Drainage density

Drainage density is a quantity used to describe physical parameters of a drainage basin. First described by Robert E. Horton, drainage density is defined as the total length of channel in a drainage basin divided by the total area, represented by the equation
The quantity represents the average length of channel per unit area of catchment and has units, which is often reduced to.
Drainage density depends upon both climate and physical characteristics of the drainage basin. Soil permeability and underlying rock type affect the runoff in a watershed; impermeable ground or exposed bedrock will lead to an increase in surface water runoff and therefore to more frequent streams. Rugged regions or those with high relief will also have a higher drainage density than other drainage basins if the other characteristics of the basin are the same.
When determining the total length of streams in a basin, both perennial and ephemeral streams should be considered. If a drainage basin contained only ephemeral streams, then the drainage density by the equation above would be calculated to be zero if the total length of streams was calculated using only perennial streams. Ignoring ephemeral streams in the calculations does not consider the behavior of the basin during flood events and is therefore not completely representative of the drainage characteristics of the basin.
Drainage density is indicative of infiltration and permeability of a drainage basin, and relates to the shape of the hydrograph. Drainage density depends upon both climate and physical characteristics of the drainage basin.
High drainage densities also mean a high bifurcation ratio.

Inverse of drainage density as a physical quantity

Drainage density can be used to approximate the average length of overland flow in a catchment. Horton used the following equation as an approximation to describe the average length of overland flow as a function of drainage density:
where is the length of overland flow and is the drainage density of the catchment.
Considering the geometry of channels on the hillslope, Horton also proposed the following equation
where is the channel slope and is the average slope of the ground in the area.

Elementary components of drainage basins

A drainage basin can be characterized by three elementary quantities: channels, the hillslope area associated with those channels, and the source areas. The channels are the well-defined segments that efficiently carry water through the catchment. Labeling these features as "channels" rather than "streams" indicates that there need not be a continuous flow of water to capture the behavior of this region as a conduit of water. According to Arthur Strahler's stream ordering system, the channels are not defined to be any single order or range of orders. Channels of lower orders combine to form higher-order channels. The associated hillslope areas are the hillslopes that slope directly into the channels. Precipitation that enters the system on the hillslope areas and is not lost to infiltration or evapotranspiration enters the channels. The source areas are concave regions of hillslope that are associated with a single channel. Precipitation entering a source area that is not lost to infiltration or evapotranspiration flows through the source area and enters the channel at the channel's head. Source areas and the hillslope areas associated with channels are differentiated by source areas draining through the channel head, while the associated hillslope areas drain into the rest of the stream. According to Strahler's stream ordering system, all source areas drain into a primary channel, by the definition of a primary channel.
Bras et al. describe the conditions that are necessary for channel formation. Channel formation is a concept intimately tied to the formation and evolution of a drainage system and influence the drainage density of catchment. The relation they propose determines the behavior of a given hillslope in response to a small perturbation. They propose the following equation as a relation between source area, source slope, and the sediment flux through this source area:
where is the sediment flux, is the slope of the source area, and is the source area. The right-hand side of this relation determines channel stability or instability. If the right-hand side of the equation is greater than zero, then the hillslope is stable, and small perturbations such as small erosive events do no develop into channels. Conversely, if the right-hand side of the equation is less than zero, then Bras et al. determine the hillslope to be unstable, and small erosive structures, such as rills, will tend to grow and form a channel and increase the drainage density of a basin. In this sense, "unstable" is not used in the sense of the gradient of the hillslope being greater than the angle of repose and therefore susceptible to mass wasting, but rather fluvial erosive processes such as sheet flow or channel flow tend to incise and erode to form a singular channel. Therefore, the characteristics of the source area, or potential source area, influence the drainage density and evolution of a drainage basin.

Relation to water balance

Drainage density is tied to the water balance equation:
where is the change in reservoir storage, is precipitation, is evapotranspiration, and are the respective groundwater flux into and out of the basin, is the groundwater discharge into streams, and is groundwater discharge from the basin through wells. Drainage density relates to the storage and runoff terms. Drainage density relates to the efficiency by which water is carried over the landscape. Water is carried through channels much faster than over hillslopes, as saturated overland flow is slower due to being thinned out and obstructed by vegetation or pores in the ground. Consequently, a drainage basin with a relatively higher drainage density will be more efficiently drained than a lower-density one. Because of the more extensive drainage system in a higher-density basin, precipitation entering the basement will, on average, travel a shorter distance over the slower hillslopes before reaching the faster-flowing channels and exit the basin through the channels in less time. Conversely, precipitation entering a lower-drainage-density basin will take longer to exit the basin due to travelling over the slower hillslope longer.
In his 1963 paper on drainage density and streamflow, Charles Carlston found that baseflow into streams is inversely related to the drainage density of the drainage basin:
This equation represents the effect of drainage density on infiltration. As drainage density increases, baseflow discharge into a stream decreases for a given basin because there is less infiltration to contribute to baseflow. More of the water entering the drainage basin during and immediately following a rainfall event exits quickly through streams and does not become infiltration to contribute to baseflow discharge. Gregory and Walling found that the average discharge through a drainage basin is proportional to the square of drainage density:
This relation illustrates that a higher-drainage-density environment transports water more efficiently through the basin. In a relatively low-drainage-density environment, the lower average discharge results predicted by this relation would be the result of the surface runoff spending more time travelling over hillslope and having a larger time for infiltration to occur. The increased infiltration results in a decreased surface runoff according to the water balance equation.
These two equations agree with each other and follow the water balance equation. According to the equations, in a basin with high drainage density, the contribution of surface runoff to stream discharge will be high, while that from baseflow will be low. Conversely, a stream in a low-drainage-density system will have a larger contribution from baseflow and a smaller contribution from overland flow.

Relation to hydrographs

The discharge through the central stream draining a catchment reflects the drainage density, which makes it a useful diagnostic for predicting the flooding behavior of a catchment following a storm event due to being intimately tied to the hydrograph. The material that overland flow travels over is one factor that influences the speed that water can flow out of a catchment. Water flows significantly slower over hillslopes compared to channels. According to Horton's interpretation of half of the inverse of drainage density as the average length of overland flow implies that overland flow in high-drainage environments will reach a fast-flowing channel faster over a shorter range. On the hydrograph, the peak is higher and occurs over a shorter range. This more compact and higher peak is often referred to as being "flashy".
The timing of the hydrograph in relation to the peak of the hyetograph is influenced by the drainage density. The water that enters a high-drainage watershed during a storm will reach a channel relatively fast and travel in the high-velocity channels to the outlet of the watershed in a relatively short time. Conversely, the water entering a low-drainage-density basin will, on average, have to travel a longer distance over the low-velocity hillslope to reach the channels. As a result, the water will require more time to reach the exit of the catchment. The lag time between the peak of the hyetograph and the hydrograph is then inversely related to drainage density; as drainage density increases, water is more efficiently drained from the basin and the lag time decreases.
Another impact on the hydrograph that drainage density has is a steeper falling limb following the storm event due to its impact on both overland flow and baseflow. The falling limb occurs after the peak of the hydrograph curve and is when overland flow is decreasing back to ambient levels. In higher-drainage systems, the overland flow reaches the channels quicker, resulting in a narrower spread in the falling limb. Baseflow is the other contributor to the hydrograph. The peak of baseflow to the channels will occur after the quick-flow peak because groundwater flow is much slower than quick-flow. Because the baseflow peak occurs after the quick-flow peak, the baseflow peak influences the shape of the falling limb. According to the proportionality put forth by Gregory and Walling, as drainage density increases, the contribution of baseflow to the falling limb of the hydrograph diminishes. During a storm event in a high-drainage-density basin, there is little water that infiltrates into the ground as infiltration because water spends less time flowing over the surface in the catchment before exiting through the central channel. Because there is little water that enters the water as infiltration, baseflow will contribute only a small part to falling limb. The falling limb is thus quite steep. Conversely, a low-drainage system will have a shallower falling limb. According to Gregory and Walling's relation, the decrease in drainage density results in an increase in baseflow to the channels and a more gradual decrease in the hydrograph.