en:hydro:greenampt
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en:hydro:greenampt [2017/10/24 22:25] – ckuells | en:hydro:greenampt [2024/04/10 10:02] (current) – external edit 127.0.0.1 | ||
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stated as follows: | 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 and $I(t)$ is the cumulative infiltration at time $t$ that is equal to $Z*(\theta_s - \theta_0)$ (conservation of mass). | 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). | ||
Line 14: | Line 10: | ||
Using the above 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)$$ |
- | \begin{equation*} | + | |
- | I = K_s*t-(h_f-h_s)*(\theta_s - \theta_0)* log_e \left( 1 - \frac{I}{(h_f-h_s)*(\theta_s-\theta_0)}\right) | + | |
- | \end{equation*} | + | |
- | $$ | + | |
- | + | ||
- | $$ | + | |
- | \begin{table} | + | |
- | \centering | + | |
- | \begin{tabular}{ l l l l } | + | |
- | with & | + | |
- | & | + | |
- | & | + | |
- | & | + | |
- | & | + | |
- | & | + | |
- | \end{tabular} | + | |
- | \end{table} | + | |
- | $$ | + | |
+ | 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: | 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 $$ |
- | \begin{equation*} | + | |
- | \frac{1}{K_s}*\left[I -(h_f-h_s)*(\theta_s - \theta_0)* log_e \left( 1 - \frac{I}{(h_f-h_s)*(\theta_s-\theta_0)}\right)\right] = t | + | |
- | \end{equation*} | + | |
- | $$ | + | |
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: | 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: |
/usr/www/users/uhydro/doku/data/attic/en/hydro/greenampt.1508876758.txt.gz · Last modified: 2024/04/10 10:14 (external edit)