Effects of surface geometry on advection–diffusion process in rough fractures

Abstract

The surface geometry of a natural fracture may feature self-affine properties, which affects the advection–diffusion process significantly. A good understanding of the underlying mechanisms of controlling such a process is of fundamental importance for a clear description of complex hydrodynamic problems. However, the effects of hydraulic tortuosity, surface roughness, scale-invariance properties, and size effect on self-affine fractures have not been fully verified. In this work, we clarify the presence of triple effects of surface geometries on the advection–diffusion process by analytical derivations, quantify their physical implications based on Poiseuille flow and Fick’s law, establish a triple-effect advection–diffusion model for rough fractures by taking into consideration the joint effect of hydraulic tortuosity (τ), surface tortuosity (τs), and stationary roughness (fσ), and then reformulate the triple-effect model into a scaling form as per the size effect in self-affine fractures. For validation, we propose a novel Weierstrass–Mandelbrot function to model self-affine fractures according to the fractal topography theory, simulate the advection–diffusion process at the pore scale by the lattice Boltzmann method, and verify the triple-effect model and its scaling form systematically. The results indicate that the advection–diffusion process is composed of effective molecular diffusion and advection-induced dispersion, with the former inversely proportional to τ2 and the latter inversely proportional to τ2τs6fσ. Moreover, τs and τ are both scaled by H−1 (H is the Hurst exponent) with the mean aperture in self-affine fractures. Theoretical analysis and numerical simulations demonstrate that our model enables the generalization of several conventional models reported in the literature.