Flow through porous media due to high pressure gradients

Abstract

While Darcy’s equations are adequate for studying a large class of flows through porous media, there are several situations wherein it would be inappropriate to use Darcy’s equations. One such example is a flow wherein the range of pressures involved is very large and high pressures and pressure gradients are at play. Here, after developing an approximation for the flow through a porous solid, that is a generalization of an equation developed by Brinkman, we study a simple boundary value problem that clearly delineates the difference between the solution to these equations and those due to the equations that are referred to as “Darcy Law”. We find that the solutions for the equations under consideration exhibit markedly different characteristics from the counterpart for the Brinkman equations or Darcy’s equations (or the Navier–Stokes equation if one neglects the porosity) in that the solutions for the velocity as well as the vorticity lack symmetry and one finds the maximum value of the vorticity occurs at the boundary near which the fluid is less viscous in virtue of the pressure being lower. We also find that for a certain range of values for the non-dimensional parameters describing the flow, boundary layers develop in that the vorticity is confined next to the boundary adjacent to which the viscosity is lower, such boundary layers being absent in the other classical cases.

Introduction

The most popular equation to describe the flow of fluids through porous solids is that due to Darcy [1]. This equation is however only valid if a plethora of conditions are met. For instance, such an approach to the flow of fluids through porous solids is incapable of predicting the stresses or strains in a solid. It merely provides the volumetric flow rate as a function of the pressure gradient. More importantly, not only are the deformations in the solid ignored and the solid treated as a rigid porous medium, the governing equations for the solid are completely ignored. Also, the frictional effects in the pores are assumed to have a very special structure and the frictional effects within the fluid are ignored. It is also tacitly assumed that the viscosity of the fluid is a constant and it does not depend on the pressure. While this might be reasonable under some operating conditions, it is definitely incorrect if the range of pressures is large. When such is the case, the viscosity can no more be considered a constant and hence the drag due to the solid, which is essentially a consequence of the frictional effects at the pore will not be that which is assumed in the derivation of Darcy’s equation.

Much of the early works on flow through porous media are applications of Darcy’s equation and experimental corroborations for the same. This early work was followed by several attempts at modifying and generalizing Darcy’s early work, Brinkman’s equation (see [2], [3], [4]) being one such example. Another important development concerning flows through porous media is due to Biot who developed a system of equations for describing the propagations of waves in solids that are infused with a fluid and suffering small deformations (see [5], [6], [7]). Copious references to studies concerning flow through porous media can be found in the treatises by Muskat [8], Polubarinova-Kochina [9] and Scheidegger [10].

Recently, Rajagopal [11] has shown how a hierarchy of models can be developed, within the context of mixture theory,¹ that is a consequence of a systematic sequence of approximations, the classical equation due to Darcy² being a consequence of the most severe of approximations. Relaxing the assumptions progressively leads to a corresponding hierarchy of a set of equations to describe the flow through porous solids. Here, we shall be concerned with one such approximation which is a generalization of the model due to Brinkman [2], [3], that takes into account the fact that the viscosity of the fluid and the frictional force (drag) due to the solid at the pores can depend upon the pressure. It is well known, even within the context of a single constituent fluid that the viscosity of the fluid can depend upon the pressure. Stokes [31] recognized that in general one cannot assume that the viscosity is independent of pressure and that in certain applications such as flows in channels and pipes, wherein the range of the pressure is not that great, viscosity could be assumed to be a constant. Stokes is very careful to make this point abundantly clear. Barus [32] recognized that the dependence of the viscosity on pressure could be dramatic and he provided a exponential dependence of the viscosity on the pressure. There was a great deal of interest in characterizing the pressure dependence of the viscosity in the years that followed and a thorough discussion of much of this work until 1930 can be found in the book by Bridgman [33]. Andrade [34] also provided a relationship between the pressure, density, temperature and viscosity that was exponential. Bridgman himself carried out numerous experiments on the pressure dependence of viscosity for various organic liquids (see [35], [33]). From these and more recent experiments (see [36], [37], [38], [39], [40], [41], [42]) it is very clear that while the viscosity can vary by a factor of as much as 10 to the power of 10%, the density changes are insignificant by comparison. For instance, when the operating pressure changes from 2 to 3 GPa the density changes by merely 4%, while the viscosity changes by 10 to the 10%. The density change due to changes in the pressure correlates well with an empirical formula provided by Dowson and Higginson [43]. Thus, when operating under a wide range of pressures it is necessary to take into account the variation of the viscosity with pressure. This fact is well recognized in the field of elastohydrodynamics (see [44]).

Darcy’s equation is an approximation for the balance of linear momentum for the fluid that takes into account the frictional resistance at the solid boundaries of the pores to the fluid that is flowing. This frictional resistance is assumed to be directly proportional to the relative velocity between the fluid and the porous solid and the constant of proportionality that is commonly referred to as the drag coefficient depends on the viscosity of the fluid. While Darcy’s law takes into account the frictional resistance, due to the solid surfaces at the pores, to the flow of the fluid, it does not take into account the frictional effects within the fluid bulk. Also, Darcy’s equation assumes that the resistance at the pores depends on the viscosity of the fluid. If the viscosity of the fluid is to be pressure dependent, it then would stand to reason that the drag coefficient would also depend upon the pressure. A generalization of Darcy’s equation that takes the frictional forces within the fluid into effect was developed by Brinkman. He was interested in an equation that would give in two limits the equation due to Darcy for flow through porous media, and the equation due to Stokes for slow flows. He thus proposed a generalization of Darcy’s equation.

Brinkman’s approximation was not carried out by starting with the balance equations for both the solid and the fluid that is flowing through and the attendant interactions between the two constituents, i.e., Brinkman did not carry out a systematic approximation within the context of mixture theory. Rather, he developed the model based on physical arguments for the flow of the fluid and the fact that the approximation had to tend to the two limits, Stokes’ equations and Darcy’s equation. Within the context of mixture theory, the following assumptions will lead to the equations that Brinkman developed.