Eigenmode Decomposition of the Diffusion Equation: Applications to Pressure-Rate Deconvolution and Reservoir Pore Volume Estimation

Abstract

SummaryThis paper proposes the application of the eigenmode decomposition to the problem of pressure-rate deconvolution. The method enabled us to write the solution of the pressure-rate deconvolution as an infinite series of eigenmodes (eigenfunctions). Each eigenmode is associated with a hydraulic diffusion time scale (eigenvalue) that depends on the intrinsic reservoir properties. The set of eigenmodes forms a basis of the solution space and the analytical solution shows some desirable properties: It is smooth, physically constrained for a closed system, and some parameters have a physical interpretation. As a result, in both single and multiwell problems, the late-time unit slope appears naturally and the curvature penalty does not have to be imposed, thus addressing current knowledge gaps. As an additional contribution, we also propose a mathematically well-defined expression in terms of eigenfunctions to the concept of investigated volume during a well test and a procedure for untangling the single-well deconvolved response from interference effects of neighboring wells under certain conditions. Applications to the analysis of drillstem tests (DST) and reservoir characterization are presented.