QUID 2017, pp. 2307-2319, Special Issue N°1- ISSN: 1692-343X, Medellín-Colombia

AN INVESTIGATION INTO CRACKS DEVELOPMENT WITH USE OF ENERGY METHOD

(Recibido el 23-06-2017. Aprobado el 27-08-2017)

Abbas Majdi Professor in the faculty of

mining engineering, Technical

faculties campus, University of

Tehran.

Soroush Ali Madadi

M.Sc. student of rock mechanics

in the faculty of mining

engineering, Technical faculties

campus, University of Tehran,

s.alimadadi@ut.ac.ir

Mohammadhossein

Khosravi

Associate professor in the

faculty of mining engineering,

Technical faculties campus,

University of Tehran.

Abstract: Since the process of initiation and propagation of cracks in the rock is under focus nowadays. Using the

fundamentals of fracture mechanics and its relationship with the injection pressure can be a suitable tool for

examining and studying the behavior of rock masses more exactly and in certain cases, estimating the in-situ stresses

in different depths of the earth. Knowing and evaluating the behavior of rock joints under the injection pressure in

rock mechanics projects are of high importance. The material resistance may include surface energy, plastic work or

other energy losses during the crack growth. Therefore, the necessity of knowing the initiation mechanism and the

way of hydraulic cracks propagation in various branches of engineering has caused the analysis of this phenomenon

and trying to obtain a suitable model for its simulation be considered strongly. In this article, we have tried to offer a

simple and efficient model for calculating the crack growth process in rock environment by employing sophisticated

analysis methods. Then, the ground model along with fluid injection is known using ABAQUS software and

employing finite element method. For progression of the research, at first, Kirsch relations were used for verification

of the way of stresses distribution around the well. The resulted acceptable error shows the proper consistency of

numerical results with the analytical relations. Then, the results obtained from the two methods were compared for

examining the accuracy of the extracted analytical relations after making sure of the simulation results. The

comparison showed that the numerical results obtained from the simulation have an acceptable consistency with the

analytical relations. Finally, the effect of injection pressure and other effective parameters have been examined.

Keywords: energy method, fracture mechanics, injection pressure, analytical methods, hydraulic fracture, finite

element method, ABAQUS

Citar, estilo APA: Majdi, A., & Madadi, S., & Khosravi, M. (2017). An investigation into cracks development with use of energy method.

Revista QUID (Special Issue), 2307-2319.

1. INTRODUCTION

Discontinuities and fractures are natural structural

flaws of rocks which are found in different scales in

rocks (from some millimeters to some thousand

meters) and determine the behavior of the rock mass.

The behavior of micro-cracks (initiation, propagation

and coalescence) has affected the behavior of large-

scale fracture phenomenon. Therefore, the

mechanism of initiation and propagation of the cracks

and also, their coalescence are different from the

flaws of the rocks in different situations (such as

different axial and lateral loading, crack slop, etc.).

(Irwin, 1958)

Knowing and evaluating the behavior of rock joints

under the injection pressure are very important in

rock mechanics projects. For example, the hydraulic

gap method which is based on the theory of fracture

mechanics and the pressure of fluid injection was

used in oil and gas industries for making desirable

cracks with the capability of proper hydraulic

direction and increasing the rate of oil and gas flow

from hydrocarbon tank with low permeability toward

the dig wells. In the area of environment engineering,

this was an effective technique in increasing the

efficiency of the methods of decontamination of in-

situ contaminated soils. The other applications of this

phenomenon include the production of the earth

thermal energy by making a hydraulic gap on hot and

dry rocks mass, and using the fluid cycle. In rock

mechanics, this method can be used to measure the

field of in-situ stresses. (Griffith, 1920)

Therefore, the necessity of knowing the initiation

mechanism and the way of hydraulic cracks

propagation in various branches of engineering has

caused the analysis of this phenomenon and trying to

obtain a suitable model for its simulation be

considered strongly. However, the importance of

hydraulic gap process in oil and gas industries to

obtain hydrocarbon reserves by increasing their

production rate and reforming the ground have been

the main motivation for making such models. (Wang,

Y, 2017)

2. OFFERING ANALYTICAL RELATIONS

FOR PREDICTING THE CRACK GROWTH

At first, we examine the offered methods on how to

solve the crack-related problems. For the cracks

without tension in local coordinates, Williams has

stated the approximate fields for displacement

components of u and v near the crack tip as follows:

Where, and are coefficients and and K

are shear module and Kolosov constant, respectively.

(Williams, 1957) Kolosov constant is as follows: (3)

(1)

2

1

1 cos cos 22 2 2 2

,2 2

1 sin sin 22 2 2 2

i

i Ii

i i

IIi

i i i iK

ru r

n i i i i

(2)

2

1

1 sin sin 22 2 2 2

,2 2

1 cos cos 22 2 2 2

i

i Ii

i i

IIi

i i i iK

rr

i i i i i

According to Irwin, the work required to propagate

the crack very little ( ) equals to the work required

for closing the crack so that it returns to its previous

length. Thus, the rate of strain energy release for

mixed state which is stated in the polar coordinates

with its source in the crack tip, is defined as

I II G G G

(4)

Where, is the general rate of energy release

separated into and components related to the

deformation states in I and II fracture mode(Wang,

Y., 2017)

Δ

Δ 00

1lim Δ ,0

2Δ

c

Ic

c r u r drc

G (5)

Δ

Δ 00

1lim Δ ,0

2Δ

c

II r rc

c r u r drc

G (6)

and are normal and shear stresses in

polar coordinates, and are the relative shear

and opening displacements between the related points

on the cracks surfaces and is the propagation of

track on its tip. The mutations of shear and opening

displacement are defined by the following relations:

, ,r r ru r u r u r (7)

, ,u r u r u r (8)

Changing the displacement fields from polar

coordinated to Cartesian coordinates is possible with

the following relations:

   ru ucos vsin (9)

   u u sin vcos (10)

and show the relative

displacements of slip and opening of crack in polar

coordinate system which is depicted in figure 1.

Figure. 1 The relative displacements of slip and

opening of crack in polar coordinate system

Therefore, supposing that the local coordinate system

of crack is in the same direction with the crack axis,

the relations are simplified and the mutations of shear

and opening displacement become as follows:

, ,ru r u r u r (11)

, ,u r r r (12)

The key idea of XFEM of enrichment is the

estimation of standard finite components with local

separations from single enrichment functions which

is selected based on the considered problem. For

crack problems, it is independent from the crack

direction. There are similar enrichment methods for

modelling the cracks which are based on the

generalized finite component method. (Horii et al

1985)

The goal is to find and coefficients

analytically and using the enrichment functions in

XFEM.

2.1. analytical expansion of Irwin integral

By inserting Williams extension into (11) and (12)

relations and keeping just the first 3 terms, we get

2 1 73

2 2 2 2

1 1 1

  , ,i i i

II

r i i i

i i i

u r u r u r m r O r

(13)

2 1 73

2 2 2 2

1 1 1

  , ,i i i

I

i i i

i i i

u r v r v r m r O r

(14)

Note that even-ordered expressions of r in the

expansion of displacement mutation are removed

from the sum because these expressions are

continuous in discontinuities and therefore, just odd-

ordered expressions are shown in (13) and (14)

relations. Also, note that the obtained sum is cut from

the formula after 5

2r order is shown. The

displacement of kinematic strain in polar coordinates

is defined as follows:

rrr

u

r

ò (15)

1ruu

r r

ò

(16)

1 1

2

rr

u uu

r r r

ò

(17)

Where , and are radial, tangential and

shear strains, respectively. Considering the plane

strain, and strains will be as follows:

1

1 1 2rr

E

ò ò (18)

1r r

E

ò (19)

By combining the stresses and strains, the

relations (18) and (19), and inserting in Williams`

answers, the normal and shear strains in the crack tip

can be defined as follows:

5 711

2 2

4

,01 1 2

i

I

i

i

Er m r O r

(20)

5 711

2 2

4

,02 1

i

II

r i

i

Er m r O r

(21)

Note that r order in relations (20) and (21) are not

changed in the extraction of strains. However, the

sum after is cut. Again, it is because of selecting

high-ordered enrichment functions and considering

XFEM formulation and the shape functions which are

mentioned below. Finally, by inserting opening

displacements in (13) and (14) and (20) and (21)

strains in the definition of strain energy release rate,

in (24), in (25) and integration, we have

11

62

0

Δ Δ Δ1 1 2

i

I

I i

i

Ec c O c

G (22)

and

11

62

0

Δ Δ Δ2 1

i

II

II i

i

Ec c O c

G (23)

Here, and are defined as relations (24) and

(25).

Δ

0

1Δ ,0

2Δ

c

I c r u r drc

G (24)

Δ

0

1Δ ,0

2Δ

c

II r rc r u r drc

G (25)

So that:

Δ 0lim ΔI Ic

c

G G (26)

Δ 0lim ΔII IIc

c

G G (27)

In limit, we obtain a simple expression

since all of the high-ordered expressions are deleted:

0 1 4Δ 01 1 2 1 1 2 4

I I I

I

E Ec m m

G (28)

0 1 4Δ 02 1 2 1 4

II II II

II

E Ec m m

G (29)

2.2. Irwin integral in XFEM

To obtain expressions for and coefficients,

imagine a general rectangular element which has a

horizontal crack with its tip at the center of the

element (figure 2).

Fig. 2. Representation of the element of crack tip

whose tip is located at the center: crack lines are

shown in red.

In polar coordinates, every point within this element

range is defined by the following coordinates:

As a result, the coordinates of the nodes in element

are and

where and are the lengths of edges in x and y

directions. The tip elements in XFEM consider the

approximate fields near the tip, thus, the

displacement field of the relation (64) can be

simplified as follows and written in polar

coordinates:

4

1 1

,Fn

I I j jI

I j

u r N u F b

(30)

In polar coordinates, standard linear shape functions

are as the relation (31):

2 2

1, 1 4 1 4                     1, ,4

4

I II

x y

x rcos y rsinN r I

h h

(31)

By replacing the relations (31) and (32) in (11) and (12), we have:

4 13 4 13

1 1 1 1

34

21 82

1

  , , , ,    

1  1 4 2 2

4

r I xI j xjI I xI j xjI

I j I j

Ix I x I

i x

u r N r u F r b N r u F r b

x rrb r b

h

(32)

and

4 13 4 13

1 1 1 1

34

21 82

1

  , , , ,   

1    1 4 2 2

4

I yI j yjI I yI j yjI

I j I j

Iy I y I

i x

u r N r u F r b N r u F r b

x rrb r b

h

(33)

Similarly, for ,0r strain, using the derivatives

of enrichment function in the equations (20) and (21),

we obtain the following expression:

3422

2 5 7 11 1321

4

2 5 7 11 1321

24

22 21

,01 1 2 1 1 3

1 4 24 22

41

1 1 2

IxI x I x I x I x I x I

I x

Ix I x I x I x I x I

I x

x I I

yI y I

I x y

xu rb rb r b r b b

hEr

x rb b rb r b b

h r

h x r y ru rb rb

r h hE

3

225 7 11 13

3422

1 4 6 8 10 1221

1 1 1 1  1 4 2

4 2 2

y I y I y I y I

Iy I y I y I y I y I y I

I x

r b r b b

x rr b b rb r b b r b

r h

(34)

Also, for strain, we have:

2 3422

2 5 7 11 132 21

3422

1 4 6 8 10 1221

34

22 5 72

1

41,0

2 1

1 1 11 4 2

4 2

x I I

r xI x I x I x I x I x I

I x y

Ix I x I x I x I x I x I

I x

IyI y I y I y I

I x

h x r y rEr u rb rb r b r b b

r h h

x rr b b rb r b b r b

r h

xu rb rb r b

h

2

11 13

4

2 5 7 11 1321

1 1 31 4 2

4 22

y I y I

Iy I y I y I y I y I

I x

r b b

x rb b rb r b b

h r

(35)

The final step is to make all of the coefficients of co-

ordered terms for the opening displacement mutation

in (32) consistent with (13) and in (33) with (14).

Similarly, the coefficients of strains in the relations

(34) and (35) are consistent with (20) and (21).

Finally, the general response for the rate of strain

energy release is obtained by the limits of finite

integral in closed form.

For the special state of , the high-ordered

terms are removed and just and

coefficients are remained which are as

follows:

4

1 1

1

1

4

I

y I

I

m b

4

1 1

1

1 

4

II

x I

I

m b

(36)

and

4 4

1 424

1 1

18 8 4

y I y II x I

I I

b bbm

(37)

421 4

4

1 8 4 8

y III x I x I

I

bb bm

(38)

Where, and coefficients are obtained from

solving algebraic equations system. Therefore, strain

energy release rate (SERR) can be calculated in

closed form and obtained without a need for post-

process special techniques:

4 42

1

1 1

4 41 4

1

1 1

1

8 2Δ 0

4 1 1 2 11

8 4 2

x Iy I

I I

I

y I y I

y I

I I

bb

Ec

b bb

G

(39)

4 421 4

1

1 1

1Δ 0

4 2 1 2 8 4 8

y Ix I x III x I

I I

bb bEc b

G

(40)

Strain intensity factors are related to SERR and can

be obtained directly from the following relations:

1   I

EK

G 2

II

EK

G (41)

Where,

and E is Young module.

3. SIMULATION PROCESS

In this section, the geometry of well and its

surrounding areas should be modeled. To do so, the

model dimensions are selected so that they don`t

affect the modeling results and be sufficiently larger

than the dimensions of well entrance. Therefore, the

dimensions of the model are selected to be 100*100

and the well diameter, 0.1 meter.

Here, a two-dimensional elastic model is used for the

comparison with the strain relations.

The mechanical characteristics of model are

simulated according to table 1.

Table 1. Input data of the software

Parameter value

Elastic module or Young module 20 GPa

Poisson ratio 0.25

Special weight of pore fluid 8.7 KN/m3

permeability 0.1 md

porosity 10%

In pro-elastic model, the internal pressure of well on

the walls is Pw = 7MPa and the hydro-static strain in

distant borders of 40 mega-pascal is Ϭ0 = 40MPa

which are applied on the model. The border

constraints for the two models are mentioned in table

2.

Table 2. Border conditions governing the process

Model type Border conditions

Linear elastic BC1: X SYMM: On the Y axis: U1 = UR2 = UR3 = 0

BC2: Y SYMM: On the X axis: U2 = UR1 = UR3 = 0

BC3: Ϭ0 = 40 MPa (Far Field Stress)

Pro-elastic BC1: X SYMM: On the Y axis: U1 = UR2 = UR3 = 0 & No Flow in X Direction

BC2: Y SYMM: On the X axis: U2 = UR1 = UR3 = 0 & No Flow in Y Direction

BC3: Pw = 7MPa

BC4: Ϭ0 = 40 MPa (Far Field Stress)

Figure 3 shows the model after applying the border

conditions.

Figure 3. General scheme of the two-dimensional

model of the digged well in linear elastic structure

under hydrostatic strain.

3.1. Examining the accuracy of made model

In this section, the accuracy of the current model is

studied by comparing the results obtained from the

numerical simulation and Korsch analytical relations.

3.1.1 Linear elastic model

Tangential and radial strains resulted from the strain

in the relations (45) and (46) can be calculated. In

these equations, pw rate should be equal to zero. The

strain meters S11 and S12 are the same as SXX and

SYY. To examine the error rate of the numerical

model, the strain rate in line with ϴ = 0 should be

calculated. The strains in numerical model are in

Cartesian coordinates. Thus, the value of S11 strain

on ϴ = 0 direction equals to the radial strain of (Ϭr)

and , the value of S22 strain on the same direction

equals to the tangential strain of (Ϭϴ). It is clear that

in ϴ = 90 direction, the values of S11 and S22 strains

will include the values of the opposite side in ϴ = 0

direction. For examining the error rate of numerical

model with analytical relations, radial and tangential

strain diagrams are drawn in ϴ = 0 direction which

are depicted in figures (4) and (5).

(43) 2 2 2

2 2 2( ) (1 )w w w

r h h w h w

R R Rp p

r r r

(44)

2 2 2

2 2 2( ) (1 )w w w

h h w h w

R R Rp p

r r r

Figure 4. Value of S22 strain in ϴ = 0 direction

(Sϴ)

Figure. 5. Value of S11 strain in ϴ = 0 direction (Sϴ)

As it is observed in the diagram of figures (4) and

(5), the numerical and analytical values are very close

to each other so that their detection is difficult. For

example, on the well walls, the value of tangential

strain from the analytical relation should be twice the

distant strain (Ϭϴ = 2* Ϭ0 = 80 MPa) which is

calculated as Ϭϴ = -72.166MPa in the numerical

model. That is, there is 9.8 percent of error which

seems acceptable.

3.1.2 Linear elastic model of well along with the

internal pressure

For examining the numerical error rate, the strain rate

in line with ϴ = 0 should be calculated. The strains in

numerical model are in Cartesian coordinates. Thus,

the value of S11 strain on ϴ = 0 direction equals to

the radial strain and the value of S22 strain on the

same direction equals to the tangential strain. It is

clear that in ϴ = 90 direction, the values of S11 and

S22 strains will include the values of the opposite

side in ϴ = 0 direction. For examining the error rate

of numerical model with analytical relations, radial

and tangential strain diagrams are drawn in ϴ = 0

direction which are depicted in figures (6) and (7).

Figure. 6. Value of S11 strain in ϴ = 0 direction (Sϴ)

Figure 7. Value of S22 strain in ϴ = 0 direction (Sϴ)

As it is observed in the diagram of figures (6) and

(7), the numerical and analytical values are very close

to each other so that their detection is difficult. For

example, on the well walls, the value of tangential

strain from the analytical relation (2) is calculated as

Ϭϴ = 73MPa, but it is calculated as Ϭϴ =

64.982MPa in numerical model. That is, there is 10.9

percent of error which seems acceptable. Of course,

the above error rate is resulted from the combination

of pore pressure and strain distribution which need

the elements with smaller dimensions and tinier

meshing which requires the stronger hardware

because of the large dimensions of the model. Of

course, the size ratio of the elements causes an error

too, because meshing is getting smaller near the well.

For calculating the strain focus near the well, the

dimensions of the elements should be small as much

as possible in order to increase the accuracy of

problem-solving.

4. EXAMINING THE PRESSURE

DISTRIBUTION AND ITS EFFECT ON

OTHER PARAMETERS

Figures (8) and (9) show how the pressure is

distributed around the well. The diagram drawn in

figures (10) and (11) show that how the pressure and

opening of the tip and opening of tip and entrance of

the crack change during the injection process.

Figure. 8. Pressure distribution inside the structure

resulted from the internal pressure

Figure 9. Pressure meter around the well (Poro-

elastic model)

Figure 10. Opening and pressure changes in terms of

time in crack tip

Figure. 11. Opening and pressure changes in

terms of time in crack entrance

As it is observed in the diagrams of figures (10) and

(11), the pressure and opening rate of crack tip and

entrance increase with the injection time which is

because of an increase in the force of injected fluid

and a decrease in the resistance of crack tip and

entrance against opening with time. But in crack tip,

considering the more resistance because of the tensile

elements, the required pressure for dominance on the

material resistance is more than that of the track

entrance.

Figures (12) and (13) show how the tangential and

radial strains are distributed around the well.

Figure 12. The meter of S11 strain around the well

(poro-elastic modeling)

Figure. 13. The meter of S22 strain around the well

(poro-elastic modeling)

Figure 14 shows the frame to frame scheme of

initiation and propagation of the crack.

Figure 14. How the crack is propagated.

5. DISCUSSION AND CONCLUSION

The main goal of this article is the improvement of

the relations of fracture mechanics by making use of

XFEM method in calculating SERR. Achieving this

goal is possible by the following cases:

The high-ordered enrichment functions are

inserted into XFEM and updated to show that if

the relations and Irwin integral are used

simultaneously, high accuracy of the calculations

is achievable.

The closed form of analytical relations for

extracting SERR are developed by making use of

the classic form of Irwin relations and the special

capabilities of WFEM. The formulae are totally in

polar coordinates and there is no need for post-

processing in SERR calculation. The suggested

solution is capable of application for arbitrary

settings of the software. The obtained results

show the strong effect of high-ordered enrichment

functions on the convergence of the crack. When

the limit of integral is toward zero, the simpler

expressions of SERR are revealed and high-

ordered terms disappear. However, we cannot

deny the direct effect of high-ordered terms on the

first order. This simple solution makes the

replacement of J integral method possible.

The suggested solution is simulated numerically

by ABAQUS software and the obtained results

show the accuracy of the suggested solution.

The results obtained from the numerical

modelling and J integral for calculation of 1k and

2k values are represented in the following tables. As

it is clear, the results obtained from the suggested

plan for high-ordered elements are very precise

which implies the accuracy of the mentioned

relations. The real values of 1k and 2k obtained by

the analytical relations are 34.0 and 4.55, respectively.

Table 3. Comparison of the calculated values of 1k using the two numerical and J integral methods with the values

obtained from the analytical relations

1k

error)%(

1 2xh

k C

error)%( 1 0k C Mesh size Element order

22.7 26.287 39.6 20.532 4989 1.2

5.4 32.147 4.4 32.512 4989 1

0.4 33.865 2.0 33.322 4989 3.2

1.1 33.620 1.1 33.620 4989 J-integral

Table 4. Comparison of the calculated values of 2k using the two numerical and J integral methods with the values

obtained from the analytical relations

2k

error)%(

2 2xh

k C

error)%( 2 0k C Mesh

size

Element

order

20.0 3.642 27.2 3.310 4989 1.2

8.1 4.182 8.6 4.159 4989 1

0.5 4.526 0.2 4.542 4989 3.2

0.8 4.513 0.8 4.513 4989 J-integral

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Wang, Y., Cerigato, C., Waisman, H. and

Benvenuti, E., 2017. XFEM with high-order

material-dependent enrichment functions for

stress intensity factors calculation of interface

cracks using Irwin’s crack closure integral.

Engineering Fracture Mechanics, 178, pp.148-

168.

M.L. Williams. On the stress distribution at the base

of a stationary crack.Journal ofApplied

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Horii, H., Nemat-Nasser S., Compression-induced

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dissertation, Columbia University).