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7deaf50402
the descriptions of the semi-implicit models are imported from dumux because I suppose that these have been modified manually and the differences should be negligible anyway...
17 lines
1.8 KiB
TeX
17 lines
1.8 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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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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This model solves equations of the form
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\[ \phi \frac{\partial (\rho_\alpha S_\alpha)}{\partial t} + \textbf{div}\, (\rho_\alpha \boldsymbol{v_\alpha}) = q_\alpha, \]
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where $ S_\alpha $ is the saturation of phase $ \alpha $ (wetting $(w) $, non-\/wetting $(n) $) and $ \boldsymbol v_\alpha $ is the phase velocity defined by the multi-\/phase Darcy equation. If a phase velocity is reconstructed from the pressure solution it can be directly inserted into the previous equation. In the incompressible case the equation is further divided by the phase density $ \rho_\alpha $. If a total velocity is reconstructed the saturation equation is reformulated into\-:
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\[ \phi \frac{\partial S_w}{\partial t} + f_w \textbf{div}\, \boldsymbol{v}_{t} + f_w \lambda_n \boldsymbol{K}\left(\textbf{grad}\, p_c + (\rho_n-\rho_w) \, g \, \textbf{grad} z \right)= q_\alpha, \] to get a wetting phase saturation or \[ \phi \frac{\partial S_n}{\partial t} + f_n \textbf{div}\, \boldsymbol{v}_{t} - f_n \lambda_w \boldsymbol{K}\left(\textbf{grad}\, p_c + (\rho_n-\rho_w) \, g \, \textbf{grad} z \right)= q_\alpha, \] if the non-\/wetting phase saturation is the primary transport variable.
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The total velocity formulation is only implemented for incompressible fluids and $ f_\alpha $ is the fractional flow function, $ \lambda_\alpha $ is the mobility, $ \boldsymbol K $ the absolute permeability, $ p_c $ the capillary pressure, $ \rho $ the fluid density, $ g $ the gravity constant, and $ q $ the source term.
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