Continuum Approach To Single Phase Flow InPorous Media

Abstract

A volume averaging technique is employed to study single phase fluid (Newtonian or non-Newtonian) flow in porous media. New terms evolve from the volume averaged governing equations: macroscopic viscosity, hydraulic dispersivity, shear factor and tortuosity. The macroscopic viscosity is the viscous diffusion coefficient for the averaged flow momentum. When fluid is Newtonian, the macroscopic viscosity reduces to the viscosity of the fluid. When the fluid is non-Newtonian, the macroscopic viscosity is the apparent viscosity of the fluid under the average microscopic shear stress and shear rate conditions. The hydraulic dispersivity is the dispersion coefficient of the flow induced diffusion. The hydraulic dispersivity increases with the flow velocity. When the flow is very strong, it is proportional to the flow velocity. The hydraulic dispersivity is the same for momentum dispersion, energy (heat) dispersion and tracer (mass) dispersion. The shear factor is the porous medium resistivity to flow. It is the result of the shear stress exerted on the porous medium matrix and the flow velocity (spatial) fluctuation in the microscopic scale. In the limit of creeping flow, the flow velocity (spatial) fluctuation has a negligible effect on the flow and the shear factor reduces to the reciprocal of permeability. When the flow is very strong, the shear factor is proportional to the flow velocity. 1 Definitions and Volume Averaging Volume averaging is performed on a Representative Elementary Volume, REV. An REV is a conceptual space unit which is the minimum volume inside the porous medium within which measurable characteristics of the porous medium Transactions on Engineering Sciences vol 9, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533 104 Advances in Fluid Mechanics become continuum quantities. The REV is so defined that an REV can be regarded as a macroscopic unit consisting of a large sum of microstructures. To help us understand the concept of the REV, let us take a brief review on the fluid continuum treatment. When the motion of fluids or physical properties of a fluid is to be deduced, one often assumes an infinitesimal volume or a size. For example, the fluid density at a given local position is the ratio of fluid mass over the volume when the volume approaches the volume of the assumed point with its centroid at the local position of concern. However, no one really cares what the size of the point is quantitatively. The fluid density is given as p(/>) = lim 4$ (1) AV->VAV The characteristic volume Vp is called the physical point (or material point) of the fluid at the mathematical point P. The concept of the mathematical point is identical to an REV. When the volume of consideration is smaller than Vp, the continuum breaks down. When the sampling volume is very small as it is suggested by the term infinitesimal used routinely in this connection in fluid mechanics texts, the fluid mass over volume ratio becomes undefined. When the sampling volume is at least of Vp, we obtain a fictitious smooth medium. When a compressible gas flows at low pressure or the molecules are large (macromolecules) and the dimension of the channel is small, the continuum concept breaks down. Under this circumstance, slip-flow can occur. That the solid matrix is immobile and is rigid or can sustain pressure and stresses makes a in porous media different from the mathematical in a pure fluid. The overall pressure and stresses for the fluid and solid matrix mixture are meaningless. However, the pressure and stresses for a fluid phase are valid measures so long as that within an REV the void space is interconnected, i.e., no barrier to prevent the filling fluid from moving around. The intrinsic phase average of a quantity, * is the local (microscopic) quantity. The deviation of a quantity, O, is the difference between the local value and the intrinsic phase average value of that quantity, 6 = e (3) Here O can be a scalar or a vector. The volume average of a quantity, O, is the average taken over the entire REV i /. O = ~ \&dV (4) V Jv The volume average value and the intrinsic phase average value are proportionally. For the flow velocity, it is given by ?=evi (5) where e is the porosity of the volume ratio, VJV. Transactions on Engineering Sciences vol 9, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533