Derive reservoir routing equation

Consider a reservoir with inflow $I$, outflow $O$ and storage $S$.

Continuity equation

$$I - O = \frac{dS}{dt}$$

Over a short interval $\Delta t$ from time 1 to time 2, use average inflow and outflow:

$$\frac{I_1 + I_2}{2}\Delta t - \frac{O_1 + O_2}{2}\Delta t = S_2 - S_1$$

Divide by $\Delta t$ and multiply by 2:

$$(I_1 + I_2) - (O_1 + O_2) = \frac{2(S_2 - S_1)}{\Delta t}$$

Collect the unknowns ($S_2$, $O_2$) on one side and the knowns on the other:

$$\left(\frac{2S_2}{\Delta t} + O_2\right) = (I_1 + I_2) + \left(\frac{2S_1}{\Delta t} - O_1\right)$$

This is the reservoir (modified Puls) routing equation. Equivalently, with $\psi = S/\Delta t + O/2$:

$$\psi_2 = \frac{I_1 + I_2}{2} + (\psi_1 - O_1)$$

Use

• $I_1, I_2$ are known from the inflow hydrograph, and $S_1, O_1$ are known from the previous step.
• A curve of $O$ against $(2S/\Delta t + O)$ is prepared from the elevation-storage and elevation-outflow data.
• For each step, the right-hand side gives $(2S_2/\Delta t + O_2)$, from which $O_2$ is read from the curve; $S_2$ then follows, and the process is repeated.