A General Analytical Solution for the Multidimensional Hydraulic Diffusivity Equation by Integral Transform Technique
Authors: Couto Paulo , Marsili Marcelo Danemberg 
Published: June 2014

Abstract
Analytical solutions for the Hydraulic Diffusivity Equation (HDE) are of great importance in reservoir engineering problems. As a common practice, they are used for numerical code validation, providing benchmark results for many situations. While numerical solutions are progressive in time and discrete in space, analytical solutions are direct and continuous. Besides, the former presents approximate results, while the latter's are exact. Furthermore, analytical solutions can provide asymptotic behaviors and parametric analysis without requiring a complete numerical solution.
This study presents the application of the Integral Transform Technique in the development of a general analytical solution for the multidimensional hydraulic diffusivity equation. This solution is valid for the transient monophasic flow through homogeneous and isotropic porous media in any orthogonal coordinate system within a regular boundary domain. A general boundary condition is considered to account for any prescribed flow or pressure at the reservoir limits. A space and time-dependent source term is included in the partial differential equation to represent either production or injection well(s). The solution methodology deals directly with time-dependent well rates and boundary conditions, sparing the use of the superposition principle.
The solution of the flow equation is presented in a generalized form from which a huge combination of boundary conditions, well locations and domain dimensions can be analyzed and its solution derived. A methodology for the application of the general solution is provided together with auxiliary tables with parameters pertinent to the solutions for the Cartesian coordinate system. The general solution is then applied in two selected case studies: a single well sealed reservoir and an inverted five-spot pattern reservoir.

Introduction
Since the advances in computing machinery technology, progressively more researchers have been concentrating efforts in the development and application of numerical methods of solution for the partial differential systems that govern diffusion-convection phenomena, such as the best know finite differences and finite elements method1. Notwithstanding, analytical methods have kept some important advantages when compared with numerical ones. Analytical methods provide exact solutions, continuous in space and time, while numerical codes work with discrete points in the domain and progressive steps in time. Analytical solutions provide straightforward parametric variation inspections without requiring a complete numerical solution. Also, analytical solutions are often treated as benchmarks for numerical code validation2. Complementarily, Cotta1 stated that " … a basic knowledge of analytical method might serve as an interface for new and powerful developments in numerical methods".
The analytical solutions of linear diffusion problems have been analyzed and compiled by Mikhailov and Özisik3 and Özisik4,5, where several different classes of heat and mass diffusion formulations are systematically solved by different analytical and approximated methods, including the Integral Transform Technique (ITT) and the Separation of Variables Method. The ITT has added great advantage over other similar techniques, such as the Laplace-Transform Technique, for it possesses easy transformation and direct inversion procedure. Although literature on the ITT has extensively addressed partial differential problems governed by heat and mass transport phenomena, applications focusing on reservoir engineering problems have been few.