Well Testing in Heterogeneous Media: A General Method to Calculate the Permeability Weighting Function
Authors: Romeu R.K. , Lara A.Q. , Noetinger B. , Renard G. 
Published: December 2007

Abstract
The perturbation method provides approximate solutions of the well pressure for arbitrarily heterogeneous media. Although theoretically limited to small permeability variations, this approach has proved to be very useful, providing qualitative understanding and valuable quantitative results for many applications. The solution is expressed by an integral equation where the permeability variations are weighted by a kernel, the permeability weighting function. As presented in previous papers, deriving such permeability weighting functions appears as a complicated calculation, available only for special cases. This paper presents a simple and general method to calculate the permeability weighting function. In the Laplace domain, the permeability weighting function is easily related to the pressure solution of the background problem. Since Laplace pressure solutions are known for many situations (various boundary conditions, stratified and composite media etc), the associated permeability weighting function can be immediately derived. Among other examples, we calculate and discuss the well pressure solution for a horizontal well producing from a heterogeneous reservoir.

Introduction
The trend for reservoir characterization has stimulated the study of well testing in more complex heterogeneous media.
Well testing in heterogeneous media has been studied by three approaches: exact analytical solutions, numerical simulations and approximate analytical solutions. Exact analytical solutions exist for a restricted class of problems, involving some simple symmetry: layered reservoir, single linear discontinuities, radially composite systems etc. Rosa and Horner computed the exact solution in the case of an infinite homogeneous reservoir containing a single circular permeability discontinuity. Most of these analytical solutions are written in the Laplace domain. Numerical methods can treat much more general situations, but have some disadvantages: their use is cumbersome, investigation is empirical and general insights are difficult to be extracted, results are inaccurate if the time and the spatial discretization were not carefully conducted. Approximate analytical solutions can be a practical way to understand the pressure behavior in geometrically complex heterogeneous media. Kuchuk et al. proposed one of these approximate methods. Another popular class of approximate analytical solutions is based on the first-order approximation obtained from perturbation methods.
This paper is related to these first-order approximate solutions of the well pressure in arbitrarily heterogeneous reservoirs. In particular, we propose an easy and general method to calculate the permeability weighting function in various flow geometries. In the next section, we define what the permeability weighting function is and review previous works in the domain. After that, we present our method to calculate the weighting permeability functions. The technique is demonstrated in three situations, including the case of flow through a horizontal well.

The Permeability Weighting Function
The perturbation method is a well known technique to solve partial differential equations involving mathematical difficulties, like variable coefficients. According to this technique, we start from an easier problem, the background problem, to modify or perturb it. The full problem is approximated by the first few terms of a perturbation expansion, usually the first two terms.
In our context, we start from considering a background medium with permeability k0 and with specified boundary conditions. The permeability k0 may vary in the space, i.e k0 (xD) What is important is that the background problem has a known exact analytical solution, PDO (XD, tD).
The full problem has the same boundary conditions of the background problem but the permeability k(XD) differs from ko(XD) in arbitrary regions of the space. Rigorously speaking, k(XD) / k0 (XD) has to be close to 1 in order to obtain sound approximations. In practice, errors tend to be small, say less than 10%, even for relatively greater contrasts, say up to 10, between these permeabilities, depending on the specific problem.
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