Richards equation

Remarks

It is the continuity equation $\partial_t \theta + \nabla \cdot q = 0$ for the water, with the flux given by the Darcy–Buckingham law $q = -K(h) \nabla (h + z)$. This law is Darcy's law with a conductivity that depends on the water content.

Example: if $K$ is constant and $\theta(h) = C h$ with a constant $C > 0$, the equation is the linear heat equation $C \partial_t h = K \Delta h$, since $\nabla \cdot e_z = 0$. Real soils have $K$ that decreases by orders of magnitude as the soil dries, so the equation is strongly nonlinear.

Non-example: the equation $\partial_t \theta = \nabla \cdot (K(h) \nabla h)$ without the term $e_z$ is not the Richards equation; it omits gravity and describes horizontal flow only.

Other common forms use $\theta$ as the unknown, with the soil-water diffusivity $D(\theta) = K \, dh / d\theta$. That form fails where the medium is saturated, since $d\theta / dh = 0$ there.