Darcy’s law for yield stress fluid flowing through a porous medium

Abstract

We measured the pressure drop vs flow rate during the flow, in a wide range of velocities, of well controlled yield stress fluids through confined packings of glass beads of different sizes. A detailed analysis of the data makes it possible to extract a general expression for the pressure drop vs flow rate curve through a porous medium as a function of the flow rate and the characteristics of the system. This general law has a form similar to the Herschel-Bulkley model describing the rheological behavior of such fluids in simple shear, i.e. it expresses as the sum of a critical (yielding) pressure drop and a flow rate dependent term. This law involves the rheological parameters of the fluid, one characteristic length of the medium, and two coefficients which only depend on the structure of the porous medium. The first coefficient is related to the path of maximum width throughout the porous medium while the second coefficient reflects the pore size distribution. The values of these coefficients were determined in the case of a granular packing.

Effective rheology of Bingham fluids in a rough channel

“…This gives the total flow rate Q as a function of pressure difference P across the length of the channel. Whatever the geometry of the channel, close to the pressure threshold P c , the flow rate scales like ( P − P c ) 2 with a prefactor function of the fluid and aperture properties.…”

“…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. Experimentally [2,9,19] they establish various laws which have quite essentially the same structure,…”

“…fluids that require that the applied stress is above a non-zero threshold-a yield stress-for flowing [1], are found in many practical applications, such as in oil industry. Indeed, emulsions [2], mud [3,4], heavy oil [5], polymeric gels such as carbopol [2] and foams [6] generate non-zero yield stress for flowing. In an oil recovery context, such fluids are injected to control premature production of water or gas.…”

“…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.…”

The Effect of Microstructure on Models for the Flow of a Bingham Fluid in Porous Media: One-Dimensional Flows

“…Another analysis of one-dimensional flow within a two-dimensional network was considered by Chen et al (2005), but this was in the context of imbibition and the displacement of a Bingham fluid by a Newtonian fluid. We would also wish to draw the readers' attention to the papers by Chevalier et al (2013Chevalier et al ( , 2014, which report on the experimental determination of a suitable Darcy's law for HerschelBulkley fluids, and the paper by Bleyer and Coussot, which is concerned with the numerical modelling of a Bingham fluid through an array of circular cylinders. Entrance effects have also been considered by Philippou et al (2016).…”