Estimates of equivalent permeability for non-Newtonian fluid flow in heterogeneous porous media

Non-Newtonian fluid flow through porous media is of great interest in petroleum engineering, environmental engineering and hydrogeology. An extensive body of literature exists on this topic; several studies are specifically concerned with evaluation of macroscopic models relating the Darcy velocity of the fluid to the pressure gradient; once a macroscopic nonlinear flow law is available, its coupling with the continuity equation results in a nonlinear second-order PDE, whose solution yields the pressure and specific discharge within the domain. At the field scale, the nonlinear problem is further complicated by intrinsic domain heterogeneity, whose interplay with non-Newtonian effects must be accounted for. In such a nonlinear framework, can the heterogeneous permeability distribution be substituted by a representative value, in analogy to the well-known linear (Darcy) case ? Here, we present a simplified approach to the derivation of an equivalent permeability for flow of a purely viscous power-law fluid with flow behavior index n in a heterogeneous porous domain under uniform flow conditions. A standard form of the flow law generalizing the Darcy’s law to non-Newtonian fluids is adopted, with the permeability coefficient being the only source of randomness. The permeability k is taken to vary as a spatially homogeneous and correlated random field, with a given probability density function f(k). Under the ergodic hypothesis, an equivalent permeability is first derived for two limit 1-D flow geometries: flow parallel to permeability variation (serial-type layers), and flow transversal to permeability variation (parallel-type layers). We then investigate whether the equivalent permeability of a 2-D or 3-D isotropic domain can be obtained by a suitable power averaging of 1-D serial and parallel results, generalizing results valid for Newtonian fluids under Darcy’s law. A comparison with an existing analytical solution (confirmed by numerical simulations in the 2-D case), demonstrates that the exponents of power averaging are functions of the flow behavior index n, in variance with the Newtonian (Darcy) case. A sensitivity analysis of the equivalent permeability expression is then performed, quantifying the influence of flow behavior index, shape of the permeability pdf, and permeability or log-permeability variance. As the latter increases, the equivalent permeability is found to decrease for 1-D serial-type layers, and to be a function of n for 1-D parallel-type layers, 2-D and 3-D cases. The equivalent permeability value is highly sensitive to the value of n for very pseudoplastic fluids with n<0.50, less so for dilatant fluids (n>1, not commonly encountered in field cases). For small n, the values of the exponents of the power averaging differ mostly from those valid in the Darcy case, suggesting strong flow channeling. For Newtonian flow, all our expressions for the equivalent permeability reduce to those derived in the past.