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6a7345fe4e
We now use the doxygen detailed description for the immiscible decoupled models, as approved by Markus.
13 lines
2.3 KiB
TeX
13 lines
2.3 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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\-In the unsaturated zone, \-Richards' equation is frequently used to approximate the water distribution above the groundwater level. \-It can be derived from the two-\/phase equations, i.\-e. \[ \frac{\partial\;\phi S_\alpha \rho_\alpha}{\partial t} - \text{div} \left\{ \rho_\alpha \frac{k_{r\alpha}}{\mu_\alpha}\; \mathbf{K} \textbf{grad}\left[ p_\alpha - g\rho_\alpha \right] \right\} = q_\alpha, \] where $\alpha \in \{w, n\}$ is the fluid phase, $\rho_\alpha$ is the fluid density, $S_\alpha$ is the fluid saturation, $\phi$ is the porosity of the soil, $k_{r\alpha}$ is the relative permeability for the fluid, $\mu_\alpha$ is the fluid's dynamic viscosity, $\mathbf{K}$ is the intrinsic permeability, $p_\alpha$ is the fluid pressure and $g$ is the potential of the gravity field.
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\-In contrast to the full two-\/phase model, the \-Richards model assumes gas as the non-\/wetting fluid and that it exhibits a much lower viscosity than the (liquid) wetting phase. (\-For example at atmospheric pressure and at room temperature, the viscosity of air is only about $1\%$ of the viscosity of liquid water.) \-As a consequence, the $\frac{k_{r\alpha}}{\mu_\alpha}$ term typically is much larger for the gas phase than for the wetting phase. \-For this reason, the \-Richards model assumes that $\frac{k_{rn}}{\mu_n}$ is infinitly large. \-This implies that the pressure of the gas phase is equivalent to the static pressure distribution and that therefore, mass conservation only needs to be considered for the wetting phase.
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\-The model thus choses the absolute pressure of the wetting phase $p_w$ as its only primary variable. \-The wetting phase saturation is calculated using the inverse of the capillary pressure, i.\-e. \[ S_w = p_c^{-1}(p_n - p_w) \] holds, where $p_n$ is a given reference pressure. \-Nota bene, that the last step is assumes that the capillary pressure-\/saturation curve can be uniquely inverted, so it is not possible to set the capillary pressure to zero when using the \-Richards model!
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