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x ct Solutions for Infinite-

Conductivity Wells

Paul Papatzacos

SPE, Rogaland Research Inst.

Summary. In pressure transient testing, the infinite-conductivity condition translates mathematically into a

uniform-pressure (or uniform-potential) condition at the well. This means the flux at different points of the well

should be determined in such a way that potential remains uniform at the well. The integral equation for

accomplishing this

is

solved analytically to yield the Laplace-transformed potential. For fractured-well problems,

this leads to a relatively fast algorithm for drawing type curves directly on a computer screen. For limited-flow

entry problems, the analytical pressure expression can be used with the method of images to treat problems in

reservoirs of finite thickness and/or areal extent.

Introduction

One traditionally thinks

of

a well as having infinite con

ductivity in the direction parallel to its axis. This assump

tion is also included as a standardoption inmost numerical

reservoir simulators. In pressure transient testing, the con

cept is important for wells with limited flow entry and

for fractured wells.

The infInite-conductivity condition translates mathemat

ically into a uniform-pressure (or rather uniform-potential)

condition at the well. This means that the flux at differ

ent points of the well is unknown a priori and must be

determined in such a way that potential is uniform at the

well. Muskat 1 recognized that this amounts to solving an

integral equation. Muskat and later Gringarten and

Ramey

2

showed how to solve the integral equation for

mulation numerically by dividing the source into small

elements, each having uniform flux. To determine these

elementary fluxes, two conditions must be applied: they

must sum to the required total rate and they must produce

a uniform potential at the well. This method was applied

by Gringarten and Ramey 3 to the limited-flow-entry

problem and by Gringarten et at. 4 to the fractured-well

problem.

In the present paper, this integral formulation is solved

analytically

to

yield the Laplace-transformed potential.

For

the fractured-well problem, the purpose of this cal

culation is to obtain an expression for the pressure that

leads to a relatively fast algorithm for drawing type

curves, both for single-well testing and for interference

testing, directly on a screen. It is usually easier to obtain

numerical results from analytical expressions than from

their nonanalytical analogs.

For the limited-flow-entry problem, on the other hand,

the purpose

of

the calculation is different. Because the

reservoir is considered to be infinite, both laterally and

in thickness,

type

curveswould not have a direct reservoir

engineering application. The interest in an analytical ex

pression in this case lies in the possibilities it offers in

conjunction with the method of images to treat problems

in reservoirs

of

fmite thickness and/or finite areal extent.

Copyright 1987 Society of Petroleum Engineers

SPE

Reservoir

E m r i n f p r i n ~

Mil

One such problem

of

interest is the calculation

of

limited

flow-entry pseudoskin

5

; another is water coning.

6

Some

details about the treatments of these problems will be

given.

Limited Flow Entry Problem

Dimensionless Variables

and

Spheroidal Coordinates.

.The situation considered is that of a line-sourcewell with

limited flow entry in a reservoir that is infinite in all three

dimensions. As shown in Fig. 1, the axes are chosen so

that the perforated wellbore covers Interval -a , +a

along the

z

axis. Note that, because of symmetry across

the xy plane, one is also solving the problem of a semi

infinite reservoir with a horizontal no-flow upper bound

ary, anda line-source well perforated from the top down

ward to the depth

of

a

kH and k y are the permeabilities in the horizontal and

vertical directions, respectively. Dimensionless coor

dinates are defined as

XD

= ky/k

H

Ih x/a ,

1a

YD = ky/k

H

1h Y/a ,

lb

and

ZD=z/a lc

and dimensionless time as

2.367 X

1 ky t

(2)

.pl c

t

a

2

The pressure potential,

.p=p P

o

zl144.0,

(3)

can easily be shown to obey the same diffusion equation

as the pressure, p. An expression for.p can

be

written

in terms of the appropriate Green function and of an

unknown source function. 3 The latter must

be

determined

217

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x

-a

y

a

z

Fig. 1-Limlted-flow-ent ry problem.

The

well Is perforated

along Interval

a, +a

on the z axis thick line .

Fig. 2-Limlted-flow-entry problem. Sections

a plane

containing

ZD

with

~ . c o n s t a n t

and a-constant surfaces.

The resulting ellipses and hyperbolas are labeled by the

corresponding and

a

values

a

Is In degrees .

in such a way that

cI>

does not vary with

z

at the wellbore.

At this point, it

is.

important to choose a system

of

coor

dinates that allows an easy representation

of

the line

source well: prolate spheroidal coordinates,

4

~ , a . , { j , re

lated to the usual Cartesian coordinates by the following

equations.

x

D =sinh sin

a.

cos

{j, :

(4a)

YD = sinh sin

a.

sin

{j,

(4b)

and

ZD

= cosh cos a. (4c)

We can now write an expression for the dimensionless

potential,

D, in terms of the unknown dimensionless

flux along the well,

qwD

(see Appendix A for a deriva

tion and for the definition

of q wD :

r r ll

D =

J

dtv

J

da. sin a.

qwD a ,tD)

o 0

e _[ R

2

/4 I

D

-I h)]

x , (6)

4J;( tD

- tv )

where

R

2

= (sinh sin

a) 2

+(cosh cos

a cos a. )2.

(8)

e

- [R ; /4 t

D

t h ]

x

4J;( tD - t v ) ; (9)

Note that

R

2

= rJ + ZD -Z V)2

and that

D

depends on

a., and tD By setting ~ O in Eq. 6, we obtain

cID

at the wellbore, which we denote by

4>wD tD ,

thus

making explicit the fact that it is uniform along the well-

Le., independent

of

the coordinate

a.:

ak

H

- - - 4 i -4 (7)

4 2j tqt

4>D

and

The sections

of

some

c o n s t a n t - ~

and constant-a. surfaces

with an arbitrary rD,ZD plane are shown in Fig. 2. The

ellipsoids become thinner as becomes smaller. The el

lipsoid for = 0 has degenerated into Segment(

- 1

,

+

1)

on the

ZD

axis. In other words, the producing interval

of

the well, given by XD =0

YD

=0 - 1 ~

~

+1 in

Cartesian coordinates, is simply given by =0 in sphe

roidal coordinates. An arbitrary point at the wellbore has

a coordinate equal to zero and an

a.

coordinate in Inter-

val

(0, 11 ).

_

2

2 ) 1 h _ n h t. . (C)

r D - XD

+

YD - SI . c; SID

a. . .

. . . . . . . . . . . . . :J

These new coordinates are restricted to .the intervals

O ~ ~ ~ o O

0 ~ a . ~ 1 I , and 0 ~ { j ~ 2 1 1 .

The surfaces with constant are rotational ellipsoids,

while the surfaces with

a.

constant are rotational hyper

boloids (oftwo sheets). The axis

of

rotation for both these

families is the ZD axis. The surfaces with {j constant are

planes containing the

ZD

axis. Because

of

the rotational

symmetry, it is useful to introduce

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In Eq. 9,

R

wis the expression obtained from Eq. 8 by

setting

=0.

One must be careful, however, because

w is known to be infinite f?r a

l i n e - s o ~ r c e

well. To

avoid infinities in the calculatIons, we wnte

2 . , 2

R = sinh sin a + cosh cos a - c o s a

10

and let go to zero. This limiting process will be

i ~ d ~

cated at the proper place later. In other

words lt

IS

assumed that the well is a thin ellipsoid that is forced to

degenerate to a line. The question arises w ~ e t h e ~ it ~ o u l d

be more realistic to model the well as a thm elbpsOld by

choosing a ,small but nonzero v ~ u e . for w thus a ~ o ~ d

ing the difficulties of taking the bmlt ThIS IS m

deed possible, but it should be kept

mmd that the

solution given here is then only a p p r o X 1 m ~ t e . because an

ellipsoidal well with w

0 has a nonvamshmg surface,

while the Green function used in Eq. 9 is correct for a

reservoir with no boundaries at finite distances. ,

We show in Appendix A that the flux,

q wD,

must

satisfy

d a

sin qwD a,tD) =

1 11

o

This is the analytical expression

of

the first condition

imposed by Muskat

and by Gringarten and R

amey2

on

the elementary fluxes-i.e.

that they add to the total rate.

Th e

second condition on qwD is expressed in Eq. 9: the

flux rate must be such as to produce a uniform potential

at the well.

Statement of the Problem.

is necessary to solve Eqs.

9 and 11 for the two unknown functions

w

and q

w

.

The time integration in Eq. 9 is a convolution, an

obvious first step is to use the Laplace

t r a n s f o r m a ~ l O n

to

eliminate one integration. Here

[j t ]

is the Laplace

transform ofj t and S is the Laplace parameter. We first

introduce the following notation:

c C [ ~ D ~ , a , t D ] = 1 P ~ , a , s ,

12

c C [ ~ w D t

D)] =1Pw S), 13

and

[qwD a,tD)]

=

U a,s).

. 14

One finds in Ref. 7 that

exp[-

R

2

/ 4tD)]

= exp -

R.[;

4 ;;t8

R

15

so by taking the Laplace transform

of

Eqs. 6, 9, and 11,

one obtains

e -R

1 P ~ , a , s = lh1d a

sin a U a ,s) ,

. .

16

o

R

SPE Reservoir Engineering,

May

1987

e-

Rw

1Pw S) =

lh

dd

sin

a

U a ,s) ,

17

o R

w

and

da

sin a U a,s) =

lis,

18

o

where

R

is given by Eq. 8 and

R

w

by Eq. 10. Now the

problem is to solve Eqs. 17

and

18

for

wand

When

the resultant expression for

U

is used in Eq.

16

and the

integration performed, one obtains an

e x p r e s s ~ o n

for the

Laplace transform

of

the dimensionless potential.

Solution. Step

1

An analytical solution

of

the prob

lem is made possible by the existence of an eigenfunction

expansion

of

exp[- R:fi ]/R. This expansion is given in

Refs. 8 and 9 for exp iTR /R and is written below for

T=i.[; and for R given by Eq.

8:

R

00

1

J

= 2 [ ;

--Son(i \lS, cos a )

R

n=O NOn

XSon i.[;, cos a R b ~ i . [ ; , I R b ~ i.[;, cosh

19

In this expression,

Smn

is the angular spheroidal wave

function while

R

and

R

are the radial spheroidal

wave functions

of

the first and third kind, respectively

see also Ref. 7 . In Eq. 19, only the

f u ~ c t i o n s

with

f i ~ s t

index equal to zero are involved. N

mn

IS the normalIZ

ing constant; i .e. , for

m=O,

da sin a SOk k.[;, cos a)Soj i.[;, cos a)=okjN

Ob

o

20

where 0

kj

is the Kronecker delta.

The following expression for SOn is needed:

00

SOn

= d ~ ~ + r i . [ ;

)P2k+r COS

a) ,

21

k=O

where P is a Legendre polynomial, and where

r=O

1 when

m

n

is even odd . The diJn i:..{;) are expansion

coefficients with known rules

of

calculation.

9

Step 2. The function U Laplace transformofthe flux,

see Eq. 14 depends only on the angular coordinate

ex

and

can be expanded in a series

of

eigenfunctions,

SOn

which

form a complete set on Interval 0,1r . There is a sim

plification because

of

the symmetry

of

the flux across the

x D,YD plane; this implies that U is

s y m m e t r i c a r o ~ n d

a=1r/2-Le.

that it is a function

of

cos

2

a. Accordmg

to Eq.

21

and the properties

of

the Legendre polynomi

als, one may then write

00

, U a,s)= 3kS02k i.[;, cos

a),

22

k=O

2 9

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XD

= cosh

p

cos

v

28a)

Infinite onductivity Vertical-

r cture

Problem

This problem s solution is exactly analogous to the

one

used previously.

The reservoir is supposed to be isotropic, with constant

p e r m e b i l i ~ k.

The fracture is ,most easily represented

in ellipticS, 1 coordinates p,v), which are related to the

Cartesian coordinates

xD YD

by

2.637

x

10

4 kt

- - - ~ - ~ - C - t X - f 2 - - - :

27c)

and

Dimensionless Variables

and

Elliptical Coordinates.

The situation is that

of

a vertical fracture. Flow is two

dimensional. The axes are shown in Fig. 3 and the frac

ture extends from f to xf along the x axis. We as

sume .an isotropic reservoir and introduce

XD =x/xf, 27a)

YD =y/xf

;

27b)

sure in finite reservoirs in situations where it is known

that such a steady-state solution exists. One case

of

in

terest consists

of

an areally square reservoir

of

finite thick

ness with a limited-flow-entry well in the middle

of

the

square and a constant-pressure condition at the lateral

boundaries. This situation arises in the calculation

of

the

critical rate to water coiting first investigated by Muskat. 1

The advantage

of

Eq.

26 is that the infinite-conductivity

condition is satisfied at the outset.

An

investigationof crit

ical rates along these lines has been performed by

Hoyland.

6

It

is

more

surprisingthatEq

26 more exactly , an

asymptotic expansion obtained from Eq. 25 for large t

D

can be used to obtain an expression for the pseudoskin

for a limited-flow-entry well in an areally infinite reser

voir,with finite thickness. The applicability

of

Eq. 26 is

not evident because dimensionless pressure is proportional

to In

tD

for large

tD

in this case, whereas Eq. 26 is a

steady-state expression. The method of images leads to

the calculation

of

an infinite sum because infinitely many

images are necessary to satisfy the no-flow boundary con

ditions at the upper and lower boundaries, and the infinite

sum turns out to be divergent. One must then identify this

mathematical divergence with the physical divergence

of

pD, which is caused by In

t D

in the limit

t D -+

The

geometrical skin is obtained as the finite

e x ~ r e s s i o n

that

remains when the divergence is subtracted. Ref. 5 dis

cusses the applicability

of

the method of images to the

problem considered. Re[ 5 shows that the method de

stroys the condition of strict uniformity of potential at the

well, but that this condition applies to a good approxima

tion for the majority

of

cases

of

interest.

2d8

iJ ;

25)

1

D =lf

In

t .

26)

e

-1

Thus t/;w is determined, and so are the k by Eq. 23),

which means that

U

is determined by Eq. 22. One finally

obtains

t

see Appendix

B :

1

= J

t/;w

k=O

R

k

t/;w s

-

J; -

( ~ , a , s ) = w ( s )

k=O

Le., an expansion in terms

of

angular functions with an

even second index. The function

U

will be determined

as soon as the coefficients

{

k are known.

Step 3. I t is now possible to solve Eqs.

17

and

18

to

find

t/;w

and

U.

Substitution

of

Eqs. 19 and 22 into Eq.

17

gives the

following relation between the coefficients

{ k

and

t

w

see Appendix

B :

Eqs. 24 and 25 constitute the solution

tothe

limited-flow

entry problem. The solution is approximate

if

w

is

given

a nonzero value, as noted above. For a line-source well,

however, the solution is exact but must be obtained by

letting w-+ 0, taking care of the fact that R i,J;, cosh

w diverges when w-+0. 9 In practice, this can be done

by inverting the Laplace transform Eq. 25) for a very

small value

of wand

using an expansion for the func

tion

R ~ ~ 1 (i...{;,

cosh

w

valid for small w.

9

This has

not been attempted in the present paper for the reasons

given in the Introduction. An analytical investigation

of

Eqs.

24

a nd 25 for small w and small

s

however, has

been carried out in Ref. 5. One product

of this investiga

tion is the long-time approximation, obtained by invert

ing term by term the small-s expansion

of

Eq. 25 10 and

dropping terms that vanish when

tD

-+

One finds that

N 0 2 k R b ~ k i . j ; ,

I R ~ i k i J ; , cosh w 23)

Substitution

of


18

gives see Ap

pendix

B

and

Applications.

Eq. 26 can now be used in conjunction

with the method

of

images to get the steady-state pres-

220

Y

D

=

sinh

p

sin

v. .

28b)


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y

- - - - - : - - - - - - - - - - - - - - - - ~

X

Fig.

3 Vertical

fracture problem. Fracture Isalong Interval

-x

+x,)

on x

, axis

thick line .

The curves with p constant areellipses, while the curves

with

v

constant are hyperbolas (see Fig. 4). The p= O el

lipse has degenerated into Segment 1 +1 along the

xD

axis and thus represents the fracture. Coordinates

p

and v vary in the intervals 0 s p s 00 and 0 S vS

2 r

The dimensionless pressure, P

D ,

can now be written

in terms of the unknown dimensionless flux at the frac

ture,

qjD

(see Appendix A):

r

D

r

21r

P D = J dt b J

dv sin

v qjD v ,tD)

o

0

e [D

/4 t

tb ]

X , (29) .

tD tb

where

kh

P D

=

4 2 ~ q

P i p (30)

and

(31)

In Eq. 30, h is the reservoir thickness. PjD t

is the

uniform dimensionless pressure at the fracture, then by

setting p= O in Eq. 29, one obtains

r

t

D r

2

PjD

=

J

dt b

J dv sin v qjD v ,tD)

o 0

e

-[ cos p-cos pl 2 /4 t D tb ]

X

(32)

tD tb


Fig. 4 Vertlcal fracture problem. Elliptical coordinates.

With p constant, lines are ellipses labeled by the values

of

p

and

with

constant, lines are branches

of

hyperbolas

labeled by the values

of

v In degrees .

Note that there is no difficulty in setting p =0 because

pressure is known to be finite at a fracture of zero thick

ness, so there is no limiting procedure involved here, as

in the case

of

the line-source well. Eq. 32 is the first

of

two equations that

qjD

must satisfy; the second is (see

Appendix A)

r

2

J

dv

sin

v qjD V ,tD)=1.

(33)

o

Statement of

the

Problem; One must solve Eqs. 32 and

33 for

qjD

and

PjD

Eq. 29 will then give the dimension

less pressure. Introducing the notation

[p D( P,v,tD)]

=1/t p,v,s), (34)

[PjD tD)] =

1/tl

s

)

, (35)

and

[qjD V,tD)]

=

V v,s), (36)

one finds that the Laplace transforms

of

Eqs. 29, 32, and

33 are

r

2

t= J dv sin v

V v

,s)K

o

.[;V (37)

o

r

2

1/tf=

J dv sin

v

V v ,s)Ko .[;lcos v-cos v I),

o

(38)

and

rll dv sin

v

V v ,s)=1/s, (39)

o

221

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whereK0 is the modified Bessel function of order

O

7

The problem reduces now to solving Eqs. 38 and 39 for

and

V.

will then be given by Eq. 37.

Solution. As in the case of the limited-flow-entry well,

an analytical solution is made possible 2Y the existence

of an eigenfunction expansion of K0

s

D . Such an

expansion is given in Ref. 8 for the Hankel function

H0 1) . Using the relation

7

one obtains, when D is given by Eq. 31,

where cem is the angular or periodic Mathieu function

and Ie

m

and Ke

m

are radial Mathieu functions.

7

There

is a normalization formula for the ce

m

functions analo

gous to Eq. 20.

7

There is also an equation like Eq.

21

for the cem; it

is written below to establish a notation that will be used

later:

00

ce2m(V,q)= q cos 2nv (41)

n=O

The Laplace transform

of

the dimensionless flux at the

fracture,

V

can beexpanded in a series

of

angular eigen

functions cem An expansion that accounts for its J sym

metry properties

V

is even about v=0,7r/2,7r,37r/2) is

V(v,s)=

Akce2k v , - ~ ; (42)

m=O 4

Le., an expansion in terms of

ce

functions, with an even

index.

It is now possible to solve Eqs.

38

and 39. The calcu

lations are exactly analogous to the ones performed in the

limited-flow-entry case, so only the final results are

reported.

The

Ak S

of

Eq. 42 are given by

2k

1/Ils) Ao -s/4

A k (43)

, 1r

Ie2k (0,s/4)Ke2k(0,s/4)

where A has been defined in Eq. 41 and

1 00 2 [ A ~ m

-s/4)]

2

s

L; ...... (44)

I/Ils) ,

m=O

Ie2m(0,s/4)Ke2m(0,s/4)

222

Finally, the Laplace transform of th e dimensionless

pressure is

1 / I p , v ~ s = 1 / I t < s

2 A ~ m ~

m=O

,4

Eq. 44 was presented by Kucuk and Brigham

11

with

another notation for the Mathieu functions.

Applications. The large

tD

approximation implied by

Eq. 45, obtained by inverting its small s expansion term

by term and dropping terms that vanish when

t

D -+ CXJ is

PD = In

e -

2p t

D

+2

In

2- f

(46)

where I is Euler s constant (0.5772), so that2ln 2- 1/2=

1.09769. Thus for p=O-Le., at the fracture : Kucuk and

Brigham s 11 result is recovered.

By assuming that the wellbore is situated at

x

D=Y

D

=

and by denoting the well radius as r w, one can extract

from Eq. 46 the following expression for the skin factor:

2r

w

s ln

p

(47)

XI

where p can be obtained from Eqs. 28:

[

1

r ~ [ l

+rh)2

-4rh

cos

2

8]

]lh

p=arc cosh

2

: (48)

The notation and definition of the inverse hyperbolic func

tion is that

of

Ref. 7, and rD and 8 are the usual polar

coordinates.

XD

=rD

cos 8 (49a)

and

YD=rD

sin 8 (49b)

Eq. 46 shows that

tD/exp 2p ,is

the proper combina

tion against which to plot

PD if

one wants curves with

different

p

values to collapse as much as possible into a

unique curve. For larger

rDi.

Eq. 48 shows that exp(2p)

is approximately equal to 4rD Eqs. 44 and 45 have been

inverted with the Stehfest algorithm.

12

Fig. 5 is a plot

ofpD vs.

rD

at a fixed angle 8 and a fixed

time.tD,

show

ing that the condition of uniform pressure at the fracture

(Le., for

8=0)

is satisfied. Table 1 gives PD vs. tD at

the fracture

(P=0 )

for different value

of

the Stehfest in

teger N. Note that

N=6

is acceptable for most practical

purposes. There is some discrepancy between the values


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Fig. 5 Vertlcal fracture problem; Po VB. the polar coor

dinate, r

for three values

of

the polar angle, 8. Note uni

form pressure at the fracture 8=0).

omenclature

a

=

half-length of interval open to flow,

ft

[m]

A ;: =

constant in Eq. 41

ce

m

=

periodic Mathieu function

7

c

t

=

total reservoir compressibility, psi

[kPa ]

d r

) =

expansion coefficient in Eq.

21

~ e x p - 2 p

D =

constant defined in Eq. 31,

dimensionless

h

=

reservoir thickness, ft [m]

= Hankel function

7

~

Iem =

Mathieu function

7

k =

permeability, md

kH:k v =

horizontal and vertical permeabilities,

md

Ke

m

=

Mathieu function

7

K

0 =

modified Bessel function

7

f =

Laplace transform

of

function f

Fig. 6 Vertlcal fracture problem;

Po VB. t

o

/exp 2p). The

elliptic angular coordinate v is

45

TABLE p0

VS

to

AT THE FRACTURE p =0 FOR

DIFFERENT VALUES OF THE STEHFEST PARAMETER, N

Pressure

Time

N=6 N=8 N=

Asymptotic

*

.01 0.1698

0.1697

0.1699

-

0.02 0.2359

0.2358 0.2357

-

0.04

0.3256 0.3253 0.3253

-

0.06 0.3914 0.3911

0 39fo

-0.08

0.4450

0.4446 0.4446

-

0.10 0.4909 0.4904

0.4904

-

0.20 0.6592

0.6585

0.6589

-

0.40

0.8693 0.8684

0.8684

-

0.60

1.0115 1.0105

1.0105

0.8423

0.80

1.1204 1.1194

1.1194

0.9861

1.00 1.2091

1.2081 1.2082 1.0977

2.00

1.5045

1.5036 1.5037 1.4443

4.00

1.8222

1.8215 1.8216

1.7908

6.00

2.0148

2.0142 2.0143

1.9936

8.00

2.1536 2.1529 2.1530

2.1374

10.00

2.2620 2.2614

2.2615

2.2490

20.00

2.6023 2.6017

2.6018 2.5956

40.00

2.9458 2.9452 2.9452 2.9421

60.00

3.1475 3.1469 3.1469 3.1449

80.00

3.2908 3.2902

3.2903

3.2887

100.00 3.4021 3.4014 3.4015 3.4003

200.00

3.7480

3.7474 3.7475 3.7468

400.00

4.0943

4.0936 4.0937 4.0934

600.00

4.2969

4.2963

4.2963

4.2962

800.00 4.4407

4.4401

4.4401

4.4400

1,000.00

4.5522 4.5516

4.5517 4.5516

2,000.00 4.8987 4.8981

4.8982 4.8981

4,000.00

5.2453

5.2447

5.2447

5.2447

6,000.00

5.4480 5.4474

5.4475

5.4474

8,000.00

5.5918 5.5912

5.5913 5.5913

Obtained from

Eq. 48.

3

e =

0

e

=

71 4

e =

71 2

2

=

1 0

1

5

1

onclusions

New theoretical results concerning infinite-conductivity

wells have been presented in the form

of

Laplace trans

forms. Some implications

of

the result for the limited

flow-entry problemhave been discussed in general terms.

Detailed calculations are presented elsewhere.

5

,6 Results

of

the inversion

of

the Laplace transform have been

presented for the vertical fracture problem. There are

some discrepancies with earlier results,4 amounting at

worst to 4 . These discrepancies are difficult to account

for because they occur between calculations that are very

different from each other, but are mild and will not intro

duce any detectable uncertainty in a type-curve analysis.

The inversion

of

the Laplace transform

of

Eq. 45 is fast

enough to allow the plotting

of

type curves directly on

a screen by an interactive computer-graphics program

without long waiting periods. Eq. 46 is new and agrees

with that obtained by Kucuk and Brigham when p

= O

1.209

shown in Table 1 and those

of

Gringarten

et ai

4

on

the

order of 4 for small times and decreasing to about 0.1

toward the bottom

of

the table. Fig. 6 shows

P

D

vs.

tD/exp 2p)

for a range

of p

values, the value

of p

being

the same. The curves are drawn by linear interpolation

between the calculated points, which are equidistant and

have a density

of

6/log cycle. Depending on the work

load, the calculation of the set

of

points needed to draw

one curve takes from 10 to

20

seconds a high-performance

32-bit minimachine.

Eq. 46 is represented by the dashed line in Fig. 6. Ta

ble 1 and.Fig. 6 show that the difference between the log

approximation Eq. 46 and the exact value is less than

about 2 when

tD >4

SPE Reservoir Engineering,

May

1987

223

8/10/2019 22. SPE-13846-PA

8/10

N = positive integer

Non ; normalIzing constant in Eq. 20

p = reservoir pressure, psi [kPa]

pD = dimensionless pressure (see Eq. 30)

PfD = PD at the fracture

Pi

=

initial pressure

Pm = Legendre polynomial

7

(see Eq. 22)

q

=

real number (see Eq. 41)

qfD

=

dimensionless flux at the fracture (see

Eqs. 29 and A-12 through A-14)

qt =

,total well volumetric flow rate, RB/D

[res m

3

/d]

q wD

= dimensionless flux at the. well (Eqs. 6

and A-5 through A-7) ,

q x,y,t = volume of oil withdrawn at point x,y

and at time

t

per unit reservoir

volume and unit time, RB/D-ft3

[hours 1 ]

q x,y,z,t

=

volume of oil withdrawn at point x,y,z

and at time t per unit reservoir

volume and unit time, RB/D-ft3

[hours 1]

r

D = d i m e ~ s i o n l e s s radial coordinate

(see Eq. 5)

r

w =

well radius, ft [m]

R = constant equal to

rB

+ ZD Zb 2 (see

Eq. 8), dimensionless

R

1

= radial spheroidal function of the first

mn

kind

7

,9

R = radial spheroidal function of the third

kind

7

,9

R

w

= R with ~ ~ (see Eq. 10),

dimensionless

s

=

Laplace parameter, dimensionless

Smn

= angular spheroidal function7,9

t =

time, hours

tD = dimensionless time

U = Laplace transform of

qw D

V = Laplace transform of

qjD

x,y,z

=

Cartesian coordinates, ft [m]

XD,YD,ZD

=

dimensionless Cartesian coordinates

xi =

fracture half-length in the x direction,

ft [m]

Z = complex variabl,e

a 3 = angular spheroidal coordinates

(see Eqs. 4)

3 k

=

coefficient in Eq. 22

Y = Euler s constant (0.5772)

oij = Kronecker delta, 1

if

i=j 0 otherwise

8

=

angular polar coordinate (see ~ q s 49)

Ak = constant defined in Eq. 43

p

=

oil viscosity, cp [Pa

s]

v

=

elliptic angular coordinate (see Eqs. 28)

= radial spheroidal coordinate (see, Eqs. 4

=

at th e well

p = elliptic radial coordinate (see Eqs. 28),

dimensionless

Po = oil mass per unit volume, lbm/ft

3

[kg/m

3

] .

224

J

= i.J;

(see Appendix B) .

7 =

real number

= porosity, fraction

4> = potential (see Eq. 3), psi [kPa]

4>

D

dimensionless potential (see Eq.

7)

eJ>i =

initial value

of 4>

psi [kPa]

cI

wD = 4> D at the well (see Eq. 9)

;

=

Laplace transform

of

4>

D

in low-flow

entry problems and

of

PD in infinite

conductivity vertical fracture

problems

;

f =

; at the fracture

;w

=

; at = w Laplace transform

of

4> wD

Subscripts

j k m n = integers

Superscripts

, =

integration variable

cknowledgments

I

gratefully acknowledge the financial support

of

Norsk

Hydro and express my thanks to Leif Larsen for much

useful advice.

References

1 Muskat,M.: Physical Principles

of

Oil Production, Inti. Human

Resources Development Corp., Boston (1981) 209.

2. Gringarten, A.C. and Ramey, H.J. Jr. : The Use of Source and

Green s Functions in Solving Unsteady-Flow Problems in

Reservoirs, SPEJ (Oct. 1973) 285-96; Trans. AIME, 255.

3. Gringarten, A.C. and Ramey, H.J. Jr.: An Approximate Infinite

Conductivity Solution for a Partially PenetratingLine-SourceWell,

SPEJ (April 1975) 140-48; Trans. AIME, 259.

4. Gringarten, A.C., Ramey, H.J. Jr., and Raghavan, R.: Unsteady

State PressureDistributions Createdby aWellWith a Single Infinite

Conductivity Vertical Fracture, SPE/ (Aug. 1974)

347 60;

Trans

AIME,257.

S Papatzacos, P.: Approximate Partial-Penetration Pseudoskin for

Infinite-Conductivity

Wells,

SPERE (May 1987) 227-34.

6. Hoyland, L.A.: Critical Rate for Water Coning in Isotropic and

Anisotropic Formations,

Cando

Tekn. thesis, Rogaland Regional

C., Stavanger, Norway (1984).

7. Abramowitz , M. and Stegun, I .A.: Handbook of Mathematical

Functions

Dover Publishing Inc., New York City (1972).

8. Morse, P.M. and Feshbach, H.: Methods of

Theoretical Physics

McGraw-Hill Book Co. Inc., New York City (1953).

9. Flammer, C.:

Spheroidal

Wave

Functions

Stanford U. Press,

Stanford, CA (1957).

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

Oxford Book Co., New York City (1959).

11. Brigham, W.E. and Kucuk, F .: T r a n s i ~ n t Flow in Elliptical

Systems,

SPE/

(Dec.1979) 401-10;

Trans.

AIME, 267.

12. Stehfest, H.: Numerical Inversion of the Laplace Transforms,

Cori1nutn ications

of

the ACM

(Jan. 1970) 13,

No.1,

Algorithm368.

13. Williams, W.E.: Partial Differential Equations Clarendon Press,

Oxford (1980).

ppendix FlowEquat ons

Flow Equation and Flux for the Limited-Flow-Entry

Problem. Equations in this Appendix are written in con

sistent units, and conversion constants are indicated when

necessary. The diffusion equation with a source term is

kH il

2

+ il

2

} + 0

2

_

cPp c

>

k

v

ox

2

oy oz k

v

ot

P

=--q x,y,z, t ,

(A-I)

k

v


8/10/2019 22. SPE-13846-PA

9/10

where q(x,y,z,t) is the volumetric rate of oil withdrawn

at Point (x,y,z,t) per uni t r ese rv oi r volume. I n ot her

w o r ~ s ,

lq(x,y,z,t)dxdydz=

-q ,

A-2)

where

q t

is the total volumetric flow rate of the well. In

tegration in Eq. A-2 can be extendedto the whole of space

because

q

wil l be different from zero only at the well.

By introduction

of

dimensionless coordinates Eq. 1),

dimensionless time Eq. 2), and dimensionless potential

Eq. 7), Eqs.

A-I

a nd A -2 b ec om e

a

2

cpD a

2

cpD a

2

cpD _ acpD =27ra

kH

i

aXE

aYE aZE

atD k

v

t

A-3),

and

r

k

v

qt

j q ~ x D Y D , Z D , t D d x D d Y D d z D= - 3 A-4)

,

kHa

To find a general expression for the function q describ

ing a line-source well as shown in F ig.

1

it is convenient

to use the so-called Dirac delta function,

O x .13

Here,

beca use the well i s along t he z axis, on e may wr ite

k

v

qt

q(XD,YD,ZD,tD)=

-3 0(XD)O(YD)qwD(ZD,tD),

kHa

. . . . . . . . . . . . . . . . . . . . . . . . . A-5)

where, on the rjght side, the constants are included for

convenience, and where the unknown function, qwD is

the dimensionless flux along the well. When qt and

a

are

expressed in customary units, the factor 1.539 X 10

4

must

be included on the r ight side. T he units f or q are then

RB/D-ft

3

.

Function

q

wD i s such that

qwD(ZD,tD)=O,

IZDI >

1 A-6)

and

+1

J

qwD(ZV,tD)dzv=1.

A-7)

1

E q. A -7 r es ul ts f rom t he co mbi na ti on of Eqs. A-4 and

A-5. By using Eqs. A-3 and A-5, one obtains

a

2

D

a

2

CPD a

2

CPD OCPD

aXE aYE aZE

OtD

Eq. A 8 can

now be solved with the help of the known

Green function

3,8

to give the dimensionless potential D :

CPD(XD,yD,ZD,tD)= dtb J

1

dzbqwD(Zb,tb)

o -1

1 [ -XE -yE -(ZD -ZV)2]

x = exp

.

4J;(tD -tv) 4(tD tv

A-9)


To

recoverEq.

6, it suffices to introduce the prolate

spheroidal coordinates given by Eq.

4

and to use zb

=cos

a

E q. A -9 t he n gi ve s E qs .

6,

a nd E q. A-7 gives E q.

9.

Flow Equation. and Flux for the Vertical-Fracture

Problem. The equations analogous to Eqs. A-3 and A-4

a re e asi ly found t o be

and

The flux

q

at the fracture can be expressed in terms of

a delta function expressing

the

fact that the fracture is an

infinitely thin sheet and an unknown dimensionless flux

at the fracture, qfD:

When

qt

h, and

x

are expressed in field units, then

the factor 1.539

X

10

4

must be included on the right side.

T he uni ts f or q are then RB/D-ft

3

. Function qjD must

satisfy the following conditions:

qjD(XD,tD)=O when

IXDI >

1 .

,

A-13)

and

+1

J

qfD(xb,tD)dx

v

=

1

A-14)

1

T he dif fer enti al e qua ti on f or PD E q. A-IO) then

becomes

A-15)

and has the solution

Using Eqs. 28 together with

xb =cos u

in Eqs. A-14

and

A-I6

yields Eqs. 29 and 33.

225

8/10/2019 22. SPE-13846-PA

10/10

ppendix

B Proof

of Equations

Fo r simplicity, the notation u=i.Jiwill be used through

out this Appendix.

Proof of Eq . 23.Substitution

of


16 gives

x [ ~

_1_

S k

u, cos a )Sok U, cos a ) R ~ u , l )

k k

X R ~ J u , cosh

where the terms involved by the a/integration have been

underlined. This integration can be performed with the

help

of

Eq. 20, giving

X r l ~ u,

I R ~ 1

u, cosh . B-1)

One first sets

w

the left side thus becoming the

a-independent function w see Eq. 17). Both sides of

the equation are then multiplied by sin a S02j U, cos a da

and integrated from 0 to 7 by use of Eq.

20 .

Eq. 23

emerges if, in addition, one uses

J I da

sin a

S02k U,

cos

a =2d

o

k

u , B-2)

o

which follows easily from Eq.

21

and from known prop

erties

of

the Legendre polynomials.

8

Proof ofEq. 24, When one makes use of Eq. B-2, sub-

stitution

of

the right side

of

Eq. 22 into Eq. 18 yields

When the right side of

Eq.

23 is substituted for the

k,

one obtains Eq. 24.

Proof of

Eq . 25. Eq.

25

is obtained immediately when

one uses Eq.

23

o r t ~

k

in Eq. B-1.

Metric onversion Factor

degrees X 1.745 329

E -0 2

rad

SPERE

Original manuscript

SPE

13846 received

in

the Societyof Petroleum Engineers office

Dec. 3,

1984. aper accepted for publication Feb.

6,

1986. Revised manuscript

re-

ceived Jan. 24,

1986.

Documents

22. SPE-13846-PA