A New Efficient Averaging Technique for Scaleup of Multimillion-Cell Geologic Models
- D. Li (Mobil Technology Co.) | B. Beckner (Mobil Technology Co.) | A. Kumar (Schlumberger)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Evaluation & Engineering
- Publication Date
- August 2001
- Document Type
- Journal Paper
- 297 - 307
- 2001. Society of Petroleum Engineers
- 5.1.5 Geologic Modeling, 5.5.3 Scaling Methods, 5.8.5 Oil Sand, Oil Shale, Bitumen, 5.8.6 Naturally Fractured Reservoir, 5.1 Reservoir Characterisation, 5.8.7 Carbonate Reservoir, 2.4.3 Sand/Solids Control, 5.6.4 Drillstem/Well Testing, 5.5 Reservoir Simulation, 3.3 Well & Reservoir Surveillance and Monitoring, 4.3.4 Scale, 1.2.3 Rock properties, 5.5.1 Simulator Development
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Increased resolution in reservoir characterization is driving the need for efficient and accurate upscaling techniques for reservoir simulation, on which reservoir performance prediction relies. Unfortunately, the existing averaging methods (i.e., harmonic, arithmetic, power law, geometric, or a combination of harmonic and arithmetic methods) are only applicable under the circumstances of perfectly layered or perfectly random heterogeneity distributions, which are rarely seen in realistic reservoir descriptions. This paper presents a new averaging method that improves these upscaling averaging methods for realistic reservoirs and can substitute for the orders-of-magnitude slower direct-simulation methods, such as pressure-solver techniques. The new averaging method first calculates the upper and lower bounds of the effective properties based on the nature of geology and then employs a new correlation, scaling, and rotation technique to estimate the effective properties for the upscaled grid. The approach not only preserves the accuracy of the time-consuming simulation methods but also retains the speed of the traditional averaging methods.
Five real sandstone and carbonate reservoir geologic models (three of which are multimillion-cell models) from Africa, North America, and South America were employed as benchmarks and working data sets to develop and validate the new technique. The technique has the advantage of handling the more irregular geometries [i.e., pinchouts, faults, and flexible simulation grids such as the Perpendicular Bisection (PEBI) grid].
Increased resolution in reservoir characterization is currently driving the need for efficient and accurate upscaling techniques. Upscaling is a technique that transforms a detailed geologic model to a coarse-grid simulation model so that the fluid-flow behaviors in the two systems are the same. Accurate upscaling consists of two inseparable parts: gridding and averaging. The former intends to capture the global geologic features of a geologic model, and the latter focuses on preserving the local geologic details within a coarse gridblock. Upscaling is necessary because available computers are usually memory-limited and are not fast enough to simulate the detailed geologic models derived from reservoir characterization. Even as computers increase in memory size and speed, accurate upscaling will always be a more cost-efficient method for simulating large, complicated reservoirs.
Averaging, one of the key components of upscaling, calculates the effective properties for a coarse simulation grid that preserves fine-grid fluid-flow dynamics (including pressure and flow rate) within the coarse gridblock. Averaging methods range from the simple averages (arithmetic, harmonic, and geometric means) to numerical simulation methods (pressure solver). Intermediate methods are, for example, power-law averaging and renormalization. Simple and intermediate methods are fast but less accurate, while numerical simulations are accurate but time-consuming. A fast and accurate averaging method is demanded for upscaling of very large geologic models.
The averaging problem is an old, unsolved problem of petroleum reservoir engineering. It is well known that the effective permeabilities for a layered, permeable medium with no crossflow are the arithmetic mean for flows parallel to the layering direction and the harmonic mean for flows perpendicular to the layering direction.1 Cardwell and Parsons2 proved that when fluid flow crosses over layers in a permeable medium, the arithmetic mean and the harmonic mean give only the upper and lower limits, respectively, for the effective permeabilities of the heterogeneous permeable medium, rather than the effective permeabilities themselves. Cardwell and Parsons also derived a pair of very useful upper and lower bounds for the effective permeability of a heterogeneous permeable medium. Even though the upper and lower bounds are strictly inside the upper and lower limits as stated by Cardwell and Parsons, they ignored the usefulness of the bounds because of their unsymmetrical forms. They concluded only that the effective permeability of a heterogeneous permeable medium lies between the arithmetic and harmonic limits. Warren and Price3 conducted several numerical experiments to investigate the effective permeability of a heterogeneous permeable medium and concluded that the effective permeability of the randomly generated 3D permeable medium equals the geometric mean of the individual permeabilities. Because of the technical limitations at that time, their conclusion actually is good only for purely uncorrelated permeability fields, which seldom exist in real petroleum reservoirs.
The most accurate way of calculating the effective permeability of a large, coarse gridblock containing many fine gridblocks is by solving flow equations with constant-pressure and no-flow boundary conditions4 or with periodic boundary conditions,5 regardless of the extensive computation required. This approach is referred to as a pressure-solver technique by many researchers because it involves solving the fine-grid pressure distribution first and then calculating the effective permeability with the pressure drop and the calculated flux. Because of computing limitations, pressure-solver techniques may not be practical for extremely large geologic models.
There are several averaging techniques intermediate between the traditional simple averaging methods and the pressure-solver techniques. The most frequently used intermediate methods are renormalization6-8 and power-law averaging.9-12 Renormalization includes a series of multiple-step calculations using an equivalent resistor-network approach. The major drawback of the renormalization technique is the use of unrealistic boundary conditions, which may result in estimation errors over 100%.13 However, some more current results14 show that use of periodic boundary conditions may improve the accuracy of the renormalization method. Power-law averaging has been used extensively in research work on upscaling in recent years.9-12 Power-law averaging is faster than the pressure-solver techniques, but it is not easy to use in practice because it requires empirical determination of the power-law averaging exponent through fine-grid simulation. The exponent can vary from one coarse gridblock to another. As a result, the use of a constant exponent for all coarse gridblocks may result in large errors.
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