Non-Darcy behavior of two-phase channel flow

We study the macroscopic behavior of two-phase flow in porous media from a phase-field model. A dissipation law is first derived from the phase-field model by homogenization. For simple channel geometry in pore scale, the scaling relation of the averaged dissipation rate with the velocity of the two-phase flow can be explicitly obtained from the model which then gives the force-velocity relation. It is shown that, for the homogeneous channel surface, Dacry's law is still valid with a significantly modified permeability including the contribution from the contact line slip. For the chemically patterned surfaces, the dissipation rate has a non-Darcy linear scaling with the velocity, which is related to a depinning force for the patterned surface. Our result offers a theoretical understanding on the prior observation of non-Darcy behavior for the multiphase flow in either simulations or experiments.

Figure 1

Examples of cell structures in periodic porous media.

Figure 2

Two-phase flow in a channel with homogeneous upper and lower boundaries.

Figure 3

Two-phase flow in a channel with chemically patterned upper and lower boundaries.

Figure 4

(left) A typical velocity contour and (right) slip-velocity profile.

Figure 5

Spacial-average velocity vs time ($ls=0.4,F=-0.01$).

Figure 6

Darcy's relation for two-phase channel flow with homogeneous boundary.

Figure 7

Some typical velocity contours and slip-velocity profile (${\theta }_s={80}^{\circ },F=-0.01$).

Figure 8

Advancing contact angle and receding contact angle (${\theta }_s={80}^{\circ },F=-0.01$).

Figure 9

Spacial-average velocity vs time (${\theta }_s={80}^{\circ },F=-0.01$).

Figure 10

Force-velocity relation for two-phase channel flow with periodic patterned boundary.

Figure 11

Scaled slip velocity vs scaled distance from MCL: the log-log scaling and the original profile (inset) [25].

Figure 12

Two-phase fluid with a moving contact line.

Figure 13

Scaled slip velocity vs scaled distance from MCL given by asymptotics [27].