  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 microns. Four theoretical models were also 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
formation factor. Comparing theoretical estimations with numerical simulations
showed that the proposed models estimate 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.
