The hydraulic efficiency of single fractures: Correcting the cubic law parameterization for self-affine surface roughness and fracture closure

Abstract

Abstract. Quantifying the hydraulic properties of single fractures is a fundamental requirement to understand fluid flow in fractured reservoirs. For an ideal planar fracture, the effective flow is proportional to the cube of the fracture aperture. Yet, real fractures are rarely planar, and correcting the cubic law in terms of fracture roughness has therefore been a subject of numerous studies in the past. Several empirical relationships between hydraulic and mechanical aperture have been proposed, based upon statistical variations of the aperture field. However, often they exhibit non-unique solutions, attributed to the geometrical variety of naturally occurring fractures. In this study, a non-dimensional fracture roughness quantification-scheme is acquired, opposing effective surface area against relative fracture closure. This is used to capture deviations from the cubic law as a function of quantified fracture roughness, here termed hydraulic efficiencies. For that, we combine existing methods to generate synthetic 3D fracture voxel models. Each fracture consists of two random self-affine surfaces with prescribed roughness amplitude, scaling exponent, and correlation length, which are separated by varying distances to form fracture configurations that are broadly spread in the newly formed two-parameter space. First, we performed a percolation analysis on 600'000 synthetic fractures to narrow down the parameter space on which to conduct fluid flow simulations. This revealed that the fractional amount of contact and the percolation probability solely depends on the relative fracture closure. Next, Stokes flow calculations are performed, using a 3D finite differences code on 6400 fracture models to compute directional permeabilities. The deviations from the cubic law prediction and their statistical variability for equal roughness configurations were quantified. The resulting 2D solution fields reveal decreasing cubic-law accordance's down to 1 % for extreme roughness configurations. We show that the non-uniqueness of the results significantly reduces if the correlation length of the aperture field is much smaller than the spatial extent of the fracture. Under this assumption, an equation was provided that predicts hydraulic efficiencies and respective fracture permeabilities with a mean error of 26.7 %. Numerical inaccuracies were quantified with a resolution test, adding another error of 7 % on top of the previous one. By this, we propose a revised parameterization for the permeability of rough single fractures, which takes numerical inaccuracies of the flow calculations into account. We show that this approach is more accurate, compared to existing formulations. It can be employed to estimate the permeability of fractures if a measure of fracture roughness is available, and it can readily be incorporated in discrete fracture network modeling approaches.

Roughness of Fracture Surfaces in Numerical Models and Laboratory Experiments

“…It is well known that surfaces of faults and fractures in rocks are rough at all scales (Brown and Scholz, 1985;Power and Durham, 1997;Candela et al, 2012). The roughness of fracture surfaces is important for a range of geological processes such as the mechanical behavior of faults (Okubo and Dietrich, 1984;Griffith et al, 2010;Candela et al, 2011a, b;Angheluta et al, 2011;Ahmadi et al, 2016) or the fluid flow in jointed rock or fault zones (Chen et al, 2000;Watanabe et al, 2008;Bisdom et al, 2016;Briggs et al, 2017;Jin et al, 2017;Zambrano et al, 2019;Kottwitz et al, 2019). However, the processes and parameters controlling the details of the fracture geometry are not fully understood yet.…”

“…It has been shown that the those measures are closely, but not perfectly, correlated to each other (Tse and Cruden, 1979;Li and Zhang, 2015). A roughness measure of particular interest due to its possible use in the parametrization of the fluid flow properties of rock fractures is the "effective surface area S" proposed by Kottwitz et al (2019), which can be considered as an extension of the "areal roughness index" defined by El-Soudani (1978) https://doi.org/10.5194/se-2021-51 Preprint. Discussion started: 19 May 2021 c Author(s) 2021.…”

Post-earthquake rapid resealing of bedrock flow-paths by concretion-forming resin

“…Hence, the cubic law will over-estimate the actual permeability and far less calcite precipitation will be needed to decrease the permeability of a real fracture with rough walls that come into contact over a significant proportion of the fracture. It is plausible that such an overestimate could be by an order of magnitude or more 56 .…”