The most general quality model is based on the ADVECTION – DISPERSION equation (mass balance of a...

Environmental quality modelsThe most general quality model is based on theADVECTION – DISPERSION equation (mass balance of a pollutant C across an elementary volume dz1 x dz2 x dz3

advection

(transport)

dispersion

+ chemical reactions (e.g. degradation) +

+ sources(z1,z2,z3,t) – sinks(z1,z2,z3,t)

321

321321

),,,(

z

Cv

z

Cv

z

Cv

t

tzzzCzzz

23

2

22

2

21

2

321 z

CD

z

CD

z

CD zzz

Special cases

The basic equation can be particularized in many different ways, depending on the specific problem.

Examples:• STATIONARY CONDITIONS: C does not change in time and thus

Note that the concentration may still change in space.

• DIFFUSION LOW with respect to transport

• SIGNIFICANT CHANGES only in 2D (e.g. aquifers) or 1D (e.g. rivers). Assuming the fluid direction to be z1, terms in z2 and/or z3 are disregarded.

• CHEMICAL REACTION TERM is known (e.g. null; C(t)=C0e-kt , first order kinetics)

Whatever the case, the possibility of solving the equation analytically is really rare. It must be solved with some numerical method (discretization in time AND space).

0),,,( 321

t

tzzzC

0321

zzz DDD

Ex. the 1-D case (river)

Consider a small, perfectly mixed volume of water, travelling along l Sp

Ip

lba

f(b)f(a)

p = pollutant concentration [unit/m3]

Sp = external pollution source [unit/(m s)]

Ip = internal pollution production/depletion [unit/(m3s)]

f = pollutant flow [unit/(m3s)]

Ex. the 1-D case (river) - 2

Mass balance (conservation) equation (law):

Sp

Ip

lba

f(b)f(a)

b

a p

b

a p

b

aAIS)b(A)a(ApA dldldl

tff

where A is the area of a volume section.


The mass balance equation can be rewritten using the Leibnitz theorem (inverting the order of integral and derivative) and the fundamental theorem of calculus as:

From which:

b

a p

b

a p

b

a

b

aAIS

AApdldldl

ldl

t

f

pp AISAA

lt

p f

f p v is the flow due to transport (v being the velocity)

is the flow due to diffusion along l (D being a dispersion coefficient)l

pD

f

Note on dispersion

D takes into account: Molecular diffusion due to the

Brownian motion of molecules (it exists also in stagnant water)

Turbulent dispersion due to turbu-lent motion (macroscopic movements like eddies)

Numerical diffusion (due to equa-tion discretization)

Hence, it’s an empiri-cal constant that needs calibration

Thus substituting the flow expressions, we get

which can be written

or, dividing by A,

The key point are thus: measure external load, determine internal sources/sinks (process)


pp AISAApAp

l

pDv

lt

pp AISAApA

l

pD

ll

v

t

p

processes internal input net ppp2

2

lD

lv

t

River models: the process

Hydraulic submodel

Biochemicalsubmodel

Thermalsubmodel

evaporation

flow

velocity

process heat

temperatureflow

velocity

sedimentation,water weeds

The biochemical submodel

Simplifications

The section is perfectly mixed Evaporation is negligible Process heat is negligible Sediments and weeds do not impact the water flow Velocity does not change temperature

Then the hydraulic and thermal sub-models can be solved independently and provide two outputs:• v(t) or v(l) and q(t) or q(l) (velocity and flow rate as

function of time or position)• T(t) or T(l) (water temperature as function of time or

position)

All the trophic chain is neglected except• The degradation of organic compounds by bacteria (using

oxygen)• The presence of bacteria

Comment on the 1-D as-sumptionAll the variables represent an average across the river cross section

PROS: simpler model shorter simulation time, easier parameter identification

CONS:less accurate model impossible to simulate unmixed sections (the model must “jump” to a well mixed section)

l z3z2l

Comment on the 1-D as-sumptionEx. The 1-D approximation cannot describe the

situation of Po river at the confluence with Ticino river (in some cases the water of the two river does not mix for few hundred meters)

In stationary conditions

Q = Q(l) does not depend on time. This means the flow changes in space, but not in time. The first de Saint Venant equation becomes:

i.e. the flow rate changes only because of discharges or abstractions. If they are measured (PRESSURES), one can compute the water balance along the river system.

Given the flow rate, one can compute the level h and the velocity v at any point along the river through some empirical formulas, such as:

N.B. a,b, , a b are parameters to be calibrated (in general, for each section l).

dQ

dlSq

)l()l( )l()l( QhaQv b

The thermal balance -1

p = Cp r T heat content (Cp specific heat, r density)

heat exchanges

The thermal balance equation has the same form with

Heat exchanges may take place:* with tributaries/effluents* with the atmosphere* with sediments

Heat generated by internal processes is usually negligible.

SedimSurfLateralpS EEE

ESedim

SurfELateralE


Lateral heat exchanges are again computed with an energy balance:

pddccl CTqTQTQE )(Lateral

Outflow(per unit length)

Tributary flow rate (per unit length)

Distributed inflow(per unit length)

Tributary temperature

Distributed inflow temperature


Surface heat exchanges depend on (constant and variable) local characteristics :

• Atmospheric radiation (e.g. geographical position)* Solar radiation (e.g. cloudiness)* Reflected radiation (e.g. water temperature)* Conductance (e.g. ait temperature)* Evaporation (e.g. wind)

evapconrifsolatm IIIIIE Surf

are difficult to evaluate.

dimSurf , SeEEIn many cases, better to substitute them with a function T(l,t) derived from a set of measures taken on each reach.

Biochemical submodel

p = DO/BOD/P/N/heavy metals/…

The basic equation can be rewritten for:

The pollutant to be modelled can be:• Non-reactive (i.e. the internal sources/sinks term = 0

and only transport and dilution are relevant)• Reactive (i.e. there are internal sources/sinks)

It may represent:• A single substance (e.g. iron, trichloroethylene) and

an equation is necessary for each substance• An aggregate indicator of various substances with

similar behavior (e.g. BOD)

Biochemical submodel - 2

For reactive substances, a specific term must be written.For instance, for a degradable substance S, degraded by a bacterial population m, the famous Michaelis-Menten formulation is:

(it’s always a sink)

Where a1 and a2 are parameters (usually, T dependent) to be estimated in each case.

IS 1S

2 Sm

- Low values of S degradation proportional to Sm

- High values of S degradation proportional to m and independent on S

Biochemical submodel - 3

For- Constant (and sufficiently high) m- Low values of S

i.e. degradation is proportional to concentration

ks is again a parameter to be calibrated.

When p=DO, the internal sources/sinks are:- Reaeration (exchange with the atmosphere) - Consumption (warning: independent on DO)- Photosynthetic production: proportional to algal density)

SkmS

SI sS

2

1

)(2 DODOk s ...21 mIS

Temperature dependence

)20(20)( TT

T20

q20

q(T)

)0(

T20

q20

q(T))0(

All the previous parameters depend on temperature (some also on the flow rate).

The dependence on temperature is usually modelled through the Arrhenius formula having again two parameters:

External load

External input of pollutants may derived from a variety of sources:• Concentrated or point (tributaries, sewage discharges, industrial

sources,…)• Distributed or non-point (runoff from agriculture or bare soil,

rainfall,…). In each section a mass balance equation holds.Ex. section A

The same holds for a unit volume and a non point source.

A

tribriver

tribtribriverriverA QQ

QCQCC

Streeter-Phelps model (1925)

Only 2 variables (BOD, DO) i.e. purely “chemical” model

Basic assumptions:- Degradation proportional to BOD concentration- DO consumption equal to BOD degradation- DO intake proportional to deficit- No dispersion

D

B

ukkl

vt

ukl

vt

)DODO(BODDODO

BODBODBOD

s21

1

Concentrated or distributed load

Oxygen addition (e.g. aerators, water falls,..)

Streeter-Phelps model - 2

))(0(BOD))0(DODO(DO)(DO

)0(BOD)(BOD

212

1

21

1ss

kkk

k

eekk

ke

e

If we assume to move along a characteristic line, i.e. following a drop of water moving with the flow, we can rewrite the equations using the flow time .t The dependence on l disappears.

In case of no external input, an analytical solution can be found

where BOD(0) and DO(0) are the values at the starting point.

D

B

ukkd

d

ukd

d

)DODO(BODDO

BODBOD

s21

1

Linear, stationary, asymptotically stable system


))(0(BOD))0(DODO(DO)(DO

)0(BOD)(BOD212

1

21

1ss

lv

kl

v

kl

v

k

lv

k

eekk

kel

el

Assuming a constant speed (at least within a certain river reach) and the two equations can be written in space instead of in flow time.

l /v

l

BOD DO

BOD(0) DOs

DO(0)

l

l = 0

v

Exponential decay Combination of two exponentials (sag curve)


DO

DOs

DO(0)

llc

dc

One can study the DO(l) function to determine the minimum: critical deficit dc at point lc

where: d(0) is the initial deficit (DOs-DO(0)) and f = k1 / k2 is called the “autopurification rate”.

clv

k

ef

d

fffk

vl

c

c

1)0(BOD

)0(BOD

)0(d)1(1ln

)1(1

This means:- The critical deficit increases with BOD(0)- The critical distance decreases with the initial deficit and initial

BODNote: BOD values are order of tens, hundreds; DO values are order of units

Streeter-Phelps model (+ & -)PLUS:- Very simple: easy to compute and calibrate (two parameters)- Linear: the superposition principle holds (the effect of two

inputs at two different locations, may be computed separately and then added)

MINUS:- For high values of BOD (initial and/or load) the computed DO

concentration becomes negative (unfeasible). This means that the degradation process becomes different (anaerobic) and this is not represented in the model

- In a “clean” river, the bacteria population may not be ready to decompose the incoming BOD and thus the real degradation is slower.

River network

To apply the model in practice, one has to subdivide the river network in a set of elementary units (reaches) that can be considered homogeneous as to:- Flow- Velocity- Temperature- Non-point loadsAt each confluence a mass balance equation must be written.

Model use

Determine the minimum treatment cost C(w) of the discharge w at a given point to guarantee a minimum DO value DO

C(w) subject to DO(l) DO 0 l L

Determine the best position to discharge a given load to maximize a water quality index W(BOD(l),DO(l))

W(BOD(l),DO(l)) subject to 0 l L

Determine the value of the unmeasured load w at point l, given a set of actual concentration measures BODi, DOi at points li.

subject to w 0.

This is an inverse problem, known as “source apportionment”.

wmin

lmax

i

2ii

2ii

wDO)l(DO)1(BOD)l(BODmin

Model use - 2

Input values (decisions, external conditions) are often (always?) known only numerically (data, assumptions). The analytical solution of model equations is not possible simulation

For simulation to be useful for decision-making:

• The model must work “around” a stable attractor

L l

t

Tdepends on boundary conditions (including inputs)

depends on initial conditions

Simulation domain

initial conditions

upstream and downstream boundary conditions

When D>0 (non negligible dispersion) the time to forget initial conditions it’s longer (backward effect).

Model use - 3

• The numerical scheme for integration of equations must be stable

The stability of a forward difference scheme for PDE is granted if the CFL (Courant-Friedrichs-Lewy) condition holds:

= Courant number <1

It means that the time discretization (time step) must be small with respect to the space discretization (in a time step the flow must travel less that a space step)

• The model must work “around” the values used for validation, but

• The input changes must have a perceivable effect on the output

• The external conditions must be representative for the decision criterion (average, extreme,…)

• The calculation must be fast enough to be repeated may times.

vl

tCr

A more refined model (Ste-hfest)Assumptions:• Negligible diffusion• No algae, no sediments (deep river)• Slow fish dynamics (hence, negligible)• Two BOD species: slowly and rapidly

degradable

State variables:- w1 rapidly degradable

BOD- w2 slowly degradable

BOD- B bacteria- P protozoa- DO oxygen

All concentrations in g/m3

A more refined model - 2

Pollutant dynamics (external input = 0)

dw1

d k11

k31w1

k32 w1

B

dw2

d k21

k33w2

k34 w2 k35w1

Bw1

k31B

Michaelis-Menten (B is relevant)

Competitive inhibition: if w1 is large, degradation of w2 is slowerIf w1 is small, degradation follows M-M

For highly polluted rivers (high values of w1 and w2 and constant B, we go back to S-P model.

When the motion law of the flow is known, the equations can be again written in terms of spatial distance from the origin.

dB

d

k31w1

k32 w1

k33w2

k34 w2 k35w1

k36

B k37

k41B

k42 BP

dP

d

k41B

k42 B k43

P


Bacteria and protozoa (external input = 0) predation by

protozoagrowth due to BOD consumption

mortality

mortality

growth due to predation of bacteriaMany parameters to estimate/evaluate.Oxygen does not enter the equations (same problem as S-P for anoxic conditions).


Pollutant dynamics (external input = 0)

dw1

d k11

k31w1

k32 w1

B

dw2

d k21

k33w2

k34 w2 k35w1

Bw1

k31B

Michaelis-Menten (B is relevant)

Competitive inhibition: if w1 is large, degradation of w2 is slowerIf w1 is small, degradation follows M-M

For highly polluted rivers (high values of w1 and w2 and constant B, we go back to S-P model.

When the motion law of the flow is known, the equations can be again written in terms of spatial distance from the origin.

PkBk

Bkk

Bkwkwk

wkk

wk

wkk

kd

d

5642

4155

54135234

23353

132

13152

s51

)DODO(DO


Oxygen dynamics reoxygenation from the atmosphere

consumption by degradation and respiration processes

consumption by predation and respiration of protozoa

Application to a real case

The Rhine river in Germany (about 500 km)

Subdivided into 12 reaches (mainly divided by tributaries) with constant flow (MQ) and lateral inflow, and constant distributed loads.

The Rhine river case

The loads are measured as COD and thus only a portion a (a parameter to be estimated) can be degraded by bacteria.

Streeter-Phelps simulation

The Streeter-Phelps model cannot represent the last reaches of the river

The Rhine river case - 2

Stehfest model simulation

There is only a marginal improvement, especially in the first reaches, to represent the actual measurements.

w1, w2, and w3 (non degradable part) are the three components of COD.

Note also that these are the CALIBRATION results. No validation has been possible due to lack of data.

However…

Rhine development plans

Let’s imagine that we want to optimally allocate new loads along the main river stream.

First, we have to define a suitable optimality indicator:

e.g. the integral of the oxygen deficit

The S-P solution

fin

o

l

ldllJ s1 DO)(DO

Almost all the new load is moved toward the end of the German stretch since, in this way, most of the deficit is produced in Holland.

old

new

Rhine development plans - 2With the Stehfest model results are different:

The new load is spread more regularly along the river and the oxygen concentration toward the end is even better than with less load!

This apparent contradiction may be explained looking at the dynamics of bacteria.

old

new

old

new

The higher bacterial population in the central reach (Rhine Valley) allows a more regular oxygen consumption.

Rhine development plans - 3The plan objective, though referring to the entire river (not just a spot value) does not take sufficiently into account the effects in the Netherland.

It is necessary to somehow evaluate the terminal condition (common problem). One possibility is to add a component of the objective based on the deficit at the border (final oxygen demand or FOD). The new objective term may have

the form:

where FOD* is a limit (untouchable) value. For low valued of FOD* the new load spreads from upstream to downstream, while for high value, the new load goes upstream only because of lack of space.

FOD*FOD

FOD2

J

Current models

Starting from the original equations many implementations have been developed mainly different for data handling and connections to GISs.

From US EPA (http://www.epa.gov/athens/wwqtsc/):

WASP – Water Quality Analysis Simulation Program

SWMM (Storm Water Management Model)

BASINS

EPD-RIV1One Dimensional Riverine Hydrodynamic and Water Quality Model

QUAL2K

Development began in 1987 (Brown and Barnwell).

By far, the most famous and used water quality model:• One dimensional.: The channel is well-mixed vertically and

laterally• Steady state hydraulics: Non-uniform, steady flow• Diurnal heat budget and temperature simulated as a function of

meteorology • Diurnal water-quality kinetics • Heat and mass inputs: Point and non-point loads and abstractions

PLUS• Carbonaceous BOD speciation (two forms, slow and fast) • Anoxia: Q2K reduces oxidation reactions to zero at low oxygen

levels• Sediment-water interactions and bottom algae• Light extinction• pH• Pathogens

NOTE: latest implementation (2009) in Excel (originally FORTRAN)

QUAL2K - 2

The Arno river and the Q2K scheme

Simulation of temperature and coliforms (importance of the selection of the hydraulic regimes)

An uncertainty (Monte Carlo) analysis is possible

QUAL2K - 3

Q2K can reproduce weirs and other hydraulic discontinuities.

Each reach is simulatedas a trapezoidal channel using Manning’s equation

QUAL2K -4

Up to 14 components. Daily input patterns.

Simulated processes

dn

upipoh

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d

s

s

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smi

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p

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e

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o

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dn

upipoh

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d

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s

s

sodcf

re

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s

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CARBON

NITROGEN

PHOSPHORUSOrganic Inorganic

Organic Ammonia Nitrate

Detritus Slow Fast Tot. inorganic

Oxygen

Algae

Integrated model: SWAT

“Soil and Water Assessment Tool” – joint study of soil use and water quality

Developed by USDA, since 1980’s Deterministic, physically based, DISTRIBUTED model (time step = 1 day) Manual: 649 pages (2009), Conferences, User Groups.

• Used for long-term studies: Assesses impacts of climate

and land management on yields of water, sediment, and agricultural chemicals

• Includes hydrology, soil temperature, plant growth, nutrients, pesticides and land management

• Uses a spatial discretization called Hydrologic Response Unit (HRU)

SWAT - 2

Land Use layer Soils layer

+ + …

A HRU is a unique combination of soil characteristics (slope, exposition, activities, rainfall,…) that are assumed constant within the unit GIS operations ArcSWAT

SWAT - 3

SWAT – 4 (ex. : water in soil)SWAT separates soil profiles into 10 layers to model inter and intra-movement between layers. The model is applied to each soil layer independently starting at the upper layer.

SWAT soil water routing feature consists of four main pathways:1. soil evaporation2. plant uptake and transpiration3. lateral flow4. percolation.• Storage routing technique

combined with a crack-flow model to predict flow through each soil layer

• Cracked flow model allows percolation of infiltrated rainfall though soil water content is less than field capacity.

• Portion that does become part of layer stored water cannot percolate until storage exceeds field capacity.

SWAT – Example GAC, EthiopiaThe Gilgel Abay Catchment (5000 km2) isthe largest of the four main sub-catchments of Lake Tana, Ethiopia

Blue Nile source

DEM 30mx30m data: 58% falls between 0-8% slope and 4% with steep slope>30% mainly in the southeast part.

DATA -1

Agriculture (around 65%) is the dominant land use in the catchment while agro-pastoral accounts 33% of the land uses , 1% is agro-forestry and 0.1 % urban.

DATA - 2

Rainfall and temperature data (outside the basin), flow data in Merawi

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

Maximum tempMinimum temp

days

T(o

c)

Model formulation• Surface runoff computation: SCS curve number procedure,

where the curve number varies with antecedent soil moisture, land cover, soil permeability

• Soil loss was calculated using MUSLE (modified soil loss equation) method which uses a runoff factor that accounts for the energy used in detaching and transporting of sediment, in addition to parameters related to land cover, soil erodibility, topographic factor & support practice factor

• SWAT allows to apply agricultural management practices (tillage, planting and growing, fertilization, irrigation …) on selected HRU

• The model assumes surface runoff interacts with the top 10mm of soil. Nutrients contained in this layer are available for transport to the main channel in the surface runoff

• The fertilization operation allows the user to specify the fraction of fertilizer that is applied to the top 10mm.

Model formulation - 2

Model formulation - 3

SWAT includes a weather generator function to fill the gaps in climate records using monthly averaged data for a number of years.The weather generator first independently generates precipitation for the day and maximum and minimum temperature, solar radiation and relative humidity are then correct them, based on the presence and absence of rain for the day.Model calibration and uncertainty (i.e. due to model input , model simplifications, and other process not included in the model) analysis were carried out

Sensitivity analysis was used to find the most sensitive parameter values for better estimations of the stream flows and twelve parameters were found to be more sensitive for flow simulation.

R2 =0.79CalibrationR2 =0.71validation

Results

Determination of critical sub-basins, producing around 60 t of sediments per hay (agricultural and steep areas)

1 2 3 4 5 6 7 8 9 10 11 1205

1015202530354045

Soya beanLentil

Months

Sedim

ent

yie

ld(t

/ha) Sediment load due to

different crops and…

…to different agricultural practices

1 2 3 4 5 6 7 8 9 10 11 120

10

20

30

40

50Mould board plowField Cultivator Conservation tillageNo tillage

Months

Sedim

ent

Yie

ld(t

/ha)

Similar results for nutrient loads.

Policy indications

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90Nitrate yieldCrop yield

percent increase of fertilization

perc

ent

incre

ase

Increased fertilization increases proportionally N load without a similar increase in yield

A decrease of cultivated area by 20% results in a decrease in sediment yield of more than by 50%.

Mould board plow tillage and Field cultivator tillage increase sediment yield by 26.6% and 2% compared to generic conservation tillage. But Field cultivator reduces sediment yield by 25% wrt conventional tillage practice

Avoiding tillage (no tillage) practice is even better for sediment yield (-25%)

Flow rate decreases by 0.4% in soybean cultivation as compared to Grain sorghum

Grain sorghum, with field cultivator and low fertilizer dose (50%) is a recommended practice getting both low sediment and nutrient release with good yield.

The most general quality model is based on the ADVECTION – DISPERSION equation (mass balance of a...

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