The Laplace Transform Finite Difference Method for Simulation of Flow Through Porous Media

Abstract

A new numerical method, the Laplace transform finite difference (LTFD) method, was developed to solve the partial differential equation (PDE) of transient flow through porous media. LTFD provides a solution which is semianalytical in time and numerical in space by solving the discretized PDE in the Laplace space and numerically inverting the transformed solution vector. The effects of the traditional treatment of the time derivative on accuracy and stability are rendered irrelevant because time is no longer a consideration. For a single time step, LTFD requires no more than eight matrix solutions and an execution time eight times longer than the analogous finite difference (FD) requirement without an increase in storage. This disadvantage is outweighed by an unlimited time step size without any loss of accuracy, a superior accuracy, and a stable, nonincreasing round off error. Thus, a problem in standard FD format may require several hundred time steps and matrix inversions between the initial condition and the desired solution time, but LTFD requires only one time step and no more than eight matrix inversions to achieve a more accurate result.