Numerical study of Bingham flow in macroscopic two dimensional heterogeneous porous media

Abstract

The flow of non-Newtonian fluids is ubiquitous in many applications in the geological and industrial context. We focus here on yield stress fluids (YSF), i.e. a material that requires minimal stress to flow. We study numerically the flow of yield stress fluids in 2D porous media on a macroscopic scale in the presence of local heterogeneities. As with the microscopic problem, heterogeneities are of crucial importance because some regions will flow more easily than others. As a result, the flow is characterized by preferential flow paths with fractal features. These fractal properties are characterized by different scale exponents that will be determined and analyzed. One of the salient features of these results is that these exponents seem to be independent of the amplitude of heterogeneities for a log-normal distribution. In addition, these exponents appear to differ from those at the microscopic level, illustrating the fact that, although similar, the two scales are governed by different sets of equations.

Minimum principle for the flow of inelastic non-Newtonian fluids in macroscopic heterogeneous porous media

“…Herschel-Bulkley: Yield stress fluids are often described by the Herschel-Bulkley rheology, τ = τ 0 + K γ n , where τ 0 is the yield stress. At Darcy's scale, the velocity field can be described by (see [22,51,52]):…”

“…The κ(ì r) field can be determined experimentally using different methods ( borehole, pumping tests). Many models have also been proposed in the literature, the most popular being parallel strata of different natures or the log-normal distribution (see for example [19][20][21][22]). This equation is 2D is also equivalent to the "cubic law" commonly used to solved the flow in heterogeneous fractures [23,24].…”