Applied Soft Computing 28 (2015) 541–549

Contents lists available at ScienceDirect

Applied Soft Computing

j ournal homepage: www.elsevier.com/locate/asoc

Evaluation of methods for estimating hydraulic parameters

b c d a,e,∗ f

S.M. Bateni , M. Mortazavi-Naeini , B. Ataie-Ashtiani , D.S. Jeng , R. Khanbilvardi

a

Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang, Jiangsu, 212013, China

b

Department of Civil and Environmental Engineering, and Water Resources Research Center, University of Hawaii at Manoa, Honolulu, HI, 96822, USA

c

Department of Civil, Surveying and Environmental Engineering, University of Newcastle, Callaghan, NSW,

d

Department of Civil Engineering, Sharif University of Technology, Tehran, Iran

e

Griffith School of Engineering, Griffith University, Gold Coast Campus, Queensland, QLD 4222, Australia

f

NOAA-Cooperative Remote Sensing Science & Technology Center (NOAA-CREST), City University of New York, NY, USA

a r t i c l e i n f o a b s t r a c t

Article history: An accurate estimation of aquifer hydraulic parameters is required for groundwater modeling and proper

Received 16 August 2013

management of vital groundwater resources. In situ measurements of aquifer hydraulic parameters are

Received in revised form 30 July 2014

expensive and difficult. Traditionally, these parameters have been estimated by graphical methods that

Accepted 17 December 2014

are approximate and time-consuming. As a result, nonlinear programming (NLP) techniques have been

Available online 24 December 2014

used extensively to estimate them. Despite the outperformance of NLP approaches over graphical meth-

ods, they tend to converge to local minima and typically suffer from a convergence problem. In this study,

Keywords:

Genetic Algorithm (GA) and Colony Optimization (ACO) methods are used to identify hydraulic

Aquifer hydraulic parameters

parameters (i.e., storage coefficient, hydraulic conductivity, transmissivity, specific yield, and leakage

Ant Colony Optimization (ACO)

factor) of three types of namely, confined, unconfined, and leaky from real time–drawdown

Genetic Algorithm (GA)

Nonlinear programming (NLP) pumping test data. The performance of GA and ACO is also compared with that of graphical and NLP

Pumping test techniques. The results show that both GA and ACO are efficient, robust, and reliable for estimating var-

ious aquifer hydraulic parameters from the time–drawdown data and perform better than the graphical

and NLP techniques. The outcomes also indicate that the accuracy of GA and ACO is comparable. Compar-

ing the running time of various utilized methods illustrates that ACO converges to the optimal solution

faster than other techniques, while the graphical method has the highest running time.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction variables. Pumping test is the most commonly used and standard

technique for inverse modeling of groundwater parameters. In this

In many parts of the world, groundwater has been used as a test, time–drawdown measurements are analyzed via the analyt-

reliable source of water for numerous purposes such as irrigation, ical solutions such as Theis, corrected Theis, and Hantush models

and domestic and industrial uses. Therefore, it is necessary to depending on the aquifer type in which the test is performed [8].

appropriately model the complex groundwater system in order to Several approaches have been proposed to solve the above-

properly manage this vital resource. To do so, an accurate specifi- mentioned inverse problem and obtain aquifer hydraulic param-

cation of aquifer hydraulic parameters such as transmissivity (T), eters. The first group of approaches is graphical. The aquifer

hydraulic conductivity (K), storage coefficient (S), leakage factor (B), hydraulic parameters estimated by the graphical technique are of

and specific yield (Sy) is needed since these parameters are com- questionable reliability since a perfect curve matching is not antic-

monly required in groundwater/aquifer flow modeling [1–7]. These ipated and the match is typically approximate [1]. The traditional

parameters can be measured by laboratory experiments or in situ graphical technique is time-consuming, burdensome, and subjec-

tests; however, the former is inaccurate, and the latter is expensive. tive [8]. The second group of approaches attempts to estimate

An alternative option for the estimation of the aquifer parame- aquifer parameters using nonlinear programming (NLP) techniques

ters is through inverse modeling. In this procedure, some variables such as the steepest descent method, conjugate gradient method,

that can be obtained more accurately (e.g., hydraulic head); then, Gauss–Newton method, Marquardt algorithm, etc. Although NLP

the aquifer parameters are inversely estimated from the measured methods often outperform the graphical approaches, they suf-

fer from a number of shortcomings. Aquifer models are typically

non-convex and nonlinear, and for such models, classical gradient-

∗ based optimization techniques may result in local optimum values

Corresponding author. Tel.: +61 755528590.

E-mail address: d.jeng@griffith.edu.au (D.S. Jeng). rather than global ones. The gradient based techniques may cause

http://dx.doi.org/10.1016/j.asoc.2014.12.022

1568-4946/© 2014 Elsevier B.V. All rights reserved.

542 S.M. Bateni et al. / Applied Soft Computing 28 (2015) 541–549

instability and convergence problems due to the numerical dif- observing the response of aquifer (drawdown) in observation .

ficulties. Moreover, their convergence to a global optima highly The hydraulic-head drawdowns from the pumping test can be used

depends on the selection of initial search points. Furthermore, these for aquifer characterization and the estimate of parameters such

techniques cannot find the local gradient for the problems with as transmissivity (T), storage coefficient (S), hydraulic conductivity

discontinuous objective function [4,9]. (K), leakage factor (B) and specific yield (Sy). Herein, the theoretical

To overcome the aforementioned shortcomings, the non- models for response of idealized aquifers to pumping are presented.

traditional methods such as evolutionary algorithms can be applied

as they can handle highly nonlinear, even non-smooth problems 2.1. Theis model

and converge to the global rather than local optimum. In this

study, two -known evolutionary algorithms namely, Genetic The well-known Theis [34] solution describes radial flow to a

Algorithm (GA) and Optimization (ACO) are proposed fully penetrated pumping well in a homogeneous, isotropic con-

to estimate the hydraulic parameters of three major types of fined aquifer of infinite horizontal extent which is pumped at

aquifers (i.e., confined, unconfined, and leaky). An objective func- a constant rate Q. Theis solution that is considered as one of

tion is formulated for the inverse estimation using the observed the fundamental breakthroughs in the development of hydrologic

hydraulic heads from pumping tests. Thereafter, GA and ACO are modeling can be shown as follows,

utilized to minimize the objective function by tuning the aquifer Q

=

hydraulic parameters. These techniques have several advantages s(r, t) W(U) (1)

4T

over gradient-based methods [10,11]: (1) They do not require

where s(r, t) is the drawdown at a radial distance r from the

a continuous objective function; (2) They often can find a near

pumping well at time t since the beginning of pumping, T is the

global optima and do not get trapped in a local-optimal solution;

2

transmissivity, S is the storage coefficient, U = r S/4Tt, and W(U)

(3) They provide a number of near global optima solutions and

is the Theis well function, which is given by

hence give users the ability to assess the solutions and make a  ∞

decision. e−x

W(U) = dx (2)

GA has been applied in a number of groundwater optimiza- x

U

tion problems. El Harrouni et al. [12] used GA to manage pumping

from wells and to estimate hydraulic parameters in heterogeneous In this study, the Theis well function W(U) is calculated via the

≤

aquifers. In a similar effort, Lingireddy [13] integrated GA and neu- Taylor series expansion for small values of U(0 < U 1) and the

ral network to obtain aquifer hydraulic parameters. Samuel and Gauss–Laguerre quadrature for large values of U(U > 1) [35].

Jha [8] and Jha et al. [4] examined effectiveness of GA in esti- Unconfined aquifers have a different response to the pumping

mating aquifer parameters. They showed that GA is a viable and test compared to confined aquifers. Therefore, the Theis model

efficient technique that often overcomes the subjectivity, high com- should be corrected in order to be used for unconfined aquifers.

putational burden and ill-posedness of traditional optimization The corrected Theis model replaces s(r, t) on the left hand side of

  2

−

methods. In a more recent attempt, Rajesh et al. [14] utilized GA, Eq. (2) with s (r, t), where s is equal to s (s /2D0), and D0 is the

graphical, and sequential unconstrained minimization approaches initial saturated thickness of the unconfined aquifer [36,37]. Theis

to retrieve unconfined aquifer hydraulic parameters. Their results and corrected Theis models are used in this study for confined and

revealed that GA outperforms the other two methods. unconfined case studies, respectively.

ACO has been successfully used in water resources optimiza-

tion problems such as minimizing the capital costs associated with 2.2. Hantush model

water distribution systems [15–18], design of irrigation networks

[19], optimal groundwater monitoring design [20], multi-purpose The leaky-aquifer theory has been evolved from the Theis equa-

reservoir operation [21,22], optimal design of open channels [23], tion. In two sets of papers, Hantush and Jacob [38] and later Hantush

underground seepage flow [24], and saltwater intrusion in the [39,40], the original differentiation between the Theis response for

coastal aquifers [25]. To the best knowledge of the authors, the confined aquifers and the one for leaky aquifers was discussed. The

only study on the estimation of aquifer parameters with ACO was analytical solution of Hantush and Jacob [38] can be couched in the

conducted by Li et al. [26], wherein the hydraulic conductivity and same form as the Theis solution (Eq. (1)) but with a more com-

storage coefficient of a confined aquifer were estimated in a syn- plicated well function. In analogy with Eq. (1), the drawdown is

thetic experimental framework. In comparison to their study, the computed via,

   

present study uses ACO to estimate the hydraulic parameters of Q r

s(r, t) = W U, (3)

three aquifer systems (confined, unconfined and leaky) from real- 4T B

world time–drawdown pumping test data. Moreover, the results

where B is the leakage factor and W(U, r/B) is known as the leaky

of ACO are compared with those of graphical and NLP techniques.

well function, which is given by

Each method is applied to 15 sets of real time–drawdown pumping

   ∞    

test data (5 sets of data for each aquifer type). These data sets are r 1 ˇ2

W U, = exp −y − dy (4) taken from [27–33]. B y y

U

This paper is organized as follows. In Section 2, the analytical

=

models of groundwater used in this study are reviewed. The graph- in which ˇ r/(2B).

ical, NLP, GA, and ACO methods are outlined in Section 3. Section 4

presents the formulation of the objective function. Section 5 pro- 3. Methods

vides the results and discussions. Finally, conclusions are reported

in Section 6. 3.1. Graphical method

As stated in the introduction, the first group of approaches for

2. Analytical models for analyzing pumping test data retrieving aquifer parameters is graphical. In order to determine

aquifer parameters such as transmissivity (T) and storage coeffi-

Pumping test is a well-known tool to characterize a system of cient (S), the pumping test is conducted and the drawdowns in

aquifers. The test is conducted by pumping water from one well and a pumped well and in several observation wells are recorded at

S.M. Bateni et al. / Applied Soft Computing 28 (2015) 541–549 543

suitable time intervals. The recorded time–drawdown data are ana-

lyzed by the standard graphical methods designed for confined,

unconfined and confined aquifers [27,29,31]. These graphical meth-

ods basically involve matching of Type curves (which vary from

the type of aquifers) with the corresponding field curves, find-

ing out a match point and then calculating aquifer parameters

using the Theis model (for confined and unconfined aquifers) or

Hantush and Jacob model (for leaky aquifers). In this study, the

time–drawdown pumping test data of confined, unconfined and

leaky aquifers were graphically analyzed using the commonly used

AquiferTest software developed by Waterloo Hydrogeologic Incor-

poration (WHI), Canada. It is a graphical oriented package, and

has a wide range of methods for analyzing pumping test data of a

variety of aquifers. For detailed information on the AquiferTest soft-

ware, the reader is referred to (http://www.swstechnology.com/

groundwater-software/pumping-test-analysis/aquifertest-pro).

3.2. Nonlinear programming (NLP)

As mentioned in the introduction, the second group of methods

for estimating aquifer parameters is based on nonlinear program-

ming techniques. In this study, the well-known GAMS (General

Algebraic Modeling System; [41]) software is employed to estimate

aquifer hydraulic parameters from pumping test data using its NLP

solvers. GAMS is a mathematical programming and optimization

modeling system that has been used widely in the water resources

and agricultural economics research communities [42–47].

GAMS has several built-in solvers for solving NLP problems,

and the program itself tries to use the best solver for each prob-

lem. One of the well-known NLP solvers in GAMS is CONOPT.

Our NLP problems in GAMS are solved using CONOPT. This solver

is mainly developed for models with smooth functions, but it

can also be used for models with a non-differentiable function.

CONOPT3, the most recent version of the solver, is a gradient

based algorithm that executes a sequential quadratic programming

(SQP) component. The SQP module utilizes the second derivative

to obtain better search directions [48]. The hydraulic parameters

in our case study problems are found via a GAMS optimization

code. Finally, the results are compared with those of the other two

methods.

3.3. Genetic Algorithm (GA)

Fig. 1. Flowchart of GA method.

The GA optimization method is based on Darwin’s theory of

evaluations. In nature, individuals which have highly adaptive

behaviors will have relatively a larger number of offspring and usually called parents, are selected based on their objective func-

consequently survive longer. GA is a robust optimization method tion values. Crossover involves a pair of chromosomes exchanging a

seeking to mathematically reproduce the mechanisms of natural portion of their bit sequence at randomly set cutting positions. The

selection [49,50]. idea behind crossover is that, by exchanging blocks of bits between

GA is a population based algorithm. A group of chromosomes two chromosomes that performed well, the GA attempts to pro-

form a population. Population size determines the number of chro- duce two new chromosomes that preserve best materials from two

mosomes in each population. In GA, parameters are presented in parents. There are a few crossover methods including one-point

form of chromosomes which, in turn, consist of genes. Traditionally, crossover, two-point crossover and uniform cross over [49]. The

GA has been developed by applying binary coding in which a chro- main difference between these methods is number of cutting pos-

mosome is represented by a string of binary bits, e.g., [00101001], itions. For instance, one-point crossover has one cutting position

that can be encoded into integer or real numbers. while two-point crossover has two cutting positions. Fig. 2 shows

A simple flowchart for GA is shown in Fig. 1. The first step in GA schematically the one-point cross over. Chromosome 1 and 2 are

is to initialize population randomly. Indeed, all chromosomes in selected as parents and then a cutting position is selected randomly

the first population are determined randomly. Then, for each chro- (Fig. 2). Finally, the genes on the right hand side of the position are

mosome, parameters are determined by decoding genes into real exchanged.

values. In the next step, chromosomes are evaluated by calculating Mutation permits introduction of new chromosomes into a

the value of objective function for each of them. population by randomly filliping a gene. A mutation probability

GA operators mimic natural processes. The main GA opera- determines frequency of random mutations to be made to indi-

tors are selection, crossover and mutation. Selection is the process vidual genes. Crossover and mutation operations do not allow

by which chromosomes are chosen for participation in the repro- GA to get trapped in local optima and search the decision space

ducing process. By applying selection operator, two chromosomes more vigorously and efficiently. As it is shown in Fig. 1, selection,

544 S.M. Bateni et al. / Applied Soft Computing 28 (2015) 541–549

Fig. 2. Schematic illustration of one point crossover.

evaluation, crossover and mutation repeat until a termination cri-

terion is met. Termination criterion is usually defined as reaching Fig. 3. Schematic of ant routes traveling between five discrete parameters.

the maximum number of generations or if there is no change in the

value of objective function after a specific number of generations.

Following Samuel and Jha [8] and Jha et al. [4], crossover and

objective function values receive less pheromone. The following

mutation probabilities of 0.8 and 0.015 are chosen. Population size

equation is used to update pheromone on segments

and number of generations are set to 80 and 100, respectively

[4,8]. The Genetic Algorithm is terminated when the maximum 

number of generations (i.e., 100) is reached. The readers are − − ≤

exp(4.6((gu T)/(gmin T))) gu T

=

referred to Goldberg [49] and Michalewicz [50] for a full description u(I) (5)

0 gu > T

of GA.

3.4. Ant Colony Optimization (ACO) where u(I) represents the intensity of pheromone on the route u in

iteration I, gu is the corresponding objective function value of route

Ant Colony Optimization (ACO) is a recently developed heuristic u, gmin is the minimum value of the objective function obtained in

optimization method. It was inspired by the fact that some kinds of the current iteration and T is a critical value of the objective func-

are blind but they can find the minimum path between their tion, above which a route receives no pheromone. T is calculated as

nest and food. This occurs because of chemical substances called follows [53]:

pheromones that ants deposit when they travel on a route.

Based on the behavior of real ants, Dorigo et al. [51] devel- g

= +

T gmin CT (6)

oped the first Ant Colony Optimization method called Ant System g

(AS) to solve the traveling salesman problem (TSP) and job-shop

scheduling problem (JSP). The results showed that ACO dominated

where CT is a constant which is determined to be 0.5, while g and

all other classical or heuristic methods solutions [52]. Since then

g are the mean and standard deviation of the objective function

ACO has been applied in a variety of engineering applications

values of the current iteration, respectively.

[16,21,25,53–55].

The main step in the algorithm is to update parameters’ intervals

Abbaspour et al. [53] developed a new approach based on

based on the amount of pheromones on each segment. The idea

the ACO concept and successfully tested the feasibility of their

behind this step is to use scoring strategy to eliminate or expand

approach by estimating unsaturated hydraulic parameters. The

segments. Once the scoring is completed, the segments which have

main difference of Abbaspour et al. [53] method with the tradi-

small scores will be deleted from both ends of the intervals. On the

tional ACO is that the initial range of parameters in their technique

other hand, if segments on either end of parameter intervals have

can be adjusted in the optimization process. For instance, the end

a higher score, the parameter could be extended in that direction,

of parameters intervals may be eliminated or expanded. This fea-

i.e., add specific number of segments in that direction.

ture makes their method more flexible to deal with the model

A score is calculated for each segment of a parameter by the

parameters range identification and to more robustly explore

following expression. Note that any single segment may be the

promising search spaces by narrowing down the search space.

crossroad of many ant routes.

In this study, the Abbaspour et al. [53] ACO algorithm is used

due to its distinct advantage compared to the traditional ACO A N (ij) (ij) method. S =   (7)

ij A N

The first step of ACO is to represent the search space as a graph. i j(ij) (ij)

For this reason, each parameter is split into a specific number of

segments with a representative value assigned to each segment.

where Sij is a score, ij is the standard deviation of pheromones, ij

Ants travel between nodes corresponding to different parameters

is accumulated pheromones on the segment j of parameter i, A is a

to define a route as illustrated in Fig. 3. In this figure, each of

constant equal to 1.0 and N is given by [53],

five parameters, V1–V5, are discretized into six segments. In addi-

tion, three sample routes are presented which show how each ant

ij

assigns a value to a parameter. N = CN (8)

ij

When ants travel on routes, the objective function of each

route is calculated based on its associated parameters. Then ants

where C is a constant set equal to 0.3. In this study, number of ants

deposit pheromone on their routes according to the obtained objec- N

is set to 100 and each parameter is discretized into 500 segments.

tive function value in a way that routes which resulted in larger

S.M. Bateni et al. / Applied Soft Computing 28 (2015) 541–549 545

fifteen sets of time–drawdown pumping test data for three aquifer

systems using graphical, GA, ACO, and NLP approaches. In this

study, the pumping test datasets associated with confined, uncon-

fined, and leaky aquifers are denoted respectively by C1–C5, U1–U5,

and L1–L5. The datasets C1, C4, U1, U2, L1, and L2 are taken from

unpublished sources, C2 and C3 are obtained from Batu [27] and

C5 is from Todd [31]. The datasets U3, U4, U5, L3, L4, and L5 are

obtained respectively from Walton [33], Fetter [29], Walton [32],

Raghunath [30], Charbeneau [28], and Fetter [29]. These datasets

in varying hydrogeologic conditions allow us to compare the appli-

cability, competence and robustness of the utilized methods in

estimating the hydraulic parameters of different types of aquifers.

The statistical performance metrics mentioned in the previous sec-

tion are used for this purpose. Finally, time–drawdown estimates

from the graphical, NLP, GA, and ACO algorithms are compared with

observations.

5.1. Confined aquifers

The datasets C2 and C3 are obtained from Batu [27], C5 dataset is

taken from Todd [31] and C1 and C4 from unpublished sources. The

estimated storage coefficient and transmissivity of five explored

confined aquifers from the graphical, NLP, GA, and ACO methods are

shown in Table 1. These hydraulic parameters are used to compute

Fig. 4. ACO algorithm for identifying hydraulic parameters of an aquifer. drawdowns in the observation well at different times via the Theis

model (Eq. (1)) and compare them with corresponding measure-

2

ments. Statistical indices (i.e., MAE, RMSE, and R ) associated with

the computed drawdowns in the observation well are presented in

4. Objective function

Table 1.

The storage coefficient and transmissivity estimates from

The objective function g is defined as the sum of squared differ-

−4 −3

the ACO vary respectively from 1.56 × 10 to 5.30 × 10 and

ence between the observed and calculated drawdowns:

2

236–5559 m /day. These values are in good agreement with those

n

= 2 obtained by the graphical, NLP, and GA methods. However, the ACO

g Min [so(t) − sc(t)] (9)

and GA results for all confined aquifers datasets (C1–C5) are bet-

t=1

ter than those of the graphical and NLP approaches (Table 1). The

where so(t) is the observed drawdown in an observation well at MAE and RMSE of ACO and GA are always less than the ones from

time t, sc(t) is the corresponding computed drawdown from a suit- the graphical and NLP methods, implying that ACO and GA can

able model (Theis, Corrected Theis or Hantush model) for the given retrieve the hydraulic parameters of confined aquifers more accu-

aquifer type, and n is the total number of observed drawdown data. rately. As shown in Table 1, for C1, C2, C3, C4, and C5 datasets,

The aim is to minimize the objective function through estimating the computed drawdown from ACO (GA) has a RMSE of 0.2737

the optimum values for hydraulic parameters of aquifers. Fig. 4 (0.2561), 0.0445 (0.0454), 0.0398 (0.0378), 0.0230 (0.0257), and

shows the flowchart of ACO algorithm for finding the hydraulic 0.0051 m (0.0052 m), which is a 14% (20%), 34% (33%), 19% (23%),

parameters of an aquifer. This flowchart is based on the ACO 56% (51%), and 41% (40%) reduction of the graphical RMSE of 0.3184,

algorithm described in Section 3.3. The iteration in the flowchart 0.0676, 0.0490, 0.0521, 0.0086 m. Analogously, for the abovemen-

continues until a termination criterion is satisfied. The algorithm tioned datasets, the MAE of estimated drawdown decreases from

−4 2

=

terminated when the criterion g 10 m is fulfilled or the max- 0.2441, 0.0586, 0.0441, 0.0338, and 0.0067 m to 0.2372 (0.2216),

imum number of iterations reaches a prescribed value of 100. The 0.0388 (0.0395), 0.0321 (0.0315), 0.0163 (0.0193), and 0.0030 m

variables Q, r, t, and So (shown in Fig. 4) are inputs for confined and (0.0033 m) by using ACO (GA) rather than the graphical approach.

leaky aquifers. For unconfined aquifers, in addition to the above- As can be seen in Table 1, the results of ACO and GA are close and

mentioned variables, D0 is required as input. comparable. In some of the confined aquifers (C2, C4, and C5) ACO

The performance of graphical, NLP, GA, and ACO methods is eval- outperforms GA, and in others (C1 and C3) GA performs better.

2

uated using the mean absolute error (MAE) and the root mean The ACO and GA methods have also the highest R value for all

square error (RMSE) metrics. These metrics are used to assess of confined aquifers datasets, showing that the drawdown esti-

the drawdown estimates from each approach. The coefficient of mates from GA and ACO are more correlated to the observations

2

determination (R ) of linear regression line between the estimated than the ones from the graphical and NLP techniques. Especially,

drawdown from the graphical, NLP, GA, and ACO models and the for the C4 aquifer, the estimated storage coefficient and trans-

observations is also used as a measure of performance. The low missivity from ACO are significantly different from those of the

2

MAE and RMSE as well as high R show that the estimated hydraulic other two methods. Given the much better performance of ACO

parameters are close to their real physical values. method (based on the statistical indices for the C4 aquifer), its

hydraulic parameters estimates are more reliable and the NLP

5. Results and discussion method has likely found a local minimum of objective func-

tion.

The AquiferTest software along with the NLP, GA, and ACO com- Overall, the statistical metrics suggest that ACO and GA are

puter programs were run to optimize aquifer parameters for five more efficient techniques for the retrieval of hydraulic parame-

time–drawdown datasets associated with each type of aquifer sys- ters of confined aquifers. Another conclusion from Table 1 is that

tems (confined, unconfined, and leaky). Thus, we analyzed a total of the errors associated with the NLP approach are less than those of

546 S.M. Bateni et al. / Applied Soft Computing 28 (2015) 541–549

Table 1

Comparison of estimated hydraulic parameters from graphical, NLP, GA, and ACO techniques for confined aquifers. The best results and its associated method are bolded.

2 2

Dataset Method Storage coefficient Transmissivity (m /day) MAE (m) RMSE (m) R

Graphical 0.002214 2090 0.2441 0.3184 0.9251

NLP 0.001240 2285 0.2766 0.2931 0.9215

Confined1 (C1)

GA 0.001998 2195 0.2216 0.2561 0.9300

ACO 0.001860 2021 0.2372 0.2737 0.9270

Graphical 0.000203 237 0.0586 0.0676 0.9861

NLP 0.000174 234 0.0402 0.0457 0.9865

Confined2 (C2)

GA 0.000166 235 0.0395 0.0454 0.9871

ACO 0.000156 236 0.0388 0.0445 0.9868

Graphical 0.000307 256 0.0441 0.0490 0.9868

NLP 0.000287 245 0.0324 0.0398 0.9873

Confined3 (C3)

GA 0.000289 247 0.0315 0.0378 0.9882

ACO 0.000287 246 0.0321 0.0398 0.9877

Graphical 0.002993 13,929 0.0338 0.0521 0.8854

NLP 0.001819 12,096 0.0293 0.0382 0.9132

Confined4 (C4)

GA 0.003891 7819 0.0193 0.0257 0.9324

ACO 0.005302 5559 0.0163 0.0230 0.9464

Graphical 0.000199 1142 0.0067 0.0086 0.9996

NLP 0.000192 1140 0.0031 0.0053 0.9997

Confined5 (C5)

GA 0.000193 1139 0.0033 0.0052 0.9997

ACO 0.000195 1136 0.0030 0.0051 0.9997

graphical method and thus NLP can estimate the storage coefficient aquifers are indicated in Table 3. Based on ACO results, the storage

−5 −3

and transmissivity more reliably than the graphical method. coefficient varies from 2.25 × 10 to 1.05 × 10 , the transmis-

2

sivity from 25.2 to 960.4 m /day and leakage factor from 221.0

to 9002.6 m. The MAE and RMSE of ACO drawdown estimates

5.2. Unconfined aquifers

are compared to those from the graphical, NLP, and GA methods

(Table 3). In analogy to the results of the confined and uncon-

Table 2 lists the estimated specific yield and hydraulic conduc-

fined aquifers (Tables 1 and 2), ACO and GA perform better than

tivity values of the five investigated unconfined aquifers from the

the graphical and NLP techniques. The errors of ACO and GA are

graphical, NLP, GA, and ACO methods. The performance metrics of

always less than those of the other two techniques. For L1, L2, L3,

each technique are also presented in Table 2. Based on the ACO

L4, and L5 aquifers, the ACO (GA) approach reduces the RMSE of

results, the specific yield of the unconfined aquifers ranges from

−4 −2

drawdown estimates from the NLP technique by 36% (47%), 27%

3.38 × 10 to 6.93 × 10 , and the hydraulic conductivity ranges

(9%), 41% (12%), 39% (16%), and 55% (63%) and from the graphi-

from 23 to 169 m/day. The retrieved specific yield and hydraulic

cal method by 47% (55%), 35% (18%), 40% (12%), 64% (50%), and

conductivity from graphical, NLP, and GA approaches are almost in

58% (66%), respectively (Table 3). Similar drop is also evident in

the same range.

the MAE associated with ACO and GA. Compared to the graphical

The ACO and GA methods perform better than the other two

and NLP methods, the ACO and GA estimates of drawdown have

techniques. For all unconfined aquifers, the MAE and RMSE values of

2

also higher values of R . These outcomes demonstrate the graphical

ACO and GA are less than the corresponding values of the other two

and NLP techniques are not as efficient as ACO and GA for esti-

methods. For U1, U2, U3, U4, and U5 datasets, the RMSE of estimated

mating hydraulic parameters of leaky aquifers. Likewise, Trinchero

drawdown from ACO (GA) is respectively 0.0194 (0.0300), 0.0801

et al. [57] showed that when various methods were applied to het-

(0.0671), 0.0314 (0.0320), 0.0265 (0.0403), 0.0103 m (0.0103 m),

erogeneous media of leaky aquifers, each method could provide

which is a factor of 67 (50), 53 (61), 37 (30), 65 (47), and 48

different estimates of hydraulic parameters. Comparing results of

(50) less than the corresponding RMSE estimates from graphical

ACO and GA reveals that none of them is superior to the other one.

approach and a factor of 40 (35), 53 (60), 3 (1), 43 (15), and 2 (5) less

For L1 and L4 aquifers, GA slightly outperforms ACO, but for L2,

than the corresponding RMSE estimates from NLP method. These

L3, and L4 aquifers, Ant Colony Optimization method gives better

results prove the capability of the ACO and GA methods in finding

results.

more accurate estimates of specific yield and hydraulic conduc-

tivity of unconfined aquifers. Compared to GA, ACO gives slightly

more accurate aquifer parameters estimates for four of the uncon-

5.4. Visual examination of retrieved time–drawdown curves

fined aquifers (U1, U3, U4, and U5 datasets). While, in one of the

unconfined aquifers (U2 dataset), GA generates better results.

The efficiency of ACO and NLP optimization techniques as well

Similar to the confined aquifer datasets, the NLP outperforms

as the graphical method in estimating the hydraulic parameters of

the graphical technique. This finding is consistent with that of Neu-

confined, unconfined and leaky aquifers is also assessed by visual

man et al. [56]. Aquifers are typically heterogeneous media and

comparison of retrieved time–drawdown curves from these three

thus their properties vary spatially such that cannot be described

techniques with observations. The comparisons are illustrated in

with certainty. Neuman et al. [56] confirmed that time–drawdown

Fig. 5. Since the time–drawdown curves from GA and ACO are very

data from randomly heterogeneous aquifers are hard to interpret

close, only the one obtained by ACO is shown herein. As can be

graphically.

seen in these plots, drawdown retrievals from ACO have always

the best agreement and correlation with the observations. In all the

5.3. Leaky aquifers cases, the second best solution is obtained from the NLP method and

the last one from the graphical method. These results are consis-

The storage coefficient, transmissivity, and leakage factor esti- tent with the outcomes of the comparisons based on the statistical

mates from graphical, NLP, GA and ACO approaches for the leaky indices in the previous section.

S.M. Bateni et al. / Applied Soft Computing 28 (2015) 541–549 547

Table 2

Comparison of estimated hydraulic parameters from graphical, NLP, GA, and ACO techniques for confined aquifers. The best results and its associated method are bolded.

2

Dataset Method Specific yield Hydraulic conductivity (m/day) MAE (m) RMSE (m) R

Graphical 0.02492 80 0.0493 0.0598 0.9665

NLP 0.01627 116 0.0296 0.0323 0.9672

Unconfined1 (U1)

GA 0.02912 71 0.0255 0.0300 0.9701

ACO 0.02476 78 0.0167 0.0194 0.9792

Graphical 0.01036 55 0.1475 0.1724 0.8309

NLP 0.00474 58 0.1496 0.1704 0.8023

Unconfined2 (U2)

GA 0.07532 31 0.0489 0.0671 0.9724

ACO 0.06928 23 0.0646 0.0801 0.9616

Graphical 0.00687 200 0.0402 0.0506 0.9463

NLP 0.00491 183 0.0288 0.0322 0.9525

Unconfined3 (U3)

GA 0.00551 195 0.2880 0.0320 0.9571

ACO 0.00651 169 0.0283 0.0314 0.9582

Graphical 0.06280 156 0.0507 0.0755 0.9713

NLP 0.03771 150 0.0390 0.0472 0.9725

Unconfined4 (U4)

GA 0.04619 134 0.0317 0.0403 0.9906

ACO 0.06886 112 0.0191 0.0265 0.9870

Graphical 0.000639 112 0.0177 0.0200 0.9626

NLP 0.000352 110 0.0089 0.0105 0.9717

Unconfined5 (U5)

GA 0.000317 115 0.0091 0.0103 0.9721

ACO 0.000338 112 0.0087 0.0103 0.9730

Table 3

Comparison of estimated hydraulic parameters from graphical, NLP, GA and ACO techniques for leaky aquifers. The best results and its associated method are bolded.

2 2

Dataset Method Storage coefficient Transmissivity (m /day) Leakage factor (m) MAE (m) RMSE (m) R

Graphical 0.0000599 860.5 1997.0 0.0065 0.0079 0.9987

NLP 0.0000674 704.8 933.4 0.0058 0.0066 0.9963

Leaky1 (L1)

GA 0.0000901 820.9 1341.6 0.0027 0.0035 0.9991

ACO 0.0000814 772.4 1692.0 0.0033 0.0042 0.9987

Graphical 0.0000140 1045.4 9985.2 0.0054 0.0072 0.9979

NLP 0.0000477 793.7 2500.0 0.0056 0.0065 0.9968

Leaky2 (L2)

GA 0.0000040 1001.5 7560.3 0.0051 0.0069 0.9974

ACO 0.0000225 960.4 9002.6 0.0039 0.0047 0.9983

Graphical 0.0010300 181.4 159.9 0.0619 0.0866 0.9881

NLP 0.0009842 228.7 408.7 0.0644 0.0878 0.9852

Leaky3 (L3)

GA 0.0011000 210.3 345.1 0.0588 0.0765 0.9804

ACO 0.0010460 183.1 221.0 0.0401 0.0519 0.9940

Graphical 0.0001960 22.9 192.9 0.0557 0.0719 0.9946

NLP 0.0001196 29.1 474.5 0.0358 0.0422 0.9956

Leaky4 (L4)

GA 0.0001256 27.6 341.1 0.0267 0.0356 0.9971

ACO 0.0001377 25.2 250.8 0.0197 0.0255 0.9984

Graphical 0.0001480 26.4 292.1 0.0515 0.06985 0.9931

NLP 0.0000840 31.5 457.2 0.0492 0.06477 0.9954

Leaky5 (L5)

GA 0.0001321 22.1 221.3 0.0202 0.02356 0.9991

ACO 0.0001270 25.4 256.1 0.0233 0.02899 0.9986

5.5. Running time of different methods

Besides the accuracy of graphical, NLP, GA, and ACO methods in

estimating aquifer hydraulic parameters, their running time is an

important issue in groundwater engineering problems. It is of our

Table 4

interest to understand which of the explored methods runs faster, Comparing running times of graphical, NLP, GA and ACO techniques for various

aquifer types. The lowest running time and its associated approach are bolded.

especially when there are a large number pumping test data from

several aquifers.

Aquifer type Method Running time (s)

The developed computer programs were executed on the WIN-

Graphical 18.9

DOWS operating system (Xeon 5128, 256 GB RAM, FSB 1066 MHz,

NLP 11.1

Confined (C1–C5)

40 GB HDD), and their running times for different aquifer types GA 13.0

are shown in Table 4. Each execution time for confined, uncon- ACO 8.2

fined, and leaky aquifers represents the mean of running times for

Graphical 20.0

C1–C5, U1–U5, and L1–L5 datasets, respectively. As illustrated, the NLP 11.8

Unconfined (U1–U5)

ACO approach converges to the optimal solution faster than other GA 14.0

ACO 9.4

methods, while the graphical technique is the slowest one. On over-

age, for the five explored confined aquifers, the running time of Graphical 46.4

NLP 25.3

ACO is 57%, 26%, 37% less than that of graphical, NLP, and GA meth- Leaky (L1–L5)

GA 28.3

ods. Corresponding values are 53%, 20%, and 33% for the unconfined

ACO 16.8

aquifers and 63%, 33%, and 42% for the leaky aquifers.

548 S.M. Bateni et al. / Applied Soft Computing 28 (2015) 541–549

Fig. 5. Comparison of time–drawdown plots from graphical, NLP, and ACO techniques for sample confined (datasets C3 and C4), unconfined (datasets U1, U2, and U4), and

leaky (dataset L1) aquifers.

6. Conclusions 11% and 33% less than the RMSEs respectively from the NLP and

graphical methods. These values are 28% and 54% for the five uncon-

In this study, the efficiency of Ant Colony Optimization (ACO), fined aquifers and 40% and 49% for the leaky aquifers. Similarly, GA

which usually ensures near-optimal or optimal solutions, is exam- performs better than the NLP and graphical methods. For the five

ined in estimating hydraulic parameters of confined, unconfined, confined aquifers, on average, the drawdown estimates from ACO

and leaky aquifers from real time–drawdown pumping test data. have a RMSE of 0.0772 m, which is a 22% decrease of the RMSE of

An objective function is formulated based on the misfit between 0.0991 m from the graphical method and a 9% reduction of the RMSE

drawdown observations from pumping test and corresponding of 0.0844 m from NLP. The results from ACO and GA are close and

drawdown estimates from Theis, corrected Theis, or Hantush mod- comparable. Results also suggest that the errors associated with

els, depending on the aquifer type. The ACO technique is utilized the NLP approach are less than those of graphical method and thus

to estimate the hydraulic parameters of various aquifer systems NLP can estimate the hydraulic parameters more accurately than

by minimizing the objective function. Performance of ACO is com- the graphical method.

pared with that of Genetic Algorithm (GA), graphical and nonlinear Comparing the running time of graphical, NLP, GA, and ACO indi-

programming (NLP) methods using fifteen sets of published and cates that ACO converges to the optimal solution faster than other

unpublished real time–drawdown data of three aquifer systems methods, while the graphical approach has the lowest convergence

(i.e., five sets for each aquifer system). rate.

The results show that ACO outperforms NLP and graphical With the presence of large memory and high speed computers

methods for all three aquifer systems and thus is a more reli- nowadays, the use of ACO approach is strongly recommended for

able technique for estimating the hydraulic parameters of confined, estimating hydraulic parameters of aquifers from the pumping test

unconfined, and leaky aquifers. On average, for the five investigated data in lieu of the subjective and burdensome graphical and NLP

confined aquifers, the RMSE of estimated drawdown from ACO is methods.

S.M. Bateni et al. / Applied Soft Computing 28 (2015) 541–549 549

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