GEOS4499
Catchment Fundamentals
Introduction
Assignment 3: Environmental Systems
Part B: Lake water balance
In Part B of this assignment you will look at a simple iterative scheme for solving a differential equation, which is the main approach we have for modelling environmental systems. The most straightforward numerical approaches involve repetition, or iteration, of a calculation to evaluate how the variables in the system changes through time (or space).
In this exercise we will look at a simple “bucket” storage model of how a lake level fluctuates through time. The hypothetical simulation in Section 1 is a modified version of an exercise from Wainright and Mulligan (2004) (see p38). Similar types of models have been used for paleo- environmental reconstructions (Bradley 1999) and in hillslope hydrology (Dunin 1976), where the bucket represents the ability of soil to hold water. This type of approach can also be used to simulate solute concentrations, which is particularly useful for quantifying exchanges between surface water and groundwater (Cook 2013) or undertaking a lake nutrient balance. In the second part of the exercise we will extend the example.
1. A hypothetical lake
Let’s start by looking at a hypothetical lake, assumed to be at the bottom of the catchment described in Exercise 3A. The depth of the lake increases due to rainfall and inflows and decreases due to evaporation and seepage (Figure 1). There is also a maximum level to which the lake can be filled, hmax (m), above which water will overflow into the next lake in the system, such that 0 ≤ h ≤ h)*+ . This simple system is described by the equation:
𝑑 h = 𝑟 2+ 𝑖 − 𝑒 = − > > 𝑘 ? h > − > @ 𝑜 ( 1 ) 𝑑𝑡 345678 A675678
where r is the rate of rainfall (m yr-1), i is the rate of water inflow from the drain or river (m yr-1), e is the rate of evaporation from the lake (m yr-1), and kh is the seepage rate from the base of the lake (infiltration to the underlying soil/aquifer), which is assumed to be a linear function of the depth of water in the lake, h (m). This seepage is controlled by the coefficient k (yr-1), which describes the permeability of the soil that makes up the lake bed. The final term o is the overflow (m yr-1), which happens when h>hmax. Before proceeding, satisfy yourself that you understand this equation and that each term in this equation has the same units, as required.
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Figure 1. Conceptual model of lake water balance
Equation (1) is a differential equation, describing the rate of change of water level over time. We
know that IJ is technically the instantaneous rate of change of lake water level. But calculus I7
tells us that we can approximate IJ by calculating the change in water level over small I7
increments of time (Euler’s method – see slides), with the most accurate approximation using the smallest increments of time.
So we have an equation for the rate of change in lake water level (Equation 1). But how do we calculate the lake level as a function of time, h(t) ? That is, how do we solve this differential equation to calculate h for any given time? To do this we will use an iterative approach. Start with an initial lake level, hi, and then calculate the change in lake level over each small increment of time (in this case dt = 1 year is fine), using the rates for each process. The water level at the next time-step is then calculated by summing the initial level and the calculated change; hi+1 = hi + dh.
Figure 2 shows the results of three different modelling scenarios, with lake level simulated for 200 years. The parameters for each scenario are shown in Table 1, but you aren’t told which scenario is which. Answer the following 5 questions (and include your answers in your write-up) before proceeding to the next section.
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Questions
1. Surface waters can be either gaining (groundwater discharge to streams and lakes), losing (recharge of aquifers by streams and lakes), or through-flow systems (Winter et al. 1998). Based on the information you have, what type of lake (gaining, losing or through- flow) is the system shown in Figure 1? What other data might you want to measure to confirm this?
2. Would the step of going from NO to h(t) be considered as differentiation or integration?
NP
Explain your answer.
3. Calculate the lake level for each scenario in Table 1 (i.e. replicate Figure 2) and work out
which set of parameters is for which model (A, B or C). If Model A is the basecase, what
type of climate variation might Models B and C represent? Explain your answer.
4. Now let’s look at the rate of change in lake level under each of these modelling scenarios.
Plot 𝒅𝒉 vs time for each model (hint: remember that this is just the change in water level 𝒅𝒕
over time). How does this plot of 𝒅𝒉 vs time relate to Figure 2? How long does it take for 𝒅𝒕
each model to reach “steady state”? Does your choice of initial water level change this?
Figure 2 Simulated lake level under three different model scenarios (A, B and C) Table 1 Model parameters for lake level simulations shown in Figure 2
Parameter
Units
Model ?
Model ?
Model ?
r
m yr-1
1
1.5
1
i
m yr-1
0.2
1.6
0.8
e
m yr-1
1
0.05
0.05
k
yr-1
0.05
0.05
0.05
hmax
m
50
50
50
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2. Simulation of water levels in Lake Imagination
Now let’s apply this approach to the set of two lakes situated at the bottom end of the catchment we analysed in Part A. In that exercise we looked at water and nutrient exports. In this case, the lake overflow water, based on the overflow criteria, will allow for water to flow from the initial lake to the downstream lake.
Given the position of upper lake in the landscape it is likely to be a through-flow lake, so we will include terms for groundwater outflow also connecting the two lakes. Let’s assume that groundwater inflow is determined by the regional groundwater gradient so that we can use a constant value. For groundwater outflow, we can use the same approach as we used above, where infiltration was a function of water level in the lake.
So our equation for the change in lake storage (m3) of the upper lake is:
𝑑𝑆655UV=𝑃+𝑄 −𝑄
−𝐸 −𝑘h 𝐴 655UV =>>>65>5?UV>>6>5>5U@V
\] ^U*_34\ 7JU ^*`U
− 𝐸 + 𝑘h 𝐴 ^A]UV =>6>>5>5U?V >>6>5>5U@V
\] U47UV34\ cVA) ^*`U Y
(2)
− 𝑘h 𝐴 (3) =>^>A]>U?V >^>A>]@UV
\] ^U*_34\ ^*`U b
34 A67(Y)
𝑑𝑡
𝑑𝑆^A]UV = 𝑃 + 𝑄 − 𝑄
𝑑𝑡
A67(Y) A67(b)
where P is the precipitation (m3/day) falling on the relevant lake (precipitation rate x lake area, A), Qin is the inflow (discharge) rate from the catchment (m3/day), Qout(1) is the overflow from the upper to the lower lake, E is the evaporation rate (m3/day) and the khA terms reflect the groundwater seepage associated with the lakes.
Now you can model the water level in the two lakes. You can use the daily rain and evaporation (data provided) to drive this model, but you will need to ensure you have consistent units across terms (i.e. all rates are /d, not a mix of /d and /y, and volumes are different from lengths!). We will assume that the sides are steep so that lake surface area is independent of water level height. Use an initial lake water level as provided in the details below.
Questions
1. Sketch the described system, setting the context and using the notation above.
2. Write out the above two equations using “difference” notation.
3. Open the LakeModel spreadsheet and finish the columns for Lake 1, using the below
settings in Table 2 (change the red numbers to match Table 2).
4. Plot the lake 1 storage and water depth. Plot the water fluxes.
5. Change the settings to see the sensitivity to leakage and maximum lake depth.
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6. Implement the columns for Lake 2, based on Eq 3 above.
7. Would increasing or decreasing the surface area of Lake 2 help stabilise the water level?
8. Use the model to demonstrate an alternate scenario of lake management – how would
this change the lake water levels?
9. How realistic is this model and what changes could you make if you were asked to
improve it?
Table 2 Model parameters for Lake Imagination water level simulation
Term
Lake 1 (Upper) Value
Lake 2 (Lower) Value
Rainfall (r)
r(t) from climate sheet
Evaporation (e)
e(t) from sheet – uses Sin equation assuming 1.5m/yr
Lake area (A)
1.0 x 107 m2
2.0 x 107 m2
Maximum lake depth (hmax)
8m
5m
Infiltration coefficient (k)
8.0×10-4 /d
1.0×10-3 /d
Initial depth (m)
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3. Lake P mass balance
Using the principles outlined above, can you undertake a mass balance of P in the lake and predict lake P concentration over time? Start by formulating your own differential equation. The input P is the mass flux (load) leaving your Exercise 3A catchment model. Assume the rain and evaporation processes don’t change lake P mass, but note they will change the lake volume and therefore the P concentration (since lake average P concentration = P mass / lake volume).
References
Bradley, R.S. (1999) Paleoclimatology: Reconstructing Climates of the Quaternary. 2nd Edition, Academic Press, London.
Cook, P.G. 2013. Estimating groundwater discharge to rivers from river chemistry surveys. Hydrological Processes 27: 3694-3707.
Dunin, F.X. (1976) Infiltration: It’s simulation for field conditions, in J.C. Roddha (ed.) Facets of hydrology, John Wiley & Sons, Chichester, 199-227.
Wainright J. and Mulligan M. (eds) (2004) Environmental Modelling: Finding simplicity in complexity, John Wiley & Sons, Chichester.
Winter T.C., J.W. Harvey, O.L. Franke, and W.M. Alley. 1998. Ground water and surface water: a single resource. U.S. Geological Survey US Geological Survey Circular 1139.
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