A simple water balance model The water balance

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НазваниеA simple water balance model The water balance
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Vishal K. Mehta

Arghyam/Cornell University

November 2006

A simple water balance model 2

The water balance 2

Key Features 3

Model Description 3

Notation 4

Equations 5

Model Input 6

Model output 6

The Hydrograph 6

Seasonal Soil Water status 7

Applications and Limitations 8

References 9


Vishal K. Mehta (Arghyam Trust/Cornell University)

Dr. M. Todd walter (Cornell University)

Dr. Stephen D. DeGloria (Cornell University)

A simple water balance model

The water balance

Constructing a water balance is one of the first tasks in understanding the water regime of a specific area. In simple terms, a water balance is a budgeting exercise that assesses the proportion of rainfall that becomes stream flow (or runoff), evapotranspiration, and drainage (or groundwater recharge). Figure 1 below represents a simple 'box' model representation of a water balance, which could be represented by Equation 1 (Thorntwaite et al, 1955;1957).

Eq 1 P = ET + RO + dSW + D; where dSW is the change in soil water over the time step.

Figure 1 A conceptual water balance

Of these, rainfall and runoff (as stream flow) are directly measured, However to close the water balance, either drainage (or groundwater recharge) or Evapotranspiration (ET) also need to be estimated/measured. Both of these are non-trivial tasks.

Water budgeting is frequently accomplished through water balance models, of which there are a vast number.

The objectives of this document and following video tutorials are:

1. To introduce the reader to a simple water balance model, namely, the Thorntwaite-Mather model, henceforth referred to as the T-M model (Thorntwaite et al, 1955;1957; Steenhuis et al, 1986);

2. To provide the reader with the tools to construct a water balance for her/his own region of interest, with the help of video tutorials and a sample Excel spreadsheet that can be downloaded and modified.

This document is intended for general instructive purposes for an audience that has some basic knowledge of water resources and associated terminology. No advanced expertise should be needed to understand and use this tutorial.

Responsibility for conscientious and socially responsible use of this material rests solely with the reader.

Thorntwaite-Mather soil water balance model

Key Features

The T-M model tracks the soil water status through time.

It is a lumped model that tracks soil water through time.

The entire watershed is treated as one unit (hence ‘lumped’).

Is appropriate for modeling at daily, weekly or monthly time steps.

A simple spreadsheet (e.g. Excel) model. No specialized software is required.

Model Description

Figure 2 Conceptual Model

Figure 2 above summarizes a simple water balance model. Water is stored in the soil reservoir until the soil water content (SW) exceeds the available water capacity (AWC), at which point the excess goes into storage (S). The monthly streamflow is a simple linear function of S. Determining the soil water budget requires keeping track of the accumulated potential water loss (APWL) and the amount of water in the soil (SW).

Watershed Storage and River Discharge:

All Excess water, i.e., water above the AWC, goes into watershed storage (S), which in-turn, feeds river discharge (Qo) from the watershed.

Eq 2

Hydrologists commonly assume that discharge is a constant fraction of watershed storage, especially for groundwater discharge into rivers – this assumption is called the linear reservoir assumption.

Eq 3

Where f is the reservoir coefficient and 0 < f < 1. If data are available, f can be empirically determined.


AWC = Available Water Capacity [depth]

(field capacity-wilting point)X(soil depth)

SW = Available Soil Water (i.e., above wilting pt.) [depth]

APWL = Accumulated Potential Water Loss (negative) [depth]

P = Net Precipitation; P - PET [depth]

P = Precipitation [depth]

PET = Potential Evapotranspiration [depth]

AET = Actual Evapotranspiration [depth]


Calculations to determine SW and APWL are performed for each time step using monthly precipitation (P) and potential evapotranspiration (PET). Excess water, i.e., net precipitation (P) in excess of the soil’s water holding capacity (AWC) leaves the soil and is stored in the watershed and eventually released to the river. Table 1 below summarizes the calculations.

Table 1 Equations

Situation in the Watershed




Soil is drying

= 0

Soil is wetting


= 0

Soil is wetting

above capacity


= 0


When P

Model Input

The required model input data are:

Rainfall (P) : Accurate and local rainfall data.

Potential Evapotranspiration (PET) : Procedures to estimate PET are on the India Water Portal, with detailed procedures in Allen at al (1998), online at


Apart from the data input, the user needs to input 3 effective parameters:

Available water capacity (AWC),

Rooting depth,

Linear reservoir coefficient 'f'

The model is sensitive to the above parameters, as the video tutorial demonstrates. Knowledge of your study area helps in informing what values are entered for the above parameters.

It is recommended that the user also has observed stream flow data, to tune the model so that modeled stream flow closely matches the observed stream flow. There is substantial risk of a very incorrect water balance without this check on model output.

Model output

The Hydrograph

Model output includes simulations of :

Stream Flow (Qo)

Actual Evapotranspiration (AET)


Soil Water status

Additionally, using Eq 1. an estimate of Drainage or Groundwater recharge can be made. Annual estimates of the water balance can be made based on the daily or monthly water balance. Monthly averages over several years of data are very useful in getting an idea of the long term seasonal water balance regime.

Figure 3 below demonstrates the model stream flow output, along with observed input rainfall data and observed stream flow data., for a watershed in south India. In this example, the annual proportions of rainfall were - 85% AET; 11% stream flow, and 4% groundwater recharge.

Figure 3 Example model hydrograph with observed rainfall and flow

As mentioned, it is very important to use local data as much as possible, and to have observed stream flow to compare with the model stream flow. In the above example, as the video tutorial explains, the rainfall is local, but the PET estimated is from coarse resolution data. Especially in mountainous areas where rainfall, as well as temperature and humidity, can be very variable, it is likely that PET estimates from coarse resolution data will be over-estimated. This is likely what is happening in Figure 2.

Seasonal Soil Water status

The output can be summarized in a very useful manner by collating it into long-term monthly averages. This allows the tracking of soil moisture status through the year, to determine periods of soil water deficit, soil water recharge, soil water utilization, and soil water surplus. These periods are described in Table 2 and demonstrated in Figure 4 below:

Table 2 : Seasonal soil water status

Soil Water Deficit (AET
mid-Nov to May in example

Soil Water Recharge (from when P>AET until accumulated (AET-P) is replenished) :

May to August in example

Soil Water Surplus : Sept to December in example

Soil Water Utilization (when P

Figure 4 Monthly average soil water balance

Applications and Limitations

The T-M model has the advantage of being one of the most simple models. It can be used to determine a general estimate of the water balance regime, for individual fields to small watersheds. However, as in all scientific investigation, this tutorial should be used responsibly and with a full knowledge of the user's specific study area. This model and its variants have been used, for example, for irrigation scheduling of individual fields, water budgeting of small watersheds, generating actual evapotranspiration estimates for comparison with other methods - to name a few applications.

Being a lumped model, in the form described, the T-M model does not provide spatially distributed predictions, nor does it perform flow routing routines.


Allen, R. G., Pereira, L.S., Raes, D., Smith, M. (1998). "Crop evapotranspiration: Guidelines for computing crop water requirements" FAO Irrigation and drainage paper 56, Rome, Italy.

Thorntwaite, C. W., J. R. Mather (1955). "The water balance." Publ. Climatol. 8(1).

Thorntwaite, C. W., J. R. Mather (1957). "Instructions and tables for computing potential evapotranspiration and the water balance." Publ. Climatol. 10(3).

Steenhuis, T.S. and W.H. Van der Molen. 1986. The Thornthwaite-Mather procedure as a simple engineering method to predict recharge. J. Hydrol. 84:221-229.


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