Experimental and numerical determination of Darcy's law for yield stress fluids in porous media

In this work, we studied experimentally and numerically the pressure–flow rate relationship for yield stress fluids in porous media. We developed and validated 3D numerical simulations of the velocity field via a lattice Boltzmann method based on the TRT scheme, and a specific experimental setup allowing yield stress fluids to flow in a closed-loop system to obtain stable rheological properties over a wide range of flow rates (4 decades). The porous medium studied experimentally is a sandstone. The flow properties were also simulated on a regular sphere packing and a random sphere packing as well as on a 3D geometry of a sandstone obtained by x-ray tomography. These different geometries allow highlighting the role of the heterogeneity of the pore structure on the flow properties. All results are expressed as the flow rate $\tilde{Q}$ versus the difference of the pressure gradient to the critical pressure $(\mathrm{\Delta }\tilde{P}-\mathrm{\Delta }{\tilde{P}}_c)$; $P_c$ defines the pressure below which there is no flow. We observed both numerically and experimentally three specific scaling regimes, already identified by Talon and Bauer [Eur. Phys. J. E 36, 139 (2013)] for a Bingham fluid in 2D porous media. We also evidenced the existence of two critical pressures: the “true” critical pressure ${\tilde{P}}_c$ defined as the pressure below which there is no flow and the “pseudo” critical pressure threshold ${\tilde{P}}_c^{\infty }$ determined by fitting the data in the high-flow-rate regime. We show that the “true” critical pressure is always lower than the “pseudo” one in heterogeneous porous media and can be equal only in the case of regular porous structures. We explain these observations using an energy minimization principle.

Figure 1

Schematic diagram of the closed-loop setup used for Carbopol injection in the porous medium. The inset displays a photograph of the real system.

Figure 2

Different porous media used in the simulation: (a) FCC packing of spheres; (b) random array of overlapping spheres; (c) 3D $\mu -\mathrm{CT}$ image of sandstone.

Figure 3

Experimental flow rate of the sandstone as a function of the pressure drop. Stars represent experimental data, and dashed lines represent the fit with Eq. (12) considering the high flow rate.

Figure 4

Numerical flow rate as function of the nondimensional pressure drop. Top left: FCC; Top right: irregular bead packing; Bottom: sandstone. Stars represent numerical data, and dashed lines represent the fit with Eq. (12) considering only the high flow rate. The open circles correspond to simulations of a Newtonian fluid (${\tau }_0=0$).

Figure 5

Streamlines inside a random packing of spheres for increasing applied pressure drop.

Figure 6

Top left: FCC sphere packing; top right: random sphere packing; bottom left: sandstone simulation; bottom right: sandstone experiments. Stars correspond to numerical or experimental data, and dashed and solid lines correspond to the power-law fit. (Please refer to the text for values of the exponents.)

Figure 7

Carbopol rheology obtained before and after the experiments by means of a rheometer and data obtained from the reference tube.

Figure 8

Comparison of the numerical and theoretical velocity profile (left) and flow rate (right) (top: parallel planes; bottom: 3D cylindrical tube).

Figure 9

Top: flow field in a wavy cosine channel for two different resolution (left: $R=16.5$; right $R=8.5$). The black dots represent the unyielded regions. Bottom: flow rate as function of the pressure drop for different resolution.

Figure 10

Flow rate as function of the pressure drop for different resolution of FCC packing.

Figure 11

Flow rate of the FCC packing as a function of the pressure difference for different values of ${\tau }_0$