Abstract
In data assimilation problems, various types of data are naturally linked to different spatial resolutions (e.g., seismic and electromagnetic data), and these scales are usually not coincident to the subsurface simulation model scale. Alternatives like upscaling/downscaling of the data and/or the simulation model can be used, but with potential loss of important information. Such alternatives introduce additional uncertainties which are not in the nature of the problem description, but the result of the post processing of the data or the geo-model. To address this issue, a novel multiscale (MS) data assimilation method is introduced. The overall idea of the method is to keep uncertain parameters and observed data at their original representation scale, avoiding upscaling/downscaling of any quantity. The method relies on a recently developed mathematical framework to compute adjoint gradients via a MS strategy in an algebraic framework. The fine-scale uncertain parameters are directly updated and the MS grid is constructed in a resolution that meets the observed data resolution. This formulation therefore enables a consistent assimilation of data represented at a coarser scale than the simulation model. The misfit objective function is constructed to keep the MS nature of the problem. The regularization term is represented at the simulation model (fine) scale, whereas the data misfit term is represented at the observed data (coarse) scale. The computational aspects of the method are investigated in a simple synthetic model, including an elaborate uncertainty quantification step, and compared to upscaling/downscaling strategies. The experiment shows that the MS strategy provides several potential advantages compared to more traditional scale conciliation strategies: (1) expensive operations are only performed at the coarse scale; (2) the matched uncertain parameter distribution is closer to the “truth”; (3) faster convergence behavior occurs due to faster gradient computation; and (4) better uncertainty quantification results are obtained. The proof-of-concept example considered in this paper sheds new lights on how one can reduce uncertainty within fine-scale geo-model parameters with coarse-scale data, without the necessity of upscaling/downscaling the data nor the geo-model. The developments demonstrate how to consistently formulate such a gradient-based MS data assimilation strategy in an algebraic framework which allows for implementation in available computational platforms.
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References
Aanonsen, S.I., Nævdal, G., Oliver, D.S., Reynolds, A.C., Vallés, B.: Review of ensemble kalman filter in petroleum engineering. SPE J. 14(3), 393–412 (2009)
Chavent, G., Dupuy, M., Lemmonier, P.: History matching by use of optimal theory. Soc. Pet. Eng. J. 15(01), 74–86 (1975)
Chung, E.T., Efendiev, Y., Jin, B., Leung, W.T., Vasilyeva, M.: Generalized multiscale inversion for heterogeneous problems. arXiv:1707.08194 (2017)
Cole, S., Lumley, D., Meadows, M., Tura, A.: Pressure and saturation inversion of 4d seismic data by rock physics forward modeling. In: SEG Technical Program Expanded Abstracts 2002, pp 2475–2478. Society of Exploration Geophysicists (2002)
Corliss, G., Faure, C., Griewank, A., Hascoet, L., Naumann, U.: Automatic differentiation of algorithms: from simulation to optimization, vol. 1. Springer Science & Business Media (2002)
Cusini, M., van Kruijsdijk, C., Hajibeygi, H.: Algebraic dynamic multilevel (adm) method for fully implicit simulations of multiphase flow in porous media. J. Comput. Phys. 314, 60–79 (2016)
Durlofsky, L.J.: Upscaling of geocellular models for reservoir flow simulation: a review of recent progress. In: 7th International Forum on Reservoir Simulation Bühl/Baden-Baden, Germany, pp 23–27 (2003)
Efendiev, Y., Galvis, J., Hou, T.Y.: Generalized multiscale finite element methods (gmsfem). J. Comput. Phys. 251, 116–135 (2013)
Emerick, A.A., Moraes, R., Rodrigues, J., et al.: Calculating seismic attributes within a reservoir flow simulator. In: Latin American & Caribbean Petroleum Engineering Conference. Society of Petroleum Engineers (2007)
Emerick, A.A., Moraes, R., Rodrigues, J., et al.: History matching 4d seismic data with efficient gradient based methods. In: EUROPEC/EAGE Conference and Exhibition. Society of Petroleum Engineers (2007)
Emerick, A.A., Reynolds, A.C.: Ensemble smoother with multiple data assimilation. Comput. Geosci. 55, 3–15 (2013)
Emerick, A.A., Reynolds, A.C.: Investigation of the sampling performance of ensemble-based methods with a simple reservoir model. Comput. Geosci. 17(2), 325–350 (2013). https://doi.org/10.1007/s10596-012-9333-z
Emerick, A.A., Reynolds, A.C., et al.: History-matching production and seismic data in a real field case using the ensemble smoother with multiple data assimilation. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers (2013)
Evensen, G.: Data assimilation: the ensemble Kalman filter. Springer Science & Business Media (2009)
Farmer, C.: Upscaling: a review. Int. J. Numer. Methods Fluids 40(1-2), 63–78 (2002)
Fossum, K., Mannseth, T.: Coarse-scale data assimilation as a generic alternative to localization. Comput. Geosci. 21(1), 167–186 (2017)
Fossum, K., Mannseth, T.: A novel multilevel method for assimilating spatially dense data. In: ECMOR XVI-16th European Conference on the Mathematics of Oil Recovery (2018)
Frederick, C., Engquist, B.: Numerical methods for multiscale inverse problems. arXiv:1401.2431 (2014)
Fu, J., Caers, J., Tchelepi, H.A.: A multiscale method for subsurface inverse modeling: Single-phase transient flow. Adv. Water Resour. 34(8), 967–979 (2011)
Fu, J., Tchelepi, H.A., Caers, J.: A multiscale adjoint method to compute sensitivity coefficients for flow in heterogeneous porous media. Adv. Water Resour. 33(6), 698–709 (2010)
Gao, G., Reynolds, A.C., et al.: An improved implementation of the lbfgs algorithm for automatic history matching. In: SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers (2004)
Gao, G., Reynolds, A.C., et al.: An improved implementation of the lbfgs algorithm for automatic history matching. SPEJ 11(01), 5–17 (2006)
Gervais-Couplet, V., Roggero, F., Feraille, M.D., Ravalec-Dupin, L., Seiler, A., et al.: Joint history matching of production and 4d-seismic related data for a north sea field case. In: SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers (2010)
Gosselin, O., Aanonsen, S., Aavatsmark, I., Cominelli, A., Gonard, R., Kolasinski, M., Ferdinandi, F., Kovacic, L., Neylon, K., et al.: History matching using time-lapse seismic (huts). In: SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers (2003)
Graham, I.G., Hou, T.Y., Lakkis, O., Scheichl, R.: Numerical analysis of multiscale problems, vol. 83. Springer Science & Business Media (2012)
Hajibeygi, H., Jenny, P.: Adaptive iterative multiscale finite volume method. J. Comput. Phys. 230(3), 628–643 (2011)
Haugen, V.E.J., Natvik, L.J., Evensen, G., Berg, A.M., Flornes, K.M., Naevdal, G., et al.: History matching using the ensemble kalman filter on a north sea field case. In: SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers (2006)
Hou, T.Y., Wu, X.H.: A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134(1), 169–189 (1997)
Jansen, J.D.: A simple algorithm to generate small geostatistical ensembles for subsurface flow simulation. Research note. Dept. of Geoscience and Engineering, Delft University of Technology, The Netherlands. https://doi.org/uuid:6000459e-a0cb-40d1-843b-81650053e093 (2013)
Jansen, J.D.: A Systems Description of Flow Through Porous Media. Springer, Berlin (2013)
Jansen, J.D.: Gradient-based optimization of flow through porous media: Version 3. Course notes. Delft University of Technology (2016).https://doi.org/10.4233/uuid:0010fdac-32ec-459b-bb9b-3e6327a85496
Jenny, P., Lee, S.H., Tchelepi, H.A.: Multi-scale finite-volume method for elliptic problems in subsurface flow simulation. J. Comput. Phys. 187(1), 47–67 (2003)
Kraaijevanger, J.F.B.M., Egberts, P.J.P., Valstar, J.R., Buurman, H.W.: Optimal waterflood design using the adjoint method. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers. https://doi.org/10.2118/105764-MS (2007)
Landrø, M.: Discrimination between pressure and fluid saturation changes from time-lapse seismic data. Geophysics 66(3), 836–844 (2001)
Le Ravalec, M., Tillier, E., Da Veiga, S., Enchéry, G., Gervais, V.: Advanced integrated workflows for incorporating both production and 4d seismic-related data into reservoir models. Oil Gas Sci. Technol.–Revue d’IFP Energies nouvelles 67(2), 207–220 (2012)
Li, Z., McWilliams, J.C., Ide, K., Farrara, J.D.: A multiscale variational data assimilation scheme: formulation and illustration. Mon. Weather. Rev. 143(9), 3804–3822 (2015)
Liu, N., Oliver, D.S., et al.: Evaluation of monte carlo methods for assessing uncertainty. SPE J. 8(02), 188–195 (2003)
Lumley, D., Meadows, M., Cole, S., Adams, D.: Estimation of reservoir pressure and saturations by crossplot inversion of 4d seismic attributes. In: SEG Technical Program Expanded Abstracts 2003, pp 1513–1516. Society of Exploration Geophysicists (2003)
MacBeth, C., Floricich, M., Soldo, J.: Going quantitative with 4d seismic analysis. Geophys. Prospect. 54(3), 303–317 (2006)
Mannseth, T., Fossum, K.: Assimilating spatially dense data for subsurface applications–balancing information and degrees of freedom. Comput. Geosci. 22(5), 1323–1349 (2018)
Moraes, R., Rodrigues, J., Hajibeygi, H., Jansen, J.D.: Multiscale gradient computation for multiphase flow in porous media. In: SPE Reservoir Simulation Conference. Society of Petroleum Engineers (2017)
Moraes, R.J.D., Rodrigues, J.R., Hajibeygi, H., Jansen, J.D.: Multiscale gradient computation for flow in heterogeneous porous media. J. Comput. Phys. 336, 644–663 (2017)
Møyner, O., Lie, K.A.: A multiscale restriction-smoothed basis method for high contrast porous media represented on unstructured grids. J. Comput. Phys. 304, 46–71 (2016)
Nocedal, J., Wright, S.: Numerical optimization. Springer Science & Business Media (2006)
Oliver, D.S., He, N., Reynolds, A.C.: Conditioning permeability fields to pressure data. In: ECMOR V-5th European Conference on the Mathematics of Oil Recovery (1996)
Oliver, D.S., Reynolds, A.C., Liu, N.: Inverse Theory for Petroleum Reservoir Characterization and History Matching. Cambridge University Press, Cambridge (2008)
Rodrigues, J.R.P.: Calculating derivatives for automatic history matching. Comput. Geosci. 10(1), 119–136 (2006)
Skjervheim, J.A., Evensen, G., Aanonsen, S.I., Ruud, B.O., Johansen, T.A., et al.: Incorporating 4d seismic data in reservoir simulation models using ensemble kalman filter. In: SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers (2005)
Strang, G.: Introduction to Linear Algebra. Wellesley-Cambridge Press, Wellesley (1993)
Tarantola, A.: Inverse Problem Theory and Methods for Model Parameter Estimation, vol. 89. SIAM (2005)
Tura, A., Lumey, D.E.: Estimating pressure and saturation changes time-lapse avo data. In: SEG Technical Program Expanded Abstracts 1999, pp 1655–1658. Society of Exploration Geophysicists (1999)
Ullmann De Brito, D., Caletti, L., Moraes, R., et al.: Incorporation of 4d seismic in the re-construction and history matching of marlim sul deep water field flow simulation model. In: SPE EUROPEC/EAGE Annual Conference and Exhibition. Society of Petroleum Engineers (2011)
Ullmann De Brito, D., Moraes, R., Emerick, A.A., et al.: The marlim field: incorporating time-lapse seismic in the assisted history matching. In: SPE Latin American and Caribbean Petroleum Engineering Conference. Society of Petroleum Engineers (2010)
Wallis, J., Tchelepi, H.A.: Apparatus, method and system for improved reservoir simulation using an algebraic cascading class linear solver. US Patent 7,684,967 (2010)
Wan, J., Zabaras, N.: A bayesian approach to multiscale inverse problems using the sequential monte carlo method. Inverse Prob. 27(10), 105,004 (2011)
Wang, Y., Hajibeygi, H., Tchelepi, H.A.: Algebraic multiscale solver for flow in heterogeneous porous media. J. Comput. Phys. 259, 284–303 (2014)
Wang, Y., Hajibeygi, H., Tchelepi, H.A.: Monotone multiscale finite volume method. Comput. Geosci. 20(3), 509–524 (2016)
Zhang, Y., Oliver, D.S., et al.: History matching using the ensemble kalman filter with multiscale parameterization: a field case study. SPE J. 16(02), 307–317 (2011)
Zhou, H., Tchelepi, H.A.: Operator-based multiscale method for compressible flow. SPE J. 13(02), 523–539 (2008)
Zhou, H., Tchelepi, H.A., et al.: Two-stage algebraic multiscale linear solver for highly heterogeneous reservoir models. SPE J. 17(02), 523–539 (2012)
Acknowledgments
This work is part of the first author’s PhD project, which was sponsored by Petrobras’ employee training program. The insightful discussions and valuable suggestions from Dr. Alexandre A. Emerick (Petrobras Research Center) are immensely appreciated. We also thank José R. P. Rodrigues (Petrobras Research Center) for the careful proofreading and improvements suggestion.
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de Moraes, R.J., Hajibeygi, H. & Jansen, J.D. A multiscale method for data assimilation. Comput Geosci 24, 425–442 (2020). https://doi.org/10.1007/s10596-019-09839-2
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DOI: https://doi.org/10.1007/s10596-019-09839-2