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Ensemble size investigation in adaptive ES-MDA reservoir history matching

  • Paulo Henrique RanazziEmail author
  • Marcio Augusto Sampaio
Technical Paper
  • 237 Downloads

Abstract

In this work, we study the ensemble size influence on an adaptive ensemble-based methodology for history matching of petroleum reservoirs. The assimilation scheme used is an adaptive ensemble smoother with multiple data assimilation (ES-MDA) in which both the total number of assimilations and the inflation factor of each iteration are defined automatically by the algorithm. This fact leads to the assumption that the predefined algorithm parameters may have influence in the total number of assimilations and the inflation factors. One main parameter that can be investigated is the number of ensemble members used in the assimilation, also called ensemble size. The ensemble size influence was analyzed by applying the adaptive ES-MDA in a synthetic large-scale reservoir model. As a result of the investigation, the ensemble size showed influence on the reduction in the uncertainty of the posterior models, but it did not show any influence on the total number of assimilations and on the inflation factor selection.

Keywords

Data assimilation History matching Reservoir simulation Ensemble smoother 

List of symbols

\(a\)

Coefficient that relates inflation factor and objective function

\({\mathbf{C}}_{\text{D}}\)

Covariance of the measurement errors

\({\mathbf{C}}_{\text{DD}}\)

Auto-covariance of the predicted data

\({\mathbf{C}}_{\text{MD}}\)

Cross-covariance between the model parameters and predicted data

\({\mathbf{d}}\)

Simulated data vector

\({\mathbf{d}}_{\text{obs}}\)

Observed data vector

\(\varvec{g}\)

Forward operator

\(i\)

Iteration index

\(j\)

Ensemble member index

\(k\)

Rock permeability

\(\varvec{K}\)

Kalman gain matrix

\({\mathbf{m}}\)

Model parameters vector

\(N_{\text{d}}\)

Number of data

\(N_{\text{e}}\)

Number of ensemble members

\(N_{\text{i}}\)

Number of iterations

\(N_{\text{m}}\)

Number of model parameters

\(O_{{N_{\text{d}} }}\)

Objective function

\(\alpha\)

Inflation factor

\(\varvec{\rho}\)

Localization matrix

\(\sigma\)

Standard deviation

\({\mathcal{N}}\left( {x,y} \right)\)

Gaussian sample with mean \(x\) and covariance \(y\)

Notes

Acknowledgements

This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. The authors would like to thank the Polytechnic School of the University of São Paulo, FAPESP (São Paulo Research Foundation) and LASG (Laboratory of Petroleum Reservoir Simulation and Management) for supporting this research and study development, and CMG (Computer Modelling Group Ltd.) for providing the reservoir simulator licenses used in this study. The authors also would like to thank the UNISIM group for providing the UNISIM benchmark data, including the forecast period of the reference model. The authors are grateful for the valuable feedback and contribution from the anonymous reviewers.

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Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  1. 1.Departamento de Engenharia de Minas e de Petróleo, Escola PolitécnicaUniversidade de São PauloSantosBrazil

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