GRC Transactions, Vol. 39, 2015

937

3D Simulation of Mixed-Mode Poroelastic Fracture Propagation for Reservoir Stimulation

Dharmendra Kumar and Ahmad Ghassemi

Mewbourne School of Petroleum & Geological EngineeringThe University of Oklahoma, Norman, OK, USA

KeywordsPoroelastic rock deformation, mechanical interaction, “stress shadowing” effect, multiple fracture propagation, mixed-mode fracture propagation

ABSTRACT

The paper presents three-dimensional numerical model for the multiple fracture propagation in the poroelastic res-ervoirs. The simulator model is developed based on the fully-coupled 3D poroelastic displacement discontinuity method for the hydro-mechanical response of the reservoir rocks and the linear elastic fracture mechanics for the fracture propa-gation. The reservoir rock mass is assumed homogenous and isotropic with constant poroelastic physical properties. The mixed-mode fracture propagation is analyzed using crack-tip opening displacement approach. Details of mathematical formulations and methodology for numerical implementation are presented first. Then, the numerical model is verified using analytical solution for a pressurized penny-shaped fracture. Finally, numerical examples for the fluid injection into a cluster of natural fractures and simultaneous propagation of pressurized multiple fractures in Westerly Granite are presented. The results demonstrate that along with the rock properties and the in-situ stress conditions, the “stress shadowing” effect which mainly depends on the spatial interval between the fractures plays a critical role in the multiple fracture propaga-tion. Also, the simulation demonstrated that mixed-mode propagation can occur under influence of “stress shadowing”.

1. Introduction

The hydraulic fracturing of a single or multiple horizontal wells is extensively used to improve the productivity of the low permeability reservoirs in the oil and gas industry. In this work, we perform a poroelastic analysis of the simultaneous fracture propagation, which might be potentially extended for the horizontal well fracturing in geothermal reservoirs. The main mechanisms involved in the hydraulic fracturing are: rock mass deformation induced by fluid pressure on the fracture surfaces, the flow of fluid within the fractures, fluid leak-off between the fracture surfaces and the reservoirs rocks, and the fracture initiation and propagation. In poroelastic reservoirs, the increase of pore pressure around the fracture surfaces due to fluid leak-off leads to an expansion of the porous matrix. Traditionally, these expansion of the poroelastic rock has been addressed by including “back-stresses” and “back-pressure” (Cleary, 1977; Vadamme et al., 1989; Vandamme and Roegiers, 1990; Clifton and Wang, 1991).

Mechanical interaction between multiple fractures is caused by “stress shadowing” effect. The term stress shadow represents variation of stresses in the region surrounding a pressurized fracture. If a second hydraulic is created parallel to an existing opened fracture and within the stress shadow region, the second fracture would experience a stress greater than the original in-situ stress. As a consequence, a higher fluid pressure would be required to propagate the second fracture and results in narrower fracture width at the same fluid pressure. The effect of stress shadow when multiple hydraulic fractures are created parallel to each other is of major interest. The spacing between the fracture surfaces and in-situ stress contrast play an important role in the mechanical interaction (Sesetty and Ghassemi, 2015). Several numerical models have been

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presented to study multiple and multistage fracturing from the single and multiple horizontal well in the elastic media based on two-dimensional models (Rafiee et al., 2012; Sesetty and Ghassemi, 2013; Bunger, 2013) or planar 3D model (Wong et al., 2013). Kumar and Ghassemi (2015) have presented a coupled 3D displacement discontinuity model for the elastic reservoirs with capabilities to simulate any number of fractures in case of simultaneous or sequential propagation schemes from a single or multiple horizontal wells. In this paper, we present a fully-coupled 3D poroelastic analysis of simultaneous propagation of multiple fractures from a horizontal well. As the main emphasis have been given to evaluate the influences of induce stresses change or “stress shadowing” effect on multiple fracture propagation, the fracture fluid flow is not considered at this time. Hence, all the numerical examples have been presented by applying a uniform fluid pressure on the fracture surfaces. The major details of the problem formulation for the case of a stationary fracture can be found in Ghassemi et al. (2013) and Safari and Ghassemi (2014). A brief mathematical description of the governing equations is provided in this paper.

2. Theory and Governing Equations

The physical problem of hydraulic fracture simulation in a poroelastic rock consists of main mechanisms: hydro-mechanical coupled response of the rock matrix, the fluid flow inside the fracture, the fluid leak-off from the fracture to the reservoir matrix, and the fracture propagation. It is worth emphasizing that deformation of the rock matrix would likely result in natural fracture slippage and/or failure of intact rock (e.g., see, Ghassemi et al., 2013; Wang and Ghassemi, 2013; Sesetty and Ghassemi, 2015; Safari and Ghassemi, 2015). Such process can be considered in our model, but are not described in this paper.

2.1. Poroelastic Deformation of Rock Matrix The three-dimensional field equations for the poroelastic rock matrix are given using the Navier’s equations with

a coupling terms and a fluid diffusion equation as (Rice and Cleary, 1976):

G ∇2ui +G

(1− 2ν)uk ,ki −α p,i = 0 (1)

∂ p∂t

−2κGB2 (1− 2ν) (1+ νu )

2

9(νu − ν) (1− 2νu )∇32 p = −

2GB(1+ νu )3(1− 2νu )

∂ε∂t

(2)

where ui is the solid displacement in the ith direction, p is the pore pressure, α is the Biot’s effective stress coefficient, G is the shear modulus, v and vu are the drained and undrained Poisson’s ratio, respectively, κ represents the reservoir rock permeability coefficient, and ɛ is the volumetric strain.

3. Numerical Implementation

The poroelastic displacement discontinuity (DD) method (an indirect boundary element method) is used for the fracture deformation modeling (Zhou and Ghassemi, 2011; Ghassemi et al., 2013).

The fracture propagation analysis is implemented using the linear elastic fracture mechanics (LEFM) approach. The detailed numerical procedure for each component are provided in following sections.

3.1. Boundary Integral Equation for Poroelastic Rock DeformationThe fracture in a poroelastic medium can be regarded as a surface across which the solid displacements and the

normal fluid flux are discontinuous. These discontinuities can be mathematically presented by a distribution of impulse point displacement discontinuity (DD) and fluid sources intensities (Df ) over time and space. If, the displacement discon-tinuities and fluid sources intensities over the fracture surface are known, the stresses and the pore pressure at any point in the reservoir can be evaluated using the integral equation method. In particular, the integral representations for the stresses and pore pressure at any point can be expressed as follows (Zhou and Ghassemi, 2011; Safari and Ghassemi, 2014):

σ ij (x,t) = σ ijknid (x − ′x ; t − ′t ) Dkn( ′x , ′t )+ σ ij

if (x − ′x ; t − ′t )Df ( ′x , ′t ){ }Γ∫0t

∫ dΓ( ′x )d ′t +σ ij0(x) (3)

p (x,t) = pijid (x − ′x ;t − ′t )Dij ( ′x , ′t )+ pif (x − ′x ;t − ′t )Df ( ′x , ′t ){ }Γ∫0

t

∫ dΓ( ′x )d ′t + p0(x) (4)

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where x and x′ are the source and field points, respectively, t is the current time and t′ is the time at which the location x′ receives the fluid flow first time, Dkn and Df represents the displacement discontinuities and fluid source intensity, respec-tively, σ ijknid , σ ijif , pijid , and pif are the instantaneous fundamental solutions for the stresses and pore pressures due to a unit impulse of the displacement discontinuity “id” in the “kn” direction and a unit impulse of the fluid source intensity “if”, and σijo(x) and p0(x) presents the in-situ stresses and the reservoir pore pressure, respectively.

3.2. Discretization of Boundary Integral Equation For numerical implementation, Eqs. (3) and (4) can be discretized in the spatial and time domains. The 4-node

quadrilateral elements are used for spatial discretization of the fracture surface. The temporal integration is performed using the time convolution algorithm (Dargush and Banerjee, 1989; Zhou and Ghassemi, 2011). For a particular time step, a constant variation of the displacement discontinuities (DD’s) and a linear variation of the fluid sources intensity (Df ) are assumed in an element. The discretized form of the boundary integral equations (3) and (4) can be given as:

σ ij (x,NΔt) =σ ijknid (x − ′x ;Δ t) Dkn( ′x ,NΔ t)

+ φ(m) σ ijif (x − ′x ;NΔ t)Df ( ′x ,NΔt)

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪dΓ( ′x )

Γe∫m=1

M∑⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

+ B(s) (x,t)+ σ ij0 (x) (5)

p(x,NΔt) =pijid (x − ′x ;Δt)Dij ( ′x ,Δt)

+ φ(m) pif (x − ′x ;Δt)Df ( ′x ,Δt)

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪dΓ( ′x )

Γe∫m=1

M∑⎡

⎣

⎢⎢

⎤

⎦

⎥⎥+ B(p) (x,t)+ p0(x) (6)

where M represents total number of elements, N is the number of time steps, Δt is the time increment, ϕ(m) represents the shape functions, B(s) and B(p) are the back-stresses and back-pressure from the previous time steps. A methodology to estimate these back-stresses and back-pressure can be found in detail as described by Ghassemi et al. (2013). To numeri-cally solve the problem, the boundary integral equation for stresses (i.e., Eq. 5) is satisfied at the centroid of each elements and the boundary integral equation for fluid pressure (i.e., Eq.6) is satisfied at the nodal positions following the point collocation technique. In process of point collocation technique, special cases occurs when the field and the collocation point coincide during integration. In these cases, different mathematical singularities occurs which requires an accurate estimation. The singularity subtraction technique as proposed by Guiggiani et al. (1992) and as implemented in Zhou and Ghassemi (2011) is used to handle the hyper-singularity issue.

For hydraulic fractures, the Eqs. (5) and (6) are subjected to complementary conditions based on intrinsic behavior of natural fractures in rock (Safari, 2013; Safari and Ghassemi, 2014). The effective stress on the natural fracture surface becomes zero when they become hydraulic fractures and there is no contact of the surfaces and two surfaces of the frac-ture are separated from each other. Hence, the resultant stresses on the fracture surfaces are due to the normal stress and fluid pressure and no shear stresses acts on the fracture surfaces. Thus, the boundary conditions for a hydraulic fracture are given as (Safari and Ghassemi, 2014):

p(x,t) = p f (x,t) ; σn(x,t) = − p f (x,t)

σs1(x,t) = 0 ; σs2(x,t) = 0 (7)

where pf represents the applied fluid pressure, σn represents the stresses acting normal to the fracture surface, and σs1 and σs2 are the in-plane and out-of-plane shear stresses acting on the fracture surface, respectively.

3.3. Fracture Propagation Analysis The fracture propagation process is analyzed in the frame work of the linear elastic fracture mechanics (LEFM),

which originates from the Griffith concept of crack stability (Griffith, 1924). Irwin (1957) had generalized the Griffith theory and demonstrated that the Griffith’s criterion for the crack growth is essentially similar to the crack growth approach based on the stress intensity factor (e.g., the crack growth will start when the stress intensity factor, K, reaches a critical values of the fracture toughness, KIC ). The maximum circumferential stress criterion as proposed by Erdogan and Sih (1963) is mostly used for hydraulic fracture propagation. This criterion suggests that the fracture will grow in a direction perpendicular to the minimum principal stress at the crack-tip. Consider, a 3D crack in the mixed-mode loading condition and introduce a local polar coordinate system (r, θ) at the crack-tip as shown in Fig. 1. In absence of effect of out-of-plane shear deformation on the fracture propagation (i.e., effect of mode-III fracture opening), the fracture will propagate in mixed-mode in the normal (n) and bi-normal plane (b). Based on the maximization of the circumferential stresses at the crack-tip, the mixed-mode fracture propagation condition follows as (Erdogan and Sih, 1963):

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KI sinθ0 + KII (3cosθ0 −1) = 0 (8)

where θ0 represents the fracture propagation angle, KI and KII are the mode-I and mode-II stress intensity factors, re-spectively. Rearrangement of above equation gives the fracture propagation angle as (Aliabadi and Rooke, 1991; Stone and Babuska, 1998):

θ0 =

0 ; KII = 0

2 tan−1 KI4KII

− sgn(KII )4 ⋅ KIKII

⎛⎝⎜

⎞⎠⎟2+8

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

; KII ≠ 0

⎧

⎨⎪⎪

⎩⎪⎪

(9)

where represents the Signum function (equal to 1, if and equal to -1, if ). The fracture propagation analysis task can be divided into two parts: the prediction of the fracture propagation, the propagation direction, and its growth rate. Details of components of propagation analysis are provided in following sections.

3.3.1. Estimation of Stress Intensity FactorsThe stress intensity factor (SIF) represents a measure of intensity of the

near crack-tip stress singularity which plays a key role in the LEFM analysis of the fracture propagation process. The magnitude of the SIF depends on the applied stresses and the geometry of the crack. The estimation of its value is a function of the location of estimation points near the crack tip. The SIF values provides a basis for analysis of the fracture propagation process within the LEFM analysis. The SIF values terms of crack tip opening displacements can be given as follows (Aliabadi and Rooke, 1991):

KI =G

4 1− ν( )2πr⋅Dn(r)

KII =G

4 1− ν( )2πr⋅Ds1(r)

KIII =G4

2πr⋅Ds2(r)

(10)

where KI, KII and KIII represents the mode-I, mode-II, and mode-III SIFs, respectively, Dn is the normal fracture opening, Ds1 and Ds2 represents the in-plane shear displacement and out-of-plane shear dis-placement, respectively, r is the distance from the estimation point to the crack-tip, G and ν represents the Shear modulus and Poisson’s ratio, respectively. In this work, the SIF values are calculated at a distance r = 0.87L (Zhang and Ghassemi, 2004; Safari and Ghassemi, 2014), where L represents the crack tip element length as shown in Fig. 2. The selection of this estimation distance is based on the agreement of the numerically calculated SIF values with the known analytical solutions. A verification of this distance assumption is provided by the same authors in Kumar and Ghassemi (2015).

3.3.2. Fracture Propagation Growth RateSecond important task of the fracture propagation analysis is to determine the optimum propagation length which

should satisfy the fracture mechanics criterion. The magnitude of fracture growth for any front node can be given using a scaling law as proposed by Mastrojannis et al. (1980):

Δai =

0 ; KIi ≤ KIC

Δa0KIi − KIC

KI ,maxi − KIC

⎛

⎝⎜⎜

⎞

⎠⎟⎟

m

; KIi > KIC

⎧

⎨⎪⎪

⎩⎪⎪

(11)

Figure 1. Representation of near crack-tip local polar coordinates.

Figure 2. Representation of a 3D crack tip element.

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where Δαi is the growth rate of the ith front node, Δα0 is the maximum growth of a front node with maximum value of effective stress intensity factor (KI ), KIC represents the mode-I fracture toughness, and m is a material dependent con-stant which is usually taken equal to 1 for the hydraulic fracture propagation. The value of the maximum fracture growth is taken equal to 10% of the initial fracture radius in this study. The effective stress intensity factor, which account for both the mode-I opening and mode-II shear displacement in the fracture propagation, is defined based on the asymptotic behavior of the circumferential tensile stress (σθθ) near the crack-tip as (Erdogan and Sih, 1963):

σθθ(r,θ) =KI2πr

+O ( r ) ; KI = KI cos3 θ02

⎛⎝⎜

⎞⎠⎟− 32KII cos

θ02

⎛⎝⎜

⎞⎠⎟sinθ0 (12)

4. Model Verification

These poroelastic deformation model has been verified previously for the stationary cracks by Zhou and Ghassemi (2011), Ghassemi et al. (2013), and Safari and Ghassemi (2014). A verification of the fracture a propagation in the poro-elastic media is presented in this study. The reservoir rock is assumed to be Westerly Granite with solid properties as listed in Table 1. For model verification, consider a pressurized planar penny-shaped fracture. At initial propagation time t = 0+, the fracture opening can be given following the classical solution by Sneddon (1946) with undrained material properties as:

w(r) =4(1− νu ) ⋅( p f −σn )a

πG1− r

a⎛⎝⎜

⎞⎠⎟

2

; r / a <1 (13)

where G and vu represents the shear modulus and undrained Poisson’s ratio, respectively, pf is the applied fluid pressure and σn is the component of in-situ stresses normal to the fracture surface, a is the fracture radius, r is the distance from any node on the fracture surface to the its center. The in-situ stress normal to the fracture surface is assumed equal to 33.39 (MPa) and a uniform fluid pressure equal to 40.29 (MPa) is applied. The initial reservoir pore pressure is assumed equal to 28.3 (MPa).The fracture is propagated using 1(min) time increment. The distribution of the fracture opening after 4 propagation steps (i.e., after 4 min) is shown in Fig. 3(a) and a comparison of analytical solution and numerical solution is presented in Fig. 3(b). The numerical fracture opening has shown a good agreement with the analytical solution.

5. Numerical Examples

Numerical examples of the fluid injection into natural fractures and simultaneous multiple fracture propagation from a horizontal well are presented in this section. The numerical model doesn’t have any limitation to simulate large

Table 1. Input parameters (Ghassemi and Zhang, 2006).

Shear modulus, G (MPa) 15.0

Drained Poisson ratio, v 0.25

Undrained Poisson’s ratio, vu 0.337

Reservoir permeability, κ (darcy) 4.053×10-7

Skempton’s coefficient, B 0.815

Mode-I fracture toughness, KIC (MPa.m0.5) 2.46

Figure 3. (a) Distribution of fracture opening, (b) comparison of analytical and numerical (initial fracture radius = 5 m, time increment = 1 min, propagation steps = 4).

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scale field problems; however, due to computational limitation, the small scale simulation examples are presented in this paper. By selecting appropriate input parameters and the initial fracture geometries, these examples can be easily extended for large scale field problems. The numerical simulations have been performed on the Dell Precision T7600 (2.30 GHz) machine and corresponding simulation times are reported for each example.

5.1. Stimulation of Natural Fractures for EGSThe model of wing crack propagation has been studies for many decades. Its role in enhanced geothermal system

(EGS) stimulation was discussed in the 2012 GRC Stimulation Workshop. Jung (2013) has studied this propagation mode in reservoir stimulation using field evidence and stress/fracture analysis. In this example, we explicitly simulated the re-sponse of the pre-existing natural fractures in a geothermal reservoir to the fluid injection into it. Ghassemi et al. (2015) presented the similar problem for fluid injection into 2D natural fractures. In this case, the problem is simulated using the elastic module by turning off the poroelastic part. The fracturing fluid is assumed incompressible and shows Newtonian behavior. The coupled problem of fracture aperture and fluid flow is iteratively solved using Picard iteration method (Yew, 1997). A cluster of two circular fractures with radius equal to 5(m) and initial opening equal to 10-6 (m) are considered. The fracture planes are inclined by 45° from the x-axis. The vertical distance (e.g., interval along z-axis) between center of the fractures is assumed equal to 18 (m) and the fractures are separated by 5(m) along x-axis. The rock physical proper-

Figure 4. (a) Distribution of fracture opening, (b) side view of propagated fracture geometry, (c) in-plane shear displacement, (d) out-of-plane shear displacement after 10 propagation steps (fluid injection time = 8 sec., simulation time = 2.46 min).

(a)

(c)

(b)

(d)

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ties are the same as listed in Table 1. The fracture system is subjected to in-situ stress condition as: σV = 42.4 (MPa), σH = 39.5 (MPa), and σh =38.0 (MPa). The fluid with viscosity equal to 0.005 N.s/m2 (5 cP) is injected at the center of each fracture with injection rate equal to 0.24 m3/s (90 bbl./min). The results of fracture opening, shear displacements, and the propagated fracture geometries after 10 propagation steps are shown in Figs. 4. As the fracture surfaces are pressurized, they start to propagate in the mixed-mode and eventually tends to align with maximum principal stress direction (e.g., z-axis). It can be observed from Fig. 4(b) that as fractures propagate, they follows wing-crack. Eventually these fractures will merge together to form a single big fracture or their wings will intersect other fractures as described by Jung (2013). The result is increase in the permeability of the reservoir. From Fig. 4(c), it can be seen that all fractures experience high shear deformation leading to shear slippage of the natural fracture surfaces.

5.2. Simultaneous Propagation of Multiple Fractures From a Horizontal Well In case of simultaneous propagation, all the fractures propagation starts at the same time. The opening of a fracture

induces additional stresses on surrounding rocks and on the adjacent fractures, which may alter the direction of maximum horizontal stress. As a results, the fracture start propagating in a non-planar manner. Few examples for multiple fractures have been presented to illustrate complex fracture interaction and effect of stress shadowing on the fracture propagation. The horizontal well is aligned with the x-axis. The minimum and maximum horizontal in-situ stresses are acting along

Figure 5. (a) Distribution of fracture opening, (b) in-plane shear displacement, (c) out-of-plane shear displacement, (d) fluid leak-off after 10 propa-gation steps (time increment = 5 min, total diffusion time = 50 min, simulation time = 61.48 min).

(a)

(c)

(b)

(d)

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x- and y- axes, respectively. The vertical in-situ stress is acting along z-axis. Two different cases depending on the in-situ stress conditions are presented in the following examples.

Case 1. Isotropic In-Situ Stress ConditionIn this case, two initially parallel planar fractures with initial radius of 1 (m) and growing in an isotropic in-situ

stress fields are considered. The initial distance between the fractures is assumed equal to 1 (m) and the initial fractures are orientated parallel to y-z plane. The isotropic in-situ stresses are assumed equal to 34.81 (MPa) and a uniform fluid pressure equal to 45 (MPa) is applied over the fracture surfaces. The initial reservoir pore pressure is assumed equal to 17.4 (MPa). As the fracture surfaces are pressurized, the fracture propagation starts radially in relation to their centers. Each fracture induces stresses in the reservoir that influences propagation of other fractures. The resulting fractures geom-etries and the distributions of the fracture opening, shear displacements and fluid leak-off after 10 propagation steps are shown in Figs. 5. Because the fractures induce equal stresses on each other, both experience mixed mode propagation and have the same growth rate but opposite propagation directions. The resultant fractures geometries are anti-symmetric and bowl-shaped. Both fractures have the same magnitude of the normal opening, shear displacements, and the fluid leak-off.

Case 2. Anisotropic In-Situ Stress ConditionIn the second case, two initially parallel planar fractures with initial radius equal to 1 (m) and growing in an anisotropic

in-situ stress fields are considered. The distance between the fractures is assumed equal to 1 (m). The rock properties are the same as listed in Table 1. The fractures are considered at a depth of 2330 (m) with in-situ stress condition as: vertical stress, σV = 60.13 (MPa), maximum horizontal stress, σH = 43.5 (MPa), minimum horizontal stress, σh =34.81 (MPa), and the reservoir pore pressure, p0 = 17.4 (MPa). In this case, the fractures are subjected to two competing types of stress components: one is the in-situ stress contrast and the other is due to stress shadowing. The “stress shadowing” effect tends to propagate the fractures in mixed mode, whereas the in-situ stress contrast tends to promote fracture propagation along the maximum principal stress direction (e.g., along z-axis in this case). As a result, a twisted non-symmetric fracture geometry as shown in Figs. 6. are generated. In this case, the fractures have propagated longer in the vertical direction as compare to the previous isotropic cases.

6. Conclusion A computational model for 3D hydraulic fracture propagation simulation in the poroelastic rock has been developed.

The coupled 3D poroelastic displacement discontinuity method for the solid deformation modeling presents a numerically and computationally efficient numerical scheme; since, only the fracture surface discretization is required. The simulta-neous propagation of multiple fractures has been presented to illustrate the coalescence of wing cracks during injection, resulting in enhanced permeability. In addition, multiple fracture propagation from horizontal wells has been studied. The

Figure 6. (a) Distribution of fracture opening (b) side view of propagated fracture geometry after 10 propagation steps (time increment = 5 min, total diffusion time = 50 min, simulation time = 60.83 min).

(a) (b)

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results shows that several factors such interval between the fracture, in-situ stress contrast and orientation of the fractures with respect to wellbore axis have strong effects on the generated fracture geometries. The stresses shadowing effect which mainly depends on the interval between the fractures shows a strong influence on the simultaneous and sequential propagation of the fractures. The deviation of the fracture is greater in case of low stress anisotropy and isotropic in-situ stress condition as compare to strongly anisotropic condition, because fractures tend to propagate in the maximum stress direction and for strong stress anisotropy the stress shadow is not dominant.

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