s, ¹ s as their FLT are singular in the limit

given by G 1 c r 1 ¼ G 1 r 1 ¼ 0 because , df r . ¼ , G 2 c r 1 , r 2 ¼ G 2 r 1 , r 2 , G 3 c r 1 , r 2 , r 3 ¼ G 3 r 1 , r 2 , r 3 , G 4 c r 1 , r 2 , r 3 , r 4 ¼ G 4 r 1 , r 2 , r 3 , r 4 ¹ G 2 c r 1 , r 2 G 2 c r 3 , r 4 ¹ G 2 c r 1 , r 3 G 2 c r 2 , r 4 ¹ G 2 c r 1 , r 4 G 2 c r 2 , r 3 ð 19Þ 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 of H ↔ 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 W corresponds to the two extreme points r 1 and r N . 2. Each black circle X corresponds to one among N ¹ 2 intermediate dummy integration variables r 2 to r N¹2 , the points being ordered from left to right. 3. Each arrow is a ‘propagator’ ==P , applied to the lag vector between two neighbouring points. 4. Each wavy line corresponds to a connected correlation function of the W or X points linked by these lines. 5. Finally, the large black circles X indicate 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, when N ¼ 2, 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 dr 2 Z dr 3 3 G 2 c r 1 ¹ r 2 ==P r 1 ¹ r 2 , s : ==P r 2 ¹ r 3 , s :== P r 3 ¹ r 4 , s G 2 c r 3 ¹ r 4 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 dr 2 Z dr 3 3 G 2 c r 1 ¹ r 2 ==P r 1 ¹ r 2 , s : ==P r 2 ¹ r 3 , s :== P r 3 ¹ r 4 , s G 2 c r 3 ¹ r 4 ¼ Z dr 2 Z dr 3 3 G 2 c ¹ r 2 ==P ¹ r 2 , s : ==P r 2 ¹ r 3 , s : ==P r 3 ¹ r 4 ¹ r 1 , s G 2 c r 3 ¹ r 4 ¹ r 1 after shifting the integration variable r 2 and r 3 by a quantity r 1 . Calling Fr 2 ¹ r 1 , s the value of the graph given by G 2 c r 1 ¹ r 2 ==P r 1 ¹ r 2 , s becomes This is a function of the argument R ¼ r 4 ¹ r 1 and 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 ==P

r, s, ¹

qq= s þ D q 2 corresponding in this 20 ¼ G 2 c r 1 ¹ r 2 ==P r 1 ¹ r 2 , s 21 ¼ Z dr 2 Z dr 3 3 F ¹ r 2 :== P r 2 ¹ r 3 , s : F r 3 ¹ r 4 ¹ r 1 , s Pumping tests in heterogeneous reservoirs 585 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 ==P

r, s as their FLT are singular in the limit

s ¼ 0, implying an 1r behaviour 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 kernel S ↔ r 1 ¹ r N , s by the following definition: S ↔ r 1 ¹ r N , 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’ in S ↔ 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 length l c , one can observe that the kernel will have a typical range equal to l c in the space domain, and a typical diffusion time over one correlation length equal to l 2 c = D 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 ¼ X ↔ q , s : 1 ↔ þ 1 s þ D q 2 qq: X ↔ q , s ¹ 1 ð 23Þ 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 ¹ 1 ¼ 1 ¹ x þ x 2 ¹ x 3 þ … þ .: X ↔ q , s : 1 ↔ þ 1 s þ D q 2 qq: X ↔ q , s ¹ 1 ¼ X ↔ q , s þ X ↔ q , s : ¹ 1 s þ D q 2 qq: X ↔ q , s þ X ↔ q , s : ¹ 1 s þ D q 2 qq: X ↔ q , s ¹ 1 s þ D q 2 : qq: X ↔ q , s þ … þ X ↔ q , s : … : ¹ 1 s þ D q 2 qq: X ↔ q , s þ … the product: ¹ 1 s þ D q 2 qq: X ↔ q , s entering N times Using the graphical definition of the kernel S ↔ , 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 of h H ↔ i that 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 of h H ↔ i that 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 h H ↔ i are recovered. This yields eqn 23. Using eqns 13 and 23, after simplification, one obtains hPi q , s ¼ 1 s þ D q 2 þ q: X ↔ q , s : q 24 or, equivalently, s þ D q 2 þ q: X ↔ q , s : q hPi q , s ¼ 1 This is the desired equation obeyed by the average pressure hPir, t. By analogy with the homogeneous case, one can observe that in the FLT space, a variable apparent diffusion coefficient equal to D 1 þ S ↔ q , s can be introduced. Returning to the space–time representation, and consider- ing the rate factor Qfc t , one obtains the following form: ] P h i r , t ]t ¼ = · D = P h i r , t þ Z t dt9 = r : Z dr9 X ↔ r9 , t9 := r9 P h i r ¹ r9 , t ¹ t9 þ Q Fc t d 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 range l c , as all the large range factors decreasing slowly at infinity are 22 586 B. Nœtinger, Y. Gautier removed by the 1P irreducibility condition. Due to the diffusive nature of the original equation, we can expect