 1991]. To reduce further or eliminate completely localized oscillations that may persist in the stabilized solution, several existing shockcapturing techniques are studied [Codina, 1993; Galeão and Dutra do Carmo, 1988; Hughes and Mallet, 1986b; Hughes, Mallet and Mizukami, 1986; Shakib et al., 1991], and a novel expression for the discontinuity-capturing diffusion is proposed. It is important to realize that the formulation presented here is very different from other methods that account for multiple-scale phenomena, such as the multiscale fi-


234

nite element method [Hou and Wu, 1997], the subgrid upscaling technique [Arbogast, 2000, 2002], and the mortar upscaling method [Peszynska et al., 2002], where the main objective is to incorporate the small-scale heterogeneity. On the other hand, the recent paper by Masud and Hughes [2002] applies the original variational multiscale formulation of Hughes [1995]; Hughes et al. [1998] to porous media flows. It is restricted, however, to the linear scalar equation describing steady-state, single-phase, Darcy flow, and the objective is to remove velocity-pressure instabilities, rather than instabilities arising from the nearly hyperbolic character of the equations. An outline of the chapter is as follows. In Section 7.2 we summarize the weak form of the problem and the associated standard Galerkin method, and describe in detail the multiple-scale approach. Special attention is given to the matrix of stabilizing coefficients and to alternative shock-capturing techniques. In Section 7.3 we present several representative numerical simulations of three-phase flows. The first application is an oil filtration problem in a relatively dry medium, and the second reproduces water-gas injection in a hydrocarbon reservoir. Numerical solutions are compared with a general, newly developed, analytical solution [Juanes and Patzek, 2002a, see also Chapter 5]. These simulations illustrate the outstanding performance of the proposed methodology. In Section 7.4 we gather the main conclusions of the chapter.

7.2

Multiscale numerical formulation

In this section, we describe a multiscale formulation for the numerical solution of the system of equations governing one-dimensional three-phase flow. We are interested in the case of small diffusion, for which the solution develops sharp features (shocks and boundary layers). The multiscale approach leads naturally to a stabilized

235


numerical method, which enhances the stability of the solution, without compromising its accuracy in the regions where the solution is smooth. The multiple-scale formalism was first proposed in Hughes [1995], and it is now recognized as a state-of-the-art method in computational fluid dynamics. In the previous chapter (see also Juanes and Patzek [2002b,d]), the formulation was applied to miscible flow of two components —described by a linear advection-diffusion equation— and immiscible flow of two phases, which is modeled by a nonlinear scalar conservation law. Here we apply and extend the formulation to the problem of three-phase porous media flow.

7.2.1

Initial and boundary value problem

The mathematical problem is defined by the one-dimensional system of conservation laws ∂t u + ∂x (f (u) − D(u)∂x u) = 0,

x ∈ Ω ≡ (0, 1),

t ∈ (0, T ],

(7.1)

where u is the vector of unknown water and gas saturations, and the expressions for the fractional flow vector f and the capillary diffusion tensor D are given in Equations (2.44)–(2.45) in Chapter 2. It is understood that the system (7.1) is in dimensionless form (see Section 2.4), and that the water and gas saturations have been normalized (see Section 2.5). We shall consider Dirichlet (essential) and Neumann (natural) boundary conditions. Let ∂Ω ≡ {0, 1} be the boundary of the domain, Γu ⊂ ∂Ω is the part of the boundary where essential conditions are imposed, and Γn ≡ ∂Ω \ Γu is the part of the


236

boundary with natural boundary conditions: ¯ on Γu , u=u ¯ (f − D∂x u)n = F

on Γn ,

(7.2) (7.3)

where n is the outward unit normal to the boundary, i.e., n = +1 at x = 1, and n = −1 at x = 0. The initial conditions u(x, t = 0) = u0 (x),

¯ ≡ [0, 1], x∈Ω

(7.4)

close the definition of the mathematical problem. Remarks 7.1. 1. The fractional flow and the diffusion tensor are (nonlinear) functions of the unknown saturations, i.e., f = f (u),

D = D(u).

(7.5)

2. The character of the system (7.1) depends on the eigenvalues and eigenvectors of the Jacobian matrix f 0 . In Juanes and Patzek [2002e,f] (see also Chapters 3 and 4) we argue that this matrix must have real and distinct eigenvalues for the solution to be physically plausible, and we derive conditions on the relative permeability functions so that this requirement is satisfied. Here, we further assume that the capillary diffusion tensor is positive semi-definite. Under these conditions, the system of equations is parabolic, and strictly hyperbolic in the limit of vanishing diffusion [Zauderer, 1983].

7.2.2

Weak form

The development of virtually all integral methods for the numerical solution of the initial and boundary value problem of the previous section starts from the weak

237


form of the mathematical problem. To this end, we define the following functional spaces: ¯ on Γu }, V := {v ∈ W : v = u

(7.6)

V0 := {v ∈ W : v = 0 on Γu },

(7.7)

where the appropriate Sobolev space W depends on the particular form of the diffusion tensor. The weak form of problem (7.1) with boundary and initial conditions (7.2)– (7.4) consists in finding u ∈ V for each fixed t ∈ (0, T ], such that (∂t u, v) + a(u, v; u) = l(v) ∀v ∈ V0 ,

(7.8)

u(x, t = 0) = u0 (x), where (∂t u, v) =

Z

∂t u · v dΩ, Z Z a(u, v; w) = − f (w) · ∂x v dΩ + D(w)∂x u · ∂x v dΩ, Ω ZΩ ¯ · v dΓ. l(v) = − F

(7.9)

Ω

(7.10) (7.11)

Γn

Remarks 7.2. 1. In the context of classical —smooth— solutions, the strong and weak forms of the mathematical problem are equivalent. The weak form is less restrictive, however, in the sense that it may have weak —discontinuous— solutions. 2. The functional spaces V and V0 are infinite-dimensional. 3. Equation (7.8) is linear in the test function v. This fact is exploited at the implementation level, for it allows to consider test functions of the form:      v1   0  (7.12) v =   +  . v2 0


7.2.3

238

Classical Galerkin method

Once the mathematical problem has been stated in weak form, it is straightforward to introduce the classical Galerkin method. Instead of considering the infinitedimensional spaces V and V0 , one employs conforming finite-dimensional spaces Vh ⊂ V and Vh,0 ⊂ V0 of piecewise polynomials, defined on a finite element mesh. The standard Galerkin approximation of (7.8) reduces to find uh ∈ Vh for each fixed t, such that (∂t uh , v h ) + a(uh , v h ; uh ) = l(v h ) ∀v h ∈ Vh,0 ,

(7.13)

and uh (x, t = 0) is the projection of the initial function u0 (x) onto space Vh . The system of ordinary differential equations (7.13) is transformed into a system of (nonlinear) algebraic equations by further discretizing the time derivative [Thomée, 1997].

7.2.4

Multiple-scale approach

It is a well-known fact that the classical Galerkin method lacks stability when diffusive effects are exceedingly small, so that the system of equations is nearly hyperbolic. The objective of the multiple-scale approach described here is to obtain a stabilized numerical formulation for this type of problems. The formulation, based on the framework originally introduced in Hughes [1995] was presented in Juanes and Patzek [2002b,d] for miscible flow of two components and immiscible flow of two phases, which involve scalar conservation laws. The numerical formulation is revisited here, and applied to systems of conservation laws. The key idea of the formulation is a multiscale split of the variable of interest u ∈ V into resolved and unresolved scales: ˜, u = uh + u

(7.14)


239

˜ is the unresolved subgrid scale. This decomwhere uh is the resolved grid scale and u position acknowledges that certain components of the solution cannot be captured by the finite element mesh. This fact is especially relevant for advection-dominated problems, where the solution develops sharp features that would require an impractical grid resolution. Decomposition (7.14) is unique if one can express the original functional space V as the direct sum of two spaces: ˜ V = Vh ⊕ V,

(7.15)

where Vh is the space of resolved scales and V˜ is the space of subgrid scales. The space V˜ is an infinite-dimensional space that completes Vh in V. This space is generally unknown, and it is the role of the subgrid model to provide a successful approximation to it. The multiscale decomposition was originally proposed for the linear advectiondiffusion equation in Hughes [1995] and [Hughes et al., 1998], and then extended to other linear [Codina, 1998, 2000; Hauke, 2002; Hauke and Garc´ıa-Olivares, 2001; Jansen et al., 1999; Masud and Hughes, 2002; Oberai and Pinsky, 1998, 2000] and nonlinear problems [Codina, 2001, 2002; Garikipati and Hughes, 1998, 2000; Hughes et al., 2000; Hughes, Mazzei and Oberai, 2001; Hughes, Oberai and Mazzei, 2001]. A common approach to deal with nonlinear problems is to linearize the equations upfront, using either a Picard or a Newton strategy [Codina, 2001, 2002]. In this work, however, we resort to the multiscale decomposition prior to any linearization. In the context of nonlinear problems, it seems natural to express the solution at a given iteration step (k) as: u(k) = u(k−1) + δu(k−1) .

(7.16)

The first term on the right hand side should be understood as an approximate solution


240

at the previous iteration level, and the second term as a correction. In principle, both terms are subject to the multiscale decomposition (7.14): (k−1)

u(k−1) = uh

˜ (k−1) , +u

(k−1)

δu(k−1) = δuh

˜ (k−1) , + δu

(7.17) (7.18)

Equation (7.17) requires that the approximate solution u(k−1) is reconstructed after every iteration. To avoid this reconstruction step, and obtain a formulation that completely decouples the resolved and unresolved scales (see below), we make the additional approximation: (k−1)

u(k−1) ≈ uh

,

(7.19)

so that the multiscale split takes the form: (k)

˜ (k−1) . u(k) ≈ uh + δ u

(7.20)

In what follows we shall drop superscripts referring to the iteration level, and simply write ˜. u ≈ uh + δ u

(7.21)

Remarks 7.3. 1. We refer to Equation (7.21) as an incremental formulation, with a multiscale decomposition of the increment. 2. The term uh in Equation (7.21) should be understood as an approximate solu˜ plays the role of a tion about which the equations are linearized. The term δ u perturbation that will allow stabilization of the solution. 3. The working assumption (7.19) makes our formulation different from that in Codina [2002] and Garikipati and Hughes [1998, 2000], where the subgrid scales


241

are tracked, and the multiscale variable is reconstructed after every step of the iterative process. The derivation of the multiscale formulation starts by invoking a multiscale split of the solution u and the test function v: ˜ ˜ ∈ V = Vh ⊕ V, u = uh + δ u

(7.22)

˜ ˜ ∈ V0 = Vh,0 ⊕ V. v = vh + v

(7.23)

Because the weak form is linear with respect to the test function v, the original mathematical problem (7.8) is split into two, a grid scale problem: ˜ ), v h ) + a(uh + δ u ˜ , v h ; uh + δ u ˜ ) = l(v h ) ∀v h ∈ Vh,0 , (∂t (uh + δ u

(7.24)

and a subscale problem: ˜ ˜ ), v ˜ ) + a(uh + δ u ˜, v ˜ ; uh + δ u ˜ ) = l(˜ (∂t (uh + δ u v ) ∀˜ v ∈ V.

(7.25)

It is important to note that the former is a finite-dimensional problem, whereas the latter is infinite-dimensional. 7.2.4.1

Subgrid scale problem

Here we derive the final form of the subscale equations, and introduce all the approximations required along the process. We start by writing the flux term as a sum of element integrals, and integrate by parts on each element: ˜, v ˜ ; uh + δ u ˜) a(uh + δ u Z X ¡ ¢ ˜ ) − D(uh + δ u ˜ )∂x (uh + δ u ˜ ) · ∂x v ˜ dΩ f (uh + δ u =− Ωe

e

=

XZ e

−

Ωe

¡ ¢ ˜ ) − D(uh + δ u ˜ )∂x (uh + δ u ˜) · v ˜ dΩ ∂x f (uh + δ u

XZ ¡ e

Γe

¢ ˜ ) − D(uh + δ u ˜ )∂x (uh + δ u ˜) n · v ˜ dΓ. f (uh + δ u

(7.26)


242

We assume continuity of the flux across interelement boundaries, so that the boundary integrals cancel each other on adjacent elements in the interior of the domain, i.e., XZ ¡ ¢ ˜ ) − D(uh + δ u ˜ )∂x (uh + δ u ˜) n · v ˜ dΓ f (uh + δ u − e

≈−

Γe

Z

Γn

(7.27)

¯ ·v ˜ dΓ ≡ l(˜ F v ).

˜ is the exact solution, or if locally The expression above is a true identity if uh + δ u mass conservative finite element spaces are employed. Otherwise, Equation (7.27) should be regarded as an approximation. We now approximate the total flux by a first-order Taylor expansion about the coarse-scale solution uh : ˜ ) − D(uh + δ u ˜ )∂x (uh + δ u ˜) f (uh + δ u = f (uh ) − D(uh )∂x uh

(7.28)

˜ − (D0 (uh )δ u ˜ )∂x uh − D(uh )∂x (δ u ˜ ) + O(|δ u ˜ |2 ). + f 0 (uh )δ u Equation (7.28) suggests defining the linearized advection-diffusion operator in conservation form: Luh v := ∂x [f 0 (uh )v − (D0 (uh )v)∂x uh − D(uh )∂x v] .

(7.29)

The operator Luh v depends in a nonlinear fashion on the approximate coarse-scale solution uh , but is linear in its argument v. We write this operator in the more suggestive (and convenient) form: Luh v := ∂x [A(uh )v − D(uh )∂x v] ,

(7.30)

where A(uh ) is a 2 × 2 “advection” operator, whose components Aij (uh ) take the following expression: Aij (uh ) :=

∂fi (uh ) X ∂Dik (uh ) − ∂x uh,k . ∂uh,j ∂u h,j k

(7.31)

243


Using Equations (7.27)–(7.30) in Equation (7.26), we write the first-order approximation of the flux term in the subgrid scale problem (7.25) as XZ ¡ ¢ ˜ dΩ ˜, v ˜ ; uh + δ u ˜) ≈ ∂x f (uh ) − D(uh )∂x uh · v a(uh + δ u Ωe

e

+

XZ e

Ωe

˜ ·v ˜ dΩ − L uh δ u

Z

Γn

(7.32)

¯ ·v ˜ dΓ F

A further approximation is to consider quasi-static subscales [Codina, 2002], i.e., ˜ ≈ 0. ∂t δ u

(7.33)

After this final assumption, and defining the grid-scale residual as ¡ ¢ R(uh ) := −∂t uh − ∂x f (uh ) − D(uh )∂x uh ,

(7.34)

the subscale problem (7.25) is written as follows: XZ e

Ωe

˜ ·v ˜ dΩ = L uh δ u

XZ e

Ωe

˜ ˜ dΩ ∀˜ R(uh ) · v v ∈ V.

(7.35)

Equation (7.35) illustrates that the subgrid scale problem is in fact a projection problem: ˜ ˜ u δu ˜ ) = Π(R(u Π(L h )), h

(7.36)

˜ is the L2 -projection onto the space of subgrid scales V. ˜ where Π Remarks 7.4. 1. The subgrid scale problem is infinite-dimensional, so one cannot expect to solve it exactly. It is necessary to resort to some kind of numerical or analytical approximation. 2. It is finally written in (7.35) as a sum of volume integrals evaluated elementwise.


244

3. To reduce dramatically the computational cost of the solution to the subscale problem, it seems appealing to localize the problem, so that it can be approximated element-by-element. The difficulty of this step stems from the fact that the boundary conditions of the local problem —values of the subscales on the inter-element boundaries— are unknown. A common modeling assumption is to use v|Γe = 0, i.e., the subscales are bubble functions that vanish on the boundaries of each element [Baiocchi et al., 1993; Brezzi et al., 1997; Franca et al., 1998; Hughes, 1995]. 4. An alternative to assumption (7.33) of quasi-static subscales would be to keep track of the value of the subscales in every element, and evaluate the time ˜ [Codina, 2002]. One of the benefits of the quasi-static subscale derivative ∂t δ u approximation —in addition to the simplicity of the formulation— is that longterm numerical solutions do not depend on the time integration strategy or the actual time step. Here, we employ an algebraic approximation to the subscales, which leads to an algebraic subgrid scale model (ASGS): ˜ ≈ τ uh R(uh ), δu

(7.37)

where τ uh is a 2 × 2 matrix of algebraic coefficients, which depend not only on the system parameters, but also on the grid scale solution uh . This approximation is substantiated by the convergence analysis of the linear case [Codina and Blasco, 2002]. It can also be justified from an asymptotic Fourier analysis [Codina and Blasco, 2002], and has proven useful in numerical tests. The matrix τ uh is known as the matrix of stabilizing coefficients or matrix of intrinsic time scales [Hughes and Mallet, 1986a], and has dimension of time. Its design, which should be ultimately dictated by stability


245

and convergence analysis, is one of the most difficult issues in the development of a stabilized numerical method. Many alternatives have been proposed, some of which are reviewed and succinctly described in Section 7.2.5. 7.2.4.2

Grid scale problem

We now revisit the grid scale equation (7.24). As in the previous section, we linearize the flux term with respect to the coarse scale solution uh : ˜ , v h ; uh + δ u ˜) a(uh + δ u Z ¡ ¢ ˜ ) − D(uh + δ u ˜ )∂x (uh + δ u ˜ ) · ∂x v h dΩ f (uh + δ u =− ZΩ ¡ ¢ =− f (uh ) − D(uh )∂x uh · ∂x v h dΩ ZΩ ¡ 0 ¢ ˜ − (D0 (uh )δ u ˜ )∂x uh · ∂x v h dΩ − f (uh )δ u ZΩ ¡ ¢ ˜ · ∂x v h dΩ + O(|δ u ˜ |2 ). D(uh )∂x δ u +

(7.38)

Ω

The first term in the final expression of (7.38) is the Galerkin term: −

Z

Ω

¡ ¢ f (uh ) − D(uh )∂x uh · ∂x v h dΩ = a(uh , v h ; uh ).

(7.39)

Writing the second integral in (7.38) as a sum of element integrals, and recalling the expression of the linearized “advection” matrix (7.31), we get: −

Z

Ω

¡

¢ ˜ − (D0 (uh )δ u ˜ )∂x uh · ∂x v h dΩ f 0 (uh )δ u XZ = e

Ωe

¡

¢ ˜ dΩ, (7.40) − AT (uh )∂x v h · δ u


246

where AT is the transpose of A, i.e., ATij = Aj i . After integration by parts element by element, the third term in (7.38) is written as Z

¡ ¢ ˜ · ∂x v h dΩ D(uh )∂x δ u Ω XZ XZ T ˜ dΩ + ˜ dΓ. (7.41) =− ∂x (D (uh )∂x v h ) · δ u (DT (uh )∂x v h )n · δ u e

Ωe

Γe

e

Defining the adjoint of the linearized advection-diffusion operator (7.30), L∗uh v := −AT (uh )∂x v − ∂x (DT (uh )∂x v),

(7.42)

and its associated boundary operator, b∗uh v := (DT (uh )∂x v)n,

(7.43)

and substituting (7.39)–(7.41) in Equation (7.38), the flux term of the grid scale equation takes the form: ˜ , v h ; uh + δ u ˜ ) = a(uh , v h ; uh ) a(uh + δ u XZ XZ ∗ ˜ dΓ + O(|δ u ˜ |2 ). (7.44) ˜ dΩ + b∗uh v h · δ u L uh v h · δ u + e

Ωe

e

Γe

Substituting the first-order approximation of (7.44) in (7.24), and considering quasistatic subscales as before, we obtain the final form of the grid scale equation: (∂t uh , v h ) + a(uh , v h ; uh ) XZ XZ ∗ ˜ dΓ = l(v h ) ∀v h ∈ Vh,0 . (7.45) ˜ dΩ + b∗uh v h · δ u L uh v h · δ u + e

Ωe

e

Γe

Remarks 7.5. 1. By direct comparison with (7.13), it is immediate to identify in Equation (7.45) the Galerkin terms and the additional stabilizing terms of the multiscale formulation.


247

2. The stabilizing terms are evaluated element by element, and consist of a volume integral and a boundary integral. The boundary contribution to the stabilizing term is neglected in the numerical simulations of Section 7.3. This simplification is sensible only if the magnitude of the diffusive effects is small, which is precisely the case of interest. 3. The grid scale equation (7.45) and the subgrid scale equation (7.35) are cou˜. pled through the value of the subscales δ u

For the simple subgrid scale

model employed here, the algebraic approximation (7.37) is substituted in Equation (7.45). 4. The subscales are proportional to the grid-scale residual, Equation (7.37). The formulation is residual-based and, therefore, automatically consistent. 5. The novel features of our formulation are the following: (a) Linearization of the equations is employed after the multiscale split. In particular, only the subscale effects are linearized, whereas the full nonlinear Galerkin term is retained in the grid scale equation. (b) The approximate solution is not reconstructed after every step in the iterative process, or even after every time step. The benefit of this working assumption is that subscale effects enter the formulation in an integral sense only.

7.2.5

Matrix of stabilizing coefficients

The description of the multiscale finite element formulation is complete up to the definition of the matrix of stabilizing coefficients τ uh . The design of this matrix is one


248

of the most difficult —and controversial— issues in the development of a stabilized formulation. This modeling step is not specific to the multiscale formulation explained here and, in fact, is shared by other stabilized formulations such as Streamline Upwind Petrov-Galerkin (SUPG) [Brooks and Hughes, 1982] and Galerkin least-squares (GLS) [Hughes et al., 1989]. Here we review briefly several options that have been considered in the literature. They all define the stabilization matrix τ for the linear advection-diffusion problem. Extension to the nonlinear problem is straightforward, after defining the linearized advection-diffusion operator (7.30). 7.2.5.1

Definition through an eigenvalue problem

This formulation was originally developed in Hughes and Mallet [1986a]. The idea is to start from the formulation of the scalar one-dimensional linear advectiondiffusion equation [Brooks and Hughes, 1982], for which it is possible to define a function τ so that the numerical solution is nodally exact, and extend it to systems of equations in multidimensions. The basis for such extension is to diagonalize the system of equations, by solving an eigenvalue problem, and transform the matrix of stabilizing coefficients accordingly. Consider the one-dimensional n×n system of linear advection-diffusion equations: ∂t u + Lu ≡ ∂t u + ∂x (Au − D∂x u) = 0,

(7.46)

where the diffusion matrix is proportional to the identity matrix, D = ²1, and the advection matrix A is assumed to be diagonalizable and have real eigenvalues. Let νi and r i be the eigenvalues and eigenvectors of matrix A, i.e., (A − νi 1)r i = 0,

i = 1, . . . , n.

(7.47)


249

We further define the n × n matrices of eigenvalues and right eigenvectors: Λ := diag(ν1 , . . . , νn ),

R := [r 1 , . . . , r n ].

(7.48)

The following identities are used throughout: Λ ≡ RT AR,

RT R ≡ RRT ≡ 1.

(7.49)

The change of variables u = Rˆ u

(7.50)

û ≡ ∂t u ˆ + Lˆ ˆ + ∂x (Λˆ ˆ ) = 0. ∂t u u − ² ∂x u

(7.51)

can be used to diagonalize (7.46):

The system (7.51) is uncoupled, and the i-th equation is a scalar linear advectiondiffusion equation: ∂t uî + ∂x (νi uî − ² ∂x uî ) = 0.

(7.52)

The goal is to define a stabilization matrix τ that treats each one of the scalar equations (7.52) in an optimal fashion. Following Codina [2000], the grid scale equation of the linear problem (7.46) can be written, after neglecting the contribution from the interelement boundary integrals, in the following convenient form (subscript h referring to the coarse scale has been omitted): Z

Ω

∂t u · v dΩ +

Z

Ω

Lu · v dΩ +

Z

Ω

L∗ v · τ Ru dΩ = 0,

(7.53)

where the adjoint operator and the grid scale residual are, respectively, L∗ v = −AT ∂x v − ∂x (² ∂x v),

(7.54)

Ru = −∂t u − Lu.

(7.55)


250

It is understood that the second term in (7.53) is integrated by parts, and that the third integral —the stabilizing term— is evaluated element by element. Under the change of variables (7.50), the Galerkin contribution to Equation (7.53) is: Z Z Z Z û · v dΩ ˆ · v dΩ + ∂t u · v dΩ + Lu · v dΩ = R∂t u RLˆ Ω Ω Ω Ω Z Z T û · RT v dΩ. ˆ · R v dΩ + Lˆ = ∂t u Ω

(7.56)

Ω

Therefore, the Galerkin contribution behaves correctly under the change of variˆ = RT v or, equivalently, ables (7.50) if the test function transforms as v v = Rˆ v.

(7.57)

We now analyze the stabilizing term of the multiscale formulation. It is not difficult to show that, under the change of variables (7.50) and (7.57), ˆ, L∗ v = R Lˆ∗ v

(7.58)

ˆ u. Ru = R Rˆ

(7.59)

Using (7.58)–(7.59), the stabilizing term in Equation (7.53) is: Z Z T ∗ ˆ u dΩ. ˆ·R L v · τ Ru dΩ = Lˆ∗ v τ R} Rˆ | {z Ω Ω =: τˆ

(7.60)

We conclude that the proper definition of the matrix of stabilizing coefficients τ for the ASGS formulation of problem (7.46) is τ = Rˆ τ RT ,

(7.61)

τˆ = diag(ˆ τ1 , . . . , τˆn ),

(7.62)

where

and each intrinsic time τî is defined for the corresponding scalar equation of the uncoupled system (7.51). The optimal definition of the intrinsic time when linear


251

finite elements are used, which is nodally exact for the steady-state advection-diffusion equation [Brooks and Hughes, 1982], takes the form: ˆ i) 1 ξ(α , τî = h 2 |νi |

(7.63)

where h is the size of the element, αi is a measure of the element Peclet number for the i-th equation, αi =

1 |νi |h , 2 ²

(7.64)

and ξˆ is a diffusion correction factor given by: 1 ˆ ξ(α) = coth(α) − . α

(7.65)

Remarks 7.6. 1. When the system (7.46) has a diffusion matrix D that is not proportional to the identity, it is not possible, in general, to diagonalize the system. In this case, the pseudo-diagonal system after the change of variables (7.50) reads: ˆ + ∂x (Λˆ ˆ ) = 0. ∂t u u − RT DR ∂x u

(7.66)

The equation above suggests using a different diffusion coefficient ²i for each componential Peclet number in Equation (7.64), given by: ²i = r Ti Dr i .

(7.67)

2. Equation (7.65) provides the optimal diffusion-correction factor, in the sense that the numerical solution to the steady-state diagonalizable advection-diffusion system is nodally exact. However, alternative definitions are possible, which dif-

252

ˆ Diffusion correction factor ξ(α)


1 0.8 0.6 0.4

ξˆ0 ξˆ1 ξˆ2 ξˆ4 ξôpt

0.2 0 0

10

5

15

20

Element Peclet number α

Figure 7.1. Comparison of alternative definitions of the diffusion correction factor ξˆ —Equations (7.65) and (7.68)–(7.71)— as a function of the element Peclet number α (after Shakib et al. [1991]). fer in their order of accuracy [Shakib et al., 1991]: ξˆ0 (α) = 1,

(7.68)

ξˆ1 (α) = min(α/3, 1), p ξˆ2 (α) = α2 /(1 + α2 ), p ξˆ4 (α) = α2 /(9 + α2 ).

(7.69) (7.70) (7.71)

A graphical comparison of the different definitions of the diffusion correction factor ξˆ as a function of the element Peclet number α is shown in Figure 7.1. 3. The formulation of the matrix τ described here applies also to the nonlinear advection-diffusion operator (7.30). In this case, the advection and diffusion matrices are not constant, but functions of the solution itself. Therefore, the matrix of stabilizing coefficients depends not only on the system parameters but also on the grid scale solution uh : τ uh = Ruh τˆ uh RTuh ,

(7.72)


253

where Ruh = [r 1 (uh ), . . . , r n (uh )].

(7.73)

We denote νi (uh ), r i (uh ), the eigenvalues and eigenvectors of the advection matrix A(uh ), and τˆ uh = diag(ˆ τ1 (uh ), . . . , τˆn (uh )),

(7.74)

where ˆ i (uh )) 1 ξ(α , τî (uh ) = h 2 |νi (uh )| 1 |νi (uh )|h αi (uh ) = , 2 ²i (uh ) ²i (uh ) = r i (uh )T D(uh )r i (uh ),

(7.75) (7.76) (7.77)

ˆ and ξ(α) is the diffusion correction factor given by Equation (7.65). Since the advection and diffusion matrices now change from point to point, one needs to solve an eigenvalue problem at each integration point. 7.2.5.2

Definition through the matrix square root

A different formulation of the matrix of stabilizing coefficients was presented in Shakib et al. [1991], in the context of the Galerkin least-squares (GLS) method. The definition of τ proposed in Shakib et al. [1991] is very general and will not be presented here. It suffices to say that it is derived in the framework of a space-time formulation of unsteady multidimensional problems with a Riemannian metric, and includes advection, diffusion, and reaction matrices. When particularized to stationary one-dimensional systems of advection-diffusion type, discretized with linear finite elements, the matrix of stabilizing coefficients takes the form: µ³ ´2 ³ c ´2 ¶−1/2 c1 2 D(uh ) + A(uh ) , τ uh = h2 h

(7.78)


254

where c1 = 4 and c2 = 2. Remarks 7.7. 1. The definition of τ given by (7.78) involves the square-root inverse, which can be computed by exploiting the Cayley-Hamilton theorem (see Marsden and Hughes [1983]), by solving an eigenvalue problem, or iteratively by resorting to some kind of Newton’s method [Shakib et al., 1991]. 2. The matrix τ defined in this way is obviously symmetric. 3. Expression (7.78) has also been justified in Codina and Blasco [2002] by means of an asymptotic Fourier analysis. 7.2.5.3

Definition through the matrix inverse

A different design of the matrix of intrinsic time scales was proposed in Codina [2000] for systems of advection-diffusion-reaction equations in multidimensions. When restricted to one-dimensional systems of advection-diffusion type, the expression of τ uh reduces to: τ uh =

³c

1 D(uh ) h2

+

´−1 c2 A(uh ) , h

(7.79)

where c1 = 4 and c2 = 2 for linear elements. This expression emanates from an analysis of the discrete maximum principle in the scalar, stationary, one-dimensional case [Codina, 1998, 2000]. Remarks 7.8. 1. The matrix τ given by (7.79) is a matrix function of the advection and diffusion matrices, which is the essential requirement for the method to provide


255

optimal stabilization for each individual scalar equation when the system is diagonalized [Codina, 2000]. 2. Expression (7.79) may be viewed as an asymptotic approximation of the previous definition of τ —Equation (7.78)— in the limit of vanishing diffusion.

7.2.6


While the multiscale formulation described above will produce stabilized numerical solutions to the three-phase flow equations (see Section 7.3), overshoots and undershoots may still remain in the neighborhood of internal and boundary layers. The reason for this localized oscillatory behavior is that the method does not guarantee monotonic solutions. One possibility to enhance the robustness of the stabilized formulation is to incorporate a discontinuity-capturing term, that will further reduce or completely eliminate spurious numerical oscillations. The basic idea of discontinuity-capturing techniques is to introduce an additional term in the grid scale equation (7.45), which is also evaluated elementwise, and satisfies the following generic design conditions [Hughes and Mallet, 1986b; Shakib et al., 1991]: 1. Consistency, which implies that the operator has to be proportional to the grid scale residual. 2. Enhanced stability, by providing extra control over the gradient of the numerical solution. 3. Accuracy, by vanishing quickly in the regions where the solution is smooth. Many of the existing discontinuity-capturing formulations can be expressed as an

Chapter 7. Multiscale finite elements for three-phase flow extra diffusion term [Codina, 1993], XZ Dsc (uh )∂x uh · ∂x v h dΩ e

256

(7.80)

Ωe

where the numerical diffusion tensor Dsc depends on the coarse scale solution. This term leads necessarily to a nonlinear method, even if the underlying equation is linear. 7.2.6.1

Classical discontinuity-capturing diffusion

It is not the purpose of this section to derive existing discontinuity-capturing formulations, and the reader is referred to the vast literature on the topic [Codina, 1993; Dutra do Carmo and Galeão, 1991; Galeão and Dutra do Carmo, 1988; Hughes and Mallet, 1986b; Hughes, Mallet and Mizukami, 1986; Johnson, 1987, 1992; Johnson and Szepessy, 1987; Johnson et al., 1990; Shakib et al., 1991; Tezduyar and Park, 1986]. Here we present four different expressions of the shock-capturing diffusion, which are inspired in the original references, and adapted to quasi-static one-dimensional problems of advection-diffusion type, discretized with linear finite elements. 1. A form of the shock-capturing diffusion based on Galeão and Dutra do Carmo [1988] is regarded in Codina [1993] as “canonical form”: 1 |R(uh )| 1. Dsc,1 = h 2 |∂x uh |

(7.81)

2. The expression above may result in numerical solutions that are a bit too diffusive. Based on an idea of Hughes, Mallet and Mizukami [1986] and Hughes and Mallet [1986b], the expression above can be modified as follows: Ã ! 2 1 |R(uh )| |R(uh )|τuh Dsc,2 = max h − , 0 1, 2 |∂x uh | |∂x uh |2

(7.82)

where the τ -norm is defined as |ϕ|2τ := ϕ · τ ϕ.

(7.83)


257

3. A similar expression to (7.81) above is proposed in Shakib et al. [1991] (termed “linear form”), and it is based on an extension of the formulation in Hughes and Mallet [1986b]. A restricted version of it reads: 1 |R(uh )|τuh 1. Dsc,3 = h 2 |∂x uh |τuh

(7.84)

4. Shakib et al. [1991] consider also another expression (termed “quadratic form”), which has features in common with the formulation in Galeão and Dutra do Carmo [1988]. When reduced to our one-dimensional model problem, it takes the form: Dsc,4 = 2

|R(uh )|2τu |∂x uh |2

h

1.

(7.85)

All the formulations of the discontinuity-capturing diffusion presented above share several properties, such as being residual-based, dimensionally consistent, and isotropic. We do not discard the possibility that using an anisotropic diffusion tensor formulation would be significantly more effective. This was precisely the conclusion in Codina [1993] for the scalar advection-diffusion equation in several space dimensions. 7.2.6.2

Novel discontinuity-capturing diffusion

The motivation for looking into alternatives to the classical formulations of shockcapturing diffusion is that the amount of numerical dissipation introduced by these methods is not sufficiently localized to the neighborhood of shocks and boundary layers. It is easy to understand the reason for this deficiency, by considering the “canonical form” (7.81) applied to a linear, quasi-steady, advection-dominated, scalar equation. In this case, the grid scale residual is R(uh ) ≈ −A∂x uh and, therefore, the shock-capturing diffusion is 1 | − A∂x uh | 1 Dsc,1 ≈ h = hA, 2 |∂x uh | 2

(7.86)


258

which is constant. This means that the formulation introduces the same amount of artificial diffusion everywhere, even though it is only required in the vicinity of sharp gradients [Juanes and Patzek, 2002b]. To remedy this undesirable behavior, we propose a discontinuity-capturing diffusion that introduces an essential difference with respect to the classical formulations of the previous paragraph: the local gradient ∂x uh is replaced by a global measure of the gradient ∼ U sc /h. In particular, we shall test the following expression: Dsc,g = Csc h

|R(uh )| 1, |U sc /h|

(7.87)

where Csc is a constant coefficient. The simulations of the next section clearly show that this formulation introduces numerical diffusion in much narrower regions of the computational domain than classical formulations.

7.3

Representative numerical simulations

In this section we present several simulations of one-dimensional three-phase flow in porous media, as described by the mathematical model of Chapter 2. For the sole purpose of testing the formulation, the capillary diffusion tensor is taken as a constant isotropic matrix, that is, we use: 



 ²w 0  D= . 0 ²g

(7.88)

As we shall see, the form of the capillary diffusion tensor may affect the detailed structure of individual shocks, but not the shock location and the global structure of the solution. The practical importance of this dependency on the form of the diffusion tensor is minimized by the fact that, because we are interested in the nearly


259

hyperbolic case, which is the most challenging to model numerically, we shall use very small values of the capillary diffusion coefficients ²w , ²g . The following relative permeability functions are used: krw = Sw2 , kro = (1 − Sw )(1 − Sg )(1 − Sw − Sg ),

(7.89)

krg = βg Sg + (1 − βg )Sg2 . These functions belong to a simple class of functions, where the water and gas relative permeabilities depend only on their own saturation, and the oil relative permeability depends on both [Stone, 1970]. The parameter βg is the endpoint-slope of the gas relative permeability function. The relevance of this parameter in the context of classical relative permeability models is discussed in Juanes and Patzek [2002e,f]. In the simulations that follow we use the value βg = 0.1. Finally, the following values of the fluid viscosity ratios are used: µg = 0.015. µo

µw = 0.4375, µo

(7.90)

The simulations presented here reproduce the conditions of the Riemann problem, which is an initial value problem on an unbounded domain defined by the system of conservation laws (7.1), together with piecewise constant initial data separated by a single discontinuity: u(x, 0) =

   u

l

  u r

if x < 0,

(7.91)

if x > 0.

We model conditions in Equation (7.91) numerically by imposing the initial condition u(x, 0) = ur on a bounded domain 0 < x < 1, and a Dirichlet boundary condition u(0, t) = ul on the left boundary. The interest in the Riemann problem is threefold. On one hand, it is particularly challenging to model numerically, since the initial

260


conditions are already discontinuous. Secondly, an analytical solution exists for the capillarity-free case, which can be used to verify the numerical solutions. Finally, it is very valuable in practical applications, because many laboratory and field experiments reproduce in fact the conditions of the Riemann problem. The general analytical solution to the Riemann problem of capillarity-free threephase flow is given in Chapter 5 of this dissertation (see also Juanes and Patzek [2002a,g]). The system of conservation laws describing three-phase flow is a 2 × 2 system, which is strictly hyperbolic for all saturation paths of interest [Juanes and Patzek, 2002e]. This implies that there are two separated waves connecting three constant states: ul (left), um (middle), and ur (right). Therefore, the solution to the Riemann problem of three-phase flow reduces to finding the intermediate constant state um as the intersection of an admissible 1-wave W1 (slow wave) and an admissible 2-wave W2 (fast wave) on the saturation triangle (see Figure 5.12 in Chapter 5): W

W

1 2 ul −→ um −→ ur .

(7.92)

Based on the analysis of the wave structure in Juanes and Patzek [2002a], a wave of the i-family connecting two constant states may only be one of the following: an i-rarefaction (Ri ), an i-shock (Si ), or an i-rarefaction-shock (Ri Si ). Since the full solution to the Riemann problem is a sequence of two waves, W1 and W2 , there are only 9 possible combinations of solutions. A schematic tree with all possible solution types is shown in Figure 5.17 in Chapter 5. Here we study two scenarios: the first one involving oil filtration in a relatively dry soil, which results in a S1 S2 solution, and the second one reproducing water-gas injection in an oil reservoir, whose solution is of type R1 S1 S2 . Both simulations are transient over a period of time, when the solution displays propagating discontinuities, and then reach quasi-steady conditions —boundary layers are present at the


261

outlet face. For each of the two problems studied, we compare the exact solution of the capillarity-free model with the numerical solution obtained using the standard Galerkin method on a very fine mesh. Then we compare the performance —on a very coarse mesh— of the classical Galerkin method with the algebraic subgrid scale method, using different formulations for the matrix of stabilizing coefficients. Different discontinuity-capturing formulations are also employed and contrasted. The comparison of stabilized formulations with the standard Galerkin method may seem a little unfair, as the test cases involve nearly hyperbolic systems, for which the classical Galerkin method is known to have unstable behavior. The motivation is to show the stabilizing effect of the new terms in the ASGS formulation, which arise from consideration of the subgrid scales. It is interesting to note that: 1. The ASGS method is in fact a Galerkin method —the coarse-scale trial and test functions belong to the same finite element space. The difference with respect to the classical Galerkin method is that the subgrid scales are modeled separately and incorporated to the coarse scale problem. 2. The computational cost of the ASGS method is essentially the same as that of the standard Galerkin method, as the former involves the calculation of just a few additional integrals, which are evaluated elementwise.

7.3.1

Oil filtration in relatively dry soil

7.3.1.1


This example reproduces filtration of a mixture of oil, water and gas through a relatively dry porous medium with some water and oil, as shown in Figure 7.2. The medium has the following initial normalized saturations: Sw = 0.15, Sg = 0.8,

262

Chapter 7. Multiscale finite elements for three-phase flow Initial saturations


Sw = 0.15

So = 0.55

So = 0.05

Sg = 0.20

Sg = 0.80

Figure 7.2. Sketch of the oil filtration problem. A mixture with high oil saturation is injected into a medium initially filled with water and gas. and So = 0.05. Fluids are injected in a proportion such that the normalized fluid saturations at the inlet face are: Sw = 0.25, Sg = 0.2, and So = 0.55. Initial saturations are homogeneous on the entire medium, and injected saturations are held constant throughout the experiment, so that the example reproduces the conditions of the Riemann problem. From a practical viewpoint, this problem could represent a contamination event in the shallow subsurface, under one-dimensional flow conditions. 7.3.1.2

Analytical solution

The exact solution to the strictly hyperbolic system of the capillarity-free problem is of type S1 S2 , that is, it consists in a sequence of two shocks. Schematically, we denote the structure of the solution as follows: S

S

1 2 ul −→ um −→ ur .

(7.93)

The left state ul = (Sw,l , Sg,l ) = (0.25, 0.2) corresponds to the injected saturations, and the right state ur = (Sw,r , Sg,r ) = (0.15, 0.8) to the initial saturations. These two states are separated by an intermediate constant state um . The analytical solution may be understood as a slow shock connecting the left and intermediate states, and a fast shock joining the intermediate and right states. Thus, the description of the analytical solution only requires finding the intermediate constant state u m and the

263

Chapter 7. Multiscale finite elements for three-phase flow G

PSfrag replacements

0

1

0 .2

ur

0.8

0 .4

0.6

0 .6

S2 0.4

0.8

S1

ul

um

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0

1

O

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Figure 7.3. Saturation path of the exact solution to the oil filtration problem. Both waves are shocks (S1 S2 solution). speed of propagation σ1 and σ2 of the slow and fast shocks, respectively. For the particular data used here, they take the following values: um = (0.615, 0.328),

σ1 = 0.156,

σ2 = 0.193.

(7.94)

A general and efficient procedure to compute the analytical solution is presented in Juanes and Patzek [2002a]. The solution —in saturation space— is shown in Figure 7.3. It is important to note that dashed lines correspond to the shock curves —set of saturation states that satisfy the Rankine-Hugoniot condition and the Lax entropy condition— and represent discontinuities in the solution. Therefore, the actual path of the shock curves on the saturation triangle is inconsequential from the point of view of the saturation profiles, and what matters is the location of the endpoints of each shock curve. In Figure 7.4 we display in a single plot the profiles of water, gas, and oil saturations against the similarity variable ζ = x/t. The solution at different times can be

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saturations

0.8 0.6

gas oil

0.4 0.2 0 0

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0.2

0.3

ζ = x/t Figure 7.4. Saturation profiles of the exact solution to the oil filtration problem. Saturations of each phase are plotted against the similarity variable ζ = x/t. obtained from one another by simple stretching. Obviously, the solution satisfies the constraint that the fluids fill up the entire pore space, which means that the sum of all three saturations adds up to one at all points. The saturations at the right boundary coincide with the initial state, and the saturations at the left boundary correspond to the injected state. This figure clearly illustrates the behavior of the displacement process: basically, the oil phase displaces the water phase, which in turn displaces gas out of the porous medium. One of the key features of the solution is the formation of a water bank —a region where the water saturation is higher than that of the initial and injected states— so that the solution is not monotonic in the traditional sense. It is also interesting to note that the slow shock involves changes in all three saturations, whereas the fast shock connects states with approximately the same oil saturation (see also Figure 7.3). 7.3.1.3

Comparison of numerical solutions

We test whether the numerical solution to the three-phase oil filtration problem with capillarity provides an accurate approximation to the analytical solution of the capillarity-free case above. Since we are interested in the nearly hyperbolic case, we


265

take small values of the capillary diffusion coefficients in Equation (7.88): ²w = 0.0005,

²g = 0.001.

(7.95)

We compute a “reference” numerical solution using the standard Galerkin method on a very fine mesh of 4000 elements. We use a Crank-Nicolson time integration technique with a constant time step of δt = 10−4 . Given this discretization and the physical parameters of the problem —in particular the speed of propagation σmax of the fast shock— we may define the following dimensionless parameters: σmax h ≈ 0.1 ²min σmax δt Co := ≈ 0.08 h P e :=

(element Peclet number),

(7.96)

(element Courant number).

(7.97)

The space and time discretization have been chosen to obtain small values of these two key parameters (P e ¿ 1, Co ¿ 1), so that the reference solution given by the classical Galerkin method is stable and accurate. The comparison between this solution and the analytical solution described above is presented in Figure 7.5 at time t = 3. The “reference” numerical solution captures correctly the global structure of the capillarity-free solution: the location of shocks and the magnitude of the intermediate constant state are predicted accurately. Further numerical simulations —using different values of the capillary diffusion coefficients and different number of elements— confirm that the standard Galerkin solution converges to the entropy solution of the capillarity-free problem. This essential property of the numerical solution, which stems from the fact that the conservation form of the equations is used [LeVeque, 1992], is in contrast to the conclusions in Todd et al. [1972], and Aziz and Settari [1979], for two-phase flow, where the nonconservation form of the equation was discretized.

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Figure 7.5. Saturation profiles of the standard Galerkin solution to the oil filtration problem on a fine mesh of 4000 elements, and comparison to the analytical solution of the capillarity-free case. Results are shown at time t = 3.

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Figure 7.6. Comparison of the saturation path obtained by the standard Galerkin method on a fine mesh of 4000 elements (dotted curve) and the exact solution (solid curve) of the oil filtration problem.


267

An interesting behavior of the numerical solution is illustrated when the saturation path is plotted on the ternary diagram (see Figure 7.6), where an apparent discrepancy between the numerical and the analytical solutions is readily observed. This “discrepancy” is restricted to the local structure of the fast shock S 2 and is, therefore, inconsequential. The detailed structure of the solution around the shock depends in an essential manner on the form of the diffusion tensor, which is ignored in the capillarity-free model and replaced by a discontinuous solution. Standard Galerkin solution. The same problem is solved using the standard Galerkin method on a coarse mesh of only 40 elements. The element Peclet number is now P e ≈ 10. A Crank-Nicolson time-stepping with δt = 0.01 is used. The associated Courant number is still very small (Co ≈ 0.08), to minimize the numerical error introduced by the time discretization. The results of this simulation are shown in Figure 7.7. The solution obtained with the classical Galerkin method on a fine mesh of 4000 elements is included for reference. Water and gas saturation profiles are plotted at two different times: t = 3 (transient conditions), and t = 8 (quasi-steady conditions). It is apparent that the standard Galerkin solution on a coarse grid lacks stability, and is polluted with spurious oscillations. The instabilities are especially severe for the long-term solution, where the oscillatory behavior spreads over most of the computational domain. Algebraic subgrid scale solutions. We present now the numerical solution to the oil filtration problem obtained with the algebraic subgrid scale (ASGS) method. We recall that the ASGS formulation —Equation (7.45)— differs from the classical Galerkin method —Equation (7.13)— in the addition of a stabilizing term, evaluated element by element. This stabilizing term involves the subgrid scales, which are mod-

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Figure 7.7. Saturation profiles of the standard Galerkin solution to the oil filtration problem on a coarse mesh of 40 elements. Results are shown at times t = 3 and t = 8. eled analytically using an algebraic approximation to the subscales —Equation (7.37). Different alternatives for the definition of the matrix τ of stabilizing coefficients were discussed in Section 7.2.5. In Figure 7.8, we plot the results obtained with the ASGS method and the definition of τ given by the eigenvalue problem (7.72) (formulation proposed by Hughes and Mallet [1986a]). The solution is much stabler than the standard Galerkin solution. The computed saturation profiles do not display global oscillatory behavior, and capture sharply the transient shocks and the stationary boundary layers. Some small overshoots and undershoots remain, however, but they are confined to the vicinity of the sharp features in the solution. The ASGS solution obtained with the τ matrix given by the matrix inverse (7.79) (formulation proposed by Codina [2000]) is shown in Figure 7.9. The solution is virtually identical to that of Figure 7.8, and the same comments apply. Stabilized solutions with shock capturing diffusion. In an attempt to reduce, or eliminate completely, the localized wiggles that remain in the solution of the stabilized ASGS method —Figures 7.8 and 7.9 above— we test several shock-capturing

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Sw Figure 7.8. Saturation profiles of the ASGS solution (τ formulation given by Hughes g andSMallet [1986a]) to the oil filtration problem on the coarse mesh. 1

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Figure 7.9. Saturation profiles of the ASGS solution (τ formulation given by Codina [2000]) to the oil filtration problem on the coarse mesh. techniques, as described in Section 7.2.6. We compare different expressions of the discontinuity-capturing diffusion applied to the same ASGS method. In this case, we choose the solution obtained with the τ matrix of Hughes and Mallet [1986a] —Figure 7.8. In Figure 7.10 we plot the results for the “canonical form” of the shock-capturing diffusion —Equation (7.81). It is found that this formulation is effective at eliminating the oscillatory behavior (compare with Figure 7.8), but at the cost of being a bit too diffusive.

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Figure 7.10. Saturation profiles of the ASGS solution to the oil filtration problem. Formulation of τ given by Hughes and Mallet [1986a]. Shock-capturing diffusion in “canonical form” —Equation (7.81). PSfrag replacements PSfrag replacements

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Figure 7.11. Saturation profiles of the ASGS solution to the oil filtration problem. Formulation of τ given by Hughes and Mallet [1986a]. Shock-capturing diffusion in “quadratic form” —Equation (7.82).

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Figure 7.12. Saturation profiles of the ASGS solution to the oil filtration problem. Formulation of τ given by Hughes and Mallet [1986a]. Proposed formulation of shockcapturing diffusion —Equation (7.87). The “quadratic form” of the discontinuity-capturing diffusion, given by Equation (7.82), yields the results shown in Figure 7.11. A single, small overshoot remains in the numerical solution: (1) at the downstream end of the fast shock during the transient phase; and (2) at the lip of the boundary layer for stationary conditions. The solution is significantly less diffusive than that of Figure 7.10. In Figure 7.12 we plot the numerical solution obtained when the novel “globalgradient form” of the discontinuity-capturing diffusion is employed —Equation (7.87)— with the following values of the parameters: U sc = (0.5, 0.5),

Csc = 2.

(7.98)

The method is able to remove the localized oscillatory behavior of the ASGS solution but is, for the parameters used, slightly too diffusive. The reason for considering the novel expression of the shock-capturing diffusion as a viable alternative to existing formulations stems from the behavior of the numerical diffusion that is actually added by each method. In Figures 7.13, 7.14, and 7.15 we plot the profile of additional diffusion introduced by the “canonical

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Figure 7.13. Profiles of shock capturing diffusion introduced by the “canonical form” —Equation (7.81)— at two different times.

Dsc

0.01

t=8

t=3 0 0

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x

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Figure 7.14. Profiles of shock capturing diffusion introduced by the “quadratic form” —Equation (7.82)— at two different times.

Dsc

0.01

t=3

0 0

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x

t=8

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1

Figure 7.15. Profiles of shock capturing diffusion introduced by the proposed formulation —Equation (7.87)— at two different times.


273

form”, the “quadratic form”, and the “global-gradient form”, respectively, at two different simulation times. The key observation is that, while the existing formulations add a significant amount of diffusion almost everywhere, the proposed formulation automatically introduces numerical dissipation only in the neighborhood of the sharp features of the solution. The latter is precisely the desired behavior of a discontinuitycapturing mechanism. Direct comparison of Figures 7.13 and 7.14 also explains, at a glance, why the “linear form” is significantly more diffusive than the “quadratic form”.

7.3.2

Water-gas injection in a reservoir

7.3.2.1


This second application involves simultaneous injection of water and gas into a porous medium that is initially filled with oil and gas (and a small amount of water), as shown in Figure 7.16. Initially, the medium has constant normalized saturations: Sw = 0.05, Sg = 0.4, and So = 0.55. Gas and water are injected in such proportion that the normalized water and gas saturations at the inlet are S w = 0.85 and Sg = 0.15, respectively. The injected saturations are assumed to be constant throughout the experiment. The values of initial and injected saturations used in this example are representative of a linear water-alternating-gas (WAG) injection process in a hydrocarbon reservoir after primary production [Christensen et al., 2001; Marchesin and Plohr, 2001]. 7.3.2.2

Analytical solution

The analytical solution to the water-gas injection problem described above is of type R1 S1 S2 , that is, the 1-wave is a rarefaction-shock and the 2-wave is a single

274

Chapter 7. Multiscale finite elements for three-phase flow Initial saturations


Sw = 0.05

So = 0

So = 0.55

Sg = 0.15

Sg = 0.40

Figure 7.16. Sketch of the water-gas injection problem. Water and gas are injected into a medium initially filled with oil and gas. shock. Schematically, we write the solution structure as R

S

S

1 1 2 ur , um −→ ul −→ u∗1 −→

(7.99)

where ul , um and ur have the same meaning as in the previous example, and u∗1 is the saturation state at which the 1-rarefaction and the 1-shock are joined —also known as the post-shock value. The variables that need to be determined to characterize fully the solution are: the intermediate constant state um , the shock speeds σ1 and σ2 , and the post-shock state u∗1 . In our particular case, u∗1 = (0.478, 0.083), um = (0.052, 0.085), σ1 = 0.712, σ2 = 1.280.

(7.100)

In Figure 7.17 we plot the analytical solution in saturation space. The solid line is the rarefaction curve —where the solution is continuous— and the dashed lines are the shock curves, which correspond to discontinuities. The fluid saturation profiles of the analytical solution to the water-gas injection problem are shown in Figure 7.18. Because the capillarity-free solution is selfsimilar, the profiles are plotted against the similarity variable ζ = x/t. The most relevant features of the solution are: 1. Oil and gas are produced by a sequence of two waves. The slow wave involves mainly displacement of oil by injected water, and the fast wave corresponds to a displacement of gas by oil.

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O

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Figure 7.17. Saturation path of the exact solution to the water-gas injection problem. The 1-wave is a rarefaction-shock and the 2-wave is a shock (R1 S1 S2 solution). 1

gas

saturations

0.8 0.6 0.4 0.2 0 0

oil water 0.5

1

1.5

2

ζ = x/t Figure 7.18. Saturation profiles of the exact solution to the water-gas injection problem. Saturations of each phase are plotted against the similarity variable ζ = x/t. 2. An oil bank —intermediate state with higher oil saturations than those of the initial and injected states— is formed. This is characteristic of water flood processes in the presence of free gas [Kyte et al., 1956; Willhite, 1986].

Chapter 7. Multiscale finite elements for three-phase flow 7.3.2.3

276

Comparison of numerical solutions

We compute a “reference” numerical solution to the water-gas injection problem with small capillary diffusion coefficients: ²w = 0.001,

²g = 0.002.

(7.101)

We use the standard Galerkin formulation on a very fine mesh of 4000 elements (h = 2.5 × 10−4 ), and a Crank-Nicolson time integration scheme with δt = 5 × 10−5 . For this space and time discretization, the element Peclet and Courant numbers are, respectively: σmax h ≈ 0.3, ²min σmax δt Co := ≈ 0.25. h P e :=

(7.102) (7.103)

In Figure 7.19 we plot the water and gas saturation profiles of the “reference” numerical solution at t = 0.5, together with the capillarity-free analytical solution. The numerical solution correctly captures the location and magnitude of the shocks, and provides an accurate representation of the rarefaction fan. As was the case in the oil filtration example, additional simulations with different space and time discretizations, and different capillary diffusion coefficients, confirm convergence of the standard Galerkin method to the entropy solution of the problem. The saturation path of the reference numerical solution is shown in Figure 7.20. In this case, the saturation path agrees very well with that of the capillarity-free solution, not only along the rarefaction, but also along shocks. The close matching along shock curves is not particularly relevant. It has to do with the fact that the shocks join states with the similar water or gas saturation, and that an isotropic capillary diffusion tensor is used.

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Figure 7.19. Saturation profiles of the standard Galerkin solution to the water-gas injection problem on a fine mesh of 4000 elements, and comparison to the analytical solution of the capillarity-free case. Results are shown at time t = 0.5.

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Figure 7.20. Comparison of the saturation path obtained by the standard Galerkin method on a fine mesh of 4000 elements (dotted curve) and the exact solution (solid curve) of the water-gas injection problem.

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Chapter Sg 7. Multiscale finite elements for three-phase flow 1 0.8

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Figure 7.21. Saturation profiles of the standard Galerkin solution to the water-gas injection problem on a coarse mesh of 40 elements. Results are shown at times t = 0.5 and t = 2. Standard Galerkin solutions. The water-gas injection problem is solved with the same physical parameters on a much coarser mesh of 40 elements and a time step δt = 0.005. The element Peclet number is now P e ≈ 30, and the element Courant number remains Co ≈ 0.25. The results are shown in Figure 7.21 at two different simulation times (t = 0.5 and t = 2), and compared with the reference numerical solution. Clearly, the standard Galerkin solution on the coarse mesh is unstable. The Galerkin solution is completely oscillatory, especially after the process reaches a quasi-steady state. Algebraic subgrid scale solutions. The numerical solution produced by the ASGS method with the τ formulation of Hughes and Mallet [1986a] is shown in Figure 7.22. The behavior of the method is remarkable, considering that a very coarse mesh of only 40 elements was used. The stabilizing term is able to remove the global oscillatory behavior of the standard Galerkin method. The solution is also extremely accurate and preserves a sharp definition of the shocks and boundary layers.

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Figure 7.22. Saturation profiles of the ASGS solution (τ formulation given by Hughes and Mallet [1986a]) to the water-gas injection problem on the coarse mesh. It should be noted, however, that other formulations of the matrix τ of stabilizing coefficients —such as those proposed in Shakib et al. [1991], or Codina [2000]— do not yield the impressive results of Figure 7.22. In some cases they even fail to converge, emphasizing the importance of an appropriate choice of τ for each particular problem. Stabilized solutions with shock capturing diffusion. Despite the effective stabilization of the ASGS method with the matrix of stabilizing coefficients given by Hughes and Mallet [1986a], some local overshooting is still present in the solution (see Figure 7.22). We make use of a discontinuity-capturing technique to remove the spurious wiggles. In Figure 7.23 we plot the numerical solution obtained after using the ASGS method above in conjunction with the proposed “global-gradient form” of the shock-capturing diffusion —Equation (7.87)— with the following parameters: U sc = (0.5, 0.5),

Csc = 2.

(7.104)

The computed solution retains exceptional accuracy in the smooth regions —the rarefaction fan and the constant saturation states— while effectively enhancing stability near the sharp gradients.

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Figure 7.23. Saturation profiles of the ASGS solution to the water-gas injection problem. Formulation of τ given by Hughes and Mallet [1986a]. Proposed formulation of shock-capturing diffusion —Equation (7.87).

Dsc

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Figure 7.24. Profiles of shock capturing diffusion introduced by the proposed formulation —Equation (7.87)— at two different times. The profile of additional diffusion introduced by the discontinuity-capturing term is plotted —at simulation times t = 0.5 and t = 2— in Figure 7.24. It is apparent that the amount of artificial diffusion is negligible everywhere, except: (1) in the vicinity of both shocks during transient conditions; (2) near the boundary layer for quasi-steady conditions. All other formulations of shock-capturing diffusion described in Section 7.2.6 are either less effective or even fail to converge.


7.4

281

Concluding remarks

We have presented a fairly general formulation for the numerical solution of nonlinear systems of conservation laws, and applied it to the equations of one-dimensional three-phase flow through porous media. Our method is based on the original framework presented in Hughes [1995], and entails a multiple-scale decomposition of the solution into resolved and unresolved scales. It is precisely the effect of the unresolved subgrid scales on the resolved grid scales that introduces a stabilizing term in the formulation. Key distinctive features of the formulation developed herein are: (1) the multiscale split is performed before any linearization of the equations, which are kept in conservation form; (2) the multiscale solution is not reconstructed from point values of coarse-scale and subgrid-scale solutions; and (3) a novel shock-capturing technique is proposed to enhance further the stability of the solution in the neighborhood of strong gradients. From the results presented in Section 7.3, we conclude that the proposed stabilized method yields numerical solutions of exceptional quality to challenging, highly nonlinear, nearly hyperbolic problems. Solutions computed on very coarse grids display excellent stability and accuracy. The algebraic subgrid model employed is quite sensitive, however, to the choice of the matrix of stabilizing coefficients τ . The definition of τ given by Hughes and Mallet [1986a], which requires the solution of an eigenvalue problem, seems to be the most applicable to the type of problems considered here. This observation is further confirmed by numerical experiments with other one-dimensional systems that become strictly hyperbolic in the limit of vanishing diffusion, such as the shallow-water equations and the Euler equations of gas dynamics. Application of the methodology to these interesting problems will be reported in subsequent publications.


282

The novel formulation of the discontinuity-capturing diffusion —named “globalgradient form”— provides an alternative to existing formulations. The simulations of Section 7.3 clearly show that, in contrast to the canonical expressions, the numerical diffusion introduced by the proposed formulation is confined to the vicinity of discontinuities in the solution.

Your vowes to her, and me, (put in two scales) Will euen weigh, and both as light as tales. — WILLIAM SHAKESPEARE, A Midsummer Night’s Dream (1590)

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Chapter 8 Closure 8.1

Conclusions

In this dissertation, we have analyzed and developed further the traditional mathematical model of three-phase flow in porous media. This model is based on the multiphase flow extension of Darcy’s equation proposed by Muskat [1949]. We intently focused on the case of small capillarity effects because, under these conditions, the behavior of the model depends exclusively on the relative permeabilities. Bell et al. [1986] found that common relative permeability models, which are widely used in the petroleum industry for the simulation of oil recovery [Stone, 1970], yield regions in the saturation space where the system has elliptic character, in an otherwise hyperbolic model. In addition, several investigations concluded that this change of character is an unavoidable consequence of three-phase flow models [Guzmán and Fayers, 1997a; Holden, 1990a; Shearer, 1988]. We have demonstrated here that this conclusion is not entirely correct, and we anticipate that this important result will give a new perspective on the study of the mathematical character of the three-phase flow equations.

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We have argued that elliptic regions are the artifacts of an incomplete mathematical model, and not a necessary consequence dictated by physics. The key observation is that the relative permeabilities are functionals of the various fluid, rock, and process descriptors, and not fixed functions of fluid saturations alone (or even saturation history). By exploiting this fundamental concept, we derived conditions that the relative permeabilities must satisfy so that the system of equations is everywhere strictly hyperbolic. It turns out that the essential requirement is a positive slope of the gas relative permeability at its endpoint saturation. This condition may be justified from the physics of multiphase fluid displacement, and it was shown here to be in good agreement with experimental data. The required conditions, derived first under negligible gravitational effects (Chapter 3), were then extended to the case of co-current three-phase flow with gravity (Chapter 4). Again, the dependence of the required conditions on the viscosity ratios and the gravity number is in accordance with the physics of multiphase flow. The fact that strict hyperbolicity may be preserved by an appropriate choice of the relative permeabilities, lead us to construct the complete analytical solution to the Riemann problem of three-phase flow. A physical assumption was made for the flux functions, namely, that the inflection loci are single connected curves corresponding to maxima of eigenvalues along integral curves. This behavior is a natural extension of the two-phase flow case, in which the flux function is S-shaped with a unique inflection point, even when gravity is included in the formulation. We have shown that the wave structure that may arise in the Riemann problem is limited to only 9 combinations of rarefactions, shocks, and rarefaction-shocks. The complete solution, which follows the theory of strictly hyperbolic systems [Ancona and Marson, 2001; Lax, 1957; Liu, 1974, 1975], was presented in Chapter 5, and efficient algorithms for

Chapter 8. Closure

285

its evaluation are given in Appendix A. In addition to revealing the wave structure, and its use as a benchmark for numerical solutions, the analytical solution developed herein has direct applicability to the interpretation of lab displacement experiments, and may be readily implemented in streamline simulators. In the second part of this dissertation (Chapters 6 and 7), we developed stabilized finite element methods for the simulation of miscible, two-phase, and three-phase flows. The essence of the formulation is to acknowledge the presence of multiple scales in the solution, and to incorporate the effects of the subgrid scales on the coarse-scale solution [Hughes, 1995]. The variational multiscale method yields stabilized numerical solutions on very coarse grids, without upsetting consistency or the order of accuracy of the approximation. On the other hand, classical numerical methods, which neglect the influence of subgrid variability, require an impractical grid resolution for either stability or accuracy purposes. The specific contributions of the formulation presented here are: (1) the multiscale formalism was applied for the first time to the simulation of multiphase flow in porous media; (2) nonlinearity of the equations was retained at the time of invoking the multiscale split, leading to an incremental formulation with a multiscale decomposition of the increment; and (3) different alternatives for the design of the matrix of intrinsic time scales τ were investigated, and a new expression for the shock-capturing diffusion was proposed and compared with existing formulations. From the numerical simulations presented here, we conclude that the formulation of τ proposed by Hughes and Mallet [1986a], which involves the solution of an eigenvalue problem, seems to be the most robust for the type of problems of interest. We conjecture that this result is related to the fact that, in the case of small capillarity studied here, the system of governing equations is (almost) strictly hyperbolic and, therefore, diagonalizable. It was also shown that the proposed form of

Chapter 8. Closure

286

the discontinuity-capturing diffusion introduces numerical dissipation in the vicinity of sharp features on a narrower region than traditional expressions. In summary, the variational multiscale formulation studied here shows great potential for the numerical solution of complex multiphase compositional flows.

8.2

Future extensions

Several issues addressed in this dissertation deserve further investigation. Some directions for future research include the following: 1. A definitive argument on whether elliptic regions are physical. In this dissertation, we have provided many reasons why elliptic regions in three-phase flow models should be regarded as nothing else than artifacts of an incomplete mathematical description. A definitive physical or mathematical argument is still lacking. Research in this direction is currently under way [Juanes et al., 2003]. 2. Initial and injected saturation states outside the three-phase flow region. Since the fluid saturations were understood to be normalized, the analytical solution to the Riemann problem derived here is valid for saturation states inside the three-phase flow region. Extension of the solution is currently being investigated, to allow for initial and/or injected states in regions of the saturation space where only one or two phases are mobile. 3. Capillarity effects. In the development of the analytical solution, capillary forces were dropped from the formulation. We plan to extend the analytical solution —in the framework of the displacement theory— to account for the local effects of capillarity on the structure of shocks, using the method of asymptotic expansions [Barenblatt, 1996; Barenblatt et al., 1990].

Chapter 8. Closure

287

4. Inversion technique for the interpretation of lab displacement experiments. A research topic of great practical interest is the development of efficient inversion algorithms —history-matching techniques— for one-dimensional displacements, based on the analytical forward model above. Such tools would permit interpretation of fluid recovery and in-situ saturation data from laboratory and field displacement experiments, and the evaluation of relative permeabilities and capillary pressures therefrom. 5. Compressibility, miscibility, and thermal effects. The developments in this dissertation are restricted to isothermal systems of three incompressible, immiscible phases (Buckley-Leverett model). This mathematical description has applicability to many relevant flow processes. Moreover, Trangenstein and Bell [1989] have shown that the black-oil model —in which miscibility of the gas and oil phases is contemplated— retain much of the mathematical structure of the Buckley-Leverett equations. There are, however, practical situations where the incompressibility and immiscibility assumptions do not hold. A fruitful, and challenging, line of research would be to relax those assumptions in the mathematical model, in order to: (1) analyze the character of the system; and (2) develop new analytical solutions. 6. Flow problems in higher dimensions, and in fractured media. In this dissertation, we concentrated mainly on one-dimensional problems, because this is the setting for the displacement theory studied in Part I. Extension of the multiscale numerical formulation to multiphase flow problems in multidimensions is an exciting task. In this context, the pressure equation needs to be solved numerically. It appears that multiscale-stabilized methods are also uniquely suited for the solution of this equation, as they can deal with the velocity-pressure in-

Chapter 8. Closure

288

stabilities of mixed formulations, and allow that equal-order interpolations be used for the pressure and velocity fields. The numerical model may then be extended to simulate multiphase flow in three-dimensional fractured media, using the general treatment of Juanes et al. [2002], applied in Molinero et al. [2002] to the simulation of single-phase flow and tracer transport. 7. Alternative time stepping techniques. In the course of this investigation, it has become apparent that transient problems display different stability issues than stationary problems. The time integration schemes used in this dissertation are simple finite difference schemes, such as backward Euler or CrankNicolson [Thomée, 1997, Chapter 1]. We have noticed that stabilization provided by the method under transient conditions is not as effective as under steady-state conditions. We have started to look into alternative time-stepping schemes, such as the time-discontinuous Galerkin method [Thomée, 1997, Chapter 12]. This method has several advantages: (1) it has good convergence and dissipation properties, (2) it is a finite element method and, thus, provides a natural setting for a space-time version of the multiscale variational method. 8. Alternative subgrid-scale models. In this dissertation, the subgrid-scale problem was modeled, rather than solved, using an algebraic subgrid scale approximation. One of the topics that is currently being investigated is the study of a different approximation to the subscales. In particular, we are interested in a numerical approximation of the subgrid scale problem with appropriate basis functions —high-order finite elements, wavelets, etc.— which are potentially capable of capturing the sharp features of the solution that the coarse mesh is unable to resolve.

289

Chapter 8. Closure

9. The variational multiscale method as a unified framework. Finally, and owing to its generality, we advocate the use of the variational multiscale method as a unifying framework for the development of numerical techniques to deal with some of the most severe and recurrent problems in subsurface flow simulation: (1) advection instabilities; (2) velocity-pressure instabilities; (3) subgrid heterogeneity; (4) a posteriori error estimation; and (5) multiscale physics.

In English, when a stop follows a vowel, . . . nothing is heard but part of the glide on to the consonant, the actual closure being formed without any breath at all. — HENRY SWEET, A Handbook of Phonetics (1877)

290

Nomenclature What follows is a partial list of symbols used in the dissertation. The same symbol may be used in different contexts to denote different variables. When appropriate, the definition is followed by the physical dimensions of the variable, even though the same symbol may be used as a dimensionless variable. The equation and page numbers refer to the first time the variable is introduced in the text.

Greek symbols βg

Endpoint slope of the gas relative permeability [dimensionless], Eq. (3.48), p. 94.

δ

Increment, p. 203.

²α

Capillary number times viscosity of the α-phase [ML−1 T−1 ], Eq. (2.36), p. 57.

Γn

Part of the boundary with Neumann boundary conditions, p. 195.

Γu

Part of the boundary with Dirichlet boundary conditions, p. 195.

Λ

Matrix of eigenvalues [LT−1 ], Eq. (7.48), p. 249.

λα

Relative mobility of the α-phase [M−1 LT], Eq. (2.6), p. 52.

λT

Total relative mobility [M−1 LT], Eq. (2.16), p. 53.

µ ¯

Geometric mean of the water and oil dynamic viscosities [ML−1 T−1 ], p. 96.

µ ˜

Viscosity ratio µw /µo in two-phase flow [dimensionless], p. 150.

µα

Dynamic viscosity of the α-phase [ML−1 T−1 ], p. 51.

Nomenclature

291

ν

Eigenvalue of the system of equations [LT−1 ], Eq. (3.6), p. 72.

νi

Eigenvalue of the i-family [LT−1 ], Eq. (3.10), p. 74.

¯ Ω

Closure of Ω, p. 195.

∂Ω

Boundary of the spatial domain, p. 195.

Ω

Spatial domain, p. 195.

˜ Π

Projection operator onto the space of subgrid scales, Eq. (7.36), p. 243.

φ

Porosity [dimensionless], p. 51.

ρα

Mass density of the α-phase [ML−3 ], p. 51.

ρD

Density ratio [dimensionless], Eq. (2.27), p. 56.

σ

Radioactive decay constant [T−1 ], Eq. (2.71), p. 66.

σi

Speed of propagation of an i-shock [LT−1 ], p. 169.

τ uh

Matrix of intrinsic time scales, Eq. (7.37), p. 244.

τ

Intrinsic time [T], p. 200.

τ

Inverse of the similarity variable ζ [L−1 T], Eq. (5.54), p. 186.

τa

Characteristic time for advection [T], p. 209.

τd

Characteristic time for diffusion [T], p. 209.

τr

Characteristic time for reaction [T], p. 209.

ϕ

Mapping from actual to reduced saturations [dimensionless], Eq. (2.49), p. 61.

Ξg

Auxiliary function to check co-current flow of gas, Eq. (4.18), p. 125.

Ξo

Auxiliary function to check co-current flow of oil, Eq. (4.15), p. 125.

Ξw

Auxiliary function to check co-current flow of water, Eq. (4.12), p. 124.

ζ

Similarity variable [LT−1 ], Eq. (5.29), p. 159.

Roman symbols A

Advection operator [LT−1 ], Eq. (7.31), p. 243.

A

Jacobian matrix of the system of equations [LT−1 ], Eq. (3.5), p. 71.

Nomenclature a

Advective velocity in miscible flow [LT−1 ], Eq. (2.81), p. 68.

a(uh )

Linearized advective velocity [LT−1 ], Eq. (6.41), p. 204.

292

a(·, ·, ·) Nonlinear form in the weak formulation, Eq. (6.12), p. 197. b∗

Boundary operator associated with L∗ [LT−1 ], Eq. (6.31), p. 201.

c

Tracer mass fraction (concentration) [dimensionless], Eq. (2.73), p. 66.

c∗

Tracer mass fraction of the source [dimensionless], Eq. (2.75), p. 66.

Cα

Capillary number of the α-phase [dimensionless], Eq. (2.29), p. 56.

Cα

Constant in the power-law expression of krα [dimensionless], Eq. (3.70), p. 102.

Co

Courant number [dimensionless], Eq. (6.54), p. 208.

D

Capillary diffusion tensor [L2 T−1 ], Eq. (2.45), p. 58.

D

Diffusion tensor in miscible flow [L2 T−1 ], Eq. (2.81), p. 68.

Dh

Physical diffusion tensor in miscible flow [L2 T−1 ], Eq. (2.78), p. 67.

Dsc

Shock-capturing diffusion tensor [L2 T−1 ], Eq. (7.80), p. 256.

Dw

Capillary diffusion in two-phase flow [L2 T−1 ], Eq. (2.69), p. 65.

Dsc

Shock-capturing diffusion [L2 T−1 ], Eq. (6.50), p. 206.

Da

Damkholer number [dimensionless], Eq. (6.56), p. 208.

˜ D

Capillary diffusion tensor as function of reduced saturations [L2 T−1 ], Eq. (2.54), p. 62.

f

Water fractional flow, fw [dimensionless], Eq. (3.3), p. 71.

¯ F

Imposed value of the volumetric flux vector at the boundary [LT−1 ], Eq. (7.3), p. 236.

F¯

Imposed value of the volumetric flux at the boundary [LT−1 ], Eq. (6.4), p. 195.

f

Vector of flux functions [LT−1 ], Eq. (2.44), p. 58.

Fi

Total advective-diffusive flux of the i-species [ML−2 T−1 ], p. 66.

Fα

Mass flux of the α-phase [ML−2 T−1 ], Eq. (2.3), p. 51.

Nomenclature

293

fα

Fractional flow of the α-phase [dimensionless], Eq. (2.18), p. 53.

foil

Dimensionless oil production rate [dimensionless], Eq. (5.55), p. 186.

f˜

Flux vector as function of reduced saturations [LT−1 ], Eq. (2.53), p. 62.

g

Gas fractional flow, fg [dimensionless], Eq. (3.3), p. 71.

g

Gravitational acceleration [MLT−2 ], p. 51.

gx

x-component of the gravitational acceleration [MLT−2 ], Eq. (2.5), p. 51.

h

Characteristic length of a finite element [L], p. 201.

h

Hydraulic head [L], Eq. (1.2), p. 6.

Hog

Auxiliary function to check the strict hyperbolicity condition along the OG edge, Eq. (3.34), p. 91.

How

Auxiliary function to check the strict hyperbolicity condition along the OW edge, Eq. (3.26), p. 88.

Hwg

Auxiliary function to check the strict hyperbolicity condition along the WG edge, Eq. (3.43), p. 92.

J

Jacobian matrix in canonical form [LT−1 ], Eq. (3.9), p. 72.

k

Absolute permeability [L2 ], p. 51.

krα

Relative permeability of the α-phase [dimensionless], p. 51.

L

Length of spatial domain [L], p. 51.

Ladv

Advective part of operator L [T−1 ], Eq. (6.32), p. 202.

L

Advection-diffusion-reaction operator [T−1 ], Eq. (6.7), p. 196.

L∗

Adjoint of operator L [T−1 ], Eq. (6.30), p. 201.

L uh

Linearized advection-diffusion-reaction operator [T−1 ], Eq. (6.42), p. 204.

l(·)

Linear form in the weak formulation, Eq. (6.13), p. 197.

M

Gravity number times oil viscosity [ML−1 T−1 ], Eq. (2.41), p. 58.

mα

Exponent in the power-law expression of krα [dimensionless], Eq. (3.70),

Nomenclature

294

p. 102. mα

Mass of the α-phase per unit bulk volume [ML−3 ], Eq. (2.2), p. 51.

n

Outward unit normal to the boundary [dimensionless], Eq. (6.4), p. 195.

ND

Gravity number [dimensionless], Eq. (2.26), p. 56.

p

Reference pressure [ML−1 T−2 ], Eq. (2.7), p. 52.

pα

Pressure of the α-phase [ML−1 T−2 ], p. 51.

Pc

Capillary pressure in two-phase flow [ML−1 T−2 ], Eq. (2.60), p. 64.

Pcα

Capillary pressure of the α-phase [ML−1 T−2 ], Eq. (2.9), p. 52.

∗ Pcα

Characteristic value of the α-phase capillary pressure [ML−1 T−2 ], Eq. (2.31), p. 57.

D Pcα

Dimensionless capillary pressure of the α-phase [dimensionless], Eq. (2.33), p. 57.

Pe

Peclet number [dimensionless], Eq. (6.55), p. 208.

q

Darcy velocity for a single fluid [LT−1 ], Eq. (1.1), p. 6.

Qi

Mass distributed source term of the i-species [ML−3 T−1 ], Eq. (2.75), p. 66.

qT

Total volumetric source term [T−1 ], Eq. (2.75), p. 66.

Qoil

Dimensionless cumulative oil production [dimensionless], Eq. (5.56), p. 186.

R(uh ) Grid-scale residual [T−1 ], Eq. (6.26), p. 200. RSi

Rarefaction-shock of the i-family, p. 173.

Ri

Rarefaction of the i-family, p. 169.

R

Matrix of right eigenvectors [dimensionless], Eq. (7.48), p. 249.

r

Eigenvector of the system of equations [dimensionless], Eq. (3.7), p. 72.

ri

Eigenvector of the i-family [dimensionless], Eq. (3.12), p. 74.

Si

Shock of the i-family, p. 170.

Sα

Saturation of the α-phase [dimensionless], p. 51.

Nomenclature

295

Sαi

Immobile saturation of the α-phase [dimensionless], Eq. (2.56), p. 62.

t

Time coordinate [T], p. 51.

T

Saturation triangle [dimensionless], Eq. (2.47), p. 59.

tD

Dimensionless time coordinate [dimensionless], Eq. (2.24), p. 56.

u

Water saturation, Sw [dimensionless], Eq. (3.2), p. 71.

¯ u

Imposed value of the unknown vector at the boundary [dimensionless], Eq. (7.1), p. 236.

u¯

Imposed value of the unknown at the boundary [dimensionless], Eq. (6.4), p. 195.

u˜

Subgrid-scale part of u, Eq. (6.18), p. 199.

u

Vector of unknown saturations [dimensionless], Eq. (2.43), p. 58.

U

Self-similar solution vector [dimensionless], Eq. (5.28), p. 159.

u0

Initial conditions of the vector u [dimensionless], Eq. (7.4), p. 236.

u0

Initial conditions of u [dimensionless], p. 195.

uh

Grid-scale part of u, Eq. (6.18), p. 199.

˜ u

Vector of reduced saturations [dimensionless], Eq. (2.49), p. 61.

v

Gas saturation, Sg [dimensionless], Eq. (3.2), p. 71.

V˜

Space of subgrid scales, Eq. (6.19), p. 199.

V

Space of trial functions, Eq. (6.8), p. 196.

V0

Space of test functions, Eq. (6.9), p. 196.

Vh

Space of grid-scale trial functions, Eq. (6.16), p. 198.

Vh,0

Space of grid-scale test functions, Eq. (6.16), p. 198.

vα

Velocity of the α-phase [LT−1 ], Eq. (2.4), p. 51.

vT

Total velocity [LT−1 ], Eq. (2.13), p. 53.

Wi

Wave of the i-family, Eq. (5.30), p. 160.

296

Nomenclature x

Spatial coordinate [L], p. 51.

xD

Dimensionless space coordinate [dimensionless], Eq. (2.23), p. 56.

z

Elevation [L], p. 51.

Subscripts ∗

Post-shock state, p. 175.

+

State at the right of a discontinuity, p. 169.

−

State at the left of a discontinuity, p. 169.

g

Gas phase, p. 51.

l

Left state of the Riemann problem, Eq. (5.27), p. 157.

m

Intermediate state of the Riemann problem, Eq. (5.42), p. 167.

o

Oil phase, p. 51.

r

Right state of the Riemann problem, Eq. (5.27), p. 157.

w

Water phase, p. 51.

,u

Partial derivative with respect to u, p. 71.

Superscripts (k)

Iteration level, p. 203.

T

Transpose (of a matrix or vector), p. 246.

Some notations will be unfortunate even when they are beautifully formatted. — DONALD E. KNUTH, The TEXbook (1984)

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In comparing various authors with one another, I have discovered that some of the gravest and latest writers have transcribed, word for word, from former works, without making acknowledgment. — PLINY THE ELDER, Natural History (77 A.D.) For your writing and reading, let that appeare when there is no neede of such vanity. — WILLIAM SHAKESPEARE, Much Ado About Nothing (1599)

345

Appendix A Algorithms for the analytical solution of the Riemann problem This appendix contains efficient algorithms for the calculation of the analytical solution to the Riemann problem of three-phase flow, described in Chapter 5. The appendix is organized in two main sections: 1. Algorithms for the calculation of the individual wave curves. 2. Algorithms for the calculation of the global solution to the Riemann problem. Obviously, these algorithms make use of those for the individual wave curves. A common feature to all the algorithms described herein is the use of a Newton iteration strategy, which results in second-order convergence. Step-by-step summaries of each algorithm are given in boxes, and the performance of the algorithms is demonstrated with representative examples.

Appendix A. Algorithms for the analytical solution of the Riemann problem

A.1

A.1.1

346

Solution algorithms for the wave curves of three-phase flow Rarefaction curves

Rarefaction curves are subsets of the integral curves, satisfying that the eigen˜ i (ξ) = (˜ value increases monotonically along the curve. Let u ui (ξ), v˜i (ξ)) be a parameterization of a rarefaction curve of the i-family, Ri (ˆ u), starting at the reference ˆ = (ˆ state u u, vˆ). From the definition of a rarefaction curve, it satisfies the initial value problem:

   u ˜ 0i (ξ) = α(ξ)r i (˜ ui (ξ)),

(A.1)

  u ˜ i (0) = u ˆ,

where α(ξ) is some scalar factor.

For a 1-rarefaction curve we choose the following parameterization: u˜1 (ξ) = uˆ + ξ. Using Equation (5.16), the initial value problem (A.1) reduces to  ν − f,u g,u r   v˜10 (ξ) = 1v = 1 = , r1u f,v ν1 − g,v   v˜1 (0) = vˆ.

(A.2)

(A.3)

For a 2-rarefaction curve we use the parameterization v˜2 (ξ) = vˆ + ξ.

Using Equation (5.17), the initial value problem (A.1) takes the form  f,v ν2 − g,v r   u˜02 (ξ) = 2u = = , r2v ν2 − f,u g,u   u˜2 (0) = uˆ.

(A.4)

(A.5)


347

A simple Runge-Kutta algorithm can be used to integrate the scalar ordinary differential equations (A.3) and (A.5).

A.1.2

Shock curves

Shock curves are subsets of the Hugoniot loci, satisfying the Lax entropy condition. Here we describe a Newton iterative procedure for the calculation of the shock curves. ˜ i (ξ) = (˜ Let u ui (ξ), v˜i (ξ)) be a parameterization of an i-shock curve, Si (ˆ u), starting ˆ = (ˆ at the reference state u u, vˆ). We use the following parameterization of the 1-shock curve: u˜1 (ξ) = uˆ + ξ.

(A.6)

For each value of the parameter ξ, u˜1 is known from (A.6), and the following system of algebraic equations needs to be solved for σ1 and v˜1 : R1 (σ1 , v˜1 ) := f (˜ u1 , v˜1 ) − f (ˆ u, vˆ) − σ1 (˜ u1 − uˆ) = 0,

(A.7)

R2 (σ1 , v˜1 ) := g(˜ u1 , v˜1 ) − g(ˆ u, vˆ) − σ1 (˜ v1 − vˆ) = 0. We define the solution vector x and the residual vector R as:      σ1   R1  x :=   , R :=   . v˜1 R2

(A.8)

We use Newton’s method to find the solution to the system (A.7), as indicated in

Figure A.1. For the 2-shock curve we use the following parameterization: v˜2 (ξ) = vˆ + ξ.

(A.9)

The iterative procedure to obtain the solution is identical to that of the 1-shock curve, but now the system needs to be solved for u˜2 and σ2 . The solution vector, residual


1. Set k = 0. Initial guess:





(0)  σ1 

x(0) = 

(0) v˜1

.

2. Evaluate residual vector and Jacobian matrix:   (k) (k) u, vˆ) − σ1 (˜ u1 − uˆ)  u1 , v˜1 ) − f (ˆ  f (˜ R(k) =  , (k) (k) (k) g(˜ u1 , v˜1 ) − g(ˆ u, vˆ) − σ1 (˜ v1 − vˆ)   ¯(k) (k) u1 − uˆ) f,v (˜ u1 , v˜1 )  ∂R ¯¯  −(˜ J (k) = =  . (k) (k) (k) ∂x ¯ −(˜ v1 − vˆ) g,v (˜ u1 , v˜1 ) − σ1 3. Solve linear system for the increment δx(k) : J (k) δx(k) = R(k) .

4. Update solution: x(k+1) = x(k) − δx(k) . 5. Check convergence: IF ||R|| < ² STOP ELSE Set k ← k + 1 and GOTO 2. Figure A.1. Newton algorithm for obtaining the 1-shock curve.

348

Appendix A. Algorithms for the analytical solution of the Riemann problem vector and Jacobian matrix are:   u˜2  x =  , σ2   u2 , v˜2 ) − f (ˆ u, vˆ) − σ2 (˜ u2 − uˆ) f (˜ R= , g(˜ u2 , v˜2 ) − g(ˆ u, vˆ) − σ2 (˜ v2 − vˆ)   u2 , v˜2 ) − σ2 −(˜ u2 − uˆ) ∂R f,u (˜ = J= . ∂x g,u (˜ u2 , v˜2 ) −(˜ v2 − vˆ)

A.1.3

349

(A.10)

Rarefaction-shock curves

Rarefaction-shock curves Ri Si (ul , ur ) are composite waves, consisting of a rarefaction curve emanating from a left state ul , and a shock curve that ends at the right state ur . Both curves join at an intermediate state u∗ , called the post-shock state. We now detail the procedure to compute 1-rarefaction-shock curves. Assume that the left state ul and the first component ur of the right state are known. A complete description of the 1-rarefaction-shock requires: the second component vr of the right state, the speed σ1 of the shock, and the post-shock value u∗ . These unknowns are obtained using the predictor-corrector algorithm of Figure A.2. A schematic of the kth iteration is shown in Figure A.3. It is important to note that the post-shock value u∗ , at which the R1 and S1 curves are connected, depends on both the left and the right states. This connection is always very smooth. In fact, it can be shown [Lax, 1957] that both curves are connected with second order tangency (same slope and curvature). This property ensures that the predictor-corrector algorithm in Figure A.2 will achieve global quadratic convergence when the initial guess is close to the solution.


350

(0)

1. Set k = 0. Initial guess vr . 2. Predictor: (i) Integrate along rarefaction curve R1 (ul ) as in Equation (A.3) until (k)

ν1 (˜ u) =

(k+1)

(ii) Set u∗

˜) σ1 (u(k) r ;u

f (ur ) − f (˜ u) = . ur − u˜

˜ at which integration terminated. =u (k+1)

3. Corrector: Solve for σ1

(k+1)

and vr

by imposing that the right state

belongs to the shock curve passing through the post-shock value, (k+1)

ur

(k+1)

∈ S1 (u∗

), using the Newton algorithm in Figure A.1.

4. Check convergence: (k+1)

IF |vr

(k)

− vr | < ² STOP

ELSE Set k ← k + 1 and GOTO 2. Figure A.2. Predictor-corrector algorithm for obtaining the rarefaction-shock curve of the 1-characteristic family. 4 u(k+1) r

ul 4 (k+1)

R1 (ul )

S1 (u∗ (k+1)

u∗

(k)

u∗

)

4 u(k) r

(k)

S1 (u∗ )

ur

Figure A.3. Schematic of the kth iteration of the predictor-corrector algorithm for obtaining a rarefaction-shock of the 1-characteristic family. The rarefaction curve and the shock curve join at the post-shock value with second-order tangency. This property allows the algorithm to achieve quadratic convergence.


351

The algorithm to compute a rarefaction-shock of the 2-characteristic family is completely analogous. In this case, the left state ul and the second component vr of the right state are known, and we use a predictor-corrector algorithm to compute the first component ur of the right state, the speed σ2 of the shock, and the post-shock value u∗ .

A.2

Solution algorithms for selected solution types of three-phase flow

Recall that the solution of the Riemann problem of three-phase flow consists in a sequence of two waves connecting three constant states: W

W

1 2 ul −→ um −→ ur .

(A.11)

The complete set of solutions (9 cases) is given in Section 5.4.3.2 of Chapter 5. In this section we describe efficient algorithms for the calculation of selected solution types. Other cases can be computed similarly.

A.2.1

S1 S2 solution

The first solution we consider is that when both waves are genuine shocks, that is, W1 ≡ S1 (ul ) and W2 ≡ S2 (ur ). These two shock curves intersect at the intermediate constant state um . A.2.1.1

Solution procedure

The unknowns are the intermediate constant state um = (um , vm ) and the shock speeds σ1 and σ2 . These 4 scalar unknowns may be found by imposing the Rankine-


352

Hugoniot jump condition on each shock: f (um , vm ) − f (ul , vl ) = σ1 (um − ul ), g(um , vm ) − g(ul , vl ) = σ1 (vm − vl ),

(A.12)

f (um , vm ) − f (ur , vr ) = σ2 (um − ur ), g(um , vm ) − g(ur , vr ) = σ2 (vm − vr ). We propose using a full Newton iterative procedure to achieve quadratic convergence. The solution vector x, the residual vector R and the Jacobian matrix J are given by:   σ  1    vm    x =  , (A.13)    um    σ2   f (um , vm ) − f (ul , vl ) − σ1 (um − ul )      g(um , vm ) − g(ul , vl ) − σ1 (vm − vl )    (A.14) R= ,   f (um , vm ) − f (ur , vr ) − σ2 (um − ur )   g(um , vm ) − g(ur , vr ) − σ2 (vm − vr )

and



−(um − ul )

f,v (um , vm )

f,u (um , vm ) − σ1

0



     −(vm − vl ) g,v (um , vm ) − σ1  g,u (um , vm ) 0   J = .   0 f (u , v ) f (u , v ) − σ −(u − u )  ,v m m ,u m m 2 m r    0 g,v (um , vm ) − σ2 g,u (um , vm ) −(vm − vr ) A.2.1.2

(A.15)

Admissibility of the solution

The solution is valid if each wave satisfies the Lax entropy condition: S1 : ν1 (ul ) > σ1 > ν1 (um ), S2 : ν2 (um ) > σ2 > ν2 (ur ).

(A.16)

Appendix A. Algorithms for the analytical solution of the Riemann problem A.2.1.3

353

Example

We show an example of a Riemann problem whose solution involves two genuine shocks. We use the relative mobilities (5.18)–(5.20), with the values of fluid viscosities in (5.23), given in Chapter 5. The left state is ul = (0.25, 0.2) and the right state is ur = (0.15, 0.8). Solution of the nonlinear system of algebraic equations (A.12) yields: um = (0.6152, 0.3278),

σ1 = 0.1560,

σ2 = 0.1930.

The schematic of the solution in the ternary diagram —saturation space— is depicted in Figure A.4. Inflection loci (dash-dotted curves) are plotted for reference also. The intermediate constant state um is located at the intersection of the two shock curves. Note that the 1-shock is admissible even though the left and right states of the discontinuity, ul and um , respectively, lie on opposite sides of the 1inflection locus. The profiles of the characteristic speeds ν1 and ν2 , and the phase saturations Sw ≡ u, Sg ≡ v, and So ≡ 1 − u − v, are plotted against the similarity variable ζ = x/t in Figure A.5. In Figure A.6 we show the evolution of the error with the number of iterations of the Newton scheme. The error at iteration k is defined as: (k−1) (k) (k−1) e(k) := ||δu(k−1) ||1 = |u(k) | + |vm − vm |. m m − um

(A.17)

The rate of convergence of the method is given by the exponent m in the following expression relating the error at iteration k + 1 with the error at iteration k: ¡ ¢m e(k+1) ≤ C e(k) ,

(A.18)

where C is a bounded positive constant. If we plot e(k+1) against e(k) in log-log scale, the convergence rate is simply the asymptotic slope of the curve. As shown in


G

PSfrag replacements

0

1

0 .2

0.8

ur 0 .4 0 .6

PSfrag replacements

0.6

S2 0.4

W 0.8

S1

ul

G

0.2

0 1

0.8

0.6

0.4

S1

0.2

O

0

1

O

um

W

S

2 Figure A.4. Schematic of the S1 S2 solution path in the ternary diagram.

ul

ν2 ν1

νi

ur

2

um

0

Sw

1

0

Sg

1

0

So

1

0 0

σ1 ζ = x/t

σ2

0.3

Figure A.5. Profiles of wave speeds and saturations for the S1 S2 solution.

354


355

0

10

−5

e(k+1)

PSfrag replacements

10

−10

10

2 1

−15

10

−20

10

−8

10

−6

10

−4

10 (k)

10

−2

0

10

e Figure A.6. Performance of the Newton iterative scheme for the S1 S2 solution, showing a quadratic rate of convergence. Figure A.6, convergence of the iterative procedure is quadratic, as expected.

A.2.2

R1 R2 solution

We consider the case when both waves are rarefaction waves, so that W1 ≡ R1 and W2 ≡ R2 . There are no discontinuities, and the solution is smooth everywhere. A.2.2.1

Solution procedure

The intermediate constant state is determined by the intersection of the two rarefaction curves. By contrast to the previous case, in which the intersecting curves were given by algebraic equations, rarefaction curves are defined by differential equations (A.3) and (A.5). We suggest a predictor-corrector strategy to find u m iteratively. The algorithm is given in Figure A.7, and the kth iteration is illustrated in Figure A.8.


(0)

(0)

356

(0)

1. Set k = 0. Initial guess um = (um , vm ). 2. Predictor: (k)

(i) Set the integration limit uˆm = um and integrate along R1 (ul ) to obtain ˆ m. point u (k)

(ii) Set the integration limit v˜m = vm and integrate along R2 (ur ) to obtain ˜ m. point u (iii) Compute eigenvectors r 1 (ˆ um ) and r 2 (˜ um ) at the integration endpoints. (k+1)

3. Corrector: new approximation um

is the intersection of two straight lines

ˆ m and u ˜ m , respectively. with orientations r 1 and r 2 emanating from points u 4. Check convergence: (k+1)

IF ||um

(k)

− um || < ² STOP

ELSE Set k ← k + 1 and GOTO 2. Figure A.7. Predictor-corrector algorithm for obtaining the R1 R2 solution. A.2.2.2


The solution is admissible if it is single-valued, that is: R1 : ν1 increases monotonically along R1 from ul to um , R2 : ν2 increases monotonically along R2 from um to ur . A.2.2.3

Example

We show an example of a Riemann problem that yields two simple rarefaction waves. We use the same relative permeability model and the same viscosities as


R2

357

4ul

r2 6

ˆm u R1 r1

¼

(k+1)

um

˜m u

(k)

um

ur4 Figure A.8. Schematic diagram of the kth iteration of the predictor-corrector procedure for a R1 R2 intersection. before. The left state is ul = (0.5, 0.5) and the right state is ur = (0.4, 0.1). Solution of the R1 R2 intersection gives the intermediate constant state: um = (0.2658, 0.3360).

In Figure A.9 we plot the solution in the ternary diagram. The intermediate constant state um is located at the intersection of the two rarefaction curves. Note that the saturation path never crosses the inflection loci. Profiles of the wave speeds and phase saturations are shown in Figure A.10. Points a < b < c < d on the x/t-axis correspond to the wave speeds ν1 (ul ) < ν1 (um ) < ν2 (um ) < ν2 (ur ). Figure A.11 shows the evolution of the error, defined in Equation (A.17), with the number of iterations of the predictor-corrector strategy. It is not surprising that convergence is quadratic, since the iterative procedure involves the eigenvectors, which are tangent to the rarefaction curves.


G

PSfrag replacements

0

1

0 .2

0.8

PSfrag replacements 0 .4

0.6

W

ul

0 .6

um

0.8

R1

0 1

0.8

0.2

0

0.2

ur

1

O

R2

0.4

R1

0.6

O

0.4

G

R2

358

W

ul ur A.9. Schematic of the R1 R2 solution path in the ternary diagram. Figure 4

ν2 ν1

νi

um

0

Sw

1

0

Sg

1

0

So

1

0

ab c

d ζ = x/t

Figure A.10. Profiles of wave speeds and saturations for the R1 R2 solution.


359

0

10

−5

e(k+1)

PSfrag replacements

10

−10

10

2 1

−15

10

−20

10

−8

10

−6

10

−4

10 (k)

10

−2

0

10

e Figure A.11. Performance of the predictor-corrector iterative scheme for the R 1 R2 solution, showing a quadratic rate of convergence.

A.2.3

R1 S1 R2 S2 solution

We now consider the case with the most complicated wave structure that may arise in the three-phase flow Riemann problem. In this case both waves are composite rarefaction-shocks: W1 ≡ R1 S1 and W2 ≡ R2 S2 . A.2.3.1

Solution procedure

The variables that need to be determined to characterize fully the solution are: the intermediate constant state um , the shock speeds σ1 and σ2 , and the post-shock states u∗1 and u∗2 of each wave. The constant state um corresponds to the intersection of the two wave curves, and the post-shock states are the points where the rarefaction curve and the shock curve of the same family are joined. Schematically, this can be represented as follows: R

S

R

S

1 1 2 2 ul −→ u∗1 −→ um −→ u∗2 −→ ur

(A.19)

The major difficulty in computing the solution is that both endpoints of the R 2 curve are unknown, so that the initial condition to start integration is not know a priori. The predictor-corrector algorithm in Figure A.12 has proven very effective.


(0)

(0)

360

(0)

1. Set k = 0. Initial guess um = (um , vm ). 2. Predictor: (k)

(i) Set the integration limit uˆm = um and determine the R1 S1 curve using the predictor-corrector algorithm in Figure A.2, to obtain the ˆ ∗1 and the endpoint u ˆ m. post-shock value u ˆ m towards ur until (ii) Integrate along the R2 curve from point u ˜ at which integration stopped. ˜ ∗2 = u ν2 (˜ u) = σ2 (˜ u; ur ). Set u (iii) Set vˆ2∗ = v˜2∗ and solve the S2 curve passing through the right state ur to ˆ ∗2 . obtain the post-shock value u (k+1)

3. Corrector: new approximation um

is the intersection of curves S1 (ˆ u∗1 )

and R2 (ˆ u∗2 ). This is done in a similar fashion to a R1 R2 intersection described in Section A.2.2. 4. Check convergence: (k+1)

IF ||um

(k)

− um || < ² STOP

ELSE Set k ← k + 1 and GOTO 2. Figure A.12. Predictor-corrector algorithm for obtaining the R1 S1 R2 S2 solution.

Appendix A. Algorithms for the analytical solution of the Riemann problem A.2.3.2

361


The solution is admissible if each of the two waves is admissible individually, that is, R 1 S1 :

R 2 S2 :

A.2.3.3

   ν1 increases monotonically along R1 from ul to u∗1 ,

  ν1 (u∗1 ) = σ1 > ν1 (um ),    ν2 increases monotonically along R2 from um to u∗2 ,

(A.20)

  ν2 (u∗2 ) = σ2 > ν2 (ur ).

Example

Using the same relative permeability model and the same viscosities as before, we solve the Riemann problem with left state ul = (0.5, 0.5) and right state ur = (0.05, 0). The R1 S1 R2 S2 solution gives: um = (0.0475, 0.3552),

u∗1 = (0.3275, 0.3618),

σ1 = 0.0805,

u∗2 = (0.0499, 0.0247),

σ2 = 7.6549.

In Figure A.13 we represent the solution as a saturation path in the ternary diagram. It is immediate to check that the solution is admissible. Each composite wave crosses the inflection locus of the corresponding characteristic family. We note that the 2shock has a very small amplitude because the right state almost coincides with the inflection locus of the 2-family. Profiles of wave speeds and phase saturations are plotted in Figure A.14. We decided to split each plot into two and use a different scale on the x/t-axis, due to the very different speeds of the 1- and 2-wave (compare the values of σ1 and σ2 above). Otherwise, we would not be able to distinguish the structure of the 1-rarefaction-shock from the plots. Points a < b < c < d now correspond to ν1 (ul ) < σ1 < ν2 (um ) < σ2 .

PSfrag replacements PSfrag replacements Appendix A. Algorithms for the analytical solution of the Riemann problem W G G 0

O

1

0 .2

R1 R2 0 .4

S1

0.8

0 .6

S2 ul

ul S1

0.2

0 1

0.8

0.6

0.4

ur

S2 0.2

O

0

1

u∗1

0.4

R1

R2

u∗2

um

u∗1

um

0.8

ur

0.6

W

u∗2

Figure A.13. R1 S1 R2 S2 solution path in the ternary diagram. 0.5

10

νi

ν2 ν1 0

0

Sw

1

0

Sg

1

0

So

1

0

. 0

a

b ζ = x/t

0.1

c

0.1

d

10

ζ = x/t

Figure A.14. Wave speeds and saturations for the R1 S1 R2 S2 solution.

362


363

The analytic program . . . costs an additional shilling. — BERNARD SHAW, How to become Musical Critic (1885)

364

Appendix B Implementation of the multiscale finite element method B.1

The variational multiscale formulation

We describe in some detail the implementation of the variational multiscale method in a finite element code, as applied to the system of equations governing one dimensional three-phase flow. The implementation was performed in the general-purpose finite element code FEAP [Taylor, 2002; Zienkiewicz and Taylor, 2000]. A detailed discussion on the practical aspects of the finite element theory and its implementation, is given in the classical books by Hughes [1987], and Zienkiewicz and Taylor [2000], including: (1) the finite element partition; (2) the global and the element points of view; (3) isoparametric interpolation; (4) the shape functions and their derivatives; (5) assembly of the element contributions, and (6) evaluation of element integrals through numerical quadrature. A succinct summary of these concepts is presented in Armero [1998].

Appendix B. Implementation of the multiscale finite element method

365

The starting point is the variational multiscale formulation —Equation (7.45) in Chapter 7— which emanates from the weak form of the mathematical problem, Equation (7.8). The grid-scale equation may be written in the following convenient form (as it will become apparent below): Z Z ¯ · v h dΓ 0= q · v h dΩ − F Ω Γn Z Z ¡ ¢ − f (uh ) + D(uh )∂x uh · ∂x v h dΩ − ∂t uh · v h dΩ − −

Ω nel Z X e=1

Ω

Ωe

˜ dΩ, L∗uh v h · u

(B.1)

∀v h ∈ Vh,0 .

˜ , which are modeled through the algebraic This equation involves the subscales u approximation: ˜ = τ uh R(uh ). u

(B.2)

All the variables are defined in Chapter 7, and will not be repeated here. Remarks B.1. 1. For the sake of generality, Equation (B.1) includes a distributed source term q. ˜ , rather than δ u ˜ , to avoid confusion 2. The incremental subscales are noted as u later in the appendix, where δ is reserved for other purposes. 3. As discussed in Chapter 7, the inter-element boundary integrals in Equation (7.45) have been neglected. It is immediate to identify the source of the different terms in Equation (B.1). The first line corresponds to the external sources (distributed and boundary contributions). The second line —together with the source terms above— is the Galerkin contribution. The third line incorporates the effect of the subgrid-scales. After these


366

observations, we write Equation (B.1) in compact form as follows: 0 = Rgal + Rsgs ,

(B.3)

where each of the terms denote, respectively, the Galerkin and subgrid-scale contributions to the residual. In the following sections of this appendix, we discuss each of these two terms separately, because it is easier to introduce the necessary concepts on the Galerkin contribution, which is the least cumbersome.

B.2

Galerkin contribution

We start our discussion from a time-discrete version of the problem. To this end, we approximate the time derivative in Equation (B.1) as follows: un+1 − unh , ∂t uh ≈ δt uh ≡ h δt

(B.4)

where the superscript n refers to the time level. The solution at a time tn+θ = (1 − θ)tn + θtn+1 , with θ ∈ [0, 1], is approximated by linear interpolation of the solution at time levels tn and tn+1 : un+θ ≡ uh (tn+θ ) ≈ (1 − θ)unh + θun+1 h h .

(B.5)

A value of θ = 1 corresponds to a backward Euler scheme, and θ = 0.5 to a CrankNicolson strategy. The time-discrete Galerkin contribution to the residual, evaluated at time tn+θ and iteration (k), reads: Z Z Z n+θ,(k) n+θ,(k) gal ¯ R (uh F · v h dΓ − δ t uh )= q · v h dΩ − · v h dΩ Ω Ω Γn ¶ Z µ n+θ,(k) n+θ,(k) n+θ,(k) − − f (uh ) + D(uh )∂x uh · ∂x v h dΩ Ω

(B.6)


B.2.1

367

Finite element residual

We now introduce the space discretization. The solution at time tn+θ and iteration (k) is expressed as n+θ,(k) uh

=

npt X

Na U n+θ,(k) , a

(B.7)

a=1

where npt is the total number of nodes of the finite element mesh, Na are the interpolation functions, and U n+θ,(k) are the nodal values of the solution. From this a point on, when the superscripts referring to the time step and the iteration level are omitted, it should be understood that they refer to time tn+θ and iteration (k), e.g., U a ≡ U n+θ,(k) . a

(B.8)

The vector of nodal values U a is a 2 × 1 vector,   1 Ua  Ua =   , Ua2

(B.9)

whose components are the water and gas saturation at node a. Remarks B.2. 1. From Equation (B.7), it is obvious that we use the same interpolation for the water and gas saturations. 2. Dirichlet boundary conditions are imposed directly —and thus the name essential boundary conditions— by requiring that the solution takes prescribed values for all nodes on that portion of the boundary: ¯ ¯ι , U ¯ι = U

for all nodes ¯ι on Γu .

(B.10)

3. The interpolation functions —or shape functions— Na , must satisfy two requirements: (1) they take a unit value at node a, and zero at all other nodes;

368


and (2) they have compact support, that is, they vanish outside the elements surrounding node a. 4. In the actual implementation, the element viewpoint is adopted, and the numbering of the nodes, and the coordinate system, are local to each element. The test functions v h are discretized similarly: vh =

npt X

W a χa ,

(B.11)

a=1

where Wa are the weighting functions, and χa are vector coefficients,   1  χa  χa =   , χ2a

(B.12)

which must vanish identically for nodes belonging to the part of the boundary where Dirichlet boundary conditions are imposed, i.e., χ¯ι = 0 ,

for all nodes ¯ι on Γu .

(B.13)

Remarks B.3. 1. In a Galerkin method, the weighting functions are chosen to be equal to the shape functions: Wa ≡ N a .

(B.14)

We denote them with a different symbol because it is illustrative at the time of analyzing the finite element equations. 2. It is essential to note that, due to the linearity of Equation (B.6) with respect to the test function v h , there is an arbitrariness to the vector coefficients χa .

Appendix B. Implementation of the multiscale finite element method Therefore, one may choose the test functions as follows:    Wi  v h = W 1i :=   , and 0   0 v h = W 2i :=   , i = 1, . . . , npt , Wi

369

(B.15)

(B.16)

except at the nodes belonging to the Dirichlet part of the boundary, where they must be identically zero. Let v h = W λi be the test function in Equation (B.6), with i = 1, . . . , npt , and λ = 1, 2. The Galerkin contribution to the finite element residual, corresponding to node i and degree-of-freedom λ, is: Z Z n+θ,(k) λ,gal λ Ri (uh )= q Wi dΩ − Ω

−

Z µ Ω

λ

Γn

F¯ λ Wi dΓ −

− f (uh ) +

2 X

D

λη

Z

Ω

δt uλh Wi dΩ

(uh )∂x uηh

η=1

¶

(B.17) ∂x Wi dΩ

Remarks B.4. 1. The solution uh (and, therefore, each of its components uλh ) is evaluated at time tn+θ and iteration (k). 2. Equation (B.17) is the (scalar) componential residual, and refers to component λ of node i. We define the 2 × 1 nodal residual vector at node i as   1  Ri  Ri =   , Ri2 and the 2npt × 1 global residual vector as:    R1   .  .  R=  . .   Rnpt

(B.18)

(B.19)


370

3. It is understood that the componential residual is evaluated from element contributions, and later assembled in the global residual vector.

B.2.2

Finite element tangent

B.2.2.1

Preliminaries

Since the solution uh may be reconstructed from the nodal values U a (a = 1, . . . , npt ) through Equation (B.7), the finite element solution at time tn is completely defined by the global vector of unknowns,   n  U1   .  .  Un =   . .   U nnpt

(B.20)

At each time step, the following problem needs to be solved: Given the solution U n at time tn , find the vector of nodal unknowns U n+1 at time tn+1 , such that the finite element residual at time tn+θ is zero, i.e., ¯n+θ Rn+θ ≡ R(U n+1 )¯ = 0.

(B.21)

Equation (B.21) is a system of nonlinear equations, which we solve iteratively using Newton’s method. Given an approximation U n+1,(k) , we update the solution: U n+1,(k+1) = U n+1,(k) + δU n+1,(k) , where the increment δU n+1,(k) is obtained by imposing ¯ n+θ ¯(k) ∂R ¯ δU n+1,(k) = 0. Rn+θ,(k+1) ≈ Rn+θ,(k) + ∂U n+1 ¯

(B.22)

(B.23)

By defining the global tangent matrix: S

(k)

¯(k) ∂Rn+θ ¯¯ := − , ∂U n+1 ¯

(B.24)


371

the increment δU n+1,(k) is the solution to the following linear system: S(k) δU n+1,(k) = Rn+θ,(k) .

(B.25)

Remarks B.5. 1. In a Newton method, the tangent needs to be re-computed, and the linear system solved, at every iteration. 2. If the linearization of the residual in Equation (B.24) is exact, and the approximate solution is close to the actual solution, the method displays second-order convergence. 3. The residual is evaluated at time tn+θ , but the linearization is performed against the unknowns at time tn+1 . 4. The global matrix tangent (B.24) is a 2npt × 2npt matrix with the following structure:



  S=  

S1,1 .. .

... Sij

S1,npt .. .

Snpt ,1 . . . Snpt ,npt

where Sij is the 2 × 2 nodal tangent matrix:   1,2 1,1 Sij Sij  Sij =  . 2,2 S S2,1 ij ij



  ,  

(B.26)

(B.27)

The (scalar) componential tangent is defined as: Sλµ ij

∂Riλ,n+θ := − µ,n+1 . ∂Uj

(B.28)

5. The tangent matrix S for three-phase flow does not have major or minor symmetries, i.e., λµ Sλµ ij 6= Sji ,

µλ Sλµ ij 6= Sij .

(B.29)


372

6. In what follows, we shall consistently use the following notation: i, j: for matrix and vector node indices (subscripts), a, b: for dummy node indices (subscripts), λ, µ: for matrix and vector component indices (superscripts), α, β, γ, η: for dummy component indices (superscripts). B.2.2.2

Storage term

The first two terms in Equation (B.17) correspond to external source terms, and do not depend on the solution. Here, we compute the contribution from the third term —storage, or temporal, term— to the componential tangent. From the definition in Equation (B.28), Sλµ,stor ij

¢ ¡ P µ Z ¶ Z λ N U ∂ δ ∂ a t a a Wi dΩ := − µ,n+1 − δt uλh Wi dΩ = ∂Uj ∂Ujµ,n+1 Ω Ω Z 1 = Wi δ λµ Nj dΩ, δt Ω

(B.30)

where δt = tn+1 − tn is the time increment, and δ λµ is the Kronecker delta, defined as    1 if λ = µ, λµ δ := (B.31)   0 if λ 6= µ. Therefore, the storage term of the 2 × 2 nodal tangent matrix has the form:   Z 1 1 0 Wi 1Nj dΩ , where 1 =  (B.32) Sstor . ij = δt Ω 0 1

Appendix B. Implementation of the multiscale finite element method B.2.2.3

373

Flux term

We now analyze the contribution from the fourth term —advection-diffusion flux term— in Equation (B.17). The contribution to the componential tangent is: Ã Z µ ! ¶ 2 X ∂ Sλµ,flux := − µ,n+1 − Dλη (uh )∂x uηh ∂x Wi dΩ − f λ (uh ) + ij ∂Uj Ω η=1 µ ¶ Z 2 X ∂ η λ λη D (uh )∂x uh dΩ = ∂x Wi µ,n+1 − f (uh ) + ∂Uj Ω η=1 (B.33) µ Z 2 λ λη X ∂f (uh ) ∂D (uh ) = ∂x W i − Nj θ + ∂x uηh Nj θ µ µ ∂uh ∂uh Ω η=1 ¶ + Dλµ (uh )∂x Nj θ dΩ. Defining the 2 × 2 “advection” matrix A through its components Aλµ , 2

∂f λ (uh ) X ∂Dλη (uh ) − ∂x uηh , A (uh ) := µ µ ∂uh ∂uh η=1 λµ

(B.34)

we may write, in its final form, the contribution to the componential tangent: Sλµ,flux ij

= −θ

Z

λµ

∂x Wi A (uh )Nj dΩ + θ Ω

Z

∂x Wi Dλµ (uh )∂x Nj dΩ,

(B.35)

Ω

and the contribution to the nodal tangent matrix: Sflux ij

B.3

= −θ

Z

∂x Wi A(uh )Nj dΩ + θ Ω

Z

∂x Wi D(uh )∂x Nj dΩ.

(B.36)

Ω

Subgrid-scale contribution

The subgrid-scale contribution to the residual is: sgs

R (uh ) = −

nel Z X e=1

Ωe

˜ dΩ, L∗uh v h · u

(B.37)

374

Appendix B. Implementation of the multiscale finite element method where L∗uh v h = −AT (uh )∂x v h − ∂x (DT (uh )∂x v h ),

(B.38)

˜ = τ uh R(uh ), u

(B.39)

¡ ¢ R(uh ) = q − δt uh − ∂x f (uh ) − D(uh )∂x uh .

(B.40)

All terms above are evaluated at time tn+θ , and iteration (k).

B.3.1

Finite element residual

Taking the test function v h = W λi in Equation (B.37), the subscale contribution to the componential residual is Riλ,sgs

=−

nel Z X e=1

2 X

Ωe α=1

where, to simplify notation, we have used

w˜iλ,α u˜α dΩ,

(B.41)

¤α £ w˜iλ,α := L∗uh W λi .

(B.42)

To fully describe the subgrid contribution to the residual, we need to express w˜ iλ,α and u˜α in computable form. From the definition of the adjoint operator in Equation (B.38), w˜iλ,α

£

=− A Ã

¤ T αλ

(uh )∂x Wi − ∂x

λα

= − A (uh ) +

³£

D

¤ T αλ

2 X ∂Dλα (uh ) η=1

∂uηh

(uh )∂x Wi !

∂x uηh

´

(B.43)

∂x Wi − Dλα (uh )∂xx Wi .

Similarly, from the definitions of the subgrid-scales in Equation (B.39), and the timediscrete grid-scale residual in Equation (B.40), u˜α =

2 X

τ αβ (uh )Rβ (uh )

β=1

=

2 X β=1

"

τ αβ (uh ) q β − δt uβ −

µX 2 η=1

Aβη (uh )∂x uηh −

2 X η=1

Dβη (uh )∂xx uηh

¶#

(B.44) .

375


The matrix of intrinsic time scales τ uh may have different expressions. Some alternatives are given in Section 7.2.5 of Chapter 7. For the purpose of this appendix, we shall restrict our attention to the expression proposed by Codina [2000]: τ uh = C−1 uh ,

where Cuh :=

c2 c1 D(uh ) + A(uh ). 2 h h

(B.45)

The implementation of other expressions is similar.

B.3.2

Finite element tangent

The contribution of the subgrid-scale term to the componential tangent is: Ã n Z ! 2 el X X ∂ Sλµ,sgs := − µ,n+1 − w˜iλ,α u˜α dΩ ij e ∂Uj e=1 Ω α=1 ! Ã (B.46) Z nel 2 α X X ∂ u ˜ ∂ w˜iλ,α α ˜ + w˜iλ,α µ,n+1 dΩ. = µ,n+1 u e ∂Uj ∂Uj e=1 Ω α=1 The expressions of the two derivatives involved in the equation above need to be written in computable form. The first one takes the form: Ã ! 2 λ,α λα 2 λα λα X ∂A (uh ) ∂ D (uh ) ∂ w˜i ∂D (uh ) η ∂x N j θ ∂x W i µ η ∂ x uh N j θ + µ,n+1 = − µ,n+1 + ∂uh ∂uh ∂uµh ∂Uj ∂Uj η=1 −

∂Dλα (uh ) Nj θ∂xx Wi , ∂uµh

(B.47) where ∂Aλα (uh ) = ∂Ujµ,n+1

Ã

2

∂ 2 f λ (uh ) X ∂ 2 Dλη (uh ) η µ − µ ∂ x uh α α ∂uh ∂uh ∂uh ∂uh η=1

!

Nj θ −

∂Dλµ (uh ) ∂x Nj θ. ∂uαh

For the evaluation of the subscale derivative, we apply the chain rule: ! Ã 2 β X ∂ u˜α ∂τ αβ (uh ) β ∂R (u ) h αβ = (uh ) . µ,n+1 R (uh ) + τ µ,n+1 ∂Ujµ,n+1 ∂U ∂U j j β=1

(B.48)

(B.49)


376

The derivative of the grid-scale residual is: 2

X ∂Aβη (uh ) 1 ∂Rβ (uh ) η βµ βµ N δ − = − j µ,n+1 µ,n+1 ∂x uh − A (uh )∂x Nj θ δt ∂Uj ∂U j η=1 +

2 X ∂Dβη (uh ) η=1

∂uµh

(B.50)

∂xx uηh Nj θ + Dβµ ∂xx Nj θ,

where ∂Aβη (uh )/∂Ujµ,n+1 is given by Equation (B.48). The derivative of the matrix of intrinsic time scales τ uh requires the derivative of the matrix inverse. Theorem B.6. Let C(ξ) be a C 1 -invertible matrix function of a parameter ξ, and let dξ C be its derivative. Then, with the usual rule of matrix multiplication, the derivative of the inverse is given by: £ ¤ ¤ £ dξ C−1 = −C−1 dξ C C−1 . Proof. By definition of the matrix inverse, CC−1 = C−1 C = 1.

Differentiating, and using the chain rule, ¡ ¢ £ ¤ £ ¤ dξ CC−1 = dξ C C−1 + C dξ C−1 = 0 Rearranging, ¤ £ ¤ £ C dξ C−1 = − dξ C C−1 , and premultiplying by C−1 , we obtain the desired result.

2


377

Recalling Equation (B.45), and using Theorem B.6, we write the final expression of the intrinsic time scales derivatives: £ ¤αβ 2 2 X X £ −1 ¤αγ ∂ C−1 (uh ) ∂τ αβ (uh ) ∂Cγη (uh ) £ −1 ¤ηβ C (uh ), (B.51) = = − C (u ) h ∂Ujµ,n+1 ∂Ujµ,n+1 ∂Ujµ,n+1 γ=1 η=1 where c1 ∂Dγη (uh ) c2 ∂Aγη (uh ) ∂Cγη (uh ) = + . h2 ∂Ujµ,n+1 h ∂Ujµ,n+1 ∂Ujµ,n+1

(B.52)

This expression completes our description of the implementation of the multiscale method for three-phase flow in a finite element code.

To a mathematician, a tensor t is a section of a certain bundle over a manifold. To an engineer or physicist, a tensor tijk is an object dressed in indices. — JERROLD E. MARSDEN and THOMAS J. R. HUGHES, Mathematical Foundations of Elasticity (1983)