A Mathematical Model of Horizontal Wells Productivity and Well ...

A Mathematical Model of Horizontal Wells Productivity and Well Testing Analysis Jing Lu

Thesis submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial ful llment of the requirements for the degree of

MASTER OF SCIENCE IN MATHEMATICS

APPROVED:

Tao Lin Robert Rogers

Shu-Ming Sun

August, 1998 Blacksburg, Virginia

Keywords: Horizontal Well, Productivity, Well Testing Copyright 1998, Jing Lu

A Mathematical Model of Horizontal Wells Productivity and Well Testing Analysis Jing Lu Department of Mathematics

ABSTRACT

This thesis presents new productivity and well testing formulae of horizontal wells. Taking a horizontal well as a uniform line source, this thesis nds velocity potential formula and the productivity formulae for a horizontal well in an ellipsoid of revolution drainage volume by solving analytically the involved three-dimensional partial dierential equations. These formulae can account for the advantages of horizontal wells, and they are more accurate than other formulae which are based on two-dimensional hypotheses. This thesis also presents new well testing formulae of horizontal wells in a single porosity system and a double porosity system. Compared with the formulae published in the literatures, our formulae, which do not use the sum of in nite series, are more reasonable and easy to be used in well testing analysis.

ACKNOWLEDGEMENTS

I can not use my words to express my deep gratitude to my advisor, Dr. Tao Lin. Whatever success I have gotten is due to his support and guidance and encouragement. His knowledge, dedication to research and ideas have been invaluable throughout the last two years and this research project was supported in part by the NSF under grant DMS-9704621. Expressions of sincere appreciations and gratitude go to Prof. Tao Lu who is the director of Center for Mathematical Sciences, Institute of Computer Application, Chengdu Branch, Academia Sinica, for drawing my attention to these problems. I would like to thank Dr. Robert Rogers and Dr. Shu-Ming Sun for kindly serving on my committee and for having critically read this manuscript and supplied helpful comments and corrections. For valuable lessons and practices I have learned in mathematics, I am greatly indebted to Professors Martin Day, Robert Wheeler, Layne Watson, Michael Renardy, and Jim Thomson, etc. It was a pleasure and a valuable professional experience to associate with all the members of Department of Mathematics. I thank all of them for their corporations, friendship and kindness which made my stay at Virginia Tech. meaningful and enjoyable. Deepest appreciations are extended to my mother for her love and many sacri ces she poured to give me the opportunity to pursue higher education.

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Contents 1 Introduction and Literature Review 2 Basic Equations 3 Velocity Potential Analysis

1 6 15

4 Productivity Formulae

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3.1 Equipotential Surfaces of Horizontal Wells : : : : : : : : : : : : : : : : : : : : : : : 16 3.2 Average Velocity Potential of Horizontal Wells : : : : : : : : : : : : : : : : : : : : : 20 4.1 Formulae for Wells at Midheight of Formation : : : : : : : : : 4.2 Well Eccentricity Problem : : : : : : : : : : : : : : : : : : : 4.2.1 Point Convergence Pressure Distribution Formulae : 4.2.2 Dimensionless Pressure Formulae of Horizontal Wells : 4.2.3 Productivity Formulae for Eccentricity Wells : : : : : 4.3 Comparisons of Productivity Formulae : : : : : : : : : : : : :

5 Well Testing Formulae for Single Porosity Reservoir

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5.1 Point Convergence Pressure Distribution Formula : : : : : : : 5.2 Dimensionless Pressure Formulae in In nite Reservoirs : : : : : 5.3 Dimensionless Pressure Formulae in Reservoirs of Finite Height 5.3.1 Reservoirs with Impermeable Boundary Conditions : : : 5.3.2 Reservoirs with Bottom Water or Gas Cap : : : : : : :

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6 Well Testing Formulae for Double Porosity Reservoir

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A Conclusions B Nomenclature

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6.1 Warren-Root Model : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 55 6.2 Laplace Transform Images of Point Convergence Pressure : : : : : : : : : : : : : : : 56 6.3 Well Testing Formulae for Wells in Finite Height Reservoirs : : : : : : : : : : : : : : 61

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List of Figures : : : : 3.1 Horizontal Well Model. : : : : : :

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: : : : : : : 3.4 Schematics of Vertical and Horizontal Well Drainage Volume. :

2.1 2.2 2.3 2.4

4.1 4.2 4.3 4.4

Element of Surface in Volume. Viscous Flow at P in Volume. : Element of Surface in Volume. Flow Geometries. : : : : : : : :

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Original Reservoir System with x/z Anisotropy and Horizontal Well. Transformed Isotropic Reservoir System with Elliptic Wellbore. : : Axis Dimensions for Transformed Wellbore. : : : : : : : : : : : : : Division of 3D Horizontal Well Problem into Two 2D Problems. : :

6.1 Warren and Root's Sketch of a Naturally Fractured Reservoir. : : :

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7 9 10 13 15 17 18 19 24 24 25 38 56

List of Tables 3.1 The ratioes of velocity potential of endpoint and midpoint. : : : : : : : : : : : : : : 21 3.2 The comparisons of the two methods of calculating average velocity potential. : : : : 21 4.1 Dimensionless pressure of endpoint under dierent boundary conditions. : : : : : : 35 4.2 The comparisons of methods to compute the productivity of a horizontal well. : : : 42 5.1 The convergence rate of the integration of K0 (z ). : : : : : : : : : : : : : : : : : : : : 50

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Chapter 1

Introduction and Literature Review A worldwide interest exists today in drilling horizontal wells to increase productivity. The production section of the horizontal well must be parallel to horizontal line. Because of its large

ow area, a horizontal well may be several times more productive than a vertical one draining the same volume. Recent interest in horizontal wells has been accelerating because of improved drilling and completion technology. This has led to increased eciency and economics in oil recovery. Increases in oil production rate and improvement in ultimate recovery has given horizontal wells the edge over vertical wells in many marginal reservoirs. However, it is more expensive to drill a horizontal well than a vertical one. Therefore, to determine the economical feasibility of drilling a horizontal well, the engineers need reliable methods to estimate its expected productivity. The advantages of horizontal wells can be classi ed under the following topics: (1)Increased productivity or injectivity; (2) Improved sweep eciency; (3) Reduced coning or viscous ngering; (4) Increased drainage area. The reduction in gas or water coning may actually be considered an improvement in vertical sweep eciency or a result of increased productivity. The advantage of horizontal wells when intersecting undrained areas is obvious, but dicult to predict. Some of the disadvantages of horizontal wells compared to vertical wells are: (1) Higher cost; (2) More dicult to log, stimulate and selectively perforate; (3) Limited recompletion alternatives for high gas or water rates; (4) Vertical permeability barriers limit vertical sweep eciency. In petroleum engineering, well productivity means well ow rate { the output of a well during a unit time length under a steady pressure drop. Well productivity formula is the formula to compute the productivity with the known well parameters and formation parameters such as well length, wellbore radius, formation thickness, reservoir uid viscosity, formation permeability, etc. Well testing is a technique to evaluate the reservoir and well parameters according to the mathematical analyses of transient pressure behavior { the pressure drop or pressure buildup measured at the wellbore. Well testing formula is the formula to describe the relationship between the time and wellbore pressure drop or pressure buildup. In this thesis, we will introduce a three dimensional (3D) model of horizontal wells, and will derive the productivity formulae and pressure drop well testing formulae that are based on the 3D model. Productivity of a horizontal well can be greater than that of vertical wells for several reasons. First, horizontal wells can be open to much greater portion of the reservoir than vertical wells. Horizontal wells can be drilled perpendicular to oriented natural fractures, and therefore, intersect with more fractures. Also, it may be possible to induce multiple hydraulic fractures in a horizontal well. The main bene t of improved productivity obviously is higher oil production rate, which must be sucient to justify the cost of drilling a horizontal well. If the main purpose for a horizontal 1

CHAPTER 1. INTRODUCTION AND LITERATURE REVIEW

2

well is to increase oil production rate, and well ow rate will not be limited by tubing or surface facilities, then productivity formula can be used to estimate the well length needed to obtain the desired oil rate. Another bene t of increased productivity is reduced the drawdown for the same withdrawal rates, possibly resulting in reduced water and /or gas production. If surface facilities are limited by gas or water handling capacity, this may mean that overall eld oil production rate can be increased. In condensate systems, increased productivity can result in reduced liquid dropout near the well. There are several potential problems that may limit the productivity increase. Skin damage may be dicult to remove from horizontal wells. Also, an eective low vertical permeability due to shales, etc., may mean that horizontal well length must be very long to obtain sucient productivity improvement. Real sweep eciency can be better in horizontal wells compared to vertical wells for favorable well orientations in pattern oods. As horizontal well length approaches a value equal to the distance between injectors and producers, areal sweep eciency theoretically will approach 100 percent. The vertical sweep eciency will depend on where the horizontal well is completed in the vertical section. The eciency could be greater or less than that of vertical wells. For example, if vertical barriers are present, vertical sweep eciency could be very poor. Sweep eciency is best evaluated by numerical simulation. For oil production from a reservoir with a gas cap and/or aquifer, horizontal well sweep eciency can be better than that of vertical wells because the gas or water crest due to horizontal wells is larger than the cone due to vertical wells. Very long horizontal wells may even allow the reservoir to produce below the critical coning rate. Horizontal wells can be completed farther away from the gas - oil contact, delaying breakthrough of gas and water. However, it may be dicult to produce oil located above the horizontal well for water - oil systems and similarly for oil below the well in gas - oil systems. For systems with both gas - oil and water - oil contacts, optimum horizontal well placement between the contacts is important. Optimum placement depends on the strength of the aquifer or gas cap, phase densities, viscosity, relative permeability and the ability to handle gas or water. Reservoir simulation is the best way to study optimum placement of the horizontal well in such systems. In general, horizontal wells are believed to perform better than their vertical counterparts in thin reservoirs, naturally fractured reservoirs ( double-porosity and discretely fractured ), reservoir with water - and gas - coning problems, and reservoirs with favorable vertical permeability anisotropy. Thus, horizontal wells should be useful in cases of thin-layered reservoirs, heavy oil, and reservoirs with gas - or water - coning problems. A convenient model to represent the pressure behavior in a horizontal well drainhole is on assumes no pressure drop in its interior during uid ow. This means that pressure is uniform along the wellbore face, and the well is said to have in nite conductivity. In practice, it is not feasible to evaluate the wellbore pressure directly from the in nite conductivity model solution, this kind of solution is then approximated with either an equivalent pressure-point or pressureaveraging technique. The main goal of this thesis is to develop necessary mathematical analyses for the horizontal well, it includes the following objectives: (1) Derive potential formula and show that horizontal wells do not have in nite conductivity by taking a horizontal well as a uniform line source in three dimensional space; (2) Derive productivity formulae and pressure drop well testing formulae of horizontal wells by using equivalent pressure point technique and pressure averaging technique.


3

For both vertical and horizontal wells, steady-state and unsteady-state pressure-transient testings are useful tools for evaluating in-situ reservoir and wellbore parameters that describe the production characteristics of a well. The use of transient well testing for determining reservoir parameters and productivity of horizontal wells has become common because of the upsurge in horizontal drilling. During the last decade, analytic solutions have been presented for the pressure behavior of horizontal wells. There have been several attempts to describe and estimate horizontal well productivity and/or injectivity indexes, sweep eciency, and several models have been used for this purpose ([26, 32, 47, 48, 72, 78]). Following the tradition of vertical well productivity models, analogous well and reservoir geometries have been considered. A widely used approximation for the well drainage is, conveniently, a parallelepiped model with no- ow or constant-pressure boundaries at the top or bottom, and either no- ow or in nite-acting boundaries at the sides. One of the earliest models was introduced rst by J.P. Borisov ([15]) in 1964, which assumed a constant pressure drainage ellipse whose dimensions depend on the well length. Later, in 1984, using Borisov's equation, F.M. Giger reported reservoir engineering aspects of horizontal drilling, developed a concept of replacement ratio, FR , which indicates the number of vertical wells required to produce at the same rate as that of a single horizontal well ([43, 44, 45]). The repalcement-ratio calculation assumes an equal drawdown for the horizontal and vertical wells. In addition, Giger studied fracturing of a horizontal well and provided a graphical solution to calculate reduction of water coning using horizontal wells ([46]). In 1987, L.H. Reiss reported a productivity-index equation for horizontal wells ([88]). In 1988, S.D. Joshi ([60]) presented an equation to calculate the productivity of horizontal wells and a derivation of that equation using potential { uid theory. That equation may also be used to account for reservoir anisotropy. To simplify the mathematical solution, Joshi reduced the threedimensional drainage problem into two two-dimensional problems. In 1989, D.K. Babu reduced a complex equation to an easy-to-use equation for calculating productivity of horizontal wells, requiring that the drainage volume be approximately box-shaped, and all the boundaries of the drainage volume be sealed ([6, 7]). In 1991, C.Q. Liu reported a two - dimensional theoretical equation to calculate oil production from a horizontal well, however the report does not include the derivation of the equation ([73]). In 1993, Z.F. Fan used conjugate transform method, got the productivity formula of a horizontal well in a reservoir with bottom water drive, his formula may be used to account for well eccentricity ( i.e., horizontal well location other than midheight of a reservoir ) ([39]). Determination of transient pressure behavior for horizontal wells has aroused considerable interest over the past 10 years. Transient pressure analysis of horizontal well is considerably more complicated than it is for vertical wells because of the potential occurrence of several transient ow periods in contrast to the occurrence of essentially one ow period for vertical well. An extensive literature survey on horizontal wells can be found. Interpretation of well tests from horizontal wells is much more dicult than interpretation of those from vertical wells because of a considerable wellbore storage eect, the 3D nature of the ow geometry and lack of radial symmetry, and strong correlations between certain parameters. Analytical solutions for the pressure behavior of uniform- ux, as well as, in nite-conductivity horizontal wells have been discussed in the literatures [1, 33, 35, 50, 51, 69, 70, 71, 91], etc. In general, the techniques explaining the pressure-transient response in horizontal wells can be grouped into two categories:(1) solutions to the pressure-transient response of horizontal drainhole based on the use of source and Green's functions and (2) solutions based on the use of integral


4

( Laplace and Fourier ) transforms ([16, 23, 66, 67, 68]). Most work dealing with the horizontal well pressure transient problem uses the instantaneous Green's function technique developed by A.C. Gringarten and H.J. Ramey to solve the 3D isotropic diusivity equation ([23, 53, 54]). P.A. Goode and R.K. Thambynayagam used nite Fourier transforms to solve the anisotropic problem for the line-source case ([50]), they presented a solution for an in nite-conductivity horizontal well located in a semi-in nite, homogeneous and anisotropic reservoir of uniform thickness and width. E. Ozkan compared the performances of horizontal wells and fully-penetrating vertical fractures ([81, 82, 83, 84]). For the horizontal wellbore, both in nite-conductivity and uniform- ux boundary conditions were used. F. Daviau also analysed the pressure behavior of horizontal wells, considering both in nite-conductivity and uniform- ux inner boundary conditions ([31]). They noted that the in nite-conductivity approximation related more closely to the real case than uniform- ux approximation. M.D. Clonts considerd the pressure response of a uniform- ux horizontal drainhole in an anisotropic reservoir of nite thickness, but in nite horizontal extension ([27]). They identi ed two possible transient ow regimes. F.J. Kuchuk extended the previous works ([31, 50, 81]) on pressure transient bevavior of horizontal wells to include the eects of gas cap and/or aquifer ([70]). They computed the pressure response at the well by averaging the pressure along the length of the well, rather than using an equivalent pressure point. A.S. Odeh and D.K. Babu noted that in nite or semi-in nite extension assumption of the reservoir in the horizontal plane, used by previous authors, could lead to the occurrence or nonoccurrence of some of the transient ow regimes. Therefore, they assumed the reservoir to be completely sealed in all three directions, identi ed four possible transient ow regimes for a horizontal well in a closed, box-shaped reservoir ([8]). R. Agullers and R.A. Beier studied the transient pressure behavior of horizontal wells in anisotropic naturally fractured reservoirs ([3, 12]). R.M. Butler, R.A. Hamm studied the gravity drainage to horizontal wells and the eect of gravity on the movement of water-oil interface for bottom water driving upwards to a horizontal well physically and theoretically ([18, 19, 20, 21, 56, 63]). F.M. Giger, P. Papatzacos, R. Suprunowicz, W.S. Huang and S.D. Joshi et al. studied the cone breakthrough, water ooding, thermal oil recovery problems for horizontal wells ([42, 46, 57, 62, 85, 95, 96, 97]). R.A. Novy pointed out that frictional losses create a pressure drop within a horizontal wellbore, thus friction can thus reduce productivity ([80]). Numerical simulation is a powerful tool for comparing the productivity of vertical and horizontal wells since it can account for heterogeneities, multi-phase ow and a variety of boundary conditions. The accuracy of numerical simulation often depends on numerous factors such as grid size, time-step size, solution methods and accuracy of the input data. In fact, reservoir description can be considered a limiting factor in accurately predicting future performance ([47, 48]). The literatures survey on horizontal wells numerical simulation can be found in [5, 9, 98]. A generalized semi-analytical productivity model, accounting for any well and reservoir con guration, has been constructed and presented recently. The model allows for the production or injection prediction of any well and reservoir con gurations in both isotropic and anisotropic media. A concern in modern reservoir management is the potential desirability of multiple horizontal laterals, frequently emanating from the same vertical well. Using the analytical solution of M.D. Clonts ([27]), D. Malekzadeh and D. Tiab have presented type curves and appropriate equations to be used in determining transmissivity and storativity from horizontal well interference test data ([58, 75]). T. Zhu discussed both multi-well as well as single well interference test analyses of horizontal well data, with the objective of obtaining estimates of the transmissivity and storativity and detection and location of reservoir boundaries ([105]). D. Malekzadeh ([76]) has presented a


5

solution for interference testing between horizontal and vertical wells, and has explained how this solution deviates from the exponential-integral solution. T. Zhu and D. Tiab ([104]) have presented analytical solutions for Multi-Point-Interference testing in a single horizontal well located in an in nite reservoir with a linear discontinuity, and transient pressure data are measured at one or more perforated horizontal sections, while uids is produced at alternate sections. A. Retnanto studied the performance of multiple horizontal well laterals in low - to medium - permeability reservoirs ([90]). Formation damage can be described as any phenomenon induced by the drilling, completion or stimulation process or by regular operations resulting in a permanent reduction in the proinjectivity of a water or gas injection well. Invasive formation damage can occur by the introduction of: (a) Foreign potentially incompatible uids into the formation; (b) Natural or arti cial solids; (c) Extraneous immiscible phases; (d) Physical mechanical damage. A few detailed discussions of mechanisms of formation damage in horizontal wells have been presented in the literatures [10, 13, 41, 89, 99, 103]. Formation damage tends to be more signi cant in horizontal vs. vertical wells for a number of reasons, some of these being: (1) Longer uid exposure time to the formation during drilling and greater potential depth of invasion in situations where sustained uid and solids losses to the formation are apparent; (2) The majority of horizontal wells remain as open hole or slotted liner completions, therefore, shallow damage, which would normally be perforated through in a typical vertical completion, may remain as an impermeable or low permeability barrier to oil or gas

ow; (3) Drawdowns applied in many horizontal wells result in selective cleanup of a small portion of the total exposed available ow area, causing the majority of the production from a relatively small fraction of the exposed wellbore face; (4) Selective stimulation in wells where slotted liners are in place is inecient. Extensive stimulation of any horizontal section is generally dicult and expensive in comparison to a vertical well, and hence many stimulation programs are ineective due to cost and/or time limitations. Most procedures which result in the contact of the formation by foreign uids or solids have a potential of permeability impairment. The most common of these would include: (a) Drilling; (b) Completion procedures; (c) Workover/kill procedures; (d) Stimulation procedures; (e) Injection procedures. The ow eciency of horizontal wells was derived by G. Renard assuming steadystate ow of an incompressible uid in a homogeneous, anisotropic medium ([89, 103]). T.P. Frick presented conceptual and mathematical descriptions of the damage along and normal to a horizontal well ([41]). D. Tiab pointed out that during the production, some sections of the horizontal well which have severe damage will not contribute to the production ([99]). The wide use of horizontal wells has required that the standard procedure for matrix acidzing be adapted to the new environment. Two major factors must be considered when designing an acid treatment for a horizontal well: the area exposed to the formation is large, and the horizontal and vertical permeabilities must be taken into account ([37, 38, 77]). Althougth productivity of horizontal wells could be two to ve times higher than productivity of vertical wells, fracturing a horizontal well may further enhance its productivity, especially when formation permeability is low. Presence of shale streaks or low vertical permeability that impedes uid ow in the vertical direction could make fracturing a horizontal well a necessity ([93]).

Chapter 2

Basic Equations This chapter will serve as an intruction to the basic equations in reservoir engineering. The theory on the ow of reservoir uid is based on the general principles of single-phase uid

owing through porous media. The driving forces for the ow of reservoir uid are reservoir uid potential gradients, temperature gradients, electrical gradients and chemical gradients. Under reservoir formation conditions, the net force Ef acting on a unit mass of reservoir uid can be given by ([14])

Ef = ,rf , rT , rE , rC;

(2.1)

where, f = reservoir uid potential, T = temperature of the reservoir uid, E = electrical potential of the reservoir uid, C = chemical potential of the reservoir uid. In this thesis, when we refer to the dimension of a physics parameter, L stands for length, its international unit is metre (m); T stands for time, its international unit is second (s); M stands for mass, its international unit is kilogram (kg). The reservoir uid potential gradient is the main driving force for reservoir uid. At a certain point the reservoir uid potential { f , i.e., the mechanical energy per unit mass of reservoir uid, and the corresponding total head hT , i.e., the mechanical energy per unit weight, for reservoir uid whose density is a function of pressure only, is given by M.K. Hubbert as ([24, 29, 30]) h = ghT =

p

Z

p0

dP + (g=g )d; c (P )

where, h = Hubbert's potential of reservoir uid [L2 T ,2 ], hT = total head of reservoir uid [L], g = acceleration due to gravity [LT ,2 ], gc = dimensionless number of g at sea level, = density of reservoir uid [ML,3], P = pressure of reservoir uid [ML,1 T ,2 ]. 6

(2.2)

CHAPTER 2. BASIC EQUATIONS

7

P

ds

R

Figure 2.1: Element of Surface in Volume. To derive an expression for potential of a uid at a point, M.K. Hubbert de ned it as the amount of mechanical energy to transform a unit mass from some reference level to an arbitrary level, d. Equation (2.2) is Hubbert's potential which is valid for both compressible and incompressible

uids. For either case, the gradient of the potential is ([5, 28, 98]) (2.3) r = 1 rP + (g=g )rd: h

c

Some authors de ne another potential function, , where = h and r = rh in which equation (2.3) becomes ([98])

r = rP + (g=gc )rd = rP + rd;

(2.4)

where,

= (g=gc ), pressure per unit distance [ML,2 T ,2 ], is also called speci c gravity, = potential of reservoir uid [ML,1 T ,2 ], P = pressure of reservoir uid [ML,1 T ,2 ]. We have the below relationship = P + d: (2.5) The net driving force for reservoir uid that results from the reservoir uid potential gradient only, can thus be expressed as Ef = ,rf : (2.6) The direction of this force Ef is perpendicular to the equipotential surfaces of the reservoir

uid. The uid will be driven in the direction of Ef , i.e., in the direction of decreasing potential. The three-dimensional ow of reservoir uid through the subsurface can be described by a combination of Darcy's equation for reservoir uid with a continuity equation ( or mass balance equation ) and equations of state for the reservoir uid and the porous medium ([24, 28, 30]).


8

We all know that Gauss' theorem ( also called the Divergence Theorem ) relates an integral over a volume, R, to an integral de ned on its surface, S , namely, Z

Z

R

Z

div~v d = ~v d~ = ~v ~nds; S

S

where ~v is a velocity vector in R, d is a dierential element of volume in R, d is a directed element of surface = ~nds, and ~n is an outward drawn unit vector normal to the scalar surface element, ds as depicted in Figure 2.1. If we consider the uid ux ~q = ~v at a point P , then Z

R

div(~v )d =

Z

S

~v ~nds:

(2.7)

Now since ~v ~nds = j~v jj~ndsj cos = dsj~vj cos where is the angle between vectors ~n and ~v, then ~v ~nds physically represents the component of the uid ux escaping from R through the element of surface ds in the direction of the outward drawn normal. Consequently, the integral of this quantity over the entire surface of R, i.e., the right-hand side of equation (2.7) represents the rate of decrease of mass from R. This can also be expressed as Z , @ () d; R

where is porosity. Therefore it follows that or combining (2.7) and (2.8)

Z

S Z

R

@t

~v ~nds = ,

@ () d; R @t

Z

div(~v )d = ,

Z

@ () d: R @t

(2.8) (2.9)

Since R is an arbitrary volume, it follows that the arguments of the integrals in equation (2.9) are identical, i.e., ) : div(~v ) = , @ (@t (2.10) Equation (2.10) is known as the continuity equation. It simply is an expression of the law of conservation of mass at a point P in R. If a source or a sink is a present at the point P , then we add a mass rate term, q~ say, to the continuity equation ([28, 64, 92]), r (~v) q~ = , @ () : (2.11)

@t

The choice of sign on the additive term is purely arbitrary. We adopt the convention that the minus sign represents a source and the plus sign a sink. Therefore the mathematical form of the mass balance in porous media is given by the continuity equation which may be written using the tensional notation as equation (2.10) or (2.11). To arrive at the basic equations that describe reservoir ow, we make use of the continuity equation (2.11), an expression for the super cial ow velocity in a porous medium ( Darcy's Law ), a mathematical expression for ow potential, and appropriate equations of state. In so doing, we take an Eulerian point of view; i.e., we focus our attention on xed points of space within the eld


9 P

v

z

Solid

Figure 2.2: Viscous Flow at P in Volume. of ow, in contradistinction to the Lagrangian method, where the coordinates of a moving particle are represented as functions of time. We furthermore invoke the basic assumptions enumerated below: (1) Flow is laminar and viscous; (2) Flow is isothermal; (3) Electrokinetic eects are negligible; (4) Diusion eects are negligible; (5) Flow is irrotational. In keeping with assumption (4), we con ne our attention to immiscible uids throughout this thesis, and we also restrict ourselves to a single-phase ow. The geometric complexity of the pores, however, does not permit formulation of the boundary conditions for the ow through a porous medium. Thus, a dierent approach must be taken. Darcy discovered a simple relationship between the velocity vector and pressure gradient for a single phase viscous ow. A basic law for uid mechanics is Newton's Law for viscous uids. Consider a uid owing along a solid interface as shown in Figure 2.2 with velocity, ~v. Newton's Law states

dF = ( ddzv )surface

where = viscosity and dF is the dierential of the viscous force. Since ow in the reservoir is very tortuous, (dv=dz )surface is very dicult to evaluate and estimations must be used. If we neglect inertial eects, then v is proportional to the ux divided by the bulk external area, A, of the porous element under consideration, i.e., v is directly proportional to q/A, similarly, dv/dz is also directly proportional to q/A. Moreover, if we could integrate over the porous surface, the result would be proportional to the bulk volume, AL, of the porous element. ( L is length of the bulk. ) This leads to the conclusion that the total viscous force F , is given by

F = Aq AL;


10

v n

ds

P

dz

Figure 2.3: Element of Surface in Volume. where is a proportionality parameter. Thus, F = vL. Now consider an element of surface in the neighborhood of the point P as shown in Figure 2.3. The component of the resisting force is the viscous force along the axis of the outward drawn normal, i.e., F = ,v ndsdz; We use a negative sign because the force is directed opposite to the outward drawn normal. The component of the driving forces result of a potential gradient at P acting on an element of surface ds, i.e., Fd = rh ndsdz: At equilibrium F = Fd , rh ndsdz = ,v ndsdz: This implies rh = ,v; or rh : v = , Now, can be expressed in terms of a rock characteristic, viz., the permeability. From laboratory experiments one deduces that = =[K ]2 where [K] is a permeability tensor. Substituting for in the above equation gives Darcy's Law ([28, 30]) ( note that rh = r ), we have

~v = , [K ] rh = , [K ] r;

(2.12)

where ~v is de ned as a volumetric ow rate across a unit cross-sectional area ( solid and uid ) averaged over a small region of space, its dimension is [LT ,1 ], the dimension of permeability K is [L2 ], the dimension of viscosity is [ML,1 T ,1 ]. This semi-empirical relationship (2.12) is used to describe single-phase ow in porous media in lieu of a momentum equation.


11

Combining equations (2.4) and (2.12), and suppose the coordinate in the vertical downward direction is z, then we can write , gg rz = , rz: c

In reservoir engineering, we often de ne the velocity potential as follows ([24, 92]): v = K = K (P + z )

(2.13)

and the dimension of velocity potential v is [L2 T ,1 ]. By equations of state, we mean relationship that relate density to pressure at a point. Reservoir uids are considered compressible and, at constant reservoir temperature, we can de ne an isothermal compressibility of uid as a positive term Cf as follows ([5, 34, 74]) (2.14) C = , 1 @V = 1 @ : f

V @P

@P

where V denotes original volume and P is pressure. Gas compressibility is signi cantly greater than those of liquid hydrocarbons, which in turn are greater than those of reservoir waters. The subscript terminology for the compressibilities of gas, oil and water is Cg ; Co ; Cw . Reservoir pore volume may change with change in uid pressure, resulting in an increased fraction of overburden being taken by reservoir rock grains. If the variation of pore volume with pressure is the rock compressibility CR , we have @ : CR = 1 @P (2.15) Liquid hydrocarbons are often assumed to exhibit constant compressibility. Equation (2.14) can be integrated to yield = 0 exp[Cf (P , P0 )]; (2.16) where the zero subscripts refer to a datum level, 0 is the density at the reference pressure P0 . If the reservoir liquid is a slightly compressible, then by Taylor's expansion,

= 0 f1 + Cf (P , P0 ) + Cf2 (P , P0 )2 =2! + g: If we neglect terms of power two and higher, we get

= 0 [1 + Cf (P , P0 )]: Similarly, we have

(2.17)

= 0 [1 + CR (P , P0)]; where o is the porosity at the reference pressure P0 . If the uid is a gas, we employ the gas law PV = nZRT = mZ=(MRT ) where m is mass, M is the molecular weight, T is temperature, and Z is gas deviation factor, and PM = ZRT ([4, 55]). Since ( [5, 55])

then

@ = 1 , 1 dZ ; Cf = 1 @P P Z dP @ = M @ ( P ): @P RT @P Z


12

Consequently,

@ ( P ); Cf = PZ @P Z for a nonideal gas. For an ideal gas Z = 1, so Cf = 1=P .

In reservoir engineering, we often de ne the total compressibility coecient as follows ([5, 34, 74]): Ct = 0 Cf + CR (2.18) The dimension of Cf ; CR and Ct is [M ,1 LT 2 ]. In summary, we may write equation of state as

= (P ):

(2.19)

We now have the following relationships: continuity equation (2.11), ow potential equation (2.2), Darcy's law (2.12) and equation of state (2.19). We emphasize that each of these equations are prescribed at some point in pore space. By combining them in an appropriate manner, one can arrive at the ow equations for single-phase uids in a porous system. Their subsequent integration over space provides a dynamic picture of global phenomena i.e., at all points in space. ( Here we use the term \integration" rather loosely, including obtaining a solution or approximate solution in some sense.) Another important point is that equations (2.11), (2.2), (2.12) and (2.19) are valid in any coordinate system. Consequently, for a given reservoir, we need only specify the coordinate system, the number of dimensions ( i.e., 1D, 2D, or 3D ), and the type of uid present to arrive at a speci c ow expression. When the entire pore space is occupied by a single phase compressible ow, after combining equations (2.10), (2.12) and (2.2), we have (2.20) r f [K ] (rP + rd)g = @ () :

@t @ , this implies that rP = (1=C )r, so equation (2.20) can be written If Cf = 1 @P f [K ] (rP + C rd)g = @ ; r f C f @t

(2.21)

f

for a system with constant porosity. In Cartesian coordinates, we have ([98]) !

!

"

#

@ Kx @ @ Ky @ @ Kz @ @ (2.22) @x Cf @x + @y Cf @y + @z Cf ( @z + Cf ) = @t : If again, the medium is homogeneous and isotropic and is constant, we get ([28, 30, 94]) K r2 = 1 @ ; = (2.23) @t Cf neglecting gravity eects. This is known as Fourier's equation or the diusivity equation. If the

uid is slightly compressible, then one can employ (2.17) to obtain an equation identical to (2.23) where is replaced by P . In equation (2.23), the constant is frequently called the diusivity coecient. We have assumed constant rock properties (K; ), constant viscosity (), negligible gravitational eects and ignored the square of the pressure gradient, and there holds ([92])


13

PLAN

ELEVATION

a)

LINEAR

b)

c)

RADIAL

SPHERICAL

Figure 2.4: Flow Geometries.

K r P = 1 @P @t ; = Cf : 2

(2.24)

Note that in addition to the assumptions already made, the validity of (2.23) and (2.24) are limited to isotropic media, horizontal formation and the uid ow obeying the Darcy's law. We know that thermodynamic behavior of gases can be described by the equation of state ([55]),

PM : = ZRT Insertion above equation into (2.20) and subsequent simpli cations results in the ow equation for ideal gases ([55]) 2 K (2.25) r2P 2 = 1 @P @t ; = C : f

The ow equations, (2.23),(2.24),(2.25) can be written in terms of rectangular ( Cartesian ), cylindrical and spherical coordinates. Since ow is normally dominant in one direction only, a suitable choice of the coordinate system may lead to a substantial mathematical simpli cation. A well is considered to produce from a symmetrical drainage area of a uniform pay thickness. Furthermore, a well is open to ow over the entire pay thickness so that the ow lines converge towards a pressure sink, the wellbore itself ( see Figure 2.4 ). Such a ow model is called the radial-cylindrical or simply the radial ow model and equation (2.24) is then reduced to ([94]) @P = K 1 @ r @P : (2.26)

@t

Cf r @r

@r

Partial well completion and/or a very thick formation may be better described by using the spherical geometry ([30]). The uid is assumed to ow towards the wellbore which is approximated by a central point. The radial form of (2.24) in spherical coordinates becomes @P = K 1 @ r @P : (2.27)

@t

Cf r2 @r

@r

Fluid ow in fractured wells of in nitely large conductivity or in long narrow channels can often be characterized by linear ow in a horizontal plane. In terms of the rectangular coordinates, the


14

one - dimensional form of the ow equation is

@P = K @ 2 P : @t Cf @x2 Two dierent physical conditions may arise at the wellbore during well testing: a) Constant production rate: AK @P ; qsf = ,B @r rw

(2.28)

(2.29)

where A = 2rw h in the case of a radial system, whereas A = 4rs2 is used in a spherical system. Note that qsf refers to the ow rate at the sandface level. b) Constant bottomhole pressure:

P = Pwf ;

r = rw ;

(2.30)

where Pwf is a constant pressure at the wellbore against which the well produces. The speci cation of the pressure behavior at the external boundary depends on the type of reservoir considered. a) Volumetric reservoir: This is a closed system characterized by no uid ow across the outer boundary. Referring to equation (2.29), the zero ow rate ( velocity ) is expressed in terms of pressure gradient as @P = 0: (2.31) @r re

b) Reservoir associated with a gas cap or bottom water is featured by constant pressure at the external boundaries. In this case the pressure distribution does not change with time and the true steady-state condition is reached; that is,

P = Pi ;

r = re :

(2.32)

c) In nitely large reservoir: This is a mathematical concept which is very useful during the initial stages of well testing when the pressure disturbance does not travel far enough to reach the reservoir boundary. (2.33) rlim !1 P = Pi : In all situations considered, it is assumed that initially the reservoir pressure is uniform

P = Pi ; t = 0 :

(2.34)

Chapter 3

Velocity Potential Analysis In this chapter, we will introduce a mathematical and physical model of horizontal wells, then we will use it to derive the velocity potential formula and calculate the average velocity potential of horizontal wells by the pressure-averaging technique and equivalent pressure-point technique. Figure 3.1 is a schematic of a horizontal well. A horizontal well of length L drains an ellipsoid, while a conventional vertical well drains a right circular cylindrical volume. Both of these wells drain a reservoir of height H , but their drainage volumes are dierent ( see Figure 3.4 ). The following hypotheses are made: (1) The reservoir is a horizontal, homogeneous, anistropic, and has constant Kx ; Ky ; Kz permeabilities, thickness H , porosity . The reservoir is in nite lateral extension, i.e., the boundaries of the reservoir in the horizontal directions are so far away that the pressure disturbance does not travel far enough to reach the boundaries during the well testing. (2) The reservoir pressure is initially constant. The pressure remains constant and equal to the initial value at an in nite distance from the well. The reservoir is bounded by top and bottom constant pressure or impermeable formations. (3) The production occurs through a well of radius Rw , represented in the model by a uniform line source located at a distance zw from the lower boundary, the length of the well is L. (4) A single-phase uid, of small and constant compressibility Cf , constant viscosity , and formation volume factor B , ows from the reservoir to the well at a constant rate Q. Fluids properties are independent of pressure. Gravity eects and wellbore-storage eects are neglected to simplify the solutions. (5) If the reservoir is an inhomogeneous anistropic naturally fractured reservoir, we use Warren-

Z=H Z

Y

H Zw X

Figure 3.1: Horizontal Well Model. 15

Z=0

CHAPTER 3. VELOCITY POTENTIAL ANALYSIS

16

Root Model, uid movement toward the wellbore occurs only in fractures ([49, 87]).

3.1 Equipotential Surfaces of Horizontal Wells In this section, we will derive the velocity potential formula of a point in 3D space and equipotential surfaces of horizontal wells. To calculate oil production from a horizontal well mathematically, the three-dimensional (3D) Laplace equation needs to be solved rst. If constant pressure at the drainage boundary and at the wellbore is assumed, the solution would give a pressure distribution within a reservoir. Once the pressure distribution is known, oil production rates can be calculated by Darcy's law, ( see equation (2.12), [28, 30]) ! v = , K r: In steady state, we have the continuity equation : ( see equation (2.10), [30] )

div(!v ) = 0; then

div(, K r) = 0:

If we use the velocity potential v , ( see equation (2.13) ) , therefore in order to derive the reasonable and reliable productivity formulae, we must solve 3D Laplace equation: ( see equation (2.24), [2] ) r2v = 0: (3.1) Suppose a mathematical point M in the 3D space, and an in nite large percolation eld around M , then the percolation velocity of a sphere surface is ([4, 28])

q ; v = Aq = 4r 2

!

the dimension of q is [L3 T ,1 ], and A is cross-sectional ow area. According to Darcy's law ([4, 64]), ! dv v = dr ; thus we obtain the velocity potential of point M in 3D space is

q + constant: v = , 4r

We consider the uniform - ux, nite - conductivity model, take the horizontal well as a uniform line source in 3D space, the total productivity is Q, thus Q=L is unit length productivity, the dimension of Q is [L3 T ,1 ], at every point along the well, the well ow rate is the same. If C (x0 ; y0 ; z0 ) is a point in 3D space, the velocity potential at point C caused by dx { a dierential segment of the horizontal well line source is

dv = , 4Qdx LR ;


17

C

A

D

B

O

Figure 3.2: The Analyses of the Velocity Potential. and

q

q

R = (x , x0 )2 + y02 + z02 = (x , x0 )2 + r02;

(3.2)

Z c Q v = , 4L dx : ,c R

(3.3)

where r02 = y02 + z02 . If we let c = L2 , then the velocity potential caused by the whole line source is

Figure 3.2 shows the analyses of the velocity potential. Point O(0; 0; 0) is the midpoint of segment AB . Point D(x0 ; 0; 0) is the projection of point C in segment AB . Let Re be the radius of triangle ABC 's circumscribed circle, considering A(,c; 0; 0), B (c; 0; 0), and let the three sides of triangle ABC be ae; eb; ce, we can show that ([40]) AC = eb = 2Re sin B; BC = ae = 2Re sin A; AB = ce = 2Re sin C; q

OA = OB = c; CD = y02 + z02 = r0 ;

and according to the Sine Law of Triangle, we have ([40]) AD = OA , OD = c , jx0 j = eb cos A = 2Re sin B cos A; BD = OA + OD = c + jx0 j = ae cos A = 2Re sin A cos B; therefore

dx = Z c q dx ,c R ,c (x , x0 )2 + r02 # " e c , x 0 +b = ln ,(c + x ) + ea 0 e = ln[ 2Re sin B (1 + cos A) ] 2R sin A(1 , cos B ) = , ln(tan A2 tan B2 );

Z

c


18

Z Y

X

Figure 3.3: The Shape of the Equipotential Surfaces. so, we have

Q ln(tan A tan B ): v = 4L 2 2

(3.4)

In the triangle, let s be half perimeter, r be the radius of inscribed circle, and let a = 12 (ae + eb), according to the Half Angle Law of Triangle ([106]), we have

s = 21 (ae + eb + ce);

e r = (s , ae)(s ,s b)(s , ce) ;

tan A2 = s ,r ae ;

Therefore we have

s

tan B2 = r e ;

s,b e r2 , c: tan A2 tan B2 = = (s ,s c) = aa + e c (s , ae)(s , b) a , c ): ln( v = 4Q L a + c

(3.5)

According to the de nition of ellipsoid of revolution ([40]), if the sum of segment AC and AB 's length is 2a, point C is on the ellipsoid of revolution whose focuses are point A and B , and a = 12 (ae + eb) is semimajor axis. So we can come to the conclusion: The equipotential surfaces of a horizontal well uniform line source in 3D space are a family of ellipsoids of revolution whose focuses are the two endpoints of the horizontal well. Figure 3.3 shows the shape of the equipotential surfaces.


19

H Re

Re

H

L

Figure 3.4: Schematics of Vertical and Horizontal Well Drainage Volume.


20

3.2 Average Velocity Potential of Horizontal Wells In this section, we will derive the average velocity potential formulae of horizontal wells by the pressure-averaging technique and equivalent pressure-point technique. Arbitrary point (x; y; z ) in 3D space is on an ellipsoid of revolution equipotential surface, and its semimajor axis is q q a(x; y; z) = 12 [ (x , c)2 + y2 + z2 + (x + c)2 + y2 + z 2 ]: (3.6) Judging by the above formula, the potential of a point on the horizontal well line source is dependent on the location of the point, because each point is on its own ellipsoid of revolution equipotential surface. However, we can derive the average velocity potential fv of a horizontal well. Z c aw (x) , c dx; ln (3.7) fv = 21c 4Q aw (x) + c ,c L where q q (3.8) aw (x) = 12 [ (x + c)2 + Rw2 + (x , c)2 + Rw2 ]: To compute the average velocity potential of the whole horizontal well, we may use the average semimajor axis of the ellipsoid of revolution equipotential surfaces. Because L Rw , thus 1 qL2 + R2 L=2 + R2 =(4L); ln(L + qL2 + R2 ) ln(2L); w w w 2 consequently, Z c 2 1 afw = 2c aw (x)dx L=2 + Rw2 =(4L) + R2Lw ln(2L=Rw ): (3.9) ,c

g is As a result, the average velocity potential of the horizontal wellbore w

Q Rw g w 4L ln[ 2L2 ln(2L=Rw )]: 2

(3.10)

The velocity potential of the two endpoints and midpoint are dependent on the semimajor axes of the ellipsoids of revolution equipotential surfaces which they are on. So, we have 2L Rw + Rw2 =(2L); Rw Rw2 =(2L); q aEw = 12 ( L2 + Rw2 + Rw ) L=2 + Rw =2 + Rw2 =(4L); q aM = (L=2)2 + Rw2 L=2 + Rw2 =L: w Thus, the velocity potential of the endpoints and midpoint are Ew 4Q L ln[Rw =(2L)];

(3.11)


21

Q 2 2 M w 4L ln(Rw =L ):

(3.12)

According to Simpsons Integral Approximation Rule ([40, 52]), the average velocity potential of the horizontal well can be approximately represented as follows Q 5 1 1 M g (3.13) w 6 (2E w + 4w ) = 4L [ 3 ln(Rw =L) , 3 ln 2]: Let I be the ratio of endpoint and midpoint's velocity potential, then E

I = Mw = 12 [1 + 1= log2 (L=Rw )] 12 ; w

(3.14)

when L ! 1, I ! 12 : Let J be the ratio of eqaution (3.13) and equation (3.10), then

J = [ 3 ln(RR2ww =L) , 3 ln 2] : ln[ 2L2 ln(2L=Rw )] 5

1

(3.15)

Table 3.1: The ratioes of velocity potential of endpoint and midpoint. Rw 0.1m 0.1m 0.1m 0.1m 0.1m L 200m 300m 400m 500m 600m I 0.545 0.543 0.542 0.541 0.539 According to Table 3.1, we can see that the endpoint's velocity potential is approximately equal to 54 percent the midpoint's velocity potential. This conclusion tells us: If we take a horizontal well as a uniform line source in 3D space, the well is not an in nite conductivity fracture, the in nite conductivity model is not exact for horizontal wells. Table 3.2: The comparisons of the two methods of calculating average velocity potential. Rw 0.1m 0.1m 0.1m 0.1m 0.1m L 200m 300m 400m 500m 600m J 0.935 0.933 0.931 0.930 0.929 According to Table 3.2, we can see that these two methods perform similarly, the dierence between them is small.

Chapter 4

Productivity Formulae In petroleum engineering, in order to determine the economic feasibility of drilling a horizontal well, the engineers need reliable methods to estimate its expected productivity { well ow rate. In this chapter, we will derive the productivity formulae of horizontal wells in three-dimensional space for steady state ow. We will show that any two-dimensional (2D) methods are not suitable for three-dimensional (3D) percolation problems unless we assume the horizontal well's length is in nite. The formulae based on 2D model can not satisfactorily re ect characteristics of horizontal wells. In a reservoir, there may be several wells working together at the same time. However, a speci c well exploits a de nite volume of reservoir uid, and this volume is called the drainage volume of that well. Let Pw be the pressure drawdown between the boundary of drainage volume and wellbore, then we have Pw = Pe , Pw = Pi , Pw ; the pressure on the external boundary Pe is assumed to be the initial pressure Pi .

4.1 Formulae for Wells at Midheight of Formation We rst consider the case in which the horizontal well is at the midheight of the formation. According to our previous equipotential surface conclusion, in order to simplify the boundary problem, we suppose the external boundary of drainage volume is also on an ellipsoid of revolution equipotential surface, so we may assume further that: The drainage volume of a horizontal well is an ellipsoid of revolution whose focuses are the two endpoints, and minor axis is the formation thickness H , the horizontal well is at the midhight of the formation ( see Figure 3.4 ). Thus, the semimajor axis of the drainage volume is q p ae = (L=2)2 + (H=2)2 = 21 L2 + H 2 : (4.1) According to equation (3.5), the velocity potential of the external boundary is "

p

#

Q ln pL2 + H 2 , L : e = 4L L2 + H 2 + L 22

(4.2)

CHAPTER 4. PRODUCTIVITY FORMULAE Let

23

p E = pL + H , L ; 2

2

L2 + H 2 + L

then, combining equations (2.13), (3.10) and (4.2), the productivity formula of a horizontal well in an ellipsoid of revolution drainage volume is Qw = 4KL2PELw2=(B ) : (4.3) ln[ R2w ln(2L=Rw ) ] In the above formula, B is formation volume factor. Formation volume factors have been given the general standard designation of B and are used to de ne the ratio between a volume of uid at reservoir conditions of temperature and pressure and its volume at standard condition : 600 F and 1 atmosphere. The factors are therefore dimensionless but are commonly quoted in terms of reservoir volume per standard volume. Thus, in this thesis, Qw means well ow rate at standard condition. p When x ! 0; 1 + x 1 + x=2, thus if L H , then p

p

L2 + H 2 + L 2L; q

L2 + H 2 , L = L( 1 + H 2 =L2 ) , L H 2 =2L;

thus equation (4.3) becomes

Qw = 4KLPHw2 =(B ) :

(4.4) ln[ 2R2w ln(2L=Rw ) ] The above formulae (4.3), (4.4) are only suitable for isotropic reservoirs { the formation's vertical permeability is equal to its horizontal permeability. In fact, in many reservoirs with anisotropic permeabilities, formation's vertical permeability is smaller than horizontal permeability. To derive the productivity formula of horizontal wells in anisotropic reservoirs, we assume that Kh = Kx = Ky ; Kv = Kz . Then equation (2.22) can be reduced to 2 2 2 Kh @ P2 + Kh @ P2 + Kv @ P2 = 0: @x @y @z

We let

s

Kv )x; xe = ( K h 4

Thus, equation (4.5) becomes

p

s

s

Kv )y; ze = ( 4 Kh )z: ye = ( K Kv h 4

2 2 2 Kv Kh( @ eP2 + @ eP2 + @ eP2 ) = 0; @x @y @z

and K; L; H become s

s

Kv )L; He = ( 4 Kh )H: f= K Kv Kh ; Le = ( 4 K Kv h p

(4.5)

(4.6)

CHAPTER 4. PRODUCTIVITY FORMULAE

24

Rw

Z Kz

X Kx

Figure 4.1: Original Reservoir System with x/z Anisotropy and Horizontal Well.

Z

, X Kx

, K , z

,

Figure 4.2: Transformed Isotropic Reservoir System with Elliptic Wellbore.


25

a (semi major axis )

b ( semi minor axis )

Figure 4.3: Axis Dimensions for Transformed Wellbore. We should use the correct equivalent wellbore radius for the anisotropic formation, which guarantees that elliptical ow eects near the well are treated correctly. We de ne permeability anisotropic factor is ([60, 89]) s

h = K Kv :

(4.7)

In the above transformed coordinates the original circular well occupies the elliptical cylinder. A horizontal well in an anisotropic system is sketched in two dimensions in Figure 4.1. Let us assume that the directional permeabilities Kx and Kz dier considerably, as they do in most reservoirs, with Kx Kz . A simple transformation of variables can be used to change all dimensions to an equivalent isotropic system. The result is shown in Figure 4.2. The wellbore is now an ellipse, as p 4 shown in Figure 4.3. The length of the majorpaxis in the z direction is Rw Kx =Kz . Similarly, the minor axis in the x direction is reduced to Rw 4 Kz =Kx . From inspectional analysis, the behavior of this transformed isotropic system is identical in every respect at all points and times to the original system. The question now is how to model this ellipic wellbore as an equivalent circle in the best way. Actually, there is no single answer to that question. From the previous work of M. Muskat, F.J. Kuchuk and W.E. Brigham ( [65, 79]), it is known that, when the ow near the well becomes steady, an elliptical well with semiaxes a and b in an isotropic formation is equivalent to a circular well of radius a+2 b . When the well ow rate is constant, the wellbore behaves as though its radius is the arithmetic average of the major and minor axes ([17, 86]), so the equivalent radius of the horizontal well is Rgw = 12 ( 1=2 + ,1=2 )Rw ; (4.8)


26

g = R . where is permeability anisotropic factor, when = 1; R w w e e e y; e z e; L; H are changed to Thus x;

xe = ,1=2 x; We de ne

ye = ,1=2 y; ze = 1=2 z:

Le = ,1=2 L;

He = 1=2 H:

F = ln[ ( +4L1)R ]; w

p

2 2 G = ln[ pKv L2 + KhH 2 , pKv L ]: Kv L + KhH + Kv L

p

(4.9) (4.10) (4.11) (4.12)

and we apply F; G and equation (4.8) to equation (4.3), then we obtain the productivity formula of a horizontal well in an anisotropic permeability reservoir

p

4 4 v3 Kh LPw =(B ) : Qw = 4 K (ln 12 + 2F + G , ln F ) p

(4.13)

It has been pointed out that, during production, the vertical permeability is more important than the horizontal permeability for horizontal wells. But the productivity formulae in [6, 7, 39, 44, 60] can not account for this viewpoint. Our formula (4.13) demonstrates it, because in formula (4.13), the exponent of Kv is 3=4 while the exponent of Kh is 1=4. Our physical system consisted of a horizontal well producing at a uniform and constant rate from an anisotropic, box-shaped drainage volume. We used the well-known line sink/source solution for the well. Thus, according to our analysis, in the immediate vicinity of the wellbore, i.e., at R = Rw , the ow is characterized by two main features: (1) the ux into the wellbore is uniform; (2) the pressure along the well varies, i.e., the well can not be an isopotential, we may use a representative average wellbore pressure. In an anisotropic formation, the isopotential lines are ellipses, the circular wellbore can not have uniform pressure along the perimeter. It is also clear that, because of friction losses and other factors, the pressure is not constant along the well ([7]). Our formulae (4.3), (4.4) and (4.13) are suitable for the horizontal wells at the midhight of a formation.

4.2 Well Eccentricity Problem In this section, we apply the three dimensional model to study the well eccentricity problem in which a horizontal well's location is not at the midheight of a reservoir. Taking a horizontal well as a uniform line source in 3D space, the coordinates of the two ends are : (,L=2; 0; zw ) and (L=2; 0; zw ). Point convergence intensity is the ow rate through a point in 3D space. If the point convergence intensity of point (x0 ; y0 ; z0 ) which is in in nite space is q, in order to obtain its steady state pressure, we have to obtain the basic solution of the partial dierential equation below ([36]), and the point (x0 ; y0 ; z0 ) becomes (x0 ; 0; zw ) ( see Figure 3.1 ). 2 2 2 Kx @ P2 + Ky @ P2 + Kz @ P2 @x @y @z


27

= qB(x , x0 )(y)(z , zw ); (4.14) where (x , x0 ), (y), (z , zw ) are Dirac functions ([106]). The dimension of the above Dirac functions such as (z , zw ) is [L,1 ], the dimension of q is [L3 T ,1 ], and the dimension of both sides of equation (4.14) is [ML,1 T ,2 ]. The total productivity of the horizontal well is Qw , its dimension is also [L3 T ,1 ]. Transformation of a mathematical model into dimensionless form is a common engineering practice. In treating pressure transient problems, the introduction of dimensionless groups reduces the number of variables and parameters which can signi cantly simplify the mathematical statement. Let , , , y z x K K K 1=2 1=2 (4.15) xD = L ( K ) ; yD = L ( K ) ; zD = L ( K )1=2 : x

y

We de ne eective permeability

z

,

K = (KxKy Kz )1=3 : Generally, there holds Kx = Ky 6= Kz , so we let Kh = Kx = Ky ; Kv = Kz :

(4.16) (4.17)

Consequently, we have ,

,

HD = HL ( KK )1=2 ; z Note that the permeability anisotropic factor is LD = ( KK )1=2 ; x

s

,

zwD = zLw ( KK )1=2 : z

(4.18)

s

h = Kx : = K Kv Kz

(4.19)

RwD ( 1=2 + ,1=2 )Rw =(2L);

(4.20)

As we stated before, because of dimensionless transformation, the circular wellbore becomes ellipse, the dimensionless equivalent radius of the well is the arithmetic mean of semimajor axis and semiminor axis of the ellipse, i.e. ([17, 86]), when = 1; RwD = Rw =L. According to equation (4.15), there hold

,

@P = @P @xD = ( 1 )( K )1=2 @P ; @x @xD @x L Kx @xD ,

@ 2 P = ( 1 )( K ) @ 2 P : @x2 L2 Kx @xD 2

Similar formulae can be derived for other derivatives, so ,

2 2 2 2 2 2 Kx @ P2 + Ky @ P2 + Kz @ P2 = ( LK2 )( @ P2 + @ P2 + @ P2 ): @x @y @z @xD @yD @zD


28

Note that (cx) = (x)=c, c is a positive constant ([106]), then ,

2

3

(xD , x0D ) = 4 (x ,L x0 ) ( KK )1=2 5 = ( K,x )1=2 L(x , x0 ); x K

and

,

2

3

(K )3=2 5 (x , x )(y )(z , z ): 4 qB(x , x0 )(y)(z , zw ) = ( qB ) D 0D D D wD 3 L (Kx Ky Kz )1=2 According to equation (4.16), there holds 2

,

3

(K )3=2 5 = 1: 4 (Kx Ky Kz )1=2 If we de ne

,

(Pi , P ) ; PD = K LqB

then equation (4.14) becomes

(4.21)

@ 2 PD + @ 2 PD + @ 2 PD = ,(x , x )(y )(z , z ): (4.22) D 0D D D wD @xD 2 @yD 2 @zD 2 We must point out that if the point convergence intensity of point (x0 ; y0 ; z0 ) is q, then the

total productivity of the horizontal well is

Qw = qLD :

(4.23)

4.2.1 Point Convergence Pressure Distribution Formulae

In this subsection, we will derive the point convergence pressure distribution formulae, i.e., dimensionless pressure formulae of the points in 3D space. We rst consider the drainage domain is between the in nite parallel planes z = H and z = 0 with the impermeable boundary condition at z = H , constant boundary condition at z = 0 ( e.g. the horizontal well is in the reservoir with bottom water ), thus

PD jzD =0 = 0;

@PD j @ND zD =HD = 0;

(4.24)

@PD is the exterior normal derivative of dimensionless pressure. where @N D Considering the simultaneous equations of (4.22) and (4.24), let the normalized orthogonal solution systems be gn (zD ), then we have

gn (zD ) = 2=HD sin[ (2n 2,H1)zD ]; q

D

(n = 1; 2; 3; :::);

(4.25)


29

because gn (zD ) satisfy the boundary conditions (4.24), and HD

Z 0

Z

HD

0

(i 6= j );

(4.26)

(n = 1; 2; 3; :::):

(4.27)

gi gj dzD = 0;

[gn ]2 dzD = 1;

Therefore, according to the properties of Dirac function and the normalized orthogonal solution systems of the partial dierential equation basic solution, there holds ([100, 101, 102])

(zD , zwD ) =

1

X

n=1

gn(zwD )gn (zD );

(4.28)

i.e.,

(zD , zwD ) =

1

X

n=1

( H2 ) sin[(2n , 1)zD )=(2HD )] sin[(2n , 1)zwD )=(2HD )]:

Let

D

PD =

1

X

n=1

Un(xD ; yD ) sin[ (2n 2,H1)zD ];

(4.29) (4.30)

D

and we de ne the two dimensional Laplace operator as follows: 2 2 Un = @ Un2 + @ Un2 :

@xD

(4.31)

@yD

Apply equations (4.31) and (4.30) to equation (4.22), then Un is the basic solution of the following equation ([100, 101, 102]) Un (xD ; yD ) , [(2n , 1)=(2HD )]2 Un (xD ; yD ) = ( 2 )(x , x )(y ) sin[(2n , 1)z =(2H )]:

HD

D

D

0

D

wD

(4.32)

D

Using integration transformation method, we have Un (xD ; yD ) = ( H2 ) sin[(2n , 1)zwD =(2HD )]f( 21 )K0 [(2n , 1)RD =(2HD )]g D = ( 1 ) sin[(2n , 1)z )=(2H )]K [(2n , 1)R =(2H )]; wD

HD

where

D

0

D

D

(4.33)

q

RD = (xD , x0D )2 + yD2 :

(4.34) Combining equations (4.30) and (4.33), if the boundary conditions are (4.24), the dimensionless pressure of the point (x0D ,0,zwD ) is

PD =

1

X

n=1

( H1 ) sin[ (2n 2,H1)zD ] sin[ (2n ,2H1)zwD ]K0 [(2n , 1)RD =(2HD )]: D

D

D

(4.35)


30

In the (4.33) and (4.35), K0 (x) is modi ed Bessel function of second kind and order zero ([106]). If the drainage domain is between the in nite parallel planes z = H and z = 0 such that the boundary condition is impermeable at z = 0 , but constant at z = H ( e.g the horizontal well is in the reservoir with gas cap ), then

PD jzD =HD = 0; is

@PD j @ND zD =0 = 0:

(4.36)

Similarly, if the boundary conditions are (4.36), the dimensionless pressure of the point (x0D ,0,zwD )

PD =

1

X

n=1

( H1 ) cos[ (2n 2,H1)zD ] cos[ (2n ,2H1)zwD ]K0 [(2n , 1)RD =(2HD )]: D

D

D

(4.37)

If the drainage domain is between the in nite parallel planes z = H and z = 0 with the boundary conditions at z = H and z = 0 are both constant ( e.g the horizontal well is in the reservoir with bottom water and gas cap ), then

PD jzD =0 = 0;

PD jzD =HD = 0:

(4.38)

Considering the simultaneous equations of (4.22) and (4.38), letting the normalized orthogonal solution systems be fn (zD ), then we have D fn(zD ) = 2=HD sin[ nz H ]; q

D

(n = 1; 2; 3; :::);

(4.39)

because fn (zD ) satisfy the boundary conditions (4.38), and HD

Z 0

HD

Z 0

is

(i 6= j );

(4.40)

(n = 1; 2; 3; :::):

(4.41)

fi fj dzD = 0;

[fn]2 dzD = 1;

Hence, if the boundary conditions are (4.38), the dimensionless pressure of the point (x0D ,0,zwD )

PD =

1

X

nzwD D ( H1 ) sin[ nz H ] sin[ H ]K0 [nRD =HD ]:

n=1

D

D

D

(4.42)

Formulae (4.35), (4.37), (4.42) are the point convergence pressure distribution formulae, i.e., dimensionless pressure formulae of the points in 3D space.

4.2.2 Dimensionless Pressure Formulae of Horizontal Wells

In this subsection, we will derive the dimensionless pressure formulae of the endpoint and midpoint of horizontal wells under dierent boundary conditions. Because the horizontal well line uniform source is located between ,L=2 x0 L=2, i.e., ,LD =2 x0D LD =2, according to the Superposition Principle of Potential, the dimensionless pressure of the endpoints and midpoint can be computed by the integration method, and the average dimensionless pressure of the whole horizontal well can be computed by the Simpsons Integral Approximation Rule.


31

From (4.34), we have q

2 jxwD , x0D j: (4.43) RwD = (xwD , x0D )2 + ywD If the boundary conditions are (4.24), in the equation (4.35), let zwD + RwD take the place of zD , and let x0D be the element of integration, along the well length LD , integrate equation (4.35), we can obtain the dimensionless pressure of the horizontal well at xwD is

PwD = i.e.,

PwD = ( H1 )f

1

X

D n=1

=

Z

then

LD = 2

,LD =2

Z 0

,LD =2

+ If we let

LD = 2

,LD =2

Z

,LD =2

PD dx0D ;

(4.44)

zwD + RwD ) ] sin[ (2n , 1)zwD ]A g; sin[ (2n , 1)2(H n 2H D

where ( note that eqauation (4.43) )

An =

LZD =2

K0 [(2n , 1)RwD =(2HD )]dx0D K0 [jxwD , x0D j(2n , 1)=(2HD )]dx0D K0 [jxwD , x0D j(2n , 1)=(2HD )]dx0D

LD =2

Z

0

(4.45)

D

K0 [jxwD , x0D j(2n , 1)=(2HD )]dx0D :

v = jxwD , x0D j(2n , 1)=(2HD ); n,1)( L2DZ,xwD )=(2HD )

8 (2 > >
> :

n,1)( L2DZ+xwD )=(2HD )

(2

K0 (v)dv +

0

9 > > =

K0 (v)dv> :

0

> ;

(4.46)

If our observation point is the endpoint of horizontal well, i.e., xwD = LD =2 or xwD = ,LD =2, then

AEn

(2

n,1)LD =(2HD )

Z D = (2n2H K0 (v)dv , 1) 0 Z 1 Z 1 2 H D K0 (v)dvg: = (2n , 1) f K0 (v)dv , 0 (2n,1)LD =(2HD )

Simplify the above formula, and note that ([52, 106]) 1

Z 0

K0 (v)dv = =2;

(4.47)

CHAPTER 4. PRODUCTIVITY FORMULAE we have where

32

AEn = (2n2H,D1) ( 2 ) , n; Z 1 n = (2n2H,D1)

n,1)LD =(2HD )

(2

If we let

then equation (4.49) becomes

(4.48)

K0 (v)dv:

(4.49)

u = (2n , 1)LD =(2HD );

(4.50)

Z 1 L D n = u K0 (v)dv: (4.51) u By the de nitions of LD ; HD ( see equation (4.18) ), u is a positive number which is greater

than 1, and we de ne

F (u) =

Z

1

u

K0(v)dv:

(4.52)

When v ! 1, according to the asymptotic expansion formula of K0 (v), there holds ([40, 52]) r K0 (v) 2v e,v ;

(4.53)

and note that when v 1, we have ep,vv e,v , thus

F (u)

= Because

Z

1 r ,v e dv

u

2v

1 ,v e dv

r

r

e,u :

Z

2 u 2

n = LuD F (u);

therefore

p

r L D n u 2 e,u = (2n ,2H1)Dp exp[,(2n , 1)LD =(2HD )]: (4.54) Note that equation (4.45) should include the in nite sum of n with respect to n. Now we

estimate the summation, since there holds equation (4.54) and zwD + RwD ) ] sin[ (2n , 1)zwD ] 1; sin[ (2n , 1)2(H 2HD D thus we let 1 X (2 n , 1) ( z + R ) (2 n , 1) z 1 wD wD wD E = H fsin[ ] sin[ ] g n ; 2H 2H D n=1

D

D

CHAPTER 4. PRODUCTIVITY FORMULAE then

p

1

33

E 3=22 (2n 1, 1) exp[,(2n , 1)LD =(2HD )] =1 p nX 1 2n,1 = 3=22 ( 2n , 1 ) n=1 p 2 ln( 1 + ); = 23=2 1 , where

X

= exp[,LD =(2HD )] < 1;

and we have used the below formula ([40, 52]) + ) = 2[ + 3 =3 + 5 =5 + ::: + 2n+1 =(2n + 1) + :::]; (jj < 1): ln( 11 , By the de nitions of ; LD ; HD and note the fact that L H , we come to the conclusion that is a positive number and ! 0, we have ) 2; ln( 11 + , thus p + ) p2,3=2 : E 232=2 ln( 11 , (4.55) Therefore, the global error is controlled by a very small positive number, when n increases, the error does not increase, it is reasonable to neglect n in equations (4.45) and (4.48), and we can simplify (4.48) as follows

AEn HD =(2n , 1):

(4.56) If our observation point is the midpoint of horizontal well, i.e., xwD = 0, by a similar method, we have (2n,1)L Z D =(2HD ) 4 H D M K0 (u)du 2HD =(2n , 1): (4.57) An = (2n , 1) When x 6= 0, there hold ([40, 52]) 1

cos(nx)=n = , ln[2 sin(x=2)];

(4.58)

cos[(2n , 1)x]=(2n , 1) = 21 ln[cot(x=2)];

(4.59)

X

n=1

1

X

n=1

0

sin(x) sin(y) = 21 [cos(x , y) , cos(x + y)];

(4.60)


34

cos(x) cos(y) = 21 [cos(x + y) + cos(x , y)]:

and note that

(4.61)

RwD 0;

2zwD + RwD 2zwD : (4.62) Combining equations (4.45), (4.56), (4.59), (4.60), and (4.62), if the boundary conditions are (4.24), the dimensionless pressure of the endpoints of the horizontal well is E = ( 1 )f PwD

1

1 sin[ (2n , 1)(zwD + RwD ) ] sin[ (2n , 1)zwD ]g; 2 n 2HD 2HD n=1 , 1 X

(4.63)

E = ( 1 )fln[cot( RwD )] , ln[cot( (2zwD + RwD ) )]g; PwD 4 4H 4H

(4.64)

E = ( 1 )fln[cot( RwD )] , ln[cot( zwD )]g: PwD 4 4H 2H

(4.65)

M = ( 1 )fln[cot( RwD )] , ln[cot( (2zwD + RwD ) )]g; PwD 2 4H 4H

(4.67)

M = ( 1 )fln[cot( RwD )] , ln[cot( zwD )]g: PwD 2 4H 2H

(4.68)

M = ( 1 )fln[cot( RwD )] + ln[cot( zwD )]g: PwD 2 4H 2H

(4.70)

M = ( 1 ) lnf sin(zwD =HD ) g: PwD 2 sin[R =(2H )]

(4.72)

D

D

D

D

Similarly, if the boundary conditions are (4.24), the dimensionless pressure of the midpoint of the horizontal well is 1 M = ( 2 )f X 1 sin[ (2n , 1)(zwD + RwD ) ] sin[ (2n , 1)zwD ]g; PwD (4.66) n=1 2n , 1 2HD 2HD D

D

D

D

In the same manner, if the boundary conditions are (4.36), the dimensionless pressure of the endpoints and the midpoint of the horizontal well are given by the two formulae below, respectively: E = ( 1 )fln[cot( RwD )] + ln[cot( zwD )]g; PwD (4.69) 4 4HD 2HD D

D

If the boundary conditions are (4.38), the dimensionless pressure of the endpoints and the midpoint of the horizontal well are given by the two formulae below, respectively: E = ( 1 ) lnf sin(zwD =HD ) g; PwD (4.71) 4 sin[RwD =(2HD )] wD

D

In summary, we have Table 4.1. Observe the above dimensionless pressures of the endpoint and the midpoint, we may come to the following conclusion which is similar to what stated in Chapter 3.


35

Table 4.1: Dimensionless pressure of endpoint under dierent boundary conditions. E Boundary Conditions PwD @P R 1 PD jzD =0 = 0; @NDD jzD =HD = 0 ( 4 )fln[cot( 4HwD )] , ln[cot( z2HwDD )]g D @PD j RwD zwD 1 PD jzD =HD = 0 ; @N D zD =0 = 0 ( 4 )fln[cot( 4HD )] + ln[cot( 2HD )]g D) g PD jzD =0 = 0 ; PD jzD =HD = 0 ( 41 ) lnf sin[sin(RzwDwD==H (2HD )] Within the given error tolerance, the dimensionless pressure of midpoint of the horizontal well is about twice as large as the dimensionless pressure of endpoints, i.e. ( see equation (3.14) ), M 2P E : PwD wD

(4.73)

According to Simpsons Integral Approximation Rule, the average dimensionless pressure of the horizontal well uniform line source is A 1 (2P E + 4P M ) 5 P E : (4.74) PwD wD 6 wD 3 wD Therefore, in a certain sense of approximate representation, the average dimensionless pressure of the whole horizontal well is 5=3 of the dimensionless pressure of endpoints.

4.2.3 Productivity Formulae for Eccentricity Wells

In this subsection, we will derive productivity formulae for eccentricity wells. By the de nition of dimensionless pressure ( see equation (4.21) ), consider the constant boundary condition, Pe = Pi , thus Pe , PwE = PwE ; (4.75) PwE means the pressure drop is measured at the endpoint of the horizontal well, and PwA means the average pressure drop of the horizontal well, we have (4.76) PwA 53 PwE : Combining equations (4.21) and (4.64), note that , ln[cot(x)] = ln[tan(x)]; when x ! 0, tan(x) ! x, and RwD 0, we have E PwD

,

LPwE = K qB

(2zwD + RwD ) )]g wD = ( 41 )fln[cot( R 4HD )] , ln[cot( 4HD RwD wD ( 41 )fln[tan( z 2HD )] , ln[tan( 4HD )]g RwD wD ( 41 )fln[tan( z 2HD )] , ln( 4HD )g;

CHAPTER 4. PRODUCTIVITY FORMULAE i.e.,

36

E ( 1 )fln[tan( zwD )] + ln( 4HD )g: PwD 4 2H R

(4.77)

M ( 1 )fln[tan( zwD )] + ln( 4HD )g: PwD 2 2H R

(4.78)

D

In the same manner, we have

D

wD

wD

Combining equations (4.18), (4.21), (4.77) and (4.78), we have ,

K LPwE =(B ) q = ln[4H 4=(R z )] : D wD )] + ln[tan( 2HwD D According to equations (4.17), (4.18) and (4.23), the total productivity of the horizontal well is

p

h Kv LPwE =(B ) Qw = ln[4H4=(K z )] : D RwD )] + ln[tan( 2HwD D

In the same manner, we have

p

M w =(B ) Qw = ln[4H2 =(KRhKv L)]+Pln[tan( zwD )] ; D wD 2HD

(4.79)

(4.80)

and PwM means the pressure drop is measured at the midpoint of the horizontal well. The total productivity Qw is the summation of every point productivity along the well, Qw is also the whole well productivity under the steady average pressure drop PwA , and note equation (4.76), there holds

p

h Kv LPwE =(B ) Qw = ln[4H4=(K z )] D RwD )] + ln[tan( 2HwD D p A KhKv LPw =(B ) = ln[4H=(R z )] D wD )] + ln[tan( 2HwD pK K L( 5 P E )=(B )D h v 3 w = ln[4H =(R z )] : D wD )] + ln[tan( 2HwD D

So, = 2:4, and we have

p

Kh Kv LPwl =(B ) ; Qw = ln[4Hl =(R z )] D wD )] + ln[tan( 2HwD D where l is a parameter, i.e.,

8 >
2; if l = M ; : 2:4; if l = A:

(4.81)

(4.82)

Equation (4.81) is our horizontal well productivity formula in bottom water reservoir whose upper boundary is impermeable and bottom boundary pressure is constant.


37

Similarly, if the horizontal well is in gas cap reservoir whose bottom boundary is impermeable and upper boundary pressure is constant, then

p

Kh Kv LPwl =(B ) : Qw = ln[4Hl =(R z )] D wD )] + ln[cot( 2HwD D

(4.83)

According to equations (4.71) and (4.72), on the analogy of the above derivation of equations, if the top and the bottom boundaries are at constant pressure ( the reservoir has gas cap and bottom water ), then the horizontal well productivity formula is

p

h Kv LPwl =(B ) ; Qw = l Ksin( zwD =HD )

lnf sin[RwD =(2HD )] g

(4.84)

l has the same meaning as in equation (4.82). When Kx = Ky = Kz , ( or Kh = Kv ), the formulae (4.81), (4.83) and (4.84) reduce to the formulae below, respectively:

Pwl =(B ) Qw = ln[4H=l(KL R )] + ln[tan( zw )] ;

(4.85)

l KLPwl =(B ) Qw = ln[4H= (R )] + ln[cot( zw )] ;

(4.86)

w

2

H

for the wells in the reservoir with bottom water whose upper boundary is impermeable and bottom boundary pressure is constant; w

2

H

for the wells in the reservoir with gas cap whose bottom boundary is impermeable and upper boundary pressure is constant; Pwl =(B ) ; Qw = l KL sin(zw =H )

(4.87) lnf sin[Rw =(2H )] g for the wells in the reservoir with gas cap and bottom water, i.e., upper boundary pressure and bottom boundary pressure are constant. Formulae (4.85), (4.86) and (4.87) are productivity formulae for eccentricity horizontal wells. We must point out the value of l varies with the point where the pressure is measured. When the pressure measuring point is the midpoint, l = 2; when the pressure measuring point is the endpoint, l = 4; at other points, l will be a dierent value, if we apply the average presssure drop, l = 2:4. And we must point that when the horizontal well is at the midheight of the pay formation, i.e., zw = H=2, we have w ) = cot( zw ) = sin(z =H ) = 1; tan( z w 2H 2H so zw w ln[tan( z 2H )] = ln[cot( 2H )] = ln[sin(zw =H )] = 0; therefore the denominators of the right sides of formulae (4.85), (4.86) and (4.87) reach their minimum values, i.e., Qw reaches maximum value. So, we have proved that in the steady state and


38

B B

A

A

A

H

B

A 2b

2a

B

Figure 4.4: Division of 3D Horizontal Well Problem into Two 2D Problems. under any boundary conditions, for maximum productivity, the horizontal well should be located at the midheight of the pay formation, as observed in oil reservoir operations ([7, 60]), and the reservoir engineering practice has showed that if the pay formation is not very thick, we may assume the horizontal well is at the midhight of the formation, and the productivity error is less than 5 percent. Taking the parameter l into account, our formulae (4.3), (4.4), (4.13) should change to the formulae below, respectively: Pwl =(B ) ; Qw = l KL2EL (4.88) 2 ln[ R2w ln(2L=R ] w) l

Qw = l KLHP2w =(B ) ;

(4.89)

4 l 3 4 Qw = l 1Kv KhLPw =(B ) : (ln 2 + 2F + G , ln F )

(4.90)

ln[ 2R2w ln(2L=Rw ) ]

p

p

4.3 Comparisons of Productivity Formulae In this section, we will recall the productivity formulae which are published in other literatures and compare them with those in this thesis. In a reservoir, there may be several wells working together at the same time. From the point of two dimensional space, a speci c well exploits a de nite area of reservoir uid, this area is called drainage area of that well. The radius of the circle whose area is equivalent to the drainage area is called drainage radius of that well. The productivity index Jh is the productivity under unit pressure drop, i.e., Jh = Qw =P . J.P. Borisov's formula : ( [15] ) 2Kh H Pw =(B ) Qw = ln(4R =L ; (4.91) eh ) + (H=L) ln[H=(2Rw )] where Reh is the drainage radius of the horizontal well. F.M. Giger's formula : ( [44] ) p2Kh LPw =(B ) ; (4.92) Qw = 1+ 1,[L=(2Reh )]2 (L=H ) lnf L=(2Reh ) g + ln[H=(2Rw )]

CHAPTER 4. PRODUCTIVITY FORMULAE where Reh is the drainage radius of the horizontal well. S.D. Joshi's formula : ( [60], see Figure 4.4 ) h H Pw =(B ) ; Qw = 2a+p42aK 2 ,L2 ] + H ln[ L L ln[ H=(2Rw )]

39

(4.93)

where a is semimajor axis of the drainage ellipse, q

a = (L=2)[0:5 + 0:25 + (2Reh =L)4 ]0:5 ; and Reh is drainage radius of the horizontal well. G. Renard's formula : ( [89] ) 2Kh H Pw =(B ) Qw = 0 ; , 1 cosh (X ) + H L ln[H=(2Rw )]

(4.94)

where X = 2a=L, a is the same as we state in (4.93), cosh,1 (x) is the inverse hyperbolic cosine function, and Rw0 = [(1 + )=(2 )]Rw ; when = 1; Rw0 = Rw . C.Q. Liu's formula : ( [73] ) 2KLPw =(B ) Qw = (4.95) zw =(2H )] g : ln[4H=(Rw )] + lnf sin[sin[ (H ,zw )=(2H )] Z.F. Fan's formula : ( [39] )

p

Kh Kv LPw =(B ) : Qw = ln[42 H= (Rw )] + ln[tan( z2Hw )]

(4.96)

In fact, when = 1, formula (4.95) becomes formula (4.96), because sin[(H , zw )=(2H )] = cos[zw =(2H )]: And it is interesting to nd that when = 1, l = 2, our formula (4.85) becomes formula (4.95) and formula (4.96). Thus we have obtained formulae (4.95) and (4.96) from a dierent approach. In [61], there is an example for us to calculate a horizontal well productivity index with dierent methods. The drainage area of the horizontal well is 325 103 m2 , the length of the well is 305 m, the wellbore's radius is 0.11 m, the crude oil's formation volume factor B is 1.34, the thickness of the pay formation is 50 m, the formation's eective permeability is 74 10,3 m2 , the viscosity is 0.62 mPa:s. The well is at the midhight of the pay formation. If we use International Units, and take the constants such as 2, 4, 2:4 and dimensional unit conversions into consideration, we apply the above formulae to compute the productivity index { Jh of this well. In the following calculations, we let

= 1; l = 2:4; K = Kh = Kv = 74 10,3 m2 :


40

(1) The method of Borisov : ( see formula (4.91) ) q

Reh = 325 103 =3:14 = 321:64m; Kh H=(B ) Jh = ln(4R =L0:)543 + (H=L) ln[H=(2Rw )] eh (0:543 74 50)=(0:62 1:34) = ln[(4 321:64) =305] + (50=305) ln[50=(2 3:14 0:11)] 3 = 1129m =(day:MPa): (2) The method of Giger : ( see formula (4.92) ) 0:543Kh L=(B ) p Jh = 1,[L=(2Reh )]2 (L=H ) lnf 1+ L= g + ln[H=(2Rw )] (2Reh ) (0:543 74 305)=(0:62 1:34) p = (305=50) ln[(1 + 1 , [305=(2 321:64)]2 =(305=(2 321:64))] + ln[50=(2 3:14 0:11)] = 1163m3 =(day:MPa): (3) The method of Joshi : ( see formula (4.93) ) q

a = (L=2)[0:5 + 0:25 + (2Reh=L)4 ]0:5 q = (305=2)[0:5 + 0:25 + (2 321:64=305)4 ]0:5 = 340:19m; Jh =

ln[

a

p

2 +

0:543Kh H=(B ) a ,L2 ] + H ln[ H=(2Rw )] L L

4 2

(0 :543 74 50)=(0:62 1:34) p (340:19)2 ,3052 ] + ( 50 ) ln[50=(2 0:11)] ln[ 2340:19+ 4305 305 = 1038m3 =(day:MPa):

=

(4) The method of Renard : ( see formula (4.94) )

X = 2a=L = 2 340:19=305 = 2:23; Jh = =

0:543Kh H=(B )

0 cosh,1 (X ) + H L ln[H=(2Rw )]

(0:543 74 50)=(0:62 1:34)

cosh,1 (2:23) + ( 50 ) ln[50=(2 3:14 0:11)] 305

= 1129m3 =(day:MPa):


41

(5) The method of Liu and Fan : ( see formulae (4.95), (4.96) ) 543KL=(B ) Jh = ln[4 H=0(:R w )] w )] + ln[tan( z 2H (0:543 74 305)=(0:62 1:34) = ln[4 50=(3:14 0:11)] + ln[tan( 3:2145025 )] = 2318m3 =(day:MPa): (6) Formula (4.3):

p E = p305 + 50 , 50 = 0:7214; 2

2

3052 + 502 + 50

(B ) Jh = 1:086KL= 2EL2

ln[ R2w ln(2L=Rw ) ] 305)=(0:62 1:34) = (1:086 742 0:72143052 ln[ 0:112 ln(2305=0:11) ] = 2097m3 =(day:MPa):

(7) Formula (4.85): 652KL=(B ) Jh = ln[4 H=0(:R w )] w )] + ln[tan( z 2H (0:652 74 305)=(0:62 1:34) = ln[4 50=(3:14 0:11)] + ln[tan( 3:2145025 )] = 2782m3 =(day:MPa): According to the above calculations, we may nd that the results calculated by 2D formulae (4.91), (4.92), (4.93), (4.94) are smaller than the results calculated by our 3D formulae (4.3), (4.85). There is a horizontal well in Liu-Hua Reservoir, South China Sea. The well and reservoir's parameters are as follows ([39]): The length of the horizontal well is 600 m; the drainage radius is 500 m; the wellbore's radius is 0.1098 m; the formation's horizontal permeability is 569 10,3 m2 ; and the vertical permeability is 280 10,3 m2 ; the formation's eective thickness is 63 m; the crude oil's formation volume factor B is 1.031; the viscosity is 65 mPa:s; the distance between the horizontal well and the bottom boundary is 56.86 m; when the producing pressure drawdown is 4.57 MPa, the actual productivity is 1288 m3 =day; when the producing pressure drawdown is 5.90 MPa, the actual productivity is 1516 m3 =day. The pressure is the wellhead pressure, it means the pressure measuring point is the endpoint. The horizontal well has additional pressure drop because of formation damage. Now, we use the above formulae to calculate the productivity of this horizonta well and compare it with the actual productivity data. Because some formulae such as (4.91), (4.92), (4.95) can not account for anisotropic permeability reservoir, we may use the eective permeability of the formation if applicable.

p

K = KhKv = 569 280 = 0:399 (m2 ); p


42

Table 4.2: The comparisons of methods to compute the productivity of a horizontal well. Formula Number (4.13) (4.91) (4.92) (4.93) (4.94) (4.95) (4.96) (4.79) Productivity when Pw = 4:57 1468 555 592 630 696 1004 994 1294 Productivity when Pw = 5:90 1896 716 764 813 898 1296 1278 1568 and

q

q

= Kh =Kv = 569=280 = 1:4255

Taking the unit conversions into consideration, we have the calculation results in Table 4.2. In Table 4.2, the unit of productivity is m3 =day, the unit of pressure drop is MPa, the rst row contains the productivity formula number; the second row contains the productivity calculated by the formula in the rst row under the pressure drop 4.57 MPa; the third row is the productivity calculated by the formula in the rst row under the pressure drop 5.90 MPa. Because this horizontal well is in the pollutant reservoir, it has additional pressure drop, the computational results should be bigger than the actual productivity ([4, 29, 30]). But we observe that in Table 4.2, only our formulae (4.13) and (4.79) satisfy this criterion. This indicates that the 3D formulae here are more reliable than those based on 2D model.

Chapter 5

Well Testing Formulae for Single Porosity Reservoir The purpose of this chapter is to derive the pressure drop well testing formulae of horizontal wells in single porosity reservoirs. Taking a horizontal well as a uniform line source in 3D space, we derive the point convergence pressure distribution formulae. Then according to the superposition principle ([2, 59]), we apply quadrature theorem to derive the uniform line source pressure distribution formulae.

5.1 Point Convergence Pressure Distribution Formula In this section, we will derive the dimensionless pressure formula of a point in a single porosity reservoir. According to mass conservation and Darcy's law ([29, 30]), if the point convergence intensity of point (x0 ; y0 ; z0 ) in in nite space is q, we have to solve the equation below to derive the point convergence pressure distribution formulae. 2 2 2 Kx @ P2 + Ky @ P2 + Kz @ P2 @x @y @z = Ct @P @t + qB(x , x0)(y , y0 )(z , z0 );

(5.1) where Ct is total compressibility coecient ( see equation (2.18) ), and the dimension of both sides of equation (5.1) is [ML,1 T ,2 ]. The initial condition and outer boundary condition are Pt=0 = Pi; P (x; y; z) = Pi ; (x; y ! 1): (5.2) We de ne xD = 2x=L; LD = 2; (5.3) s

s

x )=L; z = 2z ( Kx )=L; yD = 2y ( K D Kz Ky q 4Kx t ; P = 4L K K (P , P )=(qB ); tD = C D y z i t L2

43

(5.4) (5.5)

CHAPTER 5. WELL TESTING FORMULAE FOR SINGLE POROSITY RESERVOIR

44

s

s

x + Kx )=L; RwD = Rw ( K Ky Kz when Kx = Ky = Kz , RwD = 2Rw =L.

(5.6)

The above dimensionless forms are dierent from dimensionless forms (4.15), (4.21), (4.20). Note that if c is a positive constant, (cx) = (x)=c ([106]), consequently, equation (5.1) becomes

@ 2 PD + @ 2 PD + @ 2 PD @xD 2 @yD 2 @zD 2

D , 8(x , x )(y , y )(z , z ); = @P D 0D D 0D D 0D @t D

(5.7)

and according to (5.2), the initial condition and outer boundary condition become PD = 0; (tD = 0); PD = 0; (xD ; yD ! 1): Using Convolution Theorem and Laplace transform formulae ([52, 106]), the dimensionless pressure at point (xD ; yD ; zD ) is ([28]) Z t 2 D exp[,RD =4(tD , )] d 8 p PD = (2p)3 (tD , )3 0 i.e., (5.8) PD = p1 U (tD ) H (tD ); where

U (tD ) = exp[,qRD3 =4tD ] ; tD 2

H (tD ) = 1:

Taking the Laplace transform of the equation (5.8) with respect to tD , and note that p 1=2 ,d s 2 e 2 3=2 Lfexp[,d =(4t)]=t g = ; ,k ps

d

Lferfc( 2pk t )g = e s ;

where s is Laplace transform variable, thus we have p p 2 d b b PD = U H = pR s exp[,RD s]: D

Using inverse Laplace transform method, there holds p PD = R2 erfc[RD =(2 tD )];

where

D

q

RD = (xD , x0D )2 + (yD , y0D )2 + (zD , z0D )2 ;

(5.9) (5.10) (5.11)

and erfc(x) is complementary error function ([40]). Equation (5.10) is the dimensionless pressure formula of a point in a single porosity reservoir, i.e., point convergence pressure distribution formula.


45

5.2 Dimensionless Pressure Formulae in In nite Reservoirs In this section, we will derive the dimensionless pressure formulae for horizontal wells in in nite reservoirs. The horizontal well line source is distributed along ,1 x0D 1, y0D = z0D = 0, and dimensionless pressure of the horizontal well at xwD is

PwD =

Z 1

,1

p

2erfc[RD =(2 tD )] dx : 0D R

(5.12)

D

In order to derive the dimensionless pressure of endpoints, we let q

2 RD = (1 , x0D )2 + RwD 1 , x0D :

(5.13)

Because erfc(x) = 1 , erf (x), where erf (x) is error fuction ([40]), and RwD 0, therefore

p

p

erfc[RD =(2 tD )] dx = Z 1 dx0D , Z 1 erf [RD =(2 tD )] dx : 0D 0D RD RD ,1 ,1 RD ,1

Z 1

We have

dx0D = Z 1 q dx0D 2 ,1 RD ,1 (1 , x0D )2 + RwD

Z 1

q

2 (2 + 4 + RwD )

= ln[ RwD , ln(RwD =4); and Thus

erf (y) =

1

X

k=0

]

k

y (,1)k k!(2 k + 1) : 2 +1

p

p

erf [RD =(2 tD )] dx Z 1 erf [(1 , x0D )=(2 tD )] dx 0D 0D RD (1 , x0D ) ,1 ,1 p 1=Z tD erf (y) dy = y 0

Z 1

= p2

=ptD 1 X

1Z

0

[

k

(,1)k k!(2yk + 1) ]dy: 2

k=0

Combining the above equations, if the horizontal well is in an in nite reservoir, the dimensionless pressure of endpoints is E = ,2 ln(RwD =4) , p 4 + q4 + PwD (5.14) tD 9 t3D


46

In order to derive the dimensionless pressure of midpoint, we let q

2 RD = x20D + RwD x0D :

Thus

(5.15)

p

p

erf [RD =(2 tD )] dx Z 1 erf [x0D =(2 tD )] dx 0D 0D RD x0D ,1 p ,1 1=(2 Z tD ) erf (z) dz = z p

Z 1

,1=(2 tD ) p 1=(2 t

= p2

Z

0

D) X 1

[2

k=0

k

(,1)k k!(2zk + 1) ]dz: 2

Then for the horizontal well in an in nite reservoir, the dimensionless pressure of midpoint is M = ,2 ln(RwD =4) , p 4 + q1 + (5.16) PwD tD 9 t3D Formulae (5.14) and (5.16) show that if time is suciently long, the pressure of wellbore tends to become steady. This conclusion is consistent with well testing practice ([4, 8, 30, 91]). Formulae (5.14) and (5.16) are the dimensionless pressure formulae for horizontal wells in in nite reservoirs.

5.3 Dimensionless Pressure Formulae in Reservoirs of Finite Height In this section, we will derive the the well testing formulae in single porosity reservoirs of nite height, i.e., dimensionless Pressure formulae for horizontal wells. As Figure 3.1 shows, if the horizontal well is in a reservoir of height H , and the lateral, longitudinal lengths of the reservoir are in nite, i.e., the boundaries of the reservoir in the horizontal directions are so far away that the pressure disturbance does not travel far enough to reach the boundaries during the well testing. The well is parallel to the the top and bottom boundaries. The two endpoints of the horizontal well are (,L=2; 0; zw ); (L=2; 0; zw ), and 0 < zw < H . Dimensionless HD ; zwD are de ned as follows: s

x HD = 2H ( K Ky )=L;

s

x zwD = 2zw ( K Kz )=L:

5.3.1 Reservoirs with Impermeable Boundary Conditions

(5.17)

In this subsection, we will derive the well testing formulae for horizontal wells in nite height reservoirs with impermeable boundary conditions. If the horizontal well is in the reservoir with impermeable boundary conditions at z = 0 and z = H , we apply the method of images, and according to Prolongation Theorem ([24, 100, 101, 102]), the dimensionless pressure of point (x0D ; 0; zwD ) is ([83, 84])

CHAPTER 5. WELL TESTING FORMULAE FOR SINGLE POROSITY RESERVOIR q

1

erfc[ (RD2 + zni2 )=tD =2] q PD = f g; RD2 + zni2 i=1 n=,1 2 X

where and there holds ([52])

+ X

zn1 = zD , zwD , 2nHD ; erfc(z ) = p2

Z

z

47

zn2 = zD + zwD , 2nHD ;

1 ,u2 e du = p2z

Z

1

1 ,(zv)2 e dv:

(5.18) (5.19) (5.20)

q

When i = 1; 2, we let z = (RD2 + zni2 )=tD =2; then q

erfc[ (RD2 + zni2 )=tD =2] 1 Z 1 exp[,(R2 + z 2 )v2 =(4t )]dv: q p = D D ni tD 1 RD2 + zni2

(5.21)

According to Poisson's Summation Formula ( see [22], page 275 ), we have

p

1 2 X exp[, (a ,42tnb) ] = bt f1 + 2 exp[,(n)2 t=b2 ] cos[na=b]g: n=,1 n=1 1

+ X

Let and Thus

a = (zD zwD )v; =

1

X

n=1

b = HD v;

exp[,(n)2 tD =(HD v)2 ] cos(nzD =HD ) cos(nzwD =HD ):

PD = H2

D

11

Z 1

2 v exp[,(RD v) =(4tD )](1 + 2)dv:

(5.22) (5.23) (5.24) (5.25)

The exponential integral function Ei has the following properites

Ei (x) =

Z

et dt; ,1 t x

(5.26)

and when x < 0.01, Ei (x) can be approximated as ([52])

Ei(x) , ln(x) , ;

(5.27)

where is Euler's constant, 0.5772, and

,Ei(,x) =

1 e,xt

Z 1

t dt:

(5.28)

Let ,RD2 =(4tD ) = x, according to equation (5.28), we have Z 1 Z 1 , 21 Ei[,RD2 =(4tD )] = 12 v12 exp[,(RD v)2 =(4tD )]d(v2 ) = v1 exp[,(RD v)2 =(4tD )]dv: 1 1

CHAPTER 5. WELL TESTING FORMULAE FOR SINGLE POROSITY RESERVOIR We de ne

I0 =

11

Z 1

Because there holds ([52])

1 E [,R2 =(4t )]: 2 exp[ , ( R v ) = (4 t )] dv = , D D v 2 i D D

K0 (x) = 21

1

Z 0

exp(, , x4 ) d ; 2

48 (5.29) (5.30)

if we let x = nRD =HD , = (R4DtDv) , then d= = 2dv=v, according to equation (5.30), there holds 2

K0 (nRD =HD ) =

11

Z 0

(RD v)2 , (n)2 tD ]dv: exp[ , v 4 tD (HD v)2

Now we de ne In1 and In2 as follows: Z 1 1 exp[, (RD v)2 , (n)2 tD ]dv = K (nR =H ); In1 = 0 D D v 4tD (HD v)2 0 1 exp[, (RD v)2 , (n)2 tD ]dv: 4tD (HD v)2 0 v Therefore, when n 1; we de ne In and we have below relation Z 1 1 exp[, (RD v)2 , (n)2 tD ]dv = I , I : In = n1 n2 v 4tD (HD v)2 1

In2 =

Z 1

If we let v = 1=u, u2 = w, then du=u = dw=(2w), and In2 can be expressed as 1 exp[, (RD v)2 , (n)2 tD ]dv 4tD (HD v)2 0 v Z 1 1 exp[, RD2 , (nu)2 tD ]du = u 4tD u2 HD2 1 Z 1 1 exp[, (nu)2 tD ]du

In2 =

Z 1

HD2 Z 1 1 exp[, (n)2 tD w ]dw = 12 w HD2 1 Z 1 2 exp[, (nH) 2tD w ]dw 1 D 2 2 H ( = (n)D2 t exp[, nH)2 tD ]: D D 1

And we have

1

X

n=1

u

2 2 2 In2 ( H2 tD ) exp[, Ht2D ]=f1 , exp[, 3H 2tD ]g:

D

D

In order to derive the dimensionless pressure of endpoints, we let q

2 1 , x0D ; RD = (1 , x0D )2 + RwD

D

(5.31) (5.32) (5.33)


49

and we should integrate equation (5.18) or (5.25) with respect to x0D from ,1 to 1, according to equation (5.25), we have E = PwD

where and we have So

1 2 fJ + 2 X (Jn1 , Jn2 )[cos(nzD =HD ) cos(nzwD =HD )]g; H 0

J0 =

D

n=1

Z 1

,1

I0 dx0D ; Jn1 =

Z 1

,1

In1 dx0D ; Jn2 =

zD + zwD 2zwD ; J0 =

Z 1

,1

Z 1

,1

(5.34)

In2dx0D ;

zD , zwD = RwD :

(5.35)

I0 dx0D

Z 1 1 = 2 Ei [,RD2 =(4tD )]dx0D ,1 Z 1 1 , 2 Ei[,(1 , x0D )2 =(4tD )]dx0D ,1 Z 2 = , 12 Ei [,y2 =(4tD )]dy 0

p

= ,Ei (,1=tD ) + 2 tD

p1t Z

D

exp(,x2 )dx;

0

therefore

p

J0 = ,Ei(,1=tD ) + tD erf ( p1t ): D

(5.36)

In order to obtain the equation (5.36), we have used transformation parameters and integration by parts method ([59]). When time is suciently long ( tD is large enough ), because there hold formula (5.27) and the following formula ([52]) p p lim tD erf (1= tD ) = 2; t !1 D

so that equation (5.36) reduces to

J0 ,Ei(,1=tD ) + 2 2 , + ln tD 1:4228 + ln tD : When time is suciently short ( tD is small enough ), we have ,E (,1=t ) 0; erf ( p1 ) 1; i

so that equation (5.36) becomes

D

tD

p

J0 tD :


50

Table 5.1: The convergence rate of the integration of K0 (z ). x=1 x=2 x=3 x=4 x=5 x=6 y = ,0:328 y = ,0:097 y = ,0:031 y = ,0:010 y = ,0:003 y = ,0:001 If we let

x

Z

y = K0(z)dz , =2; 0

then we have Table 5.1. Consequently, we have

Jn1 = =

Z 1

,1

Z 1

,1

Z 1

,1

In1 dx0D K0 (nRD =HD )dx0D K0 [n(1 , x0D )=HD ]dx0D

= HnD

n=HD

2 Z

K0 (z)dz:

0

Note that 2n=HD 1, according to Table 5.1 and the formula (4.47), we have n=HD

2 Z

K0 (z )dz =2;

0

thus

Jn1 HD =2n;

and

Jn2 =

Z 1

,1

(5.37)

In2 dx0D 0:

Note that relationships (5.35) and formula (4.61), then cos(nzD =HD ) cos(nzwD =HD ) = 21 fcos(nRwD =HD ) + cos[n(zD + zwD )=HD ]g;

and there holds formula (4.58), thus we de ne

T=

1

X

(Jn1 , Jn2 ) cos(nzD =HD ) cos(nzwD =HD );

n=1

(5.38)

CHAPTER 5. WELL TESTING FORMULAE FOR SINGLE POROSITY RESERVOIR and

51

1

(Jn1 , Jn2 ) cos(nzD =HD ) cos(nzwD =HD ) n=1 1 1 HD X 2 n=1 n cos(nzD =HD ) cos(nzwD =HD ) , H4D lnf4 sin[(zD , zwD )=(2HD )] sin[(zD + zwD )=(2HD )]g X

=

, H4D lnf4 sin[RwD =(2HD )] sin(zwD =HD )g;

so

T , H4D lnf4 sin[RwD =(2HD )] sin(zwD =HD )g:

(5.39) Combining equations (5.34), (5.36), (5.38) and (5.39), if the horizontal boundary conditions at z = 0 and H are both impermeable, the dimensionless pressure of the endpoints is P E = 2 (J + 2T ): wD

HD

0

Simplify the above equation, we have E = 2 [,Ei (,1=tD ) + ptD erf ( p1 )] , lnf4 sin[RwD =(2HD )] sin(zwD =HD )g: PwD H t D

D

When time is suciently long, equation (5.40) becomes E 2 (1:4228 + ln t ) , lnf4 sin[R =(2H )] sin(z =H )g: PwD D wD D wD D H D

When time is suciently short, equation (5.40) becomes E 2 (ptD ) , lnf4 sin[RwD =(2HD )] sin(zwD =HD )g: PwD H D

In order to nd the dimensionless pressure of midpoint, we let q

2 x0D ; RD = x20D + RwD

and let

L0 =

Z 1

,1

I0 dx0D

Z 1 1 = 2 Ei [,RD2 =(4tD )]dx0D ,1 Z 1 1 , 2 Ei[,x20D =(4tD )]dx0D

,1

p = ,Ei [,1=(4tD )] + 4 tD

p1t

2

Z

D

exp(,x2 )dx

p p = ,Ei [,1=(4tD )] + 4 tD [ 2 erf ( 2p1t )]; D 0

(5.40) (5.41) (5.42)

CHAPTER 5. WELL TESTING FORMULAE FOR SINGLE POROSITY RESERVOIR i.e.,

p

L0 = ,Ei[,1=(4tD )] + 2 tD erf ( 2p1t );

(5.43)

D

and we let

Ln1 = =

Z 1

,1

Z 1

,1

Z 1

,1

52

In1 dx0D K0 (nRD =HD )dx0D K0 [nx0D =HD ]dx0D n=HD

Z = HnD K0 (z)dz ,n=HD HD (=2) 2n = HD =n:

By the analogy of the above arguments, if the horizontal boundary conditions at z = 0 and H are both impermeable, the dimensionless pressure of the midpoint is M = 2 [,Ei (,1=4tD ) + 2ptD erf ( p1 )] , 2 lnf4 sin[RwD =(2HD )] sin(zwD =HD )g: (5.44) PwD HD 2 tD When time is suciently long, equation (5.44) becomes (5.45) P M 2 (2:8091 + ln t ) , 2 lnf4 sin[R =(2H )] sin(z =H )g: wD

D

HD

wD

D

wD

D

When time is suciently short, equation (5.44) becomes M 4 (ptD ) , 2 lnf4 sin[RwD =(2HD )] sin(zwD =HD )g: PwD H D

(5.46)

Formulae (5.40), (5.41), (5.42), (5.44), (5.45), and (5.46) are dimensionless pressure formulae, i.e., the well testing formulae for horizontal wells in nite height reservoirs with impermeable boundary conditions.

5.3.2 Reservoirs with Bottom Water or Gas Cap

In this subsection, we will derive the well testing formulae for horizontal wells in nite height reservoirs with bottom water or gas cap. If the horizontal well is in the reservoir with constant boundary conditions at z = 0 and H ( the reservoir with bottom water and gas cap ), the pressure of the horizontal wellbore tends to become steady when time is suciently long ([50, 69, 71]). In such case, the dimensionless pressure at point (x0D ; 0; zwD ) is 1 X (5.47) P = 1 [ K (nR =H ) sin(nz =H ) sin(nz =H )]: D

HD n=1

0

D

D

D

D

wD

D


53

Thus, the steady dimensionless pressure of the endpoints is E PwD

Z 1 1 X 1 H f sin(nzD =HD ) sin(nzwD =HD ) K0 [n(1 , x0D )=HD ]dx0D g

,1

D n=1

1 1 1X

4 n fcos[n(zD , zwD )=HD ] , cos[n(zD + zwD )=HD ]g n 1 4 fln[2 sin(zwD =HD )] , ln[2 sin(RwD =(2HD ))]g; =1

so

E 1 lnf sin(zwD =HD ) g: PwD 4 sin[R =(2H )] wD

D the de nitions of dimensionless pressure PD in (4.21)

(5.48)

According to and (5.5), formula (5.48) is equivalent to formula (4.71). If the boundary condition at z = 0 is impermeable while the boundary condition at z = H is constant ( the reservoir with gas cap ), the pressure of the horizontal wellbore tends to become steady when time is suciently long ([50, 69, 71]). In such case, the dimensionless pressure at point (x0D ; 0; zwD ) is

PD = H1 f

1

X

D n=1

K0 [(2n , 1)RD =(2HD )] cos[(2n , 1)zD =(2HD )] cos[(2n , 1)zwD =(2HD )]g: (5.49)

We de ne

W=

then

Z 1

,1

K0 [(2n , 1)(1 , x0D )=(2HD )]dx0D ; n,1)=HD

W = (2n2H,D1) 0 2 H D (2n , 1) (=2) = HD =(2n , 1): Z (2

K0 (z)dz

Thus, according to formula (4.59), the steady dimensionless pressure of the endpoints is 1

E 1 f cos[(2n , 1)zD =(2HD )] cos[(2n , 1)zwD =(2HD )]W g PwD HD n=1 X

1 1 1X 2 n=1 2n , 1 fcos[(2n , 1)(zD + zwD )=(2HD )] + cos[(2n , 1)(zD , zwD )=(2HD )]g 14 fln[cot(zwD =(2HD ))] + ln[cot(RwD =(4HD ))]g; then we have E 1 fln[cot(R =(4H ))] + ln[cot(z =(2H ))]g: (5.50) PwD wD D wD D 4


54

Similarly, if the boundary condition at z = H is impermeable while the boundary condition at z = 0 is constant, ( the reservoir with bottom water ), the dimensionless pressure at point (x0D ; 0; zwD ) is

PD = H1 f

1

X

D n=1

K0 [(2n , 1)RD =(2HD )] sin[(2n , 1)zD =(2HD )] sin[(2n , 1)zwD =(2HD )]g; (5.51)

thus the steady dimensionless pressure at the endpoints is E 1 fln[cot(R =(4H ))] , ln[cot(z =(2H ))]g: PwD (5.52) wD D wD D 4 Considering that the de nition of PD in (4.21) is dierent from the de nition of PD in (5.5), thus we may conclude that (5.47) is equivalent to (4.42), (5.49) is equivalent to (4.37), (5.50) is equivalent to (4.69), (5.52) is equivalent to (4.65). In summary, if the horizontal well is in the reservoir with bottom water or gas cap, the pressure of the wellbore will become steady when time is suciently long. The well will produce oil at a constant rate.

Chapter 6

Well Testing Formulae for Double Porosity Reservoir In this chapter, we rst recall the Warren-Root Model. Then we will use it to nd the Laplace transform image of dimensionless pressure of the wellbore. We will also derive the pressure drop well testing formulae of horizontal wells in nite height double porosity reservoir.

6.1 Warren-Root Model If the reservoir is an inhomogeneous, anistropic naturally fractured reservoir { double porosity reservoir, we use Warren-Root Model: Reservoir uid movement toward the wellbore occurs only in fractures ([49, 87]). In 1960, Barenblatt et al. published a pioneering work on the subject of well testing in naturally fractured reservoirs. Many fractured reservoirs are characterized by an interconnected fracture network dividing the matrix rock into separate blocks. Such a system may be modelled by two overlapping continuous media, one for the medium of fractures and one for the medium of blocks. The fracture network has a high permeability and low storage capacity. In the medium of blocks the conditions are reserved. If the permeability contrast ( fracture to matrix permeability ratio ) is large, then the ow through the blocks may be disregarded and the double - porosity model may be applied. Warren and Root presented a model based on the mathematical concept of superposition of two media: A porous rock with a highly developed system of ssures can be represented as the superposition of two porous media with pores of dierent sizes. The porous media are coupled to each other by the fact that there is a liquid exchange between the two. At each point in space, we can consider two pressures and two velocities of the liquid: P1 and V1 for the liquid in the ssures and P2 and V2 for that in the blocks. The interaction of these media can be expressed by the relation: q = (P2 , P1 ); where is a characteristic of the ssured medium { interporosity ow shape factor, is density. Warren and Root based their results on the fact that in the most general case both primary and secondary porosity is present in the reservoir: An independent system of secondary porosity is superimposed on the primary or intergranular system. The obvious idealization of an intermediate porous medium is a complex of discrete volumetric elements with primary porosity which are anisotropically coupled by secondary voids. The sketch of the heterogeneous porous media is shown in Figure 6.1. 55

CHAPTER 6. WELL TESTING FORMULAE FOR DOUBLE POROSITY RESERVOIR

56

VUGS

FRACTURE ACTUAL

MATRIX RESERVOIR

MATRIX MODEL

FRACTURES RESERVOIR

Figure 6.1: Warren and Root's Sketch of a Naturally Fractured Reservoir. The material with the primary porosity is contained within a systematic array of identical rectangular parallelepipeds. The secondary porosity is contained within an orthogonal system of continuous uniform fractures which are oriented such that each fracture is parallel to one of the principal axes of permeability. Flow can occur only between the primary and secondary porosities but not through the primary porosities. Applying the continuity equation to this geometry, Warren and Root ([49, 87]) camp up with the following equations ( in dimensionless forms ) @ 2PfD + 1 @PfD = (1 , !) @PmD + ! @PfD ; (6.1) and

@rD 2

rD @rD

@tD

@tD

mD (1 , !) @P (6.2) @tD = (PfD , PmD ); where ! and are two parameters characterizing the particular reservoir under study. These two parameters are sucient distinguish a ssured reservoir from that of a homogeneous porous medium. Probably the most important contribution of this model is that it considers a variety of sedimentological cases and, the matrix contributes to the production through the fractures, i.e., the model considers the general case of a ssured reservoir. Also, a homogeneously distributed porosity is considered as a limiting case in the model when ! =1, = 1. As de ned by Warren and Root, ! and are given by: ! = Cf f =(Cf f + Cmm ); (6.3) 2 = Km rw =Kf ; (6.4) ! is the storativity coecient, is the uid transfer coecient.

6.2 Laplace Transform Images of Point Convergence Pressure In this section, we will derive Laplace transform images of the point convergence pressure of horizontal wells, i.e., Laplace transform images of pressure distribution for a continuous point source.


57

According to the mass conservation and Darcy's laws, if the point convergence intensity at point (x0 ; y0 ; z0 ) is q, taking Warren-Root Model into account, we have to solve the equations below to get the point convergence pressure distribution formulae.

m Cm @P@tm + Km (Pm , Pf ) = 0; f , Kfx @ 2 Pf , Kfy @ 2 Pf , Kfz @ 2 Pf f Cf @P @t @x2 @y2 @z 2 = Km (Pm , Pf ) , qB(x , x0 )(y , y0 )(z , z0 );

and

Pt=0 = Pi ;

Let

q

yD = 2y Kfx=Kfy =L;

q

zD = 2z Kfx =Kfz =L; LD = 2; q 4Kfx t tD = (C + ; P = 4 L Kfy Kfz (Pi , P )=(qB ): D f Cm )L2 mD + (P , P ) = 0; (1 , !) @P mD fD @t D

where

(6.6)

P (x; y; z) = Pi ; (x; y ! 1):

xD = 2x=L;

Then we have

(6.5)

fD , @ 2 PfD , @ 2 PfD , @ 2 PfD ! @P @tD @xD 2 @yD 2 @zD 2 = (PmD , PfD ) + 8(xD , x0D )(yD , y0D )(zD , z0D );

(6.7) (6.8) (6.9) (6.10)

(6.11)

! = Cf f =(Cf f + Cm m ); = Km L2 =(4Kfx ); (6.12) and ! is the storativity coecient, is the uid transfer coecient, is interporosity ow shape factor, its dimension is [L,2 ] ([49, 87]). And we point that in this chapter, the f subscripts refer to fracture medium, m subscripts refer to matrix porous medium. Taking the Laplace transform of the above equations with respect to tD , we obtain

@ 2 Pd fD + @ 2 Pd fD = sf (s)Pd + 8(x , x )(y , y )(z , z ); fD + @ 2 Pd D 0D D 0D D 0D fD 2 2 @xD @yD @zD 2 where

(6.13)

f (s) = [s!(1 , !) + ]=[s(1 , !) + ];

(6.14) and s is Laplace transform variable. The elementary solution of equation (6.13) is ([40, 101])

Pd fD

= 2 exp[, sf (s)RD ] ; p

sRD

(6.15)

CHAPTER 6. WELL TESTING FORMULAE FOR DOUBLE POROSITY RESERVOIR with

58

q

(6.16) RD = (xD , x0D )2 + (yD , y0D )2 + (zD , z0D )2 ; s s K fx fx RwD = Rw ( K + K (6.17) Kfz )=L: fy As Figure 3.1 shows, the horizontal well is in a reservoir of height H , and the lateral, longitudinal

lengths of the reservoir are in nite, i.e., the boundaries of the reservoir in the horizontal directions are so far away that the pressure disturbance does not travel far enough to reach the boundaries during the well testing. The well is parallel to the the top and bottom boundaries. The two endpoints of the horizontal well are (,L=2; 0; zw ); (L=2; 0; zw ), and 0 < zw < H . The Laplace transform image of the dimensionless pressure of the horizontal well is ([11, 25, 36])

p

Z 1 2 PfwD = s exp[,R uRD ] dx0D ; b

,1

where

(6.18)

D

u = sf (s):

(6.19) Application of the method of images, the Laplace transform image of dimensionless pressure of the wellbore is ([11, 24, 25, 84])

PbfwD =

Z 1

,1

F (s; x0D )dx0D :

(6.20)

and if the boundary conditions at z = 0 and H are both impermeable, by the method of images and the point-source solution given by equation (6.15), we have

F (s; x0D ) = 1s

1 exp[,

+ X

q

q

u(RD2 + zn2 1 )] exp[, u(RD2 + zn22)] q q + g; RD2 + zn2 1 RD2 + zn2 2

f

n=,1

(6.21)

and zn1 ; zn2 has the same meanings in (5.19),

zn1 = zD , zwD , 2nHD ;

zn2 = zD + zwD , 2nHD ;

and dimensionless HD ; zwD are de ned as follows s

x HD = 2H ( K Ky )=L;

s

x zwD = 2zw ( K Kz )=L:

(6.22)

When ! ! 1, ! 1, the double porosity reservoir would become to the single porosity reservoir ([49, 87]), and f (s) = 1; u = s: (6.23) Recall the well known Laplace transform formula ([106]), p

,k s Lferfc( 2pk t )g = e s ;

(6.24)


59

where s is Laplace transform variable, k is a constant. If we let q

k = RD2 zn2 1 ;

then combining equations (6.21), (6.23) and (6.24), leads to the formula (5.18). We may use formula (5.22) to similify equation (6.21). Recall the following Laplace transform formulae ([106]) ,dps 2 e Lfexp[,d =(4t)]=( t )g = d ; p 2 K 0 (d s) 2 Lfexp[,d =(4t)]=(bt)g = ; =

2

=

(6.25)

1 2 3 2

(6.26)

bp Lfexp[,d2 =(4t) , ct]=tg = 2K0 (d s + c); where b; c and d are constants, s is Laplace transform variable. Let

(6.27)

q

d = j 2 + (a , 2nb)2 ;

in formula (6.25), and let

(6.28)

c = (n=b)2 ;

(6.29) in formula (6.27). Multiplying both sides of equation (5.22) by exp[,j 2 =(4t)]=(1=2 t3=2 ), taking the Laplace transform of the resulting expression with respect to tD , and applying equations (6.25), (6.26), (6.27), (6.28), (6.29), we obtain the summation formula 1 + 1 exp[,ps(j 2 + (a , 2nb)2 )] 1 q X ps) + 2 X p = K s + (n=b)2 ) cos(na=b)]: (6.30) [ K 0 (j 0 (j 2 2 b j + ( a , 2 nb ) n=,1 n=1 If we use the summation formula given by (6.30), let u take the place of s, and let j = RD ; a = zD zwD ; b = HD ; (6.31) then use formula (4.61), we have cos[n(zD + zwD )=HD ] + cos[n(zD , zwD )=HD ] = 2 cos(nzD =HD ) cos(nzwD =HD ): (6.32) According to (6.30), (6.31) and (6.32), the pressure distribution for a continuous point source located at (x0D ; 0; zwD ) in a laterally in nite reservoir bounded by two impermeable planes, given by equation (6.21), can be written as: 1 X p F (s; x ) = 2 [K (R u) + 2 K (R ) cos(nz =H ) cos(nz =H )]; (6.33) D

0

sHD

0

D

0

n=1

where

D

1

D

D

wD

D

q

1 = u + (n=HD )2 : Similarly, if the boundary conditions at z = 0 and H are both at constant pressure, then the

pressure distribution for a continuous point source solution corresponding is given by

F (s; x0D ) = 1s

1 exp[,

+ X

f

n=,1

q

q

u(RD2 + zn2 1 )] exp[, u(RD2 + zn22)] q q , g; RD2 + zn2 1 RD2 + zn2 2

(6.34)


60

which, with formula (6.30), can be transformed to

F (s; x0D ) = sH1 [

1

X

D n=1

with

K0 (RD 2 ) sin(nzD =HD ) sin(nzwD =HD )];

(6.35)

q

2 = u + (n=HD )2 :

If the boundary condition at z = 0 is impermeable while the boundary condition at z = H is at constant pressure, then the pressure distribution for a continuous point source solution

corresponding is given by

q

q

exp[, u(RD2 + zn2 1 )] exp[, u(RD2 + zn2 2 )] 1 n q q F (s; x0D ) = s (,1) f + g: RD2 + zn2 1 RD2 + zn2 2 n=,1 1

+ X

If we note that ([84])

=

1

(,1)n exp[, a(x ,4t2ny) ]

+ X

1

+ X

(6.36)

2

n=,1

f2 exp[, a(x , 24nt (2y)) ] , exp[, a(x ,4t2ny) ]g; 2

2

n=,1

(6.37)

then, with the aid of formula (6.30), we can recast equation (6.36) in the form 1 X 1 F (s; x0D ) = sH f K0 (RD 3 ) cos[(2n , 1)zD =(2HD )] cos[(2n , 1)zwD =(2HD )]g; D n=1

where

(6.38)

q

3 = u + [(2n , 1)=(2HD )]2 : If the boundary condition at z = H is impermeable while the boundary condition at z =

0 is at constant pressure, then the pressure distribution for a continuous point source solution corresponding is given by 1 X 1 F (s; x0D ) = sH f K0 (RD 4 ) sin[(2n , 1)zD =(2HD )] sin[(2n , 1)zwD =(2HD )]g; D n=1

where

(6.39)

q

4 = u + [(2n , 1)=(2HD )]2 : As we have stated before, when ! ! 1, ! 1, the double porosity reservoir model becomes to a single porosity reservoir model with f (s) = 1; u = s. By the inverse Laplace transform method,

we can reduce (6.35) to (5.47), reduce (6.38) to (5.49), reduce (6.39) to (5.51). Therefore, we have obtained formulae (5.47), (5.49) and (5.51) from a dierent approach. According to formulae (6.35), (6.38) and (6.39), we come to the conclusion that when time is suciently long, the pressure of the wellbore tends to become steady. Therefore, formulae (5.48), (5.50) and (5.52) are also suitable for the horizontal wells which are in the double-porosity reservoir.


61

6.3 Well Testing Formulae for Wells in Finite Height Reservoirs In this section, we will derive the well testing formulae for horizontal wells in nite height reservoirs with impermeable boundaries. If the boundary conditions at z = 0 and H are both impermeable, according to formula (6.33), the Laplace transform image of the dimensionless pressure of the wellbore is

PfwD = sH2 b

Z 1

D ,1

p [K (RD u) + 2 0

where

1

X

n=1

K0 (RD 1 ) cos(nzD =HD ) cos(nzwD =HD )]dx0D ; (6.40) q

When x ! 0, there holds ([106])

1 = u + (n=HD )2 : K0 (x) ln(2=x) , ;

where is Euler's constant, 0.5772, thus

1 Z 1 K (R pu)dx 1 [Z 1 ln( 2p )dx , 2 ]: 0 D 0D s s R u 0D ,1

,1

D

In order to nd the dimensionless pressure of endpoints, we let q

2 RD = (1 , x0D )2 + RwD 1 , x0D ;

and

p

Jc0 (s) = 1s K0 [(1 , x0D ) u]dx0D ,1 1 s (1:4228 , ln u) ln s , ln f (s) = 1:4228 , s s s 1 : 4228 ln s ln[ s (1 , !) + ] , ln[s(1 , !)! + ] : = , + s s s s Z 1

Therefore, by the inverse Laplace transform, we have

,tD ] , E [ ,tD ]: J0 (tD ) 1:4228 + ln tD + Ei [ !(1 i 1,! , !) When u ! 0, i.e., time is very long, and n 1, we have q

1 = u + (n=HD )2 n=HD ; then when n 1, we de ne Z 1 1 Jn (s) = s K0 (RD 1 )dx0D c

,1

(6.41)


1s

Z 1

,1

62

K0 [n(1 , x0D )=HD ]dx0D

Z 2 = 1s K0 (ny=HD )dy 0

n=HD

2

Z H D K0 (z)dz = ns 0 HD =(2ns):

Therefore, when n 1, there holds

Jn (tD ) HD =(2n):

(6.42)

Similar to equations (5.38), (5.39), we have 1

X

n=1

Jn cos(nzD =HD ) cos(nzwD =HD )

, H4D lnf4 sin[RwD =(2HD )] sin(zwD =HD )g:

Therefore, if horizontal boundary conditions at z = 0 and H are both impermeable, and time is suciently long, the endpoints' dimensionless pressure of the horizontal well in a double-porosity reservoir is E = 2 f1:4228 + ln t + E [ ,tD ] , E [ ,tD ]g PfwD D i !(1 , !) i 1,! H D

, lnf4 sin[RwD =(2HD )] sin(zwD =HD )g:

(6.43)

In order to nd the dimensionless pressure of midpoint, we let q

2 RD = x20D + RwD x0D :

Similarly, if the horizontal boundary conditions at z = 0 and H are both impermeable, and time is suciently long, the midpoint's dimensionless pressure of the horizontal well in a double-porosity reservoir is M = 2 f2:8091 + ln tD + Ei [ ,tD ] , Ei [ ,tD ]g PfwD H !(1 , !) 1,! D

,2 lnf4 sin[RwD =(2HD )] sin(zwD =HD )g:

(6.44) Formulae (6.43) and (6.44) are dimensionless pressure well testing formulae for horizontal wells in nite height reservoirs with impermeable boundaries. Again, when ! ! 1, ! 1, the double porosity reservoir model becomes a single porosity reservoir model, and formula (6.43) reduces to formula (5.41), formula (6.44) reduces to formula (5.45).

Appendix A

Conclusions Now, we may come to the following conclusions: (1) The productivity formulae based on 2D model can not satisfactorily characterize horizontal wells. In order to get the reasonable and reliable productivity formulae, we must use 3D model; (2) If taking a horizontal well as a uniform line source in 3D space, the equipotential surfaces are a family of ellipsoids of revolution which focuses are the two endpoints of the well, and the well has not in nite conductivity; (3) Our well testing formulae of horizontal wells are concise and explicit, they have higher precision than the formulae in the literatures.

63

Appendix B

Nomenclature (English) A = drainage area B = formation volume factor Cf = uid compressibility CR = rock compressibility Ct = total compressibility Ei (x) = exponential integral function erf (x) = error function erfc(x) = complementary error function H = reservoir eective thickness K = absolute permeability K0 (x) = modi ed Bessel function of second kind and order zero L = well length P = pressure Q = well productivity, well ow rate Rw = wellbore radius Re = drainage radius s = Laplace transform variable t = time V = drainage volume x; y; z = lateral,longitudinal,vertical coordinate. (Greek)

= interporosity shape factor = permeability anisotropic factor (t) = Dirac function. = uid transfer coecient,fraction. = uid viscosity = density = uid potential = porosity,fraction ! = storativity coecient,fraction.

64

APPENDIX B. NOMENCLATURE Superscripts = Laplace transform e = eective A = average E = endpoint M = midpoint b

Subscripts

D = dimensionless e = external f = fracture h = horizontal i = initial m = matrix v = vertical w = well x; y; z = coordinate indicators

65

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Vita of Jing Lu

Jing Lu was born in Chengdu, China on May 8, 1969. He graduated from Chengdu No. 7 high school in 1987. He received the B.S. degree in geology at Southwest Petroleum Institute in 1991 and got the Master degree in petroleum engineering at Chengdu Institute of Technology in 1994. He worked in the Institute under Huachuan Petroleum, Natural Gas Exploration and Development Company from 1994 to 1995. He worked as a research assistant in the Institute of Math Sciences, Chengdu Branch, Academia Sinica from 1995 to 1996. He came to America in August, 1996 and studied in the Mathematics Department of Virginia Polytechnic Institute and State University. He is a member of Americian Mathematics Society and ; M; Mathematics Society.

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