Application of the renormalization group approach for permeability estimation in digital rocks

Rock permeabilities are typically calculated by applying some empirically derived relationships or expressions, which normally result in rough or erroneous values. Given advanced computing and smart techniques, rock samples acquired in boreholes or on outcrops can be imaged and broken down into small digital rocks. The renormalization group approach with isotropic block (RGAWIB) has been applied to calculate the permeability of digital rocks. The RGAWIB proposed by King can accurately derive the effective permeability if the pore structure is described correctly by the separated grid blocks. For King's method, the permeability of the grid block is assumed to be isotropic, which is not suitable for significant variance in neighboring permeabilities. In this study, we adopted discrete blocks with anisotropic permeability for the upscaling process of renormalization, which provided reasonable results for large perturbations of permeability in neighboring grid blocks. We assumed that the anisotropic block held different permeabilities in three orthogonal directions and calculated the effective permeabilities of idealized porous media and digital rocks by a renormalization group approach with anisotropic block (RGAWAB). Moreover, a permeability distribution method is presented in this paper. The original permeability distribution of an idealized porous media was calculated, along with the permeability distributions of the digital core data available in the literature in the x, y, and z directions using the method developed in this study. Then, we calculated the effective permeabilities of the digital rocks using RGAWAB in the x, y, and z directions, respectively. We compared the results with the absolute permeabilities calculated by the lattice Boltzmann method, which showed that the permeability calculation method was feasible for digital rocks.