Abstract:

In this paper we addressed and elaborated on three critical issues concerning the Laplace transform solution of the Richards equation: the convergence of the original function, the steps for inverse transform, and the applicability of the equation. By presenting theoretical counterexamples, employing integral transformations and substitution schemes as well as real engineering cases, we demonstrated that assuming the convergence of the original function can lead to significant errors. Therefore, it is necessary to assess the convergence of the infiltration function before applying the Laplace transform. We also elucidated the steps for solving the Laplace inverse transform, theoretically addressing a gap in geotechnical literature. Furthermore, we reveal an anomaly where, in certain applications, the Laplace solution may show higher moisture content in lower soil layers than in upper layers. We analyzed this phenomenon from both physical and mathematical perspectives, highlighting inherent limitations in this solution. The findings of this study may help to theoretically strengthen the framework of unsaturated soil theory.

Key words: soil mechanics, Richards Equation, inverse Laplace transform, unsaturated infiltration, solution of differential equation