# Modeling Streamflow

Hydrologic systems are complex, with processes occurring over different geographic areas characterized by highly variable parameters. In general, numerical models used for surface-water studies simulate the processes of interest as equations. Model results are difficult to confirm and often rely on the experience and judgment of the analyst; hence, it is the responsibility of the analyst to ensure that the model used produces results that are reasonable for a given watershed. The analyst also is faced with decisions regarding the length of time to be simulated. These modeling decisions require significant experience, not only as a hydrologist, but also as a user of a particular model.

Users of surface-water hydrologic modeling have two basic needs: (1) to determine the magnitude and frequency of flood flows; and (2) to determine the long-term availability of water for consumption. The two needs require different modeling approaches.

## Flood Runoff

If flood runoff is important (e.g., to determine drainage design), then only rainfall and runoff need to be modeled. In this case, relatively simple models can be used to determine a design discharge (i.e., the streamflow to be used for engineering design) and the runoff hydrograph (i.e., the plot of streamflow against time).

Two models that can be applied are the rational method and the unit hydrograph method. In both cases, short-term flow rates are the primary interest. Short-term flow rates are those that occur over a period of hours to days. For short-term times, hydrologic processes such as soil infiltration, percolation, and evapotranspiration can be ignored. Therefore, modeling approaches for short-term flows focus on the hydrograph of runoff from a single precipitation event.

Moreover, modeling of flood flow focuses on the maximum flow (discharge), or peak discharge, for an event with a particular exceedence probability. The exceedence probability is the probability of a particular event being equaled or exceeded over a given period of time, usually 1 year. The designer chooses the exceedence probability based on the perceived risk to human life or property damage if the event is exceeded.

Small design problems (those dealing with watershed drainage areas less than about 200 acres) typically use exceedence probabilities in the range of 10 percent to 20 percent because the consequences of failure are limited. For example, a structure like a bridge would be designed to withstand a stream discharge that would be exceeded less than 20 percent of the time. For such problems, it is acceptable to use simplified computational approaches that use approximate equations based on watershed properties.

#### Methods of Analysis.

In the simple computations procedures known as the rational method, the rate of maximum runoff is related to a runoff coefficient, a rainfall rate, and the drainage area through a basic mathematical formula. But if the drainage area exceeds about 200 acres, or if ponds or lakes complicate drainage, then a more complicated approach using a computer model is required. The most common method uses a unit hydrograph, which represents the runoff from the watershed as a unit pulse of runoff.

What complicates this procedure is that both a precipitation distribution (in time) as well as the watershed response (the unit hydrograph) must be determined. No longer can a single precipitation event (as represented by the rainfall intensity in the rational method) be considered. This is because there are many possible precipitation events for a given exceedence probability, each of which differs by the length of time the storm lasts. Therefore, to determine the maximum volumetric runoff intensity (the peak discharge), the designer must analyze several, perhaps many, different storm durations. This is why a computer model generally is used for this type of analysis.

In addition to the rational method and the unit hydrograph method, two other approaches are commonly used for flood analysis. If a stream gage is present at or near the location of interest, then the principles of statistics can be applied to determine the runoff and exceedence probability. This approach is called frequency analysis and can be used to estimate the flow rate. But if a stream gage is not available, then measurements at other sites can be used to create an estimate of the flow at a site of interest. The procedure used is called regionalization, and usually is accomplished by using statistical methods at several stream gages, then applying the results at the site of interest.

## Water Supply

If water supply is important, the analyst must focus on longer time periods (e.g., years or decades) compared to the short intervals used for flood analysis. Because a specific storm event is no longer the focus, all hydrologic processes come into play, and the models used to study long-term water quantity must reflect this complexity. The physical processes are represented by simplified submodels, yet the entire model must yield a reasonable solution.

The basic concept is the hydrologic budget, an accounting of the water as it moves along various pathways at the Earth's surface (in particular on the watershed of interest) in a manner that preserves (conserves) the mass of water. Components are precipitation, runoff, evapotranspiration, and movement to groundwater. The formula is P E R G = △ S , where P is incoming precipitation, E is evapotranspiration, R is runoff from the watershed, G is percolation to groundwater, and △ S (delta S) is the change in water storage in the active part of the soil profile. The terms in this equation can be expressed as either time rates or by depth or volume over some convenient period (such as 1 day). Because there are so many terms involved in the hydrologic budget, computer models are most often used for these analyses.

David B. Thompson

#### Bibliography

Viessman, Jr., Warren, and Gary L. Lewis. Introduction to Hydrology, 4th ed. New York: HarperCollins, 1996.