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14 lines
1.7 KiB
TeX
14 lines
1.7 KiB
TeX
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% This file has been autogenerated from the LaTeX part of the %
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% doxygen documentation; DO NOT EDIT IT! Change the model's .hh %
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% file instead!! %
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Adaption of the BOX scheme to the twophase flow model. This model implements two-\/phase flow of two completely immiscible fluids $\alpha \in \{ w, n \}$ using a standard multiphase Darcy approach as the equation for the conservation of momentum: \[ v_\alpha = - \frac{k_{r\alpha}}{\mu_\alpha} K \left(\text{grad} p_\alpha - \varrho_{\alpha} \boldsymbol{g} \right) \]
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By inserting this into the equation for the conservation of the phase mass, one gets \[ \phi \frac{\partial \varrho_\alpha S_\alpha}{\partial t} - \text{div} \left\{ \varrho_\alpha \frac{k_{r\alpha}}{\mu_\alpha} K \left(\text{grad} p_\alpha - \varrho_{\alpha} \boldsymbol{g} \right) \right\} = q_\alpha \;, \] discretized by a fully-\/coupled vertex centered finite volume (box) scheme as spatial and the implicit Euler method as time discretization.
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By using constitutive relations for the capillary pressure $p_c = p_n - p_w$ and relative permeability $k_{r\alpha}$ and taking advantage of the fact that $S_w + S_n = 1$, the number of unknowns can be reduced to two. Currently the model supports choosing either $p_w$ and $S_n$ or $p_n$ and $S_w$ as primary variables. The formulation which ought to be used can be specified by setting the {\ttfamily Formulation} property to either {\ttfamily TwoPCommonIndices::pWsN} or {\ttfamily TwoPCommonIndices::pNsW}. By default, the model uses $p_w$ and $S_n$.
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