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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

\displaystyle\varrho _{{\mathbb{W}}}\phi\frac{\partial S_{1}}{\partial t}+\varrho _{{\mathbb{W}}}\frac{\partial q_{1}}{\partial x} \displaystyle=M_{1}, (1a)
\displaystyle\varrho _{{\mathbb{W}}}\phi\frac{\partial S_{2}}{\partial t}+\varrho _{{\mathbb{W}}}\frac{\partial q_{2}}{\partial x} \displaystyle=M_{2}=-M_{1}, (1b)
\displaystyle\varrho _{{\mathbb{O}}}\phi\frac{\partial S_{3}}{\partial t}+\varrho _{{\mathbb{O}}}\frac{\partial q_{3}}{\partial x} \displaystyle=M_{3}, (1c)
\displaystyle\varrho _{{\mathbb{O}}}\phi\frac{\partial S_{4}}{\partial t}+\varrho _{{\mathbb{O}}}\frac{\partial q_{4}}{\partial x} \displaystyle=M_{4}=-M_{3}, (1d)

where t denotes time, x denotes position or height along the column, \phi denotes porosity, \varrho _{{\mathbb{W}}},\varrho _{{\mathbb{O}}} denote the density of water, respectively air, S_{i} the saturation, q_{i} the volume flux and M_{i} 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

\displaystyle M_{1} \displaystyle={\eta _{{2}}}\phi\varrho _{{\mathbb{W}}}\left(\frac{S_{2}-{S_{2}^{*}}}{{S_{\mathbb{W}}^{*}}-{S_{{\mathbb{W}}}}}\right)\frac{\partial{S_{{\mathbb{W}}}}}{\partial t}, (2a)
\displaystyle M_{3} \displaystyle={\eta _{{4}}}\phi\varrho _{{\mathbb{O}}}\left(\frac{S_{4}-{S_{4}^{*}}}{{S_{\mathbb{W}}^{*}}-{S_{{\mathbb{W}}}}}\right)\frac{\partial{S_{{\mathbb{W}}}}}{\partial t}. (2b)

with the parameter functions

\displaystyle{S_{\mathbb{W}}^{*}} \displaystyle=\left(1-\min{\left(S_{{\mathbb{O}\,\rm im}},(1-\epsilon _{{\rm M}}){S_{{\mathbb{O}}}}\right)}\right)\Theta\left(\partial _{t}{S_{{\mathbb{W}}}}\right)+\min{\left(S_{{\mathbb{W}\,\rm dr}},(1-\epsilon _{{\rm M}}){S_{{\mathbb{W}}}}\right)}\left[1-\Theta\left(\partial _{t}{S_{{\mathbb{W}}}}\right)\right], (3a)
\displaystyle{S_{2}^{*}} \displaystyle=\min{\left(S_{{\mathbb{W}\,\rm dr}},(1-\epsilon _{{\rm M}}){S_{{\mathbb{W}}}}\right)}\left[1-\Theta\left(\partial _{t}{S_{{\mathbb{W}}}}\right)\right], (3b)
\displaystyle{S_{4}^{*}} \displaystyle=\min{\left(S_{{\mathbb{O}\,\rm im}},(1-\epsilon _{{\rm M}}){S_{{\mathbb{O}}}}\right)}\left[1-\Theta\left(\partial _{t}{S_{{\mathbb{O}}}}\right)\right], (3c)

where the parameter S_{{\mathbb{O}\,\rm im}} is a limiting saturation for the non-percolating air, S_{{\mathbb{W}\,\rm dr}} a limiting saturation for non-percolating water and \Theta(\cdot) denotes the Heaviside step function. [681.1.8.8] Water saturation is given by {S_{{\mathbb{W}}}}=S_{1}+S_{2} and the air saturation by {S_{{\mathbb{O}}}}=S_{3}+S_{4}. [681.1.8.9] The parameter \epsilon _{{\rm M}}\approx 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

q_{4}\end{array}\right)=\Lambda\left(\begin{array}[]{c}-\partial _{x}P_{1}{-}\varrho _{{\mathbb{W}}}g\\
-\partial _{x}(P_{3}-\gamma P^{*}_{2}S_{2}^{{\gamma-1}}){-}\varrho _{{\mathbb{W}}}g+P_{{\rm a}}\partial _{x}S_{1}^{{-\alpha}}\\
-\partial _{x}P_{3}{-}\varrho _{{\mathbb{O}}}g\\
-\partial _{x}(P_{1}-\delta P^{*}_{4}S_{4}^{{\delta-1}}){-}\varrho _{{\mathbb{O}}}g+P_{{\rm b}}\partial _{x}S_{3}^{{-\beta}}\\
\end{array}\right), (4)

where P_{1},P_{3} denote the averaged pressures of the percolating phases, g the gravity acceleration, \alpha, \beta, \gamma, \delta, P_{{\rm a}}, P_{{\rm b}}, P^{*}_{2}, P^{*}_{4}, are constitutive parameters. [681.1.8.12] The parameters \alpha,\beta,P_{{\rm a}},P_{{\rm b}} are associated with capillary potentials and \gamma,\delta,P^{*}_{2},P^{*}_{4} 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 \Lambda with the coefficients

\Lambda _{{ij}}=\phi^{2}S_{i}S_{j}[{\tilde{R}}^{{-1}}]_{{ij}}, (5)

where [{\tilde{R}}^{{-1}}]_{{ij}} 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 \tilde{R}_{{11}}\approx\mu _{{\mathbb{W}}}/k and \tilde{R}_{{33}}\approx\mu _{{\mathbb{O}}}/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

\displaystyle S_{1}+S_{2}+S_{3}+S_{4}=1 (6)

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

P_{3}=P_{1}+\frac{1}{2}\left(P_{{\rm a}}S_{1}^{{-\alpha}}-P_{{\rm b}}S_{3}^{{-\beta}}\right.\left.+\gamma P^{*}_{2}S_{2}^{{\gamma-1}}-\delta P^{*}_{4}S_{4}^{{\delta-1}}\right) (7)

for the pressures of the percolating phases.