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GGR 217F fundamentals of hydrology

Assignment #2: The Water Balance Equation and Soil Moisture


The objectives of this assignment are to:

  1. Conceptualize the water balance equation for a simple environment
  2. Use physical descriptions of the processes of water transfer to calculate a mass balance
  3. To investigate how irrigation is used to supplement water inputs to the soil water balance 4. Evaluate the impacts of water reallocations on the soil water and groundwater resources
  4. Write a concise, clear report that summarizes your findings.


  1. In addition to answering the questions as asked in the assignment, please note the following requirements:
  2. You are required to show your work for all calculations. You must complete the calculations and tables in a spreadsheet application such as Excel, and show an example of the formulae used in the spreadsheet (hand-written or using a word processor). It is recommended that you link all calculations in the spreadsheet so that if you change a variable, the rest of the results will automatically update. This will save you a great deal of time.
  3. Your assignment must be printed using 11 or 12 point standard, legible font (e.g. Calibri). All written answers must be double-spaced. You may print on both sides of the page.
  4. Your assignment must show your name, student number and practical session (e.g. PRA101) on every sheet of your submitted work. It is a good idea to include this information in a header or footer on every page in your word processing document.
  5. As a requirement for successful completion of this assignment, the work presented in your submitted assignment must clearly be your own. If you do not understand this requirement, or are not clear what constitutes and differentiates individual, collaborative or plagiarized work, then ensure that you seek clarification from the professor or your teaching assistant.

Failure to observe the above requirements will result in mark penalties.


The soil water zone is one of the most important reservoirs in the hydrological cycle as it is an important interface that determines the relative amounts of input water (precipitation) that are partitioned to the outputs relative to the soil water zone (evapotranspiration, runoff, and percolation to groundwater). The conceptualization of the soil water zone as a ‘bucket’ that contains a volume of water in storage, with this storage volume changing over time in response to changes in the balance of inputs and outputs, makes understanding this important component of the hydrological cycle a bit easier.

The Water Balance

Consider a simple water balance equation for the soil water zone:


See your lecture notes or text for a detailed explanation of the terms. If the input (PRECIP) is balanced by the outputs (AE + S), then ∆ST will be equal to zero; the soil may be thought of as being in hydrologic equilibrium. In general, this is only the case over relatively long periods of time (i.e. a year or more). At shorter durations, there are definitely imbalances in the water budget that create both positive and negative changes in storage (recall the seasonal surface water balance figures from lecture/text).

The short-term changes in storage in the watershed are largely manifested as changes in soil moisture. The soil water zone is a very dynamic environment, changing in water content (∆ST ) in response to water inputs from precipitation (PRECIP), and decreasing in water content due to evaporative outputs (AE), and transfers to groundwater beneath through processes such as percolation (S). Given the episodic nature of the input (e.g. rain events), these changes in storage are highly transient; on the order of hours to days.

Describing Soil Water Storage

We can describe the capacity of the soil water zone to hold water in a number of ways.

The porosity (φ) of a soil (or other geological media) may be expressed as:

φ = Vvoid/Vsample

where Vvoid is the total volume of void space in the soil sample, and Vsoil is the total volume of the soil sample. Porosity is simply the fraction of the soil sample that can be occupied by air, water or some combination of the two.

The most common way to express the amount of water contained within a soil sample is the Volumetric Soil Moisture (VSM), which is:

VSM = Vwater/Vsample

where Vwater is the total volume of water in the sample. Be mindful of the subtle difference in the expressions for porosity and volumetric soil moisture. You will see that whereas the other two terms are always going to have a value of something less than 1, however both porosity and VSM are sometimes multiplied by 100 and expressed as a percent. Finally, it should also be obvious based on the descriptions of porosity and VSM that, at saturation (i.e. there is no air in the soil voids), that VSM is at its maximum, and that it is equal to porosity.


You have been asked to assess the need for irrigation for an agriculturally intensive region. You must perform some basic water balance calculations for the two driest months of the year to determine if irrigation is required, and how much.

Luckily, the study area is completely flat, so there is no surface runoff to be considered in the soil water balance. Therefore the S term only involves the vertical movement of surplus water back down to the groundwater zone through percolation.

Required Data

Area of the Field: 1500 hectares

Depth of Active Soil Water Zone: 1.00 metres (m)

Soil Porosity: 0.420

Field Capacity: 0.180 (expressed as VSM)

Wilting Point for the Crop: 0.100 (expressed as VSM)

VSM on Day 1 of the Study Period: 0.220

Evaportranspiration: 5.00 mm per day


For the sake of simplicity in this assignment, we will assume that the properties of the environment are homogeneous.

  • Precipitation is evenly distributed over the entire study field.
  • Evapotranspiration is evenly distributed over the entire study field and averages 5.0 mm day-1.
  • Soil moisture is evenly distributed throughout the depth of the ‘active’ soil water zone.
  • Percolation from the soil water zone to groundwater occurs only when VSM is greater than field capacity (i.e. there is gravitational water present)
    • The volume of water transferred on the daily time step is equal to the volume of water in excess of field capacity on the previous day to the exponent 0.75
    • e. volume transferred = surplus0.75
    • This relationship is empirical and does not reflect the actual physics of soil water movement

(i.e. simplified for this assignment) o Assume all values in assignment accurate to three significant digits.Tasks:

PART A: Using the data from Table 1 and information provided in this assignment, create an Excel spreadsheet that includes:

  • The precipitation volume delivered to the field each day (m3 day-1), and the total precipitation volume over the study period (60 days) (1 MARK).
  • The volume of water evapotranspired from the field each day (m3 day-1), and the total volume of water evapotranspired over the study period (60 days). (1 MARK).
  • The volume of water percolated to groundwater each day (m3 day-1), and the total volume of water percolated over the study period (60 days). (2 MARKS).
  • The volume of water stored in the soil water zone each day (m3 day-1). (2 MARKS).
  • The Volumetric Soil Moisture for each day (unitless). (1 MARK).

You must carefully consider how to construct the formulae for your spreadsheet in order to calculate the values required.


  • You will not be able to calculate a percolation rate for day 1.
  • You must calculate percolation using a conditional formula. An example of a generic conditional formula is.

IF A > B = TRUE, then A = C. IF A > B = FALSE, then A = 0”.

Excel will let you create formulas like this, and it will even ‘HELP’ you do it.

6) Produce a graph of your results. On this graph, you will present Days on the horizontal x-axis, and utilise two different vertical y-axes: Precipitation (mm/day) is to go on the left y-axis as a bar chart, and a line graph for VSM on the right y-axis. Let Excel ‘HELP’ you create a graph with two chart ‘types’ and a ‘secondary’ y-axis. The graph must be properly labeled and include a title and legend (5 marks).

PART B: Using the information provided in the required data, the calculated data, and your graph consider the following questions:

  • Is irrigation is required by this crop during the study period, and if so, on what days? (1 MARK).
  • What is the total amount of water that would be required to be applied as irrigation to offset the soil moisture deficit over the study period (in m3) (2 MARKS).


Your submission will take the form of a report on your findings. If it helps, pretend that you are producing a report for a client. It will be composed of:

  • A brief INTRODUCTION to the problem. Do not simply rewrite the introduction to the assignment (2 MARKS).
  • The METHODS that you used. Here you must describe in words the numerical approach in your MODEL. (Hint, this is also the place to put your example calculations) (5 MARKS). Your calculations performed in Parts A and B will be evaluated separately (see below).
  • The RESULTS. In this section, include all of your answers (numerical and written) from Parts A and B above. Note that your numerical solutions will be evaluated in conjunction with your spreadsheet (see below). Marks are as above in Parts A and B.
  • Your CONCLUSIONS that address the two main questions in PART B. As part of your conclusion, briefly discuss the potential weaknesses in the analyses (10 MARKS).

**As part of the successful completion of this assignment, you must also electronically submit your clear, tidy spreadsheet that you used to perform the calculations. This is how the calculations of PART A

and B will be evaluated. You must submit it through blackboard.

TOTAL MARKS: 32. Value for the term: 15%. Due Monday Feb. 22nd at 5 pm

Table 1: Precipitation data for the 60 day long study period. Also available as an excel chart.

Day Precipitation  (mm day-1)  Day Precipitation (mm day-1) 
1                   0                     31                  0 
2                   0                     32                  0 
3                   5.0                   33                  0 
4                   0                     34                  0
5                   0                     35                  0 
6                   3.0                   36                  0
7                   15.0                  37                  0
8                   0                     38                  64.0 
9                   0                     39                  15.0 
10                  0                     40                  0
11                  0                     41                  0 
12                  0                     42                  0 
13                  0                     43                  0
14                  2.0                   44                  0
15                  10.0                  45                  0 
16                  0                     46                  0 
17                  0                     47                  0 
18                  0                     48                  0
19                  0                     49                  0
20                  0                     50                  0
21                  0                     51                  0 
22                  0                     52                  12.0 
23                  0                     53                  3.0 
24                  0                     54                  0 
25                  0                     55                  0 
26                  0                     56                  0 
27                  0                     57                  0 
28                  0                     58                  1.0 
29                  0                     59                  3.0 
30                  0                     60                  2.0 
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