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The effects of uncertainty on a ground water management problem involving saltwater intrusion. Karen L. Ricciardi. Ann Mulligan. Department of Mathematics University of Massachusetts in Boston Boston, MA, USA [email protected] Woods Hole Oceanographic Institution

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The effects of uncertainty on a

ground water management

problem involving saltwater intrusion

Karen L. Ricciardi

Ann Mulligan

Department of Mathematics

University of Massachusetts in Boston

Boston, MA, USA

[email protected]

Woods Hole Oceanographic Institution

Marine Policy Center

Woods Hole, MA, USA

[email protected]


Objective

Determine a groundwater supply plan that:

* meets the demands of the community

* minimizes the risk of salt water intrusion

Difficulties:

* hydrologic parameters are uncertain

* salt water interface responds nonlinearly to

pumping changes  computational effort

De


Lower cape of massachusetts 2004 masterson
Lower CapeofMassachusetts(2004, Masterson)


Truro ma

1

2

3

5

4

6

Truro, MA

Pumping in 2004:

1. Knowles(2): 757 m3/d

2. S. Hollow(3/8): 2,158 m3/d

3. N. Truro Air Force Base 4:

312 m3/d

4. N. Truro Air Force Base 5:

312 m3/d

5. N. Unionfield: off

6. CCC-5: off

Discharge in Provincetown.

Total Supply Needs: 3,540 m3/d


Salt water interface modeling
Salt water Interface Modeling

Using the Ghyben-Herzberg approximation (1:40), the SW

interface is determined using the following iterative method.

Pumping design

Modflow

Heads at cells

SW interface

Convergence criteria met?

Yes

No

Update transmissivity



Uncertainty in layers 1 and 2
Uncertainty in Layers 1 and 2

Spatial variability (1992, Hess et al.; SGSIM)

Modeler’s uncertainty


Wells off modeler s uncertainty

1

2

3

5

4

6

Wells OFF (Modeler’s uncertainty)

0

0

0 1.5 3.0 4.5 6.0

0 1.5 3.0 4.5 6.0

0

0

0 1.5 3.0 4.5 6.0

0 1.5 3.0 4.5 6.0

0

0

0 1.5 3.0 4.5 6.0

0 1.5 3.0 4.5 6.0


Wells on modeler s uncertainty

1

2

3

5

4

6

Wells ON (Modeler’s uncertainty)

95% dryout

0

0

0 1.5 3.0 4.5 6.0

0 1.5 3.0 4.5 6.0

0

0

0 1.5 3.0 4.5 6.0

0 1.5 3.0 4.5 6.0

0

0

0 1.5 3.0 4.5 6.0

0 1.5 3.0 4.5 6.0



Management model
Management Model

objective

constraints


Results
Results

K1 = 656 m/d; K2 = 246 m/d

757 OFF

2,158  39

312  116

OFF  56

312  356

OFF  2,973


Modeler s uncertainty
Modeler’s Uncertainty

mean= 0

std= 0

mean= 43

std= 100

mean= 36

std= 38

Frequency

Pumping (m3/d)

mean= 518

std= 120

mean= 3

std= 9

mean= 2939

std= 204




Modeler s uncertainty correlation coefficients

1

2

3

5

4

6

Modeler’s UncertaintyCorrelation Coefficients


Modeler s uncertainty p values 0 05 is significant

1

2

3

5

4

6

Modeler’s UncertaintyP-values (< 0.05 is significant)


Modeler s uncertainty3
Modeler’s Uncertainty

NO UNCERTAINTY

Well 1: off

Well 2: 39 m3/d

Well 3: 116 m3/d

Well 4: 356 m3/d

Well 5: 56 m3/d

Well 6: 2,973 m3/d

MEAN VALUES OF

MULTIPLE SOLUTIONS

Well 1: off

Well 2: 43 m3/d

Well 3: 36 m3/d

Well 4: 518 m3/d

Well 5: 3 m3/d

Well 6: 2,939 m3/d



Conclusions
Conclusions

Uncertainty in the hydraulic conductivity should be considered when developing a management program where salt water intrusion may be an issue.

Examining the solutions for the scenarios representing the uncertainty allows one to ascertain information about the correlation between wells.

Examining modeler’s uncertainty using a multi-scenario approach provides a means by which it is possible to determine reliable management designs.

There are multiple designs that provide reliable solutions to the management problem.


Current work
Current Work

Equivalent solutions of the fixed supply problem

Maximum supply problem.

Spatial variability affects.

Boundary affects: Single boundary problem.

Well locations and numbers as a decision variable.



Truro ma1
Truro, MA

  • 4 layers

  • 39x85 nodes/layer (2157 active)

  • Constant head at the oceans

  • Streams are modeled as drains with conductance = 149 m2/d, head = 0.6 m

  • MODFLOW 2000: water table on; convertible boundaries

  • Steady state

  • Recharge 0.0015 ft/d



Truro ma2
Truro, MA

Head Results for Layer 1

  • Mean head values used

  • Fixed pumping

  • SS not reached, no convergence of the iterative method

26 m

18 m

9 m

0 m


Heads at wells when wells are off 100 spatially variable fields

1

1

2

2

3

3

5

5

4

4

6

6

Heads at wells when wells are OFF (100 spatially variable fields)

Well 1

Well 2

Well 4

Well 3

Well 6

Well 5


Wells on spatially variable fields

1

2

3

5

4

6

Wells ON(Spatially variable fields)



Management model1
Management Model

One perfectly homogeneous

K field for each layer.

well 1: off

well 2: variable q

well 3: off

well 4: off

well 5: variable q

well 6: variable q

total supply = 3,540 m3/d

max head = 0.91 m (not 0.86 m as in the model)

well 2: 1,076 m3/d

well 6: 2,464 m3/d


Management model2
Management Model

Objective function is:

  • Piecewise linear

  • Not differentiable

  • Minimum head varies over different wells in the feasible region

well 6

well 2

well 4


Management model3
Management Model

Constraints are:

  • Linear

  • Variable well dependence

well 6

intrusion

pumping

well 2

intrusion

well 1

intrusion


Constraints
Constraints

Too much pumping:

If q1+q2+…+q5 > 3,540,

then set q6 = q1+q2+…+q5+3,540.

This will cause the drawdown to be much larger than it would be naturally.

Salt water intrusion:

If head at welli < 0.86 m

then penalize the objective function by 1-eqi/1,000.

This will increase the value of the objective function an amount related to the pumping at the well where there is a violation.


Pattern search solver
Pattern Search Solver

1997, Torczon

2003, Kolda and Torczon

2004, Gray and Kolda (APPSPACK)


Spatial Variability

mean= 0

std= 0

mean= 57

std= 188

mean= 179

std= 130

Frequency

Pumping (m3/d)

mean= 385

std= 139

mean= 105

std= 165

mean= 2813

std= 419



Spatial variability correlation coefficients
Spatial VariabilityCorrelation Coefficients


Spatial variability p values 5 0 e 2 is significant

1

2

3

5

4

6

Spatial VariabilityP-values (< 5.0 E -2 is significant)


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