Mathematical Geology

, Volume 27, Issue 6, pp 763–787 | Cite as

Characterizing heterogeneous permeable media with spatial statistics and tracer data using sequential simulated annealing

  • Akhil Datta-Gupta
  • Larry W. Lake
  • Gary A. Pope
Article

Abstract

Characterizing heterogeneous permeable media using flow and transport data typically requires solution of an inverse problem. Such inverse problems are intensive computationally and may involve iterative procedures requiring many forward simulations of the flow and transport problem. Previous attempts have been limited mostly to flow data such as pressure transient (interference) tests using multiple observation wells. This paper discusses an approach to generating stochastic permeability fields conditioned to geologic data in the form of a vertical variogram derived from cores and logs as well as fluid flow and transport data, such as tracer concentration history, by sequential application of simulated annealing (SA). Thus, the method incorporates elements of geostatistics within the framework of inverse modeling. For tracer-transport calculations, we have used a semianalytic transit-time algorithm which is fast, accurate, and free of numerical dispersion. For steady velocity fields, we introduce a “transit-time function” which demonstrates the relative importance of data from different sources. The approach is illustrated by application to a set of spatial permeability measurements and tracer data from an experiment in the Antolini Sandstone, an eolian outcrop from northern Arizona. The results clearly reveal the importance of tracer data in reproducing the correlated features (channels) of the permeability field and the scale effects of heterogeneity.

Key words

sequential simulated annealing semianalytic tracer response integrated heterogeneity modeling 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aarts, E., and Korst, J., 1989, Simulated annealing and boltzmann machines: John Wiley & Sons, New York, 272 p.Google Scholar
  2. Bear, J., 1988, Dynamics of fluids in porous media: Dover Publications Inc., New York.Google Scholar
  3. Bruining, J., 1991, Modeling reservoir heterogeneity with fractals: Ctr. Enhanced Oil and Gas Recovery Research Rept., Univ. Texas at Austin.Google Scholar
  4. Carrera, J., and Neuman, S. P., 1986, Estimation of aquifer parameters under transient and steady state conditions: 3. Application to synthetic and field data: Water Resources Research, v. 22, no. 2, p. 228–242.Google Scholar
  5. Datta-Gupta, A., and King, M. J., 1995, A semianalytic approach to tracer flow modeling in heterogeneous permeable media: Advances in Water Resources, v. 18, no. 1, p. 9–24.Google Scholar
  6. Datta-Gupta, A., Vasco, D. W., Long, J. C. S., and Vomvoris, S., 1994, Stochastic modeling of spatial heterogeneities conditioned to hydraulic and tracer tests, 1994: Proc. Fifth Annual International Conference on High Level Radioactive Waste Management, v. 4, p. 2624–2632.Google Scholar
  7. Datta-Gupta, A., Lake, L. W., Pope, G. A., Sepehrnoori, K., and King, M. J., 1991, High resolution monotonic schemes for reservoir fluid flow simulation: In Situ v. 15, no. 3, p. 289–317.Google Scholar
  8. Deng, F. W., Cushman, J. H., and Delleur, J. W., 1993, Adaptive estimation of the log fluctuating conductivity from tracer data at the Cape Cod site: Water Resources Research, v. 29, no. 12, p. 4011–4018.Google Scholar
  9. Deutsch, C. V., and Journel, A. G., 1994, The application of simulated annealing to stochastic reservoir modeling: Advanced Technology Series, SPE, Richardson, Texas, v. 2, no. 2, p. 222–227.Google Scholar
  10. Dietrich, C. R., and Newsam, G. N., 1989, A stability analysis of the geostatistical approach to aquifer transmissivity identification: Stochastic Hydrol. Hydraul., v. 3, p. 293–316.Google Scholar
  11. Dougherty, D. E., and Marryott, R. A., 1991, Optimal groundwater management 1. Simulated annealing: Water Resources Research, v. 27, no. 10, p. 2493–2508.Google Scholar
  12. Farmer, C. L., 1989, Numerical rocks: Proc. Joint 1MA/SPE European Conference on Mathematics of Oil Recovery: Robinson College, Cambridge Univ.Google Scholar
  13. Ganapathy, S., Wreath, D. G., Lim, M. T., Rouse, B. A., Pope, G. A., and Sepehrnoori, K., 1993, Simulation of heterogeneous sandstone experiments characterized using CT scanning: SPE Formation Evaluation, v. 8, no. 4, p. 273–279.Google Scholar
  14. Johnson, D. S., Argon, C. R., McGeoch, L. A., and Schevon, C., 1989, Optimization by simulated annealing: an experimental evaluation, Part I, Graph partitioning: Operations Res., v. 37, p. 865–892.Google Scholar
  15. Journel, A. B., and Huijbregts, C. J., 1978, Mining geostatistics: Academic Press, New York, 600 p.Google Scholar
  16. Kirkpatrick, S., Gelatt, C. D., Jr., and Vecchi, M. P., 1983, Optimization by simulated annealing: Science, v. 220, no. 4598, p. 671–680.Google Scholar
  17. Kitanidis, P. K., and Vomvoris, E. G., 1983, A geostatistical approach to the inverse problem in groundwater modeling (steady state) and one-dimensional simulations: Water Resources Research, v. 19, no. 3, p. 677–690.Google Scholar
  18. Li, Dachang, and Lake, L. W., 1994, A moving window semivariance estimator: Water Resources Research, v. 30, no. 5, p. 1479–1489.Google Scholar
  19. Mantoglou, A., and Wilson, J. L., 1982, The Turning Bands Method for simulation of random fields using line generation by spectral method: Water Resources Research, v. 18, no. 5, p. 1379–1394.Google Scholar
  20. Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A., and Teller, E., 1953, Equation of state calculations by fast computing machines: Jour. Chem. Phys., v. 21, no. 6, p. 1087–1092.Google Scholar
  21. Pollock, D. W., 1988, Semianalytical computation of pathlines for finite-difference models: Ground Water, v. 26, no. 6, p. 743–750.Google Scholar
  22. Rothman, D. H., 1986, Automatic estimation of large residual statics corrections: Geophysics, v. 51, no. 2, p. 332–346.Google Scholar
  23. Rubin, Y., and Dagan, G., 1987, Stochastic identification of transmissivity and effective recharge in steady groundwater Flow. 2, case study: Water Resources Research, v. 23, no. 7, p. 1193–1200.Google Scholar
  24. Sen, M. K., and Stoffa, P. L., 1991, Nonlinear one-dimensional seismic waveform inversion using simulated annealing: Geophysics, v. 56, no. 10, p. 1624–1638.Google Scholar
  25. Sen, M. K., Datta-Gupta, A., Stoffa, P. L., Lake, L. W., and Pope, G. A., 1995, Stochastic reservoir modeling using simulated annealing and genetic algorithm: SPE Formation Evaluation, v. 10, no. 1, p. 49–55.Google Scholar
  26. Shinozuka, M., and Jan, C. M., 1972, Digital simulation of random processes and its applications: Jour. Sound and Vibrations, v. 25, no. 1, p. 111–128.Google Scholar
  27. Yang, A. P., 1990, Stochastic heterogeneity and dispersion: unpubl. doctoral dissertation. The Univ. Texas at Austin.Google Scholar

Copyright information

© International Association for Mathematical Geology 1995

Authors and Affiliations

  • Akhil Datta-Gupta
    • 1
  • Larry W. Lake
    • 2
  • Gary A. Pope
    • 2
  1. 1.Department of Petroleum EngineeringTexas A&M UniversityCollege Station
  2. 2.Department of Petroleum EngineeringThe University of TexasAustin

Personalised recommendations