Optimal Aquifer Dewatering Schemes for Excavation of Collector Line

Mehmet Tokgoz1; Koray K. Yilmaz2; and Hasan Yazicigil3

Abstract: Groundwater flow modeling and optimization techniques are used jointly to determine the optimal dewatering system design for a trench type excavation ͑with 3,000 m total length͒ to be opened below the groundwater table in the Aksaray Organized Industrial District, Aksaray, Turkey. Four dewatering systems are simulated, optimized, and compared to choose the best system design. Two of the models are designed to minimize only the total amount of pumpage ͑linear programming models͒, while the other two explicitly account for the completion cost of pumping wells as well as the total pumpage amount ͑mixed-integer programming models͒. Steady-state and transient simulation models are linked to linear and mixed-integer optimization models using response functions. Since the aquifer is unconfined, an iterative correction procedure is applied to account for the nonlinearity of the system. Optimal solutions of the models are compared on the basis of completion time of the construction, pumpage amount, number of pumping wells, and efficiency and applicability of the system in practice. Results show that the transient mixed-integer programming model depicts the most favorable solution with respect to the number of wells required, the completion time of construction, and the total volume of water pumped. DOI: 10.1061/͑ASCE͒0733-9496͑2002͒128:4͑248͒ CE Database keywords: Optimization; Aquifers; Dewatering; Excavation.

Introduction of simulation and optimization techniques. These techniques have With the aid of modern scientific research and developments in been in use in groundwater management studies since the early construction technology, new construction projects have ex- 1970s. Excellent reviews on the types of groundwater manage- panded in response to increasing world population and industri- ment models and their applications are provided by Gorelick alization. Due to the acceleration in constructional works, suitable ͑1983͒ and Yeh ͑1992͒. Applications of joint simulation and op- areas for construction have diminished, resulting in selection of timization modeling in dewatering studies are presented by less appropriate sites with foundation problems. High groundwa- Aguado et al. ͑1974͒, Aguado and Remson ͑1980͒, and Galeati ter tables are one of the most frequently encountered foundation and Gambolati ͑1988͒. problems, requiring deep excavations in pervious soils below the The purpose of this study is to develop an optimal dewatering water table. To obtain the best working conditions and slope sta- system design for an excavation in order to construct a collector bility, appropriate dewatering systems are safer and more eco- line for the drainage of the Aksaray Organized Industrial District nomical when compared with other methods for sites requiring area in central Turkey. The excavation will be opened as a trench deep excavations below the water table. The magnitude and cost temporarily, and then a drainage pipe with surrounding granular of a dewatering project depend on the size and depth of the re- filter materials will be placed into it. The objective is to reduce quired excavation and the length of time the dewatered condition the groundwater level to a reasonable depth that allows construc- must be maintained. The kind of dewatering system needed and tion of the excavation system to be made under dry conditions. its cost also depend in large measure on subsurface soil and The dewatering system is designed through application of simu- groundwater conditions, which must be completely evaluated in lation and optimization techniques using the response function advance of the start of construction. The most effective and effi- approach. Since the aquifer in which the excavation will be made cient dewatering system can be determined by the combined use is of the unconfined type, it exhibits a nonlinearity in the govern- ing groundwater flow equation. An iterative correction method is 1Geological-Geotechnical Survey Engineer, District of State of High- applied for treating this nonlinearity. ways, Technical Research Dept., 34000 Kucukyali-Istanbul, Turkey. Four dewatering systems are simulated, optimized, and com- E-mail: [email protected] pared to choose the best system design. Two of the models are 2Research Assistant, Dept. of Geological Engineering, Middle East designed to minimize only the total amount of pumpage ͑linear Technical Univ., 06531 Ankara, Turkey. E-mail: [email protected] programming models͒, while the other two explicitly account for 3Professor, Dept. of Geological Engineering, Middle East Technical the completion cost of pumping wells, along with the total pump- Univ., 06531 Ankara, Turkey ͑corresponding author͒. E-mail: age amount ͑mixed-integer programming models͒. For the steady- [email protected] state models, complete dewatering of the collector line at a single Note. Discussion open until December 1, 2002. Separate discussions stage is considered. Steady-state linear and mixed-integer optimi- must be submitted for individual papers. To extend the closing date by zation models are designed to provide optimal pumping well lo- one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and pos- cations with associated pumping rates at 120 possible well loca- sible publication on February 16, 2001; approved on September 27, 2001. tions aligned on both sides of the collector line. In transient This paper is part of the Journal of Water Resources Planning and models, optimal pumpage rates are obtained for a stage-wise, pro- Management, Vol. 128, No. 4, July 1, 2002. ©ASCE, ISSN 0733-9496/ gressive dewatering system where the stages are 200 m in length. 2002/4-248–261/$8.00ϩ$.50 per page. The dewatering period for each stage is composed of three

248 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / JULY/AUGUST 2002 months—one month for obtaining the required amount of dewatering and the following two months to maintain groundwater levels at that position during construction. All models are subject to the constraints related to the system response equations, drawdown requirements in excavation cells, and drawdown and discharge limits for pumping wells. Finally, the performance of the models are compared according to required time of completion, amount of pumpage, required number of wells, and efﬁciency and applicability in practice.

Problem and Site Characteristics

The excavation site for construction of the collector line for the Aksaray Organized Industrial District is located 15 km to the south of Aksaray city in central Turkey ͑Fig. 1͒. A trench type excavation will be made below the water table and a drainage Fig. 1. Location map of study area pipe surrounded by granular ﬁlter materials placed into it. The water table is currently 1 m below the ground surface and is to be lowered 4 m along the entire 3,000 m length of the excavation, which extends in a north-south direction ͑Fig. 2͒.

Fig. 2. Site map showing topographic surface and water table contours, borehole and investigation ditch locations, and collector line

JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / JULY/AUGUST 2002 / 249 A detailed investigation of the geology and hydrogeology of the site has been conducted by Doyuran and Erol ͑1995͒ by drill- ing boreholes, digging investigation ditches, and performing field and laboratory permeability tests. The locations of the boreholes and investigation ditches are shown in Fig. 2. The site is charac- terized by a gentle ͑0.6%͒ topographic slope with small undula- tions. Both topographic and groundwater table contours decrease from south-southwest to north in the area. The field of the Aksa- ray Organized Industrial District is covered by almost 0.50 m of top soil, underlain by gravelly silty clay and silty sand. Beneath the silty clay and silty sand cover, there exist horizontal or almost horizontal units accumulated under a lake environment. These units are the following: dirty white silty clay and clayey sand, beige-white marl, beige marly limestone, light brown–light gray– occasionally dirty white silty clay, silt and silty sand, gray coarse sand with trace amounts of gravel, conglomerate with weak lime cement, sandstone, and gray sandy limestone. Between these units, lateral and vertical transitions were determined according to the borehole and investigation ditch results. The hydrogeological conditions at the site indicate largely an unconfined aquifer system composed of various layers having vertical and horizontal transitions. The bottom elevation of aquifer is almost horizontal at an elevation of 920 m. The overall hydraulic conductivity of the system as determined from field and laboratory permeability tests vary from a minimum of 1–3 m/day at the southern part of the area to a Fig. 3. Finite difference mesh and boundary conditions maximum of about 20–40 m/day at the northern part of the area. The storativity of the system, however, was assigned a value of 0.1 based on the range of specific yield values for similar litho- logic materials given by Morris and Johnson ͑1967͒, since no The model was found to be relatively more sensitive to an in- other data were available. The calculated recharge from rainfall crease rather than a decrease in calibrated hydraulic conductivi- amounts to 9.5ϫ10Ϫ5 m/day, constituting 10% of the average ties, recharge rates, and conductances used in general head yearly precipitation ͑Tokgoz 1999͒. boundaries. The calibrated simulation model was subsequently used to develop steady-state and transient response matrices that coupled the simulation model with an appropriate optimization model. Simulation Model

An accurate model of groundwater flow is a prerequisite in de- Generation of Response Matrices veloping reliable dewatering management models. An extensive modeling study of the aquifer system is conducted using available Response functions describe the influence of a unit pulse of data on aquifer hydraulic properties, water levels, and recharge pumpage on drawdown over space and time for transient models and discharge rates. The groundwater table map given in Fig. 2 is and over space for steady-state models. Two different sets of re- considered as the steady-state head distribution of the groundwa- sponse functions are developed: one for steady-state and the other ter system, since no development has yet taken place in the aqui- for transient conditions. The steady-state drawdown response fer system. The finite-difference groundwater flow simulation function in discrete form may be written as model ͑MODFLOW͒ developed by McDonald and Harbaugh ͑1984͒ is used to simulate the unconfined flow in the aquifer NPW system, which is assumed to be isotropic and heterogeneous. s͑k͒ϭ ͚ ␤͑k, j͒Q͑ j͒ (1) ϭ The finite-difference grid lines are chosen as parallel to the j 1 collector line along the north-south axis, and the cells are squares where s(k)ϭdrawdown at well k ͑L͒; ␤(k, j)ϭaverage drawdown with 50ϫ50 m dimensions. Fig. 3 shows the finite difference at well k due to a unit pumpage at well j ͑T/L2͒; Q( j)ϭaverage mesh and the boundary conditions used in the model. The eastern volumetric rate of discharge at pumping well j ͑L3/T͒; and and northwestern boundaries are represented by no-flow boundary NPWϭtotal number of pumping wells. The steady-state response conditions, since no flux is taking place along these boundaries. functions, ␤(k, j), were generated from repeated runs of the The southern, southwestern, and northern boundaries are defined simulation model for 120 pumping wells and 60 dewatering cells as general head boundaries, since groundwater inflow and outflow along the collector line by successively subjecting each of the are taking place along these boundaries. pumping wells to a discharge 1 m3/s. The resulting drawdowns After construction of the model, calibration and sensitivity are then monitored at each of the pumping wells and dewatering studies were conducted to determine the spatial distribution and cells. The assemblage of response functions ͑a total of 21,600͒ reliability of aquifer parameters. These are reported in detail else- constitutes the steady-state response matrix that is included within where ͑Tokgoz 1999͒. At the end of several simulation runs, a the constraints of the steady-state optimization models. reasonable simulated steady-state groundwater head contour map Transient response functions in the discrete form may be writ- was obtained with a root mean square error of 0.162 m ͑Fig. 4͒. ten ͑Maddock 1972͒ as

250 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / JULY/AUGUST 2002 Fig. 4. Simulated and observed head distributions

NPW n s͑k,n͒ϭ ͚ ͚ ␤͑k, j,nϪiϩ1͒Q͑ j,i͒ (2) jϭ1 iϭ1 where s(k,n)ϭdrawdown at well k at the end of the pumping period n ͑L͒; ␤(k, j,nϪiϩ1)ϭaverage drawdown at well k at the end of pumping period n due to a unit pulse of pumpage at well j applied throughout the pumping period i ͑T/L2͒; and Q( j,i)ϭaverage volumetric rate of discharge at well j during period i ͑L3/T͒. In transient models, a progressive stage-wise dewatering system is considered. Each stage consists of a model segment having cells along three columns ͑150 m͒ and four rows ͑200 m͒͑Fig. 5͒. Thus, dewatering of the entire collector line is divided into a total of 15 segments. The dewatering period for each stage is composed of three months, with one month required for obtaining the necessary amount of drawdown and the following two months to maintain groundwater levels at that position during construction. Consequently, transient response functions, ␤(k, j,nϪiϩ1), were generated from repeated runs of the transient simulation model for each stage separately for a period of three months by successively subjecting each of the pumping Fig. 5. Finite difference mesh for stage showing locations of pumping wells and dewatering cells wells ͑a total of eight wells for each stage͒ to a discharge of 1

JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / JULY/AUGUST 2002 / 251 Fig. 6. Brief representation of iterative correction procedure m3/s for the first one-month pumping period and zero units for the 1. Generate the response matrix; final two months. Drawdowns at eight pumping wells and four 2. Obtain the optimal drawdowns and pumpage rates for the dewatering cells were then monitored over the three-month problem using the linear response matrix equation; pumping period. The response functions ͑a total of 288 for each 3. Execute the nonlinear simulation program with the optimal stage͒ were then assembled to form the transient response pumpage rates and obtain the corresponding simulated draw- matrices. downs; 4. Calculate the ratio of simulated and optimal drawdowns and the root mean square error between them; Iterative Correction Method 5. If the root mean square error exceeds 0.025 m, go to step 6; otherwise, the correction is completed; The response function approach is applicable to a linear system, 6. Multiply the corresponding element of the linear response or one that is confined. In unconfined aquifers, the changing satu- matrix with the ratio of the simulated and optimal draw- rated thickness produces a nonlinearity in the groundwater sys- downs at the observation point; and tem. Since variations in well pumpages affect the saturated thick- 7. Repeat steps 2 through 5 as necessary. ness of the aquifer, the optimization model would be using Satisfactory convergence of the optimal and simulated draw- inappropriate response functions and the calculated optimal downs was achieved at three iterations for the steady-state opti- pumpage rates and drawdowns would be incorrect. Procedures mization models ͑Fig. 7͒. Objective function values converged in such as drawdown correction factors ͑Larson et al. 1977; Heidari a similar manner, indicating a stable solution. However, as ex- 1982͒, quasilinearization ͑Willis 1984; Willis and Finney 1985͒, plained subsequently, up to 13 iterations were required for the or iterative methods ͑Danskin and Gorelick 1985͒ have been ap- transient models at high pumpage rates to achieve the required plied to remove this nonlinearity. In this study, an iterative pro- convergence. The number of iterations depends on the difference cedure similar to that proposed by Danskin and Gorelick ͑1985͒ is between the unit pumpage value used during the response matrix used to treat the system nonlinearity. In this approach, solutions to generation and the calculated optimal pumpage values. the management model are used to successively update the transmissivity distribution. Updated linear response functions are then used to generate new management solutions. This iterative pro- Optimal Dewatering System Design Models cess terminates when the transmissivity distribution is stabilized. The iterative procedure applied in this study forces the optimal The dewatering system design under consideration is to determine drawdown values obtained from optimization models to be almost the optimal pumping policies for a set of pumping wells in order identical to the externally simulated ones by updating the re- to satisfy the drawdown requirements along the proposed collec- sponse matrix of the aquifer. The procedure is continued until the tor line. The required amount of drawdown along the collector root mean square error between simulated and optimal draw- line was 4.0 m. While a number of optimization models having downs is less than or equal to 0.025 m ͑i.e., the ratio of simulated different objective functions and constraint sets can be estab- and optimal drawdowns stabilizes at or around 1.0͒. The proce- lished, two objective functions were used in the models devel- dure can be summarized as follows ͑Fig. 6͒: oped herein. One of the objective functions minimizes the total

252 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / JULY/AUGUST 2002 Fig. 7. Variation in: ͑a͒ root mean square error; ͑b͒ total optimal pumpage with iterations in steady-state linear programming model pumping rate and the other minimizes both the total well comple- groundwater levels. In addition, because of the limited thick- tion cost and the total pumping rate, while maintaining the draw- ness of the aquifer, drawdowns at the pumping well sites down limitations at both pumping and dewatering cells and the were constrained to not exceed maximum allowable values. upper and lower bounds on pumping rates. In order to enhance Thus: the decision makers’ ability to select the most efficient dewatering • At the excavation site у ϭ scheme, four different optimization models are formulated and s͑k͒ smin͑k͒ k 1,...,NOP (5) ϭ solved. Two of them were designed for steady-state conditions, where smin(k) desired level of drawdown at the dewatering while the other two were for transient conditions. cell k; and NOPϭtotal number of dewatering cells ͑a total of 60͒ within the excavation site. The required amount of drawdown was 4 m at the excavation site. Steady-State Dewatering Models • At the pumping wells р ϭ s͑k͒ smax͑k͒ k 1,...,NPW (6) ϭ Steady-State Linear Programming Model (SSLP) where smax(k) maximum allowable drawdown at the pumping ͑dewatering͒ well k; and NPWϭnumber of pumping In the first dewatering system design model, the objective is to wells ͑a total of 120͒. Maximum allowable drawdowns were ͑ minimize the total pumping rate an assumed linear surrogate for specified as 10 m at each pumping well. ͒ minimizing pumping costs from dewatering wells 3. Upper and lower limits on pumping wells: Pumping rate NPW from each well is limited by its design capacity ϭ ͑ ͒ min Z ͚ Q k (3) р ͑ ͒р ͑ ͒ ϭ kϭ1 0 Q k Qmax k k 1,...,NPW (7) ϭ where Q(k)ϭpumping rate of dewatering well k. Pumping was where Qmax(k) maximum pumping capacity of the pumping well 3 allowed at both sides of the collector line, giving a total of 120 k. The maximum pumpage rate was specified as 500 m /day for possible pumping well locations. This objective function is sub- all probable pumping wells. Because of the steady-state flow con- ject to the following: ditions, it is imperative that the probable pumping wells operate 1. Hydraulic constraints relating pumpage to drawdowns via continuously, with no allowances for shut-downs. the steady-state response matrix: The resulting linear programming model is solved iteratively NPW using the aforementioned iterative correction method via the ͑ ͒ s͑k͒ϭ ͚ ␤͑k,j͒Q͑j͒ kϭ1,...,NOW (4) CPLEX 1995 optimization program. Simulation of the optimal jϭ1 pumping policies was evaluated with the groundwater simulation where NOWϭtotal number of pumping well and dewatering model. As mentioned previously, satisfactory convergence of the cell locations ͑a total of 180͒. The other terms are defined optimal and simulated drawdowns occurred at three iterations previously in Eq. ͑1͒. ͑Fig. 7͒. The objective function value converged in a similar man- 2. Drawdown limitations: Drawdowns at the dewatering cells ner to a value of 6,056 m3/day. The optimal pumping policy with along the collector line are constrained to maintain required the initial and optimal dewatered hydraulic head distributions are

JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / JULY/AUGUST 2002 / 253 Fig. 8. Optimal pumping policy and resulting head distributions obtained from steady-state linear programming model shown in Fig. 8. The groundwater drawdowns along the collector based on the steady-state Thiem Equation ͑Anderson and Woess- line clearly show that the required dewatering of groundwater ner 1992͒. It is calculated that the difference between the actual level was achieved with those optimal pumping well locations drawdown in a pumping well and the drawdown in a model cell and associated pumpage rates ͑Fig. 8͒. The optimal solution of containing the well varies from a minimum of 0.0002 m to a this model requires 54 pumping wells with pumping rates varying maximum of 0.055 m, the average difference for 54 pumping well from as low as 0.6 m3/day to as high as the maximum design nodes being equal to 0.005 m. This difference is negligible and capacity of 500 m3/day. It has been noted that almost all of the can be ignored for all practical purposes. wells located in the northern part of the model area are pumped at the maximum capacity. This is expected because the hydraulic conductivity values in this part of the model area are larger as Steady-State Mixed-Integer Programming Model compared with the hydraulic conductivity values of other parts of (SSMIP) the area. The cost of well completion comprises an important part in the It should be noted that the drawdowns calculated by the model total cost of the groundwater dewatering system. This model is represent average drawdowns over model cells but not the local used to account for the impact of this cost on the optimal pump- drawdowns in individual pumping wells. Simulated drawdown at ing policy determination. The objective function of this model a node representing a pumping well will be less than the actual minimizes the total well completion cost and the total pumpage drawdown produced by a fully penetrating pumping well. Prickett from dewatering wells ͑1967͒ shows that the effective radius of a simulated fully penetrating pumping well is equal to 0.208a where a is the ﬁnite- NPW NPW ϭ ͑ ͒ ͑ ͒ϩ␭ ͑ ͒ difference grid spacing. Using the effective well block radius of min Z ͚ Cw k Y k ͚ Q k (8) ϭ ϭ 10.4 m, hydraulic conductivities, the optimal pumping rates, and k 1 k 1 the heads calculated by the model, an estimate of the actual draw- where Y(k)ϭzero-one integer variable such that it equals one if downs at the pumping well nodes is obtained from a formula the dewatering well k is opened and zero otherwise;

254 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / JULY/AUGUST 2002 Fig. 9. Optimal pumping policy and resulting head distributions obtained from steady-state mixed integer programming model

ϭ ͑ ͒ ␭ϭ ␭ϭ Cw(k) completion cost of well k $ ; and weighting factor hydraulic head distributions corresponding to 1 are shown in representing the relative weighting of the well completion costs Fig. 9. The differences between the actual drawdowns and the and total pumping, which is an assumed surrogate for the pump- model simulated drawdowns at the pumping well nodes were ing costs. All other terms were deﬁned previously. found to be very small, varying from a minimum of 0.007 m to a This objective function is subject to the same constraint set as maximum of 0.25 m, the average difference for 15 pumping well the SSLP model except that the upper and lower limits on pump- nodes being equal to 0.026 m. Thus, no correction for well radii ing wells are replaced with the following well completion con- was warranted. The groundwater drawdowns along the collector straints, so that potential dewatering wells must be completed line clearly show that the required dewatering of groundwater prior to operation: level was achieved with those optimal pumping well locations and their pumpage rates ͑Fig. 9͒. The optimal pumping policy ͑ ͒ ͑ ͒Ϫ ͑ ͒у ϭ Qmax k Y k Q k 0 k 1,...,NPW (9) shows that 13 out of 15 required wells are to be developed in the The resulting model is a mixed-integer programming model northern part of the model area, where hydraulic conductivities that is again solved iteratively using the iterative correction are higher. The optimal pumping rates were, however, more uni- method developed via the CPLEX ͑1995͒ program. The comple- form as compared with the SSLP model, varying from a mini- tion cost of each well is uniformly assigned a value of $2,500, mum of 382.4 m3/day to a maximum of 500 m3/day. with the remaining parameters set at the same values as in SSLP In order to determine the effects of the relative weighting fac- model. The relative weighting factor, ␭, in the objective function tor on the optimal solution, the model is resolved for ␭ϭ0, 100, is assigned various values to evaluate the response of the model. and 1,000. The trade-off curve between the objectives of minimi- The optimal solution corresponding to ␭ϭ1 requires 15 wells to zation of the total pumping rate and the total well completion cost be completed with a total pumpage rate of 7,273 m3/day. The is shown graphically in Fig. 10 as a function of relative weighting optimal pumping policy with the initial and the optimal dewatered factor. It is observed that increasing the weighting factor brings

JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / JULY/AUGUST 2002 / 255 Fig. 10. Relation between total pumpage and total well completion cost as function of relative weighting factor

the optimal solution closer to that obtained from SSLP model; balance between the desired rate of progress and the times re- that is, minimization of the total pumping rate becomes more quired for excavation, construction, and backfilling. In this study, important with higher ␭ values while the total well completion the 3,000 m length of the excavation along the collector line was cost loses its importance. Fig. 11 shows the variations in the num- divided into 15 segments, each having a length of 200 m ͑Fig. 3͒. ber of wells completed with respect to the relative weighting fac- The model segment corresponding to the first stage is shown in tor. The results indicate that, as the relative weighting factor in- Fig. 5. The dewatering period for each stage is composed of three creases, the number of wells completed also increases. The months—one month for obtaining the required amount of draw- model, however, is insensitive to the number of wells required at down and following two months to maintain the groundwater ␭ small values. The radical decrease in the number of pumping level at that position for excavation and construction. Two models wells from 54 in the SSLP model to 15 in the SSMIP model is are considered for transient conditions. The first model minimizes significant if the overriding objective is to minimize the well the total pumping rates of dewatering wells over a period of three completion costs. months, while the second model minimizes both the total pumping rate and the well completion costs. Both models are solved sequentially for each stage. Although global optimization consid- Transient-State Dewatering Models ering all stages at once should be considered, the number of con- In practice, for trench lengths greater than 120 m, a progressive straints and decision variables required in this case would be great stage-wise dewatering system is preferred whereby the entire enough to preclude a solution with the available computing tech- trench length is divided into segments or stages. While a section nology. This is especially true for the second model, in which the of the trench is dewatered, excavated, and backfilled, wells of the number of integer variables becomes 5,400 if global optimization following stage can be drilled and completed, thereby producing is considered, in contrast to 24 in the sequential optimization substantial savings in time and cost. The stage length selected is a undertaken herein.

Fig. 11. Relation between number of wells opened and relative weighting factor

256 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / JULY/AUGUST 2002 Table 1. Summary of Results Obtained from Transient-State Linear Programming Model ͑TSLP͒ Number of Total number Number of Number of Number of Total iterations for of pumping wells used in wells used in wells used in pumpage Stage number convergence wells required 1st time period 2nd time period 3rd time period ͑m3/day͒ 1 3 6 6 3 4 2,363.38 2 2 5 4 4 4 2,713.83 3 2 5 5 5 5 4,122.54 4 2 7 5 4 5 5,468.26 5 2 7 6 6 5 6,482.34 6 2 8 7 6 5 7,129.22 7 2 7 6 5 6 7,442.43 8 2 7 6 5 6 7,537.11 9 2 7 6 6 6 7,710.43 10 2 7 7 7 6 8,916.70 11 8 8 6 6 6 13,341.98 12 8 7 6 6 4 13,539.44 13 8 6 6 5 5 14,552.17 14 10 6 6 6 5 15,915.10 15 13 6 6 6 6 16,032.42 TOTAL ¯ 99 ¯¯¯133,267.35

Transient-State Linear Programming Model (TSLP) executing the simulation model under transient conditions for the pumping period of the preceding stages, plus an additional three The objective in this case is to minimize the total pumpage from months by subjecting each of the dewatering wells of the preced- dewatering wells: ing stages to their optimal discharge rates for the first pumping NPW N period and zero units for the rest of the three months. Water levels min Zϭ ͚ ͚ Q͑k,n͒ (10) at pumping wells and dewatering cells are then monitored over kϭ1 nϭ1 the last three periods to determine the drawdown limits. where Q(k,n)ϭamount of pumpage from the dewatering well k 3. Upper and lower limits on pumping wells during the period n; and Nϭtotal number of time periods ͑three months͒ considered in the analysis. This objective function is sub- р р ϭ ϭ 0 Q͑k,n͒ Qmax͑k͒ k 1,...,NPW and n 1,...,N ject to (14) 1. Hydraulic constraints that relate pumpage to drawdowns via the transient state response matrix The maximum pumpage rate, Qmax(k), of each well was specified 3 NPW n to be equal to 500 m /day as in steady-state models, but had to be s͑k,n͒ϭ ͚ ͚ ␤͑k,j,nϪiϩ1͒Q͑j,i͒ increased after the 10th stage toward the northern part of the jϭ1 iϭ1 model area in the transient-state models. This was necessary be- kϭ1,...,NOW, nϭ1,...,N (11) cause the hydraulic conductivities of the aquifer beyond stage 10 where s(k,n)ϭdrawdown at a pumping well or dewatering were high and it was impossible to achieve the required limit of cell location k at the end of pumping period n; and drawdowns at the dewatering cells with those pumping capacity NOWϭtotal number of pumping and dewatering cell loca- limitations. tions ͑a total of 12 for each stage͒. Eqs. ͑10͒–͑14͒ represent a linear programming model and are 2. Drawdown limitations: solved separately for each stage ͑a total of 15 stages͒ with chang- • At the excavation site ing initial conditions using the iterative correction method. Simu- у ϭ ϭ lation of the optimal pumping policies was evaluated with the s͑k,n͒ smin͑k,n͒ k 1,...,NOP and n 1,...,N (12) groundwater simulation model. The number of iterations for con- ϭ where smin(k,n) desired level of drawdown at the excava- vergence at each stage is given in Table 1. Satisfactory conver- tion cell k at the end of period n; and NOPϭnumber of gence of the optimal and simulated drawdowns was achieved at dewatering cells within the excavation site ͑a total of four for two to three iterations for the first 10 stages. But the number of each stage͒. iterations for convergence increased beyond the 10th stage as • At the pumping wells total pumping rate increased to achieve the required amount of р ϭ ϭ s͑k,n͒ smax͑k,n͒ k 1,...,NPW and n 1,...,N drawdowns due to higher hydraulic conductivities. The number of (13) wells opened at each stage together with their total optimal pump- ϭ where smax(k,n) maximum allowable drawdown at the ing rates are given in Table 1 and depicted graphically in Figs. 12 pumping well k at the end of pumping period n. and 13, respectively. The lowering of the groundwater table to the The maximum allowable drawdowns (smax) and the minimum desired limits along the collector line was achieved with a total of required drawdowns (smin) were updated while moving from one 99 dewatering wells. The total optimal pumping rates of stages stage to the next. This was necessary, since the system is transient vary from a minimum of 2,363 m3/day for the first stage to a and the initial conditions vary for each stage. The required draw- maximum of 16,032 m3/day for the 15th stage. The increase in down in a stage is affected by the pumping policy of the preced- total pumping rates beyond stage 10 is attributed to the higher ing stages. The limits on drawdown for a stage are updated by hydraulic conductivities observed in that part of the model area.

JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / JULY/AUGUST 2002 / 257 Fig. 12. Number of wells required in stages under transient dewa- Fig. 13. Total optimal pumpages of stages under transient dewater- tering models ing models

Fig. 14, depicting the groundwater table proﬁles along the collec- where Y(k,n)ϭzero-one integer variable such that it equals one if tor line for each stage, clearly shows that the required lowering of the dewatering well k is to be developed in time period n and zero the water table was achieved. otherwise. All other terms were deﬁned earlier. Capital invest- ment costs associated with well completion were not discounted because of the short dewatering period. Transient-State Mixed-Integer Programming Model This objective function is subject to the same constraint set as (TSMIP) the TSLP model except that the upper and lower limits on pump- This model is designed to minimize the total pumping rate and the ing wells are replaced with the following well completion con- total capital cost of well completion with the following objective straints, so that potential dewatering wells must be completed function: before they can be operated: NPW N NPW N n ϭ ͑ ͒ ͑ ͒ϩ␭ ͑ ͒ ͑ ͒ͫ ͑ ͒ͬϪ ͑ ͒у min Z ͚ ͚ Cw k Y k,n ͚ ͚ Q k,n (15) Qmax k ͚ Y k,i Q k,n 0 kϭ1 nϭ1 kϭ1 nϭ1 iϭ1

Fig. 14. Groundwater table proﬁles along collector line obtained from transient linear programming model

258 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / JULY/AUGUST 2002 Table 2. Summary of Results Obtained from Transient-State Mixed Integer Programming Model ͑TSMIP͒ Number of Total number Number of Number of Number of Total Total well iterations for of pumping wells used in wells used in wells used in pumpage completion Stage number convergence wells required 1st time period 2nd time period 3rd time period ͑m3/day͒ cost ͑dollars͒ 1 3 4 4 3 4 2,375.49 10,000 2 3 3 3 2 2 2,770.21 7,500 3 2 4 4 4 4 4,100.79 10,000 4 2 5 5 4 4 5,458.65 12,500 5 2 6 6 6 5 6,493.67 15,000 6 2 6 6 6 5 7,134.58 15,000 7 2 6 6 5 5 7,436.02 15,000 8 2 6 6 5 5 7,530.16 15,000 9 2 6 6 6 5 7,702.94 15,000 10 2 7 7 7 6 8,904.20 17,500 11 8 6 6 6 5 13,349.89 15,000 12 8 5 5 5 4 13,553.83 12,500 13 8 5 5 5 5 14,562.54 12,500 14 10 5 5 5 5 15,912.23 12,500 15 13 5 5 5 5 16,112.38 12,500 TOTAL ¯ 79 ¯¯¯133,397.58 197,500

kϭ1,...,NPW and nϭ1,...,N (16) The resulting model is a mixed integer programming model, and it is again solved for each stage iteratively using the same set Furthermore, a maximum of one dewatering well can be devel- of model parameters as in the TSLP model. Convergence of the oped at each potential site over the entire dewatering period: optimal and simulated drawdowns was similar to those of the N TSLP model. However, the number of dewatering wells opened at ͚ Y͑k,n͒р1 kϭ1,...,NPW (17) each stage was less than that of the TSLP model, giving a total of nϭ1

Fig. 15. Groundwater table proﬁles along collector line obtained from transient mixed integer programming model

JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / JULY/AUGUST 2002 / 259 Table 3. Comparison of Four Dewatering Models construction. This was achieved, however, with a 20% reduction ͑ ␭ϭ ͒ Total pumpage Completion time of in the number of pumping wells in the TSMIP model for 1 as Number of wells amount construction compared with the TSLP model. Increasing the relative weighting Model type completed ͑m3͒ ͑years͒ factor, ␭, between the objectives of minimizing the total pumping rate and the total well completion costs in mixed integer program- SSLP 54 16,578,300 7.5 ming models moves the solutions closer to the linear program- SSMIPa 15 21,237,160 8 ming models. It should be noted that all the models yield average TSLP 99 3,998,010 3.75 a drawdowns over model cells, but not the local drawdowns in TSMIP 79 4,001,940 3.75 individual pumping wells. However, the differences between ac- a ␭ϭ Solution corresponding to 1. tual drawdowns in pumping wells and the simulated ones at model cells containing the wells were found to be negligible for 79 wells ͑Tables 1 and 2͒. For ␭ϭ1, the number of wells opened all practical purposes. at each stage together with their total optimal pumping rates are given in Table 2 and depicted graphically in Figs. 12 and 13, respectively. It is noted that the significant reduction in the num- Summary and Conclusions ber of dewatering wells was achieved with a total pumping rate that is almost the same as in TSLP model ͑Fig. 13͒. Fig. 15, Using combined simulation and optimization methodology, a se- depicting the groundwater table profiles along the collector line ries of models are developed to analyze and design an efficient for each stage, clearly shows that the required lowering of the dewatering scheme for an excavation site to construct a collector water table was achieved. line in an unconfined aquifer in central Turkey. A finite-difference simulation model represented the hydraulic response of the aquifer system. The hydraulic response of the aquifer is included Discussion of Results within the constraints of the optimization models using the response function approach. The nonlinear response of the uncon- Depending on the type of formulation, application of the four fined aquifer system is handled with an iterative correction pro- SSLP, SSMIP, TSLP, and TSMIP models as described herein cedure. yields different dewatering system designs. It is important to com- Four optimal dewatering models are formulated and compared pare the results obtained from these models in order to enhance to choose the best system design. Two of the models minimize the ability of the decision makers to select the best system design. only the total amount of pumpage ͑linear programming models͒, The ultimate design adopted, however, should be based not only while the other two explicitly account for the completion cost of on the objectives ͑i.e., minimum withdrawal, minimum cost͒ con- pumping wells besides the total pumpage amount ͑mixed-integer sidered herein, but also on other factors not accounted for by the programming models͒. The models are executed under both optimization models, such as technical feasibility and environ- steady-state and transient conditions. Complete dewatering of the mental impact. For example, it may be preferable to resort to a collector line at one stage is considered in the steady-state mod- scheme with the smallest pumping rate, independently of any els, whereas in the transient models optimal pumping policy are other criteria, if there are problems with disposing all the water obtained for a stage-wise, progressive dewatering scheme. All pumped out and there are severe ͑ecological͒ limitations on the models are subjected to constraints related to the system response drawdowns propagated outside the excavated area. equations, drawdown requirements in the dewatering cells, and The results obtained from four models are summarized in drawdown and discharge limits for pumping wells. Finally, these Table 3 in terms of number of wells required, total pumping rate, dewatering system design models are compared with respect to and completion time of construction. In order to determine the completion time of construction, total pumping rates, number of time required for steady-state models to reach the steady-state pumping wells, and efficiency and applicability in practice. pumping conditions, transient simulations were made using the The results presented in the paper demonstrate the following: optimal steady-state stresses obtained from the SSLP and SSMIP 1. The dewatering system design models with the iterative cor- models. The variations in hydraulic heads with time in cells along rection procedure account for the characteristics of the un- the collector line were observed. It requires five and five and half confined aquifer system and correctly evaluate the pumpage years for the system to reach steady-state conditions for the SSLP policies while satisfying all system constraints and design and SSMIP models, respectively. This means that excavation can limitations. possibly be started after five or five and half years of pumping 2. The iterative correction method satisfactorily accounts for period with the optimal pumping policy determined from SSLP the nonlinearity of the unconfined aquifer system, requiring and SSMIP models, respectively. Adding the estimated excava- only two to three iterations in the steady-state models and at tion time ͑two months for a 200 m section͒ to these values gives low pumping rates in the transient models. The number of seven and half and eight years to complete the construction using iterations in the transient models increases with the pumping the dewatering schemes of SSLP and SSMIP models, respec- rate to achieve the required convergence, thereby necessitat- tively. ing more CPU time for solutions. The number of iterations The results show that, although steady-state models yield the depends on the difference between the unit pumpage value least number of wells as compared with transient models, the used during the response matrix generation and the calcu- completion time of construction is almost doubled and the total lated optimal pumpage values. volume of water pumped is four to five times greater than that 3. The steady-state models yield the least number of wells as obtained from the transient models. This may be of a concern if compared with the transient models. The reduction in the there are problems of disposing or possibly reinjecting all the number of wells was 45% for the linear programming mod- water pumped. Both transient models yield the same volume of els and 81% for the mixed-integer programming models. water pumped over the same pumping period to complete the This reduction in the number of wells, however, required a

260 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / JULY/AUGUST 2002 four to five fold increase in the amount of volume of water CPLEX Optimization, Inc. ͑1995͒. ‘‘Using the CPLEX Callable Library.’’ pumped and a two fold increase in the time of completion of Rep., Incline Village, Nev. construction. Danskin, W. R., and Gorelick, S. M. ͑1985͒. ‘‘A policy evaluation tool: 4. Both transient linear and integer programming models yield Management of a multiaquifer system using controlled stream re- the same volume of water pumped over the same pumping charge.’’ Water Resour. Res., 21͑11͒, 1731–1747. period to complete the construction. This was achieved, Doyuran, V., and Erol, O. ͑1995͒. Geological and geotechnical investiga- however, with a 20% reduction in the number of pumping tion of Aksaray organized industrial district, Ministry of Industry and wells required in the mixed-integer programming model Commerce, Ankara, Turkey. ͑ ͒ where equal weighting of objectives ͑i.e., minimization of Galeati, G., and Gambolati, G. 1988 . ‘‘Optimal dewatering schemes in pumping rates and well completion costs͒ is considered as the foundation design of an electronuclear plant.’’ Water Resour. Res., ͑ ͒ compared with the linear programming model. 24 4 , 541–552. ͑ ͒ 5. Increasing the relative weighting factor between the objec- Gorelick, S. M. 1983 . ‘‘A review of distributed parameter groundwater ͑ ͒ tives of minimizing the total pumping rate and the total well management modeling methods.’’ Water Resour. Res., 19 2 , 305– completion costs in the mixed-integer programming models 319. Heidari, M. ͑1982͒. ‘‘Application of linear system’s theory and linear moves the solutions closer to the linear programming mod- programming to groundwater management in Kansas.’’ Water Resour. els. Bull., 18͑6͒, 1003–1012. 6. Although the ultimate design model adopted by decision Larson, S. P., Maddock, T., and Papadopulos, S. ͑1977͒. ‘‘Optimization makers will also depend on a number of other technical, techniques applied to groundwater development.’’ Mem. Int. Assoc. financial, and environmental considerations, the transient Hydrogeol., 13, E57–E65. mixed-integer programming model depicts the most favor- Maddock, T., III. ͑1972͒. ‘‘Algebraic technological function from a simu- able solution with respect to the number of wells required, lation model.’’ Water Resour. Res., 8͑1͒, 129–134. the completion time of construction, and the total volume of McDonald, M. G., and Harbaugh, A. W. ͑1984͒. A modular three- water pumped. dimensional finite-difference groundwater flow model, Scientific, Washington, D.C. Morris, D. A., and Johnson, A. I. ͑1967͒. ‘‘Summary of hydrologic and Acknowledgments physical properties of rock and soil materials as analyzed by the Hy- drologic Laboratory of the U.S. Geological Survey 1948–1960.’’ The computer facilities provided by Middle East Technical Uni- Water Supply Paper 1839-D, U.S. Geological Survey, Washington, versity are gratefully acknowledged. The constructive comments D.C. of three anonymous reviewers are appreciated. Prickett, T. A. ͑1967͒. ‘‘Designing pumped well characteristics into elec- tric analog models.’’ Ground Water, 5͑4͒, 38–46. Tokgoz, M. ͑1999͒. ‘‘Optimal dewatering of the collector line in Aksaray References organized industrial district, Aksaray-Turkey.’’ MS thesis, Middle East Technical Univ., Ankara, Turkey. ͑ ͒ Aguado, E., and Remson, I. ͑1980͒. ‘‘Groundwater management with Willis, R. 1984 . ‘‘A unified approach to regional groundwater manage- fixed charges.’’ J. Water Resour. Plan. Manage., 106͑2͒, 375–382. ment.’’ Groundwater hydraulics, J. Rosenshein and G. D. Bennett, Aguado, E., Remson, I., Pikul, M. F., and Thomas, W. A. ͑1974͒. ‘‘Opti- eds., American Geophysical Union, Washington, D.C., 392–407. mal pumping for aquifer dewatering.’’ J. Hydraul. Div., Am. Soc. Civ. Willis, R., and Finney, B. A. ͑1985͒. ‘‘Optimal control of nonlinear Eng., 100͑7͒, 869–877. groundwater hydraulics: Theoretical development and numerical ex- Anderson, M. P., and Woessner, W. W. ͑1992͒. Applied groundwater mod- periments.’’ Water Resour. Res., 21͑10͒, 1476–1482. eling: Simulation of flow and advective transport, Academic, San Yeh, W. W.-G. ͑1992͒. ‘‘Systems analysis in ground-water planning and Diego. management.’’ J. Water Resour. Plan. Manage., 118͑3͒, 224–237.

JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / JULY/AUGUST 2002 / 261