How much water? Discharge and floods

How much water does a river carry? Hydrologists use discharge, written Q: the volume of water passing a point each second, in cubic metres per second (m³/s). It depends on how big the channel is and how fast the water moves.

The formula. Discharge is the cross-section area of the flow times its mean speed:

Q = A × v

Take a river channel that is 20 m wide and 2 m deep, with a roughly rectangular cross-section. Its flow area is A = 20 × 2 = 40 m². If the water moves at a mean 1.5 m/s, then

Q = 40 × 1.5 = 60 m³/s.

That is 60,000 litres every second, about 60 tonnes of water passing you each second. Now let it rain upstream. A bigger Q means either a larger area (the river rises) or a larger speed (it flows faster), and usually both. If the area stays fixed and Q doubles, the speed must double too: 120 ÷ 40 = 3 m/s.

Going further: what sets the speed? Engineers estimate the mean speed of open-channel flow with the Manning equation:

v = (1/n) × R^(2/3) × S^(1/2)

Here n is a roughness number (about 0.035 for a clean, winding natural channel; rough, weedy channels have bigger n), S is the slope of the water surface (metres of drop per metre of length) and R is the hydraulic radius, the flow area divided by the wetted perimeter. For our channel the wetted perimeter is 20 + 2 + 2 = 24 m, so R = 40 ÷ 24 ≈ 1.67 m. With a slope of 0.0014 (a drop of 1.4 m per kilometre):

v ≈ (1/0.035) × 1.67^(2/3) × 0.0014^(1/2) ≈ 28.6 × 1.41 × 0.0374 ≈ 1.5 m/s.

That reproduces the 1.5 m/s we used above. The takeaway: a steeper, smoother, deeper channel flows faster. This is an empirical engineering formula, good for a quick estimate.

The hydrograph. When it rains, discharge does not jump up and down at once. A hydrograph plots discharge against time. After the rain starts, the river rises (the rising limb), reaches a peak discharge, and then falls (the falling limb) back towards its normal base flow. The delay between the heaviest rain and the peak is the lag time. On bare rock, steep slopes or in towns with lots of paved ground, water runs off fast, so the lag is short and the peak is high and sharp. In forests, soils and wetlands, water soaks in and is released slowly, so the peak is lower and later.

What is a flood? A flood happens when discharge is so big that the water rises over the banks and spreads onto the land beside the river, the floodplain. Floods are a normal part of how rivers work, and the floodplain is the river's natural overflow space. Floodplains store water and slow it down, which reduces the peak further downstream. Building on them or paving them over takes this buffer away.

Floods are not only a mountain-river problem. Fast flash floods can fill a dry valley in minutes after a storm upstream, even if it is not raining where you stand. River floods on big lowland rivers rise slowly and can last for weeks. Coastal floods come from the sea. Each has different warning signs, so local official warnings matter.