Stream flow by slope-area method

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).

  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.