(deep) Lake models
The basic difference with rivers is that the horizontal dimension(s) are
comparable to depth. Thus:
• The velocity of the water is small (negligible)
• The differences along the vertical dimension are relevant
• The dynamics of higher levels of the trophic chain (phytoplankton,
zooplankton, fishes) are relevant
• Slower processes (e.g. seasonal changes) must be considered
• The underlying hydraulic model is obviously different
A lake (often artificial) can be studied as a
sequence of horizontally perfectly mixed boxes.
Stratification
The basic consequence of depth (in temperate climate) is water
stratification.
spring
wind
summer
 heating faster than mixing
warm water is less
dense, thus floats,
and needs lots of
wind to mix
same temp.
thus mixing
Stratification - 2
Stratified lakes present three distinct zones.
0
thermocline
depth (m)
epilimnium
region of
rapid
temperature
change
10
hypolimnium
20
sediments
30
0
10
20 30
temperature (°C)
Stratification - 3
A typical yearly pattern in temperate countries (dimictic lakes 
holomictic lakes)
Eutrophication
Eutrophication consequences
Loss of water quality, fish death
Eutrophication processes
DO
Photosynthesis
Periphyton Phytoplankton
(WASP7)
atmosphere
Reaeration
Periphyton is a mixture
of algae, cyanobacteria,
heterotrophic microbes
attached to submerged
surfaces
Respiration
Death&Gazing
Oxidation
Detritus
C P N
CBOD1
CBOD2
CBOD3
Dis.
Org. P
Dis.
Org. N
NH3
PO4
Nitrification
Adsorption
Mineralization
NO3
SSinorg
Denitrification
N
Settling
Phytoplankton


The growth rate of a population of phytoplankton in a
natural environment:
 is a complicated function of the species of
phytoplankton present
 involves different reactions to solar radiation,
temperature, and the balance between nutrient
availability and phytoplankton requirements
Due to the lack of information to specify the growth
kinetics for individual algal species all models
characterizes the population as a whole by the total
biomass of the phytoplankton present (measured in
terms of chlorophyll concentration)
Phytoplankton - 2
NO3
NH3
Phyt
Light
O C:N:P
Growth rate:
RG
 Gmax X T X L X N
PO4
Gmax = maximum specific growth rate constant at 20°C, 0.5 – 4.0 day-1
XT = temperature growth multiplier , dimensionless
XL = light growth multiplier, dimensionless
XN = nutrient growth multiplier, dimensionless
Phytoplankton growth
XT
 
T  20
G
Temperature multiplier
where: G = temperature correction factor for growth (1.0 – 1.1)
T = water temperature, °C
Nutrient multiplier
XN
 C Ni

Cp
 min 
,
,...
 K Ni  C Ni K p  C p 
Defines a
limiting factor
KMN
semisaturation
constant
Phytoplankton growth - 2
XL depends on the light l(z) available for photosynthesis at depth z. It may be
written using Michaelis-Menten formulation
l ( z)
X L ( z)  
kl  l ( z )
or Steele (1965) formulation
 l ( z) 
l ( z)

X L ( z)  
exp1 
ls
ls 

where ls represents an optimal (maximum) light intensity.
But
•
The incident light on a water surface varies during the day and the season
•
The light intensity naturally decreases with depth
•
The presence of phytoplankton further increases light attenuation (selfshadowing)
Light attenuation
The light intensity dependence on depth l(z) can be expressed by the BeerLambert law
l ( z )  l0 exp( ( Phyt ) z )
where l0 is the incident radiation on the surface and the function  (Phyt )
can be written as a polynomial function of phytoplankton concentration.
Typical seasonal patterns
The relation between phytoplankton and zooplankton is a typical predatorprey system.
Algal blooms occur in spring.
Other components of phyoplankton dynamics
Death rate:
RD
 k1R 1TR20  k1D  k1G Z (t )
k1R = endogenous respiration rate constant, day-1
1R
=
k1D =
k1G =
Z(t) =
temperature correction factor, dimensionless
mortality rate constant, day-1
grazing rate constant, day-1, or m3/gZ-day if Z(t) specified
zooplankton biomass time function, gZ/m3 (defaults to 1.0)
Settling rate:
RS
 vS AS / V
vS = settling velocity, m/day
AS = surface area, m2
V = segment volume, m3
Phosphorus cycle
Phytoplankton
Detr. P
Org. P
PO4
T  20
k8383

Phytoplankton P
C4 a pc  Death vs 4 
  G p  Dp   C4 a pc
t
D

Growth


Death
Settling
Mineralization

 D p fOP a pc C4  kdiss 
Death
Inorganic P
C3
vs3 (1  f d 3 )
C4
T  20
 Dp (1  fop )a pc C4  k8383
C8  G p C4 a pc 
C3
t
K mpc  C4
D
Detrital P
 C15
t
C4
C8
K mpc  C4
T  20
diss
C15 
Dissolution
vs15
D
Settling
Growth
Settling
Dissolved organic P
 C4 
 C8
T  20
T  20
 C8
 kdiss  diss C15  k83 83 


t
 K mpc  C4 
Dissolution
Mineralization
Very high number of parameters
Difficult to calibrate for a specific situation.
Ex. Phosphorus cycle parameters
Fully distributed model
It is necessary to compute all the internal processes for
each volume in a grid and model the dispersion exchanges
with the surrounding volumes (thousands of state variables)
Control of eutrophication


Reduce loads (less use of detergents or fertilizers, better treatment,…)
Artificial mixing

Superficie totale (4933000 m2)
Profondità (7,5 m)
Selective discharges
Epilimnio
Ipolimnio
Profondità (10 m)

Artificial aeration
Defining the planning/control problem
Water quality WQ is, in principle, a function of all the water components c1,…, cn in
each location z1,z2,z3 and at any time instant t.
WQ  f c1 ( z1 , z 2 , z3 , t ),..., cn ( z1 , z 2 , z3 , t ) 
In practice, we are not able to define the form of such a function.
We thus define suitable indicators WQi for each components based on some kind of
aggregation in time and/or space.
WQi  Stat ci ( z1 , z 2 , z3 , t ) 
zZ ,tT
where Stat() indicates some statistics over the spatial region Z and the time interval T.
Examples:
- Summer average oxygen concentration in the hypolimnium
- Yearly average of phytoplankton concentration in the upper 10 m
- Number of times in a year in which the nitrogen concentration in the upper 1 m
exceeds N.
Defining the planning/control problem - 2
Additionally, we can assume that the overall water quality is some aggregation of the
WQi . A common definition of the aggregation is an index formulated as a weighted
sum:
n
WQ   a iWQ i
i 1
where the weights ai express some (subjective) measure of the importance of each
factor in the assessment of water quality.
The planning/control problem can be written:
min(Cost (u ))
WQ(u )  WQ
or
where u are the decisions and Cost(u) their cost.
max WQ(u )
Cost (u )  C
Defining the planning/control problem - 3
As already noted (see slides on DPSIR), when used for planning/control, the model
works in conditions different from those used for calibration/validation.
The accuracy of the model cannot be proved.
The model can thus be used to test the effect of an input (decision, parameter,
boundary condition) on the output, to understand:
- the sense of the interaction (positive, negative)
- the entity of the interaction (larger, smaller than other input variables).
To perform such Sensitivity/Uncertainty analysis, Monte Carlo simulation is normally
used:
1. Generate random numbers for model inputs
2. Run the model with the randomized inputs
3. Store the random input values and the corresponding model outputs
4. Repeat a (high) number of times the steps (1-3)
5. Calculate correlations between model outputs and random inputs
Sensitivity/Uncertainty analysis
- Define the input to be tested
- Select a suitable distribution of
values (normal, lognormal,…)
- Generate a set of random values
- Run the model
- Analyse the output distribution
- Compute the correlation I/O
- Store the results for the
selected output
Ex. YASAIw (US EPA)
YASAIw is a free open-source framework for Monte Carlo simulation in Excel.
Three main functions:
1) To generate random model input values from a normal distribution:
= GENNORMAL(mean, stdev)
mean = nominal value for model input;
stdev = e.g. 5% of mean for sensitivity analysis
2) To save the random input values and use them in a sensitivity analysis:
= SIMOUTPUT(x, name, code)
x = cell address of the random input
name = unique name for the input
code = 1 for input “assumption”
3) To save the model output values and use them in a sensitivity analysis:
= VBAOUTPUT(x, name, code)
x = cell address of the model output
name = unique name for the output
code = 2 for output “forecast”
Sensitivity analysis of QUAL2Kw
YASAIw GENNORMAL functions
YASAIw SIMOUTPUT functions
Links to the model output sheets
Sensitivity average periphyton chlorophyll-a
to the most sensitive model inputs
Spearman's rank correlation
coefficient (Spearman's rho), is a
nonparametric measure of statistical
dependence between two variables. It
assesses how well the relationship
between two variables can be
described using a monotonic function.
An alternative approach
The solution of the planning/control problems requires the use of models to
determine the link between decisions and water quality.
However, a full quality model may not necessary, since we are just interested in
computing the link between decision and the defined water quality.
Develop a SURROGATE MODEL, i.e. a (simplified) model able to reproduce
the required function, at least for a certain range of values of u, which
means to substitute the original function WQ(u) with an approximation
WQ=fs(u).
An alternative approach - 2
The overall procedure is thus:
Define the
range of u
Simulate the
original model
Common forms for fs:
- A linear/polynomial function
- A response surface
- A neural network
- …..
It must be simple
Define a
structure
for fs
Calibrate the
parameters of fs
Validate fs
Use fs in the
optimization
procedure
The surrogate model
The surrogate model:
• It’s NOT a copy of the system
• It’s based on the original model input and output and thus does not
reproduce biochemical and physical phenomena
• It works correctly (the approximation is acceptable) only for the set of
other input used for the original model simulation (design of experiments)
• It works correctly only within the range of u for what it has been calibrated
• It’s faster to execute and thus can be repeated a high number of times
within the optimization procedure
Surrogate modelling application
Determination of the «best» number
and position of artificial aerators for an
Australian artificial reservoir.