Large-time solutions for groundwater flow problems using the relationship of small p versus large t

Abstract

An approximate solution is useful if the corresponding analytical solution is complicated and difficult to accurately evaluate. In the past, the relationship of small p versus large t was commonly applied to the Laplace domain solution and could successfully obtain a large-time solution in the groundwater area. The large-time solution usually has a simpler form than the analytical solution and is much easier for estimating the transient behavior of the groundwater flow system. However, Chen and Stone (1993) pointed out that the use of this relationship might fail to yield the correct solution in calculating the wellbore flux for the constant head test problem. Later, Mathias and Zimmerman (2003) indicated that a poor result was obtained by Gerke and van Genuchten (1993) when using the relationship of small p versus large t to derive the water transfer coefficient for the dual-porosity media problem. This note is to show the detailed mathematical derivations involved in the issues that Chen and Stone (1993) and Gerke and van Genuchten (1993) addressed and to ensure that the relationship of small p versus large t is correct to obtain a large-time solution for transient groundwater flow problems.