Abstract
Once subsurface water supplies become contaminated, designing cost-effective and reliable remediation schemes becomes a difficult task. The combination of finite element simulation of groundwater contaminant transport with nonlinear optimization is one approach to determine the best well selection and optimal fluid withdrawal and injection rates to contain and remove the contaminated water. Both deterministic and stochastic programming problems have been formulated and solved. These tend to be large scale problems, owing to the simulation component which serves as a portion of the constraint set. The overall problem of combined groundwater process simulation and nonlinear optimization is discussed along with example problems. Because the contaminant transport simulation models give highly uncertain results, quantifying their uncertainty and incorporating reliability into the remediation design results in a class of large stochastic nonlinear problems. The reliability problem is beginning to be addressed, and some strategies and formulations involving chance constraints and Monte Carlo methods are presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L.M. Abriola, “Modeling contaminant transport in the subsurface: an interdisciplinary challenge,”Reviews in Geophysics 25(2) (1987) 125–134.
D.P. Ahlfeld, J.M. Mulvey and G.F. Pinder, “Designing optimal strategies for contaminated groundwater remediation,”Advances in Water Resources 9(2) (1986a) 77–84.
D.P. Ahlfeld, G.F. Pinder and J.M. Mulvey, “Combining physical containment with optimal withdrawal for contaminated groundwater remediation,” in:Proceedings of the VI International Conference on Finite Elements in Water Resources (Springer, New York, 1986b) pp. 205–214.
D.P. Ahlfeld, J.M. Mulvey, G.F. Pinder and E.F. Wood, “Contaminated groundwater remediation design using simulation, optimization, and sensitivity theory 1. Model development,”Water Resources Research 24(3) (1988a) 431–442.
D.P. Ahlfeld, J.M. Mulvey, G.F. Pinder and E.F. Wood, “Contaminated groundwater remediation design using simulation, optimization, and sensitivity theory 2. Analysis of a field site,”Water Resources Research 24(3) (1988b) 443–452.
A.A. Bakr, “Effects of spatial variations of hydraulic conductivity on groundwater flow,” Doctoral Dissertation, New Mexico Institute of Mining and Technology (Socorro, NM, 1976).
J. Bear,Dynamics of Fluids in Porous Media (Elsevier, New York, 1972).
J. Bear and A. Verruijt,Modelling Groundwater Flow and Pollution (Reidel, 1987).
D.G. Cacuci, C.F. Weber, E.M. Oblow and J.H. Marable, “Sensitivity theory for general systems of nonlinear equations,”Nuclear Science Engineering 75 (1980) 88–110.
A. Charnes and W.W. Cooper, “Chance constraints and normal deviates,”Journal American Statistical Association 57 (1962) 134–148.
H. Cooper Jr., “The equation of groundwater flow in fixed and deformable coordinates,”Journal of Geophysical Research 7(2) (1966) 4785–4790.
G. Dagan, “Statistical theory of groundwater flow and transport: Pore to laboratory, laboratory to formation, and formation to regional scale,”Water Resources Research 22(9) (1986) 120s-134s.
J.R. Donaldson and P.V. Tryon, “Nonlinear least squares regression using STARPAC: The Standards Time Series and Regression Package,” Technical Note 1068-2, National Bureau of Standards (Boulder, CO, 1983).
L.W. Gelhar, “Stochastic subsurface hydrology from theory to applications,”Water Resources Research 22(9) (1986) 135s-145s.
P.E. Gill, W. Murray, M.A. Saunders and M.H. Wright, “Users guide for NPSOL (version 4.0): A FORTRAN package for nonlinear programming,” Report SOL 86-2, Department of Operations Research, Stanford University (Stanford, CA, 1986).
S.M. Gorelick, “A review of distributed parameter groundwater management modelling methods,”Water Resources Research 19(2) (1983) 305–319.
S.M. Gorelick, “Sensitivity analysis of optimal ground water contaminant capture curves: spatial variability and robust solutions,” in:Proceedings of the Solving Ground Water Problems with Models Conference (Denver, OC, 1987) pp. 133–146.
S.M. Gorelick, C.I. Voss, P.E. Gill, W. Murray, M.A. Saunders and M.H. Wright, “Aquifer reclamation design: The use of contaminant transport simulation combined with nonlinear programming,”Water Resources Research 20(4) (1984) 415–427.
D.B. Grove and K.G. Stollenwerk, “Chemical reactions simulated by ground-water-quality models,”Water Resources Bulletin 23(4) (1987) 601–615.
R.J. Hoeksema and P.K. Kitanidis, “An application of the geostatistical approach to the inverse problem in two-dimensional groundwater modeling,”Water Resources Research 20(7) (1984) 1003–1020.
A.G. Journel and Ch.J. Huijbregts,Mining Geostatistics (Academic Press, London, 1978).
P.K. Kitanidis, “Parametric estimation of covariances of regionalized variables,”Water Resources Bulletin 23 (1987) 557–567.
L.F. Konikow and D.B. Grove, “Derivation of equations describing solute transport in ground water,”United States Geological Survey Water Resources Investigation 77–19 (1978).
L. Lasdon, P.E. Coffman, Jr., R. MacDonald, J.W. McFarland and K. Sepehrnoori, “Optimal hydrocarbon reservoir production policies,”Operations Research 34 (1986) 40–54.
L.J. Lefkoff and S.M. Gorelick, “Design and cost analysis of rapid aquifer restoration systems using flow simulation and quadratic programming,”Ground Water 24(6) (1986) 777–790.
A. Mantoglou and J.L. Wilson, “Simulation of random fields with the turning bands method,” MIT Department of Civil Engineering Report 264 (1981).
G. de Marsily,Quantitative Hydrogeology (Academic Press, London, 1986).
B.A. Murtagh and M.A. Saunders, “MINOS 5.0 user's guide,” Report SOL 83-20, Department of Operations Research, Stanford University (Stanford, CA, 1983).
G.F. Pinder and J.D. Bredehoeft, “Application of a digital computer for aquifer evaluation,”Water Resources Research 4 (1968) 1069–1093.
G.F. Pinder and W.G. Gray,Finite Element Simulation in Surface and Subsurface Hydrology (Academic Press, Orlando, FL, 1977).
D.L. Reddell and D.K. Sunada, “Numerical simulation of dispersion in groundwater aquifers,” Hydrology Paper 41, Colorado State University (Fort Collins, CO, 1970).
J. Rubin, “Transport of reacting solutes in porous media: relation between mathematical nature of problem formulation and chemical nature of reactions,”Water Resources Research 19 (1983) 1231–1252.
F.J. Samper and S.P. Neuman, “Adjoint state equations for advective-dispersive transport,” in:Proceedings of VI International Conference on Finite Elements in Water Resources (Springer, New York, 1986) pp. 423–428.
A.E. Scheidegger, “General theory of dispersion in porous media,”Journal of Geophysical Research 66(10) (1961) 3273–3278.
J.F. Sykes, J.L. Wilson and R.W. Andrews, “Sensitivity analysis for steady-state groundwater flow using adjoint operators,”Water Resources Research 21(3) (1985) 359–371.
E. Vanmarke,Random Fields (MIT Press, Cambridge, MA, 1983).
C.I. Voss, “SUTRA: A finite element simulation model for saturated-unsaturated, fluid density-dependent groundwater flow with energy transport or chemically-reactive single-species solute transport,” United States Geological Survey Water Resources Investigation Report 84–4369 (1984).
C.I. Voss and W.R. Souza, “Variable density flow and solute transport simulation of regional aquifers containing a narrow freshwater-saltwater transition zone,”Water Resources Research 23(10) (1987) 1851–1866.
B.J. Wagner, “Optimal groundwater quality management under uncertainty,” Doctoral Dissertation, Department of Applied Earth Sciences, Stanford University (Stanford, CA, 1988).
B.J. Wagner and S.M. Gorelick, “Optimal groundwater quality management under parameter uncertainty,”Water Resources Research 23(7) (1987) 1162–1174.
N. Wanakule, L.W. Mays and L.S. Lasdon, “Optimal management of large-scale aquifers: methodology and applications,”Water Resources Research 22(4) (1986) 447–466.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Gorelick, S.M. Large scale nonlinear deterministic and stochastic optimization: Formulations involving simulation of subsurface contamination. Mathematical Programming 48, 19–39 (1990). https://doi.org/10.1007/BF01582250
Issue Date:
DOI: https://doi.org/10.1007/BF01582250