Green–Ampt infiltration model

Definition

Consider a uniform soil with saturated hydraulic conductivity $K > 0$, initial water content $\theta_i$ and saturated water content $\theta_s > \theta_i$. From time $0$ on, its surface is kept ponded with negligible depth. Set $\Delta\theta = \theta_s - \theta_i$ and let $\psi > 0$ be the suction head at the wetting front. The Green–Ampt model assumes that:

1. the soil above a sharp wetting front at depth $L(t)$ is saturated and the soil below has water content $\theta_i$;
2. the flow in the saturated zone obeys Darcy's law with conductivity $K$ and a head difference $L + \psi$ over the length $L$.

The cumulative infiltration is $F(t) = \Delta\theta \, L(t)$, and the infiltration rate $f = dF/dt$ satisfies the ordinary differential equation $$f = \frac{dF}{dt} = K \Big( 1 + \frac{\psi \, \Delta\theta}{F} \Big), \qquad F(0) = 0.$$