Influence of Landscape Coverage on Measuring Spatial and Length Properties of Rock Fracture Networks: Insights from Numerical Simulation
Influence of Landscape Coverage on Measuring Spatial and Length Properties of Rock Fracture Networks: Insights from Numerical Simulation WENZHUO CAO1 and QINGHUA LEI1
WENZHUO CAO 0
QINGHUA LEI 0
0 Department of Earth Science and Engineering, Imperial College London , London SW7 2AZ , UK
-Natural fractures are ubiquitous in the Earth's crust and often deeply buried in the subsurface. Due to the difficulty in accessing to their three-dimensional structures, the study of fracture network geometry is usually achieved by sampling two-dimensional (2D) exposures at the Earth's surface through outcrop mapping or aerial photograph techniques. However, the measurement results can be considerably affected by the coverage of forests and other plant species over the exposed fracture patterns. We quantitatively study such effects using numerical simulation. We consider the scenario of nominally isotropic natural fracture systems and represent them using 2D discrete fracture network models governed by fractal and length scaling parameters. The groundcover is modelled as random patches superimposing onto the 2D fracture patterns. The effects of localisation and total coverage of landscape patches are further investigated. The fractal dimension and length exponent of the covered fracture networks are measured and compared with those of the original non-covered patterns. The results show that the measured length exponent increases with the reduced localisation and increased coverage of landscape patches, which is more evident for networks dominated by very large fractures (i.e. small underlying length exponent). However, the landscape coverage seems to have a minor impact on the fractal dimension measurement. The research findings of this paper have important implications for field survey and statistical analysis of geological systems.
Landscape; natural fracture network; outcrop mapping; fractal dimension; length distribution
Fractures such as faults, joints and veins are
ubiquitous in crustal rocks. These naturally occurring
discontinuities often form
networks over a broad range of length scales, and
dominate the bulk behaviour of geological media
(Bonnet et al. 2001)
. It is, therefore, important to
understand and characterise the distribution of natural
fractures in geological formations, which is relevant
to a variety of engineering applications such as
groundwater remediation and geological disposal of
(Rutqvist and Stephansson 2003;
Lei et al. 2017)
However, natural fractures are often deeply buried
in the Earth’s crust. It is very difficult to perform an
accurate and complete measurement of the
three-dimensional (3D) structure of a natural fracture system.
The study of fracture network geometries is thus
usually achieved by sampling their two-dimensional
(2D) exposures at the Earth’s surface using various
field mapping techniques. Trace-line maps of fracture
networks at the metre scale are conventionally
mapped from analogue drawings at ground level or field
photographic surveys of exposed rock surfaces. Since
the 1940s, the aerial photography technique has been
employed to map geological structures for
photographs are instantaneous records of ground
details taken at heights ranging from tens of meters to
hundreds of kilometres using, e.g. airplanes
1940; Ray 1984)
, helicopters (Odling 1997), or
(Holland et al. 2009; Bertrand et al. 2015)
This technique allows researchers to capture
geological features at different length scales, especially
those large-scale structures
(Maerten et al. 2001;
Watkins et al. 2015)
. It can also help identify
anomalous areas that need further detailed field
mapping, and eliminate or reduce the survey tasks in
. For higher spatial
resolutions, remote controlled yet low cost facilities
et al. 2009)
could be utilised for the collection of
aerial images at low flying heights, e.g. tens of meters
above rock surfaces. The recent unmanned aerial
vehicles (UAV) combined with digital
photogrammetry are increasingly deployed to obtain aerial
photographs in short time intervals with high
(Bemis et al. 2014; Clapuyt et al. 2016;
Cawood et al. 2017)
. These photographs can have a
resolution at centimetre to millimetre scales, which
can significantly improve the acquisition of
geometric and structural information
(Vollgger and Cruden
. Thus, ground-based and aerial
photogrammetry in combination provides a multi-scale,
highresolution tool for collecting geological data
et al. 2014)
By analysing the samples of outcrop mapping,
many important geometrical properties of natural
fracture systems can be studied including their
density, spacing, spatial organisation, and length
(Odling 1992, 1997; Bonnet et al. 2001;
Bour et al. 2002; Lei and Wang 2016)
. However, the
quality of photogrammetric measurements is often
affected by groundcover effects due to the presence
of forests and other plant species over the Earth’s
(Belayneh et al. 2009; Le Garzic et al. 2011)
This may result in some biases in the measurement of
fracture network geometries, such as an exaggeration
of clustering behaviour, an underestimation of
fracture density, and superficial truncation of large
fractures (Lei and Wang 2016). Figure 1 shows a few
examples of fracture patterns reported in the
literature, in which the landscape coverage effects seem
(Ghosh and Daemen 1993; Rawnsley et al.
1998; Gillespie et al. 2001; Bisdom et al. 2017)
Landscapes are defined as the spatially
heterogeneous geographic areas composed of clusters of
interacting ecosystems, such as natural terrestrial
systems ranging from forests to grasslands
et al. 2001)
. Landscape patterns are formed as a result
of interactions between tectonic, lithologic and
climatic factors: tectonism constructs landscapes
through crustal movement such as uplift and warping;
lithology influences landscape evolution by
dominating erodibility of underlying rock or soil; climatic
parameters degrade landscapes through physical and
chemical erosion through wind and water
et al. 1987; Lifton and Chase 1992)
. In addition,
landscape spatial distribution is affected by biota and
human activities such as urbanisation, agriculture and
forestry (Krummel et al. 1987). The resulting
landscape is a mixture of natural and anthropogenic land
cover patches of different sizes and shapes
et al. 1987)
. The landscape patterns evolve over time
and vary in spatial distribution
(Turner and Rusher
, and their geometrical complexity may be
described by fractal geometry
Krummel et al. (1987)
first used a
perimeter-area method to evaluate the fractal
dimension of deciduous forest patterns in a region
which experienced extensive alteration of forest
cover to cropland. Natural landscapes generally
exhibit distinctive fractal features over a range of
scales from hundreds of metres to kilometres
and Chase 1992)
. Little is known about how fractal
landscape patterns were produced in a composite of
complicated natural and human processes
(Xu et al.
. However, the fractal dimension is considered
to be a very useful parameter for interpreting the
heterogeneous features and modelling the spatial
distribution of landscapes
(Gardner et al. 1987;
Turner et al. 1989)
The objective of this study is to investigate
whether and how landscape coverage and its spatial
distribution affect the measurement of the spatial and
length properties of natural fracture networks. We
consider the scenario of nominally isotropic natural
fracture systems, in which fractures are uniformly
oriented, whereas the cases of anisotropic fracture
networks with distinct fracture sets will be addressed
in our future research. The rest of the paper is
organised as follows. In Sect. 2, discrete fracture
networks governed by different combinations of
fractal dimension and power law length exponent
values are generated, and subsequently covered by
fractal landscape patterns constrained by various
fractal dimensions and coverage ratios. The statistical
properties of landscape-covered fracture network
patterns are then measured. In Sect. 3, the simulation
results about the effects of landscape fractal
dimension and coverage ratio on measuring statistical
properties of fracture networks are presented. Finally,
a short discussion is given and conclusions are drawn.
2.1. Statistical Model of Fracture Networks
Extensive field observations suggest that
fracturing occurs at all scales in the crust and creates
hierarchical structures that exhibit long-range
correlations from macroscale frameworks to microscale
(Allegre et al. 1982; Barton 1995)
. The spatial
organisation of natural fracture networks can be
characterised by the fractal dimension D, which
quantifies the manner whereby fractals cluster and
spread in the Euclidean space and can be measured
using the box-counting method
(La Pointe 1988;
Chile`s 1988; Ehlen 2000)
or the two-point correlation
(Hentschel and Procaccia 1983; Bour and
. The density distribution of fracture
lengths can be described by a statistical model as
(Bour et al. 2002; Davy et al. 2010)
nðl; LÞ ¼ aLDl a; l 2 ½lmin; lmax ;
where n(l, L)dl gives the number of fractures with
sizes belonging to the interval [l, l ? dl] (dl l) in
an elementary volume of characteristic size L, a is the
power law length exponent, a is the density term, and
lmin and lmax are the smallest and largest fracture sizes.
The density term a is related to the total number of
fractures in the system and varies as a function of
fracture orientations (Davy et al. 2010). The length
exponent a defines the relative proportion of large and
(Davy 1993; Pickering et al. 1995)
extent of the power law relation is bounded by an upper
limit lmax that is probably related to the thickness of the
crust, and a lower limit lmin that is constrained by a
physical length scale (e.g. grain size) or the resolution
of measurement (Bonnet et al. 2001). For numerical
simulations, the model size L usually meets
lmin L lmax
(Darcel et al. 2003)
In theory, D is restricted to the range [1, 2] for the
2D scenario. A D value of 2 represents a
homogeneous spatial distribution, i.e. ‘‘space filling’’. As
D decreases, the fracture pattern becomes more
clustered associated with more empty areas. In
theory, a is restricted to [1, ?] in 2D. A small
a value corresponds to a system dominated by large
fractures, while a ? ? relates to a pattern with all
fractures having an equal size (i.e. lmin). Extensive
measurements based on 2D trace maps reveal that
generally D varies between [1.5, 2] and a falls
between [1.3, 3.5]
(Bonnet et al. 2001)
. The D and
a values as well as their relationship may control the
connectivity, permeability and strength of fractured
(Darcel et al. 2003; de Dreuzy et al. 2004; Davy
et al. 2006)
. Thus, it is very important to accurately
measure the D and a values of natural fracture
networks, the observation of which, however, may be
affected by landscape variation.
In the following subsections, discrete fracture
networks defined by this power law model are
generated and then superimposed by random fractal
landscape patterns, with the statistical properties of
the original and covered fracture network patterns
further compared. Subscripts are used to distinguish
statistical properties of different categories, i.e. ‘‘f’’
for prescribed fracture network patterns, ‘‘l’’ for
landscape patterns, and ‘‘m’’ for measured fracture
network patterns affected by groundcovers.
2.2. Fracture Network Generation
The spatial distribution of discrete fracture
networks governed by a prescribed Df value could be
constructed through a multiplicative cascade process
(Darcel et al. 2003)
. This cascade process is a recursive
operation of fragmentation of the model domain into
subdomains of identical sizes. For a non-fractal pattern
(Df = 2), which has a uniform spatial distribution, the
locations of fracture barycentres are modelled using
the Poisson process. For a fractal spatial distribution of
fracture barycentres (Df \ 2), a set of non-uniform
probabilities of fracture occupancy Pi (i = 1,…,n) are
randomly permuted and assigned to subdomains at
each stage of the fragmentation cascade (Fig. 2a–c).
We only consider the mono-fractal scenario, and the set
of probabilities is related to the fractal dimension and
the scale ratio r of the fragmentation process as:
Pin¼1 rDf Pi2 ¼ 1. The number of fractures in each
subdomain is determined based on the total number of
fractures multiplied by the corresponding probability.
Within the minimum subdomains, fracture barycentres
are randomly distributed regardless of the fractal
dimension. An example of fracture barycentres with
Df = 1.5 generated from 6 cascades is given in Fig. 2c.
Fracture orientations are assigned isotropically.
Fracture lengths are sampled from a power law distribution
constrained by the length exponent af. Fractal fracture
networks governed by different af and Df values can
then be generated by synthesising the different
geometrical attributes modelled as independent random
In our numerical model, we choose the fracture
length bounds, i.e. lmax and lmin, to be 50 9 L and L/
50, respectively. The scale ratio r of the
fragmentation cascade is chosen to be 2, and the cascade
process is implemented for 6 iterations. A total of 800
fractures with random orientations are generated in
the domain. The chosen total number is much larger
than the suggested criterion of a minimum of 200
fractures to be sampled
(Bonnet et al. 2001)
thus considered sufficient for statistical analysis.
Different combinations of Df = 1.5, 1.75 and 2, and
af = 1.5, 2.5 and 3.5 are considered (9 scenarios in
total), and 10 DFN realisations are generated for each
scenario. Figure 3 shows one of the realisations of
each different case.
The spatial heterogeneity and evolution of
landscapes have been extensively studied in the field of
Gardner et al. (1987)
percolation theory to construct neutral models to
describe landscape spatial distribution in the absence
of landscape formation processes. The model
quantified landscape patterns based on the fraction of
landscape, the linear dimension of the map, and the
fractal dimension of landscape clusters. The model
was further applied to investigate the spread of
ecological disturbance across a landscape
et al. 1989)
Milne et al. (1989)
developed a spatially
neutral model and used Bayesian probabilities
conditional on twelve landscape variables to predict deer
Kubo et al. (1996)
latticestructured model to simulate the spatial expansion,
regeneration and closure of gaps in forested
ecosystems over time.
Wu et al. (2000)
variance analysis of landscapes using a multiple-scale
statistical model with spatial nested hierarchical
Thus, in this paper, we model the landscape
pattern as an assembly of small-sized patches, the
distribution of which is governed by the landscape
fractal dimension Dl and a coverage ratio c (i.e. ratio
of the groundcover area to the domain area). The
fractal landscape map is also constructed using the
multiplicative cascade process, which includes the
fragmentation cascade and the probability field
Df = 1.5
Df = 1.75
Df = 2
a f = 1.5
a f = 2.5
af = 3.5
calculation. A probability matrix comprising only
integers 1 and 0 should subsequently be mapped to
the minimal subdomains to indicate landscape
occupancy or not, depending on the eventual probability
field. Assuming that a total number of M minimal
subdomains are required to be covered for a given
landscape pattern, elements in the probability matrix
corresponding to the largest M values in the eventual
probability field are populated with a probability
value of 1, while others are assigned 0 (Fig. 2d). The
generated landscape patterns are then superimposed
onto the generated fracture network patterns, leading
to some of the fracture traces being either truncated
(if partially covered) or removed (if completely
covered). An example of a landscape pattern with
Dl = 1.5 generated from this process is shown in
We set the same scale ratio and the number of
cascade iterations as those for fracture network
generation, which corresponds to a landscape patch
resolution of 0.78 9 lmin. For each generated fracture
network realisation, ten landscape realisations are
generated for each given combination of the
landscape fractal dimension Dl = 1.5, 1.75 and 2, and the
coverage rate c = 0.1, 0.2, …, 0.5 (i.e. 15
combinations in total). Figure 4 shows the landscape-covered
fracture networks as a result of superimposing
different fractal landscape patterns onto an original
2.4. Measurement of Statistical Properties
The statistical properties of landscape-covered
fracture networks including the fractal dimension Dm
and am are further measured. The two-point
correlation function is used for measuring the fractal
dimension of fracture barycentres. The correlation
function C2(r) for N number of fracture barycentres is
C2ðrÞ ¼ 2NpðrÞ=N2
where Np(r) is the number of pairs of barycentres
whose distance is less than r
(Bour and Davy 1999)
The fractal dimension can be obtained through
regression fitting on the bilogarithmic graph.
Figure 5a shows the calculation of the fractal dimension
of an original fracture network and its
landscapecovered networks based on regression fitting and
local slope analysis. It can be seen that the fracture
network still follows a fractal spatial distribution after
The power law length exponent a can be derived
from the cumulative distribution or density distribution
of fracture lengths
(Davy 1993; Pickering et al. 1995)
Figure 5b shows an example of the density distribution
of lengths of a fracture network pattern before and after
landscape cover of different ratios. The density
distribution for different landscape coverage ratios
approximately follows a power law relationship, but
the fitted curve becomes steeper with increasing
landscape coverage, i.e. the length exponent decreases.
Figures 6 and 7 present the measured fractal
dimension Dm and length exponent am, respectively,
of landscape-covered fracture network patterns as a
result of superimposing different landscape patterns
with different fractal dimension Dl and coverage ratio
c onto the fracture networks with different underlying
fractal dimension Df and length exponent af. The
measured fractal dimension and length exponent of
the original fracture networks (i.e. c = 0) are also
presented for reference, and their slight deviation
from the input Df and af values is attributed to the
boundary censoring operation which deletes the parts
of some fractures extending outside the finite-sized
As shown in Fig. 6, the measured Dm of
landscape-covered fracture networks varies with the
increase of landscape coverage ratio, with the
variation trend determined by the properties of the original
fracture network and the landscape pattern. The
deviation of Dm from the underlying Df is found to be
in the range of |Dm - Df| \ 0.1–0.2, which seems
larger when Df is large. The measured Dm of fracture
networks superimposed by landscape patterns with
Dl = 1.5 and 1.75 are almost indistinguishable for
different coverage ratio conditions. The measured Dm
values for fracture network patterns covered by
nonfractal landscape (Dl = 2) are higher than those
covered by fractal ones (Dl = 1.5 or 1.75), especially
when the coverage ratio is high. There is no apparent
influence of the prescribed af on measured Dm values,
indicating that the measured spatial clustering of
landscape-covered fracture networks may be
independent of the underlying size scaling properties of
From Fig. 7, it can be seen that the measured
length exponent am is affected by the groundcover
effect, the extent of which depends on the properties
of the original fracture network and the landscape
pattern. An increased landscape coverage leads to an
increase in am, because the observed fractures are
shorter than the original ones due to the presence of
groundcover patches. It can be noticed that when
c = 0.1, the deviation of am from the underlying af is
quite small for all cases; however, when c C 0.2, the
difference becomes non-negligible and can even be
as large as 1–2 when half of the domain is covered,
i.e. c = 0.5. The measured am values for fracture
networks affected by non-fractal landscape systems
(Dl = 2) are much higher than those of the networks
c = 0.1
c = 0.3
c = 0.5
D l = 1.5
D l= 1.75
D l = 2
covered by fractal landscape patterns (i.e. Dl = 1.5 or
1.75), especially when the coverage ratio is high. This
is attributed to the fact that space-filling landscape
patches tend to trim more fractures, while fractally
distributed landscape patches tend to form clustered
coverage that creates higher lacunarity (exact
counterpart of clustering) in the system and thus allows
some larger fractures being preserved. Furthermore,
the influence of landscape coverage on am is more
significant for fracture networks having a small
underlying length exponent af (see the large
landscape-induced deviation of am from af when af = 1.5
in Fig. 7). This is because, when af is small, the
system is more dominated by large fractures, which
are more inclined to be affected by groundcover. The
influence of Df on am seems to be minor, suggesting
that the measured size scaling of landscape-covered
fracture networks may be independent of the
underlying spatial characteristics of the fracture networks.
In this paper, we explored the effects of landscape
coverage on measuring the spatial and length
properties of landscape-affected fracture networks for
scenarios of different fracture network parameters
(fractal dimension Df and length exponent af) and
landscape spatial distribution parameters (fractal
dimension Dl and coverage rate c). The results have
important implications for assessment of the
statistical parameters derived based on natural fracture
outcrops. To conduct field mapping, great efforts
were often devoted to search sites with good quality
exposures to keep them to a large extent free of
(Odling 1997; Odling et al. 1999;
Belayneh and Cosgrove 2004)
. However, in some
geological sites, it can be very difficult to avoid the
groundcover effects, because embryophytes can form
vegetation on the Earth’s surface over different
length scales, making the mapped outcrop containing
unknown gap zones
(Ghosh and Daemen 1993;
Rawnsley et al. 1998; Gillespie et al. 2001; Bisdom
et al. 2017)
. Especially for aerial photograph-based
mapping, small-scale fractures can be easily hidden
by forests (Lei and Wang 2016). We thus suggest that
when conducting field mapping, the distribution of
vegetation should also be measured and
characterised, especially when the landscape occupies more
than 10% of the sampling region. Based on the
landscape pattern parameters, the uncertainty of the
measured statistical properties from mapped fracture
networks can be further assessed and the underlying
fractal dimension and length exponent may be
recovered. Specifically, Df may be first estimated
based on Dl and c derived from field mapping. Then,
synthetic fracture networks associated with the Df
value and a range of potential af values can be
generated, which are further covered by landscape
patches conditioned with the measured Dl and c. The
actual af value may, therefore, be found if the am
value of the simulated landscape-covered fracture
pattern matches that of the field outcrop. It is worth
mentioning that multiple solutions to af may exist due
to the complex nature of this inverse problem. Thus, a
confidence interval may also need to be derived for
the estimated af value.
The results from our research suggest that
landscape spatial distribution may have minor effect on
measuring the fractal dimension, but can cause
significant biases in deriving the length exponent. Such
vegetation-induced biases may also be a factor that
contributes to the observed inconsistency of power
law scaling of fracture lengths over different length
(Nicol et al. 1996; de Joussineau and Aydin
2007; Davy et al. 2010)
. Actually we can also see
from Fig. 5b that the landscape coverage can lead to a
curvature at the small scale in the density distribution
of fracture lengths, which may be related to the
socalled ‘‘truncation effect’’
(Pickering et al. 1995;
Bonnet et al. 2001)
that is always present in the
length distribution plot of outcrop data
Odling 1997; Odling et al. 1999; Bour et al. 2002;
Davy et al. 2010; Le Garzic et al. 2011; Bertrand
et al. 2015; Lei et al. 2015; Lei and Wang 2016)
In the present work, fractures in each synthetic
network are assumed completely random in
orientation, aiming to provide a preliminary understanding
of the landscape cover effects on the measurement of
fracture length and spatial properties. However,
natural fracture networks in rock can often be highly
anisotropic consisting of multiple fracture sets each
linked to separate formation stage. Thus, analysing
the vegetation cover effects on the measurement of
anisotropic fracture networks will be a focus of our
future work. Furthermore, in the fracture network
model, we represented fracture traces as 1D lines of
the same width without taking into account the
variation of fracture apertures, which may scale with
fracture length following a power law
Scholz 1995; Renshaw and Park 1997; Bonnet et al.
2001; Olson 2003; Neuman 2008)
. Very large faults
with wide apertures/damage zone thickness tend to be
more observable and less affected by landscape
patches than small-scale fractures. It is worth
mentioning that the current model also assumed that
fracture network parameters are independent of
landscape parameters. However, the tectonism that is
related to the formation of superimposed fracture sets
can also have a profound effect on landscape
(Lifton and Chase 1992; Kinast et al. 2016)
high fractal dimension of landscape is usually found
in the region under a very low uplift rate, which
allows fully downcut by erosive streamflows, while a
low fractal dimension of vegetation often occurs in
the tectonic system with a high uplift rate due to the
limited degradation time of landscapes (Lifton and
Chase 1992). This implies that a correlation between
fracture network and landscape spatial distribution
patterns may exist, which needs further investigation
in the future.
To conclude, the influence of landscape coverage
on the measurement of statistical properties of nature
fracture networks was studied. Multiple realisations
of synthetic fracture networks governed by the
underlying fractal dimension and length exponent
were generated, which were then superimposed by
landscape maps with different coverage ratios and
clustering features. By comparing the
landscapecovered fracture network properties with their
underlying ones, we found that landscape cover has a
minor impact on the fractal dimension measurement,
while the length exponent tends to be overestimated
and a high discrepancy can occur in a fracture system
dominated by large fractures (i.e. a small underlying
length exponent), which are covered by space-filling
landscape patches with a high coverage ratio. The
research findings of this paper have important
implications for field mapping and statistical analysis
of geological systems.
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