Pumping optimization of coastal aquifers using probabilistic search – case study: Quaternary aquifer of El-Arish Rafah, Egypt

The increased pumping of freshwater from coastal aquifers, to meet growing demands, causes an environmental problem called saltwater intrusion. Consequently, proper management schemes are necessary to tackle such a situation and permit the optimal development of coastal groundwater basins. In this research, a probabilistic search algorithm, namely Probabilistic Global Search Lausanne (PGSL), is used to calculate optimal pumping rates of unconfined coastal aquifer. The results of using PGSL are compared with a stochastic search optimization technique, Shuffled Frog Leaping Algorithm (SFLA). The finite element method is applied to simulate the hydraulic response of the steady state homogenous aquifer. The lower and upper (LU) decomposition method is adapted to invert the conductance matrix, which noticeably decreases the computation time. The results of both the PGSL and the SFLA are verified through the application on the aquifer system underlying the City of Miami Beach in the north of Spain. Multiple independent optimization runs are carried out to provide more insightful comparison outcomes. Consequently, a statistical analysis is performed to assess the performance of each algorithm. The two optimization algorithms are then applied on the Quaternary aquifer of El-Arish Rafah area, Egypt. The results show that both algorithms can effectively be used to obtain nearly global solutions compared with other previous published results. This is an Open Access article distributed under the terms of the Creative Commons Attribution Licence (CC BY 4.0), which permits copying, adaptation and redistribution, provided the original work is properly cited (http://creativecommons.org/licenses/by/4.0/). doi: 10.2166/nh.2019.093 om https://iwaponline.com/hr/article-pdf/51/1/90/649629/nh0510090.pdf 0 Hamdy A. El-Ghandour (corresponding author) Irrigation and Hydraulics Department, Faculty of Engineering, Mansoura University, Mansoura 35516, Egypt E-mail: hamdy_elghandour@yahoo.com Emad Elbeltagi Structural Engineering Department, Faculty of Engineering, Mansoura University, Mansoura 35516, Egypt


INTRODUCTION
Groundwater contamination through saltwater intrusion threatens the health of many people living in coastal areas.
Coastal aquifers in arid and semi-arid zones which border the sea are an important source for water. Coastal areas are usually heavily urbanized zones, a fact that makes the need for freshwater even more acute. To meet growing demands, increased freshwater is extracted from these aquifers. Consequently, wells used for freshwater may be contaminated by saltwater intrusion.
In recent decades, stochastic optimization combined with flow simulation has been widely used to control saltwater intrusion and the general management of coastal groundwater.
In most previous coastal groundwater management studies,

OPTIMIZATION OF PUMPING RATES FROM COASTAL AQUIFERS
In this paper, the management strategy is settled to either maximize the economic benefit from the pumped water (Equation (1)) or maximize the total pumping of fresh water (Equation (2)). The pumping rates from a pre-existing system of wells of known locations are considered as the problem decision variables. The management strategy can be stated mathematically for each case as follows (Benhachmi et al. ): in which, f 1 and f 2 are the objective functions of maximum benefit and maximum pumping respectively; Q is the well pumping rate; B is the economic benefit for one cubic meter of fresh water; C is the cost of lifting one cubic meter of water per one meter height; L i is the ground level from a specified datum; h i is the water table from a specified datum; and nw is the number of wells. for freshwater (zone I) for saltwater (zone II) in which h is the vertically averaged piezometric head (measured from the impervious bed of the aquifer, Figure 1); k is the hydraulic conductivity; ε ¼ (ρ s À ρ f )=ρ s ; ρ s and ρ f are the densities of saltwater and freshwater; d is the elevation of mean sea level (M.S.L.) above the impervious bed of the aquifer; Q i is the rate of freshwater pumping from well i located at coordinates (x i , y i ); and δ(z): the Dirac delta function equals one if z equals zero otherwise equals zero.
In this study, for the unconfined aquifer, the following assumptions are considered ( Figure 1): (1) sharp interface between saltwater and freshwater, (2) the three-dimensional geometry flow equation is reduced to two-dimensional by Dupuit's assumption, (3) steady-state flow condition, (4) the Ghyben-Herzberg assumption is utilized to interpret the interface location, (5) wells fully penetrate the aquifer thickness, (6) horizontal impervious aquifer bed, and (7) wells are only within the fresh water zone.
Strack () presented a derivation to transform the nonlinear Equation (3a) and (3b)) to a precise linear form (Equation (4)). Consequently, the dependent variable h has to be replaced with potential ϕ, for both zones I and II (Cheng et al. ).
in which ϕ ¼ (h 2 À sd 2 )=2 for zone I, ϕ ¼ (h À d) 2 =2ε for zone II, s ¼ ρ s =ρ f , and the two zones separated at The set of constraints are ( Figure 1): in which q u is defined as the aquifer uniform rate of discharge per unit width; FE is the unconfined aquifer freshwater seepage face above the M.S.L, calculated as FEM is applied to discretize Equation (4) within the coastal aquifer JILK ( Figure 1). Then, the FEM is transformed to a set of linear equations as follows: in which [Stiff] is the stiffness/conductance square matrix with dimensions of N n ×N n , [ϕ] is the unknown vector of dimension N n ×1, and [F] is the load vector including both pumped or recharged water. Finally, Equation (11) is rearranged to simplify its use (Equation (12)) by calculating the inverse of the [Stiff] matrix using the LU-decomposition method.

PROBABILISTIC GLOBAL SEARCH LAUSANNE ALGORITHM
PGSL is a search algorithm based on sampling of the search space, without using special operators, using a probability distribution function (PDF) defined over the entire search space (Raphael & Smith b). Potential solutions are generated randomly and evaluated using a defined objective. The basic assumption is that good solutions are more likely to be found in the regions with higher probabilities. Hence, the search is concentrated in regions containing good solutions. In the beginning, each variable range (maximum valueminimum value) is divided into a number of intervals and a uniform probability is assumed within each interval. As the search progresses, intervals containing good solutions are assigned higher probability values and the search space along the variable is narrowed. As such, the search space is gradually narrowed down until convergence is achieved.
The algorithm includes four nested cycles named Sampling, Probability updating, Focusing, and Sub-domain. The PGSL involves the following steps: 1. Initialization: The PDF of each decision variable is assumed having a uniform distribution over the entire domain. 3. Determination of the global best frog position (X G ).

Population is partitioned into subsets named memeplexes
representing different cultures of frogs. Then, the following steps are carried out for each memeplex: • Determine both the best and worst frog positions (i.e. X B and X W ).
•  Table 1 shows the coordinates of the aquifer pre-existing wells, their allowable maximum and minimum pumping rates, and their corresponding ground level.

Scenario 1: maximizing the economic benefit
The management strategy is settled to maximize the economic benefits from the pumped water and to minimize the cost corresponding to lifting the water, Equation (1). The values B and C shown in Equation (1) (8) and (9) using the developed FEM simulation method, it was found that seawater had intruded into several wells (Table 4). This may be due to the different method they adopted (the theory of images) to simulate the hydraulic response in the aquifer or due to the incorrect number of mirrors used to represent the theory of images. However, the two developed models also presented a good solution for the total pumping with the SFLA outperforming the PGSL.
As such, both the PGSL and SFLA optimization algorithms are able to reach greater solutions without violating the constraints when compared to previous studies.

COMPARISON OF THE TWO OPTIMIZATION ALGORITHMS' PERFORMANCE
The methodology adopted in the comparison of the two optimization algorithms' performance begins by performing 20 consecutive trial runs corresponding to each one. Then, a comparison is carried out between both based on a set of criteria, namely: (1) the success rate, The first value corresponds to the first 13 wells while the second value is for the remaining wells.   Table 5.
From this table it is seen that the SFLA has the highest success rate, for the two scenarios of a value equal to 90% for scenario 1. This means that the SFLA is able to catch an optimal solution for the economic benefit greater than the corresponding maxi-      areas where expected returned irrigated water can recharge the aquifer. Calibrated recharge values reflect recharge due to rainfall, returned water from different water uses and possible unknown recharge from other underlying formations are not considered. As shown in Figure 6, the recharge has a maximum value of 3 × 10 -3 m/d in the north-east of the study area, while it equals 1.2 × 10 -7 m/d in the southern boundary (Abdallah ). Table 6 shows the coordinates of the different pre-existing wells in the aquifer, their allowable maximum and minimum pumping rates, and their corresponding ground level.
In this application, the parameters setting for both the SFLA and the PGSL are summarized in Table 2. After performing several trial runs for each algorithm, the largest entering the aquifer from rainfall. This is due to the non-uniform distribution of wells over the studied domain (the existing wells are not distributed over the whole studied area) and also the number of working wells located in low rainfall recharge zones ( Figure 6). The corresponding total numbers of evaluations of the objective function, Equation (1), are 113,133 and 2,000 for the SFLA and PGSL, respectively. From the results obtained, the PGSL outperforms the SFLA and, consequently, the PGSL is more efficient for large scale problems, where the best results are obtained in a smaller runtime of about 35 minutes per one run. Table 6 lists a comparison between well discharges, in each well system, corresponding to the largest total pumping of the two algorithms. Using the SFLA optimization algorithm, the total pumping is obtained from only nine wells out of the 31 wells while 19 worked wells were determined by using the PGSL optimization algorithm. Figure 7 shows the maps of the water levels contours (measured from the impervious bed) corresponding to each well system listed in Table 6 Both algorithms are then applied on the Quaternary aquifer of El-Arish Rafah, Egypt. The results obtained indicate that the performance of the SFLA is better than the PGSL in terms of the best results obtained. However, the PGSL, which is a relatively new optimization technique, outperformed other traditional stochastic optimization techniques (e.g. the GA). Also, the run time for the PGSL was significantly smaller than the SFLA. The two models can effectively and efficiently be used to solve real groundwater management problems. From the studied aquifers, it is noted that the PGSL does not give the optimum solution but the obtained results are close to the optimum. As such, the PGSL would be a suitable tool in solving large scale problems (i.e. real-life aquifers) to give an initial orientation, not a final decision.