Correlation of the two-phase flow coefficients of porous media with the rheology of shear-thinning fluids

Abstract

The macroscopic equations of the two-phase flow in porous media are generalized for the case that the displacing non-wetting phase (NWP) is a shear-thinning fluid, and a numerical scheme of inverse modeling is developed to estimate simultaneously the capillary pressure, P c( S nw), non-wetting phase, k rnw( S nw), and wetting phase, k rw( S nw), relative permeability curves, from unsteady-state experiments. The parameter estimation procedure is demonstrated by using transient experimental data of the immiscible displacement of a Newtonian wetting phase (WP) by a shear-thinning NWP from a glass-etched planar pore network. Scaling laws are developed for the width of the frontal region, by extending concepts of the gradient invasion percolation to the rate-controlled displacement of a Newtonian fluid by a power-law fluid. These scaling laws are coupled with the macroscopic fractional flow equations in order to express the width of the frontal region and macroscopic parameters of P c( S nw), k rnw( S nw) and k rw( S nw) as functions of the capillary number ( Ca), in terms of universal scaling exponents. Two parameter estimation procedures are adopted: (i) in the first approach, the NWP is treated as a mixed Meter-and-power-law fluid and a non-Darcian flow model is used; and (ii) in the second approach, the NWP is treated as a pseudo-Newtonian fluid and classical Darcy law with a constant apparent viscosity are used. The variation of the front width with Ca follows a power law, the exponent of which agrees satisfactorily with that predicted by scaling relationships. The estimated P c( S nw) and k rw( S nw) are increasing functions of Ca. The estimated P c( S nw) is a strongly increasing function of Ca, even at high values of this parameter, and almost independent on the flow model considered. The shape of k rnw( S nw) changes from concave at low Ca values, where the width of the frontal region is large, to convex at increasing Ca values, where the front width is small and the displacement is dominated by viscous forces at gradually larger scales. The variation of the parameters of P c( S nw), k rnw( S nw), k rw( S nw) with Ca agrees semi-quantitatively with the scaling relations only in the first approach, where the non-Darcian flow model is used.