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17 lines
2.1 KiB
TeX
17 lines
2.1 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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\-This model implements a non-\/isothermal two-\/phase flow for two immiscible fluids $\alpha \in \{ w, n \}$. \-Using the standard multiphase \-Darcy approach, the mass conservation equations for both phases can be described as follows\-: \[ \phi \frac{\partial \phi \varrho_\alpha S_\alpha}{\partial t} - \text{div} \left\{ \varrho_\alpha \frac{k_{r\alpha}}{\mu_\alpha} \mathrm{K} \left( \textrm{grad}\, p_\alpha - \varrho_{\alpha} \mathbf{g} \right) \right\} - q_\alpha = 0 \qquad \alpha \in \{w, n\} \]
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\-For the energy balance, local thermal equilibrium is assumed. \-This results in one energy conservation equation for the porous solid matrix and the fluids\-:
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\begin{align*} \frac{\partial \phi \sum_\alpha \varrho_\alpha u_\alpha S_\alpha}{\partial t} & + \left( 1 - \phi \right) \frac{\partial (\varrho_s c_s T)}{\partial t} - \sum_\alpha \text{div} \left\{ \varrho_\alpha h_\alpha \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left( \textbf{grad}\,p_\alpha - \varrho_\alpha \mbox{\bf g} \right) \right\} \\ & - \text{div} \left(\lambda_{pm} \textbf{grad} \, T \right) - q^h = 0, \qquad \alpha \in \{w, n\} \;, \end{align*} where $h_\alpha$ is the specific enthalpy of a fluid phase $\alpha$ and $u_\alpha = h_\alpha - p_\alpha/\varrho_\alpha$ is the specific internal energy of the phase.
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\-The equations are discretized using a fully-\/coupled vertex centered finite volume (box) scheme as spatial and the implicit \-Euler method as time discretization.
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\-Currently the model supports choosing either $p_w$, $S_n$ and $T$ or $p_n$, $S_w$ and $T$ as primary variables. \-The formulation which ought to be used can be specified by setting the {\ttfamily \-Formulation} property to either {\ttfamily \-Two\-P\-N\-I\-Indices\-::p\-Ws\-N} or {\ttfamily \-Two\-P\-Indices\-::p\-Ns\-W}. \-By default, the model uses $p_w$, $S_n$ and $T$.
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