Runoff model

A runoff model is a mathematical model describing the rainfall–runoff relations of a rainfall catchment area, drainage basin or watershed. More precisely, it produces a surface runoff hydrograph in response to a rainfall event, represented by and input as a hyetograph. In other words, the model calculates the conversion of rainfall into runoff.
A well known runoff model is the linear reservoir, but in practice it has limited applicability.
The runoff model with a non-linear reservoir is more universally applicable, but still it holds only for catchments whose surface area is limited by the condition that the rainfall can be considered more or less uniformly distributed over the area. The maximum size of the watershed then depends on the rainfall characteristics of the region. When the study area is too large, it can be divided into sub-catchments and the various runoff hydrographs may be combined using flood routing techniques.

Rainfall-runoff models need to be calibrated before they can be used.

The hydrology of a linear reservoir (figure 1) is governed by two equations.

where:
Q is the runoff or discharge
R is the effective rainfall or rainfall excess or recharge
A is the constant reaction factor or response factor with unit [1/T]
S is the water storage with unit [L]
dS is a differential or small increment of S
dT is a differential or small increment of T

Runoff equation
A combination of the two previous equations results in a differential equation, whose solution is:

This is the runoff equation or discharge equation, where Q1 and Q2 are the values of Q at time T1 and T2 respectively while T2−T1 is a small time step during which the recharge can be assumed constant.

Computing the total hydrograph
Provided the value of A is known, the total hydrograph can be obtained using a successive number of time steps and computing, with the runoff equation, the runoff at the end of each time step from the runoff at the end of the previous time step.

Unit hydrograph
The discharge may also be expressed as: Q = − dS/dT . Substituting herein the expression of Q in equation (1) gives the differential equation dS/dT = A.S, of which the solution is: S = exp(− A.t) . Replacing herein S by Q/A according to equation (1), it is obtained that: Q = A exp(− A.t) . This is called the instantaneous unit hydrograph (IUH) because the Q herein equals Q2 of the foregoing runoff equation using R = 0, and taking S as unity which makes Q1 equal to A according to equation (1).
The availability of the foregoing runoff equation eliminates the necessity of calculating the total hydrograph by the summation of partial hydrographs using the IUH as is done with the more complicated convolution method.

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