On the relationship between effective permeability and stress for unconventional rocks: Analytical estimates from laboratory measurements

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Permeability measurements show that unconventional rock is considerably stress sensitive because the matrix permeability is likely controlled by micro cracks and bedding structures in unconventional reservoirs. While there are a number of laboratory studies in the literature on the dependency of core-scale permeability on effective stress for unconventional rocks, how to determine the corresponding large-scale stress-dependency relationship (of more interest to practical applications) from the laboratory measurements has not been systematically investigated. This work proposes a method to estimate such a large-scale relationship from laboratory measurements.Based on the stochastic approach commonly used for parameter upscaling, we derived relationships between the large-scale effective permeability and the stress for the two- and three-dimensional isotropic porous media. The development is based on the empirical observation that at core scale permeability is an exponential function of effective stress. The developed large-scale relationships can be written in terms of the same mathematical form as the local-scale relationship except parameters in the large-scale relationships correspond to effective ones. The effective stress sensitivity parameter (that characterizes the stress-dependency) is simply the expected value of that at the local scale, or the arithmetic average of local values, for the two-dimensional flow problem and a function of effective stress for the three-dimensional problem.Because of its dominant two-dimensional flow along beddings (resulting from the fact that vertical permeability is significantly smaller than the horizontal one), the relationship for the two-dimensional flow case is valid for unconventional rocks. Nevertheless, we demonstrate that for typical local-scale parameter values from unconventional rocks (e.g., Barnett shale and a carbonate source rock), the relationships obtained for two- and three-dimensional problems give the essentially same results.