Investigation of the sampling performance of ensemble-based methods with a simple reservoir model

The application of the ensemble Kalman filter (EnKF) for history matching petroleum reservoir models has been the subject of intense investigation during the past 10 years. Unfortunately, EnKF often fails to provide reasonable data matches for highly nonlinear problems. This fact motivated the development of several iterative ensemble-based methods in the last few years. However, there exists no study comparing the performance of these methods in the literature, especially in terms of their ability to quantify uncertainty correctly. In this paper, we compare the performance of nine ensemble-based methods in terms of the quality of the data matches, quantification of uncertainty, and computational cost. For this purpose, we use a small but highly nonlinear reservoir model so that we can generate the reference posterior distribution of reservoir properties using a very long chain generated by a Markov chain Monte Carlo sampling algorithm. We also consider one adjoint-based implementation of the randomized maximum likelihood method in the comparisons.

[1]  D. W. Peaceman.  Interpretation of well-block pressures in numerical reservoir simulation with nonsquare grid blocks and anisotropic permeability , 1983 .

[2]  G. Evensen.  Sampling strategies and square root analysis schemes for the EnKF , 2004 .

[3]  A. Gelman,et al.  Weak convergence and optimal scaling of random walk Metropolis algorithms , 1997 .

[4]  A. Journel,et al.  Geostatistics for natural resources characterization , 1984 .

[5]  L. Hu.  Gradual Deformation and Iterative Calibration of Gaussian-Related Stochastic Models , 2000 .

[6]  J. Rosenthal,et al.  Optimal scaling for various Metropolis-Hastings algorithms , 2001 .

[7]  Albert C. Reynolds,et al.  Combining the Ensemble Kalman Filter with Markov Chain Monte Carlo for Improved History Matching and Uncertainty Characterization , 2011, ANSS 2011.

[8]  Albert C. Reynolds,et al.  Iterative Ensemble Kalman Filters for Data Assimilation , 2009 .

[9]  Dean S. Oliver,et al.  A Hybrid Markov Chain Monte Carlo Method for Generating Permeability Fields Conditioned to Multiwell Pressure Data and Prior Information , 1998 .

[10]  Xian-Huan Wen,et al.  Real-Time Reservoir Model Updating Using Ensemble Kalman Filter With Confirming Option , 2006 .

[11]  Albert C. Reynolds,et al.  Assessing the Uncertainty in Reservoir Description and Performance Predictions With the Ensemble Kalman Filter , 2005 .

[12]  Rolf Johan Lorentzen,et al.  Analysis of the ensemble Kalman filter for estimation of permeability and porosity in reservoir models , 2005 .

[13]  D. Oliver,et al.  Ensemble Randomized Maximum Likelihood Method as an Iterative Ensemble Smoother , 2011, Mathematical Geosciences.

[14]  Dean S. Oliver,et al.  THE ENSEMBLE KALMAN FILTER IN RESERVOIR ENGINEERING-A REVIEW , 2009 .

[15]  Geir Nævdal,et al.  Quantifying Monte Carlo Uncertainty in the Ensemble Kalman Filter , 2008 .

[16]  A. Reynolds,et al.  Estimation of Depths of Fluid Contacts by History Matching Using Iterative Ensemble-Kalman Smoothers , 2010 .

[17]  M. Marietta,et al.  Pilot Point Methodology for Automated Calibration of an Ensemble of conditionally Simulated Transmissivity Fields: 1. Theory and Computational Experiments , 1995 .

[18]  G. Evensen,et al.  Analysis Scheme in the Ensemble Kalman Filter , 1998 .

[19]  Alexander Y. Sun,et al.  Comparison of deterministic ensemble Kalman filters for assimilating hydrogeological data , 2009 .

[20]  A. Reynolds,et al.  Optimization Algorithms for Automatic History Matching of Production Data , 2002 .

[21]  Soroosh Sorooshian,et al.  Dual state-parameter estimation of hydrological models using ensemble Kalman filter , 2005 .

[22]  G. Evensen.  The ensemble Kalman filter for combined state and parameter estimation , 2009, IEEE Control Systems.

[23]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[24]  W. K. Hastings,et al.  Monte Carlo Sampling Methods Using Markov Chains and Their Applications , 1970 .

[25]  M. L. Wasserman,et al.  A New Algorithm for Automatic History Matching , 1974 .

[26]  Dean S. Oliver,et al.  An Iterative Ensemble Kalman Filter for Multiphase Fluid Flow Data Assimilation , 2007 .

[27]  Dilip Saha,et al.  SPIN8: a FORTRAN 77 program for automated rotation of poles , 1987 .

[28]  Dean S. Oliver,et al.  Conditioning Geostatistical Models to Two-Phase Production Data , 1999 .

[29]  Dean S. Oliver,et al.  History Matching of Three-Phase Flow Production Data , 2003 .

[30]  D. Oliver,et al.  Markov chain Monte Carlo methods for conditioning a permeability field to pressure data , 1997 .

[31]  P. Houtekamer,et al.  A Sequential Ensemble Kalman Filter for Atmospheric Data Assimilation , 2001 .

[32]  M. G. Marietta,et al.  Pilot Point Methodology for Automated Calibration of an Ensemble of Conditionally Simulated Transmissivity Fields: 2. Application , 1995 .

[33]  P. Oke,et al.  Implications of the Form of the Ensemble Transformation in the Ensemble Square Root Filters , 2008 .

[34]  Ning Liu,et al.  Assessment of Uncertainty Assessment Methods , 2001 .

[35]  Dean S. Oliver,et al.  Conditioning Permeability Fields to Pressure Data , 1996 .

[36]  Jorge Nocedal,et al.  On the limited memory BFGS method for large scale optimization , 1989, Math. Program..

[37]  M. Boucher,et al.  Interpretation of Interference Tests in a Well Field Using Geostatistical Techniques to Fit the Permeability Distribution in a Reservoir Model , 1984 .

[38]  A. Reynolds,et al.  Estimation of Initial Fluid Contacts by Assimilation of Production Data With EnKF , 2007 .

[39]  Dean S. Oliver,et al.  Memoir 71, Chapter 10: Reducing Uncertainty in Geostatistical Description with Well-Testing Pressure Data , 1997 .

[40]  A. Reynolds,et al.  History matching time-lapse seismic data using the ensemble Kalman filter with multiple data assimilations , 2012, Computational Geosciences.

[41]  A. Reynolds,et al.  Iterative Forms of the Ensemble Kalman Filter , 2006 .

[42]  Albert C. Reynolds,et al.  Monte Carlo simulation of permeability fields and reservoir performance predictions with SVD parameterization in RML compared with EnKF , 2011 .

[43]  Pavel Sakov,et al.  Relation between two common localisation methods for the EnKF , 2011 .

[44]  L. Hu,et al.  Gradual Deformation of Continuous Geostatistical Models for History Matching , 1998 .

[45]  P. Houtekamer,et al.  Data Assimilation Using an Ensemble Kalman Filter Technique , 1998 .

[46]  J. Nocedal.  Updating Quasi-Newton Matrices With Limited Storage , 1980 .

[47]  Dean S. Oliver,et al.  Evaluation of Monte Carlo Methods for Assessing Uncertainty , 2003 .

[48]  A. Lavenue,et al.  Application of a coupled adjoint sensitivity and kriging approach to calibrate a groundwater flow model , 1992 .

[49]  Jeffrey L. Anderson,et al.  A Monte Carlo Implementation of the Nonlinear Filtering Problem to Produce Ensemble Assimilations and Forecasts , 1999 .

[50]  G. Evensen,et al.  Data assimilation and inverse methods in terms of a probabilistic formulation , 1996 .

[51]  Albert C. Reynolds,et al.  Quantifying Uncertainty for the PUNQ-S3 Problem in a Bayesian Setting With RML and EnKF , 2005 .

[52]  Peter R. Oke,et al.  A deterministic formulation of the ensemble Kalman filter : an alternative to ensemble square root filters , 2008 .

[53]  J. Maerker.  Mechanical degradation of partially hydrolyzed polyacrylamide solutions in unconsolidated porous media , 1976 .

[54]  Geir Nævdal,et al.  An Iterative Ensemble Kalman Filter , 2011, IEEE Transactions on Automatic Control.

[55]  G. Evensen.  Sequential data assimilation with a nonlinear quasi‐geostrophic model using Monte Carlo methods to forecast error statistics , 1994 .

[56]  Albert C. Reynolds,et al.  Ensemble smoother with multiple data assimilation , 2013, Comput. Geosci..

[57]  J. Whitaker,et al.  Ensemble Square Root Filters , 2003, Statistical Methods for Climate Scientists.

[58]  Albert C. Reynolds,et al.  An Improved Implementation of the LBFGS Algorithm for Automatic History Matching , 2004 .

[59]  Tor Arne Johansen,et al.  Incorporating 4D Seismic Data in Reservoir Simulation Models Using Ensemble Kalman Filter , 2005 .

[60]  P. Kitanidis.  Quasi‐Linear Geostatistical Theory for Inversing , 1995 .

[61]  John F. Jordan,et al.  Reservoir Characterization: Recent Advances , 1999 .

[62]  Ning Liu,et al.  Inverse Theory for Petroleum Reservoir Characterization and History Matching , 2008 .

[63]  M. V. Krymskaya,et al.  An iterative ensemble Kalman filter for reservoir engineering applications , 2009 .

[64]  G. Chavent,et al.  History Matching by Use of Optimal Theory , 1975 .