Abstract
In the present paper, a new geostatistical parameterization technique is introduced for solving inverse problems, either in groundwater hydrology or petroleum engineering. The purpose of this is to characterize permeability at the field scale from the available dynamic data, that is, data depending on fluid displacements. Thus, a permeability model is built, which yields numerical flow answers similar to the data collected. This problem is often defined as an objective function to be minimized. We are especially focused on the possibility to locally change the permeability model, so as to further reduce the objective function. This concern is of interest when dealing with 4D-seismic data. The calibration phase consists of selecting sub-domains or pilot blocks and of varying their log-permeability averages. The permeability model is then constrained to these fictitious block-data through simple cokriging. In addition, we estimate the prior probability density function relative to the pilot block values and incorporate this prior information into the objective function. Therefore, variations in block values are governed by the optimizer while accounting for nearby point and block-data. Pilot block based optimizations provide permeability models respecting point-data at their locations, spatial variability models inferred from point-data and dynamic data in a least squares sense. A synthetic example is presented to demonstrate the applicability of the proposed matching methodology.
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References
Alcolea A, Carrera J, Medina A (2005) Pilot points method incorporating prior information for solving the groundwater flow inverse problem. Adv Water Resour 29:1678–1689
Bissel RC, Dubrule O, Lamy P, Swaby P, Lepine O (1997) Combining geostatistical modelling with gradient information for history-matching: the pilot point method. SPE ATCE, San Antonio, TX, USA, SPE 38730
Caers J (2003) Geostatistical history matching under training-image based geological constraints. SPE J 8(3):218–226
Carrera J, Neuman SP (1986) Estimation of aquifer parameters under transient and steady state conditions: 1. Maximum likelihood method incorporating prior information. Water Resour Res 22:199–210
Certes C, de Marsily G (1991) Application of the pilot point method to the identification of aquifer transmissivities. Adv Water Ressour 14(5):284–300
Chilès JP, Delfiner P (1999) Geostatistics—modeling spatial uncertainty. Wiley series in probability and statistics. New York, USA
Corey AT (1977) Mechanics of heterogeneous fluids in porous media. Water resources publications, Fort Collins, Colorado
Doherty J (2003) Groundwater model calibration using pilot points and regularization. Ground Water 41(2):170–177
Fenwick D, Thiele M, Agil M, Hussain AP, Humam F, Caers J (2005) Reconciling prior geologic information with production data using streamlines: application to a giant Middle-Eastern oil field, SPE ATCE, Dallas, TX, USA, 9–12 Oct 2005, SPE 95940
Feraille M, Manceau E, Zabalza-Mezghani I, Roggero F, Hu L-Y, Reis L (2003) Integration of dynamic data in a mature oil field reservoir model to reduce the uncertainty on production forecasting, AAPG, Salt Lake City, US
Gervais V, Roggero F (2009, in press) Integration of 4D seismic data in a history matching process using an efficient local parameterization. J Pet Sci Eng
Gervais V, Gautier Y, Le Ravalec M, Roggero F (2007) History matching using local gradual deformation. SPE/EUROPEC, London, UK, 11–14 June, SPE 107173
Goovaerts P (1997) Geostatistics for natural resources evaluation. Oxford University Press, London
Hoffman B, Caers J (2005) Regional probability perturbations for history matching. J Pet Sci Eng 46:53–71
Hu L-Y (2000) Gradual deformation and iterative calibration of Gaussian-related stochastic models. Math Geol 32(1):87–108
Hu L-Y, Jenni S (2005) History-matching of object based stochastic reservoir models. SPE J 10(3):81503 SPE
Hu L-Y, Blanc G, Noetinger B (2001) Gradual deformation and iterative calibration of sequential stochastic simulations. Math Geol 33(4):475–489. doi:10.1023/A:1011088913233
Journel AG, Huijbregts CJ (1978) Mining geostatistics. Academic Press, New York (1992)
Lavenue AM, Pickens JF (1992) Application of a coupled adjoint sensitivity and kriging approach to calibrate a groundwater flow model. Water Resour Res 28(6):1543–1569
Lavenue AM, RamaRao BS, Marsily G, de Marietta MG (1995) Pilot point methodology for automated calibration of an ensemble of conditionally simulated transmissivity fields: part 2—application. Water Resour Res 31(3):495–516
Le Ravalec-Dupin M (2005) Inverse stochastic modeling of flow in porous media—application to reservoir characterization. Editions technip, ISBN 2-7108-0864-1
Le Ravalec-Dupin M, Fenwick D (2002) A combined geostatistical and streamline-based history matching procedure. SPE ATCE, SPE 77378, San Antonio, TX, USA
Le Ravalec-Dupin M, Noetinger B, Hu L-Y, Blanc G (2001) Conditioning to dynamic data: an improved zonation approach. Pet Geosci 7:S9–S16
Marsily G, de 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 et al (eds) Geostatistics for natural resources characterization 2’ part 2. Reidel, Norwell, pp 831–849
Neuman SP (1973) Calibration of distributed parameter groundwater flow models viewed as a multi-objective decision process under uncertainty. Water Resour Res 9(4):1006–1021
Press WH, Teufolsky SA, Vetterling WT, Flannery BP (1988) Numerical recipes in C. Cambridge University Press, New York
RamaRao BS, Lavenue AM, Marsilly G, de Marietta MG (1995) Pilot point methodology for automated calibration of an ensemble of conditionally simulated transmissivity fields. Part 1. Theory and computational experiments. Water Resour Res 31(3):475–493
Roggero F, Ding DY, Berthet P, Lera O, Cap J, Schreiber PE (2007) Matching of production history and 4D seismic data—application to the Girassol field, offshore Angola. SPE ATCE, Anaheim, CA, USA, SPE 109929
Sun N-Z (1995) Inverse problems in groundwater modeling. Kluwer Academic, Dordrecht
Tarantola A (1987) Inverse problem theory—methods for data fitting and model parameter estimation. Elsevier Science, Amsterdam
Wen X-H, Deutsch CV, Cullick AS (1997) High resolution reservoir models integrating multiple-well production data. SPE ATCE, San Antonio, TX, USA, SPE 38728
Woodbury AD, Ulrych TJ (2000) A full-Bayesian approach to the groundwater inverse problem for steady state flow. Water Resour Res 36(8):2081–2093
Xue G, Datta-Gupta A (1997) Structure preserving inversion: an efficient approach to conditioning stochastic reservoir models to dynamic data. SPE ATCE, San Antonio, TX, USA, SPE 38727
Yeh WWG (1986) Review of parameter identification procedures in groundwater hydrology: the inverse problem. Water Resour Res 22:95–108
Zimmerman DA, de Marsily G, Gotway CA, Marietta MG, Axness CL, Beauheim RL, Bras RL, Carrera J, Dagan G, Davies PB, Gallegos DP, Galli A, Gomez-Hernandez J, Grindrop P, Gutjahr AL, Kitanidis PK, LaVenue AM, McLaughlin D, Neuman SP, Ramarao BS, Ravenne C, Rubin Y (1998) 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
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Le Ravalec-Dupin, M. Pilot Block Method Methodology to Calibrate Stochastic Permeability Fields to Dynamic Data. Math Geosci 42, 165–185 (2010). https://doi.org/10.1007/s11004-009-9249-x
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DOI: https://doi.org/10.1007/s11004-009-9249-x