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7/23/2019 3d Fracture Propagation Modeling
http://slidepdf.com/reader/full/3d-fracture-propagation-modeling 1/9
Sustain. Environ. Res., 25(4), 217-225 (2015) 217
*Corresponding author
Email: [email protected]
Three dimensional modelling of propagation of hydraulic fractures
in shale at different injection pressures
Vikram Vishal,1,* Nikhil Jain
2 and Trilok Nath Singh
3
1Department of Earth Sciences
Indian Institute of Technology Roorkee
Uttarakhand 247667, India2Department of Mining Engineering
Indian Institute of Technology BHU
Varanasi 221005, India3Department of Earth Sciences
Indian Institute of Technology Bombay
Mumbai 400076, India
Key Words: 3D modelling, hydraulic fractures, shale, COMSOL Multiphysics
ABSTRACT
The modeling of conjugate development of fractures and uid ow remains a signicant subject
in a diversity of rock engineering. Continuum numerical methods are paramount in the modeling of
rock engineering practice problems, merely with restrained capacities in modeling the problem of
fracture development coupled by uid ow. There exists a demand for them to be understood in details.
Driven by this, we demonstrated an approach based on a three-dimensional development of fracture
of an abstract model condensed to two-dimensional analysis comprising rocks with fractures. In the
framework of a continuum method of modeling, the contact between the fracture development anddeformation was paired with uid ow. A 3-D model was established in this case for a shale reservoir
and uid was injected at multiple pressures to understand the initiation and propagation of fractures,
as applied to the field of hydraulic fracturing. The stress, strain, displacement in the reservoir were
monitored at multiple injection pressures. Linear relations of injection pressures were observed with
these parameters. A detailed insight with quantication of the values is given into the subject based on
the ndings of this study.
INTRODUCTION
With a rapid rise in energy demands and with
continuous development in industry, exploration andgeneration of new sources of energy has gathered
momentum in the recent years. Traditionally, coal has
been one of the main resources of energy of the world.
However, oil and gas reservoirs have become very
popular over last few decades. More recently, tight
sands and shale gas reservoirs also gained the attention
of industry for power generation as they exhibit
tremendous potential resource for future development,
and analysis of these systems is proceeding briskly.
With a fast pace decline in conventional petroleum
reserves, unconventional resources have gained a
progressively significant role in the energy industry
over past few years and turning to be an important
element in years to come.
Gases in shale are stored as the free gas in bothfracture and matrix pores and as absorbed gas on the
surface of micro-pores [1,2]. Modeling and simulation
of shale gas reservoir presents an unusual problem.
These reservoirs have trenchant properties, such as [3]:
In some of the reservoirs, nearly 50% of the gas
content is absorbed gas from organic materials.
The ow is not easy to comprehend due to the
extremely low matrix permeability of nano-
Darcy levels.
Due to the presence of both natural and induced
fractures, complex fracture network distribution
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Vishal et al., Sustain. Environ. Res., 25(4), 217-225 (2015)
in rocks. Here, a finite element model showing the
variation of stress-strain and displacement of the rock
and fractures when subjected to hydraulic fracturing
at different injection pressures was constructed fordifferent conditions in reservoirs and with variation
in cohesive forces and internal friction angle. The
fluid-solid coupling was used to show the variation
in behavior of rock and to study fracture initiation
and propagation. A demonstration of Biot equations
for the rock was considered as the base equation for
construction and evaluation of model followed by
pressure equation of the fracture and the discussion
of numerical solution of the combined system of
equations.
1. Problem Description
Experimental data of fractured core in a reservoir
was adopted along with few other parameters value
from an Indian shale reservoir to bring forth the
fractured porous media geometry within the numerical
simulations using COMSOL Multiphysics [11].
We examined the response of few parameter
variations in a transverse section of a fractured bore
well, drilled in a layer of rock strata containing shale.
A study of horizontal fracturing at intermediate depths
of a reservoir is shown in the article assuming the
minimum vertical stress. The paper also presents the behavior of stress and strain surrounding the fracture
and the bore well. The variations in bore-hole pressure
and in displacement are also included in the study.
2. Governing Equations and Boundary Conditions
The fluid flow was governed on the basis of
modied Darcy’s law in a poroelastic rock model. The
structural deformation was governed by
where σ denotes the stress tensor, and the directional
components of the gradient in uid pressure, p, make
up a vector forces, F , k stands for permeability and µ is
for uid dynamic viscosity.
where, Q is the mass source term.Darcy's law accepts that the fluctuation of the
velocity field when a fluid makes it to the porous
medium is stimulated by the uid pressure gradient.
system may be formed. The elementary enabling
technology is reactivation of hydraulic fracturing
by narrow, calcite-natural fractures.
Hydraulic fracturing and horizontal drillinghave been main pillar technologies in economical
and successful extraction of natural gas from shale
reservoirs. In petroleum industry, for enhanced recovery
of gas and oil, hydraulic fracturing has been most
familiar simulation method. The hydraulic fractures
developed in rocks through the artificial stimulation
of underground reservoirs exercise a fundamental
inuence on various mechanical and transport attributes
of the rocks, including the elastic modulus, anisotropy,
elastic wave velocities and permeability [4,5]. These
concepts have been utilized during development and
analysis of uid ow through hydraulic fractures.
NUMERICAL MODELING USING FINITE
ELEMENT METHOD
A nite element advancement method is suggested
in this paper for the modeling of hydraulic fracturing
in 3-D. The model is established on Biot poroelasticity
equations for the contortion of the rock caused by
gradients in fluid pressure. By means of “fracture
porosity”, with special regridding being absent, in this
case, the fracture is constituted on the same regular
grid as the rock. In a rock, the volume fraction is
represented by fracture porosity. With respect to both
uid pressure and displacement, it is possible to build
a uniform element formulation for the rock and the
fracture by means of fracture porosity [6 ,7]. This
model has a touchstone based on strain of element,
where propagation of fracture occurs, when the strain
computed at the centre of element exceeds a limit.
As ind ica ted by Dioda to , the re ex is t s a
classification into ‘explicit discrete fracture
formulation’, ‘discrete fracture networks’ and ‘two
continuum formulations’ and here we carry on with
the micro-scale only [8]. With coupled analytical
mechanistic modeling, Bai et al. took benet of nite
element simulations for porous media having fractures.
Porous media flow physics was employed for both
fracture and matrix for these simulations [9].
There are various new implementations in 3D
model equated to its 2D preprocessor. The failure
measure in 3D is established on strain in the centre
of each element, whereas the 2D model has a failure
criterion grounded on the strain of sides of an element,
bonding to connect the nodes [10]. In the following,
we demonstrate a coupled fracture and ow modelingapproach in the fabric of an uninterrupted numerical
method. This is centered on interpreting the basic
processes of increasing fractures and linked uid ow
218
F =∇− σ (1)
(2)
(3)
(4)
[ ] Qu =∇ ρ
F u D =∇∇− ][ε =∇u ε σ D=
( )[ ] 0/ =∇−∇ pµ κ
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3. Model Defnition
The model geometry is a block with different
layers of different thickness varying in the verticalstratification. A cross section of a bore well with few
induced fractures by a cut section of the block is shown
in Fig. 1. The dimension of block is 500*400*600
cu. feet with diameter of cut section of bore hole is
2ft. There are multiple fractures originating at around
300 ft depth from the top surface of the block. These
fractures are taken as linear fractures for the sake of
simplicity in calculations. The block model is mainly
constructed on two materials: sandstone in the second
top and bottom layers and shale sandwiched between
two sandstone layers, while the top most layer is of
soil with a hole boundary that receives uid pressure.
4. Application of COMSOL Multiphysics
The model were established based on Biot
poroelast icity concepts and open hole multilatera l
well models from geo-mechanics and subsurface ow
modules of COMSOL Multiphysics 4.2a. We applied
the poroelasticity physics along with stationary study to
solve the given equations. We only used poroelasticity
for porous matrix and controlled flow for fluid
migration in fractures and the well; elsewhere we used
Darcy law (esdl) [14-17]. A controlled ow inside thefracture was assumed using the following governing
equation:
There was a major challenge of meshing of small
sized micro level elements in the model in 3D and
required high level of computing. The rectangular
parallelepiped box model helped developing certain
layers, depicting different rocks and fractures in the
intermediate shale reservoir. The elements of bore hole
In the above mentioned equation, u is the fluidDarcy velocity, k is the medium permeability, µ is the
dynamic viscosity of fluid, pf is the pressure gra-
dient, ρ is the fluid density, g is the gravitational
acceleration and is the unit vector in the direction
over which the gravity would take effect. Inputs and
few variables used in the model are mentioned in
Tables 1 and 2.
The terms of the fail parameters and ‘fail’
expression as mentioned in Table 2 are poro.sp1, poro.
sp2, poro.sp3 which denote principal stresses, p_r is
pressure in reservoir, pf is the pressure of injected uid,
C1 and C2 are the calibration constant of the model and phi is the friction angle in degrees [12].
The mathematical form as described of 3D
Coulomb failure criterion relates rock failure, three
principal stresses (σ1, σ2 and σ3) and the uid pressures
are as follows:
where, S o is the coulomb cohesion and ϕ is the
Coulomb friction angle.
On calibration, fail = 0 indicates the onset of rock
failure; fail < 0 denotes failure; and fail > 0 predicts
stability.
Since the model here solves for the change in pres-
sure caused by pumping at high pressure as well as the
stresses and strains and displacements that the pressure
change triggers, it calculates the expression by using
the change in pressure than its absolute value [13].
219
(5)
(7)
(8)
(6)
D∇
Table 1. Input values along with descriptions and units
Input Parameters Description Values Units p_w Pressure by uid 2.80 E + 06 [Pa]
So coulomb cohesion force 1.28 E + 07 [Pa]Phi Friction angle 24 [deg]C1 Calibration constant 1 14.7 Unit lessC2 Calibration constant 2 40 Unit less pf Pressure in fracture 5.10 E + 06 [Pa]
p_r Pressure in reservoir 8.60 E + 06 [Pa]
Table 2. Variables present in the model
N 2 * So * cos(phi)/(1 - sin(phi)) Fail parameter 1Q (1 + sin(phi))/(1 - sin(phi)) Fail parameter 2Fail (((poro.sp3 + C1 * (p_r - pf)) - Q * (poro.sp1 + C1 * (p_r - pf)) + N * (1 + (poro.sp2 -
poro.sp1)/(poro.sp3 - poro.sp1)))/C2) [1/psi]
Fail expression
fail = (σ 3 + p) – Q (σ 1 + p) +
N (1 + (σ 2 – σ 1 )/(σ 3 – σ 1 ))
Q = ((1 + sinϕ )/(1- sinϕ ))
N = ((2 cosϕ )/(1 – sinϕ ))S o
(5)0)( =∇∇
pk low
µ
∇
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Several studies on modeling of various natural and
engineering phenomena have gathered momentum in
recent years [13, 25-27]. In context of unconventional
reservoirs, validation of natural systems is often done
by simultaneous experimental and simulation studies
[28-33].
RESULTS AND DISCUSSION
The model was computed with the given set of
parameters and reservoir conditions and keeping all
other parameters same/constant, the fluid injection
pressure for fracture propagation was varied in the
established model. A different trend of displacement
behavior was wi tnessed, at near frac ture ti ps of
certain common set of fractures subjected to different
fluid pressure at different instances. Other physical-
mechanical properties (Table 4) of the reservoir rock
(shale) like Young’s modulus, cohesion, etc. along with
principal stresses, principal strains, elastic strain energyvaried due to change in injection pressure, starting at 10
MPa and increased by 2 MPa in the established model
in subsequent runs. A line joining the fracture tips of
three fractures of comparable geometry, lying on one
side of well was taken into consideration for plotting
the results of changes in hydro-mechanical behavior of
the rock as shown in Fig. 1b.
around the fractures were taken mostly symmetrical in
size. The shape of fractures are conical, all connected
to the bore well. The cut layered section of the whole
block model is taken for analysis of stress-strain and
displacement due to hydraulic pressure build up inside
well and fracture. The meshing has been fine with
variation of high quality to light low quality mesh
present at sharp corners to broad faces respectively
and the statistics of mesh is given in Table 3 [18,19].
In view of the scale and resolution as required for our
flow model, it is decided that an acceptable standard
finite element framework provided in COMSOL
Multiphysics is used to perform the implementation
[20]. Numerical modeling is widely used to understand
the fluid flow behavior in different unconventional
reservoirs [21-24].
The study of few fractures from the developed
model is discussed in this paper. The basic required
data and parameters were provided to the model as
inputs; shown in Table 1 and boundary conditions wereevaluated based on those parameters and variables
assigned during modeling.
The fail expression showed the conditions of
fracture generation and propagation depending upon
fracture pressure [pf], target reservoir pressure [p_r],
uid pressure [p_w], coulomb cohesion force [So] and
internal friction angle [phi] as major affecting values.
Fig. 1. (a) Mesh generation in the block having shale layer sandwiched between two sandstone layers and a top soil
covering as seen in 2D, (b) block model showings the red line (marked by arrow) along which the properties
variations are calculated. The line is joining the fracture tips or nearby points.
220
Table 3. Mesh statistics of the model
Property ValueMinimum element quality 4.109e-6Average element quality 0.6758
Tetrahedral elements 65934Triangular elements 9588
Edge elements 1281Vertex elements 83
Table 4. Physical-mechanical properties of material given
as input to the model
Property Name Value UnitYoung’s modulus E 30e9 Pa
Poisson’s ratio Nu 0.15 1Bulk modulus K 8.9e9 PaShear modulus G 1.38e10 PaDilation angle psid 0.1744 rad
a b
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Vishal et al., Sustain. Environ. Res., 25(4), 217-225 (2015)
tips. It is evident that passage of uids induces a rise in
the stresses in the rock. The stress was measured in the
direction perpendicular to the orientation of the fracture
length to understand the role of fluid in expansion ofthe fractures.
Stress increased up to 9.8 and 11.2 MPa in the
fractures in the second principal direction (Fig. 3).
Corresponding to the stress, deformation of rocks took
place and peak values of strain were attained. Stress
and strain in both principal directions other than that
acting parallel to fracture length are comparatively
less. However, high pressures inuence the rock on all
sides which indicates possible expansion other than
propagation along the fracture length. The case of
maximum injection pressure was studied thoroughly
for understanding a critical scenario of propagation offractures at high injection pressure of 20 MPa (Fig. 4).
While breakdown pressure is immediately transferred
at the fracture tips, pressure also builds up in the rock
strata in between the fractures. These high pressures
cause deformation in rock and fractures are initiated
and propagated.
The displacement curve (in Fig. 5) shows peaks
at the nearby fracture points through which the line
passes, that decline gradually and rise again before it
reaches another fracture point. This denotes the maxima
of displacements occurring in fracture at its tip during
propagation of cracks at breakdown pressure equal toinjection pressure. Red color in the legend also shows
the variation of displacement in the model subjected
to very high pressure. The average total volume
displacement is 0.50292 mm at {x, y: 3, -0.097} on
the selected plane (front face of fractures). While the
total displacement of the fracture tips was estimated
at approximately 1.162 to 1.186 mm, it is a reection
of deformation of rock matrix as a consequence of
fracture propagation.
A detail analysis in the propagation of fractures
The sudden dip or rise in the graphs is an
indication that the evaluation line is at or very near to
fracture tip. The x-axis in the Figs. 2-5 is the evaluation
line with y-axis being a variable property, here, principal strain, stress, injection fluid pressure and
total displacement. The horizontal and vertical axes
denote the length or distance for evaluating the size and
location of any point on the model (like normal axes)
and on the right side of right axes a legend or colors
are there with a different scale for evaluating say, total
surface displacement.
The negative principal stresses (Fig. 3) indicates
the compression by rock and the sudden positive values
indicate the interaction of negative and positive value
of rock and fluid pressure respectively, where fluid
pressure is very high. Thus expansion and propagationof fractures can be a possibility as shown by the
principal strain and stress graphs in Figs. 2 and 3.
Figures 2 and 3 represent the variation in strain and
stresses respectively along the line joining the fracture
221
Fig. 2. Second principal strain experienced by rock at left
side fracture tips joined by a hypothetical line as
shown in Fig. 1.
Fig. 3. Second principal stress experienced by rock at leftside fracture tips joined by a hypothetical line as
shown in Fig. 1 (Negative sign indicating reverse
direction).
Fig. 4. Injection fluid pressure of 20MPa as breakdown
pressure as experienced at left side fracture tips
joined by a hypothetical line as shown in Fig. 1.
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Vishal et al., Sustain. Environ. Res., 25(4), 217-225 (2015)
sharp near the fracture tips while it becomes gentle
away from the tips. It may be noted that there is
negligible deformation at 100 m away from the fracture
tip. This is particularly important to understand that the
rock strata are overall stable. Further, these contours
are vital to define the precise location of the fracture
to prevent any sort of propagation of fractures in the
adjoining rock strata. A clear indication of the same
is that, had the rock been fractured very close to the
overlying strata, the deformation contour lines wouldhave interfered leading to propagation of fracture in the
overlying strata. From the displacement graph (Fig. 5)
and contour diagram (Fig. 8), it stands that at the tip of
fracture high pressure exists which allow it to further
propagate in order to release the in-situ stresses.
The legend in the contour diagram indicates the
magnitude of displacement at every point lying on a
was done. The advantage of constructing 3-D
numerical models lies in the fact that the changes in
the rock dimensions can be clearly visualized and
quantied in all three principal directions. Red arrows
marked in Fig. 6 are representative displacement
vectors and it is clearly visible that deformation takes
place in rock strata parallel to the fracture arc as well
as in the perpendicular direction with both vertical
and horizontal components. It may be emphasized
that the zone near fracture tips experience maximum
displacement as indicated in red in Fig. 7. The
implication of such zones is that the rock is weakenedand underwent deformation. Any further rise in
injection pressure could lead to a fast propagating
facture.
The movement of rocks at the breakdown
pressure is not linear and straight in direction of uid
flow but so in transverse direction also aiding the
fracture expansion with propagation (Fig. 8). A clearer
demarcation of the displacement zone was identified
using contour lines. The gradient of displacement is
Fig. 6. Displacement eld in the model.
Fig. 7. Total volume displacement (in mm) in the model at20 MPa injection pressure in the mid reservoir.
222
Fig. 5. Total displacement at left side fracture tips joined
by a hypothetical line as shown in Fig. 1.
Fig. 8. Total surface displacement as illustrated by contourlines, each indicating a certain displacement value
in mm, at 20 MPa injection pressure in middle
reservoir.
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CONCLUSIONS
The current modeling approach mentioned in
the paper anticipates the conversion from stochastic
micro level crack generation to visible or macro level
localized fracturing unitedly with the development
of fluid flow in low-permeability rocks during
process of hydro-mechanical contact. The model was
established on the Biot equations for coupled fluid
ow and deformations in the rock, and a nite element
expression for the fluid pressure in the fracture. The porosity-permeability formulation allowed for a unied
representation of both the fracture and rock on the same
regular nite element grid.
Fracture extension forced by hydraulic pressure
was probed from the view point of coupled fracture-
ow interactions. It was assumed that a fracture event
happens instantaneously and that the fluid volume in
the fracture remains the same after an event of bond
breaking. The pressure drop in the fracture that follows
the breaking of a bond was computed with a procedure.
The behavior of rock is unique during different stages
of fracturing and fracture propagation and initiation.The fracture initiates at breakdown high pressure and
then with pore pressure the fracture propagates. With
time if fracture gets closed then with re-fracturing
pressure, it re-opens.
The trends shown with variation of principal
stresses and strains at tips of fracture with different
injection pressure indicates the rock behavior and its
tensile and fracturing property at different pressures
under in-situ condition. The modeling results indicate
that the chosen model is adequately efficient of
reproducing the development of hydraulic fracturing
and fluid flow in a physically naturalistic mode. This
formulation is able enough of symbolizing the two
critical pressures: Fracture initiation and breakdown
given color line. The shale reservoir block is subjected
to very high pressure: 10 to 20 MPa as compared to
nearby layers which are subjected to pressure of around
2.8 MPa. The values of displacement pore pressure andDarcy flow velocity were computed at each injection
pressure. All these parameters showed a direct linear
relationship with injection pressure (Figs. 9-11). The
value of total displacement increased from 0.452 mm
at 10 MPa injection pressure to 1.188 mm at 20 MPa
pressure. Rise of approximately 9.4 MPa occurred
as the injection pressure was increased from 10 to 20
MPa. Further the Darcian flow velocity increased by
nearly 136% as the injection pressure was increased
from its initial value to final value. Although high
injection pressure was applied, fluid flow in rocks
was at a relatively low velocity as compared to thereservoirs due to extremely low permeability of the
shale.
Fig. 11. Variation of Darcy’s ow velocity with injection
pressure.
223
Fig. 9. Variation of maximum displacement at different
injection pressure.
Fig. 10. Variation of maximum pore pressure with
injection pressure.
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Vishal et al., Sustain. Environ. Res., 25(4), 217-225 (2015)
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225
Discussions of this paper may appear in the discus-
sion section of a future issue. All discussions should
be submitted to the Editor-in-Chief within six months
of publication.
Manuscript Received: September 25, 2014
Revision Received: January 28, 2015
and Accepted: March 30, 2015