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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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