Dirac Delta and Gaussian Distribution for Solving the Three-Dimensional Transient Groundwater Flow Equation

Abstract

This paper presents an analytical solution to the three-dimensional transient groundwater flow equation in a homogeneous, isotropic, and infinite confined aquifer using the Fourier transform. The governing equation is solved based on two initial conditions: a Dirac Delta distribution representing an instantaneous point source, and a Gaussian distribution representing a smooth, localized initial perturbation. To verify the accuracy of the analytical solution of the Gaussian initial condition, a numerical solution using the Finite Difference Method with a Forward Time Centred Space scheme is performed with comparisons made at time various times. Errors are quantified using the  norm. A grid convergence study confirms second-order spatial convergence which is consistent with the theoretical truncation error of the central difference scheme. The analytical solutions provide exact deterministic benchmarks for three-dimensional transient groundwater flow problems establishing a foundation for future stochastic extensions.