limitations, and guidance is offered for appropriate use of the algorithm.
The primary postulate of this algorithm is that the rela- tionship between concentration and head difference con-
straints is unique and reliable. This assumption forms the basis for the updating method described in eqn 18. The use
of this assumption has two subsidiary assumptions. First, it is assumed that changing a head difference constraint does
not significantly affect those concentration values that are not associated with that head difference constraint. That is,
only a single gradient affects each concentration. Second, all concentrations that are affected by a given head differ-
ence constraint are affected in a consistent manner—for example, the concentrations increase or decrease together.
The validity and significance of these two assumptions are discussed in the examples below.
6.1 Influence of multiple head difference constraints on a single concentration
Intuitively, the assumption that a single concentration is only affected by a single head difference constraint might
be violated in circumstances where concentration at a single location is affected by pumping at several wells. Hence, one
might expect that significant competition effects might cause failure of the algorithm. While no proof can be offered
that this will never be a problem, our experience indicates that this will not be the problem that might be imagined.
For example, consider the problem depicted in Fig. 6. The problem set-up is such that the concentrations at locations g,
h, i, j, k, and l should be significantly impacted by pumping at wells 2 and 3. However, these concentrations are linked to
gradients I and II which are closely associated with well 1. Despite this apparent potential for interaction, a solution is
achieved after 53 iterations as depicted in Fig. 6b and 6c. That interference is occurring can be seen from examination
of Table 3 which shows the gradients and pumping rates associated with wells 1 and 3 along with the concentrations
at location h over the first five iterations. At iteration 3, gradient I is driven to a value of 0.8109. This increase in
gradient produces an increase in pumping at well 1 and a decrease in concentration at h. Having overshot the desired
decrease the concentration is now below the standard, the algorithm reduces gradient I at iteration 4 to get an increase
in concentration at h to just meet the standard. However, at the same iteration gradient III has increased causing well 3
to increase its pumping. This causes a further reduction in pumping due to the interference between pumping at well 3
and concentration at h. After one additional iteration, 5, the gradient increases again at I seeking to increase the concen-
tration. The new hydro-chemical regime produced by large pumping at 3 responds and the concentration increases.
Because the algorithm only utilizes information from the past step, it is able to recover from a change in the type of
response that is encountered at the concentration response point.
6.2 Multiple concentrations influencing a single head difference constraint
Another possible problem is the association of several dis- tant concentration constraints with a single head difference
constraint. The presumption made in the association of a concentration constraint with a head difference constraint
is that there exists a significant physical connection between the two and that all concentration constraints connected to
the same head difference constraint have a similar response to changes in that constraint. The appropriate level of
significance is a matter of judgment, however, multiple
Table 1. Convergence behavior for single well test problem
Iteration Gradient I
Well 1 Concentration a
1 0.0100
¹ 11 311.7
25.9518 2
0.0095 ¹
11 308.9 25.9545
3 3.9585
¹ 33 157.6
11.0713 4
5.5694 ¹
42 070.2 7.8665
5 7.0103
¹ 50 042.0
5.8580 6
7.6259 ¹
53 447.8 5.1859
7 7.7961
¹ 54 389.5
5.0187 8
7.8151 ¹
54 494.6 5.0004
9 7.8155
¹ 54 496.7
5.0000
Table 2. Convergence behavior for multiple well containment problem
Iteration Gradient I
Gradient II Gradient III
Well 12 Well 35
Concentration a Concentration b
Concentration c 1
0.0100 0.0100
0.0100 ¹
66 098.0 4.4073
4.4073 20.9961
4.44053 2
0.0095 0.0095
0.0095 ¹
66 080.4 ¹
66 081.8 4.4055
20.9982 4.4036
3 0.1757
3.7775 0.1769
0.0 ¹
347 575.8 0.0054
0.0034 0.0018
4 0.0000
2.8808 0.0000
0.0 ¹
293 935.9 0.0769
0.0626 0.0384
5 0.0000
0.0000 0.0000
¹ 65 773.1
¹ 65 774.6
4.3686 21.0336
4.3667 6
0.000 2.2025
0.0000 0.0
¹ 23 367.6
0.5292 0.5714
0.4156 7
0.000 1.7258
0.000 0.0
¹ 224 854.8
1.16181 2.3357
1.9750 8
0.000 1.0060
0.000 0.0
¹ 181 798.3
2.6305 8.6080
11.3810 9
0.0000 1.4201
0.1000 0.0
¹ 206 565.6
2.5626 4.8891
4.7512 10
0.000 1.4077
0.0962 0.0
¹ 205 827.1
2.5932 5.0148
4.9067 11
0.000 1.4092
0.0940 0.0
¹ 205 914.2
2.5896 4.9999
4.8882 12
0.0000 1.4092
0.1076 0.0
¹ 205 913.9
2.5897 5.0000
4.8883 13
0.0000 1.4092
1.1076 0.0
¹ 205 913.9
2.5897 5.0000
4.8883
Groundwater transport management 601
concentration constraints that are driven by substantially different physical phenomena linked to the same head
difference constraint can produce problems in performance of the algorithm. An example of this behavior is provided in
the test problem depicted in Fig. 7. Here, concentration constraints are linked as shown in Fig. 7a. After 19 itera-
tions, the algorithm begins to oscillate in the solution selected at each iteration. This behavior is depicted in
Table 4 for iterations 20–40. In even numbered iterations gradient I is lowered to satisfy constraint i. The pumping
rate at well 1 is dropped and constraint i is nearly satisfied. However, the decrease in pumping rates causes an increase
in concentration at constraint c, where the constraint is violated. In odd numbered iterations, gradient I is increased
to satisfy constraint c. The pumping rate is increased, and the violation of constraints is reversed. The other head
difference constraints and pumping rates do not change appreciably
during these
iterations. The
apparently counter-intuitive response to pumping at constraints c and
i is a result of the fact that they are in two different hydro- chemical regimes of the plume. On the upgradient side,
increasing pumping at well 1 causes contraction of the plume and a reduction of concentrations at c. On the down-
gradient side, increased pumping causes movement of a high concentration zone closer to constraint i. The plume
that is produced during even numbered iterations is shown in Fig. 7b. The plume produced in odd numbered iterations
is shown in Fig. 7c. It is clear that we are seeking a single head difference constraint and associated pumping rate to
satisfy conditions in two distinctly different portions of the plume. Because of this difference in response to pumping at
the two concentration constraints, no single gradient value or associated pumping rate can satisfy both constraints,
and the algorithm fails. Hence, care must be taken to avoid assigning concentration constraints to head difference
constraints that are not expected to respond in a similar fashion.
6.3 Non-monotonic convergence
