The Water Balance Equation: A Fundamental Tool in Hydrology

Every drop of water on Earth is constantly on the move – falling as rain, seeping into the ground, flowing through rivers, or evaporating back into the atmosphere. But how do hydrologists keep track of all this movement? The answer lies in the water balance equation, a mathematical tool rooted in one of the most fundamental principles in science: the conservation of mass. Whether you’re managing a farm’s irrigation system or planning flood control for an entire river basin, this equation is your starting point.

What is the water balance equation? 🔗

At its core, the water balance equation is an accounting system for water. It tracks all the water entering a defined area (inflows), all the water leaving that area (outflows), and any change in the amount of water stored within it. The general form of the equation is:

Change in Storage = Inflows – Outflows

Or, expressed more formally:

ΔS = P – ET – Q

Where P is precipitation, ET is evapotranspiration, Q is runoff (streamflow), and ΔS represents the change in water storage within the system. This equation can be adapted and expanded depending on the type and scale of the hydrological system being studied – from a small agricultural plot to an entire continent.

The principle behind it is simple: water doesn’t just appear or vanish. Every drop entering a system must either leave it or remain stored somewhere within it. This idea, grounded in the law of conservation of mass, makes the water balance equation universally applicable across hydrology, agriculture, and water resource engineering.

Key components of the water balance equation 🔗

To use the water balance equation effectively, you need to understand each of its individual components. Let’s break them down.

Precipitation (P) 🔗

Precipitation is the primary input of water into most hydrological systems. It includes all forms of water falling from the atmosphere – rain, snow, sleet, and hail. In very arid environments, even dew and fog can contribute to the water supply. Precipitation is measured using rain gauges and weather stations and is typically expressed in millimetres (mm) of water depth over a given area.

The amount and pattern of precipitation directly influence every other component in the water balance. Seasonal variations, storm intensity, and long-term climate trends all play a role in determining how much water enters a system at any given time.

Evapotranspiration (ET) 🔗

Evapotranspiration is often the largest output component in the water balance equation. It combines two processes: evaporation, where water changes from liquid to vapour from wet surfaces like soil and water bodies, and transpiration, where plants release water vapour through their leaves.

Hydrologists distinguish between two types of ET. Potential evapotranspiration (PE) is the amount of water that would evaporate if an unlimited supply of water were available – essentially the atmospheric demand for water. Actual evapotranspiration (AE) is the water that actually evaporates and transpires, depending on how much moisture is available. When soil moisture runs low, actual ET falls below potential ET, and plants begin to experience water stress.

Surface runoff (Q) 🔗

When rainfall exceeds the soil’s capacity to absorb water – known as the infiltration capacity – the excess flows over the land surface as runoff. This water eventually reaches streams, rivers, and lakes. Runoff is influenced by factors such as soil type, slope, vegetation cover, and the degree of urbanisation. Paved surfaces in cities, for instance, generate significantly more runoff than forested hillsides because water cannot infiltrate through concrete.

Runoff has two components: direct runoff (quickflow), which reaches stream channels rapidly during or after a storm, and baseflow, which is the slower contribution from groundwater seeping into streams over a longer period.

Groundwater and subsurface flow 🔗

Not all precipitation that infiltrates the soil stays near the surface. Some of it percolates deeper, recharging groundwater aquifers. This deep percolation is an important outflow in the water balance of the root zone, particularly in agricultural contexts. In some areas, groundwater can also flow back upward through capillary rise, contributing moisture to the root zone – especially where the water table is shallow.

Change in storage (ΔS) 🔗

Storage refers to the water held in the soil, in groundwater aquifers, in surface water bodies, or even as snow and ice. The change in storage over a given period is simply the difference between what was stored at the beginning and what remains at the end. If more water enters the system than leaves, storage increases. If more water leaves, storage decreases.

In soil moisture terms, the upper limit of storage is the field capacity – the maximum water a soil can hold against gravity. Fine-textured soils (like clay) hold more water than coarse-textured soils (like sand). When the soil is at field capacity and precipitation still exceeds evapotranspiration, the result is a surplus, which generates runoff. When evapotranspiration exceeds precipitation and the soil dries out completely, the result is a deficit.

Scales of application 🔗

One of the most powerful features of the water balance equation is its flexibility. It can be applied at virtually any spatial or temporal scale.

Plot or field scale 🔗

At the plot scale, the water balance is commonly used in agriculture. Here, the control volume is typically the root zone of a crop, and the equation accounts for precipitation, irrigation, crop evapotranspiration, deep percolation, and changes in soil moisture. This is the scale at which irrigation scheduling decisions are made – a topic discussed in more detail below.

Catchment or watershed scale 🔗

At the catchment scale, the equation tracks all water moving through a defined drainage area. A catchment (or watershed) is a region where all precipitation drains to a common outlet point, such as a river mouth. At this scale, the equation often takes the form:

ΔS + ΔG + ΔW = P – ET – Q

Where ΔS, ΔG, and ΔW represent changes in soil water, groundwater, and surface water storage respectively. Catchment-scale water balances are essential for flood forecasting, drought monitoring, and long-term water resource planning.

Global and continental scale 🔗

At the largest scales, the water balance helps scientists understand how water is distributed across the planet. At the global scale, the equation simplifies considerably because, over long periods, the total volume of water on Earth remains essentially constant. Precipitation over land is balanced by evapotranspiration and streamflow returning water to the oceans. Research using data from over 1,000 global catchments has shown that arid and semi-arid regions require longer time periods before the storage-change term becomes negligible – a key consideration in climate studies.

The water balance equation in agriculture 🔗

For farmers and irrigation managers, the water balance equation is not just a theoretical concept – it’s a practical, daily tool. The agricultural application focuses on the soil water balance within the crop root zone, tracking water additions (rain and irrigation) against water losses (crop water use, runoff, and deep percolation).

Irrigation scheduling using the water balance method 🔗

The water balance approach to irrigation scheduling works much like managing a bank account. You start with a known soil moisture “balance,” add deposits (rainfall and irrigation), and subtract withdrawals (crop evapotranspiration). The goal is to keep the balance above a critical threshold where plants begin to experience stress.

The equation used for daily irrigation scheduling is typically:

Dc = Dp + ETc – P – Irr

Where Dc is the current day’s soil water deficit, Dp is the previous day’s deficit, ETc is crop evapotranspiration, P is precipitation, and Irr is irrigation applied. When the deficit reaches the maximum allowable depletion (MAD) – usually about 50% of the total available water in the root zone – it’s time to irrigate.

The University of Minnesota Extension recommends this method as a reliable way to schedule irrigation, especially when combined with in-field soil moisture sensors for periodic calibration. The Colorado State University Extension also provides detailed guidance on using crop coefficients and weather data to estimate daily ETc for this purpose.

Why it matters for crop production 🔗

Applying too much water wastes resources and can push nutrients and chemicals below the root zone through deep percolation, potentially contaminating groundwater. Applying too little water causes crop stress and reduces yields. The water balance method helps farmers find the right balance – applying the right amount of water at the right time. This is particularly important in regions facing water scarcity, where every litre of irrigation water must be used efficiently.

Assumptions and limitations 🔗

While the water balance equation is conceptually straightforward, applying it in practice comes with challenges.

Measurement difficulties 🔗

Not all components of the equation are easy to measure directly. Evapotranspiration, for example, is typically estimated rather than measured. Surface runoff and deep percolation can be especially difficult to quantify at the field level. In many practical applications, these terms are simplified or estimated indirectly.

The steady-state assumption 🔗

For long-term analyses, hydrologists often assume that the change in storage is negligible – meaning that over a sufficiently long period, what goes in roughly equals what comes out. While this assumption works well for humid catchments over multi-year periods, research has shown it can introduce significant errors in arid and semi-arid regions where storage changes remain important even over decades.

Spatial variability 🔗

Real landscapes are not uniform. Soil types, vegetation cover, slope, and land use vary within any given area. A single water balance equation applied uniformly over a large, heterogeneous catchment may miss important local variations. This is why advanced water balance modelling software divides catchments into smaller sub-units, each with its own set of parameters, to capture spatial variability more accurately.

Water balance and climate change 🔗

Climate change is fundamentally altering the water cycle, making water balance analysis more important than ever. As temperatures rise, evapotranspiration rates increase, potentially reducing the water available for runoff and groundwater recharge. Changes in precipitation patterns – more intense storms in some regions, prolonged droughts in others – directly affect every term in the equation.

For agriculture, this means that historical irrigation schedules may no longer be adequate. Regions that once relied on snowpack as a natural water storage mechanism are seeing earlier snowmelt and more rainfall instead of snow, shifting the timing of water availability. Water balance studies help quantify these shifts and guide adaptation strategies, from selecting drought-tolerant crops to redesigning irrigation infrastructure.

A simple worked example 🔗

Consider a small agricultural field that receives 80 mm of rainfall in a month. The crop uses 60 mm through evapotranspiration. Deep percolation accounts for 10 mm, and there is no irrigation or surface runoff. Using the basic equation:

ΔS = P – ET – Deep Percolation
ΔS = 80 – 60 – 10 = 10 mm

This means the soil moisture increased by 10 mm over the month. If the soil started at 70% of its field capacity, this additional 10 mm moves it closer to full capacity. In the following dry month, the farmer can use this stored water as a buffer before irrigation becomes necessary – precisely the kind of decision the water balance equation supports.

What do you think? How could smallholder farmers in water-scarce regions adopt the water balance approach for irrigation management without access to advanced weather stations? And in the context of rapidly changing climate patterns, how frequently should water balance assessments be updated to remain useful for long-term planning?

References

1. https://en.wikipedia.org/wiki/Water_balance
2. https://extension.umn.edu/irrigation/evapotranspiration-based-irrigation-scheduling-or-water-balance-method
3. https://www.sciencedirect.com/topics/agricultural-and-biological-sciences/water-balance
4. https://geo.libretexts.org/Bookshelves/Geography_(Physical)/The_Physical_Environment_(Ritter)/10:_The_Hydrosphere/10.03:_The_Water_Balance
5. https://extension.colostate.edu/resource/irrigation-scheduling-the-water-balance-approach/
6. https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/2020WR027392
7. https://aquaticecodynamics.github.io/hydrology-workbook/E5.html