### Description

The ABCD water balance model is a simple hydrologic model for simulating streamflow in response to precipitation and potential evapotranspiration developed by Thomas (1981).

The model is comprised of two storage compartments: soil moisture and groundwater. The soil moisture gains water from precipitation and loses water to evapotranspiration (ET), surface runoff and groundwater recharge. The groundwater compartment gains water from recharge and loses water as discharge. The total streamflow is the sum of surface runoff from the soil moisture and groundwater discharge.

### Input Data

The model runs on a daily time step and requires input timeseries of precipitation, minimum and maximum air temperature, and observed streamflow. The air temperature data are used to compute PET using the method described by Shuttleworth (1993). More information about the input data can be found on the Setup page.

### Parameters

There are four parameters governing the model behavior:

• a controls the amount of runoff and recharge that occurs when the soils are under-saturated.
• b controls the saturation level of the soils.
• c defines the ratio of groundwater recharge to surface runoff.
• d controls the rate of groundwater discharge.

### Mass Balance Diagrams

The model equations are represented by the interactive visualizations shown below. Each visualization demonstrates how the model computes the storage terms and fluxes for each time step. The diagrams are based on a mass balance:

$$Storage_{t} + \sum{Outflow_t} = Storage_{t-1} + \sum{Inflow_t}$$

This equation means that the sum of remaining storage and total outflow must equal the sum of initial storage and total inflow.

For the soil moisture compartment, the mass balance equation is:

$$Soil Moisture_t + ET_t + Runoff_t + Recharge_t = Soil Moisture_{t-1} + Precip_t$$

For the groundwater compartment, the mass balance equation is:

$$Groundwater_t + Discharge_t = Groundwater_{t-1} + Recharge_t$$

### Soil Moisture

a:
b:
c:
PET:

#### Show Equations

$$\begin{array} \\ W_t &= \text{Available Soil Water} \\ Y_t &= \text{Evapotranspiration Potential} \\ P_t &= \text{Precipitation} \\ S_t &= \text{Soil Moisture} \\ PET_t &= \text{Potential Evapotranspiration} \\ ET_t &= \text{Actual Evapotranspiration} \\ DR_t &= \text{Direct Runoff} \\ GR_t &= \text{Groundwater Recharge} \end{array}$$
$$\begin{array} \\ W_t &= S_{t-1} + P_t = S_t + ET_t + GR_t + DR_t \\ Y_t &= S_t + ET_t = \frac{W_t+b}{2a} - \sqrt{\left(\frac{W_t+b}{2a}\right)^2 - \frac{bW_t}{a}} \\ S_t &= Y_te^{-PET_t/b} \\ ET_t &= Y_t\left(1-e^{-PET_t/b}\right) \\ DR_t &= (1-c)*(W_t - Y_t) \\ GR_t &= c*(W_t - Y_t) \end{array}$$

### Groundwater

d:

#### Show Equations

$$\begin{array}\\ G_t &= \text{Groundwater Storage} \\ GR_t &= \text{Groundwater Recharge} \\ GD_t &= \text{Groundwater Discharge} \end{array}$$
$$\begin{array} \\ G_t + GD_t &= G_{t-1} + GR_t \\ G_t &= \frac{1}{1+d} (G_{t-1} + GR_t)\\ GD_t &= d G_t \end{array}$$

### References

• Thomas, H. A. (1981). Improved Methods for National Water Assessment. Report, Contract: WR15249270. Washington, D.C.: US Water Resource Council.
• Shuttleworth, W. J. (1993). Chapter 4: Evaporation. In D. R. Maidment (Ed.), Handbook of Hydrology (pp. 4.1-4.53). New York: McGraw-Hill, Inc.