A General Analytical Solution for the Multidimensional Transient Linear Hydraulic Diffusivity Equation in Heterogeneous and Anisotropic Porous Media

Abstract

Abstract A general analytical solution for the multidimensional transient linear hydraulic diffusivity equation is presented and its application is discussed in this work. The analytical solution is valid for heterogeneous and anisotropic porous media in any orthogonal coordinate system within a regular boundary domain. A source term, dependent on position and time, was included in the partial differential equation to represent production and/or injection wells. A general boundary condition is considered to account for a constant pressure boundary, a no-flow boundary (symmetry or impermeable boundary) or any prescribed flow boundary. The Generalized Integral Transform Technique (GITT) was used to obtain the analytical solution of the proposed problem. This technique transforms a complex partial differential equation into a system of simpler ordinary differential equations that can be solved by analytical, numerical or hybrid methods. Here, we consider a fully analytical solution to the problem. The general solution is then applied to a one-phase tank model in a homogeneous and isotropic porous medium for validation and analysis. A wide variety of applications can be envisioned and discussed for the proposed general solution: well testing in regular and irregular domains, analysis of local reservoir heterogeneities, analysis of reservoir anisotropy degree, analysis of stream lines in naturally fractured and hydraulic fractured reservoirs, formation damage quantification, reservoir shape factor calculations, oil recovery in different waterflood patterns, amongst many others. These applications are discussed and analyzed in the light of the presented general analytical solution. Introduction Pressure and flow rate are two important quantities to be determined in reservoir engineering problems. While the pressure at any point in the reservoir is completely defined by its numerical value (it is a scalar quantity), the flow rate is defined by its value and direction. Experience has shown1 that there is flow rate when the pressure distribution in not uniform within the reservoir. The magnitude and direction of the flow rate depend on the pressure distribution and the flow is always in the direction of decreasing pressure.