Pressure-Saturation Formulation
The algebraic relations in Equation 8-19 and Equation 8-20 allow for a reduction of the number of dependent variables. An often used way is to eliminate N − 1 phase pressures and one of the saturations. This results in a so-called pressure-saturation formulation (Ref. 3). The procedure in the Multiphase Flow in Porous Media interface is to pick one of the phases, let us say phase , then express the volume fraction, or saturation, , of this phase in terms of the saturations of the other phases and in addition to use the pressure, , of this phase to define the other phase pressures:
(8-21)
(8-22)
The equations for the saturations si, (i ≠ ic) are given by Equation 8-17 and Equation 8-18. To arrive at an equation for , the conservation equations of all phases are summed. This results in
(8-23)
where the total mass source Qtot is given by
(8-24)
The equations for the saturations si are solved in the Phase Transport in Porous Media interface (Equation 8-17 and Equation 8-18). The Equation 8-23 for the pressure field , needed as an input to the Phase Transport in Porous Media interface, is solved for in the Darcy’s Law interface: the Multiphase Flow in Porous Media multiphysics coupling interface replaces the equation
(8-25)
which is originally implemented in the Darcy’s Law interface, with Equation 8-23 by adding the following terms to the left-hand side of Equation 8-25:
(8-26)
and by setting the right-hand side Qm to be equal to Qtot. The volume averaged density ρtot is given by
(8-27)
Furthermore, the Multiphase Flow in Porous Media multiphysics coupling interface couples the user inputs for the porosity and permeability of the porous matrix in the Darcy’s Law interface to the corresponding user input fields in the Phase and Porous Media Transport Properties feature, and couples the pressure field computed for in the Darcy’s Law interface to the user input field for the pressure of the phase computed from the volume constraint.
Note that when the hydraulic conductivity of a domain is specified in the Darcy’s Law interface (instead of the permeability), the coupled Phase and Porous Media Transport Properties feature is supplied with a permeability that is computed using a reference kinematic viscosity of 1.004*10−6  m2·s−1 (kinematic viscosity of water at 293.15  K).
In the other direction, the multiphysics coupling node provides the averaged density (denoted by ρtot in Equation 8-26 above) and effective viscosity to the Darcy’s Law interface.