Scale-dependent permeability and formation factor in porous media: Applications of percolation theory

Abstract

Understanding porous media properties and their scale dependence have been an active subject of research in the past several decades in hydrology, geosciences and petroleum engineering. The scale dependence of flow in porous media is attributed to small- and large-scale heterogeneities, such as pore size distribution, pore connectivity, long-range correlations, fractures and faults orientations, and spatial and temporal variations. The main objective of this study was to investigate how permeability (k) and formation factor (F) vary with sample dimension at small scales by means of a combination of pore-network modeling and percolation theory. For this purpose, the permeability and formation factor were simulated in twelve three-dimensional pore networks with different levels of pore-scale heterogeneities. Simulations were carried out at five different network sizes, i.e., 1130, 2250, 3380, 4510 and 6770 µm. Four theoretical models were developed based on percolation theory to estimate the scale dependence of permeability and formation factor from the pore-throat radius distribution. In addition, two other theoretical scale-dependent permeability models were proposed to estimate permeability at different scales from the pore-throat radius distribution and/or formation factor. Comparing theoretical estimations with numerical simulations showed that the proposed models estimated the scale dependence of permeability and formation factor reasonably. The calculated relative error (RE) ranged between −3.7 and 3.8% for the permeability and between 0.21 and 4.04% for the formation factor in the studied pore-networks.