Numerical Investigations on the Effect of Fracture Length Distribution on the Representative Elementary Volume of 3D Discrete Fracture Networks

Abstract

Determination of the representative elementary volume (REV) of fractured rock masses based on equivalent permeability ( $K$ ) is significantly dependent on the geometric characteristics of fractures. In this work, a series of numerical simulations were performed to analyze the relationship between geometric characteristics of fractures and the REV size, in which fracture length follows a power-law distribution. A method to evaluate the $K$ of a three-dimensional (3D) discrete fracture network (DFN) by extracting the equivalent pipe network (EPN) model from the DFN model was utilized and verified. The results show that $K$ of the 3D DFN model has an exponential relationship with the power exponent ( $a$ ) of fracture length distribution and the evaluation of $K$ agrees well with that reported in previous studies, confirming the reliability of the EPN model for calculating seepage properties of complex 3D DFN models. When the side length of submodels ( $L_n$ ) is small, the $K$ varies significantly due to the influence of random number seeds used to generate fracture length, location, and orientation. The $K$ holds a constant value after $L_n$ exceeds some specific value. The critical model scale is determined as the REV size, and the corresponding volume of the 3D DFN model is represented by $V_{\text{REV}}$ . The $V_{\text{REV}}$ varies within a narrow range when $a\le 4.0$ . When $a=4.5$ , the $V_{\text{REV}}$ rapidly increases to more than 3.4 times than that when $a=4.0$ . The fluid flow becomes more inhomogeneous due to the small nonpersistent fractures that dominate the preferential flow paths when $a$ exceeds a certain value (i.e., 4.5). The $K$ at the REV size decreases exponentially with the increment of $a$ . This tendency can be explained by the decrease of the average intersection length ( $L_i$ ) with the increment of $a$ , which is a geometric parameter for reflecting the connectivity of the fracture network.