The Use of Source and Green's Functions in Solving Unsteady-Flow Problems in Reservoirs

Abstract

Abstract Although it is an old method, the use of Green's functions to solve unsteady flow problems in reservoir engineering is not widely practiced. The reason is that it is difficult to find the appropriate Green's junction. In this study, tables of instantaneous Green's and source functions were prepared that can be used with the Newman's prepared that can be used with the Newman's product method to generate solutions for a wide product method to generate solutions for a wide variety of reservoir flow problems. New solutions for infinite conductivity sources were also prepared. Introduction The transient flow of a slightly compressible fluid in a homogeneous and anisotropic porous medium D, bounded by a surface Se (Fig. 1), is described by the diffusivity equational derived from the continuity equation and Darcy's law? Assuming constant permeabilities, porosity, and fluid viscosity and small pressure gradients everywhere, and neglecting the effect of gravity, the diffusivity equation can be written as Equation (1) where x, y and z are the principal axes of permeability, and the coefficients Nx, Ny, Nz are permeability, and the coefficients Nx, Ny, Nz are the principal diffusivities. When Nx = Ny = Nr (cylindrical systems), the diffusivity equation can be written as Equation (2) The diffusivity constants are given by Equation (3) Many techniques have been used for solving Eqs. 1 and 2. Most of these were first used for solving heat conduction problems and have since been applied by different authors to petroleum engineering. In the literature, most problems were solved either with Laplace transforms or Fourier transforms. One useful method employs Lord Kelvin's instantaneous point source solution. Another method that is of value although very rarely used is the Green's function method. This study presents the point source solution as part of a more general theory of Green's functions. part of a more general theory of Green's functions. This theory is applied in combination with other techniques to yield immediate solutions to difficult flow problems, some of which either have not been published or have been solved by long analytical published or have been solved by long analytical methods or sophisticated numerical techniques only.