Combined use of BEM and genetic algorithms in groundwater flow and
mass transport problems
K. L. Katsifarakis1, D. K. Karpouzos2 and N. Theodossiou3
Division of Hydraulics and Environmental Engineering,
Dept. of Civil Engineering, A.U.Th.
GR-54006 Thessaloniki, Macedonia, GREECE
E-mail: klkats@civil.auth.gr1, dimkarp@civil.auth.gr2
niktheo@vergina.eng.auth.gr3 .
Abstract
The boundary element method (BEM) can be used very efficiently, in solving
groundwater flow problems. Genetic algorithms (Gas), on the other hand,
constitute a very efficient optimization tool. In this paper, BEM and Gas have
been combined to find optimal solutions in 3 classes of commonly encountered
groundwater flow and mass transport problems: a) Determination of
transmissivities in zoned aquifers (inverse problem), based on a restricted number
of field measurements b) Minimization of pumping cost from any number of
wells under various constraints and c) Hydrodynamic control of a contaminant
plume, by means of pumping and injection wells. Application examples show
that the proposed combination is very efficient in optimizing development and
protection of groundwater resources.
Keywords
Groundwater flows; Groundwater pollution control; Boundary elements; Genetic
algorithms; Optimization
1. Introduction
Fresh water availability is a basic prerequisite for development of human
activities. Water shortages occur quite often in many areas of the world, calling
for optimal management of both surface and ground water resources.
Groundwater resources are more abundant at the global scale. Moreover,
their quality is usually better, since they are naturally more protected. It should be
mentioned though, that, once polluted, their restoration is more difficult.
The need for optimal use and protection of groundwater resources calls for
improved simulation and optimization techniques. Such a technique is presented
in this paper. It consists of a combination of a boundary element code and a
genetic algorithm. The former is used to simulate groundwater flow, while the
latter serves as optimization tool.
The aforementioned technique can be used in a wide range of problems. The
boundary element code can simulate flows in aquifers consisting of any number
of zones of different transmissivities, while the genetic algorithm requires only
proper definition of the function, which should be minimized or maximized.
The two codes, which constitute the proposed technique, are briefly outlined
in the following paragraphs. Then their use is illustrated, through application
examples to three problems, commonly encountered in groundwater hydraulics,
namely: a) Determination of transmissivities in zoned aquifers (inverse problem),
based on a restricted number of field measurements b) Minimization of pumping
cost from any number of wells under various constraints and c) Hydrodynamic
control of a contaminant plume, by means of pumping and injection wells.
2. The boundary element code
A boundary element code is used to calculate hydraulic head and/or velocity
values at selected points of the flow field. The boundary element code (BEM) is
probably the most versatile method in solving steady-state groundwater flow
problems. In many such problems, hydraulic heads φ and/or velocities V are
required in very few points of the flow field only. This requirement fits perfectly
with the main feature of BEM, in which discretization is restricted to only the
external and internal field boundaries. The resulting values of φ and q = φ/n on
the boundaries, permit calculation of φ and V at each internal point separately.
Thus, calculation of φ and V values at irrelevant points of the flow field (e.g. grid
nodes) is completely avoided. Moreover, velocities are calculated directly (not
through differences of adjacent φ values). Finally, an equally important
advantage of BEM is that wells are described very accurately as concentrated
2
«loads», i.e. without distributing well flow rates to grid elements. This property
of BEM allows precise calculations at the vicinity of wells, which is the most
important field area in a large number of practical problems.
The proposed boundary element code has been extensively tested in
applications to zoned aquifers e.g.[1]. It is based on constant boundary elements,
for the following reasons: a) They simulate real flow field boundaries such as
constant head and impermeable boundaries quite accurately b) They permit
analytical calculation of the coefficients of the unknowns' system and c) Their
accuracy is satisfactory [2].
3. The genetic algorithm code
Genetic algorithms are a mathematical tool, with a very wide range of
applications, e.g. [3]. They are particularly efficient in optimization problems,
especially when the respective objective functions exhibit many local optima or
discontinuous derivatives. Of late, they are becoming popular in the field of
groundwater hydraulics, too [4,5].
There are already extensive textbooks, e.g. Goldberg [6] and Michalewicz
[7], which deal with the theoretical background, the computational details and
applications of genetic algorithms (and other evolutionary techniques). Their
main concepts, together with the features of the particular code, which is used in
this paper, are briefly described in the following paragraphs.
Genetic algorithms are essentially a mathematical imitation of a biological
process, namely that of evolution of species. To solve a problem, they start with a
number of random, potential solutions of that problem. These solutions, which
are called chromosomes, constitute the population of the first generation. In
binary genetic algorithms, each chromosome is a binary string of predetermined
length.
Each chromosome of the first generation undergoes evaluation, by means of
a pertinent function or process. This process depends entirely on the specific
application of genetic algorithms. Then, the second generation is produced, by
means of certain operators, which imitate biological processes and apply to the
chromosomes of the first generation. The main genetic operators are: a) selection
3
b) crossover and c) mutation. Many other operators have also been proposed and
used.
Selection is used first. It leads to an intermediate population, in which better
chromosomes have, statistically, more copies. These copies replace some of the
worst chromosomes. Then, the other operators apply to a number of randomly
selected members of this intermediate population. The result is an equal number
of new chromosomes, i.e. new solutions, which replace the old ones. Thus, the
next generation is formed.
The
whole
process,
i.e.
evaluation-selection-crossover-mutation-other
operators, is repeated for a predetermined number of generations. It is anticipated
that, at least in the last generation, a chromosome will prevail, which represents
the optimal (or at least a very good) solution to the examined problem.
The features of the genetic operators, which have been included in the
proposed code, are outlined in the following paragraphs.
3.1 Selection
Selection can be accomplished in many ways. The most common processes are:
a) The biased roulette wheel and b) The tournament method. The latter has been
preferred, because it applies equally well to maximization and to minimization
problems, while the former applies naturally to maximization problems only.
Selection through the tournament method starts with determination of the
respective selection constant KK (which has been set equal to 3, in all
applications). Then it proceeds in the following way: KK chromosomes are
randomly selected from the population of the current generation, and their fitness
values are compared to each other. The chromosome with the best (largest or
smallest) fitness value passes to the intermediate population. This process is
repeated PS times, PS being the population size. In this way, the intermediate
population is formed. Moreover, in our genetic code, the best chromosome of
each generation is separately passed to the new generation.
3.2 Crossover
Crossover applies to pairs of chromosomes, which are binary strings of length
4
SL. Two chromosomes, which are named parents, are randomly selected from the
intermediate population. An integer number XX, between 0 and (SL-1), is also
randomly selected. Then each parent binary string is cut to 2 pieces, immediately
after position XX. The first piece of each parent is combined with the second
piece of the other. In this way, two new chromosomes are formed, which are
called offspring and substitute their parents in the next generation.
Crossover aims at combining the best features of both parents to one
offspring. All chromosomes of the intermediate population have equal probability
of undergoing crossover. But this probability is actually larger for the better
chromosomes of the parent generation, because they have got more copies in the
intermediate population.
3.3 Mutation
Mutation applies to characters (genes), which form the chromosomes. In binary
genetic algorithms, the gene, which is selected for mutation, is changed from 0 to
1 and vice versa. This process aims at: a) extending the search to more areas of
the solution space (mainly in the first generations) and b) helping local
refinement of good solutions (mainly in the last generations). The mutation
probability is equal for all genes of all chromosomes. Its magnitude depends on
the chromosome length SL, but generally it is much smaller than the respective
crossover probability, because the latter refers to chromosomes, not to genes.
3.4 Antimetathesis
Many additional operators have been proposed in the literature, to further
improve performance of genetic algorithms. A number of them are problem
specific, while others are of general use. In this paper, one more operator, of
general use, has been included. This operator has been proposed by Katsifarakis
and Karpouzos [8]. It applies to pairs of successive positions (genes) of a
chromosome. Any position (except for the last one) can be selected, with equal
probability pa. If the value of the selected gene equals to 1, it is set to 0, while that
of the following gene is set to 1. The opposite happens if the value of the selected
gene is 0. More explicitly, the following happen, with regard to gene pairs:
5

11 becomes 01

00 becomes 10

10 becomes 01

01 becomes 10
In the first two cases, the operator is equivalent to mutation at the selected
position. In the last two though, it is equivalent to mutation of both genes.
Morover, it can be interpreted as a limiting case of the inversion operator.
The new operator has been called antimetathesis, based on its function (when
different from mutation). This name is in line with tradition in genetic algorithm
terminology, which calls for terms of greek origin. Antimetathesis and mutation
are used interchangeably (in the even and odd generations respectively). It has
been anticipated that this combination is the most effective, both in refinement of
good solutions and in exploring different areas of the solution space.
The contribution of antimetathesis in refining a good solution can be seen
through the following example: A genetic algorithm is used to find the optimum
value of function F(x), x being an integer between 0 to 1000. Let’s assume that
this optimum occurs for x = 82, and that a good approach, i.e. x = 81, has already
been obtained. In binary form 82 = [0001010010], while 81 = [0001010001],
since each chromosome should have 10 digits, to be able to represent values up to
1000. Comparing the 2 chromosomes, one can see that mutation can not improve
the solution. Antimetathesis, though, can lead to the optimum, if applied to the
9th position of the chromosome. On the contrary, mutation can produce the
optimum value, starting from x = 83. So, in this sense, mutation and
antimetathesis are complementary to each other.
Antimetathesis is also complementary to mutation in leading search to
different solutions, for the following reason: The jump, or change caused by
mutation, is always equal to a power of 2. The change introduced by
antimetathesis is equal to 2i - 2i-1. Thus, the solution space can be searched more
thoroughly, if the two operators are used interchangeably.
3.5 Handling constraints
In many applications, optimization is subject to constraints. This means that
6
chromosomes, which result from genetic operators, may represent infeasible
solutions. The usual way to deal with constraints, is to include penalty functions
in the evaluation process. Each penalty function affects the fitness value of
chromosomes, which violate the respective constraint, increasing it in
minimization problems and decreasing it in maximization ones. Other approaches
include rejection of infeasible chromosomes and modification of genetic
operators.
Constraint handling depends essentially on the particular problem. For this
reason, it is further discussed in conjunction with two application examples.
4. The complete numerical tool
Boundary elements have already been combined with genetic algorithms, in the
field of groundwater hydraulics. El Harrouni et al. [9, 10] have focused their
attention on heterogeneous aquifers, while Karpouzos and Katsifarakis [11] and
Katsifarakis et al. [12] have worked mainly on zoned aquifers.
To construct the complete simulation and optimization tool, which is
described in this paper, the boundary element code has been divided in two parts.
The first, which includes data input and some preliminary calculations, is
executed only once. The second part has to be executed for every chromosome of
every generation, since it forms the basis of the chromosome evaluation
procedure.
5. Application examples
5.1 Determination of transmissivities in zoned aquifers
To simulate flow and mass transport in aquifers, one should know the pertinent
flow parameters, such as transmissivity. The difficulty of obtaining these
parameters depends on the features of the flow field, e.g. [13, 14]. Little work has
been done in zoned aquifers [11]. The code, which has been described sofar, can
be used very successfully in this task. This is illustrated through application to the
aquifer of figure 1, which consists of 4 zones of different transmissivities.
7
Φ = 15
F (200, 2000)
E (1300, 2000)
10 l/s
T3
W3 (450, 1700)
3
K (1300, 1300)
D (2000, 1300)
G (0, 1300)
15 l/s
T4
2
T2
W2 (700, 1050)
I (1350, 925)
20 l/s
4
25 l/s
H (0, 600)
W4 (1650, 650)
T1
1
W1 (450, 500)
C (2000, 350)
A (0, 250)
Φ=0
B (1300, 0)
Figure 1. Aquifer with 4 zones of different transmissivities
The outer boundary of the aquifer consists of two impermeable boundaries,
namely CDE and FGHA, and two constant head boundaries, namely ABC and
EF, with φ = 0 and φ = 15, respectively. Four wells, one in each zone, pump at
constant flow rates, which appear, together with well coordinates, in figure 1.
Hydraulic head, under steady state flow conditions, is measured at four control
points, which also appear in figure 1 (as small triangles). The respective
coordinates are shown in table 1.
8
Table 1. Coordinates and hydraulic heads at control points
xi
yi
φmi
500
480
-8.07
650
1050
-20.7
500
1600
-26.08
1700
700
-11.69
In our example, field measurements are substituted by virtual ones, which
have been derived in the following way: The direct problem is solved for the
hydraulic head φmi at the control points, using the transmissivity values, which
appear in table 2. The results are shown in column 3 of table 1.
Table 2 Real transmissivities of the aquifer zones (in m2/s)
T1 = 0.002
T2=0.0005
T3 = 0.0001
T4 = 0.001
The transmissivities, which appear in table 2, are considered as the real ones,
i.e. those which should be calculated, by means of the proposed code. To achieve
this, the following adjustments have been made, regarding the chromosome
structure and the evaluation process:
Each chromosome is a binary string, which represents a combination of 4
transmissivity values (one for each aquifer zone). These values are multiplied by
a large number, in our case 106. The resulting integer numbers of the decimal
system can be transformed easily to binary ones and vice versa. The search space
for each transmissivity extends from 0 to 10000 (0.01106). This means that it is
assumed (e.g. based on a geological field survey or other preliminary estimates)
that the maximum possible transmissivity value Tmax is less than 0.01. The choice
of Tmax is deliberately large (compared to the “real” transmissivities), in order to
investigate the efficiency of the code.
Tmax determines the chromosome length, in the following way: In order to
exceed 10000, a binary number should have at least 14 digits. Thus, the
9
chromosome length equals 56, since the aquifer has got 4 zones, while Tmax is
actually set at 16383.
Evaluation of each chromosome includes solution of the direct problem,
using the respective transmissivity combination. Thus hydraulic heads φi at the
control points are calculated. Then the fitness of the chromosome V(x) is
calculated as:
V(x) =  φi - φmi 
(1)
that is as the difference between measured and calculated hydraulic heads.
Obviously, V(x) = 0 for the exact solution. So, in this kind of applications, the
best fitness value is known a priori.
To apply the boundary element sub-code, the external and internal field
boundaries have been divided into 28 and 19 boundary elements respectively.
Their length does not exceed 300m, nor the double of their distance from wells
(and control points).
Finally, the parameters, which have been used in the genetic algorithm subcode, appear in table 3.
Table 3 Parameters of the genetic algorithm sub-code
Population size PS
40
Number of generations
130
Chromosome length ChrL
56
Crossover probability CP
0.40 to 0.45
Mutation/antimetathesis probability MP
0.016 to 0.018
Mutation probability has been determined on the basis of chromosome
length. The product MP  ChrL is kept close to but less than 1.
5.1.1 Results
10
The program has been run many times, in order to derive a statistically sound
estimate of its efficiency. In all runs, the error in the final results was less than
5%, which is quite acceptable. The evolution of the fitness value for a typical run
appears in the diagram of figure 2, while the respective transmissivity values
appear in table 4.
60
50,45
fitness value
50
40
30
20
13,13
7,79 6,15
5,59 4,99 4,84 4,86
3,34
10
1,75 0,48 0,41
0,05 0,04
0
1
10
20
30
40
50
60
70
80
90
100
110
120
generations
Figure 2 Evolution of the fitness value of each generation's best chromosome
Table 4 Fitness of the best chromosome and respective transmissivity values
Generation Fitness value
11
T1
T2
T3
T4
1
50.44526
7427
135
214
6976
10
13.13388
5377
263
106
747
20
7.79283
5459
375
91
744
30
6.15175
5203
373
96
879
40
5.59182
5515
377
96
843
50
4.98727
4929
381
97
811
60
4.83562
4928
383
97
819
130
70
4.85829
4888
382
97
819
80
3.33774
2025
447
101
849
90
1.74577
2020
509
101
865
100
0.47694
1998
501
101
977
110
0.40646
1993
501
101
1009
120
0.04582
1998
499
100
1001
130
0.03941
1999
499
100
1001
5.1.2 Field measurement errors
In the previous tests, exact values for the virtual field measurements have been
used. In practice, though, there is always some error, of 1-2 cm. If we assume an
error of 5 cm in each measurement, the best fitness value will be 0.20 (4∙0.05)
instead of 0. It is reasonable, then, to assume that the additional error in
transmissivity determination will be negligible, since, as shown in table 4, with
fitness values as large as 0.40, the error in transmissivities does not exceed 1%.
To confirm this assumption, exact φmi values, which appear again in the first
column of table 5, have been replaced by the erroneous ones, which are shown in
the second column. Typical results appear in column 3. The error in
transmissivity estimation is again smaller than 5%.
Table 5. Typical transmissivities for field measurements with reasonable errors
Correct φmi
Erroneous φmi
Typical results
-8.07
-8.12
2050
-20.7
-20.65
495
-26.08
-26.13
100
-11.69
-11.63
999
It should be mentioned, though, that in our application, hydraulic head
drawdowns at the control points are significant (between 10 and 40m), due to
12
rather small distance from their respective wells and to adequate well flow rates.
5.1.3 Effect of the distance between wells and control points
To further investigate the role of the distance between wells and control points,
coordinates of the latter were changed. The new distances exceeded 400 m. The
code has been run again several times. It always yielded quite satisfactory results
for the transmissivity of zone 3. But the error in the other 3 zones was very often
unacceptably large. The respective chromosome fitness values were large, too,
serving as an error warning.
A careful investigation, through the solution of the direct problem, revealed
that hydraulic head in all control points was actually dominated by the effect of
well 3, due to the small transmissivity of the respective zone.
A practical guideline resulting from this investigation, is that control points
should be close to the respective wells. Measurements at the wells are not
recommended, because of the inaccuracy in the definition of well radius and
water level oscillations.
Finally, it has been checked whether it is possible to determine zone
transmissivities separately, using the respective wells and control points only.
Numerical experiments have shown that this is not a good practice (except for the
zone with the smallest transmissivity). If, for instance, we use the well and
piezometer of zone 1, we may end up with T1= 1.01310-3 m2/s (instead of the
correct 210-3 m2/s). The reason for this failure is that there are many
transmissivity combinations, which may produce the correct piezometric head
value at the control point. Accurate determination of T1 is not guaranteed even if
one more piezometer is added in zone A.
5.2 Minimization of pumping cost from a group of wells
Minimization of groundwater pumping cost, through proper distribution of the
total flow rate to a number of existing wells, is a problem that arises quite often in
groundwater resources management. It can also be regarded as an environmental
problem, namely that of minimization of energy consumption and of the
respective environmental impact. From the mathematical point of view, it is a
13
typical optimization problem, and typical optimization techniques, e.g. linear and
non-linear programming, have been used for its solution. The objective function,
which should be minimized, is:
N
F  C   H i Qi
(2)
i 1
where C is a constant and N is the number of the wells, while Qi and Hi are the
flow rate and the distance between ground level and water level respectively, at
well i, i.e.
Hi =Elevi - φi
(3)
The basic constraint is that the sum of the well flow rates should be equal to
the water demand Qd, which is known a priori. Additional constraints may
include upper bounds of water level drawdowns at the wells [8] or at protected
areas of the flow field.
Φ = 20
F (300, 2100)
F (1800, 2100)
T3 = 0.0005
W9
W8
W7
H2 (500, 1500)
G (0, 1500)
D (200, 1500)
H1 (700, 1500)
W5
W6
W2
3
T1 = 0.0001
4
W4
2
W1
T2 = 0.001
1
A (0, 250)
W3
B (500, 0)
Figure 3.
Φ=0
C (200, 0)
System of 9 wells in an aquifer with 3 zones of different
transmissivities
The efficiency of the proposed code in this group of problems is
14
demonstrated, through application to the aquifer of figure 3. This aquifer bears 3
zones of different transmissivities, with the following values:

T1 = 0.0001 m2/s

T2 = 0.001 m2/s

T3 = 0.0005 m2/s
Aquifer's external boundary consists of two impermeable parts, namely CDE and
FGA and two constant head parts, namely ΑΒC and EF, with hydraulic head φ =
0 and φ=20, respectively.
Nine wells are available to pump a total groundwater flow rate of 250 l/s. As
shown in fig. 3, two of them are located in zone 1, four in zone 2 and three in
zone 3. Ground elevations (Elev) at the locations of the wells, with reference to
φ=0 plane, appear in table 6, together with the respective coordinates.
Table 6. Coordinates and ground elevations at the locations of the wells
well
xi
yi
Elevi
well
xi
yi
Elevi
1
250
650
5
6
1600
1200
20
2
350
1100
5
7
1500
1800
30
3
1100
300
15
8
1000
1800
35
4
1200
800
15
9
500
1800
35
5
1250
1200
18
In order to implement the proposed code, the following adjustments have been
made, regarding the chromosome structure and the evaluation process: Each
chromosome is a binary string, which represents a combination of 9 well flow
rate values Qi. To allow for the total flow rate (i.e. for 250 l/s) to be pumped from
a single well, 8 genes are needed for each Qi. Thus, for 9 wells, the chromosome
length SL is equal to 72.
Evaluation of each chromosome includes the following steps:
15
a) Chromosome adjustment to fulfill the main constraint. Each Qi corresponds to
8 genes (digits), so it may obtain any value from 0 to 255. Therefore SQ, i.e. the
sum of the 9 Qi may vary from 0 to 2295. According to the main constraint,
though, SQ should be equal to Qd = 250. To fulfill the constraint, each Qi is
multiplied by the factor Qd/SQ. In this way, proportions between well flow rates
are preserved.
b) Calculation of the hydraulic head φi at the location of each well, using the
respective set of adjusted well flow rates, by means of the boundary element subcode.
c) Calculation of Hi for each well, by means of equation 3. Then, the fitness value
VB is calculated directly from equation 2 (in which the constant C has been set to
1). As in the previous application example, the fitness of each chromosome
increases, as the value of VB decreases.
To implement the boundary element sub-code, the external and internal field
boundaries have been divided in 25 and 11 boundary elements respectively, as
shown in figure 3.
Finally, the parameters, which have been used in the genetic algorithm subcode, appear in table 7.
Table 7 Parameters of the genetic algorithm sub-code
Population size PS
40
Number of generations
130
Chromosome length ChrL
72
Crossover probability CP
0.40 to 0.45
Mutation/antimetathesis probability MP
0.013
5.2.1 Typical results
The code has been run many times, in order to derive a statistically sound
estimate of its efficiency. In all of them, the fitness value of the best chromosome
of the last generation, ranged between 26.34 and 26.41. Results of a typical run
appear in table 8. It includes generation number, the fitness value VB of the best
chromosome and the respective 9 well flow rates.
16
Table 8. Best chromosome’s fitness value and respective well flow rates (l/s)
Gen
VB
Q1
Q2
Q3
Q4
Q5
Q6
Q7
Q8
Q9
1
37.7833 14.88 21.22
3.90
10
27.3948
8.03
4.86
47.18 51.80 35.99 36.72 18.97 15.81 30.64
20
26.7135
7.18
4.94
50.49 47.80 32.54 34.78 24.01 23.56 24.69
30
26.4395
5.05
4.81
54.33 36.78 34.86 34.86 25.72 25.24 28.37
40
26.3914
4.99
4.75
57.51 36.60 34.22 34.46 25.43 24.95 27.09
50
26.3505
5.14
4.40
59.20 37.92 31.31 31.56 27.15 25.68 27.64
60
26.3465
5.12
4.39
60.98 37.32 31.22 31.46 26.10 25.85 27.56
70
26.3457
5.10
4.37
60.74 38.14 31.10 31.34 26.00 25.75 27.45
80
26.3457
5.11
4.38
60.80 37.94 31.37 31.37 26.02 25.78 27.24
90
26.3457
5.10
4.37
60.68 38.11 31.31 31.31 25.97 25.73 27.43
100 26.3457
5.10
4.37
60.68 38.11 31.31 31.31 25.97 25.73 27.43
110 26.3457
5.10
4.37
60.68 38.11 31.31 31.31 25.97 25.73 27.43
120 26.3457
5.10
4.37
60.68 38.11 31.31 31.31 25.97 25.73 27.43
130 26.3456
5.12
4.39
60.73 37.80 31.22 31.46 26.10 25.85 27.32
44.39 37.56 53.17
6.10
26.10 42.68
5.2.2 Introduction of additional constraints
In many cases of practical interest, it is required to keep Hi smaller than a certain
value, either at the wells or at certain vulnerable areas of the flow field. Such
constraints can be taken into account rather easily, by incorporating a penalty
function to the evaluation process. In this way, chromosomes that violate the
constraints are not rejected, but their fitness decreases.
As an example, a constraint regarding the shaded area of figure 3, has been added
to the previous problem, which is expressed as:
Hi < 60
(4)
To check whether this constraint is fulfilled, 4 control points on the boundary
of the shaded area are used. These points appear as small triangles in fig.3.
17
Control point 1 is the closest to well 3, control point 2 to well 1, control point 3 to
wells 2, 4, 5, 8 and 9 and control point 4 to well 6. If Hi exceeds 60 at any of
these points, the quantity:
PEN = 10(Hi – 60)2
(5)
is added to the fitness value VB of the respective chromosome. In this way, the
penalty depends strongly on the degree of violation of the constraint.
Table 9. Typical results for the constrained case (Qi in l/s, Hj in m)
case
constrained Hj
unconstrained Hj
erroneous Tj
fitness value
29.0876
26.3456
26.472
Q1
6.78
5.12
5.12
Q2
4.91
4.39
4.35
Q3
59.64
60.73
65.06
Q4
21.75
37.80
40.98
Q5
19.88
31.22
32.79
Q6
14.97
31.46
32.27
Q7
37.65
26.10
22.80
Q8
44.90
25.85
22.54
Q9
39.52
27.32
24.08
H1
44.36
49.00
H2
55.69
64.38
H3
60.03
70.49
H4
58.40
68.43
Results of a typical run, including fitness value of the best chromosome, the
respective well flow rates Qi and the values of Hj at the control points, appear in
18
column 2 of table 9. In the third column of the same table, typical results, which
have been obtained without the constraint, are presented for comparison
purposes.
It can be seen that, as a result of the constraint, the largest Hi value decreased
drastically, from 70.49m to 60.03m. Fulfillment of the constraint, though,
resulted in an increase of 10.4% in the pumping cost, which is proportional to the
fitness value.
5.2.3 Errors in transmissivity values
In the previous tests, it has been assumed the transmissivities of aquifer zones are
known exactly. In practice, though, approximations of the order of 10% are
considered as satisfactory.
If the degree of over or underestimation is the same for all transmissivities,
the optimal combination of well flow rates remains practically unchanged,
because of the proportional change of all φi values. So such errors do not affect
the optimization process, despite the resulting changes in chromosome fitness
values. The worst error combination is overestimation of some transmissivities
and underestimation of others. To investigate this case, T2 has been set to 0.0011
m2/s (instead of 0.001 m2/s), while T3 has been set to 0.00045 m2/s (instead of
0.0005 m2/s). Results of a typical run, including the “corrected” (i.e. for the real
transmissivities) fitness value of the best chromosome, together with the
respective well flow rates, appear in table 9; column 4. It can be seen that,
differences from the reference case (column 3 of table 9) are rather small, while
the increase of cost is 0.5% only. The conclusion is that reasonable errors in
transmissivity values have acceptable impact upon the proposed optimization
process.
5.3 Hydrodynamic control of a contaminant plume
Groundwater is in many cases the major source for water supply. Moreover,
aquifer restoration requires time - and money- consuming procedures. Thus,
groundwater contamination, which may be caused by disposal of pollutants into
natural water systems, is a very important environmental problem.
19
Several restoration techniques, which can be expressed as management
problems, have been developed through the years involving hydrodynamic
control and in-situ remediation. Greenwald and Gorelick [15] provide a detailed
literature review on the subject by classifying the available methodologies
according to their general goals. In their works Theodossiou et al. [16] and
Mylopoulos et al [17] presented optimization techniques for aquifer cleanup,
aiming at the minimization of treatment costs, expressed as linear or nonlinear
objective functions. In the former work, a classical one-stage optimization
approach was used, whereas in the latter a sequential optimization procedure was
proposed. According to this procedure, at the end of each time period the
effectiveness of the calculated optimum pumping strategy is checked, by
updating the plume boundary through the application of a solute transport model.
Moreover, Latinopoulos et al. [18] examined the effects of the transmissivity
changes upon the optimal solutions of the management problem.
The proposed combination of a boundary element and a genetic algorithm
code has been already used by Katsifarakis et al. [12] to study hydrodynamic
control of a contamination plume in a homogeneous aquifer. In that application,
pumping contaminated water from 2 wells, located in the interior of the plume
was taken into account. Here, the efficiency of the proposed code is further
investigated, through application to the aquifer of figure 4, which consists of 2
zones of different transmissivities.
The external boundary of the aquifer consists of two impermeable parts
(namely BCD and FGA) and two constant head parts (namely AB and DEF) with
hydraulic head φ = 0 and φ = 30m respectively. The aquifer is confined, with
thickness b = 40m and porosity n = 0.2. The transmissivities of the two zones are
T1 = 0.002 m2/s and T2 = 0.02 m2/s.
As shown in fig. 4, a contaminant plume has been formed, which moves
towards boundary AB. Coordinates and water velocities (in m/s) at 6 points of the
plume boundary, which can be used as control points, are shown in table 10. The
velocities have been calculated by means of the boundary element code.
20
Φ=30
F (0,3000)
30
29
28
E(1800,3000)
25
26
24
23
27
31
3
32
T2=0.02
2
33
W3
34
G(0,1600)
14
15
13
16
12
D(2400,2610)
W1
4
22
5
21
W2
1
6
20
19
11
10
9
T1=0.002
C(2400,1200)
8
7
17
4
3
18
2
6
5
Φ=0
1
A(0,0)
Figure 4. Contaminant plume in an aquifer with 2 zones of different
transmissivities
Table 10. Coordinates and velocities at the plume boundary.
21
point
xi
yi
Vx 106
Vy 106
1
1800
2000
+0.1193
-7.2698
2
1600
2200
-0.0957
-6.6934
3
1600
2400
-0.4292
-6.4113
4
1800
2400
-0.8161
-6.8059
5
2000
2200
-0.4807
-7.6382
6
2000
2000
+0.0284
-7.6909
B(2400,600)
Three wells are available, to prevent further contamination of the aquifer.
Their coordinates appear in table 11, together with the respective ground
elevations (Elev), with reference to φ=0 plane. Well 1 is located upflow, relative
to the plume and can be used to reduce water velocities towards AB, by pumping
clean water. Well 2 is located at the interior of the plume and can be used to
pump contaminated water. Part or all of this water can be returned to the aquifer
after treatment, through well 3, which is located downflow, relative to the plume.
Table 11. Coordinates and ground elevations at the locations of the wells
well
xi
yi
Elevi
1
1800
2600
40
2
1800
2200
40
3
1600
1900
35
The optimization task is to minimize pumping cost from the first two wells.
As in the previous application example, the function, which should be minimized,
is:
2
F   H i Qi
(6)
i 1
For injection well 3, which is not included in the cost function, two constraints
are imposed:

φ3  Elev3 (to avoid pressurized injection)

Q3  Q2 (to avoid use of additional water resources)
A set of additional constraints is needed to express the aim of the study, i.e.
containment of contamination inside its current boundaries, or even reduction of
the contaminated area. These constraints can be expressed mathematically as
inequalities, involving water velocity values Vj at suitable points of the boundary
of the contamination plume, e.g. at the aforementioned 6 control points. More
precisely, the following constraints have been introduced:
22

Vy(1)  0

Vx(2)  0

Vx(3)  0

Vy(3)  0

Vy(4)  0

Vx(5)  0

Vx(6)  0

Vy(6)  0
5.3.1 The evaluation procedure
In order to implement the proposed code, the following adjustments have been
made, regarding the chromosome structure and the evaluation process:
Each chromosome is a binary string, which represents a combination of 3
well flow rates Qi. To determine its length an upper bound for each well flow rate
Qmax should be defined. To this end, one well pumping from an infinite field,
with the same hydraulic features as zone 2 of the original one (i.e. b = 40m, n =
0.2, T = 0.02m2/s) has been considered. Its flow rate Q should equal 110 l/s, to
produce velocities of about 7.69110-6 m/s at a distance of 283 m. In this
calculation, the velocity at control point 6 and its distance from well 2 were taken
into account. To allow for a reasonable safety margin, Qmax is set to 150 l/s. Thus,
8 genes are needed for each Qi, and the chromosome length SL equals 24.
Evaluation of each chromosome starts with a comparison between Q2 and Q3.
If the absolute value of the latter is larger, it is set equal to Q2. Thus, the
respective constraint is fulfilled. Then, evaluation proceeds with the following
steps:
a) Calculation of the hydraulic head φi at the location of each well and of the
velocities Vx and Vy at the 6 control points, using the respective set of well flow
rates, by means of the boundary element sub-code.
b) Calculation of Hi for each well and of the fitness value VB, the latter from
equation 6.
c) Check for constraint violations. As far as velocities are concerned, a penalty
23
equal to 1.6 is added to VB, for each constraint violation. The number 1.6 has
been derived using the notion of the respective infinite field, as in the derivation
of Qmax. The penalty is double, if the constraint on φ3 is violated.
To implement the boundary element sub-code, the external and internal field
boundaries have been divided in 28 and 6 boundary elements respectively, as
shown in figure 4.
Finally, the parameters, which have been used in the genetic algorithm subcode, appear in table 12.
Table 12. Parameters of the genetic algorithm sub-code
Population size PS
40
Number of generations
100
Chromosome length ChrL
24
Crossover probability CP
0.40 to 0.45
Mutation/antimetathesis probability MP
0.04
5.3.2 Typical results
The program has been run several times. Typical results appear in the second
column of table 13. It includes the fitness value VB of the best chromosome, the
respective Qi and φi values at the wells and the velocities at the control points. It
can be seen that all constraints are fulfilled and that well 1 is not used at all.
Table 13. Best chromosome’s fitness value and respective Qi, φi, Vxj and Vyj
values
VB
2.2186
2.0665
Q1
0.0
0.0
w
Q2
120.0
52.0
e
Q3
120.0
52.0
l
φ1
28.96
29.10
l
φ2
21.51
25.26
s
φ3
34.60
30.46
24
c
Vx1*106
+9.6840
+4.2640
o
Vy1*106
+9.8055
+0.1295
n
Vx2*106
+11.6875
+5.0103
Vy2*106
+1.5644
-3.1150
Vx3*106
+5.2859
+2.0473
Vy3*106
-7.4000
-6.8398
Vx4*106
+0.6905
-0.1632
Vy4*106
-14.5630
-10.1673
Vx5*106
-8.4126
-3.9178
Vy5*106
-4.6006
-6.3219
Vx6*106
-0.1842
-0.0637
Vy6*106
+0.0701
-4.3278
1.6
1.3
t
r
o
l
p
o
i
n
t
s
penalty value
5.3.3 Influence of the penalty value
To investigate the role of the penalty value, it has been reduced to 1.3 (from 1.6)
and the program has been run again. Typical results appear in column 3 of table
13. It can be seen that well flow rates are clearly smaller, but one constraint,
namely Vy(6)  0, is not fulfilled. So, the respective solution is not acceptable.
The conclusion is that with inelastic constraints, penalty values should be
large enough to ensure their fulfillment.
5.3.4 Computer time requirement
The time required to run the respective program is comparatively large. This is
due to the repetitive use of the boundary element code.
6. Final remarks
The combination of boundary elements and genetic algorithms, which has been
described, offers an attractive and dependable alternative to classical optimization
techniques in the field of groundwater hydraulics. Genetic algorithms which are
based on simple mathematics, can be easily adapted to each specific problem,
25
while versatility of BEM can be fully exploited.
The relative drawback of the proposed scheme, i.e. comparatively large
computer time requirement, is offset by simplicity of input data preparation,
which saves a lot of time for the user.
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