Macroscopic model for unsteady generalized Newtonian fluid flow in homogeneous porous media

A macroscopic model for unsteady incompressible isothermal non-Newtonian flow in homogeneous porous media, taking into account inertial and slip effects at solid–fluid interfaces, is derived in this work. The development is carried out considering a general Newton’s law of viscosity for the fluid phase. Using the classical volume averaging method, the seepage velocity is shown to be solenoidal. The macroscopic momentum equation is derived in the Laplace domain, employing a simplified version of the volume averaging method, which calls upon Green’s formulas and adjoint problems for Green’s function pairs for the velocity and pressure. In the Laplace domain, the macroscopic momentum equation takes the form of Darcy’s law corrected by a term that accounts for the initial flow condition. Once transformed back into the time domain, this equation provides the macroscopic velocity that depends on two terms. The first one is under the form of a time convolution between the macroscopic pressure gradient and the time derivative of an apparent permeability tensor. The second one is a memory term that accounts for the effect of the initial flow conditions. These two effective quantities are determined from the solution of a single closure problem that naturally results from the derivations. The model is consistent with the unsteady model in the Newtonian case and simplifies to the steady versions of some non-Newtonian macroscopic flow models. The macroscopic model is validated with pore-scale simulations performed in 2D model porous structures, considering a Carreau fluid. The impact of inertia and non-Newtonian effects on the dynamics of the macroscopic coefficients is highlighted.