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14 lines
2.3 KiB
TeX
14 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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This model implements a variant of the Richards equation for quasi-\/twophase flow. In the unsaturated zone, Richards' equation is frequently used to calculate the water distribution above the groundwater level. It can be derived from the twophase equations, i.e. \[ \frac{\partial\;\phi S_\alpha \rho_\alpha}{\partial t} - \mathbf{div} \left\{ \rho_\alpha \frac{k_{r\alpha}}{\mu_\alpha}\;K \mathbf{grad}\left[ p_\alpha - g\rho_\alpha \right] \right\} = q_\alpha, \] 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, $K$ 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 twophase 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 $\frac{k_{rn}}{\mu_n}$ tends to infinity. This implies that the pressure of the gas phase is equivalent to a static pressure and can thus be specified externally 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 inverted uniquely, so it is not possible to set the capillary pressure to zero when using the Richards model!
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