COASTAL AQUIFER MANAGEMENT: monitoring, modeling, and case studies - Chapter 11 potx

Inappropriate management of coastal aquifers may lead to the intrusion of saltwater into freshwater wells, destroying them as sources of freshwater supply.. One of the goals of coastal a

CHAPTER 11

Pumping Optimization in Saltwater-Intruded Aquifers

A.H.-D Cheng, M.K Benhachmi, D Halhal, D Ouazar, A Naji,

K EL Harrouni

1 INTRODUCTION

Coastal aquifers serve as major sources for freshwater supply in many countries around the world, especially in arid and semiarid zones Many coastal areas are heavily urbanized, a fact that makes the need for freshwater even more acute [Bear and Cheng, 1999] Inappropriate management of coastal aquifers may lead to the intrusion of saltwater into freshwater wells, destroying them as sources of freshwater supply One of the goals of coastal aquifer management is to maximize freshwater extraction without causing the invasion of saltwater into the wells

A number of management questions can be asked in such considerations For existing wells, how should the pumping rate be apportioned and regulated so as to achieve the maximum total extraction? For new wells, where should they be located and how much can they pump? How can recharge wells and canals be used to protect pumping wells, and where should they be placed? If recycled water is used in the injection, how can we maximize the recovery percentage? These and other questions may

be answered using the mathematical tool of optimization

Efforts to improve the management of groundwater systems by computer simulation and optimization techniques began in the early 1970s [Young and Bredehoe, 1972; Aguado and Remson, 1974] Since that time, a large number of groundwater management models have been successfully applied; see for example Gorelick [1983], Willis and Yeh [1987], and many

other papers published in the Journal of Water Resources Planning and Management, ASCE, and the Water Resources Research Applications of

these models to aquifer situations with the explicit threat of saltwater intrusion in mind, however, are relatively few [Cumming, 1971; Cummings

and McFarland, 1974; Shamir et al., 1984; Willis and Finney, 1988; Finney

et al., 1992; Hallaji and Yazicigil, 1996; Emch and Yeh, 1998; Nishikawa,

1998; Das and Datta, 1999a, 1999b; Cheng et al., 2000] In terms of

management objectives, some of these studies have addressed relatively complex settings such as mixed use of surface and subsurface water in terms

of quantity and quality, water conveyance, distribution network, construction and utility costs, etc However, saltwater intrusion into wells has been dealt with in simpler and indirect approaches, for example, by constraining drawdown or water quality at a number of control points, or by minimizing the overall intruded saltwater volume in the entire aquifer The explicit modeling of saltwater encroachment into individual wells resulting in the

removal of invaded wells from service is found only in Cheng et al [2000]

This chapter reviews some of the earlier considerations of pumping optimization in saltwater-intruded aquifers under deterministic conditions, and furthermore, introduces the uncertainty factor into the management problem The resultant methodology is applied to the case study of the City

of Miami Beach in the northeast Spain

2 DETERMINISTIC SIMULATION MODEL

The first step of modeling is to have a physical/mathematical model Depending on the available data input from the field problem and the desirable outcome of the simulation, models of different levels of complexity, ranging from the sharp-interface model to the density-dependent miscible transport model, can be used [Bear, 1999] For the method of solution, it can range from simple analytical solutions [Cheng and Ouazar, 1999] to the various finite-element- and finite-difference-based numerical solutions [Sorek and Pinder, 1999] In principle, any of the above models and methods can be used; in reality, however, the selection of the model is dependent on the tolerable computer CPU time, as both the optimization and the stochastic modeling can be computational time consuming

In our case, the Genetic Algorithm (GA) has been chosen as the optimization tool Due to the large number of individual simulations needed

in the GA, the simulation model needs to be highly efficient in order to stay within a reasonable amount of computation time For this reason, the sharp interface analytical solution is chosen, which is briefly described in the following

Figures 1(a) and (b) respectively give the definition sketch of a confined and an unconfined aquifer The aquifers are with homogeneous

hydraulic conductivity K and constant thickness B in the confined aquifer

case Distinction has been made between two zones—a freshwater only zone (zone 1), and a freshwater–saltwater coexisting zone (zone 2) Following the

Figure 1: Definition sketch of saltwater intrusion in (a) a confined aquifer,

and (b) an unconfined aquifer

work of Strack [1976], the Dupuit-Forchheimer hydraulic assumption is used

to vertically integrate the flow equation, reducing the solution geometry from

three-dimensional to two-dimensional (horizontal x-y plane) Steady state is

assumed The Ghyben-Herzberg assumption of stagnant saltwater is utilized

to find the saltwater–freshwater interface With the above common

assumptions of groundwater flow, the governing equation for the system is

the Laplace equation:

where ∇ is the Laplacian operator in two-spatial dimensions (x and y), and 2

the potential φ is defined differently in the two zones

saltwater invaded zone

freshwater zone

pumping well inactive well

1[ ( 1) ] for zone 22( 1)

( ) for zone 22( 1)

f

f

h sd s

s

φφ

ρρ

as the saltwater and freshwater density ratio, and other definitions are found

in Figure 1

In our problem, we consider a semi-infinite coastal plain bounded by

a straight coastline aligned with the y-axis (Figure 2) Multiple pumping

wells are located in the aquifer with coordinates ( , )x y and discharge i i Q i

There is a uniform freshwater outflow rate q The aquifer can be confined or

unconfined Solution of the potential φ for this problem can be found by the

method of images and has been given by Strack [1976] (see also Cheng and

With the above solution, the toe location of saltwater wedge x is found toe

where the potential takes the value φtoe,

Since φtoe is some known number evaluated from Eq (7), Eq (6) can be

solved for x for each given y value using a root finding technique toe

3 OPTIMIZATION UNDER DETERMINISTIC CONDITIONS

The management objective of the coastal pumping operation is to

maximize the economic benefit from the pumped water less the utility cost

for lifting the water For simplicity, we assume that the value of water and

the utility cost are both linear functions of discharge Q The objective is to i

maximize the benefit function Z with respect to the design variables Q i

In the above B is the economic benefit per unit discharge, p C is the cost p

per unit discharge per unit lift height, L is the ground elevation at well i, i

and h is the water level in well i It should be remarked that although a i

relative simple model is used for the right-hand side of Eq (8), it can be

generalized to a realistic microeconomic model involving supply and demand without complicating the solution process

The pumping operation is subject to some constraints First, the discharge of each well must stay within the certain limits set by the operation conditions such as the minimum feasible pumping rate, maximum capacity

of the pump, restriction on well drawdown, etc This can be written as

We note that the second condition in the above allows the well to be shut down Second, it is required that saltwater wedge does not invade the pumping wells

x i toe<x i at y= y i; for all active wells (10) where x stands for the toe location in front of well i i toe

Since genetic algorithm can only work with unconstrained problems,

it is necessary to convert the constrained problem described by Eqs (8)-(9)

to an unconstrained one This is accomplished by the adding penalty to the objective function for any violation that takes place:

x < We notice that the constraint Eq (9) is not x

included in Eq (11) because it is automatically satisfied by setting the population space in genetic algorithm

4 GENETIC ALGORITHM

Conventional optimization techniques, such as the linear and nonlinear programming, and gradient-based search techniques are not suitable for finding global optimum in space that is discontinuous and contains a large number of local optima, which are the prevalent conditions for the optimization problem defined above To overcome these difficulties,

a genetic algorithm (GA) has been introduced and successfully applied

[Cheng et al., 2000] GA is a probabilistic search based optimization

technique that imitates the biological process of evolution [Holland, 1975] Its application to groundwater problems started in the mid-1990s [McKinney

and Lin, 1994; Ritzel et al., 1994; Rogers and Fowla, 1994; Cienlawski et al., 1995], and since that time it has found many applications (See Ouazar

and Cheng [1999] for a review.)

A brief illustration of the GA solution procedure applied to the

current problem is given below Given the solution space of Q defined by i

Eq (9), we discretize it in order to reduce the number of trial solutions from

infinite to a finite set As an example, if each discharge is constrained

between 100≤Q i ≤500 m³/day, and the desirable accuracy of the solution is

5 m³/day (which is a rather crude resolution), then for each Q there exist 82 i

possible discrete values (including the zero pumping rate) If there are 10

wells in the field, then the total number of possible combinations of pumping

rate is 8210 =1.4 10× 19 One of the combinations is the optimal pumping

solution we look for This search space is so huge that if we spend 1 sec of

CPU time to conduct a single simulation to check its benefit, it will take

11

4 10× years to complete the work The search space of a typical field

problem in fact is greater than the above Hence we must follow some

intelligent rules in the search; this is where the GA comes in

GA seeks to represent the search space by binary strings In the

above example, it is sufficient to represent all possible combinations of

pumping rate by a 64-bit binary string (264=1.8 10× 19) To seed an initial

population, a random number generator is used to flip the bits between 0 and

1 to create individuals in the form of 01101…10111 (64 digits long), each

one corresponding to a distinct set of pumping rates Typically a relatively

small number of individuals, say 10 to 20, are created to fill a generation

Individuals are then tested for their fitness to survive by running the

deterministic simulation as described above The fitness is determined by the

objective function given as the right-hand side of Eq (11)

Once the fitness is determined for each individual in the generation,

certain evolutional-based probabilistic rules are applied to breed better

offspring For example, in a simple genetic algorithm (SGA), three rules,

selection, crossover, and mutation, are used [Michalewicz, 1992] First, the

selection process decides whether an individual will survive by “throwing a

dice” using a probability proportional to the individual’s fitness value

Second, the GA disturbs the resulting population by performing crossover

with a probability of p In this operation, each binary string (individual) is c

considered as a chromosome Segments of chromosome between individuals

can be exchanged according to the predetermined probability Third, to

create diversity of the solution, GA further perturbs the population by

performing mutation with a probability of p In this operation, each bit of m

the chromosome is subjected to a small probability of mutation by allowing

it to be flipped from 1 to 0 or the other way around After these steps, a new

generation is formed and the evolution continues The process is terminated

2 3 4

15

0 1000 2000 3000 4000

Figure 3: Pumping wells in a coast and saltwater intrusion front

by a number of criteria, such as no improvement observed in an number of generations, or reaching a pre-determined maximum number of generation The reader can consult the above-cited references for more detail

5 EXAMPLE OF DETERMINISTIC OPTIMIZATION

This test case was examined in Cheng et al [2000] Assume an unconfined aquifer with K = 40 m/day, q = 40 m²/day, d = 15 m, ρs = 1.025 g/cm³, and ρf = 1 g/cm³ Figure 3 gives an aerial view of the coast and the locations of 15 pumping wells The well coordinates are shown in columns (2) and (3) of Table 1 Each well is bounded by a maximum and a minimum well discharge, as indicated in columns (4) and (5) In this optimization problem, only the benefit from the pumped volume is considered, and the utility cost is neglected The objective function (11) is modified to

Table 1: Optimal pumping well solution

The GA described earlier is used for optimization In the first attempt, the

optimization was conducted by assuming all 15 wells are in operation The

search space for each well is defined between min

i

Q and max

i

Q with increment size of roughly 1 m³/day and also the zero discharge If a well is invaded, a

penalty is imposed with an empirical penalty factor r to discourage such i

events If the well is shut down, Q= , the program detects it and no penalty 0

is applied for invasion This allows the inactive wells to be intruded in order

to increase pumping

After three runs of GA with different seeding of initial population,

the best solution gives the total discharge of 3,610 m³/day The optimal

solution shows that eight wells are in operation and seven are shut down The

fact that so many wells are shut down is not surprising, as an estimate based

on a simple analytical solution [Cheng et al., 2000] shows that the well field

is too crowded and some wells can be taken out of action

The program was run on a Pentium 450MHz microcomputer It was

terminated when the maximum number of generations was reached, for about

6 hours of CPU time Since an near optimal solution may not have been

reached, a second search is conducted using a refined strategy In the second

search, only cases with any combinations of seven, eight, and nine wells in

operation are admitted into the search space Wells not selected do not exist and can be invaded This strategy much reduces the size of the search space and better solution is obtained The best solution is a seven-well case as shown in column (6) of Table 1 The toe location in front of the wells is shown in column (7) The total pumping rate is 3,891 m³/day The saltwater intrusion front is graphically demonstrated in Figure 3, with the well locations marked We notice that two of the inactive wells, 4 and 12, are intruded by saltwater

6 STOCHASTIC SIMULATION MODEL

The solution presented above assumes deterministic conditions, i.e., all aquifer data are known with certainty This is not true in reality as hydrogeological surveys are expensive and time consuming to conduct; hence hydrogeological data are rare The optimization model needs to take this reality into consideration

The first step of conducting a stochastic optimization is to have a stochastic simulation model This can be accomplished by applying the second order uncertainty analysis of Cheng and Ouazar [1995] to the deterministic model given as Eq (6) Based on the approximation of Taylor series, the statistical moments of toe location can be related to the moments

of uncertain parameters as [Naji et al., 1998]

1,2

where x , q , and K are respectively the mean toe location, the mean toe

freshwater outflow rate, and the mean hydraulic conductivity; 2

x

σ , 2

q

σ , and 2

K

σ are respectively the variance of toe location, freshwater outflow rate, and hydraulic conductivity; and x toe(q K is the toe location evaluated using the , )mean parameter values In the above, we have neglected the covariance σqK

by assuming that it is small The above equations state that in order to obtain the mean toe location and its standard deviation, we first need to calculate the toe location using the mean parameter values, i.e., x toe(q K This is , )obtained from the deterministic solution by solving Eq (6) using the given

q and K values Next, we need to find the partial derivatives of toe location

with respect to q and K This is found by perturbing the q and K values by

small amounts in Eq (6) In other words, Eq (6) is solved for the toe

location using values of q ± ∆ and K q ± ∆ and the difference in K x is toe

found Utilizing finite difference approximation, the partial derivative

7 CHANCE CONSTRAINED OPTIMIZATION

The optimization problem described in Sections 2 through 5 is based

on deterministic conditions In the event of input data uncertainty, a

stochastic optimization is necessary The chance-constrained programming

[Charnes and Cooper, 1959; 1963] is used for this purpose This optimization

model allows us to use stochastic parameters as input data and produces an

output prediction based on desirable reliability level

Charnes and Cooper [1959, 1963] studied chance constrained

programming by transforming a stochastic optimization problem into a

deterministic equivalent The chance-constrained programming can

incorporate reliability measures imposed on the decision variables This

methodology has been applied to solve a number of groundwater

management problems Tung [1986] developed a chance-constrained model

that takes into account the random nature of transmissivity and storage

coefficient Wagner and Gorelick [1987] presented a modified form of the

chance constrained programming to determine a pumping strategy for

controlling groundwater quality Hantush and Marino [1989] presented a

chance-constrained model for stream-aquifer interaction Morgan et al

[1993] developed a mixed-integer chance-constrained programming and

demonstrated its applicability to groundwater remediation problems

Chance-constrained groundwater management models have also been applied to

design groundwater hydraulics [Tiedman and Gorelick, 1993] and quality

management strategies [Gailey and Gorelick, 1993] Chan [1994] developed

a partial infeasibility method for aquifer management Datta and Dhiman

[1996] utilized a chance-constrained model for designing a groundwater

quality monitoring network Wagner [1999] employed the

chance-constrained model for identifying the least cost pumping strategy for

remediating groundwater contamination Sawyer and Lin [1998] considered

the combination of uncertainty in the cost coefficients and constraints of the

groundwater management model

For the present problem we assume that the freshwater outflow rate q and the hydraulic conductivity K are random variables, causing the toe

location in front of each well toe

i

x to be uncertain The constraint given by

Eq (10) needs to be modified to a probabilistic one:

Prob( toe ) ; for all active wells

i i

where R is the desirable reliability level of prediction set by the water

manager The chance constraint converts the above probabilistic constraint into a deterministic one:

1( ) toe ; for all active wells

x , and F−1( )R is the value of the standard normal cumulative

probability distribution corresponding to the reliability level R The

chance-constrained optimization problem is then defined by the objective function

Eq (8), which is subject to the constraints Eqs (9) and (16)

In order to apply GA for the solution of the optimization problem,

we need to convert the constrained problem to an unconstrained one Similar

to the deterministic problem, this is accomplished by imposing penalty for the violation of the chance constraint Eq (16):

2 1

8 CASE STUDY—MIAMI BEACH, SPAIN

The above-proposed optimization model has been tested and applied

to a few hypothetical as well as real cases [Benhachmi et al., 2003a, b] Here,

we report the case study of the city of Miami Beach in northeast Spain

A large fraction of the total population of Spain (about 80% of its 6

million inhabitants) lives along the Catalonia coast [Bayó et al., 1992] This

concentration of population creates large freshwater demands for domestic consumption, in addition to the agricultural, industrial, and tourism needs Aquifers along the coast have been subjected to intensive exploitation;