# 3.0 Water Balance GEOG370 Hydrology Spring 2013 D.L. Peters

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1 3.0 Water Balance

2 Water Balance approach modeling framework for simplifying, describing & quantifying the hydrological budget of water inputs & outputs of components of the hydrological cycle specific to an area (e.g., region, watershed) and time interval (e.g., annual, monthly, weekly.) ultimately, water balance is climate driven via variations in precipitation (M) & temperature (E) other local factors (vegetation, soils, land use, seasonality )

3 Water Balance Equation Amount of water entering & leaving a control volume during a time period (Δt) as: inflow (I) outflow (O) = change in storage (ΔS) I O = ΔS or I Δt O Δt ΔS Δt (balance eqn) (difference eqn) (differential eqn) = and i o = ds dt Assumption: density taken as constant (1000 kg m -3 ) express as volume (m 3 ) Conservation of mass

4 Water Balance Equation In a drainage basin, each portion of the budget is a specific process/component of the cycle, P + G in 123 inflow, I Q + ET + G out outflow, O = ΔS what dimensions/units could be used? depth (L) over area (L 2 ) = volume (L 3 ) over time rate of water movement (LT -1 or L 3 T -1 ) e.g., mm P per month, m 3 s -1 for river discharge

5 Water Balance Equation If assume continuity storage change ΔS = 0, inputs = outputs from year to year (average over long time e.g. 30 years) assumes no significant climate changes, surficial changes, anthropogenic impacts, or major Δstorage Time averaging applied to develop conservation equation

6 Water Balance Equation But - Temporal Variability Inputs, storages, and outputs are all distributed variables Quantities can vary with time Thus concept of time variability is inherent to ch concept of the system

7 Water Balance Equation But - Spatial Variability Rates of Inputs and outputs and many other hydrologically relevant properties vary spatially over geographic region (e.g, watershed) Description of spatial variability notable precipitation are based on measurements made over time at discrete points (precipitaion gauges)

8 Space & Time Scales Hydrologic Processes

9 Applications of the Water Balance Equation accounting of major inputs, outputs & delayed components over specific spatial & temporal scale annual water budget from monthly climate averages determine times of moisture use, deficit, recharge, surplus water supply/recharge planning assessing human impacts why is this useful? e.g. why are wetlands drying up in northern Alberta

10 Limitations of the water balance equation Very difficult to model accurately..one of the most challenging Rubik Cubes of nature, one in which the squares change colours, shapes and sizes as they are being moved around by the different forces, and in which even the structural setup changes with time Klemes (1988), A hydrological perspective

11 Example: Evaporation pan Water balance components? P E = ΔS (mm) E = P - ΔS (mm)

12 Water Balance of Jemis Lake Q F + P O + P S + E O +G = ds/dt (including snow/ice growth and melt) DEM Jemis Fieldwork to Validate Algorithms Evaporation & Precipitation Groundwater Flow 0.80 m deep when full Low relief - little surface runoff to lake

13 Example: Lake Athabasca What are the water balance components of this system?

14 Example : Global Water Balance Model

15 Watershed balance model

16 Thornthwaite Monthly Water Balance The Thornthwaite water balance (Thornthwaite, 1948; Mather, 1978; 1979) uses an accounting procedure to analyze the allocation of water among various components of the hydrologic system. Inputs to the model are monthly temperature and precipitation. Outputs include monthly potential and actual evapotranspiration, soil moisture storage, snow storage, surplus, and runoff.

17 Precipitation inputs Precipitation from local climate station, Rain & Snow When monthly temperature is < or = 0 o C all P is considered to be snow and PET is set to zero. Snow accumulates until mean monthly temperature exceeds 0 o C. SNOW m = (1 F m ) P m F m is a melt factor see handout for calculation monthly temperature > 0 o C, P is considered to be liquid (RAIN) and snowmelt occurs. RAIN m = F m P m P m as monthly water inputs (W m ) W m = W total water inputs, m define as time interval W m = RAIN m (rain or snow) + MELT m (delayed delivery) MELT M = F M (PACK M-1 + SNOW M )

18 Evapotransipiration Output Combined water loss as vapour via evaporation (E) & transpiration (T) from plants ET depends on: potential ET or PET m = ƒ (surface air temp, daylength, solar azimuth) * e (T ) = 924 D Ta a a Hamon (1963) PET m actual ET m = ƒ (PET m, available moisture from SOIL m, veg) ET m = W m + SOIL m-1 - SOIL m

19 Soil moisture Key source or sink of moisture for ET process controls available water for AET m via both evaporation from land surfaces and plant transpiration SOIL m = min {[(W m PET m ) + SOIL m-1 ] or SOIL max } SOIL max = FC (Z rz ) 100mm for topsoil in other words SOIL m will be the lesser of values in argument ΔSOIL m = SOIL m-1 {1 exp[ (PET m W m )/SOIL max ]}

20 Soil moisture deficit Soil moisture deficit (-ΔSOIL) loss of SOIL water to satisfy ET (utilization) when W m < PET m deficit conditions result as PET m > ET m (i.e., DEFIC m = PET m -ET m ) unsatisfied PET m demand is withdrawn from SOIL m to reach ET m ET m = W m + ΔSOIL m and is limited by SOIL max (e.g., 100 mm)

21 Soil moisture recharge Soil moisture recharge (+ΔSOIL m ) input of water to SOIL when W m > PET m SOIL is recharged first (up to e.g., 100 mm)

22 Soil moisture surplus Excess surplus moisture is termed runoff and in the water balance equation is assumed to become streamflow (Q) excess as runoff, Q m = W m -ET m - ΔSOIL m

23 Thornthwaite Water Balance Model Victoria Airport ANNUAL WATER BALANCE FOR [click to INSERT LOCATION] by: [click to INSERT NAME, STUDENT #] P PACK W SOIL ET SURP PET DEFIC AMOUNT (mm) jan feb mar apr may jun jul aug sep oct nov dec MONTH

24 Thornthwaite Water Balance Model Montreal ANNUAL WATER BALANCE FOR [click to INSERT LOCATION] by: [click to INSERT NAME, STUDENT #] P PACK W SOIL ET SURP PET DEFIC AMOUNT (mm) jan feb mar apr may jun jul aug sep oct nov dec MONTH

25 Thornthwaite Water Balance Model Florida ANNUAL WATER BALANCE FOR [click to INSERT LOCATION] by: [click to INSERT NAME, STUDENT #] P PACK W SOIL ET SURP PET DEFIC AMOUNT (mm) jan feb mar apr may jun jul aug sep oct nov dec MONTH

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