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 Ann Mulligan Woods Hole Oceanographic Institution Marine Policy Center Woods Hole, MA, USA

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 Cape of Massachusetts (2004, Masterson)

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

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

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)

Wells ON (Modeler’s uncertainty) % dryout

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 [1] 0.02 Well Well Well Well Well [1] Only feasible in 5% of the scenarios

Management Model constraints objective

Results Pumping (m 3 /d) Head m Well 1Off0.86 Well Well Well Well Well 62, K 1 = 656 m/d; K 2 = 246 m/d 757  OFF 2,158    356 OFF  56 OFF  2,973

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

Modeler’s Uncertainty

Modeler’s Uncertainty Correlation Coefficients well 2well 3well 4well 5well 6 well 1n/a well well well well

well 2well 3well 4well 5well 6 well 1n/a well E E E E-5 well E E E -7 well E E-9 well E -1 Modeler’s Uncertainty P-values (< 0.05 is significant)

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

Risk (50% reliable) well 1well 2well 3well 4well 5well

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 m 2 /d, head = 0.6 m MODFLOW 2000: water table on; convertible boundaries Steady state Recharge ft/d

Truro, Massachusetts LayerElevation (m) Hydraulic conductivity (m/d) to -1.5 with topographic relief to to to

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) Well Well 2 Well 3 Well 4 Well 5 Well 6

Wells ON (Spatially variable fields)

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 [2] 0.06 Well Well Well Well Well [2] Only feasible in 21% of the scenarios.

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

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

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

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.

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 mean= 2813 std= 419 mean= 105 std= 165 mean= 385 std= 139 Pumping (m 3 /d) Frequency

Spatial Variability

Spatial Variability Correlation Coefficients well 2well 3well 4well 5well 6 well 1n/a well well well well

well 2well 3well 4well 5well 6 well 1n/a well E E E E-3 well E E E -7 well E E -2 well E -7 Spatial Variability P-values (< 5.0 E -2 is significant)