Continuum approach to single phase flow in porous media

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.