On the determination of a generalized Darcy equation for yield stress fluid in porous media

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<h2>Abstract</h2> Flows of Non-Newtonian fluids have different applications in porous media. Indeed, slurries, colloidal suspensions, emulsions, foams or heavy oil which present complex rheologies are commonly injected in porous media for various applications (Oil recovery, hydraulic fracturing, soil consolidation, etc.) Among the large number of different non-Newtonian fluids an important class of behaviour is represented by the yield-stress fluids, viz. fluids that require a minimum of stress to flow. Yield stress fluids are usually modelled as a Bingham fluid or by the Herschel-Bulkley equation. In the present work, we initially use a Lattice-Boltzmann TRT scheme to determine the flow structure in a synthetic porous medium. We then determined a generalized Darcy equation by evaluating the flow for different applied pressure drops dP. As expected, one can determine a critical pressure drop dPc below which there is no flow. Above this threshold, we observe three different flowing regimes as function of the distance to the critical pressure dP-dPc. Regime I corresponds to the situation where the 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. The relation between flow rate and the difference in pressure to the critical yield pressure becomes quadratic. Finally, Regime III corresponds to the situation where all the fluid is sheared. In which case one has a linear relationship between the flow rate and the applied pressure. In this presentation, we will study the crucial importance of heterogeneities in these flow regimes. In particular, we will show that even if these regimes are the consequence of disorder, some statistical laws have exponents that are independent of disorder. Finally, we will propose different models, based on the pore network, in order to better understand the origin of theses exponents and the influence of the disorder.