Runoff Model (reservoir) - Linear Reservoir

Linear Reservoir

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

1. flow equation: Q = A.S, with units, where L is length (e.g. mm) and T is time (e.g. hr, day)
2. continuity or water balance equation: R = Q + dS/dT, with units

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
S is the water storage with unit
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:

• Q2 = Q1 exp { −A (T2 − T1) } + R

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.

Determining the response factor A
When the response factor A can be determined from the characteristics of the watershed (catchment area), the reservoir can be used as a deterministic model or analytical model, see hydrological modelling.
Otherwise, the factor A can be determined from a data record of rainfall and runoff using the method explained below under non-linear reservoir. With this method the reservoir can be used as a black box model.

Conversions
1 mm/day corresponds to 10 m3/day per ha of the watershed
1 l/sec per ha corresponds to 8.64 mm/day or 86.4 m3/day per ha