Absolute permeability calculations in micro-computed tomography models of sandstones by Navier-Stokes and lattice Boltzmann equations

Abstract

This paper investigates the calculation features of sandstones absolute permeability coefficients on their digital images using the lattice Boltzmann equations (LBE) and Navier-Stokes equations (NSE). The main attention in this work is given to comparison of numerical solutions of NSE and LBE, and to study the effects of grid refinement and grid coarsening on the calculated permeability coefficient. 3D digital images of sandstones were obtained using X-ray computed tomography. The permeability coefficients were calculated in sandstone samples from Imperial College London open library as well as from Ashalchinskoe (Tatarstan, Russia) and Vostochno-Birlinskoe (Ulyanovsk region, Russia) oil fields. When using LBE, a multi-relaxation time collision operator was applied. It was shown that permeability coefficients, calculated using LBE, of original digital images are 20–30% higher than when using NSE. The study of grid refinement was performed at the refinement levels ranges from 2 to 10. In this paper, the level of grid refinement characterizes the multiplicity of the grid step splitting. Strong dependence of the permeability coefficients on the grid refinement level was revealed for each mathematical model. The NSE solutions are significantly less sensitive to the refinement level in comparison with LBE solutions and the discrepancy between them decreases with increase in the refinement level. The grid independence of the NSE solution is achieved at grid refinement level 3, whereas for LBE even refinement level 10 is not enough for this. Issues arising from grid coarsening are discussed and solutions are developed at coarsening resolutions in two and three times. It was found that grid coarsened in two times is valid for single-phase flow simulations and for calculation of permeability coefficients with good accuracy of <10% error compared to original grid. This result allows one to reduce the grid dimension in 23 times which can significantly economy computational cost. For grid coarsened in three times, the calculated flow characteristics deviate by more than 10% from the initial values, but deviations decrease with increasing permeability.