We have simulated the effect of fracture characteristics on permeability (k), mean of first passage time distribution (<T>) of non-reactive solutes, and megascopic dispersivity (αME). We used a power law for fracture length distribution and a fBm for fracture aperture spatial distribution, which can be characterized by length exponent (a) and Hurst exponent (H) respectively. A 2D random lattice at percolation threshold is employed for orientation and length of fractures, and the range of 1.4<a<2.2 is considered. The results show that <T>increases with a and decreases with H, whereas k is not significantly affected by a and H. The difference of <T>due to H decreases as a becomes larger. αMEdecreases with a and increases with H. The difference of αME?due to H decreases with a, and αME approaches a constant as a becomes larger. Finally, <T>is proportional to Lx, where x is the exponent that increases from 1.02 to 1.18 with a, which is a result considerably different from that of classical percolation theory.


Flow and solute transport in a fractured medium has been a major topic researched among petroleum engineers, hydrologists, and environmental engineers. According to recent observations of fractured media, fracture length distributions follow a power law(1) such as Equation (1) (Available in full paper) where n(l)dl is the number of fractures having a length in the range [l, l+dl], and a is the length exponent varying generally between 1 and 3. Accumulated experimental and theoretical studies also leave little doubt that rock fractures are fractal, and that at the field scales, the fractured medium may be at percolation threshold(2).

Although geometrical properties of, and flow in the fractured medium that has power law fracture length distribution at percolation threshold have been studied recently(3, 4), few researches have been performed about solute transport considering the effect of a fracture aperture spatial distribution in such a medium. In this study we assume that a fracture aperture spatial distribution follows fractional Brownian motion (fBm) characterized by Hurst exponent (H), and we simulate numerically the effect of a and H on permeability (k), mean of first passage time distribution (<T>), and megascopic dispersivity (αME).


We need five steps to numerically simulate the flow and solute transport in a medium considered in this study:

  • generation of a fBm fracture aperture spatial distribution,

  • generation of a power law fracture length distribution,

  • calculation of flow field and k,

  • simulation of solute transport,

  • calculation of first passage time (FPT; T) and dispersivity (α).

Generation of a fBm Fracture Aperture Spatial Distribution

We consider a fracture as a flat plate with variable aperture along the direction of length of the fracture. Since a spatial distribution of permeability can be assumed to follow a fBm(5), we assume that a spatial distribution of fracture aperture also obeys a fBm. A fBm is a stochastic process bH (r)(6) with the properties Equation (2) (Available in full paper)

This content is only available via PDF.
You can access this article if you purchase or spend a download.