PTC simulations, stochastic optimization and safety strategies for groundwater pumping management: case study of the Hersonissos Coastal Aquifer in Crete
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Abstract
Recently, the wellknown Princeton Transport Code (PTC), a groundwater flow and contaminant transport simulator, has been coupled with the ALgorithm of Pattern EXtraction (ALOPEX), a realtime stochastic optimization method, to provide a freshwater pumping management tool for coastal aquifers, aiming in preventing saltwater intrusion. In our previous work (Proceedings of INASE/CSCCWHH 2015, Recent Advances in Environmental and Earth Sciences and Economics, pp 329–334, 2015), the PTCALOPEX approach was used in studying the saltwater contamination problem for the coastal aquifer at Hersonissos, Crete. Extending these results, in the present study the PTCALOPEX approach is equipped with a nodal safety strategy that effectively controls saltwater front’s advancement inside the aquifer. In cooperation with an appropriate penalty system, the performance of PTCALOPEX algorithm is studied considering several pumping and weather condition scenarios. This study also establishes pumping/well scenarios that ensure the needed volume of fresh water to the local community without risking saltwater contamination.
Keywords
ALOPEX stochastic optimization PTC code Coastal aquifers Saltwater intrusion Pumping managementIntroduction
Saltwater Intrusion (SWI) in coastal aquifers poses a significant threat to the quality of groundwater reserves. SWI is mainly attributed to unrestrained freshwater withdrawals that disturb the natural freshwater–saltwater equilibrium in groundwater coastal systems. To protect groundwater reserves and design sustainable water management strategies for coastal aquifers, researchers have been intensively studying SWI via the combined use of mathematical models, numerical simulations and optimization algorithms (see for example Alimohammadi and AfsharUnit 2005; Ospina et al. 2006; Stratis et al. 2015; Strack 1989; Karterakis et al. 2007; Mantoglou 2003).
It is well known that the nonlinear behavior of unconfined coastal aquifers, pertaining to the response of the hydraulic head to the pumping conditions, results in nonlinear optimization. To circumvent this problem, earlier optimization formulations were based on modified versions of linear (Shamir et al. 1984; Uddameri and Kuchanur 2007 among others) or nonlinear programming (Gorelick et al. 1984; Willis and Finney 1988; Reinelt 2005; Mantoglou et al. 2004). In the last two decades, there has been an increase in developing and applying heuristic and evolutionary algorithms for solving waterresources management problems, including SWI. The advantage of using these algorithms is that they can handle non differentiable and discontinuous functions, as well as integer variables, but at the expense of increased computational effort (Yeh 2015). Heuristic and evolutionary algorithms have been used with success in various implementations that deal with the SWI management problem (Cai et al. 2001; Karterakis et al. 2007; Mantoglou and Papantoniou 2008; Dhar and Datta 2009; Kourakos and Mantoglou 2011; Karatzas and Dokou 2015). Detailed information regarding simulation–optimization models and their application to coastal groundwater optimization may be found in the recent reviews of Sreekanth and Datta (2015) and Ketabchi and AtaieAshtiani (2015).
SWI is being modeled by two main simulation approaches: the Sharp Interface and Dispersive Interface (densitydependent) simulation models. Both approaches have distinct advantages and disadvantages in the context of developing simulation–optimization models for coastal aquifer management (Sreekanth and Datta 2015). For a detailed analysis of the theoretical developments and implementation of these two modeling approaches the interested reader is directed to Werner et al. (2013). In this work, the sharp interface approach was used. Thus, it is expected that the validity of the solutions obtained here are subject to the conditions imposed by the assumptions of the sharp interface combined with the Ghyben–Herzberg relationship. In general, the sharp interface model is suitable when the transition zone is narrow relative to the scale of the problem. This can be affected by many parameters. According to LlopisAlbert and PulidoVelazquez (2014) the transition zone width remains relatively narrow for lower values of dispersion (longitudinal and transverse dispersivity) and hydrogeological parameters (transmissivity or horizontal hydraulic conductivity) and higher values of vertical hydraulic conductivity and recharge.
The objective of this work is to assess SWI intrusion for the Hersonissos aquifer, located in Crete, Greece, and provide, in the sequel, sustainable management alternatives. The SWI phenomenon at this aquifer has been studied previously by a number of researchers. In Papadopoulou et al. (2009) (see also Papadopoulou et al. 2010), the finite difference MODFLOW and the finite element PTC models are employed to simulate saltwater intrusion and compare the numerical results to the ones obtained by geostatistical techniques (Kriging). In Karterakis et al. (2007), the PTC simulator is coupled with a differential evolution (DE) algorithm to maximize the total extracted freshwater volume from five preselected pumping locations (production wells) while satisfying minimum hydraulic head constraints at specified locations, ensuring no further intrusion of seawater. The same approach was saken in Dokou and Karatzas (2012), using sequential linearization to reduce the computational cost. The Hersonissos aquifer has been also studied in Voudouris et al. (2004), by making use of geostatistical techniques (Kriging and Ordinary Kriging).
ALOPEX stochastic unconstrained optimization originates at the area of neurophysiology (cf. Harth and Tzanakou 1974) and, since then, has been applied with success in many realtime applications (see for example Zakynthinaki and Saridakis (2003), and the references therein). Recently, in Stratis et al. (2015), the dynamics of the algorithm were studied in depth for the problem of saltwater intrusion of coastal aquifers. The determination of the algorithm’s feedback and noise amplitudes and the introduction of an effective penalty system, to enforce problem’s constraints, revealed its potential to successfully treat the problem of pumping management in coastal aquifers.
The approach employed in our previous work (Stratis et al. 2015) was to combine the groundwater simulation model PTC with the newly introduced constrained version of the ALOPEX stochastic optimization technique. The objective was to maximize groundwater withdrawal in the existing pumping well network while avoiding saltwater to enter a safe zone around the active wells in the region.

The adaptation of an appropriate penalty system, which, in cooperation with the unconstrained version of ALOPEX used here (ALOPEX V), deals with the problem of controlling the advancement of saltwater front inside the aquifer,

The creation of a safety net, namely a polygonal grid of guarding nodes, delimiting a salinization free safety area around each active pumping location,

Several pumping scenarios, with different positions of active pumping locations and different recharging values from natural resources (rain, nearby rivers, etc.),

The study of presenting solutions able to resolve the local community’s current needs of freshwater pumping volume, by suggesting new pumping active locations.
Methodology
Groundwater simulation model: PTC
PTC employs a hybrid splitting algorithm for solving the fully threedimensional system. The domain is discretized into approximately parallel horizontal layers, within each of which a finite element discretization is employed allowing accurate representation of irregular domains. The vertical connection of the layers is represented by finite differences. This hybrid finite element and finite difference coupling provides the opportunity to divide the computations into two steps during a given time iteration (splitting algorithm). At the first step, all horizontal equations are solved while at the second step, the vertical equations which connect the layers are solved (cf. Babu et al. 1997). This model has been successfully used in several previous studies (e.g., Aivalioti and Karatzas 2006; Koukadaki et al. 2007; Dokou and Pinder 2011).
Sharp interface approach
Study area and numerical model development
Regarding the saltwater intrusion toe location, according to the Ghyben–Herzberg relationship, \(h_f\) was estimated at \(2.5\ \mathrm{m}\), given that the depth of the studied aquifer is about \(100\ \mathrm{m}\) (based on boring log information). Thus, the contour of \(100+2.5=102.5\ \mathrm{m}\) represents the hydraulic head isolevel limit below which a zone is considered intruded by saltwater.
Pumping management
Safety strategy
We remark that the above value, although in our present implementation is kept uniform for all active wells, can be set differently for every active pumping location, accordingly to a local protection strategy.
Constrained ALOPEX for pumping management

In both penalty phases, the values of those \(Q_i\) needed to be rectified, are being, ultimately, reduced or increased by a percentage, which in our implementation (as in Stratis et al. 2015) has been set to \(5\,\%\). This is a \(5\,\%\) policy, applied by the parameter \(\delta =0.05\).
 In phase one, the current pumping rate \(Q_i^{(k)}\) of the \(i{\mathrm{th}}\) well, obtained in (6), is being rectified in case it violates the local maximum, the local minimum or the global maximum pumping rate constraints. Namely, in case \(Q_i^{(k)}>\overline{Q}_i\ \text{ or }\ Q_i^{(k)}<\underline{Q}_i\), the value of \(Q_i^{(k)}\) is being rectified byfor \(i=1,\ldots ,M\). And furthermore, having defined the quantities$$\begin{aligned} Q_i^{(k)} = (1\delta )\overline{Q}_i\ \text{ or }\ Q_i^{(k)} = (1+\delta ) \underline{Q}_i \end{aligned}$$(7)with \(S_0^{(k)}=S(Q_1^{(k1)},\ldots ,Q_{M}^{(k1)})\), if \(\tilde{S}>0\), then the value of \(Q_i^{(k)}\) is being further rectified by$$\begin{aligned} S_{i1}^{(k)}&:=\sum _{j=1}^{i1} Q_j^{(k)} + \sum _{j=i}^M Q_j^{(k1)}\quad \text{ and }\nonumber \\ \tilde{S}&:=S_{i1}^{(k)}+\Delta Q_i^{(k)}  \overline{Q}_A \end{aligned}$$(8)for \(i=1,\ldots ,M\).$$\begin{aligned} Q_i^{(k)} = Q_i^{(k)}  (1+\delta ) \tilde{S} \end{aligned}$$(9)
 In phase two, the enforcement of the toeconstraint is being achieved in two cycles. In cycle one, only the pumping rates of the active wells at risk, meaning \(x_{\tau ,i} > x_id_s\), are being rectified by the penalty equationfor \(i=1,\ldots ,M\). Then, in cycle two, if the previous local rectification did not manage to protect the wells in danger, a global rectification is applied during which the penalty presented in equation (10) is being enforced on all aquifer wells.$$\begin{aligned} Q_i^{(k)} = (1\delta ) Q_i^{(k)}, \end{aligned}$$(10)
 In order to create an effective stopping criterion for the optimization procedure, we use a combination of the standard deviation \(\sigma _{\hat{k}}\) of the objective function’s values in the last \(\hat{k}\) iterations (in the present work \(\hat{k}=20\)) and the differential of the objective function’s mean value \(\mu _{\hat{k}}\), in a window of the last \(2\hat{k}\) iterations. We define the windowed mean value \(\mu _{\hat{k}}\) and standard deviation \(\sigma _{\hat{k}}\) aswhere k is the current iteration, with \(k2\hat{k}>0\). Then, assuming that no constraint is being violated, the satisfaction of the following criterion$$\begin{aligned} \mu _{\hat{k}}&=\frac{1}{\hat{k}} \sum _{i=k\hat{k}}^{k} P(\mathbf Q ^{(i)})\quad \text{ and }\nonumber \\ \sigma _{\hat{k}}&=\sqrt{\frac{1}{\hat{k}} \sum _{i=k\hat{k}}^{k}({P(\mathbf Q ^{(i)})\mu _{\hat{k}}})^2} \end{aligned}$$(11)for positive small tolerances \(\varepsilon _1\) and \(\varepsilon _2\) (in the present work both set equal to \(10^{2}\)), effectively terminates the optimization procedure.$$\begin{aligned} \left( \sigma _{\hat{k}}< \varepsilon _1 \right) \wedge \left( {\mid } \mu _{\hat{k}}  \mu _{2\hat{k}} {\mid }< \varepsilon _2\right) \end{aligned}$$(12)
Numerical simulations
In this section we include the results from characteristic numerical simulations performed for the freshwater aquifer located at the Municipality of Hersonissos, \(25\ \mathrm{km}\) east of Heraklion, at the Greek island of Crete.

Dry Case \(N=0\ \mathrm{mm}/\mathrm{year}\), while the sideways (subsurface) recharge, characterized by the Neumann boundary conditions, is as defined in Fig. 2,

Wet Case \(N=500\ \mathrm{mm}/\mathrm{year}\) with percentage of infiltration set at \(30\,\%\), while the sideways recharge is as described in Fig. 2 increased by \(20\,\%\).
Pumping Capabilities (\(m^3/days\)) of Active Locations
i  1  2  3  4  5 

\(\overline{Q}_i\)  1800.00  2520.00  576.00  2520.00  146.00 
\(\underline{Q}_i\)  540.00  756.00  172.80  756.00  43.80 
In Table 1 we have included the maximum \(\overline{Q}_i\) pumping capabilities of all five active locations, while the corresponding minimum ones have been set to satisfy \(\underline{Q}_i=0.3*\overline{Q}_i\). Finally, we note that in all numerical simulations the pumping rates \(Q_i,\ i=1,\ldots ,M\) are considered to be numbered in a toptobottom fashion, namely \(y_1 \ge \ldots \ge y_M\).
Profile DRY_5: dry case, five (5) active pumping locations
ALOPEX/PTC performance: Profile DRY_5
Problem parameters  Total optimal values  Stopping criterion optimal values 

k (\(\#\) iter.)  404  79 
\(P(\mathbf Q ^{(k)})\)  0.9756  0.97153 
\(Q_1^{(k)}\)  1787.02  1735.11 
\(Q_2^{(k)}\)  1368.66  1340.61 
\(Q_3^{(k)}\)  560.20  561.66 
\(Q_4^{(k)}\)  2519.43  2515.91 
\(Q_5^{(k)}\)  138.09  123.45 
\(S(\mathbf Q ^{(k)})\)  6373.40  6276.74 
\(t_k\)  378.5125  74.0161 
Profile DRY_4: dry case, four (4) active pumping locations
ALOPEX/PTC performance: Profile DRY_4
Problem parameters  Total optimal values  Stopping criterion optimal values 

k (\(\#\) iter.)  331  64 
\(P(\mathbf Q ^{(k)})\)  0.96042  0.95591 
\(Q_1^{(k)}\)  0.00  0.00 
\(Q_2^{(k)}\)  1392.37  1395.91 
\(Q_3^{(k)}\)  570.67  547.20 
\(Q_4^{(k)}\)  2513.18  2456.94 
\(Q_5^{(k)}\)  127.87  138.46 
\(S(\mathbf{Q ^{(k)}})\)  4604.09  4538.51 
\(t_k\)  304.6019  58.8958 
Indeed one may readily verify that, omitting the values associated to well No. 1 from Table 2, the rest of the reported results remain practically identical to the results included in Table 3. Therefore, the presence or absence of well No. 1 from the optimization process does not affect the optimal values of the rest of the active locations. Nevertheless, well No. 1 is being kept active in our simulations for literature compatibility reasons.
Profile WET_5: wet case, five (5) active pumping locations
ALOPEX/PTC performance: Profile WET_5
Problem parameters  Total optimal values  Stopping criterion optimal values 

v (\(\#\) iter.)  198  63 
\(P(\mathbf Q ^{(k)})\)  0.99999  0.99654 
\(Q_1^{(k)}\)  1777.16  1518.21 
\(Q_2^{(k)}\)  2514.42  2426.24 
\(Q_3^{(k)}\)  545.27  545.98 
\(Q_4^{(k)}\)  2511.35  2502.93 
\(Q_5^{(k)}\)  136.80  123.41 
\(S(\mathbf Q ^{(k)})\)  7485.00  7116.77 
\(t_k\)  178.7071  56.8614 
Indeed, as both Fig. 6a, b suggest, all pumping locations, except naturally the top one, remain safe from saltwater intrusion, since the saltwater interface (contour line at \(h=102.5\) m) is not even close to the safety distance of \(d_s\)=180 m from all protected active locations. ALOPEX drives in less than 50 iterations (see Fig. 6c) the objective function to its global maximum and remains close to it, for the rest of the process, within very small amplitude fluctuations. Similarly, inspecting Fig. 6d, it can be noticed that all control variables \(Q_i\) reach within a few iterations maximum performance.
Profile DRY_6: dry case, six (6) active pumping locations
ALOPEX/PTC performance: Profile DRY_6A
Problem parameters  Total optimal values  Stopping criterion optimal values 

k (\(\#\) iter.)  189  58 
\(P(\mathbf Q ^{(k)})\)  0.96473  0.9416 
\(Q_1^{(k)}\)  1710.00  1790.87 
\(Q_2^{(k)}\)  2499.29  2111.83 
\(Q_3^{(k)}\)  1099.91  1016.55 
\(Q_4^{(k)}\)  227.50  286.30 
\(Q_5^{(k)}\)  2504.28  2247.74 
\(Q_6^{(k)}\)  114.34  140.54 
\(S(\mathbf Q ^{(k)})\)  8155.32  7593.83 
\(t_k\)  337.5716  103.5934 
ALOPEX/PTC performance: Profile DRY_6B
Problem parameters  Total optimal values  Stopping criterion optimal values 

k (\(\#\) iter.)  256  35 
\(P(\mathbf Q ^{(k)})\)  0.93066  0.91743 
\(Q_1^{(k)}\)  1710.00  1710.00 
\(Q_2^{(k)}\)  1263.83  1303.97 
\(Q_3^{(k)}\)  181.44  194.16 
\(Q_4^{(k)}\)  1728.71  1805.58 
\(Q_5^{(k)}\)  87.26  81.42 
\(Q_6^{(k)}\)  2393.37  2012.97 
\(S(\mathbf Q ^{(k)})\)  7364.61  7108.10 
\(t_k\)  500.9508  68.4894 
These new positions are marked by Well 2 and Well 6 on Fig. 7a, b, respectively, and their corresponding maximum and minimum pumping rates were set to \((\overline{Q}_2,\underline{Q}_2)=(2500,750)\ \mathrm{m}^3/\mathrm{day}\) and \((\overline{Q}_6,\underline{Q}_6)=(2500,750)\ \mathrm{m}^3/\mathrm{day}\). All simulation results obtained for this case are summarized in Tables 5, 6 and in Fig. 7.
Inspecting, now, Tables 5 and 6, it can be readily verified that both case scenarios resulted in significantly increased total volume of optimal pumping rates. The differences observed between the total volume of the optimal pumping rates, as well as the distribution of the pumping volume among the active locations in both case scenarios, suggest that the location of the extra artificial well significantly affects the final optimal pumping setup.
In conclusion, the above simulation results suggest that the Hersonissos aquifer might be able to deliver increased pumping volume capability by following the described strategy. Further investigation in this direction is required.
Finally, we report that all experiments were conducted on an Intel quadcore i7, 3.4 Mhz PC, with 8 Gb DDR3 RAM, using PTC and MATLAB environments.
Conclusion
In this study, we combine the PTC groundwater simulation model with the newly introduced constrained version of the ALOPEX stochastic optimization technique, in an attempt to maximize groundwater withdrawal in the existing pumping well network without risking saltwater contamination of the active pumping locations. For this, we introduce a nodal safety strategy that controls saltwater front’s advancement inside the aquifer and prohibits saltwater intruding a safety zone around the active pumping locations in the region. The results, for both dry and wet case scenarios considered, revealed that the ALOPEX stochastic optimization method cooperates effectively with PTC, proving a new simple and effective tool for studying saltwater intrusion in coastal aquifers. We strongly believe that the reported results justify and encourage further investigation on the performance of the method.
Notes
Acknowledgments
The present research work has been cofinanced by the European Union (European Social Fund ESF) and Greek national funds through the Operational Program Education and Lifelong Learning of the National Strategic Reference Framework (NSRF)—Research Funding Program: THALES (Grant Number: MIS379416). Investing in knowledge society through the European Social Fund.
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