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A New Approach for Numerical Modeling of Hydraulic Fracture Propagationin Naturally Fractured ReservoirsR. Keshavarzi, Young Researchers Club, Science and Research Branch, Islamic Azad University, Tehran, Iran;S. Mohammadi, School of Civil Engineering, University of Tehran, Tehran, Iran

Copyright 2012, Society of Petroleum Engineers

This paper was prepared for presentation at the SPE/EAGE European Unconventional Resources Conference and Exhibition held in Vienna, Austria, 20-22 March 2012.

This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not beenreviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, itsofficers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission toreproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

Abstract

Hydraulic fracturing of a naturally fractured reservoir is a challenge for petroleum industry, as fractures can have complex

growth patterns when propagating in systems of natural fractures that leads to significant diversion of hydraulic fracture paths

due to intersection with natural fractures which causes difficulties in proppant transport. In this study, an eXtended Finite

Element Method (XFEM) model has been developed to account for hydraulic fracture propagation and interaction with natural

fracture in naturally fractured reservoirs including fractures intersection criteria into the model. It is assumed that fractures are

 propagating in an elastic medium under plane strain and quasi-static conditions. The results indicate that hydraulic fracture

diversion before and after intersecting with natural fracture is strongly controlled by the in-situ horizontal differential stress

and the orientation of the natural fractures as well as hydraulic fracture net pressure. It is observed that hydraulic fracture net

 pressure increase leads to decreasing induced fracture diversion and in-situ horizontal differential stress decrease results in

increasing induced fracture diversion before intersecting with natural fracture. In addition, potential debonding of sealed

natural fracture in the near-tip region of a propagating hydraulic fracture before fractures intersection has been modeled which

is one of the phenomena that has been rarely taken into account, as debonding of natural fracture before fractures intersection

is of great importance that may lead to diverting the induced fracture into double-deflection in natural fracture and can explain

hydraulic fracture behaviors due to interaction with natural fracture at different conditions. Also, it’s been observed that at low

angles of approach with low to high differential stress, the induced hydraulic fracture opens the natural fracture while at high

to medium angles of approach, natural fracture opening and crossing both are observed depending on the differential stress.

Comparison of the numerical and experimental studies results has shown good agreement.

Introduction

Hydraulic fracture growth through naturally fractured reservoirs presents theoretical, design, and application challenges since

hydraulic and natural fracture interaction can significantly affect hydraulic fracturing propagation. Although hydraulic

fracturing has been used for decades for the stimulation of oil and gas reservoirs, a thorough understanding of the interaction

 between induced hydraulic fractures and natural fractures is still lacking whereas the interaction between pre-existing natural

fractures and the advancing hydraulic fracture is a key challenge especially in unconventional gas reservoirs, because withoutfractures, it is not possible to recover hydrocarbons from these reservoirs. Meanwhile, natural fracture systems are important

and should be considered for optimal stimulation. For naturally fractured formations under reservoir conditions, natural

fractures are narrow apertures which are around  10-5

to 10-3

  m wide and have high length/width ratios (>1000:1) (Liu

2005).Typically natural fractures are partially or completely sealed but this does not mean that they can be ignored while

designing well completion processes since they act as planes of weakness reactivated during hydraulic fracturing treatments

that improves the efficiency of stimulation (Gale et al. 2007). The problem of hydraulic and natural fractures interaction has

 been widely investigated both experimentally (Lammont and Jessen 1963; Blanton 1982; Warpinski and Teufel 1987; Zhou et

al. 2008; Athavale and Miskimins 2008) and numerically (Jeffrey and Zhang 2009; Rahman et al. 2009; Dahi Taleghani 2009;

Chuprakov et al. 2010; McLennan et al. 2010; Min et al. 2010; Akulich and Zvyagin 2008). Many field experiments also

demonstrated that encountering a propagating hydraulic fracture and natural fractures may lead to arrest of fracture

 propagation, fluid flow into natural fracture, creation of multiple fractures and fracture offsets (Stadulis 1995; Britt and Hager

1994; Rodgerson 2000, Jeffrey et al. 2010) which will result in a reduced fracture width. This reduction in hydraulic fracture

width may cause proppant bridging and consequent premature blocking of proppant transport (so-called screenout)

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(Economides 1995;  Potluri et al.2005) and finally treatment failure. Although various authors have provided fracture

interaction criteria (Blanton 1982; Warpinski and Teufel 1987; Renshaw and Pollard 1995) but still determining the induced

fracture growth path due to interaction with pre-existing fracture and getting a viewpoint about variable or variables which

have a decisive impact on hydraulic fracturing propagation in naturally fractured reservoirs is still unclear and highly

controversial. However, experimental studies have suggested that horizontal differential stress, angle of approach and

treatment pressure are the parameters affect hydraulic and natural fracture interaction (Blanton 1982; Warpinski and Teufel

1987; Zhou et al. 2008) but a comprehensive analysis of how different parameters influence the fracture behavior has not been

fully investigated to date. For the purpose of this study, an eXtended finite element method (XFEM) is applied to simulatehydraulic fracture propagation and intersection. The motivations behind applying XFEM are the desire to avoid remeshing in

each step of the fracture propagation, being able to consider arbitrary varying geometry of natural fractures and the

insensitivity of fracture propagation to mesh geometry which improves the accuracy of the solution (Mohammadi 2008). The

main objective of this paper was to investigate hydraulic fracturing propagation in naturally fractured reservoirs and the

dominant factors governing the diversion of hydraulic fractures in the presence of natural fractures, debonding of natural

fracture before intersection with hydraulic fracture and hydraulic and natural fracture intersection through a 2D XFEM

numerical model.

Interaction between Hydraulic and Natural Fractures

The interaction between pre-existing natural fractures and the advancing hydraulic fracture is a key issue leading to complex

fracture patterns. Large populations of natural fractures are sealed by precipitated cements which are weakly bonded with

weak mineralization and adhesion that even if there is no porosity in the sealed fractures, they may still serve as weak paths forthe growing hydraulic fractures (Gale et al. 2007). In this way, experimental studies (Blanton 1982; Warpinski and Teufel

1987; Zhou et al. 2008) suggested several possibilities that may occur during hydraulic and natural fractures interaction.

Blanton (Blanton 1982) conducted some experiments on naturally fractured Devonian shale as well as blocks of hydrostone in

which the angle of approach and horizontal differential stress were varied to analyze hydraulic and natural fracture interaction

in various angles of approach and horizontal differential stresses. He concluded that any change in angle of approach and

horizontal differential stress can affect hydraulic fracture propagation behavior when it encounters a natural fracture which

will be referred to as opening, arresting and crossing. Warpinski and Teufel (Warpinski and Teufel 1987) investigated the

effect of geologic discontinuities on hydraulic fracture propagation by conducting mineback experiments and laboratory

studies on Coconino sandstone having pre-existing joints. They observed three modes of induced fracture propagation which

were crossing, arrest by opening the joint and arrest by shear slippage of the joint with no dilation and fluid flow along the

 joint. In 2008 (Zhou et al. 2008) some laboratory experiments were performed to investigate the interaction between hydraulic

and natural fractures. They also observed three types of interactions between hydraulic and pre-existing fractures which were

the same as Warpinski and Teufel’s observations. The above referenced experimental studies have investigated the initial

interaction between the induced fracture and the natural fracture, however, in reality may be the hydraulic fracture is arrested

 by natural fracture temporarily but with continued pumping of the fluid, the hydraulic fracture crosses at the intersection

(Fig.1a) or turns into the natural fracture and opens it (Fig.1b) depending on the fluid pressure distribution. In some cases the

hydraulic fracture may get arrested if the natural fracture is long enough and favorably oriented to accept and divert the fluid.

Also, potential debonding of sealed natural fracture in the near-tip region of a propagating hydraulic fracture before fractures

intersection is one of the phenomena that has been rarely taken into account which can somehow explain hydraulic and natural

fracture behaviors before and after intersection. Debonding of natural fracture prior to intersection with hydraulic fracture is

due to tensile stress exerted ahead of hydraulic fracture tip and if this stress is large enough, it debonds the sealed natural

fracture (Lawn, 2004).

Fig. 1— Schematic illustration for hydraulic and natural fracture interaction: (a) hydraulic fracture crosses the natural fracture and (b)

hydraulic fracture turns into natural fracture and opens it.  

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Fracture Propagation and Interaction Criteria

Various authors (Blanton 1982; Warpinski and Teufel 1987) have provided analytical and empirical equations and gave

several relations for predicting the interaction between a natural fracture and hydraulic fracture based on experimental studies

 but they only considered the initial interaction between the induced fracture and the natural fracture. In this study, a linear

elastic fracture mechanics (LEFM) criteria has been applied for fracture propagation and interaction. Fracture propagation in

LEFM is a function of opening and shearing mode stress intensity factors ( K  I and  K  II , respectively), which are measures of

stress concentration at the tip of the crack. The two stress intensity factors are combined in the energy release rate fracture

 propagation criterion used in this research. The energy release rate, G, is related to the stress intensity factors through Eq. 1(Mohammadi 2008):

)(1   22

 II  I   K  K 

 E G   +

!=   (1)

 

where E’  =  E for plane stress ( E is Young's modulus) and E’  = E /(1- !2 ) for plane strain (where  ! is the Poisson's ratio). If the

energy release rate is greater than a critical value, Gc, the fracture will propagate critically. The direction of hydraulic fracture

 propagation will be calculated by Eq. 2 (Sukumar and Prévost 2003):

!!!

"

#

$$$

%

&+!!

"

#$$

%

&±=   8

4

12tan

2

1-

 II 

 I 

 II 

 I 

c

 K 

 K 

 K 

 K '    (2)

 

During hydraulic and natural fracture interaction at the intersection point the hydraulic fracture has more than one path to

follow which are opening or crossing. The most likely path is the one that has the maximum energy release rate. So, at the

intersection point energy release rate is calculated for both opening, Gopening  , and crossing , Gcrossing , and if (Gopening  /Gcrossing )>1

opening will occur and hydraulic fracture turns into natural fracture and opens it, while if (Gopening  /Gcrossing )< 1 crossing takes

 place and hydraulic fracture crosses the natural fracture.

Extended Finite Element Method FormulationTo model hydraulic fracture propagation and interaction with natural fracture, one of the most efficient and newest numerical

methods called extended finite element method (XFEM) has been applied. XFEM was developed in 1999 (Moës et al. 1999) to

help the shortcomings of the conventional finite element method and has been used to model the propagation of various

discontinuities such as cracks and fractures. The difficulties of conventional finite element method do not exist in simulationsmodeled by the extended finite element since the crack is not modeled as a geometric entity and it does not need to conform to

element edges. Similarly, crack propagation can be modeled without any remeshing (Mohammadi 2008). This method was

originally developed to enrich the displacement field near a crack tip in the standard finite element method (Belytschko and

Black 1999), and later was extended to modeling discontinuities (such as fractures) and arbitrary branched and intersecting

fractures. Therefore, XFEM is particularly well-suited for modeling hydraulic fracture propagation and interaction with natural

fracture. In XFEM approximation, the displacement field at any point is divided into two parts as shown in Eq. 3 (Mohammadi

2008).

!!! !

!"! !

!"#  (3) 

Where uh, u

 FE  and u

 ENR are approximated displacement field, conventional (continuous) and enriched (discontinuous) parts of

the displacement approximation, respectively. Eq. 3 can be rewritten as below (Mohammadi 2008),

!!!   !   !!   !   !! !

!

!!!

!!   !   !   !   !!

!

!!!

 

where u j is the vector of regular degrees of nodal freedom in the finite element method,  N  is a shape function, ak  is the added

set of degrees of freedom to the standard finite element model and "  (x) is the discontinuous enrichment function defined for

the set of nodes that the discontinuity has in its influence domain. Eq. 4 can be rearranged in order to model crack surfaces and

tips in the extended finite element method as below (Mohammadi 2008),

(4)

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!!!   !   !!   !   !! !

!

!!!

!!   !   !   !   !   ! !   !   !!   !!

!

!!!

 

!   !!   !   !!!! !!

!!!   !!

!!

!"

!!!

!"!

!!!

 

!   !!   !   !!!! !!

!!!   !!

!!

!"

!!!

!"!

!!!

 

where m is the set of nodes that have the crack face (but not the crack tip) in their support domain, while mt 1 and mt 2 are the

sets of nodes associated with crack tips 1 and 2 in their influence domain, respectively; u j  are the nodal displacements

(standard degrees of freedom). H( #  ) is crack face enrichment function and ah , !!! and !!

! 

are vectors of additional degrees of

nodal freedom for modeling crack faces and the two crack tips, respectively, and !!

!  (x), i =1, 2 represent mf crack tip

enrichment functions. So, after displacement approximation, strain and stress can be computed in a fractured body.

Numerical ResultsModeling hydraulic fracturing process by itself is a complicated phenomenon due to the heterogeneity of the earth structure,

in-situ stresses, rock behavior and the physical complexities of the problem, hence if natural fracture is added up to the

 problem its getting much more complex in both field operation and numerical aspects. For simplicity, it is assumed that rock is

a homogeneous isotropic material and the fractures are propagating in an elastic medium under plane strain and quasi-static

conditions by a constant and uniform net pressure throughout the hydraulic fracture system. A schematic illustration for the

 problem has been presented in Fig. 2 which shows that hydraulic fracture propagates toward the natural fracture and intersects

with it at a specific angle of approach, $ , and in-situ horizontal differential stress, (% 1 -% 3).

Fig. 2— Schematic of hydraulic fracture intersecting pre-existing natural fracture. 

So, a 2D XFEM code has been developed to model hydraulic fracture propagation in naturally fractured reservoirs and

interaction with natural fractures. For this purpose, firstly Warpinski and Teufel’s (Warpinski and Teufel 1987) experiments

have been modeled to see how much the results of the developed XFEM model for hydraulic and natural fracture interaction,

are compatible with them. Table. 1 presents the results of XFEM code which can be compared with Warpinski and Teufel’s

(Warpinski and Teufel 1987) experiments. As shown in Table. 1, the results of XFEM code indicate that at high to medium

(5)

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angles of approach, natural fracture opening and crossing both are observed depending on the differential stress while at low

angles of approach with low to high differential stress, the predominant case during hydraulic and natural fracture interaction

is opening which are in good agreement with Warpinski and Teufel’s (Warpinski and Teufel 1987) experiments.

TABLE 1— COMPARISON OF XFEM CODE RESULTS WITH WARPINSKI AND TEUFEL’S (WARPINSKI AND TEUFEL

1987) EXPERIMENTS 

 Angle ofapproach

(!o)

Max.horizontal

stress(psi)

min.horizontal

stress(psi)

Horizontaldifferential

stress(psi)

Experiments results(Warpinski and Teufel

1987)Gopening /Gcrossing   XFEM results

30 1000 500 500 Opening 3.46 Opening30 1500 500 1000 Opening 2.05 Opening30 2000 500 1500 Shear slippage 1.29 Opening60 1000 500 500 Opening 1.948 Opening60 1500 500 1000 Opening 1.201 Opening60 2000 500 1500 Crossing 0.785 Crossing90 1000 500 500 Opening 1.013 Opening90 1500 500 1000 Crossing 0.833 Crossing90 2000 500 1500 Crossing 0.598 Crossing

Meanwhile debonding of natural fracture prior to hydraulic and natural fracture intersection could also be modeled which is a

complicated and very interesting phenomena that has been rarely investigated. Fig. 3 presents pre-existing fracture debonding

 before intersection with hydraulic fracture at approaching angles of 30o, 60

o, 90

o in Warpinski and Teufel’s (Warpinski and

Teufel 1987) experiments.

Fig. 3— Debonded zones of natural fracture before intersecting with hydraulic fracture at 30o(horizontal differential stress=1500 psi),

60o

(horizontal differential stress=1000 psi), 90o

(horizontal differential stress=1500 psi): the upper images show the coordinates of

hydraulic and natural fracture relative to each other where the debonded zones are highlighted in red and the images below them are

the numerical deformed configurations (magnified by 3). 

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Also, it is clearly observed that as hydraulic fracture is advancing toward natural fracture, debonded zone in each step may

differ from the previous one. Fig. 4, shows the debonded zone at hydraulic and natural fracture intersecting point. As shown in

Fig. 4, at medium to low angles of approach the debonded zone at the intersecting point is in such a way that the most likely

case is hydraulic fracture diversion into natural fracture while the debonded zone at high angles of approach is much more

limited which makes crossing the most probable case.

Fig. 4— Debonded zones of natural fracture at the intersecting point with hydraulic fracture at 30o

(horizontal differential stress=1500

psi), 60o

(horizontal differential stress=1000 psi), 90o

(horizontal differential stress=1500 psi): the upper images show the coordinates

of hydraulic and natural fracture relative to each other where the debonded zones are highlighted in red and the images below them

are the numerical deformed configurations (magnified by 3). 

Fig. 5 shows opening or crossing of natural fracture by hydraulic fracture after intersection, for different natural fracture

orientations and horizontal differential stresses in Warpinski and Teufel’s (Warpinski and Teufel 1987) experiments.

Fig. 5— The results of hydraulic and natural fracture interaction after intersection: the left image is a natural fracture with the

orientation of 30o(horizontal differential stress=1500 psi), the middle image is a natural fracture with the orientation of 60

o(horizontal

differential stress=1000 psi) and the right image shows a natural fracture with the orientation of 90o (horizontal differential

stress=1500 psi). 

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So, it is clearly observed that how well hydraulic and natural fracture interaction can be demonstrated just by considering and

 paying attention to the debonded zone in each step. Warpinski and Teufel’s (Warpinski and Teufel 1987) experiments were

 performed just a few steps before hydraulic and natural fracture intersection but in reality hydraulic and natural fracture

interaction should be investigated in a reservoir scale model. For this purpose, a natural fracture which is 10m long and 10m

far from the wellbore with the orientation of the 30o, 60

o, 90

o  has been modeled that interacts with advancing hydraulic

fracture. Differences in horizontal stresses are made by varying the maximum horizontal stress from 1159.4 to 2898.5 psi (8 to

20 MPa) and maintaining a constant minimum horizontal stress of 500 psi (3.45 MPa) and the fracturing fluid pressure is

2898.5 psi (20 MPa). Also, Table. 2 presents the properties of the reservoir rock used in the model.

TABLE 2— RESERVOIR ROCK PARAMETERS USED IN THE MODEL 

)psi(4*106

(E)Young!s modulus

0.25Poisson!s ratio (")

)MPa.m1/2

(0.75Fracture toughness

 

The results from the developed XFEM model indicate that when the orientation of natural fracture is 30o

debonding of natural

fracture due to hydraulic fracture propagation toward it, begins around 5m far from the natural fracture while for the

orientation of 60o and 90

o, debonding starts about 2.5m far from the natural fracture (Fig. 6).

Fig. 6— Beginning of debonding (highlighted in red) in reservoir scale XFEM model for a natural fracture with orientation of 30o

(horizontal differential stress=2173.9 psi), 60o

(horizontal differential stress=1159.4 psi), 90o

(horizontal differential stress=2173.9 psi).

As soon as debonding of natural fracture is getting started a diversion in hydraulic fracture propagation before intersecting

with natural fracture, begins which is a function of debonded zone length and position. Fig. 7 shows how hydraulic fracture

diversion occurs before intersecting with natural fracture. As shown in Fig. 7, some part of the natural fracture that has been

debonded in the previous step may become closed in the next step due to stresses exerted by hydraulic fracture and the

debonded zone is not necessarily symmetrical relative to hydraulic fracture tip. Fig. 8 presents opening or crossing of natural

fracture by hydraulic fracture after intersection, for different natural fracture orientations and horizontal differential stresses in

a reservoir scale XFEM model. In addition, it is clearly observed that in a specific horizontal differential stress, any increase inhydraulic fracture net pressure (difference between fracturing fluid pressure and minimum horizontal stress) leads to

decreasing induced hydraulic fracture diversion before intersecting with natural fracture (Fig. 9). As shown in Fig. 9, there is a

60o oriented natural fracture in 2898.5 psi maximum horizontal stress and 724.6 psi minimum horizontal stress (2173.5 psi

horizontal differential stress), in case the fracturing fluid pressure is 2898.5 psi (2173.5 psi net pressure), the diversion of

hydraulic fracture is more than the case that the fracturing fluid pressure is 2173.5 psi (1449.2 psi net pressure) or 1449.2 psi

(724.6 psi net pressure). In other words, under a higher net pressure there is a higher driving force at the hydraulic fracture tip,so perturbation of the stress field induced by the natural fracture and debonded zone appears to be less influential to divert the

hydraulic fracture. After hydraulic and natural fracture intersection, the hydraulic fracture may be diverted into natural fracture

and opens it or crosses the natural fracture without any significant diversion. Table. 3 presents the results of hydraulic and

natural fracture interaction after intersection, for the developed   reservoir scale XFEM model. According to Table. 3 andXFEM model results, hydraulic fracture tends to cross pre-existing natural fracture only under high differential stresses and

high angles of approach while at intermediate and low differential stresses and low angles of approach the hydraulic fracture

tends to open the pre-existing natural fracture.

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Fig. 7— Diversion of the hydraulic fracture before intersecting with the natural fracture in reservoir scale XFEM model: upper imagesare related to a natural fracture with orientation of 30o (horizontal differential stress=2173.9 psi), middle images are related to a

natural fracture with orientation of 60o(horizontal differential stress=1159.4psi) and below them are the images related to a natural

fracture with orientation of 90o(horizontal differential stress=2173.9 psi). 

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Fig. 8— The results of hydraulic and natural fracture interaction after intersection, in a reservoir scale XFEM model: the left

image is a natural fracture with the orientation of 30o(horizontal differential stress=2173.9 psi), the middle image is a natural fracture

with the orientation of 60o

(horizontal differential stress=1159.4psi) and the right image shows a natural fracture with the orientation

of (horizontal differential stress=2173.9 psi).

Fig. 9— Comparing the diversion of hydraulic fracture before intersecting with a 60o oriented natural fracture in a different net

pressures and specific horizontal differential stress. Net pressures=2173.9 psi, 1449.2 psi and 724.6 psi for the left, middle and right

images respectively at 2173.9 psi horizontal differential stress.

TABLE 3— RESULTS OF HYDRAULIC AND NATURAL FRACTURE INTERACTION FOR THE

DEVELOPED RESERVOIR SCALE XFEM MODEL 

Orientation ofnatural fracture

(#o)

Max.

horizontal

stress

(psi)

min.

horizontal

stress

(psi)

Horizontal

differential

stress

(psi)

G opening / G crossing   XFEM results

30 2898.5 724.6 2173.9 1.36 Opening

30 1884 724.6  1159.4 2.1 Opening30 1159.4 724.6  434.8 3.613 Opening60 2898.5 724.6  2173.9 0.735 Crossing60 1884 724.6  1159.4 1.236 Opening60 1159.4 724.6  434.8 1.966 Opening90 2898.5 724.6  2173.9 0.529 Crossing90 1884 724.6  1159.4 0.709 Crossing90 1159.4 724.6  434.8 1.077 Opening

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ConclusionsA new geomechanical approach has been proposed to demonstrate the hydraulic fracture propagation in naturally fractured

reservoirs through XFEM model. An energy criterion has been applied to predict hydraulic fracture path due to interaction

with natural fracture. To show the efficiency of the developed XFEM code, firstly the results obtained from XFEM model

have been compared with experimental studies which shows good agreement. It’s been concluded that natural fracture most

 probably will divert hydraulic fracture at low angles of approach while at high horizontal differential stress and angles of

approach of 60 or greater, the hydraulic fracture crosses the natural fracture. Meanwhile, the growing hydraulic fracture exertslarge tensile stress ahead of its tip which leads to debonding of sealed natural fracture before intersecting with hydraulic

fracture that is a key point to demonstrate hydraulic and natural fracture behaviors before and after intersection. Then, a

reservoir scale XFEM model has been developed to investigate the hydraulic fracture propagation and interaction with natural

fracture which indicates that debonding of natural fracture starts several meters before intersecting with hydraulic fracture and

also hydraulic fracture diversion takes place even before intersecting with natural fracture. Debonding of sealed natural

fracture and hydraulic fracture diversion are strongly controlled by the in-situ horizontal differential stress and the orientation

of the natural fracture as well as hydraulic fracture net pressure. It can also be concluded that higher net pressure decreases

hydraulic fracture diversion in a particular horizontal differential stress, hence higher net pressure is needed when fracturing

naturally fractured reservoirs since any abrupt change or diversion in hydraulic fracture propagation increases the risk of

 premature screenout. The findings in this study can be used to explain different observed behaviors of hydraulic fracturing in

naturally fractured reservoirs as well as activation of natural fractures and the potential conditions that may lead to hydraulic

fracturing operation failure.

Nomenclature

"  (x) = the discontinuous enrichment function defined for the set of nodes that the discontinuity has in its influence domain

 ! = Poisson's ratio

ak  = the added set of degrees of freedom to the standard finite element model

 E = Young's modulus

 F l  = crack tip enrichment functions

G = strain energy release rate

 H( #  ) = crack face enrichment function

 K  I = opening mode stress intensity factor

 K  II  = shearing mode stress intensity factor N   = shape function

u ENR

 = enriched (discontinuous) part of the displacement approximation

u FE

= conventional (continuous) part of the displacement approximation

uh = approximated displacement field 

u j = the vector of regular degrees of nodal freedom in the finite element method

XFEM = Extended Finite Element Method

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