s ¼ 0 which corresponds to the static limit. For large wave vectors one can compute q.S s· q

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 =hPir ¹ r9, t ¹ t9 by =hPir, t in eqn 25 under the integral sign, producing the following approximation: ] P h i r , t ]t ¼ = · D = P h i r , t þ = r : X ↔ q ¼ 0 , s ¼ 0 := r P h i r , t þ Q Fc t 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 eff ¼ D 1 þ X ↔ q ¼ 0 , s ¼ 0 ¼ D 1 þ h H ↔ i q ¼ 0 , s ¼ 0 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 ¹ D = r9 P h i r , t ¹ R t dt R dr9 S ↔ r9 , t9 : = r9 P h i r ¹ r9 , t ¹ t9 can itself be interpreted as the average flux vector at point r at time t 12 . 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. ] P h i r , t ]t ¼ ¹ = r : V r , t þ Q Fc t d r d t The last approximation leading to eqn 26 thus becomes very easy to interpret using the preceding expression of the flux Vr, t. Another interesting property of the kernel S ↔ 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 length l c than 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 expansion 15 . 7 A SECOND-ORDER CALCULATION When computing the S ↔ kernel up to the order j 2 , we only need compute the value of the graph . We obtain X ↔ r9 , t9 j 2 ==P r9 , t9 C r9 28 where j 2 Cr9 denotes the covariance function of the permeability field which is assumed to be isotropic, with Cr9 ¼ 0 ¼ 1. Computing the FLT of S ↔ , for q ¼ 0 and s ¼ 0, using Parseval’s equality and the isotropy hypothesis, in two dimensions, we obtained X ↔ q ¼ 0 , ¼ ¹ 1 2D j 2 1 29 Using eqn 21, the effective diffusion tensor is thus given by D eff ¼ D 1 ¹ 1 2D j 2 1 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 medium 11,16 . More generally, a wave vector dependant effective dif- fusivity may be introduced by the definition D eff

q, s ¼ 0 ¼

D 1 þ S ↔ q,s ¼ 0 which corresponds to the static limit. For large wave vectors q, one can compute q.S ↔

q, s· q

using eqn 28 and the convolution theorem. We obtain lim q→` q: X ↔ q , s : q ¼ ¹ 1 D j 2 q 2 31 This value is independent of s, meaning that in the large q 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 l c , the permeability is equal to kr ¼ 0. The solution to the diffusion equation behaves thus locally as p r ¼ mQ= 2pk r ¼ ln r . After averaging, we get , p r . ¼ mQ= 2pk h ln r where k h denotes the harmo- nic averaging of the permeability k h ¼ , k r ¼ ¹ 1 . ¹ 1 . Applying the Fourier Transform to the expression of hpri for large wave vectors q 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. Pumping tests in heterogeneous reservoirs 587 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: P r ¼ 0 , t ¼ Q 4pD fmc t 3 t 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: P r ¼ 0 , t ¼ Q 4pD eff fc t 3 t 33 where the equivalent diffusivity D eff is as defined in Sections 4, and 6. It is thus straightforward to procure an effective permeability through the relation k eff ¼ fmc t .D eff Here we will provide an heuristic proof. To begin with, the basic idea is to study the behaviour of the ensemble average pressure hPr ¼ 0, ti. 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 in r ¼ 0 to obtain the expression of hPr ¼ 0, ti, but the 1t singularity gives rise to a diverging integral in the Fourier inversion. To circumvent it, we compute the Fourier–Laplace transform of thPr ¼ 0, ti which is given by ¹ ] P r ¼ , s h 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 r ¼ , s ]s ¼ 4pD well test ¹ 1 ¼ D eff ¹ 1 34 Using the inverse Fourier transform, we must try to evaluate s ]P r ¼ , s ]s ¼ s 4p 2 Z dq 1 s þ D q 2 þ q: X ↔ q , s : q 2 þ q · ] X ↔ q , s ]s ·q s þ D q 2 þ q: 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