Exploring Direct Sampling and Iterative Spatial Resampling in History Matching

We explore multiple methods to history match a porosity field by using both production data and a secondary data set. In this paper, seismic impedance is used as secondary data but any data set with identical dimensions as the primary data can be used. The task is formulated as an inverse problem where the production data is estimated through an upscaled flow simulator. Co-located permeability values are obtained though an empirical relationship from the well logs. The secondary seismic data is treated in two ways. Firstly, it is used as conditioning data to further constrain the porosity simulation. This means that the production data is the only data that has a likelihood function. In the second method, porosity and seismic impedance are simulated simultaneously. This corresponds to a likelihood function with combined production and seismic data. If the second method is used, a more constrained porosity field is obtained in which channels follow the seismic more closely, whereas when the first method is used the channels are slightly thinner and more spread out. Because increased amount of conditioning data yields a more constrained a posteriori distribution it is likely that the optimization takes requires less steps to reach an optimum compared with the case where both seismic and production data is matched.

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

[2]  P. C. Shah,et al.  Reservoir History Matching by Bayesian Estimation , 1976 .

[3]  P. Sabatier.  On geophysical inverse problems and constraints , 1976 .

[4]  Wafik B. Beydoun,et al.  Reference velocity model estimation from prestack waveforms: Coherency optimization by simulated annealing , 1989 .

[5]  A. Tarantola,et al.  Monte Carlo estimation and resolution analysis of seismic background velocities , 1991 .

[6]  Klaus Mosegaard,et al.  A SIMULATED ANNEALING APPROACH TO SEISMIC MODEL OPTIMIZATION WITH SPARSE PRIOR INFORMATION , 1991 .

[7]  R. M. Srivastava,et al.  Multivariate Geostatistics: Beyond Bivariate Moments , 1993 .

[8]  Michael S. Bahorich,et al.  3-D Seismic Discontinuity For Faults And Stratigraphic Features: The Coherence Cube , 1995 .

[9]  B. Kennett.  On the density distribution within the Earth , 1998 .

[10]  M. Sambridge.  Geophysical inversion with a neighbourhood algorithm—II. Appraising the ensemble , 1999 .

[11]  E. Candès,et al.  Curvelets: A Surprisingly Effective Nonadaptive Representation for Objects with Edges , 2000 .

[12]  Benoit Noetinger,et al.  Gradual Deformation and Iterative Calibration of Sequential Stochastic Simulations , 2001 .

[13]  Dean S. Oliver,et al.  History Matching of Three-Phase Flow Production Data , 2001 .

[14]  Sebastien Strebelle,et al.  Conditional Simulation of Complex Geological Structures Using Multiple-Point Statistics , 2002 .

[15]  N. Bohr.  MONTE CARLO METHODS IN GEOPHYSICAL INVERSE PROBLEMS , 2002 .

[16]  E. Candès,et al.  Continuous Curvelet Transform : I . Resolution of the Wavefront Set , 2003 .

[17]  L. Hu,et al.  An Improved Gradual Deformation Method for Reconciling Random and Gradient Searches in Stochastic Optimizations , 2004 .

[18]  E. Candès,et al.  New tight frames of curvelets and optimal representations of objects with piecewise C2 singularities , 2004 .

[19]  Niels Bohr,et al.  Monte Carlo sampling of solutions to inverse problems , 2004 .

[20]  Lexing Ying,et al.  3D discrete curvelet transform , 2005, SPIE Optics + Photonics.

[21]  E. Candès,et al.  Continuous curvelet transform: II. Discretization and frames , 2005 .

[22]  Alexandre Boucher,et al.  Evaluating Information Redundancy Through the Tau Model , 2005 .

[23]  Laurent Demanet,et al.  Fast Discrete Curvelet Transforms , 2006, Multiscale Model. Simul..

[24]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[25]  J. Caers,et al.  The Probability Perturbation Method: A New Look at Bayesian Inverse Modeling , 2006 .

[26]  Kurt J. Marfurt,et al.  3D volumetric multispectral estimates of reflector curvature and rotation , 2006 .

[27]  Paul Switzer,et al.  Filter-Based Classification of Training Image Patterns for Spatial Simulation , 2006 .

[28]  K. Marfurt,et al.  Curvature attribute applications to 3D surface seismic data , 2007 .

[29]  F. Herrmann,et al.  Non-linear primary-multiple separation with directional curvelet frames , 2007 .

[30]  F. Herrmann,et al.  Compressed wavefield extrapolation , 2007 .

[31]  M. Kenward,et al.  An Introduction to the Bootstrap , 2007 .

[32]  Jef Caers,et al.  Comparing the Gradual Deformation with the Probability Perturbation Method for Solving Inverse Problems , 2007 .

[33]  Kurt J. Marfurt,et al.  Seismic Attributes for Prospect Identification and Reservoir Characterization , 2007 .

[34]  T. Mukerji,et al.  Quantifying Spatial Trend of Sediment Parameters In Channelized Turbidite, West Africa , 2007 .

[35]  Jingjing Zheng,et al.  Coherence Cube Based On Curvelet Transform , 2008 .

[36]  L. Hu,et al.  Extended Probability Perturbation Method for Calibrating Stochastic Reservoir Models , 2008 .

[37]  Felix J. Herrmann,et al.  Non-parametric seismic data recovery with curvelet frames , 2008 .

[38]  T. Mukerji,et al.  Robust scheme for inversion of seismic and production data for reservoir facies modeling , 2009 .

[39]  Gregoire Mariethoz,et al.  The Direct Sampling method to perform multiple‐point geostatistical simulations , 2010 .

[40]  J. Caers,et al.  Stochastic Simulation of Patterns Using Distance-Based Pattern Modeling , 2010 .

[41]  G. Mariéthoz,et al.  Bayesian inverse problem and optimization with iterative spatial resampling , 2010 .