New Analytical Solutions for Predicting Pressure Distribution and Transient

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New Analytical Solutions for PredictingPressure Distribution and Transient

Behavior in Wedges andTruncated Wedges

N.M. Anisur Rahman, SPE, Fekete Associates Inc., and Ramon G. Bentsen, SPE, U. of Alberta

Summary

In most situations, predicting the pressure distribution and tran-sient behavior becomes very difficult because of a lack of neces-sary mathematical solutions. However, analytical solutions are of-ten used for such purposes, even though it may be necessary tocompromise the geometry of the actual reservoir to obtain a solu-tion. In this study, an integral-transform technique (ITT) has beenused to develop new analytical solutions for pressure distributionin wedges and truncated wedges.

Introduction

Reservoirs of various shapes are encountered in practice. Evidence

of wedges and truncated wedges has been found in Texas, Loui-siana, and elsewhere. The advantages of using analytical solutions,when available, have been recognized for many complicated prob-lems involving the flow of heat and fluid. Because analytical so-lutions provide good insight into many reservoir problems as wellas benchmarks for many complicated situations in reservoirs, theyhave proven to be useful tools. But efforts to develop analyticalsolutions for some transient-flow problems have been limited be-cause of a lack of appropriate mathematical tools. In the literature,analytical solutions for wedges are available. Application of thesesolutions very often forces the use of an approximate value of thewedge angle instead of the actual value. As a result, a considerableamount of error may be introduced into the predicted pressureresponses. Moreover, no solutions for truncated wedges have beenreported in the literature.

Rahman and Bentsen1

have introduced the idea of using theITT for the purpose of developing comprehensive solutions totransient-flow problems in porous media in a Cartesian coordinatesystem. Numerous problems are described in the cylindrical coor-dinate system for the purpose of attaining convenient methods tosolve them. The solution methodology described in Ref. 1 is notentirely applicable to solving problems in the cylindrical coordi-nate system, because the radial space variable in a cylindricalcoordinate system involves Bessel functions. Moreover, the eigen-value problem (EVP) owing to the radial space variable depends uponthe EVP owing to the angular space variable, if it exists. Therefore,Rahman and Bentsen2 have used the ITT for developing comprehen-sive solutions analytically in a cylindrical coordinate system.

Chen and Raghavan3 have developed solutions in the Laplace-space domain for pressure distribution in wedges, removing someof the restrictions on the wedge angles. These solutions are validfor a wedge angle of n / m, where n and m are positive integers,which are prime to each other. Previously, using the method of images,4 solutions were available when the wedge angle could beexpressed as / m, where m is a positive integer. Despite Chen andRaghavan’s substantial contribution to improving the situation, theinconveniences associated with history matching and nonlinear

regression in real problems cannot be eliminated altogether. Ap-plication of these solutions often forces the use of an approximatevalue of the wedge angle instead of the actual value. In otherwords, a milder restriction on the wedge angle is still in placedespite the presentation of this work. Charles et al.5 suggest thatappropriate analytical solutions would improve the current level of understanding of flow in wedges. It can be shown that considerableamounts of error in estimating rock and fluid properties may beincurred if the available approximate solutions are used.

Solutions for wedges and truncated wedges, which consider thepossibility of restricted flow in 3D because of partially penetratingor partially completed wells, are presented in this study. Despitehaving addressed specific issues, these solutions are, of course,

subsets of the general solutions for transient flow as presented inRef. 2. All the partial differential equations, including the initialand boundary conditions, and the solutions are written in termsof a potential () which is the energy per unit volume. This ap-proach is taken to conserve the generality of solutions incorporat-ing the effects of gravity. However, the potential terms can bereplaced simply by the pressure terms when the contribution of gravity is negligible.

Development of General Solutions for Wedgesand Truncated Wedges

Governing Differential Equations. The diffusivity equation indimensionless potential for the flow of fluid owing to productionthrough a well at a dimensionless rate of q D(t D) in a homogeneousand isotropic porous medium comprising the domain ℜ in a cy-

lindrical coordinate system (Fig. 1) is given by2

2 D + Qt D =

D

t D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

The source function, Q(t D), in Eq. 1 considers the production tobe taking place through a line-sink, vertical well with the produc-tion not necessarily taking place over its entire length. This allowsone to use the forthcoming solutions to examine limited-entry orpartially penetrating wells. The composition of the left side of Eq.1 and the initial condition to which the governing equation issubject both depend upon the number and types of dimensions.The source terms and the corresponding initial condition can bechosen from Table 1, subject to the coordinate system shown inFig. 1. When the dependence of the solution upon the z directionis considered, as in 3D problems, the completed zone of the pro-ducing well needs to be specified, as illustrated in Fig. 2. As canbe determined by inspection from the corresponding source termsin Table 1, the completed zones are specified by the Heavisideunit-step function. The initial conditions are considered to be uni-form; that is, there is a uniform potential over all of the boundedregions before any production begins.

Corapcioglu et al.6 have recognized different types of boundaryconditions in reservoirs. Thus, Eq. 1 can be applied in wedges(Fig. 3) and truncated wedges (Fig. 4) subject to the various pos-sible sets of boundary conditions, as listed in the second and thirdcolumns of Table 2, in all directions of flow. In each direction of flow, the boundary conditions can be chosen from one of the ninepossible sets mentioned in the second and third columns of Table

Copyright © 2003 Society of Petroleum Engineers

This paper (SPE 86195) was revised for publication from paper SPE 71585, prepared forpresentation at the 2001 SPE Annual Technical Conference and Exhibition, New Orleans,30 September–3 October. Original manuscript received for review 27 November 2001.Revised manuscript received 12 March 2003. Manuscript peer approved 14 April 2003.

280 September 2003 SPE Journal



2. Once the boundary conditions have been recognized, the typesof boundary conditions, as mentioned in the sixth column of thesame table, can characterize them. The nomenclature for a particularset of boundary conditions in each direction is illustrated in Fig. 5.

The parameters for the conditions at boundaries of the thirdkind (Cauchy type),

1

, 1

, 2

, and 2

, can be constructed in sucha way that the equations take care of the appropriate situationsbecause of partial communication, or the presence of skin, at theboundaries. These parameters also can be set equal to 0 or 1 toobtain the conditions at the boundaries of the first (Dirichlet type)or second kind (Neumann type). The first-kind condition ariseswhen the dimensionless potential at a boundary is specified, andthe second-kind condition arises when the dimensionless fluidflow rate through a boundary is specified, including a no-flowsituation. Also, the time-dependent, nonhomogeneous functions,

f 1(t D) and f 2(t D), as in conditions of the third kind, allow for speci-fying any time-dependent conditions at the boundaries. When ac-counting for the various boundary conditions, the parameters

1,

2

, 1

, 2

, f 1

(t D), and f 2

(t D) need to be estimated beforehand if theconditions are of the third kind. These parameters are used later forcomputing the eigenvalues, norms, and eigenfunctions in each di-

rection. Thus, the mathematical formulation described in Eq. 1,along with its boundary conditions, deals directly with time-dependent variations of production rate and boundary conditions.In specifying a time-dependent, dimensionless production rate,q D(t D), the Heaviside unit-step function may be used. With thiskind of representation, a varying production rate is approximatedas a series of small, constant-rate stair steps. Mathematically, it isconvenient to deal with a time-dependent production rate or aboundary condition in this way.

Eigenvalue Problems. The homogeneous version of an actualflow problem can be divided into a number of EVPs. Table 2shows the conversion of the boundary conditions in the actual

problems into the corresponding forms needed for the EVPs.Fluid-flow situations can be split into a number of EVPs for 2Dand 3D representations. A system, which is subject to 2D flow in(r, ), results in two EVPs, where the r -direction EVP dependsupon the -direction EVP. Moreover, if it exists, the -directionEVP is always independent. Similarly, a system subject to 3D flowresults in three EVPs, two of which are independent. The solutionsfor all problems considered here are functions of at least r and (Table 1). Each EVP needs to be considered independently, whileharmonizing the boundary conditions of the original problem.

The eigenfunctions, norms, and eigenvalues in each directionare taken, for the corresponding boundary conditions, from Tables3 through 5.

The r -direction transforms are also termed Hankel transformsin the literature. These transforms are taken either from Table 3, if there is no inner boundary in the r direction as in wedges, or fromTable 4, if there is an inner boundary in the r direction as intruncated wedges (Fig. 6). When using Table 2 for a case with noinner boundary, the second and forth columns are to be ignored.The boundary condition for this situation is specified in terms of the outer boundary condition only. The ideas for specifying bound-ary conditions, as in Fig. 5, can be used here as well by disregard-ing the component owing to the inner boundary conditions (Im).For instance, to specify a condition of the third kind along ther -direction for a wedge system, one uses O3.

In situations where o<2 , the -direction transforms are es-sentially the rectangular transforms, and they can be taken fromTable 5. The z-direction EVP, if it exists, turns out to be indepen-

dent of those in the other directions. These transforms are available inTable 5. Schematics of the - and z-direction EVPs are shown inFig. 7, illustrating the rectangular nature of the transforms.

The method of constructing combinations of eigenfunctionsand norms of the entire systems, utilizing the appropriate valuesfrom the preceding tables, is shown in Table 6.

General Solution. Following the procedures of the ITT, the ana-lytical solution to Eq. 1, subject to an initial condition and bound-ary conditions of the I3O3 type, can be written as2

Dr , t D = l,m,n=1

l,m,nr e

−l,m,n t D

vl,m,n

0

t D

wl,m,n el,m,n

d , . . . . . (2)

where

wl,m,n =ℜ

r D Q l,m,nr ℜ + ℜ

GS . . . . . . . . . . . . . . . . . . (3)

Eq. 2 is the general solution to Eq. 1 for the I3O3 type boundaryconditions. In Eqs. 2 and 3, the subscripts l, m, and n are usedwhenever there is 3D flow. But for 2D flow in (r, ), the subscriptsl and m are used (Table 6). The second (boundary) integral on theright side of Eq. 3 is caused by nonhomogeneous boundary con-ditions. The components of this integral are explained in AppendixA. However, the boundary integral does not exist for problems thatinvolve homogeneous conditions at all of the boundaries, because,in such cases, the f 1 and f 2s vanish.

Fig. 1—Cylindrical coordinate system.

281September 2003 SPE Journal



Constructing Solutions to Particular Problems

In the preceding section, the general solution to transient-flowproblems has been presented. Illustrative examples to demonstratethe application of this general solution in deriving solutions forspecific problems are presented in Appendix B. The general solu-tion presented as Eq. 2 utilizes nonhomogeneous boundary condi-tions of the I3O3 type. Fig. 8 shows the major steps involved inconstructing a solution and computing numerical values for agiven set of time and space variables. The following subsectionsexplain how to deal with specific issues of constructing the solu-tions to particular problems from the general solution.

Dirichlet-Type Condition at Boundaries. When the condition ata boundary is of the Dirichlet type, the solution as given by Eq. 2remains the same. But the constituent parts of Eq. 3 need to beadjusted because of the type of boundary conditions as discussedin Appendix A.

Neumann-Type Condition at Boundaries. If the boundary con-ditions in any direction are of the Neumann type (I2O2), the so-lution given by Eq. 2 needs to be augmented, because zero is alsoan eigenvalue in that particular direction. This means that an ad-ditional term, corresponding to the zero eigenvalue, has to beadded to the solution presented in Eq. 2. This kind of change canbe incorporated independently into the - and z-direction EVPsonly. In such a situation, all ’s are equal to 0. Two examples inAppendix B illustrate this idea.

While utilizing the O2 conditions in a wedge system or theI2O2 conditions in a truncated wedge system, caution should beexercised for the r -direction EVPs, as pointed out in the footnotesfound in Tables 3 and 4, respectively. It has been mentioned thatthe solution needs to be augmented when m0. There can beonly one of two possible reasons why the condition m0 is valid:first, the -direction EVP does not exist (i.e., D does not dependupon ); and second, the I2O2 conditions prevail in the -direction

EVP. The first reason is not possible in a wedge or a truncatedwedge system. Therefore, in an r -direction EVP, whenever it issubject to the O2 condition in a wedge system or the I2O2 con-ditions in a truncated wedge system, a zero eigenvalue needs to beconsidered whenever the second reason is involved.

Cauchy-Neumann Mixed Conditions at Boundaries. If the con-ditions at the boundaries in a particular direction are mixed, withboth Cauchy and Neumann types (e.g., I3O2), the expression for D in Eq. 2 remains unchanged. However, appropriate values forthe and s have to be constructed in Eq. 3.

Dirichlet-Neumann Mixed Conditions at Boundaries. When theboundary conditions in a particular direction are of the I2O1 orI1O2 type, the composition of the boundary integral, if it exists,because of nonhomogeneous condition(s), in Eq. 3 needs to beadjusted, keeping the solution in Eq. 2 the same. This matter isdiscussed further in Appendix A.

Computational Aspects of the Solutions

A complete process, starting from developing the analytical solu-tion for a particular problem to getting numerical values, is pre-sented in Fig. 8. A typical exercise with this algorithm shows thatmost of the computational load is devoted to computing the nu-merical values of the eigenvalues where applicable. Appendix Cpresents a discussion of the computational aspects of eigenvalues,norms, and eigenfunctions.

Discussion

New solutions to problems involving all possible combinations of conditions involving the first, second, and third kinds on theboundaries of wedges and truncated wedges are presented for pre-dicting pressure distribution and transient behavior. All the wedgeand truncated wedge geometries are idealistic as can be repre-sented by the cylindrical coordinate system (Fig. 1). This meansthat closed-form solutions, in terms of the time and space vari-ables, can be written down for all 243 possible combinations of theconditions at the boundaries, including constant-potential, no-flow, and mixed-boundary conditions in wedges subject to 3Dflow. Similarly, the solutions for all 729 possible combinations of boundary conditions can be considered in truncated wedges sub-

ject to 3D flow. Also for 2D flow, all 27 possible solutions to theproblems of wedges, and the 81 possible solutions to the problemsof truncated wedges, are derivable using the method outlined here.The presentation of the solutions is compact and comprehensive.This allows any possible situation in the field to be considered. Incontrast, existing solutions do not possess this kind of generality.The solutions in Ref. 3 have considered different combinations of boundary conditions, keeping the top and bottom boundary surfacesimpermeable. Time derivatives of the dimensionless potential can befound directly without going through a numerical procedure. This

Fig. 2—Specification of a partially completedwell in a 3D situation.

Fig. 3—A wedge system in which production is occurringthrough a partially penetrating well.

Fig. 4—A truncated-wedge system in which production isoccurring through a partially penetrating well.




facilitates the use of derivative analysis in well tests for identifyingdifferent flow regimes and computing different parameters.

There is no need to use any numerical techniques for inversionwhen using the ITT for computing transient responses and theirderivatives. When a boundary condition is of the third kind, themethod of images, with the line-source solution, cannot be used.6

Thus, the application of the ITT as a means to solve differentialequations for fluid flow through porous media is more tractable com-pared to many other methods of solution that are currently in use.

From the compact form of the new solutions, it is evident thatparticular solutions can be derived for wedge-shaped reservoirssubject to 3D flow without any restriction on the wedge angle.Examples in Appendix B have proved this contention. As the newsolutions are good for any real values of dimensions, including thewedge angles, it has been demonstrated with examples in Appen-

dix B that the solutions for truncated wedges are more general thanare those for wedges. Therefore, this constitutes a considerableimprovement over the solutions that are currently available in theliterature. Transient responses can be developed for different as-pects of the wedge and truncated wedge systems by applying thenew solutions. Thus, the new solutions will facilitate characteriza-tion of transient behavior. A few cases are presented in AppendixB, showing the versatility of the solutions for wedge-shaped res-ervoirs with fully and partially penetrating wells.

The boundary conditions are considered in a generalized waythat accounts for time-dependent situations. A computationallyefficient algorithm for time-dependent production rates and bound-ary conditions can be developed. Therefore, it is possible to in-clude the effects of wellbore storage and skin in the solutions.Being presented in real time and space variables, the solutions are

readily programmable into software packages.

Conclusions

1. The ITT can be used to develop new, comprehensive solutionsto transient-flow problems for wedges and truncated wedges.

2. In some cases, the new solutions are free from the restrictive cri-teria for application that pertain to those available in the literature.

Nomenclature

B formation volume factor

Bi i2+i

2l21−

m2

l2 Li

2, i=1, 2

ct total compressibility, Lt2 /mFig. 5—Specification of boundary conditions in each direction(r =r, , or z ).




f ri , f i , f zi nonhomogeneous terms in boundary conditions

for r, , z coordinates, respectively, i1, 2

F i 1− m

2

l2 Li

2, i=1, 2

g acceleration caused by gravity, L/t2

G constituent of boundary integral in Eq. 3 caused

by nonhomogeneous boundary conditions,

Appendix A

h pay thickness or location of outer boundary in z

direction, L

h D h/r wh1, h2 extent of producing interval, Fig. 2, L

h1 D , h2 D h1 / r w , h2 / r w , respectively

H ( z-) Heaviside unit-step function, equal to 1 for z

and 0 for z<

J ( z) Bessel function of the first kind of order

J ( z) derivative of J ( z) with respect to r D

Fig. 6—Schematics illustrating the domains in r -direction EVP.




k permeability, L2

L, L1

, L2 locations of boundaries in EVPs ( L and L

2 are

outer boundaries, where applicable)

nr , n , n z norms of r, , z direction EVP, respectively

p(r, , z, t ) pressure as function of r, , z and t, m/Lt2

p D dimensionless pressure,2 kh

q p B pi− pr , t

Fig. 7—A schematic showing a general coordinate r (where r = and z D ).

Fig. 8—Summary of the steps involved in the method of solution.




pi initial pressure, m/Lt2

q(t ) production rate as a function of time, L3 /t

q D(t D) q(t )/ q p

q p reference production rate, L3 /t

Q(t D) source term in diffusivity equation, Table 1

r, , z cylindrical coordinates

r arbitrary point with definite coordinates

r space variable in an EVP for or z, Fig. 7

r D r/r wr w wellbore radius, L

r X , areal location of producing well

r XD r X / r w Ro locations of outer boundary parallel to

z plane

RoD Ro / r wℜ bounded region under consideration

s skin factor as communication resistance at a

boundary

S boundary surface in for domain ℜ

S 2

lY (l L1)+

2Y (l L2)

t time, t

t D dimensionless time,kt

ct r w2

U 1

l J (l L1)+

1 J (l L1)

V 2

l J (l L2)+

2 J (l L1)

W 1

lY (l L1)+ 1Y (l L1)

Y ( z)

Bessel function of the second kind of order Y ( z) derivative of Y ( z) with respect to r D

z D z / r w sum of squares of eigenvalues in r and z

directions ( -direction eigenvalue is not

contributing to directly) (e.g., l2+n

2 for 3D

problems)

ri, i, zi parameters as coefficients of derivative of

dependent variable in boundary conditions for r,

, z coordinates, respectively, i1, 2

(r −a) Dirac delta function, equal to 0 when r a

ri, i, zi parameters as coefficients of dependent variable

in boundary conditions for r, , z coordinates,

respectively, i1, 2

o location of outer boundary in directionl, m, n lth, mth, nth eigenvalue for r, direction, z

direction problem, respectively

viscosity, m/L

norm of entire system, Table 6

density of fluid, m/L3

dummy variable for t D porosity

(r , t ) potential, p(r , t )– gz,m/L2

D dimensionless potential,2 kh

q p B i−r , t

i initial potential, m/L2

eigenfunction of entire system, Table 6

r , , z eigenfunctions of r -, -, z-direction EVP,

respectivelyℜ differential element in the region ℜ (e.g., dr D d

in 2D and dr Dd dz D in 3D problems)

Sr i ith boundary in

r direction within region ℜ

Subscripts

D dimensionless

i 1 for inner and 2 for outer boundaries

l, m, n lth, mth, nth eigenvalue, eigenfunction, or

parameter in r , , z direction, respectively

Acknowledgments

The Bangladesh U. of Engineering and Technology and the U. of Alberta provided the computer time.

References

1. Rahman, N.M.A. and Bentsen, R.G.: “Use of an Integral-Transform

Technique for Comprehensive Solutions to Transient-Flow Problems in

Homogeneous Domains,” paper 2000-42 presented at the 2000 Cana-

dian International Petroleum Conference, Annual Technical Meeting of

the Petroleum Society, Calgary, Alberta, 6–8 June.

2. Rahman, N.M.A. and Bentsen, R.G.: “Comprehensive Solutions for

Transient-Flow Problems in 3D Homogeneous Domains,” paper SPE

68139 presented at the 2001 SPE Middle East Oil Show and Confer-

ence, Bahrain, 17–20 March.

3. Chen, C.-C. and Raghavan, R.: “Computing Pressure Distribution in

Wedges,” SPEJ (March 1997) 24.

4. Carslaw, H.S. and Jaeger, J.C.: Conduction of Heat in Solids,

secondedition, Oxford U. Press (1959) 277–79.

5. Charles, D.D., Rieke, H.H., and Purushothaman, R.: “Well-Test Char-

acterization of Wedge-Shaped, Faulted Reservoirs,” paper SPE 56685

presented at the 1999 SPE Annual Technical Conference and Exhibi-

tion, Houston, 3–6 October.

6. Corapcioglu, M.Y., Borekci, O., and Haridas, A.: “Analytical Solutions

for Rectangular Aquifers with Third-Kind (Cauchy) Boundary Condi-

tions,” Water Resources Research (April 1983) 19, No. 2, 523.

Appendix A—Components of the BoundaryIntegral Owing to Nonhomogeneous Terms

The boundary integral in Eq. 3 is defined as

ℜ

GS =

i

1,2−1

ir D r l f ri

ri

S ri

+ T r

1

r D2i

1,2−1

i m f i

i

S i

+ T r 1 T 1i

1,2−1i z n f zi

zi

S zi

, . . . . . . . . . . . . . . . . . (A-1)

where

f ri = f riT 1 T z1, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-2)

f i = f iT z1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-3)

In Eqs. A-1 through A-3, i1 means the inner boundary andi2 means the outer boundary in a particular direction. Theseequations are valid for 3D flow. However, if the flow in a par-ticular direction does not exist, the corresponding T -function istaken as unity. For example, if the z-direction flow does not exist,then f zi0 and T z[1]1. A T -function of an arbitrary function F (

r )

in anyr direction is defined as

T r [F

r =

ℜ

F r

r

r d

r , . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-4)

wherer r, , or z.

When the boundary conditions in a particular direction are of the I1O1 type, Eq. A-1 needs to be modified. For example, if theboundaries in the r direction have the dimensionless potential

specified (i.e., r i0 and r i1), the term rl

ri

h as t o b e r e-

placed by1

r i

d r l

dr Din Eq. A-1.

If the boundary conditions in a particular direction are of theI2O1 type, Eq. A-1 needs to be modified too. For instance, in ther direction, if the production rate at the inner boundary is speci-fied (i.e., r 11 and r 10) and the dimensionless potential atthe outer boundary is specified (i.e., r 10 and r 11), the

term rl

ri

in Eq. A-1 has to be replaced by rl

r 1

for i1, and by

1

r 2

d r l

dr Dfor i = 2. Such replacements will be reversed in case of

the I1O2 type.When there is no inner boundary in the r direction (while using

Table 3), the summation term with S ri in Eq. A-1 turns out to be just a single term with i2.




Appendix B—Illustrative Examples forConstructing Solutions

Here, examples are presented to illustrate how the solutions fortransient-flow problems subject to particular sets of boundary con-ditions can be constructed using the tables. In all these examples,the effects of gravity are neglected, and dimensionless pressure isused as the dependent variable in place of dimensionless potential.In these examples, because the -direction EVP exists, the Besselfunction(s) in the r -direction EVP will be of order m (the eigen-value in the direction).

2D Flow in a Wedge System. Here, production is occurring

through a fully penetrating, line-sink well at a rate of q D1 froma wedge system with dimensions RoD , o , and h D. The vertical wellis located at (r XD , X ). The dimensionless pressure at each outerboundary is kept at zero. This is a 2D problem in (r, ). Thus, thefollowing outer boundary conditions arise:

p D0 at 0, o, and at r D RoD.

Therefore, one can obtain the following information from Table 1:

Number of EVPs: 2 (r and directions);

Qt D =2 r D − r XD − X

r D.

r-direction information (from Tables 2 and 3):Type of outer boundary condition: O1 with f r 20 at L2

RoD (no inner boundary exists); r l J (l r D);Norm: n l0.5 R2

oD J 2 (l RoD);

Equation for eigenvalues l:

J l RoD = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-1)

-direction information (from Tables 2 and 5):Type of boundary conditions: I1O1 with f 10 at L10,and f 20 at L2 o;Eigenfunction: msin m Norm: nm o /2;Equation for eigenvalues m:

sin m o = 0, which leads to m = m

o . . . . . . . . . . . . . . (B-2)

Eq. B-2 provides eigenvalues in the direction. The positivevalues for m have to be known from Eq. B-2 before Eq. B-1 canbe solved for l, the eigenvalues in the r direction, because thisoperation involves a Bessel function of the first kind of order m.

Therefore, one obtains the following expressions for the entirewedge system:

l,m rl m; l,mnrl nm; l,m2l .

It appears that m (eigenvalue in the direction) does not contrib-ute directly to the value of l,m, but l depends upon m, as in Eq.B-1. Using Eq. 3, one gets

wl,m2 J (lr XD)sin m X .

Thus, Eq. 2 becomes

p D(r D , , t D)

8

RoD2 o

l=1

m=1

J lr XDsin m X

l,m J 2l RoD

J lr Dsin m 1 − e−l,m t D.

. . . . . . . . . . . . . . . . . . . . . . . . . (B-3)

As is obvious from the solution presented for the wedge sys-tem, eventually the transient-pressure responses lead to a steady-state solution.

3D Flow in a Wedge System. In this case, production is occurringthrough a partially penetrating, line-sink well at a rate of q D1

from a wedge system with dimensions RoD , o , and h D. The ver-tical well is located in the areal plane at (r XD , X ), and completedbetween h1 D and h2 D within the pay thickness. The top and bottomboundary surfaces are considered closed, while the dimensionlesspressure at each of the other outer boundaries is maintained at zero.This is a 3D problem in (r, , z). Thus, the following outer bound-ary conditions arise:

p D = 0 at = 0, o, and at r D = RoD, and p D

z D

= 0 at z D = 0, h D.

Therefore, one can obtain the following information from Table 1:Number of EVPs: 3 (r, , and z directions);

Qt D =2 h Dr D − r XD − X H z D − h2 D − H z D − h1 D

h2 D − h1 Dr D.

r -direction information (from Tables 2 and 3):Type of outer boundary condition: O1 with f r 20 at

L2 RoD (no inner boundary exists); r l J (lr D);Norm: n l0.5 R2

oD J 2 (l RoD);


J l RoD = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-4)

-direction information (from Tables 2 and 5):Type of boundary conditions: I1O1 with f 10 at L10,

and f 20 at L2 o;Eigenfunction: msin m ;Norm: nm o /2;Equation for eigenvalues m:

sin m o = 0, which leads to m = m o. . . . . . . . . . . . . . . (B-5)

The positive values for m have to be known from Eq. B-5before Eq. B-4 can be solved for l , the eigenvalues in the r direction, because this operation involves a Bessel function of thefirst kind of order m.

z-direction information (from Tables 2 and 5):Type of boundary conditions: I2O2 with f z10 at L10,and f z20 at L2h D (0 is also an eigenvalue);

For zero eigenvalue:

Eigenfunction: zn01;Norm: n zn0h D.

For nonzero eigenvalues:Eigenfunction: zncos n z D;Norm: n znh D /2;Equation for eigenvalues n:

sin n h D = 0, which leads to n = n h D. . . . . . . . . . . . . . . (B-6)

Therefore, one obtains the following expressions for the entiresystem:

l,m,0 r l m z n0; l,m,0nr l n m n z n0; l,m,0l2;

l,m,n r l m z n; l,m,nnr l n m n z n; l,m,n l2 + n

2.

Using Eq. 3, one gets

wl,m,02 h D J (l r XD)sin m X for the zero eigenvalue, and

wl,m,n =2 h D J vl r XDsin m X sinnh2 D − sinnh1 D

h2 D − h1 Dn

for the non-

zero eigenvalues in the z direction.

Thus, Eq. 2 becomes

p Dr D, , z D, t D =

8

RoD2 o

l=1

m=1

J l r XDsin m X

l,m J 2l RoD

J lr Dsin m 1 − e−l

2t D




+16

h2 D − h1 D RoD2 0

l=1

m=1

n=1

J l r XDsin m X sinnh2 D − sinnh1 D

nl,m,n J 2l RoD

J 0lr Dsin m cosn z D1 − e−l,m,n t D. . . . . . . . . . . . . . . . . (B-7)

The first term on the right side of Eq. B-7 is caused by the zeroeigenvalue in the z direction. As h1 D→0 and h2 D→h D in Eq. B-7,the situation resembles that for a fully penetrating well as in thefirst example. Thus, the second term in Eq. B-7 vanishes, and thesolution degenerates to that for a 2D situation, as in Eq. B-3,because of these limiting values. It is obvious from the solution inB-7 for a wedge system that eventually the transient-pressure re-

sponses lead to a steady-state solution.

2D Flow in a Truncated Wedge System. Here, production isoccurring through a fully penetrating, line-sink well at a rate of q D1 from a truncated wedge system with dimensions RiD , RoD , o , and h D. The vertical well is located at (r XD , X ). Except for thetwo closed boundaries in the r direction, the dimensionless pres-sure at the other outer boundaries is kept at zero. This is a 2D problemin (r, ). Thus, the following outer boundary conditions arise:

p D

r D= 0 at r D = RiD, RoD, and p D = 0 at = 0, o.

Therefore, one can obtain the following information from Table 1:Number of EVPs: 2 (r, directions);

Qt D =2 r

D − r

XD −

X

r D.

r-direction information (from Tables 2 and 4):Type of boundary conditions: I2O2 with f r 10 at L1

RiD , and f r 20 at L2 RoD;

r lY (l RoD) J (lr D)−J (l RoD)Y (lr D);

Norm: nl =2

2

F 2 J 2l RiD − F 1 J

2l RoD

l2 J

2l RiD

;


J l RiDY l RoD − J l RoDY l RiD = 0. . . . . . . . . . . . . (B-8)

-direction information (from Tables 2 and 5):Type of boundary conditions: I1O1 with f 10 at L10,

and f 20 at L2 o;Eigenfunction: m sin m ;Norm: nm o /2;Equation for eigenvalues m:

sin m o = 0, which leads to m = m o. . . . . . . . . . . . . . . (B-9)

The positive values for m have to be known from Eq. B-9before Eq. B-8 can be solved for l , because this operation in-volves a Bessel function of the first kind of order m. Notice thatthe zero eigenvalue in the r direction is not considered, because thezero eigenvalue in the direction does not exist.


l,m rl m; l,mnrl nm; l,ml2.

It appears that m (eigenvalue in the direction) does notcontribute directly to the value of l,m; moreover, l dependsupon m , as in Eq. B-8. Using Eq. 3, one gets

wl,m2 [Y (l RoD) J (lr XD)−J (l RoD)Y (lr XD)]sin m X.

Thus, Eq. 2 becomes

p Dr D , , t D =

2 3

ol=1

m=1

Y l RoD J lr XD − J l RoDY l r XDsinm X

F 2 J 2l RiD − F 1 J

2l RoD

× J 2l RiDY l RoD J lr D − J l RoDY lr D

× sin m 1 − e−l,m t D. . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-10)

As is obvious from the solution presented for the truncatedwedge system, eventually the transient-pressure responses lead toa steady-state solution.

3D Flow in a Truncated Wedge System. Here, production isoccurring through a partially penetrating, line-sink well at a rate of q D1 from a truncated wedge system with dimensions RiD , RoD , o , and h D. The vertical well is located in the areal plane at (r XD , X ), and completed between h1 D and h2 D in the total pay thickness.The top and bottom boundaries are considered closed, while thedimensionless pressure at each of the other boundaries is main-tained at zero. This is a 3D problem in (r, , z). Thus, the followingouter boundary conditions arise:

p D = 0 at = 0, o , and at r D = RiD , RoD , and

p D

z D

= 0 at z D = 0, h D.

Therefore, one can obtain the following information from Table 1:Number of EVPs: 3 (r, , and z directions);

Qt D =2 h Dr D − r XD − X H z D − h2 D − H z D − h1 D

h2 D − h1 Dr D.

r-direction information (from Tables 2 and 4):Type of outer boundary condition: I101 with f r 1 = 0 at L1 =

RiD , and f r 2 = 0 at L2 = RoD ;

r l = Y l RoD J l r D − J l Rod Y l r D;

Norm: nl =2

2

J 2l RiD − J

2l RoD

l2 J

2l RiD;


J l RiDY l RoD − J l RoDY l RiD = 0. . . . . . . . . . . (B-11)

-direction information (from Tables 2 and 5):Type of boundary conditions: I1O1 with f 10 at L10,and f 20 at L2 o;Eigenfunction: m sin m ;Norm: nm o /2;Equation for eigenvalues m:

sin m o = 0, which leads to m = m o. . . . . . . . . . . . . (B-12)

The positive values for m have to be known from Eq. B-12before Eq. B-11 can be solved for l , the eigenvalues in the r direction, because this operation involves a Bessel function of thefirst kind of order m.

z-direction information (from Tables 2 and 5):Type of boundary conditions: I2O2 with f z10 at L10,and f z20 at L2h D (0 is also an eigenvalue);

For zero eigenvalue:Eigenfunction: zn01;Norm: n zn0h D.

For nonzero eigenvalues:Eigenfunction: zncos n z D;Norm: n znh D /2;

Equation for eigenvalues n:

sin n h D = 0, which leads to n = n h D. . . . . . . . . . . . . (B-13)

Therefore, one obtains the following expressions for the entiresystem:

l,m,0 r l m zn0; l,m,0nrl nm n zn0; l,m,0 2l ;

l,m,n r l m zn; l,m,nnrl nm n zn; l,m,n l2 + n

2.


wl,m,02 h D[Y (l RoD) J (lr XD)− J (l RoD)Y (lr XD)]sin m X

for the zero eigenvalue, and




wl,m,n =2 h DY l RoD J l r XD − J l RoDY l r XD

h2 D − h1 Dn

× sin m X [sinnh2D − sinnh1D] for the nonzero eigen-values in the z direction.

Therefore, Eq. 2 becomes

p Dr D , , z D ,t D =

2 3

ol=1

m=1

J

2l RiDY l RoD J l r XD − J l RoDY lr XDsin m X

J 2l RiD − J

2l RoD

× Y l RoD J l r D − J l RoDY l r D sin m 1 − e−

l

2t D

+

4 3

h2 D − h1 D ot =1

m=1

n=1

Y l RoD J l r XD − J l RoDY l r XD

nl,m,n J 2l RiD − J

2l RoD

× l2

J 2l RiDsin m X sinnh2 D − sinnh1 D

× Y l RoD J l r D − J l RoDY l r D sin m cosn z D

× 1 − e−l,m,n t D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-14)

The first term on the right side of Eq. B-14 is caused by the zeroeigenvalue in the z direction. As h1 D→0 and h2 D→h D , the welltends to be fully penetrating. For this case, the second term in Eq.B-14 vanishes, and the solution degenerates to that for a 2D situ-

ation. It is obvious from the previous solution for the truncatedwedge system that eventually the transient-pressure responses leadto a steady-state solution. The solution in Eq. B-14 degenerates tothe solution for the wedge system in B-3 when RiD→0. Thus, asolution for a truncated wedge system is more general than that fora wedge system under identical conditions.

A Wedge System Subject to Partial Communication at anOuter Boundary. A wedge system with dimensions RoD , o , andh D is considered to be communicating partially at the outer bound-ary in the r direction because of the presence of a thin (boundary)skin. The resistance to communication of fluid at this boundary isspecified by a skin factor s. The other two boundaries in the direction are closed to any fluid communication. Production isoccurring through a fully penetrating, line-sink well at a rate of

q D

1. The well is located at (r XD , X ). This is a 2D problem in (r, ). Thus, the following outer boundary conditions arise:

p D

r D+

p D

s = 0 at r D = RoD , and

p D

= 0 at = 0, o.

Therefore, one can obtain the following information from Table 1:Number of EVPs: 2 (r and directions);

Qt D =2 r D − r XD − X

r D.

r-direction information (from Tables 2 and 3):Type of outer boundary condition: O3 with 21, 2 1/ s, f r 20 at L2 RoD (no inner boundary exists);Eigenfunction: r l J (lr D);

Norm: nl = 0.5 RoD2 s2 + l2 RoD2 − m2

l2

J 2l RoD;


l J l RoD + J l RoD s = 0. . . . . . . . . . . . . . . . . . . . . . . (B-15)

-direction information (from Tables 2 and 5):Type of boundary conditions: I2O2 with f 10 at L10,and f 20 at L2 o (0 is also an eigenvalue);

For zero eigenvalue:Eigenfunction: m01;Norm: nm o;

For nonzero eigenvalues:Eigenfunction: m cos m ;Norm: nm o /2;Equation for eigenvalues m:

sin m o = 0, which leads to m = m o. . . . . . . . . . . . . . (B-16)


l,0 rl m0; l,0nrl nm0; l,m0 l

2; l,m rl m; l,mnrl nm; l,ml

2.


wl,02 J (lr XD) for the zero eigenvalue, andwl,m2 J (lr XD)sin m X for the nonzero eigenvalues in

the direction.

Therefore, Eq. 2 becomes

p Dr D , ,t D =

4

ol=1

J 0l r XD J 0l r D

J 02l RoD RoD

2 s

2 + l2 RoD

2 1 − e

−l2

t D

+8

ol=1

m=1

J 0l r XD J 0l r Dcos m X cos m

J 2l RoD RoD

2 s2 + l

2 RoD2 − m

2 1 − e

−t 2

t D.

. . . . . . . . . . . . . . . . . . . . . . . (B-17)

As s→, the solution in Eq. B-17 tends to the solution for aclosed wedge system. However, this solution in its present formleads to a steady-state solution.

Appendix C—Computation of Eigenfunctions,Norms, and Eigenvalues

When computing the dimensionless potential responses at a givendimensionless time and location, a set of eigenvalues needs to becalculated beforehand. The equations for computing eigenvaluesare presented in Tables 3 through 5. In most of the cases, compu-tation of positive roots as eigenvalues in the r direction involvesthe solution of transcendental equations using a suitable numericaltechnique. Newton’s method for solving nonlinear equations hasbeen found to be effective for this purpose. The r -direction equa-tions are nonlinear because they involve Bessel functions. Thecharacteristics of these functions are discussed in the literature.The computation of eigenvalues for the eigenvalue problems in the

and z directions, with sets of boundary conditions I2O2, I2O1,I1O2, and I1O1, is trivial. But the rest of the listed problems mayinvolve the solution of transcendental equations using a suitablenumerical technique.

Once an eigenvalue is known, the corresponding values of thenorm and the eigenfunction can be computed readily using theappropriate formulas listed in Tables 3 through 5. Table 6 showsthe combination of eigenvalues and eigenfunctions for multidimen-sional problems. Typically, 30 to 60 sets of eigenvalues have beenfound to be adequate to obtain accuracy up to 10-12 in computing thedimensionless potential (or pressure) and/or its time derivative.

N.M. Anisur Rahman is an R&D Engineer at Fekete AssociatesInc., Calgary. e-mail: [email protected]. Previously, heworked at the Bangladesh U. of Engineering and Technology

and the U. of Alberta. His research interests include reservoircharacterization and general reservoir engineering. Rahmanholds BS and MS degrees in mechanical engineering from theBangladesh U. of Engineering and Technology, and a PhD de-gree in petroleum engineering from the U. of Alberta. RamonG. Bentsen is a Professor Emeritus at the School of Mining andPetroleum Engineering, U. of Alberta, Edmonton, Canada. e-mail: [email protected]. Bentsen has worked forSchlumberger Surenco in Venezuela and Colombia, and withMcCullough Tool Company and Imperial Oil Ltd. in Alberta.Bentsen holds a BS degree in petroleum engineering from theU. of Oklahoma, and MS and PhD degrees in petroleum andnatural gas engineering, respectively, from PennsylvaniaState U.


New Analytical Solutions for Predicting Pressure Distribution and Transient

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