Summary
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 well pressure solution using this method is expressed by an integral equation where the permeability variations are weighted by a kernel, the permeability weighting function. As discussed in previous papers, deriving such permeability weighting functions appears to be a complicated calculation, available only for special cases. In this article we present 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 derived immediately. Among other examples, we calculate and discuss the well pressure solution for a horizontal well that is 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 that involve some simple symmetry: layered reservoirs, single linear discontinuities, radial composite systems etc.1 Rosa and Horne2 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.3 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 article is related to these first-order approximate solutions of 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 work in the domain. After that, we present our method to calculate the permeability weighting functions. The technique is demonstrated in three situations, including the case of flow through a horizontal well.
Permeability Weighting Function
The perturbation method is a well known technique by which 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 k0 may vary in space, i.e., k0(x→D). What is important is that the background problem has a known exact analytical solution, pD0(x→D,tD).
The full problem has the same boundary conditions of the background problem but the permeability k(x→D) differs from k0(x→D) in arbitrary regions of space. Strictly speaking, k(x→D)/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 up to, say, 10 between these permeabilities, depending on the specific problem.
The dimensionless well pressure of the full problem, pwD(tD), is approximated by the sum of two terms:
p w D ( t D ) ≅ p w D 0 ( t D ) + p w D 1 ( t D ) , ( 1 )
where pwD0 is the solution of the background problem, which is known, and pwD1 corresponds to the effect of the variation of the permeability. This second term is computed by
p w D 1 ( t D ) = ∫ − ∞ + ∞ Δ k D ( x → D ) W ( ∙ x D , t D ) d ∙ x D , ( 2 )
the terms of which will be explained.
The dimensionless permeability variation ΔkD may be alternatively defined by
Δ k D ( ∙ x D ) = l n ( k ( x → D ) / k 0 ( x → D ) ) , ( 3 a )
Δ k D ( ∙ x D ) = 1 − ( k 0 ( x → D ) / k ( x → D ) ) , ( 3 b )
Δ k D ( ∙ x D ) = ( k ( x → D ) / k 0 ( x → D ) ) − 1 , ( 3 c )
or other equivalent first-order approximations. These three expressions have the same first-order terms of their Taylor series, and produce very close results for k(x→D)/k0(x→D) near 1. However, these definitions are not equally robust for greater permeability contrasts.
First Order Framework for Gauge k-Vortices