Advertisement

Mathematical Geosciences

, Volume 45, Issue 2, pp 225–252 | Cite as

Reservoir Description with Integrated Multiwell Data Using Two-Dimensional Wavelets

  • Abeeb A. AwotundeEmail author
  • Roland N. Horne
Article

Abstract

Inferring reservoir data from dynamic production data has long been done through matching the production history. However, proper integration of available production history has always been a challenge. Different production history data such as well pressure and water cut often occur at different scales making their joint inversion difficult. Furthermore, production data obtained from the same well or even the same reservoir are often correlated making a significant portion of the dataset redundant. Thirdly, the massiveness of the data recorded from wells in a large reservoir over a long period of time makes the nonlinear inversion of such data computational demanding. In this paper, we propose the integration of multiwell production data using wavelet transform. The method involves the use of a two-dimensional wavelet transformation of the data space in order to integrate multiple production data and reduce the correlation between multiwell data. Multiple datasets from different wells, representing different production responses (pressure, water cut, etc.), were treated as a single matrix of data rather than separate vectors that assume no correlation amongst datasets. This enabled us to transform the multiwell production data into a two-dimensional wavelet domain and subsequently select the most important wavelets for history match. By minimizing the square of the Frobenius norm of the residual matrix we were able to match the calculated response to the observed response. We derived the relationship that allows us to replace a conventional minimization of the sum of squares of the l 2 norms of multi-objective functions with the minimization of the square of the Frobenius norm of the integrated data. The usefulness of the approach is demonstrated using two examples. The approach proved very effective at reducing correlation between multiwell data. In addition, the method helped to reduce the cost of computing sensitivity coefficients. However, the method gave poor prediction of water cut when the datasets were not scaled before inverse modeling.

Keywords

Parameter estimation Data integration Wavelet transform Adjoint sensitivity Reparameterization 

Notes

Acknowledgements

We acknowledge the support received from the Saudi Aramco Research Collaboration Fellowship and the Stanford University Petroleum Research Institute for Innovation in Well Testing (SUPRI-D) toward completing this work.

References

  1. Al-Harbi M, Cheng H, He Z, Datta-Gupta A (2005) Streamline-based production data integration in naturally fractured reservoirs. SPE Reserv Eval Eng 10:426–439 Google Scholar
  2. Awotunde AA, Horne RN (2011a) A multiresolution analysis of the relationship between spatial distribution of reservoir parameters and time distribution of well-test data. SPE Reserv Eval Eng 14:345–356 Google Scholar
  3. Awotunde AA, Horne RN (2011b) A wavelet approach to adjoint state sensitivity computation for steady-state differential equations. Water Resour Res 47:1–21 CrossRefGoogle Scholar
  4. Awotunde AA, Horne RN (2012) An improved adjoint-sensitivity computations for multiphase flow using wavelets. SPE J 17:402–417 Google Scholar
  5. Aziz K, Settari A (1979) Petroleum reservoir simulation. Applied Science Publishers, London Google Scholar
  6. Caers J (2003) Geostatistical history matching under training-image based geological constraints. SPE J 8:218–226 Google Scholar
  7. Carter RD, Kemp LF, Piece AC, Williams DL (1974) Performance matching with constraints. SPE J 14:187–196, trans., AIME 257 Google Scholar
  8. Chavent GM, Dupuy M, Lemonnier P (1975) History matching by use of optimal theory. SPE J 15:74–86 Google Scholar
  9. Chu L, Schatzinger RA, Tham MK (1998) Application of wavelet analysis to upscaling of rock properties. SPE Reserv Eval Eng 1:75–81 Google Scholar
  10. Chui CK (1992) An introduction to wavelets. Academic Press, San Diego. Google Scholar
  11. Gill PE, Murray W, Wright MH (1981) Practical optimization. Academic Press, New York Google Scholar
  12. Golub GH, Van Loan CF (1996) Matrix computations. John Hopkins University Press, Baltimore Google Scholar
  13. Griva I, Nash SG, Sofer A (1996) Linear and nonlinear optimization. SIAM Press, Philadelphia Google Scholar
  14. He Z, Yoon S, Datta-Gupta A (2002) Streamline-based production data integration with gravity and changing field conditions. SPE J 7:423–436 Google Scholar
  15. Hu LY (2000) Gradual deformation and iterative calibration of Gaussian-related stochastic models. Math Geol 32:87–108 CrossRefGoogle Scholar
  16. Hu LY (2002) Combination of dependent realizations within the gradual deformation method. Math Geol 34:953–963 CrossRefGoogle Scholar
  17. Huang X, Kelkar M (1996) Integration of dynamic data for reservoir characterization in the frequency domain. Paper SPE 115795 presented at the SPE annual technical conference and exhibition, Denver, Colorado, USA, 6–9 October 1996 Google Scholar
  18. Jacquard P, Jain C (1965) Permeability distribution from field pressure data. SPE J 5:281–294, trans., AIME 234 Google Scholar
  19. Johansen K, Caers J, Suzuki S (2007) Hybridization of the probability perturbation method with gradient information. Comput Geosci 11:319–331 CrossRefGoogle Scholar
  20. Journel AJ (1974) Geostatistics for conditional simulation of ore bodies. Econ Geol 69:673–687 CrossRefGoogle Scholar
  21. Levenberg K (1944) A method for the solution of certain non-linear problems in least squares. Q Appl Math 2:164–168 Google Scholar
  22. Liu M, Liu YZ, Guo T, Cai L (2006) Interwell heterogeneity analysis by dynamic correlation method. Paper SPE 104399 presented at the SPE international oil and gas conference and exhibition, Beijing, China, 5–7 December 2006 Google Scholar
  23. Lu P, Horne RN (2000) A multiresolution approach to reservoir parameter estimation using wavelet analysis. Paper SPE 62985 presented at the SPE annual technical conference and exhibition, Dallas, 1–4 October 2000 Google Scholar
  24. Mallat S (1989) A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans Pattern Anal Mach Intell 11:674–693 CrossRefGoogle Scholar
  25. Marquardt D (1963) An algorithm for least-squares estimation of nonlinear parameters. SIAM J Appl Math 11:431–441 CrossRefGoogle Scholar
  26. Nakashima T, Durlofsky L (2010) Accurate representation of near-well effects in coarse-scale models of primary oil production. Transp Porous Media 83:741–770 CrossRefGoogle Scholar
  27. Nocedal J, Wright SJ (2006) Numerical optimization, 2nd edn. Springer, New York Google Scholar
  28. Panda MN, Mosher CC, Chopra AK (2001) Reservoir modeling using scale-dependent data. SPE J 6:157–170 Google Scholar
  29. Peaceman DW (1977) Fundamentals of numerical reservoir simulation. Elsevier, New York Google Scholar
  30. Percival D, Walden AT (2000) Wavelet methods for time series analysis. Cambridge University Press, Cambridge, England Google Scholar
  31. Sahni I, Horne RN (2005) Multiresolution wavelet analysis for improved reservoir description. SPE Reserv Eval Eng 8:53–69 Google Scholar
  32. Sahni I, Horne RN (2006) Generating multiple history-matched realizations using wavelets. SPE Reserv Eval Eng 9:217–226 Google Scholar
  33. Strebelle S, Payrazyan K, Caers J (2003) Modeling of a deepwater turbidite reservoir conditional to seismic data using principal component analysis and multi-point geostatistics. SPE J 8:227–235 Google Scholar
  34. Yoon S, Malallah AH, Datta-Gupta A, Vasco DW, Behrens RA (2001) A multiscale approach to production data integration using streamline models. SPE J 6:182–192 Google Scholar

Copyright information

© International Association for Mathematical Geosciences 2013

Authors and Affiliations

  1. 1.King Fahd University of Petroleum & MineralsDhahranSaudi Arabia
  2. 2.Stanford UniversityStanfordUSA

Personalised recommendations