Green & Ampt equation

The penetration depth of the infiltrating wetting front is at any moment in time . If we assume that the wetting front is a sharp Dirac delta-function, Darcy's law can be stated as follows:

where is the hydraulic conductivity and is the cumulative infiltration at time that is equal to (conservation of mass).

Using the above relation for to eliminate and performing the integration yields,

with infiltration amount in [cm], hydr. conductivity in [cm/h], wetting front pressure head (negative) in [cm], water pressure at surface (ponding) in cm, moisture content at saturation, antecedent moisture.

In order to solve this implicit equation, we need to bring to one side of the equation:

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

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]")