# On the Density Variability of Poissonian Discrete Fracture Networks, with application to power-law fracture size distributions

### Philippe Davy

### Caroline Darcel

### Romain Le Goc

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.

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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 *p*_{32} can
be calculated from one-dimensional fracture intersections along scanlines or
boreholes using some conversion factor (Mauldon, 1994; Terzaghi, 1965;
Wang, 2005). Most of these studies focus on characterizing the mean density
of a fractured system with poor interest to the underlying variability.
However, fracture density variability is an indicator for fracture
clustering, which may have dramatical impact on connectivity (Darcel et
al., 2003b; La Pointe, 1988; Manzocchi, 2002). Darcel et al. (2013)
proposed an analytical solution to quantify at any scale, the standard
deviation associated to fracture frequency *p*_{10}, considering
fracture-borehole crossing as a one-dimensional Poisson point process
(random positions with fixed density). They show that the standard deviation
associated to the number of intersections per unit length *p*_{10} is
inversely proportional to the square root of the measurement scale. This
solution was also demonstrated by Lu et al. (2017), who validated
it numerically computing *p*_{10} values on boreholes crossing
three-dimensional Poissonian DFN models of fixed fracture density. In this
paper, we aim to develop analytical solutions for three-dimensional
Poissonian DFN models to quantify the range of variability of any kind of
fracture density, of any dimension. Particularly, we propose solutions for
three-dimensional density variability that cannot be obtained directly from
the field. We show that this variability depends on the measurement scale
and the fracture size distribution, but not on the orientation distribution.
These solutions are validated by numerical simulations, computing the
associated fracture densities mean and variance at various scales for
three-dimensional Poissonian Discrete Fracture Networks.

In the following, we will calculate the variability of the fracture
densities *p*_{Dd} such as defined by Dershowitz and Herda (1992) and
Mauldon and Dershowitz (2000). *D* and *d* refers to the dimensions of the
embedding system Σ and of the fracture trace in
Σ, respectively, so that *d*≤*D*. For instance, *p*_{10}
is the number of intersection points (*d*=0) along a line (*D*=1) per unit
line size. We denote by *μ*_{f} the contribution of a fracture *f* to
*p*_{Dd} (*μ*_{f} is a *d*-dimension measure) and *s* the scale at which
*p*_{Dd} and *μ*_{f} are calculated so that:

First, we calculate the spatial variability of the measure *μ*_{f}(*s*)
calculated in a *D*-sphere of size *s* embedded in a larger space
Σ of volume *V*. In *D*-dimension, a fracture is
represented by an object of dimension *D*−1 (a point in a scanline, a line
in a 2-D outcrop, or a surface in the 3-D space) with a size *S*_{f}. The
proportion of the total volume occupied by the fracture – i.e., the volume
where the measure *μ*_{f}(*s*) is not nil – is:

We then assume that, when it is not nil, the measure *μ*_{f}(*s*) derives
from a distribution with a mean $\stackrel{\mathrm{\u203e}}{{\mathit{\mu}}_{f,s}}$ and a standard
deviation ${\mathit{\sigma}}_{{\mathit{\mu}}_{f,s}}$. For the total system, we can calculate the
first and second order moment of *μ*_{f}(*s*):

The variance *σ*^{2}(*μ*_{f}) is written as:

We now sum the contribution of all fractures to calculate $\mathit{\mu}\left(s\right)={\sum}_{f}{\mathit{\mu}}_{f}\left(s\right)$. With the Poisson's
hypothesis, all fractures are independent events and the variance *σ*^{2}(μ) is the sum of the variance of each fracture:

And finally, the variability of the density measure *p*_{Dd} is ${\mathit{\sigma}}^{\mathrm{2}}\left({p}_{Dd}\right)={\mathit{\sigma}}^{\mathrm{2}}\left(\mathit{\mu}\right)/{s}^{\mathrm{2}D}$. In the following, we
compute the dimensionless ratio between the variance *σ*^{2}(*p*_{Dd}) and the square of the mean $\stackrel{\mathrm{\u203e}}{{p}_{Dd}}$:

In the fractal theory, *λ*(⋅) is called *lacunarity* (Plotnick
et al., 1996). This is a scale dependent measure, whose analysis gives an
idea of the scaling of fracture network textural heterogeneity and
potentially on different regimes (Plotnick et al., 1996; Roy et al.,
2014).

For geological environments, fracture networks are characterized by a wide
distribution of fracture sizes. We denote by *n*(l) the density
distribution of fracture sizes, and *n*(l)*d**l* is the number of
fractures of size in the range $[l,l+dl]$ per system volume (Davy, 1993).
The total number of fractures in a system of volume *V* is $N=V\int n\left(l\right)dl$ . We then assume that the measure *μ*_{f}(*s*) and the
occupied volume ratio *P*_{f}(*s*) only depends on the fracture size *l* so
that ${\mathit{\mu}}_{f}\left(s\right)=\mathit{\mu}(l,s)$ and ${P}_{f}\left(s\right)=P(l,s)$.
Equation (6) can now be written as a function of the fracture size
distribution:

In the following, we especially focus on the power-law size distribution, which have been found to adequately fit natural systems (Bonnet et al., 2001; Bour, 2002; Bour and Davy, 1999; Odling, 1997):

with *a* the power-law exponent, *α* the density term.

## 2.1 1-D measurements

Fracture abundance is often quantified on boreholes using the
one-dimensional fracture frequency *p*_{10} measurement, defined as the
number *N* of crossing fractures per unit borehole length *L*. If a fracture
intersecting this borehole is also part of a subsample of size *s*, the
associated measure $\mathit{\mu}(l,s)=\mathrm{1}$. The ratio of subsamples intersected by this
fracture is $P\left(l,s\right)=s/L$. Considering that *s*≪*L*, then
$P\left(l,s\right)\ll \mathrm{1}$, and the lacunarity of such measurement is the
one of a one-dimensional Poisson point process (Darcel et
al., 2013; Lu et al., 2017):

## 2.2 3-D measurements

In three-dimensional systems, the fractures *d*-measures *μ*_{f} are
either their occurrence (*d*=0), surface (*d*=2) or surrounded sphere
(*d*=3). *μ*_{f}(*s*) such as defined in the previous section is the part
of the measure *μ*_{f} that is included into a domain *S* of size *s*,
that is the product of *μ*_{f} by the occurrence probability of the
fracture to be included into the domain *S*, Π_{f,s} (also noted Π(*l*,*s*) following the notation of the previous section). We define Π_{f,s}
as the ratio between the fracture surface included into the domain *S* to
the total fracture surface. If the fracture size is larger than *s*, both
the average value of Π_{f,s} and its standard deviation are
proportional to the surface *s*^{2}. If $P\left(l,s\right)\ll \mathrm{1}$, that is
for large systems, Eqs. (4) and (7) requires to know the expression of
${\left(\stackrel{\mathrm{\u203e}}{\mathrm{\Pi}\left(l,s\right)}\right)}^{\mathrm{2}}+{\mathit{\sigma}}^{\mathrm{2}}\left(\mathrm{\Pi}\left(l,s\right)\right)$. The scaling of ${\left(\stackrel{\mathrm{\u203e}}{\mathrm{\Pi}\left(l,s\right)}\right)}^{\mathrm{2}}+{\mathit{\sigma}}^{\mathrm{2}}\left(\mathrm{\Pi}\left(l,s\right)\right)$ and
*P*(*l*,*s*) are given in Table 1, with *β*_{s} and *β*_{f} shape factors depending on domain *S* and
fractures geometries respectively. This binary model will allow us to
simplify the analytical solutions, although we may have significant errors
when *l*∼*s*.

### 2.2.1 Fracture number density *p*_{30}

The *p*_{30} measure counts the number of fractures *N* per unit volume *V*.
Fracture occurrence can be measured as $\mathit{\mu}\left(l,s\right)=\mathrm{\Pi}\left(l,s\right)$. The corresponding lacunarity ${\mathit{\lambda}}_{{p}_{\mathrm{30}}}$ under our
hypothesis is then given by:

If we consider that fractures have all the same size *l*_{f}, then:

If we consider now that the network follows a power-law size distribution
with minimum and maximum fracture size *l*_{min} and *l*_{max}, the *p*_{30}
lacunarity equation is defined by three regimes:

### 2.2.2 Fracture intensity *p*_{32}

Fracture intensity *p*_{32} measures the total fracture surface per unit
volume. The contribution of each fracture of size *l* in the domain *S* is
$\mathit{\mu}\left(l,s\right)=\mathrm{\Pi}\left(l,s\right){\mathit{\beta}}_{f}{l}^{\mathrm{2}}$. The
corresponding lacunarity ${\mathit{\lambda}}_{{p}_{\mathrm{32}}}$ is then given by:

Following the same reasoning as for *p*_{30} density, if we consider that
all fractures have the same size *l*_{f}, we find that the *p*_{32}
lacunarity ${\mathit{\lambda}}_{{p}_{\mathrm{32}}}$ follows Eq. (11). If we consider now that the
network follows a power-law size distribution, the *p*_{32} lacunarity
equation is defined by three regimes:

### 2.2.3 Percolation parameter *p*

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 $\mathit{\mu}(l,s)=({\mathit{\pi}}^{\mathrm{2}}/\mathrm{8})\cdot \mathrm{\Pi}\left(l,s\right){l}^{\mathrm{3}}$ (De Dreuzy et al., 2000). The
corresponding lacunarity *λ*_{p} is then given by:

If we consider that all fractures have the same size *l*_{f}, then:

If fracture size distribution follows a power-law size distribution, the percolation parameter lacunarity becomes:

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 *L*=400, and other
fracture parameters (orientations, size…) are picked up in the
corresponding distribution independently of each other. The fracture shapes
are disks, and the diameters *l* are either constant or power-law
distributed. In order to minimize finite-size effects, fracture centers are
generated in a cubic box system of size $\left(L+{l}_{max}\right)$ with
*l*_{max} the largest fracture. We here define two fracture network sets
with constant size: (1) ${l}_{f}=\mathrm{20}\ll L$ and (2) ${l}_{f}=L=\mathrm{400}$. We also
define four fracture network sets (3, 4, 5, 6) following power-law size
distribution with ${l}_{min}=\mathrm{1}$, ${l}_{max}=\mathrm{100}$, and where the power-law
exponent *a* is within the range [2,5]. All network
sets are simulated for different fracture intensities ${p}_{\mathrm{32}}\in \mathit{\{}\mathrm{0.1},\mathrm{0.2},\mathrm{0.3},\mathrm{0.4},\mathrm{0.5}\mathit{\}}$. We test two different fracture orientation
distributions: (a) uniform, all orientations are equally represented, (b) Fisher distribution *f*(*θ*,*κ*) (Fisher, 1953) with a
mean pole *θ* defined by a strike and dip angle both of 45^{∘}, and a dispersion factor *κ*=15. For statistical analysis, we perform
50 realizations of each fracture sets. Figure 1 shows
examples of generated networks and their associated size distributions.

## 3.1 1-D measurements

We compute the *p*_{10} lacunarity curves on 81 boreholes crossing the whole
domain, equally spaced and parallel to the *z* direction, divided in
subsamples of size $s\in [\mathrm{0.02}L,L]$ with no overlap. The general *p*_{10}
lacunarity curve of the system is obtained by stacking the contribution of
each borehole. Figure 2 shows that the lacunarity
curves associated to the generated Poissonian DFN models with power-law size
distribution and mean density $\stackrel{\mathrm{\u203e}}{{p}_{\mathrm{32}}}$ follow Eq. (9) as predicted by Darcel et al. (2013) and Lu et al. (2017). When *s*∼*L*,
the lacunarity curve drops down because of finite-size effects. Lacunarity
curves are not the same for the uniform and the Fisher orientation
distributions, because the $\stackrel{\mathrm{\u203e}}{{p}_{\mathrm{10}}}$ densities are not the same.
Indeed, for the same *p*_{32} density, the *p*_{10} value measured on a
borehole depends of the relative orientation between fractures and the
borehole. For uniform orientation ${p}_{\mathrm{10}}={p}_{\mathrm{32}}/\mathrm{2}$, and for the defined
Fisher orientation *p*_{10}=0.75*p*_{32} which can be found from Wang (2005) formulae.

## 3.2 3-D measurements

To construct the experimental three-dimensional density lacunarity curves,
we divide the cubic domain of size *L* in *L*^{3}∕*s*^{3} subdomains with no
overlap at different scales *s*. On each subdomain, the three-dimensional
fracture densities defined in Sect. 2.2 (*p*_{30}, *p*_{32}, *p*) are
computed numerically in order to obtain the associated mean and standard
deviation over the whole domain at scale *s*, giving access to the
corresponding lacunarities. For the analytical analysis, we consider that
the fracture disk shape factor is ${\mathit{\beta}}_{f}=\mathit{\pi}/\mathrm{4}$, and that the
subdomain shape factor is *β*_{s}=1. Figure 3 focuses on the *p*_{32}
lacunarity of constant size fracture networks with total fracture intensity
$\stackrel{\mathrm{\u203e}}{{p}_{\mathrm{32}}}=\mathrm{0.5}$. For the case where the fracture size *l*_{f}=20, Eq. (10) shows two asymptotes for the experimental lacunarity curves for *s*≪*l*_{f} and for *s*≫*l*_{f}, respectively. When *s*∼*l*_{f}, the equations
overpredict the lacunarity by a factor of at most 3.

Figure 4 focuses on various 3-D density lacunarity
curves for networks following power-law size distributions. When the study
scale *s* is larger than the minimum fracture size *l*_{min}, the *p*_{30}
lacunarity curves (Fig. 4a) evolves as a three-dimensional binomial process
($\sim {s}^{-\mathrm{3}}$), which is coherent with the fact that fractures are
positioned randomly in space. The fracture intensity lacunarity ${\mathit{\lambda}}_{{p}_{\mathrm{32}}}$ and percolation parameter lacunarity *λ*_{p} (Fig. 4b and
c respectively), are divided in three different regimes. When we
investigate the dependence upon the scale $s\in \left[{l}_{min},{l}_{max}\right]$, their scaling clearly depends of the power-law size distribution
exponent *a*. For large *a* values (*a*≥5), the *p*_{32} lacunarity curve
scales as $\sim {s}^{-\mathrm{3}}$ because small fractures are dominant. For this range
of study scale, we can notice a slight discrepancy between the analytical
solutions and the numerical experiments. The lower *a* the larger this
discrepancy. This can be explained by the fact that for low *a* values, the
number of fractures whose size *l*∼*s* is always significant.
Nevertheless, these analytical solutions reproduce well the observed scaling
of fracture densities lacunarity. At scale $s\sim {l}_{max}$, the difference
of *p*_{32} lacunarity between networks following either a power-law size
distribution of exponent *a*=2 or *a*=5 vary up to a factor 10^{3}.
Finally, Fig. 4d analyses ${\mathit{\lambda}}_{{p}_{\mathrm{32}}}$ at fixed scale *s*=20 for
different bulk $\stackrel{\mathrm{\u203e}}{{p}_{\mathrm{32}}}$ values, highlighting that fracture density
lacunarity is inversely proportional to the total fracture density. Finally,
for any fracture density, all lacunarity curves are identical whatever the
fracture orientation distribution (uniform or Fisher), which proves the
orientation independency.

We propose here analytical solutions to quantify the density variability
associated to Poissonian fracture networks, using the dimensionless variance
parameter *λ* that we call lacunarity. Solutions are demonstrated for
any kind of fracture density of any dimension, with application to uniform
and power-law size distribution. The application to other fracture size
distribution (exponential, log-normal…) is straightforward.
These analytical solutions were validated numerically by computing the
density variability on Poissonian DFN models for different bulk density
values, fracture size and orientation distributions. We show that the
variability of three-dimensional densities is dependent on the study scale
and the fracture size distribution, but not on the orientation distribution.
Moreover, we show that for networks following power-law size distributions,
the scaling of the variability of the fracture intensity *p*_{32} is
strongly dependent of this power-law exponent. This suggest that the
variability of *p*_{32} density cannot be estimated from borehole *p*_{10}
measurements, as it is done to quantify the mean *p*_{32} density over the
whole network using simple stereological rules. These solutions, which are
developed for purely stochastic networks, can thus be used as a reference to
estimate three-dimensional fracture density variability on the field, which
is out of reach of our investigation technics. Further work is ongoing to
quantify the fracture density variability for networks showing fractal
correlations (Darcel et al., 2003b), which may also be quantified
from the field. We expect the clustering effect associated to such networks
to increase significantly the variability of fracture densities and its
scaling.

The Python script used for generation and analysis of Discrete Fracture Networks in this paper can be found at https://github.com/elavoine/DFNDensityVariability (last access: 30 May 2019).

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.

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