Power-Law Averaging for Inference of Effective Permeability

Abstract Power law averaging was developed to scale fine grid permeability models to effective permeability models on a coarse grid for flow simulation. Direct calculation of effective permeability with selected boundary conditions replaced the need for heuristic scaling procedures such as power law averaging. New areas of application have emerged for power law averaging. First, successful inversion of well test and production data require techniques to simultaneously account for small scale data, coming from core and log measurements, with large scale data coming from well test and production data. The power law formalism can be used to transform the permeability data coming from different scales so that the transformed permeability averages linearly, which is a requirement of geostatistical techniques. Second, the effective permeability in sandstone/shale systems can be calculated with the volume fraction of shale and the constituent permeability values, provided that the directional averaging exponents can be calibrated to the geological setting. The theory behind power law averaging is revisited and new areas of application are developed. Introduction Power law averaging was developed to upscale fine scale realizations to coarse scale models for flow simulation1,2,3; however, with increases in computing power, upscaling is easily performed with quick flow simulators instead of approximative scaling relations. We will revisit power law averaging and describe possible applications in modern reservoir characterization. Among other things, well log data provide a measure of porosity and the volume fraction of shale. The porosity data can be used directly, but when permeability measurements are sparse, permeability must be based on the combined spatial characteristics of the shale and sandstone. The power law averaging method provides a way to calculate directional permeability values that account for the orientation of the shales. Figure 1 shows schematically how the VSH log data can be transformed to a range of horizontal and vertical permeabilities based on different ωvalues in the power law transformation. Another problem in modern geostatistics is the integration of small-scale core-based permeability with large-scale production data. See Figure 2 for a schematic illustration of the different scales at which data is collected and how they are combined to create multiple realizations at an intermediate scale. The problem is the vast difference in scale and the highly non-linear averaging of permeability. To further complicate this situation, modelling is often performed at an intermediate scale between the core and production data. Gaussian techniques require the data to be transformed to a Gaussian distribution, but permeability does not average linearly after Gaussian transformation; however, a power law transform of permeability provides values that average linearly and permits the data to be simultaneously accounted for in modelling via a direct simulation approach. When unstructured grids are used the data must be linear with scale and power law averaging provides a means to do this. Figure 3 shows an example of an unstructured grid with cells that vary in size. Modern flow simulators are tending towards unstructured grids. Power law transformation will permit direct modelling of different volumes with block kriging.

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