A composite criterion to predict subsequent intersection behavior between a hydraulic fracture and a natural fracture

Engineering Fracture Mechanics 209 (2019) 61–78

Contents lists available at ScienceDirect

Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech

A composite criterion to predict subsequent intersection behavior between a hydraulic fracture and a natural fracture

T

⁎

Yu Zhaoa,b, Yong Fa Zhanga, , Peng Fei Hea a

School of Civil Engineering, Chongqing University, Chongqing 400045, China Key Laboratory of New Technology for Construction of Cites in Mountain Area (Chongqing University), Ministry of Education, Chongqing 400045, China

b

A R T IC LE I N F O

ABS TRA CT

Keywords: Hydraulic fracture Approaching process Intersection modes Analytical composite criterion Approaching distance

Hydraulic fracturing is a highly effective technique applied to enhance the productivity of unconventional reservoirs. The fracture induced by hydraulic fracturing operation may cross, open or get arrested by the pre-existing discontinuities in naturally fractured formations, which greatly influences the efficiency of resource extraction. In order to predict the subsequent behaviors and fully understand the stress stability during the process of a hydraulic fracture approaching a natural fracture, we propose a new composite criterion through coupling the fluid flow and rock elastic deformation. This new criterion can simultaneously predict three intersection behaviors based on the critical stress conditions of slipping, opening and crossing. Besides, the criterion is validated by comparing the predicted results to three independent experiments and published intersection criteria, and good agreement with the experiments as well as sufficient advantages over the previous criteria is observed Finally, the perturbation of different parameters including minimum horizontal stress, fracture toughness and approaching distance is introduced to the composite criterion by analysing the NF stress field and parametric sensitivity which further conforms this criterion’s applicability. The results also demonstrate that the variation of the introduced parameters will rezone the intersection scope and then change the geometry of the critical slippage and opening curves, which can promote or inhabit the occurrence of the predicted intersection behaviors to different degrees.

1. Introduction Hydraulic fracturing has been extensively used as a stimulation technique to enhance the productivity of unconventional oil and gas reservoir, which also plays an essential part in rock engineering, nuclear waste management, geothermal energy extraction and fluid underground storage [1–4]. Problems arising from hydraulic fracturing involves the nonlinear coupling of rock deformation and fluid flow, the nonlocal character of the fracture elastic response, the time dependence of fracture propagation and the interacting interference between the pre-existing and induced fractures. These problems will result in the offsetting of fracture path and leak-off of fracturing fluid and then complicate the analysis of hydraulic fracturing [5]. Whereas the economic production from these reservoirs tends to depend on the efficiency of hydraulic fracturing stimulation treatment [6]. So fully modeling the hydraulic fracturing is necessary and significant in optimizing the fracturing parameters and stimulating reservoirs production.

⁎

Corresponding author. E-mail address: [email protected] (Y.F. Zhang).

https://doi.org/10.1016/j.engfracmech.2019.01.015 Received 24 October 2018; Received in revised form 9 January 2019; Accepted 9 January 2019 Available online 14 January 2019 0013-7944/ © 2019 Elsevier Ltd. All rights reserved.

Engineering Fracture Mechanics 209 (2019) 61–78

Y. Zhao et al.

Nomenclature σβy Symbols τβxy σγ, βx

HF NF KI KIC

hydraulic fracture natural fracture with frictional interface stress intensity factor mode I fracture toughness of rock (critical stress intensity factor) β intersection angle between HF and NF PHF internal fluid pressure within HF PNF total normal stress preventing NF from opening σh minimum horizontal in situ stress σH maximum horizontal in situ stress t injection time l(t)/l half-length of HF Pnet(x, t) net fluid pressure inside HF Q0 injection rate σxx normal stress in the direction of x-axis induced by net pressure σyy normal stress in the direction of y-axis induced by net pressure τxy shear stress induced by net pressure σβx total principal stress applied parallel to NF

σγ,βy

τγ,β

σPnet, βx

σPnet, βy

τPnet, β T0 rc Δl

interface (along the direction of βx axis) total normal stress applied perpendicularly on NF interface (along the direction of βy axis) total shear stress applied on NF interface normal stress component generated by in situ stress filed along the direction of βx axis on NF interface normal stress component generated by in situ stress field along the direction of βy axis on NF interface shear stress component generated by in situ stress field on NF Interface normal stress components along the direction of βx axis generated by induced stress field on NF interface normal stress components along the direction of βy axis generated by induced stress field on NF interface shear stress components generated by induced stress field on NF interface rock tensile strength critical radius of nonlinear region where the stresses on the natural interface are maximized HF approaching distance

Early research efforts concentrated on modeling the propagation of single planar hydraulic fracture (HF) in linear elastic, impermeable and homogeneous rock. As the horizontal in situ stress in the overlying and underlying layers is much larger than that in the reservoir layer, the growth of HF height is expected to decline and ultimately stop, and therefore the phenomenon of constant fracture height appears [7,8]. When the fracture length is sufficiently small compared to the fracture height, this hydraulic fracturing problem can be theoretically simplified to a plane-strain model, well-known as KGD and PKN models [9–11]. These models typically rely on the simplification of fracturing problem either with respect to the fracture width profile or the fluid pressure distribution. On this basis, through coupling the elasticity equations and lubrication equations, Desroches et al. [12] derived a zero-toughness singular solution of the fracture width near the HF tip region where the stress singularity is weaker than that of inverse square root in linear elastic fracture mechanics. The followers Garagash [5,13,14] and Adachi [15] focused on the particular near-tip behavior of the HF and derived the global solution of plane-strain hydraulic fracturing problem supposing that the strong fluid-solid coupling was mainly confined to the fracture near-tip region. Further, Detournay [16] developed a theoretical framework to predict the propagation modes of the HF and identified the fluid viscosity and solid fracture toughness dominant regimes of the energy dissipation process. In terms of toughness-dominated regime, the solution to the stress field near an HF can be approximated and the viscosity of fracturing fluid is assumed to be zero or infinitely small. However, the simple HF geometric assumption of a straight and bi-wing planar feature is untenable because of the pre-existing geological discontinuities including fissures, veins, joints, faults and bedding planes (assumed to be a natural fracture with frictional interface and thereafter called NF) in naturally fractured formations [17]. The HF will inevitably intersect with multiple NFs and therefore result in complicated fracture networks [18]. Plenty of microseismic measurements and field observations [19–21] also suggest that the creation of complicated fracture networks is a common occurrence during the process of hydraulic fracturing treatments. Based on experiment and field analysis [22–24], three intersection modes (crossing, opening and slippage) were mainly observed when the HF propagating in naturally fractured formations as sketched in Fig. 1. The case of “Crossing”, “Opening” and “Slippage” means that the HF crosses the NF, the NF slips upon contact with HF, and the HF propagates along the dilated NF, respectively. The

Crossing

Arrested

Opening

Fig. 1. Intersection modes between an HF and a NF [24]. 62

Engineering Fracture Mechanics 209 (2019) 61–78

Y. Zhao et al.

occurrence of each intersection behavior depends on the stress conditions and material mechanic properties to a large extent [25]. In reality, the minimum horizontal stress contrast defines the lower limit for the propagation capacity of the HF where the pressure for initiating a fracture needs to exceed the sum of the minimum horizontal stress and the rock tensile strength [7]. Combining the rock failure and fracture propagation condition along with certain specific assumptions, several criteria have been developed to predict which mode will dominate the fracture behavior [22,24,26,27]. For example, Renshaw and Pollard [27] established a compressional crossing criterion to evaluate a new fracture re-initiation under the assumption that no slippage occurs in NF all along. But Renshaw and Pollard’ model was confined to an orthogonal intersection angle with the NF and merely focused on the co-action of fracture tip stress and far field in situ stress. The fluid pressure effect induced by the propagating HF and the case of opening were not considered, although Renshaw and Pollard’ model was later extended to non-orthogonal intersection by Gu and Weng [24]. Blanton [22] developed a crossing and opening criterion by simplifying the profiles of normal and shear tractions at the

(a) Geometry of the intersection model

(b) limit state of the HF tip ( l 0) Fig. 2. Schematic diagram of a hydraulic fracture approaching a natural fracture. 63

Engineering Fracture Mechanics 209 (2019) 61–78

Y. Zhao et al.

NF, which also neglected the effect of the stress induced by the propagating HF. According to Warpinski and Teufel [26] and Yao et al. [28], the stress field produced by fluid pressure inside HF poses obvious positive effect on reducing the arresting threshold whereas passive influence on the critical opening limit. But neither of the two models take into account of the effect of fluid flow (i.e. injection rate and fluid viscosity) which is known to be crucial from experimental and field observations [29]. Based on these results, Chuprakov et al. [30] quantitatively described the NF activation and presented a multi-parameter Open-T model which is sensitive to fracture toughness, injection rate, HF length and NF permeability. Nevertheless, the Open-T model merely studied the subsequent behaviors after an HF intersected with a NF, where the stress singularity diminished and resulting comparably small stress component near the HF tip. Hence the Open-T model becomes inapplicable in dealing with the case of an HF approaching a NF. To elaborate the intersection mechanism of a finite-length HF propagating towards a NF, Llanos et al. [31] reformulated Renshaw and Pollard’s intersection geometry and obtained a reduced form of the control equations by means of scaling the parameters of the fracture length and the in situ stress. However, Llanos’s calculation model merely considers the situation that the HF approaches the NF perpendicularly. Thus the non-orthogonal intersection should be generalized due to the arbitrary distribution of NFs in fractured formations. Besides, the coupling of fluid flow and rock elastic deformation was not involved in the reduced equations, which makes this model limited for complete analysis. Generally, the review of previous criteria above mainly focused on the behavior after intersection but omitted the approaching intersection process simultaneously coupling fluid flow and rock elastic deformation. In this paper, we pay our attention to presenting a composite criterion to predict subsequent behaviors based on stress conditions before the HF comes into contact with the NF. This criterion is validated by comparing the predicting results with previously published experimental observations and intersection criteria. Furthermore, this new criterion includes the sensitivity of the HF approaching distance missed in previous intersection models, with which the new criterion is discussed in detail. 2. Composite criterion 2.1. Problem statement and assumptions As shown in Fig. 2(a), a plane-strain intersection model is developed considering a uniformly pressured finite-length HF (green elliptical region) approaching an unbounded frictional NF (red rough line) at an arbitrary angle β. The internal fracturing fluid inside the HF is incompressible and inviscid. And the initial stress field is dominated by far-field in situ stress components σH and σh which are respectively parallel and perpendicular to the HF in the Oxy reference plane. The yellow arrow points out the injection location and injection rate Q0. For convenience, a local coordinate system (O′ βx βy) is established associated with the NF, where the βx axis is codirected with the NF and the βy axis corresponds to the direction of the approaching HF and intersects with the βx axis at the NF midpoint. During the process of an HF approaching a NF, the stress components acting on the NF are influenced by the co-action of the fluid flow within the HF on both sides of the injection point and the far field in situ stress. As the HF extends and moves closer to the NF, the stress levels on the NF increase and become closer to the critical failure thresholds (rock tensile strength or NF cohesional strength). Assuming a limit state where the HF is infinitely close to but not contact with the NF (Fig. 2(b)), the existing stress singularity near the HF tip will greatly influence the stress distribution on the NF, and certain points on the NF may have reached or even exceeded the critical failure limit. In general, the goal of the work is to predict the intersection behaviors (including crossing, slippage and opening) between the HF and the NF based on known parametric conditions and assumed geometric model. To ensure the applicability of the presented model, some key assumptions are necessary to be illustrated: (1) The rock matrix is assumed to be homogenous, isotropic, linear elastic and impermeable. And the mechanic properties on both sides of the NF is identical with the rock matrix; (2) No fluid lag zone exists in the HF, which means the induced stress field will be completely generated by the fluid pressure (or more precisely by the net fluid pressure); (3) The HF width is not considered, so the concerned HF propagation problem is limited to one dimensional motion of a twodimensional plane strain; (4) The right tip of the HF is infinitely close to but not contact with the NF. Thus, the HF tip is not blunted, and the stress singularity still exists; (5) The possible behaviors of crossing, slippage and opening between the HF and the NF are assumed mutually independent. The mixed behaviors of opening after crossing and crossing after slippage are not considered. 2.2. Solution of net pressure inside the HF For the initial assumption of inviscid fracturing fluid, the propagation of HF can be identified as the toughness-dominated regime [15,16]. The half-length of the HF is designated l(t); the HF width is w(x, t); Pnet denotes the net pressure within the HF which only depends on the x coordinate (with origin at the injection point, O); and the fluid injection time is t. According to Detournay [16], the unified form of theoretical solution to a single fracture is defined by means of dimensionless analysis: 64

Engineering Fracture Mechanics 209 (2019) 61–78

Y. Zhao et al.

l (t ) = L (t )·γ [ρ (t )] w (x , t ) = ε (t )·L (t )·Ω(ξ , t ) Pnet (x , t ) = ε (t )·E′·Π(ξ , t )

(1)

where ξ = x/l(t) (0 ≤ ξ ≤ 1) refers to the scaled position which defines a moving system of coordinates (with regard to the fixed system of coordinates x); ε(t) denotes a small dimensionless parameter that guarantees the variation range of Ω and Π from zero to infinity; L represents the length scale; γ, Ω, and Π are the dimensionless fracture half-length, opening and net pressure, respectively; and ρ(t) is the dimensionless evolution parameter [15]. Based on the scaling of Eq. (1), the elasticity equation can be expressed as

Ω=−

4γ π

∫0

1

1 − ξ2 +

ln

1−

ξ2

1 − s2

Π(s, t ) ds

1 − s2

−

(2)

The fluid lubrication equation can be written as

̇ ̇ ⎛ ε ṫ + Lt ⎞ Ω − Lt ξ ∂Ω + ρṫ ⎜⎛ ∂Ω − ξ dγ ∂Ω ⎟⎞ = 1 ∂ ⎜⎛Ω3 ∂Π ⎞⎟ L⎠ L ∂ξ γ dρ ∂ξ ⎠ Gm γ 2 ∂ξ ⎝ ∂ξ ⎠ ⎝ε ⎝ ∂ρ ⎜

⎟

(3)

The fluid mass balance becomes

2γ

∫0

1

Ωdξ = Gv

(4)

The HF propagation condition (KI = KIC) is

Ω = G k γ1/2 (1 − ξ )1/2 ,

1−ξ≪1

(5)

In the toughness-dominated regime, the dimensionless groups of Gv and Gk both equal 1, and Gm is a function expression about the μ′ known parameters (Gm = 3 ′ , μ′ represents the magnitude of fluid viscosity) according to [15]. The classic zero-viscosity solution of a ε Et toughness-dominated fracture is derived by Garagash [5,14], and the related dimensionless parameters are expressed as follows:

8 ·KIC , 2π

K′ =

E′ = 1/3

K ′4 ⎞ ε (t ) = ⎜⎛ 4 ⎟ ⎝ E ′ Q0 t ⎠ γ=

2 π 2/3

,

E 1 − v2

(6) 2/3

,

E′Q0 t ⎞ L (t ) = ⎛ ⎝ K′ ⎠

Ω(ξ ) = π −1/3·(1 − ξ 2)1/2 ,

[0, 1]

Π=

(7)

π 1/3 8

(8)

where E′ is defined as plane-strain elastic modulus, v denotes Poisson’s ratio, KIC is the rock fracture toughness and Q0 represents the injection rate. Finally, the hydraulic fracturing solution of a single HF is obtained by combining Eqs. (1) and (6)–(8):

E′ 2/3 l (t ) = γ ·⎛ ⎞ ·(Q0 t )2/3 ⎝ K′ ⎠

(9)

K ′ 2/3 w (ξ , t ) = Ω(ξ )·⎛ ⎞ ·(Q0 t )1/3 ⎝ E′ ⎠

(10)

K ′ 4/3 Pnet (ξ , t ) = E′·Π(ξ )·⎛ ⎞ ·(Q0 t )−1/3 ⎝ E′ ⎠

(11)

h

h

Pnet=PHF ı h

ıH

y x

PHF

ıH

ıH ıh Total stress

ıH ıh Far-field stress

Induced stress

Fig. 3. Schematic of a single HF stress decomposition (stress superposition principle). 65

Engineering Fracture Mechanics 209 (2019) 61–78

Y. Zhao et al.

2.3. Slippage condition for the NF The HF propagates in a bi-directional compressive stress state and is subject to the combined action of far-field in situ stress (σH and σh) and inner fluid pressure (Pf) as shown in Fig. 3. According to the stress superposition principle [32], the total stress on the NF interface should be equivalent to the superposition of the far field stress and the induced stress. The far field stress keeps uniform and is generated by in situ stress, while the induced stress originates from the net pressure (Pnet) within the HF and is generated by its inner fluid flow. According to Westergaard’s analyses [33], the induced stress field produced by the net fluid pressure within the HF in Oxy reference plane is ∗ ∗ r0 ⎧σ = P ⎡ 2 cos(θ − θ0) + 2 sin θ0 sin θ − sin θ1× ⎤ − Pnet net ⎡ 2r ∗ ⎤ × ∗ + θ − θ ) − sin θ sin(θ∗ + θ − θ ) ⎥ ⎪ xx ⎣ ⎦ ⎢ sin( θ 1 0 2 2 0 ⎦ ⎣ ⎪ ⎪ ∗ − θ ) − 2 sin θ sin θ∗ + sin θ × 2 cos( θ r0 0 0 1 ⎤ σyy = Pnet ⎡ 2r ∗ ⎤ × ⎡ ⎨ ∗ + θ − θ ) + sin θ sin(θ∗ + θ − θ ) ⎥ ⎣ ⎦ ⎢ sin( θ 1 0 2 2 0 ⎦ ⎣ ⎪ ⎪ r0 × [sin θ1 cos(θ∗ + θ1 − θ0) − sin θ2 cos(θ∗ + θ2 − θ0)] ⎪ τxy = Pnet ⎡ ⎣ 2r ∗ ⎤ ⎦ ⎩

(12)

where

− π ⩽ {θ0 , θ1, θ2, θ∗} ⩽ π , 1 2

r∗ =

r1 r2 , θ∗ =

r0 =

x2

r1 =

(x − l)2 + y 2 , θ1 = tan−1 ( x − l ),

r2 =

(x + l)2 + y 2 , θ2 = tan−1 ( x + l ),

+

y2

(θ1 + θ2 ) , y

, θ0 = tan−1 ( x ), y

y

The stress components of the HF induced stress applied to the NF, inclined by the angle β with respect to the direction of σxx, are expressed as follows: σxx + σyy

σxx − σyy

cos(2β )+ τxy sin(2β ); + ⎧ σPnet, βx = 2 2 ⎪ σxx + σyy σxx − σyy σ cos(2β )− τxy sin(2β ); = − 2 2 ⎨ Pnet, βy σxx − σyy ⎪ τP , β = − 2 sin(2β )+ τxy cos(2β ); ⎩ net

(13)

The stress components projected on NF (inclined at angle β) from far-field stresses σH and σh are given by σ +σ

σ −σ

H H h h ⎧ σγ , βx = 2 + 2 cos 2β ⎪ σ +σ σ −σ σγ , βy = H 2 h − H 2 h cos 2β ⎨ σ H − σh ⎪ τγ , β = − sin 2β 2 ⎩

(14)

The total normal and shear stress of the combined stress field can be obtained by superposing Eqs. (5) and (6):

σ = σPnet, βy + σγ , βy ⎧ βy ⎨ ⎩ τβxy = τPnet, β + τγ , β

(15)

When the normal stress on the NF is compressive, the failure of the interface can be judged by Mohr-Coulomb criterion (supposing compressive stress to be negative):

|τβxy| ⩾ c − μσβy

(16)

where μ is the friction coefficient, c denotes the cohesion of the NF, σβy and τβ are the total normal and shear stresses on the NF interface, respectively. The onset of slip is estimated analytically using the geometry illustrated in Fig. 1(b). If Eq. (16) holds, the NF may undergo shear failure at different degrees such that the growth of the HF will probably get arrested by the NF. Once the HF is arrested by the NF, the pre-existing stress singularity at HF tip will diminish and the stress field in the vicinity of the HF tip will make much of a difference. 2.4. Opening criterion for the NF The critical fluid pressure required to open the NF and move the propagation ahead [28] is

PNF = σβ y +

KIC πl

η

(17) (18)

PNF ⩽ PHF 66

Engineering Fracture Mechanics 209 (2019) 61–78

Y. Zhao et al.

where η = γNF/γRock, γNF and γRock are the surface energy of the NF and the intact rock matrix, respectively. The NF will be probably opened by the fracturing fluid inside the HF under the opening condition (Eq. (18)) which defines the upper limit of the NF opening in combined stress field. Upon contact, the penetrating fracturing fluid may flow along either side of the NF midpoint, which highly depends on the intersection angle [7,34]. 2.5. Crack initiation condition According to the linear elastic fracture mechanics, the stress field of a fracture tip [27] is approximated as K

θ

θ

3θ

I cos 2 (1 − sin 2 sin 2 ) ⎧ σx = σH + 2πr ⎪ KI θ θ 3θ σy = σh + 2πr cos 2 (1 + sin 2 sin 2 ) ⎨ ⎪ τ = KI sin θ cos θ cos 3θ xy 2 2 2 2πr ⎩

(19)

where KI is the stress intensity factor, r and θ are the polar coordinates with origin at HF tip, and θ = β (or β–π). As the HF tip constantly approaches the NF, the stress projected on the NF becomes increasingly great so that the inelastic deformation is produced within a certain range of the HF tip, which implies the existence of a nonlinear region adjacent the approaching intersection point in limit state (Fig. 1(b)). Within the nonlinear region, the rock may stay in elastoplastic or plastic state and the highly accumulated elastic strain energy near the HF tip has been dissipated to some extent in the form of heat and surface energy, which makes the linear elastic fracture mechanics invalid for distances less than the critical radius (rc) to the HF tip. Therefore, the effective stresses within nonlinear region is always assumed to be comparable or less than the stresses at rc [24,35–36]. Similar to the notion of Gu and Weng [24], which considered the inelastic behavior of the HF tip prior to contacting with the NF, the mechanical condition for a new fracture initiation on the opposite side of the NF interface is to achieve equilibrium between the maximum principle tensile stress and the rock tensile strength T0. Hence the critical initiating condition is given by

σ1 =

σx + σy 2

+

⎛ ⎝

σx − σy 2

2

2 ⎞ + τxy = T0 ⎠

(20)

With known T0 and KIC (critical stress intensity factor), the nonlinear region critical radius rc is derived combining Eqs. (19) and (20). Thus the initiation position can be determined by checking the stress acting on the NF starting from the distance of rc away from the NF midpoint. It follows then that crossing will occur if the stresses of a certain points outside nonlinear region (r ≥ rc) dissatisfy the shear failure condition (Eq. (16)) and the opening condition (Eq. (18)):

|τβxy | < c − μσβy;

PNF > PHF

(21)

2.6. Composite criterion for the intersection behaviors Numerous researches [24,26,30] have demonstrated that the in situ stress difference and intersection angle are the predominant factors to be considered in the analysis of HF-NF intersection mechanism. Thus, we pay attention to the sensitivity of the stress difference of far-field maximum and minimum horizontal stresses and the intersection angle on condition that other related parameters including fracture toughness, friction coefficient and initial fracture length remain constant.

Fig. 4. Diagram of different intersection modes based on composite criterion where the dimensionless stress constant Δ = (σH − σh)/σh. 67

Engineering Fracture Mechanics 209 (2019) 61–78

Y. Zhao et al.

Combining the slippage, opening and initiation conditions expressed in Eqs. (16), (18) and (21), the possible intersection behaviors can be predicted mathematically with coded computational program, and the corresponding critical slippage as well as the opening curves can also be drawn out. As delineated in Fig. 4, the intersection behaviors are classified into 4 types, namely, crossing zone, slippage zone (no opening), opening zone and opening zone (no slippage). According to the composite criterion proposed in this paper, the diagram of stress difference versus intersection angle was divided into four possible intersection behaviors in corresponding color areas: (1) Crossing zone of pink area: Under the condition of satisfying no slippage and no opening, HF will directly pass through NF; (2) Slippage zone of green area: Under the condition of satisfying slippage and no opening, shear slippage will occur in NF; (3) Opening zone (no slippage) of orange area: Under the condition of no slippage and opening, the NF is dilated and the HF will propagate along the direction of NF extending. (4) Opening zone of blue area: Under the condition of satisfying slippage and opening, the HF is opened and will propagate along NF. The division of these zones is directly related to the critical shear failure condition and the critical opening condition of the NF. Therefore, the scope of each zone will also be directly related to the mechanical parameters of the two critical conditions. 3. Validation 3.1. Comparison with laboratory experiments To validate the applicability of the composite prediction, the results of possible intersection behaviors based on the presented analytical model are compared to three independent laboratory experiments: Blanton experiments, Zhou experiments and Gu experiments [22,37,23,24]. It should be noted that the value of η in Eq. (17) can be assumed to be zero for the surface energy of NF is infinitely small compared with that of rock. 3.1.1. Blanton experiments Blanton [22,37] preformed the fracture intersection experiments using hydrostone blocks which were fractured by a pre-existing surface under different angles of approach (30°, 45°, 60° and 90°) and stress states (−σh = 5 or 10 MPa and the principal differential stresses varied from 2 to 15 MPa). The pre-fractured surface was frictional and cohesiveless with a friction coefficient of 0.75. The tensile strength of hydrostone was 3.1 MPa. And the Mode I fracture toughness was 0.176 MPa m1/2. The fracturing fluid was injected at constant flow rate of 0.82 cm3/s through a wellbore simulated by a steel pipe. In addition, the half-length of the HF was equal to 0.06 m. The fluid viscosity is assumed to be sufficiently small in a toughness-dominated HF, so its effect becomes negligible. Table 1 lists the results of Blanton experiments [37] and the corresponding calculation outcomes from the new criterion. 3.1.2. Zhou experiments Zhou et al. [23] reported similar experiments about the HF-NF intersection on cement-sand model blocks with scaled proportion to real rock in triaxial compressive stress state. The pre-fractures were created by 3 types of paper with cohesion of 3.2 MPa and friction coefficient of 0.38, 0.89 and 1.21, respectively. The interaction angles in each block between the HF and NF were varied systematically of 30°, 60° and 90°. The Mode I fracture toughness of model blocks is 0.59 MPa m1/2, and the half-length of HF was designed as 0.06 m. The vertical stress remain invariable at 20 MPa, meanwhile the horizontal stress difference was changed from 3 to 10 MPa. 3.1.3. Gu experiments In the recent experiments conducted by Gu et al. [24], Calton sandstone (T0 = 4.054 MPa, KIC = 1.6 MPa m1/2) was cut into rectangle blocks and prepared for the intersection test at true triaxial-stress condition. A discontinuity interface characterized with friction efficient of 0.615 and negligible cohesion were prefabricated in the blocks at specified angles ranging from 45° to 90°. Silicone Table 1 Blanton experiments (μ = 0.75, η = 0) [37]. β

σH (MPa)

σh (MPa)

rc (10−4m)

NF slipping

PHF (MPa)

PNF (MPa)

PHF-PNF

Experimental results

Criterion results

90° 60° 60° 60° 45° 45° 45° 45° 45° 30° 30°

−14 −12 −20 −14 −20 −18 −16 −14 −10 −20 −19

−5 −10 −5 −5 −5 −5 −5 −5 −5 −5 −10

0.9470 0.4840 1.268 1.268 1.187 1.188 1.189 1.192 1.199 1.003 1.003

No No Yes Yes Yes Yes Yes Yes Yes Yes Yes

5.4053 10.405 5.4053 5.4053 5.4053 5.4053 5.4053 5.4053 5.4053 5.4053 10.405

11.853 7.5390 12.883 8.3827 8.1815 7.1835 6.1862 5.1897 3.2009 3.3792 4.8169

<0 >0 <0 <0 <0 <0 <0 >0 >0 >0 >0

Crossing Opening Crossing Arrested Arrested Arrested Arrested Arrested Opening Arrested Opening

Crossing Opening Arrested Arrested Arrested Arrested Arrested Opening Opening Opening Opening

68

Engineering Fracture Mechanics 209 (2019) 61–78

Y. Zhao et al.

oil (viscosity of 1 Pa s) was injected at a constant rate of 0.5 cm3/s. The vertical stress remain invariable at 27.58 MPa, meanwhile the horizontal stress difference was changed from 0.69 to 10.35 MPa with unchangeable minimum horizontal stress of 6.89 MPa. And the initial half-length of HF was 0.076 m. In general, the comparison results in Tables 1–5 indicate that the calculation outcomes obtained from the composite criterion show good agreement with the experimental results except several particular cases (highlighted with bold font). The disagreement between the experimental and predicting results arises from the inappropriate range of nonlinear zone which implies that the actual nonlinear region at the crack tip is relatively larger. The reason for this change is the co-action of stress singularity and plastic zone ahead of the HF tip which can enlarge the radius of the nonlinear zone. In this case, the initially calculated points at rc no longer follow the linear elastic mechanics. So the inapplicable cases emerged. We drew the conclusion that when the radius rc of the nonlinear region in Blanton experiments is greater than 1.65 × 10−4 m, the calculated outcomes will coincide with the experimental results, which further verified the reason of the disagreement. 3.2. Comparison with previous intersection criteria Considerable research effort have made and concentrated on establishing effective predicting criterion for the HF-NF intersection behavior through theoretical and numerical analysis [22,26,28,38–40]. To explore the discrepancy and advantage, we conduct a comparison between the composite criterion and a series of previous criteria (Blanton’s criterion, Gu and Weng criterion and Yao criteria) under the same parametric conditions. It is worth noting that the imperial unit of psi is used for the convenience of comparing the predicting results with the original data in Figs. 5, 6(a) and 7(a). Fig. 5 shows the boundary of different areas of the predicting intersection behavior corresponding to different analytical criteria in the parameter space of the intersection angles β and the horizontal in situ stress difference. The initial material parameters are selected from Blanton experiments [22], where Devonian shale is chose as the rock sample with tensile strength T0 of 5.67 MPa (823 psi), fracture toughness KIC of 1.59 MPa m1/2, Young modulus E of 10 GPa and Poisson ratio ν of 0.22, NF cohesion c of 0 MPa, NF friction efficient μ of 0.75, minimum horizontal in situ stress σh of 5 MPa (725 psi), initial HF length l of 0.06 m, and injection rate Q of 0.82e−6 m3/s. (1) Blanton’s criterion In Blanton’s experiments and analysis of HF-NF intersection [22,37] an intersection criterion was developed and validated through dynamic triaxial fracturing test and field observation. The crossing will occur and the initiation of a new fracture can progress provided that the treating pressure within the HF exceeds the superposition of the rock tensile strength and the stress acting parallel to the NF. After mathematical simplification, the final form of the crossing criterion is introduced by the following:

σH − σh >

−T0 cos 2β − b sin 2β

(22)

where b = 0.2 as an asymptote for an unbounded interface at the NF [22]. As delineated in Fig. 5(a) (red dash line), the lower and upper branches of Blanton criterion curve are almost parallel to the bottom segment of slippage curve and the top part of opening curve, respectively. This result implies that the two criteria share approximate criterion form for a relatively large and small intersection angle. However, based on the initial assumptions and mechanical conditions, the subsequent behavior predicted by Blanton criterion was either crossing or opening, but the case of slippage and the effect of induced stress field were not taken into consideration, which results in the criterion merely relating to three parameters (stress difference, intersection angle and b). Hence Blanton criterion was to some extent insufficient for predicting all of the intersection behaviors. (2) Gu and Weng criterion (Gu & Weng criterion) Gu and Weng [24] considered an HF approaching a frictional interface prior to contact at non-orthogonal angles and proposed a crossing criterion based on linear elastic stress analysis near the fracture tip. This criterion can be applied to determine the occurrence of crossing provided that the accumulated fluid pressure is sufficient to initiate a new crack on the opposite side of the NF before the superposition stress consisting of the remote in situ stress and the HF-tip stress field along the interface reaches the critical shear Table 2 Zhou-1 experiment (μ = 0.38, η = 0) [23]. β (°)

σH (MPa)

σh (MPa)

rc (m)

NF slipping

PHF (MPa)

PNF (MPa)

PHF-PNF

Experimental results

Criterion results

90° 90° 60° 60° 30° 30°

−8 −8 −10 −8 −10 −8

−3 −5 −3 −3 −3 −3

0.0020 0.0012 0.0026 0.0026 0.0021 0.0021

No No No No Yes Yes

4.3593 6.3593 4.3593 4.3593 4.3593 4.3593

7.6363 7.0646 6.4670 4.9670 6.1692 0.1169

<0 <0 <0 <0 <0 >0

Crossing Crossing Crossing Opening Arrested Opening

Crossing Crossing Crossing Crossing Arrested Opening

69

Engineering Fracture Mechanics 209 (2019) 61–78

Y. Zhao et al.

Table 3 Zhou-2 experiment (μ = 0.89, η = 0) [23]. β (°)

σH (MPa)

σh (MPa)

rc (m)

NF slipping

PHF (MPa)

PNF (MPa)

PHF-PNF

Experimental results

Criterion results

90° 90° 60° 60° 60° 30° 30° 30°

−10 −10 −10 −13 −8 −10 −8 −13

−5 −3 −3 −3 −5 −5 −5 −3

0.0011 0.0019 0.0026 0.0026 0.0026 0.0012 0.0012 0.0021

No No No No No Yes Yes Yes

6.3593 4.3593 4.3593 4.3593 6.3593 6.3593 6.3593 4.3593

9.0646 9.5846 6.4670 8.7170 4.4710 5.8476 8.4760 1.3669

<0 <0 <0 <0 >0 >0 <0 >0

Crossing Crossing Crossing Crossing Opening Opening Opening Arrested

Crossing Crossing Crossing Crossing Opening Opening Arrested Opening

Table 4 Zhou-3 experiment (μ = 1.21, η = 0) [23]. β (°)

σH (MPa)

σh (MPa)

rc (m)

NF slipping

PHF (MPa)

PNF (MPa)

PHF-PNF

Experimental results

Criterion results

90° 90° 60° 60° 30° 30°

−8 −13 −13 −10 −13 −8

−3 −3 −3 −3 −3 −3

0.0020 0.0019 0.0026 0.0026 0.0021 0.0021

No No No No Yes Yes

4.3593 4.3593 4.3593 4.3593 4.3593 4.3593

7.6363 12.6345 8.7170 6.4670 1.3669 0.1169

<0 <0 <0 <0 >0 >0

Opening Crossing Opening Opening Opening Opening

Crossing Crossing Crossing Crossing Opening Opening

Table 5 Gu experiment (μ = 0.615, η = 0) [24]. β (°)

σH (MPa)

σh (MPa)

rc (m)

NF slipping

PHF (MPa)

PNF (MPa)

PHF-PNF

Experimental results

Criterion results

90° 90° 75° 75° 45° 45°

−13.79 −7.58 −17.24 −8.27 −17.24 −8.27

−6.89 −6.89 −6.89 −6.89 −6.89 −6.89

0.0044 0.0049 0.0054 0.0055 0.0054 0.0055

No No No No No No

10.1640 10.1640 10.1640 10.1640 10.1640 10.1640

14.0960 8.0874 15.8320 7.5093 7.4406 3.0238

<0 >0 <0 <0 >0 >0

Crossing No Crossing Crossing No Crossing No Crossing No Crossing

Crossing Opening Crossing Opening Opening Opening

resistance. As shown in Fig. 5(a) (black dash dot line), Gu & Weng criterion was plotted as the boundary between the area of crossing and no-crossing. And the crossing area is relatively reduced compared with the crossing region separated by the two critical curves of the composite criterion. As the intersection angle exceeds 67.5°, the lower branch of Gu & Weng crossing curve becomes closer to xaxial than that of proposed opening curve but farther from the x-axial than that of the proposed slippage curve. These distinctions are attributable to the neglected effect of the inner fluid pressure inside the HF in Gu & Weng criterion. Moreover, field and laboratory observations have demonstrated that opening is a common occurrence when investigating the intersection between the HF and NF [23,30,37]. The opening zone (no slippage) depicted in orange color in Fig. 4 also designates that the NF may also be dilated in the case of no slippage occurring, which will subsequently impede the HF to penetrate across the NF. (3) Yao criteria In Yao's paper [28], it is assumed that an HF gets blunted upon contacting a NF and will temporarily stay in the NF for a while and then breakout to propagate in a mechanically favorable direction depending on the orientation of natural fracture relative to the stress field. On basis of energy conservation and slip stability analysis, the opening and slippage criterion in term of stress difference was derived and correspondingly expressed as [28]:

σH − σh >

σH − σh >

KIC (1 −

η)

πl sin2 β

(23)

c − μKIC/ πl sin β (cos β − μ sin β )

(24)

Yao criterion is delineated in Fig. 5(b) compared with the presented composite criterion. It can be seen that Yao’ slippage curve is approximately manifested in the regulation of quadratic function, which means that the difficulty for slippage to occur decreases initially and then increases progressively with the increase of intersection angle. This significantly differs from the monotonically steep decrease of the slippage curve proposed herein. Furthermore, the opening curve of Yao outlines a larger crossing area than that of the composite criterion as the intersection angle decreases away from 90°. On the other hand, neither the opening condition nor the shear failure condition in Yao criterion takes into account of the stress 70

Engineering Fracture Mechanics 209 (2019) 61–78

Y. Zhao et al.

(a) Compared to Blanton and Gu & Weng criteria

(b) Compared to Yao criteria Fig. 5. HF-NF criteria plotted as the boundary of the intersection areas: (a) Compared to Blanton and Gu & Weng criteria; (b) Compared to Yao criterion (σh = −5 MPa).

field generated by the fluid pressure within the HF. This difference is the key reason for the mismatch between Yao criterion and proposed composite criterion. 4. Discussion Based on Eqs. (9)–(21), the composite criterion should be a function of horizontal in situ stress, initial fracture length, intersection angle, fracture toughness and approaching distance. As mentioned above, the composite criterion curves of the stress difference versus the intersection angle were drawn and delineated at constant minimum horizontal stress and fracture toughness, and infinitesimally small approaching distance, which initially and geometrically restrict the composite criterion curve to be explored in depth. For further understanding and intensively studying the composite criterion, it’s necessary to introduce the parametric perturbation of minimum horizontal in situ stress, fracture toughness and approaching distance, and conduct stress field analysis adjacent to the intersection. For convenience, we choose the parameters of Blanton experiments [22] for calculating and analyzing the stress field within the interval range of (−0.14 m, 0.14 m)βx-βy on the NF interface. Moreover, a slip function f is introduced based on the slippage condition, which is expressed as:

f = |τβxy| − c + μσβy, (σβy < 0)

(25)

4.1. Initial constant minimum horizontal in situ stress The critical curves of composite criterion considering the influence of varying the minimum horizontal stress at constant maximum horizontal stress (σH = −20 MPa or 2900 psi) are depicted in Fig. 6(a). And the slippage analysis of a certain points at the intersection angle β = 30° (solid black dots) and β = 75° (red forks) with the stress difference of 4, 8, 12, 16 MPa are also respectively shown in Fig. 6(b) and (c). To facilitate the contrast analysis, the critical curves of stress difference versus intersection angle at fixed minimum horizontal stress (σh = −5 MPa) are delineated in Fig. 7(a), and the slippage stability of the analyzed points marked in 71

Engineering Fracture Mechanics 209 (2019) 61–78

Y. Zhao et al.

(a) Composite criterion at constant maximum horizontal stress

(b) =30°

(c) =75°

Fig. 6. Schematic of composite criterion disturbed by varying minimum horizontal stress (a) and stress field analysis of certain points at angle of 30° (b) and 75° (c) on NF interface, respectively (σH = −20 MPa or 2900 psi).

Fig. 7(a) are shown in Fig. 7(b–c). It should be specified that the critical curves in Figs. 6 and 7 are both calculated and drawn under identical parametric conditions aside from initial valve of the horizontal stress. Besides, due to the existence of the stress singularity at the intersection, the starting points for calculation are determined by the radius of nonlinear region ± rc and sixty points are symmetrically extracted from the natural frictional interface on both sides of the intersection in βx-βy coordinate system. Comparing the curves in Figs. 6(a) and 7(a), it can be seen that as the variation of horizontal stress changes from decreasing σh to increasing σH, the slippage curve deflects symmetrically along the line β = 60° while the opening curve hardly exhibits significant changes in shape. On the other hand, there is an evident discrepancy in the curvature of opening curves in Fig. 6(a) compared with Fig. 7(a). The discrepancy mainly embodies as the decrease opening tendency and subsequently imposes greater difficulty for opening to occur. In addition, it’s also easy to find that slippage becomes easier as the intersection angle β decreases from 90° according to 72

Engineering Fracture Mechanics 209 (2019) 61–78

Y. Zhao et al.

(a) Composite criterion at constant minimum horizontal stress

(b) slip stability of solid black circles for =30°

(c) slip stability of red forks for =75°

Fig. 7. Schematic of composite criterion disturbed by varying maximum horizontal stress (a) and stress field analysis of certain points at angle of 30° (b) and 75° (c) on NF interface, respectively (σh = −5 MPa or 725 psi).

Fig. 6(a), which is completely opposite to the slip tendency in Fig. 7(a). For further analysis of the NF stability, we solely concentrate on the slippage curves in Figs. 6 and 7. As Figs. 6 and 7 show, we can predict that the solid black dots will probably slip whereas the red forks can remain stable regardless of the stress difference and initial horizontal stress value in line with their relative position to the slippage curves. This indicates that varying the minimum horizontal stress does not affect the final predicted intersection results at some intersection angles, thus it is feasible for the composite criterion to be used for the verification of the laboratory experiments as the minimum horizontal stress varies. Figs. 6(b–c) and 7(b–c) present the slip function distribution on the NF interface when the HF approaches the NF at the angle of 30° (b) and 75° (c). The slip functions under different stress conditions share similar distribution characteristics of steep convex profile, which is manifested as the larger values of slip function for the closer distance to the intersection. And the slip function values 73

Engineering Fracture Mechanics 209 (2019) 61–78

Y. Zhao et al.

of the calculation points on the negative NF interface (βx < 0) are less than those on positive NF interface (βx > 0). This difference means slippage is probably to occur on the positive NF rather than on the negative NF interface. In addition, the values of slip function partially or entirely exceeds zero for β = 30° in Figs. 6(b) and 7(b), which implies the tangential stress applied along the NF is greater than the shear resistance and therefor further substantiates the slippage occurrence. As for β = 75°, the values of slip function all stay below zero in spite of the initial horizontal stress. This result ensures sufficient stability to inhibit slippage on the NF interface. For clarity, the zero value of slip function is marked out with dash lines shown in these figures. 4.2. Fracture toughness The opening and slippage curves of the composite criterion with fracture toughness of 0.59, 1.09, 1.59 and 2.09 MPa/m1/2 are plotted in Fig. 8. It’s easy to find that a small contrast of fracture toughness makes significant changes to the slippage curves whereas a relatively slight effect on the opening curve. Specifically, as the fracture toughness increases, the scope of crossing zone extends especially at lower stress difference, while the slippage zone decreases despite the intersection angle and the stress difference. This discrepancy signifies that the occurrence of crossing becomes easier at larger values of the fracture toughness. Furthermore, the opening curves have two-stage variation with the increase of fracture toughness: (i) distinguishable changes of the opening curves in the action of different fracture toughness are found in the angle range of 26–54° (shadow area); (ii) the opening curves favor to coincide as the intersection increase from 54°. Then we come to the conclusion that the effect of fracture toughness on the open curve is limited to certain range of intersection angle. Fig. 9 shows the slippage stability and stress distribution of the analyzed point which is marked with hollow black dots in Fig. 8. Easy to find that for different fracture toughness, the values of slip function of the analyzed point increase progressively as the calculated points move closer to the HF tip and finally presents a steep increase at the intersection. In addition, the slip function on positive βx axis is greater than that on negative βx axis, indicating that the incipient slippage probably takes place on the right side of the NF rather than on the left. Totally speaking, the slip function increase with the increase of fracture toughness, which is contrary to the variation laws in the coordinate interval of (−0.14 m, −0.08 m) on the NF interface. Thus the fracture toughness has a nonlinear effect on the slip function as the coordinate of the NF varies. Note that the increase of slip function means its absolute value decreases and the possibility of the NF slippage increases. Further, the variation of the total normal stress with the increase of the fracture toughness on both sides of βx = −0.02 m is entirely contrary. The higher the fracture toughness gets, the smaller the total normal stress on the left side of βx = −0.02 m is, whereas the greater the normal stress on the right of βx = −0.02 m becomes. However, the contrast of the normal stress on the right of βx = −0.02 m is comparably small over that on the right side under different fracture toughness. A reasonable explanation for this result is that the calculated points at the positive βx axis is far from the HF, where the net pressure within the HF has an inappreciable stress component on the positive NF interface. As shown in Fig. 9(c), the total shear stress appears approximately symmetrical distribution on both sides of the NF midpoint and has a positive correlation with fracture toughness. Hence the slippage stability of NF is mainly dominated by the total normal stress. 4.3. Approaching distance Note that the parameter of the approaching distance is introduced as a small perturbation between the right-tip of HF and the intersection point as shown in Fig. 2(b). The aim of introducing approach distance (Δl) to the criterion is to investigate whether the slippage will occur before the HF tip intersects with the NF. The magnitude of approaching distance reflects the degree of HF tip close to the intersection. For the convenience of comparison, the HF half-length is assumed to be 6 m, and four sets of approaching distance (0, 0.03, 0.07, 0.22 and 1.5 m) are taken into consideration.

Fig. 8. Schematic of composite criterion when introducing variation in fracture toughness (σh = −5 MPa). 74

Engineering Fracture Mechanics 209 (2019) 61–78

Y. Zhao et al.

(a) Slip function

(b) Total normal stress

(c) Total shear stress Fig. 9. Slippage stability and stress distribution on the NF interface at analyzed point for intersection angle β = 66°, σH = −20 MPa and σh = −5 MPa.

Fig. 10 displays the critical opening and slippage curves under different approaching distances. Affected by the increasing approaching distance, the critical opening curves (Fig. 10(a)) are drawn close to the coordinate axis, and the angle interval, within which the occurrence of opening is more prone even at relatively small stress difference, is surely reduced. Thus it might become more difficult for the NF opening especially at greater intersection angles. Actually, the opening mechanism can occur only under the prerequisite of the HF contacting HF. But here we merely concentrate on possible behaviors before the intersection. Thus, the opening curves are delineated through comparing the fluid pressure (PHF) inside the HF with the total normal stress (σβy) acting perpendicularly to the NF, which defines the upper limit for opening to occur. As shown in Fig. 10(b), any small perturbation of the approaching distance makes significant changes on the interval of

75

Engineering Fracture Mechanics 209 (2019) 61–78

Y. Zhao et al.

Fig. 10. The critical opening (a) and slippage (b) curves under different approaching distances (σh = −5 MPa).

intersection angles based on the presented composite criterion. And the larger value of approaching distance indicates the more symmetrical shape of the slippage curves. The form of asymmetrical curves for Δl = 0.03 m is due to the combined action of fluid pressure and far-field in situ stress. And as the approaching distance increases, the HF goes far away from the NF. So the influence of fluid pressure diminished gradually and the remote in situ stress starts to exert a dominating effect on the slippage. The shape of critical slippage curve for Δl = 1.5 m is similar to Yao’s depicted in Fig. 5, which illustrates that the stress effect of net pressure is slight enough that it can be simply replaced by remote in situ stress. Thus, the composite criterion can be seen as a more general form of Yao criterion during process of an HF approaching a NF. The slip function curves in Fig. 11 show different regularities under varied intersection angles. In detail, the slip function curves are symmetrically distributed on both sides of the NF midpoint for β = 90°. While β equals 60° or 30°, the values of slip function on the left side are evidently greater than that on the right. This discrepancy is because the left side (negative βx axis) is closer to the HF than the right side, leading to a higher influence degree, which appears a larger variation magnitude of the slip function on the left. In summary, the chance of slippage of the NF increases gradually during the approaching process. When the intersection angle increases from 30° to 90°, the slip function value decreases. And the natural fracture becomes less likely to slip, which indicates that the crossing behavior may more easily occur. 5. Conclusion In this paper, a composite criterion for separating opening, slippage and crossing zone is proposed based on dimensionless stress difference versus intersection angle analysis. The new criterion considers the coupling effects of fluid pressure and in situ stress on the intersection modes between the HF and the NF. Following this criterion, the theoretical predicting results were compared with experimental data as well as the previous criteria at the same parametric conditions. Reasonable accuracy is observed, which perfectly illustrates the criterion’s applicability and its potential value in engineering. Furthermore, the variation of introduced parameters will more or less change the geometry of the critical curves, which may rezone the intersection scope and therefor promote or inhibit the occurrence of predicting behaviors. A high contrast in the magnitude of maximum horizontal stress (σh = constant) will result in NF slipping, whereas in the magnitude of minimum horizontal stress (σH = constant) may inhibit the occurrence of slippage. The fracture toughness has a minor effect on opening curves at a 76

Engineering Fracture Mechanics 209 (2019) 61–78

Y. Zhao et al.

Fig. 11. The slip function under different approaching distance for the intersection angle β = 30° (a), 60° (b) and 90° (c).

specified angle range, whereas a relatively obvious impact on the critical slippage curve in spite of the intersection angle and the stress difference. After introducing the approaching distance, the angle interval of the opening occurrence is limited to small stress difference and the slippage curves are gradually dominated by the far-field in situ stress. Other interesting conclusions involving with parametric and stability analysis of a certain point are summarized as follows: (1) The slip function, normal and shear stress distribution along NF interface is highly dependent on the βx coordinate. And the incipient points of most likely slipping or opening are concentrated prior to the intersection; (2) The right side (positive βx coordinate) on the NF is more likely to slip than the left; (3) Increasing stress difference may not always enhance the slip stability of NF interface. It should also be related to the magnitude of intersection angle; 77

Engineering Fracture Mechanics 209 (2019) 61–78

Y. Zhao et al.

(4) The slip function and normal stress exhibit nonlinear relationship with the increase of negative βx coordinate under different fracture toughness. But in general, the increase of fracture toughness will increase the possibility for the NF to slip; (5) During the process of a hydraulic fracture approaching a natural fracture, the slip probability of the natural fracture increases gradually. When the intersection angle increases from 30° to 90°, a hydraulic fracture crosses a natural fracture more easily. Acknowledgements This work is supported by the National Natural Science Foundation of China (Nos. 51374257 and 50804060). References [1] Lisjak A, Kaifosh P, He L, et al. A 2D, fully-coupled, hydro-mechanical, FDEM formulation for modelling fracturing processes in discontinuous, porous rock masses. Comput Geotech 2017;81:1–18. [2] Yoon JS, Zimmermann G, Zang A. Discrete element modeling of cyclic rate fluid injection at multiple locations in naturally fractured reservoirs. Int J Rock Mech Min Sci 2015;74:15–23. [3] Sun RJ. Theoretical size of hydraulically induced horizontal fractures and corresponding surface uplift in an idealized medium. J Geophys Res 1969;74(25):5995–6011. https://doi.org/10.1029/jb074i025p05995. [4] Chen Z. Finite element modelling of viscosity-dominated hydraulic fractures. J Petrol Sci Eng 2012;88–89(2):136–44. https://doi.org/10.1016/j.petrol.2011.12. 021. [5] Garagash. Hydraulic fracture propagation in elastic rock with large toughness. Proceedings of 4th North American rock mechanics symposium, the Netherlands. 2000. p. 221–8. [6] Weng Xiaowei. Modeling of complex hydraulic fractures in naturally fractured formation. J Unconvent Oil Gas Resour 2015;9:114–35. [7] Sarmadivaleh M, Rasouli V. Modified Renshaw and Pollard criteria for a non-orthogonal cohesive natural interface intersected by an induced fracture. Rock Mech Rock Eng 2013;47(6):2107–15. [8] Yew CH, Weng X. Mechanics of hydraulic fracturing. Houston: Gulf Professional Publishing; 2014. [9] Khristianovic SA. Zheltov YP. Formation of vertical fractures by means of highly viscous liquid. Proceedings of the fourth world petroleum congress, Rome. 1955. p. 579–86. [10] Perkins T, Kern L. Widths of hydraulic fractures. J Pet Tech Trans AIME 1961;222:937–49. [11] Geertsma, de Klerk. A rapid method of predicting width and extent of hydraulic induced fractures. J Petrol Technol 1969;21(12):1571–81. [12] Desroches J, Detournay E, Lenoach B, et al. The crack tip region in hydraulic fracturing. Proc R Soc 1994;447(1929):39–48. [13] Garagash. Near-tip processes of fluid-driven fractures PhD thesis Minneapolis University of Minnesota; 1998. [14] Garagash Detournay. Viscosity-dominated regime of a fluid-driven fracture in an elastic medium. Symposium on analytical and computational fracture mechanics of non-homogeneous materials, IUTAM 2002. 2002. p. 25–9http://doi.org/10.1007/978-94-017-0081-8. [15] Adachi JI. Fluid-driven fracture in permeable rock PhD thesis Minneapolis University of Minnesota; 2001. [16] Detournay E. Propagation regimes of fluid-driven fractures in impermeable rocks. Int J Geomech 2004;4(1):35–45. https://doi.org/10.1061/(asce)15323641(2004)4:1(35). [17] Wu R, Kresse O, Weng X, Cohen C-E, Gu H. Modeling of intersection of hydraulic fractures in complex fracture networks. SPE hydraulic fracturing technology conference. Society of Petroleum Engineers; 2012https://doi.org/10.2118/152052-ms. [18] Kresse O, Weng X, Wu R, Gu H. Numerical modeling of hydraulic fractures intersection in complex naturally fractured formations. American Rock Society Association; 2012http://doi.org/2012. 10.1007/s00603-018-1539-5. [19] Maxwell SC, Urbancic TI, Steinsberger NP, Zinno R. Microseismic imaging of hydraulic fracture complexity in the Barnett shale. Paper SPE 77440, SPE annual technical conference and exhibition, San Antonio, Texas; 2002, 29 September–2 October. 2002http://doi.org/10.2523/77440-ms. [20] Fisher MK, Davidson BM, Goodwin AK, Fielder EO, Buckler WS, Steinberger NP. Integrating fracture mapping technologies to optimize stimulations in the Barnett shale. Paper SPE 77411, SPE annual technical conference and exhibition, San Antonio, Texas, USA; 2002, 29 September–2 October. 2002http://doi.org/ 10.2118/77441-ms. [21] Warpinski NR, Kramm RC, Heinze JR, Waltman CK. Comparison of single- and dual-array microseismic mapping techniques in the Barnett shale. Paper SPE 95568, SPE annual technical conference and exhibition, Dallas, Texas; 2005, 9–12 October. 2005http://doi.org/10.2523/95568-ms. [22] Blanton. Propagation of hydraulically and dynamically induced fractures in naturally fractured reservoirs. SPE unconventional gas recovery symposium. Society of Petroleum Engineers; 1986. [23] Zhou J, Chen M, Jin Y, Zhang G. Analysis of fracture propagation behavior and fracture geometry using tri-axial fracturing system in naturally fractured reservoirs. Int J Rock Mech Min Sci Geo-Mech Abstr 2008;45:1143–52. [24] Gu H, Weng X, Lund J, Mack M, Suarez-Rivera R. Hydraulic fracture crossing natural fracture at non-orthogonal angles: a criterion and its validation. SPE hydraulic fracturing technology conference. Society of Petroleum Engineers; 2011http://doi.org/10.2118/139984-pa. [25] Taleghani AD, Olson JE. How natural fractures could affect hydraulic-fracture geometry. SPE J 2013;19(1):161–71. https://doi.org/10.2118/167608-pa. [26] Warpinski NR, Teufel LW. Influence of geologic discontinuities on hydraulic fracture propagation. SPE J Petrol Technol 1987;39(2):209–20. https://doi.org/10. 2118/13224-pa. [27] 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 Geo-Mech Abstr 1995;32(3):237–49. [28] Yao Y, Wang W, Keer LM. An energy based analytical method to predict the influence of natural fractures on hydraulic fracture propagation. Eng Fract Mech 2017. https://doi.org/10.1016/j.engfracmech.2017.11.020. [29] Beugelsdijk LJL, Cjd Pater, Sato K. Experimental hydraulic fracture propagation in a multi-fractured medium. SPE Asia Pacific conference on investigated modelling for asset management. Society of Petroleum Engineers; 2000https://doi.org/10.2118/59419-ms. [30] Chuprakov D, Melchaeva O, Prioul R. Injection-sensitive mechanics of hydraulic fracture interaction with discontinuities. Rock Mech Rock Eng 2014;47(5):1625–40. [31] Llanos EM, Jeffrey RG, Hillis R, Zhang X. Hydraulic fracture propagation through an orthogonal discontinuity: a laboratory, analytical and numerical study. Rock Mech Rock Eng 2017;6:1–18. [32] Valko P, Economides MJ. Hydraulic fracture mechanics. Chichester: John Wiley & Sons; 1995. [33] Westgaard HM. Bearing pressures and crack. J Appl Math Mech 1939;6:A49–53. [34] Song CP, Yiyu LU, Xia BW, et al. Effect of natural fractures on hydraulic fracture propagation in coal seam. J Northeastern Univ (Natural Sci) 2014;35(5):756–60. [35] Barenblatt GI. The mathematical theory of equilibrium cracks in brittle fracture. Adv Appl Mech 1962;7:55–129. [36] Dugdale DS. Yielding of steel sheets containing slits. J Mech Phys Solids 1960;8(2):100–4. [37] Blanton TL. An experimental study of interaction between hydraulically induced and pre-existing fractures. Society of petroleum engineers unconventional gas technology symposium, Pittsburgh. 1982https://doi.org/10.2523/10847-ms. [38] Dahi A, Olson JE. Numerical modeling of multistranded-hydraulic-fracture propagation: accounting for the interaction between induced and natural fractures. Soc Petrol Eng 2011;16:575–81. https://doi.org/10.2118/124884-pa. [39] Li ZC, Li LC, Li M, et al. A numerical investigation on the effects of rock brittleness on the hydraulic fractures in the shale reservoir. J Nat Gas Sci Eng 2018;50:22–32. https://doi.org/10.1016/j.jngse.2017.09.013. [40] Liu HL, Li LC, Li ZC, et al. Numerical modelling of mining-induced inrushes from subjacent water conducting karst collapse columns in northern China. Mine Water Environ 2017. https://doi.org/10.1007/s10230-017-0503-z.

78