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RESEARCH PAPERS

The Ensemble Kalman Filter for Continuous Updating of Reservoir Simulation Models

[+] Author and Article Information
Yaqing Gu

Mewbourne School of Petroleum and Geological Eng., The University of Oklahoma, 100 East Boyd Street, SEC T301, Norman, OK 73019yaqing.gu-1@ou.edu

Dean S. Oliver

Mewbourne School of Petroleum and Geological Eng., The University of Oklahoma, 100 East Boyd Street, SEC T301, Norman, OK 73019dsoliver@ou.edu

J. Energy Resour. Technol 128(1), 79-87 (Jul 27, 2005) (9 pages) doi:10.1115/1.2134735 History: Received January 07, 2005; Revised July 27, 2005

This paper reports the use of ensemble Kalman filter (EnKF) for automatic history matching. EnKF is a Monte Carlo method, in which an ensemble of reservoir state variables are generated and kept up-to-date as data are assimilated sequentially. The uncertainty of reservoir state variables is estimated from the ensemble at any time step. Two synthetic problems are selected to investigate two primary concerns with the application of the EnKF. The first concern is whether it is possible to use a Kalman filter to make corrections to state variables in a problem for which the covariance matrix almost certainly provides a poor representation of the distribution of variables. It is tested with a one-dimensional, two-phase waterflood problem. The water saturation takes large values behind the flood front, and small values ahead of the front. The saturation distribution is bimodal and is not well modeled by the mean and variance. The second concern is the representation of the covariance via a relatively small ensemble of state vectors may be inadequate. It is tested by a two-dimensional, two-phase problem. The number of ensemble members is kept the same as for the one-dimensional problem. Hence the number of ensemble members used to create the covariance matrix is far less than the number of state variables. We conclude that EnKF can provide satisfactory history matching results while requiring less computation work than traditional history matching methods.

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Copyright © 2006 by American Society of Mechanical Engineers
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Figures

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Figure 1

Water saturation profiles before and after first application of the Kalman correction at 110days

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Figure 2

The transformed variable and water saturation profiles after first application of the Kalman correction at 110days

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Figure 3

110days, saturation profiles after application of the Kalman correction

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Figure 4

The observed and computed water and oil production rates from all ensemble members after Kalman correction, without iteration

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Figure 5

The observed and computed water and oil production rates from all ensemble members after Kalman correction, with iteration

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Figure 6

RMS error of porosity (left) and natural logarithm permeability (right) with and without iteration

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Figure 7

The true porosity (left) and permeability (right) fields

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Figure 8

Oil production rates at the 4 producers after Kalman correction

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Figure 9

The permeability field for model 1 before (left) and after (right) kalman correction at 10days

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Figure 10

The final porosity (left) and permeability (right) fields for model 1 after 200days of Kalman correction

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Figure 11

10days, water saturation at the middle column before (left) and after (right) Kalman correction

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Figure 12

Material balance checking after 200days of data assimilation

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Figure 13

Comparison of the water injection rate in each model with the data

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