Analytical Study of the Interaction Between Hydraulic Fracture and the Natural Interface

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Table of Contents

Executive Summary

Acknowledgements

Table of Contents

Table of Figures

Nomenclature………………………………………………………

Chapter 1: Hydraulic Fracturing

1.1 Introduction…………………………………………………..

1.2 Hydraulic fracturing……………………………………………..

1.3 Interaction between hydraulic fracture and the natural interface……………..

1.4 Thesis Structure

Chapter 2: Literature Review

2.1 Analytical Solutions

2.1.1 Blanton’s Criterion (1986)

2.1.2 Warpinski and Teufel’s criteria (1987)

2.1.3 Renshaw and Pollard criteria

2.1.4 Modified Renshaw and Pollard

2.1.5 Further expansion of Modified Renshaw and Pollard

2.2 Laboratory Experiments

2.3 Numerical Simulations

Chapter 3: Analytical Criterion

Chapter 4: Conclusions

4.1 Conclusion

4.2 Recommendations for further work

References

Table of Figures

Figure 1 Schematic of a hydraulic fracture approaching a natural interface (H. Gu, 2012) and the slippage zone.

Figure 2 Potential interaction mechanisms between a hydraulic fracture and the natural interface.

Figure 3 Blanton’s crossing criterion: Graph illustrating the relationship between differential stress and angle of approach for different values of b. Rock tensile strength of 510 psi is used.

Figure 4 Schematic of the in-situ stresses acting on the plane of a natural fracture………

Figure 5 Warpinski and Teufel’s criterion: Plot of differential stress versus angle of approach to illustrate the interaction modes.

Figure 6 Graphical representation of the crossing interaction mechanism, modelling the modified Renshaw and Pollard criterion for a cohesive interface.

Nomenclature

Symbol: Meaning:
θ

/

β

Angle of approach
μf’ Apparent friction coefficient
v(x0) Coefficient that can be calculated using equation 2.5
b Coefficient used in equation 2.2 that can be evaluated using equation 2.3
μf Friction coefficient
l Half the length of the open section of the fracture
c Length of slippage zone
x0 Location of re-initiation
σHMax Maximum horizontal stress
σhMin Minimum horizontal stress
Pn Net pressure (overpressure)
σn Normal stress acting in the direction of the natural interface as a result of in-situ stresses
σH’ Normal stress in the direction of the natural fracture
P Pressure
τ0 Shear strength of the natural fracture plane
τ Shear stress acting in the direction of the natural interface
στ’ Shear stress in the direction of the natural fracture
σT Sum of stresses acting parallel to the natural fracture as a result of in-situ stresses
T0 Tensile rock strength
μf” Total friction coefficient

Chapter 1: Hydraulic Fracturing

1.1 Introduction

Hydraulic fracturing has become an indispensable technique for the development of unconventional oil and gas reservoirs. The drilling technique was first introduced in 1947 with the process becoming commercial by 1950 (King, 2012). Hydraulic fracturing, a common part of the well completion process, is used to stimulate unconventional resources such as shale oil and tight gas. Production of oil and gas from unconventional wells has become a more prominent source of energy due to the rapid decline in production from conventional resources over the past two decades (Sarmadivaleh, 2012). Hydraulic fracturing has a broad range of applications. Some of these include; disposal of waste during the drilling process (i.e. Drilling cuttings), geothermal energy extraction, fault reactivation in mining and the measurement of in situ stresses (J. Adachi, 2007).

When a reservoir formation is hydraulically fractured, this creates a fracture network. It is important to ensure that the fracture propagation does not exceed the reservoir interval, such as the cap rock. The propagation pathway of the fracture can be significantly affected by the presence of a natural fracture, and can be both beneficial or detrimental (Hassan Fatahi, 2017). Shale formations commonly contain pre-existing natural fractures, and when a natural fracture interacts with an induced hydraulic fracture, this causes a complex fracture network (H. Gu, 2012). Hence, it is important to gain a thorough understanding of the effects of the hydraulic fracturing process. This is discussed in further detail in the following sections.

This thesis examines the interaction mechanisms mentioned above with the objective to understand the effects of how a hydraulic fracture interacts with a naturally fractured reservoir. It also aims to present an analytical method to predict the propagation pathway the hydraulic fracture will take, upon interacting with the natural interface. The relevant numerical and experimental lab tests will also be outlined to validate the analytical model. Chapter one aims to briefly outline the hydraulic fracturing process, followed by the interaction mechanisms involved. The objectives of this research and the significance of this study are then proposed and lastly the structure of this thesis is summarised.

1.2 Hydraulic fracturing

Hydraulic fracturing is a method of reservoir stimulation used to enhance the oil and gas recovery from unconventional wells to a profitable amount. This technique is most commonly used in tight gas, shale gas and coal seam gas (CSG) reservoirs, where a low matrix permeability prohibits commercial production of gas. The process of hydraulic fracturing begins with perforating the casing using perforating guns. This creates ‘finger-like’ holes in the formation for the fracturing fluid to enter through as well as they provide access for the natural gas to enter the wellbore. The viscous fracturing fluid is referred to as a ‘pad’ and is pumped into the wellbore at a constant flow rate (Taleghani, 2009). This results in an increase in the downhole pressure, thus inducing a crack in the rock. The pressure at which the crack is induced is referred to as the initiation pressure. When the breakdown pressure; the maximum pressure of the wellbore, is greater than the downhole pressure, the fracture propagates through the formation via the perforations, creating a fracture network (Sarmadivaleh, 2012).

Following this, a slurry is injected into the wellbore at several stages. The slurry is comprised of mainly water, with some proppants such as sorted sand, and a small amount of chemical additives. The sand particles remain between the pores of the formation once the fracturing fluid is removed. The purpose of the sand is to ensure the induced fractures remain open once the pressure is reduced to enable the production of oil and gas through a conductive path. At this point, the slurry breaks down to a lower viscosity and is easily returned to the wellbore, creating a highly conductive path for the oil and gas to travel through resulting in an enhanced recovery. A plug is then set and new perforations are made at the next interval of the casing (Taleghani, 2009). For each new interval, a different concentration of proppant is used, with the last stage generally containing the highest concentration (Sarmadivaleh, 2012). This method allows the enhancement of recovery to extend to greater lateral lengths of the well. While the fractures themselves are approximately 0.25 inches or less, the effective horizontal length of the well may extend to 3000 feet (Taleghani, 2009).

A hydraulic fracture will propagate towards the direction of least resistance; therefore, it grows in the direction perpendicular to the smallest of the three principal stresses. Figure 1 below illustrates the interaction between a hydraulic fracture approaching a natural interface. The smallest principle stress is generally in the horizontal plane, as seen in Figure 1, indicating that the fracture is restricted by the vertical plane. This is common for most reservoir depths of interest in the petroleum industry. Hence, the fracture will propagate in the direction normal to the minimum horizontal stress. The direction of propagation is significantly influenced by; the rock mechanical properties, in-situ stresses, rheology of the fracturing fluid and the presence of discontinuities (Taleghani, 2009). Discontinuities are classified as any mechanical break or plane of weakness in rock mass. These include; bedding planes, faults, foliation, fractures, joints, rock cleavage and shear zones (B.B.S. Singhal, 2010). In the following section, the effects of a natural fracture present in the formation on hydraulic fracturing propagation will be explored.

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+(l+c)

+l

Slippage zone

-l

Slippage zone

-(l+c)

Figure 1 Schematic of a hydraulic fracture approaching a natural interface (H. Gu, 2012) and the slippage zone.

1.3 Interaction between hydraulic fracture and the natural interface

Natural fractures can form due to tectonic movement after deposition or from the pressure formed when hydrocarbons were created (King, 2012). It is generally assumed that natural fractures form in the vertical direction in-situ, hence it is most efficient to drill horizontal wells (Norbeck). The interaction between a hydraulic fracture and the natural interface is dependent on in-situ stresses, mechanical properties of the rock and the existing natural fracture, angle of approach and hydraulic fracture treatment parameters, including the fracturing fluid properties and injection rate (H. Gu, 2012) In general, we consider three main interaction mechanisms, arrest or slippage, opening and crossing (Sarmadivaleh, 2012). These are discussed further in the section below.

To simplify the model, we break down the process into two steps. Firstly, when analysing a hydraulic fracture at a natural interface, it is important to consider the stress field generated by the hydraulic fracture at the fracture tip (H. Gu, 2012). According to laboratory experiments carried out by Warpinski in 1985, this is caused by the fluid lag between the fracture tip and the fluid front inside a hydraulic fracture (Warpinski, 1985). At this point, the natural fracture is already under the influence of the stress field generated by the hydraulic fracture. The net fluid pressure; the difference between the fracturing fluid and the in-situ stresses, at the intersection point is considered zero. Hence, when analysing this process, we consider the mechanical interactions between the induced fracture and the natural interface without considering fluid flow (H. Gu, 2012). Numerous outcomes may occur as a result of a hydraulic fracture propagating towards a natural interface. These include arrest (slippage) and crossing, and are depicted in Figure 2 below.

Secondly, the fluid pressure increases once the fluid front meets the intersection point of the natural fracture. For the slippage mechanism, if the fluid pressure is greater than the normal compressive stress, it is possible for the fluid to flow into and dilate the natural fracture. If this is the case, and the flow of fluid into the fracture continues, the dilated fracture becomes a part of the fracture network as the induced fracture turns and propagates along the dilated natural interface (H. Gu, 2012).

If a hydraulic fracture crosses the natural interface without opening the fracture or a change in direction, it is considered a planar fracture. This is caused when the fluid pressure is less than the normal stress on the natural fracture. Enhanced leak-off may occur if the rock is permeable. Conversely, if the fluid pressure if greater than the normal stress, the induced fracture will not cross the natural interface but instead dilate and propagate along the natural fracture, and a complex fracture network may form (H. Gu, 2012).

The offsetting interaction mechanism has also been included in Figure 2. This mechanism works by first initiating a fracture and opening the natural interface, followed by reinitiating along the interface at a point other than at the interaction site (Sarmadivaleh, 2012).

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Figure 2 Potential interaction mechanisms between a hydraulic fracture and the natural interface (Sarmadivaleh, 2012).

Natural fractures are often unproductive due to a lack of connectivity between the interfaces. Hence, the use of hydraulic fracturing to connect the natural fractures and therefore obtain a productive reservoir (Norbeck). For certain reservoirs such as lenticular sands, it is necessary to hydraulically fracture as many of the natural fractures as possible with the intention to connect them to a single borehole (Sarmadivaleh, 2012). However, the location of natural fractures is often unknown and difficult to predict. Consequently, the location of where to hydraulically fracture is arbitrary. The current method of selecting the hydraulic fracture initiation points is to equally space them in segments along the horizontal well. Upon execution of this method, it can be detrimental as the induced fractures may damage the natural fracture system or completely miss the fracture network (Norbeck).

1.4 Thesis Structure

Several criteria have been developed to predict the behaviour of the induced fracture at the natural interface. A criterion based on ‘compressional crossing’ was proposed by Renshaw and Pollard (1995), which looks at a crossing interaction at orthogonal angles. This criterion has been modified for non-orthogonal angles and non-orthogonal angles with a non-cohesive interface. This is suggestive that the criterions available are not yet fully developed for all possible cases in the reservoir. These criteria were proposed over a decade ago, and as such, newly developing ideas have been presented with experimental and numerical data to validate them. Hence, the purpose of this research is to close the gap between the existing literature and the newly surfaced information. Lab experiments and numerical simulations have been used to validate the proposed analytical solutions.

This thesis comprises of five chapters. Chapter 2 reviews the literature already presented for this topic. It explores the analytical solutions used to determine the interaction between a hydraulic fracture and the natural interface.

Chapter 3 presents a new analytical model along with the relevant experimental and laboratory data to validate the model.

Chapter 2: Literature Review

2.1 Analytical Solutions

2.1.1 Blanton’s Criterion (1986)

In 1986 Blanton developed a criterion demonstrating the relationship between differential stress and angle of approach. The proposed solution considers the hydraulic fracture to initially arrest upon interaction with the natural interface. At this point, the pressure will increase and continue to increase at the intersection point until opening of the natural fracture or re-initiation occurs on the other side. The criterion for crossing states that crossing will occur when the pressure required for re-initiation is less than the opening pressure of the fracture. The crossing condition requires a pressure value that is greater than the sum of stresses acting parallel to the fracture plus the tensile stress (Blanton, 1986). This can be expressed by the following equation:

P>σT+T0

(2.1)

σT

is dependent on far-field stresses, fracture pressure, geometry of the interaction zone, frictional slippage and the opening along the natural fracture, therefore is considered a complex term. The slippage zone has been illustrated in Figure 1 above. Blanton proposed a dimensionless relationship for the crossing criterion with respect to the parameters that affect

σT:

σHMax-σhMinT0> -1cos2θ-bsin2θ

(2.2)
The following equation is used to calculate b:

b=12cvx0-(x0-l)μf

(2.3)

Where,

c=width of slippage zone

l=half the length of the open section from -l to+l of the open section

μf=the friction coefficient

x0=re-initiation point

The values for

x0and

v(x0)can be determined as follows:

x0= 1+c2+eπ2μf1+eπ2μf0.5

(2.4)

and

vx0=1π[x0+lln⁡x0+l+cx0+l2+x0-lln⁡x0-l-cx0-l2+cln⁡x0+l+cx0-l-c2

(2.5)

To evaluate

σT, an expression has been developed by superimposing two stress field states. For

σ1, at an angle of

θ, the far-field stresses acting on the plane of a natural fracture will yield

σT1. Conversely, for

σ2, the far-field stresses acting on the plane of a natural fracture is zero and a distribution of shear stresses on the natural fracture are found to produce

σT2. The resultant shear stress is zero, which is due to the shear stress along the open section of the fracture being equal in magnitude but opposite in sign to the shear stress at

σ1.

The value of b in equation 2.3 above is an important parameter in determining the crossing criterion. As c approaches zero, indicating no slippage, the value of b tends to infinity, hence the crossing criterion will be satisfied. On the contrary, if the value of c is significantly increased, b will drop significantly as the two values are inversely proportional (

b∝1c) (Blanton, 1986). The following equation outlines the term b as c approaches infinity:

b=12πln⁡1+1+eπ2μf0.5 1-1+eπ2μf0.5 2

(2.6)

Using equation 2.6 to determine b, the criterion has been calculated (equation 2.2) and plot in Figure 3 below, for the values of b at 0.2, 0.4 and 0.6. As seen by the graph, as the value of b decreases, the mode of opening decreases and the crossing criterion becomes more prominent. For angles less than 30 degrees, the opening interaction is preferred and for angles greater than 60 degrees the crossing mechanism is favoured.

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Figure 3 Blanton’s crossing criterion: Graph illustrating the relationship between differential stress and angle of approach for different values of b. Rock tensile strength of 510 psi is used (Sarmadivaleh, 2012).

According to laboratory results presented by Llanos (2006), Blanton’s criterion requires higher normal stresses for crossing to occur. Blanton’s criterion does not consider the stress field exerted by the hydraulic fracture on the interface and assumes a simplified shear stress distribution along the interface (Sarmadivaleh, 2012).

2.1.2 Warpinski and Teufel’s criteria (1987)

Warpinski and Teufel’s criteria is developed based on an estimation of the pore pressure distribution of the natural fracture plane followed by the conditions under which the joint will dilate or undergo shear slippage. Four simple cases are modelled to predict the pore pressure distribution of any practical situation (N.R. Warpinski, 1987). These models are used to determine the overpressure along the discontinuity and is defined as the difference between fracture pressure and the minimum horizontal stress. According to Warpinski and Teufel’s model, shear slippage will occur when the shear stress is greater than the normal stress acting in the direction of the natural fracture plane, as illustrated in Figure 4 below. Considering Coulomb’s failure criterion (John Conrad Jaeger, 2011) derived from the linear friction law for shear failure along the natural fracture, the relationship for shear slippage can be expressed as:

τ>τ0+μf(σn-p)

(2.7)

where,

τ0=inherent shear strength of the interface (psi)

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Figure 4 Schematic of the in-situ stresses acting on the plane of a natural fracture.

Considering the principle stresses and the orientation of the joint,

θ, with respect to

σ1, equation 2.7 can be re-arranged to give:

σHMax-σhMin<2pn(1-cos2θ)

(2.8)

As seen in Figure 5 below, a graph has been generated to demonstrate the relationship between differential stress and angle of approach. For this model, three net pressures

Pnof 0.1, 0.7 and 1.0 MPa have been used. Equation 2.8 is represented by the dashed curves indicating the opening mode.

Figure 5 Warpinski and Teufel’s criterion: Plot of differential stress versus angle of approach to illustrate the interaction modes (Sarmadivaleh, 2012).

Warpinski and Teufel then developed an expression for the deviatoric stress required for arrest at the interface:

σHMax-σhMin≥ 2τ0-2pnμfsin2θ+ μfcos2θ-μf

(2.9)

Equation 2.9 has also been plot in Figure 5 above, and is represented by the solid curves for the same three values of net pressure, 0.1, 0.7 and 1.0 MPa. This model corresponds to the crossing interaction mode (N.R. Warpinski, 1987; Sarmadivaleh, 2012).

2.1.3 Renshaw and Pollard criteria

A criterion based on ‘compressional crossing’ was proposed by Renshaw and Pollard (1995), which predicts whether a fracture will cross a frictional interface at orthogonal angles. The criterion assumes crossing will occur if there is no slip at the interface. This occurs due to a transfer of tensile stress from re-initiation of the fracture on the opposite side of the interface. If the frictional strength (shear stress) at the interface is less than the induced tensile strength, slippage will occur and the fracture will arrest upon interaction with the natural fracture. As Renshaw and Pollard’s crossing criterion assumes that no slippage or opening will occur at the interface, the frictional strength must be greater than the induced tensile strength.

Some further assumptions made in relation to the stresses, pressure, orientation and materials of the fracture include:

  • The approaching fracture intersects the least compressive stress at orthogonal angles.
  • The propagation of the fracture is unaltered by the frictional interface.
  • The fluid pressure in the approaching fracture is small (energetically more favourable), to ensure the fracture will cross the interface rather than deflect into the interface. For this to occur the fluid pressure must be less than the remote compressional stress acting across the interface.
  • The substances present around the interface are linear elastic, homogenous and isotropic.

Hence, using linear elastic fracture mechanics (John Conrad Jaeger, 2011), the stresses near the tip of the approaching fracture can be determined. The crossing criterion is stated below as:

-σHMaxT0-σhMin>1+μf3μf

(2.10)

The criterion proposed by Renshaw and Pollard is only valid for a hydraulic fracture intersecting a frictional interface at orthogonal angles, and for surfaces with the similar elastic properties (C.E. Renshaw, 1995). As mentioned above, the criterion assumes no slip at the interface, and hence crossing will occur. This contradicts other experimental data that states crossing may occur after slip and leak-off into the natural fracture. Consequently, Renshaw and Pollard’s criterion requires a larger estimation of the discontinuity strength needed to permit crossing (Ella Maria Llanos, 2006).

2.1.4 Modified Renshaw and Pollard

Sarmadivaleh and Rasouli (2013) proposed a modified criterion of Renshaw and Pollard’s criteria, based on non-orthogonal cohesive fractures. They attempt to present a more general case of a non-cohesive fracture propagating at non-orthogonal angles. The criteria are divided into three parts; interface with shear strength, non-orthogonal angle of approach and the general case for non-orthogonal cohesive natural fracture. To account for the cohesion, an imaginary friction coefficient parameter was introduced.

Interface with shear strength:

The initial shear condition proposed by Renshaw and Pollard (1995) is modified with the consideration of cohesion. An interface with cohesion is less likely to slip in comparison to a non-cohesive surface. Thus, the initial shear condition:

στ'< -μfσH’

(2.11)

becomes,

στ'<τ0-μfσH’

(2.12)

where,

στ’=shear stress in the direction of the natural fracture

σH’=normal stress in the direction of the natural fracture

When a hydraulic fracture is approaching a natural fracture, it induces a stress field at the fracture tip. Considering the case of an orthogonal angle of approach of the hydraulic fracture, the shear stress applied is simply a function of the shear stress induced by the stress field of the hydraulic fracture. Conversely, for the case of a non-orthogonal interface, the stresses are a function of both in-situ stresses and the induced stress generated by the fracture tip. The following equation was developed by Sarmadivaleh and Rasouli for the crossing criterion, originally proposed by Renshaw and Pollard (1995):

-σHMax(T0-σhmin)>1+μf”3μf”

(2.13)

where,

μf”=μf’+μf         (2.14)

μf’=τ0σHMax-11+μf         (2.15)

Equation 2.13 has been represented graphically below for different ratios of shear strength to maximum horizontal stress. When the ratio is 0, the interaction represents Renshaw and Pollard’s model as seen by equation 2.10 above. It is evident that the region above the curves represents the crossing mechanism. As seen by the equations above, as the ratio of shear strength to maximum horizontal stress increases, there is a greater chance of the crossing interaction occurring. As a result, when the ratio increases, the crossing criterion becomes less dependent on the friction coefficient.

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Figure 6 Graphical representation of the crossing interaction mechanism, modelling the modified Renshaw and Pollard criterion for a cohesive interface (Sarmadivaleh, 2012).

Non-orthogonal angle of approach:

Sarmadivaleh and Rasouli then considered a more general case: a non-orthogonal angle of approach with a cohesionless interface. For this model, the far-field in-situ stresses must be transformed and the following equation results:

-σn(T0-σT)>1-sinθ2sin⁡3θ2+1μfcos⁡θ2sin⁡θ2cos⁡θ2cos⁡3θ2+α  1+sin⁡θ2sin⁡3θ2

General case for non-orthogonal cohesive natural fracture:

Considering a general case for a non-orthogonal angle of approach with a cohesive interface, the general form is expressed as:

-σn(T0-σT)>1-sinθ2sin⁡3θ2+1μf”cos⁡θ2sin⁡θ2cos⁡θ2cos⁡3θ2+α  1+sin⁡θ2sin⁡3θ2

where,

μf”=μf’+μf

μf’=τ0σn1-sin⁡θ2sin⁡3θ21-sin⁡θ2sin⁡3θ2+ 1μfcos⁡θ2 sin⁡θ2cos⁡θ2cos⁡3θ2+α-1

2.1.5 Further expansion of Modified Renshaw and Pollard

Gu and Weng (2010) also proposed a modified criterion of Renshaw and Pollard. This criterion is not expressed as an explicit equation and is evaluated numerically making it difficult to apply to practical situations. The criterion is valid for non-orthogonal angles of approach with a cohesive interface. The following equation was proposed:

τ0μ-σHMax(T0-σhMin)>0.35+0.35μ1.06

(2.19)

2.2 Field Studies

Summary of field studies carried out

2.3 Laboratory Experiments

Summary of experiments carried out

2.4 Numerical Simulations

Summary of numerical simulations

Chapter 3: Analytical Criterion

Chuprakov, Melchaeva and Prioul OpenT Interaction Criterion:

Chuprakov et al. (2013), proposed a “blunted dislocation discontinuity approach” to predicting the interaction between hydraulic fracture and the natural interface. This analytical model considers both the strength and energy criterion, known as “mixed criterion” (D. Chuprakov, 2013).  The criterion accounts for key parameters that are neglected in previous models presented by Blanton (1986), Warpinski and Teufel (1987), Renshaw and Pollard (1995), Gu and Weng (2010) and Sarmadivaleh and Rasouli (2013). The new criterion includes the following key parameters that are relevant to practical situations:

  • Chuprakov et al. consider the fracture contact ‘problem’ using a rigorous elastic solution.
  • For hydraulic fracture re-initiation across the natural interface, an energy-related condition is used to evaluate the crack initiation.
  • All angles of interaction between the hydraulic fracture and the natural interface can be evaluated using the criterion.
  • Takes into consideration the effects of fluid flow such as, viscosity and fluid flow rate.
  • Accounts for the permeability of the natural fracture and the possibility of fluid penetration into the natural interface.

Chapter 4: Experimental studies

New experimental studies to validate criterions

Chapter 5: Numerical simulations

New numerical simulations to validate criterions

Chapter 6: Discussion and Results

Discussion/ comparison of results and methods (analytical, experimental and numerical)

Chapter 7: Conclusion

7.1 Conclusion

7.2 Recommendations for further work

References

B.B.S. Singhal, R. P. G. (2010). Applied Hydrogeology of Fractured Rocks (2 ed.). Netherlands: Springer Netherlands.

Blanton, T. L. (1986). Propagation of Hydraulically and Dynamically Induced Fractures

in Naturally Fractured Reservoirs Society of Petroleum Engineers, SPE Paper No. 15261.

C.E. Renshaw, D. D. P. (1995). An experimentally verified criterion for propagation across unbonded frictional interfaces in brittle, linear elastic materials. International Journal of Rock Mechanics Mining Science and Geomechanics, 3.

D. Chuprakov, O. M., R. Prioul. (2013). Injection-sensitive mechanics of hydraulic fracture interaction with discontinuities. Rock Mechanics and Rock Engineering, 47(5), 1625-1640. doi:10.1007/s00603-014-0596-7

Ella Maria Llanos, R. J., Richard Hills, Xi Zhang. (2006, 5-8 November 2006). Factors Influencing Whether Induced Hydraulic Fractures Cross

Pre-Existing Discontinuities. Paper presented at the AAPG International Conference and Exhibition, Perth, Western Australia.

H. Gu, X. W., J. Lund, M. Mack, U. Ganguly, R. Suarez-Rivera. (2012). Hydraulic Fracture Crossing Natural Fracture at Nonorthogonal Angles: A Criterion and Its Validation. 20-26.

Hassan Fatahi, M. H., Mohammad Sarmadivaleh. (2017). Numerical and experimental investigation of the interaction of natural

and propagated hydraulic fracture. Journal of Natural Gas Science and Engineering, 37, 409-424. doi:10.1016/j.jngse.2016.11.054

J. Adachi, E. S., A. Peirce, J. Desroches. (2007). Computer simulation of hydraulic fractures. International Journal of Rock Mechanics and Mining Science, 44, 739-757.

John Conrad Jaeger, N. G. W. C., Robert W Zimmerman. (2011). Fundamentals of Rock Mechanics (4 ed.). Malden, MA: Blackwell Publishing.

King, G. E. (2012). Hydraulic Fracturing 101: What Every Representative, Environmentalist, Regulator, Reporter, Investor, University Researcher, Neighbor, and Engineer Should Know About Hydraulic Fracturing Risk. Journal of Petroleum Technology, 64(04).

N.R. Warpinski, L. W. T. (1987). Influence of Geologic Discontinuities on Hydraulic Fracture Propagation. Journal of Petroleum Technology, 39(02), 209-220. doi:10.2118/13224-pa

Norbeck, J. H. Identification and Characterisation of Natural Fractures While Drilling Underbalanced. (Master of Science (Engineering)), Colorado School of Mines, Colorado.

Sarmadivaleh, M. (2012). Experimental and numerical study of interaction of a pre-existing natural interface and an induced hydraulic. Curtin University Perth, WA.

Taleghani, A. D. (2009). Analysis of hydraulic fracture propagation in fractured

reservoirs: an improved model for the interaction between

induced and natural fractures. (Doctor of Philosophy), University of Texas at Austin, Austin, Texas. Retrieved from https://repositories.lib.utexas.edu/bitstream/handle/2152/18381/dahitaleghania23003.pdf?sequence=2&isAllowed=y

Warpinski, N. R. (1985). Measurement of Width and Pressure in a Propagating Hydraulic Fracture. Society of Petroleum Engineers Journal, 25(01), 46-54. doi:10.2118/11648-pa

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