Porous Media

In multiphase flows, the porous viscous resistance and the inertial resistance depend on both the porous material and the phase material.

For Mixture Multiphase (MMP) flows through porous media, the momentum equation for the mixture velocity $v_{m}$ can be solved either using the physical velocity formulation Eqn. (1852) or the superficial velocity formulation Eqn. (1843).

The porous viscous resistance force $P_{v}$ and porous inertial resistance force $P_{i}$ are specified for the mixture velocity $v_{m}$ .

Given the porous viscous resistance $P_{v}^{i}$ of phase $i$ , the corresponding resistance for the mixture can be computed by a mass-weighted harmonic average:

Figure 1. EQUATION_DISPLAY

P_{v} = {(\sum_{i} Y_{i} {P_{v}^{i}}^{- 1})}^{- 1}

(2904)

where $Y_{i}$ is the mass fraction of phase $i$ . For porous inertial resistance, the phase resistances $P_{i}^{i}$ lead to the mixture inertial resistance:

Figure 2. EQUATION_DISPLAY

P_{i} = {(\sum_{i} \frac{Y_{i} | v_{m} |}{| v_{i} |} {P_{i}^{i}}^{- 1})}^{- 1}

(2905)

Phase Slip in Porous Media

It is assumed that, in each phase $i$ , the flow is dominated by diffusive effects, and that the only relevant effects are due to the pressure gradient, gravity, and user sources. Therefore the phase velocity can be written as:

Figure 3. EQUATION_DISPLAY

v_{i} = - \frac{1}{χ} P_{v, i}^{- 1} \nabla p + \frac{ρ_{i}}{χ} P_{v, i}^{- 1} g + \frac{1}{χ} P_{v, i}^{- 1} f_{u, i}

(2906)

where $v_{i}$ is the physical velocity of phase $i$ , $P_{v, i}$ is the porous viscous resistance of phase $i$ , and $f_{u, i}$ is the user source applied to phase $i$ . This equation is a generalized version of Darcy's law.

Physical Velocity Porous Media Modeling

From Eqn. (2906), the phase slip velocity can be computed as:

Figure 4. EQUATION_DISPLAY

v_{i j} = v_{i} - v_{j} = (- P_{v, i}^{- 1} + P_{v, j}^{- 1}) \frac{1}{χ} \nabla p + (ρ_{i} P_{v, i}^{- 1} - ρ_{j} P_{v, j}^{- 1}) \frac{1}{χ} g + \frac{1}{χ} (P_{v, i}^{- 1} f_{u, i} - P_{v, j}^{- 1} f_{u, j})

(2907)

Superficial Velocity Porous Media Modeling

For porous media modeling using the superficial velocity formulation, the superficial velocity for the mixture $v_{s} = χ v$ is used. To preserve the relation between phase and mixture velocity (that is, $\sum_{i} \frac{α_{i} ρ_{i}}{ρ_{m}} v_{i} = v$ ), the superficial phase velocity $v_{s, i} = χ v_{i}$ .

From the superficial phase velocity, the following expression for superficial phase slip velocity is obtained:

Figure 5. EQUATION_DISPLAY

v_{s, i j} = v_{s, i} - v_{s, j} = (- P_{v, i}^{- 1} + P_{v, j}^{- 1}) \nabla p + (ρ_{i} P_{v, i}^{- 1} - ρ_{j} P_{v, j}^{- 1}) g + (P_{v, i}^{- 1} f_{u, i} - P_{v, j}^{- 1} f_{u, j})

(2908)