Dear All,

I am trying to perform incompressible Direct Numerical Simulations (DNS) of a polymer flow in a porous medium of an array of cylinders (see the picture). The top and bottom are periodic boundary conditions(BC), the left side is an inlet, the right is an outlet. At the outlet, we set constant velocity Uin and homogeneous BC for the pressure dp/dx=0, at the outlet we set homogeneous BC for the velocity, and p=0. Due to the heating of cylinders, the fluid can have different viscosity in space (for example near the wall the viscosity decreases).

My goal is to extract macroparameters, like porosity, permeability, etc. And I faced the problem of permeability calculation for both one-phase flow and two-phase flow.

I have 2 questions:

1) the first one is related to the one-phase flow.

In this case, the classic permeability definition is

K = mu*U_fil/|grad p|

The mu is the viscosity, U_fil is the filtration velocity, p is the pressure. Due to the constant volumetric rate, we have U_fil=Uin. The pressure gradient can be estimated as

|grad p| = (\int_0^H p(inlet)dy - \int_0^H p(outlet)dy)/H, where we averaged along the vertical line at inlet and outlet, \int means integral from 0 to the height H.

But what to do with variable viscosity?

2) In the case of multiphase flow considering capillary effects, the velocities in the fluid 1 and 2 are

u1 = - K*f1(s)\grad p 1/mu1,

u2 = - K*f2(s)\grad p 2/mu2,

where mu1, mu2 are corresponding viscosities, f1, f2 -are relative permeabilities, p1, p2 are pressures that are different for the sake of considering capillary effects.

How I understood, u1, u2 are calculated from integration in phases 1 and 2, correspondingly.

u1 = \int_0_H phi*(1 - chi)*u1(x,y)dy/H, u2 = \int_0_H (1 - phi)*(1 - chi)*u2(x,y)dy/H,

where phi and chi are volume fractions of fluid 1 and solid, respectively. Here I noticed that velocity can change along Ox. How to calculate p1

Here several questions arise. How to extract f1, f2?

Detailed information in https://en.wikipedia.org/wiki/Relative_permeability

Best regards,

Evgenii