An Application of Green Function Technique and Ewald's Algorithm for Well Test Analysis

The objective of well test analysis is to estimate the reservoir properties from the pressure response. Generally this is achieved by solving inverse problem, mathematical model of reservoir generates the pressure response to the actual one closely. The mathematical model demands solution of single-flow equation (SFE). The required time for inverse problem depends on efficiency of regression scheme and count rate of SFE (direct problem). The abstract focuses on the solution of direct problem only. There are various solutions of SFE can be used in well test analysis. Often pressure response is approximated by analytical solution for uniform infinite reservoir tough this assumptions restrict domain of its applicability. The method of Laplace transformation is used especially for tests with transient production rate. However this method requires complex calculations for inverse transformation. Finite difference solution also can be obtained but it is time-consuming. SFE can be solved through Green function method and it could be applied for various reservoir geometry and arbitrary production rates. Solution is presented as a series of over eigen values of differential operator. However these series converge conditionally and summation involve big number of terms. Therefore the summation is time-consuming and restricts application of Green function for inverse problems in well test analysis. The same problems are well-known in quantum theory of solid state, the algorithm for fast summation of such series was proposed by Ewald [1]. In the presents work we use Ewald’s algorithm for well test analysis. We have considered synthetic examples of tests with various boundary conditions and reservoir geometry. The algorithm provides faster procession compared to finite differences solution. We successfully applied the Ewald's procedure for interpretation of real pressure buildup data with accounting sandface flow for vertical and horizontal wells. [1] Ewald P.P. Ann Physic 64 1921