Numerical simulation of complex fracture growth during ...

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ngineering 60 (2008) 86–104www.elsevier.com/locate/petrol

Journal of Petroleum Science and E

Numerical simulation of complex fracture growth during tightreservoir stimulation by hydraulic fracturing

Md. Mofazzal Hossain a,⁎, M.K. Rahman b

a Australian School of Petroleum, The University of Adelaide, Adelaide, SA-5005, Australiab School of Oil and Gas Engineering, The University of Western Australia, Perth, WA 6009, Australia

Received 29 July 2006; accepted 22 May 2007

Abstract

The success or failure of hydraulic fracturing technology is largely dependent on the design of fracture configurations andoptimization of treatments compatible with the in-situ conditions in a given reservoir. The petroleum industry continues to facechallenges with this technology in the field, such as premature screen-outs, high treating pressures, complexities with multiplefractures propagation, complex fracture propagation from the deviated wellbore, etc. As these challenges persist betterunderstanding of hydraulic fracture behavior for various reservoir conditions is still an important topic for research. Since themechanism of hydraulic fracture growth involves the rock stress field and fluid flow field, the modeling work of fracture growthrequires the treatment of coupled fluid flow and structural deformation phenomena. In this context, this paper briefly, summarizesan existing numerical tool for fracture growth analysis based on coupled fluid flow and structural deformation phenomena. Solidmodels have been developed to simulate different field conditions and then solved by using this numerical tool. The fieldconditions include different stress regimes, fracture geometry and fracture and well orientations. Results for different conditionshave been presented and discussed to provide guide lines for better planning and design of hydraulic fracturing. The key finding isthat if the well orientation and fracture configuration are not compatible with the in-situ stresses, complex fracture growthdiminishes the likelihood of success and exhibits some of the above mentioned symptoms during treatments in the field.© 2007 Elsevier B.V. All rights reserved.

Keywords: Hydraulic fracture stimulation; Natural and multiple fracture stimulation; Fracture orientation; Stress regime; In-situ stress; Coupledfluid and structural deformation

1. Introduction

Stimulation of tight reservoirs by hydraulic fracturinghas been established as a very successful technology forimproving the petroleum production performance. Re-cently, this technology has also been extended to various

⁎ Corresponding author.E-mail addresses: [email protected],

[email protected] (M.M. Hossain), [email protected](M.K. Rahman).

0920-4105/$ - see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.petrol.2007.05.007

unconventional applications, such as completion of highpermeability unconsolidated formations (FracPacking),geothermal energy resources extraction, waste re-injec-tion, produced water re-injection, coal bed methane gasproduction, etc. The success of fracture stimulation islargely dependent on the size, shape and the propagationbehavior of the created hydraulic fracture. A fractureinitiated from a deviated wellbore becomes subject to acomplex stress state and leads to the development of acomplex geometry of the propagated fracture. This resultsin a fracture of very limited width. The limited width and

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http://dx.doi.org/10.1016/j.petrol.2007.05.007

87Md.M. Hossain, M.K. Rahman / Journal of Petroleum Science and Engineering 60 (2008) 86–104

the tortuous fracture geometry hinder the flow of injectedproppant inside the fracture, which results in a lowproppant concentration and hence inadequate fractureconductivity. The ultimate consequence is significantreduction of reservoir productivity, making the wholestimulation process unsuccessful. A recent trend istherefore to develop coupled non-planar fracture modelsand their use for interesting parametric studies (Hossain,2001; Dong and de Pater, 2001, 2002; Garcia and Teufel,2005; Rungamornrat et al., 2005) to understand thecomplex fracture growth. This paper is a contributiontowards this trend. In this study, an attempt has beenmadeto develop insights to explain how the complex fracturegeometry grows under different stress andwell conditions,and their effects on the injection pressure, using a flow-deformation coupled numerical tool, HYFRANC3D. Thistool was developed by the Cornell Fracture Group (http://fac.cfg.cornell.edu; Carter et al., 2000). The tool considerscomplex fracture geometry, non-linear coupling betweenequations that characterize fluid flow in the fracture,structural deformation and possible interaction with othertypes of fractures (e.g. multiple fractures, natural fractures,etc.).

A vast literature exists that reports the progress ofbasic understanding and optimization of this technol-ogy (Settari and Cleary, 1984; Rahim et al., 1995;Mohaghegh et al., 1999; Soliman and Boonen, 2000;Hossain et al., 2000; Rahman et al., 2001; Queipo et al.,2002; Rahman and Joarder, 2006; Bohloli and de Pater,2006; Rahman et al., 2007) and different fracturepropagation models (Mendelsohn, 1984a,b; Veatch andMoschovidis, 1986; Settari and Cleary, 1986; Gidleyet al., 1989; Valko and Economides, 1995; Yew, 1997).Over the years, the works related to the development offracture models advanced rapidly from 2D analyticalmodels, for instance PKN andKGDmodels, to complex3D numerical models. Warpinski et al. (1994) includesbrief descriptions and a comparison of predictions for anumber of simulators, including 2D and pseudo-3Dmodels. All of these fracture propagation models arebased on a similar criterion, which assumes that fracturepropagation takes place when certain deformationparameters reach a critical value. For example, accord-ing to the linear elastic theory, the fracture is assumed tostart propagating once the tensile stress at the fracturetips reaches the tensile strength of the rock. Theexistence of a stress singularity at or near the fracturetip, however, makes the linear elastic fracture mechan-ics (LEFM) theory adequate to predict the fracturegrowth behavior accurately. In LEFM, such a stresssingularity is addressed using its strength, termed asstress intensity factor.

Since the mechanism behind hydraulically initiatedfracture heavily involves a rock stress field and a fluidflow field, the problem requires the treatment ofcoupled fluid flow and structural deformation phe-nomena. It is to be noted that a strong non-linearityexists between the solid structure and the moving fluid.Therefore, the mathematical formulation of an overallfracture propagation model requires coupling of a setof complex equations, hence the development ofsophisticated numerical tools based on finite elementor boundary element methods. Since the propagationof fracture is mainly controlled by the stress singularityat the fracture tip, it is sufficient to consider theproblems on the fracture boundary rather then thewhole region as considered in finite element methods.Hence, the boundary element method is usuallyconsidered to be more suitable to solve for structuralresponses due to rock mass, in-situ stresses andfracturing fluid pressure. On the other hand, the fluidflow equation can be more conveniently solved by afinite element method. Therefore, the overall compu-tation time to solve a fluid pressure driven fracturepropagation problem can be minimized significantlyby combining these two numerical methods. Based onthis principle, a hydraulic fracture propagation simu-lator (HYFRANC3D) was developed (Carter et al.,2000). This simulator has been used in this study toinvestigate the complex fracture growths underdifferent conditions.

In the rest of this paper, we briefly introduce theboundary element method as it applies to modeling thecoupled hydraulic fracturing problem and explain thenumerical solution technique. We examine a number ofcase studies, review their results and present finalconclusions.

2. Boundary element method

The boundary element method (BEM) is one of themost powerful numerical techniques developed in theshadow of the finite difference and finite elementmethods. The application of BEM is perfectlyfeasible to two and three-dimensional fracture pro-blems; however, the code requires a multi-domaindiscretization capability. This capability facilitates themodeling of two fracture surfaces in separate sub-domains (Luchi and Rizzuti, 1987). The stressintensity factors in 3D fracture geometries can beaccurately calculated by a standard BEM code. Theapplication of BEM for solving three-dimensionalfracture problems is well documented in the literature(Rizzo, 1967; Cruse, 1969; Lachat and Watson, 1976;

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http://fac.cfg.cornell.edu

Fig. 1. Schematic of the fracture region showing that the fluid entersand leaves a chosen domain A through ∂A in the fracture domain Ω.(after Carter et al., 2000).

88 Md.M. Hossain, M.K. Rahman / Journal of Petroleum Science and Engineering 60 (2008) 86–104

Brebbia, 1978; Brebbia et al., 1984; Luchi andRizzuti, 1987).

The basic boundary integral equation that providesthe relationship between the displacement U, and thetraction T, at a surface Γ, of a homogeneous isotropicmedia, or a sub-domain into which the body has beendivided, can be written (Brebbia et al., 1984; Luchi andRizzuti, 1987; Banerjee, 1994) as:

CijðA1ÞUjðA1Þ þZCFijðA1;A2ÞUjðA2ÞdC

¼ZCGijðA1;A2ÞTjðA2ÞdC ð1Þ

where A1 and A2 are the points on the boundarysubsurface and the boundary surface, respectively. Fij

(A1,A2), Gij(A1,A2) are the functions representingdisplacements and tractions, respectively, in the jdirection at point A2 corresponding to a unit point loadacting in the i direction applied at A1, and Cij is acoefficient function. These functions can be obtainedfrom the singular solution for given boundary conditions(Banerjee, 1994). The traction T represents a stressnormal to a surface, T=σn, where n is unit outwardnormal. Eq. (1) disregards the body force term andapplies to media that follow the linear elastic materialbehavior.

In order to use the boundary integral equationdetailed in Eq. (1) for modeling of hydraulic fracturingproblems, the fluid pressure term needs to be included intraction, T. Detailed formulations of such problems aredescribed in the following section.

2.1. Mathematical modeling of hydraulic fracture

The modelling of hydraulic fracturing problem isdifferent in the sense that the traction, T, contains thepressure term arising due to fluid flow through thefracture. The displacement, U, is the resultant of thatfrom the fluid flow effect and the structural elasticresponse. The total displacement gives the fractureaperture that can be expressed as:

w ¼ w0 þ wp ð2Þ

where wo is the aperture contribution from theexternal stress and wp is that from the fluid pressure.The value of wp can be expressed as wp=λp, where λis the influence coefficient. Following the basicboundary integral equation (Eq. (1)) and applyingthe principle of fluid flow through parallel plates (i.e.q ¼ � w3

12lgrad p) and mass conservation, the final form

of the integral equation for the coupled problem ofhydraulic fracturing can be expressed as (Carter et al.,2000):

ZXdp

AwAt

dXþZX

w3

12lðgrad p: grad dpÞdX

þZCdp bV 4=3dC ¼ QðtÞdpðOÞ ð3Þ

in which the bulk fracture is described by a sub-domain Ω in a 3D space F and Γ is the boundarybetween Ω and the front region of fracture (Fig. 1); pis the pressure inside the fracture; w is the totalfracture aperture; q is the flow rate or influx into thefracture; t is the flow time and Q is the source or sinkstrength of any point O inside a fracture region.

The first two integral terms in Eq. (3) represent theusual solution of a hydraulic fracture problem. The thirdterm is a contour integral to incorporate the asymptoticsolution for fluid flow. The right hand side of thisequation accommodates the boundary condition of asource term at any point (Carter et al., 2000).

It is to be noted that the asymptotic solution of fluidflow involves a strong non-linear coupling between themoving fluid and the solid deformation, particularly inthe vicinity of the hydraulically induced fracture tip. Thenon-linear coupling makes an exact matching singular-ity between the pressure and the elasticity equations(SCR Geomechanics Group, 1993; Desroches et al.,1994). The nature and the strength of such a singularityin the fracture depend on fluid properties, formationproperties and the fracture propagation speed. Theircoupled formulation led to the development of LinearElastic Hydraulic Fracturing (LEHF) theory. Accordingto LEHF, the solution of crack tip fields is representedby fracture width w(ρ) and expressed for the Newtonianfluid and impermeable formation as, w(ρ)=βV1/3 where


b ¼ 2ð37=6Þ lE V

� �1=3q2=3, in which ρ is the curvilinear

distance measured on the fracture surface between anypoint and fracture front and μ is the fracturing fluidviscosity and V is fracture propagation speed.

The HYFRANC3D simulator works based on thecoupled Eq. (3), where the structural responses are solvedby BEM and the fluid flow part is solved by FEM. Thedetailed finite element solution and nodal discretization ofEq. (3) have been presented by Carter et al. (2000).

3. Numerical solution

Since the problem is both time and space dependent,the problem can be solved either by searching ageometry of the propagated fracture for a given injectionperiod, or by searching the injection period for a givengeometry of the propagated fracture. The first approachrequires more computational time than the secondapproach and hence the first approach has not beenincorporated in HYFRANC3D. As the fracture initiationmechanism is not incorporated in HYFRANC3D, thesecond approach requires an assumed initial fracture asthe starting point which then is modified by subsequentanalyses incorporating mixed mode fracture propaga-tion theory. The solution process consists of thefollowing operations:

1. Create the boundary element mesh on an assumedfracture geometry (e.g. penny shaped crack) for agiven time stage

2. Apply the relevant pressure boundary condition tothe wellbore (if any)

3. Solve for elasticity by boundary elementmethod (BEM)4. Solve for the coupled hydraulic fracture problem,

making use of the previous solution (obtained for the(n−1)th time stage)

5. Allow next stage of fracture propagation6. Repeat steps 1–5 for the new fracture

Following the above solution process, numericalanalysis is performed using BEM based code for 3DHydraulic Fracture Analysis called HYFRANC3D. Asmentioned earlier, HYFRANC3D combines the bound-ary element analysis for structural response with thefinite element analysis for fluid flow. The boundaryelement solution is provided by an external programcalled BES (see http://www.cfg.cornell.edu). The over-all solution methodology consists of the following steps:

1. Extend the fracture up to an arbitrary distance basedon given boundary condition and discretize theextended fracture,

2. Solve the elastic structural deformation problemusing BES,

3. Solve the fluid flow equation in an iterative fashionuntil the solution converges to an equilibrium fluidpressure.

The solution process within this loop is complex. Toinitiate the iterative fracture propagation process, aninitial fracture configuration is necessary that satisfiesthe fracture propagation criterion. This initial fractureconfiguration can be estimated approximately usinganalytical linear elastic fracture mechanics principles,or by a number of trials. Once the first step of fracturepropagation is complete using the initial configuration,this new solution can be used as the starting point for thenext step of propagation. This process continues insubsequent steps of propagation. The iterative solutionproceeds at two levels for given initial values of fractureopening, pressure, and fracture front speed. First, for agiven time step, the fluid flow equation is solvediteratively using reasonable tolerance on the fractureopening to judge whether the solution has converged.Once this is satisfied, the global mass-balance and thefracture tip speed are checked. The total volume ofinjected fluid minus the leaked-off volume should equalthe fracture volume. Ignoring the fluid lag effect at thetip, the fluid speed at the fracture front is assumed to beequal to the fracture front speed. The consequences offluid lag effect in 3D hydraulic fractures are wellreported by Advani et al. (1997). However, SCRGeomechanics Group (1993) and Carter et al. (2000)have argued, based on their numerical results, that thefluid lag effect on hydraulic fracture width and pressureis negligible.

Both the equations for fluid flow and total volumebalance express the satisfaction of global mass-balance.If these two equations are not satisfied, the time step isadjusted and the fluid flow equation is solvediteratively again. This process continues until thesolution has converged or the number of globaliterations has exceeded the user supplied maximumvalue. In case of permeable formations, an additionalset of iterations for the first stage is considered to rampup the leak-off from the impermeable to the finalpermeable solution.

For each stage, the fracture is allowed to propagatestep-wise as limited extensions of the previous fractureas shown in Fig. 2.

The extension is estimated based on the assumptionthat the local extension of the fracture ΔL(x) at a point xalong the fracture front is proportional to the speed ofthe fluid at that point v(x). The local extension is scaled

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by a maximum propagation length L0, specified by theuser as:

DLðxÞ ¼ vðxÞmaxðvÞ L0: ð4Þ

The ideal maximum propagation length proves to be inthe order of 10% of the fracture penetration into theformation. The ratio of speed and propagation length islater used to determine the initial time step between twosubsequent models, asΔL=vΔt. The direction of fracturepropagation, θ(x), at each point is determined accordingto the maximum circumferential stress criterion (Des-roches and Carter, 1996; Rahman et al., 2000) for mixedmode fracture propagation and evaluated in a planenormal to the fracture front at each point and given by:

coshðxÞ2

KIcos2 hðxÞ

2� 32KIIsinhðxÞ

� �zKIC ð5Þ

where KI,KII, KIC are the stress intensity factors for modeI (opening), mode II (shearing) and fracture toughnessrespectively. KI and KII are estimated using the displace-ment correlation techniques (Ingraffea and Manu, 1980;Ingraffea, 1987; Luchi and Rizzuti, 1987) as functions ofthe nodal displacements. The existence of mode III(tearing) is not very prevalent in hydraulic fracturesemanating from typical well configurations subject to in-situ reservoir stresses. Therefore, this mode has not beenconsidered in this study.

4. Case studies

Numerical simulations of a number of cases havebeen performed to investigate the behavior of thepropagated fracture geometry from the wellbore underdifferent stress regimes. The effects of perforationorientation and multiple fractures are also investigated.

It is assumed that the fluid is Newtonian and there isno leak-off of fluid to the formation. Both of these

Fig. 2. Schematic of fracture propagation process.

assumptions have primary effects on the fracture widthand hence effects on fracture volume and injectionpressure. However, it is strongly believed that modelingof non-Newtonian fluid behavior and fluid leak-off willnot change the general trends of findings in this study.

4.1. Model configuration

A penny shaped fracture of 4 cm radius is consideredto be initiated hydraulically from the center of awellbore of 40 cm diameter. The wellbore is assumedto be located at the center of a block of 200 cm×200 cm×200 cm dimensions (Fig. 3). Since the modelis considered to be equilibrium in a 3D stress system,analyses of half the block is sufficient due to symmetricboundary conditions. The applied far-field stresses areshown in Fig. 3, with the borehole axis aligned with oneof the far-field stress directions. The formation isassumed to be linearly elastic with Young's modulus10 GPa, Poison's ratio 0.25, and fracture toughness5 MPa√m.

Fig. 4 shows the boundary element (meshed) modeland applied boundary conditions for a non-perforatedwell. The far-field in-situ reservoir stresses: maximumhorizontal stress σH; minimum horizontal stress σh; andvertical stress σv correspond to the wellbore localstresses σx, σy and σz (Fig. 3), respectively. Makingappropriate correspondence between the in-situ stressesand the local stresses, the actual well orientation can besimulated. For example, Fig. 4 can be considered as avertical wellbore, when the wellbore axis is parallel tothe σv direction (i.e. σz=σv). Similarly, the wellbore ishorizontal along σh when σz is set to σH. This techniquehas been used to avoid repetition of modeling of thegeometry.

The fracture propagation study has been carried outinto two categories:

(i) the structural response with constant fracturepressure

(ii) the coupled fluid and structural response withconstant injection rate

The objective of the first category is to understandthe behaviour of the fracture geometry when a fracturepropagates along preferred and non-preferred directions.The assumption of constant pressure allows the rapidgrowth of propagated fracture geometry with minimumcomputational effort. On the other hand, the objective ofthe second category is to understand the behaviour ofthe fracture pressure when a fracture propagates alongpreferred and non-preferred directions.


4.2. Results and discussion

4.2.1. The structural response for non-perforatedwellbore

Numerical analysis was performed for a longitudinalfracture (i.e. the fracture plane is parallel to the wellboreaxis) and a transverse fracture (i.e. the fracture plane isperpendicular to the wellbore axis). These fractureswere initiated from a non-perforated wellbore subject tothree different stress regimes (normal faulting, σvNσHNσh; reverse faulting, σHNσhNσv and strike-slip,σHNσvNσh). The geometric and mechanical propertiesand the applied stresses are shown in Table 1.

A constant pressure equal to the wellbore pressure isapplied inside the fracture; the magnitude of thispressure is set equal to the fracture closure stress. Theactual closure stress depends on the fracture orientationand the stress regime considered for a particular case.For example, in the case of longitudinal fractures in theσH direction in normal faulting stress regimes, themagnitude of closure stress is taken equal to σh.

From an iterative solution of fracture propagation,the stress intensity factors, KI and KII, at the differentlengths of propagation, are obtained. The stress intensityfactors for longitudinal and transverse fractures ema-

Fig. 3. Schematic of the model showing the boun

nating from the vertical wellbore under normal faulting,strike-slip and reverse faulting stress regimes are plottedin Figs. 5, 6 and 7, respectively.

It can be observed from Figs. 5 and 6 that the stressintensity factor KI increases for longitudinal fracturesfrom the vertical wellbore in normal faulting and strike-slip stress regimes whereas the value of KII is verynegligible and remains constant during fracture propa-gation. This indicates that the opening mode dominatesthe fracture propagation process, which results in aplanar fracture. This conforms to the general consensusthat when a fracture initiates with appropriate orienta-tion in the preferred direction, which is a longitudinalfracture in the maximum horizontal stress direction in avertical wellbore under normal faulting and strike-slipstress regimes, it propagates in its own plane.

Reverse trends for KI and KII are observed in Figs. 5and 6 for transverse fractures emanating from thevertical wellbore subject to the same stress regimes. Thisis because the transverse fracture from the verticalwellbore in the normal faulting and strike-slip stressregimes is not favorable for propagation. However, aperfectly transverse fracture should propagate in planeremaining perfectly perpendicular to σv though requir-ing high pressure to overcome high σv. This is because

dary conditions and necessary dimensions.

Fig. 4. Illustration of boundary element model showing half of the model to represent a wellbore whose axis is parallel to σz stress direction with astarter fracture at the center of the block (enlarged view) aligned in the σx stress direction.


the shear component, and hence KII, is zero for such acondition. In reality, a perfect transverse fracture (inother words a perfect planar fracture perpendicular toany stress direction) is not possible in highly heteroge-

Table 1Geometric dimensions and rock properties considered for the analysis

Geometric dimensions

Block dimension 200 cm×200 cm×200 cmFracture shape Penny shapedFracture radius (half length) 4 cmFracture length 8 cmWellbore diameter 20 cm

Applied stresses

Stress regime σv

Normal faulting 80 MPaReverse faulting 60 MPaStrike-slip 70 MPa

Applied constant fluid pressur

Fracture type Normal faultingLongitudinal fracture 60 MPaTransverse fracture 80 MPa

neous reservoir rock. As soon as the fracture goesslightly out of plane, the shear component startsdeveloping to reorient the fracture further towards thepreferred direction for fracture propagation with

Mechanical properties

Elastic modulus (E) 10 GpaPoisson's ratio 0.25Leak-off coefficient 0.0Fracture toughness 5.0 MPa√m

σH σh

70 MPa 60 MPa80 MPa 70 MPa80 MPa 60 MPa

e (MPa)

Reverse faulting Strike-slip faulting70 MPa 60 MPa60 MPa 60 MPa

Fig. 5. Stress intensity factors, KI and KII at different propagated fracture radii of longitudinal and transverse fractures for a non-perforated verticalwellbore in a normal faulting stress regime.


minimum resistance. To mimic this fracture behavior inheterogeneous rock, the transverse fracture in the 2ndstep of propagation was made very slightly out of plane.In these cases, KI decreases and the corresponding valueof KII increases as the fracture propagates further.Apparently, this is in contrast with the study of Ingraffea(1987) in which it is stated that a fracture finding itselfunder substantial mode II loading does not long remainin high KII /KI domain of interaction, rather it quicklychanges its trajectory to minimize or eliminate the KII

components. The authors believe that Ingraffea's

Fig. 6. Stress intensity factors, KI and KII at different propagated fracture radwellbore in a strike-slip stress regime.

conclusion is correct when the fracture is alreadydeviated by a certain degree with the preferred directionof fracture propagation. In other words, the fracture hasalready reached the substantial mode II loading. In sucha case, as soon as the fracture starts propagating, it startsaligning itself with the preferred direction throughturning only and consequently, the value of KII startsdecreasing. On the contrary, the authors createdtransverse fracture is absolutely in the opposite phaseof the preferred direction and plane, and has not reachedthe substantial mode II loading state. The alignment of

ii of longitudinal and transverse fractures for a non-perforated vertical

Fig. 7. Stress intensity factors, KI and KII at different propagated fracture radii of longitudinal and transverse fractures for a non-perforated verticalwellbore in a reverse faulting stress regime.


this fracture with the preferred direction and planerequires a total of 90° turning/twisting, which requires asubstantial increase of the KII component initially. Oncethe critical mode II loading state is past, the stressintensity factors KI and KII would behave during furtherpropagation in the way as Ingraffea explained. However,driving the fracture beyond the critical phase mayrequire a significant increase in pressure inside thefracture. To extend the current investigation up to thatphase was not possible due to the requirement of animpracticably long computational time. Only the initial

Fig. 8. Illustration of twisted transverse fracture emanating fro

turning and twisting trend of the fracture is shown inFig. 8. In conclusion, the gradual increase inKII in Figs. 5and 6 for the vertical fractures can be justified withoutany contradiction to Ingraffea's statement, which wascorrect in the context described by Ingraffea (1987).

On the other hand, in a reverse faulting stress regimethe minimum principal stress direction lies in thevertical direction. Therefore, a transverse fracture insuch a case should behave similar to the longitudinalfracture in normal and strike-slip stress regimes, and alongitudinal fracture in this case similar to the transverse

m the vertical wellbore in normal faulting stress regime.

Fig. 9. Illustration of twisted longitudinal fracture emanating from the vertical wellbore in reverse faulting stress regime.


fracture, for very similar reasons presented in theforegoing paragraph. This common sense expectationis perfectly supported by the trends of stress intensityfactors KI and KII in Fig. 7 and by the geometry ofpropagated fractures in Fig. 9.

In elemental sense, the increase or decrease of KI andKII depends on the relative magnitudes of nodal

Fig. 10. Boundary element model of perforated vertical wellbore

deformations along and perpendicular to the fracturearound the fracture tip. These deformations, in turn, arecomplex functions of the current fracture geometry,directions and magnitudes of the applied stresses withrespect to the fracture direction and the fluid pressureinside the fracture. Further details can be found in thepaper by Ingraffea and Manu (1980). The key finding of

, where the perforation is aligned with the direction of σH.

Fig. 11. Maximum principal stress distribution along the perforation from the wellbore wall.


this study is, however, if the fracture is already along thepreferred direction and the constant fracture pressure ismaintained, the value of KI increases. This indicates thatthe propagation of this fracture is possible withgradually decreasing fracture pressure. On the otherhand, if the fracture is along the non-preferred directionand the constant fracture pressure is maintained, thevalue of KII increases. In combination with Eq. (5), thisindicates that the further propagation of this fracture atsome stage may not be possible if the fracture pressure isnot increased. This is the direct consequence of out ofplane fracture growth.

Fig. 12. Tangential stress distribution along the

4.2.2. Structural response for perforated wellboreThis model has the same dimensions as the non-

perforated wellbore model. In addition, a perforation hasbeen included from which the fracture has initiated. Theperforation tunnel is assumed to be a cone having a facediameter of 1.3 cm, end diameter of 0.65 cm and alength of 18 cm, at the center of the block. A starterfracture of radius 2.5 cm is assumed on each side of theperforation.

The meshed model with a perforation aligned withthe maximum horizontal stress is shown in Fig. 10. Thismodel was used to predict the possible location of

base of perforation on the wellbore wall.

Fig. 13. Illustration of longitudinal and transverse fractures emanatingfrom the perforated vertical wellbore, when the perforation is orientedin the direction of σH.


tension at the interface of the wellbore wall and theperforation from which a fracture is likely to initiate forgiven stresses and fluid pressure.

The wellbore is vertically situated in a normalfaulting stress regime having the stress magnitude: σx=σH=70 MPa, σy=σh=60 MPa and σz=σv=80 MPa.The wellbore pressure is gradually increased until atensile stress develops in the formation.

Three potential locations are investigated for tensilestress from where the fracture is more likely to initiateand propagate. Location-1 is designated by 0°, which isthe mid-depth of the perforation. Location-2 is at 45° upfrom Location-1 and Location-3 is a further 45° up,which is at the top of the perforation. The stress state at180° off-phase from each location is identical. Theminimum principal stress distribution at these threelocations along the perforation length is plotted inFig. 11. It is observed from Fig. 11 that the minimumprincipal stress is more tensile (−ve is tension and +ve iscompression) in Location-3 compared with the othertwo locations. This indicates that the probability ofinitiating the fracture is more likely from Location-3than the two other locations and will initiate at the baseof the perforation. This is highlighted in Fig. 12, wherethe minimum tangential stress along the perforation faceis found at the top or bottom of the perforationcircumference (in Fig. 12, when tangential distance is90°, i.e. Location-3). This simple study indicates that thelongitudinal fracture is likely to happen from a verticalwellbore in a normal faulting stress regime even after thewellbore is perforated. However, the possibility ofmultiple fractures from different locations can not bedismissed as all three locations are in tension. Theirpropagation behavior is therefore studied further.

4.2.3. Fractured perforated wellboreTo investigate the effect of fracture orientation with

respect to in-situ stress direction with the three differentstress regimes, a longitudinal and a transverse fracture of2.5 cm radius are created as starter fractures on bothsides of a perforation face at the vertical wellbore(Fig. 13). The applied stresses, pressures and mechan-ical properties are the same as used for the non-perforated wellbore as given in Table 1. The perforationis considered aligned with the direction of σH for all thethree stress regimes.

Fig. 14 presents the stress intensity factors, KI andKII, at different fracture propagation radii in the normalfaulting stress regime for both longitudinal and trans-verse fractures. The figure shows the same trends ofstress intensity factors with slightly different numericalvalues, as that with the non-perforated wellbore.

Therefore, the fracture propagation behavior is alsoexpected to be similar to that for the non-perforatedwellbore. This is also manifested by the turning–twistingtrend of a transverse fracture in Fig. 15. Similar resultsare expected for strike-slip and reverse faulting stressregimes and therefore they are not studied further withperforated wellbores.

4.3. Coupled fluid and structural response

From the above case studies, it is now clearlyestablished that the fracture initiated in the non-

Fig. 14. Stress intensity factors, KI and KII, at different propagated fracture radii of longitudinal and transverse fractures for a perforated verticalwellbore in a normal faulting stress regime.


preferred direction develops complex geometry throughturning and twisting during propagation. This isexpected to result in flow constriction and high treatingpressure. In order to investigate the nature of pressureresponse during complex hydraulic fracture propaga-tion, a case of coupled fluid flow and structural responseproblem has been studied. A horizontal wellbore subjectto the normal faulting stress regime is considered toalign along the σh direction for this case. A longitudinalfracture is initiated from the preferred as well as non-preferred directions. The effect of multiple fractures onthe treating pressure and fracture volume is alsoinvestigated.

Fig. 15. Propagated transverse fracture initiated from the perforation aligniregimes.

4.3.1. Fracture in the preferred vs non-preferreddirection

Fig. 16 represents the horizontal wellbore orientedalong the σh direction. The applied in-situ stresses arethe stress along the wellbore axis, σh=44 MPa, thevertical stress, σv=77 MPa and the other horizontalstress, which is orthogonal to the wellbore axis, σH=56 MPa. The fluid is considered to be a Newtonian onehaving a viscosity of μ=100 cp. A constant flow rateof 7.5 cm3/s is simulated in the fracture mouth. Thefluid leak-off is ignored. The mechanical properties ofthe formation are used as presented in Table 1. A pennyshaped longitudinal fracture was initiated along the

ng in the σH direction for vertical wellbore in normal faulting stress

Fig. 16. Details of boundary element models of a horizontal well aligned along the minimum horizontal stress (half of block is considered due tosymmetry).


preferred direction first, i.e. along σv. The height(diameter) of the initiated fracture mouth at thewellbore wall was 10 cm, which was then propagatedstepwise. Similar operations were performed with aninitiated fracture along the non-preferred direction, i.e.along σH. Pressures developed at the fracture mouthduring fracture propagation and the overall fracture

Fig. 17. Comparison of propagation pressures and volume of fracture

volumes for both cases are plotted in Fig. 17,representing the first case by M1 and the second caseby M2. Results in Fig. 17 clearly show that the fractureinitiated along the non-preferred direction (M2)requires very high pressure for propagation and thefracture volumes tends to reduce after a small extent offracture propagation.

s initiated at preferred (M1) and non-preferred (M2) direction.

Fig. 18. Detail finite element meshing, and tortuous fracture plane during propagation of fracture emanating from the horizontal wellbore in the non-preferred direction obtained from HYFRANC3D analysis.


The visualization of complex fracture growth alongthe non-preferred direction (Fig. 18) allows one tocomprehend the physical phenomena that lead toincreasing injection pressure and decreasing fracturevolume. Clearly, the fracture has turned and twisted toorient itself with the preferred plane. On the other hand,the fracture initiated along the preferred direction (M1)

Table 2Summary of laboratory stress conditions and treatment parameters

Test no. Fracture radius (mm) Fractureconfiguration

Fracturetest material

DC-3 10 of each fracture Multiple ParallelFracture

Silica, Cement

DC-4 10 Single completion Silica, cementa Total injection rate.b Injection rate into each fracture.

is more favorable for propagation in terms of bothpropagation pressure and fracture volume.

The pressure response as a function of fracturepropagation in this small-scale problem studied has,however, clearly demonstrated that the complex fracturegrowth in the field may lead to the requirement of atreatment pressure which may be beyond the practical

Injection rate (Q)(cm3/s)

Fracture viscosity (μ)(Pa s)

Stresses appliedto block(MPa)

σx σy σz

0.011 a 81 7 6 5

0.024 b 53 7 6 5

Fig. 19. Illustration of single transverse fracture model with boundary conditions analyzed by HYFRANC3D and propagated fracture's configuration.

Fig. 20. Illustration of multiple parallel fracture model and divergence of propagated fractures.


Fig. 21. Mode I and mode II stress intensity factors at different propagated fracture radii of single and multiple parallel fractures.


pump capacity. Also the fracture volume, and henceflow conduit, may eventually offer very little predictedproduction benefits. These two together clearly explainthe reasons for screen-outs and non-productive out-comes of many practical fracture treatments.

4.3.2. Single fracture vs multiple fractureIn this case study, single and multiple parallel

transverse fractures are modeled. Laboratory testswere conducted by Crosby (1999) for these fractureconfigurations. The laboratory stress conditions andtreatment parameters are summarized in Table 2. Thenumerical models for the single and multiple fracturesand the applied boundary conditions are illustrated inFigs. 19 and 20 respectively. Note that only half of theblock was discretized to take the advantages ofsymmetric boundary conditions. Fluid leak-off in bothmodels was set to zero again.

Fig. 22. Comparison of propagation pressures and fractu

The stress intensity factors (KI and KII) at the fracturetip for both single and multiple parallel fractures areplotted in Fig. 21 as functions of the propagated fractureradius. The corresponding fracture propagation pressureand the fracture volume are plotted in Fig. 22.

From Fig. 21, it is observed that the value ofKI for thesingle fracture considerably increases with the propa-gated fracture radius keeping the very insignificant KII

constant. The multiple parallel fractures have shown thereverse effect, particularly the increasing nature of KII.This indicates turning and twisting of multiple fracturesduring propagation. This effect has inflicted a divergentnature of propagation of multiple parallel fractures asshown in Fig. 20. As a consequence of this non-planarfracture growth, the injection pressure has increased andthe fracture volume has decreased as functions offracture propagation (Fig. 22). These findings are alsosupported by the experimental results of Crosby (1999).

re volume of parallel multiple and single fractures.


5. Conclusions

Based on the results and discussions, the followingmajor conclusions can be made.

• It is a very general perception that the fractureinitiated in the non-preferred direction and planeturns and twists during propagation and tends to bealigned with the preferred direction and plane.However, if a perfectly planar fracture is perfectlyoriented along the non-preferred direction, theoreti-cally the fracture should propagate in plane thoughmay require higher pressure than that for the fracturein the preferred direction. In the field, however, therock formation is extremely heterogeneous which ismore likely to induce out of plane fracture growth.Whenever the fracture propagation is simulatedalong the non-preferred direction in this study, aslightly out of plane fracture configuration isintroduced at the early propagation stage to mimicthe field nature. The mixed mode fracture propaga-tion model then convoluted the non-planar fracturegrowth further, as expected in the non-preferreddirection.

• A longitudinal fracture from a vertical wellboresubject to a normal faulting or strike-slip stressregime is favorable for propagation along themaximum horizontal stress direction without turningand twisting. A transverse fracture is favorable forpropagation from such a well if the stress regime isreverse faulting. Fractures initiated otherwise turnand twist during propagation and lead to complexgeometry. This behavior remains unchanged even ifthe wellbore is perforated along the preferreddirection. In a number of cases studied, these havebeen successfully simulated by the numerical tool,HYFRANC3D, used in this study. The tool hasrightly facilitated the coupled fluid flow anddeformation analysis capability that is crucial toaccurately model the fluid driven propagatingbehavior of hydraulic fractures.

• There is a possibility of initiation of multiplefractures at the base of the perforation although thefracture at a certain location dominates the others.The top as well as the bottom of the perforation basealong the maximum horizontal stress is such a likelylocation for longitudinal fracture initiation from avertical wellbore subject to a normal faulting stressregime.

• Any perforation which is at off-phase with thepreferred direction causes fracture turning andtwisting.

• Regardless of the causes of multiple fractures andfracture turning and twisting during propagation,they are the source of high treating pressures andreduced fracture volume. These have been clearlysimulated for the small-scale case problems studiedin this work. The results, however, have clearlyindicated that the complex fracture growth from thenon-preferred direction is the most likely reason forpremature screen-outs in many fracture treatments inthe field. Also the resulting reduced fracture volumeis one of the major reasons for reduced productivity.

• It is very important to optimize the well trajectory,perforation direction and fracture configurations for agiven stress condition in the field to avoid thetreatment failures related to complex fracture growth.Brief guidance includes: (1) oriented perforationalong σH direction for vertical wells subject tonormal faulting stress regimes; (2) longitudinalfracturing in vertical wells subject to reverse faultingstress regime is very unlikely to be successful; (3)multiple transverse fracturing of vertical wellssubject to reverse faulting stress regimes andhorizontal wells along σh direction are likely to besuccessful; however, optimizing the transversefracture spacing is crucial to avoid non-planartransverse fracture growth which potentiallydiminishes the predicted treatment benefits.

Acknowledgements

The authors gratefully acknowledge the contributionsof A/Prof. Sheikh Rahman, School of PetroleumEngineering, University of New South Wales and theEnglish editing service of Ms Therese Ellis, School ofOil and Gas Engineering, University of WesternAustralia. Finally, the two anonymous reviewers deservethanks for their constructive comments which improvedthis paper.

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