---
title: "Stream flow by slope-area method"
subject: "Engineering Hydrology"
chapter: 4
key: "semester_5/engineering_hydrology:slope-area-method-explain"
source: https://www.bctengineer.com/q/semester_5/engineering_hydrology/slope-area-method-explain
site: BCT Engineering
---

# Stream flow by slope-area method

Engineering Hydrology · Chapter 4: Surface Runoff

- Asked in: 2070 Chaitra, 5 marks; 2067 Shrawan (old course), 6 marks; 2066 Magh (old course), 8 marks (asked 3 times)
- Video: [Slope Area Method of Stream Flow Measurement by Dr. Upain Kumar Bhatia](https://www.youtube.com/watch?v=NdtMFAQxuY0) — ACET CIVIL TECH
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/engineering_hydrology/slope-area-method-explain
- Whole chapter: https://www.bctengineer.com/md/semester_5/engineering_hydrology/chapter-4

## Question

Explain the stream flow computation by slope area method.

## Short answer

Peak flood discharge is obtained by applying Manning’s equation to high-water marks at two cross-sections. Friction slope is computed from water-surface fall and velocity-head difference, iterated until discharge converges.

## Answer

The **slope-area method** is an indirect method used to estimate the peak flood discharge in a reach of a natural channel when a current meter cannot be used (high flood, no gauge). It uses the high-water marks left by the flood and the channel geometry, and applies Manning's equation.

#### Principle
For steady non-uniform flow, the discharge is
$$
Q = K\sqrt{S_f},\qquad K = \frac{1}{n}AR^{2/3}
$$
where $K$ is the conveyance, $n$ is Manning's roughness, $A$ the flow area, $R$ the hydraulic radius and $S_f$ the friction slope. The friction slope is not equal to the water surface slope when the channel is non-uniform, so it is found from the energy equation between sections 1 (upstream) and 2 (downstream).

```text
  energy line ------\
                     \___ h_f
  water surface ---\   \____
                    \_ fall h
  bed  ------------------------
     [1]<----- L ----->[2]
```

$$
h_f = (h_1 - h_2) + \left(\frac{\alpha_1 v_1^2}{2g} - \frac{\alpha_2 v_2^2}{2g}\right) - h_e
$$
$$
h_e = k_e\left|\frac{\alpha_1 v_1^2}{2g} - \frac{\alpha_2 v_2^2}{2g}\right|
$$
where $h_1 - h_2$ is the fall of water surface, $\alpha$ the velocity-head coefficient and $k_e$ the eddy loss coefficient (0.3 for a gradual expansion, 0.1 for a gradual contraction).

#### Procedure
1. Select a straight, uniform reach, free of large boulders and bends, with well-defined high-water marks on both banks. Take two or more cross-sections normal to the flow.
2. Survey the cross-sections and the high-water mark levels, and measure the reach length $L$.
3. Compute $A$, $P$, $R$ for each section and estimate $n$ from the bed material and vegetation.
4. Compute $K_1, K_2$ and the mean conveyance $K_m=\sqrt{K_1K_2}$.
5. Assume $S_f = (h_1-h_2)/L$ and find a first estimate $Q = K_m\sqrt{S_f}$.
6. Find $v_1=Q/A_1$, $v_2=Q/A_2$, the velocity heads and $h_e$. Compute $h_f$, then a new $Q=K_m\sqrt{h_f/L}$.
7. Repeat until $Q$ no longer changes.

#### Merits and limits
- It needs no flow measurement during the flood, and is cheap and quick.
- Accuracy depends mostly on $n$ and the quality of the high-water marks; the error can be 10 to 25 %.
- It applies only to steady flow in a stable reach. It is used to extend the rating curve to peak stage.

## Check yourself

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

1. When is the slope-area method used?
   A. For low flow
   B. For sediment
   C. For rainfall
   D. To estimate peak flood discharge indirectly from high-water marks when a current meter cannot be used
   Answer: D. It is an indirect method using channel geometry and flood marks with Manning's equation. (Key point: Purpose)

2. Which formula is used?
   A. Q = A/S
   B. Q = K sqrt(S_f) with K = (1/n) A R^(2/3)
   C. Q = C(G-a)^b
   D. Q = P - E
   Answer: B. Manning's equation expressed through conveyance K and friction slope S_f. (Key point: Principle)

3. Why is the friction slope not simply the water-surface slope?
   A. n is unknown
   B. For non-uniform channels velocity head changes and eddy loss must be considered via energy equation
   C. Slope is constant
   D. Water surface is not measured
   Answer: B. h_f = fall + change in velocity head - h_e. (Key point: Friction slope)

4. How is Q finalised?
   A. Average of two dischargers
   B. Use rating curve
   C. One calculation
   D. Iterate: assume S_f, find Q, compute velocity heads and h_f, recompute Q until it stops changing
   Answer: D. The velocity heads depend on Q, so iteration is required. (Key point: Iteration)

## Related in Engineering Hydrology

- [Slope-area discharge, 1300 m reach](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Aslope-area-3500-3250) — Engineering Hydrology, chapter 4, asked 2×
- [Slope-area flood discharge, 7 km reach](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Aslope-area-45-50-7km) — Engineering Hydrology, chapter 4, asked 2×
- [Slope-area discharge with three sections](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Aslope-area-3-section-260m) — Engineering Hydrology, chapter 4
- [Principle and procedure of slope-area method](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Aslope-area-principle-procedure) — Engineering Hydrology, chapter 4
- [Slope-area discharge, 10 km reach](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Aslope-area-306-920) — Engineering Hydrology, chapter 4
- [Stream flow from current meter data](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Acurrent-meter-0-035-0-74n) — Engineering Hydrology, chapter 4, asked 2×

## Related in other subjects

- [Slope area method of discharge](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Aslope-area-method) — Hydro Power, chapter 2
- [Stream flow from 0.2d and 0.8d measurements](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Astream-flow-measurement-data) — Hydro Power, chapter 2
- [Methods of stream flow measurement](https://www.bctengineer.com/question.md?key=semester_6%2Fenergy_resources%3Aflow-measurement-methods-micro-hydro) — Energy Resources, chapter 5
- [Moment area: slope and deflection under 50 kN](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Amoment-area-12-ss-defl-4m-d) — Theory of Structures I, chapter 4