3 Definition of the Model

[681.1.8.1] In this section, a brief summary of the mathematical model is given. [681.1.8.2] The following equations are based on volume, mass and momentum balance equations analogous to the foundations of the traditional theory (see e.g. [9] for a succinct but detailed parallel development). [681.1.8.3] It is assumed that both fluids are incompressible and immiscible and that the lateral dimensions of the column are small with respect to the capillary fringe. [681.1.8.4] Hence, a one dimensional description is an appropriate approximation. [681.1.8.5] The mass balances for the four fluid phases (percolating water is identified by the index 1, non-percolating water by 2, percolating oil by 3 and non-percolating oil by 4) read as

where t denotes time, x denotes position or height along the column, ϕ denotes porosity, ϱW,ϱO denote the density of water, respectively air, Si the saturation, qi the volume flux and Mi the mass exchange term of the phase i. [681.1.8.6] The mass exchange term accounts for the fact that percolating and non-percolating phases of the same fluid exchange mass by break-up and coalescence. [681.1.8.7] The mass exchange terms take the form

with the parameter functions

where the parameter SO⁢im is a limiting saturation for the non-percolating air, SW⁢dr a limiting saturation for non-percolating water and Θ(⋅) denotes the Heaviside step function. [681.1.8.8] Water saturation is given by SW=S1+S2 and the air saturation by SO=S3+S4. [681.1.8.9] The parameter ϵM≈0 is a mathematical regularization parameter. [681.1.8.10] It allows to simulate also primary processes. [681.1.8.11] The volume fluxes of the phases take the form

where P1,P3 denote the averaged pressures of the percolating phases, g the gravity acceleration, α, β, γ, δ, Pa, Pb, P2*, P4*, are constitutive parameters. [681.1.8.12] The parameters α,β,Pa,Pb are associated with capillary potentials and γ,δ,P2*,P4* with the energy stored in the interface between the non-percolating phases and the surrounding percolating phases of the other fluid [9]. [681.1.8.13] A generalized mobility matrix is denoted by Λ with the coefficients

where R~-1i⁢j denote the components of the inverse of the viscous coupling parameter matrix. [681.1.8.14] A comparison with the classical two-phase Darcy equations yields that R~11≈μW/k and R~33≈μO/k, where k denotes the permeability of the porous medium. [681.1.8.15] The system of equations is closed with the volume conservation for incompressible fluids and incompressible porous media

plus a special form of the general self-consistent closure condition [11, 5]

for the pressures of the percolating phases.