Water Resources Planning and Management Daene C. McKinney Water Quality.

Water Resources Planning and Management

Daene C. McKinney

Water Quality

Water Quality Management• Critical component of overall water management in a basin• Water bodies serve many uses, including

– Transport and assimilation of wastes – Assimilative capacities of water bodies can be exceeded WRT intended

uses • Water quality management measures

– Standards• Minimum acceptable levels of ambient water quality

– Actions• Insure pollutant load does not exceed assimilative capacity while

maintaining quality standards– Treatment

Water Quality Management Process

• Identify – Problem– Indicators – Target Values

• Assess source(s)• Determine linkages

– Sources Targets

• Allocate permissible loads

• Monitor and evaluate• Implement

Physical Processes Controlling Flux

• Advection– Solutes carried along by flowing water

• Diffusion– Transport by molecular diffusion

• Dispersion– Transport by mechanical mixing

Models

• Advection + dispersion - major processes by which dissolved matter is distributed throughout a water body (e.g., river)

C = concentration (M/L3)V = Average velocity in reach (L/T)D = Longitudinal dispersion coefficient (L2/T)t = timex = longitudinal distance

Advection term

Dispersion term

Source term

Reaction term

Eq. 10.10 in text

Steady-state Model• Steady-state

– Where k = decay rate (1/T) kC

x

CV

x

CD

2

20

2

41

V

kDm

– Solution is

– W = loading (M/T) at x = 0

Eq 10.12 in text

Advection - Dispersion

Advection Dominated Flow

Water Quality Example

• W1, W2 = Pollutant loads (kg/day)• x1, x2 = Waste removal efficiencies (%)• P2

max, P3max = Water quality standards (mg/l)

• P2, P3 = Concentrations (mg/l)• Q1, Q2, Q3 = Flows (m3/sec)• a12, a13, a23, = Transfer coefficients

Water Quality ExampleParamet

erUnits Value

Q1 m3/s 10

Q2 m3/s 12

Q3 m3/s 13

W1 kg/day 250,000

W2 kg/day 80,000

P1 mg/l 32

P2max mg/l 20

P3max mg/l 20

a12 - 0.25

a13 - 0.15

a23 - 0.60


0.11 x79.128.1 21 xx

8.01 x

0.12 x


• Cost of treatment at 1 greater than cost at 2 (bigger waste load at 1)

• Marginal cost at 1 greater than marginal cost at 2, c1 > c2 for same level of treatment


2,10.10.0

79.128.1

8.0

toSubject

Minimize

21

1

2211

ix

xx

x

xcxc

i

Cost of treatment at 1 >= cost at 2marginal cost at 1, c1, >= marginal cost at 2, c1, for the sameamount of treatment.

Example

• Irrigation project– 1800 acre-feet of water per year

• Decision variables

– xA = acres of Crop A to plant?

– xB = acres of Crop B to plant?

1,800 acre feet = 2,220,267 m3

400 acre = 1,618,742 m2

Crop A Crop B

Water requirement (Acre feet/acre) 3 2

Profit ($/acre) 300 500

Max area (acres) 400 600

Example

2

4

6

8

10

2 4 6 8 10xA (hundreds acres)

x B (

hu

nd

red

s ac

res)

xB< 600

xA> 0 xA< 400

3xA +2 xB < 1800

xB > 0

Example

2

4

6

8

10


x B (

hu

nd

red

s ac

res)

xB< 600

xA> 0 xA< 400

xB > 0

Z=3600=300xA +500xB

Z=2000=300xA +500xB

Z=1000=300xA +500xB

(200, 600)

GAMS CodePOSITIVE VARIABLESxA, xB;

VARIABLESobj;

EQUATIONS objective, xAup, xBup, limit;

objective.. obj =E= 300*xA+500*xB;xAup.. xA =L= 400.;xBup.. xB =L= 600.;limit.. 3*xA+2*xB =L= 1800;

MODEL Calibrate / ALL /;SOLVE Calibrate USING LP MAXIMIZING obj;

Display xA.l;Display xB.l;

Marginal, Lagrange multiplier, shadow price, dual variable

GAMS Output LOWER LEVEL UPPER MARGINAL

---- EQU objective . . . 1.000---- EQU xAup -INF 200.000 400.000 .---- EQU xBup -INF 600.000 600.000 300.000---- EQU limit -INF 1800.000 1800.000 100.000

LOWER LEVEL UPPER MARGINAL

---- VAR xA . 200.000 +INF .---- VAR xB . 600.000 +INF .---- VAR obj -INF 3.6000E+5 +INF .

Marginal

Marginals

• Marginal for a constraint = Change in the objective per unit increase in RHS of that constraint.– i.e., change xB

– Objective = 360,000– Marginal for constraint = 300– Expect new objective value = 360,300

600Bx 601Bx

New Solution LOWER LEVEL UPPER MARGINAL

---- EQU objective . . . 1.000---- EQU xAup -INF 199.333 400.000 .---- EQU xBup -INF 601.000 601.000 300.000---- EQU limit -INF 1800.000 1800.000 100.000

LOWER LEVEL UPPER MARGINAL

---- VAR xA . 199.333 +INF .---- VAR xB . 601.000 +INF .---- VAR obj -INF 3.6030E+5 +INF .

180023 BA xxNote: Adding 1 unit to xB adds 300 to the objective, but constraint 3 says

and this constraint is “tight” (no slack) so it holds as an equality, therefore xA must decrease by 1/3 unit for xB to increase by a unit.

Unbounded Solution

00

400

toSubject

500300Maximize

BA

A

BA

xx

x

xxZ

2

4

6

8

10


x B (

hu

nd

red

s ac

res)

xA> 0 xA< 400

xB > 0

unbounded

Take out constraints3 and 4, objective can Increase without bound

Infeasibility

00

300023

600

400

toSubject

500300Maximize

BA

BA

B

A

BA

xx

xx

x

x

xxZ

2

4

6

8

10


x B (

hu

nd

red

s ac

res)

xB< 600

xA> 0 xA< 400

3xA +2 xB > 3000

xB > 0

Change constraint 4to >= 3000, then no intersection of constraints exists and no feasible solution can be found

Multiple Optima

BA xxZ 200300Objective

Change objective coefficient to 200, then objective has same slope as constraint and infinite solutions exist 2

4

6

8

10


x B (

hu

nd

red

s ac

res)

xB< 600

xA> 0 xA< 400

xB > 0

Z=1800=300xA +200xB

Infinite solutions on this edge

3xA +2 xB < 1800

Water Resources Planning and Management Daene C. McKinney Water Quality.

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