3d fracture propagation modeling

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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



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




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

∇




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




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.




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.




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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Singh, Permeability of sub-critical carbon dioxide

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

3d fracture propagation modeling

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