Numerical Methods for Contaminant Transport in Porous ... - CiteSeerX

Numerical Methods for Contaminant Transport in Porous Media Richard E. Ewing Suzanne L. Weekes

Abstract. The transport of contaminants in porous media is a complex process that may involve from one to three phases and possibly several components appearing in dierent phases based upon reservoir conditions. Model equations for these ow regimes are described. All models can be put into the form of convection/diusion/reaction equations coupled to mass balance and momentum balance equations. Discretization techniques that address aspects of conservation of mass and accurate and stable treatment of transport are discussed. x1. Introduction

In order to understand the fate and transport of contaminants in groundwater systems, mathematical and numerical models must be developed, which can describe the essential aspects of the ow and transport. Dierent models must be used at dierent length scales, since highly localized phenomena may drastically impact macro-scale ow processes. Once the location of a contaminated plume is determined, one needs accurate models to design and optimize a remediation strategies for the contamination. The partial dierential equation models used in the simulators are convection-dominated. Mixed nite element methods are described to treat the strong variation in coecients arising from heterogeneities. An operatorsplitting technique is then used to address the disparate temporal scales of convection, diusion, and reaction terms. Convection is treated by time stepping along the characteristics of the associated pure convection problem, and diusion is modeled via a Galerkin method for single phase ow and a Petrov-Galerkin technique for multiphase regimes. Eulerian-Lagrangian techniques, MMOC (modi ed method of characteristics) described by Douglas and Russell [18] or Ewing et al. [36] or ELLAM (Eulerian-Lagrangian Localized Adjoint Methods) introduced by Celia et al. [9], eectively treat the advection-dominated processes. Extensions of ELLAM to the multiphase regime appear in [23]. Also higher-order Godunov type schemes are discussed as new possibilities to accurately approximate the transport process in a stable manner. Accurate approximations of the uid velocities needed in the Eulerian-Lagrangian time-stepping procedure are obtained by mixed nite element methods. When reaction terms are present, their rapid eects in relation to convection or diusion must be treated carefully since they strongly aect the conditioning of the resulting systems.

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R.E. Ewing and S.L. Weekes

Even at the nest grid level available using eectively scaled models, many phenomena cannot be resolved. Understanding and modeling of these localized phenomena require the use of adaptive or local grid re nement. Usual implementation of local grid re nement techniques destroys the eciency of large-scale simulation codes. Techniques which involve a relatively coarse macro-mesh are the basis for domain decomposition methods [6, 22] and associated parallel solution algorithms. Accuracy, eciency of implementation, and adaptivity of these techniques have been discussed [21, 33, 34]. In Section 2, we present model equations for various ow regimes. In Section 3, we discuss mixed nite element methods used to obtain accurate uid velocities in the presence of heterogeneities in the medium. In Section 4, we present various operator-splitting numerical schemes to model the transport mechanisms. In Section 5, we brie y discuss the need for adaptive re nement methods to treat highly localized processes.

x2. Model Equations

2.1 Single-Phase Flow

We consider the multiphase ow processes involved in the transport of contaminants in groundwater. The simplest and the most popular is the model of fully saturated incompressible porous media. In this case the water (or the liquid) phase occupies the whole pore space and the ow is due to the nonuniform pressure distribution. The mathematical formulation is based on the mass balance equation and Darcy's law (see, e.g., [5], [54]):

r (u) = q and u = ? K (rp ? g); in ; (1) where u is the volumetric ux of water, q is the source or sink of uid, is the uid density, K is the absolute permeability tensor, is the dynamic

uid viscosity, p is the uid pressure, and g is the acceleration vector due to gravity. Darcy's law provides a relation between the volumetric ux in the mass conservation equation and the pressure in the uid. This relation is valid for viscous dominated ows which occur at relatively low velocities. The transport of a contaminant that is dissolved in the water is described by the following equation:

@ (c) + r (uc) ? r (Drc) + c = G(c); @t

in ; t > 0: (2)

Here c is the concentration of the contaminant, D is the dispersion tensor, is some reaction rate, = is the water content, is the porosity, and G is a source/sink term. When the uid does not have large velocity variations and is assumed to satisfy Fick's Law, the form of the diusion/dispersion tensor D is given by

Numerical Methods for Contaminant Transport in Porous Media

3

D = dmI + juj [dl E (u) + dt(I ? E (u))]

(3)

Eij (u) = ui u2j ; juj

(4)

where

dm is the molecular diusion coecient, and dl and dt are the longitudinal and transverse dispersion coecients, respectively. In general, dl 10dt, but this may vary greatly with dierent soils, fractured media, etc. Also, the viscosity in Equation (1) is assumed to be determined by some mixing rule. In addition to Equations (1) and (2), initial and boundary conditions are speci ed. The ow at injection and production wells is modeled in Equations (1) and (2) via point or line sources and sinks.

2.2 Multiphase Flow

When either an air or vapor phase or a non-aqueous phase liquid contaminant (NAPL) is present, the equations describing two phase, immiscible ow in a horizontal porous medium are given by

@ (w Sw ) ? r K w krw r(p ? g) = q ; x 2 ; t 2 J; (5) w w w w @t w @ (aSa ) ? r K a kra r(p ? g) = q ; x 2 ; t 2 J; a a a a @t a

(6)

where the subscripts w and a refer to the water and air phases respectively, Si is the saturation, pi is the pressure, i is the density, kri is the relative permeability, i is the viscosity, and qi is the external ow rate, each with respect to the ith phase. The saturations sum to unity. Thus, one of the saturations can be eliminated; let Sw = 1 ? Sa . The pressure between the two phases is described by the capillary pressure

pc (Sw ) = pa ? pw :

(7) Although formally, the equations presented in Equations (1) and (2) seem quite dierent from those in Equations (5) and (6), the latter system may be rearranged in a form which very closely resembles the former system. In order to use the same basic simulation techniques in our sample computations to treat both miscible and immiscible displacement, we will follow the ideas of Chavent [10]. Let in R3 represent a porous medium. The global pressure p and total velocity v formulation of a two-phase water (w) and air (a) ow model in

is given by the following equations [11]:

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@(p) + q(x; S ; p); Sa ca dp + r v = ? w dt @t

v = ?K(rp ? G );

x 2 ; t > 0;

x 2 ; t > 0;

(8) (9)

w + r (f v ? K q g ? D(S ) rS ) = ?S @(p) + q ; @S w a w w w w @t w (10) @t x 2 ; t > 0:

The global pressure and total velocity are de ned by

Z S a ? w dpc 1 1 p = 2 (pw + pa ) + 2 d d and v = vw + va : (11) Sc where pc (Sc) = 0. Further, d=dt (@=@t) + va =Sa r, = w + a is the total mobility, i = kri =i , i = w; a, is the mobility for water and air, and K is the absolute permeability tensor. The gravity forces G and capillary diusion term D(S ) are expressed as (12) G = w w + a a g and D(S ) = ?K f dpc

w

a w dS

w

and the compressibility ca and fractional ow of water fw are de ned by (13) c = 1 da and f = w : a

a dpa

w

We note that in this formulation, the only diusion/dispersion term is capillary mixing described by Equation (12). We are currently developing macrodispersion terms to model ow in heterogeneous media at a larger scale.

2.3 Compositional Model

We next consider multiphase ow with mass interchange between phases in porous media. Speci cally, the uid phases will be gas, oil (non-aqueous phase) contaminant, and water, and we shall use the subscripts g, o, and w to refer to them.

S r S r S Kk @ g so o sw w ? r brg (rpg ? g g) @t b + b + b g o w g g r K k r K k so ro sw rw + b (rpo ? o g) + b (rpw ? w g) = qg ; oo w w S K k r S r K k @ o ro v g v rg @t b + b ? r b (rpo ? og) + b (rpg ? g g) = qo ; o g o o g g @ Sw ? r Kkrw (rp ? g) = q ; @t w bw w bw w w

(14)


5

where b is the -phase formation volume factor, rso is the gas solubility in oil, and rsw is the gas solubility in water. A total velocity and global pressure formulation can also be given for this compositional model [49]. Thus, all models in this section can be considered in the same convection/diusion/reaction form.

x3. Mixed Methods

The systems in (1), (8){(10), or the analogue from (14), are solved sequentially. We will discuss the system (8){(9) here. An approximation for v is rst obtained at time level t = tn from a solution of (9) with the mobility evaluated from the value of Sw at time level tn?1 . Equations (8) and (9) can be solved via a mixed nite element methods for the uid velocity. Since the transport term in (10) is governed by the uid velocity, accurate simulation requires an accurate approximation of the velocity v. Because the lithology in the reservoir can change abruptly, causing rapid changes in the

ow capabilities of the rock, the coecient K in (9) and (10) can be discontinuous. In this case, in order for the ow to remain relatively smooth, the pressure changes extremely rapidly. Standard procedures of solving (8) and (9) are to eliminate the velocity and solve the remaining second-order equation as a parabolic partial dierential equation for pressure; the dierentiation of K can produce very poor approximations to the velocity v. In this section, a mixed nite element method for approximating v and p simultaneously, via the coupled system of rst-order partial dierential equations (8) and (9), will be discussed. We de ne certain function spaces and notation. Let W = L2( ) be the set of all functions on whose square is nitely integrable. Let H (div; ) be the set of vector functions v 2 L2( ) 2 such that r v 2 L2 ( ) and let V = H (div; ) \ fv n = 0 on ? = @ g : R R Let (v; w) = vw dx, hv; wi = ? wv ds, and kvk2 = (v; v) be the standard L2 inner products and norm on and ?. We obtain the weak solution form of (8) and (9) by dividing each side of (8) by K (inverting the tensor K), multiplying by a test function u 2 V , and integrating the result to obtain ?(K)?1v; u = (p; ru) + (G; u); u 2 V: (15) The right-hand side of (15) was obtained by further integration by parts and use of the boundary conditions. Next, multiplying (8) by w 2 W and integrating the result, we complete our weak formulation, obtaining dp @(p) (r v; w) = (q; w) ? S c ; w ? ;w : (16) a a dt

@t

For a decreasing sequence of mesh parameters h > 0, we choose nite{dimensional subspaces Vh and Wh with Vh V and Wh W and seek a solution pair (Uh ; Ph) 2 Vh Wh satisfying

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?(K)?1V ; u ? (P ; div u ) = (G; u ); h h h dp h @(p) h

(divVh; wh) = (q; wh) ? Sa ca dt + @t ; wh ;

uh 2 Vh ; (17) wh 2 Wh : (18)

Equations (17) and (18) lead to a saddle-point problem when the compressibility is small requiring care in solution. Preconditioning or ecient iterative methods are essential. Eective preconditioners are presented in [1, 12, 24, 27, 30, 38, 48, 47] and ecient multigrid techniques are being developed [7, 37]. Another need is to develop algorithms, codes, and analyses for mixed methods on general domains. Codes have been written dividing a general skewed brick into either ve or six tetrahedra. Then preconditioners have been developed to eectively solve the resulting methods [28, 56].

x4. Operator-Splitting Techniques

4.1 Characteristic Methods

In nite dierence simulators, the convection is stabilized via upstream-weighting techniques which often leads to severe unphysical diusive eects. In a nite element setting, we use a possible combination of a modi ed method of characteristics and Petrov-Galerkin techniques to treat the transport separately in an operator-splitting mode. In miscible or multicomponent ow models, the convective part is a linear function of the velocity. An operator-splitting technique has been developed to solve the purely hyperbolic part by time stepping along the associated characteristics [18, 35, 59]. This technique was termed a modi ed method of characteristics (MMOC) approach. This technique is a discretization back along the \characteristic" generated by the rst order derivatives from (2). Although the advection-dominance in the original (2) makes it nonself-adjoint, the form with directional derivatives is self-adjoint and discretization techniques for self-adjoint equations can be utilized. This modi ed method of characteristics can be combined with either nite dierence or nite element spatial discretizations. In immiscible or multiphase ow, the convective part is nonlinear. A similar operator-splitting technique to solve this equation needs reduced time steps because the pure hyperbolic part may develop shocks. An operatorsplitting technique has been developed for immiscible ows [14, 19] which retains the long time steps in the characteristic solution without introducing serious discretization errors. The splitting of the convective part of (10) into two parts: f m(S )+b(S )S , is constructed [19] such that f m(S ) is linear in the shock region, 0 S S1 1, and b(S ) 0 for S1 S 1.


7

The operator splitting is de ned by the following set of equations: Sw + d f m(S ) rS ' d S = 0; ' @@t (19) w w dS d w w + r (bm (S )S ) ? r (D(S )rS ) = ?S @'(p) + q ; (20) ' @S w w w w w @t w @

tm t tm+1 , together with proper initial and boundary conditions. As noted earlier, the saturation Sw is coupled to the pressure/velocity equations,

which will be solved by mixed nite element methods [17, 26, 35]. For a fully developed shock, the characteristic solution of (19) will always produce a unique solution and, as in the miscible or single-phase case, we may use long time steps t without loss of accuracy. Unfortunately, the modi ed method of characteristics techniques mentioned above generally do not conserve mass. Also, the proper method for treating boundary conditions in a conservative and accurate manner using these techniques is not obvious. Recently, Celia, Russell, Herrera, and Ewing have devised Eulerian-Lagrangian localized adjoint methods (ELLAM) [9, 53], a set of schemes that are de ned expressly for conservation of mass properties. The ELLAM formulation was motivated by localized adjoint methods [8, 51], which are one form of the optimal test function methods discussed in [2, 14, Lu = f; x 2 or (x; t) 2 ; (21) denote a partial dierential equation in space or space-time. Integrating against a test function ', we obtain the weak form

Z

Lu'd! =

Z

f'dw:

(22)

If we choose test functions ' to satisfy the formal adjoint equation L ' = 0 and ' = 0 on the boundary, except at @ certain nodes or edges denoted by Ii , then integration by parts (the divergence theorem in higher dimensions) yields Z XZ uL 'd! = f'd!: (23) i

Ii

Various dierent test functions can be used to focus upon dierent types of information. Herrera has built an extensive theory around this concept; see [51] for references. The theory is quite general and can deal with situations where distributions do not apply, such as when both u and ' are discontinuous. As in the work of Demkowitz and Oden [16], we want to localize these test functions to maintain sparse matrices. Certain choices of space-time test functions which are useful for linear equations of the form (3) have been described by Demkowitz and Oden [16] and Russell [58]. For examples of nonlinear applications of the form (10), see Ewing [23], Herrera and Ewing [52], and Dahle, Ewing, and Russell [15].

8


Dierences between ELLAM and MMOC for linear partial dierential operators have been discussed by Russell [58] and Russell and Trujillo [60]. These comparisons also apply in the nonlinear problems considered here. The latter reference also contains excellent discussions of the errors involved in numerical integration along the characteristics via various tracking algorithms when the coecients are spatially dependent, and for the terms arising when the adjoint equation is not completely satis ed. Since one motivation for considering ELLAM instead of MMOC techniques was to obtain more accurate treatment of the boundary conditions, we next extend our previous treatment of these terms for constant coecients to the nonlinear case. In summary, we have presented the extensions of the ELLAM ideas discussed by Celia et al. [9], Ewing [23], Herrera et al. [53, 52], and Russell [59, 58] to the nonlinear equations needed to model multiphase ow. The onepoint integration rules in time make this an extension of the MMOC ideas with more accurate treatment of the boundary conditions. More accurate temporal integration rules involving more complex approximation procedures are under development. Recently ELLAM techniques have been extended to a wide variety of applications [15, 40, 39, 42, 43, 41, 44, 45, 63, 64, 65]. Optimal order error estimates have been developed for advection [40], advection-diusion 45, 64, advection-reaction [40, 39, 43, 41, 44, 65], and advection-diusion-reaction [42, 63] systems.

4.2 Godunov Methods

In this section, we consider the approximation of convection equations,

Ut + r F(U ) = Q;

(24)

via a Godunov-type upwind scheme. Upwind schemes, whose origin may go back as far as 1952, have as their germ, the idea that the physical processes of the equation must be introduced into the numerical discretization. In Godunov's scheme, at each time step the conserved properties are considered to be piecewise constant on the grid cells, resulting in local Riemann problems. Solving these local problems exactly introduces the physical properties of the conservation law at a high level of approximation. This results in a scheme which maintains the conservative nature of the equation that it models and captures shock fronts moving at the correct, physical speeds. Godunov's method is only rst-order accurate in space so shock pro les which are produced are smeared. Second-order versions, while resolving solutions better than rst-order schemes have a dispersive property which produces spurious oscillations in regions of rapid transitions. In order to circumvent this problem, `limiters' are used to control the gradients of the computed solution [61]. At solution extrema, the limited higher-order schemes actually revert to the rst-order method, relying on monotonicity principles


9

to prevent oscillations. Godunov's scheme and all of its derivatives [62, 13, 50] are classi ed as Godunov-type methods. We give now a general description of such schemes. Consider an unstructured mesh with polygonal grid cells. Integrating (24) over the j th grid cell 4j at time t gives

@ Z U (x; t) dx = ? Z F(U ) n dx + Z Q(x; t) dx: j; @t 4j @ 4j @ 4j

(25)

This expresses the conservation property that the rate at which the amount of U found in the volume 4j changes is equal to the rate at which U leaves the volume through the boundary of 4j , as determined by the ux function F (U ), plus other external rate eects such as well production or injection, denoted by Q. Indicate the cell average of a quantity over 4j by a single subscript j . In the numerical schemes these averages are simultaneously regarded as the values at the cell centres. From (25) e(j ) Z @ U (x; t) = ? 1 X @t j k4j k =1 Aj; F(U ) nj; dAj; + Qj ;

(26)

where Aj; denotes the th face of 4j , nj; is the unit outer normal to this face and e(j ) = number of faces of 4j ( = number of vertices of 4j in two dimensions). Godunov-type methods proceed directly from conservation and we take a semi-discrete approach in discretizing (26). There are three stages in the process: the reconstruction stage, the solution stage, and the evolution stage. In the rst stage, a piecewise continuous pro le for U is reconstructed out of the cell average data. Next, using this reconstruction, approximations to the ux integrals in (26) are made. This second stage is the physical step of the procedure and requires studying the wave structure of the solution to the conservation law at the cell interfaces. These approximations give a spatial discretization of (26) and result in a coupled system of ordinary dierential equations in time for the conserved quantity U . In the last step, evolution in time is achieved by applying a standard numerical solver for these ordinary dierential equations. We reconstruct an approximate pro le of U at time t which we call U (x). This pro le depends on the values of U at the cell centres and satis es the conservation requirement that 1 Z U (x) dx = U : (27)

k4j k 4j

j

In Godunov's method, U is piecewise constant and naturally takes the value Uj on 4j . This approximation is accurate only up to rst order and results in a scheme which is rst order in space. To obtain higher-order methods, higher order polynomial reconstructions [62, 13, 50] are made relying on data from wider and wider stencils of neighbor cells.

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We restrict our discussion to linear reconstructions for second order accuracy. Letting xj denote the centroid of 4j , the reconstruction must be U (x) = U (xj ) + rUj (x ? xj ); (28) for any constant gradient vector, to satisfy (27). The gradient vector chosen should be the best estimate of the solution gradient in 4j that can be determined from the data in the neighboring cells. For example, on a uniform, one-dimensional grid, the best estimate to the slope is the central dierence approximation. In addition, the linear reconstruction should be exact for linear data. To prevent the introduction of non-physical oscillations into the solution, a monotonicity constraint is imposed on the scheme. This says that the reconstructed pro le should not indicate new extrema which were not necessarily implied by the pointwise data in the cell centres. The gradient in (28) is altered or limited to achieve this end. In cells containing solution extrema then, the gradient is limited to nought giving the constant pro le of the rst-order method. Once there are data which can be assigned to the faces Aj; of 4j , we move to the solution stage where the ux integral at the cell interfaces is approximated. The reconstruction U is generally discontinuous at the faces Aj; , so there are two sets of data viewing from the inside and the outside of ? ; U + respectively, the uxes in (26) are approximated cell j . Calling these Uj; j; as follows: e(j ) Z X

=1 Aj;

F(U ) nj; dAj;

e(j ) X

=1

? ? ; U + jA j : Fj; Uj; j; j;

(29)

Here Fj; is a two-point, Lipschitz-continuous, monotone numerical ux that is consistent with F nj; meaning that

Fj; (a; a) = F nj;(a): (30) Also F (a; b) is a non-decreasing function of a and a non-increasing function of b with the additional conservation requirement that F (a; b) = (?F )(b; a): (31) In the work in this paper, the Godunov ux is used. The Godunov solution is the exact solution at the interface of the one-dimensional Riemann problem ? ; U + , and is given by the formula: with left and right states Uj; j;

min

(w) a < b; Fj; (a; b) = maxw2[a;b] FF nnj; (32) a b: j; (w) w2[b;a] ? ; U + which are determined by the reconstructed pro le, Thus the states Uj; j;

distinguish one scheme from the other.

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Having computed the ux values at the cell interfaces, the evolution stage is completed by solving the system of o.d.e.s formed by (26) and (29), e(j ) ?U ? ; U + jA j + Q W (U ); @ U (x; t) = ? 1 X F j j j j; j; j; j; @t k4j k =1

(33)

via the forward Euler scheme if one is performing the rst order Godunov's method or, for second order accuracy, via the second-order TVD Runge-Kutta scheme: U~j = Uj + tW (Uj ); U~ j = U~j + tW (U~j ); (34) ~ 1 Vj = 2 U j + Uj :

We use Vj to denote the approximate solution to (24) with initial data Uj , at a time t later.

4.3 Gradients and Limiters For the original gradient rUj on 4j in (28), we employ a least squares

reconstruction. The idea of using a least-squares t was introduced in [3] and is further studied for three dimensional work in [66]. We choose the linear polynomial with value Uj at the mass centre xj of the j th grid cell 4j that minimizes the error between the projected values computed at the distance one cell centres and the values of U there. The error is measured in the energy or L2 norm. Figure 1 depicts the 1 *

U C*

*

UD

Uj

U * A

*U B

Fig. 1.

Towards a least squares reconstruction on grid cell 4j shown shaded.

situation for a typical quadrilateral element 4j and its distance one neighbors. The cell-averaged and simultaneously pointwise data is given by UA ; ; UD

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at the points xA ; ; xD . We nd the least squares solution rUj to the over-constrained system of equations below

0 UA ? Uj 1 0 xA ? xj 1 Uj C C B@ xB ? xj CA rUj = BBB UUCB??Uv jC xC ? xj A @ xD ? xj

UD ? Uj :

(35)

Clearly, this procedure yields the exact linear construction from data with linear variation. If the gradients used for the reconstruction of the approximate solution to (24) at time tn are too large, additional extrema may be introduced into the numerical solution producing spurious oscillations when the solution is advanced to the next time level. In high resolution schemes, the gradients are limited until they satisfy a monotonicity constraint to prevent this from occurring. Our monotonicity constraint states that the values produced by the recon? , should not lie beyond the maximum struction at the face centres, called Uj; and minimum values at the cell centres of the neighbouring centroids. and neighboring cells. To this end, we apply a limiter j 2 [0; 1] so that the (28) is replaced by U (x) = Uj + j rUj (x ? xj ): (36) De ne Ujmax = max(Uj ; Uneighbours) and Ujmin = min(Uj ; Uneighbours). Then should be just large enough to ensure that Ujmin U (4j ) Ujmax : (37) We compute

8 U max?Uj > ? ? Uj > 0; min 1; Uj? ?Uj U > j; < j; j = > min 1; Ujmin ?Uj U ? ? U < 0; ; ? ?Uj j j; Uj; > : ?

1 Uj; ? Uj = 0 at each of the faces of 4j , and de ne j = min( ij );

(38)

(39)

Note that if Ujmax = Uj or Ujmin = Uj which is the case when Uj is an extrema, j 0 and the reconstructed pro le on 4j is U (x) = Uj . Once the monotonicity constraint is enforced, it is easy to prove that, under the appropriate CFL constraint, the maximum principle holds. This principle states that a maximum cannot increase and a minimum cannot decrease with time. This guarantees that with the limiters used, the numerical scheme will not produce unphysical oscillations in its solutions. The proof of the following theorem is given in [49]:


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Theorem 1. Consider Vj computed according to (33), (34) with states ? ; U + ; U min and U max de ned as above. Then the maximum principle Uj; j j; j that

Ujmin Vj Ujmax ;

(40)

in the absence of a source term (Qj =0), holds under the CFL condition

where

t e(1j ) ;

(41)

Pe(j) jA j ! =1 j; dF : = max dU 1 j k4j k

(42)

x5. Local-Re nement Techniques It seems natural to relate the size of the coarse domains to the solution of the pressure-velocity equation [19], since the velocity varies slowly and de nes a natural long length scale compared to the variation of the saturation S at a front. A simple local error estimate which determines if a coarse-grid block must be re ned, is given in [19]. Normally, local re nement must be performed if a uid interface is located within the coarse-grid block in order to resolve the solution there. A slightly dierent strategy is to make the region of local re nement big enough such that we can use the same re nements for several of the large time steps allowed by the method. The local gridre nement strategy combined with the operator splitting is de ned in the literature [14, 19]. The solution at groups of the coarse-grid vertices and the local re nement calculations may be sent to separate processors to achieve a high level of parallelism in the solution process. The dicult problem with these techniques is the communication of the solution between the ne and coarse grids. The use of local grid re nement in large-scale simulators often destroys the vectorization and ecient solution capabilities of the codes. Patch approximation techniques coupled with domain decomposition iterative solution methods [6] have proven to be very eective for developing accurate and ecient local grid re nement in the context of existing simulators. Mass balance considerations are very important for accuracy. Approximation techniques for cell-centered nite dierence methods appearing in [33, 34] are discussed and compared with other methods. These techniques can be extended to local time-stepping schemes [25, 29, 32] and to algorithms for mixed nite element methods [31, 46]. Mixed methods are being incorporated into existing nite dierence simulators to address the need for accurate approximation for uid velocities in the context of heterogeneous media.

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x6. Numerical results In this section, we present a computational example which uses some of the ideas and techniques described in the preceding sections. The problem considered is the standard quarter ve spot simulation of two immiscible, incompressible uids. The governing equations are given by equations (8){(10) with the air phase (a) being replaced by a non-aqueous liquid phase (n). We take the absolute permeability tensor K to be constant and isotropic, so we set K = I. The porosity is taken also to be constant. We consider a horizontal system where the gravity eects are neglected. For the example presented: krw = Sw2 ; krn = (1 ? Sw )2 and n =w = 4. Initially, the well is lled with the non-aqueous uid, which is Sw = 0 throughout the reservoir. Water is injected through a well at the lower right corner of the computational domain at a constant rate, and uid is produced at the same rate through a well in the upper left corner. The computation is performed on a triangular nite element grid shown in Figure 2. 2

Fig. 2.

Finite element grid and well arrangement for quarter ve spot simulations..

For the approximation of the pressure system, we use lowest order Raviart-Thomas pairs of nite dimensional spaces [57]. The velocity space Vh is made up of piecewise linear functions with continuous normal components on the triangle edges, while the pressure space Wh consists of piecewise constant functions. This is alternated with the higher-order Godunov method described in Sections 4.2 and 4.3 for the saturation equation.

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Figure 3 shows the water saturation contours after 0.43 pore volumes have been injected. The graphs in Figure 4 display the NAPL recovery and cumulative NAPL recovery results as a function of pore volume injected (PVI). These results are from data on the coarser grid. 3 800 TRIANGULAR ELEMENTS

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NAPL Recovery

0.8

0.6

0.4

0.6 0.5 0.4 0.3 0.2

0.2

0.1 0 0

0.2

0.4

0.6 PVI

Fig. 4.

0.8

1

0 0

0.2

0.4

0.6

0.8

PVI

NAPL recovery results from calculations on 800 element grid..

1

16


x7. Conclusion

We see that a global pressure formulation can be eectively utilized for both single and multiphase ow models. This allows a common basis for accurately approximating the global velocity (or mass ux) via mixed nite element techniques. These methods conserve mass locally and accurately treat large discontinuities in ow properties due to lithological or facies changes in the soil or rock. Characteristic methods can be utilized eectively to stabilized transport equations when the characteristics are easy to approximate. When characteristic methods are not as approximate, higher-order Godunov techniques oer strong possibilities to stabilize transport without introducing too many dispersive diculties. We are currently developing codes based upon these techniques. Finally adaptive grid re nement or time-stepping procedures may be necessary to treat highly localized phenomena in space or time.

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[email protected] http://www.isc.tamu.edu/ ewing/

Suzzane L. Weekes Department of Mathematics Texas A&M University College Station, Texas 77843-3368

[email protected] http://www.math.tamu.edu/ suzanne.weekes/