fix formula syntax for the coupled models

This commit is contained in:
Andreas Lauser
2009-06-10 12:08:30 +00:00
committed by Andreas Lauser
parent 74b2d5bd23
commit a2f89656f9
8 changed files with 0 additions and 26 deletions

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Single phase isothermal flow model is implemented for compressible flow. \begin{align*} \phi \frac{\partial \varrho}{\partial t} + \vec{\nabla} \cdot (- \varrho \frac{\bar{\bar{K}}}{\mu} ( \nabla p -\varrho \vec{g})) = q \end{align*} which is discretized by this model using vertex centered finite volume (box) scheme as spatial and the implicit Euler method as time discretization. However, the model can also be used for incompressible single pahse flow modeling, when in problem file a fluid with constant density is chosen.

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This implements an isothermal two phase two component model.
Depending on the value of the \char`\"{}Formulation\char`\"{} property, the primary variables are either \$p\_\-w\$ and \$S\_\-n;X\$ or \$p\_\-n\$ or \$S\_\-w;X\$. By default they are \$p\_\-w\$ and \$S\_\-n\$

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This implements a non-isothermal two-phase two-component model with Pw and Sn/X as primary unknowns. You can use Pn and Sw/X as primary variables if you set the Formulation property to pNsW.

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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: \[ v_\alpha = - \frac{k_{r\alpha}}{\mu_\alpha} K \left(\grad p_\alpha - \varrho_{\alpha} \boldsymbol{g} \right) \]
By inserting this into the equation for the conservation of the phase mass, one gets \[ \phi \frac{\partial \varrho_\alpha S_\alpha}{\partial t} - \Div \left\{ \varrho_\alpha \frac{k_{r\alpha}}{\mu_\alpha} K \left(\grad p_\alpha - \varrho_{\alpha} \boldsymbol{g} \right) \right\} = q_\alpha \;, \] which is discretized by this model using the fully-coupled vertex centered finite volume (box) scheme as spatial and the implicit Euler method as time discretization.
By using constitutive relations for the capillary pressure $p_c = p_n - p_w$ and relative permability $k_{r\alpha}$ 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 chosing 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 {\tt Formulation} property to either either {\tt \doxyref{TwoPIndices::pWsN}{p.}{structDune_1_1TwoPIndices_7bab78f10df58319eafe8b79a1d28553}} or {\tt \doxyref{TwoPIndices::pNsW}{p.}{structDune_1_1TwoPIndices_b2b28761c782605cf080f9b6f9b618b5}}. By default, the model uses $p_w$ and $S_n$.

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This model implements a non-isothermal two-phase flow of two completely immiscible fluids $\alpha \in \{ w, n \}$. Using the standard multiphase Darcy approach the mass conservation equations for both phases can be described as follows: \begin{eqnarray*} && \phi \frac{\partial (\sum_\alpha \varrho_\alpha S_\alpha )}{\partial t} - \sum_\alpha \Div \left\{ \varrho_\alpha \frac{k_{r\alpha}}{\mu_\alpha} \mbox{\bf K} (\grad p_\alpha - \varrho_{\alpha} \mbox{\bf g}) \right\} - \sum_\alpha q_\alpha^\kappa = \quad 0 \qquad \alpha \in \{w, n\} \end{eqnarray*} For the energy balance local thermal equilibrium is assumed which results in one energy conservation equation for the porous solid matrix and the fluids: \begin{eqnarray*} && \phi \frac{\partial \left( \sum_\alpha \varrho_\alpha u_\alpha S_\alpha \right)}{\partial t} + \left( 1 - \phi \right) \frac{\partial (\varrho_s c_s T)}{\partial t} - \sum_\alpha \Div \left\{ \varrho_\alpha h_\alpha \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left( \grad \: p_\alpha - \varrho_\alpha \mathbf{g} \right) \right\} \\ &-& \Div \left( \lambda_{pm} \grad \: T \right) - q^h \qquad = \quad 0 \qquad \alpha \in \{w, n\} \end{eqnarray*}
the equations are discretized by this model using the fully-coupled vertex centered finite volume (box) scheme as spatial and the implicit Euler method as time discretization.
Currently the model supports chosing 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 {\tt Formulation} property to either either {\tt \doxyref{TwoPNIIndices::pWsN}{p.}{structDune_1_1TwoPIndices_7bab78f10df58319eafe8b79a1d28553}} or {\tt \doxyref{TwoPIndices::pNsW}{p.}{structDune_1_1TwoPIndices_b2b28761c782605cf080f9b6f9b618b5}}. By default, the model uses $p_w$, $S_n$ and $T$.

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Implementation of a decoupled formulation of a two phase two component flow processin porous media.
This implementation is written for a gas-liquid system with two components. An IMPES-like method is used for the sequential solution of the problem. Capillary forces and diffusion are neglected. Isothermal conditions and local thermodynamic equilibrium are assumed. Gravity is included. For the physical description of gas and liquid derivations of the classes \doxyref{Gas\_\-GL}{p.}{classDune_1_1Gas__GL} and \doxyref{Liquid\_\-GL}{p.}{classDune_1_1Liquid__GL} have to be provided. The template parameters are the used grid class and the desired number type (usually double) The pressure equation is given as \[ -\frac{\partial V}{\partial p}\frac{\partial p}{\partial t}+\sum_{\kappa}\frac{\partial V}{\partial C^{\kappa}}\nabla\cdot\left(\sum_{\alpha} {\bf v_\alpha} \varrho_\alpha X_\alpha^\kappa \right)=\sum_{\kappa}\frac{\partial V}{\partial m^{\kappa}}q^{\kappa}\] See paper SPE 99619 or \char`\"{}Analysis of a Compositional Model for Fluid Flow in Porous Media\char`\"{} by Chen, Qin and Ewing for derivation. The transport equation is \[ \frac{\partial C^\kappa}{\partial t} = - \nabla \cdot \sum{{\bf v_\alpha} \varrho_\alpha X_\alpha^\kappa} + q^\kappa \]
The pressure equation is given as $ -\frac{\partial V}{\partial p}\frac{\partial p}{\partial t}+\sum_{\kappa}\frac{\partial V}{\partial m^{\kappa}}\nabla\cdot\left(\sum_{\alpha}C_{\alpha}^{\kappa}\mathbf{v}_{\alpha}\right)=\sum_{\kappa}\frac{\partial V}{\partial m^{\kappa}}q^{\kappa}$ See paper SPE 99619 for derivation. The transport equation is $ \frac{\partial C^\kappa}{\partial t} = - \nabla \cdot \sum{C_\alpha^\kappa f_\alpha {\bf v}} + q^\kappa $

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In the unsaturated zone Richards` equation can be used. Gas has resistance against the water flow in porous media. However, viscosity of air is about 1\% of the viscosity of water, which makes it highly mobile compared to the water phase. Therefore, in Richards` equation only water phase with capillary effects are considered, where pressure of the gas phase is set to a reference pressure (${p_n}_{ref}$).
\begin{align*} \varrho \hspace{1mm} \phi \hspace{1mm} \frac{\partial S_w}{\partial p_c} \frac{\partial p_c}{\partial t} - \nabla \cdot (\frac{kr_w}{\mu_w} \hspace{1mm} \varrho_w \hspace{1mm} K \hspace{1mm} (\nabla p_w - \varrho_w \hspace{1mm} \vec{g})) \hspace{1mm} = \hspace{1mm} q \\ ,where \hspace{1mm} p_w = {p_n}_{ref} - p_c \end{align*} Here $ p_w $, $ p_c $, $ p_n $ denotes water pressure, capillary pressure, non-wetting phase reference pressure repectively.
To overcome convergence problem $ \frac{\partial S_w}{\partial p_c} $ is taken from old iteration step.