diff --git a/dumux/porousmediumflow/richards/model.hh b/dumux/porousmediumflow/richards/model.hh index bb0e4f084251841e37bf36abeff43961476abf98..00bdb7d97a456a4733540228fbf7e92a56e00e20 100644 --- a/dumux/porousmediumflow/richards/model.hh +++ b/dumux/porousmediumflow/richards/model.hh @@ -26,43 +26,27 @@ q_w, \f] * - * is frequently used to - * approximate the water distribution above the groundwater level. - * - * It can be derived from the two-phase equations, i.e. - \f[ - \phi\frac{\partial S_\alpha \varrho_\alpha}{\partial t} - - - \nabla \cdot \left\lbrace - \varrho_\alpha \frac{k_{r\alpha}}{\mu_\alpha}\; \mathbf{K} \; - \left(\nabla - p_\alpha - \varrho_\alpha \textbf{g} - \right) - \right\rbrace - = - q_\alpha, - \f] - * * where: * * \f$ \phi \f$ is the porosity of the porous medium, - * * \f$ S_\alpha \f$ represents the saturation of phase \f$ \alpha \f$, - * * \f$ \varrho_\alpha \f$ is the mass density of phase \f$ \alpha \f$, - * * \f$ k_{r\alpha} \f$ is the relative permeability of phase \f$ \alpha \f$, - * * \f$ \mu_\alpha \f$ is the dynamic viscosity of phase \f$ \alpha \f$, + * * \f$ S_w \f$ represents the water saturation, + * * \f$ \varrho_w \f$ is the water density, + * * \f$ k_{rw} \f$ is the relative permeability of the water phase, + * * \f$ \mu_w \f$ is the dynamic viscosity of the water phase, * * \f$ \mathbf{K} \f$ is the intrinsic permeability tensor, - * * \f$ p_\alpha \f$ is the pressure of phase \f$ \alpha \f$, + * * \f$ p_w \f$ is the liquid water pressure, * * \f$ \mathbf{g} \f$ is the gravitational acceleration vector, - * * \f$ q_\alpha \f$ is a source or sink term. + * * \f$ q_w \f$ is a source or sink term. * + * It can be derived from the two-phase flow equations. * In contrast to the full two-phase model, the Richards model assumes * gas as the nonwetting 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 \f$1\%\f$ of the viscosity of liquid water.) As a - * consequence, the \f$\frac{k_{r\alpha}}{\mu_\alpha}\f$ term - * typically is much larger for the gas phase than for the wetting + * consequence, the mobility (\f$\frac{k_{r}}{\mu}\f$) is + * typically much larger for the gas phase than for the wetting * phase. For this reason, the Richards model assumes that - * \f$\frac{k_{rn}}{\mu_n}\f$ is infinitely large. This implies that + * gas phase mobility is infinitely 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. @@ -72,9 +56,9 @@ * saturation is calculated using the inverse of the capillary * pressure, i.e. \f[ - S_w = p_c^{-1}(p_n - p_w) + S_w = p_c^{-1}(p_g - p_w) \f] - * holds, where \f$p_n\f$ is a given reference pressure. Nota bene, + * holds, where \f$p_g\f$ is a given reference gas 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 diff --git a/dumux/porousmediumflow/richardsextended/model.hh b/dumux/porousmediumflow/richardsextended/model.hh index 0c04bd0a8c057190965e09f6827bf01b6753f01f..49922b3f1a6c4df0ec12feeb97c96f3f01019929 100644 --- a/dumux/porousmediumflow/richardsextended/model.hh +++ b/dumux/porousmediumflow/richardsextended/model.hh @@ -37,63 +37,14 @@ * * \f$ \mathbf{K} \f$ is the intrinsic permeability tensor, * * \f$ p_w \f$ is the pressure of the wetting phase, * * \f$ \mathbf{g} \f$ is the gravitational acceleration vector, - * * \f$ \bf D_{n,pm}^{w} \f$ is the diffusivity of water in the non-wetting phase, + * * \f$ \bf D_{n,pm}^{w} \f$ is the effective diffusivity of water in the non-wetting phase, * * \f$ X_n^w \f$ is the mass fraction of water in the non-wetting phase, * * \f$ q_w \f$ is a source or sink term in the wetting phase, * - * is frequently used to - * approximate the water distribution above the groundwater level. - * - * It can be derived from the two-phase equations, i.e. - \f[ - \phi\frac{\partial S_\alpha \varrho_\alpha}{\partial t} - - - \nabla \cdot \left\lbrace - \varrho_\alpha \frac{k_{r\alpha}}{\mu_\alpha}\; \mathbf{K} \; - \left( \text{\nabla} - p_\alpha - \varrho_\alpha \textbf{g} - \right) - \right\rbrace - = - q_\alpha, - \f] - * - * where: - * * \f$ \phi \f$ is the porosity of the porous medium, - * * \f$ S_\alpha \f$ represents the saturation of phase \f$ \alpha \f$, - * * \f$ \varrho_\alpha \f$ is the mass density of phase \f$ \alpha \f$, - * * \f$ k_{r\alpha} \f$ is the relative permeability of phase \f$ \alpha \f$, - * * \f$ \mu_\alpha \f$ is the dynamic viscosity of phase \f$ \alpha \f$, - * * \f$ \mathbf{K} \f$ is the intrinsic permeability tensor, - * * \f$ p_\alpha \f$ is the pressure of phase \f$ \alpha \f$, - * * \f$ \mathbf{g} \f$ is the gravitational acceleration vector, - * * \f$ q_\alpha \f$ is a source or sink term. - * - * In contrast to the full two-phase model, the Richards model assumes - * gas as the nonwetting 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 \f$1\%\f$ of the viscosity of liquid water.) As a - * consequence, the \f$\frac{k_{r\alpha}}{\mu_\alpha}\f$ term - * typically is much larger for the gas phase than for the wetting - * phase. For this reason, the Richards model assumes that - * \f$\frac{k_{rn}}{\mu_n}\f$ is infinitely 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. - * - * The model thus chooses the absolute pressure of the wetting phase - * \f$p_w\f$ as its only primary variable. The wetting phase - * saturation is calculated using the inverse of the capillary - * pressure, i.e. - \f[ - S_w = p_c^{-1}(p_n - p_w) - \f] - * holds, where \f$p_n\f$ 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! + * additionally models water vapor diffusion in the gas phase. + * The model is derived based on the two-phase flow equations + * based on the assumption that the gas phase does not move but + * and remains at constant pressure. */ #ifndef DUMUX_RICHARDSEXTENDED_MODEL_HH diff --git a/dumux/porousmediumflow/richardsnc/model.hh b/dumux/porousmediumflow/richardsnc/model.hh index 737f318cd671b74bdb660bf744cc032769334485..813666b637968139d80e210a8b5ee39f2dd35810 100644 --- a/dumux/porousmediumflow/richardsnc/model.hh +++ b/dumux/porousmediumflow/richardsnc/model.hh @@ -10,7 +10,8 @@ * \brief Base class for all models which use the Richards, * n-component fully implicit model. * - * In the unsaturated zone, Richards' equation + * This extension of Richards' equation, allows for + * the wetting phase to consist of multiple components: *\f{eqnarray*} && \frac{\partial (\sum_w \varrho_w X_w^\kappa \phi S_w )} {\partial t} @@ -31,38 +32,9 @@ * * \f$ \mathbf{K} \f$ is the intrinsic permeability tensor, * * \f$ p_w \f$ is the pressure of the wetting phase, * * \f$ \mathbf{g} \f$ is the gravitational acceleration vector, - * * \f$ \bf D_{w,pm}^{k} \f$ is the diffusivity of component \f$ \kappa \f$ in the wetting phase, + * * \f$ \bf D_{w,pm}^{k} \f$ is the effective diffusivity of component \f$ \kappa \f$ in the wetting phase, * * \f$ X_w^k \f$ is the mass fraction of component \f$ \kappa \f$ in the wetting phase, - * * \f$ q_w \f$ is a source or sink term in the wetting phase, - * - * is frequently used to - * approximate the water distribution above the groundwater level. - * - * In contrast to the full two-phase model, the Richards model assumes - * gas as the nonwetting 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 \f$1\%\f$ of the viscosity of liquid water.) As a - * consequence, the \f$\frac{k_{r\alpha}}{\mu_\alpha}\f$ term - * typically is much larger for the gas phase than for the wetting - * phase. For this reason, the Richards model assumes that - * \f$\frac{k_{rn}}{\mu_n}\f$ is infinitely 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. - * - * The model thus chooses the absolute pressure of the wetting phase - * \f$p_w\f$ as its only primary variable. The wetting phase - * saturation is calculated using the inverse of the capillary - * pressure, i.e. - \f[ - S_w = p_c^{-1}(p_n - p_w) - \f] - * holds, where \f$p_n\f$ 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! + * * \f$ q_w \f$ is a source or sink term in the wetting phase. */ #ifndef DUMUX_RICHARDSNC_MODEL_HH