Isothermal water flows in low porosity porous media in presence of vapor–liquid phase change

Abstract

In this article we consider a mathematical model for a low porosity porous medium saturated by water, present both as the liquid and the vapor phase. In the isothermal case we propose a new formulation using a single nonlinear parabolic–hyperbolic equation for the fluid mixture density $X$. We present the derivation of the unified model and a number of numerical simulations based on regularization and Kirchhoff’s transform.

Introduction

We study the flow of a water–vapor mixture saturating low porosity rocks. Such systems are present in many applications, particularly in geology. The assumption of low porosity allows to consider the situation in which the temperature field is constant, because it is stabilized by the dominating volume fraction of the rocks. Therefore, the isothermal configuration can be justified even in the presence of phase change. In the classic approach, once the temperature is fixed, the state of the water is determined by the pressure. Under the assumption of thermodynamical equilibrium, by comparison with phase change pressure, corresponding to the given temperature, we find out whether the water is in the liquid or in the vapor state, or both phases are simultaneously present. Accordingly, mass balance is expressed by PDEs of different nature. The backdraw of such an approach is that it involves the explicit analysis of the interfaces, which is a hard task to achieve, especially in multidimensional problems.

There is a recent surge of interest for such questions in the case of multi-phase multi-component flows through porous media. Contrary to case of two immiscible incompressible phases, for which there is an extensive literature (see e.g.  [1], [2], [3] and references therein), the situation where we deal with at least one compressible phase remains with incomplete mathematical theory. For partial results we refer to  [4]. In  [5] constitutive laws for compressible phase were adapted to Chavent’s global pressure.

One of the main difficulties is that in general we should deal with systems of degenerate parabolic PDEs and it is not clear how to guarantee a priori that saturations remain in the physical range. See  [4] for remarks. Furthermore, there is appearance of single-phase zones occupied by the fluid which is over- (or under-) saturated. In an oversaturated zone the two-phase flow equations degenerate and cannot be longer used, which provokes serious numerical problems. The presence of several components clearly complicates the situation. Motivated by recent applied needs, Panfilov introduced a new formulation, extending the concept of phase saturation so that it may be negative and larger than one. His method allowed using the existing numerical simulators of two-phase flow for modeling single-phase zones. For details we refer to  [6], [7]. A similar approach, but extending different unknowns, was presented by Bourgeat et al. in  [8].

Our situation is much simpler, since we consider water as the only component and in isothermal conditions. Our aim is to describe the physical model simply by a scalar degenerate parabolic equation for the liquid–vapor mixture density, a quantity possessing a natural definition. Of course it applies only to our particular situation, although some generalization may be possible (for instance to non-isothermal cases).

The plan of the paper is the following: In Section  2 we present derivation of the model. In the Section  2.6 we define precisely the notion of the entropy solution and the regularization procedure, allowing us to define the numerical approximation. In Section  3, we present numerical simulations obtained by using the Comsol Multiphysics and the Scilab based numerical code.