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Computational Geosciences

, Volume 19, Issue 5, pp 1089–1107 | Cite as

Population-based sampling methods for geological well testing

  • Hamidreza Hamdi
  • Yasin Hajizadeh
  • Mario Costa Sousa
ORIGINAL PAPER

Abstract

In this paper, the application of a population-based sampling algorithm, i.e., differential evolution, in the geological well testing of a multi-layered faulted reservoir model is discussed. In this sense, the available multiple well test datasets are used to calibrate the geological model parameters rather than fitting simplified analytical models. In the exercise studied in this paper, the parameter space includes a range of geostatistical, petrophysical, and structural parameters. The differential evolution algorithm starts with an initial random population from the ranges of input variables and progresses with successive evaluation of the static models’ transient tests. The static models’ input parameters are perturbed to generate new populations, which can finally match the truth model well test derivative with lower misfits. The ensemble of population models (samples) along with the misfit values are used to highlight the value of well test data in reducing the uncertainty in the parameter space. A Bayesian framework is employed to implement the Markov chain Monte Carlo (McMC) methods to estimate the posterior distributions of the parameters. The results are confirmed by the sample-based Sobol sensitivity indices, which rank the influential parameters. To reduce the computational cost of the McMC and sensitivity indices, a cross-validated proxy model (i.e., Multivariate Adaptive Regression Spline) is constructed. The effect of different variants of differential evolution algorithm on the geological well test matching is also discussed. This paper provides a workflow for quantitative integration of well test data into the reservoir characterization workflow.

Keywords

Geological well testing Markov chain Monte Carlo Proxy model Differential evolution 

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References

  1. 1.
    Kuchuk, F.J., Hollaender, F., Onur, M., Ramakrishnan, T.S.: Pressure transient formation and well testing:Convolution, Deconvolution and Nonlinear Estimation Elsevier Science Ltd (2010)Google Scholar
  2. 2.
    Landa, J.L., Kamal, M.M., Jenkins, C.D., Horne, R.N.: Reservoir characterization constrained to well test data: a field example. Paper presented at the SPE Annual Technical Conference and Exhibition, Denver, Colorado 6-9 October (1996)Google Scholar
  3. 3.
    Hamdi, H.: Illumination of channelised fluvial reservoirs using geological well testing and seismic modelling. Unpub. PhD Thesis, Heriot-Watt University, p 247 (2012)Google Scholar
  4. 4.
    Corbett, P.W.M., Hamdi, H., Gurav, H.: Layered fluvial reservoirs with internal fluid cross flow: a well-connected family of well test pressure transient responses. Pet. Geosci. 18, 231–238 (2012)CrossRefGoogle Scholar
  5. 5.
    Corbett, P.W.M., Geiger-Boschung, S., Borges, L.P., Garayev, M., Gonzalez, J.G., Valdez, C.: Limitations in numerical well test modelling of fractured carbonate rocks. Paper presented at the SPE EUROPEC/EAGE Annual Conference and Exhibition, Barcelona, Spain, 01 (2010)Google Scholar
  6. 6.
    Corbett, P.W.M., Mesmari, A., Stewart, G.: A method for using the naturally-occurring negative geoskin in the description of fluvial reservoirs (1996)Google Scholar
  7. 7.
    Massonnat, G.J., Bandiziol, D.: Interdependence between geology and well test interpretation. Paper presented at the SPE annual technical conference and exhibition, Dallas, Texas, 01 (1991)Google Scholar
  8. 8.
    Zheng, S., Corbett, P., Stewart, G.: The impact of variable formation thickness on pressure transient behavior and well test permeability in fluvial meander loop reservoirs. Paper presented at the SPE Annual Technical Conference and Exhibition, Denver, Colorado, 01 (1996)Google Scholar
  9. 9.
    Hamdi, H., Ruelland, P., Bergey, P., Corbett, P.W.M.: Using geological well testing in the improved selection of appropriate reservoir models. Petroleum Geoscience (2013)Google Scholar
  10. 10.
    Landa, J.L.: Integration of well testing into reservoir characterization. In: Kamal, M.M (ed.) : Transient well testing, vol. 23, p 849. Society of Petroleum Engineers, USA (2009)Google Scholar
  11. 11.
    Corbett, P.W.M.: Petroleum geoengineering: integration of static and dynamic models, vol 12.DISC No. 12. EAGE/SEG (2009)Google Scholar
  12. 12.
    Bourdet, D.: Well test analysis—the use of advanced interpretation models. Elsevier (2002)Google Scholar
  13. 13.
    Zheng, S.Y., Corbett, P.W.M., Emery, A.: Geological interpretation of well test analysis: a case study from a fluvial reservoir in the Gulf of Thailand. J. Pet. Geol. 26(1), 49–64 (2003). doi: 10.1111/j.1747-5457.2003.tb00017.x CrossRefGoogle Scholar
  14. 14.
    Boutaud de la Combe, J.-L., Akinwumni, O., Dumay, C.D., Tachon, M.: Use of DST for effective dynamic appraisal: case studies from deep offshore West Africa and associated methodology. In: Paper presented at the SPE Annual Technical Conference and Exhibition, Dallas, Texas, 01 (2005)Google Scholar
  15. 15.
    Ehlig-Economides, C.A., Joseph, J.A., Ambrose Jr. R.W., Norwood, C.: A modern approach to reservoir testing (includes associated papers 22220 and 22327). SPE J. Pet. Technol. 42(12) (1990). doi:  10.2118/19814-pa
  16. 16.
    Gok, I., Onur, M., Kuchuk, F.J.: Estimating formation properties in heterogeneous reservoirs using 3D interval pressure transient test and geostatistical data. Paper presented at the SPE Middle East Oil and Gas Show and Conference, Kingdom of Bahrain, 01 (2005)Google Scholar
  17. 17.
    Robertson, E., Corbett, P.W.M., Hurst, A., Satur, N., Cronin, B.T.: Synthetic well test modelling in a high net-to-gross outcrop system for turbidite reservoir description. Pet. Geosci. 8(1), 19–30 (2002). doi:  10.1144/petgeo.8.1.19 CrossRefGoogle Scholar
  18. 18.
    Bard, Y.: Nonlinear Parameter Estimation. Academic Press, NY (1974)Google Scholar
  19. 19.
    Gilman, J.R., Ozgen, C.: Reservoir simulation: history matching and forecasting. Society of petroleum engineers, Richardson, TX (2013)Google Scholar
  20. 20.
    Poli, R., Kennedy, J., Blackwell, T.: Particle swarm optimization. Swarm Intell. 1(1), 33–57 (2007). doi: 10.1007/s11721-007-0002-0  10.1007/s11721-007-0002-0 CrossRefGoogle Scholar
  21. 21.
    Hajizadeh, Y., Christie, M.A., Demyanov, V.: Ant colony optimization for history matching. Paper presented at the EUROPEC/EAGE Conference and Exhibition, Amsterdam, The Netherlands, 8-11 (2009)Google Scholar
  22. 22.
    Storn, R., Price, K.: Differential evolution—a simple and efficient adaptive scheme for global optimization over continuous spaces. In: Technical Report TR-95-012. Berkeley (1995)Google Scholar
  23. 23.
    Oliver, D., Chen, Y.: Recent progress on reservoir history matching: a review. Comput. Geosci 15(1), 185–221 (2011). doi: 10.1007/s10596-010-9194-2 CrossRefGoogle Scholar
  24. 24.
    Evensen, G.: Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J. Geophys. Res. Oceans 99(C5), 10143–10162 (1994). doi: 10.1029/94JC00572 CrossRefGoogle Scholar
  25. 25.
    Bazargan, H., Christie, M., Tchelepi, H.: Efficient Markov chain Monte Carlo sampling using polynomial chaos expansion. Paper presented at the SPE Reservoir Simulation Symposium, The Woodlands, Texas, USA 18-20 FebruaryGoogle Scholar
  26. 26.
    Heidari, L., Gervais, V., Ravalec, M.L., Wackernagel, H.: History matching of petroleum reservoir models by the Ensemble Kalman Filter and parameterization methods. Comput. Geosci. 55(0), 84–95 (2013). doi:  10.1016/j.cageo.2012.06.006 CrossRefGoogle Scholar
  27. 27.
    Lu, F., Morzfeld, M., Tu, X., Chorin, A.J.: Limitations of polynomial chaos expansions in the Bayesian solution of inverse problems. J. Comput. Phys. 282, 138–147 (2015). doi: 10.1016/j.jcp.2014.11.010 CrossRefGoogle Scholar
  28. 28.
    Hajizadeh, Y., Christie, M.A., Demyanov, V.: Application of differential evolution as a new method for automatic history matching. Paper presented at the Kuwait International Petroleum Conference and Exhibition, Kuwait City, Kuwait,14-16 DecemberGoogle Scholar
  29. 29.
    Wan, Z., Igusa, T.: Adaptive sampling for optimization under uncertainty. In: Proceedings of the 4th international symposium on uncertainty modelling and analysis, College Park, MD. p. 423. IEEE computer society, 943696 (2003)Google Scholar
  30. 30.
    Wetter, M., Wright, J.A.: A comparison of deterministic and probabilistic optimization algorithms for nonsmooth simulation based optimization. Build. Environ. 39(8), 989–999 (2004)CrossRefGoogle Scholar
  31. 31.
    Nissen, V., Propach, J.: On the robustness of population-based versus point-based optimization in the presence of noise. IEEE Trans. Evol. Comput. 2(3), 107–119 (1998). doi: 10.1109/4235.735433 CrossRefGoogle Scholar
  32. 32.
    Price, K., Storn, R.M., Lampinen, J.: Differential evolution: a practical approach to global optimization. Springer, Berlin (2005)Google Scholar
  33. 33.
    Bourdet, D., Whittle, T.M., Douglas, A.A., Pirard, Y.M.: A new set of type curves simplifies well test analysis. World Oil. 196(6), 95–106 (1983)Google Scholar
  34. 34.
    Ferraro, P., Verga, F.: Use of evolutionary algorithms in single and multi-objective optimization techniques for assisted history matching (2009)Google Scholar
  35. 35.
    Barker, J.W., Cuypers, M., Holden, L.: Quantifying uncertainty in production forecasts: another look at the PUNQ-S3 Problem. SPE J. 6(4), 433–441 (2001). doi: 10.2118/74707-pa CrossRefGoogle Scholar
  36. 36.
    Erbaş, D., Christie, M.: Comment la stratégie de l’échantillonnage affecte-t-elle les estimations d’incertitude ? Oil & Gas Science and Technology -. IFP Rev. 62(2), 155–167 (2007)Google Scholar
  37. 37.
    Alpak, F.O., Kats, F.v.: Stochastic history matching of a deepwater turbidite reservoir. Paper presented at the SPE Reservoir Simulation Symposium, The Woodlands, Texas, 2-4 (2009)Google Scholar
  38. 38.
    Kruschke, J.: Doing bayesian data analysis: A tutorial with R and Bugs. Academic Press (2010)Google Scholar
  39. 39.
    Gamerman, D.: Markov chain Monte Carlo: stochastic simulation for Bayesian inference. Chapman & Hall, London (1997)Google Scholar
  40. 40.
    Shonkwiler, R.W., Mendivil, F.: Explorations in monte carlo methods. Springer, Berlin (2009)CrossRefGoogle Scholar
  41. 41.
    Tong, C.: PSUADE. In: Center for applied scientific computing lawrence livermore national laboratory, livermore, CA (2013)Google Scholar
  42. 42.
    Geman, S., Geman, D.: Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. Pattern analysis and machine intelligence. IEEE Trans. PAMI 6(6), 721–741 (1984). doi: 10.1109/TPAMI.1984.4767596 CrossRefGoogle Scholar
  43. 43.
    Koziel, S., Leifsson, L.: Surrogate-based modeling and optimization. Springer, Berlin (2013)Google Scholar
  44. 44.
    Forrester, A., Sobester, A., Keane, A.: Engineering design via surrogate modelling: A practical guide. Wiley, New York (2008)Google Scholar
  45. 45.
    Friedman, J.H.: Multivariate adaptive regression splines. Ann. Stat 19(1), 1–67 (1991). doi: 10.2307/2241837 CrossRefGoogle Scholar
  46. 46.
    Zhan, C.-S., Song, X.-M., Xia, J., Tong, C.: An efficient integrated approach for global sensitivity analysis of hydrological model parameters. Environ. Model Softw. 41(0), 39–52 (2013). doi: 10.1016/j.envsoft.2012.10.009 CrossRefGoogle Scholar
  47. 47.
    Balshi, M.S., McGuire, A.D., Duffy, P., Flannigan, M., Walsh, J., Melillo, J.: Assessing the response of area burned to changing climate in western boreal North America using a multivariate adaptive regression splines (MARS) approach. Glob. Chang. Biol. 15(3), 578–600 (2009). doi: 10.1111/j.1365-2486.2008.01679.x CrossRefGoogle Scholar
  48. 48.
    Leathwick, J.R., Rowe, D., Richardson, J., Elith, J., Hastie, T.: Using multivariate adaptive regression splines to predict the distributions of New Zealand’s freshwater diadromous fish. Freshw. Biol. 50(12), 2034–2052 (2005). doi: 10.1111/j.1365-2427.2005.01448.x CrossRefGoogle Scholar
  49. 49.
    Hamdi, H., Hajizadeh, Y., Azimi, J., Sousa, M.C.: Sequential Bayesian optimization coupled with differential evolution for geological well testing. Paper presented at the 76th EAGE Conference and Exhibition 2014 Amsterdam, the Netherlands,16–19 (2014)Google Scholar
  50. 50.
    Cheng, M.-Y., Cao, M.-T.: Accurately predicting building energy performance using evolutionary multivariate adaptive regression splines. Appl. Soft Comput. 22(0), 178–188 (2014). doi: 10.1016/j.asoc.2014.05.015 CrossRefGoogle Scholar
  51. 51.
    Hamdi, H., Jamiolahmady, M., Corbett, P.W.M.: Modeling the interfering effects of gas condensate and geological heterogeneities on transient pressure response. SPE J. 18(4), 656–669 (2013). doi:  10.2118/143613-pa CrossRefGoogle Scholar
  52. 52.
    Deutsch, C.V.: Geostatistical reservoir modelling. Oxford University Press, New York (2002)Google Scholar
  53. 53.
    Doyen, P.: Seismic reservoir characterization: an earth modelling perspective. EAGE publications (2007)Google Scholar
  54. 54.
    Dubrule, O.: Geostatistics for seismic data integration in Earth models. Distinguished instructor series 6. Society of Exploration Geophysics, Tulsa, USA (2003)Google Scholar
  55. 55.
    Deutsch, C.V., Journel, A.G.: GSLIB: geostatistical software library and user’s guide. Oxford University Press, New York (1992)Google Scholar
  56. 56.
    Corbett, P.W.M., Hamdi, H., Gurav, H.: Layered fluvial reservoirs with internal fluid cross flow: a well-connected family of well test pressure transient responses. Pet. Geosci. 18, 219–229 (2012)CrossRefGoogle Scholar
  57. 57.
    Chiles, J.P., Delfiner, P.: Geostatistics: modeling spatial uncertainty, vol. 713 of wiley series in probability and statistics. Wiley, New Jersey (2012)CrossRefGoogle Scholar
  58. 58.
    Oliver, M.A., Webster, R.: Basic steps in geostatistics: the variogram and kriging. Springer, Berlin (2015)Google Scholar
  59. 59.
    Pyrcz, M.J., Deutsch, C.V.: Geostatistical reservoir modeling. Oxford University Press, London (2014)Google Scholar
  60. 60.
    Pedersen, M.E.H.: Good parameters for differential evolution. In: vol. Technical Report HL1002 (2010)Google Scholar
  61. 62.
    Hajizadeh, Y.: Population-based algorithms for improved history matching and uncertainty quantification of petroleum reservoirs Heriot-Watt University (2011)Google Scholar
  62. 63.
    Hamdi, H., Behmanesh, H., Clarkson, C.R., Costa Sousa, M.: Using differential evolution for compositional history-matching of a tight gas condensate well in the Montney Formation in western Canada. Journal of Natural Gas Science and Engineering (in press) (2015)Google Scholar
  63. 64.
    Tvrdik, J.: Differential evolution: competitive setting of control parameters. In: Proceedings of the International Multiconference on Computer Science and Information Technology, 207–213 (2006). http://www.citeulike.org/user/andizuend/article/8501230
  64. 65.
    McKay, M.D., Beckman, R.J., Conover, W.J.: A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21(2), 239–245 (1979). doi:  10.2307/1268522 Google Scholar
  65. 66.
    Breiman, L.: Random forests. Mach. Learn. 45(1), 5–32 (2001). doi: 10.1023/A:1010933404324 CrossRefGoogle Scholar
  66. 67.
    Kohavi, R.: A study of cross-validation and bootstrap for accuracy estimation and model selection. Paper presented at the Proceedings of the 14th international joint conference on Artificial intelligence- volume 2,Montreal, Quebec, CanadaGoogle Scholar
  67. 68.
    Elisseeff, A., Pontil, M.: Leave-one-out error and stability of learning algorithms with applications. In: Suykens, J., Horvath, G., Basu, S., Micchelli, C., Vandewalle, J. (eds.) Learning Theory and Practice. IOS Press, Amsterdam (2002)Google Scholar
  68. 69.
    Homma, T., Saltelli, A.: Importance measures in global sensitivity analysis of nonlinear models. Reliab. Eng. Syst. Saf 52(1), 1–17 (1996). doi: 10.1016/0951-8320(96)00002-6 CrossRefGoogle Scholar
  69. 70.
    Sobol’, I.M.: On the distribution of points in a cube and the approximate evaluation of integrals. USSR Comput. Math. Math. Phys. 7(4), 86–112 (1967). doi: 10.1016/0041-5553(67)90144-9 CrossRefGoogle Scholar
  70. 71.
    van Riel, N.A.W.: Dynamic modelling and analysis of biochemical networks: mechanism-based models and model-based experiments. Brief. Bioinform 7(4), 364–374 (2006)CrossRefGoogle Scholar
  71. 72.
    Saltelli, A.: Sensitivity analysis for importance assessment. Risk Anal. 22(3), 579–590 (2002). doi:  10.1111/02724332.00040 CrossRefGoogle Scholar
  72. 73.
    Shukhman, B.V., Sobol’, I.M.: Integration with quasirandom sequences: numerical experience. Int. J. Mod. Phys. C 06(02), 263–275 (1995). doi: 10.1142/S0129183195000204 CrossRefGoogle Scholar
  73. 74.
    Bollen, K., Stine, R.: Direct and indirect effects: classical and bootstrap estimates of variability. Sociol. Methodol. 20, 115–140 (1990) http://www.citeulike.org/user/ctacmo/article/553224 CrossRefGoogle Scholar
  74. 75.
    Tong, C., Graziani, F.: A Practical global sensitivity analysis methodology for multi-physics applications. In: Graziani, F. (ed.) Computational Methods in Transport: Verification and Validation, vol. 62. Lecture Notes in Computational Science and Engineering, 277-299. Springer Berlin Heidelberg (2008)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Hamidreza Hamdi
    • 1
  • Yasin Hajizadeh
    • 1
  • Mario Costa Sousa
    • 1
  1. 1.University of CalgaryCalgaryCanada

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