Digital Image-Based Stress–Permeability Relationships of Rough Fractures Using Numerical Contact Mechanics and Stokes Equation

Flow in fractures is sensitive to their geometrical surface characteristics. The surface can undergo deformation if there is a change in stress. Natural fractures have complex geometries and rough surfaces which complicates the modelling of deformation and fluid flow. In this paper, we present a computational model that takes a digital image of a rough fracture surface and provides a stress–permeability relationship. The model is based on a first-principle contact mechanics approach at the continuum scale. Using this first principle approach, we investigate numerically the effect of fracture surface roughness and shifting of surfaces on the permeability evolution under applied stress and compare the results with laboratory experiments. A mudrock core fracture surface was digitalized using an optical microscope, and 2D cross sections through fracture surface profiles were taken for the modelling. Mechanical deformation is simulated with the contact mechanics based Virtual Element Method solver that we developed within the MATLAB Reservoir Simulation Toolbox platform. The permeability perpendicular to the fracture cross section is determined by solving the Stokes equation using the Finite Volume Method. A source of uncertainty in reproducing laboratory results is that the exact anchoring of the two opposite surfaces is difficult to determine while the stress–permeability relationship is sensitive to the exact positioning. We, therefore, investigate the sensitivity to a mismatch in two scenarios: First, we assess the stress–permeability of a fracture created using two opposing matched surfaces from the rock sample, consequently applying relative shear. Second, we assess the stress–permeability of fractures created by randomly selecting opposing surfaces from that sample. We find that a larger shift leads to a smaller drop in permeability due to applied stress, which is in line with a previous laboratory study. We also find that permeability tends to be higher in fractures with higher roughness within the investigated stress range. Finally, we provide empirical stress–permeability relationships for various relative shears and roughnesses for use in hydro-mechanical studies of fractured geological formations.