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

The non-uniform flow velocity distribution plays a critical role in enhancing the longitudinal dispersion of the transported solute in a fluid system. For inert solute diffused in constrained fracture systems, the resulting longitudinal diffusion coefficient is the summation of the molecular diffusion coefficient of the solute and the diffusion coefficient induced by the advection process. The validity of this relationship has been verified and shown to be of fundamental significance in aerodynamics. Solute transportation is affected by several factors. Among them, the surface geometry significantly affects the advection-diffusion process owing to its tortuous nature accompanied by self-affine properties. To this end, a thorough understanding of the underlying mechanism responsible for the advection-diffusion process is of fundamental importance in describing and solving complex hydrodynamic problems and their practical implications.

The previous investigations to address the effects of surface geometry were based on either experimental analysis, numerical simulations or analytical derivations. However, these approaches have one or two limitations that compromise the quality and accuracy of the outcomes. For instance, most experimental analysis ignores the scale-invariance and multi-scale effects that hinder comprehensive understanding of the underlying mechanisms. Today, analytical-numerical coupling solutions have attracted significant attention from the research community because the analytical derivation part can help develop a mathematical framework for describing the mechanism while the numerical simulation quantifies the effects of the associated control factors. Additionally, the fractal theory and lattice Boltzmann method (LBM) provide the required technical and theoretical support for deriving, simulating and analyzing the solutions. They are commonly used to model the self-affine geometries of the natural fractures and control mechanism of the transport processes, respectively. Nevertheless, the effects of various properties such as hydraulic tortuosity, size, scale-invariance and surface roughness on the self-affine structures are not fully explored and verified.

To address the above challenge, a team of researchers from Henan Polytechnic University: Junling Zheng (PhD candidate), Xiaokun Liu, Yi Jin, Jiabin Dong and Qiaoqiao Wang investigated the effects of surface geometry of rough fractures on the advection-diffusion process using analytical-numerical solutions. First, a mathematical framework based on the Taylor-Aris equations was derived to aid exploration and understanding of the triple-effects of surface geometries. Based on the Fick and Poiseuille flow laws, the physical implications of the triple effects were quantified. Next, the triple-effect model was redesigned into a scaling form accounting for the effects of three main parameters: hydraulic tortuosity, stationary roughness and surface tortuosity. Finally, the self-affine fractures were modeled based on the fractal theory using a newly proposed Weierstrass–Mandelbrot function, and the developed model was validated via LBM simulations. The work is currently published in the journal, Chemical Engineering Journal.

The authors reported that the advection-diffusion process consists of the advection-induced dispersion and effective molecular diffusion. The latter was inversely proportional to the square of the hydraulic tortuosity, while the latter is inversely proportional to the product of the sixth power of surface tortuosity, square of hydraulic tortuosity and stationary roughness. Additionally, the hydraulic and surface tortuosity were scaled by H – 1 with a mean aperture in the self-affine fractures (where H denotes the Hurst exponent). Furthermore, the surface geometry control mechanisms were effectively clarified, and the demonstrated mathematical framework was useful in fitting real data to the model.

In summary, the authors reported the presence of the triple-effects as well as the surface and hydraulic tortuosity and stationary roughness effects of surface geometry on the advection-diffusion process. The established model provided a proper definition of the parameters involved with sound physical meanings and their implications. Overall, the presented model could generalize various conventional models available in the literature. In a statement to Advances in Engineering, the authors said that the study facilitate future investigation in more complex advection-diffusion processes.