Analytical Upscaling of Permeability for Three-Dimensional Grid Block
- Jaedong Lee (U. of Tulsa) | Ekrem Kasap (U. of Tulsa) | M.G. Kelkar (U. of Tulsa)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- March 1996
- Document Type
- Journal Paper
- 59 - 68
- 1996. Society of Petroleum Engineers
- 5.5.3 Scaling Methods, 5.1.5 Geologic Modeling, 4.3.4 Scale, 5.1 Reservoir Characterisation, 5.5 Reservoir Simulation, 5.6.5 Tracers, 2.4.3 Sand/Solids Control
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With modem conditional simulation methods, detailed reservoir descriptions can be constructed on a very small scale. Many methods also allow detailed descriptions which include basic geologic features such as cross beds and laminations. Unfortunately, present flow simulators are not capable of incorporating these small scale descriptions due to computational limitations. The small scale values, especially, the permeabilities, need to be upscaled appropriately to the size suitable for flow simulations.
Many methods, some numerical and some analytical, have been reported in the literature to upscale the permeability values. Numerical methods are computationally expensive, whereas, many analytical methods are restricted to two dimensions. This paper presents an analytical method to estimate the upscaled permeability tensor in three dimensions using small scale permeability values. The method is based on dividing the upscaled block into eight smaller blocks and using cross flow and noncross flow averaged permeabilities based on manipulation of Darcy's law. The method is validated by comparing the results with numerical simulator.
For successful field development and reservoir management, field scale numerical simulations of flow performances are required. Reservoir performance simulators require input data on a scale that is much larger than the scale of laminations and cross beddings. These small scale heterogeneities in the reservoir have been observed to influence reservoir performance. To account for the effect of small scale heterogeneities, effective permeability of grid block needs to be properly defined. An effective permeability preserves the ratio of the fluid flux and the potential drop across a heterogeneous block and an equivalent block.
Several methods are presented in the literature for calculating effective permeability of a grid block. These methods can generally be divided into numerical and analytical methods. Numerical methods are used to handle complex heterogeneities, but may be computationally demanding. on the other hand, analytical methods are usually restricted by simplifying assumptions but less expensive than numerical methods in terms of computational cost.
Haldorsen and Lake used the stream line model for calculating effective permeability of sand/shale distributions. Begg and King presented a method to capture several stochastic sand/shale distributions in both two and three dimensions. Begg et al. extended Begg and King's statistical streamline method to layered media with different anisotropic sand permeabilities and shale statistics in each layer. King proposed real-space renormalization technique which uses a resistor network as a model for the permeability. Williams extended King's method by providing two schemes for calculating effective properties; small cell methods and large cell methods.
Although practical, the disadvantage of the above described analytical methods is their inability to capture the presence of induced cross flow due to the presence of heterogeneities. The general form of permeability is a tensor and consequently the effective - upscaled - permeability is also a tensor. Tensorial representation of permeability becomes important when the driving force is not aligned with one of the principal directions of the permeability, thereby the velocity vector along each principal axis is affected by transverse pressure gradients. Aasum shows following two base cases which require full tensor effective permeability representation:
1. Heterogeneous or homogeneous depositional structures (layers) are oriented at angles other than those parallel and perpendicular to the simulation coordinate axes.
2. Heterogeneities are not centered within the grid block.
White and Horne showed the necessity of permeability tensor in the case of local permeability variation. Kasap and Lake calculated an effective permeability tensor for a rectangular system with a perturbation. The perturbation is a region with different permeability value than the rest of the system. Aasum et al. extended this method to calculate effective permeability tensor of a simulator grid block under generalized anisotropic conditions. Although the method accounts for generally anisotropic systems, it is restricted to two dimensional systems.
A property is anisotropic if its value depends on the direction in which it is measured. Permeability anisotropy is caused by the orientation and shape of the asymmetric grains in the porous media. Permeability anisotropy causes fluid to flow in a direction different from that in the direction of bulk flow.
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