t kernel of the integro-differential equation

Use of the Fourier–Laplace transform and of diagrammatical methods to interpret pumping tests in heterogeneous reservoirs Benoıˆt Nœtinger Yann Gautier Institut Franc¸ais du Pe´trole, He´lioparc Pau-Pyre´ne´es, 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. The Nth 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 c t total compressibility Pa ¹ 1 Dr diffusivity k r = Fmc t at point r m 2 s ¹ 1 D arithmetic average of the local diffusivity Dr m 2 s ¹ 1 D eff effective diffusivity m 2 s ¹ 1 kr permeability at point r m 2 l c correlation length m r position vector m t time s q Fourier parameter m ¹ 1 s Laplace parameter s ¹ 1 Pr, t time derivative of the pressure at point r at time t Pa-s ¹ 1 Q rate at the well per unit of length m 2 s ¹ 1 Vr, t average local flow rate at point r at time t Pa-m s ¹ 2 d . Dirac delta function F porosity m fluid viscosity Pa-s j 2 permeability variance m 4 S ↔

r,t kernel of the integro-differential equation

Advances in Water Resources, Vol. 21, pp. 581–590, 1998 q 1998 Elsevier Science Limited All rights reserved. Printed in Great Britain 0309-17089819.00 + 0.00 P I I : S 0 3 0 9 - 1 7 0 8 9 7 0 0 0 1 4 - 6 581 To whom correspondence should be addressed. 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 authors 2–4 , 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 time t, 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 response 5 : 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 King 6 and Christakos 7 . 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 wells 8 . 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 582 B. Nœtinger, Y. Gautier pressure variations is given by 1,9,10 : ]P r , t ]t ¼ = · D r =P r , t þ Q Fc t d r d t 1 Here, Dr ¼ krfmc t denotes the diffusivity. Q represents the fluid flow rate per unit of time and per unit of well length and the two delta functions represent the well. The parameters kr, f, m and c t denote, 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 Pr, t considered in the present work corresponds to the time derivative of the pressure considered in standard tests 9 . Its SI unit is thus Pa-s ¹ 1 . To solve this evolution problem, we choose an initial condition given by Pr, t ¼ 0 ¼ 0, and we assume that the pressure vanishes at infinity for all times t. In this case, a unique solution exists. As the time Dirac function dt 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 value Pr ¼ 0, t, from which we would like to obtain information on the diffusivity Dr. Here, the diffusivity Dr is assumed to be a random function and our first goal is to find an equation obeyed by the mean pressure field hPr, ti, where the averaging symbol h … i means an ensemble averaging over the realiza- tions of the random functions Dr 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 Z ` dt exp ¹ st Z dr e iq:r P 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 coefficient D , the FLT of the diffusion equation with Q=fc t ¼ 1 gives P q , s ¼ 1 s þ D q 2 3 One should note that P .,. 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: P r ¼ , t ¼ Q 4pD Fc t 3 t 4 It is useful to introduce the so-called ‘pressure moments’ 10 which are nth order tensors by m n t ¼ Z dr r n P r , t 5 Using the FLT and expanding the space-dependant expo- nential in series, we obtain the following results: m m t ¼ i n ] n P q , t ]q n q ¼ 0 and m m s ¼ i n ] n P q , s ]q n q ¼ 0 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 zone 10 . Considering the homo- geneous case, one produces m 2 t ¼ 2D 1 t 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 = · D þ jdf r =P r , t þ Q fc t d r d t 8 Here, j is the mean square deviation of the local diffusivity and the random function dfr is assumed to be stationary, of zero mean, and to have known statistical properties. To simplify the analysis, we chose Qfc t ¼ 1. Assuming that the solution to this equation can be written as a power series P r , t ¼ S ` N ¼ 0 P N r , t where P N is of order j N , we solve the diffusion equation by iteration giving rise to an infinite hierarchy of diffusion problems defined by ]P N þ 1 r , t ]t ¼ = · D =P N þ 1 r , t þ j=· df r =P N r , t 9 The formal solution to this equation yields the N þ 1th order term by means of the formula P N þ 1 r , s ¼ j Z dr 1 =P r ¹ r 1 , s : df r 1 =P N r 1 , s Pumping tests in heterogeneous reservoirs 583 and thus by induction we get P N r , s ¼ j N Z dr 1 Z dr 2 … Z dr N 3 =P r ¹ r 1 , s df r 1 : ==P r 1 ¹ r 2 , s df r 2 … df r N ¹ 1 : ==P r N ¹ 1 ¹ r N , s df r N := P r N , s ð 10Þ The Nth order term gives the effect of N diffusivity fluctua- tions at N points. Computing the average of this Nth order term requires the expressions of the averages of the products j N , df r 1 df r 2 … df r N . which are the N- body correlators Gr 1 , r 2 , … , r N . Using the stationary hypothesis, we can write the following equality: Gr 1 , r 2 ; … ; r N Þ ¼ Gr 2 ¹ r 1 ; … ; r N ¹ r 1 Þ . Eqn 10 can be thus transformed as follows by integrat- ing N ¹ 2 times over the intermediate integration variables r 2 , … , r N¹1 : P N r , s ¼ j N Z dr 1 Z dr N =P r ¹ r 1 , s : Z dr 2 … dr N ¹ 1 G r 2 ¹ r 1 , …, r N ¹ 1 ¹ r 1 , r N ¹ r 1 ==P r 1 ¹ r 2 , s : ==P r 2 ¹ r 3 , s … ==P r N ¹ 2 ¹ r N ¹ 1 , s : ==P r N ¹ 1 ¹ r N , s ÿ:= P r N , s 11 Shifting all the intermediate integration variables r 2 , … , r N¹1 of a quantity r 1 , we obtain an expression of the form P N r , s ¼ j N Z dr 1 Z dr N =P r ¹ r 1 , s : H ↔ N : r 1 ¹ r N , s := P r N , s 12 We recognize two successive convolution products. As this expression is valid for N . 1, and by introducing the FLT to transform convolution products, one can sum the whole series, resulting in the following: P h i q , s ¼ 1 s þ D q 2 ¹ q s þ D q 2 : , H ↔ . q , s : q s þ D q 2 13 the second-order tensor H ↔ being defined as: H ↔ r 1 ¹ r n , s ¼ X ` N ¼ 2 j N H ↔ N r 1 ¹ r n , s 14 After averaging, the whole effect of the heterogeneities will be contained in the averaged h H ↔ i second-order tensor. It is useful to note that, using eqns 6 and 13, the average second moment can be given by m 2 s ¼ ¹ ] 2 P h i q , s ]q 2 q ¼ 0 ¼ 2 D s 2 1 ↔ þ H ↔ q ¼ 0 , s s 2 B 1 C A 15 If H ↔ q ¼ 0 , s ¼ 0 has a well-defined limit when s goes to zero, we obtain, in the time domain, an Einstein-like relation: , m 2 . t ¼ 2 D 1 t þ H h i ↔ q ¼ 0 , s ¼ 0 t 16 Comparing with the homogeneous medium result [eqn 7], we may define an effective diffusion tensor D eff of D eff ¼ D 1 þ H h i ↔ q ¼ 0 , s ¼ 0 17 Permitting the Laplace parameter s approach 0 is equiva- lent to computing the steady-state limit. It is interesting to check that h H ↔ i q ¼ 0 , s ¼ 0 may be identified term by term with the perturbation series arising from classical steady-state equivalent permeability calculations 11 . 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 Indelmann 12 . 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 King 6 and Christakos and coworkers 7,13 , 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 the Nth order correlation function under the form of a cumulant expansion: G N r 1 , r 2 , …, r N ¼ G N c r 1 , r 2 , …, r N þ X ∪I a ¼ I Y a G Card I a c I a 18 The index ‘c’ indicates connected correlation functions. I denotes N points {r 1 , r 1 , … , r N } and I a 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 Ref. 14 . As an example, the first four cumulants are 584 B. Nœtinger, Y. Gautier 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