Skip to main content

Identification and Control Problems in Petroleum and Groundwater Modeling

  • Chapter
Control Problems in Industry

Abstract

The petroleum industry has well-established partial differential equation models for multi-phase fluid flow through porous media, but the use of control-theoretic methods for optimization of petroleum recovery is fairly new. The approaches discussed in this survey could lead to a significant payoff to the petroleum industry through improved efficiency in petroleum recovery operations. There is also significant potential for the application of these methods in groundwater remediation.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. Alemán-Gomez, T.R. Ramamohan, J.C. Slattery, A statistical structural model for unsteady-state displacements in porous media, paper SPE 13265 , presented at the 1984 SPE Annual Technical Conference and Exhibition, Houston, Texas, September 16-19, 1984.

    Google Scholar 

  2. K. Aziz and A. Settari, Petroleum Reservoir Simulation , Applied Science Publishers LTC, London, 1979.

    Google Scholar 

  3. H.T. Banks, Computational issues in parameter estimation and feed-back control problems for partial differential equations, Physica D 60 (1992), 226–238.

    Article  MathSciNet  MATH  Google Scholar 

  4. H.T. Banks, Kunisch, Estimation Techniques for Distributed Parameter Systems , Birkhäuser, 1989.

    Book  MATH  Google Scholar 

  5. Y. Bard, Nonlinear Parameter Estimation , Academic Press, New York City, 1974.

    MATH  Google Scholar 

  6. D.P. Bertsekas, Constrained Optimization and, Lagrange Multiplier Methods , Academic Press, New York, 1982.

    MATH  Google Scholar 

  7. E.P. Box and G.C. Tiao, Bayesian Inference in Statistical Analysis , Addison-Wesley, Reading, Massachusetts, 1973.

    MATH  Google Scholar 

  8. S.E. Buckley, M.C. Leverett, Mechanisms of fluid displacement in sands, Trans. AIME 146 (1942), 107–116.

    Google Scholar 

  9. S. Chang, W. Yeh, A proposed algorithm for the solution of the large-scale inverse problem in groundwater, Water Resour. Res . 12 (1976), 365–374.

    Article  Google Scholar 

  10. G. Chavent, M. Dupuy, P. Lemonnier, History matching by use of optimal control theory, SPEJ (February 1975), 74–86;Trans. AIME 259.

    Google Scholar 

  11. G. Chavent, Identification of distributed parameter systems: About the output least squares method, its implementation and identifiability, Identification and System Parameter Estimation, Proceedings 5th IFAC Symp . (R. Isermann, ed.), Darmstadt, FRG, Pergamon, 85–97.

    Google Scholar 

  12. G. Chavent, M. Dupuy, P. Lemonnier, History matching by use of optimal control theory, SPE Journal , (February 1975), 74–86.

    Google Scholar 

  13. W.H. Chen, G.R. Gavalas, J.H. Seinfeld, M.L. Wasserman, A new algorithm for automatic history matching, SPEJ (December 1974), 593–608;Trans. AIME 257.

    Google Scholar 

  14. W.H. Chen, J.H. Seinfeld, Estimation of spatially varying parameters in partial differential equations, Int. J. Control 15 (1972), 487–495.

    Article  MATH  Google Scholar 

  15. R.E. Collins, Flow Fluids Through Porous Materials , The Petroleum Publishing Company, Tulsa, Oklahoma, 1976.

    Google Scholar 

  16. R.L. Cooley, Incorporation of prior information on parameters into nonlinear regression groundwater flow models, I. Theory, Water Resour. Res . 18(4) (1982), 965–976.

    Article  Google Scholar 

  17. G. Dagan, Stochastic modeling of groundwater flow by unconditional and conditional probabilities, I. Conditional simulation and the direct problem, Water Resour. Res . 18(4) (1982), 813–833.

    Article  Google Scholar 

  18. J.P. Delhomme, A variability and uncertainty in groundwater flow parameters: a geostatistical approach, Water Resour. Res . 15(4) (1979), 269–280.

    Article  Google Scholar 

  19. J.E. Dennis, R.B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations , Prentice Hall, Inc., Englewood Cliffs, 1983.

    MATH  Google Scholar 

  20. N. Distefano, A. Rath, An identification approach to subsurface hydrological systems, Water Resour. Res . 11 (1975), 1005–1012.

    Article  Google Scholar 

  21. D.C. Dobson, Estimates on resolution and stabilization for the linearized inverse conductivity problem, Inverse Problems 8 (1992), 71–81.

    Article  MathSciNet  MATH  Google Scholar 

  22. E.L. Donaldson, G.W. Dean, Two- and three-phase relative permeability studies, RI 6826 , U.S. Bureau of Mines, 1966.

    Google Scholar 

  23. R.E. Ewing Determination of coefficients in reservoir simulation, Numerical Treatment of Inverse Problems for Differential and Integral Equations (P. Deuflhardt, E. Hairer, eds.), Birkhäuser, Berlin, 1982, 206–226.

    Google Scholar 

  24. R.E. Ewing, Problems arising in the modeling of processes for hydrocarbon recovery, inThe Mathematics of Reservoir Simulation, SIAM Frontiers in Applied Mathematics 1 (R.E. Ewing, ed.), SIAM, Philadelphia, Pennsylvania, 1983.

    Google Scholar 

  25. R.E. Ewing, J.H. George, Identification and control for distributed parameters in porous media flow, Distributed Parameter Systems, Lecture Notes in Control and Information Sciences 75 (M. Thoma, ed.), Springer-Verlag, May 1985, 145–161.

    Chapter  Google Scholar 

  26. R.E. Ewing, T. Lin, A class of parameter estimation techniques for fluid flow in porous media, Advances in Water Resources 14(2) (1991), 89–97.

    Article  MathSciNet  Google Scholar 

  27. R.E Ewing, J.S. Sochacki, T. Lin, Interface conditions for acoustic wave propagation, Mathematical and Numerical Aspects of Wave Propagation Phenomena (G. Cohen, I. Halpern, P. Joly, eds.), SIAM Publications, Philadelphia, Pennsylvania, 1991, 155–164.

    Google Scholar 

  28. R.E. Ewing, M.S. Pilant, J.G. Wade, A.T. Watson, A.T., Parameter estimation in petroleum and groundwater modeling, IEEE Computational Science and Engineering 1(3) (Fall 1994), 19–31.

    Article  Google Scholar 

  29. G.R. Gavalas, P.C. Shah, J.H. Seinfeld, Reservoir history matching by Bayesian estimation, SPEJ (December 1976), 337–350;Trans. AIME 261.

    Google Scholar 

  30. J. Glimm, B. Lindquist, Scaling laws for macrodispersion, Computational Methods in Water Resources, IX (T. Russell, R. Ewing, C. Brebbia, W. Gray, G. Pinder, eds.), 1992, 35–51.

    Google Scholar 

  31. S.M. Gorelick, A review of distributed parameter groundwater management modeling methods, Water Resources Research 19(2) (1983), 305–319.

    Article  Google Scholar 

  32. S.M. Gorelick, C.I. Voss, P. Gill, M. Murray, M. Saunders, M. Wright, Aquifer reclamation design: The use of contaminant transport simulation combined with nonlinear programming, Water Resources Research 20(4) (1984), 415–427.

    Article  Google Scholar 

  33. C.W. Groetsch, C.W., Inverse problems in the mathematical sciences , Vieweg, 1993.

    MATH  Google Scholar 

  34. P.C. Hansen, Analysis of discrete ill-posed problems by means of the L-curve, SIAM Review 34(4) (1902), 561–580.

    Article  Google Scholar 

  35. M. Honarpour, L. Koederitz, A.H. Harvey, Relative Permeability of Petroleum Reservoirs , CRC Press, Boca Raton, Florida, 1986.

    Google Scholar 

  36. J. Jaffre, J. Roberts, INRIA LeChesney, France1991 (private communication).

    Google Scholar 

  37. E.F. Johnson, D.P. Bossier, V.O. Naumann, Calculations of relative permeability from displacement experiments, Trans AIME 216 (1959), 370–378.

    Google Scholar 

  38. S.C. Jones, W.O. Roszelle, Graphical techniques for determining relative permeability from displacement experiments, JPT (May 1978), 807–817;Trans. AIME 265.

    Google Scholar 

  39. P.D. Kerig, A.T. Watson, Relative permeability estimation from displacement experiments: An error analysis, SPERE (March 1986), 175–182;Trans. AIME 282.

    Google Scholar 

  40. P.D. Kerig, A.T. Watson, A new algorithm for estimating relative permeabilities from displacement experiments, SPERE (February 1987), 103–112;Trans. AIME 283.

    Google Scholar 

  41. P.K. Kitanidis, E.G. Vomvoris, A geostatistical approach to the inverse problem in groundwater modeling (steady state) and one- dimensional simulations, Water Resour . Res. 19(3) (1983), 667–690.

    Google Scholar 

  42. C. Kravaris, J. Seinfeld, Identification of parameters in distributed parameter systems by regularization, SIAM J. Cont. Optim . 23 (1985), 217–241.

    Article  MathSciNet  MATH  Google Scholar 

  43. L.D. Landau, E.M. Lifshitz, Fluid Mechanics , Pergamon Press, New York, 1959.

    Google Scholar 

  44. T. Lee, J.H. Seinfeld, Estimation of two-phase petroleum properties by regularization, J. Computational Physics 69 (1987), 397–419.

    Article  MathSciNet  MATH  Google Scholar 

  45. Lions, J.L., Optimal Control of Systems Governed by Partial Differential Equations , Springer, 1971.

    MATH  Google Scholar 

  46. C.-Y. Lin, J.C. Slattery, Three-dimensional randomized network model for two-phase flow through porous media, AIChE J . 28 (1982), 311–324.

    Article  Google Scholar 

  47. S.F. McCormick, J.G. Wade, Multigrid solution of linearized, regularized least-squares problems in electrical impedance tomography, Inverse Problems 9(6) (1993), 697–713.

    Article  MathSciNet  MATH  Google Scholar 

  48. G.M. Mejia, A.T. Watson, P.C. Richmond, Accuracy of estimates of two-phase flow functions from displacement experiments, (1992), preprint.

    Google Scholar 

  49. G.M. Mejia, A.T. Watson, P.C. Richmond, A method for estimating three-phase flow functions, (1992), preprint.

    Google Scholar 

  50. J.J. More, D.C. Sorensen, Computing a trust region step, SIAM J. Statistical Computation 4(3) (1983), 553–572.

    Article  MathSciNet  MATH  Google Scholar 

  51. S.P. Neuman, A statistical approach to the inverse problem of aquifer hydrology. III. Improved solution method and added perspective, Water Resources Research 16(2) (1980), 331–346.

    Article  Google Scholar 

  52. S.P. Neuman, A. Fogg, E. Jacobson, A statistical approach to the inverse problem of aquifer hydrology, II. Case study, Water Resour. Res . 16 (1980),33–58.

    Article  Google Scholar 

  53. S.P. Neuman, S. Yakowitz, A statistical approach to the inverse problem of aquifer hydrology, I. Theory, Water Resources Research 15(4) (1979), 845-60.

    Google Scholar 

  54. J.E. Nordtvedt, G. Mejia, P. Yang, A.T. Watson, Estimation of capillary pressure and relative permeability from centrifuge experiments, SPE Reservoir Engineering , (to appear).

    Google Scholar 

  55. J.S. Osoba et al ., Laboratory measurements of relative permeability, Trans. AIME 192 (1951), 47–56.

    Google Scholar 

  56. E.J. Peters, S. Khataniar, The effect of instability on relative permeability curves obtained by dynamic-displacement method, SPEFE (December 1987), 469–474;Trans AIME 283.

    Google Scholar 

  57. M.S. Pilant, W. Rundell, A uniqueness theorem for determining conductivity from overspecified boundary data, Journal of Math. Anal. Appl . 136(1) (1988), 20–28.

    Article  MathSciNet  MATH  Google Scholar 

  58. M.S. Pilant, W. Rundell, An iteration method for the determination of an unknown boundary condition in a parabolic initial-boundary value problem, Proc. Edin. Math. Soc . 32 (1989), 59–71.

    Article  MathSciNet  MATH  Google Scholar 

  59. M.S. Pilant, W. Rundell, A method for identifying nonlinear terms in parabolic initial boundary value problems, Advances in Water Resources , Computational Mechanics Publications, 14(2) (1991), 83–88.

    Article  MathSciNet  Google Scholar 

  60. M.J.D. Powell, Variable metric methods for constrained optimization, Mathematical Programming: The State of the Art, Bonn 1982 (A. Bachem, M. Grötschel,B. Korte, eds.), Springer-Verlag, Berlin, 1983, 288–311.

    Google Scholar 

  61. L.A. Rapoport, W.J. Leas, Properties of linear waterfloods, Trans. AIME 198 (1953), 139-48.

    Google Scholar 

  62. P.C. Richmond, Estimating Multiphase Flow Functions in Porous Media From Dynamic Displacement Experiments , Ph.D. Dissertation, Texas A&M University, 1988.

    Google Scholar 

  63. P.C. Richmond, A.T. Watson, Estimation of multiphase flow functions from displacement experiments, SPE Reservoir Engineering 5 (1990), 121–127.

    Google Scholar 

  64. T. Russell, R. Ewing, C. Brebbia, W. Gray, and G. Pinder, eds., Computational Methods in Water Resources , IX. Vol. 1: Numerical Methods in Water Resources, Elsevier, London, 1992.

    Google Scholar 

  65. T. Russell, R. Ewing, C. Brebbia, W. Gray, and G. Pinder, eds., Computational Methods in Water Resources , IX. Vol. 2: Mathematical Modeling in Water Resources, Elsevier, London, 1992.

    Google Scholar 

  66. F. Santosa, W.W. Symes, An Analysis of Least Squares Inversion , Society of Exploration Geophysicists, Tulsa, Oklahoma, 1989.

    Book  Google Scholar 

  67. D.N. Saraf, J.P. Batycky, C.H. Jackson, D.B. Fisher, An experimental investigation of three-phase flow to water/oil/gas mixtures through water-wet sandstones, paper SPE 10761 , presented at the SPE California Regional Meeting, San Francisco, March 24–26, 1982.

    Google Scholar 

  68. A.M. Sarem, Three-phase relative permeability measurements by unsteady-state method, Soc. Pet. Eng. J . (September 1966), 199–205.

    Google Scholar 

  69. J.H. Seinfeld, C. Kravaris, Distributed parameter identification in geophysics - petroleum reservoirs and aquifers, Distributed Parameter Control Systems (S.G. Tzafestas, ed.), Pergamon, 1982.

    Google Scholar 

  70. P.C. Shah, G.R. Gavalas, J.H. Seinfeld, Error analysis in history matching: The optimal level of parameterization, SPEJ (June 1978), 219-28;Trans. AIME 265.

    Google Scholar 

  71. J.S. Sochacki, J.H. George, R.E. Ewing, S.B. Smithson, Interface conditions for acoustic and elastic wave equations, Geophysics 56 (1991), 168–181.

    Article  Google Scholar 

  72. E.V. Spronsen, Three-phase relative permeability measurements using the centrifuge method, paper SPE 10688 , presented at the Third Joint Symposium on Enhanced Oil Recovery, Tulsa, Oklahoma, April 4–7, 1982.

    Google Scholar 

  73. T.M. Tao, A.T. Watson, Accuracy of JBN estimates of relative permeability: Part 1. Error analysis, SPEJ (April 1984), 209–214.

    Google Scholar 

  74. T.M. Tao, A.T. Watson, Accuracy of JBN estimates of relative permeability: Part 2. Algorithms, SPEJ (April 1984), 215–223.

    Google Scholar 

  75. G.M. Valazquez, A Method for Estimating Three-Phase Flow Functions , Ph.D. Dissertation, Texas A&M University, College Station, Texas, May 1992.

    Google Scholar 

  76. C.R. Vogel, J.G. Wade, A modified Levenberg-Marquardt algorithm for large-scale inverse problems, Computation and Control III (K. Bowers, J. Lund, eds.), Birkhäuser, Boston, 1993, 367–378.

    Chapter  Google Scholar 

  77. C.R. Vogel, J.G. Wade, Analysis of cost ate discretizations in parameter estimation for linear evolution equations, SIAM Journal on Control and Optimziation , (to appear).

    Google Scholar 

  78. C.R. Vogel,J.G. Wade, Iterative SVD-based methods for ill- posed problems, SIAM Journal on Scientific and Statistical Computing 15(3) (1994), 736–754.

    Article  MathSciNet  MATH  Google Scholar 

  79. J.G. Wade, A convergence theory for fully Galerkin approximations of parabolic PDE in inverse problems, Journal of Mathematical Systems, Estimation and Control 4(2) (1994).

    Google Scholar 

  80. M.L. Wasserman, A.S. Emanuel, J.H. Seinfeld, Practial applications of optimal-control theory to history-matching multiphase simulator models, SPEJ (August 1975), 347–55;Trans. AIME 259 .

    Google Scholar 

  81. A.T. Watson, J.H. Seinfeld, G.R. Gavalas, P.T. Woo, History matching in two-phase petroleum reservoirsSPEJ (December 1980), 521–532.

    Google Scholar 

  82. A.T. Watson, J.M. Gatens III, W.J Lee, Z. Rahim, An analytical model for history matching naturally fractured reservoir production data, SPE Reservoir Engineering 5 (1990), 384–388.

    Google Scholar 

  83. A.T. Watson et al ., A regression-based method for estimating relative permeabilities from displacement experiments, SPERE (August 1988), 953–958.

    Google Scholar 

  84. A.T. Watson, P.C. Richmond, P.D. Kerig, T.M. Tao, A regression- based method for estimating relative permeabilities from displacement experiments, SPE Reservoir Engineering 3 (1988), 953–958.

    Google Scholar 

  85. A.T. Watson, J.G. Wade, R.E. Ewing, Parameter and system identification for fluid flow in underground reservoirs, in Inverse Problems and Optimal Design in Industry (H.W. Engl, J. McLaughlin, eds.), Teubner, Stuttgart, 1994.

    Google Scholar 

  86. A.T. Watson, H.S. Lane, J.M. Gatens III, History matching with cumulative production data, Journal of Petroleum Technology , 42 (1990), 96–100.

    Article  Google Scholar 

  87. A.T. Watoson, P.C. Richmond, P.D. Kerig, T.M. Tao, A regression- based method for estimating relative permeabilities from displacement experiments, SPE Reservoir Engineering , 3 (1988), 953–958.

    Google Scholar 

  88. S. Yakowitz, L. Duckstein, Instability in aquifer identification: Theory and cases, Water Resour. Res . 16(6) (1980), 1045–1061.

    Article  Google Scholar 

  89. P.-H. Yang, A.T. Watson, Automatic history matching with variable-metric methods, SPERE (August 1988), 995–1001.

    Google Scholar 

  90. P.-H. Yang, A.T. Watson, A Bayesian methodology for estimating relative permeability curves, SPE Reservoir Engineering (May 1991), 259–265.

    Google Scholar 

  91. W. Yeh, Review of parameter identification procedures in groundwater hydrology: The inverse problem, Water Resour. Res . 22(2) (1986), 95–108.

    Article  Google Scholar 

  92. W. Yeh, Y.S. Yoon, K.S. Lee, Aquifer parameter identification with kriging and optimum parameterization, Water Resour. Res . 19(1) (1983), 225–233.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Birkhäuser Boston

About this chapter

Cite this chapter

Ewing, R.E., Pilant, M.S., Wade, J.G., Watson, A.T. (1995). Identification and Control Problems in Petroleum and Groundwater Modeling. In: Lasiecka, I., Morton, B. (eds) Control Problems in Industry. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-2580-5_6

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-2580-5_6

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-1-4612-7589-3

  • Online ISBN: 978-1-4612-2580-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics