Determining effective permeability at reservoir scale: numerical simulations and critical path analysis

Abstract

Determining the effective permeability (keff) of geological formations has broad applications to site remediation, aquifer discharge or recharge, hydrocarbon production, and enhanced oil recovery. However, due to the presence of heterogeneity across scales, accurate calculation of keff requires precise characterization of reservoirs. In the literature, stochastic, theoretical, and numerical methods have been proposed to determine the value of keff. In this study, we propose applications from critical path analysis (CPA), an upscaling technique from statistical physics. More specifically, we postulate that permeability at the mode of permeability probability function should represent the effective permeability of a reservoir. To validate this hypothesis, we construct two- and three-dimensional random (uncorrelated) geologic formations based on permeability measurements from the Borden site and assume that the permeability distribution conforms to the log-normal probability density function. The log-normal distribution with different geometric means (4.5×10-12≤k_g≤1.0×109 m2) and standard deviations (0.05≤σ≤6) is used to generate 10 different formations. We apply the COMSOL platform to numerically simulate 2 and 3D flow and determine the keff in such formations. We also calculate the keff using several other approaches proposed in the literature, such as perturbation theory, renormalization group theory, and effective-medium approximation. Comparing the numerically determined keff values with the theoretically estimated ones demonstrates that the CPA provides accurate estimations in both two and three dimensions. Although the CPA estimates the keff with RMSLE = 0.50 more accurate than the other approaches in two dimensions, the renormalization group theory with RMSLE = 0.90 provides slightly better estimations than the CPA with RMSLE = 1.14 in three dimensions. Results show that although perturbation theory and the effective-medium approximation provide reasonable keff estimations in formations with σ < 2, they substantially overestimate the effective permeability in highly heterogeneous formations. We found that CPA provided a powerful platform to estimate effective permeability at the reservoir scale in uncorrelated formations. However, further investigations are still required to evaluate its predictability in formations with spatial correlations.