×

A pattern-matching method for flow model calibration under training image constraint. (English) Zbl 1421.86026

Summary: Modern geostatistical modeling techniques are developed to simulate complex geologic connectivity patterns (e.g., curvilinear fluvial systems) at the grid-level using a training image (TI) that encodes multiple-point statistical (MPS) information. A challenging aspect of using MPS methods is conditioning the resulting models on nonlinear flow data. We develop a pattern-matching method for calibration of MPS-based facies models subject to the TI constraint. Since the exact statistical information in the TI can only be expressed empirically, flow data conditioning and pattern matching are carried out in two iterative steps, using an alternating-direction algorithm. Flow data integration is formulated through a regularized least-squares by taking advantage of learned \(k\)-SVD sparse parametrization and \(l_1\)-norm sparsity-promoting regularization methods. The TI constraint is enforced through a MPS-based pattern-matching algorithm that uses the identified model calibration solution to generate a corresponding facies model that is consistent with the TI. The pattern-matching algorithm uses a local search template to scan the TI to find facies patterns with smallest distances from the corresponding local patterns in the parameterized approximate solution. The identified patterns for each location in the model are stored and used to estimate local conditional probabilities for assigning the facies types to each grid cell. The resulting solution is passed to the flow data conditioning step as a regularization term to perform the next iteration. The process is repeated until the MPS facies model provides an acceptable match to the data. Numerical experiments are presented to evaluate the performance of the pattern-matching method for calibration of complex facies models.

MSC:

86A32 Geostatistics
62H35 Image analysis in multivariate analysis
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Arpat, G.B., Caers, J.: A multiple-scale, pattern-based approach to sequential simulation. In: Geostatistics Banff 2004, pp 255-264. Springer, Netherlands (2005)
[2] Bhark, E.W., Jafarpour, B., Datta-Gupta, A.: A generalized grid connectivity – based parameterization for subsurface flow model calibration. Water Resour. Res. 47, 10 (2011) · doi:10.1029/2010WR009982
[3] Bregman, N.D., Bailey, R.C., Chapman, C.H.: Crosshole seismic tomography. Geophysics 54(2), 200-215 (1989) · doi:10.1190/1.1442644
[4] Caers, J., Hoffman, T.: The probability perturbation method: A new look at Bayesian inverse modeling. Math. Geol. 38(1), 81-100 (2006) · Zbl 1119.86312 · doi:10.1007/s11004-005-9005-9
[5] Candès, E.J., Wakin, M.B.: An introduction to compressive sampling. IEEE Signal Process. Mag. 25(2), 21-30 (2008) · doi:10.1109/MSP.2007.914731
[6] Cardiff, M., Kitanidis, P. K.: Bayesian inversion for facies detection: an extensible level set framework. Water Resour. Res. 45, 10 (2009) · doi:10.1029/2008WR007675
[7] Carrera, J., Neuman, S.P.: Estimation of aquifer parameters under transient and steady-state conditions, 1. Maximum likelihood method incorporating prior information. Water Resour. Res. 22(2), 199-210 (1986) · doi:10.1029/WR022i002p00199
[8] Chavent, G, Bissell, R.: Indicators for the refinement of parameterization. In: Tanaka, M., Dulikravich, G.S. (eds.) Inverse Problems in Engineering Mechanics 1998. (Proceedings of the third International Symposium on Inverse Problems ISIP 98 held in Nagano, Japan), pp 309-314. Elsevier (1998)
[9] Deutsch, C.V., Wang, L.: Hierarchical object-based stochastic modeling of fluvial reservoirs. Math. Geol. 28(7), 857-880 (1996) · doi:10.1007/BF02066005
[10] Doherty, J.: Ground water model calibration using pilot points and regularization. Ground Water 41(2), 170-177 (2003) · doi:10.1111/j.1745-6584.2003.tb02580.x
[11] Franssen, H., Alcolea, A., Riva, M., Bakr, M., van der Wiel, N., Stauffer, F., Guadagnini, A.: A comparison of seven methods for the inverse modeling of groundwater flow. Application to the characterization of well catchments. Adv. Water Resour. 32, 851-872 (2009) · doi:10.1016/j.advwatres.2009.02.011
[12] Gavalas, G.R., Shah, P.C., Seinfeld, J.H.: Reservoir history matching by Bayesian estimation. Soc. Petrol. Eng. J. 16(06), 337-350 (1976) · doi:10.2118/5740-PA
[13] Golmohammadi, A., Jafarpour, B.: Simultaneous geologic scenario identification and flow model calibration with group-sparsity formulations. Adv. Water Resour. 92, 208-227 (2016) · doi:10.1016/j.advwatres.2016.04.007
[14] Golmohammadi, A., Khaninezhad, M.R.M., Jafarpour, B.: Group-sparsity regularization for ill-posed subsurface flow inverse problems. Water Resour. Res. 51(10), 8607-8626 (2015) · doi:10.1002/2014WR016430
[15] Hakim-Elahi, S., Jafarpour, B.: A distance transform for continuous parameterization of discrete geologic facies for subsurface flow model calibration. Water Resour. Res. 53(10), 8226-8249 (2017) · doi:10.1002/2016WR019853
[16] Hill, M.C., Tiedeman, C.R.: Effective Groundwater Model Calibration: With Analysis of Data, Sensitivities, Predictions, and Uncertainty. Wiley (2006)
[17] Hu, L.Y., Chugunova, T.: Multiple-point geostatistics for modeling subsurface heterogeneity: a comprehensive review. Water Resour. Res. 44, 11 (2008)
[18] Hu, L.Y., Jenni, S.: History matching of object-based stochastic reservoir models. SPE J. 10(03), 312-323 (2005) · doi:10.2118/81503-PA
[19] Jacquard, P., Jain, C.: Permeability distribution from field pressure data. Soc. Pet. Eng. J., 281-294 (1965)
[20] Jafarpour, B., Khodabakhshi, M.: A probability conditioning method (PCM) for nonlinear flow data integration into multipoint statistical facies simulation. Math. Geosci. 43(2), 133-164 (2011) · Zbl 1207.86010 · doi:10.1007/s11004-011-9316-y
[21] Jafarpour, B., McLaughlin, D.B.: Reservoir characterization with the discrete cosine transform. SPE J. 14 (01), 182-201 (2009) · doi:10.2118/106453-PA
[22] Jafarpour, B., Goyal, V.K., McLaughlin, D.B., Freeman, W.T.: Compressed history matching: Exploiting transform-domain sparsity for regularization of nonlinear dynamic data integration problems. Math. Geosci. 42(1), 1-27 (2010) · Zbl 1185.94011 · doi:10.1007/s11004-009-9247-z
[23] Khaninezhad, M.M., Jafarpour, B., Li, L.: Sparse geologic dictionaries for subsurface flow model calibration: Part I. Inversion formulation. Adv. Water Resour. 39, 106-121 (2012) · doi:10.1016/j.advwatres.2011.09.002
[24] Khaninezhad, M.-R., Golmohammadi, A., Jafarpour, B.: Discrete regularization for calibration of geologic facies against dynamic flow data. Water Resour. Res, 54. https://doi.org/10.1002/2017WR022284https://doi.org/10.1002/2017WR022284 (2018)
[25] Khodabakhshi, M., Jafarpour, B.: A Bayesian mixture-modeling approach for flow-conditioned multiple-point statistical facies simulation from uncertain training images. Water Resour. Res. 49(1), 328-342 (2013) · doi:10.1029/2011WR010787
[26] Kitanidis, P.K., Vomvoris, E.G.: A geostatistical approach to the inverse problem in groundwater modeling (steady state) and one- dimensional simulations. Water Resour. Res. 19(3), 677-690 (1983) · doi:10.1029/WR019i003p00677
[27] Lee, J., Kitanidis, P. K.: Bayesian inversion with total variation prior for discrete geologic structure identification. Water Resour. Res. 49(11), 7658-7669 (2013) · doi:10.1002/2012WR013431
[28] Liu, E., Jafarpour, B.: Learning sparse geologic dictionaries from low-rank representations of facies connectivity for flow model calibration. Water Resour. Res. 49, 7088-7101 (2013). https://doi.org/10.1002/wrcr.20545 · doi:10.1002/wrcr.20545
[29] McLaughlin, D, Townley, L R: A reassessment of the groundwater inverse problem. Water Resour. Res. 32(5), 1131-61 (1996). https://doi.org/10.1029/96WR00160 · doi:10.1029/96WR00160
[30] Oliver, D S, Reynolds, AC, Liu, N: Inverse Theory for Petroleum Reservoir Characterization and History Matching. Cambridge University Press (2008)
[31] Roweis, S.T., Saul, L.K.: Nonlinear dimensionality reduction by locally linear embedding. Science 290 (5500), 2323-2326 (2000) · doi:10.1126/science.290.5500.2323
[32] Rudin, L. I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D: Nonlin. Phenom. 60(1), 259-268 (1992) · Zbl 0780.49028 · doi:10.1016/0167-2789(92)90242-F
[33] Sahni, I., Horne, R.N.: Multiresolution wavelet analysis for improved reservoir description. SPE Reserv. Eval. Eng. 8(01), 53-69 (2005) · doi:10.2118/87820-PA
[34] Sarma, P., Durlofsky, L.J., Aziz, K.: Kernel principal component analysis for efficient, differentiable parameterization of multipoint geostatistics. Math. Geosci. 40(1), 3-32 (2008) · Zbl 1144.86004 · doi:10.1007/s11004-007-9131-7
[35] Strebelle, S.: Conditional simulation of complex geological structures using multiple-point statistics. Mathem. Geol. 34(1), 1-21 (2002) · Zbl 1036.86013 · doi:10.1023/A:1014009426274
[36] Tikhonov, A.: Solution of incorrectly formulated problems and the regularization method. In Soviet. Math. Dokl 5, 1035-1038 (1963) · Zbl 0141.11001
[37] Vo, H.X., Durlofsky, L.J.: A new differentiable parameterization based on principal component analysis for the low-dimensional representation of complex geological models. Math. Geosci. 46(11), 775-813 (2014) · Zbl 1323.86048 · doi:10.1007/s11004-014-9541-2
[38] Zhou, H, Gómez-Hernández, J.J., Li, L.: Inverse methods in hydrogeology: evolution and recent trends. Adv. Water Resour. 63, 22-37 (2014). https://doi.org/10.1016/j.advwatres.2013.10.014. ISSN 0309-1708 · doi:10.1016/j.advwatres.2013.10.014
[39] Zhou, H., Gómez-Hernández, J.J., Li, L.: A pattern- search- based inverse method. Water Resour. Res. 48, 3 (2012) · doi:10.1029/2011WR011195
[40] Zimmerman, D. A., de Marsily, G., Gotway, C.A., Marietta, M.G., Axness, C.L., Beauheim, R.L., Bras, R.L., et al.: A comparison of seven geostatistically based inverse approaches to estimate transmissivities for modeling advective transport by groundwater flow. Water Resour. Res. 34(6), 1373-1413 (1998) · doi:10.1029/98WR00003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.