**Abstract**

A 2-D numerical study was carried out, using a fully coupled rock deformation and fluid flow hydraulic fracturing model, on fracture network formation by advancing, widening and interconnecting discrete natural fractures in a low-permeability rock, some of which are small enough to be considered as a flaw that acts as a fracture seed. The model also includes fractures connecting into one another to form a single hydraulic fracture. In contrast to previous fracture network models, fracture extension and fluid flow behavior, frictional slip, and fracture interaction are all explicitly addressed in this model. Incompressible Newtonian fluid is injected at a constant total rate into fractures to study viscous fluid effects on the network formation. The algorithm for flow division and coalescence is validated through some examples.

Numerical results show that the incremental crack propagation that connects isolated natural fracture sets depends on the current stress state and the fracture arrangement. The newly created connecting fracture segments increase local conductivity since they are oriented along a path that is easier to open when pressurized by fluid and provide a new path for fluid flow. However the hydraulic fracture growth process is retarded by some of the resulting geometric changes such as intersections and offsets, and the growth-induced sliding that can impose a barrier to further fracture growth and fluid flow into parts of the network. Such barriers may eventually result in a fracture branch initiating and growing that results in a relatively shorter and more conductive path through a fracture network zone.

We consider a specific fracture arrangement consisting of around 20 conductive pre-existing fractures to study the effective behavior of the hydraulic fracture growth through a natural fracture network. Mechanical responses have been studied for two different fracture and flow scenarios depending on the fluid entry details: one fracture system assumes each of four entry

© 2013 Zhang and Jeffrey; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

fractures has one quarter of the total injection rate and the other system is defined to maintain the influx rates into each inlet fracture so that the pressure across all four inlet fractures is equal (but not necessarily constant in time). For the latter case, a preferential flow pathway is developed as a result of hydraulic fracture growth and the overall permeability of the fracture system increases rapidly after this hydraulic fracture path develops. The former injection condition results in development of more evenly distributed advancing fractures that provide a more homogeneous flow pattern.

the fractures are either uniform [7] or based on the steady-state solution for flow in a porous medium [8]. The effect of fluid viscosity on pressure distribution has therefore not been considered in these fracture network models. Moreover, some simplifications in the fracture geometry changes, such as the details of interactions at intersections and in regions where fractures close, are used in these models and these simplications affect fluid flux distribution. Uncoupling of deformation and fluid flow, as used in some models, limits the application of

Development of Fracture Networks Through Hydraulic Fracture Growth in Naturally Fractured Reservoirs

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141

Fracture models that do not consider flow in the matrix, are applicable to low-permeability rocks [9]. In general, the fracture aperture is very narrow and would be different in each fracture branch intersecting at the same point, and these differences are expected to be significant in their effect on fluid movement through the intersections. In contrast to the stressdriven uniform fracture nucleation in porous rocks subject to a tensile loading environment, the propagation of a hydraulic fracture through a network of pre-exsiting fractures is depend‐ ent on local stress states around fracture tips, which in turn depends on the fluid flow and pressure along the non-planar hydraulic fracture path. Under most circumstances, one dominant hydraulic fracture is generated and the entire injected fluid rate is carried to the outlet through this preferential flow path, while most shorter natural fractures that are intersected by this main fracture remain closed or act as dead end branches [9]. Models that include the coupling of rock deformation and viscous fluid flow provide a means of studying the fracture development and the evolution of distinct preferential flow paths and develop‐

At intersections of two or more fractures, the kinematic deformation transfer between slip and opening of the fractures can induce additional fracture aperture changes. In addition, the viscous effect becomes stronger for narrow channel widths, which are commonly associated with intersections and offsets. The importance of tracking the details of fracture geometry lies in the fact that although the pressure losses may diminish after a long time, initial fracture geometry details may strongly affect the final fracture patterns. Studies that neglect viscous fluid effects, by using uniform pressure or steady-state transport and deformation models, will, in many cases miscalculate the stresses and flow rates, thus producing incorrect fracture and flow patterns as time-dependent pressure responses are not determined accurately.

Mechanical interaction among fractures has not received sufficient attention in the literature involving discrete fracture models especially for cases involving pressurized viscous fluids sufficient to result in hydraulic fracture growth. Any inaccuracy in the calculated fracture pathway may cause incorrect flux redistribution at intersections, as fluid flow behaviour is strongly dependent on local width. Due to intrinsic complexity of the problem, numerical methods appear to be the only approach able to explicitly solve the nonlinear and nonlocal fracture-fluid and fracture-fracture interaction in such fracture network. The Distinct Element Method (DEM) and the Finite Element Method (FEM) have been used for this purpose [10,11]. However, the fracture pathways are confined along the element edges in classical FEM models andthe out-of-plane fracturepropagationisdifficultto accurately simulatedbecause itrequires remeshing.WehavedevelopedaBoundaryElementMethod(BEM) basedprogramfortreating this coupled problem [12,13]. The validation of the code has been carried out for various simple

the results obtained.

ment of a dominant fracture.

#### **1. Introduction**

The understanding of fluid movement through fractured rock masses is essential to improving the success of reservoir stimulations for energy resources such as shale gas and geothermal energy and for stimulation by hydraulic fracturing of any naturally fractured rock mass. As we know, fractures play an important role in the flow of fluid through rock masses by building connected networks that channel flow. These networks develop either through enhanced conductivity of existing fractures or by new fracture growth that connects existing fractures. There are many studies in the literature devoted to characterising the fracture system connec‐ tivity in relation to fracture orientation, size and conductivity (see the comprehensive review [1]). Besides these static factors, the time dependent evolution of fluid pressure and stress states can generate different fracture patterns that act to assist or inhibit further fluid flow, in particular forming a preferential fluid pathway in the presence of viscous fluid flow [2, 3]. Under some circumstances, the fracture growth driven by pressurized fluids can propogate a hydraulic fracture to connect two isolated fracture clusters. Cross-cutting fractures in the connected region are filled with fluid and pressurized but may open or not. Therefore fluid movement in a fractured rock mass will involve both new fracture growth and permeability enhancement of existing fractures. Clearly, using an equivalent porous continuum model to represent fluid flow and fracture growth would be inaccurate, especially when the flow is dominated by hydraulic fracture processes. The approach here is to study the full coupled process in order to determine the parameters that control fracture development. Simplifica‐ tions and averaging methods can eventually be realistically employed without degrading the ability to predict fracture growth.

Discrete fracture models, so named because these models treat fractures as discrete entities, are applicable whenever the process involves fractures growth and flow where details such as opening, shearing or growth of the fractures are being studied. If many fractures are consid‐ ered, these discrete models become computationally demanding. Such a system is very heterogeneous and localized in both fracture growth and fluid flow. Early numerical models treated the hydraulic fractures as single planar fractures and did not consider fracture interaction [4-6]. The emergent behaviours associated with fracture propagation under a tensile displacement boundary condition have been described as straight paths using a subcritical failure criterion and a propagation speed exponent. Recently, the effect of curving fractures on fluid flow in both the fractures and the matrix under a tensile displacement boundary condition has been considered [7,8]. However, the pressure distributions used inside the fractures are either uniform [7] or based on the steady-state solution for flow in a porous medium [8]. The effect of fluid viscosity on pressure distribution has therefore not been considered in these fracture network models. Moreover, some simplifications in the fracture geometry changes, such as the details of interactions at intersections and in regions where fractures close, are used in these models and these simplications affect fluid flux distribution. Uncoupling of deformation and fluid flow, as used in some models, limits the application of the results obtained.

fractures has one quarter of the total injection rate and the other system is defined to maintain the influx rates into each inlet fracture so that the pressure across all four inlet fractures is equal (but not necessarily constant in time). For the latter case, a preferential flow pathway is developed as a result of hydraulic fracture growth and the overall permeability of the fracture system increases rapidly after this hydraulic fracture path develops. The former injection condition results in development of more evenly distributed advancing fractures that provide

The understanding of fluid movement through fractured rock masses is essential to improving the success of reservoir stimulations for energy resources such as shale gas and geothermal energy and for stimulation by hydraulic fracturing of any naturally fractured rock mass. As we know, fractures play an important role in the flow of fluid through rock masses by building connected networks that channel flow. These networks develop either through enhanced conductivity of existing fractures or by new fracture growth that connects existing fractures. There are many studies in the literature devoted to characterising the fracture system connec‐ tivity in relation to fracture orientation, size and conductivity (see the comprehensive review [1]). Besides these static factors, the time dependent evolution of fluid pressure and stress states can generate different fracture patterns that act to assist or inhibit further fluid flow, in particular forming a preferential fluid pathway in the presence of viscous fluid flow [2, 3]. Under some circumstances, the fracture growth driven by pressurized fluids can propogate a hydraulic fracture to connect two isolated fracture clusters. Cross-cutting fractures in the connected region are filled with fluid and pressurized but may open or not. Therefore fluid movement in a fractured rock mass will involve both new fracture growth and permeability enhancement of existing fractures. Clearly, using an equivalent porous continuum model to represent fluid flow and fracture growth would be inaccurate, especially when the flow is dominated by hydraulic fracture processes. The approach here is to study the full coupled process in order to determine the parameters that control fracture development. Simplifica‐ tions and averaging methods can eventually be realistically employed without degrading the

Discrete fracture models, so named because these models treat fractures as discrete entities, are applicable whenever the process involves fractures growth and flow where details such as opening, shearing or growth of the fractures are being studied. If many fractures are consid‐ ered, these discrete models become computationally demanding. Such a system is very heterogeneous and localized in both fracture growth and fluid flow. Early numerical models treated the hydraulic fractures as single planar fractures and did not consider fracture interaction [4-6]. The emergent behaviours associated with fracture propagation under a tensile displacement boundary condition have been described as straight paths using a subcritical failure criterion and a propagation speed exponent. Recently, the effect of curving fractures on fluid flow in both the fractures and the matrix under a tensile displacement boundary condition has been considered [7,8]. However, the pressure distributions used inside

a more homogeneous flow pattern.

140 Effective and Sustainable Hydraulic Fracturing

ability to predict fracture growth.

**1. Introduction**

Fracture models that do not consider flow in the matrix, are applicable to low-permeability rocks [9]. In general, the fracture aperture is very narrow and would be different in each fracture branch intersecting at the same point, and these differences are expected to be significant in their effect on fluid movement through the intersections. In contrast to the stressdriven uniform fracture nucleation in porous rocks subject to a tensile loading environment, the propagation of a hydraulic fracture through a network of pre-exsiting fractures is depend‐ ent on local stress states around fracture tips, which in turn depends on the fluid flow and pressure along the non-planar hydraulic fracture path. Under most circumstances, one dominant hydraulic fracture is generated and the entire injected fluid rate is carried to the outlet through this preferential flow path, while most shorter natural fractures that are intersected by this main fracture remain closed or act as dead end branches [9]. Models that include the coupling of rock deformation and viscous fluid flow provide a means of studying the fracture development and the evolution of distinct preferential flow paths and develop‐ ment of a dominant fracture.

At intersections of two or more fractures, the kinematic deformation transfer between slip and opening of the fractures can induce additional fracture aperture changes. In addition, the viscous effect becomes stronger for narrow channel widths, which are commonly associated with intersections and offsets. The importance of tracking the details of fracture geometry lies in the fact that although the pressure losses may diminish after a long time, initial fracture geometry details may strongly affect the final fracture patterns. Studies that neglect viscous fluid effects, by using uniform pressure or steady-state transport and deformation models, will, in many cases miscalculate the stresses and flow rates, thus producing incorrect fracture and flow patterns as time-dependent pressure responses are not determined accurately.

Mechanical interaction among fractures has not received sufficient attention in the literature involving discrete fracture models especially for cases involving pressurized viscous fluids sufficient to result in hydraulic fracture growth. Any inaccuracy in the calculated fracture pathway may cause incorrect flux redistribution at intersections, as fluid flow behaviour is strongly dependent on local width. Due to intrinsic complexity of the problem, numerical methods appear to be the only approach able to explicitly solve the nonlinear and nonlocal fracture-fluid and fracture-fracture interaction in such fracture network. The Distinct Element Method (DEM) and the Finite Element Method (FEM) have been used for this purpose [10,11]. However, the fracture pathways are confined along the element edges in classical FEM models andthe out-of-plane fracturepropagationisdifficultto accurately simulatedbecause itrequires remeshing.WehavedevelopedaBoundaryElementMethod(BEM) basedprogramfortreating this coupled problem [12,13]. The validation of the code has been carried out for various simple cases involving both viscous fluid and uniform pressure. Additionally, the program treats rock deformation and fluid flow as a whole in that the field variables are obtained in a single framework, instead of the one-way coupling scheme as used in Reference [11].

The purpose of this paper is to provide some initial results for hydraulic fracture growth through a natural fracture network. Fracture growth is allowed and is based on a local failure criterion[14] and fracture coalescence can take place to form a path through an existing network of natural fractures. The numerical treatment of fracture coalescence has been detailed in [12, 13]. The rock mass is assumed to be impermeable and the fluid flow is confined to occur along the pre-existing or newly created fractures. The fracture nucleation sites are embedded as the pre-existing secondary fractures with small sizes to reflect the tensile strength heterogeneities existing along the fracture surface [15]. For this plane-strain model, the strain in the out-ofplane direction is assumed to be zero and the fracture should be visualised as extending uniformly a significant distance in this direction.

### **2. The model**

The basic governing equations and the boundary conditions are provided in our previous work (see References [12,13,16]), dealing with the hydraulic fracture model that fully couples mechanisms of rock deformation and viscous fluid flow. The basis of the model is briefly described here for the sake of completeness.

We allow for the fracture surfaces to be rough and tortuous, which imparts a hydraulic aperture for the closed fracture allowing fluid flow, but causing no stress and deformation changes. Fluid volumetric flux *q* in a closed fracture segment is described by:

$$q = -\frac{\varpi^3}{\mu'} \frac{\partial p\_f}{\partial \mathbf{s}} \tag{1}$$

<sup>3</sup> () ()

1

c m

its contribution to fluid flow is retained as provided in Eq. (3).

*n*

 s

 t

s

*s*

t

portions, we have *σ<sup>n</sup>* = *pf* .

*<sup>∞</sup>* , *σyy*

The global mass balance requires

*<sup>∞</sup>* and *σxy*

to the plane strain Young's modulus, *E'*, where *<sup>E</sup>* '= *<sup>E</sup>*

denoted as *σxx*

v

*<sup>p</sup> w w <sup>f</sup> ts s*

12

2

paper. When fracture surfaces are separated, the change of the hydraulic aperture ceases, but

but fluid flow in the closed fracture segment is based on the pressure diffusion equation

<sup>1</sup> ( )0 *f f p p t ss* v

where *χ*<sup>1</sup> is the compressibility of the fracture with units of Pa-1 and it is set as 10<sup>−</sup><sup>8</sup>

The nonlocal elastic equilibrium equations for a system of N fractures are given as:

<sup>1</sup> <sup>11</sup> <sup>12</sup> <sup>0</sup> <sup>1</sup>

*r*


*xx x x*

*<sup>N</sup> <sup>l</sup>*

åò

*r <sup>N</sup> <sup>l</sup>*

=

*r*

=

åò

*r*


*xx x x*

( , ) ( ) [ ( , ) ( , ) ( , ) ( , )]

( , ) ( ) [ ( , ) ( , ) ( , ) ( , )]

*t G s w s t G s v s t ds*

*t G s w s t G s v s t ds*

where the coordinates of a location of a material point are denoted as *x* =(*x*, *y*) in a twodimensional Cartesian reference framework, *t* is time and *v* is the shear displacement discon‐ tinuity, ℓ*r*is the length of fracture *r*. *σn*is the normal stress and *τs* is the shear stress carried by the fracture because of its frictional strength, obeying Coulomb's frictional law characterized by the coefficient of friction *λ*, which limits the shear stress | *τ<sup>s</sup>* | ≤*λσn*, that can act in parts of fractures that are in contact, but vanishes along the separated parts. Along the opened fracture

In addition, σ1 and τ1 are the normal and shear stresses, respectively, along the fracture direction at location x caused by the far-field stress, whose normal and shear components are

> <sup>0</sup> <sup>0</sup> ( ) *<sup>f</sup> <sup>l</sup>* åò *w ds Q t* + = v

And the fluid front in the hydraulic fracture will be, in general, not coincident with the fracture tip. The fluid front location is found by using Eqs. (1) and (2) with the following equation

*<sup>∞</sup>* . *G*ij are the hypersingular Green's functions, which are proportional

(6)

(1 − *ν* <sup>2</sup> ) .

<sup>1</sup> <sup>21</sup> <sup>22</sup> <sup>0</sup> <sup>1</sup>

 v

m

æ ö ç ÷ ç ÷ è ø

Development of Fracture Networks Through Hydraulic Fracture Growth in Naturally Fractured Reservoirs

¶ ¶+ ¶ + <sup>=</sup> ¶¶ ¶ (3)

¶ ¶ ¶ - = ¶ ¶¶ ¢ (4)

Pa-1 in this

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143

(5)

and in an opened fracture portion it is

$$q = -\frac{(w+\varpi)^3}{\mu'} \frac{\partial p\_f}{\partial s} \tag{2}$$

where *μ* ′ =12*μ* with *μ* being fluid dynamic viscocity. *w* and *ϖ* are mechanical opening and hydraulic aperture along the fracture surface and are functions of time and location. The former is determined by the stress condition given below, but the latter obeys an evolution equation linearly proportional to fluid pressure change [13]. It is noted that the initial value of hydraulic aperture is denoted as *ϖ*0, which is a reflection of the fracture surface roughness and tortuosity.

Fluid flow in the opened fracture portion is based on the lubrication equation:

Development of Fracture Networks Through Hydraulic Fracture Growth in Naturally Fractured Reservoirs http://dx.doi.org/10.5772/56405 143

$$\frac{\partial(\mathcal{w}+\boldsymbol{\sigma})}{\partial t} = \frac{\partial}{\partial \mathbf{s}} \left( \frac{(\boldsymbol{w}+\boldsymbol{\sigma})^3}{12\,\mu} \frac{\partial p}{\partial \mathbf{s}} \boldsymbol{f} \right) \tag{3}$$

but fluid flow in the closed fracture segment is based on the pressure diffusion equation

$$\frac{\partial p\_f}{\partial t} - \frac{1}{\chi\_1 \mu'} \frac{\partial}{\partial s} (\sigma^2 \frac{\partial p\_f}{\partial s}) = 0 \tag{4}$$

where *χ*<sup>1</sup> is the compressibility of the fracture with units of Pa-1 and it is set as 10<sup>−</sup><sup>8</sup> Pa-1 in this paper. When fracture surfaces are separated, the change of the hydraulic aperture ceases, but its contribution to fluid flow is retained as provided in Eq. (3).

The nonlocal elastic equilibrium equations for a system of N fractures are given as:

$$\begin{aligned} \sigma\_n(\mathbf{x},t) - \sigma\_1(\mathbf{x}) &= \sum\_{r=1}^N \int\_0^{l\_r} [\mathcal{G}\_{11}(\mathbf{x},s)\mathbf{w}(s,t) + \mathcal{G}\_{12}(\mathbf{x},s)\mathbf{v}(s,t)]ds \\ \sigma\_s(\mathbf{x},t) - \tau\_1(\mathbf{x}) &= \sum\_{r=1}^N \int\_0^{l\_r} [\mathcal{G}\_{21}(\mathbf{x},s)\mathbf{w}(s,t) + \mathcal{G}\_{22}(\mathbf{x},s)\mathbf{v}(s,t)]ds \end{aligned} \tag{5}$$

where the coordinates of a location of a material point are denoted as *x* =(*x*, *y*) in a twodimensional Cartesian reference framework, *t* is time and *v* is the shear displacement discon‐ tinuity, ℓ*r*is the length of fracture *r*. *σn*is the normal stress and *τs* is the shear stress carried by the fracture because of its frictional strength, obeying Coulomb's frictional law characterized by the coefficient of friction *λ*, which limits the shear stress | *τ<sup>s</sup>* | ≤*λσn*, that can act in parts of fractures that are in contact, but vanishes along the separated parts. Along the opened fracture portions, we have *σ<sup>n</sup>* = *pf* .

In addition, σ1 and τ1 are the normal and shear stresses, respectively, along the fracture direction at location x caused by the far-field stress, whose normal and shear components are denoted as *σxx <sup>∞</sup>* , *σyy <sup>∞</sup>* and *σxy <sup>∞</sup>* . *G*ij are the hypersingular Green's functions, which are proportional to the plane strain Young's modulus, *E'*, where *<sup>E</sup>* '= *<sup>E</sup>* (1 − *ν* <sup>2</sup> ) .

The global mass balance requires

cases involving both viscous fluid and uniform pressure. Additionally, the program treats rock deformation and fluid flow as a whole in that the field variables are obtained in a single

The purpose of this paper is to provide some initial results for hydraulic fracture growth through a natural fracture network. Fracture growth is allowed and is based on a local failure criterion[14] and fracture coalescence can take place to form a path through an existing network of natural fractures. The numerical treatment of fracture coalescence has been detailed in [12, 13]. The rock mass is assumed to be impermeable and the fluid flow is confined to occur along the pre-existing or newly created fractures. The fracture nucleation sites are embedded as the pre-existing secondary fractures with small sizes to reflect the tensile strength heterogeneities existing along the fracture surface [15]. For this plane-strain model, the strain in the out-ofplane direction is assumed to be zero and the fracture should be visualised as extending

The basic governing equations and the boundary conditions are provided in our previous work (see References [12,13,16]), dealing with the hydraulic fracture model that fully couples mechanisms of rock deformation and viscous fluid flow. The basis of the model is briefly

We allow for the fracture surfaces to be rough and tortuous, which imparts a hydraulic aperture for the closed fracture allowing fluid flow, but causing no stress and deformation changes.

> 3 *f p*

<sup>3</sup> ( ) *<sup>f</sup> w p*

hydraulic aperture along the fracture surface and are functions of time and location. The former is determined by the stress condition given below, but the latter obeys an evolution equation linearly proportional to fluid pressure change [13]. It is noted that the initial value of hydraulic aperture is denoted as *ϖ*0, which is a reflection of the fracture surface roughness and tortuosity.

v

m

Fluid flow in the opened fracture portion is based on the lubrication equation:

*s*

=12*μ* with *μ* being fluid dynamic viscocity. *w* and *ϖ* are mechanical opening and

v

m

*s*

¶ = - ¢ ¶ (1)

<sup>+</sup> ¶ = - ¢ ¶ (2)

Fluid volumetric flux *q* in a closed fracture segment is described by:

*q*

*q*

framework, instead of the one-way coupling scheme as used in Reference [11].

uniformly a significant distance in this direction.

142 Effective and Sustainable Hydraulic Fracturing

described here for the sake of completeness.

and in an opened fracture portion it is

**2. The model**

where *μ* ′

$$\sum \int\_{0}^{l\_f} (w + \varpi) ds = Q\_0 t \tag{6}$$

And the fluid front in the hydraulic fracture will be, in general, not coincident with the fracture tip. The fluid front location is found by using Eqs. (1) and (2) with the following equation

$$\dot{I}\_f = q(l\_{f'}t) / w(l\_{f'}t) \tag{7}$$

**3. A simple test problem**

specified.

Figure 1(d).

**4. Random fracture geometry**

To validate the model on coalescence of fluid flow branches, we present results for one specific case, as shown in Figure 1, where two hydraulic fractures with a spacing 0.8 m driven by the same fluid source (located at the plane with x=-1.0 m on the left of the natural fracture) intersect the natural fracture orthogonally. And the sum of their inflow rates is a constant. A new fracture site is located on the natural fracture that is 2.4 m in length, and it is assumed to be a pre-existing flaw at 0.12 m long. It is anticipated that after intersection, the fluid will invade the natural fracture until it reaches the new fracture in Figure 1(a), which is subject to a less compressive stress. As the new fracture becomes the weakest point to continue crack growth, it will propagate when the fluid pressure reaches a sufficient value, as shown in Figure 1(c). It should be noted that when the middle section of the natural fracture between two hydraulic fractures is filled with fluid, a sudden increase in the pressure occurs as indicated in Figure 1(a) and (b) for two close time steps, in the same way as injection into a closed container. At the time shown in Figure 1(c) the natural fracture appears to be full of fluid, the whole fracture system experiences a similar pressure level, high enough to cause the new fracture to propa‐ gate. The material constants used for this problem are listed in Table. 1 if not otherwise

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The variations of flow rates into each branch with time are provided in Figure 1(d). The two hydraulic fractures reach the natural fracture at time t=6.98 s and the fluid front reaches the new fracture site at *t*=10.85 s. In Figure 1(a), the inflow into the new fracture is only from the top hydraulic fractures up to the onset of coalescence of two fluid branches with the natural fracture at *t*=12.6 s. There is a short-time period where *Q*5 occurs before fluid branch coalescence and this flux is represented by *Q*<sup>4</sup> later on. Subsequently, the value of *Q*6 increases rapidly to a higher level (*Q*<sup>6</sup> / *Q*0=0.8). This implies that most of injected fluid is entering the new fracture and this promotes its growth. The fluid rates at the injection fractures all meet the continuity requirement. The larger value of *Q*3 compared to *Q*4 indicates that the hydraulic fracture closer to the fracture nucleation site is contributing more in fluid flux to sustain the new fracture growth. As for the loss of geometric symmetry, it is interesting to note that the outflux from the top hydraulic fracture is also larger than its counterpart since *Q*<sup>1</sup> >*Q*2 clearly shown in

In this section, we present more complex cases where several high-angle joints are defined in a random distribution ahead of hydraulic fractures that grow from a left entry zone (along the plane x=-1 m) toward the right (through the plane x=2 m). The remote stress conditions and some geometric parameters are provided in the caption of Figure 2. The fracture injection occurs into four individual fractures located on the far left, as shown in Figure 2. The fracture segments coloured green denote the initial existing fracture configuration. Each existing fracture consists of a single fracture or of several connected fractures of different sizes and

The fracture growth is based on using the maximum hoop stress criterion, with the maximum mixed-mode stress intensity factor reaching a critical value [14]

$$\cos\frac{\Theta}{2}(K\_I\cos^2\frac{\Theta}{2} - \frac{3}{2}K\_{II}\sin\Theta) = K\_{lc} \tag{8}$$

where *KI* and *KII* are calculated stress intensity factors, *KIc*is tensile mode fracture toughness and *Θ* is the fracture propagation direction relative to the current fracture orientation. The predicted orientation follows the maximum tensile stress direction, and the near-tip stresses are approximated by the analytical LEFM solutions [14].

The problem must be completed by specifying the imposed boundary conditions at the wellbore, that is, the sum of injection rates of hydraulic fractures connected into the wellbore or to the entry zone, should be equal to the given injection rate *Q*0. At the fracture tip the displacement discontinuities are zero, *w*(ℓ*r*, *t*)=0 and *v*(ℓ*r*, *t*)=0. In addition, the entire system is assumed to initially be stationary and unsaturated.

The numerical scheme for the above nonlinear and nonlocal coupled problem has been detailed in our previous paper [16] based on the Displacement Discontinuity Boundary Element Method. The hydraulic fractures and other geological discontinuities like joints and faults, and natural fractures are discretised with constant displacement elements. The model solves the hydraulic fracture problem simultaneously including the effects of viscous fluid flow and coupled rock deformation. The solutions are consistent with existing results. Also, a fracture can intersect another one in its path and a new fracture can be nucleated from a position on the natural fracture. In particular, the new fracture seeds are pre-defined in this paper along some natural fractures. The interested reader is referred to our previous papers for the details on implementation of fracture growth and coalescence [12,16]. One important check here is the satisfication of fluid mass conservation after fluid branch coalescence, due to redirection of newly-created fractures.


**Table 1.** Material properties
