Porous media fluid motion

The first question is addressed in the second section through a discussion of porous media characteristics and the distribution of fluids in porous media. Fluid motion in porous media is discussed in the third section, and partitioning and reactive processes are discussed in the fourth section. 

The governing flow equation describing flow through as porous medium is known as Darcy s law, which is a relationship between the volumetric flow rate of a fluid flowing linearly through a porous medium and the energy loss of the fluid in motion. 

All these different mechanisms of mass transport through a porous medium can be studied experimentally and theoretically through classical models (Darcy s law, Knudsen diffusion, molecular dynamics, Stefan-Maxwell equations, dusty-gas model etc.) which can be coupled or not with the interactions or even reactions between the solid structure and the fluid elements. Another method for the analysis of the species motion inside a porous structure can be based on the observation that the motion occurs as a result of two or more elementary evolutions that are randomly connected. This is the stochastic way for the analysis of species motion inside a porous body. Some examples that will be analysed here by the stochastic method are the result of the particularisations of the cases presented with the development of stochastic models in Sections 4.4 and 4.5. 

A fluid s motion is a function of the properties of the fluid, the medium through which it is flowing, and the external forces imposed on it. For onedimensional steady laminar flow of a single fluid through a homogeneous porous medium, the relationship between the flow rate and the applied external forces is provided by Darcy s law ... 

The third problem is known as the Saffinan-Taylor instability of a fluid interface for motion of a pair of fluids with different viscosities in a porous medium. It is this instability that leads to the well-known and important phenomenon of viscous fingering. In this case, we first discuss Darcy s law for motion of a single-phase fluid in a porous medium, and then we discuss the instability that occurs because of the displacement of one fluid by another when there is a discontinuity in the viscosity and permeability across an interface. The analysis presented ignores surface-tension effects and is thus valid strictly for miscible displacement. ... 

Problem 9-22. Flow in a Brinkman Medium. Fluid flow in a packed bed or porous medium can be modeled as flow in a Brinkman medium, which we may envision as a bed of spherical particles. Each particle in the bed (there are n particles per unit volume) exerts a drag force on the fluid proportional to fluid velocity relative to the particle given by Stokes law, i.e., ( —Gtt/hiu, where a is the characteristic size of a bed particle). Thus the equations describing the fluid motion on an averaged scale (averaged over many bed particles, for example) are... 

A problem that is somewhat analogous to the instability of an accelerating interface occurs when two superposed viscous fluids are forced by gravity and an imposed pressure gradient through a porous medium. This problem was analyzed in a classic paper by Saffman and Taylor.16 If the steady state is one of uniform motion with velocity V vertically upwards and the interface is horizontal, then it can be shown that the interface is stable to infinitesimal perturbations if... 

In many applications, we may wish to describe the motion of two fluids, which may be miscible (i.e., the interfacial tension is zero, though the fluids may have different viscosities or other properties) or immiscible (oil and water, for example) in a porous medium. At the microscale, within the pores, there is a well-defined interface separating two immiscible... 

Problem 12-17. Buoyancy-Driven Instability of a Fluid Layer in a Porous Medium Based on Darcy s Law. We consider the classical Rayleigh-Benard problem of a fluid layer that is heated from below, except in this case, the fluid is within a porous medium so that the equations of motion are replaced with the Darcy equations, which were discussed in Subsection Cl of this chapter. Hence the averaged velocity within the porous medium is given by Darcy s law... 

Problem 12-18. Buoyancy-Driven Instability of a Fluid Layer in a Porous Medium Based on the Darcy-Brinkman Equations. A more complete model for the motion of a fluid in a porous medium is provided by the so-called Darcy Brinkman equations. In the following, we reexamine the conditions for buoyancy-driven instability when the fluid layer is heated from below. We assume that inertia effects can be neglected (this has no effect on the stability analysis as one can see by reexamining the analysis in Section H) and that the Boussinesq approximation is valid so that fluid and solid properties are assumed to be constant except for the density of the fluid. The Darcy Brinkman equations can be written in the form... 

A porous medium consists of a packed bed of solid particles in which the fluid in the pores between particles is free to move. The superficial fluid velocity V is defined as the volumetric flow rate of the fluid per unit of cross-sectional area normal to the motion. It is the imbalance between the pressure gradient (VP) and the hydrostatic pressure gradient (pg) that drives the fluid motion. The relation that includes both viscous and inertial effects is the Forscheimer equation [47]... 

On the other hand, when a macroscopic pressure gradient VP is applied to the porous medium, the fluid percolates through it with a Darcy velocity U. Additionally, the electrolyte flowing in the interstices affects the equilibrium ion distribution within the Debye layer, so that these ions are also set into motion. This results in a macroscopic electric current density I flowing through the porous medium in the absence of any external electric field. 

Some simple and useful examples are worked out in Sec. IV. First, the simultaneous motions of fluid and charges through a spatially periodic porous medium are analyzed linearization enables us to summarize the results by means of four electro-osmotic tensors that relate the fluxes of charge and matter to the potential and pressure gradients. The cases of a semi-finite void space limited by a plane solid wall and of plane or circular channels are briefly addressed the symmetry properties deduced from the general Onsager theorem are verified these simple configurations are further used to check the numerical routines. 

Contact angle hysteresis is one mechanism that hinders motion of fluid drops in a porous medium. Find an expression for the pressure drop... 

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