1. Linear Reservoir

A watershed or drainage basin. https://handwiki.org/wiki/index.php?curid=1476532

Figure 1. A linear reservoir. https://handwiki.org/wiki/index.php?curid=2047242

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

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

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:

• Q2 = Q1 exp{−A (T2 − T1)} + R[1 − exp{−A (T2 − T1)}]

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.^[2]

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/s per ha corresponds to 8.64 mm/day or 86.4 m³/day per ha

2. Non-Linear Reservoir

Figure 2. A non-linear reservoir. https://handwiki.org/wiki/index.php?curid=2054191

Figure 3. The reaction factor (Aq, Alpha) versus discharge (Q) for a small valley (Rogbom) in Sierra Leone. https://handwiki.org/wiki/index.php?curid=1177930

Figure 4. Actual and simulated discharge, Rogbom valley. https://handwiki.org/wiki/index.php?curid=1177930

Figure 5. Rainfall and recharge, Rogbom valley. https://handwiki.org/wiki/index.php?curid=1077313

Figure 6. Non-linear reservoir with pre-reservoir for recharge. https://handwiki.org/wiki/index.php?curid=1823612

Contrary to the linear reservoir, the non linear reservoir has a reaction factor A that is not a constant,^[3] but it is a function of S or Q (figure 2, 3).

Normally A increases with Q and S because the higher the water level is the higher the discharge capacity becomes. The factor is therefore called Aq instead of A.
The non-linear reservoir has no usable unit hydrograph.

During periods without rainfall or recharge, i.e. when R = 0, the runoff equation reduces to

• Q2 = Q1 exp { − Aq (T2 − T1) }, or:

or, using a unit time step (T2 − T1 = 1) and solving for Aq:

• Aq = − ln (Q2/Q1)

Hence, the reaction or response factor Aq can be determined from runoff or discharge measurements using unit time steps during dry spells, employing a numerical method.

Figure 3 shows the relation between Aq (Alpha) and Q for a small valley (Rogbom) in Sierra Leone.
Figure 4 shows observed and simulated or reconstructed discharge hydrograph of the watercourse at the downstream end of the same valley.^{[4] [5]}

3. Recharge

Figure 7. Runoff from the water balance. https://handwiki.org/wiki/index.php?curid=2088814

The recharge, also called effective rainfall or rainfall excess, can be modeled by a pre-reservoir (figure 6) giving the recharge as overflow. The pre-reservoir knows the following elements:

• a maximum storage (Sm) with unit length [L]
• an actual storage (Sa) with unit [L]
• a relative storage: Sr = Sa/Sm
• a maximum escape rate (Em) with units length/time [L/T]. It corresponds to the maximum rate of evaporation plus percolation and groundwater recharge, which will not take part in the runoff process (figure 5, 6)
• an actual escape rate: Ea = Sr·Em
• a storage deficiency: Sd = Sm + Ea − Sa

The recharge during a unit time step (T2−T1=1) can be found from R = Rain − Sd
The actual storage at the end of a unit time step is found as Sa2 = Sa1 + Rain − R − Ea, where Sa1 is the actual storage at the start of the time step.

The Curve Number method (CN method) gives another way to calculate the recharge. The initial abstraction herein compares with Sm − Si, where Si is the initial value of Sa.

4. Nash Model

The Nash Model uses a cascade of linear reservoirs to predict streamflow. Software for this model exists.^[6] https://handwiki.org/wiki/index.php?curid=1472224

The Nash model ^[7] uses a series (cascade) of linear reservoirs in which each reservoir empties into the next until the runoff is obtained. For calibration, the model requires considerable research.

5. Software

Figures 3 and 4 were made with the RainOff program,^[8] designed to analyse rainfall and runoff using the non-linear reservoir model with a pre-reservoir. The program also contains an example of the hydrograph of an agricultural subsurface drainage system for which the value of A can be obtained from the system's characteristics.^[9]

The SMART hydrological model^[10] includes agricultural subsurface drainage flow, in addition to soil and groundwater reservoirs, to simulate the flow path contributions to streamflow.

Vflo is another software program for modeling runoff. Vflo uses radar rainfall and GIS data to generate physics-based, distributed runoff simulation.

The WEAP (Water Evaluation And Planning) software platform models runoff and percolation from climate and land use data, using a choice of linear and non-linear reservoir models.

The RS MINERVE software platform simulates the formation of free surface run-off flow and its propagation in rivers or channels. The software is based on object-oriented programming and allows hydrologic and hydraulic modeling according to a semi-distributed conceptual scheme with different rainfall-runoff model such as HBV, GR4J, SAC-SMA or SOCONT.