On the determination of a generalized Darcy equation for yield-stress fluid in porous media using a Lattice-Boltzmann TRT scheme

Abstract

Simulating flow of a Bingham fluid in porous media still remains a challenging task as the yield stress may significantly alter the numerical stability and precision. We present a Lattice-Boltzmann TRT scheme that allows the resolution of this type of flow in stochastically reconstructed porous media. LB methods have an intrinsic error associated to the boundary conditions. Depending on the schemes this error might be directly linked to the effective viscosity. As for non-Newtonian fluids viscosity varies in space the error becomes inhomogeneous and very important. In contrast to that, the TRT scheme does not present this deficiency and is therefore adequate to be used for simulations of non-Newtonian fluid flow. We simulated Bingham fluid flow in porous media and determined a generalized Darcy equation depending on the yield stress, the effective viscosity, the pressure drop and a characteristic length of the porous medium. By evaluating the flow in the porous structure, we distinguished three different scaling regimes. Regime I corresponds to the situation where fluid is flowing in only one channel. Here, the relation between flow rate and pressure drop is given by the non-Newtonian Poiseuille law. During Regime II an increase in pressure triggers the opening of new paths and the relation between flow rate and the difference in pressure to the critical yield pressure becomes quadratic: $q \propto \left( {\tilde d_p - \tilde d_{p_c } } \right)^2$. Finally, Regime III corresponds to the situation where all the fluid is flowing. In this case, $q \propto \left( {\tilde d_p - \tilde d_{p_c } } \right)$.

Graphical abstract