====== 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:

<code S |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]")
</code>