Taking into account air pressure variations in subsoil air-water

The Richards equation is widely used in hydrogeology for modelling flow in an unsaturated porous media. It is a simplified version of the general two-phase model, which assumes that air pressure is invariant. To fully take into account the presence of air which may be trapped in the subsoil, it will be necessary to consider the non simplified two-phase flow. The aim of this work is to examine different numerical methods for trading these two models and to compare the results obtained using 3-dimensional experiments. In our study we assume that water is incompressible and that air is slightly compressible. We model both air and water flows using generalized Darcy's law together with mass conservation. Under the hypothesis that the capillary pressure depends only on water saturation, the two-phase flow is described by a system of two nonlinear equations, the saturation equation and the pressure equation, whose unknows are water saturation and air pressure. Richards's equation is obtained, up to a change of variable, by supposing that air pressure is uniformly equal to atmospheric pressure. Thus the flow is described by the saturation equation alone, as for a single-phase flow. Without this hypothesis, the coupling between the air equation and the pressure equation is strong: the pressure equation contains indeed a saturation diffusion term, and because of air compressibility the pressure equation is not stationary. A formulation for two-phase flow using the global pressure (a device introduced for petroleum engineering, generally for incompressible flows) is discussed. The idea is to eliminate the saturation diffusion term in the pressure equation by introducing a new artificial variable, homogeneous to a pressure, using the assumption that the capillary pressure remains low. The mathematical properties of the new system are interesting, but we will see that it is not valid for low saturations and so we turn our attention to a standard two-phase model. For time discretization of the two-phase equations we use a variation of the IMPES method (implicit pressure explicit saturation): we decouple the air and pressure equations. For the spatial discretization, we use finite volumes for convection, and mixte finite elements of lowest order for diffusion. Convection terms are upwinded with a Godunov scheme. The treating of the diffusion term is difficult because it varies rapidly in space. We will compare two different methods for handling this term. One is the classical method using harmonic averages of the diffusion coefficients, the other uses implicit upwinding to evaluate these coefficients. We use a Newton method to solve the nonlinear systems. Numerical 3-D simulation of infiltration under rainfalls will illustrate the presentation.