A methodology for the derivation of non-Darcian models for the flow of generalized Newtonian fluids in porous media

Abstract

A methodology is presented for the development of analytic models that can suitably describe the flow of generalized Newtonian fluids in porous media. First, the volume averaging technique is employed for the derivation of generalized non-Darcian models concerning the creeping flow of inelastic non-Newtonian fluids in porous media. Two macroscopic properties are involved in these models: the effective permeability and average viscosity. The approximate Rabinowitch–Mooney equation is used to obtain analytical solutions for the flow of Meter and mixed Meter-and-power law fluid models through a pore of arbitrary cross-sectional shape. The analytical solutions are integrated into numerical simulators of the one-phase flow of the foregoing fluids in two-dimensional pore networks. The effective medium theory (EMT) is coupled with pore network simulations in order to replace the disordered porous medium with an effective medium that is described by two anisotropic hydraulic conductances and two effective hydraulic pore radii. Finally, closed expressions are developed to relate the superficial flow velocity with the pressure gradient and the macroscopic properties with the effective hydraulic pore radii, rheological parameters and pressure gradient. The new flow models predict satisfactorily numerical simulations of the flow of non-linear fluids in theoretical pore networks, and experiments of the flow of non-Newtonian oils in two-dimensional artificial glass-etched pore networks. Any observed discrepancies are attributed to the use of a small number of pore structure parameters for the homogenization of the porous medium, as well as to uncertainties embedded into the estimated values of rheological parameters.