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1 INTRODUCTION
An Enhanced Geothermal System (EGS) seeks at ex-
tracting heat from deep underground by circulating a
fluid between injection and production wells. In most
EGS projects, fluid flow occurs essentially along
fractures as the matrix permeability is small (Genter
et al. 2010). In general, the low permeability of such
reservoirs prevents production at economical rates.
The enhancement of fracture permeability to facilitate
fluid circulation is required. One approach to enhance
and create the permeability is to perform hydraulic
stimulations. In order to design and assess EGS res-
ervoir stimulation strategies, development of a geo-
logical model with representation of lithological do-
mains and characterization of the fracture network is
required. This characterization includes a correct sta-
tistical distribution of fractures and fracture set prop-
erties. Such a geological model is crucial for geome-
chanical simulations during the life-time of an EGS
reservoir, too.
The existence of fractures within the target EGS
rock mass plays a controlling role on the mechanical
behavior during reservoir stimulation. Valley and
Evans (2007) have identified more or less continuous,
local stress variations from analyzing wellbore fail-
ures along deep wells at the Soultz-sous-Forêts EGS
in France, and attributed these stress changes to the
existence of natural fractures cut by the deep bore-
holes. Similar observations were made at the Basel
EGS project in Switzerland (Valley and Evans 2009).
This paper aims at studying the distribution of frac-
ture spacing in the crystalline basement section below
Basel that was penetrated by the Basel-1 well.
Knowledge of the spacing distribution is a crucial
step towards populating a reservoir model with syn-
thetic fractures consistent with the observations ob-
tained from deep boreholes. For our analysis we used
the fracture data from Ziegler et al. (2015). Ziegler et
al. (2015) have characterized the natural fractures and
fracture zones characteristics (location, orientation,
set spacing, fracture zone thickness and internal struc-
ture) in the crystalline basement (“Deep Heat Min-
ing” project, Switzerland).The analysis was per-
formed using an acoustic televiewer log and other
logging data (e.g., bulk density and p-wave velocity).
For a complete description of the dataset and interpre-
tation methodology we refer to Ziegler et al. (2015).
Amongst the various fracture attributes such as per-
sistence, spacing, aperture etc., this study focuses on
analyzing the spacing distribution of fractures using
fractal geometry. A fractal analysis of fracture inter-
sections with the borehole is expected to improve our
understanding of the distribution of fractures in the
crystalline basement. We aim at quantifying the frac-
tal dimension of fracture sets spacing in the crystal-
line basement along the Basel-1 well using the Can-
tor's Dust method. We also studied the statistical
distribution of fracture spacing in different sets to find
any probable power-law distributions. The relation
between the power law exponents and fractal dimen-
sions is discussed, too.
Preliminary fractal analysis of fracture spacing inferred from an acoustic
televiewer log run in the Basel-1 geothermal well (Switzerland)
M. J. A. Moein
Geological Institute, ETH Zurich, Switzerland
B. Valley
Center for Hydrogeology and Geothermics, University of Neuchâtel, Switzerland
M. Ziegler
Geological Institute, ETH Zurich, Switzerland
ABSTRACT: Development of a geological model for an Enhanced Geothermal System (EGS) reservoir re-
quires a characterization of the fracture network. Previous studies have identified fractures from an acoustic
image log run in the crystalline basement section of the Basel-1 well and described fracture sets and fracture
zone characteristics. Using the Cantor’s Dust method, we investigated the fractal behavior of four major natural
fracture sets intersected by the Basel-1 well. These sets are differently distributed between 2600 m and 5000 m
(i.e., the logged depth) along the well below the rotary table of the rig. The obtained fractal dimensions (D)
vary between 0.35 and 0.60 for the four fracture sets. Similar D values ranging for sets 1–3 in the interval
between 2600 m and 3000 m emphasize the role of the geological setting in the distribution of fractures. More-
over, the spacing distribution of sets 2 and 3 show a clear power-law distribution.
2 FRACTAL BEHAVIOR OF FRACTURES
The fractal geometry is a branch of mathematics that
deals with the quantification of repeating patterns on
different scales. Fractals have been widely used in
different areas of geoscience to understand these pat-
terns. Using this geometry, it is possible to quantify
the scaling of fracture attributes. In addition, it has
been used to cluster earthquakes and quantify induced
seismicity (Smalley et al. 1987, Enescu and Ito 2001,
Hainzl 2004). Scaling and spatial clustering of frac-
tures are important parameters that control the organ-
ization of the flow within a fractured rock reservoir.
Barton and Zoback (1992) have suggested that the
self-similarity of fractures in the Earth’s crust exists
on a very wide range of scales ranging from millime-
ters to hundreds of meters. Self-similar distributions
show scale invariant characteristics. There are numer-
ous observations in the literature reporting the fractal
behavior of different fracture attributes such as trace
length, spacing, RQD (Rock Quality Designation,
Deere and Deere (1988)), aperture, surface roughness
etc. (Velde et al. 1990, Power and Tullis 1991, Barton
and Zoback 1992, Boadu and Long 1994).
3 FRACTAL ANALYSIS OF FRACTURES
INTERSECTING A BOREHOLE
The intersection of fracture planes with a borehole
centerline can be represented as points along a single
dimension. If the distribution of fracture intersections
follows a fractal behavior, the following equation will
be valid for all r:
𝑁 = 𝑐 ∙ 𝑟 −𝐷 (1)
where N is the number of divisions that cover all of
the fractures in equally spaced divisions of length r,
D is the fractal dimension, and c is a constant. The
fractal dimension D is in the range 0 < D < 1. Here,
we will apply the Cantor’s Dust method to obtain the
fractal dimension of different fracture sets intersect-
ing the borehole.
3.1 Cantor’s Dust method (Cantor 1872)
This method tries to cover the intersection of fracture
planes with the borehole by successively smaller in-
tervals in each step. If the fracture intersections in a
set follow a fractal distribution, the number of inter-
vals containing fractures follow a linear trend with
slope m on a log-log plot. The slope m of this line
equals to 1−D. A detailed description to obtain fractal
dimensions with the Cantor’s Dust method is given
by Velde et al. (1991).
3.2 Power-law statistical distributions
It is assumed that the true spacing of fractures in spe-
cific sets follows a power-law distribution. Valley
(2007) has studied different statistical parameters us-
ing different distributions for different fracture sets in
the Soultz-Sous-Forêts EGS. He has extracted power-
law exponents for different sets by fitting a line to the
linear section of plots showing log(cumulative num-
ber) vs. log(spacing) and obtained a wide range of ex-
ponents ranging between 0.63 and 1.06 for four frac-
ture sets in the GPK3 and GPK4 wells. While
lognormal distribution shows a better fit to the spac-
ing distributions derived from borehole imaging at
Soultz (both wells), the spacing distribution obtained
from core fracture logs shows a better fit to the power-
law distribution (Genter et al. 1997). This deviation is
mainly due to the bias imposed by the censoring ef-
fect. The censoring effect is due to the resolution limit
where short fractures are not completely observed
(Bonnet et al. 2001). Moreover, the spacing might be
influenced by truncation effect due to incomplete
sampling of long fractures.
4 GEOLOGICAL SETTING AND AVAILABLE
DATASETS
The crystalline section below Basel was penetrated by
the Basel-1
well between 2507 m and 5000 m (all
depths given in this article are measured along hole
from the rotary table). A
total of 1164 natural frac-
tures (certain and uncertain) were identified in the
logged interval between 2600 m and 5000 m (Ziegler
et al. 2015, Fig. 1). Out of these, 1035 fractures were
grouped into six possible sets based on their orienta-
tion. Some of the sets may be conjugate pairs. In this
study we do not take the uncertainty rating into ac-
count and studied all fractures that were grouped into
the largest sets 1–4. The number of fractures in sets 5
and 6 is too small to perform a statistical study. The
number of fractures in different sets and their mean
orientation (dip direction and dip angle) are listed in
Table 1. The distribution of four fracture sets along
the borehole is displayed in Figure 1a, while Figure
1b and 1c show the p-wave velocity and smoothed
density. It can be seen that the fracture frequency in
the first 400 m is distinctly higher than in the rest of
the borehole and p-wave velocity is less. This was
suggested to likely reflect a paleo-weathered zone
since the interface with the sediments lies at ~2507 m
(Ziegler et al. 2015).
Figure 1. a) Histograms of all fractures (1164 fractures) and set 1–4 vs. depth in 10 m bins. b) Smoothed p-wave velocity versus depth
using a moving average filter with a window length of 100 m. c) Smoothed density log using a moving average filter with a window
length of 100 m.
Table 1. Number of fractures in sets and fracture set mean ori-
entation
5 RESULTS
5.1 Cantor’s Dust method analysis
Table 2 summarizes the results of fractal dimensions
using Cantor’s Dust method (see section 3.1). Set 4
has the highest D value, while set 3 has the lowest.
Set 3 occurs in two intervals, between about 2600–
2957 m and 4000–4261 m (Fig. 1a).The fractal anal-
ysis of the different intervals shows different fractal
dimensions of 0.45 (2600–2957 m) and 0.59 (4000–
4261 m) for set 3 (Table 3). This might be due to the
slightly different geological context for these two in-
tervals: the upper interval is located within the more
fractured section of the rock mass presenting also
lower p-wave velocity (see Fig. 1b) while the lower
interval is located at a depth with smaller fracture
density and no velocity anomaly. Indeed, the first 400
m interval reflects a rock mass which is different from
deeper sections (see section 4). The fractal dimen-
sions of sets 1–3 in this interval are reported in (only
a few fractures of set 4 occur in the interval 2600–
3000 m). The similar D values suggest the same dis-
tribution for the three fracture sets in the top 400 m of
the logged section. At least, set 1 and set 2 may be a
conjugate pair conform to the Hercynian direction
(i.e., striking NW). This might reflect the same origin
of the fractures.
Table 2. Fractal dimension of sets 1–4 along the Basel-1 well
Fracture set
D
[-]
1
0.54
2
0.48
3
0.35
4
0.60
Table 3. Fractal dimension of set 3 in different intervals
Fracture set
D
[-]
1
0.54
2
0.48
3
0.35
4
0.60
Table 4. Fractal dimension of different sets in the top 400 m of
the logged section
Fracture set
D
[-]
1
0.46
2
0.44
3
0.45
5.2 Power-law distribution of spacing
The exponent of power-law distribution of fracture
spacing were determined similarly to Valley (2007)
(see section 3.2). For this analysis, we computed true
fracture spacing by considering the mean set orienta-
tion which is oblique to the borehole and correcting
the spacing measured along the borehole. Although
the spacing distributions are not entirely linear on a
log-log plot, the linearity was sufficient for sets 2 and
3 over about two orders of magnitude. We picked the
longest range of spacing with linear distribution and
determined its slope m (Figs 2 and 3). According to
the discussion in section 3.2, the observed spacing
distribution is affected by the censoring and trunca-
tion effects. For set 2 and 3, there is a good agreement
between the power-law exponents and the fractal di-
mension (actually 1–D) obtained with the Cantor’s
Dust method as can be seen in Table 5. The spacing
distribution of set 1 and set 4 do not show a power-
law behavior and no clear linear section is seen. Fig-
ure 4 shows the spacing distribution of set 1 as an ex-
ample.
Figure 2. True spacing distribution of set 2 showing a clear
power-law distribution.
Fracture
set
Number of
fractures
Mean dip
direction
[°]
Mean dip
angle
[°]
1
348
250
66
2
297
62
62
3
173
195
61
4
152
308
68
Figure 3. True spacing distribution of set 3 showing a power-law
distribution for more than 2 orders of magnitude.
Figure 4. True spacing distribution of set 1, showing no evident
power-law distribution.
Table 5. Power-law exponent of sets 2 and 3 compared with the
fractal dimension given as 1–D.
Fracture set
Power-law
exponent (|m|)
[-]
1−D
[-]
2
0.57
0.52
3
0.61
0.65
6 DISCUSSION
Based on the Cantor’s Dust method the intersections
of different fracture sets with the Basel-1 borehole re-
veal fractal dimensions varying between 0.35 and
0.60. Low fractal dimensions (closer to zero) indicate
highly clustered sets (sets 2 and 3), while higher val-
ues correspond to more evenly distributed fracture
sets (sets 1 and 4). The fractal analysis of the three
fracture sets (sets 1–3), occurring in the upper 400 m
of the logged crystalline section, shows very similar
fractal dimensions. This can be interpreted as same
clustering behavior, which might reflect an influence
of similar formation processes. At least, set 1 and set
2 may be a conjugate pair of tectonic fractures con-
form to the Hercynian direction (i.e., striking NW;
Table 1). As discussed in section 4, the uppermost
portion of the crystalline basement rocks has been ex-
humed close to the Earth’s surface before Permian,
Mesozoic and Cenozoic sedimentary rocks were de-
posited (Ziegler et al., 2015 and references therein).
The fracture frequency distribution and p-wave ve-
locities in this section are visibly different from the
deeper parts of the penetrated crystalline basement.
Different lithology, mechanical properties, rock tex-
ture and mineralogy, loading history etc., may lead to
different distributions of fractures. Different fractal
dimensions for different subintervals of the fracture
set 3 might indicate such an effect with a stronger
clustering possibly due to the overprint of near sur-
face effects for the shallower interval (Table 3). How-
ever, the variation of fractal dimension with increas-
ing depth for different sets is a topic for further
investigations.
Although the statistical distribution of fracture sets
is highly affected by censoring and truncation effects,
power-law distributions were observed in two sets
(sets 2 and 3) for about two orders of spacing magni-
tudes. We found that the exponents are very close to
1−D, with D previously obtained from Cantor’s Dust
analysis. Thus, it is possible to obtain the fractal di-
mension of a fracture set by using its statistical distri-
bution if it follows a power-law.
7 CONCLUSIONS
The fractal dimension of different sets intersecting the
Basel-1 geothermal well, utilizing the Cantor’s Dust
method, varies between 0.35 and 0.60. Low fractal di-
mensions correspond to low degree of clustering
along the borehole. The fractal dimensions of three
present sets in the first 400 m might reflect the same
origin and fracture types. At least, set 1 and set 2 may
be a conjugate pair of tectonic fractures. Different li-
thology, mechanical properties, rock texture and min-
eralogy, loading history etc., may lead to a different
distribution of fractures. Moreover, the power-law
exponent of the spacing distributions of two sets is
very close to 1−D, obtained by Cantor’s Dust analy-
sis. It shows the possibility of obtaining the fractal di-
mension of fracture sets by using their statistical dis-
tributions (if these follow power-laws).
ACKNOWLEDGEMENTS
The research leading to these results has received
funding from the European Community’s Seventh
Framework Program under grant agreement No.
608553 (Project IMAGE). A special gratitude is
given to Dr. Keith Evans for his guidance and con-
structive criticism in performing this research.
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