General Method to Calculate the Permeability Weighting Function for Well Testing in Heterogeneous Media

Abstract

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.