diff --git a/dumux/porousmediumflow/1p/model.hh b/dumux/porousmediumflow/1p/model.hh index 9567ad7a05e361cfc788ab90d1972356b9d3c970..974f746043cb0f88a51e07d71992262c509be8e1 100644 --- a/dumux/porousmediumflow/1p/model.hh +++ b/dumux/porousmediumflow/1p/model.hh @@ -12,11 +12,20 @@ * Single-phase, isothermal flow model, which uses a standard Darcy approach as the * equation for the conservation of momentum. For details on Darcy's law see dumux/flux/darcyslaw.hh. * - * Furthermore, it solves the mass continuity equation: + * Furthermore, it solves the mass continuity equation * \f[ \phi \frac{\partial \varrho}{\partial t} + \text{div} \left\lbrace - \varrho \frac{\textbf K}{\mu} \left( \textbf{grad}\, p -\varrho {\textbf g} \right) \right\rbrace = q, * \f] +* where: + * * \f$ \phi \f$ is the porosity of the porous medium, + * * \f$ \varrho \f$ is the mass density, + * * \f$ \textbf{K} \f$ is the intrinsic permeability tensor, + * * \f$ \mu \f$ represents the dynamic viscosity, + * * \f$ p \f$ is the pressure, + * * \f$ \textbf{g} \f$ is the gravitational acceleration vector, + * * and \f$ q \f$ is a source or sink term. + * * * The model supports compressible as well as incompressible fluids. */ diff --git a/dumux/porousmediumflow/1pnc/model.hh b/dumux/porousmediumflow/1pnc/model.hh index 74eab408b965aa7e4f68187a0bcbd2a7e3be95ba..e44629d578927dab000be18b306ad3f64d7016b2 100644 --- a/dumux/porousmediumflow/1pnc/model.hh +++ b/dumux/porousmediumflow/1pnc/model.hh @@ -17,7 +17,7 @@ \f[ \phi\frac{\partial \varrho}{\partial t} - \text{div} \left\{ \varrho \frac{\textbf K}{\mu} \left(\textbf{grad}\, p - \varrho {\textbf g} \right) - \right\} = q \;, + \right\} = q. \f] * * The transport of the components \f$\kappa \in \{ w, a, ... \}\f$ is described by the following equation: @@ -25,8 +25,19 @@ \phi \frac{ \partial \varrho X^\kappa}{\partial t} - \text{div} \left\lbrace \varrho X^\kappa \frac{{\textbf K}}{\mu} \left( \textbf{grad}\, p - \varrho {\textbf g} \right) - + \varrho D^\kappa_\text{pm} \textbf{grad} X^\kappa \right\rbrace = q. + + \varrho D^\kappa_\text{pm} \textbf{grad} X^\kappa \right\rbrace = q, \f] + * + * where: + * * \f$ \phi \f$ is the porosity of the porous medium, + * * \f$ \varrho \f$ is the mass density, + * * \f$ X^\kappa \f$ is the mass fraction of component \f$ \kappa \f$, + * * \f$ \textbf{K} \f$ is the intrinsic permeability tensor, + * * \f$ \mu \f$ represents the dynamic viscosity, + * * \f$ p \f$ is the pressure, + * * \f$ \textbf{g} \f$ is the gravitational acceleration vector, + * * \f$ {\bf D_{pm}^\kappa} \f$ is the diffusivity in the porous medium, + * * and \f$ q \f$ is a source or sink term. * * The model is able to use either mole or mass fractions. The property useMoles can be set to either true or false in the * problem file. Make sure that the according units are used in the problem setup. useMoles is set to true by default. diff --git a/dumux/porousmediumflow/1pncmin/model.hh b/dumux/porousmediumflow/1pncmin/model.hh index db3ff0612148ac0b338a9010592b05d2674b48a0..3c853041f9dfc8931a936858c071ab5f62d1a1cf 100644 --- a/dumux/porousmediumflow/1pncmin/model.hh +++ b/dumux/porousmediumflow/1pncmin/model.hh @@ -16,33 +16,56 @@ * For details on Darcy's law see dumux/flux/darcyslaw.hh. * * By inserting Darcy's law into the equations for the conservation of the -* components, one gets one transport equation for each component +* components, one gets one transport equation for each component, * \f[ \frac{\partial ( \varrho_f X^\kappa \phi )} {\partial t} - \text{div} \left\{ \varrho_f X^\kappa \frac{k_{r}}{\mu} \mathbf{K} (\text{grad}\, p - \varrho_{f} \mathbf{g}) \right\} - \text{div} \left\{{\bf D_{pm}^\kappa} \varrho_{f} \text{grad}\, X^\kappa \right\} -- q_\kappa = 0 \qquad \kappa \in \{w, a,\cdots \} +- q_\kappa = 0 \qquad \kappa \in \{w, a,\cdots \}, * \f] +* where: +* * \f$ \phi \f$ is the porosity, +* * \f$ \varrho_f \f$ is the mass density of the fluid, +* * \f$ X^\kappa \f$ is the mass fraction of component \f$ \kappa \f$ in the fluid, +* * \f$ k_{r} \f$ is the relative permeability, +* * \f$ \mu \f$ represents the dynamic viscosity, +* * \f$ \mathbf{K} \f$ is the intrinsic permeability tensor, +* * \f$ p \f$ is the pressure, +* * \f$ \mathbf{g} \f$ is the gravitational acceleration vector, +* * \f$ {\bf D_{pm}^\kappa} \f$ is the diffusivity in the porous medium, +* * \f$ q_\kappa \f$ is a source or sink term. * * The solid or mineral phases are assumed to consist of a single component. -* Their mass balance consist only of a storage and a source term: +* Their mass balance consists of only a storage and a source term, * \f[ - \frac{\partial ( \varrho_\lambda \phi_\lambda )} {\partial t} = q_\lambda + \frac{\partial ( \varrho_\lambda \phi_\lambda )} {\partial t} = q_\lambda, * \f] -* +* where: +* * \f$ \varrho_\lambda\f$ represents the mass density of the solid phase, +* * \f$ \phi_\lambda \f$ is the porosity of the solid, +* * \f$ q_\lambda \f$ is a source or sink term representing. * * The primary variables are the pressure \f$p\f$ and the mole fractions of the * dissolved components \f$x^k\f$. The primary variable of the solid phases is the volume * fraction -\f$\phi_\lambda = \frac{V_\lambda}{V_{total}}\f$. +\f$\phi_\lambda = \frac{V_\lambda}{V_{total}}\f$, +* +* where: +* * \f$ V_\lambda \f$ is the volume of phase \f$ \lambda \f$, +* * \f$ V_{\text{total}} \f$ is the total volume of the system. * * The source an sink terms link the mass balances of the n-transported component to the * solid phases. The porosity \f$\phi\f$ is updated according to the reduction of the initial * (or solid-phase-free porous medium) porosity \f$\phi_0\f$ by the accumulated volume -* fractions of the solid phases: -* \f$ \phi = \phi_0 - \sum (\phi_\lambda)\f$ +* fractions of the solid phases, +* \f$ \phi = \phi_0 - \sum (\phi_\lambda),\f$ +* +* where: +* * \f$ \phi \f$ represents the remaining porosity in the system, +* * \f$ \phi_0 \f$ is the initial porosity, +* * \f$ \phi_\lambda \f$ denotes the volume fraction of phase \f$ \lambda \f$. * Additionally, the permeability is updated depending on the current porosity. */ diff --git a/dumux/porousmediumflow/2p/model.hh b/dumux/porousmediumflow/2p/model.hh index 534154f545a283f95ef7567f12c8520541e5e43c..ea762c5ec9bf7a793e4255aa0a8b64bad6137300 100644 --- a/dumux/porousmediumflow/2p/model.hh +++ b/dumux/porousmediumflow/2p/model.hh @@ -21,8 +21,18 @@ - \text{div} \left\{ \varrho_\alpha \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left(\textbf{grad}\, p_\alpha - \varrho_{\alpha} \mathbf{g} \right) - \right\} - q_\alpha = 0 \;, + \right\} - q_\alpha = 0, \f] + *where: + * * \f$ \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. * * By using constitutive relations for the capillary pressure \f$p_c = * p_n - p_w\f$ and relative permeability \f$k_{r\alpha}\f$ and taking diff --git a/dumux/porousmediumflow/2p1c/model.hh b/dumux/porousmediumflow/2p1c/model.hh index 2ad2fad615434184658f43f1ef79f1570f5b30e8..aadf24482aae9322d7a2e33bd1b1e84bf8b2fd51 100644 --- a/dumux/porousmediumflow/2p1c/model.hh +++ b/dumux/porousmediumflow/2p1c/model.hh @@ -24,8 +24,18 @@ * phase mass, one gets \f[ \phi \frac{\partial\ \sum_\alpha (\rho_\alpha S_\alpha)}{\partial t} \\-\sum \limits_ \alpha \text{div} \left \{\rho_\alpha \frac{k_{r\alpha}}{\mu_\alpha} -\mathbf{K} (\mathbf{grad}p_\alpha - \rho_\alpha \mathbf{g}) \right \} -q^w =0 +\mathbf{K} (\mathbf{grad}p_\alpha - \rho_\alpha \mathbf{g}) \right \} -q^w =0, \f] + * where: + * * \f$ \phi \f$ is the porosity of the porous medium, + * * \f$ S_\alpha \f$ represents the saturation of phase \f$ \alpha \f$, + * * \f$ \rho_\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^w \f$ is a source or sink term. * * By using constitutive relations for the capillary pressure \f$p_c = * p_n - p_w\f$ and relative permeability \f$k_{r\alpha}\f$ and taking diff --git a/dumux/porousmediumflow/2p2c/model.hh b/dumux/porousmediumflow/2p2c/model.hh index fab70a8dc390f2a21e0ef9f37f1172e27cf4d664..4b388d2ccbce567017f5033a6f4e3fd512f460d7 100644 --- a/dumux/porousmediumflow/2p2c/model.hh +++ b/dumux/porousmediumflow/2p2c/model.hh @@ -14,22 +14,30 @@ * \f$\kappa \in \{ \kappa_w, \kappa_n \}\f$, where \f$\kappa_w\f$ and \f$\kappa_n\f$ are * the main components of the wetting and nonwetting phases, respectively. * The governing equations are the mass or the mole conservation equations of the two components, - * depending on the property <tt>UseMoles</tt>. The mass balance equations are given as: + * depending on the property <tt>UseMoles</tt>. The mass balance equations are given as * \f[ \phi \frac{\partial (\sum_\alpha \rho_\alpha X_\alpha^\kappa S_\alpha)}{\partial t} - \sum_\alpha \text{div} \left\{ \rho_\alpha X_\alpha^\kappa v_\alpha \right\} - \sum_\alpha \text{div} \mathbf{F}_{\mathrm{diff, mass}, \alpha}^\kappa - - \sum_\alpha q_\alpha^\kappa = 0 \qquad \kappa \in \{\kappa_w, \kappa_n\} \, , \alpha \in \{w, n\}, + - \sum_\alpha q_\alpha^\kappa = 0 \qquad \kappa \in \{\kappa_w, \kappa_n\} \, , \alpha \in \{w, n\}. \f] - * using the mass fractions \f$X_\alpha^\kappa\f$ and the mass densities \f$\rho_\alpha\f$, while - * the mole balance equations use the mole fractions \f$x_\alpha^\kappa\f$ and molar - * densities \f$\varrho_{m, \alpha}\f$: + * The mole balance is given as * \f[ \phi \frac{\partial (\sum_\alpha \varrho_{m, \alpha} x_\alpha^\kappa S_\alpha)}{\partial t} + \sum_\alpha \text{div} \left\{ \varrho_{m, \alpha} x_\alpha^\kappa v_\alpha \right\} + \sum_\alpha \text{div} \mathbf{F}_{\mathrm{diff, mole}, \alpha}^\kappa - - \sum_\alpha q_\alpha^\kappa = 0 \qquad \kappa \in \{\kappa_w, \kappa_n\} \, , \alpha \in \{w, n\}. + - \sum_\alpha q_\alpha^\kappa = 0 \qquad \kappa \in \{\kappa_w, \kappa_n\} \, , \alpha \in \{w, n\}, \f] + * where: + * * \f$ \phi \f$ is the porosity of the porous medium, + * * \f$ S_\alpha \f$ represents the saturation of phase \f$ \alpha \f$, + * * \f$ \rho_\alpha \f$ is the mass density of phase \f$ \alpha \f$, + * * \f$ X_\alpha^\kappa \f$ is the mass fraction of component \f$ \kappa \f$ in phase \f$ \alpha \f$, + * * \f$ x_\alpha^\kappa \f$ is the mole fraction of component \f$ \kappa \f$ in phase \f$ \alpha \f$, + * * \f$ v_\alpha \f$ is the velocity of phase \f$ \alpha \f$, + * * \f$ \mathbf{F}_{\mathrm{diff, mass}, \alpha}^\kappa \f$ represents the diffusive mass flux of component \f$ \kappa \f$ in phase \f$ \alpha \f$, + * * \f$ q_\alpha^\kappa \f$ is a source or sink term. + * * Boundary conditions and sources have to be defined by the user in the corresponding * units. The default setting for the property <tt>UseMoles</tt> can be found in the 2pnc model. * diff --git a/dumux/porousmediumflow/2pnc/model.hh b/dumux/porousmediumflow/2pnc/model.hh index 2f06005de08e273ab6527db1940b71e96cbfa322..c4eeee6390be88cf38a4df96750a5a591d6e8ced 100644 --- a/dumux/porousmediumflow/2pnc/model.hh +++ b/dumux/porousmediumflow/2pnc/model.hh @@ -18,7 +18,7 @@ * For details on Darcy's law see dumux/flux/darcyslaw.hh. * * By inserting Darcy's law into the equations for the conservation of the - * components, one gets one transport equation for each component + * components, one gets one transport equation for each component, * \f{eqnarray*}{ * && \frac{\partial (\sum_\alpha \varrho_\alpha X_\alpha^\kappa \phi S_\alpha )} * {\partial t} @@ -28,13 +28,27 @@ * \nonumber \\ \nonumber \\ * &-& \sum_\alpha \text{div} \left\{{\bf D_{\alpha, pm}^\kappa} \varrho_{\alpha} \text{grad}\, X^\kappa_{\alpha} \right\} * - \sum_\alpha q_\alpha^\kappa = 0 \qquad \kappa \in \{w, a,\cdots \} \, , - * \alpha \in \{w, g\} + * \alpha \in \{w, g\}, * \f} + * where: + * * \f$ \phi \f$ is the porosity of the porous medium, + * * \f$ S_\alpha \f$ represents the saturation of phase \f$ \alpha \f$, + * * \f$ \rho_\alpha \f$ is the mass density of phase \f$ \alpha \f$, + * * \f$ X_\alpha^\kappa \f$ is the mass fraction of component \f$ \kappa \f$ in phase \f$ \alpha \f$, + * * \f$ x_\alpha^\kappa \f$ is the mole fraction of component \f$ \kappa \f$ in phase \f$ \alpha \f$, + * * \f$ v_\alpha \f$ is the velocity of phase \f$ \alpha \f$, + * * \f$ {\bf D_{\alpha, pm}^\kappa} \f$ is the diffusivity of component \f$ \kappa \f$ in phase \f$ \alpha \f$, + * * \f$ q_\alpha^\kappa \f$ is a source or sink term. * * The solid or mineral phases are assumed to consist of a single component. - * Their mass balance consist only of a storage and a source term: + * Their mass balance consists of only a storage and a source term, * \f$\frac{\partial \varrho_\lambda \phi_\lambda )} {\partial t} - * = q_\lambda\f$ + * = q_\lambda,\f$ + * + * where: + * * \f$ \varrho_\lambda \f$ is the mass density of the solid phase \f$ \lambda \f$, + * * \f$ \phi_\lambda \f$ is the porosity of the solid, + * * \f$ q_\lambda \f$ is a source or sink term. * * By using constitutive relations for the capillary pressure \f$p_c = * p_n - p_w\f$ and relative permeability \f$k_{r\alpha}\f$ and taking diff --git a/dumux/porousmediumflow/2pncmin/model.hh b/dumux/porousmediumflow/2pncmin/model.hh index af3b475b279164f48fc5fdeffd201c934527e591..ed403351ba2a42759479d7ff8fc0a796e5ee78ec 100644 --- a/dumux/porousmediumflow/2pncmin/model.hh +++ b/dumux/porousmediumflow/2pncmin/model.hh @@ -18,7 +18,7 @@ * For details on Darcy's law see dumux/flux/darcyslaw.hh. * * By inserting Darcy's law into the equations for the conservation of the - * components, one gets one transport equation for each component + * components, one gets one transport equation for each component, * \f{eqnarray*}{ * && \frac{\partial (\sum_\alpha \varrho_\alpha X_\alpha^\kappa \phi S_\alpha )} * {\partial t} @@ -28,13 +28,28 @@ * \nonumber \\ \nonumber \\ * &-& \sum_\alpha \text{div} \left\{{\bf D_{\alpha, pm}^\kappa} \varrho_{\alpha} \text{grad}\, X^\kappa_{\alpha} \right\} * - \sum_\alpha q_\alpha^\kappa = 0 \qquad \kappa \in \{w, a,\cdots \} \, , - * \alpha \in \{w, g\} + * \alpha \in \{w, g\}, * \f} * + * where: + * * \f$ \phi \f$ is the porosity of the porous medium, + * * \f$ S_\alpha \f$ represents the saturation of phase \f$ \alpha \f$, + * * \f$ \rho_\alpha \f$ is the mass density of phase \f$ \alpha \f$, + * * \f$ X_\alpha^\kappa \f$ is the mass fraction of component \f$ \kappa \f$ in phase \f$ \alpha \f$, + * * \f$ x_\alpha^\kappa \f$ is the mole fraction of component \f$ \kappa \f$ in phase \f$ \alpha \f$, + * * \f$ v_\alpha \f$ is the velocity of phase \f$ \alpha \f$, + * * \f$ {\bf D_{\alpha, pm}^\kappa} \f$ is the diffusivity of component \f$ \kappa \f$ in phase \f$ \alpha \f$, + * * \f$ q_\alpha^\kappa \f$ is a source or sink term. + * * The solid or mineral phases are assumed to consist of a single component. - * Their mass balance consist only of a storage and a source term: + * Their mass balance consists of only a storage and a source term, * \f$\frac{\partial ( \varrho_\lambda \phi_\lambda )} {\partial t} - * = q_\lambda\f$ + * = q_\lambda,\f$ + * + * where: + * * \f$ \varrho_\lambda \f$ mass density of the solid phase \f$ \lambda \f$, + * * \f$ \phi_\lambda \f$ is the porosity of the solid, + * * \f$ q_\lambda \f$ is a source or sink term. * * By using constitutive relations for the capillary pressure \f$p_c = * p_n - p_w\f$ and relative permeability \f$k_{r\alpha}\f$ and taking diff --git a/dumux/porousmediumflow/3p/model.hh b/dumux/porousmediumflow/3p/model.hh index 9416a6b37d914f38ced9e758d0b7e8684a1878a9..93355b96a1044ce7cbb6988848e7ee96e039bfc3 100644 --- a/dumux/porousmediumflow/3p/model.hh +++ b/dumux/porousmediumflow/3p/model.hh @@ -22,8 +22,18 @@ - \text{div} \left\{ \varrho_\alpha \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left(\textbf{grad}\, p_\alpha - \varrho_{\alpha} \mathbf{g} \right) - \right\} - q_\alpha = 0 \;. + \right\} - q_\alpha = 0, \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. * * The model uses commonly applied auxiliary conditions like * \f$S_w + S_n + S_g = 1\f$ for the saturations. diff --git a/dumux/porousmediumflow/3p3c/model.hh b/dumux/porousmediumflow/3p3c/model.hh index ca7c8add70f6512ede88cab7e25ca2effc26d51d..4896a2b9ba917a56e06c7304df2700fe37e11ef3 100644 --- a/dumux/porousmediumflow/3p3c/model.hh +++ b/dumux/porousmediumflow/3p3c/model.hh @@ -28,8 +28,18 @@ \nonumber \\ && - \sum\limits_\alpha \text{div} \left\{ D_\text{pm}^\kappa \frac{1}{M_{\kappa}} \varrho_{\alpha} \textbf{grad} X^\kappa_{\alpha} \right\} - - q^\kappa = 0 \qquad \forall \kappa , \; \forall \alpha + - q^\kappa = 0 \qquad \forall \kappa , \; \forall \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$, + * * \f$ \rho_\alpha \f$ is the mass density of phase \f$ \alpha \f$, + * * \f$ X_\alpha^\kappa \f$ is the mass fraction of component \f$ \kappa \f$ in phase \f$ \alpha \f$, + * * \f$ x_\alpha^\kappa \f$ is the mole fraction of component \f$ \kappa \f$ in phase \f$ \alpha \f$, + * * \f$ v_\alpha \f$ is the velocity of phase \f$ \alpha \f$, + * * \f$ {\bf D_{\alpha, pm}^\kappa} \f$ is the diffusivity of component \f$ \kappa \f$ in phase \f$ \alpha \f$, + * * \f$ M_\kappa \f$ is the molar mass of component \f$ \kappa \f$ + * * \f$ q_\alpha^\kappa \f$ is a source or sink term. * * Note that these balance equations are molar. * diff --git a/dumux/porousmediumflow/3pwateroil/model.hh b/dumux/porousmediumflow/3pwateroil/model.hh index 397b2007f6c98dbb8af14cbce644e0202d582f8f..3a167f89f3ebb7a0028258e853b6a9e015dece2e 100644 --- a/dumux/porousmediumflow/3pwateroil/model.hh +++ b/dumux/porousmediumflow/3pwateroil/model.hh @@ -29,9 +29,20 @@ \nonumber \\ && - \sum\limits_\alpha \text{div} \left\{ D_\text{pm}^\kappa \varrho_\alpha \frac{1}{M_\kappa} \textbf{grad} X^\kappa_{\alpha} \right\} - - q^\kappa = 0 \qquad \forall \kappa , \; \forall \alpha + - q^\kappa = 0 \qquad \forall \kappa , \; \forall \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$, + * * \f$ \rho_\alpha \f$ is the mass density of phase \f$ \alpha \f$, + * * \f$ X_\alpha^\kappa \f$ is the mass fraction of component \f$ \kappa \f$ in phase \f$ \alpha \f$, + * * \f$ x_\alpha^\kappa \f$ is the mole fraction of component \f$ \kappa \f$ in phase \f$ \alpha \f$, + * * \f$ v_\alpha \f$ is the velocity of phase \f$ \alpha \f$, + * * \f$ {\bf D_{\alpha, pm}^\kappa} \f$ is the diffusivity of component \f$ \kappa \f$ in phase \f$ \alpha \f$, + * * \f$ M_\kappa \f$ is the molar mass of component \f$ \kappa \f$ + * * \f$ q_\alpha^\kappa \f$ is a source or sink term. + * * Note that these balance equations are molar. * * The model uses commonly applied auxiliary conditions like diff --git a/dumux/porousmediumflow/mineralization/model.hh b/dumux/porousmediumflow/mineralization/model.hh index 2577e9874d0204fd2eceb1de83a54c918971ffd9..d23ae48e6b9edd807bb9057729a3f57107bca782 100644 --- a/dumux/porousmediumflow/mineralization/model.hh +++ b/dumux/porousmediumflow/mineralization/model.hh @@ -12,8 +12,15 @@ * components. * * The solid or mineral phases are assumed to consist of a single component. - * Their mass balance consist only of a storage and a source term: - * \f$\frac{\partial ( \varrho_\lambda \phi_\lambda )} {\partial t} = q_\lambda\f$ + * Their mass balance consists of only a storage and a source term, + * \f[ + * \frac{\partial ( \varrho_\lambda \phi_\lambda )} {\partial t} = q_\lambda, + * \f] + * + * where: + * * \f$ \varrho_\lambda \f$ is the mass density of the solid phase \f$ \lambda \f$, + * * \f$ \phi_\lambda \f$ is the porosity of the solid, + * * \f$ q_\lambda \f$ is a source or sink term. */ #ifndef DUMUX_MINERALIZATION_MODEL_HH diff --git a/dumux/porousmediumflow/mpnc/model.hh b/dumux/porousmediumflow/mpnc/model.hh index 8ff3a7249be66a0eddf1727993f58fc751b35a64..f3ea0e7504450c50134441a7a20471e8b8b5b5c0 100644 --- a/dumux/porousmediumflow/mpnc/model.hh +++ b/dumux/porousmediumflow/mpnc/model.hh @@ -22,7 +22,7 @@ * * By inserting this into the equations for the conservation of the * mass of each component, one gets one mass-continuity equation for - * each component \f$\kappa\f$ + * each component \f$\kappa\f$, * \f[ \sum_{\kappa} \left( \phi \frac{\partial \left(\varrho_\alpha x_\alpha^\kappa S_\alpha\right)}{\partial t} @@ -35,10 +35,21 @@ \right) = q^\kappa \f] - * with \f$\overline M_\alpha\f$ being the average molar mass of the + * with \f$\overline M_\alpha\f$ being the average molar mass of * phase \f$\alpha\f$: \f[ \overline M_\alpha = \sum_\kappa M^\kappa * \; x_\alpha^\kappa \f] * + * Additionally: + * * \f$ \phi \f$ is the porosity of the porous medium, + * * \f$ S_\alpha \f$ represents the saturation of phase \f$ \alpha \f$, + * * \f$ \rho_\alpha \f$ is the mass density of phase \f$ \alpha \f$, + * * \f$ X_\alpha^\kappa \f$ is the mass fraction of component \f$ \kappa \f$ in phase \f$ \alpha \f$, + * * \f$ x_\alpha^\kappa \f$ is the mole fraction of component \f$ \kappa \f$ in phase \f$ \alpha \f$, + * * \f$ v_\alpha \f$ is the velocity of phase \f$ \alpha \f$, + * * \f$ {\bf D_{\alpha, pm}^\kappa} \f$ is the diffusivity of component \f$ \kappa \f$ in phase \f$ \alpha \f$, + * * \f$ \overline M_\alpha \f$ is the average molar mass of phase \f$ \alpha \f$ + * * \f$ q_\alpha^\kappa \f$ is a source or sink term. + * * For the missing \f$M\f$ model assumptions, the model assumes that * if a fluid phase is not present, the sum of the mole fractions of * this fluid phase is smaller than \f$1\f$, i.e. diff --git a/dumux/porousmediumflow/nonisothermal/model.hh b/dumux/porousmediumflow/nonisothermal/model.hh index b04d686674d223e97920c95de4269fad8a0920ff..7d4593ac2bcb1efaa4463e26066392d2337474dc 100644 --- a/dumux/porousmediumflow/nonisothermal/model.hh +++ b/dumux/porousmediumflow/nonisothermal/model.hh @@ -17,7 +17,7 @@ * * For the energy balance, local thermal equilibrium is assumed. This * results in one energy conservation equation for the porous solid - * matrix and the fluids: + * matrix and the fluids, \f{align*}{ \phi \frac{\partial \sum_\alpha \varrho_\alpha u_\alpha S_\alpha}{\partial t} & + @@ -30,12 +30,25 @@ \left( \textbf{grad}\,p_\alpha - \varrho_\alpha \mathbf{g} \right) \right\} \\ & - \text{div} \left(\lambda_{pm} \textbf{grad} \, T \right) - - q^h = 0. + - q^h = 0, \f} - * where \f$h_\alpha\f$ is the specific enthalpy of a fluid phase - * \f$\alpha\f$ and \f$u_\alpha = h_\alpha - - * p_\alpha/\varrho_\alpha\f$ is the specific internal energy of the - * phase. + * where: + * * \f$ \phi \f$ is the porosity of the porous medium, + * * \f$ S_\alpha \f$ represents the saturation of phase \f$ \alpha \f$, + * * \f$ \rho_\alpha \f$ is the mass density of phase \f$ \alpha \f$, + * * \f$ h_\alpha \f$ is the specific enthalpy of phase \f$ \alpha \f$, + * * \f$ u_\alpha \f$ is the specific internal energy of phase \f$ \alpha \f$, + * * \f$ \lambda_{pm}\f$ is the heat conductivity in the porous medium, + * * \f$ T \f$ is the temperature, + * * \f$ \rho_s \f$ is the mass density of the solid phase, + * * \f$ c_s \f$ is the heat capacity of the solid, + * * \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^h \f$ is a source or sink term. + * */ #ifndef DUMUX_NONISOTHERMAL_MODEL_HH diff --git a/dumux/porousmediumflow/richards/model.hh b/dumux/porousmediumflow/richards/model.hh index af9151801da9f7a9bbffeb1309122ef8a99a8db4..e1f00351fa61de68a19517c568d7363e058be614 100644 --- a/dumux/porousmediumflow/richards/model.hh +++ b/dumux/porousmediumflow/richards/model.hh @@ -16,13 +16,14 @@ - \text{div} \left\lbrace \varrho_w \frac{k_{rw}}{\mu_w} \; \mathbf{K} \; - \left( \text{\textbf{grad}} + \left( \textbf{grad} p_w - \varrho_w \textbf{g} \right) \right\rbrace = q_w, \f] + * * is frequently used to * approximate the water distribution above the groundwater level. * @@ -32,21 +33,24 @@ - \text{div} \left\lbrace \varrho_\alpha \frac{k_{r\alpha}}{\mu_\alpha}\; \mathbf{K} \; - \left( \text{\textbf{grad}} + \left(\textbf{grad} p_\alpha - \varrho_\alpha \textbf{g} \right) \right\rbrace = q_\alpha, \f] - * where \f$\alpha \in \{w, n\}\f$ is the fluid phase, - * \f$\kappa \in \{ w, a \}\f$ are the components, - * \f$\rho_\alpha\f$ is the fluid density, \f$S_\alpha\f$ is the fluid - * saturation, \f$\phi\f$ is the porosity of the soil, - * \f$k_{r\alpha}\f$ is the relative permeability for the fluid, - * \f$\mu_\alpha\f$ is the fluid's dynamic viscosity, \f$\mathbf{K}\f$ is the - * intrinsic permeability, \f$p_\alpha\f$ is the fluid pressure and - * \f$g\f$ is the potential of the gravity field. + * + * 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 diff --git a/dumux/porousmediumflow/richardsextended/model.hh b/dumux/porousmediumflow/richardsextended/model.hh index 91ae346ebb53756a11c8d61c0acbecd756ba38fe..4d76a3039a82b2ac994934b4c7b00b41833e6273 100644 --- a/dumux/porousmediumflow/richardsextended/model.hh +++ b/dumux/porousmediumflow/richardsextended/model.hh @@ -27,6 +27,20 @@ = q_w, \f] + * where: + * * \f$ \phi \f$ is the porosity of the porous medium, + * * \f$ S_w \f$ represents the saturation of the wetting-phase, + * * \f$ \varrho_w \f$ is the mass density of the wetting phase, + * * \f$ \varrho_n \f$ is the mass density of the non-wetting phase, + * * \f$ k_{rw} \f$ is the relative permeability of the wetting phase, + * * \f$ \mu_w \f$ is the dynamic viscosity of the wetting phase, + * * \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$ 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. * @@ -43,14 +57,17 @@ = q_\alpha, \f] - * where \f$\alpha \in \{w, n\}\f$ is the fluid phase, - * \f$\kappa \in \{ w, a \}\f$ are the components, - * \f$\rho_\alpha\f$ is the fluid density, \f$S_\alpha\f$ is the fluid - * saturation, \f$\phi\f$ is the porosity of the soil, - * \f$k_{r\alpha}\f$ is the relative permeability for the fluid, - * \f$\mu_\alpha\f$ is the fluid's dynamic viscosity, \f$\mathbf{K}\f$ is the - * intrinsic permeability, \f$p_\alpha\f$ is the fluid pressure and - * \f$g\f$ is the potential of the gravity field. + * + * 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 diff --git a/dumux/porousmediumflow/richardsnc/model.hh b/dumux/porousmediumflow/richardsnc/model.hh index 7cd5c9475fd099a10043721526f7e4fe1d03d5bf..b6768784636caac922eb25e69596e7069c1aa14c 100644 --- a/dumux/porousmediumflow/richardsnc/model.hh +++ b/dumux/porousmediumflow/richardsnc/model.hh @@ -20,8 +20,21 @@ \nonumber \\ \nonumber \\ &-& \sum_w \text{div} \left\{{\bf D_{w, pm}^\kappa} \varrho_{w} \text{grad}\, X^\kappa_{w} \right\} - \sum_w q_w^\kappa = 0 \qquad \kappa \in \{w, a,\cdots \} \, , - w \in \{w, g\} + w \in \{w, g\}, \f} + * where: + * * \f$ \phi \f$ is the porosity of the porous medium, + * * \f$ S_w \f$ represents the saturation of the wetting phase, + * * \f$ \varrho_w \f$ is the mass density of the wetting phase, + * * \f$ k_{rw} \f$ is the relative permeability of the wetting phase, + * * \f$ \mu_w \f$ is the dynamic viscosity of the wetting phase, + * * \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$ 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. * diff --git a/dumux/porousmediumflow/solidenergy/model.hh b/dumux/porousmediumflow/solidenergy/model.hh index b1ad4448de8f17e1ccf9b01370f1472db2c20159..0ab596483652b9c7abe625167975d520a86379de 100644 --- a/dumux/porousmediumflow/solidenergy/model.hh +++ b/dumux/porousmediumflow/solidenergy/model.hh @@ -15,8 +15,14 @@ \frac{ \partial n c_p \varrho T}{\partial t} - \text{div} \left\lbrace \lambda_\text{pm} \textbf{grad} T \right\rbrace = q, \f] - * where \f$n\f$ is the volume fraction of the conducting material, \f$c_p\f$ its specific heat capacity, - * \f$\varrho\f$ its density, \f$T\f$ the temperature, and \f$\lambda\f$ the heat conductivity of the porous solid. + * where: + * * \f$ n \f$ represents volume fraction of the conducting material, + * * \f$ c_p \f$ is the specific heat capacity at constant pressure, + * * \f$ \varrho \f$ is the mass density, + * * \f$ \lambda_\text{pm} \f$ is the heat conductivity in the porous medium, + * * \f$ T \f$ is the temperature, + * * \f$ q \f$ is the heat source term. + * */ #ifndef DUMUX_SOLID_ENERGY_MODEL_HH diff --git a/dumux/porousmediumflow/tracer/model.hh b/dumux/porousmediumflow/tracer/model.hh index 9278a68dc97d630aac1a4ff36d11f9e7c767ba0a..c9b4242c29cdb6380f1ce877ff35574bcf68357f 100644 --- a/dumux/porousmediumflow/tracer/model.hh +++ b/dumux/porousmediumflow/tracer/model.hh @@ -21,8 +21,15 @@ \f[ \phi \frac{ \partial \varrho X^\kappa}{\partial t} - \text{div} \left\lbrace \varrho X^\kappa {\textbf v_f} - + \varrho D^\kappa_\text{pm} \textbf{grad} X^\kappa \right\rbrace = q. + + \varrho D^\kappa_\text{pm} \textbf{grad} X^\kappa \right\rbrace = q, \f] +* where: + * * \f$ \phi \f$ is the porosity of the porous medium, + * * \f$ \varrho \f$ is the mass density, + * * \f$ X^\kappa \f$ is the mass fraction of component \f$ \kappa \f$, + * * \f$ \textbf{v}_f \f$ is the velocity of the fluid, + * * \f$ {\bf D_{pm}^\kappa} \f$ is the diffusivity in the porous medium, + * * \f$ q \f$ is a source or sink term. * * The model is able to use either mole or mass fractions. The property useMoles can be set to either true or false in the * problem file. Make sure that the according units are used in the problem setup. useMoles is set to true by default.