The calculation of stream discharge (\(Q\)), a measurement fundamental to hydrology, reveals the volume of water flowing past a specific point over a set period of time. This discharge is calculated by the relationship: \(Q\) equals the cross-sectional Area (\(A\)) of the stream multiplied by the average Velocity (\(V\)) of the water. This formula, \(Q = A \times V\), quantifies the stream’s flow, often expressed in cubic meters per second (\(\text{m}^3/\text{s}\)) or cubic feet per second (\(\text{ft}^3/\text{s}\)). Determining this number requires careful, systematic measurements of the stream channel’s shape and the water’s speed, since natural streams rarely have uniform channels.
Measuring the Stream’s Cross-Section
The first step in calculating discharge is to accurately determine the cross-sectional Area (\(A\)) of the flowing water, which represents the stream’s width multiplied by its average depth. Since natural stream beds are highly irregular, simply measuring the total width and a single depth point will produce an inaccurate area. The professional method involves establishing a straight transect line across the stream perpendicular to the flow and dividing the entire cross-section into multiple vertical segments, sometimes called panels.
For each segment, the width is measured, and a specific depth reading is taken at the vertical line that divides it. The stream’s total cross-sectional area is then calculated by summing the area of all these individual segments. This segmented approach, known as the mid-section method, accounts for the varying depths from the shallow edges to the deepest part of the channel, providing a representative average depth and thus a highly accurate Area (\(A\)).
Techniques for Determining Water Speed
The second part of the discharge calculation is determining the average water Velocity (\(V\)), as water speed varies significantly from the surface to the streambed. Water velocity is slowest near the bottom and banks due to frictional drag and typically fastest in the upper third of the water column. To account for this variation, hydrologists must use specialized techniques to find a representative average velocity for each vertical segment.
For a quick, basic estimate, the float method can be employed, which involves timing a buoyant object as it travels a measured distance. Since a floating object only measures the surface velocity, which is usually faster than the overall average, a correction factor must be applied. The surface velocity is typically multiplied by a coefficient ranging from 0.66 to 0.85 to estimate the average velocity. The float method is simple and cost-effective but is significantly less accurate due to the influence of wind and surface turbulence.
A more professional and accurate approach uses a current meter, a device with rotating cups or a propeller that measures the speed of the water at a specific point. For shallow streams, the mean velocity in a vertical segment can be estimated by taking a single measurement at 60% (0.6) of the total depth below the surface. This “six-tenths rule” is based on the finding that the average velocity within a vertical column consistently occurs at this depth. For deeper streams, an even more precise measurement is achieved by averaging the velocities taken at 20% (0.2) and 80% (0.8) of the depth below the water surface.
Final Calculation and Reliability Checks
Once the cross-sectional Area (\(A\)) has been calculated from the segmented depth measurements and the average Velocity (\(V\)) has been determined for each segment, the final discharge (\(Q\)) can be calculated. The total discharge for the stream is found by summing the individual discharge values for every vertical segment. Each segment’s discharge is calculated by multiplying its segment area by its corresponding average velocity, and the sum of these products gives the stream’s total discharge.
It is important to maintain unit consistency, as the final discharge is a volumetric flow rate. If the area is measured in square meters (\(\text{m}^2\)) and the velocity in meters per second (\(\text{m}/\text{s}\)), the resulting discharge will be in cubic meters per second (\(\text{m}^3/\text{s}\)). Conversely, using feet and seconds will yield cubic feet per second (\(\text{ft}^3/\text{s}\) or cfs).
The reliability of the final discharge number depends heavily on proper site selection and measurement technique. The measurement site, or reach, should be a relatively straight section of the stream with a uniform channel, free of large obstructions or excessive turbulence. Repeating the measurements at slightly different locations within the same reach and averaging the results can help mitigate potential errors caused by localized disturbances. Poor site selection, such as a location near a sharp bend, can lead to inaccurate velocity readings and an unreliable final discharge calculation.