Why Laplace Transforms?
Many physical processes are described by the diffusion equation including temperature distributions, diffusion of chemicals, and fluid flow in porous media. My particular interest is in modeling oil reservoirs, so the diffusion or diffusivity equation is of fundamental importance. In cases where a reservoir can be considered uniform and homogeneous, the time variable in the partial differential equation can be transformed using Laplace transforms, eliminating the time derivatives, and converting a partial differential equation into an ordinary differential equation, which is fairly easily solved. Problems involving Laplace transforms also arise in computing statistical functions and other areas of math. For reference, the diffusivity equation used for radial flow of a single fluid near a well in a homogeneous reservoir can be written as follows, where pD represents the dimensionless pressure, rD represents the dimensionless radial distance from a well, and tD represents dimensionless t
