MATHEMATICAL ANALYSIS OF A DUPUIT-RICHARDS MODEL

Analyse mathématique d'un modèle Dupuit-Richards

Safaa Al Nazer (1) , Carole Rosier (1) , Munkhgerel Tsegmid (2)

1 LMPA - Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville
2 MUST - Mongolian University of Science and Technology

This article is concerned with a model which is an alternative to the 3D-Richards equation to describe the flow of water in shallow aquifers. The model couples the two dominant types of flow existing in the aquifer. The first is described by the classic Richards problem in the upper capillary fringe. The second results from Dupuit's approximation after vertical integration of the conservation laws between the bottom of the aquifer and the saturation interface. The final model consists of a strongly coupled system of parabolic-type pde which are defined in a time-dependent domain. First, we show how taking the low compressibility of the fluid into account eliminates the nonlinearity in the time derivative of the Richards equation. Then, the general framework of parabolic equations is used in non-cylindrical domains to give a global in time existence result to this problem.

Keywords: Dupuit-Richards equations; free boundary problems; global existence; weak solution; fluid flow modeling

Domains: Mathematics [math]; Analysis of PDEs [math.AP]; Physics [physics]; Mechanics [physics]; Fluid mechanics [physics.class-ph]

Safaa Al Nazer, Carole Rosier, Munkhgerel Tsegmid. MATHEMATICAL ANALYSIS OF A DUPUIT-RICHARDS MODEL. Electronic Journal of Differential Equations, 2022, 2022 (6), pp.1-22. ⟨hal-03551653⟩