the study fracture evolution of coal and rock mass under

117 Copyright © Canadian Research & Development Center of Sciences and Cultures

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Advances in Petroleum Exploration and DevelopmentVol. 9, No. 2, 2015, pp. 117-120DOI:10.3968/6978

The Study Fracture Evolution of Coal and Rock Mass Under Hydraulic Fracturing

WANG Tingting[a]; ZHAO Wanchun[a],*; FENG Xiaohan[a]; YAN Yongqiang[b]

[a] Northeast Petroleum University, Daqing, China.[b] Yushulin Oilfield Development Co. Ltd, Daqing, China.*Corresponding author.

Supported by the Academic Backbone of Heilongjiang Province University Youth Support Program (1253G011).

Received 2 May 2015; accepted 5 June 2015Published online 30 June 2015

AbstractThe hydraulic fracturing technology is the main technical means of coalbed methane. However, it is hard to describe the fracture formation mechanism and evolution law in the process of fracturing. It caused the present studies restrict the effective mining of coalbed methane. This article is mainly study the process of fracture cracking and extending based on the angle of energy. It introduces the theory of entropy to analyse the micro defect evolution under hydraulic fracturing, and builds up the evolution model of the micro fracture number, angle, length and opening based on the theory of entropy. Then it analyses the main controlling factors of the fracture evolution. It will provide a new research approach for the law of hydraulic fracturing evolution.Key words: Entropy theory; Hydraulic fracturing; Damage evolution

Wang, T. T., Zhao, W. C., Feng, X. H., & Yan, Y. Q. (2015). The study fracture evolution of coal and rock mass under hydraulic fracturing. Advances in Petroleum Exploration and Development, 9(2), 117-120. Available from: URL: http://www.cscanada.net/index.php/aped/article/view/6978 DOI: http://dx.doi.org/10.3968/6978

INTRODUCTIONThe hydraulic fracturing has been widely used at home and abroad in the study of the coalbed methane development because of the characteristics of its

economy, effectivity and perfect technology. In the study of fracture evolution, Huang Wenxi and Ding Jinli et al.[1] from Tsinghua University using the thick and thin cylindrical specimens and the soil materials studied the necessary conditions of specimens hydraulic fracturing. From the study of field test, M. K. Hubbert and D. G. Willis[2] focused on the problem of rock drilling hydraulic fracturing pressure, make sure that the direction of crack surface is always perpendicular to the minimum principal stress direction. Tsay[3] overcame the difficulties of the mesh regeneration in the finite element and used the numerical collection method combined with the crack opening displacement method predicting the fracture propagation rules successfully. M. M. Hossain and other scholars[4-6] used the method of elastic mechanics studying the influence of the fracture initiation and extend from the fracture pressure, direction of fracture and perforation under the two conditions of any orientation perforation or not perforation for wellbore. But under the process of hydraulic fracturing, there is no accurate and reasonable method to make a full description because of the complex of the fracture distribution and the different of strata stress distribution. It’s always been the bottleneck of coal and rock mechanical response and the system space distribution.

Comentropy is named by engineer Shannon in American bell telecom test in 1948. He made the entropy defined in the classical thermodynamics introduce to information theory as a measure of average information. It made entropy breakthrough the limitation of the thermodynamics and infiltrate into the different areas[7]. It is always accompanied by a certain energy conversion when we are in the process of rock mass hydraulic fracturing[8]. So this article will introduce the entropy change theory to the study of fracture initiation and propagation under hydraulic fracturing. It will build up the model of rock damage and fracture evolution under hydraulic fracturing in order to better reveal the

118Copyright © Canadian Research & Development Center of Sciences and Cultures


fracture formation mechanism and process of coal rock. It will provide the reliable theory basis for optimizing construction parameters and fracturing process and improving the fracturing effect.

1. THE COMENTROPY DEFINITION OF COAL AND ROCK FRACTURINGAssuming the fracture is tension fissure. The growth of strain energy for any micro fracture is that,

, 0

2a

F j jQ G dA= ∫

,F jj

QQ

λ =

lnλ λ= −j j jS

. (1)Define,

, 0

2a

F j jQ G dA= ∫

,F jj

QQ

λ =

lnλ λ= −j j jS

. (2)

Where: Q is the total strain energy of the system; λj ≥ 0 ( j = 1,2,...,n) is the distribution of strain energy in the system of the rock mass.

Define the comentropy function of system sj is that[9],

, 0

2a

F j jQ G dA= ∫

,F jj

QQ

λ =

lnλ λ= −j j jS (0 ≤ λ j≤ 1). (3)From the Figure 1, we can express the fracture behavior

of rock mass fracturing process based on comentropy. In the Formula 3, the comentropy function will increase with the rise of λj from zero. When it reaches the top, it begins to decrease. When λj tends to 1, sj tends to zero. As is shown by Figure 1, this change accords with the law of comentropy change in the process of fracture propagation.

jλ

jS

0 1

Figure 1The Change Curve of Comentropy

The comentropy of any fracture propagation is that,

0 02 2

ln= − ∫ ∫a a

j jj

G dA G dAS

Q Q. (4)

For the energy and material exchange with the outside world, and the condition of fracture evolution is ordered, the comentropy of hydraulic fracturing system is the lowest. From Formula 4, we can find the condition of fracture initiation for any facture is that,

S

=0j

jσ∂

∂

1 , 2 ,D i v i IK k K k K K= ⋅ + ⋅ ⋅

( )

,

,

( , , )( )

D i

v i

I i

i b f

K f w aK g v

K p a

p p G θ

σ

π

δ σ

= =

=

= + −

2221

3 4 22

12 ( )

VG k k KE EV D V

βν −= +

[ ] ( )

1

, 1 , 1 2 , , 1 , 11

1 σ ρ−

− − −=

= − ⋅ + ⋅ − ⋅

∑i

m i m i m i m i m ii

N F N F N N

,

,

.

. (5)

Assuming the fracture evolution relatives to the rock mass characteristics surrounding the fracture and the

speed of evolution. Define the stress intensity factor K of the fracture dynamic evolution is that,

S=0j

jσ∂

∂

1 , 2 ,D i v i IK k K k K K= ⋅ + ⋅ ⋅

( )

,

,

( , , )( )

D i

v i

I i

i b f

K f w aK g v

K p a

p p G θ

σ

π

δ σ

= =

=

= + −

2221

3 4 22

12 ( )

VG k k KE EV D V

βν −= +

[ ] ( )

1

, 1 , 1 2 , , 1 , 11

1 σ ρ−

− − −=

= − ⋅ + ⋅ − ⋅

∑i


N F N F N N

,

,

.

. (6)Where,

S=0j

jσ∂

∂

1 , 2 ,D i v i IK k K k K K= ⋅ + ⋅ ⋅

( )

,

,

( , , )( )

D i

v i

I i

i b f

K f w aK g v

K p a

p p G θ

σ

π

δ σ

= =

=

= + −

2221

3 4 22

12 ( )

VG k k KE EV D V

βν −= +

[ ] ( )

1

, 1 , 1 2 , , 1 , 11

1 σ ρ−

− − −=

= − ⋅ + ⋅ − ⋅

∑i


N F N F N N

,

,

.

(7)

Where: KD,i is the damage function of the fracture growing; Kv,i is the fracture dynamic growth rate function; KI is the static stress intensity factor of fracture growth; k1, k2 are both the dimensionless impact factors of KD,i and Kv,i; v is the dynamic wave velocity of fracture growth; ve is the surface wave velocity of rock; pi is the effective stress of the fracture surface; pb is the fracturing pump pressure; Gf is the gravity of the fracturing fluid; σθ is the circumferential stress of fracture.

At the moment, the relationship between the extend strain energy release rate G of any micro fracture and the stress intensity factor K is that,

22

213 4 2

2

12 ( )

VG k k KE EV D V

βν −= +

[ ] ( )

1

, 1 , 1 2 , , 1 , 11

1 σ ρ−

− − −=

= − ⋅ + ⋅ − ⋅

∑i


N F N F N N

( ) ( )

,i

,1

, , 11

,

2c

1 ln1 ln

m ii

m ii

m i m i

m i

N

N

s sF

s

FD

ρ

σσσ

=

−

=

−=

=−

∑

[ ] ( )

( ) 1, , 1

, 1 , 1 , , 1 11c

,1

ln111 ln

θδ σσ

−−

− − −=

=

− = − + ⋅ − −

∑∑

im i m i

m i m i m i m i ii

m ii

p NN F N N N

D N

( )

( )

( )( )

1,, , 1 , 1

, , 1 , , 1 11, c

,1

2

,,,. 1 1

, 1 , 11, c

,1

ln111 ln

ln1 ln

θ

θ

δ σσ

δ σσ

−− −

− − −=

=

−

− =− −−

=

− − = − + ⋅ − −

−= − + ⋅

−

∑∑

∑

∑

im im i m i m i

m i m i m i m i iim i

m ii

i

m im im i i

m i m iim i

m ii

ps s NN N N N

s D N

NpsN N

s D N

,

,

.

.

. (8)

2. THE EVOLUTION MODEL OF COAL AND ROCK FRACTURING PARAMETERS

2.1 The Evolution Model of Micro Fracture Bifurcation NumberAssuming the rock has m natural fractures and the energy released promotes new micro fractures in the process of loading. When fracturing pump i stage loading, the number of micro fractures is that,

2221

3 4 22

12 ( )

VG k k KE EV D V

βν −= +

[ ] ( )

1

, 1 , 1 2 , , 1 , 11

1 σ ρ−

− − −=

= − ⋅ + ⋅ − ⋅

∑i


N F N F N N

( ) ( )

,i

,1

, , 11

,

2c

1 ln1 ln

m ii

m ii

m i m i

m i

N

N

s sF

s

FD

ρ

σσσ

=

−

=

−=

=−

∑

[ ] ( )

( ) 1, , 1

, 1 , 1 , , 1 11c

,1

ln111 ln

θδ σσ

−−

− − −=

=

− = − + ⋅ − −

∑∑

im i m i

m i m i m i m i ii

m ii

p NN F N N N

D N

( )

( )

( )( )

1,, , 1 , 1

, , 1 , , 1 11, c

,1

2

,,,. 1 1

, 1 , 11, c

,1

ln111 ln

ln1 ln

θ

θ

δ σσ

δ σσ

−− −

− − −=

=

−

− =− −−

=

− − = − + ⋅ − −

−= − + ⋅

−

∑∑

∑

∑

im im i m i m i


m ii

i

m im im i i

m i m iim i

m ii

ps s NN N N N

s D N

NpsN N

s D N

,

,

.

.

.(9)

Where,

2221

3 4 22

12 ( )

VG k k KE EV D V

βν −= +

[ ] ( )

1

, 1 , 1 2 , , 1 , 11

1 σ ρ−

− − −=

= − ⋅ + ⋅ − ⋅

∑i


N F N F N N

( ) ( )

,i

,1

, , 11

,

2c

1 ln1 ln

m ii

m ii

m i m i

m i

N

N

s sF

s

FD

ρ

σσσ

=

−

=

−=

=−

∑

[ ] ( )

( ) 1, , 1

, 1 , 1 , , 1 11c

,1

ln111 ln

θδ σσ

−−

− − −=

=

− = − + ⋅ − −

∑∑

im i m i

m i m i m i m i ii

m ii

p NN F N N N

D N

( )

( )

( )( )

1,, , 1 , 1

, , 1 , , 1 11, c

,1

2

,,,. 1 1

, 1 , 11, c

,1

ln111 ln

ln1 ln

θ

θ

δ σσ

δ σσ

−− −

− − −=

=

−

− =− −−

=

− − = − + ⋅ − −

−= − + ⋅

−

∑∑

∑

∑

im im i m i m i


m ii

i

m im im i i

m i m iim i

m ii

ps s NN N N N

s D N

NpsN N

s D N

,

,

.

.

(10)

Where: Nm,i is the evolution number of micro fracture; ρi is the density of micro fracture; F1 is the relative increment of comentropy in the process of fracture evolution, stand for the survival status of fracture evolution; F2(σ) is the impact ability of the fracturing load stress, dimensionless; σ is the load stress of hydraulic fracturing; D is the damage variable of rock mass.

119 Copyright © Canadian Research & Development Center of Sciences and Cultures

WANG Tingting; ZHAO Wanchun; FENG Xiaohan; YAN Yongqiang (2015). Advances in Petroleum Exploration and Development, 9(2), 117-120

By arrange,

2221

3 4 22

12 ( )

VG k k KE EV D V

βν −= +

[ ] ( )

1

, 1 , 1 2 , , 1 , 11

1 σ ρ−

− − −=

= − ⋅ + ⋅ − ⋅

∑i


N F N F N N

( ) ( )

,i

,1

, , 11

,

2c

1 ln1 ln

m ii

m ii

m i m i

m i

N

N

s sF

s

FD

ρ

σσ

σ

=

−

=

−=

=−

∑

[ ] ( )

( ) 1, , 1

, 1 , 1 , , 1 11c

,1

ln111 ln

θδ σ

σ

−−

− − −=

=

− = − + ⋅ − −

∑∑

im i m i

m i m i m i m i ii

m ii

p NN F N N N

D N

( )

( )

( )( )

1,, , 1 , 1

, , 1 , , 1 11, c

,1

2

,,,. 1 1

, 1 , 11, c

,1

ln111 ln

ln1 ln

θ

θ

δ σ

σ

δ σ

σ

−− −

− − −=

=

−

− =− −−

=

− − = − + ⋅ − −

−= − + ⋅

−

∑∑

∑

∑

im im i m i m i


m ii

i

m im im i i

m i m iim i

m ii

ps s NN N N N

s D N

NpsN N

s D N

,

,

.

.

.

(11)For the i stage loading, the evolve number of micro

fracture is that,

2221

3 4 22

12 ( )

VG k k KE EV D V

βν −= +

[ ] ( )

1

, 1 , 1 2 , , 1 , 11

1 σ ρ−

− − −=

= − ⋅ + ⋅ − ⋅

∑i


N F N F N N

( ) ( )

,i

,1

, , 11

,

2c

1 ln1 ln

m ii

m ii

m i m i

m i

N

N

s sF

s

FD

ρ

σσσ

=

−

=

−=

=−

∑

[ ] ( )

( ) 1, , 1

, 1 , 1 , , 1 11c

,1

ln111 ln

θδ σσ

−−

− − −=

=

− = − + ⋅ − −

∑∑

im i m i

m i m i m i m i ii

m ii

p NN F N N N

D N

( )

( )

( )( )

1,, , 1 , 1

, , 1 , , 1 11, c

,1

2

,,,. 1 1

, 1 , 11, c

,1

ln111 ln

ln1 ln

θ

θ

δ σσ

δ σσ

−− −

− − −=

=

−

− =− −−

=

− − = − + ⋅ − −

−= − + ⋅

−

∑∑

∑

∑

im im i m i m i


m ii

i

m im im i i

m i m iim i

m ii

ps s NN N N N

s D N

NpsN N

s D N

,

,

.

.

.

(12)The Formula 12 is the standard evolution model of

coal rock micro fracture number.

2.2 The Evolution Model of Micro Fracture Bifurcation Direction

θ θ

α

α x

y

……

……

Figure 2The Bifurcation Characteristics of Fracture

The schematic diagram of bifurcate fracture, the stress intensity factor is that,

( )( )π( cos )

( , )2i i i

a b ibK G pa

α θθ

+ +=

3 ( , / )AG b aθ

0 02 2

ln= − ∫ ∫a a

i ii

G dA G dAS

Q Q ( )( )

( )( )

222

13 4 20

2

222

13 4 20

2

π( cos )12 ( , )2 ( ) 2

π( cos )12 ( , )2 ( ) 2

ln

α θβν θ

α θβν θ

+ + − + = −

+ + − + ⋅

∫

∫

a

i i

i

a

i i

a b iV bk k G p dAE EV D V a

SQ


Q

S =0i

iθ∂∂

. (13)

Where:

( )( )π( cos )( , )

2i i i

a b ibK G pa

α θθ

+ +=

3 ( , / )AG b aθ

0 02 2

ln= − ∫ ∫a a

i ii

G dA G dAS

Q Q ( )( )

( )( )

222

13 4 20

2

222

13 4 20

2

π( cos )12 ( , )2 ( ) 2

π( cos )12 ( , )2 ( ) 2

ln

α θβν θ

α θβν θ

+ + − + = −

+ + − + ⋅

∫

∫

a

i i

i

a

i i


SQ


Q

S =0i

iθ∂∂

is the stress intensity factor coefficient before the micro fracture bifurcates.

Introduce the Formula 4 is that,

( )( )π( cos )( , )

2i i i

a b ibK G pa

α θθ

+ +=

3 ( , / )AG b aθ

0 02 2

ln= − ∫ ∫a a

i ii

G dA G dAS

Q Q ( )( )

( )( )

222

13 4 20

2

222

13 4 20

2

π( cos )12 ( , )2 ( ) 2

π( cos )12 ( , )2 ( ) 2

ln

α θβν θ

α θβν θ

+ + − + = −

+ + − + ⋅

∫

∫

a

i i

i

a

i i


SQ


Q

S =0i

iθ∂∂

. (14)

According to the Formula 13, we can find the relationship between comentropy and bifurcation point is that,

( )( )π( cos )( , )

2i i i

a b ibK G pa

α θθ

+ +=

3 ( , / )AG b aθ

0 02 2

ln= − ∫ ∫a a

i ii

G dA G dAS

Q Q ( )( )

( )( )

222

13 4 20

2

222

13 4 20

2

π( cos )12 ( , )2 ( ) 2

π( cos )12 ( , )2 ( ) 2

ln

α θβν θ

α θβν θ

+ + − + = −

+ + − + ⋅

∫

∫

a

i i

i

a

i i


SQ


Q

S =0i

iθ∂∂

(15)For any bifurcation stage, it need satisfy that,

( )( )π( cos )( , )

2i i i

a b ibK G pa

α θθ

+ +=

3 ( , / )AG b aθ

0 02 2

ln= − ∫ ∫a a

i ii

G dA G dAS

Q Q ( )( )

( )( )

222

13 4 20

2

222

13 4 20

2

π( cos )12 ( , )2 ( ) 2

π( cos )12 ( , )2 ( ) 2

ln

α θβν θ

α θβν θ

+ + − + = −

+ + − + ⋅

∫

∫

a

i i

i

a

i i


SQ


Q

S =0i

iθ∂∂

. (16)

The Formula 16 is the model of fracture bifurcation angle based on comentropy.

3. CASE STUDYChoose a coal block of Hegang Dongxing coal mine. Its basic parameters are the following. The maximum horizontal main stress sH is 32 MPa. The minimum horizontal main stress sh is 28.5 MPa. The formation pressure P0 is 11.4 MPa. The modulus of elasticity E is 2,800 MPa. The critical damage strength sc is 30 MPa. The fluid compressibility factor a is 0.8. The poisson’s ratio is 0.20. The porosity of fracture and substrate are boss 0.15. The fracturing fluid density is 2.16 g/cm3. The fracturing fluid velocity is 4.0 m3/min. The hole radius is 0.12 m. The angle of internal friction is 0.35.

3.1 The Influence of the Hole Fracture Evolution With the Net Fluid PressureThe Figures 3-6 are the relation charts about the net fluid pressure influent the micro fracture growing number, length, opening and bifurcation angle. As is shown by these charts, with the net fluid pressure increasing, the micro fracture growing number, length and opening are all have a certain degree of increase. But the micro fracture bifurcation angle has a degree of decline. According to the analysis of calculation condition, the first bifurcation angles are almost between 50° and 60°.

0

100

200

300

400

500

600

20 30 40 50 60 70 80 90The net fluid pressure（MPa）

The

mic

ro f

ract

ure

grow

ing

num

ber

(pie

ce)

Figure 3The Chart of Micro Fracture Growing Number

00.020.040.060.080.1

0.120.140.160.180.2

10 11 12 13 14 15 16 17 18The net fluid pressure（MPa）

The

mic

ro fr

actu

regr

owin

g le

ngth

(m)

Figure 4The Chart of Micro Fracture Growing Length

120Copyright © Canadian Research & Development Center of Sciences and Cultures


0

1

2

3

4

5

6

20 30 40 50 60 70 80 90The net fluid pressure（MPa）

The

mic

ro fr

actu

regr

owin

g op

enin

g（m

m）

Figure 5The Chart of Micro Fracture Growing Opening

55

55.5

56

56.5

57

57.5

58

58.5

30 35 40 45 50 55 60The net fluid pressure（MPa）

The

angl

e w

ith th

em

ain

frac

ture

Figure 6The Chart of the Angle With the Main Fracture

CONCLUSIONIn this passage, we have built up the evolution model of micro fracture number, angle, length and opening based on the theory of entropy according to analyse the fracture evolution law in the process of hydraulic fracturing. It provides the theoretical foundation for the fracture evolution rules.

For the geologic parameter of one block of Dongxing Coal mine and combine the theoretical models have been finished, we analyse the main controlling factors of fracture damage evolution under hydraulic fracturing. Then we get the curves of relationship between micro fracture evolution number, angle, length, opening and the net fluid pressure, the modulus of elasticity, the poisson’s ratio.

Following the increasing of the net fluid pressure, the micro fracture growing number, length and opening are all increasing, and the bifurcation angle becomes lower. The first bifurcation angles are almost between 50° and 60°, and the micro fracture largest opening is 5.061 mm. Following the increasing of the modulus of elasticity, the micro fracture growing length, opening and bifurcation angle are all declining. Following the increasing of the poisson’s ratio, the micro fracture growing length and opening are both having a certain degree of decrease, and the bifurcation angle becomes larger.

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[2] Hubbert, M. K., & Willis, D. G. (1957). Mechanics of hydraulic fracturing. Trans AIME, 210, 153-168.

[3] Tsay, R. J., Chiou, Y. T., & Chuang, W. L. (1999). Crack growth prediction by manifold method. Journal of Engineering Mechanics, 125(8), 884-890.

[4] Hossain, M. M., Rahman, M. K., & Rahman, S. S. (2000). Hydraulic fracture initiation and propagation: roles of wellbore trajectory, perforation and stress regimes. Journal of Petroleum Science and Engineering, 27, 129-149.

[5] Zhang, G. Q., & Chen, M. (2009). Complex fracture shapes in hydraulic fracturing with orientated perforations. Petroleum Exploration and Development, 36(1), 103-107.

[6] Jiang, X., Chen, M., & Zhang, G. Q. (2009). Impact of oriented perforation on hydraulic fracture initiation and propagation. Chinese Journal of Rock Mechanics and Engineering, 28(7), 1321-1326.

[7] Jaynes, E. T. (1982). On the rationale of maximum entropy methods. Proceedings of IEEE, 70, 939-952.

[8] Deng, G. Z., & Zhu, W. S. (1997). The basic characteristics of rock mass nonlinear unloading and the entropy change. Journal of Xi’an Mining Institute, 17(4), 332-335．

[9] Xu, C. H., & Ren, Q. W. (2004). Criterion of entropy catastrophe of stability of surrounding rock. Rock and Soil Mechanics, 25(3), 437-440.

the study fracture evolution of coal and rock mass under

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