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: [Formula: see text]. Finally, Regime III corresponds to the situation where all the fluid is flowing. In this case, [Formula: see text].

Effective rheology of Bingham fluids in a rough channel

“…Presently, only three flowing regimes have been observed numerically (see [15]). The first one, in the limit P ∼ P c , is linear ( i.e., Q ∼ ( P − P c )) and corresponds the flow channelization along one channel.…”

“…There have been a number of experimental [2,9], numerical [10][11][12][13][14][15][16] and theoretical [17,18] studies of the flow of yield stress fluids in porous media. One of the main objective is to derive an generalized Darcy equation for yield stress fluids relating mean flow rate, the pressure gradient and a critical pressure gradient.…”

“…In particular, n = 1 defines the Bingham fluid. However, numerical simulations and porenetwork models [15,17,20] demonstrate the situation may be much more complex that Equation (1) may be expected when the flow occurs in non-trivial geometries. Three different flow regimes,…”

“…Besides offering interesting insights into the complexitites of the flow of yield stress fluids even in this seemingly simple case, it is of direct relevance in connection with flow both in two and three-dimensional porous media flow. Close to the yield threshold, there will be only one single one-dimensional channel where there is flow irrespecive of the dimensionality of the porous medium (see e.g., [11,15,17,21]). …”

Obstructed and channelized viscoplastic flow in a Hele-Shaw cell

“…Subsequent studies in the area have explored the viscoplastic version of the Saffman-Taylor instability (Pascal 1981;Coussot 1999) and the dynamics of displacement fronts (Bittleston, Ferguson & Frigaard 2002;Pelipenko & Frigaard 2004). The latest developments have been directed at bridging the gap between viscoplastic flow dynamics on the pore scale and a macroscopic analogue of Darcy's law (Chevalier et al 2013(Chevalier et al , 2014Talon & Bauer 2013;Bleyer & Coussot 2014).…”