Generalized Newtonian fluid flow in porous media

Single-fluid-phase porous medium systems are typically modeled at an averaged length scale termed the macroscale, and Darcy's law is typically relied upon as an approximation of the momentum equation under Stokes flow conditions. Standard approaches for modeling macroscale single-fluid-phase flow of generalized Newtonian fluids (GNFs) extend the standard Newtonian model based upon Darcy's law using an effective viscosity and assuming that the intrinsic permeability is invariant with respect to fluid properties. This approach results in a need to perform an experiment for a non-Newtonian fluid, the introduction of effective parameters that are not tied to known microscale physics, and uncertainty regarding the dependencies of the fitted empirical parameter on system properties. We use the thermodynamically constrained averaging theory (TCAT) to examine the formulation and closure of a macroscale model for GNF flow that is consistent with microscale conservation principles and the second law of thermodynamics. A direct connection between microscale and macroscale quantities is used to formulate an expression for interphase momentum transfer for GNF flow in porous medium systems. Darcy's law is shown to approximate momentum transfer from the fluid phase to the solid phase. Momentum transfer is found to depend on the viscosity at the solid surface, which is only invariant for Newtonian flow. TCAT is used to derive a macroscale equation for the hydraulic resistance based on accessible fluid and solid properties. This hydraulic resistance may be used in the same way that hydraulic conductivity is typically used to model flow at the macroscale, and it includes parameters that can be calculated a priori, without the need to carry out microscale simulations, or experiments, for any GNF. The TCAT approach is validated for four model isotropic and anisotropic media and five Cross-model fluids. The traditional shift factor and effective viscosity are related to the newly derived TCAT model, shedding new light on this common empirical approach. The results from this work form a basis for the modeling of GNF flow in porous medium systems under Stokes flow, which is predictive given the rheological properties of the GNF and the resistance observed for Newtonian flow.

Figure 1

Geometries used include: (a) a set of parallel slits; (b) a BCC array of spheres with the blue box representing the domain; (c) a polydisperse, randomly packed set of spheres; and (d) a BCE array of ellipsoids with the blue box representing the domain.

Figure 2

Parallel slit flow: (a) magnitudes of relevant terms in Eq. (28); and (b) comparison of normalized terms.

Figure 3

BCC flow: (a) magnitudes of relevant terms in Eq. (28); and (b) comparison of normalized terms.

Figure 4

Simulation results for: (a) a set of parallel slits; (b) BCC array of spheres; (c) randomly packed sphere domain; (d) BCE array of ellipsoids for Fluid 3; and (e) BCE array of ellipsoids for Fluid 4.