Modelling Flow through Porous Media under Large Pressure Gradients

The most interesting and technologically important problems in the study of flow through porous media involve very high pressures and pressure gradients in the flow do- main such as enhanced oil recovery and carbon dioxide sequestration. The popular Darcy or Brinkman models do not take into account the changes in the fluid properties (like viscosity) due to high pressures and temperatures, or the deformation of the solid itself as the fluid flows through it. We focus on the pressure dependence of viscosity and show that its significance in these problems cannot be neglected.

Mixture theory is employed as the tool to develop models for this task. The popular models due to Darcy and Brinkman (and their generalizations) are derived using a general thermodynamic framework which appeals to the criterion of maximal rate of entropy production. Such a thermodynamic approach has been used with great success to describe various classes of material response and here we demonstrate its use within the context of mixture theory. One such generalization of the Brinkman model takes into account the variation of the viscosity and the drag coefficient with the pressure and is used in the problems studied subsequently.

We then consider the steady flow of a fluid through a porous slab, driven by a large pressure gradient, and show that the traditional approach that ignores the variation of the viscosity and drag with the pressure greatly over-predicts the mass flux taking place through the porous structure. While incorporating the pressure dependence of viscosity and drag leads to a ceiling flux, the traditional approaches lead to a continued increase in the flux with the pressure difference.

The effect of inhomogeneities and anisotropy of the porous medium is investigated by modifying the previous problem to prescribe the drag coefficient as a piecewise constant, positive definite second order tensor. Finally, we allow for the possibility that the flow is unsteady, the viscosity and drag are dependent on the pressure and consider the flow of a fluid due to a pulsatile forcing pressure at one end of a rigid, homogenoues, isotropic solid while the other end is open to the atmosphere. In contrast to certain non-Newtonian fluids where the volumetric flux is enhanced by pulsating the pressure gradient about a non-zero mean value, we find that pulsations in the pressure diminish the volumetric flux in case of the flow through a porous medium when the fluid viscosity is considered to be pressure dependent.