Abbas Majdi Soroush Ali Madadi Mohammadhossein Khosravi · 2018-01-19 · QUID 2017, pp. 2307-2319,...
Transcript of Abbas Majdi Soroush Ali Madadi Mohammadhossein Khosravi · 2018-01-19 · QUID 2017, pp. 2307-2319,...
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,
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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und Plastizitaet, pages 551-590,1958.
Griffith, A.A., The phenomenon of rupture and flow
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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
Mechanics, pages 101-106, 1957.
Horii, H., Nemat-Nasser S., Compression-induced
microcrack growth in brittle solids: axial
splitting and shear failure, J. Geophy Res.,
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brittle and cohesive fracture (Doctoral
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