Differences

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--- en:hydro:greenampt [2017/10/24 21:51] – ckuells
+++ en:hydro:greenampt [2024/04/10 10:02] (current) – external edit 127.0.0.1
@@ Line 1: / Line 1: @@
 ====== Green & Ampt equation ======
-At any time, $t$, the penetration of the infiltrating wetting front will be $Z$. Darcy's law can be
+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]$$
-\begin{equation*}
-q = \frac{dI}{dt} = -K_s * \left[\frac{h_f-(h_s+Z)}{Z}\right]
-\end{equation*}
-$$
-where $K_s$ is the hydraulic conductivity corresponding to the surface water content, and $I(t)$ is the
+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).
-cumulative infiltration at time $t$, and is equal to $Z*(\theta_s - \theta_0)$.
-Using this relation for $I(t)$ to eliminate $Z$ and performing the integration yields,
+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)$$
-\begin{equation*}
-I = K_s*t-(h_f-h_s)*(\theta_s - \theta_0)* log_e \left( 1 - \frac{I}{(h_f-h_s)*(\theta_s-\theta_0)}\right)
-\end{equation*}
-$$
-$$
+with $I(t)$ infiltration amount in [cm], $K_s$ hydr. conductivity in [cm/h], $h_f$ wetting front pressure head (negative) in [cm], $h_s$  water pressure at surface (ponding) in cm, $\theta_s$ moisture content at saturation, $\theta_0$ antecedent moisture.
-\begin{table}
-  \centering
-  \begin{tabular}{ l l l l }
-with &   $I(t)$ & infiltration amount & $[cm]$   \\
-     &   $K_s$  & hydr. conductivity  & $[cm/h]$  \\
-     &   $h_f$  & wetting front pressure head (negative) & $cm$  \\
-     &   $h_s$  & water pressure at surface (ponding)    & $cm$  \\
-     &   $\theta_s$ & moisture content at saturation     & $-$   \\
-     &   $\theta_0$ & antecedent moisture                & $-$   \\
-\end{tabular}
-\end{table}
-$$
-In order to solve this equation, we need to bring $I(t)$ to one side of the equation:
+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 $$
-\begin{equation*}
-\frac{1}{K_s}*\left[I -(h_f-h_s)*(\theta_s - \theta_0)* log_e \left( 1 - \frac{I}{(h_f-h_s)*(\theta_s-\theta_0)}\right)\right] = t
-\end{equation*}
-$$
-The R-program to calculate infiltration amounts with Green & Ampt looks like this:
+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+ |Greem-Ampt.R>
+<code S |Green-Ampt.R>
-        <<GreenAmpt, fig=TRUE, height=4.0, echo=FALSE>>=
         I    <- seq(0,100,by=1.0)
         t0   <- 0.05
@@ Line 55: / Line 29: @@
         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>