RICHARDS Mode

Governing Equations

RICHARDS mode applies to single phase, variably saturated, isothermal systems. The governing mass conservation equation is given by

(1)\[\frac{{{\partial}}}{{{\partial}}t}\left(\porosity s\eta\right) + {\boldsymbol{\nabla}}\cdot\left(\eta{\boldsymbol{q}}\right) = Q_w,\]

with Darcy flux \({\boldsymbol{q}}\) defined as

(2)\[{\boldsymbol{q}} = -\frac{kk_r(s)}{\mu} {\boldsymbol{\nabla}}\left(P-\density gz\right).\]

Here, \(\porosity\) denotes porosity [-], \(s\) saturation [m\(^3\) m\(^{-3}\)], \(\eta\) molar water density [kmol m\(^{-3}\)], \(\density\) mass water density [kg m\(^{-3}\)], \({\boldsymbol{q}}\) Darcy flux [m s\(^{-1}\)], \(k\) intrinsic permeability [m\(^2\)], \(k_r\) relative permeability [-], \(\mu\) viscosity [Pa s], \(P\) pressure [Pa], \({\boldsymbol{g}}\) gravity [m s\(^{-2}\)]. Supported relative permeability functions \(k_r\) for Richards’ equation include van Genuchten, Books-Corey and Thomeer-Corey, while the saturation functions include Burdine and Mualem. Water density and viscosity are computed as a function of temperature and pressure through an equation of state for water. The source/sink term \(Q_w\) [kmol m\(^{-3}\) s\(^{-1}\)] has the form

(3)\[Q_w = \frac{q_M}{W_w} \delta({\boldsymbol{r}}-{\boldsymbol{r}}_{ss}),\]

where \(q_M\) denotes a mass rate in kg/m\(^{3}\)/s, and \({\boldsymbol{r}}_{ss}\) denotes the location of the source/sink.