Semianalytical Approach to the Computation of the Laplace Transform of Source Functions

The method of “Sources and Sinks” has has found wide application to the solution of the diffusion equation for linear conduction of heat in infinite, semi-infinite and finite solids; see, for example, (Carslaw and Jaeger, 1988). Since reservoir flow of a slightly compressible, constant viscosity fluid is governed by the diffusion equation, fundamental solutions of the one-dimensional diffusion equation are routinely used to build solutions to three-dimensional flow problems using the Newmann product method (Newmann, 1936). These solutions can be rapidly generated for constant-rate production (or injection) to a uniform-flux line or plane in an anisotropic homogeneous reservoir; (see for example, (Gringarten and Rame, 1973)). However, for variable-rate, variable-flux or complex-geometry (heterogeneous reservoir) problems, solutions are most easily generated by manipulating Laplace transforms of the product solutions and inverting to the time domain using a numerical inversion procedure such as the Stehfest algorithm (Stehfest, 1970). Raghavan and Ozkan (1994) presented complete analytical expressions for the Laplace transforms of the 3-D source functions. For flow in rectangular parallelepiped reservoirs, their expressions involve triple infinite series.