Mathematical Geosciences

, Volume 42, Issue 1, pp 1–27 | Cite as

Compressed History Matching: Exploiting Transform-Domain Sparsity for Regularization of Nonlinear Dynamic Data Integration Problems

  • Behnam JafarpourEmail author
  • Vivek K. Goyal
  • Dennis B. McLaughlin
  • William T. Freeman


In this paper, we present a new approach for estimating spatially-distributed reservoir properties from scattered nonlinear dynamic well measurements by promoting sparsity in an appropriate transform domain where the unknown properties are believed to have a sparse approximation. The method is inspired by recent advances in sparse signal reconstruction that is formalized under the celebrated compressed sensing paradigm. Here, we use a truncated low-frequency discrete cosine transform (DCT) is redundant to approximate the spatial parameters with a sparse set of coefficients that are identified and estimated using available observations while imposing sparsity on the solution. The intrinsic continuity in geological features lends itself to sparse representations using selected low frequency DCT basis elements. By recasting the inversion in the DCT domain, the problem is transformed into identification of significant basis elements and estimation of the values of their corresponding coefficients. To find these significant DCT coefficients, a relatively large number of DCT basis vectors (without any preferred orientation) are initially included in the approximation. Available measurements are combined with a sparsity-promoting penalty on the DCT coefficients to identify coefficients with significant contribution and eliminate the insignificant ones. Specifically, minimization of a least-squares objective function augmented by an l 1-norm of DCT coefficients is used to implement this scheme. The sparsity regularization approach using the l 1-norm minimization leads to a better-posed inverse problem that improves the non-uniqueness of the history matching solutions and promotes solutions that are, according to the prior belief, sparse in the transform domain. The approach is related to basis pursuit (BP) and least absolute selection and shrinkage operator (LASSO) methods, and it extends the application of compressed sensing to inverse modeling with nonlinear dynamic observations. While the method appears to be generally applicable for solving dynamic inverse problems involving spatially-distributed parameters with sparse representation in any linear complementary basis, in this paper its suitability is demonstrated using low frequency DCT basis and synthetic waterflooding experiments.


History matching Compressed sensing Regularization Parameterization Sparsity Facies characterization 


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  1. Aanonsen SI (2005) Efficient history matching using a multiscale technique. In: Proc 2005 SPE reservoir simulation symposium, Paper SPE 92758, Houston Google Scholar
  2. Abacioglu Y, Oliver DS, Reynolds AC (2001) Efficient history matching using subspace vectors. Comput Geosci 5:151–172 CrossRefGoogle Scholar
  3. Abdelhalim MB, Salama AE (2003) Implementation of 3D-DCT based video encoder/decoder system. In: International symposium on signals, circuits and systems, pp 389–392 Google Scholar
  4. Ahmed A, Natarajan T, Rao KR (1974) Discrete cosine transform. IEEE Trans Biomed Eng C 23:90–93 CrossRefGoogle Scholar
  5. Aziz K, Settari A (1979) Petroleum reservoir simulation. Applied Science Publishers LTD, London Google Scholar
  6. Bear J, Verruijt A (1987) Modeling groundwater flow and pollution. Reidel, Dordrecht. p 414 Google Scholar
  7. Bissell R (1994) Calculating optimal parameters for history matching. In: Proc fourth European conference on the mathematics of oil recovery, Roros, Norway Google Scholar
  8. Bloomfield P, Steiger W (1983) Least absolute deviations: theory, applications, and algorithms. Progr probab statist, vol 6. Birkhäuser, Boston Google Scholar
  9. Brouwer DR, Nævdal G, Jansen JD, Vefring EH, van Kruijsdijk CPJW (2004) Improved reservoir management through optimal control and continuous model updating. Paper SPE 90149 presented at the SPE annual technical conference and exhibition, Houston, TX Google Scholar
  10. Caers J, Zhang T (2004) Multiple-point geostatistics: a quantitative vehicle for integrating geologic analogs into multiple reservoir models. AAPG Mem 80:383–394 Google Scholar
  11. Candès EJ, Tao T (2006) Near optimal signal recovery from random projections: universal encoding strategies? IEEE Trans Inf Theory 52(12):5406–5425 CrossRefGoogle Scholar
  12. Candès EJ, Romberg J, Tao T (2006) Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans Inf Theory 52(2):489–509 CrossRefGoogle Scholar
  13. Chavent G, Bissell R (1998) Indicators for the refinement of parameterization. In: Tanaka M, Dulikravich GS (eds) Proceedings of the third international symposium on inverse problems ISIP 98 held in Nagano, Japan. Inverse problems in engineering mechanics. Elsevier, Amsterdam, pp 309–314 Google Scholar
  14. Chen SS, Donoho DL, Saunders MA (2001) Atomic decomposition by basis pursuit. SIAM Rev 43(1):129–159 CrossRefGoogle Scholar
  15. de Marsily G, Lavedan G, Boucher M, Fasanino G (1984) Interpretation of interference tests in a well field using geostatistical techniques to fit the permeability distribution in a reservoir model. In: Verly G, David M, Journel AG, Marechal A (eds) Geostatistics for natural resources characterisation. Reidel, Dordrecht, pp 831–949 Google Scholar
  16. Donoho DL (2006) Compressed sensing. IEEE Trans Inf Theory 52(4):1289–1306 CrossRefGoogle Scholar
  17. Donoho DL, Tanner J (2009) Counting faces of randomly-projected polytopes when the projection radically lowers dimension. J Am Math Soc 22(1):1–53 CrossRefGoogle Scholar
  18. Donoho DL, Vetterli M, DeVore RA, Daubechies I (1998) Data compression and harmonic analysis. IEEE Trans Inf Theory 44(6):2435–2476 CrossRefGoogle Scholar
  19. Donoho DL, Elad M, Temlyakov VM (2006) Stable recovery of sparse overcomplete representations in the presence of noise. IEEE Trans Inf Theory 52(1):6–18 CrossRefGoogle Scholar
  20. ECLIPSE reservoir simulator (2006) Manual and technical description, Schlumberger GeoQuest. Houston Google Scholar
  21. Fletcher AK, Rangan S, Goyal VK (2009) Necessary and sufficient conditions for sparsity pattern recovery. IEEE Trans Inf Theory 55(12), to appear. arXiv:0804.1839v1
  22. Gavalas GR, Shah PC, Seinfeld JH (1976) Reservoir history matching by Bayesian estimation. Soc Petrol Eng J 16(6):337–350 Google Scholar
  23. Gonzalez RC, Woods RE (2002) Digital image processing, 2nd edn. Prentice-Hall, Upper Saddle River Google Scholar
  24. Grimstad AA, Mannseth T, Nævdal G, Urkedal H (2003) Adaptive multiscale permeability estimation. Comput Geosci 7:1–25 CrossRefGoogle Scholar
  25. Jahns HO (1966) A rapid method for obtaining a two-dimensional reservoir description from well pressure response data. Soc Petrol Eng J 6(12):315–327 Google Scholar
  26. Hansen P (1992) Analysis of discrete ill-posed problems by means of the L-curve. SIAM J Sci Comput 14(6):1487–1503 CrossRefGoogle Scholar
  27. Jacquard P, Jain C (1965) Permeability distribution from field pressure data. Soc Petrol Eng J 281–294 Google Scholar
  28. Jafarpour B, McLaughlin DB (2008) History matching with an ensemble Kalman filter and discrete cosine parameterization. Comput Geosci 12(2):227–244 CrossRefGoogle Scholar
  29. Jafarpour B, McLaughlin DB (2009) Reservoir characterization with discrete cosine transform. Part 1: Parametrization—Part 2: History matching. SPE J 14(1):182–201 Google Scholar
  30. Jafarpour B, Goyal VK, Freeman WT, McLaughlin DB (2009) Transform-domain sparsity regularization for inverse problems in geosciences. Geophysics 74(5):R69–R83 CrossRefGoogle Scholar
  31. Jain AK (1989) Fundamentals of digital image processing. Prentice-Hall, Upper Saddle River Google Scholar
  32. Karhunen K (1947) Über lineare Methoden in der Wahrscheinlichts-keitsrechnung. Ann Acad Sci Fenn Ser A I 37:3–79 Google Scholar
  33. Li R, Reynolds AC, Oliver DS (2003) History matching of three-phase flow production data. SPE J 8(4):328–340 Google Scholar
  34. Liu DC, Nocedal J (1989) On the limited memory method for large scale optimization. Math Program B 45(3):503–528 CrossRefGoogle Scholar
  35. Loéve MM (1978) Probability theory, 4th edn. Springer, New York. 2 vols Google Scholar
  36. Natarajan BK (1995) Sparse approximate solutions to linear systems. SIAM J Comput 24(2):227–234 CrossRefGoogle Scholar
  37. Nocedal J, Wright SJ (2006) Numerical optimization, 2nd edn. Springer, New York Google Scholar
  38. Oliver SD, Reynolds AC, Liu N (2008) Inverse theory for petroleum reservoir characterization and history matching. Cambridge University Press, Cambridge Google Scholar
  39. Rao KR, Yip P (1990) Discrete cosine transform: algorithms, advantages, applications. Academic Press, Boston Google Scholar
  40. Reynolds AC, He N, Chu L, Oliver DS (1996) Reparameterization techniques for generating reservoir descriptions conditioned to variograms and well-test pressure data. Soc Petrol Eng J 1(4):413–426 Google Scholar
  41. Rodrigues JRP (2006) Calculating derivatives for automatic history matching. Comput Geosci 10:119–136 CrossRefGoogle Scholar
  42. Sahni I, Horne R (2005) Multiresolution wavelet analysis for improved reservoir description. SPE Reserv Eval Eng 8:53–69 Google Scholar
  43. Sarma P, Durlofsky LJ, Khalid A, Chen WH (2006) Efficient real-time reservoir management using adjoint-based optimal control and model updating. Comput Geosci 10:3–36 CrossRefGoogle Scholar
  44. Sarma P, Durlofsky LJ, Aziz K (2008) Kernel principal component analysis for efficient, differentiable parameterization of multipoint geostatistics. Math Geosci 40(1):3–32 CrossRefGoogle Scholar
  45. Tarantola A (2004) Inverse problem theory. Methods for model parameter estimation. SIAM, Philadelphia Google Scholar
  46. Thompson AM, Brown J, Kay J, Titterington DM (1991) A study of methods of choosing the smoothing parameter in image restoration by regularization. IEEE Trans Pattern Anal Mach Intell 13(4):326–339 CrossRefGoogle Scholar
  47. Tibshirani R (1996) Regression shrinkage and selection via the lasso. J R Stat Soc, Ser B 58(1):267–288 Google Scholar
  48. Tikhonov AN, Arsenin VI (1977) Solution of ill-posed problems. Winston, Washington Google Scholar
  49. Wainwright MJ (2009) Sharp thresholds for high-dimensional and noisy sparsity recovery using 1-constrained quadratic programming (lasso). IEEE Trans Inf Theory 55(5):2183–2202 CrossRefGoogle Scholar
  50. Wu Z, Reynolds AC, Oliver DS (1999) Conditioning geostatistical models to two-phase production data. Soc Petrol Eng J 3(2):142–155 Google Scholar

Copyright information

© International Association for Mathematical Geosciences 2009

Authors and Affiliations

  • Behnam Jafarpour
    • 1
    Email author
  • Vivek K. Goyal
    • 2
  • Dennis B. McLaughlin
    • 3
  • William T. Freeman
    • 2
  1. 1.Department of Petroleum EngineeringTexas A&M UniversityCollege StationUSA
  2. 2.Department of Electrical Engineering and Computer ScienceMassachusetts Institute of TechnologyCambridgeUSA
  3. 3.Department of Civil and Environmental EngineeringMassachusetts Institute of TechnologyCambridgeUSA

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