# Discharge (hydrology)

In hydrology, discharge is the volume rate of water flow, including any suspended solids (e.g. sediment), dissolved chemicals (e.g. CaCO3(aq)), or biologic material (e.g. diatoms), which is transported through a given cross-sectional area. Frequently, other terms synonymous with discharge are used to describe the volumetric flow rate of water and are typically discipline dependent. For example, a fluvial hydrologist studying natural river systems may define discharge as streamflow, whereas an engineer operating a reservoir system might define discharge as outflow, which is contrasted with inflow.

## Theory and calculation

GH Dury and MJ Bradshaw are two hydrologists who devised models showing the relationship between discharge and other variables in a river. The Bradshaw model described how pebble size and other variables change from source to mouth; while Dury considered the relationships between discharge and variables such as slope and friction.

The units that are typically used to express discharge include m³/s (cubic meters per second), ft³/s (cubic feet per second or cfs) and/or acre-feet per day. For example, the average discharge of the Rhine river in Europe is 2,200 cubic metres per second (78,000 cu ft/s) or 190,000,000 cubic metres (150,000 acre·ft) per day.

A commonly applied methodology for measuring, and estimating, the discharge of a river is based on a simplified form of the continuity equation. The equation implies that for any incompressible fluid, such as liquid water, the discharge (Q) is equal to the product of the stream's cross-sectional area (A) and its mean velocity ( $\bar{u}$), and is written as: $Q=A\,\bar{u}$

where

• $Q$ is the discharge ([L3T−1]; m3/s or ft3/s)
• $A$ is the cross-sectional area of the portion of the channel occupied by the flow ([L2]; m2 or ft2)
• $\bar{u}$ is the average flow velocity ([LT−1]; m/s or ft/s)

## Catchment discharge

The catchment of a river above a certain location is determined by the surface area of all land which drains toward the river from above that point. The river's discharge at that location depends on the rainfall on the catchment or drainage area and the inflow or outflow of groundwater to or from the area, stream modifications such as dams and irrigation diversions, as well as evaporation and evapotranspiration from the area's land and plant surfaces. In storm hydrology, an important consideration is the stream's discharge hydrograph, a record of how the discharge varies over time after a precipitation event. The stream rises to a peak flow after each precipitation event, then falls in a slow recession. Because the peak flow also corresponds to the maximum water level reached during the event, it is of interest in flood studies. Analysis of the relationship between precipitation intensity and duration, and the response of the stream discharge is aided by the concept of the unit hydrograph which represents the response of stream discharge over time to the application of a hypothetical "unit" amount and duration of rain, for example 1 cm over the entire catchment for a period of one hour. This represents a certain volume of water (depending on the area of the catchment) which must subsequently flow out of the river. Using this method either actual historical rainfalls or hypothetical "design storms" can be modeled mathematically to confirm characteristics of historical floods, or to predict a stream's reaction to a predicted storm.

The relationship between the discharge in the stream at a given cross-section and the level of the stream is described by a rating curve. Average velocities and the cross-sectional area of the stream are measured for a given stream level. The velocity and the area give the discharge for that level. After measurements are made for several different levels, a rating table or rating curve may be developed. Once rated, the discharge in the stream may be determined by measuring the level, and determining the corresponding discharge from the rating curve. If a continuous level-recording device is located at a rated cross-section, the stream's discharge may be continuously determined.

Flows with larger discharges are able to transport more sediment downstream.