Effective Permeability Estimation for 2-D Fractal Permeability Fields
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Effective Permeability Estimation for 2-D Fractal Permeability Fields1 Tayfun Babadagli2 Hurst exponents (H) of the distribution of permeability at micro (pore) scale were measured as close to 0.1 for sandstone and limestone samples. Based on these observations and previously reported H values for field scale permeability distribution ranging between 0.6 and 0.9, square permeability fields at different scales varying between 1 and 100 ft were generated for the H values of 0.1, 0.5, and 0.9. The study also considered different permeability fields and number of grids ranging from 10 to 500 md and from 8 × 8 to 64 × 64, respectively. The effective permeability of fractally distributed 2-D fields was calculated using different averaging techniques and compared to the actual (equivalent) permeability obtained through numerical simulation. The geometric mean and power averaging techniques as well as the perturbation theory yielded the most reasonable agreement between the actual and calculated effective permeabilities. The accuracy of these techniques increases with increasing average model permeability. It was also observed that as the H decreases, the permeability values obtained were higher than the actual values. Two extreme values of the number of grids (8 × 8 and 64 × 64) yielded the highest error percentages. Thus, the optimum number of grids was found to be 16 × 16 and 32 × 32 depending on the average permeability of the model. The exponent of the power law model was correlated to the fractal dimension of the permeability field for 8 × 8 and 64 × 64 grids. While a good correlation exists for 8 × 8 number of grids, no correlation was obtained for 64 × 64. Hence, an alternate model was proposed for 8 × 8 grids but for grid numbers higher than 32 × 32, no technique was found suitable for averaging of the fractal permeability fields. KEY WORDS: effective permeability; upscaling; averaging techniques; fractal distribution of permeability; grid size and dimension; Hurst exponent; fractal dimension; micro scale.
INTRODUCTION Permeability mapping is one of the most critical aspects of modeling subsurface reservoirs. In preparation of the permeability maps, core level permeability data need to be up-scaled to a grid scale to reduce the number of grids to an optimal number. This entails accurate and reliable averaging of permeability. 1 Received
7 November 2003; accepted 5 March 2005; Published online: 18 April 2006. of Civil and Environmental Engineering, School of Mining and Petroleum, University of Alberta, 3-112 Markin CNRL/Natural Resources Engineering Facility, Edmonton, Alberta, T6G 2W2, Canada; e-mail: [email protected].
2 Department
33 C 2006 International Association for Mathematical Geology 0882-8121/06/0100-0033/1
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Babadagli
To date, many different averaging techniques were proposed for permeability up-scaling. Permeability, unlike porosity and saturation, which are additives, cannot easily be averaged using these techniques (Richardson and others, 1987). Classical averaging techniques (static tech
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