11 Water Balance
Suresh Jain
Objectives
- To understand the concept and significance of water balance in the hydrological cycle and the environment
- To understand the concept of principle of conservation and the water balance operational at global and catchment scales
- To understand the mathematical model used to determine the water balance at different scales
- To understand the applications of water balance equation for the assessment and evaluation of the conditions of a catchment
1. Introduction
Water is a dynamic entity, is always in constant motion in the environment and has the ability to change its phase as per the variations in the ambient atmosphere. The hydrologic cycle is a natural cycle that directs and influences the flow of water in the environment and forms a fundamental concept in hydrology. It is a complicated relationship between precipitation and runoff. The water balance concept forms an integral part in hydrological estimates as it helps in understanding the paths and storage of water resources in a system. It helps in examining the hydrologic cycle for any period of time and also estimating the impact of changes in any of the components, i.e., inflows or outflows have on the overall balance of the system. It is an efficient method to study the impacts of anthropogenic activities on the water balance of any subsystem of the hydrologic cycle for any size of area, and at any time period. Considering the importance of the concept of water balance, this module aims at increasing the understanding of the basic concepts of water balance of the readers.
2. Water Balance: Basic Concepts
2.1. Principle of Conservation
The principle of conservation also understood as the principle of conservation of mass states that the mass of a closed system, i.e., a system that restricts transfer of mass and energy, the total mass of the system always remains constant and hence the quantity of ‘mass’ remains conserved. It is based on the fact that mass can neither be created nor destroyed. This principle finds immense applications in understanding the flows of mass and energy in physical systems operating in the atmosphere. It states that certain tangible physical properties of a system remain conserved in a closed or isolated system. The principle of conservation helps in predicting the physical changes occurring in a system at a macroscopic level without getting in details of the physical and chemical changes occurring at the microscopic level.
2.2. Water Budget Equation
The principle of conservation can be effectively applied to the water systems of the atmosphere to comprehensively study the hydrologic cycle. The hydrologic balance is expressed by the equation of continuity which balances the total inflow and outflow of water in a water body. The water balance model for a water system can be conceptually understood by Figure 1.
The water flows represented in the above figure can be expressed in the form of a simple mathematical model. This is the difference in the inflow and outflow of water in a given watershed as the total change in storage as explained in equation 1.
Where, I = Mass inflow; O= Mass outflow; and ΔS = change in mass storage.
The simplified form of water budget equation for a catchment that balances the water inflows and outflows can be expressed by equation 2.
Where, P = Precipitation; Q = Runoff; E = Evaporation; and ΔS = change in storage
This equation shows that water is entering (inflow) in a system in the form of precipitation and leaving (outflow) in terms of evaporation and surface runoff. Therefore water budget equation has important application in the hydrological assessments as it helps the hydrologists to trace the pathways and changes in water storage in a system. The water budget equation is applicable over any space and time scales. The water budget equation can be altered in terms of inflows and outflows to accurately reflect the characteristics of any water system. The water balance equation can be applied at the global as well as a system scale such as lake, stream, river and groundwater by altering the factors for inflows and outflows. These systems are the subjects of detailed discussion in the ensuing sections of this module.
3. Global Water Balance
Global water balance is the water budget that is prevalent in the atmosphere at the global scale, i.e., including all the continents of the world. The world is known to have ~1386 million cubic kilometers of water. Majority of the water (around 96.5%) present in the world is saline in nature and is stored in the oceans. However, around 1% of the water present on land is also saline. Accounting the total saline water prevailing in the atmosphere, it is estimated that only 35.0 Mm3 of fresh water is available of which ~68% is contained in a frozen state in the polar regions and on mountain tops and glaciers. The distribution of water on the earth surface is presented in Table 1.
The water present in various forms across the world maintains a balance and the overall quantity of water remains conserved. The global annual water balance is presented in Table 2.
The amount of water that evaporates over the oceans is estimated to be 9% higher than the amount of water that is restored back in the oceans as precipitation. However, it is observed that the amount of precipitation received by land is higher in comparison to that of oceans. Thus, the differential of water amounting to 0.047 Mm3 is replaced by the surface runoff and groundwater outflow to the oceans in order to maintain the balance of water in the system. This represents the overall balance in the water system that is maintained by different forms of water bodies and different processes occurring in the environment.
4. System Water Balance
4.1. Water Balance of a Lake
A lake is a slow moving water body that is formed due to the accumulation of water in surface depressions. The lakes have two predominant motions of water, i.e., vertical movement also defined as the movement in columns (surface to bottom) and horizontal movement also defined as the movement through the length of the lakes (inflow to outflow). The water balance of a lake takes into account all the movements undertaking within this system and is determined using the mathematical relation presented in equation 3.
Where, Qin = Input flux; Qout = Output flux; V = Volume of lake (m3); dV/dt = Rate of volume change with time; h = depth of water (m); A = Area of lake (m2)
In order to estimate the total storage, it is very important to identify the pathways of inflow and outflow in a lake. Figure 2 (a & b) provides a typical example of the key pathways of movement of water in a lake.
The key components of the water balance equation of a lake are precipitation, evaporation, evapotranspiration, surface runoff and groundwater flow. The water balance equation for a lake thus gets modified as per the parameters defines in the above figures and can be represented by equation 4.
Where,
P = precipitation;
R = surface runoff;
G = net ground water flow out of the catchment;
E = evaporation;
T = transpiration; and
ΔS = change in water storage.
EXAMPLE 1: The surface elevation of a lake having an area of 5000 ha was 105 m above datum on 1 January, 2008. The following dynamics of water flow were observed in the lake during the month:
i. Inflow from surface run-off = 6 m3/s
ii. Rainfall = 145 mm
iii. Outflow from the lake = 6.5 m3/s
iv. Evaporation = 6.1 cm
Considering yourself as a hydrologist estimate the water surface elevation of the lake at the end of the month of January.
SOLUTION: Water Budget equation for the lake can be written as:
Input Volume – Output Volume = Change in storage
(?∆? + ? × ?) − (?∆? + ? × ?) = ∆S
Where, I = average inflow rate; Q = average outflow rate; P = precipitation; E = evaporation; A = surface area of the lake; and S = change in lake storage volume; Δt = 1 month = 2.592 Ms
EXAMPLE 2: Heavy rains were experienced by a catchment of area 137 ha of the order of 11 cm in a span of 80 minutes. One of the stream in the catchment removed the incoming water at a rate of 2.8 m3/s for 12 hours. The stream expelling the water was rendered dry after the event of storm. Estimate the effect of transpiration, evaporation and infiltration on the balance of the water system. Also, estimate the runoff coefficient of this catchment area.
SOLUTION: Water Budget equation in a time Δt for the catchment area can be written as:
? − ? − ? − ? − ? = ∆S
In this case, Δt = runoff duration = 12 hours
Considering that the change in storage = ΔS = 0
(G + E + T) = water not available to runoff due to infiltration, evaporation and transpiration = Losses = L
Assuming that groundwater storage does not contribute to runoff in the stream.
Thus, ? − ? = L
Where, P = Precipitation inputs in twelve hours = 1,50,700 m3
R = runoff volume = outflow volume in catchment in a span of 12 hours = 1, 20,960 m3
Losses = ? − ? = 29,740 ?3
Now, Runoff coefficient = Runoff/ Rainfall = 100800/143850 = 0.8
EXAMPLE 3: WATER BALANCE OF RIVER BASIN: The Hongru river basin (HRB) is located in the southwest zone of the Huaihe river basin between 113º38΄-115º30΄N and 32º30΄-33º30΄E. It is known to be one of the largest braches of this river and covers a drainage area of 12,380 km2 and a length of 325 km. It has been estimate that the long term average annual rainfall received by HRB amounts to 915 mm, average long term runoff equals 244 mm while the evapotranspiration from the basin amounts to 650 mm. Water is withdrawn from the basin for three major purposes as follows:
Long term average Industrial Water Supply = 116 x 106 m3
Long term average Agricultural Water Supply = 3 x 106 m3
Long term average Domestic Water Supply = 2 x 106 m3
You have been assigned a job by the government of China to assess the major issues aligned with the river basin so that an efficient strategy can be adopted to cater to the needs of the growing population in the adjacent regions of the basin. Analyzing the data available for the basin, identify a critical concern that should be addressed at a top priority in the HRB.
SOLUTION: In order to identify the key concern aligned with the Hongru river basin (HRB), the first step is to estimate the water balance of the basin.
The Water Balance Equation for the river basin can be expressed by the following equation:
? = ? + ? ± ∆? ± ∆V
Where,
P = Precipitation
Q = Runoff from the basin
E = Evapotranspiration
ΔW = Water Withdrawals
ΔV = Water stored in reservoir or lake
Estimating the total water that is stored in the reservoir of the river basin = ΔV
∆ = P− ( Q + E − ∆W )
∆ = (11.3 × 109) − ((3.02 × 109) + (8.3 × 109) + (121 × 106))
∆ = 11.3 × 109 − 11.4 × 109 = −0.1 m 3
The estimation of the water storages revealed that the net outflow is more than the net inflow in the basin. As per the estimations, it is expected the HRB would face major water shortage problems which can be attributed to the rapid population growth and industrial developments.
4.2. Mathematical Forms of water balance for different zones:
a. Water Balance for unsaturated zones:
The water balance of an unsaturated zone is predominantly governed by the sub-surface flows that determine the soil moisture. Thus, the soil-moisture budget can be established for an unsaturated zone on the concept of water balance equation by measuring the inputs through precipitation and its distribution in the plants, evaporation and soil moisture. Figure 3 presents a pictorial presentation of an unsaturated zone, i.e., the zone where soil-water balance is prevalent that determines the pathways and storage of water.
Where, I = Infiltration rate of water into unsaturated zone (mm/d)
E = Evapotranspiration rate of water from unsaturated zone (mm/d)
G = Rate of capillary rise from the saturated zone (mm/d)
R = Rate of percolation to the saturated zone (mm/d)
ΔW = Change in soil water storage in the unsaturated zone (mm)
Δt = computation interval of time (d)
EXAMPLE 4:
A catchment having an area of 290 km2 has been recorded to receive heavy rains for a period of 7 hours with an intensity of 1.8 cm/hr. The total runoff volume was estimated to be 18.25 x 106 m3. The surface of the catchment was observed to be highly favorable for infiltration. Considering the hydrological observations, estimate the average infiltration rate of the basin.
SOLUTION: The average infiltration rate of the basin can be estimated using the water budget equation for this system.
Total Rainfall = Intensity of rainfall x Duration = 1.8 x 7 = 12.6 cm
Volume of runoff = 18.25 x 106 m3
Area of the basin = 290 km2 = 290 x 106 m2
Depth of runoff = Volume of runoff/Area of the basin = 0.062 m = 6.2 cm
Estimating the Infiltration (I) using the water budget equation as described below since other storages are not operative in the present case:
? = ? − Q
Where, P = Precipitation andQ = Runoff
I= 12.6 − 6.2 = 6.4 cm
Average infiltration rate = 6.4/7 = 0.91 cm/h = 9.1 mm/hr
b. Water Balance for Land Surfaces:
The mathematical equation for estimating the water balance for land surfaces is described in equation 6.
Where, I = Rate of infiltration into the unsaturated zone (mm/d)
P = Precipitation for time interval Δt (mm)
El= Evaporation from the land surface (mm/d)
Qi = Lateral inflow of surface water into the catchment area (A) (m3/d)
Qo = Lateral outflow of surface water into the catchment area (A) (m3/d)
A = Water balance area (A2)
ΔWs = Change in surface water storage in the unsaturated zone (mm)
c. Ground Water Balance:
The groundwater zone is a complex system comprising of three regions which accommodate the predominant flows of water, i.e., zone of aeration, zone of saturation and the water table. The three regions can be briefly understood as under:
- Zone of Aeration: This is the uppermost zone which is nearest to the surface made of soil and rock. This is the zone through which the excess surface water flows and the gaps between the soil is filled with both air and water
- Water Table: Water table is the region that separates the zone of aeration and the zone of saturation. It is primarily known as the region prevalent on top of the saturation zone
- Zone of Saturation: The zone underneath the zone of aeration is known as the saturation zone. This zone is characterized by rocks and pores that are filled up to saturation level by subsurface water
A diagrammatic representation of the various zones in a groundwater system present below the land surface layer is provided in Figure 4.
EXAMPLE 5: An area of 380 m2 received a rainfall of 8 cm during a specific period. The sub-surface flows occurring in the region resulted in an average increase of 0.82 m in the groundwater level. However, a net groundwater outflow for that period was estimated to be 12 m3 and the total pumping was 0.82 m3. Estimate the specific yield of the aquifer.
SOLUTION: The water budget equation for a ground water system can also be represented by the following equation
Where,
hr = Average rise in water level in the region
Sy = Specific Yield
Vp = Annual draft in the region for unit area
f = Proportion of the annual draft withdrawn in a specified period
Net Outflow = Groundwater Outflow from the region
Thus, for the above case estimating the annual change in groundwater storage ΔS
Where,
Rr = Recharge from rainfall; Wr = Waste water recharge; Tp = Draft from groundwater Og = Groundwater outflows
∆ S = 380 (0.08) + 0.0 − 0.82 − 12
∆? = 17.58 ?3
Now, Specific yield of the aquifer = Sy
?? = ∆? (???? × ℎ? )
?? = 17.58⁄(380 × 0.82) = 0.056
Thus, Specific yield of aquifer = Sy = 0.056
d. Integrated Water Balance Equation:
The integrated water balance equation combines all the three water balances operating at the land surfaces, unsaturated zones and groundwater. The integrated water balance equation helps in describing the entire hydrologic cycle that accounts for all the major inflows and outflows of water in the atmosphere. It provides a holistic picture of the variations that occur in the pathways of water flows and total storage in the atmosphere by simultaneously analyzing the water balances occurring at smaller scales of the hydrologic cycle. Figure 5 provides a description of the integrated water movement occurring in the atmosphere.
5. Summary
In this module we learnt about:
- Principle of conservation
- Water balance in the hydrologic cycle
- Global Water balance
- Basic Concepts of water balance equation
- Mathematical Representations of water Balance of different sub systems of hydrologic cycle
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