Rock Mech Rock Eng DOI 10.1007/s00603-016-1097-7
ORIGINAL PAPER
3D Numerical Modeling of the Propagation of Hydraulic Fracture at Its Intersection with Natural (Pre-existing) Fracture Ali Naghi Dehghan1 • Kamran Goshtasbi2 • Kaveh Ahangari1 • Yan Jin3 Aram Bahmani4
•
Received: 8 November 2015 / Accepted: 17 September 2016 Springer-Verlag Wien 2016
Abstract A variety of 3D numerical models were developed based on hydraulic fracture experiments to simulate the propagation of hydraulic fracture at its intersection with natural (pre-existing) fracture. Since the interaction between hydraulic and pre-existing fractures is a key condition that causes complex fracture patterns, the extended finite element method was employed in ABAQUS software to simulate the problem. The propagation of hydraulic fracture in a fractured medium was modeled in two horizontal differential stresses (Dr) of 5e6 and 10e6 Pa considering different strike and dip angles of pre-existing fracture. The rate of energy release was calculated in the directions of hydraulic and pre-existing fractures (Gfrac =Grock ) at their intersection point to determine the fracture behavior. Opening and crossing were two dominant fracture behaviors during the hydraulic and pre-existing fracture interaction at low and high differential stress conditions, respectively. The results of numerical studies were compared with those of experimental models, showing a good agreement between the two to validate the accuracy of the models. Besides the horizontal differential stress, strike and dip angles of the natural (pre-existing)
& Ali Naghi Dehghan
[email protected] 1
Department of Mining Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran
2
Department of Mining Engineering, Faculty of Engineering, Tarbiat Modares University, Tehran, Iran
3
College of Petroleum Engineering, China University of Petroleum, Beijing 102249, China
4
Department of Mechanical and Mechatronics Engineering, University of Waterloo, University Avenue 200, Waterloo, ON N2L3G1, Canada
fracture, the key finding of this research was the significant effect of the energy release rate on the propagation behavior of the hydraulic fracture. This effect was more prominent under the influence of strike and dip angles, as well as differential stress. The obtained results can be used to predict and interpret the generation of complex hydraulic fracture patterns in field conditions. Keywords Hydraulic fracture Natural (pre-existing) fracture 3D Numerical modeling Fracture intersection Fracture behavior Strike and dip angles Horizontal differential stress Energy release rate List of Symbols asJ Unknowns associated with node J for enrichment function s C Cohesive strength of natural fracture plane E Young’s modulus of elasticity E0 Plane-strain modulus of elasticity e Void ratio G Energy release rate Gc Critical energy release rate Grock Rock fracture energy Gfrac Energy required to overcome the cement strength in the natural fracture k Permeability K Stress intensity factor Kc Fracture toughness KI Mode I stress intensity factor KII Mode II stress intensity factor KIII Mode III stress intensity factor Kf Coefficient of friction (internal friction coefficient of natural fracture plane) NI Shape function at node I
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N~J nenr S Ss U uC uE uI To a b c Dr m rh rH rV rc / Ws
Shape function for enrichment at node J Number of enrichment types Set of all nodes in the domain Set of nodes enriched by Ws Displacements Continuous displacement field Discontinuous displacement field Nodal unknowns Tensile strength of the specimen containing the pre-existing fracture Angle between the direction of hydraulic fracture propagation and the pre-existing fracture dip Angle between the direction of hydraulic fracture propagation and the pre-existing fracture strike Unit weight of the cement specimen Horizontal differential stress Poisson’s ratio Minimum principal stress in horizontal direction Maximum principal stress in horizontal direction Principal stress in vertical direction Unconfined compressive strength Porosity Enrichment functions
1 Introduction Hydraulic fracturing is a major stimulating technique to improve oil and gas recovery from naturally fractured reservoirs. In many areas, the direction of current in situ tectonic stress has not varied since the formation of natural fractures (Laubach et al. 2004). Therefore, hydraulic fracture may be propagated in the direction parallel and/or subparallel to the natural fracture planes with which it intersects. In other regions, natural fracture planes are formed by the stress regimes, which are totally different from those of today. The angles of natural fracture planes (strike and dip) may be oblique, orthogonal, and/or coalescence to the hydraulic fracture path, affected by the propagating behavior of hydraulic fracture and geometry of reservoirs (Dehghan et al. 2015a, b, 2016). Natural fractures have normally narrow apertures around 10-5 to 10-3 m (Liu 2005), sealed by calcite or quartz cements (Gale et al. 2007). Thus, understanding the propagation behavior and geometry of hydraulic fractures and their interaction with natural fractures are crucial in the designing, monitoring, and assessing of hydraulically induced fractures and their impacts on the wellbore production enhancement. According to the principles of fracture mechanics, the stress intensity factors at the fracture tip can be used as the fundamental parameter to characterize the fracture initiation and propagation due to the hydraulic fracture. Stress
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intensity factors at the tip reach their critical values at the crack at the onset of wellbore fracturing. Therefore, the initiation and trajectory of the fracture can be predicted by crack tip stress intensity factors (i.e., KI, KII, and KIII). The literature shows that there are copious experimental and numerical researches. The fracture toughness has been examined in different types of rocks by researchers in pure mode I (Aliha et al. 2013), pure mode II (Ayatollahi and Aliha 2008), and mixed mode I/II (Saghafi et al. 2010). Aliha et al. (2015) investigated mixed mode I/III fracture resistance of different brittle and quasi-brittle materials, such as marble, graphite, etc. These mixed modes can occur for a pre-existing fracture due to the hydraulic fracturing process. The principals of fracture mechanics include several well-known criteria named MTS (Erdogan and Sih 1963), G (Hussain et al. 1974), SED (Sih 1974), GMTS (Smith et al. 2001), CZM (Go´mez et al. 2009), and MTSED (Ayatollahi and Saboori 2014), effective in predicting the crack behavior. However, the mentioned principals cannot be very effective by themselves, in terms of hydraulic fracture, due to the interactions between existing cracks, in situ stresses, unstable loading conditions, etc. Many researchers have worked on the problem of interaction between hydraulic and natural fractures to understand the factors effective in the propagation of hydraulic fractures. Amongst such researchers are Lamont and Jessen (1963), Blanton (1986), Warpinski and Teufel (1987), Renshaw and Pollard (1995), Beugelsdijk et al. (2000), Lhomme et al. (2002), Casas and Miskimins (2006), Zhou et al. (2008), Athavale and Miskimins (2008), Jeffrey et al. (2009), Zhou et al. (2010), Yan et al. (2011), Olson et al. (2012), Liu et al. (2014), Fan and Zhang (2014), Sarmadivaleh and Rasouli (2014, 2015), Zhu et al. (2015), Fallahzadeh et al. (2015), Dehghan et al. (2015a, b, 2016), Yushi et al. (2016). These studies have mostly indicated that in situ stresses, angle of approach, interfacial friction coefficient, fluid injection rate, and fracturing fluid viscosity are the most effective factors in the propagation of hydraulic fracture. Under different conditions, the progressive hydraulic fracture may cross natural fracture at their interaction point, or it might be arrested by opening and/or shear slippage of natural fracture. Zhou et al. (2008) investigated the effect of the shear strength of pre-fracture on the propagation of hydraulic fracture, using three types of paper, rice, printer, and wrapping, with the friction coefficients (Kf ) of 0.38, 0.89, and 1.21, respectively. The measured interface cohesion (c) of the papers was 3.2 MPa. The cement blocks are of dimensions 300 9 300 9 300 mm and have a tensile strength of 3 MPa. The results obtained for type two prefracture (printer paper) were in good agreement with those of Blanton (1986), Warpinski and Teufel (1987), and Potluri et al. (2005), excluding one discrepancy at the
3D Numerical Modeling of the Propagation of Hydraulic Fracture at Its Intersection with…
intersection angle of 60. This means that horizontal differential stress and angles of approach were the most effective factors in the fracture behavior when hydraulic fracture intersected the pre-fractures. For the first time, Dehghan et al. (2015a, b) studied the effects of the strike and dip angles of natural fracture planes on the propagation of hydraulic fracture before, during, and after interaction with the natural fracture in different in situ stress conditions. Since then, they have not been considered in assessing the interaction between natural fractures and advancing hydraulic fracture. Considering dip and strike angles, two approaching angles are created at the intersection point between hydraulic and preexisting fractures. Thus, ‘‘dip’’ is the angle of approach between the direction of hydraulic fracture propagation and the dip of the pre-existing fracture plane. By ‘‘strike’’, it is meant as the angle of approach between the direction of hydraulic fracture propagation and the strike of the preexisting fracture plane. The schematic views of these angles are presented in Fig. 2b as well. Accordingly, similar to Zhou et al. (2008), printer paper with a medium aperture of 0.11 mm and a friction coefficient of 0.89 was used in each cement specimen (Kf ¼ 0:89 and c ¼ 3:2 MPa) to simulate natural fracture. According to Dehghan et al. (2015a, b), the strike and dip of natural fracture have a significant effect on the propagation behavior and geometry of hydraulic fracture. That is, they could be changed from crossing and vertical to arrest and tortuous with decreasing of the strike and dip of natural fracture, respectively. Dehghan et al. (2016) investigated the hydraulic fracture initiation and propagation mechanisms in a pressurized wellbore, considering non-fractured and fractured reservoirs. Their objective was to gain a better understanding of the effects of pre-existing fracture on and far from the wellbore wall, as well as modeling differential stress on the initiation and propagation pressure and fracture geometry. According to the reported findings, (1) the presence of prefracture in the wellbore wall, (2) high differential stress, and (3) medium to high dip and strike of pre-fracture in the far-wellbore region are dominant factors in the decrease of initiation and propagation pressure in the hydraulically induced fracture (due to the decrease of stress state concentration around the wellbore) and increase of the crossing fracture behavior. Moreover, relatively large differences were observed between the fracture initiation pressure (FIP) and fracture propagation pressure (FPP) values obtained in lab experiments and those yielded by analytical solutions. The authors posited that this discrepancy arises due to incorporating some simplifications as assumptions in the analytical solutions (i.e., rock is an elastic, isotropic, homogeneous medium, etc.), in addition
to failure of considering the compressibility and viscosity of fracturing fluid and the wellbore pressurization rate. Extensive theoretical and numerical investigations have been conducted on the interaction between hydraulic and natural fractures (Zhang and Jeffrey 2006; Akulich and Zvyagin 2008; Hossain and Rahman 2008; Thiercelin 2009; Xu et al. 2009; Zhang and Jeffrey 2009; Rahman et al. 2009; Chuprakov et al. 2010; Dershowitz et al. 2010; Rogers et al. 2010, 2011; Dahi-Taleghani and Olson 2011; Keshavarzi and Mohammadi 2012; Gu et al. 2012; Wu et al. 2012; Kresse et al. 2011, 2013; Zhou and Hou 2013; Behnia et al. 2014, 2015; Funatsu et al. 2015). The authors of some of these studies focused on the mechanical interaction when a hydraulic fracture intersects the pre-existing fracture, while others considered the fluid flow in the hydraulic and natural fractures. Numerous fracture models have emerged, such as 2D analytical models (PKN and KGD), considering the reservoir as a homogeneous medium. These have also led to complex 3D numerical models (P3D and PL3D), such as those developed by Simonson et al. (1978), Carter et al. (2000), Economides and Nolte (2000), Adachi et al. (2007), Meyer and Bazan (2011), Nagel et al. (2011), and Nagel and Sanchez-Nagel (2011). Most of the P3D models (referred to as ‘‘cell-based’’ and ‘‘lumped’’ models) offer semi-analytical solutions. The calculation concepts incorporated in these approaches, while crude, are highly efficient. Therefore, P3D models are often used in the design of hydraulic fracture treatment (Economides and Nolte 2000; Adachi et al. 2007). On the other hand, the boundary integral method is applied to solve the mechanical problem in moving triangular mesh PL3D models (Peirce and Siebrits 2001) and fixed rectangular mesh (Siebrits and Peirce 2002), based on 3D linear elastic theory. In this model, the fracture front is no longer regular, but rather strongly depends on the material property and closure stress. Thus, this approach is more realistic compared with the previous P3D model. Since 1D or 2D flow is considered in all the aforementioned fracture propagation models, several researchers have attempted to develop numerical 3D models for simulating hydraulic fracturing more realistically and accurately using fully coupled fluid and solid deformation (Garcia and Teufel 2005; Hossain and Rahman 2008; Li et al. 2012; Nagel et al. 2013; Zhou and Hou 2013). In their work, Hossain and Rahman (2008) attempted to elucidate how complex 3D fracture geometry is propagated under different stress and well conditions. The authors also studied their effects on the injection pressure, using a flowdeformation coupled numerical tool HYFRANC3D that combines the boundary element method (BEM) for structural response with the finite element method (FEM) for
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fluid flow. They classified the propagation of hydraulic fracture into two categories: (1) structural response with constant fracture pressure and (2) coupled fluid and structural response with constant injection rate. The objective of introducing these categories was to evaluate the behavior of fracture geometry and fracture pressure, respectively, when a fracture is propagated in the preferred and nonpreferred directions. More recently, Li et al. (2012) introduced a 3D FEM to consider the coupled effects of seepage, damage, and stress field, using a 3D Rock Failure Process Analysis Parallel code (RFPA3D-Parallel). RFPA3D-Parallel is an extension of 2D Rock Failure Process Analysis (RFPA2D), based on an improved flow-stress (strain)-damage (FSD) model (Tang 1997; Tang et al. 2002; Liang et al. 2004). In a rock sample, the numerical simulation of the hydraulic fracturing process was conducted by applying this model on three coupled processes, namely (1) mechanical deformation of solid media by fluid pressure, (2) flow of fluid into the fracture, and (3) fracture propagation. In their study, Nagel et al. (2013) performed numerical investigations to evaluate fundamental geomechanical characteristics of hydraulic fracture propagation in a naturally fractured rock mass. The authors used a series of continuum and discrete element model (2D DEM and 3D DEM) simulations in mechanical-only and fully coupled hydro-mechanical modes. They posited that the main factors in controlling the extent of the interaction between hydraulic fracture and natural fracture are: (1) conductivity and orientation of the natural fracture network relative to the stress field; (2) mechanical properties of natural fracture, such as internal friction angle, degree of cementation, fracture stiffness, and initial aperture; (3) in situ stress and pore pressure conditions; and (4) operational parameters, such as fluid injection rate, fluid viscosity, and injected fluid volume. In a work published in the same year, Zhou and Hou (2013) introduced a new approach, which they integrated into the numerical simulator FLAC3D. In the presented method, fracture propagation was considered in a 3D geometric model under a 3D stress state with a fully hydromechanical coupling effect between fracture and matrix. This methodology was verified through a laboratory fracture simulation, which allowed the measured and calculated results to be compared. The authors claimed that the new approach could be used to model the propagation of a single fracture and study its influence on the adjacent rock formation and neighboring fractures. Various cases (i.e., 1/2, 1/8, etc.) have been modeled partially in several numerical studies for hydraulic fracture propagation to accelerate the calculation, considering the symmetric boundary conditions (Hossain and Rahman 2008; Zhou and Hou 2013).
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Several numerical methods have been presented for modeling fracture propagation, including the incremental crack growth method based on GMTS criteria (Aliha et al. 2010), displacement discontinuity method (DDM) (Behnia et al. 2014), cohesive zone method (CZM) (Khoramishad et al. 2010), discrete element model (DEM) (Nagel et al. 2013), FEM (Li et al. 2012), and extended finite element method (XFEM) (Giner et al. 2009; Chen et al. 2012). Among these methods, XFEM is of particular interest, as it is posited to enhance the solution accuracy due to remeshing avoidance in each step of the fracture propagation. The work reported here focuses on the investigation of dominant factors effective in the passage and/or diversion of hydraulic fracture in a medium with natural (pre-existing) fracture. For this purpose, the development of 3D numerical models is attempted, aiming to enhance the understanding of the interaction between hydraulic and pre-existing fractures at their intersection point. This approach would also facilitate comparison of the obtained results with those yielded by extant experimental models. Accordingly, the extended finite element method (XFEM) is employed in ABAQUS software to make the 3D numerical computations more tractable. In this regard, a static mesh is considered as fracture propagation, thus eliminating the need for re-meshing and avoiding the associated computational expense.
2 Interaction Between Hydraulically Induced and Natural Fractures The interaction between hydraulic fracture and natural fracture is a significant condition, leading to complex patterns of hydraulic fracture propagation. Numerous populations of natural fractures in existence are sealed by precipitated cements (quartz, calcite, etc.). Even with no porosity in the sealed fractures, they may still serve as planes of weakness for propagating hydraulic fractures (Gale et al. 2007). Three types of interaction might occur in propagating hydraulic fracture in the naturally fractured reservoirs, as shown in Fig. 1. First, when natural fractures have no influence, hydraulic fracture may be propagated in the direction parallel to the maximum horizontal stress (rH ). This may be due to the high cement strength of natural fracture compared to the matrix strength of surrounding rock, unfavorable geometry of natural fracture planes, and/or insufficient fracturing pressure at the intersection point for overcoming the normal stress (rn ) perpendicular to the natural fracture plane. Second, hydraulic fracture is arrested and the fluids are completely truncated into the natural fracture plane. In this situation, natural fracture is opened or sheared due to the sufficient energy of propagating hydraulic fracture for overcoming the normal
3D Numerical Modeling of the Propagation of Hydraulic Fracture at Its Intersection with… Fig. 1 Three possible scenarios of interaction between hydraulic and natural fractures: a crossing, b opening, c coalescence of opening and crossing (DahiTaleghani and Olson 2011)
stress acting on the natural fracture, the strength of cements, and the friction between the surfaces of natural fracture. Third, hydraulic fracture turns into natural fracture and makes it open, and then goes outside through a flaw or a weak surface existing on the surface of natural fracture. This fact primarily depends on the orientation (strike and dip) of natural fracture relative to the stress field (Dehghan et al. 2015a, b, 2016). Blanton (1986), Warpinski and Teufel (1987), and Renshaw and Pollard (1995) have experimented and developed several criteria for determining the interaction between hydraulic fracture and natural fracture. This criterion only considers the initial interaction between induced and natural fractures. Therefore, in this research, linear elastic fracture mechanics (LEFM) are considered for predicting the propagation of hydraulic fracture and its interaction with natural (pre-existing) fracture. LEFM, the basic theory of fracture, was introduced by Griffith (1921) and completed by Irwin (1958). Fracture propagation in LEFM theory is the function of opening (I), shearing (II), and tearing (III) mode stress intensity factors (KI, KII, and KIII, respectively), the values of stress concentration at the crack tip (Lawn 2004). In this study, the mentioned factors are combined with the fracture propagation energy release rate and defined as follows (Irwin 1958): G¼
2 ðKI2 þ KII2 þ KIII ð1 þ tÞÞ 0 E
ð1Þ
where G is the energy release rate; KI, KII, and KIII are stress intensity factors; E0 ¼ E(E = Young’s modulus) for plane stress conditions and E0 ¼ E=ð1 t2 Þ(t = Poisson’s ratio) for plane strain conditions. If the energy release rate exceeds the critical value (Gc ), then the fracture grows critically. Hydraulic fracture has more than one path of propagation at its intersection point with natural fracture, which can be the opening and crossing of natural fracture. The most likely path for hydraulic fracture would be that of the maximum energy release rate (Freund and Suresh 2003). Thus, the energy release rate is computed for two likely
paths of opening (Gfrac ) and crossing (Grock ) at the intersection point of hydraulic and natural fractures, where Grock is the rock fracture energy (corresponding to fracture toughness) and Gfrac is the energy required for overcoming the cement strength of natural fracture. If Gfrac is greater than or equal to Grock , opening will happen, and hydraulic fracture is diverted to natural fracture and opens it. However, if Gfrac is less than Grock , crossing will occur and hydraulic fracture is propagated in the direction of maximum principle stress (rH ), without diverting the fluid along the natural fracture.
3 Extended Finite Element Method (XFEM) Crack growth modeling has been conducted in the finite element method, by applying various re-meshing strategies (Bouchard et al. 2000; Patza´k and Jira´sek 2004). However, re-meshing is computationally burdensome due to the transfer of data between different meshes. Mo¨es et al. (1999) developed the extended finite element method (XFEM) and addressed this inefficiency. Therefore, XFEM is used to model the propagation of discontinuity, like fractures. In this approach, the fracture is propagated independently from the mesh configuration, as discontinuity is permitted to cross the elements. Basically, this method was developed to enrich crack displacement discontinuity and near-tip fields in the standard finite element method (Belytschko and Black 1999). Then, it was extended to model discontinuities, arbitrary branched and intersecting fractures. Thus, XFEM is particularly well suited for modeling the propagation of hydraulic fracture and its interaction with the existing discontinuities, including natural fractures and joints already existing in the media. XFEM approximation is conducted based on decomposing the displacement field into conventional (continuous) (uC ) and enriched (discontinuous) (uE ) components as follows (Mo¨es et al. 1999): u ¼ uC þ uE
ð2Þ
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A. N. Dehghan et al. Vertical
σV
(a)
(b) Wellbore Hydraulic fracture
Top
Wellbore Pre-existing fracture
σh
30 cm
Hydraulic fracture (HF) iniaon Intersecon point of HF with NF
β
α
Longitudinal
Natural (Pre-exisng) fracture (NF)
Horizontal : Angle of Dip : Angle of Strike
y
z
x
Fig. 2 Schematic representation of the testing block: a geometry of fractured block and directions of in situ stresses, b intersection of a hydraulic fracture with strike and dip of a natural (pre-existing) fracture
The conventional (continuous) component is approximated by the classic finite element shape function as: X NI ðxÞuI ð3Þ uC ¼
4 Numerical Simulation of Hydraulic Fracture Propagation 4.1 Geometry and Model Setup
I2S
where S is the set of all nodes in the domain, NI is the shape’s function for the continuous component, and uI is the nodal unknown. The enrichment part of the displacement approximation is given as: uE ¼
nenr X X
N~J ðxÞws ðxÞasJ
ð4Þ
s¼1 J2Ss
where nenr is the number of enrichment types; N~J the shape functions for enrichment; Ws the enrichment functions; Ss the set of nodes enriched by Ws ; asJ the unknowns associated with node J for enriching function s. For the purpose of numerical efficiency and simplicity, Eq. 4 is rewritten as follows: uE ¼
nenr X X
N~J ðxÞ ws ðxÞ wsI ðxÞ asJ
ð5Þ
s¼1 J2Ss
It should be mentioned that displacement could disappear from the nodal point area by the enrichment function (Ws ), shifted via the nodal value (WsI ). Hence, to consider a particular displacement discontinuity behavior, the interpolation shape function in Eq. 5 can have contradictory order as the continuous part of displacement. In order to avoid re-meshing in each step of the fracture propagation, various geometries of discontinuity and insensitivity of fracture propagation to the mesh geometry are arbitrarily considered in applying XFEM for modeling hydraulic fracture propagation.
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Numerical modeling of hydraulic fracture propagation is a complicated phenomenon due to the heterogeneity of earth structure, indeterminate in situ stresses, rock behavior, and physical complexity of the problem. Thus, experimental and numerical investigations become much more complex if natural fracture is added to the hydraulic fracture problem. In this research, 3D modeling is conducted for hydraulic fracture propagation in a fractured medium based on the experimental studies carried out on the fractured artificial specimens (Dehghan et al. 2015a, b, 2016). Hydraulic fracture tests are conducted on several cubic blocks of dimension 300 mm per side, constructed by cement and quartz sand in a 1:1 mass ratio (Fig. 2a). A sheet of printer paper is cast into each block in order to simulate natural fracture. The interfacial coefficient of friction is 0.89 and the cohesion is 3.2 MPa for the surface of the printer paper. Two approaching angles are simultaneously considered between hydraulic and natural fractures by creating dip (a) and strike (b) for pre-existing fracture in the block (Fig. 2b). Tables 1 and 2 present the geometric parameters and hydro-mechanical properties of the block, respectively. Dimensional analyses have been performed to achieve a reasonable correspondence between the results of lab experiments and field scale operations, considering scaling laws. So far, several scaling laws have been presented to produce representative test results, extrapolated properly to the field conditions (de Pater et al. 1994; Detournay 2004;
3D Numerical Modeling of the Propagation of Hydraulic Fracture at Its Intersection with… Table 1 Geometric parameters of the fractured block Block dimensions
Pre-existing fracture dimensions
Pre-existing fracture angles ()
Width, x (cm)
Height, y (cm)
Depth, z (cm)
Length (cm)
Thickness (mm)
Width (cm)
Dip (a)
Strike (b)
30
30
30
20
0.11
20
30–90
30–90
Table 2 Basic hydro-mechanical properties of the test block Parameters
Symbols
Unit
Values
Unconfined compressive strength (UCS)
rc
Pa
25.88e6
Poisson’s ratio
v
–
0.15
Young’s modulus
E
Pa
7.43e9
Tensile strength
T0
Pa
3e6
Fracture toughness
KIC
MPa m1/
0.75
Porosity
/
%
Void ratio
e
–
2
Unit weight
c
kg/cm
Permeability
K
cm/s
1.85 1.88e-2 3
2300 1e-4
Bunger 2005; Sarmadivaleh and Rasouli 2015). The experiments are scaled through these laws in terms of energy rates associated with fluid flow, fracture opening, and rock separation. Considering low injection rate in the laboratory, highly viscous fluids and materials should be applied with low fracture toughness. Besides, regarding the conductivity of pre-existing fracture in the block, another scale factor is also considered for the stresses due to the dependence of fracture aperture on the average stress level. Therefore, in situ stresses simulation is a key point in the experiments of hydraulic fracturing. Laboratory experiments are performed in a normal-faulting stress regime. Based on scaling analysis, low and high values are considered for horizontal differential stress (Dr): maximum horizontal stress (rH ) and minimum horizontal stress (rh ). More details on the scaling analysis as well as information about how to scale the testing variables can be found in Zhou et al. (2008, 2010) and Dehghan et al. (2015a, b, 2016). Table 3 illustrates a summary of the experimental conditions and results for hydraulic fracture tests of synthetic rock samples. Anisotropy of in situ stresses around the wellbore and the geometry of natural fracture are the main effective factors in the initiation and propagation of hydraulic fracture (Dehghan et al. 2016). In this research, hydraulic fracture propagation is investigated considering a pre-existing fracture with different strike and dip angles, located at a certain distance from the borehole wall (initiation point of hydraulic fracture). Accordingly, the interaction between hydraulic and pre-existing fractures is assessed by creating a 3D
Inside diameter of wellbore (mm)
6
geometry of models, nominating boundary conditions, and applying fracture pressure uniformly to the initiation point. In situ stresses are applied to the models as the boundary conditions. By considering a normal-faulting stress regime, maximum stress (rV ) is vertical with a constant value of 20e6 Pa and horizontal differential stress is changed from 5e6 to 10e6 Pa. For simplicity, it is assumed that the host rock is composed of homogeneous and isotropic material. Besides, the hydraulic fracture progresses under constant and uniform pressure in an elastic medium under plane strain. In other words, a constant pressure equal to the wellbore pressure is applied inside the fracture. The magnitude of this pressure is considered to be equal to that of the fracture initiation pressure (FIP) obtained from lab experiments (Table 3). The initiation of a fracture depends generally on its orientation (Dehghan et al. 2016), on the stresses around the wellbore caused by initial geostress, and the changes of stress due to the fracture growth (Warpinski and Branagan 1989). For symmetric reasons, the models’ size can also be reduced to 1/4 of the test blocks, leading to faster calculation. Figure 3 shows the geometry of the new test blocks with 15 cm width (x direction), 30 cm height (y direction), and 15 cm depth (z direction). A pre-existing notch (the yellow line with dimensions 1 9 2 cm2 in the figure) is located in the model (x and z directions) as the initiation and propagation point of hydraulic fracture. A 3D model of the test specimen with different strike and dip angles was meshed in ABAQUS software, using 21,000 solid C3D20 elements. Figure 3 displays the typical 3D mesh pattern generated for the test specimen, as well as the zoomed view of the initial notch region. Hydraulic fracture is vertically propagated in the direction of maximal principal stress, with regard to the governing stress regime and loading conditions (Fig. 4). 4.2 Results and Discussion The main purpose of this research is to study the propagation behavior of hydraulic fracture in the interaction with pre-existing fracture under normal stress regime conditions. Hydraulic fracture is propagated as the structural response to constant fracture pressure in a fractured medium (Table 3). This constant pressure allows the rapid growth of propagated fracture geometry
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A. N. Dehghan et al. Table 3 Summary of hydraulic fracture tests conducted on the fractured specimens (Dehghan et al. 2016) Dr (rH rh ) MPa
Fracture initiation pressure (FIP) (MPa)
Fracture propagation pressure (FPP) mean (MPa)
Fracture propagation results
5
24.40
13.73
Crossed
5
25.10
14.62
Crossed
5
5
26.43
15.84
Opened
10
5
5
24.06
13.79
Opened
10
5
5
25.69
17.39
Shear slippage
20
10
5
5
24.42
18.53
Shear slippage
90
20
10
5
5
24.70
14.32
Opened
30
60
20
10
5
5
26.46
18.67
Opened
1-9
30
30
20
10
5
5
25.65
19.50
Opened
2-1
90
90
20
14
4
10
10.51
4.81
Crossed
2-2
90
60
20
14
4
10
10.27
5.56
Crossed
2-3
90
30
20
14
4
10
09.63
6.05
Shear slippage
2-4
60
90
20
14
4
10
12.96
5.27
Crossed
2-5 2-6
60 60
60 30
20 20
14 14
4 4
10 10
11.15 12.76
9.32 9.43
Crossed Crossed
2-7
30
90
20
14
4
10
10.37
9.05
Crossed
2-8
30
60
20
14
4
10
10.24
9.37
Crossed
2-9
30
30
20
14
4
10
10.54
9.84
Opened
Test no.
Pre-existing fracture
In situ stresses
Dip (a) ()
Strike (b) ()
rV MPa
rHmax MPa
rhmin MPa
1-1
90
90
20
10
5
1-2
90
60
20
10
5
1-3
90
30
20
10
1-4
60
90
20
1-5
60
60
20
1-6
60
30
1-7
30
1-8
Fig. 3 Geometric model generated by ABAQUS software for simulating the laboratory test
Dip Strike Pre-existing fracture
Hydraulic fracture initiation
with minimum computational effort (Hossain and Rahman 2008). The energy release rate is an effective factor in determining the behavior of hydraulic fracture at its intersection point with pre-existing fractures (Gfrac =Grock ).
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Numerical modeling of hydraulic fracture propagation is carried out at the intersection point (different values of strike and dip) considering the horizontal differential stresses (Dr) of 5e6 and 10e6 Pa.
3D Numerical Modeling of the Propagation of Hydraulic Fracture at Its Intersection with…
Dip 30°
Dip 60°
Dip 90°
2.50
Crossing Opening
Gfrac / Grock
2.00
Boundary line
1.50
1.00
0.50
0.00 20
30
40
50
60
70
80
90
100
Strike angles (degree)
Fig. 5 Propagation behavior of hydraulic fracture at low differential stress condition (Dr ¼ 5e6 Pa)
Fig. 4 Hydraulic fracture propagation in the direction of maximum horizontal stress (rH ) from a perforated vertical wellbore (Hossain and Rahman 2008)
4.2.1 Hydraulic Fracture Propagation at Low Differential Stress Condition (Dr ¼ 5e6 Pa) The behavior and geometry of hydraulic fracture is investigated through 3D numerical modeling, analyzing hydraulic fracture and its interaction with natural (pre-existing) fracture at low differential stress condition of 5e6 Pa. Table 4 presents the results obtained from numerical simulations and experimental tests. Two types of behavior, opening and crossing behavior, are observed in the interaction between hydraulic and pre-existing fractures. Figure 5 shows the change of hydraulic fracture behavior for different values of Gfrac =Grock versus strike and dip of the pre-existing fracture. If the Gfrac =Grock ratio decreases to less than 1.0 (Gfrac =Grock \1), hydraulic fracture crosses the pre-existing fracture by increasing the strike and dip
angles relative to the orientation of hydraulic fracture propagation (maximum horizontal stress). However, hydraulic fracture is arrested by opening the pre-existing fracture (dominant fracture behavior) while decreasing the strike and dip angles and increasing the Gfrac =Grock ratio to more than 1.0. If the dip angle (a) is high (90) and the strike angle (b) is reduced from 90 to 30, the behavior of hydraulic fracture propagation is changed from crossing into opening. Figure 6 shows the crossing behavior of hydraulic fracture for a model of pre-existing fracture with strike and dip 90. The growth of fracture during hydraulic fracturing is presented in Fig. 7 in four stages. According to the figure, the angle of growing fracture direction (h) continuously changed, as several parameters are effective in changing the fracture trajectory angle during the fracture path. These parameters are diverting pure mode (I) to mixed modes (I/II, I/III, II/III), permeability of material, and initial void. Decreasing the strike angle of pre-existing fracture to 30 results in the diversion of hydraulic fracture towards
Table 4 The results of numerical modeling and laboratory experiments under low differential stress condition (5e6 Pa) Model no.
Natural (pre-existing) fracture angles ()
Gfrac =Grock
Numerical results
*Experimental results
Dip (a)
Strike (b)
L1
90
90
0.820
Crossing
Crossing
L2
90
60
0.954
Crossing
Crossing
L3
90
30
1.051
Opening
Arrest (opening)
L4
60
90
1.034
Opening
Arrest (opening)
L5
60
60
1.235
Opening
Arrest (shear slippage)
L6
60
30
1.447
Opening
Arrest (shear slippage)
L7
30
90
1.383
Opening
Arrest (opening)
L8
30
60
1.692
Opening
Arrest (opening)
L9
30
30
2.074
Opening
Arrest (opening)
* After Dehghan et al. 2015a, b
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(b)
Pre-exisng fracture
(a)
Fig. 6 a Crossing of pre-existing fracture under dip and strike angles of 90, b cross-section of growing hydraulic fracture plane (model L1)
Fig. 7 Crossing status of hydraulic fracture propagation at its intersection with pre-existing fracture: a start of fracture, b expansion of the fracture before intersection, c intersection point and start of crossing, d occurrence of crossing
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3D Numerical Modeling of the Propagation of Hydraulic Fracture at Its Intersection with…
Intersection area & fracture diversion
(a)
(b)
Fig. 8 a Hydraulic fracture arrested by opening the pre-existing fracture, b a cross-section of the stress distribution on the hydraulic fracture propagation plane (model L3)
and into the pre-existing fracture at the intersection point (Fig. 8). The fracture mode is opening (mode I) at the onset of hydraulic fracture initiation and propagation. However, it is changed into mixed modes at its intersection point with pre-existing fracture, due to the opening of pre-existing fracture and the diversion of hydraulic fracture. In other words, mode I of loading condition occurs if the angle between hydraulic fracture and maximum horizontal stress is approximately zero (before meeting the interaction point), i.e., KII = KIII = 0. Besides mode I, other mixed modes (i.e., I/II, I/III, and II/III) can also occur due to sliding and tearing along the pre-existing fracture sides, considering the diversion of hydraulic fracture into preexisting fracture. No significant change is observed in the propagation behavior of hydraulic fracture by decreasing the pre-existing fracture strike from 90 to 30 if the dip angle (a) of
pre-existing fracture decreases to 60 and 30. In all models (L4–L9), after interaction with pre-existing fracture, hydraulic fracture is arrested and then propagated by opening the pre-existing fracture, until it reaches the tip of pre-existing fracture and is then stopped. Figure 9 presents the arresting of hydraulic fracture propagation through opening pre-existing fracture in the model with strike and dip angles of 30. Hydraulic fracture is propagated along the maximum horizontal stress (rH ) direction from its initiation. The growth of hydraulic fracture is stopped for a while in intersecting with pre-existing fracture and then the pre-existing fracture is opened by increasing the pressure at the intersection point, leading to the diversion of hydraulic fracture. Experimentally, the hydraulic fracture propagation behavior is also similar to that of numerical models at low differential stress condition (Table 4). Indeed, the fracture
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(a)
(b)
Fig. 9 a Diverting path of hydraulic fracture propagation at its intersection with pre-existing fracture, b cross-section of hydraulic fracture plane during propagation and diversion into a pre-existing fracture with dip and strike angles of 30 (model L9)
(b)
(a) Wellbore
HF propagation
Pre-existing fracture
Pre-existing fracture plane
Fig. 10 Behavior of hydraulic fracture propagation in laboratory experiments: a crossing model L2 (dip 90, strike 60), b opening model L7 (dip 30, strike 90)
behavior is changed from crossing to opening by decreasing the dip and strike angles of pre-existing fracture. Figure 10a, b presents the crossing and opening behaviors of hydraulic fracture propagation in two different test blocks, respectively. 4.2.2 Hydraulic Fracture Propagation at High Differential Stress Condition (Dr ¼ 10e6 Pa) The dip and strike angles of pre-existing fracture are the same in this case as the previous one. Here, the propagation behavior of hydraulic fracture is investigated at its intersection with pre-existing fracture only under higher differential stress condition. In this regard, the maximum and minimum horizontal principal stresses are changed from
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10e6 and 5e6 Pa, respectively, to 14e6 and 4e6 Pa, respectively. The vertical principal stress (rV ) is 20e6 Pa and vertical stress is the maximum principal stress with the constant value of 20e6 MPa. Table 5 presents the results obtained from numerical models and laboratory experiments. Figure 11 shows the changes of Gfrac =Grock under various pre-existing fracture dip and strike angles at high differential stress condition. Based on the figure, pre-existing fracture is opened by hydraulic fracture under low dip and strike angles and Gfrac =Grock ratio values greater than 1.0 (Gfrac =Grock [ 1). This ratio is reduced to less than 1.0 (Gfrac =Grock \1) with the increase of dip and strike angles of pre-existing fracture relative to the growth and direction of the fracture. Consequently, crossing occurs as the
3D Numerical Modeling of the Propagation of Hydraulic Fracture at Its Intersection with… Table 5 The results of numerical modeling and laboratory experiments under high differential stress condition (10e6 Pa) Model no.
Natural (pre-existing) fracture angles ()
Gfrac =Grock
Numerical results
*Experimental results
Dip (a)
Strike (b)
H1
90
90
0.559
Crossing
Crossing
H2
90
60
0.685
Crossing
Crossing
H3
90
30
1.021
Opening
Arrest (shear slippage)
H4
60
90
0.837
Crossing
Crossing
H5
60
60
0.970
Crossing
Crossing
H6 H7
60 30
30 90
1.202 0.989
Opening Crossing
Crossing Crossing
H8
30
60
1.206
Opening
Crossing
H9
30
30
1.540
Opening
Arrest (opening)
* After Dehghan et al. 2015a, b
Dip 30°
Dip 60°
Dip 90°
2.00
Crossing Opening
Gfrac / Grock
1.50
Boundary line
1.00
HF propagation
0.50
Pre-existing fracture
0.00 20
30
40
50
60
70
80
90
100
Strike angles (degree)
Fig. 11 Propagation behavior of hydraulic fracture at high differential stress condition (Dr ¼ 10e6 Pa)
approximate dominant behavior of hydraulic fracture propagation. Pre-existing fracture with dip 90 and strike 30 is opened at the intersection point; then, hydraulic fracture is propagated along the pre-existing fracture. If the strike angle increases from 60 to 90, the hydraulic fracture, after intersecting, may cross pre-existing fracture, and is propagated in the line with maximum horizontal stress. In this condition, the propagation behavior of hydraulic fracture at its intersection with pre-existing fracture is consistent with both laboratory experiments and numerical models, considering low differential stress. In case of decreasing pre-existing fracture dip (from 90 to 60) and increasing the Gfrac =Grock ratio, hydraulic fracture recrosses the pre-existing fracture with medium and high strike angles of 60 and 90, respectively, and is arrested by pre-existing fracture with low strike angle. The pre-existing fracture with dip and strike angles of 60 and 30, respectively (model H6, Table 5), is crossed by hydraulic fracture in the experimental model (Fig. 12). However, in the numerical model, hydraulic fracture
Fig. 12 Hydraulic fracture crosses the pre-existing fracture with dip of 60 and strike of 30, and is then propagated along the line with maximum horizontal stress (experimental model H6)
cannot cross the pre-existing fracture at low or high differential stresses (Fig. 13). Following the interpretation of hydraulic fracture propagation behavior in a fractured medium, pre-existing fracture dip decreasing from 60 to 30 is subsequently investigated for the strike angles of 30 to 90. Regarding the dip of 30, hydraulic fracture crosses the pre-existing fracture with the strike angle of 90. However, if pre-existing fracture strike decreases to 60 and/or 30, the hydraulic fracture is arrested at its intersection point with pre-existing fracture. The pre-fracture with dip 30 and strike 90 is crossed by hydraulic fracture due to the high rate of energy release in the hydraulic fracture direction relative to the pre-existing fracture (Gfrac =Grock \1). In such cases, no diversion occurs in the main direction of hydraulic fracture (Fig. 14). In models H8 and H9, the strike angles decrease to 60 and 30, respectively, at the intersection point; the energy release rate increases along the pre-existing fracture relative to the hydraulic fracture path (Gfrac =Grock [ 1); the pre-existing fracture is opened;
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(a)
(b)
Fig. 13 a Opening the pre-existing fracture in mixed mode and diversion or offset of hydraulic fracture path, b cross-section of hydraulic fracture plane diverted into the pre-existing fracture with dip of 60 and strike of 30 (numerical model H6)
HF propagation Pre-existing fracture
(a)
(b-1)
(b-2)
Fig. 14 Crossing the pre-existing fracture with dip of 30 and strike of 90 (model H7): a the experimental simulations, b-1, 2 numerical simulations; b-2 a cross-section of the propagating hydraulic fracture plane in crossing mode
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3D Numerical Modeling of the Propagation of Hydraulic Fracture at Its Intersection with…
(a)
(b)
Fig. 15 a Hydraulic fracture diverted into pre-existing fracture, b a cross-section of hydraulic fracture plane and its propagation along a preexisting fracture with dip of 30 and strike of 60 (numerical model H8)
σH
HF propagation Pre-existing fracture
σh
Fig. 16 The induced fracture is initiated in the direction normal to the smallest horizontal principal stress (rh ), then propagated in the line with maximum horizontal stress (rH ), and after interacting with pre-existing fracture (dip 30, strike 60) crossed it without opening and fluid flow along the pre-existing fracture until the fluid reached the model boundary (experimental model H8 test)
and the direction of hydraulic fracture propagation is diverted into the pre-existing fracture (Fig. 15). According to the figure, the effect of differential stress on the propagation behavior of hydraulic fracture gradually decreases with the decrease of dip angles. Consequently, the dominant behavior of hydraulic fracture is starting to open. The propagation behavior of hydraulic fracture in model H8 (dip 30 and strike 60) in the numerical modeling is different from that of experimental simulation (Fig. 16).
Regarding the geometry and propagation of fracture in a fractured case, opening mode fracture (mode I) takes place when hydraulic fracture behavior is crossing at the intersecting point without any propagation along the other paths. Moreover, if hydraulic fracture is turned into preexisting fracture and propagated along it, besides opening mode (mode I), shearing and tearing modes (modes II and III) may occur during its propagation. In other words, the direction of hydraulic fracture propagation is converted into pre-existing fracture; therefore, the fracture mode may change from pure opening mode (I) into mixed modes (I/II, I/III, and II/III). Based on the fracture mechanics theories, a pre-existing fracture with different conditions could have a pivotal role on the trajectory with respect to the primary fracture at the interaction point. Hydraulic fracture growth is observed at its intersection point with pre-existing fracture and is changed into fracture mode and also its growth angle by opening the pre-existing fracture (Fig. 17). The results are obtained from the numerical modeling of hydraulic fracture propagation in a fractured medium (at the intersection with a pre-existing fracture), showing almost good agreement with those of experimental tests. Like high differential stress conditions (10e6 Pa), the fracture behaviors in models H6 and H8 are different from those of hydraulic fracture propagation in the experimental case. The fact can be justified by: (1) dissipated energy fractions due to solving the problems and complications by XFEM; (2) an interval existed between void ratio and the
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Fig. 17 Various stages of propagation (opening case) of hydraulic fracture at its intersection with a pre-existing fracture, due to the change of pure mode I to different mixed modes: a intersection and
start of opening, b expansion of the fracture, c more expansion of the fracture to meet the model’s boundary
material’s porosity in the real material and the software definition; (3) numerical errors in XFEM equations in software codes. Horizontal differential stress (Dr) and angles of preexisting fracture dip (a) and strike (b) are two factors effective in the energy release rate (in the direction of hydraulic and pre-existing fractures, Grock and Gfrac , respectively), used to determine the behavior and geometry of hydraulic fracture propagation. At low differential stress condition, due to high stress concentration in the surrounding rock, higher fluid pressure is needed to reach the critical values of stress intensity factors (KI, KII, and KIII) for initiation and propagation at the tip of hydraulic fracture, particularly at the onset of interaction with pre-existing fracture. The propagation of hydraulic fracture at the intersection point depends on the strike and dip angles of pre-existing fracture. Hydraulic fracture crosses or opens the pre-existing fracture, along which it is propagated. If the strike and dip angles of pre-existing fractures are low and medium at the intersection point with hydraulic fracture, stress intensity factors and, subsequently, energy release rate (G) increase along the pre-existing fracture relative to the hydraulic fracture propagation path (Gfrac =Grock [ 1). Consequently, hydraulic fracture opens the pre-existing fracture and is propagated along it. However, if these angles are high, the energy release rate (G) increases at the tip of growing hydraulic fracture with respect to the direction of pre-existing fracture at the intersection point (Gfrac =Grock \1). Therefore, pre-existing fracture is crossed by hydraulic fracture. If differential stress increases, the stress intensity factor reaches its critical value along the hydraulic fracture under lower fluid pressure. At the intersection point between hydraulic and pre-existing fractures, when dip
and strike angles are medium and high, the energy release rate (G) is higher along the hydraulic fracture path compared to that in the direction of pre-existing fracture (Gfrac =Grock \1). In such cases, hydraulic fracture crosses the pre-existing fracture. The energy release rate decreases in the direction of induced fracture with the decrease of dip and strike angles of pre-existing fracture (Gfrac =Grock [ 1). In contrast, the energy release rate increases in the direction of pre-existing fracture due to its higher interaction with hydraulic fracture. Consequently, increasing the energy release rate in the direction of preexisting fracture results in the opening of pre-existing fracture and diverting the growth of hydraulic fracture from its main path. The effect of strike angle is almost lower on the hydraulic fracture propagation compared to that of dip angle, at low differential stress condition, due to the high interaction between hydraulic and pre-existing fractures. Hydraulic fracture behavior is affected by changing the dip angle from 90 to 60 and 30 or from 30 to 60 and 90. No significant change is observed in the behavior of hydraulic fracture at the dip angles of 30 and 60 by increasing and/or decreasing the strike angles of pre-existing fracture with various dip angles. Besides dip angle, the strike angle also has significant effects on the propagation behavior and geometry of hydraulic fracture at high differential stress condition (10 MPa), considering lower interaction between hydraulic and pre-existing fractures. It should be mentioned that, in the experiments, dip angle has a lower effect on the hydraulic fracture propagation. The complexity of propagation behavior of hydraulic fracture is lower under high differential stress condition compared to that of low differential stress condition. In fact, dip and strike angles of
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pre-existing fractures are significantly effective in the propagation of hydraulic fracture. According to the linear elastic fracture mechanics (LEFM), the stress field around the crack tip is proposed by Williams (1957) as a set of infinite series expansions. The first term is singular and depends on stress intensity factors (SIF) (i.e., KI, KII, and KIII) and the other terms are non-singular. The first non-singular term is called T-stress in the parallel-to-the-crack component of stresses and is independent of the distance from the crack tip. The higher order terms are near the crack tip in the series expansion. Moreover, the so-called parameters and the fracture energy can be considered as pivotal parameters in modeling the hydraulic fracture propagation, concerning different crack opening modes and maximum principal stresses. As mentioned earlier, different methods have been presented for estimating the propagation path of hydraulic fracture. However, some of these methods have limitations for considering all crack parameters in the modeling. Non-singular terms of Williams’ series cannot be defined in the XFEM modeling of ABAQUS. Therefore, the prediction of the crack growth path has a minor difference in comparison with the experimental results. According to Aliha et al. (2010, 2015), an incremental method can better estimate the mixed mode fracture path, through considering T-stress based on GMTS criterion, compared to the XFEM method in ABAQUS. This can be attributed to more precise estimation of the crack tip stress field used in the GMTS criterion relative to the maximum principal stress considered in the XFEM method. In this research, 3D models of XFEM are simplified by ignoring T-stress and other non-singular terms of stress fields around the crack tip.
5 Conclusion The results obtained from this research are summarized as follows: •
•
•
The numerical results obtained from 3D modeling of the interaction between induced and pre-existing fractures are validated through experimental simulations of hydraulic fracture propagation in naturally fractured reservoirs. The horizontal differential stress value and the strike and dip angles of the natural (pre-existing) fracture are two factors influencing the behavior and geometry of hydraulic fracture propagation. The energy release rate at the intersection point (Gfrac =Grock ) is another dominant factor affecting hydraulic fracture propagation. This factor depends on the differential stress and natural fracture strike and dip.
•
•
•
•
•
Opening and crossing are two dominating fracture behaviors occurring at low and high differential stress conditions, respectively. At low differential stress condition, if dip or strike of pre-existing fractures decreases, the Gfrac =Grock ratio increases and, consequently, hydraulic fracture propagation behavior changes from crossing to opening. The Gfrac =Grock ratio decreases with the increase of differential stress and pre-existing fracture dip and strike angles, due to the decrease in the stress concentration in the fracture’s surrounding zone and the interaction between induced and pre-existing fractures. The effects of strike and dip angles on the hydraulic fracture propagation behavior increases with the increase of horizontal differential stress. The 3D models developed in this research may be used to understand the significant changes in the propagation behavior of hydraulic fracture in the naturally fractured reservoirs, particularly at the intersection point between hydraulic and natural fractures, which is not considered in the conventional models.
Acknowledgments The authors would like to particularly thank Prof. Giovanni Barla for his valuable and helpful comments on this paper. This work was supported by the National Natural Science Foundation of China (grant no. 51174217).
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