Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol21.Issue8.1998:

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Use of the Fourier–Laplace transform and of

diagrammatical methods to interpret pumping

tests in heterogeneous reservoirs

**Benoıˆt Nœtinger* & Yann Gautier**

Institut Français du Pe´trole, He´lioparc Pau-Pyreńeés, 2, Avenue Pierre Angot, 64000 Pau, France (Received 5 November 1996; accepted 13 May 1997)

Advances in computer power and in reservoir characterization allow simulation of pressure transients in complex reservoirs generated stochastically. Generally, interpretation of these transients gives useful information about the reservoir hydraulic properties: a major goal is to interpret these transients in the stochastic context. First we ensemble average the pressure over all the random permeability field realizations to derive an equation which drives the ensemble averaged pressure. We use the Fourier transform in space and the Laplace transform in time, in conjunction with a perturbation series expansion in successive powers of the permeability fluctuations to obtain an explicit solution. TheNth order term of this series involves the hydrodynamic interaction between N permeability heterogeneities and after averaging we obtain an expansion containing correlation functions of permeability fluctuations of increasing order.

Next, Feynman graphs are introduced allowing a more attractive graphical interpretation of the perturbation series. Then series summation techniques are employed to reduce the graph number to be summed at each order of the fluctuation expansion. This in turn gives useful physical insights on the homogenization processes involved. In particular, it is shown that the sum of the so-called ‘one-particle irreducible graphs’ gives the kernel of a linear integro-differential equation obeyed by the ensemble average pressure. All the information about the heterogeneity structure is contained in this renormalized kernel, which is a limited range function.

This equation on its own is the starting point of useful asymptotic results and approximations. In particular it is shown that interpretation of pumping tests yields the steady-state equivalent permeability after a sufficiently long time for an infinite reservoir, as expected.q_{1998 Elsevier Science Limited. All rights reserved.} Key words:averaging, Feynman graphs, Fourier–Laplace transform, heterogeneous medium, pumping tests.

NOMENCLATURE

ct total compressibility (Pa¹1)

D(r) diffusivity(k(r))=(Fmct)at pointr(m2s¹1)

D0 arithmetic average of the local diffusivityD(r) (m2s¹1)

Deff effective diffusivity (m2s¹1)

k(r) permeability at pointr(m2₎

lc correlation length (m)

r position vector (m) t time (s)

q Fourier parameter (m¹1

)

s Laplace parameter (s¹1)

P(r,t) time derivative of the pressure at pointrat timet (Pa-s¹1₎

Q rate at the well (per unit of length) (m2_s¹1₎

V(r,t) average local flow rate at pointrat timet (Pa-m s¹2₎

d(.) Dirac delta function

F porosity

m fluid viscosity (Pa-s)

j2 permeability variance (m4)

↔

(r,t) kernel of the integro-differential equation

P I I : S 0 3 0 9 - 1 7 0 8 ( 9 7 ) 0 0 0 1 4 - 6

581

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1 INTRODUCTION

Pressure transient analysis is a useful tool to characterize subsurface properties at a scale larger than core plugs (see Ref.1). It helps engineers gain information about reservoir volume, permeability and about the connectivity between two different wells, both in an oil industry or hydrogeology context. The basic principle of a well test is to record the pressure variations at some wells given that one fluid (oil, water, etc.) is pumped into another well at a given flow rate. As the pressure obeys a diffusion equation (direct problem), known analytical solutions can be used to match the model parameters with the observed data (inverse problem). Many commercial software packages use sophisticated non-linear regression procedures to achieve this task.

The increasing interest in geostatistical modelling of heterogeneous reservoirs allowing the generation of equi-probable possible reservoir images requires elaboration of new interpretation methods for pressure transients. Two important problems must be examined. First, as the local permeability has large and complex spatial variations, no exact analytical solutions are known. As the direct problem has no simple solution, it seems pointless to devise inversion procedures. The second problem is of a more conceptual nature: what is the information provided by a well test in a geostatistical context and how can we include it in a geo-statistical description? Using a probabilistic approach implies, of course, a rather important change in the inter-pretation methods.

The answer to this last question depends on the degree of external information available (of geological nature for example). In the case where the local permeability mean and covariance function are well known and fixed, we would like to generate well test conditioned equiprobable images, i.e. if we perform a pressure transient test on any of these images, we would like to match the observed data. This problem was addressed by some authors2–4, who have used fast approximate solutions to the direct problem, as well as simulated annealing techniques to generate con-strained images. In other cases, geostatistical parameters are not well known, and we would expect to fix them using pressure transient analysis, leading to a sort of ‘secondary inversion’ problem.

In both cases, the approximate solution to the direct prob-lem must be found, by answering the following:

1. At a given timet, what is the relationship between the pressure response and the local permeability map? Do characteristic times and lengths exist?

2. It has been observed that many heterogeneous reservoirs have a homogeneous-like response5: does the well test ‘self average’ the reservoir and how can we quantify this process? Is it possible to define an apparent equivalent permeability of the medium and its relationship with the steady-state equivalent permeability?

To address both problems, we will employ a perturbation

formalism, seeking the form of the equation driving the averaged pressure (i.e. pressure averaged over all the per-meability map realizations). We will obtain an integro-differential linear equation characterized by a memory kernel which depends on the permeability map covariance function. In the Fourier–Laplace space, the transformation of this kernel can be interpreted as a wave-vector and Laplace parameter-dependant diffusion coefficient. To procure this, we will use the Fourier–Laplace transform as well as Feynman graph sumation techniques (the Dyson equation), following a technique proposed by King6 and Christakos7. It will be shown that the kernel of this integro-differential equation is the sum of specific functions of the many body covariance of the permeability field. An interesting fact is that for media characterized by a finite correlation length, this kernel appears to have the same limited spatial range, a physically appealing result. This property gives an understanding of how the integro-differential equation degenerates into a diffusion-like equation for long times and large length-scales, giving rise to a single effective diffusion coefficient which may be identified with the steady-state value.

To obtain explicit results, the preceding theory is truncated at the second order, and classical formulae are recovered: the equivalent diffusivity being equal to the geo-metric mean of the local values up to this order. The large wave-vector behaviour of the kernel is examined and it is shown that in this limit the equivalent diffusivity tends towards the harmonic average of the local values. This result can be explained by a simple physical interpretation, in connection with the previously studied problem of defin-ing an apparent equivalent permeability in radial flows near wells8. Finally, returning to the conventional interpretation of transient well tests, we show that the steady-state equiva-lent permeability is recovered for long time tests.

This paper is organized as follows. In the next section, we give the basic hypothesis notations and equations. Next, the Fourier–Laplace transform is presented, along with the homogeneous solutions and the use of so-called pressure moments. Then, we present a perturbation expansion method, and the averaging scheme. Next, Feynman graphs are introduced allowing us to recast series expansion in another form which yields the desired mean equation directly. We perform a second-order approximation and we show how the steady-state equivalent permeability may be recovered. After that, we illustrate how classical interpretation of well tests about heterogeneous and iso-tropic reservoirs gives rise to the steady-state equivalent permeability of the reservoir.

2 BASIC EQUATIONS, NOTATIONS AND DEFINITIONS

We consider a two-dimensional heterogeneous reservoir produced by a single well located at the coordinate origin. Under classical hypothesis, the equation driving the

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pressure variations is given by1,9,10: ]_P(r,t)

]_t ¼=·(D(r)=P(r,t))þ Q

Fct

d(r)d(t) (1) Here,D(r)¼k(r)/fmctdenotes the diffusivity.Qrepresents

the fluid flow rate per unit of time and per unit of well length and the two delta functions represent the well. The parametersk(r),f,m andctdenote, respectively, the rock

permeability, porosity and the fluid viscosity and compres-sibility. The porosity is assumed to be constant. Using a spatial delta function for the source term, we assume implicitly that the well radius is negligible: such an approximation is quite correct for time-scales greater than typical diffusion times over the well radius. Note that the delta function with respect to time is the derivative of the Heaviside function. This implies that the pressure P(r, t) considered in the present work corresponds to the time derivative of the pressure considered in standard tests9. Its SI unit is thus Pa-s¹1.

To solve this evolution problem, we choose an initial condition given by P(r,t ¼0) ¼0, and we assume that the pressure vanishes at infinity for all timest. In this case, a unique solution exists. As the time Dirac functiond(t) is the derivative of the Heaviside function, we see that we are working directly with the so-called pressure derivative, of great use in well test interpretation methods. A typical field measurement is the valueP(r¼0,t), from which we would like to obtain information on the diffusivityD(r).

Here, the diffusivity D(r) is assumed to be a random function and our first goal is to find an equation obeyed by the mean pressure field hP(r, t)i, where the averaging symbolh…imeans an ensemble averaging over the realiza-tions of the random funcrealiza-tionsD(r) weighted by their prob-ability measure.

3 FOURIER–LAPLACE TRANSFORM (FLT) This transformation is useful in obtaining simple results. It is defined by

P(q,s)¼

0 dt exp(¹st)

dreiq:rP(r,t) (2) The properties of this transformation are well known. To simplify notation, arguments (r,t) are reserved for the real space–time representation, and (q,s) for the FLT. In some cases, we will use only the Fourier (or Laplace) part of the transform, and again the notation convention will give the chosen transformation. In particular, for the homogeneous case with a constant diffusion coefficientD0, the FLT of the

diffusion equation withQ=fct¼1 gives P0(q,s)¼

1 sþD0q2

(3) One should note that P0(.,.) corresponds to the Green’s

function of the diffusion problem. In the space and time

domain we obtain the well-known formula, which is the basis of well test interpretation:

P0(r¼0,t)¼ Q

4pD0Fct3t

(4) It is useful to introduce the so-called ‘pressure moments’10 which arenth order tensors by

mn(t)¼ Z

dr rnP(r,t) (5)

Using the FLT and expanding the space-dependant expo-nential in series, we obtain the following results:

m_m(t)¼in] n_P₍_q_,_t₎

]qn

_q_¼₀

andm_m(s)¼in] n_P₍_q_,_s₎

]qn

_q_¼₀

(6) Note that without the Laplace transform, one can obtain the moments using the same scheme directly in the time domain. Of particular interest is the second moment (a second-order tensor) which gives an estimation of the size of the investigated zone10. Considering the homo-geneous case, one produces

m₂(t)¼2D01t (7)

This is a useful and well-known result (see Ref. 10 for a derivation).

4 THE PERTURBATION EXPANSION AND SOME PRELIMINARY RESULTS

In order to set up a perturbation expansion scheme, let us write the original diffusion problem as follows:

]_P(r,t)

]_t ¼=·((D0þjdf(r))=P(r,t))þ Q

fct

d(r)d(t) (8)

Here,jis the mean square deviation of the local diffusivity and the random functiondf(r) is assumed to be stationary, of zero mean, and to have known statistical properties. To simplify the analysis, we chose Q/fct¼1. Assuming that

the solution to this equation can be written as a power series P(r,t)¼ S

N¼0P

N₍_r_,

t) where PN is of order jN, we solve the diffusion equation by iteration giving rise to an infinite hierarchy of diffusion problems defined by

]_PNþ1(_r,_t)

]_t ¼=·(D0=P Nþ1₍_r_,

t))þj=_·(_d_f(_r)=_PN(_r,_t))

(9) The formal solution to this equation yields the (Nþ 1)th order term by means of the formula

PNþ1(r,s)¼j

(4)

and thus by induction we get PN(r,s)¼jN

Z dr1

Z dr2…

dr_N3 =_P0(_r_¹_r1,_s)_d_f(_r1): ==_P0(_r₁_¹_r2,_s)_d_f(_r2)…_d_f(_r

N¹1): ==_P0(_r

N¹1¹rN,s)df(rN):=P0(rN,s) ð10Þ TheNth order term gives the effect ofNdiffusivity fluctua-tions atNpoints. Computing the average of thisNth order term requires the expressions of the averages of the products jN,_d_f(_r1)_d_f(_r2)…_d_f(_r

N). which are the N -body correlators G(r1, r2,…, rN). Using the stationary

hypothesis, we can write the following equality: G(r1,

r₂;…;r_NÞ ¼G(r2¹r1;…;rN¹r1Þ.

Eqn (10) can be thus transformed as follows by integrat-ing (N¹2) times over the intermediate integration variables (r2,…,rN¹1):

PN(r,s)

¼jN

Z dr₁

dr_N=_P₀(r¹r₁,s): Z

dr2…dr_N¹1G(r2¹r1, …,rN¹1

¹r₁,r_N¹r₁)==_P₀(r₁¹r₂,s)): ==_P

0(r2¹r3,s))…==P0(rN¹2¹rN¹1,s)): ==_P0(_r

N¹1¹rN,s))ÿ:=P0(rN,s)

(11) Shifting all the intermediate integration variables (r2,…,

rN¹1) of a quantityr1, we obtain an expression of the form

PN(r,s)

¼jN

Z dr₁

dr_N=_P₀(r¹r₁,s):H

↔

N_:

(r1¹rN,s):=P0(rN,s)

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We recognize two successive convolution products. As this expression is valid forN._{1, and by introducing the FLT}

to transform convolution products, one can sum the whole series, resulting in the following:

h i(q,s)¼ 1

sþD0q2

¹ q

sþD0q2

:,_H↔ .(_q,_s): q

sþD0q2

(13) the second-order tensorH↔ being defined as:

↔

(r1¹rn,s)¼ X` N¼2

jNH

↔

N₍_r

1¹rn,s) (14) After averaging, the whole effect of the heterogeneities will be contained in the averagedhH↔isecond-order tensor.

It is useful to note that, using eqns (6) and (13), the average second moment can be given by

₍

s)¼ ¹ ] 2

h i(q,s) ]q2

_q_¼₀

¼2 D0

s2 1 ↔

H↔(q¼0,_s)

s2 0

B @

C A

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If H↔(q¼0,_s_¼₀) _{has a well-defined limit when} _s _goes

to zero, we obtain, in the time domain, an Einstein-like relation:

,m₂.(_t)¼2(D01tþh iH ↔

(q¼0,_s_¼₀)_t) ₍₁₆₎

Comparing with the homogeneous medium result [eqn (7)], we may define an effective diffusion tensorDeffof

D_eff¼D01þh iH ↔

(q¼0,_s_¼₀) ₍₁₇₎

Permitting the Laplace parameter sapproach 0 is equiva-lent to computing the steady-state limit. It is interesting to check that hH↔i(q¼0,_s_¼₀) _{may be identified term by}

term with the perturbation series arising from classical steady-state equivalent permeability calculations11. This means that the steady-state equivalent permeability emerges quite naturally as a long-time limit of a transient analysis.

5 AVERAGING PROCEDURE AND FEYNMAN GRAPHS

So far, we have obtained a systematic scheme to express the average solution of the random diffusion problem, but what we really require is an expression for an effective equation driving this average solution. Eqn (13) gives the FLT of Green’s function of this equation, so it would appear suf-ficient to compute the inverse FLT to obtain this master equation, as was proposed by Indelmann12. In this section, we will give an explicit direct expression of this equation using summation techniques that are commonly used in particle physics. Such methods were first introduced in our domain by King6 and Christakos and coworkers7,13, in a steady-state context. These methods are very interesting to define up-scaled parameters and to understand the averaging process of diffusion. In particular, this will allow us to identify some long-time apparent parameters to their steady-state value. Another interesting aspect of this formu-lation is that generally the equation driving the problem is more local than the solution, so boundary condition effects are more easily accounted for (as an example, the diffusion equation is local, while its Green’s function is not!).

To proceed, let us write theNth order correlation function under the form of a cumulant expansion:

GN(r₁,r₂, …,r_N)¼GNc(r1,r2, …,rN)

þ X

∪Ia¼I

Y a

GCard(c Ia)(Ia)

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The index ‘c’ indicates connected correlation functions. I denotes N points {r1, r1,…, rN} and Ia denotes any partition of this set in a parts. In the gaussian case, all cumulants having N . _{2 are equal to 0 (see for example}

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given by

G1c(r1)¼G1(r1)¼0(because,df(r). ¼0),

G2c(r1,r2)¼G2(r1,r2), G3c(r1,r2,r3)¼G3(r1,r2,r3),

G4c(r1,r2,r3,r4)¼G4(r1,r2,r3,r4)

¹G2c(r1,r2)G2c(r3,r4)¹G2c(r1,r3)G2c(r2,r4)

¹G2c(r1,r4)G2c(r2,r3) ð19Þ

The cumulant expansion is nothing more than a systematic scheme for isolating the specific correlation between N points from products of lower-order correlations. Returning to our series expansion, and inserting eqn (18), the original Nth order term of the series expansion ofH↔will break down each term into several elements containing products of con-nected correlation functions. We now obtain a new series expansion which can be written graphically using Feynman graphs by

These graphs are only a more attractive way of writing the series expansion using the following rules:

1. Each white circle Wcorresponds to the two extreme pointsr1andrN.

2. Each black circleXcorresponds to one amongN¹2 intermediate dummy integration variables r2torN¹2,

the points being ordered from left to right.

3. Each arrow is a ‘propagator’==_P₀, applied to the lag vector between two neighbouring points. 4. Each wavy line corresponds to a connected

correlation function of theWorX points linked by these lines.

5. Finally, the large black circlesXindicate that we are dealing with a pth order connected correlation func-tion (or cumulant), where p is the number of wavy lines that intercept the circle. This function depends on the p position vectors of the W or X points involved.

Note that due to the Laplace transform with respect to the time argument, no time integration is required.

By definition, whenN¼2, we have no intermediate black integration points; in particular the graph is given by

Another example is the graph which

corresponds to the following integral in Laplace

representation: Z

dr2

dr33G2c(r1¹r2)==P0(r1¹r2,s): ==_P0(r₂¹r3,s):==_P0(r₃¹r4,s)G2c(r₃¹r4)

One should be careful with the role of the contraction points (.) between tensors.

In eqn (20), the last four graphs correspond to the fourth-order term of the original series expansion, broken down into four elements due to the expression of the fourth-order correlation function [eqn (19)]. The first three graphs correspond in fact to lower-order correlations, and the last graph to the fourth-order cumulant.

6 SUMMATION OF THE IRREDUCIBLE GRAPHS AND DERIVATION OF AN EFFECTIVE EQUATION It can be noted that among all these graphs, some of them

can be broken down into two separate parts by cutting one arrow. These graphs may be expressed in a very simple manner as a function of lower-order graphs. For example, consider the last example of graph . Using its expression given in the previous section, we obtain

Z dr₂

dr₃3_G2_c(r₁¹r₂)==_P₀(r₁¹r₂,s): ==_P₀(r₂¹r₃,s):==_P₀(r₃¹r₄,s)G2c(r3¹r4)

Z dr₂

dr₃3_G2_c(¹r₂)==_P₀(¹r₂,s): ==_P₀(r₂¹r₃,s):

==_P0(r₃¹(r₄¹r1),s)G2c(r₃¹(r₄¹r1))

after shifting the integration variabler2andr3by a quantity

r1. CallingF(r2¹r1,s) the value of the graph

given by G2c(r1¹r2)==P0(r1¹r2,s)becomes

This is a function of the argumentR¼r4¹r1and under

this form, we recognize a convolution product of three functions. Thus the FLT of this function appears to be the product of two lower-order graphs ( ) FLT, times an extra factor. This factor is the Fourier transform of ==_P₀(r, s), (¹qq=(sþD0q2)) (corresponding in this

(20)

¼G2c(r1¹r2)==P0(r1¹r2,s) (21)

Z dr₂

dr₃3F(¹r₂)

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example to the arrow linking points 2 and 3). One could check that it is again the case for any graph that can be broken down into two parts by cutting one single arrow: the proof follows exactly the same reasoning. It is sufficient to replace the two extreme graphs by any other more complex graph. It would be appealing to remove these factors ==_P₀_(r, _s_{) as their FLT are singular in the limit}

s¼0, implying an 1/rbehaviour in real space giving rise to important boundary conditions effects.

To circumvent this problem, and to reduce the number of graphs to be computed, it is classical practice in field theory to define a new kernelS↔(r1¹rN,s) by the following

definition:S↔(r1¹rN,s) represent the sum of all ‘1P

irre-ducible graphs’ (i.e. the graphs that cannot be broken down into two parts by splitting a single arrow). Graphically, we obtain

Readers familiar with field theoretical methods will recog-nize the ‘self-energy’, or ‘mass operator’ inS

↔

typical range equal tolcin the space domain, and a typical

diffusion time over one correlation length equal tol2c=D0 in

the time domain.

We may now derive the useful identity, also classical in the field theoretical framework (Dyson equality):

H↔

D E

(q,s)¼X ↔

(q,s): ↔1 þ 1

sþD0q2

qq:X ↔

(q,s)

¹1

ð23Þ

This identity may be derived by formally expanding the inverse of the operator on the right-hand side of the above equality in a geometric series (1 þ x)¹1

¼ 1 ¹

xþx2¹x3þ…þ.:

X↔

(q,s): ↔1 þ 1

sþD0q2

qq:X ↔

(q,s)

¹1

¼X

↔

(q,s)þX ↔

(q,s): ¹1

sþD0q2

qq:X ↔

(q,s)

þX

↔

(q,s): ¹1

sþD0q2

qq:X ↔

(q,s) ¹1 sþD0q2

:qq:X ↔

(q,s)

þ…þX

↔

(q,s):…: ¹1

sþD0q2

qq:X ↔

(q,s)

þ…(the product: ¹1

sþD0q2

qq:X ↔

(q,s)entering N times)

Using the graphical definition of the kernelS↔, one can see that the first term of the second line of this equality gives the sum of all the 1P irreducible graphs, the second term gives the sum of all the graphs ofhH↔ithat may be cut into only two 1P irreducible parts. More generally, the Nth order term gives the contribution of the graphs arising in the expansion ofhH↔ithat may be broken down into exactly N 1P irreducible parts. All the possible permutations will be recovered once. Considering the whole sum, we can see that all the original graphs of the series defining hH↔i are recovered. This yields eqn (23).

Using eqns (13) and (23), after simplification, one obtains hPi(q,s)¼ 1

sþD0q2þq:

X↔

(q,s):q

(24)

or, equivalently, sþD0q2þq:

X↔

(q,s):q

hPi(q,s)¼1

This is the desired equation obeyed by the average pressure hPi(r,t). By analogy with the homogeneous case, one can observe that in the FLT space, a variable apparent diffusion coefficient equal to D01þS

↔

(q,s) can be introduced. Returning to the space–time representation, and consider-ing the rate factor Q/fct, one obtains the following form:

]_{h i}_P (r,t)

]_t ¼=·(D0=h iP (r,t))þ Zt

0dt

=_r: Z

dr9X

↔

(r9,_t9):=_r₉h i_P (r¹r9,_t¹t9)

þ Q

Fc_td(r)d(t)

(25) The subscript in the gradient operator indicates the variable with respect to which the gradient is evaluated when there is a possible ambiguity. This equation has the form of an integro-differential equation with a memory kernel

S ↔

(r9,_t9). This memory kernel depends only on the perme-ability connected correlation functions of any order and can be computed in a systematic way using eqn (22). This for-mulation of the problem in terms of an effective equation is more adapted to obtain additionnal approximations, parti-cularly in the long-time, long distance limit than the first form given by eqn (13).

In particular, in typical cases the memory kernel↔S(r9,_t9) must be quite a localized function of spatial rangelc, as all

the large range factors decreasing slowly at infinity are (22)

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removed by the 1P irreducibility condition. Due to the diffusive nature of the original equation, we can expect that for large times and large lengthscales, the mean pres-sure will have typical variations over lengthscales given by eqn (16) very much larger than ↔S(r9,_t9)_{. We can} thus approximate =hPi(r ¹ r9_{, t} _¹ _t9_{) by} =hPi(r, t) in eqn (25) under the integral sign, producing the following approximation:

]h i_P(r,t)

]_t ¼=·(D0=h iP(r,t))

þ=

r: X↔

(q¼0,_s_¼₀):=

rh iP (r,t)

þ Q

Fct

d(r)d(t)

(26)

This equation is a diffusion-like equation corresponding to a large time and large distance approximation of the origi-nal integro-differential equation. Once again, it appears an effective diffusion tensor equal to

D_eff¼D01þ

X↔

(q¼0,_s_¼₀)_¼_D

þhH

↔

i(q¼0,_s_¼₀)

(27)

We used eqn (23) to obtain the last equality, which shows that this result is consistent with eqns (16) and (17) of the effective diffusivity tensor obtained using the second-moment approach. Such an equality could not be easily obtained without using eqn (25).

Another interesting physical feature of eqn (25) is that the vectorial quantity V(r, t) defined by V(r,t)¼

¹D0=r9h iP(r,t)¹ Rt

0 dt R

dr9_S↔(_r9,_t9):=

r9h iP(r¹r9,t¹t9)

can itself be interpreted as the average flux vector at pointr at timet12. This corresponds to a generalized form of Dar-cy’s law in which the local flow rate is a linear function operating on the whole pressure gradient field variations. Using this interpretation, eqn (25) appears to be no more than the usual transient mass balance equation associated with this flux, i.e.

]_{h i}_P (_r,_t)

]_t ¼ ¹=r:V(r,t)þ Q

Fct

d(r)d(t)

The last approximation leading to eqn (26) thus becomes very easy to interpret using the preceding expression of the fluxV(r,t).

Another interesting property of the kernelS ↔

was derived for an infinite medium. The preceeding results and the general framework which we have set up show that this affirmation is probably correct. A rigorous derivation would certainly involve a considerable amount of work.

Analogous conclusions were given by Rubin and Dagan by means of a second-order perturbation expansion15.

7 A SECOND-ORDER CALCULATION

When computing the S↔kernel up to the orderj2, we only need compute the value of the graph . We obtain

X↔

(r9,_t9)j2==_P₀(r9,_t9)_C(r9) (28) where j2C(r9_{) denotes the covariance function of the} permeability field which is assumed to be isotropic, with C(r9_¼₀₎_¼_{1. Computing the FLT of} S↔, forq¼0 and s¼0, using Parseval’s equality and the isotropy hypothesis, in two dimensions, we obtained

X↔

(q¼0,₀)_{¼ ¹} 1

2D0

j21 ₍₂₉₎

Using eqn (21), the effective diffusion tensor is thus given by

D_eff¼D01¹

1 2D0

j21 ₍₃₀₎

This is a well-known result, as it corresponds to the steady-state second-order expression for the effective permeability of a heterogeneous stationary and isotropic medium11,16.

More generally, a wave vector dependant effective dif-fusivity may be introduced by the definitionDeff(q,s¼0)¼

D01þS

↔

(q,s¼0) which corresponds to the static limit. For large wave vectorsq, one can compute (q.S↔ (q, s)·q) using eqn (28) and the convolution theorem. We obtain

lim q→`q:

X↔

(q,s):q¼ ¹ 1

j2q2 (31)

the diffusion equation behaves thus locally as p(r)¼mQ=(2pk(r¼0))ln(r). After averaging, we get

,_p(_r). ¼mQ=(2pkh)ln(r) wherekh denotes the

harmo-nic averaging of the permeability kh¼ ,k(r¼0)

¹1_.¹1_.

Applying the Fourier Transform to the expression ofhp(r)i for large wave vectorsq yields a result which agrees with eqn (31). This result should be compared with previous results of Matheron, who showed that for steady-state radial flow, the definition of an equivalent permeability is ambiguous, depending on the ratio of the radius of the well with respect to the medium typical size. This illustrates that caution is required when computing some limiting behaviours.

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8 CONCERNING THE APPARENT

PERMEABILITY GIVEN BY WELL TESTS

We will now take a look at a more classical analysis of well tests. In the homogeneous case, it is recognized that well tests are interpreted according to the following formula:

P0(r¼0,t)¼

Q 4pD0fmct3t

(32) Let us now assume that the reservoir is of infinite extent, heterogeneous and of stationary and isotropic heterogene-ities. Our present goal is to show that for large times, we have the following behaviour:

P0(r¼0,t)¼

Q 4pDefffct3t

(33) where the equivalent diffusivity Deff is as defined in

Sections 4, and 6. It is thus straightforward to procure an effective permeability through the relationkeff¼fmct.Deff

Here we will provide an heuristic proof. To begin with, the basic idea is to study the behaviour of the ensemble average pressurehP(r¼0,t)i. The difference with the pre-viously given formulation is the averaging symbol: well tests are performed in one realization so the aim is to con-sider that the pressure diffusion takes into account the full set of heterogeneities in a single realization for sufficiently long times. This regime is likely to occur when the investi-gation radius—defined using the moment approach—is larger than the correlation length.

It would be thus sufficient to invert eqn (24) inr¼0 to obtain the expression ofhP(r¼0,t)i, but the 1/tsingularity gives rise to a diverging integral in the Fourier inversion. To circumvent it, we compute the Fourier–Laplace transform ofthP(r¼0,t)iwhich is given by ¹]_h_P(_r_¼₀,_s)_i=]_s_{. As we} are seeking the long-time behaviour of this function and as we expect a constant, it is most logical to examine the quantity

lim s→0 s

3 ]P(r¼0,s)

]_s

¼(4pDwell test)¹1¼(Deff)¹1 (34) Using the inverse Fourier transform, we must try to evaluate

s]P(r¼0,s) ]_s ¼

s 4p2

Z dq

(sþD0q2þq:

X↔

(q,s):q)2

q· ]X

↔ (q,s) ]_s ·q

(sþD0q2þq:

X↔

(q,s):q)2

2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 (35)

It would be appealing to replace↔S(q,s) byS↔(q¼0,s¼0), and to compute the desired limit directly, but the integrals are not absolutely convergent and somewhat more calcula-tion is needed. Rescaling the wave vector q by q¼qps=(D0)q9_{, we get a clearer form for the first term} of the preceding equation:

s 4p2

dq 1

(sþD0q2þq:

X↔

(q,s):q)2

¼ 1

4p2

dq 1

(1þq2_þ_q_:

X↔

s D0 r

q,s

:q)2

(36)

Now, we can safely consider the limit s ¼ 0 under the integral sign, as the integral is absolutely convergent for largeq wave vectors. We thus obtain

D0 lim

s→0

1 4p2

dq 1

1þq2_þ_q_:

X↔

s D0 r

q,s

D0 :q 0 B B B @ 1 C C C A 2 0 B B B B B B B B B B B B B @ 1 C C C C C C C C C C C C C A ¼ 1

4p2

dq 1

1þq2þq: X↔

(0,0) D0 :q 0 B B B @ 1 C C C A 2 (37) AsS ↔

(q¼0,s¼0) is isotropic, after integration, we pro-duce the desired result:

D_{well test}¼D01þ

X↔

(q¼0,_s_¼₀) ₍₃₈₎

Now, we must show that the limit of the second term of the right-hand side I2of eqn (35) is equal to zero in the lows

limit. To achieve this task, we will explicitly use the second-order expansion of the↔S(q,s) kernel whose expres-sion in the real space is↔S(r9,_t9)_¼_j2==_P

0(r9,t9)C(r9), and

its FLT is given by X↔

(q,s)¼ ¹j2

dq9q9q9 1 sþD0q92

C(q¹q9) (39)

Using this result, along with the explicit form of the second term, we get the following expression, after a rescaling of the wave vectors qandq9_{by the factor} _s=D0

(9)

I2¼

s 4pD0

Z dq

dq9qq:q9q9

3 1

1þq2þq: X↔

(

s=D0 p

q,s) D0

B B B @

C C C A 2

3_C( _s=D0 p

(q¹q9)) 1

(1þq92₎2

(40)

q þ q0_{. The integration over} _q _{is thus feasible and after} tedious calculation, we are led to evaluate the limit for smallsof the following quantity:

I2¼s

0 dq f(q)C(

s=D0 p

in which the functionf(q) is regular near zero, and behaves as 1/qfor large q.

Assuming that the correlation functionC(r) has a limited rangelcin real space, we can restrict theqintegration over

wave-vectors of modulus such thatq,p_D₀=s=lc. With this

natural cut-off, it may be shown that the integral behaves as sln(s) whensgoes to zero.

This means that, up to this order, the well test determined permeability coincides with the stationary value for a suf-ficiently long time test. It must be stressed that the limit is reached quite slowly, and that the well test interpretation method, in practice, could yield an apparent value quite different from the theoretical one.

9 DISCUSSION AND CONCLUSIONS

Using a systematic perturbation expansion and Feynman graphs, we derived an exact integro-differential equation which drives the average pressure in stationary hetero-geneous reservoirs. This equation is characterized by a memory kernel↔S(r,t) which contains all the information about the heterogeneities’ covariance functions. Such an equation was first recognized by Cushman and Ginn17, Neuman and Orr18, Christakos and coworkers7,13, and was also recently proposed by Indelman12, but this latter author did not give the interpretation of the memory kernel in terms of irreducible graphs. An important factor is the limited range of the memory kernel induced by the limited range of the permeability correlations, which allows us to show that at long times, the effective equation driving the mean pressure is a diffusion-like equation. In this case, the steady-state equivalent permeability of the medium is recovered. As the S↔(r, t) kernel is quite local, boundary conditions effects must be important only at a typical distance lc

from the boundaries, implying a robustness of the present approach when considering real finite size problems.

To obtain a better understanding of the homogenization effects and of the emergence of a simplified description by means of an effective equation, it would be useful to com-pute the covariance functions between local pressure varia-tions, i.e. the quantities _hP(r,t)P(r9,_t)_{i ¹}_h_P(_r,_t)_i_h_P(_r9,_t)_i_. Due to ergodicity effects, such quantities could vanish for sufficiently long times. This would imply that the average pressure which was the subject of this paper will be almost surely observable for sufficiently long times. This would provide better knowledge of the homogenization process performed by well testing and would give insights about the characteristic time scales and lengthscales. This study would help us to complete the proof of the convergence of the well test apparent permeability towards its steady-state value. King6 and Christakoset al.7 showed that diagram-matical methods can handle such tasks. We expect that for an infinite reservoir, this variance would tend to zero for large times t. This implies that at long times the average description becomes ‘almost sure’, giving us a dynamical image of ergodicity effects. This work, involving rather tedious calculations, is currently in progress.

Finally, we assumed thatS↔(r,t) had a simple behaviour in the long time–long distance, i.e. that the limiting behaviour in the FLT domainS↔(r,t)(q¼0,s¼0) was well defined. This hypothesis should break down in the case of fractal-like medium having correlations at all scales. Our formal-ism, used in conjunction with renormalization-group meth-ods could lead to the derivation of scaling laws for the variations in the effective permeability.

ACKNOWLEDGEMENTS

The referees are gratefully acknowledged for their useful comments.

REFERENCES

1. Horne, R. N., Modern Well Test Analysis. Petroway, Palo Alto 1990.

2. Deutsch, C. V. and Journel, A. G., Annealing techniques applied to the integration of geological and engineering data. Report for the Stanford Center for Reservoir Forecast-ing, Stanford University, 1992.

3. Feitosa, G.S., Lifu, Chu, Thompson, L.G. & Reynolds, A.C. Determination of permeability distribution from well test pressure data.SPE, 1993,26407.

4. Sagar, R. K., Reservoir description by integration of well test data and spatial statistics. Ph.D. Report, University of Tulsa, 1993.

5. Butler, J.J. Jr Pumping tests in nonuniform aquifers: the radially symmetric case. Journal of Hydrology, 1988, 101 15–30.

6. King, P. The use of field theoretic methods for the study of flow in a heterogeneous porous medium.Journal of Physics A: Mathematics Generalized, 1987,203935–3947. 7. Christakos, G., Miller, C.T. & Oliver, D. Stochastic

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perturbation analysis of groundwater flow. spatially variable soils, semi infinite domains and large fluctuations.Stochastic Hydrology and Hydraulics, 1993,7(3) 213–239.

8. Matheron, G. Composition des perme´abilite´s en milieu poreux he´te´roge`ne: me´thode de schwydler et re`gles de ponde´ration.Revue de l’Institut Franc¸ais du Pe´trole, 1967, 22(3) 443–466.

9. Daviau Interpe´tetion des enais de puits: les me´thodes nou-velles. Editions Technip, Paris, 1986.

10. Nœtinger, B. A pressure moment approach for helping pres-sure transient analysis in complex heterogeneous reservoirs. SPE, 1993,26 466.

11. Dagan, G., Flow and Transport in Porous Formations. Springer, New York, 1989.

12. Indelman, P. Averaging of unsteady flows in heterogeneous media of stationary conductivity. Journal of Fluid Mechanics, 1996,31039–61.

13. Christakos, G., Hristopulos, D.T. & Miller, C.T. Stochastic diagrammatic analysis of groundwater flow in heterogeneous

porous media.Water Resources Research, 1995,31(7) 1687– 1703.

14. Amit, J.,Field Theory, the Renormalization Group and Cri-tical Phenomena. World Scientific, Singapore, 1984. 15. Rubin, Y. & Dagan, G. Stochastic analysis of boundaries

effects on head spatial variability in heterogeneous aquifers 1. Constant head boundary.Water Resources Research, 1988, 24(10) 1689–1697.

16. Nœtinger, B. The effective permeability of an heterogeneous porous medium. Transport in Porous Media, 1994,1599– 127.

17. Cushman, J.H. & Ginn, T.R. Nonlocal dispersion in porous media with continuously evolving scale of heterogeneity. Transport in Porous Media, 1993,13(1) 123–138.

18. Neuman, S.P. & Orr, S. Prediction of steady-state flow in nonuniform geologic media by conditional moments: exact nonlocal formalism, effective conductivities and weak approximation. Water Resources Research, 1993, 29(2) 341–364.

(1)

given by

G1c(r1)¼G1(r1)¼0(because,df(r). ¼0), G2c(r1,r2)¼G2(r1,r2), G3c(r1,r2,r3)¼G3(r1,r2,r3), G4c(r1,r2,r3,r4)¼G4(r1,r2,r3,r4)

¹G2c(r1,r2)G2c(r3,r4)¹G2c(r1,r3)G2c(r2,r4)

¹G2c(r1,r4)G2c(r2,r3) ð19Þ The cumulant expansion is nothing more than a systematic scheme for isolating the specific correlation between N points from products of lower-order correlations. Returning to our series expansion, and inserting eqn (18), the original Nth order term of the series expansion ofH↔will break down each term into several elements containing products of con-nected correlation functions. We now obtain a new series expansion which can be written graphically using Feynman graphs by

These graphs are only a more attractive way of writing the series expansion using the following rules:

1. Each white circle Wcorresponds to the two extreme pointsr1andrN.

2. Each black circleXcorresponds to one amongN¹2 intermediate dummy integration variables r2torN¹2, the points being ordered from left to right.

3. Each arrow is a ‘propagator’==_P₀, applied to the lag vector between two neighbouring points. 4. Each wavy line corresponds to a connected

correlation function of theWorX points linked by these lines.

Note that due to the Laplace transform with respect to the time argument, no time integration is required.

By definition, whenN¼2, we have no intermediate black integration points; in particular the graph is given by

Another example is the graph which

corresponds to the following integral in Laplace

representation: Z

dr₂ Z

dr₃3_G2_c(r₁¹r₂)==_P₀(r₁¹r₂,s): ==_P0(r₂¹r3,s):==_P0(r₃¹r4,s)G2c(r₃¹r4)

One should be careful with the role of the contraction points (.) between tensors.

6 SUMMATION OF THE IRREDUCIBLE GRAPHS AND DERIVATION OF AN EFFECTIVE EQUATION It can be noted that among all these graphs, some of them

can be broken down into two separate parts by cutting one arrow. These graphs may be expressed in a very simple manner as a function of lower-order graphs. For example,

consider the last example of graph .

Using its expression given in the previous section, we obtain Z

dr₂ Z

dr₃3_G2_c(r₁¹r₂)==_P₀(r₁¹r₂,s): ==_P₀(r₂¹r₃,s):==_P₀(r₃¹r₄,s)G2c(r3¹r4)

¼ Z

dr₂ Z

dr₃3_G2_c(¹r₂)==_P₀(¹r₂,s): ==_P₀(r₂¹r₃,s):

==_P0(r₃¹(r₄¹r1),s)G2c(r₃¹(r₄¹r1))

after shifting the integration variabler2andr3by a quantity r1. CallingF(r2¹r1,s) the value of the graph

given by G2c(r1¹r2)==P0(r1¹r2,s)becomes

This is a function of the argumentR¼r4¹r1and under this form, we recognize a convolution product of three functions. Thus the FLT of this function appears to be the product of two lower-order graphs ( ) FLT, times an extra factor. This factor is the Fourier transform of ==_P0(r, s), (¹qq=(sþD0q2)) (corresponding in this (20)

¼G2c(r1¹r2)==P0(r1¹r2,s) (21)

¼ Z

dr₂ Z

dr₃3F(¹r₂)

(2)

To circumvent this problem, and to reduce the number of graphs to be computed, it is classical practice in field theory to define a new kernelS↔(r1¹rN,s) by the following definition:S↔(r1¹rN,s) represent the sum of all ‘1P irre-ducible graphs’ (i.e. the graphs that cannot be broken down into two parts by splitting a single arrow). Graphically, we obtain

Readers familiar with field theoretical methods will recog-nize the ‘self-energy’, or ‘mass operator’ inS

↔

(see for exam-ple Ref.14, and references therein). As this kernel is the sum of irreducible graphs, assuming that all the connected cor-relation functions have a limited range equal to the correla-tion lengthlc, one can observe that the kernel will have a typical range equal tolcin the space domain, and a typical diffusion time over one correlation length equal tol2c=D0 in the time domain.

We may now derive the useful identity, also classical in the field theoretical framework (Dyson equality):

H↔

D E

(q,s)¼X

↔

(q,s): ↔1 þ 1 sþD0q2

qq:X

↔

(q,s)

¹1

ð23Þ This identity may be derived by formally expanding the inverse of the operator on the right-hand side of the above equality in a geometric series (1 þ x)¹1 ¼ 1 ¹ xþx2¹x3þ…þ.:

X↔

(q,s): ↔1 þ 1 sþD0q2

qq:X

↔

(q,s)

¹1

¼X

↔

(q,s)þX

↔

(q,s): ¹1 sþD0q2

qq:X

↔

(q,s)

þX

↔

(q,s): ¹1 sþD0q2

qq:X

↔

(q,s) ¹1 sþD0q2

:qq:X

↔

(q,s)

þ…þX

↔

(q,s):…: ¹1 sþD0q2

qq:X

↔

(q,s) þ…(the product: ¹1

sþD0q2 qq:X

↔

(q,s)entering N times)

Using eqns (13) and (23), after simplification, one obtains hPi(q,s)¼ 1

sþD0q2þq: X↔

(q,s):q

(24)

or, equivalently, sþD0q2þq:

X↔ (q,s):q

hPi(q,s)¼1

↔

(q,s) can be introduced. Returning to the space–time representation, and consider-ing the rate factor Q/fct, one obtains the following form:

]h i(_P r,t)

]_t ¼=·(D0=h i(P r,t))þ Zt

0dt 9

=_r: Z

dr9X

↔

(r9,_t9):=_r₉h i(_P r¹r9,_t¹t9)

þ Q

Fc_td(r)d(t)

↔

In particular, in typical cases the memory kernel↔S(r9,_t9) must be quite a localized function of spatial rangelc, as all the large range factors decreasing slowly at infinity are (22)

(3)

removed by the 1P irreducibility condition. Due to the diffusive nature of the original equation, we can expect that for large times and large lengthscales, the mean pres-sure will have typical variations over lengthscales given by eqn (16) very much larger than ↔S(r9,_t9). We can thus approximate =hPi(r ¹ r9_{, t} _¹ _t9_{) by} =hPi(r, t) in eqn (25) under the integral sign, producing the following approximation:

]h i(_P r,t)

]_t ¼=·(D0=h i(P r,t)) þ=

X↔

(q¼0,_s_¼0):=

rh i(P r,t)

þ Q

Fct

d(r)d(t)

(26)

D_eff¼D01þ X↔

(q¼0,_s_¼0)_¼_D

þhH

↔

i(q¼0,_s_¼0)

(27)

Another interesting physical feature of eqn (25) is that the vectorial quantity V(r, t) defined by V(r,t)¼

¹D0=r9h i(P r,t)¹

0 dt R

dr9_S↔(_r9,_t9):=

r9h i(P r¹r9,t¹t9)

]h i(_P _r,_t)

]_t ¼ ¹=r:V(r,t)þ Q Fct

d(r)d(t)

The last approximation leading to eqn (26) thus becomes very easy to interpret using the preceding expression of the fluxV(r,t).

Another interesting property of the kernelS

↔

(r9, t9) is that, owing to its more localized nature than the original H↔ tensor, eqn (25) must be robust when considering finite size domains implying Dirichlet or Neuman boundary con-ditions. In particular, it is reasonable to think that if the boundaries local radii of curvature are greater than the cor-relation lengthlcthan eqn (25) could still work, although it was derived for an infinite medium. The preceeding results and the general framework which we have set up show that this affirmation is probably correct. A rigorous derivation would certainly involve a considerable amount of work.

Analogous conclusions were given by Rubin and Dagan by means of a second-order perturbation expansion15.

7 A SECOND-ORDER CALCULATION

When computing the S↔kernel up to the orderj2, we only need compute the value of the graph . We obtain

X↔

(q¼0,0)_{¼ ¹} 1

2D0

j21 ₍₂₉₎

Using eqn (21), the effective diffusion tensor is thus given by

D_eff¼D01¹ 1 2D0

j21 ₍₃₀₎

This is a well-known result, as it corresponds to the steady-state second-order expression for the effective permeability of a heterogeneous stationary and isotropic medium11,16.

More generally, a wave vector dependant effective dif-fusivity may be introduced by the definitionDeff(q,s¼0)¼ D01þS

↔

(q,s¼0) which corresponds to the static limit. For large wave vectorsq, one can compute (q.S↔ (q, s)·q) using eqn (28) and the convolution theorem. We obtain

lim

q→`q:

X↔

(q,s):q¼ ¹ 1 D0

j2q2 (31)

This value is independent ofs, meaning that in the largeq limit (small lengthscales), transient effects are negligible. This gives an apparent diffusivity equal to the harmonic mean of the local diffusivities. This result has a simple physical meaning: consider a well located at r ¼ 0. For lengthscales very much smaller than the correlation length lc, the permeability is equal to k(r¼ 0). The solution to the diffusion equation behaves thus locally as p(r)¼mQ=(2pk(r¼0))ln(r). After averaging, we get ,_p(_r). ¼mQ=(2pkh)ln(r) wherekh denotes the harmo-nic averaging of the permeability kh¼ ,k(r¼0)

¹1_.¹1_. Applying the Fourier Transform to the expression ofhp(r)i for large wave vectorsq yields a result which agrees with eqn (31). This result should be compared with previous results of Matheron, who showed that for steady-state radial flow, the definition of an equivalent permeability is ambiguous, depending on the ratio of the radius of the well with respect to the medium typical size. This illustrates that caution is required when computing some limiting behaviours.

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8 CONCERNING THE APPARENT

PERMEABILITY GIVEN BY WELL TESTS

We will now take a look at a more classical analysis of well tests. In the homogeneous case, it is recognized that well tests are interpreted according to the following formula:

P0(r¼0,t)¼

Q 4pD0fmct3t

P0(r¼0,t)¼

Q 4pDefffct3t

(33) where the equivalent diffusivity Deff is as defined in Sections 4, and 6. It is thus straightforward to procure an effective permeability through the relationkeff¼fmct.Deff Here we will provide an heuristic proof. To begin with, the basic idea is to study the behaviour of the ensemble average pressurehP(r¼0,t)i. The difference with the pre-viously given formulation is the averaging symbol: well tests are performed in one realization so the aim is to con-sider that the pressure diffusion takes into account the full set of heterogeneities in a single realization for sufficiently long times. This regime is likely to occur when the investi-gation radius—defined using the moment approach—is larger than the correlation length.

It would be thus sufficient to invert eqn (24) inr¼0 to obtain the expression ofhP(r¼0,t)i, but the 1/tsingularity gives rise to a diverging integral in the Fourier inversion. To circumvent it, we compute the Fourier–Laplace transform ofthP(r¼0,t)iwhich is given by ¹]_h_P(_r_¼0,_s)_i=]_s_{. As we} are seeking the long-time behaviour of this function and as we expect a constant, it is most logical to examine the quantity

lim

s→0 s

3 ]P(r¼0,s) ]_s

¼(4pDwell test)¹1¼(Deff)¹1 (34) Using the inverse Fourier transform, we must try to evaluate

s]P(r¼0,s) ]_s ¼

s 4p2

Z dq

1 (sþD0q2þq:

X↔

(q,s):q)2 þ

q· ]X

↔

(q,s) ]_s ·q (sþD0q2þq:

X↔

(q,s):q)2 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 (35)

It would be appealing to replace↔S(q,s) by↔S(q¼0,s¼0), and to compute the desired limit directly, but the integrals are not absolutely convergent and somewhat more calcula-tion is needed. Rescaling the wave vector q by q¼qps=(D0)q9_{, we get a clearer form for the first term} of the preceding equation:

s 4p2

dq 1

(sþD0q2þq: X↔

(q,s):q)2

¼ 1

4p2 Z

dq 1

(1þq2_þ_q:

X↔ s D0 r

q,s

:q)2

(36)

Now, we can safely consider the limit s ¼ 0 under the integral sign, as the integral is absolutely convergent for largeq wave vectors. We thus obtain

D0 lim

s→0

1 4p2

dq 1

1þq2_þ_q:

X↔ s D0 r

q,s

D0 :q 0 B B B @ 1 C C C A 2 0 B B B B B B B B B B B B B @ 1 C C C C C C C C C C C C C A ¼ 1 4p2 Z

dq 1

1þq2þq: X↔ (0,0) D0 :q 0 B B B @ 1 C C C A 2 (37) AsS ↔

(q¼0,s¼0) is isotropic, after integration, we pro-duce the desired result:

D_{well test}¼D01þ X↔

(q¼0,_s_¼0) ₍₃₈₎

Now, we must show that the limit of the second term of the right-hand side I2of eqn (35) is equal to zero in the lows limit. To achieve this task, we will explicitly use the second-order expansion of the↔S(q,s) kernel whose expres-sion in the real space is↔S(r9,_t9)_¼_j2==_P

0(r9,t9)C(r9), and its FLT is given by

X↔

(q,s)¼ ¹j2 Z

dq9q9q9 1 sþD0q92

C(q¹q9) (39) Using this result, along with the explicit form of the second term, we get the following expression, after a rescaling of the wave vectors qandq9_{by the factor} _s=D0

p :

(5)

I2¼ s 4pD0

j2 Z

dq Z

dq9qq:q9q9

3 1

1þq2þq: X↔

( s=D0 p

q,s) D0

:q 0

B B B @

C C C A 2

3_C( _s=D0 p

(q¹q9)) 1 (1þq92₎2

(40)

To examine the behaviour of this quantity, we first let the Laplace parametersgo to zero in the integrand. However, this leads to the evaluation of two logarithmically divergent integrals. To obtain well-defined results, we return to eqn (34) and change the integration variableq9by writingq9¼ q þ q0_{. The integration over} _q _{is thus feasible and after} tedious calculation, we are led to evaluate the limit for smallsof the following quantity:

I2¼s Z`

0 dq f(q)C( s=D0 p

in which the functionf(q) is regular near zero, and behaves as 1/qfor large q.

Assuming that the correlation functionC(r) has a limited rangelcin real space, we can restrict theqintegration over wave-vectors of modulus such thatq,p_D₀=s=lc. With this natural cut-off, it may be shown that the integral behaves as sln(s) whensgoes to zero.

9 DISCUSSION AND CONCLUSIONS

from the boundaries, implying a robustness of the present approach when considering real finite size problems.

To obtain a better understanding of the homogenization effects and of the emergence of a simplified description by means of an effective equation, it would be useful to com-pute the covariance functions between local pressure varia-tions, i.e. the quantities _hP(r,t)P(r9,_t)_{i ¹}_h_P(_r,_t)_i_h_P(_r9,_t)_i. Due to ergodicity effects, such quantities could vanish for sufficiently long times. This would imply that the average pressure which was the subject of this paper will be almost surely observable for sufficiently long times. This would provide better knowledge of the homogenization process performed by well testing and would give insights about the characteristic time scales and lengthscales. This study would help us to complete the proof of the convergence of the well test apparent permeability towards its steady-state value. King6 and Christakoset al.7 showed that diagram-matical methods can handle such tasks. We expect that for an infinite reservoir, this variance would tend to zero for large times t. This implies that at long times the average description becomes ‘almost sure’, giving us a dynamical image of ergodicity effects. This work, involving rather tedious calculations, is currently in progress.

ACKNOWLEDGEMENTS

The referees are gratefully acknowledged for their useful comments.

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flow in a heterogeneous porous medium.Journal of Physics

A: Mathematics Generalized, 1987,203935–3947. 7. Christakos, G., Miller, C.T. & Oliver, D. Stochastic

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Hydrology and Hydraulics, 1993,7(3) 213–239.

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porous media.Water Resources Research, 1995,31(7) 1687–

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Cri-tical Phenomena. World Scientific, Singapore, 1984. 15. Rubin, Y. & Dagan, G. Stochastic analysis of boundaries

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approximation. Water Resources Research, 1993, 29(2)

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