Two-Phase Fluid

Two-phase porous flow describes two immiscible fluids, such as water and gas or water and oil, sharing the same pore space. Because both phases compete for the available pores, each phase’s mobility depends on saturation. The equations below give the standard continuum formulation; they do not prescribe a particular relative-permeability curve or imply that this formulation is used by every OpenFDEM fluid solver.

Saturation and capillary pressure

For a two-phase system, the phase saturations satisfy

\[S_w+S_n=1,\]

where \(S_w\) and \(S_n\) are the wetting- and non-wetting-phase saturations. Capillary pressure is the pressure difference between the phases, conventionally defined as

\[p_c(S_w)=p_n-p_w,\]

where \(p_w\) and \(p_n\) are the wetting- and non-wetting-phase pressures. A constitutive relation \(p_c(S_w)\) closes the pressure- saturation relation and reflects pore-scale interface and wettability effects.

Phase-specific Darcy law and mass balance

The Darcy flux for phase \(\alpha\in\{w,n\}\) is

\[\mathbf{q}_{\alpha} =-\frac{\mathbf{k}\,k_{r\alpha}(S_\alpha)}{\mu_\alpha} \left(\nabla p_\alpha-\rho_\alpha\mathbf{g}\right),\]

where \(\mathbf{k}\) is the absolute permeability tensor, \(k_{r\alpha}\) is the dimensionless relative permeability, \(\mu_\alpha\) and \(\rho_\alpha\) are phase viscosity and density, and \(\mathbf{g}\) is gravity. The phase mass balances are

\[\frac{\partial(\phi\rho_\alpha S_\alpha)}{\partial t} +\nabla\cdot(\rho_\alpha\mathbf{q}_\alpha)=m_\alpha, \qquad \alpha\in\{w,n\},\]

where \(\phi\) is porosity and \(m_\alpha\) is a mass source per unit bulk volume for that phase. These balances are coupled through the saturation constraint, the capillary-pressure relation, and the relative-permeability functions.

Relative permeability and closure

Relative permeability accounts macroscopically for the reduction in a phase’s conducting pore space due to the presence of the other phase. It usually varies between zero and one and is specified as a function of saturation, often together with residual saturations. One illustrative Corey-type closure uses the effective wetting saturation

\[S_e=\frac{S_w-S_{wr}}{1-S_{wr}-S_{nr}}, \qquad k_{rw}=S_e^{n_w}, \qquad k_{rn}=(1-S_e)^{n_n},\]

where \(S_{wr}\) and \(S_{nr}\) are residual wetting- and non-wetting-phase saturations, and \(n_w,n_n\) are empirical exponents. This is an example, not a universal law: measured or calibrated capillary-pressure and relative-permeability curves should be used when available. Hysteresis, compressibility, dissolution, and dynamic interfacial effects require additional constitutive assumptions.

Model limits

If capillary effects are negligible, the phase pressures may be approximated as equal; if one phase is absent, the model reduces to single-phase flow. Saturation fronts can be sharp, so the predicted solution may depend on constitutive curves, mesh resolution, and time-step selection. The choice of primary unknowns (for example, one phase pressure and one saturation) is a numerical formulation decision and does not change the underlying conservation laws.