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# Green & Ampt equation

The penetration depth of the infiltrating wetting front is $Z$ at any moment in time $t$. If we assume that the wetting front is a sharp Dirac delta-function, Darcy's law can be stated as follows:

$$q = \frac{dI}{dt} = -K_s * \left[\frac{h_f-(h_s+Z)}{Z}\right]$$

where $K_s$ is the hydraulic conductivity and $I(t)$ is the cumulative infiltration at time $t$ that is equal to $Z*(\theta_s - \theta_0)$ (conservation of mass).

Using the above relation for $I(t)$ to eliminate $Z$ and performing the integration yields,

$$I = K_s*t-(h_f-h_s)*(\theta_s - \theta_0)* log_e \left( 1 - \frac{I}{(h_f-h_s)*(\theta_s-\theta_0)}\right)$$

with $I(t)$ infiltration amount in [cm], $K_s$ hydr. conductivity in [cm/h], $h_f$ wetting front pressure head (negative) in [cm], $h_s$ water pressure at surface (ponding) in cm, $\theta_s$ moisture content at saturation, $\theta_0$ antecedent moisture.

In order to solve this implicit equation, we need to bring $I(t)$ to one side of the equation:

$$\frac{1}{K_s}*\left[I -(h_f-h_s)*(\theta_s - \theta_0)* log_e \left( 1 - \frac{I}{(h_f-h_s)*(\theta_s-\theta_0)}\right)\right] = t $$

We can then insert $I$ can calculate $t$ - we calculate the time that corresponds to a given infiltration amount. An R-code to calculate infiltration amounts with Green & Ampt looks like this:

- |Green-Ampt.R
I <- seq(0,100,by=1.0) t0 <- 0.05 ts <- 0.25 hs <- 0.0 # cm hf <- -12.0 # cm Ks <- 8.0 # cm/hour t <- 1/Ks*((I-(hf-hs)*(ts-t0))*log(1-(I/((hf-hs)*(ts-t0))))) # hours plot(t,I,xlim=c(0,6),ylim=c(0,25),xlab="t [hour]", ylab="I in [cm]")