This paper presents analytical solutions to estimate at any scale the fracture density variability associated to stochastic Discrete Fracture Networks. These analytical solutions are based upon the assumption that each fracture in the network is an independent event. Analytical solutions are developed for any kind of fracture density indicators. Those analytical solutions are verified by numerical computing of the fracture density variability in three-dimensional stochastic Discrete Fracture Network (DFN) models following various orientation and size distributions, including the heavy-tailed power-law fracture size distribution. We show that this variability is dependent on the fracture size distribution and the measurement scale, but not on the orientation distribution. We also show that for networks following power-law size distribution, the scaling of the three-dimensional fracture density variability clearly depends on the power-law exponent.

Characterizing fracture networks in geosciences is a key challenge for many
industrial projects such as deep waste disposal, hydrogeology or petroleum
resources, because it may change the mechanical (Davy et al., 2018;
Grechka and Kachanov, 2006) and hydrological (Bogdanov et al., 2007; De
Dreuzy et al., 2001a, b) behaviour of the rock mass. Fractures being
ubiquitous and at all scales, the description of these physical properties
is often far beyond the reach of a continuum approach (Jing, 2003).
Discrete Fracture Networks are computational models explicitly representing
the geometry of fractures in a network, and can be used as a basis for
physical simulations (mechanical strength, flow, transport…)
(see Jing, 2003; Lei et al., 2017, for a review). Considering
the scarce nature of geological data, statistical methods have been widely
used to generate DFN models, where all fracture geometrical attributes are
treated as independent variables from probability distribution derived from
the field. Indeed, fracture networks are often described from size
distribution, sets of orientations, location, and densities (Dershowitz
and Einstein, 1988). Unfortunately, the difficulty of access and resolution
to volumetric data makes it difficult to directly measure three-dimensional
fracture densities. Stereological analysis proposes theoretical relationship
to calculate the 3-D density from 1-D or 2-D measurements under some
assumptions (Berkowitz and Adler, 1998; Darcel et al., 2003a; Warburton,
1980). For example, the total fracture surface per unit volume

In the following, we will calculate the variability of the fracture
densities

For geological environments, fracture networks are characterized by a wide
distribution of fracture sizes. We denote by

Fracture abundance is often quantified on boreholes using the
one-dimensional fracture frequency

In three-dimensional systems, the fractures

Scaling of parameters

The

Fracture intensity

Theoretical (dash) and experimental (dots)

The percolation parameter gives an idea of the connectivity of the network
(Bour and Davy, 1997, 1998; Robinson, 1983). Fundamentally, it is total
excluded volume around fractures per unit volume so that for disk-shaped
fractures the associated measure

Fracture intensity lacunarity curves

Fracture densities lacunarity curves

In this section, we aim at validating the analytical solutions developed in
Sect. 2, with numerical experiments on Discrete Fracture Networks (DFN). We
generate some very simple Poissonian DFN models where the position of each
fracture is set randomly within a cubic system of size

We compute the

To construct the experimental three-dimensional density lacunarity curves,
we divide the cubic domain of size

Figure 4 focuses on various 3-D density lacunarity
curves for networks following power-law size distributions. When the study
scale

We propose here analytical solutions to quantify the density variability
associated to Poissonian fracture networks, using the dimensionless variance
parameter

The Python script used for generation and analysis of Discrete Fracture
Networks in this paper can be found at

EL, PD and CD conceived the idea of this study. EL and PD developed the analytical solutions. EL and RLG developed the code for generation and analysis of DFNs. EL took the lead in writing manuscript, all authors providing critical feedback.

The authors declare that they have no conflict of interest.

This article is part of the special issue “European Geosciences Union General Assembly 2019, EGU Division Energy, Resources & Environment (ERE)”. It is a result of the EGU General Assembly 2019, Vienna, Austria, 7–12 April 2019.

The authors acknowledge Svensk Kärnbränslehantering AB, the Swedish Nuclear Fuel and Waste Management Company for the funding of this work.

This paper was edited by Thomas Nagel and reviewed by two anonymous referees.