Solute transport along a single fracture in a porous rock: a simple analytical solution and its extension for modeling velocity dispersion

Abstract

A simple and robust solution is developed for the problem of solute transport along a single fracture in a porous rock. The solution is referred to as the solution to the singleflow-path model and takes the form of a convolution of two functions. The first function is the probability density function of residence-time distribution of a conservative solute in the fracture-only system as if the rock matrix is impermeable. The second function is the response of the fracture-matrix system to the input source when Fickian-type dispersion is completely neglected; thus, the effects of Fickian-type dispersion and matrix diffusion have been decoupled. It is also found that the solution can be understood in a way in line with the concept of velocity dispersion in fractured rocks. The solution is therefore extended into more general cases to also account for velocity variation between the channels. This leads to a development of the multi-channel model followed by detailed statistical descriptions of channel properties and sensitivity analysis of the model upon changes in the model key parameters. The simulation results obtained by the multi-channel model in this study fairly well agree with what is often observed in field experiments-i.e. the unchanged Peclet number with distance, which cannot be predicted by the classical advectiondispersion equation. In light of the findings from the aforementioned analysis, it is suggested that forced-gradient experiments can result in considerably different estimates of dispersivity compared to what can be found in naturalgradient systems for typical channel widths.

Channel network concept: an integrated approach to visualize solute transport in fractured rocks

“…It is worth emphasizing that the dispersion caused by flow channeling in a rock volume is shown to increase in proportion to the observation distance, due primarily to the effect of velocity dispersion (Liu et al 2017). This is embodied in the CNM; therefore, in more general scenarios, where essential spatial heterogeneities in fractured rocks, i.e., deterministic and stochastic fractures, fracture zones and tunnels, are introduced into the system, the CNM can explain the apparent scale dependency of dispersion, provided conductive channels connect to form effective transport paths with different flow rates.…”

“…The inconsistency between the calibrated Pe = 1 and Pe = 5 together with the unknown physical explanation behind this inconsistency make the calibrated parameters t w , Pe, and SF questionable. Not to mention that these values are valid only under the test conditions and should not be used to, e.g., extend the model results to other distances or flow rates (Liu et al 2017).…”

“…In this paper, various phenomena that give rise to dispersion in channels in fractures are briefly discussed. It is followed by studying the Taylor dispersion in tapered channels, which are deemed to better describe flow paths in fractured crystalline rocks (Liu et al 2017). Following that, by some simple calculations, it is illustrated that man-made field experiments that are commonly used to assess dispersion properties are subject to inherent difficulties because in experiments with larger flow rates, dispersion can be dominated by other mechanisms than what will prevail under natural flow conditions.…”

Breakthrough Investigation of Advective and Diffusive Transport in a Porous Matrix with a Crack

“…al. [14] derived an analytical solution for what they call velocity dispersion, i.e., dispersive effects caused by the velocity distributions. The topic is more recently been re-examined using domain-random walk numerical method [15].…”

Anomalous Solute Transport in a Cylindrical Two-Zone Medium with Fractal Structure

“…The transfer from macropores to the environment is described by the diffusion equation. Such types of models were analyzed, in particular, in [1][2][3][4][5][6][7][8]. In the second group of models, the geometry of macropores and their environment is not considered explicitly, but instead, channels of various sizes and surrounding rocks are considered to be whole and are studied from a macroscopic point of view.…”