Computational Geosciences

, Volume 10, Issue 3, pp 303–319 | Cite as

On optimization algorithms for the reservoir oil well placement problem

  • W. Bangerth
  • H. Klie
  • M. F. Wheeler
  • P. L. Stoffa
  • M. K. Sen


Determining optimal locations and operation parameters for wells in oil and gas reservoirs has a potentially high economic impact. Finding these optima depends on a complex combination of geological, petrophysical, flow regimen, and economical parameters that are hard to grasp intuitively. On the other hand, automatic approaches have in the past been hampered by the overwhelming computational cost of running thousands of potential cases using reservoir simulators, given that each of these runs can take on the order of hours. Therefore, the key issue to such automatic optimization is the development of algorithms that find good solutions with a minimum number of function evaluations. In this work, we compare and analyze the efficiency, effectiveness, and reliability of several optimization algorithms for the well placement problem. In particular, we consider the simultaneous perturbation stochastic approximation (SPSA), finite difference gradient (FDG), and very fast simulated annealing (VFSA) algorithms. None of these algorithms guarantees to find the optimal solution, but we show that both SPSA and VFSA are very efficient in finding nearly optimal solutions with a high probability. We illustrate this with a set of numerical experiments based on real data for single and multiple well placement problems.


reservoir optimization reservoir simulation simulated annealing SPSA stochastic optimization VFSA well placement 


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  1. 1.
    Arbogast, T., Wheeler, M.F., Yotov, I.: Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences. SIAM, J. Numer. Anal. 34(2), 828–852 (1997)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Aziz, K., Settari, A.: Petroleum reservoir simulation. Applied Science, London (1979)Google Scholar
  3. 3.
    Bangerth, W., Klie, H., Matossian, V., Parashar, M., Wheeler. M.F.: An autonomic reservoir framework for the stochastic optimization of well placement. Cluster Computing 8(4), 255–269 (2005)CrossRefGoogle Scholar
  4. 4.
    Becker, B.L., Song, X.: Field development planning using simulated annealing-optimal economic well scheduling and placement. In: SPE Annual Technical Conference and Exhibition, Dallas, Texas, SPE 30650, October 1995Google Scholar
  5. 5.
    Bittencourt, A.C., Horne, R.N.: Reservoir development and design optimization. In SPE Annual Technical Conference and Exhibition, San Antonio, Texas, SPE 38895, October 1997Google Scholar
  6. 6.
    Carlson, M.: Practical reservoir simulation. PennWell Corporation (2003)Google Scholar
  7. 7.
    Centilmen, A., Ertekin, T., Grader, A.S.: Applications of neural networks in multiwell field development. In: SPE Annual Technical Conference and Exhibition, Dallas, Texas, SPE 56433, October 1999Google Scholar
  8. 8.
    Chunduru, R.K., Sen, M., Stoffa, P.: Hybrid optimization methods for geophysical inversion. Geophysics 62, 1196–1207 (1997)CrossRefGoogle Scholar
  9. 9.
    Fanchi, J.R.: Principles of Applied Reservoir Simulation. Boston: Butterworth-Heinemann Gulf Professional Publishing, 2nd edn. (2001)Google Scholar
  10. 10.
    Gerencsér, L., Hill, S.D., Vágó, Z.: Optimization over discrete sets via SPSA. In Proceedings of the 38th Conference on Decision and Control, Phoenix, AZ, 1999, 1791–1795 (1999)Google Scholar
  11. 11.
    Gerencsér, L., Hill, S.D., Vágó, Z.: Discrete optimization via SPSA. In: Proceedings of the Americal Control Conference, Arlington, VA, 2001, 1503–1504 (2001)Google Scholar
  12. 12.
    Goldberg, D.: Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley (1989)Google Scholar
  13. 13.
    Guyaguler, B.: Optimization of well placement and assessment of uncertainty. PhD thesis, Stanford University, Department of Petroleum Engineering (2002)Google Scholar
  14. 14.
    Guyaguler, B., Horne, R.N.: Uncertainty assessment of well placement optimization. In: SPE Annual Technical Conference and Exhibition, New Orleans, Louisiana, September, October 2001. SPE 71625Google Scholar
  15. 15.
    Helmig, R.: Multiphase Flow and Transport Processes in the Subsurface. Springer, Berlin (1997)Google Scholar
  16. 16.
    Houck, C., Joines, J.A., Kay, M.G.: A genetic algorithm for function optimization: A Matlab implementation. Technical Report TR 95-09, North Carolina State University (1995)Google Scholar
  17. 17.
    Hyne, N.: Nontechnical Guide to Petroleum Geology, Exploration, Drilling and Production. Pennwell Books, 2nd edn. (2001)Google Scholar
  18. 18.
    Ingber, L.: Very fast simulated reannealing. Math. Comput. Model. 12, 967–993 (1989)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Klie, H., Bangerth, W., Wheeler, M.F., Parashar, M., Matossian, V.: Parallel well location optimization using stochastic algorithms on the grid computational framework. In: 9th European Conference on the Mathematics of Oil Recovery, ECMOR, Cannes, France, EAGE August 30–September 2, 2004Google Scholar
  20. 20.
    Lacroix, S., Vassilevski, Y., Wheeler, J., Wheeler, M.F.: Iterative solution methods for modelling multiphase flow in porous media fully implicitly. SIAM J. Sci. Comput. 25(3), 905–926 (2003)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Lacroix, S., Vassilevski, Y., Wheeler, M.F.: Iterative solvers of the Implicit Parallel Accurate Reservoir Simulator (IPARS). Numer. Linear Algebra Appl. 4, 537–549 (2001)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Lagarias, J.C., Reeds, J.A., Wright, M.H., Wright, P.E.: Convergence behavior of the Nelder–Mead simplex algorithm in low dimensions. SIAM J. Optim. 9, 112–147 (1999)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Lu, Q.: A parallel multi-block/multi-physics approach for multi-phase flow in porous media. PhD thesis, University of Texas at Austin, Austin, Texas (2000)Google Scholar
  24. 24.
    Lu, Q., Peszyńska, M., Wheeler, M.F.: A parallel multi-block black-oil model in multi-model implementation. In: 2001 SPE Reservoir Simulation Symposium, Houston, Texas, SPE 66359 (2001)Google Scholar
  25. 25.
    Lu, Q., Peszyńska, M., Wheeler, M.F.: A parallel multi-block black-oil model in multi-model implementation. SPE J. 7(3), 278–287 SPE 79535 (2002)Google Scholar
  26. 26.
    Mattax, C.C., Dalton, R.L.: Reservoir simulation. In: SPE Monograph Series, volume 13, Richardson, Texas (1990)Google Scholar
  27. 27.
    Mitchell, M.: An Introduction to Genetic Algorithms. The MIT Press (1996)Google Scholar
  28. 28.
    Nelder, J.A., Mead, R.: A simplex method for function minimization. Comput. J. 7, 308–313 (1965)Google Scholar
  29. 29.
    Nocedal, J., Wright, S.J.: Numerical Optimization. Springer Series in Operations Research. Springer, New York (1999)Google Scholar
  30. 30.
    Pan, Y., Horne, R.N.: Improved methods for multivariate optimization of field development scheduling and well placement design. In: SPE Annual Technical Conference and Exhibition, New Orleans, Louisiana, 27–30, SPE 49055 September 1998Google Scholar
  31. 31.
    Parashar, M., Klie, H., Catalyurek, U., Kurc, T., Bangerth, W., Matossian, V., Saltz, J., Wheeler, M.F.: Application of grid-enabled technologies for solving optimization problems in data-driven reservoir studies. Future Gener. Comput. Syst. 21(1), 19–26 (2005)CrossRefGoogle Scholar
  32. 32.
    Parashar, M., Wheeler, J.A., Pope, G., Wang, K., Wang, P.: A new generation EOS compositional reservoir simulator. Part II: Framework and multiprocessing. In: Fourteenth SPE Symposium on Reservoir Simulation, Dallas, Texas, 31–38. Society of Petroleum Engineers June 1997Google Scholar
  33. 33.
    Parashar, M., Yotov, I.: An environment for parallel multi-block, multi-resolution reservoir simulations. In: Proceedings of the 11th International Conference on Parallel and Distributed Computing and Systems (PDCS 98), 230–235, Chicago, IL, International Society for Computers and their Applications (ISCA) Sep. 1998Google Scholar
  34. 34.
    Pardalos, P.M., Resende, M.G.C. eds.: Handbook of Applied Optimization, pp. 808–813. Oxford University Press (2002)Google Scholar
  35. 35.
    Peszyńska, M., Lu, Q., Wheeler, M.F.: Multiphysics coupling of codes. In: L.R. Bentley, J.F. Sykes, C.A. Brebbia, W.G. Gray, G.F. Pinder, eds., Computational Methods in Water Resources, 175–182. A. A. Balkema (2000)Google Scholar
  36. 36.
    Powell, M.J.D.: The NEWUOA software for unconstrained optimization without derivatives. Technical Report DAMTP 2004/NA08, University of Cambridge, England (2004)Google Scholar
  37. 37.
    Rian, D.T., Hage, A.: Automatic optimization of well locations in a north sea fractured chalk reservoir using a front tracking reservoir simulator. In: SPE International Petroleum & Exhibition of Mexico, Veracruz, Mexico, SPE 28716, October 2001Google Scholar
  38. 38.
    Russell, T.F., Wheeler, M.F.: Finite element and finite difference methods for continuous flows in porous media. In: R.E. Ewing, eds., The Mathematics of Reservoir Simulation, pp. 35–106. SIAM, Philadelphia (1983)Google Scholar
  39. 39.
    Sen, M., Stoffa, P.: Global Optimization Methods in Geophysical Inversion. Elsevier (1995)Google Scholar
  40. 40.
    Spall, J.C.: Multivariate stochastic approximation using a simultaneous perturbation gradient approximation. IEEE Trans. Autom. Control. 37, 332–341 (1992)MATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Spall, J.C.: Adaptive stochastic approximation by the simultaneous perturbation method. IEEE Trans. Autom. Contr. 45, 1839–853 (2000)MATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    J.C. Spall. Introduction to Stochastic Search and Optimization: Estimation, Simulation and Control. Wiley, New Jersey (2003)MATHGoogle Scholar
  43. 43.
    Wang, P., Yotov, I., Wheeler, M.F., Arbogast, T., Dawson, C.N., Parashar, M., Sepehrnoori, K.: A new generation EOS compositional reservoir simulator. Part I: Formulation and discretization. In: Fourteenth SPE Symposium on Reservoir Simulation, Dallas, Texas, pp. 55–64. Society of Petroleum Engineers, June 1997Google Scholar
  44. 44.
    Wheeler, M.F.: Advanced techniques and algorithms for reservoir simulation, II: The multiblock approach in the integrated parallel accurate reservoir simulator (IPARS). In: J. Chadam, A. Cunningham, R.E. Ewing, P. Ortoleva, M.F. Wheeler, eds., IMA Volumes in Mathematics and its Applications, Volume 131: Resource Recovery, Confinement, and Remediation of Environmental Hazards. Springer (2002)Google Scholar
  45. 45.
    Wheeler, M.F., Wheeler, J.A., Peszyńska, M.: A distributed computing portal for coupling multi-physics and multiple domains in porous media. In: L.R. Bentley, J.F. Sykes, C.A. Brebbia, W.G. Gray, G.F. Pinder, eds., Computational Methods in Water Resources, 167–174. A. A. Balkema (2000)Google Scholar
  46. 46.
    Yeten, B.: Optimum deployment of nonconventional wells. PhD thesis, Stanford University, Department of Petroleum Engineering (2003)Google Scholar
  47. 47.
    Yeten, B., Durlofsky, L.J., Aziz, K.: Optimization of nonconventional well type, location, and trajectory. SPE J. 8(3), 200–210 SPE 86880 (2003)Google Scholar
  48. 48.
    Yotov, I.: Mortar mixed finite element methods on irregular multiblock domains. In: J. Wang, M.B. Allen, B. Chen, T. Mathew, eds., Iterative Methods in Scientific Computation, IMACS series Comp. Appl. Math., vol. 4, 239–244. IMACS, (1998)Google Scholar
  49. 49.
    Zhang, F., Reynolds, A.: Optimization algorithms for automatic history matching of production data. In: 8th European Conference on the Mathematics of Oil Recovery, ECMOR, Freiberg, Germany, EAGE, September 2002Google Scholar

Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

  • W. Bangerth
    • 1
    • 2
  • H. Klie
    • 3
  • M. F. Wheeler
    • 3
  • P. L. Stoffa
    • 4
  • M. K. Sen
    • 4
  1. 1.Department of MathematicsTexas A&M UniversityCollege StationUSA
  2. 2.Institute for GeophysicsThe University of Texas at AustinAustinUSA
  3. 3.Center for Subsurface Modeling, Institute for Computational Engineering and SciencesUniversity of Texas at AustinAustinUSA
  4. 4.Institute for Geophysics, John A. and Katherine G. Jackson School of GeosciencesUniversity of Texas at AustinAustinUSA

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