SPE-173342-MS

Application of Extended Finite Element Method (XFEM) to SimulateHydraulic Fracture Propagation from Oriented Perforations

Jay Sepehri, PinnacleA Halliburton Service; Mohamed Y. Soliman and Stephen M. Morse, Texas TechUniversity

Copyright 2015, Society of Petroleum Engineers

This paper was prepared for presentation at the SPE Hydraulic Fracturing Technology Conference held in The Woodlands, Texas, USA, 35 February 2015.

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

Abstract

Understanding fracture initiation and propagation from perforated wellbores is essential to designing aperforation scheme to achieve an efficient hydraulic fracture stimulation treatment. The effect ofperforation design on hydraulic fracture propagation has been extensively studied using experimental andanalytical methods. Because the experimental investigation of hydraulic fracture is complicated, expen-sive, and often returns limited results, numerical methods can be applied as an efficient way to simulatefracture propagation from perforations. An Extended Finite Element Method (XFEM) was used to developa model to investigate the effects of various parameters on fracture propagation from a set of perforations.These parameters included perforation orientation, perforation length, stress anisotropy, and elasticproperties of the formation. Fracture propagation patterns from the XFEM model were first matchedagainst published experimental studies and exhibited good agreement. The model was then used tobroaden the study of perforation effects. Results of the modeling proved the effects of perforationorientation and length on hydraulic fracture propagation pattern. Horizontal stress anisotropy and rockmechanical properties were observed to strongly influence fracture propagation. It was also observed that,when two or more perforations are positioned at different orientation angles at the same depth, a fracturetends to propagate from the less deviated perforation. In these cases, the more deviated perforation candevelop a short fracture, following a propagating pattern that could be caused by stress shadowing/interference. Stress interference between two perforations positioned closely together results in eitherperforation breakdown or fracture propagating away from one another. The simulation results from thisstudy offer methods to enhance perforation design for hydraulic fracture treatment, particularly in the caseof high stress anisotropy and high uncertainty in a preferred fracture plane. Analyzing competingperforations suggests that a technique based on this concept can be applied when high uncertainty existsregarding the direction of the principal horizontal stresses through increasing perforation density.

IntroductionBecause of common well stability issues, isolating wells from unwanted zones, and several otheroperational considerations, wellbores are usually cased and then perforated; therefore, the majority ofhydraulic fracturing treatments are performed through perforations. Perforating provides the channel of

communication between the wellbore and the reservoir. In hydraulic fracture treatments, perforations canserve as an initial fracture to help fracture initiation and control propagation direction. Perforations playan important role in complex fracture geometries surrounding the wellbore. Initiation of a single widefracture and avoiding multiple T-shaped and reoriented fractures from a wellbore is one of the primaryobjectives of using perforation techniques.

The objective in terms of the perforating design for fracturing should be choosing perforatingparameters that minimize near-wellbore (NWB) pressure drops during both fracturing operations andproduction. Some of these NWB effects are perforation friction, microannulus pinch points from gunphasing misalignment, multiple competing fractures, and fracture tortuosity caused by a curved fracturepath (Romero et al. 1995). An optimal perforation for fracture propagation would have a minimuminjection pressure, initiate only a single fracture, and generate a fracture with minimum tortuosity at anachievable fracture initiation pressure (Behrmann and Nolte 1999).

Several studies have investigated the effect of perforations on hydraulic fracture initiation andpropagation. The first work on the effect of perforation on hydraulic fracturing was presented by Daneshy(1973), who showed that the direction of induced hydraulic fracture is not dictated by perforationorientation. This work showed that, in many cases, fluid can travel from the perforation through the areabetween the casing and formation to initiate a fracture in the direction of maximum horizontal stress.

In an experimental work, El Rabaa (1989) examined effect of different parameters on perforationdesign and showed that multiple fractures could be created when the perforated interval was longer thanfour times wellbore diameters. Soliman (1990) showed that fracture reorientation can affect the analysisof microfrac and minifrac tests in horizontal wells. Hallam and Last (1991) recommended that, withoutknowledge of the stress directions, a low phasing angle should be used. In another series of laboratoryexperiments (Abass et al. 1994 and 1996), the effects of oriented perforations in vertical and horizontalwells on hydraulic fracturing treatments and sand control were studied. Results of such experimentsshowed the relation between the breakdown pressure and hydraulic fracture width and perforationorientation.

Experimental investigation of hydraulic fracture is a complicated and expensive process and haslimited results; therefore, numerical methods can be applied as an efficient way to simulate suchphenomena. Numerical simulation of the hydraulic fracturing process is essential to understanding thecomplex mechanics of hydraulic fracturing. However, because of the strong nonlinearity in coupling ofthe fluid flow inside the fracture and fracture propagation, numerical simulation of hydraulic fracturingis a very complex problem. This problem gets even more complex considering the existence of naturalfractures, fluid leak off to the formation, and multilayers with varying physical and mechanical properties.Great effort has been devoted to the numerical simulation of hydraulic fracturing with the first three-dimensional (3D) modeling efforts beginning in the late 1970s. Significant progress has been madedeveloping two-dimensional (2D), pseudo3D, and 3D numerical hydraulic fracture models during the lastseveral years.

The finite element method (FEM) has been used to model hydraulic fracture propagation in hetero-geneous rocks, which can include nonlinear mechanical properties and can be subjected to complexboundary conditions. However, the FEM requires remeshing after each time step for the mesh to conformexactly to the fracture geometry as the fracture propagates. This can result in higher computation time andsometimes convergence problems.

By adding special enriched shape functions and additional degrees of freedom to the standard finiteelement approximation, the XFEM overcomes the drawbacks associated with the use of the conventionalFEM and enables the fracture to be represented without meshing fracture surfaces. Therefore, the fracturegeometry is completely independent of the mesh configuration. As a result, remeshing is not necessary,which results in faster simulation of the fracture propagation. Since the introduction of this method, manynew formulations and applications have appeared in the literature. The XFEM has been used to investigate

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hydraulic fracture problems by many authors (Lecampion 2009; Dahi-Taleghani and Olson 2011; Weberet al. 2013; Chen 2013).

This study focuses on the implementation of XFEM to simulate propagation of hydraulic fractures froma perforated cased hole. Finite element software with XFEM capability was implemented to investigatethe effects of perforation orientations and the NWB stress regime on induced hydraulic fractures. Thisstudy demonstrates the capabilities of XFEM for solving the complex problems of hydraulic fracturepropagation and observing the effects of perforation orientation on induced fractures. The results from thisstudy verified with some experimental observations in the literature help provide a better understandingof the problem and the contributing parameters to help improve hydraulic fracturing treatments.

Extended Finite Element MethodXFEM was developed in 1999 (Mos et al. 1999) as a technique to enhance the capabilities of theconventional FEM. XFEM is particularly convenient when modeling discontinuities. Compared to FEM,XFEM provides many improvements for the numerical modeling of fracture propagation. In the traditionalformulation of the FEM, a fracture has to follow element edges. In contrast, the fracture geometry in theXFEM does not need to be aligned with the element edges, which provides great flexibility. The XFEMis based on the enrichment of the FEM by adding extra degrees of freedom to the nodes of the elementscut by the fracture. In this way, the fracture is included in the numerical model without the need to modifythe domain discretization because the mesh is generated completely independent of the fracture. There-fore, only a single mesh is necessary for any fracture length and orientation. In addition, nodessurrounding the fracture tip are enriched with functions that reproduce the asymptotic fracture tipbehavior. This enables the modeling of the fracture within the fracture-tip elements and increases theaccuracy in terms of the computation of the stress intensity factors.

When a fracture is added to the finite element domain, two different sets of elements affected by thefracture can be identified: Elements that are completely cut by the fracture and elements that are partiallycut and contain fracture tips. Nodes at these elements are said to be enriched by extra degrees of freedom(DOF). Those elements, which are completely cut by fractures, are enriched by multiplying their shapefunction into Heaviside function H(x), which has unit value and changes the sign as it crosses the crackand conveys discontinuity across fracture faces.

(1)

For those elements that are not completely fractured and contain fracture tips, the Heaviside functioncannot be used to approximate the displacement field in the entire element. For the fracture tip elements,the enrichment functions are applied. These functions were originally introduced by Fleming et al. (1997)for use in the element free Galerkin method and adapted by Belytschko and Black (1999) for use in XFEMformulation. These four functions describe the fracture tip displacement field. The first function isdiscontinuous at the fracture tip.

(2)

In this equation r, are polar coordinates at the fracture tip. The above functions can reproduce theasymptotic Mode I and II displacement fields and represent the near-tip singular behavior. By using theenrichment functions in Eq. 2, four different additional degrees of freedom in each direction for each node

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are added to those related to the finite element formulation. Fig. 1 shows graphically the two types ofenriched nodes in the XFEM domain.

Based on what has been discussed about the enrichment functions, the following expression for theXFEM displacement approximation can be formulated

(3)

(4)

Where, J is the set of nodes whose elements is completely cut by the fracture and therefore enrichedwith the Heaviside function H(x), K1 and K2 are the sets of nodes containing fracture tips 1 and 2, andfracture tip enrichment functions are and . ui are the standard degrees of freedom, and

are the vectors of additional nodal degrees of freedom for modeling fracture faces and the twofracture tips, respectively. Without any enrichment, Eq. 4 reduces to the classical finite element approx-imation:

(5)

Model SetupThe model to study the effect of perforation orientation is based on some experimental studies (Abass etal. 1994 and 1996). Their laboratory experiments were designed to investigate the effect of perforationorientation in vertical and horizontal wells on the hydraulic fracturing treatment. Dimensions andparameters in this study are selected to be similar to the ones in those experiments to be able to crossreference some of the results.

Rock samples used in those experiments were rectangular blocks of hydrostone (gypsum cement) withdimensions of 6 6 10 in. These blocks were created from mixing water and hydrostone with a weightratio of 32/100, respectively. A wellbore was drilled in the center of the block in the direction of the 10-in.side. The wellbore was cased and perforated. A series of perforation orientations was considered: 0,

Figure 1Enriched nodes in XFEM domain (Giner et al. 2008).

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15, 30, 45, 60, 75, and 90 from the preferredfracture plane (PFP). All samples were confined ina triaxial loading vessel and the principal stresses of3,000 psi vertical, 2,500 psi maximum horizontal,and 1,400 psi minimum horizontal stresses wereapplied. No pore fluid was present within the sam-ple blocks (Abass et al. 1996).

Fig. 2 illustrates a schematic of core sample andTable 1 lists the physical and mechanical properties.

Because the samples used for those experimentsare cut to observe the fracture propagation pattern, itis a good idea to consider only a 2D model, asshown in Fig. 2 (right). The 2D model used can savesimulation time and the results can be compared to available photographs taken from the sample afterbeing cut.

Perforations are created to serve as communication channels between wellbore and formation and tohelp decrease breakdown pressure and ease fracture propagation. There are several methods to create aperforation in a well drilled and cased in the laboratory. For example, Bunger et al. (2004) created a starterfracture by inserting a rod into the injection tube and striking it firmly with a hammer. Another methodcould be by rotating a bent, sharpened rod inside the drill hole to scratch out the material. Unfortunately,no information is provided on how the wells were perforated in the experiment targeted in the study byAbass et al. (1994, 1996). In the model used throughout this study, perforations are modeled as initialfractures extended from wellbore in opposite direction (180 phasing). Two wings are considered to havethe same length and zero initial width and positioned at angle measured from maximum horizontal stressdirection and increasing counterclockwise. No casing or cement is considered in this model; therefore,fractures are initiated and propagated from the perforation tip. Depending on the perforation length andorientation, it will take a distance of few wellbore radiuses for the propagating fracture to exit the NWBregion and reorient in the direction of the PFP.

Numerical ResultsAfter the model described was constructed, sensitivity analyses were performed to obtain a betterunderstanding of hydraulic fracture propagation from perforations and to examine different parameters

Figure 2Schematic of the simulation model.

Table 1List of parameters used in the simulation model.

Dimension 6 6 10 in.

Wellbore radius 0.25 in.

Perforation length 0.25 in.

Youngs modulus 2.07 106 psi

Poissons ratio 0.21

Min. horizontal stress 1400 psi

Max. horizontal stress 2500 psi

Applied vertical stress 3000 psi

Sample permeability 39 md

Porosity 26.5%

Uniaxial compressive strength 8032 psi

Tensile strength (Brazilian) 807.6 psi

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and design configurations without the need to go through experimental design. In this study, the effectsof perforation angle, perforation length, rock mechanical properties, stress anisotropy, and nearbydiscontinuities on fracture propagation from cased hole are investigated and analyzed.

Perforation AngleGenerally, fractures tend to initiate when the applied pressure overcomes the tensile stress at the wellboreand propagate in the direction of the PFP. When the wellbore is cased and perforated, perforation can helpthe initiation of fracture by serving as initial fractures, but the created fracture will not necessarily followthe orientation provided by the perforation. Once the NWB region is passed, the initiated fracture willfollow the path of least resistance, which is perpendicular to direction of minimum horizontal stress. Toinvestigate this effect, models with perforation at different angles are built and run. The orientation angleis measured from the direction of the PFP (which is parallel to the direction of maximum horizontal stress)in a positive x direction. In all cases, perforations are of the same length, equal to one wellbore radius,and other properties are kept the same. The results are presented in (Figs. 3 and 4). The result fromorientation angles 45, 60 and 75 and 90 are compared to some photographs from experimental studiesavailable.

As illustrated in Figs. 3 and 4, there is a very good match between the results from the numerical modelin this study and the experimental models in terms of fracture reorientation pattern. This demonstratesconfidence in the model to perform further simulation. It should be remembered that the experimentalstudy includes a cement layer behind the casing and, in some cases, including 75 and 90, injected fluid

Figure 3Fracture reorientation at 45 and 60 compared to the experimental model (photos from Abass et al. 1994).

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has breached through the microannulus fractures inthe cement to find a shorter path toward the PFP.Because the numerical model is an ideal modelwithout defect and the effect of loose cement andcasing are not included, that phenomena will notoccur and fractures initiate only from perforations.

Reorientation RadiusAs fracture propagates away from wellbore, the NWB effects are diminished and fracture reorients topropagate in the preferred direction. The reorientation pattern is affected by stress distribution around the

Figure 4Fracture reorientation at 75 and 90 compared to the experimental model (photos from Abass et al. 1994).

Table 2Results of perforation angle for different reorientation radii.

Orientation Angle () Reorientation Radius (x rw)

0 0

15 1.2

30 1.8

45 4

60 5

75 5.5

90 7.5

Figure 5Effect of orientation angle on reorientation radius.

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wellbore. It is estimated that drilling a well will alter initial stress around wellbore over a distance ofapproximately two to three times the wellbore radius. Therefore, it takes some distance for fracture to exitthe shadow of this stress change. The distance from wellbore to the point where fracture is completelyaligned with PFP or perpendicular to minimum horizontal stress is called reorientation radius. Thisdistance varies with orientation angle and is larger for more deviated perforations. Table 2 and Fig. 5summarize the results of a set of simulations where the reorientation has been measured for differentorientation angles.

The reorientation radius has been estimated graphically; but, as expected, demonstrates an increasingtrend when the orientation angle increases. This is attributed to the effect of stress distribution around thewellbore. As Figs. 6 through 8 demonstrate, tangential stress around the wellbore is increasing withincreasing . This is also the case at any radius around wellbore. Therefore, it is expected that the NWB

Figure 6Tangential stress around wellbore increases with increasing angle.

Figure 7Tangential Stress at Perforation tip for different angles (L rw).

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effects will extend farther as deviates from thedirection of maximum horizontal stress, which im-plies that, in higher angles, it takes longer for thefracture to exit the NWB stress regime.

Breakdown PressureThe previous simulation set was run several times toobtain the breakdown pressure. Different pressureswere applied to the model to find the smallest pres-sure at which the fracture starts to propagate. Theresults are presented in Table 3 and plotted in (Fig.9). As the results of this sensitivity analysis show,the breakdown pressure increases as the perforation angle increases. This is because higher pressure isrequired to overcome the tangential stress at the wellbore in higher angles. Fig. 7 shows plotted tangentialstress for a wellbore of a radius rw 0.00635 m under the maximum and minimum horizontal stress of2,500 and 1,400 psi. At the tip of perforation (R 2rw), this stress increases, as plotted in Figs. 6 and 7.

Initial Perforation LengthPerforation length is one is the primary parameters to be considered when designing hydraulic fracturingfrom cased holes. Typical perforating guns have penetrations of 1 to 30 in. The length of the actualperforation downhole is a function of the standoff of the perforating gun from the casing. Less standoffmeans a longer perforation tunnel, while more standoff results in a shorter perforation length. For thisstudy, because it deals with a small scale model, perforation length is limited to be in the range of onequarter to twice wellbore radius. Fig. 10 compares results of simulation for different perforation lengths.Perforations are all 45 and other parameters are kept constant, except the perforation length changingfrom 0.25rw to 2rw.

The initial perforation path affects the direction of fracture growth. As can be observed from Fig. 10,the longer the perforation length, the longer it takes for the fracture to reorient. This argument is true forthe cases where fractures are only allowed to initiate from the tip of perforations. If casing with cementwith low bounding property were considered in the model and fracture could propagate throughmicrofractures in the cement, the result would have been different. In that case, the fracture might chooseto propagate through those microfractures because of the friction loss and pressure drop in longperforations.

Figure 8Fracture reorientation affected by stress distribution around wellbore.

Table 3Results of breakdown pressure for different reorientationradius.

Orientation Angle () Reorientation Radius (x rw)

0 3.2

15 4.2

30 4.8

45 5.4

60 6.8

75 6.6

90 7.2

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Fig. 11 compares the tangential stress for differ-ent perforation lengths. As perforation length in-creases (larger L/rw), the perforation tip is locatedwithin areas with lower tangential stress, whichmeans stress to overcome at the tip of perforation islower, and this results in the fracture propagatingeasier.

Mechanical PropertiesYoungs modulus is one of the mechanical proper-ties examined in this model to observe its effect onfracture propagation. Youngs modulus has littleeffect on initial stress distribution, but has signifi-cant effect on fracture propagation pattern. Increas-ing Youngs modulus increases the reorientation

Figure 9Change in breakdown pressure with perforation angle.

Figure 10Effect of initial perforation length.

Figure 11Variation of tangential stress with perforation length.

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radius. By increasing the Youngs modulus from 0.2106 to 6106 psi, the reorientation radius hasincreased 1.5 fold. Fig. 12 presents the effect of Youngs modulus.

Stress AnisotropyStress anisotropy, the difference between maximum and minimum horizontal stress, plays an importantrole in fracture propagation pattern. Simulations were performed for different horizontal stresses at a fixedorientation angle of 45 and Fig. 13 displays the results. As stress anisotropy decreases (increasing Sh),there is an increasing trend in reorientation radius. The results from this simulation suggest that, informations with higher stress anisotropy, the role of perforation orientation becomes more significant.Higher anisotropy results in rapid changes in fracture direction, and this means increasing tortuosity andlimiting fracture width. Therefore, in higher anisotropy, there is a smaller tolerance on perforationdeviation angle. This should be considered when perforating for hydraulic fracture treatments.

Varying Rock PropertiesIn this scenario, rock mechanical properties of some sections of the rock sample were considered to bedifferent and simulations were run again. Fig. 14 presents results. Fracturing from perforations in stiffer

Figure 12Effect of Youngs modulus on fracture reorientation.

Figure 13Effects of stress anisotropy (?S SH-Sh).

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rock propagates and reorients before the perforation on the opposite side even initiates a fracture. Theresult in Fig. 14 suggests that, if the perforation is positioned at a more fracturable part of the rock, itcreates a higher chance of initiating and propagating a fracture and formations with varying properties candevelop asymmetric one wing fractures.

Competing PerforationsAll previous simulations were performed with two perforations positioned at 180 phasing configuration.Other configurations, such as 120 and 60 are also applied in the field. They are of course on a spiral orhelix configuration. In this model, all of the perforations are applied on a horizontal 2D model.Fig. 15presents the results for two perforations with smaller phasing.

When two or more perforations are on a plane (assuming perforation at the same depth instead of ona helix configuration), there is competition between the fractures to propagate. As Fig. 15 demonstrates,when two perforations exist (15/45, 30/60, or 45/90), a fracture will propagate from the perforationpositioned at a smaller angle from the PFP. This could be because fracture initiation and propagation is

Figure 14Effect of varying rock properties.

Figure 15Effect of two competing perforations.

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easier from less deviated perforations. In reality, fracture initiated from perforations at close angle maylink up and form one fracture, which would be in the direction of the less deviated perforation.

Also, as can be observed in the case of 30/60 perforations, the initiated fractures propagate away fromone another because of stress interference, which probably results in early screenout of the perforationwith higher deviation angle. The same discussion can be made for three competing fracture in 120phasing, as Fig. 16 depicts.

The results obtained from simulating competing perforations can have many applications whendesigning perforation for hydraulic fracturing. In cases where perforation are not aligned with the PFPbecause of a lack of information regarding PFP orientation or because of operational limitations, using theconcept of two or more competing perforations can help increase the chance of developing fractures in thePFP direction and decreasing tortuosity. Also, by decreasing perforation phasing, there are more chancesfor more fractures in an orientation close to the PFP. Although there are always concerns regarding casingstability in lower phasing angles or higher shot densities, the idea of competing perforation can still beused by increasing operational efficiency.

It should be noted that perforations cannot be placed too closely to one another. For example,positioning two perforations at 30 and 45 can result in perforation breakdown, as the results in Fig. 17suggest. In this case, the stress interference from two perforations will force fracture propagating awayfrom each other, which would not be a favorable phenomenon. Even if one of these perforations couldresult in a fracture propagating, as shown in Fig. 17 on the right, the resulted tortuosity can be very severearound the wellbore, which could eventually reduce wellbore production performance.

ConclusionsA numerical model was developed using XFEM to investigate hydraulic fracture propagation fromoriented perforations. The primary goal in this study was to highlight and verify the effect of parametersconsidered to affect this process as well as emphasize promising capabilities of XFEM for solving

Figure 16Three competing perforations at 120 phasing.

Figure 17Two competing perforations positioned too closely.

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hydraulic fracturing simulation problems. The results for fracture propagation patterns were confirmed tomatch the results from published experimental studies and showed good agreement. Then, the verifiedmodel was used to study the effects of different parameters involved in fracture propagation through aperforated wellbore. Perforation angle from the PFP, perforation length, rock mechanical properties, andhorizontal stress anisotropy were observed to affect the fracture propagation pattern. It was observed thatfractures propagated from more deviated perforations require higher breakdown pressure and tend toreorient to the PFP direction over larger distances. Changing formation Youngs modulus has little effecton initial stress distribution around wellbore, but directly affects fracture reorientation radius. In forma-tions with higher stress anisotropy, fractures propagated from perforations reorient over a shorter distance.This effect is stronger for more deviated perforations. Perforations positioned at the part of formation withlarger Youngs modulus had a greater chance of initiating and propagating a fracture. Therefore, rockswith varying properties might develop asymmetric one-wing fractures, even if perforated at 180 phasing.

Competition exists between perforations with different orientation angles to develop hydraulic frac-tures. In these cases, fractures tend to propagate from less deviated perforations and those more deviatedcan develop short fractures, propagating away from one another because of stress interference and/or cancease to propagate over a short distance, which could result in perforation screenout. However, stressinterference between two perforations positioned closely to each other can result in either perforationbreakdown or fractures propagating away from one another.

The findings of this study suggest that, because horizontal stress anisotropy plays an important role infracture reorientation radius and consequently tortuosity, in formations with high stress anisotropy, caremust be taken to identify the PFP and avoid perforation in orientations highly deviated from this plane.Results from competing perforations analysis suggest that this technique can be applied when highuncertainty exists regarding the direction of maximum and minimum horizontal stress. This technique canbe applied by increasing shot density or a new design of perforation gun to apply multiple perforationsat the same depth.

Nomenclature

h, Sh Minimum horizontal stressH, SH Maximum horizontal stressr, Polar coordinatea,b Vectors for added degrees of freedomDOF Degree of freedomE Youngs modulusF Crack tip enrichment functionH(x) Crack face enrichment functionL Perforation lengthNi Shape functionsPFP Preferred fracture planerw Wellbore radiusuFEM Finite element part of displacement approximationuENR Enriched part of displacement functionXFEM Extended Finite Element Method

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Application of Extended Finite Element Method (XFEM) to Simulate Hydraulic Fracture Propagation ...IntroductionExtended Finite Element MethodModel SetupNumerical ResultsPerforation AngleReorientation RadiusBreakdown PressureInitial Perforation LengthMechanical PropertiesStress AnisotropyVarying Rock PropertiesCompeting Perforations

Conclusions

References