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Sparsity-Promoting Solution of Subsurface Flow Model Calibration Inverse Problems

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Advances in Hydrogeology

Abstract

Identification of heterogeneous hydraulic aquifer properties from limited dynamic flow measurements typically leads to underdetermined nonlinear inverse problems that can have many solutions, including solutions that are geologically implausible and fail to predict future performance of the system. The problem is usually regularized by incorporating implicit or explicit prior information to stabilize the solution techniques and to obtain plausible solutions. A meaningful regularization must be informed by the physics of the problem, distinct properties of the formation under investigation, and other available sources of information (e.g., outcrop, well logs, and seismic). This chapter proposes sparsity as an intrinsic property of spatially distributed aquifer hydraulic properties that can be used to regularize the solution of the related ill-posed inverse problem. Inspired by recent advances in sparse signal processing, formalized under the compressed sensing paradigm, proper sparsifying bases are introduced to describe aquifer hydraulic conductivity distribution. Such descriptions give rise to a sparse reconstruction formulation of the subsurface flow model calibration inverse problem, which can be efficiently solved following recent algorithmic developments in sparse signal processing. The compressed sensing paradigm specifies the conditions under which unique solutions to underdetermined linear system of equations exist and can be computed efficiently. Sparsity is a fundamental notion in compressed sensing, and is often present in many natural images. In particular, sparsity is prevalent in describing many spatially correlated aquifer properties. The practical implications of compressed sensing are as far reaching as the solution of underdetermined system of equations is in science and engineering. This chapter introduces the guidelines set forth by sparse reconstruction techniques and the compressed sensing paradigm and incorporates them to formulate and solve ill-posed groundwater model calibration inverse problems.

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References

  • Aanonsen S (2008) Efficient history matching using a multiscale technique. SPE Reservoir Eval Eng 11(1):154–164

    CAS  Google Scholar 

  • Aharon M, Elad M, Bruckstein A (2006) K-SVD (capitalized): An algorithm for designing overcomplete dictionaries for sparse representation. IEEE Trans Signal Process [see also IEEE Trans Acoust Speech Signal Process] 54(11):4311–4322

    Article  Google Scholar 

  • Ahmed A, Natarajan T, Rao KR (1974) Discrete cosine transform. IEEE Trans Biomed Eng C23:90–93

    Google Scholar 

  • Aziz K, Settari A (1979) Petroleum reservoir simulation. Springer, New York

    Google Scholar 

  • Bear J, Verruijt A (1987) Modeling groundwater flow and pollution (theory and applications of transport in porous media). Applied Science Publishers, London

    Book  Google Scholar 

  • Bhark E, Jafarpour B, Datta-Gupta A (2011) A generalized grid-connectivity-based parameterization for subsurface flow data assimilation. Water Resour Res 47,W06517:32 PP

    Google Scholar 

  • Britanak PC, Yip P, Rao K (2007) Discrete cosine transform: general properties, fast algorithms, and integer approximation. Academic Press

    Google Scholar 

  • Carrera J, Alcolea A, Medina A, Hidalgo J, Slooten LJ (2005) Inverse problem in hydrogeology. Hydrol J 13(1):206–222

    Google Scholar 

  • Chartrand R, Yin W (2008) Iteratively reweighted algorithms for compressive sensing. IEEE International Conference on Acoustics 2008, Speech and Signal Processing. ICASSP 3869–3872

    Google Scholar 

  • Chavent G, Bissell R (1998) Indicators for the refinement of parameterization. In: Tanaka M, Dulikravich GS (eds) Inverse Problems in Engineering Mechanics (Proceedings of the International Symposium on Inverse Problems in Engineering Mechanics 1998 (ISIP’98), Nagano, Japan) Elsevier, pp 309–314

    Google Scholar 

  • Chen SS, Donoho DL, Saunders MA (1998) Atomic decomposition by basis pursuit. SIAM J Sci Comput 20:33–61

    Article  CAS  Google Scholar 

  • Constable S, Parker R, Constable C (1987) Occams inversion: a practical algorithm for generating smooth models from electromagnetic sounding data. Geophysics 52:289–300

    Article  Google Scholar 

  • Couvreur C, Bresler Y (2000) On the optimality of the backward greedy algorithm for the subset selection problem. SIAM J Matrix Anal Appl 21:797–808

    Article  Google Scholar 

  • Daubechies I (1988) Multiresolution approximation and wavelet orthonormal bases of l 2(r). Trans Am Math Soc LXI:909–996

    Google Scholar 

  • Daubechies I, Defrise M, De Mol C (2004) An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Comm Pure Appl Math 57:1413–1457

    Article  Google Scholar 

  • Daubechies I, Fornasier M, Loris I (2008) Accelerated projected gradient method for linear inverse problems with sparsity constraints. J Fourier Anal Appl 14,5–6:764–792

    Article  Google Scholar 

  • Deutsch CV, Journel AG (1998) GSLIB - Geostatistical software library and user’s guide. Oxford University Press, New York

    Google Scholar 

  • Doherty J (2003) Groundwater model calibration using pilot points and regularisation. Ground Water 41(2):170–177

    Article  CAS  Google Scholar 

  • Donoho DL (2006) Compressed sensing. IEEE Trans Inform Theory 52(4):1289–1306

    Article  Google Scholar 

  • Donoho DL, Elad M (2003) Optimally sparse representation in general (nonorthogonal) dictionaries via l 1 minimization. Proc Natl Acad Sci USA 100(5):2197–2202 (electronic)

    Google Scholar 

  • Efron B, Hastie T, Johnstine I, Tishirani R (2004) Least angle regression. Ann Stat 32:407–499

    Article  Google Scholar 

  • Elad M, Matalon B, Zibulevsky M (2007) Coordinate and subspace optimization methods for linear least squares with non-quadratic regularization. Appl Comput Harmonic Anal 23(3):346–367

    Article  Google Scholar 

  • Figueiredo MAT, Nowak RD (2003) An em algorithm for wavelet-based image restoration. IEEE Trans Image Process 12:906–916

    Article  Google Scholar 

  • Gavalas GR, Shah PC, Seinfeld JH (1976) Reservoir history matching by bayesian estimation. Soc Petrol Eng J 16(6):337–350

    Google Scholar 

  • Gorodnitsky IF, Rao BD (1997) Sparse signal reconstruction from limited data using focuss: A re-weighted norm minimization algorithm. IEEE Trans Signal Process 45:600–616

    Article  Google Scholar 

  • Grimstad AA, Mannseth T, Nvdal G, Urkedal H (2003) Adaptive multiscale permeability estimation. Comput Geosci 7:1–25

    Article  Google Scholar 

  • Hill MC, Tiedeman CR (2007) Effective groundwater model calibration. Wiley

    Google Scholar 

  • Jacquard P, Jain C (1965) Permeability distribution from field pressure data. Transactions of the American Institute of Mining, Metallurgical and Petroleum Engineers. (Abbreviated Trans. AIME). Soc Pet Eng J 5(4):281–294

    Google Scholar 

  • Jafarpour B (2011) Wavelet reconstruction of geologic facies from nonlinear dynamic flow measurements. IEEE Trans Geosci Rem Sens 49(5):1520–1535

    Article  Google Scholar 

  • Jafarpour B, McLaughlin DB (2008) History-matching with an ensemble kalman filter and discrete cosine parameterization. Comput Geosci 12,2:227–244

    Article  Google Scholar 

  • Jafarpour B, McLaughlin DB (2009a) Reservoir characterization with discrete cosine transform. part-1: parameterization. SPE Journal, Soc Pet Eng J 14(1):182–201, http://www.onepetro.org/mslib/servlet/onepetropreview?id=SPE-106453-PA

    Google Scholar 

  • Jafarpour B, McLaughlin DB (2009b) Reservoir characterization with discrete cosine transform. part-2: history matching. SPE Journal, Soc Pet Eng J 14(1):182–201, http://www.onepetro.org/mslib/servlet/onepetropreview?id=SPE-106453-PA

  • Jafarpour B, Goyal VK, McLaughlin DB, Freeman WT (2009) Transform-domain sparsity regularization for inverse problems in geosciences. Geophysics 74(45):R69–R83, DOI:10.1190/1.3157250

    Article  Google Scholar 

  • Jafarpour B, Goyal VK, McLaughlin DB, Freeman WT (2010) Compressed history matching: exploiting transform-domain sparsity for regularization of nonlinear dynamic data integration problems. Math Geosci 42:1–27, doi: 10.1007/s11004-009-9247-z

    Article  Google Scholar 

  • Karlovitz LA (1970) Construction of nearest points in the l p , p even and l ∞ norms. J Approx Theory 3:123–127

    Article  Google Scholar 

  • Kitanidis K, Vomvoris EG (1983) A geostatistical approach to the inverse problem in groundwater modeling (steady state) and one-dimensional simulations. Water Resour Res 19(1):677–690

    Article  Google Scholar 

  • Kreutz-Delgado K, Murray JF, Rao BD, Engan K, Lee TW, Sejnowski TJ (2003) Dictionary learning algorithms for sparse representation. Neural Comput 15(2):349–396

    Article  Google Scholar 

  • Li L, Jafarpour B (2010) Effective solution of nonlinear subsurface flow inverse problems in sparse bases. Inverse Probl 26(10),105016:24 PP

    Google Scholar 

  • Mallat S (1989) Multiresolution approximation and wavelet orthonormal bases of l 2(r). Trans Am Math Soc 315:69–88

    Google Scholar 

  • Mallat S (2009) A wavelet tour of signal processing 3rd edn. Academic Press

    Google Scholar 

  • Mallat S, Zhang Z (1993) Matching pursuits with time-frequency dictionaries. IEEE Trans Signal Process 41:3397–3415

    Article  Google Scholar 

  • McLaughlin DB, Townley LR (1996) A reassessment of the groundwater inverse problem. Water Resour Res 32(5):1131–1161

    Article  Google Scholar 

  • Mohammad-Khaninezhad MR, Jafarpour B, Li L (2012a) Sparse geologic dictionaries for subsurface flow model calibration: Part i. inversion formulation. Adv Water Resour 39:106–121

    Article  Google Scholar 

  • Mohammad-Khaninezhad MR, Jafarpour B, Li L (2012b) Sparse geologic dictionaries for subsurface flow model calibration: Part ii. robustness to uncertainty. Adv Water Resour 39:122–136

    Article  Google Scholar 

  • Mohammad-Khaninezhad MR, Jafarpour B (2013) Estimating reservoir connectivity from dynamic data under geologic uncertainty. SPE J (in revision)

    Google Scholar 

  • Natarajan BK (1995) Sparse approximate solutions to linear systems. SIAM J Comput 24(2):227–234

    Article  Google Scholar 

  • National Research Council (2000) Seeing into the earth: Noninvasive characterization of the shallow subsurface for environmental and engineering application. Technical report, Washington, DC

    Google Scholar 

  • Oliver DS, Reynolds AC, Liu N (2008) Inverse theory for petroleum reservoir characterization and history matching. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Portniaguine O, Zhdanov MS (1999) Focusing geophysical inversion images. Geophysics 64(3):874–887

    Article  Google Scholar 

  • Rao BD, Kreutz-Delgado K (2003) An affine scaling methodology for best basis selection. IEEE Trans Signal Process 47:187–200

    Article  Google Scholar 

  • Sahni I, Horne RN (2005) Multiresolution wavelet analysis for improved reservoir description. SPEREE 8(1):53–69

    Article  CAS  Google Scholar 

  • Santosa F, Symes WW (1986) Linear inversion of band-limited reflection seismograms. SIAM J Sci Stat Comput 7:1307–1330

    Article  Google Scholar 

  • Tikhonov AN, Arsenin VY (1977) Solution of ill-posed problems. Winston, distributed by Halsted Press, New York

    Google Scholar 

  • Tonkin M, Doherty J (2005) A hybrid regularized inversion methodology for highly parameterized models. Water Resour Res 41,W10412:170–177

    Google Scholar 

  • Tropp JA, Gilbert AC (2007) Signal recovery from partial information via orthogonal matching pursuit. IEEE Trans Inf Theory 53:4655–4666

    Article  Google Scholar 

  • Yeh TCJ, Lee CH, Hsu KC, Illman WA, Barrash W, Cai X, Daniels J, Sudicky E, Wan L, Li G, Winter CL (2008) A view toward the future of subsurface characterization: Cat scanning groundwater basins. Water Resour Res W03301, doi:10.1029/2007WR006375

    Google Scholar 

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Authors and Affiliations

  1. Department of Chemical Engineering and Materials Science, University of Southern California, 925 Bloom Walk, HED 313, Los Angeles, CA, 90089-1211, USA

    Behnam Jafarpour

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  1. Behnam Jafarpour
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Correspondence to Behnam Jafarpour .

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Editors and Affiliations

  1. , Civil & Environmental Engineering, California State University, North State College Blvd. 800, Fullerton, 92831, California, USA

    Phoolendra K. Mishra

  2. Repository Performance Department, Sandia National Laboratories, Carlsbad, New Mexico, USA, n/a, Carlsbad, 88220, New Mexico, USA

    Kristopher L. Kuhlman

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Jafarpour, B. (2013). Sparsity-Promoting Solution of Subsurface Flow Model Calibration Inverse Problems. In: Mishra, P., Kuhlman, K. (eds) Advances in Hydrogeology. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6479-2_4

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