Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol21.Issue7.1998:

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Solution of the groundwater transport

management problem by sequential relaxation

**David P. Ahlfeld*, Antigoni Zafirakou**

& R. Guy Riefler

Research Center for Groundwater Remediation Design, Department of Civil and Environmental Engineering, U-37, University of Connecticut, Storrs, CT 06269, USA

(Received 3 March 1995; revised 26 December 1995; accepted 29 April 1997)

A heuristic algorithm is presented for problems which are formulated to find an optimal groundwater remediation strategy with constraints on confined groundwater flow and contaminant transport. The problem is simplified by decoupling the transport constraints from the hydraulic constraints to produce a linear hydraulic control optimization problem. The solution is obtained by an iterative process in which the constraints on hydraulic gradient are updated, using information from transport simulation, and the hydraulic control problem is solved repeatedly. In effect, the transport simulation is used to calibrate the head difference constraint values of the hydraulic control problem. The algorithm is described in detail and its convergence is demonstrated on several examples. The advantages and limitations of the algorithm are discussed.q_{1998 Elsevier Science Limited. All rights reserved}

Key words: groundwater, transport management, sequential relaxation, optimization.

1 INTRODUCTION

The remedial design problem consists of selecting the loca-tions of wells and associated rates of pumping so as to satisfy hydraulic control and solute concentration criteria while minimizing overall system cost. Numerous manage-ment models have been proposed which couple numerical optimization methods with groundwater flow and contami-nant transport simulation models (for example, Refs.1–3). One of the more computationally challenging types of management models are those where concentrations are constrained as a function of pumping rates at remediation wells. The relationship between concentration and pumping

is non-linear and non-convex. Formulations with

constrained concentrations are generally computationally intensive.4,5

In this paper, a heuristic algorithm is described and demonstrated for solving formulations in which concentra-tions are constrained. The method is based on decoupling the concentration constraints from the remaining, relatively easy, hydraulic control problem. The full problem is solved

using an iterative technique. At each iteration the hydraulic constraints are adjusted. The adjustment is based on simula-tion of contaminant transport using the optimal pumping rates from the previous hydraulic control problem solution. Iterations cease when optimal pump rates converge.

The motivation for the work presented here is to exploit the relatively low computational costs associated with the hydraulic control problem while incorporating the impact of the hydraulic control solution on contaminant transport into the decision process. The method will ultimately produce a remedial strategy which is feasible and optimal with respect to the hydraulic control problem and satisfies the constraints on concentration. The approach can be viewed as calibrating the constraints of the hydraulic control problem by examin-ing the response of the transport solution.

Another algorithm which has involved combined use of both flow and transport modeling without direct solution of the contaminant remediation problem is that by Atwood and Gorelick.6 These authors presented a scheme in which a central extraction well, with specified location and rate, was used and a set of hydraulic control wells around the perimeter of the plume were selected to produce a capture zone. In their approach, the hydraulic control wells had to be outside the plume perimeter. A transport model was used to predict the approximate location of the plume over time for

591

*Corresponding author.

Present address: Department of Civil and Environmental Engineering, Tufts University, Medford, MA 02155, USA.

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purposes of defining candidate well locations. The hydraulic control model was solved and the resulting impact on trans-port was simulated. Modifications were made to the candi-date well locations to insure that the hydraulic control wells remained outside of the plume boundary.

Datta and Peralta7 proposed a regional groundwater management model in which a convective transport model was used to modify the solution of a regional hydraulic control model. In their case, the derivative of concentration in a cell with respect to head in that same cell was calculated analytically from the finite difference approximation. The impact of head changes on concentration in adjacent cells was not directly incorporated.

2 MODELING GROUNDWATER FLOW AND CONTAMINANT TRANSPORT

The system under study is represented by the well known three-dimensional, coupled model for areal groundwater flow and transport of a non-decaying, adsorbing, desorbing solute in a confined aquifer, and consists of three partial differential equations with appropriate boundary conditions:

=_·K·=_hþ X

k¼1

qkd(xk, yk, zk)¼0 (1)

v¼ ¹K·=h (2)

=·vD·=c¹v·=c¹

Xnp

k¼1

qk(c¹c0k)d(xk, yk, zk)¼vR

]c ]t

(3) where

K¼hydraulic conductivity tensor (l/t),

h¼hydraulic head (l),

np¼number of pumps,

qk ¼pump rate for pump located at point (xk, yk, zk) (l3/t),

d(xk, yk, zk) ¼dirac delta function evaluated at point (xk, yk, zk) (1/l3),

v¼average Darcy velocity vector (l/t),

v¼porosity of aquifer medium (dimensionless),

c¼contaminant concentration (m/l3),

c0 k ¼ contaminant concentration in pumped fluid at

pump point k (m/l3),

R¼retardation coefficient (dimensionless),

D¼hydrodynamic dispersion tensor (m2/t).

3 OPTIMIZATION FORMULATION FOR HYDRAULIC CONTROL

The method to be described here is applicable to problems in which constraints are placed on both heads and concen-trations. Ahlfeld and Heidari3provide a detailed derivation

of head constrained problems. For purposes of demon-strating the approach the following optimization formula-tion is considered

minimizeX

j[J

bjqj (4)

such that

hi$hpi,l i[L (5)

h_i#hp

i,u i[U (6)

hi1¹hi2$di i[D (7)

qj#qpj j[J (8)

ci,T(q)#cpi i[I (9)

where q represents the vector of all extraction rates at candidate well locations, qjis the extraction rate at location

j, and J is the set of candidate well locations made available

to the optimizer. At optimality, the model sets a given pump rate to a positive value indicating withdrawl at that well or zero if no pumping is needed.

The objective function (eqn (4)) provides a measure of cost wherebjis a cost coefficient on pumping at location j. Constraint eqn (5) and (6) place bounds on the head at node

i, where hi , l* and hi , u* are specified lower and upper bound values, and hiis the head at location i that results from the stresses described by q as simulated by eqn (1). The sets L and U indicate the node points at which, respectively, lower and upper bounds are placed on the heads. Constraint eqn (7) is used to force a gradient in the hydraulic flow field, either horizontally or vertically. Here, di is a specified bound on the difference in heads at node points i1 and i2 (hi 1and hi 2). D is the set of indices which identify node pairs at which requirements on the differences in heads are imposed. Constraint eqn (8) limits extraction rates to the bound expressed by qj*.

Finally, constraint eqn (9) requires that the concentration be less than or equal to a prespecified value at the end of the planning horizon at observation points within the system. Here, ci* is the specified maximum concentration bound at node i, ci, T(q) is the concentration at node i at the last time period of the simulation as a function of pump rates, and I is the set of nodes at which system concentration behavior will be observed.

Because of the linear form of eqn (1), constraint eqn (5)– (8) are linear functions of pumping. Thus, if constraint eqn (9) is neglected the resulting problem is a linear program3 which can be solved with the widely used simplex method.8 The largest cost associated with solving the linear hydraulic control problem is the calculation of the response matrix. The response matrix consists of derivatives of hydraulic head with respect to pumping. The response matrix is often computed by perturbation which requires a flow simu-lation for each candidate well.3,9The response matrix will not change during execution of the sequential relaxation

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algorithm. At each iteration only the right hand side of the linear program is changed. Repeated solution of the linear program involves relatively small computational cost. The computational advantage described here depends on the linear form of all the constraint equations except those involving concentration. Therefore, the aquifer thickness must be assumed independent of head; however, additional linear constraints can be added to the formulation presented here.

4 HEURISTIC ALGORITHM FOR OPTIMAL REMEDIAL DESIGN

The formulation described above is solved through a proce-dure involving decoupling and iteration. If the concentration constraint (eqn (9)) is relaxed the resulting problem is a linear program which is solved with modest computational effort. Given a particular solution to the hydraulic control problem, a simulation of the effect of the particular pumping scheme on contaminant transport is performed. If the con-centration constraints do not satisfy certain criteria then the constraints on head difference (which are synonymously referred to as gradient constraints) are adjusted and the hydraulic control problem is resolved.

The functional relationships can be identified by defining

L(d) as the linear programming operator which solves the

relaxed problem given by eqns (4)–(8) mapping the head difference constraint vector, d, to pumping rates (all con-straints other than eqn (7) are assumed fixed), and S(q) as the simulation model operator which solves eqns (1)–(3) and maps pumping rates to concentrations.

When viewed as a sequential process, it can be seen that the concentration vector, c, is a function of the head differ-ence constraints posed in the hydraulic control problem. In functional form this can be written as

q¼L(d) (10)

c¼S(q) (11)

c¼S(L(d)) (12)

The key to the algorithm presented here is determination of new head difference constraints after evaluation of eqn (12). It is presumed that a robust relationship between head difference constraints and concentration constraints is present. This relationship can be assured by careful placement of candidate wells, head difference constraints and concentration constraints in close proximity, so that each concentration constraint can be unambiguously asso-ciated with a head difference constraint. The guiding requirement for constraint placement is to insure that modification of head difference constraints will impact pumping rates at candidate wells which, in turn, will impact concentrations at the associated concentration con-straint locations. If a head difference concon-straint does not

affect the optimal pumping (e.g. a head difference straint that is always satisfied), then concentration con-straints associated with that head difference constraint cannot be controlled.

The algorithm development proceeds by defining, for each head difference location i, a unique set of concentra-tion observaconcentra-tion points, Ii, such that Ii∩Ij ¼ 0 for all i not

equal to j and I1∪I2∪ …Ii∪ …In¼I where n is the

number of head difference constraint pairs in D. A first order Taylor Series expansion of c, as described in eqn (12), about the head difference constraint value is per-formed. The vector c can be approximated in terms of its individual elements by

ckþ1

l ¼c k lþ

]cl

]d_i(d

kþ1

i ¹dik) l[Ii, i¼1, …, n (13)

where the superscripts indicate the iteration level. Note that it is implicitly assumed in eqn (13) that the concentration depends only on the ith head difference constraint. The criteria for selection of di

kþ1

is that the concentration that results from imposition of the new head difference con-straint value will satisfy concon-straint (9). This criteria is approximated by imposing the requirement on eqn (13) that dik

þ1_{be selected so that}

cklþ1#cpl (14)

Substituting eqn (13) into eqn (14) and rearranging yields

cklþ

]_c_l ]di

(dkþ1

i ¹d k

i)#cpl (15)

dkiþ1#d k i þ

(cp

l ¹c k l)

]cl=]di

(16) If the partial derivative in eqn (16) is approximated by

]cl

]di

¼ (c

k l¹c

k¹1

l )

(dk i ¹d

k¹1

i )

(17) then eqn (16) can be used to provide a proposed change in head difference constraint based on each of the associated concentration constraints. Hence, for the ith head difference constraints the proposed changes are given by

dkþ1

i,l #d k i þ

(cp

l¹c k l)

(ck l¹ck

¹1

l )

(dk i ¹d

k¹1

i ) l[Ii (18)

eqn (18) yields multiple proposed head difference con-straint values, di,lk

þ1_{, if the head difference constraint has}

multiple concentration constraints with which it is asso-ciated. In this circumstance, a prioritization method must be devised. Considering the inequality described in eqn (18), four cases can arise. They can be examined by con-sidering two factors: the sign of(cp

l ¹ c k

l)and the sign of

the gradient in eqn (17). To facilitate this consideration define

D¼ (c p

l ¹c k l)

(ck l¹ck

¹1

l )

(dk i ¹dk

¹1

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Proposed head difference constraint changes which derive from locations where the concentration constraint are not satisfied, i.e. cl . cl*, are given priority over all other proposed constraint changes regardless of the sign of eqn (17). To satisfy all concentration constraints at the next iteration, the maximum head difference change proposed is selected. Thus, the ith head difference constraint is chosen to satisfy

dikþ1¼d k i þmax

l[Ip

lDlsign(D) (20) where the set Ii* is defined so that l[Iipif and only if l[Ii

and cl.cpl. Note that the fact that eqn (19) may take on a

positive or negative sign is incorporated.

If all the concentration constraints associated with the ith head difference constraint are satisfied as inequalities, then the head difference constraint can be relaxed until at least one of the concentration constraints is binding. The direc-tion of change of the head difference constraint needed to achieve a binding concentration constraint will depend on the sign of eqn (17). The magnitude of the change is deter-mined by the minimum proposed head difference change, since the observation point which generates the minimum proposed head difference is estimated to be the first to reach its bound. Thus, when cl # cl* for all l[Ii, the ith

con-straint head difference is chosen to satisfy

dikþ1¼d k i þmin

l[Ii

lD_lsign(D) (21) The algorithm can be stated concisely using the following definitions;

qk ¼vector of pump rates at iteration k,

ck¼vector of concentrations at locations from set I at iteration k,

dk ¼vector of head difference requirements at loca-tions from set D at iteration k,

d0 ¼ an initial guess of the head difference

require-ments, and

G(c)¼the head difference updating operator, summar-ized in eqns (19)–(21), which maps concentration to updated head difference constraint values.

The algorithm proceeds as follows; Step 1: Initialization

set q0¼` set d0¼d0

set c0¼S(L(d0)) set d1¼d0þdd

set k¼1

Step 2: Solution of hydraulic control model set qk¼L(dk)

Step 3: Test for convergence if_kqk ¹qk¹1_{` ,} _d_stop

else continue

Step 4: Solution of transport model set ck¼S(qk)

Step 5: Update head difference requirements set dkþ1

¼G(ck) Step 6: Iterate

set k¼kþ1 ; go to Step 2

Here dd is a perturbation of the initial head difference

constraint value, k…k is a suitable norm of the indicated quantity, and d is a specified convergence criteria. Eqns (19)–(21) define, in scalar form, the vector relation required in Step 5 of the algorithm.

It should be noted that the test for convergence is imposed on the pumping rates. Since pumping rates are the primary decision variables of the formulation, their use in testing for convergence is a natural choice. Examination of eqns (19)– (21) reveals that head difference constraints are not modi-fied, if all the concentration constraints are satismodi-fied, and at least one concentration constraint associated with each head difference constraint is binding. If the head difference con-straints are not modified, then the new pumping rates that are derived from Step 2 will be unchanged from the previous iteration. Hence, convergence of the optimal pumping rates implies convergence of the head difference constraints, and in turn, convergence of the concentration constraints. Cases may arise where all concentration constraints associated with a head difference constraint are satisfied as inequal-ities. Evaluation of eqn (21) would then dictate a change in the head difference constraint value. However, if the modified head difference constraint is already satisfied, then the optimal pumping rates may not change at the next iteration and the algorithm will terminate. As long as concentration constraints are satisfied, such termination is acceptable.

One complication that may arise in execution of this algorithm is that a linear program created by updating of the head difference constraints in Step 5 may prove infeasible when Step 2 is evaluated in the next iteration. A direct method for accommodating this problem is to allow the updating algorithm to select head differences according to the rules discussed above and attempt to solve the control problem. If the resulting control problem is infeasible, then all gradient changes can be reduced by a factor. To quantify this, define an updating factor alpha, with 0#a#1, so that the updated gradients are calculated as

dkþ1¼dkþa(dkþ1¹dk) (22)

where ˆdkþ1 is the new head difference constraint used in step 2 of the algorithm. If the problem in step 2 is infeasible, we seek to maximize alpha such that the con-straints are feasible.

5 DEMONSTRATION OF THE ALGORITHM The algorithm described above has been implemented in a single code which couples and repeatedly executes the respective flow and transport equations. In the tests to be conducted here these equations are solved using the codes

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MODFLOW10for three-dimensional groundwater flow and MT3D11 for three-dimensional solute transport. Both of these codes employ the finite difference method. Testing of the algorithm that is presented here has been performed on a hypothetical problem solved on the 50003_{2300 foot} finite difference grid shown in Fig. 1. A uniform grid spacing of 100 feet is used over the entire domain. Two model layers are represented, each with a thickness of 100 feet, although, the model parameters in each layer are iden-tical and, for the results presented here, all constraints and pumping wells are located in the upper layer. The hydraulic conductivity is a uniform 10 feet/day, the porosity is 0.25, the non-pumping regional gradient is 0.02 and the disper-sivities are 45 and 17.5 feet, respectively, in the longitudinal and transverse directions. A plume is generated on this grid from a Dirichlet source with simulation conducted over a 10 year period. The resulting plume, which will serve as initial conditions for subsequent optimization runs, is shown in Fig. 2.

In a first example, designed to demonstrate the behavior of the algorithm, a single pumping well is placed in the domain along with a head difference constraint and concen-tration observation point. The arrangement of these problem

elements are shown in Fig. 3(a), where the open circle repre-sents the candidate well location, the heavy line reprerepre-sents the location of the head difference constraint, the asterisk represents the location of the concentration observation point and the dashed line indicates the head difference con-straint to which the concentration concon-straint is associated. The candidate well, head difference constraint and observa-tion point are all numbered for convenience.

The concentration constraint imposed at the observation point is 5 concentration units after 10 years of remedial pumping. The algorithm converges after nine iterations from an initial head difference constraint of 0.01 and initial concentration at the concentration observation point of 26. The convergence behavior of the gradients required in each hydraulic control problem, the pumping rates, and the resulting concentrations are shown in Table 1. The first iteration uses the specified initial guess of the gradient value of 0.01. The second iteration uses a user-defined perturbed value of the initial gradient to produce the first derivative of the form of eqn (17). Once begun, all the variables in the algorithm follow a rapid, monotonic convergence to the solution where the constraint is satisfied to five digits of accuracy. The steady hydraulic head

Fig. 1. Cell-centered finite different grid for test problem.

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distribution after the optimal pumping has been imposed and the resulting state of the plume after 10 years is shown in Fig. 3(b) and 3(c) respectively. The head distribu-tion is depicted with contours at 5 foot intervals as labeled on the figure. The concentration distribution is represented by iso-concentration contours at concentration units of 1, 5, 10, 50, 100 as labeled.

The ability of the algorithm to solve problems with

different arrangements of candidate wells and constraints is demonstrated in the next two test problems. A common design problem is the selection of a set of wells within a contaminant plume that produce containment of the plume. This problem is typified by the constraint setup depicted in Fig. 4(a), where 35 candidate wells are placed within the plume, and three hydraulic head difference constraints are set at the down gradient end of the plume. One concentration

Fig. 3. Single well test problem: (a) problem set-up; (b) hydraulic head contours at 5 foot intervals using optimal pumping; (c)

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observation point is associated with each hydraulic head difference constraint as shown. The resulting problem is solved successfully in 13 iterations. As can be seen in Fig. 4(b) and 4(c), well 35 is selected and produces a scheme which meets the specified standard at the observation points. This problem is symmetric about the longitudinal axis of the plume. An alternate solution would consist of pumping at well 12 which would produce a plume that is symmetric to

that shown here. The choice of pumping wells can be seen by examining the iterative development of the solution for this problem depicted in Table 2. The algorithm begins with symmetric head difference constraints and a nearly symmetric solution with pumping at both wells 12 and 35. However, because of round-off errors in computation of the hydraulic head solution the pumping is slightly different at the two wells and the concentration at the symmetric points

Fig. 4. Multiple well containment problem: (a) problem set-up; (b) hydraulic head contours at 5 foot intervals using optimal pumping; (c)

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a and c are slightly different. This difference tips the solu-tion towards use of well 35. Note that at iterasolu-tion 5 the gradients are set to zero in an attempt by the algorithm to increase the concentrations at the observation points to their standard. This over-compensation produces violations at observation point b which is corrected in subsequent iterations. It should also be noted that in these test problems,

a minimum head difference constraint value of 0.0 is established. In addition, a rule is imposed that introduces a specified increase in head difference constraint, when any of the constraints associated with a head difference con-straint at 0.0 are violated. This rule is implemented in iteration 9 of this test problem.

Another example is depicted in Fig. 5. Here, the scheme

Fig. 5. Longitudinal well containment problem: (a) problem set-up; (b) hydraulic head contours at 5 foot intervals using optimal pumping;

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consists of pumping along the longitudinal axis of the plume to prevent transverse spreading of contaminant. The head difference constraints are associated with clusters of con-centration constraints located on the sides of the plume. The associations are indicated by the dashed lines connecting head difference constraints with clusters of concentration

constraint locations on Fig. 5(a). The wells are placed away from the centerline of the plume to produce a non-symmetric solution. This problem is readily solved in 14 iterations with pumping rates of 12 746, 11 524 and 27 004 cubic feet per day respectively at wells 1, 2 and 3. The hydraulic head and concentration surfaces produced by

Fig. 6. Head difference constraint interference problem: (a) problem set-up; (b) hydraulic head contours at 5 foot intervals using optimal

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the optimal pumping solution are depicted in Fig. 5(b) and 5(c), respectively. Concentration constraints at observation points f and r are binding at the solution. The head difference constraints that control the pumping at wells 1 and 3 are in turn controlled by the nearby binding concentra-tion constraints. The pumping at well 2 is due in large part to the specified minimum head difference constraint of 0.0.

6 DISCUSSION OF POTENTIAL LIMITATIONS OF THE ALGORITHM

Fig. 7. Multiple concentration constraint interference problem: (a) problem set-up; (b) concentration contours at 1, 5, 10, 50 and 100

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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. 6(b) and 6(c). 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 ¹11 311.7 25.9518

2 0.0095 ¹11 308.9 25.9545

3 3.9585 ¹33 157.6 11.0713

4 5.5694 ¹42 070.2 7.8665

5 7.0103 ¹50 042.0 5.8580

6 7.6259 ¹53 447.8 5.1859

7 7.7961 ¹54 389.5 5.0187

8 7.8151 ¹54 494.6 5.0004

9 7.8155 ¹54 496.7 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 ¹66 098.0 4.4073 4.4073 20.9961 4.44053

2 0.0095 0.0095 0.0095 ¹66 080.4 ¹66 081.8 4.4055 20.9982 4.4036

3 0.1757 3.7775 0.1769 0.0 ¹347 575.8 0.0054 0.0034 0.0018

4 0.0000 2.8808 0.0000 0.0 ¹293 935.9 0.0769 0.0626 0.0384

5 0.0000 0.0000 0.0000 ¹65 773.1 ¹65 774.6 4.3686 21.0336 4.3667

6 0.000 2.2025 0.0000 0.0 ¹23 367.6 0.5292 0.5714 0.4156

7 0.000 1.7258 0.000 0.0 ¹224 854.8 1.16181 2.3357 1.9750

8 0.000 1.0060 0.000 0.0 ¹181 798.3 2.6305 8.6080 11.3810

9 0.0000 1.4201 0.1000 0.0 ¹206 565.6 2.5626 4.8891 4.7512

10 0.000 1.4077 0.0962 0.0 ¹205 827.1 2.5932 5.0148 4.9067

11 0.000 1.4092 0.0940 0.0 ¹205 914.2 2.5896 4.9999 4.8882

12 0.0000 1.4092 0.1076 0.0 ¹205 913.9 2.5897 5.0000 4.8883

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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. 7(b). The plume produced in odd numbered iterations is shown in Fig. 7(c). 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

In conventional gradient based algorithms, the objective function (if minimized) is expected to decrease mono-tonically at each iteration. In fact, the proof of this descent is a necessary (but not sufficient) condition for proof of convergence.8However, the present algorithm contains no guarantee that such descent will occur. Instead, the algo-rithm can be viewed as one in which the head difference constraints are adjusted to best coincide with the concentra-tion constraints. The initial constraint space is bounded by linear hydraulic head difference constraints and non-linear

Table 3. Initial iterations for head difference constraint interference problem

Iteration Gradient I Gradient III Well 1 Well 3 Concentration h

1 0.0100 0.0100 ¹10 429.8 ¹10 301.9 6.0151

2 0.0095 0.0095 ¹10.429.8 ¹10 299.4 6.0158

3 0.8109 0.0095 ¹14 249.1 ¹9915.1 4.2864

4 0.4802 1.0095 ¹16 379.3 ¹14 282.7 1.3545

5 0.8914 2.3288 ¹15 358.5 ¹22 604.3 5.6478

Table 4. Iterations for the multiple concentration constraint interference problem showing oscillation of head difference constraint values.

Iteration Gradient I Gradient III Gradient IV Well 1 Well 2 Well 3 Concentration c Concentration i

20 39.3390 0.6063 5.5601 ¹188 433.3 ¹5146.3 ¹402.5 6.2879 3.5690

21 62.5194 0.6793 0.5590 ¹293 581.3 ¹212.6 ¹149.8 5.0017 14.8626

22 42.2761 0.6678 0.5590 ¹201 764.0 ¹4803.6 ¹4102.5 6.873 4.2135

23 62.5515 0.6777 0.5590 ¹293 726.8 ¹196.2 ¹144.2 5.0004 14.8659

24 43.7731 0.6762 0.5590 ¹208 554.9 ¹4504.8 ¹3807.4 5.9922 5.3748

25 62.5592 0.6774 0.5590 ¹293 761.5 ¹193.3 ¹142.7 5.0001 14.8664

26 43.0313 0.6773 0.5590 ¹205 190.6 ¹4681.1 ¹3.951.7 6.0384 4.7169

27 62.5604 0.6774 0.5590 ¹293 767.2 ¹192.8 ¹142.5 5.0000 14. 8664

28 43.5760 0.6774 0.5590 ¹207 661.1 ¹4556.6 ¹3845.4 6.0043 5.1760

29 62.5605 0.6774 0.5590 ¹293 767.8 ¹192.8 ¹142.5 5.0000 14.8663

30 43.2311 0.6774 0.5590 ¹206 096.9 ¹4636.0 ¹3912.6 6.0259 4.8675

31 62.5605 0.6774 0.5590 ¹293 767.8 ¹192.8 ¹142.5 5.0000 14.8664

32 43.4873 0.6774 0.5590 ¹207 258.9 ¹4577.1 ¹3862.7 6.0098 5.0895

33 62.5606 0.6774 0.5590 ¹293 768.2 ¹192.8 ¹142.5 5.0000 14.8664

34 43.3127 0.6774 0.5590 ¹206 467.2 ¹4.6172 ¹3896.7 6.1207 4.9302

35 62.5606 o.6774 0.5590 ¹293 768.1 ¹192.8 ¹142.5 5.0000 14.8664

36 43.4480 0.6774 0.5590 ¹207 080.5 ¹4586.1 ¹3870.3 6.0123 5.0518

37 62.5607 0.6774 0.5590 ¹293 768.5 192.7 ¹142.5 5.0000 14.8664

38 43.3470 0.6774 0.5590 ¹206 622.6 ¹4609.3 ¹3890.3 6.0186 4.9596

39 62.5606 0.6774 0.5590 ¹293 768.2 ¹192.8 ¹142.5 5.0000 14.8664

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concentration constraints. During the course of the algo-rithm the concentration constraints are relaxed and the sim-plified problem is solved. The resulting solution is tested to see if any of the relaxed concentration constraints are violated or not binding. The head difference constraints are then translated within the constraint space by adjustment of the head difference constraint values to better coincide with the concentration constraints in the vicinity of the solu-tion. Our extensive testing has indicated that situations may arise in which small changes in head difference constraint values produce large rearrangements of pumping and sig-nificant changes in concentration. In these cases the objec-tive function may not decrease, and the solution may move further from satisfaction of the concentration constraints. However, the algorithm generally recovers from these cir-cumstances and eventually converges to a solution. 6.4 Computational performance

The primary advantage of the approach taken in this algo-rithm is computational efficiency. Repeated solution of the linear program involves relatively small computational cost, since only the right hand side of the linear program is modified. The only significant computational costs are initial calculation of the response matrix and repeated evaluation of the transport equation which is performed once in each iteration of the algorithm. Other approaches to solving eqns (4)–(9) have included direct solution by gradient based non-linear optimization methods, the use of dynamic programming, genetic algorithms, simulated annealing and neural networks. In each of these cases the computational time is dominated by the need to repeatedly solve eqn (3). While the limited testing performed to date on the present algorithm is insufficient to confirm computa-tional superiority, some general observations can be made. The examples presented here that successfully converged required less than 53 iterations with one evaluation of eqn (3) per iteration. Ahlfeld et al.4report on an application that is solved using gradient based optimization combined with adjoint sensitivity analysis that requires over 400 evalua-tions of eqn (3). Rogers and Dowla12report an application solved using neural networks that requires over 200 evalua-tions of eqn (3). Marryott et al.13 report an application solved with simulated annealing that required over 2000 evaluations of eqn (3), although they report that subsequent improvements to the algorithm can reduce this substantially. Culver and Shoemaker5 utilize dynamic programming for solution of multiple time period problems. While providing flexibility in formulation, these algorithms are generally very computationally intensive, requiring many times the computational effort required for the present algorithm. Comparison of these reported computational results must be treated with caution due to differences in computational costs for other portions of the algorithms and differences in the applications reported; however, it is apparent that the present algorithm has the potential to solve problems of this type with reasonable computational effort.

7 CONCLUSION

A new heuristic algorithm has been proposed for solving problems which are formulated to find an optimal ground-water remediation strategy with constraints on confined groundwater flow and contaminant transport. The algorithm is based on solving the hydraulic control problem repeatedly while modifying the head difference constraints. These constraints are modified in response to the impact of optimal pumping on simulated concentration. The algorithm requires that each concentration constraint be associated with a single head difference constraint, so that a relation-ship can be developed between the head difference con-straint and the concentration concon-straint.

The algorithm has been demonstrated to successfully converge on several test problems. These problems encom-pass several classes of possible arrangements of locations of candidate wells, head difference constraints and concentra-tion constraints for which the algorithm is intended. The algorithm can fail when multiple concentration constraints that are associated with a single head difference constraint experience significantly different responses to changes in pumping. Interference by multiple wells in the relationship between concentration and gradient constraints can also cause failure of the algorithm in some circumstances. How-ever, the use of only prior iterate information assists the algorithm in recovering from such interference events. The computational performance of this algorithm shows promise; however, problems described above with problem set-up, interference and convergence should cause caution to be exercised in the implementation of this approach.

ACKNOWLEDGEMENTS

This work was funded in part by support from Dupont Corporation, NSF Grant #BES-9311559, and from the Research Center for Groundwater Remediation Design. References to software are for identification purposes only.

REFERENCES

1. Gorelick, S.M. A review of distributed parameter ground-water management modeling methods. Water Resources Research, 1993, 19(2).

2. Willis, R. and Yeh, W. W.-G., Groundwater Systems Plan-ning and Management. Prentice Hall, Englewood Cliffs, NJ, 1987.

3. Ahlfeld, D.P. and Heidari, M. Applications of optimal hydraulic control to ground-water systems. Journal of Water Resources Planning and Management, 1994, 120, 350–365.

4. Ahlfeld, D.P., Pinder, G.F. and Mulvey, J.M. Contaminated groundwater remediation design using simulation, optimiza-tion, and sensitivity theory 2. Analysis of a field site. Water Resources Research, 1988, 24(3).

5. Culver, T.B. and Shoemaker, C.A. Dynamic optimal control for groundwater remediation with flexible management per-iods. Water Resources Research, 1992, 28(3), 629–641.

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6. Atwood, D.F. and Gorelick, S.M. Hydraulic gradient control for groundwater contaminant removal. Journal of Hydrology, 1985, 76, 85–106.

7. Datta, B. and Peralta, R.C. Optimal modification of regional potentiometric surface design for groundwater contaminant containment, Transactions of the Amer. Soc. of Agric. Engrs., 1986, 29(6), 1611–1622.

8. Luenberger, D. G., Linear and Nonlinear Programming. Addison-Wesley, Reading, MA, 1984.

9. Ahlfeld, D.P., Page, R.H. and Pinder, G.F. Optimal ground-water remediation methods applied to a superfund site: from formulation to implementation. Groundwater, 1995, 33. 10. McDonald, M. G. and Harbaugh, A. W., A modular

three-dimensional finite-difference ground-water flow model.

Techniques of Water-resources. Investigations of the United States Geological Survey, U.S. Geological Survey, 1988. 11. Zheng, A. C., A Modular Three-dimensional Transport

Model for Simulation of Advection, Dispersion, and Chemi-cal Reactions of Contaminants in Groundwater Systems. S.S. Papadopulos and Assoc., Inc., Bethesda, MD, 1992. 12. Rogers, L.L. and Dowla, F.U. Optimization of groundwater

remediation using artificial neural networks with parallel solute transport modeling. Water Resources Research, 1994, 30(2), 457–481.

13. Marryott, R.A., Dougherty, D.E. and Stollar, R.L. Optimal groundwater management 2. Application of simulated annealing to a field-scale contamination site. Water Resources Research, 1993, 29(4).

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6 DISCUSSION OF POTENTIAL LIMITATIONS OF THE ALGORITHM

Several limitations to the algorithm are possible. These relate to limitations of the general nature of the relationship between gradients and concentrations and poor problem set-up. Several examples are provided to demonstrate these Fig. 7. Multiple concentration constraint interference problem: (a) problem set-up; (b) concentration contours at 1, 5, 10, 50 and 100

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limitations, and guidance is offered for appropriate use of the algorithm.

6.1 Influence of multiple head difference constraints on a single concentration

6.2 Multiple concentrations influencing a single head difference constraint

Iteration Gradient I Well 1 Concentration a

1 0.0100 ¹11 311.7 25.9518

2 0.0095 ¹11 308.9 25.9545

3 3.9585 ¹33 157.6 11.0713

4 5.5694 ¹42 070.2 7.8665

5 7.0103 ¹50 042.0 5.8580

6 7.6259 ¹53 447.8 5.1859

7 7.7961 ¹54 389.5 5.0187

8 7.8151 ¹54 494.6 5.0004

9 7.8155 ¹54 496.7 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 ¹66 098.0 4.4073 4.4073 20.9961 4.44053

2 0.0095 0.0095 0.0095 ¹66 080.4 ¹66 081.8 4.4055 20.9982 4.4036

3 0.1757 3.7775 0.1769 0.0 ¹347 575.8 0.0054 0.0034 0.0018

4 0.0000 2.8808 0.0000 0.0 ¹293 935.9 0.0769 0.0626 0.0384

5 0.0000 0.0000 0.0000 ¹65 773.1 ¹65 774.6 4.3686 21.0336 4.3667

6 0.000 2.2025 0.0000 0.0 ¹23 367.6 0.5292 0.5714 0.4156

7 0.000 1.7258 0.000 0.0 ¹224 854.8 1.16181 2.3357 1.9750

8 0.000 1.0060 0.000 0.0 ¹181 798.3 2.6305 8.6080 11.3810

9 0.0000 1.4201 0.1000 0.0 ¹206 565.6 2.5626 4.8891 4.7512

10 0.000 1.4077 0.0962 0.0 ¹205 827.1 2.5932 5.0148 4.9067

11 0.000 1.4092 0.0940 0.0 ¹205 914.2 2.5896 4.9999 4.8882

12 0.0000 1.4092 0.1076 0.0 ¹205 913.9 2.5897 5.0000 4.8883

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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. 7(a). 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