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33 lines
4.2 KiB
TeX
33 lines
4.2 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 solves equations of the form \[ \phi \left( \rho_w \frac{\partial S_w}{\partial t} + \rho_n \frac{\partial S_n}{\partial t}\right) + \textbf{div}\, \boldsymbol{v}_{total} = q. \] \-The definition of the total velocity $\boldsymbol{v}_{total}$ depends on the choice of the primary pressure variable. \-Further, fluids can be assumed to be compressible or incompressible (\-Property\-: {\ttfamily \-Enable\-Compressibility}). \-In the incompressible case a wetting $(w) $ phase pressure as primary variable leads to
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\[ - \textbf{div}\, \left[\lambda \boldsymbol K \left(\textbf{grad}\, p_w + f_n \textbf{grad}\, p_c + \sum f_\alpha \rho_\alpha \, g \, \textbf{grad}\, z\right)\right] = q, \]
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a non-\/wetting ( $ n $) phase pressure yields \[ - \textbf{div}\, \left[\lambda \boldsymbol K \left(\textbf{grad}\, p_n - f_w \textbf{grad}\, p_c + \sum f_\alpha \rho_\alpha \, g \, \textbf{grad}\, z\right)\right] = q, \] and a global pressure leads to \[ - \textbf{div}\, \left[\lambda \boldsymbol K \left(\textbf{grad}\, p_{global} + \sum f_\alpha \rho_\alpha \, g \, \textbf{grad}\, z\right)\right] = q. \] \-Here, $ p_\alpha $ is a phase pressure, $ p_ {global} $ the global pressure of a classical fractional flow formulation (see e.\-g. \-P. \-Binning and \-M. \-A. \-Celia, ''\-Practical implementation of the fractional flow approach to multi-\/phase flow simulation'', \-Advances in water resources, vol. 22, no. 5, pp. 461-\/478, 1999.), $ p_c = p_n - p_w $ is the capillary pressure, $ \boldsymbol K $ the absolute permeability, $ \lambda = \lambda_w + \lambda_n $ the total mobility depending on the saturation ( $ \lambda_\alpha = k_{r_\alpha} / \mu_\alpha $), $ f_\alpha = \lambda_\alpha / \lambda $ the fractional flow function of a phase, $ \rho_\alpha $ a phase density, $ g $ the gravity constant and $ q $ the source term.
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\-For all cases, $ p = p_D $ on $ \Gamma_{Dirichlet} $, and $ \boldsymbol v_{total} \cdot \boldsymbol n = q_N $ on $ \Gamma_{Neumann} $.
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\-The slightly compressible case is only implemented for phase pressures! \-In this case for a wetting $(w) $ phase pressure as primary variable the equations are formulated as \[ \phi \left( \rho_w \frac{\partial S_w}{\partial t} + \rho_n \frac{\partial S_n}{\partial t}\right) - \textbf{div}\, \left[\lambda \boldsymbol{K} \left(\textbf{grad}\, p_w + f_n \, \textbf{grad}\, p_c + \sum f_\alpha \rho_\alpha \, g \, \textbf{grad}\, z\right)\right] = q, \] and for a non-\/wetting ( $ n $) phase pressure as \[ \phi \left( \rho_w \frac{\partial S_w}{\partial t} + \rho_n \frac{\partial S_n}{\partial t}\right) - \textbf{div}\, \left[\lambda \boldsymbol{K} \left(\textbf{grad}\, p_n - f_w \textbf{grad}\, p_c + \sum f_\alpha \rho_\alpha \, g \, \textbf{grad}\, z\right)\right] = q, \] \-In this slightly compressible case the following definitions are valid\-: $ \lambda = \rho_w \lambda_w + \rho_n \lambda_n $, $ f_\alpha = (\rho_\alpha \lambda_\alpha) / \lambda $ \-This model assumes that temporal changes in density are very small and thus terms of temporal derivatives are negligible in the pressure equation. \-Depending on the formulation the terms including time derivatives of saturations are simplified by inserting $ S_w + S_n = 1 $.
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\-In the \-I\-M\-P\-E\-S models the default setting is\-:
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\begin{itemize}
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\item formulation\-: $ p_w-S_w $ (\-Property\-: {\ttfamily \-Formulation} defined as {\ttfamily \hyperlink{a00056_a04294fbcf0af5328016a160dbd8bfff9}{\-Decoupled\-Two\-P\-Common\-Indices\-::pw\-Sw}})
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\item compressibility\-: disabled (\-Property\-: {\ttfamily \-Enable\-Compressibility} set to {\ttfamily false})
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\end{itemize}
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In the IMPES models the default setting is: \begin{itemize}
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\item formulation: Property: {\ttfamily Formulation} defined as {\ttfamily DecoupledTwoPCommonIndices::pwSw}
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\item compressibility: disabled Property: {\ttfamily EnableCompressibility} set to {\ttfamily false}
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\end{itemize}
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