1. Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of Massachusetts in Boston Boston, MA, USA Karen.Ricciardi@umb.edu Ann Mulligan Woods Hole Oceanographic Institution Marine Policy Center Woods Hole, MA, USA amulligan@whoi.edu

2. 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

3. Lower Cape of Massachusetts (2004, Masterson)

4. 1 2 3 4 5 6 Truro, MA Pumping in 2004: 1. Knowles(2): 757 m 3 /d 2. S. Hollow(3/8): 2,158 m 3 /d 3. N. Truro Air Force Base 4: 312 m 3 /d 4. N. Truro Air Force Base 5: 312 m 3 /d 5. N. Unionfield: off 6. CCC-5: off Discharge in Provincetown. Total Supply Needs: 3,540 m 3 /d

5. 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 Update transmissivity SW interface Convergence criteria met? YesNo

6. How does uncertainty affect the management of coastal water supply?

7. Uncertainty in Layers 1 and 2 Modeler’s uncertainty Spatial variability (1992, Hess et al.; SGSIM)

8. Wells OFF (Modeler’s uncertainty) 1 2 3 4 5 6 0 1.5 3.0 4.5 6.0 0 0 00 0 0

9. Wells ON (Modeler’s uncertainty) 1 2 3 4 5 6 95% dryout 0 1.5 3.0 4.5 6.0 0 0 0 0 0 0

10. Head (m) with “known” K Modeler’s Uncertainty Pumping (m 3 /d) Mean Head (m) S.D.Pumping (m 3 /d) Mean Head (m) S.D. Well 10.8600.840.207570.82 [1] 0.02 Well 22.3702.290.4621580.990.10 Well 31.6301.580.333121.270.25 Well 41.2901.250.293120.980.22 Well 52.6202.540.5102.300.45 Well 62.7702.680.4902.630.47 [1] Only feasible in 5% of the scenarios

11. Management Model constraints objective

12. Results Pumping (m 3 /d) Head m Well 1Off0.86 Well 2392.27 Well 31161.47 Well 43561.02 Well 5562.43 Well 62,9730.86 K 1 = 656 m/d; K 2 = 246 m/d 757  OFF 2,158  39 312  116 312  356 OFF  56 OFF  2,973

13. Modeler’s Uncertainty mean= 0 std= 0 mean= 43 std= 100 mean= 36 std= 38 mean= 518 std= 120 mean= 3 std= 9 mean= 2939 std= 204 Pumping (m 3 /d) Frequency

14. Modeler’s Uncertainty

16. Modeler’s Uncertainty Correlation Coefficients well 2well 3well 4well 5well 6 well 1n/a well 21.000.600.25-0.13-0.68 well 3 1.000.520.20-0.79 well 4 1.000.21-0.86 well 5 1.00-0.17 1 2 3 4 5 6

17. well 2well 3well 4well 5well 6 well 1n/a well 205.3 E -42.0 E -14.9 E -14.4 E-5 well 3 03.7 E -32.9 E -12.6 E -7 well 4 02.8 E -12.6 E-9 well 5 03.8 E -1 Modeler’s Uncertainty P-values (< 0.05 is significant) 1 2 3 4 5 6

18. Modeler’s Uncertainty MEAN VALUES OF MULTIPLE SOLUTIONS Well 1: off Well 2: 43 m 3 /d Well 3: 36 m 3 /d Well 4: 518 m 3 /d Well 5: 3 m 3 /d Well 6: 2,939 m 3 /d NO UNCERTAINTY Well 1: off Well 2: 39 m 3 /d Well 3: 116 m 3 /d Well 4: 356 m 3 /d Well 5: 56 m 3 /d Well 6: 2,973 m 3 /d

19. Risk (50% reliable) well 1well 2well 3well 4well 5well 6 03896558702499 03019158702561 02309158702632 01779158702685 01339158702729 0719160402774 0073666352765

20. 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

21. 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

22. Thank you.

23. Truro, MA 4 layers 39x85 nodes/layer (2157 active) Constant head at the oceans Streams are modeled as drains with conductance = 149 m 2 /d, head = 0.6 m MODFLOW 2000: water table on; convertible boundaries Steady state Recharge 0.0015 ft/d

24. Truro, Massachusetts LayerElevation (m) Hydraulic conductivity (m/d) 1 7.6 to -1.5 with topographic relief 61.0 2-1.5 to -24.422.9 3-24.4 to -61.015.2 4-61.0 to -152.40.3

25. 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

26. Heads at wells when wells are OFF (100 spatially variable fields) 1 2 3 4 5 6 Well 1 1 2 3 4 5 6 Well 2 Well 3 Well 4 Well 5 Well 6

27. Wells ON (Spatially variable fields) 1 2 3 4 5 6

28. Head (m) with “known” K Spatial Variability Pumping (m 3 /d) Mean Head (m) S.D.Pumping (m 3 /d) Mean Head (m) S.D. Well 10.8600.890.047570.42 [2] 0.06 Well 22.3702.400.0221581.040.13 Well 31.6301.650.043121.340.05 Well 41.2901.310.053121.020.05 Well 52.6202.650.0202.420.06 Well 62.7702.800.0202.760.02 [2] Only feasible in 21% of the scenarios.

29. 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 m 3 /d max head = 0.91 m (not 0.86 m as in the model) well 2: 1,076 m 3 /d well 6: 2,464 m 3 /d

30. Management Model Objective function is: Piecewise linear Not differentiable Minimum head varies over different wells in the feasible region well 6 well 4 well 2

31. Management Model Constraints are: Linear Variable well dependence pumping well 2 intrusion well 6 intrusion well 1 intrusion

32. Constraints Too much pumping: If q 1 +q 2 +…+q 5 > 3,540, then set q 6 = q 1 +q 2 +…+q 5 +3,540. This will cause the drawdown to be much larger than it would be naturally. Salt water intrusion: If head at well i < 0.86 m then penalize the objective function by 1-e qi/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.

33. Pattern Search Solver 1997, Torczon 2003, Kolda and Torczon 2004, Gray and Kolda (APPSPACK)

34. Spatial Variability mean= 0 std= 0 mean= 57 std= 188 mean= 179 std= 130 mean= 2813 std= 419 mean= 105 std= 165 mean= 385 std= 139 Pumping (m 3 /d) Frequency

35. Spatial Variability

36. Spatial Variability Correlation Coefficients well 2well 3well 4well 5well 6 well 1n/a well 21.000.25-0.120.31-0.56 well 3 1.000.060.70-0.79 well 4 1.00-0.03-0.38 well 5 1.00-0.78

37. well 2well 3well 4well 5well 6 well 1n/a well 201.9 E -15.2 E -11.0 E -11.7 E-3 well 3 07.8 E -12.7 E -53.7 E -7 well 4 08.9 E -14.0 E -2 well 5 05.0 E -7 Spatial Variability P-values (< 5.0 E -2 is significant) 1 2 3 4 5 6