Skip to main content

Advertisement

SpringerLink
Log in

Population-based sampling methods for geological well testing

  • ORIGINAL PAPER
  • Published:
Computational Geosciences Aims and scope Submit manuscript

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.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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)

  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)

  3. Hamdi, H.: Illumination of channelised fluvial reservoirs using geological well testing and seismic modelling. Unpub. PhD Thesis, Heriot-Watt University, p 247 (2012)

  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)

    Article  Google Scholar 

  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)

  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)

  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)

  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)

  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)

  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. Corbett, P.W.M.: Petroleum geoengineering: integration of static and dynamic models, vol 12.DISC No. 12. EAGE/SEG (2009)

  12. Bourdet, D.: Well test analysis—the use of advanced interpretation models. Elsevier (2002)

  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

    Article  Google Scholar 

  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)

  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. 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)

  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

    Article  Google Scholar 

  18. Bard, Y.: Nonlinear Parameter Estimation. Academic Press, NY (1974)

    Google Scholar 

  19. Gilman, J.R., Ozgen, C.: Reservoir simulation: history matching and forecasting. Society of petroleum engineers, Richardson, TX (2013)

  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

    Article  Google Scholar 

  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)

  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)

  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

    Article  Google Scholar 

  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

    Article  Google Scholar 

  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 February

  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

    Article  Google Scholar 

  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

    Article  Google Scholar 

  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 December

  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)

  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)

    Article  Google Scholar 

  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

    Article  Google Scholar 

  32. Price, K., Storn, R.M., Lampinen, J.: Differential evolution: a practical approach to global optimization. Springer, Berlin (2005)

    Google Scholar 

  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. Ferraro, P., Verga, F.: Use of evolutionary algorithms in single and multi-objective optimization techniques for assisted history matching (2009)

  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

    Article  Google Scholar 

  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. 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)

  38. Kruschke, J.: Doing bayesian data analysis: A tutorial with R and Bugs. Academic Press (2010)

  39. Gamerman, D.: Markov chain Monte Carlo: stochastic simulation for Bayesian inference. Chapman & Hall, London (1997)

    Google Scholar 

  40. Shonkwiler, R.W., Mendivil, F.: Explorations in monte carlo methods. Springer, Berlin (2009)

    Book  Google Scholar 

  41. Tong, C.: PSUADE. In: Center for applied scientific computing lawrence livermore national laboratory, livermore, CA (2013)

  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

    Article  Google Scholar 

  43. Koziel, S., Leifsson, L.: Surrogate-based modeling and optimization. Springer, Berlin (2013)

  44. Forrester, A., Sobester, A., Keane, A.: Engineering design via surrogate modelling: A practical guide. Wiley, New York (2008)

  45. Friedman, J.H.: Multivariate adaptive regression splines. Ann. Stat 19(1), 1–67 (1991). doi:10.2307/2241837

    Article  Google Scholar 

  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

    Article  Google Scholar 

  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

    Article  Google Scholar 

  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

    Article  Google Scholar 

  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)

  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

    Article  Google Scholar 

  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

    Article  Google Scholar 

  52. Deutsch, C.V.: Geostatistical reservoir modelling. Oxford University Press, New York (2002)

    Google Scholar 

  53. Doyen, P.: Seismic reservoir characterization: an earth modelling perspective. EAGE publications (2007)

  54. Dubrule, O.: Geostatistics for seismic data integration in Earth models. Distinguished instructor series 6. Society of Exploration Geophysics, Tulsa, USA (2003)

  55. Deutsch, C.V., Journel, A.G.: GSLIB: geostatistical software library and user’s guide. Oxford University Press, New York (1992)

    Google Scholar 

  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)

    Article  Google Scholar 

  57. Chiles, J.P., Delfiner, P.: Geostatistics: modeling spatial uncertainty, vol. 713 of wiley series in probability and statistics. Wiley, New Jersey (2012)

    Book  Google Scholar 

  58. Oliver, M.A., Webster, R.: Basic steps in geostatistics: the variogram and kriging. Springer, Berlin (2015)

    Google Scholar 

  59. Pyrcz, M.J., Deutsch, C.V.: Geostatistical reservoir modeling. Oxford University Press, London (2014)

    Google Scholar 

  60. Pedersen, M.E.H.: Good parameters for differential evolution. In: vol. Technical Report HL1002 (2010)

  61. Hajizadeh, Y.: Population-based algorithms for improved history matching and uncertainty quantification of petroleum reservoirs Heriot-Watt University (2011)

  62. 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)

  63. 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. 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. Breiman, L.: Random forests. Mach. Learn. 45(1), 5–32 (2001). doi:10.1023/A:1010933404324

    Article  Google Scholar 

  66. 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, Canada

  67. 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. 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

    Article  Google Scholar 

  69. 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

    Article  Google Scholar 

  70. 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)

    Article  Google Scholar 

  71. Saltelli, A.: Sensitivity analysis for importance assessment. Risk Anal. 22(3), 579–590 (2002). doi: 10.1111/02724332.00040

    Article  Google Scholar 

  72. 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

    Article  Google Scholar 

  73. 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

    Article  Google Scholar 

  74. 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)

Download references

Author information

Authors and Affiliations

  1. University of Calgary, 2500 University Dr NW, Calgary, Alberta, T2N 1N4, Canada

    Hamidreza Hamdi, Yasin Hajizadeh & Mario Costa Sousa

Authors
  1. Hamidreza Hamdi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yasin Hajizadeh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mario Costa Sousa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hamidreza Hamdi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hamdi, H., Hajizadeh, Y. & Costa Sousa, M. Population-based sampling methods for geological well testing. Comput Geosci 19, 1089–1107 (2015). https://doi.org/10.1007/s10596-015-9522-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10596-015-9522-7

Keywords

Search

Navigation