Multiobjective Design of Groundwater Monitoring Network Under ...

Water Resour Manage (2012) 26:1809–1825 DOI 10.1007/s11269-012-9988-1

Multiobjective Design of Groundwater Monitoring Network Under Epistemic Uncertainty Anirban Dhar & Rajvardhan S. Patil

Received: 22 May 2010 / Accepted: 13 January 2012 / Published online: 28 January 2012 # Springer Science+Business Media B.V. 2012

Abstract A methodology is proposed for optimal design of groundwater quality monitoring networks under epistemic uncertainty. The proposed methodology considers spatiotemporal pollutant concentrations as fuzzy numbers. It incorporates fuzzy ordinary kriging (FOK) within the decision model formulation for spatial estimation of contaminant concentration values. A multiobjective monitoring network design model incorporating the objectives of fuzzy mass estimation error and spatial coverage of the designed network is developed. Nondominated Sorting Genetic Algorithm-II (NSGA-II) is used for solving the monitoring network design model. Performances of the proposed model are evaluated for hypothetical illustrative system. Evaluation results indicate that the proposed methodology perform satisfactorily under uncertain system conditions. These performance evaluation results demonstrate the potential applicability of the proposed methodology for optimal groundwater contaminant monitoring network design under epistemic uncertainty. Keywords Monitoring network design . Optimization . Groundwater pollution

1 Introduction Management of groundwater pollution is a major concern of modern times. Industrial development and modern practices have worsened the groundwater pollution scenario. However, due to lack of funds it is hard to implement proper monitoring strategies at the field level. Generally the amount of information available from the water supply wells and field are very limited. Process control, performance measurement requires time consuming A. Dhar (*) : R. S. Patil Department of Civil Engineering, Indian Institute of Technology Kharagpur, Kharagpur, WB 721302, India e-mail: [email protected] A. Dhar e-mail: [email protected] R. S. Patil e-mail: [email protected]

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and costly data collection effort. Tracking the transient pollutant plume is a challenging task due to the uncertainties in predicting the complex subsurface flow and transport processes. The contaminant plume is not known due to its inherent epistemic uncertainty (Datta et al. 2009a; Datta et al. 2011). The design of groundwater monitoring network requires proper definition of objectives. The objective can be defined as monitoring the actual state of groundwater systems, i.e., estimatimation of the state variables in space and detection of trends. The general problem of groundwater monitoring network design can be stated as optimization of design objective(s) subject to budgetary limitation. If no uncertainty is found in predicting the pollutant plumes and no economical restrictions are there, then there would be no need for optimal monitoring network design. An efficient and optimally designed monitoring network can save significant amount of long term monitoring cost. Comprehensive review of groundwater monitoring network design can be found in Loaiciga et al. (1992), ASCE Task Committee (2003), and Dhar and Datta (2010). Due to dynamic nature of the pollutant plumes, monitoring network design is a time varying problem. Only a few studies have addressed the issue of designing monitoring networks based on multiple time steps, e.g., Mugunthan and Shoemaker (2004), Herrera and Pinder (2005), Dhar and Datta (2007), Chadalavada and Datta (2008). Uncertainty based monitoring network designs are very limited in literature,e.g., Meyer and Brill (1988), Datta and Dhiman (1996), Mugunthan and Shoemaker (2004), Herrera and Pinder (2005), Zhang et al. (2005), Wu et al. (2006), Dhar and Datta (2007), Chadalavada and Datta (2008), Datta et al. (2009b). Uncertainty can be categorized into two basic types: aleatory uncertainty and epistemic uncertainty. It is impossible to reduce the aleatory uncertainty with higher amount of information. However, reduction is possible in case of epistemic uncertainty. Monitoring network design can be classified as an approach for reducing epistemic uncertainty about the subsurface system. Fuzzy representation of pollutant concentration is one of the approaches of handling the epistemic uncertainty. Fuzzy techniques are widely used in the field of subsurface hydrology to assess the vulnerability of pollution (Ozbek and Pinder 2006). Use of fuzzy set theory with imprecise parameters (Dou et al. 1995), neuro-fuzzy techniques (Dixon 2004; Affandi and Watanabe 2007; Duarte and Rosario 2007; Kholghi and Hosseini 2009), fuzzy logic approach (Muhammetoglu and Yardimci 2005; Afshar et al. 2007; Bisht et al. 2009), fuzzy modeling and fuzzy relation analysis (Qin et al. 2006; Huang et al. 2007) are common for subsurface systems analysis. Classical geostatistical spatial interpolation techniques are helpful for finding out deterministic attribute values (e.g., concentration) at unknown locations based on the available information. Moreover, spatial interpolation techniques with capability of handing the imprecise information are important from monitoring network design point of view. Imprecise information based interpolation techniques are reported in Bardossy et al. (1987), Diamond (1988), Diamond (1989), Bardossy et al. (1990a), Bardossy et al. (1990b), Huang et al. (1998), Bandemer and Gebhardt (2000), Sunila et al. (2004). Multiobjective formulations involving different design objectives for monitoring network design problems are presented in Reed and Minsker (2004), Kollat and Reed (2007), Kollat et al. (2008), Dhar and Datta (2009). These studies focus on the improvement of the objective function(s). However, less attention has been given to individual solutions. The objective of the groundwater monitoring network design is to provide the sufficient information with minimal cost to fulfill the management goals. In reality, feedback information are needed for effective implementation of any operation policy. Compliance monitoring is required to judge effectiveness of the prescribed management strategy. To account for the epistemic uncertainties involved in the design process a fuzzy kriging based groundwater monitoring network design methodology is proposed. This design framework combines

Multiobjective Design of Groundwater Monitoring Network

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groundwater flow and transport simulation in conjunction with imprecise spatial interpolation and optimization model. The fuzzy ordirnary kriging algorithm (Diamond 1989) is implemented as logical external module to the optimization algorithm.

2 Monitoring Network Design Model Groundwater system is complex system due to lack of knowledge about systems parameter (s) which can be termed as epistemic uncertainty. Epistemic uncertainty can be dealt within the framework of interval analysis, fuzzy set theory, possibility theory, evidence theory and imprecise probability theory. In the present work fuzzy set theory is utilized to frame the epistemic uncertainty of groundwater system. 2.1 Fuzzy Representation of Pollutant Concentration Fuzzy number is a fuzzy set with convex membership function, which assumes a maximum value of 1 and minimum value zero. Spatiotemporal (x, t) concentration value can be assumed to be a fuzzy numbers and associated membership function can be represented as triangular fuzzy membership function (Fig. 1). Triangular fuzzy membership function requires specification of lower, modal and upper values. The lower (c), modal (cm) and upper (c) values are defined in terms of 5, 50, and 95 percentile values (A percentile is the value of a variable below which a certain percent of observations fall). A fuzzy number ec (concentration value) is a special fuzzy subset on the set < of real numbers which satisfy the following conditions:

& &

There exists a cðx; tÞ 2 0.5, ≤10

173

10

Unknown

10

> 10, ≤20

48

10

Unknown

9 8

> 20, ≤30 > 30, ≤40

23 9

4 2

Unknown Unknown

7

> 40, ≤50

4

1

1

6

> 50, ≤60

2

1

1

5

> 60, ≤70

3

1

1

4

> 70, ≤80

0

0

0

3

> 80, ≤90

0

0

0

2

> 90, ≤100

0

0

0

1 Total

> 100, ≤110

1 263

1 30

1 30

is representing its spatial correlation structure is an exponential model (Herrera and Pinder 2005):

h ð11Þ g F ðhÞ ¼ σ2F 1 exp lF where, F(x) ln K(x) variance of F(x) σ2F correlation scale. lF Ababou (1988) concluded that for the variable F(x), the effects of local integration over discretization of length Δx can be avoided if λF and the variance of F(x), σ2F satisfy the following relationship: η¼

lF 1 þ σ2F $x

Fig. 4 Pollution zones for monitoring time step: 01

ð12Þ


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In the present work integral scale is assumed to be equal to 15 m (λF). The value of σF is taken as 1. From Eq. 12 it can be shown that Δx should be always less than 7.5 m. Thus our assumption regarding discretization, i.e., Δx05 m is valid. Also, longitudinal dispersivity value is taken as 1 m. The aquifer parameters are given in Table 1. During evaluation process, it is assumed that within planning horizon of 3 years remediation work can be completed. Monitoring time step of 1 year is proposed to evaluate the performance of the remediation work. Sometimes information are needed from all different level of pollution zones to see the effectiveness of remediation strategy. In the present study, restrictions are imposed on selection of wells from different pollution zones (Table 2) and referred to as Type A. Type B restrictions are less rigid than Type A (Table 2). Different zones are shown in Figs. 4, 5 and 6 for three monitoring time steps. To evaluate the performance of the developed methodology four different scenarios (A2, B2, A4, B4) are considered. If the modal concentration in a particular cell is greater than 0.5 mg/l it is defined as a polluted cell. Numbers of polluted cells are found to be equal to 263 (NP1 ) for 1st year, 598 (NP2 ) for 2nd year and 703 (NP3 ) for 3rd year. In all cases 30 (N w ) monitoring wells are selected out of 263 potential locations. The resulting scenarios are solved using NSGA-II (Deb 2001). Semivariograms for different time steps are presented in Table 3. Cell indices (i.e., row, column numbers) are used as coordinates for obtaining the theoretical semivariograms. Thus, lag distance h is actually h/Δx (as Δx0Δy).


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Table 3 Derived theoretical semivariograms for different monitoring time steps Monitoring time step

Semivariogram Lower

Modal

Upper

01

3 Nug(0)+4 Pow(0.55)

15 Nug(0)+48 Pow(0.55)

160 Nug(0)+190 Pow(0.55)

02

1.9 Nug(0)+1.10 Pow(0.55)

10 Nug(0)+18 Pow(0.55)

75 Nug(0)+68 Pow(0.55)

12 Nug(0)+15 Pow(0.55)

54 Nug(0)+55 Pow(0.55)

03

1.5 Nug(0)+1 Pow(0.55) 0 ;h ¼ 0 • g 1 ðhÞ ¼ a1 Nugð0Þ ¼ a1 ; h > 0

• g 2 ðhÞ ¼ a2 Powðb2 Þ ¼ a2 hb2 ; h 0; 0 < b2 2 • gðhÞ ¼ g 1 ðhÞ þ g 2 ðhÞ

3.1 Scenario A2 Scenario-A2 refers to Type-A restriction with 2 objectives. It considers the objectives of minimizing fuzzy mass estimation error at the end of first monitoring time step and maximizing the inter-well distance. Mathematically fuzzy mass is calculated as zeroth moment using fuzzy concentration for all polluted cells (NP1 ) of the aquifer at the end of first monitoring time step. Fuzzy concentrations at all unmonitored polluted location are estimated using fuzzy kriging. Figure 7 shows the nondominated front for Scenario-A2. In order to validate the results two points A2-1 (1.651340, 3661.856179) and A2-3 (2.833587, 4208.897739) are chosen on the final front. In solution A2-1, 1.651340 percent of fuzzy mass estimation error is present, while the total inter-well distance is 3661.856179. If the total inter-well distance is

Fig. 7 Nondominated front for scenario-A2


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Fig. 8 Configuration of monitoring wells for solution A2-3

increased to 4208.897739 (solution A2-3), consequently estimation error will increase to a value of 2.833587. It is evident that for any improvement in one objective, the other has to be sacrificed, as expected from a multiobjective problem with conflicting objectives. A typical well configuration corresponding to A2-3 is shown in Fig. 8. 3.2 Scenario B2 Conditions and objectives for Scenario-B2 are same as Scenario-A2 expect the type of restriction imposed on selection of wells. It considers Type-B restriction. Figure 9 shows the nondominated front for Scenario-B2. It is evident that with less restrictive type-B constraint, scenario-B2 gives less mass estimation compared to scenario-A2. In order to validate the results two points B2-4 (0.378836, 5334.371138) and B2-19 (2.547480, 6035.013738) are chosen on the final front. In solution B2-4, 0.378836 percent of fuzzy mass estimation error

Fig. 9 Nondominated front for scenario-B2

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Fig. 10 Configuration of monitoring wells for solution B2-19

is present, while the total inter-well distance is 5334.371138. If the total inter-well distance is increased to 6035.013738 (solution B2-19), consequently estimation error will increase to a value of 2.547480. A typical well configuration corresponding to B2-19 is shown in Fig. 10. 3.3 Scenario A4 Real optimality of any policy depends on the time horizon considered. Monitoring for single time step (i.e., for shorter time horizon) can result in solutions which are myopic in nature, and not optimal for whole planning horizon. However, it is difficult to prescribe a single monitoring plan for planning horizon, due to transient nature of the pollutant plume. Scenario A4 considers the objectives of minimizing fuzzy mass estimation error at the end of all three monitoring time step and maximizing the total inter-well distance. Scenario-A4 refers to Type-A restriction with 4 objectives. Mathematically, fuzzy mass is calculated as

Fig. 11 Parallel axis plot for scenario-A4


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Fig. 12 Configuration of monitoring wells for solution A4-7

zeroth moment using fuzzy concentration for all polluted cells (NP1 ,NP2 ,NP3 ) of the aquifer at the end of all monitoring time steps. Figure 11 shows the parallel axis plot for resulting objective functions from multiobjective optimization solutions. It consists of 4 different axes denoting different objective function values. Solution presented in Fig. 11 shows mutual conflict of the objective functions by intersecting nature of the lines in between two axes. In order to validate the results two points A4-1 (1.270628, 72.688474, 107.702587, 2594.182377) and A4-7 (7.639051, 64.053881, 87.796822, 3979.667022) are chosen from the final results. In solution A4-1, 1.270628%, 72.688474%, 107.702587% of fuzzy mass estimation error are present for 1st, 2nd and 3rd year, while the total inter-well distance is 2594.182377. If the total inter-well distance is increased to 3979.667022 (solution A4-7), consequently estimation error will increase to 7.639051%, 64.053881%, 87.796822% for three different monitoring time steps. A typical well configuration corresponding to A4-7 is shown in Fig. 12.

Fig. 13 Parallel axis plot for scenario-B4

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Fig. 14 Configuration of monitoring wells for solution B4-14

3.4 Scenario B4 Scenario-B4 is similar to scenario-A4 except the type of restriction. Scenario-B4 refers to TypeB restriction with 4 objectives. Figure 13 shows the parallel axis plot for resulting objective functions from multiobjective optimization solutions. In order to validate the results two points B4-1 (0.369466, 72.474376, 100.544520, 4377.500949) and B4-14 (10.619083, 67.248951, 103.783015, 5193.476565) are chosen from the final results. In solution B4-1, 0.369466%, 72.474376%, 100.544520% of fuzzy mass estimation error are present for 1st, 2nd and 3rd year, while the total inter-well distance is 4377.500949. If the total inter-well distance is increased to 5193.476565 (solution B4-14), consequently estimation error will increase to 10.619083%, 67.248951%, 103.783015% for three different monitoring time steps. A typical well configuration corresponding to B4-14 is shown in Fig. 14.

4 Discussions and Conclusions The primary objective of this study is to develop a methodology for monitoring of contaminated aquifers under epistemic uncertainty. Spatial interpolation is an important component for monitoring network design problem. The developed methodology utilizes fuzzy kriging linked nonlinear formulation for the monitoring network design problem. In this study, the fuzzy kriging linked NSGA-II algorithm is run for a population size of 24 and for 500 generations. Instead of using binary decision variables, real coded version of NSGA-II is used, as this version shows much faster convergence compared to the binary coded one. Moreover, by using real coded version constraint (8) can be avoided. It reduces the number of variables from NP1 to Nw. Different parameter values associated with the optimization algorithm NSGA-II are shown in Table 4. These parameters are varied for testing the sensitivity Table 4 NSGA-II parameters

Parameter

Value

Population size

24

Crossover probability

0.9

Distribution index for crossover Mutation probability

10.0 0.1

Distribution index for mutation

20.0


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of the solutions. But no significant improvement is observed. Linked simulation optimization carries the curse of computational burden with it. Each run, i.e., 24×500 function evaluations took around 2/4 days (varies for different scenarios) in Intel® Core™ 2 Duo CPU E8400 @ 3.00 GHz with 2 GB RAM. Computational complexity dictated the choice of the number of generations. Thus the resulting front may not be the Pareto optimal front or true nondominated front, but a near nondominated one. The objective of monitoring may generally differ based on site specific requirements. The solutions for the design methodology are dependent on the objectives specified. Due to independent nature of objectives, it is useful to analyze the system within multiobjective optimization framework. These limited results and evaluations should help the decision maker to choose the proper solution for design of the monitoring network. Sometime result specifies less than 30 monitoring wells. This may be due to the fact that the configuration is optimal in terms of objective function values. Dhar and Datta (2009) have showed that mass estimation error is not only dependent on the configurations but also on the number of wells. Performance evaluation results of this methodology show substantial improvement in the performance of the monitoring network design in terms of handling uncertainty in complex groundwater system. Multiobjective problem with spatial interpolation constraints results in a nonlinear formulation of the design problem. Nonlinear formulations faces problem with discrete variables, non-convexity, and discontinuity (Dhar and Datta, 2009). They inherit the possibilities of the solutions getting stuck to some local optima. However, these results show potential applicability of the developed methodology. Monitoring network design is essentially a mixed-integer/integer problem. Smaller changes in the concentration values can influence the resulting solutions. Use of percentile values eliminates the chances of deviation. Thus, fuzzy approach is more robust in avoiding sensitivity problem compared to conventional deterministic approach. A methodology for groundwater monitoring network design is developed to the epistemic uncertainty in complex groundwater system. Limited performance evaluations show utility and applicability of the approach. The developed methodology is essentially generic in nature and should be applicable to various types of groundwater contamination problem with minor modifications.

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