Simultaneous Estimation of Geologic and Reservoir State Variables Within an Ensemble-Based Multiple-Point Statistic Framework
Tóm tắt
Assessment of uncertainty due to inadequate data and imperfect geological knowledge is an essential aspect of the subsurface model building process. In this work, a novel methodology for characterizing complex geological structures is presented that integrates dynamic data. The procedure results in the assessment of uncertainty associated with the predictions of flow and transport. The methodology is an extension of a previously developed pattern search-based inverse method that models the spatial variation in flow parameters by searching for patterns in an ensemble of reservoir models. More specifically, the pattern-searching algorithm is extended in two directions: (1) state values (such as piezometric head) and parameters (such as conductivities) are simultaneously and sequentially estimated, which implies that real-time assimilation of dynamic data is possible as in ensemble filtering approaches; and (2) both the estimated parameter and state variables are considered when pattern searching is implemented. The new scheme results in two main advantages—better characterization of parameters, especially for delineating small scale features, and an ensemble of head states that can be used to update the parameter field using the dynamic data at the next instant, without running expensive flow simulations. An efficient algorithm for pattern search is developed, which works with a flexible search radius and can be optimized for the estimation of either large- or small-scale structures. Synthetic examples are employed to demonstrate the effectiveness and robustness of the proposed approach.
Tài liệu tham khảo
Aanonsen S, Nævdal G, Oliver D, Reynolds A, Valles B (2009) The ensemble Kalman filter in reservoir engineering—a review. SPE J 14(3):393–412
Abdollahifard MJ, Faez K (2013) Stochastic simulation of patterns using Bayesian pattern modeling. Comput Geosci 17(1):99–116
Alcolea A, Renard P (2010) Blocking moving window algorithm: conditioning multiple-point simulations to hydrogeological data. Water Resour Res 46:W08511
Arpat GB (2005) Sequential simulation with patterns. PhD thesis, Stanford University
Caers J (2002) Geostatistical history matching under training-image based geological model constraints. In: SPE annual technical conference and exhibition, SPE 77429
Caers J (2003) Efficient gradual deformation using a streamline-based proxy method. J Pet Sci Eng 39(1–2):57–83
Chen Y, Zhang D (2006) Data assimilation for transient flow in geologic formations via ensemble Kalman filter. Adv Water Resour 29(8):1107–1122
Evensen G (2003) The ensemble Kalman filter: theoretical formulation and practical implementation. Ocean Dyn 53(4):343–367
Fu J, Gómez-Hernández JJ (2009) A blocking Markov chain Monte Carlo method for inverse stochastic hydrogeological modeling. Math Geosci 41(2):105–128
Gómez-Hernández JJ, Sahuquillo A, Capilla JE (1997) Stochastic simulation of transmissivity fields conditional to both transmissivity and piezometric data. 1. Theory. J Hydrol 203(1–4):162–174
Gu Y, Oliver D (2006) The ensemble Kalman filter for continuous updating of reservoir simulation models. J Energy Resour Technol 128(1):79–87
Guardiano F, Srivastava R (1993) Multivariate geostatistics: beyond bivariate moments. In: Soares A (ed) Geostatistics-Troia. Kluwer Academic, Dordrecht, pp 133–144
Hamill T, Whitaker J, Snyder C (2001) Distance-dependent filtering of background error covariance estimates in an ensemble Kalman filter. Mon Weather Rev 129:2776–2790
Harbaugh AW, Banta ER, Hill MC, McDonald MG (2000) MODFLOW-2000, the US Geological Survey modular ground-water model. US Geological Survey, Branch of Information Services, Reston, VA, Denver, CO
He J, Sarma P, Durlofsky LJ (2013) Reduced-order flow modeling and geological parameterization for ensemble-based data assimilation. Comput Geosci 55:54–69
Hendricks Franssen H, Kinzelbach W (2008) Real-time groundwater flow modeling with the ensemble Kalman filter: joint estimation of states and parameters and the filter inbreeding problem. Water Resour Res 44(9):W09408
Hoffman BT, Caers J (2005) Regional probability perturbations for history matching. J Pet Sci Eng 46(1–2):53–71
Hu LY, Chugunova T (2008) Multiple-point geostatistics for modeling subsurface heterogeneity: a comprehensive review. Water Resour Res 44(11):W11413
Hu LY, Zhao Y, Liu Y, Scheepens C, Bouchard A (2012) Updating multipoint simulatings using the ensemble Kalman filter. Comput Geosci 51:7–15
Huang Y, Srinivasan S (2012) Efficient conditional simulation of spatial patterns using a pattern-growth algorithm. Geostatistics Oslo 2012:209–220
Jafarpour B, Khodabakhshi M (2011) A probability conditioning method (PCM) for nonlinear flow data integration into multipoint statistical facies simulation. Math Geosci 43(2):133–164
Kashib T, Srinivasan S (2006) A probabilistic approach to integrating dynamic data in reservoir models. J Pet Sci Eng 50(3):241–257
Li L, Zhou H, Hendricks Franssen HJ, Gómez-Hernández JJ (2012) Modeling transient groundwater flow by coupling ensemble Kalman filtering and upscaling. Water Resour Res 48(1):W01537
Li L, Srinivadan S, Zhou H, Gómez-Hernández J (2013) A pilot point guided pattern matching approach to integrate dynamic data into geological model. Adv Water Resour. doi:10.1016/j.advwatres.2013.10.008
Mariethoz G, Renard P, Caers J (2010a) Bayesian inverse problem and optimization with iterative spatial resampling. Water Resour Res 46(11):W11530
Mariethoz G, Renard P, Straubhaar J (2010b) The direct sampling method to perform multiple-point geostatistical simulaitons. Water Resour Res 46:W11536
Meerschman E, Pirot G, Mariethoz G, Straubhaar J, Meirvenne M, Renard P (2013) A practical guide to performing multiple-point statistical simulations with the direct sampling algorithm. Comput Geosci 52:307–324
Ramarao BS, LaVenue de AM, Marsily G, Marietta MG (1995) Pilot point methodology for automated calibration of an ensemble of conditionally simulated transmissivity fields. 1. Theory and computational experiments. Water Resour Res 31(3):475–493
Rezaee H, Mariethoz G, Koneshloo M, Asghari O (2013) Multiple-point geostatistical simulation using the bunch-pasting direct sampling method. Comput Geosci 54:293–308
Straubhaar J, Renard P, Mariethoz G, Froidevaux R, Besson O (2011) An improved parallel multiple-point algorithm using a list approach. Math Geosci 43(3):305–328
Strebelle S (2002) Conditional simulation of complex geological structures using multiple-point statistics. Math Geol 34(1):1–21
Sun AY, Morris AP, Mohanty S (2009) Sequential updating of multimodal hydrogeologic parameter fields using localization and clustering techniques. Water Resour Res 45(7):W07424
Tahmasebi P, Hezarkhani A, Sahimi M (2012a) Multiple-point geostatistical modeling based on the cross-correlation functions. Comput Geosci 16(3):779–797
Tahmasebi P, Sahimi M, Mariethoz G, Hezarkhani A (2012b) Accelerating geostatistical simulations using graphics processing units (GPU). Comput Geosci 46:51–59
Wen X, Chen W (2006) Real-time reservoir model updating using ensemble Kalman filter with confirming option. SPE J 11(4):431–442
Zhang T, Switzer P, Journel A (2006) Filter-based classification of training image patterns for spatial simulation. Math Geol 38(1):63–80
Zhou H, Gómez-Hernández J, Hendricks Franssen H, Li L (2011) An approach to handling non-Gaussianity of parameters and state variables in ensemble Kalman filtering. Adv Water Resour 34(7):844–864
Zhou H, Gómez-Hernández J, Li L (2012) A pattern-search-based inverse method. Water Resour Res 48(3):W03505