Pressure transient response of stochastically heterogeneous fractured reservoirs

$: Pressure transient response of stochastically heterogeneous fractured reservoirs$
Transport in Porous Media 11: 263-280, 1993. 263 �9 1993 Kluwer Academic Publishers. Printed in the Netherlands.

Pressure Transient Response of Stochastically Heterogeneous Fractured Reservoirs

CAROL BRAESTER and DAVID G. Z E I T O U N * Department of Civil Engineering, Technion, Haifa 32000, Israel

(Received: 24 June 1991; revised: 3 June 1992)

Abstract. A stochastic model for flow through inhomogeneous fractured reservoirs of double porosity, based on Barenblatt et al.'s continuum approach, is presented. The fractured formation is conceptualized as an interconnected fracture network surrounding porous blocks, and amenable to the continuum approach. The block permeability is negligible in comparison to that of the fractures, and therefore the reservoir permeability is represented by the permeability of the fracture network. The fractured reservoir inhomogeneity is attributed to the fracture network, while the blocks are considered homogeneous. The mathematical model is represented by a coupled system of partial differential random equations, and a general solution for the average and for the correlation moments of the fracture pressure are obtained by the Neumann expansion (or Adomian decomposition). The solution for pressure is represented by an infinite series and an approximate solution for radial flow, is obtained by retaining the first two terms of the series. The purpose of this investigation is to get an insight on the pressure behavior in inhomogeneous fractured reservoirs and not to obtain type curves for determination of reservoir properties, which owing to the nonnniqueness of the solution, is impossible. For the present analysis we assumed an ideal reservoir with cylindrical symmetric inhomogeneity around the well. Fractured rock reservoirs being practically inbomogeneous, it is of interest to compare the pressure behavior of such reservoirs, with Warren and Roots's solution for (ideal) homogeneous reservoirs, used as a routine for determining the fractured reservoir characteristic parameters ). and o), using the results of well tests. The comparison of the results show that inhomogeneous and homogeneous reservoirs exhibit a similar pressure behavior. While the behavior is identical, the same drawdown or a build-up pressure curve may be fitted by different characteristic dimensionless parameters 2 and ~o, when attributed to an inhomogeneous or a homogeneous reservoir. It is concluded that the ambiguity in determining the fractured reservoir 2 and co, makes questionable the usefulness of determination of these parameters. Computations were also carried out to determine the correlation between the fracture pressure at the well and the fracture pressure at different points.

Key words. Fractured reservoirs, double-porosity models.

Nomenclature

a coefficient in the sink/source function L characteristic length c compressibility nl unit normal vector E { } ensemble average p pressure G(,) Green function /3 Laplace transform of p h thickness of reservoir ro autocorrelation distance k permeability rw well radius

trend component of k q rate of flow /~ fluctuated component of k u flux

*Currently at the Department of Mechanical and Civil Engineering, University of Vermont, U.S.A.

264 c. BRAESTER AND D. G. ZEITOUN

t time p autocorrelation function x Cartesian coordinates ~b porosity c~ geometrical parameter ~ trend component of 4~ Fi part of the boundary ~ fluctuated component of ~b 2 (ar~kl/k2) dimensionless parameter V nabla operator (1) ~)2C2/(~)1C1-~-~,)2C2) dimensionless

parameter Subscripts # dynamic viscosity 1 medium of blocks (matrix) cr,k standard deviation governing the 2 medium of fracture

variation of/~2 ~ standard deviation governing the vari-

ation of ~2

1. Introduction

The fractured reservoirs considered in this study are conceptualized as an interconnected fracture network surrounding porous blocks, and are amenable to the continuum approach.

Solutions to problems of flow through fractured reservoirs requires the knowledge of the spatial distribution of the reservoir properties. Continuum and discrete models are used for representing the properties of fractured reservoirs.

The discrete approach requires a detailed description of the reservoir properties at the scale of the individual blocks and fractures. Such a description of real reservoirs is obviously impossible, and therefore discrete representations are limited to theoretical investigations. The continuum approach dealing with properties averaged over Representative Elementary Volumes, circumvents the need of such a detailed description. Due to the scale of the fracture microscopic inhomogeneity, much larger than that of the blocks, the Representative Elementary Volumes at which reservoir properties have to be defined, is dictated by the scale of the fracture network. This scale, which includes at least several numbers of blocks, prevents laboratory determination of the fractured reservoir properties, and imposes in situ well tests. Prediction of reservoir performances, say by reservoir simulators, requires that rock properties by defined over the entire reservoir domain. Practically, the number of wells at which the reservoir properties are determined is small in comparison to the number of points at which the rock properties have to be defined, and simple or geostatistical extrapolation methods are used for extrapolating the data at hand over the whole reservoir.

A deterministic continuum mathematical model of single-phase flow through double porosity homogeneous fractured reservoirs was formulated by Barenblatt et al. (1960). The constituent media, fractures and blocks, are considered overlapping continua. The equations of conservation of mass are written separately for each medium; a sink/ source term in each equation expresses the transfer of fluid between fractures and blocks.

Analytical solutions to Barenblatt et al. (1960) equations were presented by Barenblatt (1963), Warren and Root (1963), Odeh (1965), Kazemi et al. (1969), deSwaan (1976), Najureta (1976), and Da Prat et al. (1981).

PRESSURE TRANSIENT RESPONSE OF RESERVOIRS 265

Warren and Root (1963) solutions to Barenblatt et al. (1960) equations for radial flow, include two characteristic dimensionless parameters of the fractured reservoir:

arwkl/k2 and co = c~2c2/(c~1Cl +q~2c2), where a is a parameter related to the specific surface of the block, rw is the radius of the well, k is permeability, ~ is porosity, c is total compressibility; subscripts 1 and 2 denote the medium of the blocks and fractures, respectively. Note that in Warren and Root's (1963) solution, all fractured reservoir properties, i.e., fracture and block permeabilities and porosities as well as the parameter a are considered constant.

For pumping at constant rate of flow from an infinite reservoir, initially at constant pressure, the pressure vs. time behavior (p, log t), as resulting from Warren and Root's (1963) solution, is characterized by the following three typical stages: (i) a first stage during which the liquid is removed mainly from the fractures, and the pressure follows a straight line with a slope related to the fracture network properties; (ii) an intermediate stage, characterized by a curve of variable slope, and reflecting the onset of fluid flow from the blocks into the fractures; and (iii) a third stage during which fluid is removed from both fractures and blocks, and the pressure follows a straight line, with a slope corresponding to an equivalent homogeneous reservoir. Due to the low permeability of the blocks in comparison to that of the fractures, the permeability of the homogeneous equivalent is practically equal to the permeability of the fracture network. Under such conditions, the two straight lines corresponding to the early stages and to the late stages of the drawndown (or of the build-up) are of equal slopes. The intercept with the time axis of the first and second straight lines are related to the porosity-compressibility product of the medium of fractures and of the homogeneous equivalent, respectively (Braester, 1984).

Kazemi (1969) numerically solved the same problem considered by Warren and Root (1963) by using a discrete representation of the fractures and blocks.

The Warren and Root (1963) solution is used as a routine for determining the characteristic parameters of the fractured reservoir 2 and co. The parameter 2 characterizes to the time of transition from the first straight line, in the (p, log 0-plot, to the second straight line of the asymptotic behavior, and is related to the characteristic size of the block, through the parameter a. The parameter co is related to the displacement between the two straight lines.

Fractured rock reservoirs being practically inhomogeneous, it is of interest to study the behavior of such reservoirs.

This investigation presents a stochastic model of flow through inhomogeneous fractured reservoirs, based on Barenblatt et al.'s (1960) continuum approach. The inhomogeneity of the fracture network, such as fracture length, frequency, and orientation, and the inhomogeneity in the individual fracture properties, such as opening and filling material, results in large variabilities of the macroscopic fracture permeability and porosity. On the other hand, the medium of the blocks may be regarded as an ordinary porous medium with approximate constant properties. Subsequently, in the present stochastic model, the spatial variability in the reservoir

266 C. BRAESTER AND D. G. ZEITOUN

properties is attributed to the medium of fractures, while the block properties are considered constant.

The permeability and the porosity of the fracture network are considered random functions of position, defined by the average, constant standard deviation, and autocorrelation function characterized by the integral scale. All other model parameters are assumed to be constant. The statistical parameters are considered to be known, and we focus our investigation on the influence of spatial variation of fractures permeability and porosity on the pressure behavior.

A solution of the random partial differential equations is obtained by the Laplace transform with respect to time, written as a solution of a stochastic integral equations, and Neumann expansion (Adomian, 1983, 1986), resulting in an infinite series expansion for the pressure in the fractures.

An approximate solution, obtained by retaining the first two terms of the series, is presented for radial flow in an ideal reservoir with cylindrical symmetric inhomogeneity around the well. The equations are solved under identical boundary and initial conditions as considered by Warren and Root (1963), and results are presented for several values of 2 and o2 for coefficients of variation for permeability in the range 0.15 to 0.30.

The influence of inhomogeneity in fracture properties is analyzed by comparing the pressure behavior (p, log t) with the Warren and Root (1963) solution for constant reservoir properties.

We also investigated the correlation between the fracture pressure at the well and at different points, as function of time. Correlation of pressures gives an indication on the extent of the area within fractured rock parameters may be extrapolated.

2. The Stochastic Mathematical Model

Let us consider Barenblatt's model equations for horizontal flow of a slightly compressible fluid. Similarly to Warren and Root (1963), the flow through the blocks is neglected.

Under these assumptions, the governing equations of flow are

0pl ak, ~blc1~7-, + ---~* (Pl - P2) -- 0, (1)

u ~ #

0p2 _ 2 c 2 - ~ 1- V'(kz Vpz) - akt (Pl - P2) = 0, (2)

# #

where k is permeability, # is dynamic viscosity, p is pressure, ~b the porosity, c is compressibility and a is a parameter, proportional to the specific surface of the block, and controlling the transfer of liquid between fractures and blocks. Subscripts 1 and 2 denote the medium of blocks and fractures, respectively.

The considered initial condition is constant reservoir pressure, i.e.,

t --- 0: pl(x, 0) = pz(x, 0) = Po. (3)


The boundary conditions, written in a general formal, are

kZVk2"ni q- o~i(P2 - P2i) = u~' on Fi, (4) //

where ni is the unit normal vector (positive outward), along the boundary Fi, part of the entire boundary F.

r = U , r , , (5)

F, 0 Fj = ~3, (6)

P2, and u* are prescribed functions of space and ai determines the nature of the boundary condition on F~: prescribed flux u* (~ = 0), prescribed fracture pressure P2, (~ = oo), and mixed conditions for intermediate ~i values.

The permeability k2 and the porosity q~2 are decomposed into constant trends k-2, q52 and fluctuated components /~/(x), ~2(x), representing the spatial variation of fracture permeability and porosity around the average value, i.e.,

k2 = k-2 +/~2, (7)

(~2 = (~2 -']- (~2" (8)

Similarly to Warren and Root (1963), we consider the following dimensionless parameters

/ .

Po = u ~ ( P o - p(x, t)), (9)

X x o = (10 )

= a~22 L2, (11) 2

t D = (C1(~1 "q- C2(~2)1/,/L2 , (12)

(~2 C2 co - clq51 + c2~b2' (13)

where L and u are characteristic length and specific flux, respectively (e.g., in radial flow, L = rw and u = q/2nLh).

Substitution of the dimensionless parameters defined by Equations (9) to (13) into Equations (1) and (2) yields the following stochastic model equations:

(1 - c o ) ~ + 2 ( p l . - pzo ) = 0, (14) o t D

268 C. B R A E S T E R A N D D. G. Z E I T O U N

2(pl/~ PZD)

= VDVk" VDP2D,

C2 (]~2

where

/3k = ~22'

/~b - - C1(~ 1 "[- C2q~ 2

Written in a dimensionless form, the initial conditions are

t D : O: p l D ( X D , O) = p Z D ( X o , O) = 0

and the boundary conditions are

- Vpa'ni - f l i (P2 - Di) = R i + VkVDP2o'rli on Fi,

where

u* R i - -

a#L ~,- ~ ,

D i = ;uL(PO - Pi).

Applying the Laplace transform

plD(xo , t~) = /~ID(XD, S) e sty' ds,

p3v(x~, to) = O2o(x~, s) e stD ds,

to Equat ions (14) and (15) we obtain

"~ff2D /~ID -- S(1 -- CO) + 2'

[1 + 6k(XD)]vZ/~2D -- SEfo(S) + Z3,(Xo)]/~2D = -- VD13k'Vo/~2o

subject to the boundary conditions

- V p z o ' n i - - f l i ( P z D - - Oi) = Ri + ~kVD~ao'nl on Fi,

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)


where

2 + s(1 - c@co fo(s) = (28)

,~ + s(1 - co)

Local changes in the opening of the fractures or in the properties of the filling material may lead to drastic changes in the fracture network permeability, while the fracture porosity may remain unaffected. Therefore, the fracture permeability and the fracture porosity may be considered independent parameters. Subsequently, it is assumed that Vk(X) and f~(x), representing the relative variation of porosity and permeability respectively, are statistically independent. These two parameters are considered to be homogeneous zero mean random fields. In such a description, at each point Xfk(X ) and gr are random variables.

In a second order representation, a random field can be characterized by average, standard deviation and by the autocorreIationfunction, describing the spatial depend- ency between the values of fractures parameters at different points in space, i.e.,

E{~k(X~)} = 0,

E{~k(~)e~(~o)} = av~pv~(IXlD -- xZDI), (29)

= 0,

%p (Ixlo - x2DI), (30)

E{ek(X1D)V()(X2D)} = O, ( 3 1 )

where E{ } denotes the ensemble average, i.e., the average over the set of realizations of the random fields.

One of the most important characteristics of the autocorrelation function is the autocorrelaton distance defined as

r0~, = p~(r) dr, (32) --CO

ro,o = pv~(r) dr, (33) --cO

where p(r) is the autocorrelation function and ~ is the scalar distance between the points in space. Note that when ro goes to infinity, the random field degenerates into a single random variable.

From the physical point of view, the scales of length involved in the problem are: (i) the radius of the well (rw), (ii) the size of the Representative Elementary Volume (rs), over which the model continuum parameters are defined and (iii) the autocorrelation distance ro.

The size of the Representative Elementary Volume is generally much larger than rw; ro is at least in the same order of magnitude as rs. Then, if rw corresponds to the length of reference of the problem, both fk(Xo) and ~3o(xD) behave in practice like

270 C. BRAESTER AND D. G. Z E I T O U N

random variables. The study of the influence of the autocorrelation distance, in the present investigation will consider two extreme values.

2.1. M E T H O D OF SOLUTION

The Neumann expansion (Adomian, 1983, 1986) procedure requires the knowledge of the first term of the series solution po(xD) corresponding to the solution of Equations (1) and (2) for constant fracture permeability and porosity (k-2 and q52). These partial differential equations may be solved by classical methods.

The term p,, n ~> 1 of the Neumann series, is calculated by an iteration procedure, using the relationship between p,, n/> 1 and the preceding term of the series p,_ 1 (Appendix A). Practically, to obtain a solution the series is truncated after a few terms.

p,(x) = f v G(xD, tP)F(Ok(~O), Ve~(~), P,-1 (~), S) dV(tp) +

+ f r 6(x,, ~)~aVDp.- ~" n, dF, (34)

where

F(~a(~'), ~(~9), P,-1 (t)), s)

= ~k(~)V~p.-1 (4,) + s~(4))p._l (~) + v ~ k - V,~p._l (~), (35)

where G(xD, ~) represents the Green function associated with the differential operator V 2 - sfo(s) and the boundary conditions expressed by Equation (27).

G(xD, ~) is the solution of the following equation

V2 G - sfo(s)G=6(xD -- O) (36)

subject to the boundary condition

- V G ' n i + f l i G = O onFi. (37)

In the following, the average fracture pressure and the second-order moments are computed for small variation of the parameters. In these computations the terms of the order of magnitude of O(a~k) for the permeability variation and O(a~) for the porosity variation are neglected. It may be shown, that in this case

E{p2o(x) = po(x) + E{p2~(x)} + O(a~p) (38)

and

E{(p2D(XID) -- E{pZD(XID)})(P2D(X2D) -- E{pZD(XzD)})}

= (PlD(XlD)PlD(X2D)} + O(o'v4p),

where O(o-4p) equals variability.

(39)

4 O(av 4) or O(o-v~), depending on the investigated parameter


The computation of these moments requires the computation of E{pz(x)} and E{pI(XaD)PI(XzD)}. These expressions were derived from the general expression of pl(xD) and pZ(XD), in terms of Po and autocorrelation function (Appendix B).

pl(x) = fv G(xD, $)F(fk(0), 6,(~), Po($), S) dV($) +

+ fr G(XD, ~)t3kVoPo" ni dF, (40)

pz(x) = fv G(xD, O)F(~k(O), ~r Pl(O), S) dV(O ) +

+ frG(X~ @)kVDPl"ni dr. (41)

The inverse Laplace transform of the above solution enables the computation of the fracture pressure.

3. A Solution for Radial Flow

The present stochastic approach was used to obtain an approximate solution for radial flow, when retaining the first two terms of the infinite series expansion. Identical boundary and initial conditions as considered by Warren and Root (1963) were assumed, i.e., initial conditions

t = 0: pl(x, 0) = p2(x,0) = Po

and boundary conditions

(42)

k2 @2 q r = rw: - - - (43)

lz Or 27zrwh'

r --* oe: P2 = 0 (44)

expressing production from a well, at constant rate of flow (q), from a reservoir of infinite extent, initially at constant pressure.

For radial flow, Equation (26) yields

D~2o = O, (45)

where D denotes the differential operator

~2 1 D - Or~ + rD OrD sf(s, #k, re). (46)


The correspondent boundary conditions are

~ P 2 D - - 1 r D = 1: 8r~- - 1 + /Tk(1 ~ ' (47)

ro --* o~: i02D = 0. (48)

The Neumann expansion method (Adomian, 1983, 1986), used for solving this equation, requires the knowledge of the following deterministic functions:

(i) po(ro), which in the present case correspond to the well-known Warren and Root (1963) solution,

K ~ sx/s~~176 (49) po(r D, S) = s x / s f o(s)K t ( ~ ) ,

where Ko, K1 are the modified Bessel functions of zero order and first order respectively, and

(ii) the Green adjoint operator, solution to the differential equation

d e G 1 dG dr~ -I- rD dr D s fo(s)G = (~(rD -- O), (50)

8G ro = 1: ~ = o, (51)

ro ~ oe" G = 0. (52)

The analytical solution of the Green function G equation, achieved by classical techniques, is

wl(rD)w~(r G(rD, 0) = ; 1 <<, ro <~ 0, (53)

c(r

G(ro, 0) = Wl(t))w2(rD); rD <~ 0, c(O)

w2(O) = Ko(Z~), (54)

Ko ()~rd)11 (X) -- K t (Z)Io (zro) wl(ro) =

K I (z)

K I (Z) c(O) - z ~ ' (55)

where Z = x~o(S) . ~k(re) and f~(rD) are assumed independent zero mean one-dimensional random

function of the radial distance. Distributions of fracture permeability of fractured reservoirs of double porosity have not been investigated. Without any evidence of such distributions, in the present study we will assume normal distributions.


For each realization of the stochastic fields, on a cylinder of radius ro the values of the relative variation of permeability G(rD) are the same, while at different radial distances they are different. The same is also true for the relative variation values of

the porosity f~(ro).

3.1. INVESTIGATED CASES A N D RESULTS

As an example we present calculations of the pressure variation in time for the following values of 2 and co: (i) 2 = 10 -8 and co = 10-3; (ii) 2 = 10 2 and co = 10-3; (iii) 2 = 10 -s and co = 0.4.

In all cases we considered the relatively low coefficients of variations of reservoir fracture permeability and porosity of 0.15 and 0.30.

The parameter rov~ is a measure of the variation of the permeability with respect to the radial distance; small rov~ expresses drastic changes in permeability while large rov~ expresses smooth changes. The same is true for ro~, expressing the changes in porosity. In the first case we considered a correlation distance for permeability rov~ and porosity ro~ equals to 7.0rw while in the last two cases we considered correlation distances for permeability rovk and porosity rov~ equal to 0.05rw.

The calculated average fracture pressure of the reservoir with stochastic fracture permeability and porosity was compared with the Warren and Root (1963) solution for a deterministic ideal reservoir with constant properties.

The influence of the porosity variability was found to be negligible; no significant differences between the Warren and Root (1963) solution and the present one have been found.

The results of the calculations for the variation of pressure in time are presented in Figures 1, 2, and 3.

The difference between the Warren and Root (1963) solution and the present solution increases with the coefficient of variation cvk. However, after a relatively short period of time the flow is governed by the average reservoir properties. As G(x), representing the relative variation of permeability is considered a zero mean random fields, after some period of time the pressure reaches the asymptotic behavior corresponding to the Warren and Root (1963) deterministic solution.

Test results are, a priori valid only at the well, and it is of practical interest to determine the extent of the area over which the characteristic fractured reservoir parameters, determined by well tests, may be extrapolated.

The extent of this area may be determined by calculating the correlation between the fracture pressure at the well and at different points rD; note that for constant permeability and porosity, this area is infinite.

Figure 4 presents the autocorrelation function (p) vs. the distance from the well (ro) at different times, for 2 = 10 -2, co = 5 x 10 - 2 and rovk = 0.05rw.

For relatively low time values, i.e., during the first stages of the test, the results show a bad correlation between the fracture pressure at the well and at other

274

8

~r >;

~ 5 _/ ~ ~.~ = o .~5

cv~: = 0.00 (Warren and Root solution)

I I I I H [ i l I11111[ I I T l l H t [ I , , , , , . I

IO IO 2 IO 3 IO 4 Dimensionless tiIne tD

Fig. 1. D r a w d o w n pressurc var ia t ion in t ime for ,~ = 10 - s , e0 = 10 -3 and ro,~ = 7.0rw, as a func-

tion of cvk.

C. B R A E S T E R A N D D. G. Z E I T O U N

7

Also 6 , \ = 2 . 5 x i 0 -a t m d w = O /

Warren and Root solution (cvk = 0 ) / for a homogeneous r e s e r v o i r /

I0 10 2 I0 3 10 4 10 5

Dimensionless time to

Fig. 2. D r a w d o w n pressure var ia t ion in t ime for = 10 -2, co = 10 -3 and ro~. = 0.05rw as a func-

tion of cvk.

7 Alao A = 4 . 0 x l 0 - a a n d ~ a = l x I0 -~ /

6 Warrcn and Root solution (cv~ = 0 ) /

,o ;3 ~ 3

s 2 ~'X,'c,~ = 0.30

\ , ~ l~,ool, 8olutioll)

0 l I H I I I t [ [ l l [ I [ [ I I l l l l t l l I I ( t l l l l [ I I ~ I I I L l l

tO I0 2 I0 3 tO 4 I0 5

Dimensionless time tD

Fig. 3. D r a w d o w n pressure var ia t ion in t ime for 2 = 10 - s , e) = 0,4, and r0~ = 0.05r~, as a func-

tion of cv k.

I.O ~ tl) = 109

t~ = I0 ~

0 2. 4 6 8 I0 12 Dimensionless radius rD

Fig. 4. The au tocor re la t ion function vs. d imen-

sionless radius and time, for 2 = 10 -2, (9 =

5 x 10 -2 and roy ~ = 0.05r~,.

~. 0.8 I 0 .6

0 . 4 o

-<

0 .2

different points, while for large time values the pressures are fully correlated. This means that for the considered value ro~k = 0.05rw and for relatively small time values, extrapolation of the characteristic parameters of the inhomogeneous fractured rock reservoirs is limited to the narrow region around the well. For rovk = 7.0rw it was found that the pressures are practically correlated even for small time values.

4. Discuss ion and Conclusions

It should be pointed out that the purpose of this investigation is to investigate the behaviour of the inhomogeneous fractured reservoir of double porosity, and not to


provide type curves for determination of double-porosity reservoirs, which is

considered impossible because of the nonuniqueness of the solution.

To achieve a solution for radial flow we assumed an ideal reservoir with

cylindrical symmetric inhomogeneity around the well. This assumption, enables a solution for the pressure in the fractures, and a comparison with the Warren and Root (1963) solution for an ideal homogeneous double-porosity reservoir. The results

are consistent with the behaviour of the inhomogeneous reservoirs in general (Zeitoun and Braester, 1991).

The inspection of the pressure behavior in time of reservoir with inhomogeneous

fracture permeability reveals a similar behavior to that of a homogeneous reservoir with constant fracture permeability. However, the same curve (p, log t) is fitted with different parameters 2 and co, when attributed to homogeneous or to inhomogeneous

reservoirs. This is true even for the small coefficients of variation of fracture permeability considered, in the range of 0.15 to 0.30.

For example, the curve in Figure 2 corresponding to an inhomogeneous fractured reservoir with 2 = 10 -2, co = 10 -3 and cvk = 0.30 can be fitted by Warren and Root's (1963) solution corresponding to a homogeneous reservoir with 2 = 2.5 x 10 -3 and 09=0 .

Also the curve in Figure 3 corresponding to an inhomogeneous fractured reservoir with 2 = 10 -8, co = 0.4 and cvk = 0.30 can be fitted by the Warren and Root (1963) solution corresponding to a homogeneous reservoir with 2 = 4 x 10 -3 and o9 = 10 -2.

If we admit that the fractured reservoirs encountered in nature are inhomogeneous, then the routine of determining the fractured reservoir characteristic properties 2 and o9 by using the results of well tests, under the assumption of homo-

geneous fractured reservoirs, is ambiguous and therefore irrelevant. This is true even for the relatively small coefficients of variation for permeability, in the range of 0.15

to 0.30, considered in the present study (Figures 1 to 3). One of the purposes of the determination of the parameter 2 is to relate it to the

characteristic size of the fractured reservoir block. Representing the fractured

reservoir as a discrete medium of fractures and blocks in an alternating horizontal disposition, similar to Kazemi (1969), Braester (1984), showed that the block size is not sensitively reflected in the parameter 2 of the drawdown or build-up well tests. This study shows that determination of both 2 and o9 is ambiguous.

As is well known, the transition of pressure to the homogeneous behavior is relatively short, and not noticed except in a narrow region around the well. The

transient to the homogeneous behavior is not of interest for one phase flow and the only reason for the determination of the characteristic parameters of the fractured reservoir is to relate to the performances of two-phase flow reservoir.

The impossibility to relate the parameter 2 to the size of the block and the ambiguity in the determination of 2 and 09, as resulting from the present study makes questionable the usefulness of determination of these parameters using standard well test routines.


Appendix A: The Neumann Expansion Method

According to the Neumann expansion method (Adomian, 1983, 1986), the random differential equation is separated into a deterministic and a stochastic part, as follows

VZ/~zD -- sfo(s)~2u

= - - ~ k ( X D ) V 2 p 2 D "a t- SV&(XD)P2 D + V D ~ k" VDP2D. (56)

The considered boundary conditions, expressed in a general form are

- V~zo 'n i - /3 i (~2D -- Di) = Ri + 6kVo/~2D'nl on F~. (57)

Then, the solution of equation (eq:43) is written as a sum of undefined functions

+oo

= p , . (58) i = 0

Since the functions Pl i = 1, 2, ... are undefined, it is possible to make the following identification

For n = 0

V2po - sfo(s)po = 0, (59)

- Vpo'nl - fii(Po - Of) = Ri on Fi (60)

and for n ~> 1

vZf in - s fo(s) f i zD = - - ~ k ( X D ) V 2 f f n - 1 "J7 S~&(XD)I2n- 1 -~- VDVk~ (61)

subject to the boundary conditions

- V p . ' n i - fllP. = ~kVDp.-1"hi on Fi. (62)

The solution is calculated by recurrence, using the first term series solution of the deterministic function po(xD).

The calculation of p., n >~ 1, requires a solution of Equations (45) and (46), relating between p., n ~> 1 and the preceding term of the series p._ 1.

To obtain a solution to Equation (45) we write the solution in an integral form, as

follows

p.(x) = f v G(xD, ~)F(~k(~O), ~(~O), p ._l (~O), s) dV(~) +

+ fr G(xo, ~9)fkVDp.- 1" ni dF, (63)

where

F(ek(O), e4)(O), p . -1 (O), s)

-- ~k(O)V2p.-l(O) + SOo($)p.-a(O) + VD~k'VDp.-I(O), (64)


and G(xD, 0) represents the Green function associated with the differential operator V~, - sfo(s) and the boundary conditions of Equation (45).

G(xD, 0) is the solution of the following equation

V~G - sfo(s)G = 6(xD -- ~9), (65)

- V G ' n ~ + f l i G = O onFi (66)

The integral representation of the solution expressed by Equation (34) enables the determination of the order of magnitude of E{p,} which is O(a~k} or O(a~o) depending on the investigated parameter variability.

As a result, for the computation of the average fracture pressure, the series solution may be truncated. The order of magnitude of the truncated series depends on the expected accuracy and on the other of magnitude of the parameter variability.

However, considering normal random fields, the odd moments vanish and a truncation of the series after the second term corresponds to an order of magnitude of the neglected terms of O(a~), in case of permeability variability or O(av4~), in case of porosity variability (Lin, 1968).

In the following, the computation of the average fracture pressure and the second-order moments are performed for small variation of the parameters. In these computations, the terms in the order of magnitude of O(o-4~) or O(o'v%) are neglected. It can be shown that in this case

E{p2D(*) = po(X) + E{p2(x)} + O(~,) , (67)

E{(P2D(Xlo) -- E{p2D(XlD})(Pzo(XzD) -- E{ P2D(X2D})} (68)

4 = E{pl(XID)pa(X2D)} + O(o'~p), (69)

where O(a~p) equals O(av 4) or O(a~,), depending on the investigated parameter variability.

The calculation of these moments requires the computation of E{pz(x)} and E{ Pl (xlD)Pl (XzD)}.

The mathematical expressions of these two terms are derived in terms of Po and autocorrelation function in Appendix B, and are obtained from the general expression of pl(xD) and p2(xD).

Pa (x) = f v G(xD, O)F(~k(O), Ve,(O), Po(O), S) dV(O) +

+ f r G(xD, t~)~kVDpo'nl dF, (70)


p2(x) = fv 6(xD, 0)F(fk(0), ~,(0), P1(0), s) dV(0) +

+ f r G(xo, O)~kVDpl "nl dF. (71)

Appendix B: Formulae for the Radial Flow

B.1. THE AVERAGE FRACTURE PRESSURE

For radial flow the mathematical expression of the Laplace transform of the average fracture pressure at the wall may be written as follows

(a) For variation of the permeability only

n=6 2 E{!32o(ro)} = a~ ~ In(rD), (72)

n=l

where

Ii(rD) = s2( fo(s ) ) 2 G(ro, O)G(O, 02)P(0 - I//2)Po(02)dO d02, (73) d l

I2(rD) = S(fo(S))2 1 G(rD, 0)G(0, 02) (0 - 02) d 0 d 0 2 , (74) 1

I3(ro) = sa(fo(s)) G(rv, 0)G(0, 1.)p(0 - 1.) d0, (75) 1

14(ro) = sE(fo(s)) 1 1 G(ro, 01 (0, 02t (0 - 02)Po(021 d0 d02, (76)

I5(rD) = 1 a G(ro, O)~-r (O, 02)~Trr(O - 02) (021d02d02, (77)

/6(rD) = G(rD, 0)G(0, 1.) (0 - 1.) d0. (78) 1

(b) For the variation of the porosity only:

E{/~2o(ro)} = o~(J1 (rD)), (79)

where

Jl(ro) = s 2 G(rD, 0)G(0, 02)P(0 - 02) Po(02)d0 d02. (80)


B.2. SECOND-ORDER MOMENT OF THE FRACTURE PRESSURE

For radial flow the mathematical expression of the Laplace transform of the second order moments of the fracture pressure between two different points 01 and 02 may be written as follows:

(a) For the variation of the permeability only

2( )) E{P2v(rl)pzo(r2)} = ark Al(rl,rz) + 2.* A,(rl, r2) + A6(ri,r2), (81) n

where

Al(r,, r2)

= s2(fo(S)) 2 G(rl , 0)G(r2, 02)/9(0 - 02)Po(02)Po(0) dO d0e , (82) 1 1

A2 (rl, r2 )

A3(rl, r2) = -- s(fo(s)) a(rl , 0)G(r> 1.)p(0 - 1.)Po(0) dO, (84) 1

&(rl , r2)

; ~ ~ 1 d2pz't" 02) ? r ~ (02) ddPr~ (0) d0 d02 . (85) = 6(r~, 0 ) 6 ( r 2 , 0 2 ) U ; r ~ , -

As(r l , r2 )=f~G(r l ,0 )G(r2 ,1 . )~r (O-1 . )~r (O)d 0, (86)

A6(rl, r2) = G(rl, 1.) G(r2, 1.). (87)

(b) For the variation of the porosity only

E{~2D(rl)}~gD(r2)} = 2 a,r (rl, r2)), (88)

where

B1 (rl, r2) = s 2 G(rl, 0)G(r2, 02)0(0 - 02) P0(02) dO d0z. (89) 1 1

References

Adomian, G., 1986, Nonlinear Stochastic Operator Equations, Academic Press, New York. Adomian, G., 1983, Stochastic Systems, Academic Press, New York. Barenblatt, G. I., Zheltov, Yu. P., and Kochina, I. N., 1960, Basic concepts in the theory of seepage of

homogeneous liquids in fissured rocks, PrikL Mat. i Mekh. Soviet Appl. Math. Mech. 24(5), 852-64.


Barenblatt, G. I., 1963, On certain boundary-value problems for the equations of seepage of a liquid in fissured rocks, PriM. Mat. i Mekh. on some boundary-value problems for equations of fluid seepage in fractured rocks (in Russian), 27(2), 348-350.

Braester, C., 1984, Influence of block size on the transition curve for a drawdown test in a naturally fractured reservoir, Soc. Petrol Eng. J. 24(5), 498 504.

Da Prat, G., Cinco-Ley, H., and Ramey, H. J. Jr., 1981, Decline curve analysis using type curve for two porosity systems, Soc. Petrol En 9. J. 21(3), 354~62.

deSwaan, O. A., 1976, Analytic solutions for determining naturally fractured reservoir properties by well testing, Soc. Petrol Eng. J., 117-22.

Kazemi, H., 1969, Pressure transient analysis of naturally fractured reservoirs, with uniform fracture distribution, Trans. AIME Vol. 246, Soc. Petrol Eng. J., 451 62.

Kazemi, H., Seth, M. S., and Thomas, G. W., 1969, The Interpretation of interference tests in naturally fractured reservoirs with uniform fracture distribution, Soc. Petrol Eng. J., Trans. AIME, Vol. 246, 463-72.

Lin, Y. K., 1968, Probabilistic Theory of structural Dynamics, MacGraw-Hill, New York. Najurieta, H. L., 1976, A Theory for the pressure transient analysis in naturally fractured reservoirs, paper

SPE 6017 presented at the 1976 Annual Technical Conference and Exhibition, New Orleans, Oct. 3-6. Odeh, A.S., 1965, Unsteady-state behavior of naturally fractured reservoirs, Soc. Petrol Eng. J., Trans.

AIME Vol. 234, 60-64. Warren, J. E. and Root, P. J., 1963, The behavior of naturally fractured reservoirs, Soe. Petrol En 9. J.,

Trans. AIME Vol. 228, 245-55. Zeitoun, D. G. and Braester, C. B., 1991, A Neumann expansion approach to flow through heterogeneous

formations, Stochastic Hydrol. Hydraul. 5(3), 207-226.

Pressure transient response of stochastically heterogeneous fractured reservoirs

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