RICHARDS Mode

Governing Equations

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

(1)$$\frac{\partial }{\partial t}\left(\varphi s\eta \right)+\mathbf{\nabla }\cdot \left(\eta \mathit{q}\right)=Q_w,$$

with Darcy flux $\mathit{q}$ defined as

(2)$$\mathit{q}=-\frac{k{k}_r(s)}{\mu }\mathbf{\nabla }(P-\rho gz).$$

Here, $\varphi$ denotes porosity [-], $s$ saturation [m${}^3$ m${}^{-3}$], $\eta$ molar water density [kmol m${}^{-3}$], $\rho$ mass water density [kg m${}^{-3}$], $\mathit{q}$ Darcy flux [m s${}^{-1}$], $k$ intrinsic permeability [m${}^2$], $k_r$ relative permeability [-], $\mu$ viscosity [Pa s], $P$ pressure [Pa], $\mathit{g}$ 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)$$Q_w=\frac{q_M}{W_w}\delta (\mathit{r}-{\mathit{r}}_{ss}),$$

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