---
title: "Derive reservoir routing equation"
subject: "Engineering Hydrology"
chapter: 7
key: "semester_5/engineering_hydrology:reservoir-routing-equation-derive"
source: https://www.bctengineer.com/q/semester_5/engineering_hydrology/reservoir-routing-equation-derive
site: BCT Engineering
---

# Derive reservoir routing equation

Engineering Hydrology · Chapter 7: Flow Routing

- Asked in: 2071 Shrawan, 4 marks
- Video: [Reservoir Routing: Level Pool (Modified Puls) Method | Theory & Example](https://www.youtube.com/watch?v=BAcwWzot6ds) — Water Edu Hub
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/engineering_hydrology/reservoir-routing-equation-derive
- Whole chapter: https://www.bctengineer.com/md/semester_5/engineering_hydrology/chapter-7

## Question

Starting from the continuity equation, obtain the equation of reservoir routing.

## Short answer

Integrate continuity over Δt using average inflow and outflow, giving I1+I2−O1−O2 = 2(S2−S1)/Δt. Rearrange to (2S2/Δt + O2) = (I1+I2) + (2S1/Δt − O1), the modified Puls routing equation.

## Answer

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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The derivation starts from:
   A. Manning's equation
   B. Continuity I - O = dS/dt
   C. Bernoulli
   D. Darcy
   Answer: B. Continuity over dt with average flows gives the equation. (Key point: Starting point)

2. Over dt, flows are:
   A. Averaged: (I1+I2)/2 and (O1+O2)/2
   B. Taken as I2 and O2
   C. Taken as I1 only
   D. Ignored
   Answer: A. Trapezoidal averaging gives the finite form. (Key point: Averaging)

3. The final routing equation is:
   A. (2S2/dt - O2) = (I1+I2) + (2S1/dt + O1)
   B. S2 = I2 - O2
   C. O2 = I2
   D. (2S2/dt + O2) = (I1+I2) + (2S1/dt - O1)
   Answer: D. Unknowns S2, O2 are on the left. (Key point: Final equation)

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