Effects of the nonlinear gradient term on the transient pressure solution for a radial flow system

Journal of Petroleum Science and Engineering, 8 ( 1993 ) 241-256

241

Elsevier Science Publishers B.V., Amsterdam

Effects of the nonlinear gradient term on the transient pressure solution for a radial flow system Chayan Chakrabarty, S.M. Farouq Ali and W.S. Tortike Department of Mining, Metallurgical and Petroleum Engineering, University of Alberta, Edmonton, Alta. T6G 2G6, Canada (Received May 1, 1992; revised version accepted July 8, 1992 )

ABSTRACT Chakrabarty, C., Farouq Ali, S.H. and Tortike, W.S., 1993. Effects of the nonlinear gradient term on the transient pressure solution for a radial flow system. J. Pet. Sci. Eng., 8: 241-256. This study presents new analytical pressure solutions for high pressure-gradient flow of a single-phase, slightly compressible fluid under transient conditions. A nonlinear partial differential equation is generally used to describe the pressure behavior during such flows through porous media. The nonlinearity of this equation arises from the presence of the quadratic pressure-gradient term in the diffusion equation. In order to obtain a standard linear diffusion equation so that closed-form analytical solutions can be developed, the original nonlinear equation has traditionally been linearized by assuming the pressure gradient to be small throughout the reservoir at all times. However, during certain operations such as hydraulic-fracturing, high-drawdown flows, slug testing, large-pressure pulse testing, etc., the assumption of a small pressure gradient may not be justified and the nonlinear pressure-gradient term must be taken into account. Moreover~ in an age of increased sophistication of reservoir flow analysis and prediction methods and improved resolution of pressuremeasurement devices, the effects of the quadratic pressure-gradient term on the transient pressure behavior must be understood quantitatively. In this paper, analytical dimensionless pressure solutions of the nonlinear diffusion equation are derived by using the Laplace transform. Constant-rate and constant-pressure inner boundary conditions and infinite, closed and constantpressure outer boundary conditions have been considered. For the constant-rate inner boundary condition, solutions are presented for both injection and production problems by taking into account the presence of wellbore storage. Deviations from existing linear solutions are identified and are related to a dimensionless group, o~, which is proportional to the fluid compressibility. It is shown that for constant-rate inner boundary condition,and infinite or constant-pressure outer boundary conditions, the linear pressure solutions are within 5% of the corresponding nonlinear solutions for magnitudes of c~ less than 0.01 (within the dimensionless time range considered). However, for a closed outer boundary, the wellbore pressure predicted by the linear solution may be significantly smaller than that predicted by the nonlinear solution at large times. Analytical steady- and pseudosteady-state solutions are also presented and compared with the corresponding linear solutions. It has been shown, with examples, that significant errors may be incurred by using the linear pressure solutions in some cases.

Introduction The partial differential equation (PDE) describing the pressure distribution during the flow of single-phase, slightly compressible fluids through porous media is nonlinear Correspondence to: C. Chakrabarty, Department of Mining, Metallurgical and Petroleum Engineering, University of Alberta, Edmonton, Alta. T6G 2G6, Canada.

0920-4105/93/$06.00

(Matthews and Russell, 1967). A standard linear-diffusion equation in terms of pressure cannot be obtained unless the quadratic term in the pressure-gradient expression is ignored. Thus, a nonlinear equation can be avoided by neglecting the nonlinear term and the resulting linear equation can be solved for a variety of reservoir flow conditions. Such an approach has generally been taken to solve many reservoir-engineering problems, particularly in the

© 1993 Elsevier Science Publishers B.V. All rights reserved.

242

area of pressure-transient analysis, by assuming the pressure gradient to be small throughout the reservoir at all times. The assumption of small pressure gradients, however, may introduce significant errors in the pressure solution during certain operations such as hydraulic fracturing, large-drawdown flows, slug testing, drill-stem testing and large-pressure pulse testing. With the advent of ultrasensitive pressure-measurement devices, interpretation of small deviations of recorded pressures, pressure derivatives, pressure/pressure-derivative ratios, etc., from the corresponding linear solutions for buildups and drawdowns is routinely undertaken. It has been shown that part of such observed deviations can be attributed to the approximation made in linearizing the nonlinear PDE, rather than to possible anomalies in the physical system (Odeh and Babu, 1988 ). It is also to be noted that for highcompressibility systems, the nonlinear term may not be negligible in comparison to the other terms in the original PDE. Further discussion on this aspect of the problem is presented later in this paper. Odeh and Babu (1988) reported analytical solutions to the nonlinear PDE for three different cases. No wellbore storage was considered, and for radial flow system only a constant-rate inner boundary and an infinite reservoir were considered. They showed that for the cases studied, the error in implementing the linear solution is small, and thus, is acceptable for most reservoir engineering applications. Finjord and Aadnoy ( 1989 ) presented steady-state and approximate pseudosteadystate solutions and Finjord (1987) presented an approximation scheme to present wellbore pressure solutions for the nonlinear flow problem. Wang and Dusseault ( 1991 ) presented an analytical pressure solution of the nonlinear PDE for a single set of boundary conditions, by expressing the PDE in terms of density and,thereby, linearizing the PDE (Muskat, 1946). In the present study, the radial nonlinear

PDE is solved for a variety of boundary conditions to analyze the pressure distribution around a finite-radius wellbore. Solutions are presented for constant-rate (both production and injection, with and without wellbore storage) and constant-pressure inner boundary conditions, with the outer boundary conditions being infinite, closed or constant-pressure. Some of these solutions are then compared with the corresponding linear solutions. Exact steady- and pseudosteady-state pressure solutions of the nonlinear PDE are derived for a constant-rate inner boundary condition (both production and injection cases). Comparison of the nonlinear late-time solutions with the corresponding linear solutions are presented with the use of examples. Pressure solutions of the nonlinear PDE

No wellbore storage For a radial flow system, the PDE describing the single-phase flow of a slightly compressible fluid in a homogeneous and isotropic porous medium is (Matthews and Russell, 1967):

02p ~_l OP+c{OP'~, Or2 r Or \Orl

-

~lZct OP k Ot

(1)

In the linearized analysis, the pressure-gradient-squared term in Eq. 1 is discarded by assuming small fluid-compressibility and small pressure-gradients throughout the system (Matthews and Russell, 1967 ). For a constant-rate inner-boundary condition, the following dimensionless variables are defined:

PD--

2zkh [pi--p(r,t)] qP

(2)

ro=r/rw

(3)

kt tu -- ¢/UC,r2w

(4)

For a constant-pressure inner-boundary con-

243

dition, the dimensionless pressure is defined as"

P=Pw at r=rw

(12)

Outer boundary condition (infinite reservoir):

taD_

Pi - p ( r , t )

By defining the parameter: a=-

(5)

P i -- Pw

following dimensionless

q/tc 2nkh

(6a)

(6b)

for constant-pressure inner boundary, Eq. 1 can be expressed, in terms of the dimensionless variables, as:

OPD rD OrD

0 2pD

1

( OPD'~2 0PD \ ~rD,] -- OtD

Letting XD =

exp ( a p D )

(8 )

Equation 7 can be rewritten as: 02XD 10XD OXD OrE + rD OrD -- OtD

(9)

The transformation described by Eq. 8 is identical to the one proposed by Odeh and Babu ( 1988 ). A similar transform was also used by Kikani and Pedrosa (1991 ). Equation 9 can be solved for a variety of boundary conditions, as will be shown subsequently. The initial and boundary conditions in terms of the dimensional variables are as follows: Initial condition:

p(r,t) =Pi at t=O

(13)

Outer boundary condition (closed/zero flux condition ):

OP-o at r=rc Or

(10)

P=Pi at r=re

2nkh Op at It Or

r=rw

(11)

where the positive sign is for production and the negative sign is for injection. Inner boundary condition (constant-pressure case)"

(15)

The initial and boundary conditions, when expressed in terms of the transformed variable xD and dimensionless variables rD and tD, can be written, respectively, as: xD(rD,0) = 1

(16)

OXD -- ~ O~XD at rD = 1 OrD

( 17 )

XD(rD = 1,tD) = e "

(18)

XD(OV,tD) = 1

(19)

OXD --0 at ro=reD OrD

(20)

XD(reo,tD) = 1

(21)

The top sign at the right-hand side of Eq. 17 corresponds to production and the bottom sign to injection; identical notations will be used for subsequent equations of a similar type. The Laplace transform Of XD with respect to to is defined as: )?D(rD,S) =

L oXD(rD,tD)exp

(--StD)dtD (22)

Inner boundary condition (constant-rate case) q=±

(14)

Outer boundary condition (constant-pressure condition:

for constant-rate inner boundary, and

a = -c(p~ - P w )

p(~,t)=pi

The Laplace transform of the boundary conditions and Eq. 9 using the initial condition is undertaken. Equation 9 is thus transformed to the following ordinary differential equation: d2XD

l

d~ D

drZD t rD drD =SXD--1

(23)

244

The solution to Eq. 23 is:

qP pe --Pwc=APo=2---~-~ln ( reD)

XD(rD,S) =AIo(rDx/-S) + BKo(rDx/SS) +1_

The corresponding drawdown value for a pseudosteady-state system is given by:

s (24) where the constants A and B can be determined using the appropriate boundary conditions. The nonlinear pressure solutions for various boundary conditions, without considering wellbore storage effects, are presented in Appendix A.

With wellbore storage For a constant production/injection rate at the wellbore, the inner boundary condition (including wellbore storage effects ) can be expressed in terms of dimensionless quantities as:

OPD rD=l OPD ~D=

CD0~D

-T-O~DD

qu _½] pe-Pwf=Apo=~-~ [ln(reD)

(25)

1

(29)

Denoting the drawdown values (for the nonlinear case) obtained from Eqs. D2 and D5 for the steady- and pseudosteady-state conditions, respectively, a s A p n , the ratio of Apn to Apo for these two conditions can be expressed as follows:

Apn In[ I-T-a ln(reD) ] Apo-

-aln(reD)

(30)

for the steady-state system, and

3p~ =1

(28)

ln(XeD/XwD)

(31)

Ap0-- --a[ln(reD)-- ½] for the pseudosteady-state system.

where: C

CD 2ztq)cthr2w -

-

(26)

Equation 25 can be expressed in terms of the transformed variable ~D as: _+dXDdrDrD=,=(CDs-a)~DI'D=I --CD

(27)

The nonlinear pressure solutions for constantrate production/injection problems with wellbore storage effects are presented in Appendix B. The linear pressure solutions with and without wellbore storage are presented in Appendix C for the sake of comparison.

Steady- and pseudosteady-state solutions Appendix D presents late-time nonlinear pressure solutions for the cases of constant wellbore production and injection rates. Using the linear solution, the wellbore pressure drawdown, &-pwf, for a steady-state system, is given by:

Results and discussion

Solutions with and without wellbore storage The nonlinear wellbore-pressure and pressure-derivative solutions obtained analytically in this study were compared with the corresponding linear solutions for the case of a constant production rate; similar comparisons using the other solutions presented in this study can also be easily undertaken. The analytical solutions were inverted numerically from the Laplace transform solutions using the Stehfest algorithm (Stehfest, 1970). The nonlinear solutions are characterized by the parameter, a, which is a measure of the difference between the linear and nonlinear solutions. Figures 16 present the solutions for various outer boundary conditions with the nonlinear solutions being characterized by three different values of a, viz., - 10 -2, - 1 0 - 4 and - 1 0 - 6 . For finite reservoirs (Figs. 3-6 ), we also use a

245 10--

I

¢"1

-

~

8-

&= - 1 0

6-

o o~=-I0_2

4-

& ot =-10-

0.1 10-I

100

l0 t

l 02

103

104

l05

106

107

tD

Fig. 1. Comparison of linear and nonlinear solutions: infinite reservoir.

3.

-

Linear Solution a=-lO

e-, "m

--6

0 o~=-10 2 x l 0 -t-

-4 -2

ZX

o~=-10

0.1 '

1 0 -1

' ''""I

10°

' '''""I

l0 t

' '''""I

10 2

' '''"'q

' ''""'I

10 3

' '''""I

104

I0 s

' '"""I

'

106

'""'

10 7

tD

Fig. 2. Comparison of linear and nonlinear pressure derivatives: infinite reservoir. o f - 10- 3 and - 10- ]. It is to be noted that the magnitude o f o~ would typically be o f the order of 10 -3 for oil (for a constant-rate innerb o u n d a r y condition). However, this value might be higher or lower depending on the flow

rate, fluid compressibility, viscosity, permeability and formation thickness. F r o m Figs. 1 and 2 it is seen that at early times, the linear and nonlinear solutions are almost identical so long as the magnitude o f a

246

10000

Linear Solution -6 ct = -10

1000

t~ 100

o

o~=-10

o

c~=-10

A

tx=-10

[]

c~=-lO

-4

-3 -2 -1

10

reD = 500

1 02

10 a

1 04

10 s

106

107

10 a

1 09

tD Fig. 3. Comparison of linear and nonlinear pressure solutions: closed outer boundary. 10000

Linear Solution 1 000

!

~

loo-

lo,

]

o

o~= -10 -6 -4 a=-lO

o

(z=-10 - 3

A A A A

-2

A

a=-lO

[]

t x = - l O -1

red

102

1 03

104

10 s

1 06

= 500

107

tD

10 8

1 09

Fig. 4. Comparison of linear and nonlinear pressure derivatives: closed outer boundary.

is equal to or less than l 0-2. A similar observation m a y also be m a d e for reservoirs with closed or constant pressure outer b o u n d a r y

conditions (from Figs. 3 - 6 ) . However, for o r = - 1 0 - ' , the nonlinear solutions m a y be quite different from the linear solutions even

247 10

4 2-

Linear Solution

1

4-

0-,

¢

2"

°

0.1

°

O

cz=-10 2-

zx

c( = - 1 0

[]

(,=-10

/ -1

2-

0.01

'

'''""1

1 02

'

'''""1

1 0a

I

' ''"'"1

1 04

I I IIIII I

'

~''""1

1 06

1 0s

1

'

'''U"l

07

'

~''""1

1 0a

'

'''""

1 09

1 0 ~°

tD Fig. 5. Comparison of linear and nonlinear pressure solutions: constant-pressure outer boundary.

1

4-

reD = 500 , Linear Solution 0.1-

~

"1~-I~

( z = - 1 0 --6

al

,

o

=_,0

IS] 0.01

' '+u'"l 1 02

1 0a

~ =-10 -1

' '"U"l 1 04

' ''u'"l 1 0s

' ''UU'l

' ''"'"l

1 06

1

' ''+'"'l 07

1 0a

' ''"'"1

%

1 09

10

10

tD Fig. 6. Comparison of linear and nonlinear pressure derivatives: constant-pressure outer boundary.

at early times for finite reservoirs. At late times (tD > 3 × 10 7 ), the nonlinear solutions for a = - l0 -2 are significantly lower than the corresponding linear solutions for the closed

outer boundary case (Figs. 3 and 4 ). It is clear that the smaller the magnitude o f or, the larger is the time when the linear and nonlinear solutions diverge. Also, it can be shown that for

248

t000

iiiiiiiiiiiiiiiiiiiiii iiii

64-

2.

e,,

too ]

/

64-

/• 2+

10

/¢

.," ,'"

s"

~ . . - - ' " " ,"

./"

.,.... . J.-° ~ .,..,,,,,J

,,,""

....

/"

re~:+000;~=-'0

" " " reD = 500; Ct = - 1 0 ............reD = 5000' ct = - 1 0

,•" ,°,°"

-2

-2

' reD = 500; Ct = - I 0

8

........ I

106

....

10 7

""1

'

10 8

''"'"1

'

10 9

''"'"1

'

1010

'''""1

'

1011

' ......

I

........

1012

I

........

l 0 t3

10 la

tD Fig. 7 . L a r g e - t i m e n o n l i n e a r p r e s s u r e s o l u t i o n s for d i f f e r e n t r e s e r v o i r sizes: closed o u t e r b o u n d a r y .

CD 1

C~

ool/oi

++=_1o-2

0.001 ~ / ~ "

[]

cz = - 1 0 --4

0.0001 10 °

101

102

10 a

104

10 s

106

107

1 0a

tD Fig. 8. Wellbore p r e s s u r e s o l u t i o n s for linear a n d n o n l i n e a r cases with wellbore storage: infinite reservoir.

magnitudes o f a equal to or less than 10-2 and for tD < 3 × 10 7, the difference between the lin-

ear and nonlinear solutions is less than 5%. It can be seen from Fig. 3 that for a closed outer

249

102 10 -1 1 ~ 1 ~ G

1

'ill;< 1 0°

I

I

I

1 02

1 04

1 06

1 08

tD Fig. 9. Pressurederivativesfor linearand nonlinear caseswith wellborestorage:infinite reservoir.

10

j"

S

1 04

reD = 500

1 03

CD

/

~

x

#

t,~t~ax

1 02

1 0~

1 0°

1 0 -1

-2

~rl 0_0.~~s~-~'_~n~"

/X o~=-10~

~ 1 0 4 ~

[]

1 0 -2

' 1 02

'''""1

' 1 03

'''""1

' '''""1 1 04

' '''""1 1 0s

' 1 06

'''""1

' 1 07

o~=-10 ' ''""1 1 08

'

' ''""1 1 09

'

'''"' 1 0 ~°

tD Fig. 10. Wellborepressure solutions for linearand nonlinearcaseswith wellborestorage:closedouter boundary,

250

10000 red = 500

1000

100

CD

10 102 A

(t=-lO

[3

~=-10

-2 -4

0.1

~1L04 0.01 1 02

1 03

1 04

10 s

1 0s

1 07

1 08

1 09

tD Fig. 11. Pressure derivatives for linear and nonlinear cases with wellbore storage: closed outer boundary.

10

CD 6 4-

/

reD = 500

2

1_ 6

3

/ []

.1

! ! illll t

1 04

~ ,~HH

I

10 s

~ 'i""i'('rt'hI

~

,~,,z

10 s

I

A

o~ = - 1 0

~3

o~ = - 1 0

....... ~ '~'~Hq

1 07

'

1 08

-2 --4

~"'i''~"~l

109

~

~ T~H~

101°

tD Fig. 12. Wellbore pressure solutions for linear and nonlinear cases with wellbore storage: constant-pressure outer boundary.

251

10

reD = 500

•••1.•

C~

Zk

~ = -10 -2

.

C~

2-

0.1 102 4

103

2-

104

- ~ ! ~ _

0.01 1 04

1 05

1 06

107

1 08

109

10 lo

tD Fig. 13. Pressure derivatives for linear and nonlinear cases with wellbore storage: constant-pressure outer boundary.

boundary, the nonlinear pressure-solution at large times exhibits a much slower rise than the linear solution. In order to examine the largetime behaviour of the nonlinear solutions for a closed outer-boundary system more carefully, Fig. 7 is presented. From this figure it is clear that for pseudosteady-state flow, the linear pressure drop with time can occur at a much lower rate than that predicted by the linear solution, especially when the magnitude of a is high. Similar observations can also be made by examining the linear and nonlinear wellbore pressure and pressure-derivative solutions in the presence of wellbore storage. These solutions are presented in Figs. 8-13. Here, the nonlinear solutions are characterized by a = - 10 -2 and - 1 0 -4. At early times, the characteristic unit-slope line can be seen in the log-log plots. It is also obvious that the linear and nonlinear solutions are almost the same at early times for the three different cases of outer-boundary conditions. But, for the closed outerboundary condition, for

tD > approximately 3 × 10 7, the flowing wellbore pressure for a = - 10 -2 is significantly higher at any given time than that predicted by the linear solution.

Steady- and pseudosteady-state solutions for constant-rate case At late times, for a constant-rate innerboundary condition, the pressure solution goes to steady- or pseudosteady-state depending on whether the outer-boundary condition is constant-pressure or closed, respectively. Unlike the case of the linear solutions, the nature of the nonlinear solutions would also depend on whether a constant-rate production or injection is carried out at the wellbore (Odeh and Babu, 1988). Values of the ratios Apn to APo represented by Eqs. 30 and 31 can be obtained for given values of red and a. Table 1 shows the ratios calculated for steady- and pseudosteady-state (for constant-rate production) systems for various values of reD and a. It can be seen that for a given value of reD, the ratio

252 TABLE 1 Ratios ofdrawdown values for the linear and nonlinear cases for steady- and pseudosteady-states

red

Io~l

Stead-state Ratio

=dpn 3po

Pseudosteady-state Ratio

=Apn ,Jpo

500

10 -2 10 -3 10 -4 10 -5

0.97016 0.99691 0.99969 0.99997

1.02741 1.00265 1.00026 1.00003

1000

10 -2 10 -3 10 -4 10 - s

0.96697 0.99656 0.99965 0.99997

1.03112 1.00299 1.00029 1.00003

5000

10 -z 10 -3 10 -4 10 -5

0.95969 0.99577 0.99957 0.99996

1.03990 1.00380 1.00038 1.00004

would approach unity as a decreases in magnitude. Also, for a given value of a, the ratio would approach unity as red decreases. Thus, calculations of wellbore pressure drawdown, based on the linear solutions, would incur errors which increase with increasing magnitudes of a and reD. Also, it can be noted that by considering the linear solutions, the drawdown (for production ) would always be overpredicted for a steady-state case and underpredicted for a pseudosteady-state case. The most important observation that can be made by examining Table 1 is that the error in the "linear" Ap calculation would rarely exceed 4-5% and is, thus, acceptable for engineering calculations. These errors are applicable for the simple flow system examined in this study. For other more complex systems (for example, for the flow of a non-Newtonian, power-law-type fluid through porous media), these errors may be much higher under certain situations. Moreover, as is shown subsequently by considering an example from Dake (1978), significant errors can be incurred in certain calculations using the linear drawdown solutions. We consider a producing well that

has been stimulated by steam-soaking. The goal is to find the increase in the productivity index ( P I ) of the well by assuming steady-state flow conditions. It has also been assumed that the zone around the wellbore can be divided into two parts: one with rw
(32)

Assuming q = 1 0 0 m3/day, k = 1 0 . 1 3 mD, h = 10 m and c = 2 × 10- 9 P a - 1, the P I ratio increase is calculated, using Eq. 32, to be 1.61. Thus, for this set of values, the calculation using the linear solution would be optimistic by about 157%. We consider another fictitious example. After completion a well has been stimulated so that the permeability out to a distance ofrh has increased to ten times its original permeability. The viscosity, compressibility and thickness are the same in the high- and low-permeability zones. For such a case, the P I ratio increase using the linear solution, with the same values ofrw, rh and re as considered in the

253

previous example, can be shown to be 3.175. However, considering q = 1 0 0 m3/day, k = 10.13 m D (original permeability), h = 10 m,/~ = 100 cP and c = 2 × 10 - 9 P a - 1, the PI ratio increase can be calculated to be only 1.905. Thus, the "linear" calculation is too high by about 67%. If the compressibility value is assumed to be twice of what has been considered before, the error would be approximately 101%. We have shown examples where the linearized analysis can lead to serious errors. For a closed outer-boundary condition, the linear solution would underpredict the flowing wellbore pressure (for production) for large magnitudes of a, at large times (tD> 3X 107). For a constant-rate inner-boundary condition, the parameter, 04 can take on large values if the fluid compressibility, flow rate or viscosity is high, or if the formation capacity is low. One implication of this observation can be understood by considering the following case. The Lloydminster heavy-oil reservoirs in Canada have been known to exhibit an abnormally high primary production performance and one of the factors attributed to this behaviour is the presence of occluded gas in the heavy oil (at gas saturations below the critical value ) giving rise to high compressibilities of the flowing fluid (Smith, 1988 ). Thus, the nonlinear pressure-gradient term in Eq. 1 must be considered in the flow prediction and interpretation methods for such a reservoir. Also, the fact that the nonlinear solution shows that for high flow rates there might be a significant correction to

the flowing wellbore pressure, suggests that the nonlinear term in Eq. 1 must be taken into account when non-Darcy effects in oil flow are estimated. A similar observation was also made by Finjord and Aadnoy ( 1989 ).

Conclusions The nonlinear wellbore-pressure solutions obtained analytically in this study compare quite well with the corresponding linear solutions at early times, with and without the presence of wellbore storage. Also, for infinite and constant-pressure outer boundary conditions, the linear and nonlinear wellbore pressure solutions compare very well with errors < 5% (up to atD less than 109, for magnitudes of ot equal to or less than 0.01). For a closed outerboundary condition, the solutions may diverge quite significantly, especially for large magnitudes of a and at large times. For a better understanding and proper interpretation of the deviation of field-observed pressure data from the theoretical values, one should be aware of these errors in the linear solutions. Nonlinear steady- and pseudosteady-state pressure solutions have also been presented in this study. It is seen that the error in the calculation of the wellbore pressure drawdown using the linear solutions would be less than 5%. However, it has been shown with examples that use of the linear solutions may be extremely unsatisfactory for certain reservoir engineering calculations and thus, should be, if at all, applied with caution.

Appendix A. Nonlinear pressure solutions (no wellbore storage) Constant-rate inner-boundary, infinite reservoir: + 0~Ko(rox~) 1 "XD(rD'S)= Sx~s K I ( x~S ) T otSKo ( x~s ) + S

(A1)

Constant-rate inner-boundary, closed outer boundary: tx{KI (renx~S)lo(rnx~S) + I t (rcDX~S)Ko(rDx/S)} s{Kt(reD~S)[x~S[i(x~S)+-Odo(x/SS)]+It(reD~S)[--x/SKl(x~S)+-OlKo(x/S)]}

.~D(rD,S) = ~_

Constant-rate inner boundary, constant-pressure outer boundary:

+l_ S

(A2)

254 1

-~D{rD.s) = +--S{Ko(reD&)[ ~/SII { ~ ) -t-alp (x/S) ] + Io (reDX/S) [ ~fsK, {V/s) -T- o~Ko(-v~)] } +-s

(A3)

Constant-pressure inner boundary, infinite reservoir:

XD(rms) = ~

Sl~o( x~ S)

Ko(rD.¢~) +-

(A4)

S

Constant-pressure inner boundary, closed outer boundary: )~D(rD,S) --

(e"--l )[Kl(r~Dx/~)lo(rDx/s)+Ko(rr~&)l,(reD&) . /S[KI(reD.,fS)Io(x/~)+Ko(v/S)I,(reDx~S) ]

] 1 +S

{A5)

Constant-pressure inner boundary, constant-pressure outer boundary:

XD{ro, s) = ( e " - 1)[lo(reD~fS)Ko{rD~s) --lo(rox/s)Ko(reDx/S)] + ! stIo(r~DvS}Ko(~)--Io(,ffs)Ko(r~D.~SS)] s

(A6)

The plus-minus or minus-plus signs in Eqs. A1-A3 refer to production or injection cases, with the top sign being for production and the bottom one for injection. This notation will be followed in all subsequent appendices in this paper.

Appendix B. Nonlinear pressure solutions (with wellbore storage) Constant-rate inner boundary, infinite reservoir:

~Ko(rD~)

l )~D(rD, s) =st (CDS-a)Ko(\,~) +_.~ssK,{xfs) ] +-s

(BI)

Constant-rate inner boundary, closed outer boundary:

~D(ro,s) =

/a [Ko(rD~s)I, (reD.,fS) +Io( rDX/s)K, (reDv~s) ] +--1 s[(CDS--O~)AI __+N/SzJ2 /- ] S

(B2)

where:

A~ = Io (v/s)K, (red V~') + Ko(v/s)l, (reDVFS)

(B3)

and A2 =K, (. ~ ) I , ( reDw£S) - K l (r¢D~ss)l~ (.~ss)

(B4)

Constant-rate inner boundary, constant-pressure outer boundary: XD (rD,S)

o~[ (Io(r~D.c~)Ko(rDxfs)--Ko(reDVS)Io(rD~S) ] + -l S st ( C o s - a ) & + , ~ & ]

(B5)

where

A3 = Ko( \,/S )Io( reDv~ ) -- Io( ~/ss) Ko( reDV~ )

(B6)

and Z~4

=I, (x/S)Ko(r~DV~) + K, (~/S)/o (r~Dv':SS)

(BT)

Appendix C. Linear pressure solutions (no wellbore storage) The linear PDE, governing the single-phase radial flow of a slightly compressible fluid in a homogeneous system, can be expressed in dimensionless form as:

OZpD + 1 Opt) Opt) Or~) rD Orb -- OtD

(C1)

255 Pressure solutions to Eq. CI, expressed in Laplace space for different cases, are presented below. Constant-rate inner boundary, infinite reservoir:

Ko(rDx~) 10D(rD'S) =$3/2KI (x/S)

(c2)

Constant-rate inner boundary, closed outer boundary:

_ IO(rDw/S)K,(reDx~S)+Ko(rDxflS)I,(reDx/S) pD ( rD'S ) --S3/2 { I ~(reDx/s)K, ( x~SS) -- K, ( r~ox~ss ) l , ( x/~ ) }

(C3)

Constant-rate inner boundary, constant-pressure outer boundary:

lo( reDxfS )Ko( rDx/S ) -- Ko( reD~S )Io( ro,/S) /TD(rD'S) =sS/2{ lo( r~Dv~ )K, ( x~ss) + Ko( reDxfSS)I, (x/S)}

(C4)

Constant-pressure inner boundary, infinite reservoir: /~D( rD,S ) -- KO(rD x/S)

(c5)

sKoG~)

Constant-pressure inner boundary, closed outer boundary: ~-

/-

/_

ll (reD\/'S)Ko(rDV's)Ki (reDxf-s)lo(rDw"s) pD(rD'S) =S{I, (r~Dv~)Ko (V~) + K~ (reo.,/~)lo (x/SSS)}

(C6)

Constant-pressure inner boundary, constant-pressure outer boundary: /TD(rD,S) = l°( r¢Dx/~ ) K°( rDv~ ) -- Ko( r~D,$'S ) Io( rox/s )

S{Io(r~Dv/ S)Ko( v S)--Ko( reow~)lo(~¢' S~j

(c7)

With wellbore storage Constant-rate inner boundary, infinite reservoir:

pD ( rD,S ) =

K° ( rD x~S )

s[v~K, (v~) + CDSKo(V~)]

(C8)

Constant-rate inner boundary, closed outer boundary:

/~D(rD,S) =

Ii (red x/~)Ko(rovfs) +KI (reDV~)lo(rDx/S)}

(C9)

Constant-rate inner boundary, constant-pressure outer boundary:

&(ro,s)

lo (red V s)Ko(roV/s) --Ko(r~DV~)Io(rD~/SS) S [ CDSZJ 3 "~"N/TSSz]4]

(c~0)

where A~, A2, 63 and z~4 a r e given by Eqs. B3, B4, B6 and B7, respectively.

Appendix D. Nonlinear steady- and pseudosteady-state pressure solutions

Steady-state solutions For steady-state condition, noting that:

OXD/OtD =0

(DI)

and using conditions given by Eqs. 17 and 21, we get the following dimensionless pressure distribution:

1 ln[lT-oeln(rD) ] PD(rD) = a L1-T-a ln(reD)J

(D2)

256

Pseudosteady-state solutions For pseudosteady-state condition during constant-rate production/injection, we have:

OXD/OtD = +_OeflXD

( D3 )

where:

fl= Z/r2D

(D4)

Using Eq. D3 along with Eq. 20, the nonlinear pressure solutions for pseudosteady-state condition can be written as: 1

PD (rt) ) --PwD = - - In {XD/XwD }

( D5 )

OL

where: XD Jl ( r e D \ / / ~ ) Yo(rD X / ~ ) -- Y, ¢reDx/-~fl)Jo(rdx/~-fl) XwD-- J,(rcpv ~7~fl)Yo(x/'~-~)- r l ( r c D x / ~ ) J o ( , ¢ ~ ) }

(D6)

for the production case, and,

XD I,(reDw~)Ko(FD~) + KI(reD~fl)Io(FDxf-~fl) Xwo-

I, ( r e D , ¢ ' ~ ) K o ( , f ~ ) +K, ( r c D x / - ~ ) l o ( x / - ~ ) }

(D7)

for the injection case.

Nomenclature C

C

c~ C~

h

~,Ko Ii, Ki Jo, Yo JI, ,]2 k l P PD Pi

Pw q r FD Fe

reD t tD )(D

0

Fluid compressibility, kPa-~ [psi_ ~] Wellbore storage, m3/kPa [it3/psi] Dimensionless wellbore storage Total compressibility, k P a - 1 [ psi- ~] Formation thickness, m [It] Modified Bessel functions of order zero Modified Bessel functions of order one Bessel functions of order zero Bessel functions of order one Permeability, m 2 [d] Laplace variable Pressure, kPa [psi] Dimensionless pressure Initial pressure, kPa [psi] Constant wellbore pressure, kPa [psi] Constant flow rate, m3/d [ B / D ] Radius, m [ft] Dimensionless radius External radius, m [itl Dimensionless external radius Time, hours Dimensionless time Transformed dimensionless variable, given by Eq. 8 Laplace transform ofxD Parameter given by Eq. 6 Constant defined by Eq. D 4 Viscosity, Pa.s [cP] Porosity, fraction

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