A stochastic analysis of macroscopic dispersion in fractured media

Abstract

A stochastic modeling technique has been developed to investigate mass transport in a network of discrete fractures. The model is based on the repetitive generation of realizations of a fracture network from probability distributions, describing the fracture geometry, and on a solution for mass transport within each network, using a particle‐tracking technique. The system we work with consists of two orthogonal fracture sets of finite length, oriented at various angles with respect to the direction of the mean hydraulic gradient. Emphasis is placed on describing the character of dispersion, which develops as a consequence of fracture interconnectivity, and on testing the validity of the conventional diffusion‐based model of dispersion in describing transport in fractured media. Results show that mass distributions have a complex form. Marked longitudinal dispersion can develop even a short distance from a source. The distribution of mass in the direction of flow has a consistent negative skew. This pattern of dispersion arises from the limited number of pathways for mass to migrate through the network. Controlling factors in the transport process are the orientation of the fracture sets with respect to the mean hydraulic gradient, the difference in the mean flow velocity in the two fracture sets, and the standard deviation in velocity for fracture set 1. Transport patterns can change greatly as the orientation of the hydraulic gradient changes with respect to the two fracture sets. A conventional diffusion‐based model of dispersion cannot characterize transport in these fracture networks. A skewed spatial distribution of mass is observed much more frequently than a Gaussian distribution. When the mean velocities in the two fracture sets are not equal, the form of mass spreading is described by a more general, skewed distribution that accounts for the bias in the probability of mass moving along one fracture set over another. There is a tendency for mass to form a more symmetric distribution as the orientation of the two fracture sets is rotated toward a 45° angle with respect to the direction of the mean hydraulic gradient. Furthermore, constant dispersivity values or simple dispersivity functions are not definable because of the sensitivity of transport to the local velocity field in the fracture network.

Analytical solutions for solute transfer characteristics at continuous fracture junctions

“…The conventional continuum mod½l is simple, but it has its limits because of the inability to correctly account for the effect of natural discontinuity geometry and to determine the irregular range and degree of contamination. Thus the simulation using the discrete fracture system or the hybrid discrete continuum system has become increasingly popular [ Previous works on the transport of solute at discontinuity junctions applied the complete mixing theory in which the mass flux of solute is proportional to the outlet fluid flux [Krizek et al, 1972;Schwartz et al, 1983;Schwartz and Smith, 1988] or the streamline-routing theory in which the mass flux is determined only by the discharge patterns in related fractures [ Robinson and Gale, 1990]. The diffusional-mixing theory, in which the solute mixing is accelerated by diffusion, is realistic, but it has not been applied in most discrete fracture network models.…”

Simulation of a field scale tritium tracer experiment in a fractured, weathered shale using discrete-fracture/matrix-diffusion and equivalent porous medium models

“…Studies of discrete fracture models conducted in the 1980's (Endo et al, , Schwartz et al, 1983, Long et al, 1982and Long and Wfiherspoon, 1985) described the importance of effective porosfty (which they define as the porosfty contributed by fractures) and fracture orientation, but matrix transport parameters were considered insignificant and were not included in their models. On the other hand, other studies (Foster, 1975;Day 1977;Sudicky and Frind, 1982;Malowszewski andZuber, 1985 andHarrison et al, 1992;Sudicky and McLaren, 1992) established the importance of matrix diffusion in fractured, high porosity materials by indicating its effect on delaying solute breakthrough.…”

Processes, mechanisms, parameters, and modeling approaches for partially saturated flow in soil and rock media; Yucca Mountain Site Characterization Project

“…In statistical network models, measured or empirical distributions of fracture geometry are used to generate realizations of fracture network patterns (Long et al, 1982;Schwartz et al, 1983). The debate in the literature is whether it is meaningful to use the concept of REV in describing the flow and transport through the fracture network, and whether we can still use a macroscopic perme_,.,ility tensor to model the fracture system.…”