Formal derivation of seawater intrusion models

In this paper, we consider the flow of fresh and saltwater in a saturated porous medium in order to describe the seawater intrusion. Starting from a formulation with constant densities respectively of fresh and of saltwater, whose velocity is proportional to the gradient of pressure (Darcy's law), we consider the formal asymptotic shallow water limit as the ratio between the thickness and the horizontal length of the porous medium tends to zero. In this limit, we derive the Dupuit-Forchheimer condition and as a consequence reduced models of Boussinesq type both in the cases of unconfined and confined aquifers

A finite volume scheme for a seawater intrusion model with cross-diffusion

International audienceWe consider a finite volume scheme for a seawater intrusion model. It is based on a two-point flux approximation with upwind mobilities. The scheme preserves at the discrete level the main features of the continuous problem: the non-negativity of the solutions, the decay of the energy and the control of the entropy and its dissipation. Moreover the scheme converges towards a weak solution to the problem. Numerical results are provided to illustrate the behavior of the model and of the scheme

Convergence of a positive nonlinear control volume finite element scheme for an anisotropic seawater intrusion model with sharp interfaces

International audienceWe consider a degenerate parabolic system modelling the flow of fresh and saltwater in an anisotropic porous medium in the context of seawater intrusion. We propose and analyze a nonlinear Control Volume Finite Element scheme. This scheme ensures the nonnegativity of the discrete solution without any restriction on the mesh and on the anisotropy tensor. Moreover It also provides a control on the entropy. Based on these nonlinear stability results, we show that the scheme converges towards a weak solution to the problem. Numerical results are provided to illustrate the behavior of the model and of the scheme

A gravity current model with capillary trapping for oil migration in multilayer geological basins

International audienceWe propose a reduced model accounting capillary trapping to simulate oil migration in geological basins made of several rock types. Our model is derived from Darcy type models thanks to Dupuit approximation and a vertical integration in each geological layer. We propose a time-implicit finite volume scheme which is shown to be unconditionally stable and to admit discrete solutions. Numerical outcomes are then provided in order to illustrate the behavior of our reduced model

A gravity current model with capillary trapping for oil migration in multilayer geological basins

We propose a reduced model accounting capillary trapping to simulate oil migration in geological basins made of several rock types. Our model is derived from Darcy type models thanks to Dupuit approximation and a vertical integration in each geological layer. We propose a time-implicit finite volume scheme which is shown to be unconditionally stable and to admit discrete solutions. Numerical outcomes are then provided in order to illustrate the behavior of our reduced model

Finite speed of propagation and waiting time for a thin film Muskat problem

International audienceFinite speed of propagation is established for non-negative weak solutions to a thin film approximation of the two-phase Muskat problem. The temporal expansion rate of the support matches the scale invariance of the system. Moreover, we determine sufficient conditions on the initial data for the occurrence of waiting time phenomena