Is the permeability of naturally fractured rocks scale dependent?

The equivalent permeability, k(eq) of stratified fractured porous rocks and its anisotropy is important for hydrocarbon reservoir engineering, groundwater hydrology, and subsurface contaminant transport. However, it is difficult to constrain this tensor property as it is strongly influenced by infrequent large fractures. Boreholes miss them and their directional sampling bias affects the collected geostatistical data. Samples taken at any scale smaller than that of interest truncate distributions and this bias leads to an incorrect characterization and property upscaling. To better understand this sampling problem, we have investigated a collection of outcrop-data-based Discrete Fracture and Matrix (DFM) models with mechanically constrained fracture aperture distributions, trying to establish a useful Representative Elementary Volume (REV). Finite-element analysis and flow-based upscaling have been used to determine k(eq) eigenvalues and anisotropy. While our results indicate a convergence toward a scale-invariant k(eq) REV with increasing sample size, k(eq) magnitude can have multi-modal distributions. REV size relates to the length of dilated fracture segments as opposed to overall fracture length. Tensor orientation and degree of anisotropy also converge with sample size. However, the REV for k(eq) anisotropy is larger than that for k(eq) magnitude. Across scales, tensor orientation varies spatially, reflecting inhomogeneity of the fracture patterns. Inhomogeneity is particularly pronounced where the ambient stress selectively activates late-as opposed to early (throughgoing) fractures. While we cannot detect any increase of k(eq) with sample size as postulated in some earlier studies, our results highlight a strong k(eq) anisotropy that influences scale dependence.