Modeling of shallow aquifers in interaction with overland water

摘要

In this paper, we present a new class of efficient models for water flow in shallow un-confined aquifers, providing an alternative to the classical but less tractable 3d-Richards model. Its derivation is guided by two objectives: to obtain a model that has low computational cost and yields relevant results on every time scale. Thus, we keep track of two types of flow that occur in such a context and are dominant when the ratio of thickness to longitudinal length is small: the first is dominant on a small time scale and is described by a vertical 1d-Richards problem; the second corresponds to a large time scale, when the evolution of the hydraulic head becomes independent of the vertical variable. These two types of flow are appropriately modeled by a one-dimensional and two-dimensional system of PDE boundary value problems, respectively. They are coupled at an artificial level below which the Dupuit hypothesis holds true (i.e., the vertical flow is instantaneous) so that the global model is mass conservative. Tuning the artificial level, which can even depend on an unknown of the problem, we obtain the new class of models. Using asymptotic expansions, we prove that the 3d-Richards model and each model in the class behave identically on every considered time scale (short, intermediate, and large) in thin aquifers. The results are illustrated by numerical simulations, and it is demonstrated that they fit well with those obtained by the original 3d-Richards model even in non-thin aquifers.