Karen L. Ricciardi

The effects of uncertainty on aground water management

problem involving saltwater intrusion

Department of MathematicsUniversity of Massachusetts in Boston

Boston, MA, USAKaren.Ricciardi@umb.edu

Ann MulliganWoods Hole Oceanographic Institution

Marine Policy CenterWoods Hole, MA, USAamulligan@whoi.edu

Objective

De

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

Lower Capeof

Massachusetts

(2004, Masterson)

1

23

45

6

Truro, MAPumping in 2004:1. Knowles(2): 757 m3/d2. S. Hollow(3/8): 2,158 m3/d3. N. Truro Air Force Base 4:

312 m3/d4. N. Truro Air Force Base 5:

312 m3/d5. N. Unionfield: off6. CCC-5: off

Discharge in Provincetown.Total Supply Needs: 3,540 m3/d

Salt water Interface ModelingUsing the Ghyben-Herzberg approximation (1:40), the SWinterface is determined using the following iterative method.

Pumping design Modflow

Heads at cells

Update transmissivity

SW interface

Convergence criteria met?

Yes No

How does uncertainty affect the management of

coastal water supply?

Uncertainty in Layers 1 and 2

Modeler’s uncertainty

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

Wells OFF (Modeler’s uncertainty)

1

2 345

60 1.5 3.0 4.5 6.0 0 1.5 3.0 4.5 6.0

0 1.5 3.0 4.5 6.00 1.5 3.0 4.5 6.0

0 1.5 3.0 4.5 6.0 0 1.5 3.0 4.5 6.0

0

0

00

0

0

Wells ON (Modeler’s uncertainty)

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2 345

6

95% dryout

0 1.5 3.0 4.5 6.00

0 1.5 3.0 4.5 6.00

0 1.5 3.0 4.5 6.00

0 1.5 3.0 4.5 6.00

0 1.5 3.0 4.5 6.00

0 1.5 3.0 4.5 6.00

Head(m) with “known” K

Modeler’s UncertaintyPumping(m3/d)

Mean Head (m)

S.D. Pumping(m3/d)

Mean Head (m)

S.D.

Well 1 0.86 0 0.84 0.20 757 0.82[1] 0.02Well 2 2.37 0 2.29 0.46 2158 0.99 0.10Well 3 1.63 0 1.58 0.33 312 1.27 0.25Well 4 1.29 0 1.25 0.29 312 0.98 0.22Well 5 2.62 0 2.54 0.51 0 2.30 0.45Well 6 2.77 0 2.68 0.49 0 2.63 0.47

[1] Only feasible in 5% of the scenarios

Management Model

ii headminmax

dm

i

i

dm

m3

3

0pumping86.0head540,3supply

constraints

objective

Results

Pumping(m3/d)

Headm

Well 1 Off 0.86

Well 2 39 2.27

Well 3 116 1.47

Well 4 356 1.02

Well 5 56 2.43

Well 6 2,973 0.86

K1 = 656 m/d; K2 = 246 m/d757 OFF

2,158 39312 116

312 356

OFF 56

OFF 2,973

Modeler’s Uncertaintymean= 0std= 0

mean= 43std= 100

mean= 36std= 38

mean= 518std= 120

mean= 3std= 9

mean= 2939std= 204

Pumping (m3/d)

Freq

uenc

y

Modeler’s Uncertainty

Modeler’s Uncertainty

Modeler’s UncertaintyCorrelation Coefficients

  well 2 well 3 well 4 well 5 well 6

well 1 n/a n/a n/a n/a n/a

well 2 1.00 0.60 0.25 -0.13 -0.68

well 3   1.00 0.52 0.20 -0.79

well 4     1.00 0.21 -0.86

well 5       1.00 -0.17

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2 345

6

  well 2 well 3 well 4 well 5 well 6

well 1 n/a n/a n/a n/a n/a

well 2 0 5.3 E -4 2.0 E -1 4.9 E -1 4.4 E-5

well 3   0 3.7 E -3 2.9 E -1 2.6 E -7

well 4     0 2.8 E -1 2.6 E-9

well 5       0 3.8 E -1

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

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6

Modeler’s Uncertainty

MEAN VALUES OFMULTIPLE SOLUTIONS

Well 1: offWell 2: 43 m3/dWell 3: 36 m3/dWell 4: 518 m3/dWell 5: 3 m3/dWell 6: 2,939 m3/d

NO UNCERTAINTY

Well 1: offWell 2: 39 m3/dWell 3: 116 m3/dWell 4: 356 m3/dWell 5: 56 m3/dWell 6: 2,973 m3/d

Risk (50% reliable)

well 1 well 2 well 3 well 4 well 5 well 6

0 389 65 587 0 2499

0 301 91 587 0 2561

0 230 91 587 0 2632

0 177 91 587 0 2685

0 133 91 587 0 2729

0 71 91 604 0 2774

0 0 73 666 35 2765

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.

Conclusions

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.

Current Work

Thank you.

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, Massachusetts

Layer Elevation (m) Hydraulic conductivity (m/d)

1 7.6 to -1.5 with topographic relief 61.0

2 -1.5 to -24.4 22.9

3 -24.4 to -61.0 15.2

4 -61.0 to -152.4 0.3

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)

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2 345

6

Well 11

2 345

6

Well 2

Well 3 Well 4

Well 5 Well 6

Wells ON (Spatially variable fields)

1

2 345

6

Head(m) with “known” K

Spatial Variability

Pumping(m3/d)

Mean Head (m)

S.D. Pumping(m3/d)

Mean Head (m)

S.D.

Well 1 0.86 0 0.89 0.04 757 0.42[2] 0.06Well 2 2.37 0 2.40 0.02 2158 1.04 0.13Well 3 1.63 0 1.65 0.04 312 1.34 0.05Well 4 1.29 0 1.31 0.05 312 1.02 0.05Well 5 2.62 0 2.65 0.02 0 2.42 0.06Well 6 2.77 0 2.80 0.02 0 2.76 0.02

[2] Only feasible in 21% of the scenarios.

Management Model

One perfectly homogeneous K field for each layer.

well 1: offwell 2: variable qwell 3: offwell 4: offwell 5: variable qwell 6: variable q

total supply = 3,540 m3/dmax head = 0.91 m (not 0.86 m as

in the model)well 2: 1,076 m3/dwell 6: 2,464 m3/d

Management Model

Objective function is:• Piecewise linear• Not differentiable• Minimum head varies

over different wells in the feasible region

well 6

well 4

wel

l 2

Management Model

Constraints are:• Linear• Variable well

dependencepumping

well 2 intrusion

well 6intrusion

well 1intrusion

ConstraintsToo 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 mthen 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

1997, Torczon

2003, Kolda and Torczon

2004, Gray and Kolda (APPSPACK)

Spatial Variability

mean= 0std= 0

mean= 57std= 188

mean= 179std= 130

mean= 2813std= 419

mean= 105std= 165

mean= 385std= 139

Pumping (m3/d)

Freq

uenc

y

Spatial Variability

Spatial VariabilityCorrelation Coefficients

  well 2 well 3 well 4 well 5 well 6

well 1 n/a n/a n/a n/a n/a

well 2 1.00 0.25 -0.12 0.31 -0.56

well 3   1.00 0.06 0.70 -0.79

well 4     1.00 -0.03 -0.38

well 5       1.00 -0.78

  well 2 well 3 well 4 well 5 well 6

well 1 n/a n/a n/a n/a n/a

well 2 0 1.9 E -1 5.2 E -1 1.0 E -1 1.7 E-3

well 3   0 7.8 E -1 2.7 E -5 3.7 E -7

well 4     0 8.9 E -1 4.0 E -2

well 5       0 5.0 E -7

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

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6