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Research Papers

Upscaling of the Geological Models of Large-Scale Porous Media Using Multiresolution Wavelet Transformations

[+] Author and Article Information
M. Reza Rasaei

Institute for Petroleum Engineering, University of Tehran, Tehran 11365-4563, Iran

Muhammad Sahimi1

Mork Family Department of Chemical Engineering and Materials Science, University of Southern California, Los Angeles, CA 90089-1211moe@iran.usc.edu

1

Corresponding author.

J. Heat Transfer 131(10), 101007 (Jul 29, 2009) (12 pages) doi:10.1115/1.3167544 History: Received October 04, 2008; Revised February 03, 2009; Published July 29, 2009

To model fluid flow and energy transport in a large-scale porous medium, such as an oil or a geothermal reservoir, one must first develop the porous medium’s geological model (GM) that contains all the relevant data at all the important length scales. Such a model, represented by a computational grid, usually contains several million grid blocks. As a result, simulation of fluid flow and energy transport with the GM, particularly over large time scales (for example, a few years), is impractical. Thus, an important problem is upscaling of the GM. That is, starting from the GM, one attempts to generate an upscaled or coarsened computational grid with only a few thousands grid blocks, which describes fluid flow and transport in the medium as accurately as the GM. We describe a powerful upscaling method, which is based on the wavelet transformation of the spatial distribution of any static property of the porous medium, such as its permeability, or a dynamic property, such as the spatial distribution of the local fluid velocities in the medium. The method is a multiscale approach that takes into account the effect of the heterogeneities at all the length scales that can be incorporated in the GM. It generates a nonuniform computational grid with a low level of upscaling in the high permeability sectors but utilizes high levels of upscaling in the rest of the GM. After generating the upscaled computational grid, a critical step is to calculate the equivalent permeability of the upscaled blocks. In this paper, six permeability upscaling techniques are examined. The techniques are either analytical or numerical methods. The results of computer simulations of displacement of oil by water, obtained with each of the six methods, are then compared with those obtained by the GM.

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Copyright © 2009 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 1

The DB4 scaling (top) and wavelet (bottom) functions

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Figure 2

A sample of the geostatistical permeability field used in the simulations

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Figure 3

Comparison of the reservoir’s average pressure as computed with the upscaled model using the analytical methods of upscaling the permeabilities and in the GM. The analytical methods are the Arith-Harm averages, the analogy with ENs, and the IWT. Upscaling was done uniformly with the thresholds ϵs=ϵd=1.

Grahic Jump Location
Figure 4

Same as in Fig. 3 but using the numerical methods of upscaling the permeabilities, which are the PS, the WPS, and the unbiased methods

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Figure 5

Comparison of the rate of oil production as computed with the upscaled model using the analytical methods of upscaling the permeabilities and in the GM. The analytical methods are the Arith-Harm averages, the analogy with ENs, and the IWT. Upscaling was done uniformly with the thresholds ϵs=ϵd=1.

Grahic Jump Location
Figure 6

Same as in Fig. 5 but using the numerical methods of upscaling the permeabilities, which are the PS, the WPS, and the unbiased methods

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Figure 7

Comparison of the water cuts (rate of water production) as computed with the upscaled model using the analytical methods of upscaling the permeabilities and in the GM. The analytical methods are the Arith-Harm averages, the analogy with ENs, and the IWT. Upscaling was done uniformly with the thresholds ϵs=ϵd=1.

Grahic Jump Location
Figure 8

Same as in Fig. 7 but using the numerical methods of upscaling the permeabilities, which are the PS, the WPS, and the unbiased methods

Grahic Jump Location
Figure 9

Comparison of the reservoir’s average pressure as computed with the upscaled model using the analytical methods of upscaling the permeabilities and in the GM. The analytical methods are the Arith-Harm averages, the analogy with ENs, and the IWT. Upscaling was done nonuniformly with the thresholds ϵs=ϵd=0.8.

Grahic Jump Location
Figure 10

Same as in Fig. 9 but using the numerical methods of upscaling the permeabilities, which are the PS, the WPS, and the unbiased methods

Grahic Jump Location
Figure 11

Comparison of the rate of oil production as computed with the upscaled model using the analytical methods of upscaling the permeabilities and in the GM. The analytical methods are the Arith-Harm averages, the analogy with ENs, and the IWT. Upscaling was done nonuniformly with the thresholds ϵs=ϵd=0.8.

Grahic Jump Location
Figure 12

Same as in Fig. 1 but using the numerical methods of upscaling the permeabilities, which are the PS, the WPS, and the unbiased methods

Grahic Jump Location
Figure 13

Comparison of the water cuts (rate of water production) as computed with the upscaled model using the analytical methods of upscaling the permeabilities and in the GM. The analytical methods are the Arith-Harm averages, the analogy with ENs, and the IWT. Upscaling was done nonuniformly with the thresholds ϵs=ϵd=0.8.

Grahic Jump Location
Figure 14

Same as in Fig. 1 but using the numerical methods of upscaling the permeabilities, which are the PS, the WPS, and the unbiased methods

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