Elsevier

Journal of Hydrology

Volume 361, Issues 1–2, 30 October 2008, Pages 52-63
Journal of Hydrology

Optimal design of pumping networks in coastal aquifers using sharp interface models

https://doi.org/10.1016/j.jhydrol.2008.07.022Get rights and content

Summary

A method for optimal design of pumping networks in coastal aquifers based on genetic algorithms and numerical solution of the governing differential equations of freshwater flow, is developed. The objective is to optimize the well locations subject to constraints that protect the aquifer from saltwater intrusion. Two are developed. The first is based on simultaneous optimization of well locations and pumping rates using genetic algorithms (GA). The second method follows two stages where the first stage concerns optimization of well locations and the second stage concerns optimization of pumping rates. The last method uses GA for searching the well positions and at each generation of GA, a simpler optimization with respect to the pumping rates is performed using nonlinear programming. The formulation of the constraints is based on numerical simulation of freshwater flow equations obtained from the sharp interface and the Ghyben–Herzberg approximations and the single potential formulation of Strack. The optimization methodologies are applied effectively for determining the optimal design of a pumping network and assessment of pumping rates in an island coastal aquifer.

Introduction

Spatial and time distribution of freshwater often does not agree with the demand. Coastal regions and islands constitute unique cases where the water needs during the summer months are especially high due to tourism while freshwater resources are limited due to droughts. In order to meet the increasing needs for freshwater, coastal aquifers are intensively pumped causing saltwater intrusion. Therefore it is important to develop appropriate management models for optimal design of pumping networks and for assessing the maximum pumping rates in coastal aquifers while protecting them from saltwater intrusion.

The problem of pumping optimization using aquifer simulation models has been extensively investigated in the literature (see, e.g. Mayer et al. (2002)). Gorelick, 1983, Ahlfeld and Heidari, 1994 present a management approach based on groundwater simulation models and optimization methods. Many different objective functions and sets of constraints have been proposed depending on the problem. Shamir et al., 1984, Hallaji and Yazicigil, 1996, Cheng et al., 2000, Mantoglou, 2003, Mantoglou et al., 2004 aim at maximization of the total pumping rate while Das and Datta (1999a) aim at the minimization of the salinity of pumped water. Emch and Yeh (1998) include the pumping cost in the objective function. It is also possible to consider multiple objectives, constituting a multiobjective optimization problem (Shamir et al., 1984, Emch and Yeh, 1998, Das and Datta, 1999b).

Often the purpose of optimization is to maximize the total pumping rate from a number of wells while controlling the saltwater intrusion into the aquifer. The constraints restrain the pumping rate between a minimum and a maximum value (e.g. Hallaji and Yazicigil, 1996, Emch and Yeh, 1998, Das and Datta, 1999a, Das and Datta, 1999b, Cheng et al., 2000). Constraints imposed by Cheng et al., 2000, Mantoglou, 2003 control the location of the toe. Additional constraints may include maintaining water levels, flow potential or salt concentration of the pumped water at desired levels.

The management methods require the use of aquifer simulation models. A common approach for simulation of flow in coastal aquifers is based on the sharp interface approximation and the Ghyben–Herzberg relation (Bear, 1979). Emch and Yeh, 1998, Cheng and Ouazar, 1999, Cheng et al., 2000, Mantoglou, 2003, Mantoglou et al., 2004 presented coastal aquifer management models based on the sharp interface approximation. Strack’s (1976) flow potential is used in order to trace the toe of saltwater lens (Cheng and Ouazar, 1999, Mantoglou, 2003, Mantoglou et al., 2004). More complex seawater intrusion models consider the transport processes that occur in the mixing zone (Das and Datta, 1999a, Das and Datta, 1999b). The differential equations of flow are solved analytically or numerically in order to calculate the aquifer response for various pumping scenarios.

Most research articles in the groundwater management literature focus on pumping optimization for known well locations. Gorelick et al., 1984, Wang and Ahlfeld, 1994 examine optimal design of pumping networks. The optimization method employed depends on whether the optimization formulation is linear (Ahlfeld and Heidari, 1994, Hallaji and Yazicigil, 1996, Mantoglou, 2003, Mantoglou et al., 2004) or nonlinear (Gorelick et al., 1984, Shamir et al., 1984, Wang and Ahlfeld, 1994, Hallaji and Yazicigil, 1996, Emch and Yeh, 1998, Mantoglou, 2003, Mantoglou et al., 2004). The selected method also depends on the continuity of the objective function, the existence of derivatives of the objective function, the existence of local minimums, etc. Park and Aral (2004) examined the problem of optimization of well locations and pumping rates in coastal aquifers using the analytical solutions of Strack, 1976, Cheng et al., 2000. However, these analytical solutions are applicable only in aquifers of semi-infinite dimensions. Mantoglou (2003) developed analytical solutions based on the method of images that are better suited for orthogonal aquifers of finite dimensions.

This paper presents a methodology for optimal design of pumping networks and assessment of pumping rates in coastal aquifers using genetic algorithms and numerical simulation governing equations of flow. The modeling methodology is based on the sharp interface approximation and the Ghyben–Herzberg relation. The flow equations are expressed using the Strack (1976) potential. The governing flow equations are solved numerically using finite differences which overcomes the semi-infinite aquifer geometry limitations required by the analytical solutions of Cheng et al., 2000, Park and Aral, 2004 and the orthogonal geometry limitations of Mantoglou (2003) The management model aims at optimization of the well locations and maximization of the total pumping rate subject to constraints that protect the aquifer from saltwater intrusion. The governing differential equations are solved using finite difference method (MODFLOW). Two optimization methods are developed. The first is based on simultaneous optimization of well locations and pumping rates using genetic algorithms (GA). The second method combines genetic algorithms and nonlinear programming (Sequential Quadratic Programming – SQP) in a two stage optimization. The well position optimization is carried out using GA while at each generation of GA a simpler optimization with respect to the pumping rates is performed using SQP.

Genetic algorithms are stochastic search methods that mimic the metaphor of natural biological evolution (see Holland, 1975, Goldberg, 1989, Fogel, 1994). They operate on a population of potential solutions applying the principle of survival of the fittest to produce improved approximations to a solution. A new set of approximations is created at each generation by selecting individuals according to their fitness and breeding them together using operators borrowed from natural genetics. This process leads to the evolution of populations of individuals that are better suited to their environment than their parents, just as in natural adaptation.

Genetic algorithms differ substantially from more traditional search and optimization methods. The most significant difference is that GA search a population of possible solutions in parallel rather than a single solution. Additionally, GA do not require calculation of derivatives of the objective function. Therefore genetic algorithms can solve general classes of optimization problems even when the objective function is not continuous and the derivatives of the objective function do not exist. Optimization based on genetic algorithms is very flexible and, if the number of generations is large, it converges to the global rather than local minimum.

The proposed optimization methodology is applied to a coastal unconfined aquifer in the Greek island of Kalymnos for optimal design of a pumping network and for assessing the pumping rates while protecting the wells from saltwater intrusion.

Section snippets

Governing equations of flow and saltwater intrusion in coastal aquifers

Mantoglou et al. (2004) proposed a numerical aquifer model based on the sharp interface approximations and the flow potential of Strack (1976), assuming that the salt–fresh water interface encounters the aquifer base at a toe (Fig. 1). This requires that the aquifer is of large extend compared to its depth. The model is based on the sharp interface approximation which is appropriate in regional scale problems where the transition zone is narrow relative to the scale of the problem. In the

Simultaneous optimization of well locations and pumping rates using genetic algorithms

In the first method, the optimization is performed in one stage where the well positions are optimized simultaneously with the pumping rates. Let k be the number of wells pumping the coastal aquifer with pumping rates Qi; i = 1,  ,k. Let (xwi, ywi); i = 1,  ,k represent the coordinates of the wells, where xwi is the distance from the coast and ywi is the distance from the south boundary. When the well locations are not known, the objective is to determine the optimal location of the wells that yield

Two stage optimization of well locations and pumping rates

The second method of optimization follows two stages where the first stage concerns optimization of the location of wells and the second stage concerns optimization of pumping rates. The first stage uses genetic algorithms and the second stage uses nonlinear programming methods (Sequential Quadratic Programming) as follows.

The first optimization stage concerning optimization of well locations is expressed asmaximizex:Qtot=i=1kQi(x)where the decision variables in (6) include the well

Application to an aquifer of orthogonal shape

An aquifer having a simple rectangular geometry (Fig. 4) is selected in order to test and compare the two optimization methods. The recharge rate at the higher elevations of the aquifer is N = 150 mm/year distributed over an area of A = 9 km2 while the recharge rate at the lower elevations of the aquifer is N = 30 mm/day. The sea boundary constitutes a constant head boundary whereas the north and south boundaries are assumed impermeable. The aquifer thickness from the base to the sea surface is taken as

Application to coastal aquifer in a Greek island

The simulation and optimization methodology developed above is applied to an unconfined aquifer located at Vathi valley in the Greek island of Kalymnos (Fig. 7). As discussed in Mantoglou et al. (2004), the aquifer consists of high permeability limestone which outcrops at the highland areas while the impermeable bottom of the aquifer is composed by schist. The geologic formations found in the valley are highly permeable scree, medium permeability alluvium deposits and almost impermeable tuff

Summary and conclusions

A method for optimal design of pumping networks in coastal aquifers was developed and utilized in a coastal aquifer in the Greek island of Kalymnos. The objective is to optimize the well locations and to maximize the total pumping rates subject to constraints that protect the aquifer from saltwater intrusion.

The method is based on genetic algorithms and nonlinear optimization subject to constraints that limit the saltwater intrusion into the aquifer. The simulation model is based on the sharp

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