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.

ISWFoam: a numerical model for internal solitary wave simulation in continuously stratified fluids

A numerical model, ISWFoam, for simulating internal solitary waves (ISWs) in continuously stratified, incompressible, viscous fluids is developed based on a fully three-dimensional (3D) Navier–Stokes equation using the open-source code OpenFOAM®. This model combines the density transport equation with the Reynolds-averaged Navier–Stokes equation with the Coriolis force, and the model discrete equation adopts the finite-volume method. The k–ω SST turbulence model has also been modified according to the variable density field. ISWFoam provides two initial wave generation methods to generate an ISW in continuously stratified fluids, including solving the weakly nonlinear models of the extended Korteweg–de Vries (eKdV) equation and the fully nonlinear models of the Dubreil–Jacotin–Long (DJL) equation. Grid independence tests for ISWFoam are performed, and considering the accuracy and computing efficiency, the appropriate grid size of the ISW simulation is recommended to be 1/150th of the characteristic length and 1/25th of the ISW amplitude. Model verifications are conducted through comparisons between the simulated and experimental data for ISW propagation examples over a flat bottom section, including laboratory scale and actual ocean scale, a submerged triangular ridge, a Gaussian ridge, and slope. The laboratory test results, including the ISW profile, wave breaking location, ISW arrival time, and the spatial and temporal changes in the mixture region, are well reproduced by ISWFoam. The ISWFoam model with unstructured grids and local mesh refinement can effectively simulate the evolution of ISWs, the ISW breaking phenomenon, waveform inversion of ISWs, and the interaction between ISWs and complex topography.

Viscous flow around three-dimensional macroscopic cavities in a granular material

Viscous flow around spherical macroscopic cavities in a granular material is investigated. The Stokes equation inside and the Darcy–Brinkman equation outside the cavities are considered. In particular, the interaction of two equally sized cavities positioned in tandem is examined in detail, where the asymptotic effect of the other cavity is taken into account. The present analysis gives a reasonable estimate on the volume flow into the cavity and the local enhancement of stresses. This is applicable to predict the microscale waterway formation in that material, onset of landslides, collapse of cliffs and river banks, etc.