Rock Mechanics and Rock Engineering (2020) 53:739–753

https://doi.org/10.1007/s00603-019-01942-1

ORIGINAL PAPER

Brittleness Index: From Conventional to Hydraulic Fracturing Energy

Model
Runhua Feng1   · Yihuai Zhang2 · Ali Rezagholilou1 · Hamid Roshan3 · Mohammad Sarmadivaleh1

Received: 30 June 2018 / Accepted: 1 August 2019 / Published online: 20 August 2019
© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Abstract
The success of a hydraulic fracturing (HF) operation is strongly dependent on brittleness of the formation. Several models
based on mechanical testing, mineral composition, and sonic log data have been proposed to quantify the brittleness of
formations known as brittleness index (BI). The limitations of these conventional BI models, in particular, the consistency
and applicability at field scale with the complex in situ conditions are poorly understood. We therefore developed a novel BI
model based on the hydraulic fracture propagation energy (HFPE) criterion. A set of hydraulic fracturing (HF) experiments
was conducted on samples with different mineralogy at true tri-axial stress conditions for model definition, i.e., the scaling
law was employed to ensure that hydraulic fracture propagation resembles the field conditions. To assess the performance
of the proposed BI model in predicting the rock fracability, the obtained results of the model on different samples were
compared with conventional models. It was shown that the predictions of the conventional BI models are predominantly
related to the failure characteristics of the rock rather than its fracability. We showed that such BI models cannot assess the
hydraulic fracturing feasibility where complex failure mechanisms (i.e., splitting, shear, friction, etc.), geological condition
(i.e., in situ stress) and operational factors (i.e., injection rate, fluid viscosity, etc.) are involved. The new proposed BI model,
however, successfully predicted the fracability of different rock types.

Keywords  Brittleness index model · Hydraulic fracturing · Hydraulic fracture propagation energy (HFPE) · Fracability ·
Failure mechanism
Abbreviations BHP Bottom hole pressure
BI Brittleness index HFPE Hydraulic fracture propagation energy
BIs Brittleness indices
List of Symbols
HF Hydraulic fracturing
E Young’s modulus
TTSC True tri-axial stress cell
v Poisson’s ratio
UCS Uniaxial compressive strength
Ф Internal friction angle
TCS Tri-axial compressive strength
σT Tensile strength
BTT Brazilian tensile test
KIC Fracture toughness
SCB Semi-circular bend
εelastic Elastic strain
S1–4 Type of sample (1–4)
εtotal Total strain
σv Vertical stress
σH Maximum horizontal stress
σh Minimum horizontal stress
* Runhua Feng
runhua.feng@student.curtin.edu.au ∅ Porosity
μ Viscosity
1
School of WASM: Minerals, Energy and Chemical T Temperature
Engineering, Curtin University, 26 Dick Perry Ave, Us Elastic potential energy
Kensington, WA 6151, Australia
Uy Surface energy
2
The Lyell Centre, Heriot-Watt University, Up Plastic deformation energy
Edinburgh EH14 4AS, Scotland, UK
EI Injection energy
3
School of Minerals and Energy Resources Engineering, ΔW Change of elastic potential energy
UNSW Australia, Kensington, Sydney 2052, Australia

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740 R. Feng et al.

EHF The energy required for hydraulic fracture Gale 2009; Jin et al. 2015). For example, Bishop (1967)
propagation introduced a BI model based on the peak (Tf) and residual
EG The energy required to create new fracture strength (Tr) of the rock obtained from the tri-axial stress
faces loading. Hajiabdolmajid et al. (2003) later suggested that
Ed Aseismic deformation energy according to the developed plastic strain must be considered in the model
fracture opening development and proposed a new strain-based BI model. It
Ed (Ductile) Aseismic deformation energy according to is, however, documented that both the strength-dependent
fracture opening in ideally ductile rock and strain-dependent BI models have significant uncertain-
Ed (Semi) Aseismic deformation energy according to ties and limitations. For instance, the rocks with the identical
fracture opening in semi-brittle rock strength can exhibit different displacements along the same
I Additional energy loss stress path (Hajiabdolmajid et al. 2003; Zhang et al. 2016a).
Iv Viscous dissipation energy Hucka and Das (1974) defined their BI model as the ratio of
Il Fluid leak-off energy elastic deformation energy to the total deformation energy
EAES Seismic dissipated energy obtained from the stress–strain response of the rock. They
Ef Dissipated energy caused by friction of micro- suggested that combining the uniaxial compressive strength
shear plane (UCS) with tensile strength (σT), and the internal friction
EH Dissipated energy caused by new surface crea- angle (Ф) can quantify the BI more accurately (Table 1).
tion of microshear failure Despite this effort, there is no consistency amongst these
ER Dissipated energy through ultrasonic waves formulations leading to rather questionable practical applica-
Q(t) Fluid injection rate tions of these models.
Pb Breakdown pressure The more practical approach based on petrophysical
Pe BHP pressure when hydraulic fracture reaches analysis was introduced by Rickman et al. (2008). In this
the boundary approach, BI was obtained using dynamic Young’s modu-
P(t) Variable BHP with time during fracture lus (E) and Poisson’s ratio (v) extracted from the acoustic
propagation log data. The method has been particularly attractive, as
tb Time of breakdown no core sample is required. Field implementation of the
tf Time at end of hydraulic fracture propagation model, however, indicated that BI of quartz-rich shale
(candidate) and limestone (caprock) formations cannot
be differentiated using this approach (Perez Altamar
1 Introduction and Marfurt 2014). Altindag and Guney (2010) later
proposed a BI model by correlating the specific energy
Hydraulic fracturing (HF) has been widely used in many (corresponding to UCS and σ T ) and brittleness. This
engineering applications, e.g., reservoir stimulation (Feng BI model has been widely applied to drilling and tun-
et al. 2019; Zeng et al. 2019; Lu et al. 2019; Holt et al. neling applications; however, its application to hydraulic
2015), geothermal extraction (Legarth et al. 2005), and geo- fracturing operation is yet to be assessed. Tarasov and
sequestration (Zhang et al. 2016b; Al-Khdheeawi et al. 2017; Potvin (2013) defined the brittleness of the rock based
Middleton et al. 2015; Lebedev et al. 2017). Brittleness is on the post-peak energy release (correlating the load-
the critical mechanical criterion that identifies the fracabil- ing and unloading elastic modulus) in tri-axial loading.
ity of a formation (Tarasov and Potvin 2013). Despite the This model considered both the pre- and post-failure pro-
success of the hydraulic fracturing in increasing the reser- cesses to describe the deformation range from brittle to
voirs’ productivity (Rickman et al. 2008; Yao 2012; Zhang ductile regime. However, it is hard to obtain the precise
et al. 2016a), the formation fracability at relatively higher stress–strain response for the model during the post-peak
level of stresses is poorly understood. This is mainly due to stage (Zhang et al. 2016a). Jin et al. (2015) proposed a
the complexity of formation fractibility at higher ductility modified mineralogical approach and reported a good
imposed by higher in situ stresses. Therefore, new BI models agreement of the model with the experimental data. The
are required to address this complexity (Medlin and Masse extrapolation of such non-physical correlation beyond its
1986; Papanastasiou 1997; Jiang et al. 2017). measurement range can be, however, erroneous. In addi-
The conventional brittleness index (BI) models have been tion, mineralogy alone is very unlikely to give a good
mainly developed based on (a) rock mechanical responses estimate of the BI, since the other factors such as grain
(Bishop 1967; Hucka and Das 1974; Hajiabdolmajid et al. size and skeleton cementation have profound effect on
2003; Altindag and Guney 2010; Tarasov and Potvin 2013), brittleness (Luan et al. 2014). In a current effort, Papa-
(b) rock acoustic responses (Rickman et al. 2008) and (c) nastasiou and Atkinson (2015) presented a new definition
mineralogical composition (Jarvie et al. 2007; Wang and of BI for hydraulic fracturing based on dislocation theory,

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Brittleness Index: From Conventional to Hydraulic Fracturing Energy Model 741

Table 1  Lists of brittleness indices investigated in this study

BI models Description Test method References

BI1 =
𝜀el
εel—recoverable strain UCS test Hucka and Das (1974)
𝜀total
εtotal—total strain
BI2 =
Wre Wre—recoverable energy As above Hucka and Das (1974)
Wtotal
Wtotal—total energy
(0.8−�)−1 Edyn—dynamic Young’s modulus Acoustic measurement Rickman et al. (2008)
(E vdyn −0.4
)
1 dyn
BI3 = +
2 8−1 0.15−0.4 νdyn—dynamic Poisson’s ratio
∅—porosity
BI4 =
UCS−𝜎T σc—unconfined compressive strength Brazilian and UCS test Hucka and Das (1974)
UCS+𝜎T
σT—maximum tensile strength
As above As above Altindag and Guney (2010)
√
UCS×𝜎T
BI5 = 2

BI6 = sin(𝛷) Ф—internal friction angle TCS test Hucka and Das (1974)
BI7 = 1 −
(𝜎1 −𝜎3 ) σ1—maximum principle stress TCS test Papanastasiou and Atkinson (2015)
2c cos 𝛷−(𝜎1 +𝜎3 ) sin 𝛷
σ3—minimum principle stress
Ф—internal friction angle
c—cohesion
BI8 = 1 −
Ed Ed—deformation energy Hydraulic fracturing test Defined in this paper
Ed +EG
EG—surface energy

which combines the rock strength with in situ stresses. 2 Methodology

This model is based on Mohr–Coulomb criterion and was
derived from analytical model of hydraulic fracturing in 2.1 Sample Preparation
weak formations, however, the model has not been vali-
dated experimentally. Due to destructive nature of mechanical testing and the fact
Despite the use of conventional BI models to predict that test repeatability is crucial for the comparison purposes
the fracability of the formations, their applicability for between different models and to eliminate the variabilities
actual hydraulic fracturing assessment is unknown. The from natural samples, synthetic specimens were prepared
complex failure mechanisms during hydraulic fracturing and used (Guo et al. 1993; Feng et al. 2018a). Synthetic rocks
(i.e., tension, shear, and microcracking), geological con- have been long used to calibrate and validate rock constitutive
dition (i.e., in situ stress, faulting), and operational vari- models (Heath 1965; Guo et al. 1993; Holt et al. 1993; Luan
ability (i.e., injection rate, fluid viscosity, etc.) all imply et al. 2016; Sarmadivaleh and Rasouli 2015; Wygal 1963;
that the brittleness should be defined based on the frac- Younessi et al. 2012). In a recent study, Luan et al. (2016)
turing fluid as well as the mechanical properties. It has prepared synthetic shale samples by mixing, packing, and
been well documented that the propagation of hydraulic curing the raw materials including superfine quartz, calcite,
fracture at field scale is governed by the physical pro- kaolinite and organic carbon. A similar approach was used
cesses including (a) viscous fluid flow; (b) elastic and/ in this study to resemble the unconventional formation rocks
or plastic deformation; (c) fracture penetration and (d) (He et al. 2018). The specimens were made of superfine sil-
fluid leak-off through fracture faces (Papanastasiou 1997; ica sand (0.1–0.25 mm), kaolinite (~ 0.004 mm) and calcite
Detournay 2016). Thus, energy criterion for hydraulic (~ 0.004 mm), i.e., Portland cement was added to bond the
fracture propagation, which integrates the complete fail-
ure mechanisms and physical processes, is a more appro-
priate alternative way of analysis. An energy criterion for
hydraulic fracturing propagation was therefore developed
in this study and tested with hydraulic fracturing experi-
ment. Consequently, a new brittleness index model ­(BI8)
was proposed based on this energy criterion (Table 1),
i.e., the so-called HFPE criterion. The performance of
the proposed model to estimate fracability was also com-
pared with main conventional BI models, i.e., B ­ I1 to ­BI7
as shown in Table 1.
Fig. 1  Minerals of silica sand (a), kaolinite (b), and calcite (c)

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742 R. Feng et al.

particles (Fig. 1). Four types of samples with different mineral

compositions were used, see Table 2 ­(S1 is quartz-rich, S ­ 2 is
clay-rich, ­S3 is calcite-rich and ­S4 is averagely mixed). The
water sorption characteristics of kaolinite group clay minerals
have been investigated by White and Pichler (1959). Water
sorption of kaolinite was considered to ensure that the mix-
ture including cement was homogeneous and excess free water
was thus avoided (Roshan and Asef 2010). Different types
and quantities of mineral powders [i.e., cement and clay ratio
(W/C)] were tested against a set of water ratios from 1 to 2.
Hence, W/C of 1.25 was selected for the samples of S ­ 1, ­S3, and
­S4 and 1.8 for ­S2 due to having a higher amount of kaolinite.
The procedure of sample preparation is summarized in
Fig. 2, i.e., (a) mixing and screening, (b) casting, and (c)
curing. As shown in Fig. 2c, the large and small cylindrical
samples with the diameters of 15 and 3.8 cm and the height-
to-diameter ratio of 2 were, respectively, prepared. Additional
cube samples with 10 cm length and semi-circular samples
with diameter of 15 cm and the thickness of 10 cm were also
prepared and used in the experiments. The samples were taken
out of the mold after 1 day and were immediately moved to
humidifier for 2 weeks curing at the temperature of 50 °C and
Fig. 2  Sample preparation process: a screening (mixed minerals and
90% humidity (Fig. 2c). cement); b casting; and c curing

2.2 Rock Mechanical Testing

Due to ultra-low permeability of synthetic samples com-
Uniaxial compressive strength (UCS) tests (Fig. 3a) were per- posed of fine particles of cement, clay and calcite (Yuan
formed on the samples to obtain the mechanical properties et al. 2018), the porosity ( ∅ ) measurement using conven-
required for the BI models (Table 1) and scaling analysis of tional experimental techniques is difficult, thus the porosity
HF experiment (Feng et al. 2018b). The measured mechanical ( ∅ ) was alternatively estimated using an empirical equation
properties were uniaxial compressive strength (UCS), Young’s (Han et al. 1986):
modulus (E), Poisson’s ratio (ν) and overall stress–strain
5.59 − 2.18C − Vp
response (Abdolghafurian et al. 2017; Roshan et al. 2017a, b; �= , (3)
Wang et al. 2018a). In addition, compressional (Vp) and shear 6.93
wave velocity (Vs) (km/s) measured by acoustic core holder where C is the weight percentage of clay, calcite, and cement
(Fig. 4) were used to calculate the dynamic Young’s Modulus content of the sample and Vp is the measured compressive
(Edyn) and Poisson’s ratio (Fjar et al. 2008; Zhang et al. 2018): wave velocity. Edyn, vdyn and ∅ were used for ­BI3 calculation.
( ) Brazilian tensile test (BTT) is an indirect method to obtain
𝜌Vs2 3Vp2 − 4Vs2 the maximum tensile strength of the rock (σT) (He and Hay-
Edyn = ( ) , (1) atdavoudi 2018). The measured tensile strength from BTT
Vp2 − Vs2 along with the measured UCS were used as input for B ­ I4 and
­BI5 (Table 1). The tri-axial compressive strength (TCS) tests
(Fig. 3b) were also carried out to extract the internal friction
angle (Ф) and cohesion (Co) based on the Mohr–Coulomb
( )
Vp2 Vp2 − 2Vs2
vdyn = ( ) . (2)
2 Vp2 − Vs2
Table 2  Composition of four Silica (%) Kaolinite (%) Calcite (%) Cement (%) Water/cement Density (g/cm3)
types of synthetic samples used
in the study S1 (quartz-rich) 52.5 22.5 0.0 25 1.25 1.73
S2 (clay-rich) 22.5 52.5 0.0 25 1.80 1.46
S3 (calcite-rich) 15.0 7.5 52.5 25 1.25 1.62
S4 (average mixed) 30.0 22.5 22.5 25 1.25 1.66

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Brittleness Index: From Conventional to Hydraulic Fracturing Energy Model 743

Fig. 3  Schematic of rock mechanical testing setup: a uniaxial compressive strength (UCS) test; b tri-axial compressive test (TCS); c Brazilian
tensile test (BTT) and d semi-circular bending test (SCB)

( )2 ( )3
a a a
Yk = 4.47 + 7.4 − 106 + 433.3 , (5)
D D D
where P is the applied force, a is the length of the notch,
D is the diameter and t is the thickness of the sample, Yk is
the geometrical factor estimated by third-order polynomial
in Eq. (5). In the SCB test, the diameter (D) of the sample
was 15 cm, length of the notch (a) is 1.9 cm, the distance
between two supporting rod (2S) is 12 cm and thickness is
10 cm.
Fig. 4  Schematic of the acoustic core holder (Nabipour 2013)
2.3 Hydraulic Fracturing Test

failure criterion (Roshan et al. 2017a, b), i.e., required by B

­ I6 2.3.1 Scaling Analysis
model. In addition, the semi-circular bending test (SCB) was
conducted to obtain the fracture toughness (KIC) (Chong et al. The propagation of a hydraulic fracture is dominated by two
1987) which was needed for the scaling law (Detournay 2004) competing dissipative mechanisms associated with viscos-
of the hydraulic fracturing test. The equations of KIC proposed ity of the fluid (µ) and material toughness (KIC), and two
by Chang et al. (2002) were thus used: storage mechanisms related to the fluid storage within the
fracture and the leak-off into the permeable solid (Detournay
2016). It is noteworthy that the fracture propagation at field
√
P 𝜋a
KIC = Yk , (4)
Dt

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744 R. Feng et al.

scale is toughness dominated at the early stage of propaga- is the crack length. Equation (6) emphasizes that for a crack
tion but rapidly turns into viscous-dominated process (Mack propagation, the minimum released strain energy rate ( dcs  )
dU

and Warpinski 2000) before it goes back to toughness domi- should be above or equal to the crack resistance (right-hand
nated again in the last stage of its extension (Detournay et al. side of Eq. 6).
2007). To simulate a valid field-scale hydraulic fracturing On the other hand, the energy criterion of hydraulic frac-
test with bench-scale experiment, the scaling law is thus ture propagation can be expressed as (Goodfellow et al.
applied to ensure that the hydraulic fracture propagates with- 2015):
out being influenced by the boundaries (Detournay 2016).
The dimensionless form of physical parameters defined by
EHF EAES
⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ (7)
scaling law can ensure that the laboratory-scale experiment EI + ΔW = EG + Ed + I + Ef + EH + ER ,
reproduces similar results to those of field scale (De Pater
et al. 1994; Detournay 2004; Detournay et al. 2007; Sar- where EI is the fluid injection energy and ∆W is the change
madivaleh and Rasouli 2015). Based on previous experi- in elastic potential energy. On the right-hand side of Eq. (7),
mental works on fracture propagation (Sarmadivaleh and the associated output energy can be divided into two main
Rasouli 2015), we used a relatively high viscous Newtonian parts: hydraulic fracture components (EHF) and seismic dis-
fluid (i.e., honey) with the injection rate of 0.2 cc/min for sipated energy (EAES). EHF consists of EG (the energy needed
10 cm samples (with known mechanical properties). This in to create new fracture faces), Ed (aseismic deformation
turn ensures that the dimensionless toughness number stays energy according to fracture opening/dilation), and I (any
below 1 to have only viscous-dominated fracture propaga- additional energy loss). The I itself includes the viscous dis-
tion. The temperature and moisture content play a major role sipation energy (Iv) and fluid leak-off loss (Il). It is noted that
in the rheology of honey (Abu-Jdayil et al. 2002; Bhandari majority of energy is consumed by EG, Ed and I processes
et al. 1999). Therefore, the viscosity of honey within the during fracturing (Shlyapobersky 1985; Zhao et al. 2018).
temperature range of 22–32 °C was measured and presented Furthermore, the radiated seismic energy, EAES accounts
in Fig. 5. for friction of microshear plane (Ef), new surface creation
of microshear failure (EH) and energy radiated as ultra-
2.3.2 Energy Criterion sonic waves (ER). This radiated seismic energy (EAES) dur-
ing hydraulic fracture propagation is ≪ 1% of the injection
Griffith’s energy criterion (Griffith and Eng 1921) derived energy based on both field observations (Boroumand and
based on linear elastic fracture mechanics (LEFM) has been Eaton 2012; Maxwell et al. 2008; Warpinski et al. 2012)
widely applied to simulate the fracture propagation in brit- and laboratory measurements (Goodfellow et al. 2015). It is
tle materials. Orowan (1954) and Irwin (1957) modified therefore intuitive to assume EAES ≈ 0.
the energy criteria by considering the irreversible energy In addition, the change in elastic potential energy ∆W
mechanism: is negligible in the formulation of energy budget due to a)
insignificant seismic dissipated energy (Goodfellow et al.
≥
dUs dUy dUp
+ , (6) 2015) and b) no (or negligible) normal stress acting on the
dc dc dc fracture plane (change in the elastic potential energy per-
where Us is the elastic potential energy (or strain energy), Uy pendicular to fracture plane, EFGH area in Fig. 6). The fluid
is the surface energy, Up is plastic deformation energy and c leak-off is also neglected due to ultra-low permeability of the
samples (i.e., below 0.01 mD).
With above assumptions, Eq. (7) can be simplified:

20
EI = EG + Ed + Iv . (8)
Viscosity of Honey (Pa.s)

It is known that the increase in rock deformation with

16
µ = 552,492×T-3.438 ductile behavior increases the energy required for fracture
R² = 0.9963
12 propagation (Fischer-Cripps 2007; Wang 2015; Yao 2012).
Considering the above statement, we examine the two
8
extreme ends of Eq. (8) where the rock is ideally brittle or
4 ductile. It is seen from Fig. 6 that Ed dominates the energy
consumption in fracture propagation for an ideally ductile
0
20 22 24 26 28 30 32 sample (i.e., BI = 0, A–D in Fig. 6), which generates an
Temperature of Honey (Celsius) exceptionally short wide fracture (Yao 2012). On the other
hand, Ed approaches zero for an ideal brittle case (i.e., BI = 1,
Fig. 5  Measurement of honey viscosity versus temperature A–B–C in Fig. 6) which produces a long narrow fracture

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Brittleness Index: From Conventional to Hydraulic Fracturing Energy Model 745

Fig. 6  Schematic of typical Typical laboratory hydraulic fracture

pressure curve during hydraulic
Brile fracture propagaon A D
fracturing test
Ducle fracture propagaon

CDEF: maximum viscous dissipaon (Iv)

energy perpendicular to fracture plane

EFGH: change in elasc potenal

Injecon rate
Pressure

B C
Normal stress on fracture plane plus the viscous dissipaon

E F
Normal stress on fracture plane
G H
Closure Pressure
Time When fracture reaches to
the specimen boundaries

(Yao 2012). Therefore, we formulate the new brittleness t1

∫t0 (11)
index based on these extreme cases, where BI is related to EI = Q P(t) dt.
 . The maximum value for Iv can be measured when
Ed
E +E
d G
fracture reaches the boundaries of the specimen (EFCB area As plastic deformation increases the effective fracture
in Fig. 6) and can be subtracted from the energy balance toughness, less fracture surface area is created at the same level
shown in Eq. (8) to define the new BI model. With this anal- of injection energy thus causing higher net pressure (Papa-
ogy, the new BI model can be expressed: nastasiou 1999). Yao (2012) performed a numerical analysis
Ed based on cohesive fracture mechanics for brittle and ductile
BI8 = 1 − . (9) rock. The results showed that (a) in an ideally brittle rock,
Ed + EG
the BHP dramatically decreases once it reaches the critical
Quantifying the aseismic deformation energy (Ed) is tech- value of formation breakdown and (b) for the ideally ductile
nically difficult and the results can be highly uncertain, i.e., rock, BHP increases gradually after fracture initiation and a
Ed was estimated to be 18–94% of the total injection energy large plastic strain is developed. This phenomenon was also
(EI) in the laboratory exercises and 15–80% in the field exer- observed in the field operations (Jiang et al. 2017).
cises (Boroumand and Eaton 2012; Goodfellow et al. 2015; Thus for an ideally ductile rock, the injection energy was
Maxwell et al. 2008; Warpinski et al. 2012). To mitigate mainly used to produce the rock deformation during the frac-
these uncertainties, the overall energy during hydraulic frac- ture propagation—from formation breakdown (tb) to the end
turing (HF) was therefore investigated. The overall energy time of hydraulic fracture propagation (tf). This means that
is directly related to the injection energy (EI) that can be additional injection energy (in addition to the energy stored
represented by the integration of hydraulic horsepower over in fracture until breakdown) is required to propagate the
the time interval of fracturing period (t0–t1) (Goodfellow fracture (in this case, fracture propagation pressure is equal
et al. 2015): to the maximum pressure, see Fig. 6). Consequently, for the
hydraulic fracture propagation in ideally ductile rock, Ed can
t1
be approximated as:
∫t0 (10)
EI = P(t) Q(t) dt,
tf

∫tb (12)
Ed(Ductile) = Q (Pb − Pe ) dt,
where P(t) is the bottom hole pressure (BHP) during the test
and Q(t) is the fluid injection rate which is constant during
fracturing operation. Thus, Eq. (10) can be rewritten as:

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746 R. Feng et al.

where Pb is the breakdown pressure, Pe represents the 2.3.3 Hydraulic Fracturing Experimental Setup
wellbore pressure when fracture reaches the boundary, tb
is the time at formation breakdown and tf is the time when The schematic of 10 cm cubic sample used for HF experi-
hydraulic fracture reaches its boundary. Zero is a reason- ment is shown in Fig. 7, i.e., the length of open hole used
able approximation for Ed in ideally brittle rock (BHP dra- for injection was almost one-third of the sample’s length.
matically decreases to fracture propagation pressure once it The schematic of the setup used for hydraulic fracturing
reaches the formation breakdown, see Fig. 6). Therefore, for experiment is additionally shown in Fig. 8. The main com-
hydraulic fracture propagation in semi-brittle rock, Ed can ponents of the system are: (a) pumping system to supply the
be approximated as: pressure along with accumulator for fracturing fluid injec-
tion, (b) fracturing system to apply the stresses (σv, σH and
σh) on specimen (Liu et al. 2019a, b) through TTSC swing
tf

∫tb (13)
Ed(Semi) = Q (P(t) − Pe ) dt.
(Feng and Sarmadivaleh 2019; Wang et al. 2019) and (c)
data acquisition system to record the pressures and control
Using Eqs. (12) and (13) with an analogy to Eq. (9), the pumps, i.e., extra hand pumps and oil tank were used for
the BI can be represented as: pressurization/depressurization and filling up/refilling the
system.
Ed(Semi) To perform the hydraulic fracturing test, the left/right side
BI8 = 1 − (14) of the accumulator was filled with fracturing fluid (honey),
Ed(Ductile)
respectively. The pressure applied by oil was transmitted to
which is schematically represented by areas shown in Fig. 6: honey by the piston inside the accumulator during injection
process. The needle valve (Vi) precisely controlled the flow
SABC
BI8 = 1 − (15) rate to balance the upstream and downstream pressures. This
SABCD
provided a choking effect when fracture was initiated and
Equation (15) should be therefore a good estimation of therefore restricted rapid fracture growth (Bunger 2005).
Eq. (9) and was thus used to calculate the BI in this study. The tests were conducted with 8 MPa stress in perpendic-
ular-to-wellbore direction (σv and σH) and zero in wellbore
path (σh) to avoid any premature rock failure during loading
(Wang et al. 2018b) (Fig. 7). The pressures of the injection

Fig. 7  Schematic of 10 cm cubic sample used for hydraulic fracturing tests

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Brittleness Index: From Conventional to Hydraulic Fracturing Energy Model 747

Fig. 8  Schematic of hydraulic fracturing setup. a Pumping system; b fracturing system; and c data acquisition system. PT pressure transducer,
PG pressure gauge, V valve

side (upstream) and the wellbore side (downstream) were 3.2 Hydraulic Fracturing Experiment
recorded by pressure transducers (Keller X30) and saved in
Control Centre Series 30 software (CCS 30). As shown in Fig. 12, the morphology of hydraulic fractures
for all testing samples is radial (penny shape), which satisfies
the assumption of scaling analysis proposed by Detournay
3 Results and Discussion (2016). The fracture propagation is transverse (perpendicu-
lar to the wellbore and minimum horizontal stress) and is
3.1 Rock Mechanical Testing consistent for all samples. The pressure diagnostics from
injection and wellbore sides during hydraulic fracturing test
As abovementioned, four different types of samples were are further shown and interpreted in Fig. 13, i.e., both pres-
tested namely S ­ 1 (quartz-rich sample), S
­ 2 (clay-rich sam- sure and temperature from injection and wellbore sides were
ple), ­S3 (calcite-rich sample) and ­S4 (mixed average). The collected with an interval of 1 s. The data were smoothed by
failure patterns of samples after four mechanical tests (a— the moving average method to improve the signal-to-noise
UCS, b—TCS, c—BTT and d—SCB) are presented in ratio (Sarmadivaleh and Rasouli 2015). It is noted that the
Fig. 9. Shear failure is observed in all samples including an pressure data obtained from wellbore side are more reliable,
additional natural sample (NS used as a reference) (Fig. 9). since the micro-scale valve near the injection side caused a
The results of BTT and SCB are also shown in Fig. 9c, d slight delay in the acquisition. Finally, the wellbore pres-
indicating developed cracks along the vertical line of the sure for fracture initiation (Pi), formation breakdown (Pb)
samples, caused by tension as expected. The stress–strain and end of propagation pressure (Pe) are obtained (Fig. 13).
curve of ­S1 from UCS test is additionally shown in Fig. 10, As mentioned previously, the deformation energy can be
i.e., the elastic strain and polynomial function fitting the interpreted by pressure response corresponding to HF test
axial stress–strain curve are shown on the graph, where the (Fig. 14), i.e., the values of ­BI8 for quartz-rich (­ S1), clay-rich
length of DE and DF represent the elastic and total strain. ­(S2), calcite-rich (­ S3) and mixed average (­ S4) are 0.73, 0.4,
The recoverable and total energy can be represented as the 0.36 and 0.54, respectively.
area under BCF and ACF, respectively. The internal friction
angle and cohesion are also obtained based on Mohr–Cou- 3.3 Brittleness Indices (BIs)
lomb criterion (Fig. 11). The obtained mechanical properties
are additionally listed in Table 3, which are later used in BI The results of brittleness index models (­ BI1–8) are listed in
calculation and scaling law. Table 4. It is seen from this table that the magnitude of all BI

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748 R. Feng et al.

Fig. 9  Failure patterns of the samples after a UCS; b TCS; c BTT and d SCB tests

elastic to plastic transition based on the stress–strain data

thus inducing a subjective estimation (Fig. 10). Moreover,
the values of UCS of rocks are usually one order of magni-
tude greater than their tensile strength (σT). Considering the
­BI4 model, the values of all four types of samples ­(S1–4) are
very close to each other but deviates from other BI models
(Fig. 15). Based on the first-order assessment with above
discussion, the models of B ­ I1, ­BI2, and B
­ I4 are not recom-
mended for brittleness evaluation for practical applications
at this stage.
From models of ­BI3 and ­BI6, quartz-rich sample ­(S1)
Fig. 10  Stress vs strain curve obtained from UCS testing on sample
exhibited the highest brittleness, followed by mixed average
­(S1)
­(S4), calcite-rich (­ S3), and the lowest brittleness for clay-rich
­(S2). However, the brittleness of ­S4 is larger than that of S ­1
based on B ­ I5 model. The contradiction can be explained by
the parameters representing the failure mechanism in differ-
ent BI models. ­BI3 was derived based on dynamic Young’s
modulus (E) and Poisson’s ratio (v) to have direct field appli-
cations (Rickman et al. 2008). However, it is evident that
distinct shale interval may exhibit analog E or ν due to the
variety of mineral composition, but significantly different
failure mechanisms. In addition, ­BI5 was defined based on
uniaxial compressive strength (UCS) and tensile strength
(σT) and ­BI6 was developed based on the internal friction
Fig. 11  Fitted Mohr–Coulomb criterion to the TCS data of sample angle obtained from tri-axial loading. These tests induce
­(S1) different failure modes and damage mechanisms can be
different.
models vary from 0 to 1 except the values of ­BI5 which are Holt et al. (2015) demonstrated that the fracturing pro-
significantly higher than 1. The B
­ I5 model therefore requires cess can be dominated by either tensile or shear failure due
substantial modification to give within range results. Fur- to the complexity of geological conditions. The magnitude
thermore, it is noteworthy that the magnitude of B ­ I1 and of ­BI7 is quite low (0.1–0.22) compared with that of other
­BI2 is highly dependent on the selected inflection point for models. This significant difference can be explained by

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Brittleness Index: From Conventional to Hydraulic Fracturing Energy Model 749

Table 3  Mechanical properties of the specimens and the measurement methods

Mechanical properties Quartz-rich ­(S1) Clay-rich ­(S2) Calcite-rich ­(S3) Mixed average (­ S4) Test method

UCS, psi (MPa) 2248 ± 200 (15.5) 1407 ± 120 (9.7) 1740 ± 140 (12) 2102 ± 200 (14.5) UCS
v 0.11 ± 0.02 0.27 ± 0.03 0.14 ± 0.01 0.18 ± 0.02 As above
E, psi (Gpa) 1.67 × 106 ± 4 × 105 (11.5) 9.7 × 106 ± 8 × 104 (6.7) 1.2 × 106 ± 3 × 105 (8.5) 1.51 × 106 ± 1 × 105 (10.4) As above
Ф (°) 44.4 30.8 38.6 43.7 TCS
Co, psi (MPa) 471 (3.25) 404 (2.79) 374 (2.58) 445 (3.07) As above
Vp, km/s 2667 1826 2253 2035 Acoustic
Vs, km/s 1740 1130 1283 1300 As above
∅ 0.22 0.21 0.11 0.21 As above
σT, psi (MPa) 161 ± 10 (1.11) 95.7 ± 5 (0.66) 159 ± 12 (1.1) 200 ± 15 (1.38) BTS
KIC, psi√in (Mpa√m) 293 ± 15 (0.32) 178 ± 12 (0.20) 223 ± 10 (0.25) 192 ± 20 (0.21) SCB

Fig. 12  Fractured samples corresponding to hydraulic fracturing test: a ­S1 (quartz-rich), b ­S2 (clay-rich), c ­S3 (calcite-rich) and d ­S4 (average
mixed)

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750 R. Feng et al.

Fig. 13  Variation of wellbore 2000 14 0.07 9.8

Injection Side
pressure and injection history Pi
for sample (­ S4) from hydraulic 12 Wellbore side Pb 0.05

Pressurization rate (MPa/s)

Pressurization rate (psi/s)

fracturing test 1600 Pressurization rate(Injection)
10 0.03 4.8

Pressure (MPa)
Pressurization rate(Wellbore)

Pressure (psi)
1200 8 0.01
-0.2
6 -0.01
800
4 Pe -0.03
-5.2
400
2 -0.05
Constant injection rate=0.2 mL/min
0 0 -0.07 -10.2
0 1000 2000 3000 4000 5000
Time(s)

Fig. 14  Deformation energy (green shadow) corresponding to the hydraulic fracturing tests. a Quartz-rich (­S1), b clay-rich (­S2), c calcite-rich
­(S3) and d mixed average ­(S4). Note that the time of initial calibration for injection and wellbore side are different for every case

the formulation of B ­ I7. This model is sensitive to a high discrepancies between the results of B ­ I8 with BIs in cases
level of stresses or in other words the high level of ductility of ­S2 and ­S3. The presence of calcite mineral in the rock is
in relatively weak rocks (refer to UCS and σT in Table 3) often assumed to increase the brittle behavior of the rock and
which should lead to superficially lower BI prediction. This therefore assists increasing the stimulated reservoir volume
is also consistent with the simulation work accomplished by (SRV) during hydraulic fracturing treatment (Jin et al. 2014;
Papanastasiou (1999) and Papanastasiou et al. (2016). The Wang and Gale 2009). However, the tri-axial testing results
applicability of ­BI7, however, needs further investigation for showed that the plastic deformation of calcite crystals can
hard formations. be considerable at evaluated pressure (even at room tem-
The results obtained from B ­ I8 indicate that quartz-rich perature) (Evans et al. 1990; Wong and Baud 2012). Our
sample ­(S1) exhibited the highest brittleness, followed by results from ­BI8 model also show a consistent trend later
mixed average ­(S4), clay-rich ­(S2), and finally the lowest where calcite-rich shale ­(S3) exhibits a high level of ductil-
brittleness for calcite-rich (­ S3). This in turn indicates the ity which should potentially hinder the hydraulic fracture

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Brittleness Index: From Conventional to Hydraulic Fracturing Energy Model 751

Table 4  Brittleness index
results

Fig. 16  Brittleness indices ­
(BI3, ­BI7 and ­
BI8) versus investigated
samples
Fig. 15  Brittleness index obtained from different models

Therefore, commonly used BI analytical models were

propagation. This is also in good agreement with predictions used to estimate the fractibility of synthetic samples of sil-
of ­BI3 and B
­ I7 models (Fig. 15). It is also noted that the ica-rich ­(S1), clay-rich ­(S2), calcite-rich ­(S3) and mixed aver-
­BI3 and ­BI7 have consistent trends (Fig. 16). Quantitatively age samples (­ S4). The bench-scale HF tests were then carried
the BI is overestimated by B­ I3 model but underestimated by out under true tri-axial stress condition (TTSC) on the cubic
­BI7 model when compared with the proposed model ­(BI8) samples of 10 cm. A novel BI model was later proposed
(Fig. 16). based on the hydraulic fracture propagation energy (HFPE).
By comparing the results from proposed and conventional
BI models, it was revealed that the conventional models per-
4 Conclusion form poorly in quantifying rock fracability. We showed that
the novel ­BI8 model can successfully predict the rock fract-
Despite the rapid development of brittleness index (BI) mod- ibility and can be used for field applications, as it integrates
els, the performance of these models has not been compre- different failure mechanisms (i.e., tensile, shear, friction,
hensively evaluated with the experimental data (Hucka and etc.), geological conditions (i.e., in situ stress) and opera-
Das 1974; Rickman et al. 2008; Altindag and Guney 2010; tional parameters (i.e., injection rate, fluid viscosity, etc.).
Jin et al. 2014; Papanastasiou and Atkinson 2015; Zhang
et al. 2016a). While few studies investigated the role of brit-
tleness through theoretical and numerical investigations of
hydraulic fracturing process (Holt et al. 2011; Papanastasiou References
1997; Yao 2012), limited numbers of proposed BI models
have been validated by hydraulic fracturing (HF) experi- Abdolghafurian M, Feng R, Iglauer S, Gurevich B, Sarmadivaleh M
(2017) Experimental comparative investigation of dynamic and
mental data.

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752 R. Feng et al.

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