engineering fracture mechanicsdownload.xuebalib.com/jo3gcucrsr4.pdf · an energy based analytical...

Engineering Fracture Mechanics 189 (2018) 232–245

Contents lists available at ScienceDirect

Engineering Fracture Mechanics

journal homepage: www.elsevier .com/locate /engfracmech

An energy based analytical method to predict the influence ofnatural fractures on hydraulic fracture propagation

https://doi.org/10.1016/j.engfracmech.2017.11.0200013-7944/� 2017 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: [email protected] (Y. Yao).

Yao Yao a,⇑, Wenhua Wang a, Leon M. Keer ba School of Mechanics and Civil Engineering, Northwestern Polytechnical University, Xi’an 710072, People’s Republic of ChinabDepartment of Civil and Environmental Engineering, Northwestern University, 2145 Sheridan Rd., Evanston, IL 60208, USA

a r t i c l e i n f o

Article history:Received 17 August 2017Received in revised form 9 November 2017Accepted 12 November 2017Available online 14 November 2017

Keywords:Hydraulic fractureAnalytical methodNatural fractureEnergy

a b s t r a c t

Hydraulic fracturing is a widely applied stimulation method to enhance the productivity ofunconventional resources. The hydraulic fracturing operation in naturally fractured reser-voirs is complex, and the fractures can intersect a natural interface such as a bedding plane.The hydraulic fracture may either cross or be arrested by slippage without dilation, and thefracture plane can be opened upon arriving the interface. In the current study, a theoreticalapproach to predict the fracture extension encountering a natural fracture under far fieldstresses is developed, based on the Griffith stability criterion. The critical fluid pressurerequired to cross the interface, open the natural fracture, or make slippage take place isobtained. New criteria to separate opening zone, arrest and crossing zone are proposedbased on stress field difference. The theoretical predictions are compared with experimen-tal data and show reasonable accuracy.

� 2017 Elsevier Ltd. All rights reserved.

1. Introduction

Hydraulic fracturing treatment plays an essential role in unconventional shale gas development. The fracturing fluid ispumped into the reservoir at high pressure, providing a path for oil or natural gas towards the production well. Over the pastdecades, several models were proposed to simulate the nucleation and propagation of hydraulic fractures, such as the KGDand PKN models [10,22]. With the development of computational technology, numerical models were employed to simulatefractures with more complex geometries based on linear elastic fracture mechanics, which generally provides reasonablepredictions for hard rocks [26,14,7,21,33,34].

In the naturally fractured oil or gas reservoirs, the assumption that hydraulic fracture is a straight and bi-wing planar fea-ture is untenable because of the existence of natural fractures, faults, bedding planes and stress contrasts. Hydraulic fracturepropagation can be affected by the natural fractures in a shale gas reservoir. The interaction between hydraulic and naturalfractures is a complex process. Experimental analysis [2,38,27] shows three main modes when the interaction betweenhydraulic fracture and natural fracture occurs. The first mode is that the hydraulic fracture crosses the natural fracturedirectly. The second mode is when the natural fracture opens and the hydraulic fracture propagates along the natural frac-ture. The third mode denotes that the hydraulic fracture is arrested before reaching the interface and slippage takes place.Some effective criteria for this competition were developed to predict which mode will dominate the fracture behavior,based on linear-elastic fracture mechanics [2] (Renshaw and Pollard, 1995) [30,15].

http://crossmark.crossref.org/dialog/?doi=10.1016/j.engfracmech.2017.11.020&domain=pdfhttps://doi.org/10.1016/j.engfracmech.2017.11.020mailto:[email protected]://doi.org/10.1016/j.engfracmech.2017.11.020http://www.sciencedirect.com/science/journal/00137944http://www.elsevier.com/locate/engfracmech

Nomenclature

c half-length of a line crackc0 cohesion of the interfaceDc length of crack tip moving aheadE modulus of elasticityE0 modulus of elasticity for plane strain conditionG0 critical energy release rate of rock matrixGin critical energy release rate of interface fillerGIC critical energy release rateKIC critical stress intensity factorQ heat generatedwðxÞ width of line crackW1 elastic energyDW0 Strain energy increased without far field stressDW 00 strain energy increased crossing NF with far field stressDW 000 strain energy increased as NF opens with far field stressDW1 elastic energy increased without far field stressDW 01 elastic energy increased with far field stressDW 001 elastic energy increased as NF opens with far field stressDWp work done by constant pressure without far field stressDW 0p work done by constant pressure crossing NF with far field stressDW 00p work done by constant pressure as NF opens with far field stressc surface energyc0 surface energy of rock matrixcin surface energy of interface fillerd thickness of infinite domaing ratio of cin and c0h intersection anglel friction coefficientm Poisson’s ratiorH maximum horizontal in-situ stressrh minimum horizontal in-situ stressrn normal stress of NFrt tensile strengths shear stressp constant pressure in crackpn net pressurephf critical fluid pressure crossing NFpnf critical fluid pressure as NF openspsp critical fluid pressure as slippage takes placeDv crack volume increasedDvhf crack volume increased with far field stressDvnf crack volume increased as NF opens with far field stress

Y. Yao et al. / Engineering Fracture Mechanics 189 (2018) 232–245 233

The interaction between hydraulic and natural fractures could induce a hydraulic fracture network and enhance the com-plexity of the problem. The interaction is affected by the approach angle, in-situ stresses, rock mechanical properties, prop-erties of the natural fractures and the treatment parameters, including fracturing fluid properties, injection rate and others. Itis essential to determine whether a hydraulic fracture crosses or is captured by natural fractures, since this behavior controlsthe geometry of the resulting fracture network.

Large amounts of numerical, theoretical, and experimental research has been performed on the interaction betweenhydraulic and natural fractures [19,2] (Renshaw and Pollard, 1995) [30,1,25,35,36,38,4,16,20]. Most studies focus on themechanical interaction when a hydraulic fracture reaches a pre-existing fracture, where the fluid flow in the hydraulicand natural fractures is investigated.

Blanton [2] developed a method considering the coefficient of friction to predict whether a hydraulic fracture will prop-agate across the interface or show pressure induced slip under different conditions. Experiments on both Devonian shale andhydrostone indicate that the morphology of hydraulic fractures is strongly influenced by natural fractures. The experimentson hydrostone in particular show that hydraulic fractures tend to cross pre-existing fractures under higher differential

234 Y. Yao et al. / Engineering Fracture Mechanics 189 (2018) 232–245

stresses and larger angles of approach. Under intermediate or low differential stresses with approach angles along the pre-existing direction, the hydraulic fractures could open the pre-existing fracture and divert the fracturing fluid or arrest thehydraulic fracture propagation.

Based on the linear-elastic fracture mechanics solution for stresses near the fracture tip, Renshaw and Pollard (1995) pro-posed another approach to predict the fracture propagation direction when a frictional interface is orthogonal to theapproaching fracture. However, the intersection angle between a hydraulic fracture and natural fracture can range between0 and 90� in the field. The Renshaw and Pollard criterion has been extended to predict intersection at nonorthogonal anglesby Gu and Weng [12]. On the other hand, Lamont and Jessen [19], and Daneshy [6] have demonstrated that the propagatinghydraulic fracture in a fractured reservoir could either cross or turn into the natural fracture. In some cases, the fracture willturn into the natural fracture for a short distance and then breakout again to propagate in a mechanically more favorabledirection, depending on the orientation of natural fracture relative to the stress field.

Warpinski and Teufel [30], and Zhou et al. [38] observed three types of interactions between hydraulic and natural frac-tures: namely crossing, opening and dilating the natural fracture. The hydraulic fracture can be arrested by shear slippage ofthe natural fracture without dilation when fluid flows along the natural fracture. Most of the experimental and field studieson the mechanism of hydraulic fracture propagation [3,38] have shown that the in-situ stresses, horizontal stress difference,angle of approach, interfacial friction coefficient, fluid injection rate and fracturing fluid viscosity are the key parametersaffecting hydraulic fracture propagation. Warpinski and Teufel [30] developed a method that governs the arresting modebased on differential stresses. However, the surface energy that generates the new facture is not considered, which ignoresthe variation of joint strength characteristics when the joint growth situation changes.

Several analytical methods have been developed to predict the interaction mechanism of induced or natural fractures[8,5,24,31]. They investigated the effects of different parameters, including in-situ stresses, approach angle, rock mechan-ical properties, hydraulic fracture treatment parameters and the properties of natural fractures such as cohesion and fric-tion angle of the interface. Fracture development on a weak interface ahead of a fluid-driven crack has been simulated toinvestigate the effects of delaminating and shear zone on the interface [9], and the secondary crack may be inducedbecause of friction along the natural fracture [13]. However, the surface energy of the pre-existing interface was usuallyignored.

In the current study, the critical fluid pressures to cross the interface, open the natural fracture and allow slippage to takeplace are predicted, and a new criterion to describe the fracture propagation direction is proposed. The analytical predictionis compared with experimental data and shows good agreement.

2. Critical fluid pressure for induced fracture and opening natural fracture

It is well accepted that a fluid lag exists between the fracture tip and the fluid front inside a hydraulic fracture fromlaboratory and field observations. When the fracture tip reaches the interface of natural fracture, the fluid front remainsbehind due to fluid lag. Although the effects of fluid flow are limited, the natural fracture is under the influence of thestress field around the fracture tip. Gu and Weng (2010) developed a criterion to predicate crossing type interactionconsidering the fluid lag effect. When the fracture tip cannot cross the interface and the fluid front reaches the naturalfracture, the stress concentration at the fracture tip will vanish and fluid pressure at the intersection point rises. Thenatural fracture may open or slippage will take place. A new approach to predict differential stress and approach angleis proposed in the process when the fluid front reaches the interface. The effect of fluid lag is ignored in the currentstudy. It is assumed that the initial interaction with the natural fracture is blunted and arrested and the fracture fluidcan penetrate into the intersection. The subsequent derivation is based on the effect of fluid pressure with fracturepropagation.

Griffith’s analysis of crack stability theory [11,29] assume a slice of thickness d from an infinite domain in the absence offar-field stress, containing a three dimensional extension of a ‘‘line crack” with half-length c, as shown in Fig. 1.

By assuming the linear crack is of elliptical shape, the width is given by Eq. (1). If the crack tip moves ahead by Dc, theincreased volume can be calculated by Eq. (2) [29].

wðxÞ ¼ 4pE0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

c2 � x2p

0 6 x 6 c ð1Þ

Fig. 1. Propagation of a linear crack.


Dv ¼ 2pdpcDcE0

ð2Þ

Supposing that the crack is opened by a constant pressure p, the work done by the inner fracturing fluid pressure is theproduct of pressure p and the increased volume Dv , and which is defined as DWp. As the crack propagates, energy is absorbedby the deformation of the rock matrix. This part of energy is stored in the medium as strain energy, which is denoted as DW0.The other part of the entire work DWp is to create new surfaces. Griffith postulated that if the specific surface energy c char-acterizes the energy consumption while a unit area of new surface is created and is a material property, then the energyrequired to create new surfaces is 2cdDc. If there is surplus energy available, it will be dissipated as heat Q. The energy bal-ance equation could be written as:

DWp ¼ DW0 þ 2cdDc þ Q ð3Þ

Based on the Griffith stability criterion [29], the crack is stable if:

DWp < DW0 þ 2cdDc ð4Þ

When the far-field stress is not zero, the elastic energy W1 is required to be defined. It represents the product of crack

volume and far-field stress perpendicular to the propagation direction. If the fracture propagation direction is perpendicularto the minimum horizontal principal stress as shown in Fig. 2, the elastic energy is changed and DW1 is:

DW1 ¼ Dvrh ð5Þ

However, the existence of far-field stress makes the configuration more complex. To apply the Griffith stability theory

with far-field stress, the simulation should be decomposed into two parts. Thus, an associated problem is created, as shownin Fig. 3 [29]. Fig. 3b shows a scenario similar to the Griffith stability problem except for the difference of inner pressure, andthe increased crack volume that can be obtained. In Fig. 3c, it is assumed that rh exists in the far field and rh is not moved tothe crack surface. Fig. 3c represents that the elastic energy is accumulated by overcoming the far-field stress with increasingcrack volume.

When the crack tip moves ahead by Dc, the changed volume is expressed by Eq. (2), here p� rh takes the place of p asfollows:

Dvhf ¼ 2pðp� rhÞdcDcE0 ð6Þ

From Eq. (6):

DW 0p ¼ pDvhf ¼2ppðp� rhÞdcDc

E0ð7Þ

DW 00 ¼ðp� rhÞDvhf

2¼ pdðp� rhÞ

2cDcE0

ð8Þ

DW 01 ¼ Dvhfrh ð9Þ

The generated heat is small and can be neglected. Thus, the new stability criterion can be postulated as:

DW 0p ¼ DW 00 þ DW 01 þ 2c0dDc ð10Þ

The critical energy release rate GIC is related to the stress intensity factors through Irwin’s relation:

Fig. 2. Schematic of a hydraulic fracture that propagates perpendicular to the minimum horizontal principal stress.

Fig. 3. Schematic of configuration with far-field stress rh .


K2IC ¼ GICE0 ð11Þ

where E0 ¼ E=ð1� v2Þ for plane strain conditions, m is the Poisson’s ratio and GIC ¼ 2c0 according to the fracture mechanics.

From Eqs. (10) and (11), the critical fluid pressure required to drive the crack tip moving ahead can be calculated by:

phf ¼ rh þKICffiffiffiffiffiffi

pcp ð12Þ

The propagation of hydraulic fracture is assumed as planar Mode I. According to the fracture toughness criterion [17], ifthe stress intensity factor is greater than the critical value KIC , the fracture will propagate. Assuming the load on the fractureface in a bi-directional compression plane is phf , the critical stress intensity factor [32,11] is:

KIC ¼ ðp� phf Þffiffiffiffiffiffi

pcp ð13Þ

From Eq. (13), the critical pressure can be obtained. Then, the Griffith stability criterion can be extended to predict whenthe hydraulic fracture will propagate along the natural fracture direction.

When the hydraulic fracture encounters the natural fracture as shown in Fig. 4, the hydraulic fracture may open the nat-ural fracture and propagate along the weak interlayer, and the width will change correspondingly. Assuming that the cracktip moves ahead by Dc, if it propagates along the natural fracture, the changed volume can be determined by:

Dvnf ¼ 2pðp� rnÞdcDcE0 ð14Þ

The shear and normal stress acting on the plane of the natural fracture can be obtained from the 2D stress resolution [18]:

rn ¼ rH þ rh2 �rH � rh

2cos 2h ð15Þ

s ¼ rH � rh2

sin 2h ð16Þ

Then,

DW 00p ¼ pDvnf ¼2ppðp� rnÞdcDc

E0ð17Þ

DW 000 ¼ðp� rnÞDvnf

2¼ pdðp� rnÞ

2cDcE0

ð18Þ

DW 001 ¼ Dvnfrn ð19Þ

From Eq. (10), the stability criterion can be obtained:

DW 00p ¼ DW 000 þ DW 001 þ 2cindDc ð20Þ

From the above equations, the critical pressure required to open the natural fracture and move the fracture ahead can be

determined as:

pnf ¼ rn þKICffiffiffiffiffiffi

pcp ffiffiffigp ð21Þ

Fig. 4. Schematic of hydraulic fracture encountering a natural fracture.


where g denotes cin=c0 and equals to Gin=G0, cin is the surface energy of the interface filler, c0 is the surface energy of rockmass matrix. If the bedding plane consists of the same rock as the mass matrix, it is assumed that the ratio of surface energyequals to the ratio of tensile strength of bedding plane and rock mass. It should be noted that tensile strength could beobtained from experiments and is easier to be determined, compared with the surface energy.

3. Validation of the critical fluid pressure

Hydraulic fracture propagates along the direction with least required energy, and if pnf is smaller than phf , then the nat-ural fracture will open. On the contrary, the hydraulic fracture may cross the interface. By comparing Eqs. (12) and (21), thecrossing criterion can be obtained, as expressed in Fig. 5. The curved line means that different fluid pressure is required toopen natural fracture at various approaching angles. Various curves are displayed with respect to different values of the ratiog. The dashed line represents the required fluid pressure to propagate along the maximum horizontal in-situ stress. Thecurve above the dashed line represents that the fluid pressure of opening a natural fracture is larger than that of hydraulicfracture, which means that crossing will take place. Conversely, the curve below the dashed line represents that the fluidpressure required to open natural fracture is smaller than that of the hydraulic fracture, which means the natural fracturewill open. The applied principal stresses are rH ¼ 10 MPa, rh ¼ 8 MPa. The calculated fracture toughness is 0:59 MPa m1=2and the initial crack length is 0.06 m.

In Zhou et al.’s study [38], different types and thickness of papers were cast into the block as pre-fractures. The block ismade of cement and sand with a compressive strength of 28.34 MPa. The angle in each block between the hydraulic fractureand the pre-fracture were varied systematically. The interaction angles were 30�, 60� and 90�, respectively. The ratio g can be

Fig. 5. the required fluid pressures with respect to different g.


regarded as zero for the surface energy of paper, which is much smaller than that of rock. Other related parameters areKIC ¼ 0:59 MPa m1=2 and c ¼ 0:06 m. The comparison is displayed in Table 1.

In the experiments performed by Gu et al. [15], samples were cut at specified angles 45�, 75� and 90�. Then samples werereassembled using epoxy. The block samples are made of Colton sandstone with a tensile strength of 4:054 MPa. The fracturetoughness is 0:7 MPa m1=2, and the initial crack length is 0:076 m. The comparison is displayed in Table 2.

To check applicability of the proposed method, the analytical results are compared with experiments by Zhou et al. [38]and Gu et al. [15]. In Tables 1 and 2, the bold cells show the cases that the criterion did not match the experimental results.Owing to the existence of plastic zones ahead of the crack tip, the actual fluid pressure to cross an interface is relatively lar-ger, and hence the values are in the acceptable range. In general, the theoretical predictions show good agreement comparedwith the experimental data.

4. Opening and arresting criteria on differential stresses

When propagating towards the fracture interface, the hydraulic fracture may neither cross the interface nor open the nat-ural fracture. There are two patterns: The first case denotes that the hydraulic fracture is arrested before encountering theinterface. The second case is when slippage occurs and water conductivity path forms, but hydraulic fracture re-initiateslater on the other side of the interface as the inner fluid pressure increases. The first case is not considered in the currentstudy. For the second case, some theoretical and numerical methods were developed. If the hydraulic fracture is linked tothe weak layer and pressure in the weak layer cannot open the natural fracture, slippage may occur [2,38]:

s > l � ðrn � pÞ þ c0 ð22Þ

The critical value of the fracturing fluid pressure is:

psp ¼ rn �s� c0l

ð23Þ

where l is the coefficient of friction and c0 denotes cohesion of the interface.If a natural fracture arrests an approaching fracture, the crack tip will be blunted and the stress singularity will diminish

as the hydraulic fracture propagates into natural fracture; then fracture propagation stops and the fluid enters into the weaklayer. As the pressure increases, slippage will occur if psp is smaller than pnf and phf . Thus, the following arresting criterion byslippage can be developed:

rH � rh >c0 � lKICffiffiffiffipcp

sin h cos h� l sin2 hð24Þ

Warpinski and Teufel [30] considered the Coulomb failure criterion for slippage along the natural fracture and proposed arelationship that governs the arresting mode based on the differential stresses:

rH � rh > c0 � lpnsin h cos h� l � sin2 h

ð25Þ

In Eq. (25), the arresting of hydraulic fracture is dependent on the net pressure, hydraulic fracture length, approach angle,friction coefficient and cohesion of the interface. From Fig. 6, the scope of arresting mode extends as net pressure increases.This is especially the case when the right side of Eq. (25) is zero, and all the zones that belong to the curve above the X-axiscould induce the arresting mode. However, the natural fracture may open or the hydraulic fracture may cross the interface ifthe net pressure is high enough. In fact, whether slippage takes place depends on the property of natural fracture such asfriction coefficient and cohesion of the interface. The jointing crevasse grows as long as the fluid pressure is larger thanthe critical fluid pressure psp, which means, the potential of slippage has been determined for a special weak layer. The roleof fluid pressure is to achieve the potential.

As the pressure at the intersection continues to increase, the opening and crossing modes may initiate, and re-initiationcan occur later in the slippage zone or at the end of the natural fracture. It should be noted that re-initiation is not consideredin the current study. When the fracturing fluid meets the interface, the natural fracture will open if the fluid pressure pnf isless than the pressure phf required to drive the crack tip to move along the initial direction with increasing fluid pressure. Theopening criterion in terms of differential stress can be expressed as:

rH � rh < KICffiffiffiffiffiffipcp �1� ffiffiffigp

sin2 hð26Þ

Eq. (26) depends on the hydraulic fracture length, intensity factor of rock mass matrix, approach angle and the ratio of cin=c0.Fig. 7 shows comparison of the proposed opening criterion to the experimental results [3] with respect to various ratios of g,where the rock mass matrix is Devonian Shale and the weak mixture is hydrostone. The value g is determined to be 0.04 inEq. (26). Fracture toughness of Devonian shale is 1:57 MPa m1=2 and the initial crack length is 0.06 m. In Fig. 7, with increas-

Table 1Comparison between Zhou et al.’s experiments [38] and the developed approach.

h ð�Þ rH (MPa) rh (MPa) g phf � pnf l Our result Experiment result Zhou [38]90 �10 �5 0 � 0.89 Crossing Crossing90 �10 �3 0 � 0.89 Crossing Crossing60 �10 �3 0 � 0.89 Crossing Crossing60 �13 �3 0 � 0.89 Crossing Crossing60 �8 �5 0 � 0.89 Crossing Dilated30 �10 �5 0 + 0.89 Dilated Dilated30 �8 �5 0 + 0.89 Dilated Dilated30 �13 �3 0 � 0.89 Crossing Arrested90 �8 �3 0 � 0.38 Crossing Crossing90 �8 �5 0 � 0.38 Crossing Crossing60 �10 �3 0 � 0.38 Crossing Crossing60 �8 �3 0 � 0.38 Crossing Dilated30 �10 �3 0 � 0.38 Dilated Arrested30 �8 �3 0 + 0.38 Dilated Dilated

Table 2Comparison between Gu et al.’s experiments (2011) and the developed approach.

h ð�Þ rH (MPa) rh (MPa) g phf � pnf l Our result Experiment result Zhou [38]90 �13.78 �6.89 0 � 0.615 Crossing Crossing90 �7.58 �6.89 0 + 0.615 No Crossing No Crossing75 �17.24 �6.89 0 � 0.615 Crossing Crossing75 �8.27 �6.89 0 + 0.615 No Crossing No Crossing45 �17.24 �6.89 0 � 0.615 Crossing No Crossing45 �8.27 �6.89 0 + 0.615 No Crossing No Crossing

Fig. 6. Regions of arresting mode based on Warpinski and Teufel’s criterion for four different net pressures; pn = 0.1, 0.7, 1.0, 1.1 MPa, respectively.


ing approach angle, the differential stress decreases. The curve of opening criterion can describe the opening zonesaccurately.

To further prove the validity of Eq. (26), the analytical prediction is compared with another experimental study by Blan-ton [2]. The rock mass matrix is Devonian Shale and weak mixture is hydrostone in the experiments. Fracture toughness ofDevonian shale is 1:57 MPa m1=2 and the initial crack length is 0.06 m. The parameter g is 0.04. The figure indicates a goodagreement of theoretical predictions to the experimental results. As shown in Fig. 8, a larger g represents a more difficultopening mechanism, and the opening zone decreases as g increases.

In the developed model, the tendency of variation of opening with surface energy is different from Blanton’s work. InFig. 9, from the experimental analysis of Blanton [2], the opening mechanism is easier when the surface energy of interfacefiller increases, which is not realistic. More energy is required to open the natural fracture if the surface energy is larger. Thusthe opening of a natural fracture with larger surface energy requires lower differential stress.

In the particular case (approach angel equals 90�), as shown in Fig. 10, the interface opens easier as the G of interface fillerdecreases under certain differential stress. Thus, when G of the interface filler increases, the differential stress reduces if thenatural fracture opens, and the opening zone is reduced. The opening zone will vanish if the interface filler properties are the

Fig. 7. Comparison of the proposed opening criterion to the experimental results [3].

Fig. 8. Comparison of the proposed opening criterion to experimental results from Blanton [2]

Fig. 9. Opening criterion of Blanton [2] for G = 35, 70 and 105 N=m, from left to right.


Fig. 10. Configuration of hydraulic fracture encountering an interface filler for approach angle equals 90�.


same as mass matrix, and therefore the opening criterion in Eq. (26) is reasonable. In Fig. 11, according to Eq. (26), fracturesimulation of the purple point for g ¼ 0:05 should cross, simulation of the red point for g ¼ 0:05 should not cross and thesimulation of red point for g ¼ 0:5 should cross.

Numerical simulation is performed to verify the proposed method by using the commercial finite element software ABA-QUS. The finite element model is plane strain and the size is 105� 105 m, the fluid injection point A is in the middle of theedge and vertical to the X axis. The displacement of U1 of the left edge and displacement of U1, U2 of the other edges areconstrained. The element type of the model is CPE4P. The hydraulic fracture model is displayed in Fig. 12. Parameters of cor-responding elements are adjusted to represent the natural fracture according to different values of g. The change of values isshown in Table 3. Mechanical parameters of the rock matrix are obtained from the shallow shale in Longmaxi, South China[20]. Permeability of the rock is set as 5� 10�9 m2, and leak-off is 5:8� 10�10 m=ðkPa sÞ. The water based fluid parametersare given in Table 4.

It is assumed that the highlighted elements represent the natural fracture in Figs. 13–15, and the approach angle is set tobe 90� to ignore the effects of the arrest zone. The critical stress difference is 4.8 MPa when g is 0.05 and 1.811 MPa when g is0.5. In the numerical analysis, stress differences are 4 MPa and 8 MPa, respectively. The simulation results are presentedbelow (see Table 5 for the load conditions in the numerical simulation).

Fig. 11. Change of differential stresses with the angle of approach.

Fig. 12. Hydraulic fracture model using XFEM.

Table 3Mechanical parameters of shale adopted in the numerical simulation.

E (GPa) m rt (MPa) KIC (MPa m1=2) GIC (N=m) Porosity

Rock 14.06 0.367 11.67 1.20 89 0.33NF 14.06 0.367 g � 11:67 ffiffiffigp � 1:2 g � 89 0.33

Table 4Fluid parameters in the numerical simulation.

Injection rate (m3=s) Gap flow (kPa s) Unit weight (kN=m3) Injection time (s)

8� 10�5 1� 10�6 7.8 240

a. Crack tip before NF b. Crack tip in NF

Fig. 13. Hydraulic fracture process no crossing NF when stress difference is 4 MPa and g is 0.05.


a. Crack tip before NF b. Crack tip close to NF

c. Crack tip cross NF d. Crack tip far from NF

Fig. 14. Hydraulic fracture process crossing NF when stress difference is 4 MPa and g is 0.5.


When the stress difference is 8 MPa, the crack tip can cross natural fracture. Stress distribution at the crack tip shows abutterfly pattern and symmetric when crack is far from the natural fracture. When crack tip is close to natural fracture, thestress distribution is disturbed by the natural fracture, which is influenced by the stress shadow affect. When the stress dif-ference is 4 MPa, the crack tip cannot cross the natural fracture. In the numerical analysis, elements of weaker parametersare assumed as the natural fracture. When the crack tip intersects the weaker elements, the hydraulic fracture propagatesalong the maximum principal stress direction. When the crack tip comes from the weaker elements under a stress differenceof 4 MPa, the crack tip arrests at the interface because hydraulic fracture propagates along the maximum principal stressdirection in any element. However, at the right interface, the fracture should propagate along natural fracture, which couldlead to convergence problems and is consistent with the theoretical prediction: opening of the natural fracture. The resultsindicate that the opening zone decreases as the value of g increases.

From Eqs. (24) and (26), Fig. 16 can be obtained, which shows regions of different interaction modes with respect to dif-ferential stress. In Fig. 16, the ratio of surface and approach angles dominate the interaction mode, in which case theapproach angle plays a more important role. It is noted that hydraulic fractures cross the pre-fractures only at high horizon-tal differential stress at approach angles of 60� or larger. Hydraulic fractures open the pre-existing fractures only at low hor-izontal differential stress or low approach angles because the fluid pressure in the hydraulic fracture is sufficient to open thepre-fractures. On the other hand, hydraulic fractures will be arrested by shear slippage of the pre-fracture only at high dif-ferential stress and approach angles range from 30� to 60�.

5. Conclusion

Natural fracture plays a significant role in hydraulic fracture propagation and pressure response. An energy based theo-retical method is developed to predict the fluid pressures crossing the interface and opening the natural fracture, and thecorresponding opening and arresting criteria are proposed. Good agreement is obtained between the analytical predictionsand experimental results. The analysis illustrates that mechanical properties (including surface energy, the friction coeffi-cient and cohesion) of the interface and approach angle have a strong influence on the interaction mode, and the interactionmode is sensitive to the approach angle. From the numerical analysis, crossing is easier to occur when the approach angle is

Table 5Loading condition in the numerical simulation.

g rv (MPa) rH (MPa) rh (MPa)

Fig. 13 0.05 16 16 12Fig. 14 0.5 16 16 12Fig. 15 0.05 16 20 12

Fig. 16. Regions of different interaction modes.

a. Crack tip before NF b. Crack tip close to NF

c. Crack tip cross NF d. Crack tip far from NF

Fig. 15. Hydraulic fracture process crossing NF when stress difference is 8 MPa and g is 0.05.



close to 90� under high differential stresses. When the intersection angle decreases from 90�, the interface slip is more likelyto occur under high differential stresses at medium angle ranges from 30� to 60�. The opening mode will be retained withsmall differential stresses. At a lower intersection angle range, opening of the natural fracture will occur. By comparing withexperimental results, the proposed method can predict the complex fracture network and the corresponding fluid pressurewith reasonable accuracy.

Acknowledgement

This work was supported by the National Natural Science Foundation of China (No. 11572249, 11772257).

References

[1] Beugelsdijk LJL, Pater CJ, Sato K. Experimental hydraulic fracture propagation in a multi-fractured medium. In: SPE Asia Pacific Conference, society ofpetroleum engineers; 2000.

[2] Blanton. An experimental study of interaction between hydraulically induced and pre-existing fractures. In: SPE unconventional gas recoverysymposium, society of petroleum engineers; 1982.

[3] Blanton. Propagation of hydraulically and dynamically induced fractures in naturally fractured reservoirs. In: SPE unconventional gas recoverysymposium, society of petroleum engineers; 1986.

[4] Chuprakov DA, Akulich AV, Siebrits E, Thiercelin M. Hydraulic fracture propagation in a naturally fractured reservoir. SPE J 2011;26(1):88–97.[5] Dahi-Taleghani A, Olson JE. Numerical modeling of multistranded hydraulic fracture propagation: accounting for the interaction between induced and

natural fractures. SPE J 2009;16(3):575–81.[6] Daneshy AA. Hydraulic fracture propagation in the presence of planes of weakness. In: SPE-European Spring Meeting, society of petroleum engineers;

1974.[7] Dean RH, Schmidt JH. Hydraulic-fracture predictions with a fully coupled geomechanical reservoir simulator. SPE J 2009;14(4):707–14.[8] Gale JFW, Reed RM, Holder J. Natural fractures in the Barnett shale and their importance for hydraulic fracture treatments. AAPG Bull 2007;91

(4):603–22.[9] Galybin AN, Mukhamediev SA. Fracture development on a weak interface ahead of a fluid-driven crack. Eng Fract Mech 2014;129:90–101.[10] Geertsma J, De Klerk F. A rapid method of predicting width and extent of hydraulically induced fractures. J Petrol Tech 1969;1(12):1571–81.[11] Griffith AA. The phenomena of rupture and flow in solids. Philos Trans Roy Soc Lond 1920;A221:163–98.[12] Gu X, Weng X. Criterion for fractures crossing frictional interfaces at non-orthogonal angles. In: 44th US rock mechanics symposium and 5th US-

Canada Rock mechanics symposium, American Rock Mechanics Association; 2010.[13] Goldstein RV, Osipenko NM. Initiation of a secondary crack across a frictional interface. Eng Fract Mech 2015;140:92–105.[14] Shet C, Chandra N. Analysis of energy balance when using cohesive zone models to simulate fracture processes. J Eng Mater Tech 2002;124:440–50.[15] Gu X, Weng X, Lund JB. Hydraulic fracture crossing natural fracture at nonthogonal angles: a criterion and its validation. SPE J 2011;27(1):20–6.[16] Hou B, Chen M, Li ZM, et al. Propagation area evaluation of hydraulic fracture networks in shale gas reservoirs. Petrol Expl Dev 2014;41(6):833–8.[17] Irwin GR. Analysis of stress and strains near the end of a crack traversing a plate. J Appl Mech 1957;24:361–4.[18] Jaeger JC, Cook NGW, Zimmerman RW. Fundamentals of rock mechanics. Oxford: Blackwell; 2007.[19] Lamont N, Jessen FW. The effects of existing fractures in rocks on the extension of hydraulic fractures. J Pet Technol 1963;15(2):203–9.[20] Zhi Li, Changgui Jia, Chunhe Yang, Yijin Zeng, Yintong Guo. Propagation of hydradulic fissures and bedding planes in hydraulic fracturing of shale. Chin

J Rock Mechan Eng 2014;34(1):12–20.[21] Mokryakov. Analytical solution for propagation of hydraulic fracture with Barenblatt’s cohesive tip zone. Int J Fract 2011;169(2):159–68.[22] Nordgren. Propagation of a vertical hydraulic fracture. Soc Petrol Eng J 1972;12(4):306–14.[24] Olson, Bahorich B, Holder. Examining hydraulic fracture – natural fracture interaction in hydrostone block experiments. In: Hydraulic fracturing

technology conference, society of petroleum engineers; 2012.[25] Potluri NK, Zhu D, Hill AD. The effect of natural fractures on hydraulic fracture propagation. In: SPE European formation damage conference, society of

petroleum engineers; 2005.[26] Settari A, Cleary MP. Development and testing of a pseudo-three-dimensional model of hydraulic fracture geometry. Soc Petrol Eng Prod Eng 1986;1

(6):449–66.[27] Song CP, Lu YY, Xia BW, Hu K. Effects of natural fractures on hydraulic fracture propagation of coal seams. J Northeastern Univ 2014;35(5):756–60.[29] Valko P, Economides MJ. Hydraulic fracture mechanics. Wiley; 1996.[30] Warpinski NR, Teufel LW. Influence of geologic discontinuities on hydraulic fracture propagation. J Pet Tech 1987;39(2):209–20.[31] Wang W, Olson, Prodanovic M. Natural and hydraulic fracture interaction study based on semi-circular bending experiments. In: The Unconventional

Resources Technology Conference, AAPG Combined Publications Database; 2013.[32] Westgaard HM. Bearing pressures and cracks. J Appl Mech 1939;61:49–53.[33] Yao Y. Linear elastic and cohesive fracture analysis to model hydraulic fracture in brittle and ductile rocks. Rock Mech Rock Eng 2012;45(3):375–87.[34] Yao Y, Liu L, Keer LM. Pore pressure cohesive zone modeling of hydraulic fracture in quasi-brittle rocks. Mech Mater 2015;83:17–29.[35] Zhang X, Jeffrey RG. The role of friction and secondary flaws on deflection and reinitiation of hydraulic fractures at orthogonal pre-existing fractures.

Geophys J Int 2006;166(3):1454–65.[36] Zhang X, Jeffrey RG. Defection and propagation of fluid-driven fractures at frictional bedding interfaces: a numerical investigation. J Struct Geol

2007;29(3):396–410.[38] Zhou J, Chen M, JIN Y, et al. Analysis of fracture propagation behavior and fracture geometry using a triaxial fracturing system in naturally fractured

reservoirs. Int J Rock Mech Min Sci 2008;45(7):1143–52.[39] Renshaw CE, Pollard DD. An experimentally verified criterion for propagation across unbounded frictional interfaces in brittle, linear elastic-materials.

Int J Rock Mech Min Sci 1995;32(3):237–49.

http://refhub.elsevier.com/S0013-7944(17)30864-0/h0020http://refhub.elsevier.com/S0013-7944(17)30864-0/h0025http://refhub.elsevier.com/S0013-7944(17)30864-0/h0025http://refhub.elsevier.com/S0013-7944(17)30864-0/h0035http://refhub.elsevier.com/S0013-7944(17)30864-0/h0040http://refhub.elsevier.com/S0013-7944(17)30864-0/h0040http://refhub.elsevier.com/S0013-7944(17)30864-0/h0045http://refhub.elsevier.com/S0013-7944(17)30864-0/h0050http://refhub.elsevier.com/S0013-7944(17)30864-0/h0055http://refhub.elsevier.com/S0013-7944(17)30864-0/h0065http://refhub.elsevier.com/S0013-7944(17)30864-0/h0070http://refhub.elsevier.com/S0013-7944(17)30864-0/h0075http://refhub.elsevier.com/S0013-7944(17)30864-0/h0080http://refhub.elsevier.com/S0013-7944(17)30864-0/h0085http://refhub.elsevier.com/S0013-7944(17)30864-0/h0090http://refhub.elsevier.com/S0013-7944(17)30864-0/h0095http://refhub.elsevier.com/S0013-7944(17)30864-0/h0100http://refhub.elsevier.com/S0013-7944(17)30864-0/h0100http://refhub.elsevier.com/S0013-7944(17)30864-0/h0105http://refhub.elsevier.com/S0013-7944(17)30864-0/h0110http://refhub.elsevier.com/S0013-7944(17)30864-0/h0130http://refhub.elsevier.com/S0013-7944(17)30864-0/h0130http://refhub.elsevier.com/S0013-7944(17)30864-0/h0135http://refhub.elsevier.com/S0013-7944(17)30864-0/h0145http://refhub.elsevier.com/S0013-7944(17)30864-0/h0150http://refhub.elsevier.com/S0013-7944(17)30864-0/h0160http://refhub.elsevier.com/S0013-7944(17)30864-0/h0165http://refhub.elsevier.com/S0013-7944(17)30864-0/h0170http://refhub.elsevier.com/S0013-7944(17)30864-0/h0175http://refhub.elsevier.com/S0013-7944(17)30864-0/h0175http://refhub.elsevier.com/S0013-7944(17)30864-0/h0180http://refhub.elsevier.com/S0013-7944(17)30864-0/h0180http://refhub.elsevier.com/S0013-7944(17)30864-0/h0190http://refhub.elsevier.com/S0013-7944(17)30864-0/h0190http://refhub.elsevier.com/S0013-7944(17)30864-0/h9000http://refhub.elsevier.com/S0013-7944(17)30864-0/h9000

本文献由“学霸图书馆-文献云下载”收集自网络，仅供学习交流使用。

学霸图书馆（www.xuebalib.com）是一个“整合众多图书馆数据库资源，

提供一站式文献检索和下载服务”的24 小时在线不限IP

图书馆。

图书馆致力于便利、促进学习与科研，提供最强文献下载服务。

图书馆导航：

图书馆首页文献云下载图书馆入口外文数据库大全疑难文献辅助工具

http://www.xuebalib.com/cloud/http://www.xuebalib.com/http://www.xuebalib.com/cloud/http://www.xuebalib.com/http://www.xuebalib.com/vip.htmlhttp://www.xuebalib.com/db.phphttp://www.xuebalib.com/zixun/2014-08-15/44.htmlhttp://www.xuebalib.com/

An energy based analytical method to predict the influence of natural fractures on hydraulic fracture propagation1 Introduction2 Critical fluid pressure for induced fracture and opening natural fracture3 Validation of the critical fluid pressure4 Opening and arresting criteria on differential stresses5 ConclusionAcknowledgementReferences

学霸图书馆link:学霸图书馆

engineering fracture mechanicsdownload.xuebalib.com/jo3gcucrsr4.pdf · an energy based analytical...

Documents