**4. Random fracture geometry**

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

Of course, some numerical difficulties exist in the simulation of the process of connecting two intersecting fractures [16]. A mesh sensitivity analysis has been carried out so that the change in fracture orientation does not affect the numerical accuracy as the fracture direction has to be such that the intersecton occurs at the the end of an element on the fracture. A fracture connection event that results in a strong deflection from the calculated direction can influence the subsequent fluid flow. In these cases, the natural fractures must be re-meshed and the job must be run again. In the numerical simulation, the individual element size in the vicinity of intersection point is chosen with care, so that the fracture connection can be completed as a

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For the sake of convenience in discussion, the above two injection types are respectively named as type I and type II injection. Figure 2 shows the results at the time of breakthrough when the hydraulic fracture emerges on the right side (x=2 m) of the network of natural fractures, for these two injection types. After fracture reorientation to the direction normal to the minimum principal stess, the fracture development through the network zone is complete. As expected, the type I injection condition results in more fracture growth paths and intersections across the network, while the type II condition results in a localized path, with only one hydraulic fracture continuing to grow past the network on the right side. In Figure 2(b) there are two separate unconnected sets of fractures from top to bottom, both of which connect to two of the initial fractures on the left which are connected to (in this case) the equal pressure fluid source. The influxes into the bottom fracture set decrease in time and more of the total volume injected enters the upper fracture set. The upper fracture set is more conductive. We define the plane *x*>2 m as the exit zone. There is one outlet fracture in this case. However, there are two fluid outlets or extraction sites in Figure 2(a) for type I injection. On the other hand, for the type II injection, the new fracture segments are mainly created along the upper fracture path.

Figure 3 shows the outlet flux variations in time for the two cases used in Figure 2. For type I injection, only around 43 percent of injected fluid has passed through the fracture system to the outlet at the end of simulation period and the other 57 percent is contained in fracture branch inflation or growth in the network. It is also found in Figure 3 that in this case the outlet flux (sum of outlet 1 and outlet 2 for case (a)) increases at a very slow rate at the large time. More fractures under type I injection are connected with each other as a result of crack growth near the fluid source and some fractures continue to grow after the main conductive channels are developed. Fracture segments not opened also store some fluid because of their initial conductivity. The rate that fluid is stored in the fracture network differs between the type I and type II injection conditions. For type II injection, more than 55 percent of injected fluid exits the outlet fracture at the end of the simulation and this outlet flux is increasing at a very large rate as shown in Figure 3 so that the rate of fluid volume stored in the system will become

In addition, the breakthrough time for these two injection types is provided in this figure. The type II injection has a much earlier breakthrough time so that the fracture growth through the

extremely small soon in light of this trend.

fracture network is more rapid.

smooth path.

**Figure 1.** The pressure profiles at three specific time instants: (a) t=12.6 s, (b) t=13.08s and (c) t=18.5 s and (d) the evolution of influxes around the intersection points and the fracture nucleation site for the case of one natural frac‐ ture and two hydraulic fractures that approach the natural fracture at a right angle. The fracture nucleation site on the natural fracture is located at 0.2 m above the mid plane of the two hydraulic fractures. The applied stress components are σ*xx* <sup>∞</sup> =5 MPa, σ*yy* <sup>∞</sup> =4 MPa and σ*xy* <sup>∞</sup> =0. The initial hydraulic aperture 0 for the natural fracture is 0.03 mm.

orientations. The hydraulic fractures driven by pressure act to connect these separate fractures to form a conductive path from left to right and the newly created fractures are coloured in red in Figure 2. Two different injection boundary conditions are used in the computations. One is associated with even distribution of the injection rate into the four entry fractures on the left, and the second condition specifies that the pressure at each of these four initial fractures is equal. The latter condition is physically reasonable if we assume a wellbore lies along the yaxis of Figure 2, while the former condition would require isolation and injection into each fracture at the same rate.

Of course, some numerical difficulties exist in the simulation of the process of connecting two intersecting fractures [16]. A mesh sensitivity analysis has been carried out so that the change in fracture orientation does not affect the numerical accuracy as the fracture direction has to be such that the intersecton occurs at the the end of an element on the fracture. A fracture connection event that results in a strong deflection from the calculated direction can influence the subsequent fluid flow. In these cases, the natural fractures must be re-meshed and the job must be run again. In the numerical simulation, the individual element size in the vicinity of intersection point is chosen with care, so that the fracture connection can be completed as a smooth path.

For the sake of convenience in discussion, the above two injection types are respectively named as type I and type II injection. Figure 2 shows the results at the time of breakthrough when the hydraulic fracture emerges on the right side (x=2 m) of the network of natural fractures, for these two injection types. After fracture reorientation to the direction normal to the minimum principal stess, the fracture development through the network zone is complete. As expected, the type I injection condition results in more fracture growth paths and intersections across the network, while the type II condition results in a localized path, with only one hydraulic fracture continuing to grow past the network on the right side. In Figure 2(b) there are two separate unconnected sets of fractures from top to bottom, both of which connect to two of the initial fractures on the left which are connected to (in this case) the equal pressure fluid source. The influxes into the bottom fracture set decrease in time and more of the total volume injected enters the upper fracture set. The upper fracture set is more conductive. We define the plane *x*>2 m as the exit zone. There is one outlet fracture in this case. However, there are two fluid outlets or extraction sites in Figure 2(a) for type I injection. On the other hand, for the type II injection, the new fracture segments are mainly created along the upper fracture path.

Figure 3 shows the outlet flux variations in time for the two cases used in Figure 2. For type I injection, only around 43 percent of injected fluid has passed through the fracture system to the outlet at the end of simulation period and the other 57 percent is contained in fracture branch inflation or growth in the network. It is also found in Figure 3 that in this case the outlet flux (sum of outlet 1 and outlet 2 for case (a)) increases at a very slow rate at the large time. More fractures under type I injection are connected with each other as a result of crack growth near the fluid source and some fractures continue to grow after the main conductive channels are developed. Fracture segments not opened also store some fluid because of their initial conductivity. The rate that fluid is stored in the fracture network differs between the type I and type II injection conditions. For type II injection, more than 55 percent of injected fluid exits the outlet fracture at the end of the simulation and this outlet flux is increasing at a very large rate as shown in Figure 3 so that the rate of fluid volume stored in the system will become extremely small soon in light of this trend.

orientations. The hydraulic fractures driven by pressure act to connect these separate fractures to form a conductive path from left to right and the newly created fractures are coloured in red in Figure 2. Two different injection boundary conditions are used in the computations. One is associated with even distribution of the injection rate into the four entry fractures on the left, and the second condition specifies that the pressure at each of these four initial fractures is equal. The latter condition is physically reasonable if we assume a wellbore lies along the yaxis of Figure 2, while the former condition would require isolation and injection into each

**Figure 1.** The pressure profiles at three specific time instants: (a) t=12.6 s, (b) t=13.08s and (c) t=18.5 s and (d) the evolution of influxes around the intersection points and the fracture nucleation site for the case of one natural frac‐ ture and two hydraulic fractures that approach the natural fracture at a right angle. The fracture nucleation site on the natural fracture is located at 0.2 m above the mid plane of the two hydraulic fractures. The applied stress components

(a) (b)

1 **Time = 18.52 seconds**

**x (m)**

1 **Time = 12.6 seconds**

146 Effective and Sustainable Hydraulic Fracturing

**10 MPa**

**10 MPa**

**y (m)**


*Qi*

0.0

<sup>∞</sup> =0. The initial hydraulic aperture 0 for the natural fracture is 0.03 mm.

0.2

0.4

0.6

*Q2*

*Q1*

0.8

1.0

/*Q0*


0

0.5

**x (m)**

*Q3*

*Q6*

*Q5*

*Q4 Q5*

t=6.98 s t=10.85 s

1 **Time = 13.08 seconds**

**10 MPa**


Time (s) 6 8 10 12 14 16 18 20 22 24

t=12.7 s

*Q3 Q2*

*Q6*

*Q4*

*Q1*


**y (m)**


**y (m)**


are σ*xx*

<sup>∞</sup> =5 MPa, σ*yy*


0

0.5


0

0.5

(c) (d)


**x (m)**

<sup>∞</sup> =4 MPa and σ*xy*

fracture at the same rate.

In addition, the breakthrough time for these two injection types is provided in this figure. The type II injection has a much earlier breakthrough time so that the fracture growth through the fracture network is more rapid.

Time (s) 30 40 50 60 70 80 90 100

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

**Figure 3.** Evolution of normalised outlet flux that is defined by the entrance flow rate of the outlet fractures by the injection rates for two cases provided in Figure 2. There are two outlets for case (a) and they are numbered from the

The rapidly increasing trend in outlet flux is also reflected in the opening profiles as shown in Figure 4(a) with the wider open fracture path corresponding to the path that carries the most fluid. The wider fracture channels are localised along the preferential pathway along the upper fracture set to the right-top fracture outlet under type II injection. It is found that at the given time instant, up to 85 percent of the injected fluid is pumped into this preferential path and the fractures not included in this path are all static at late times in the simulation. At the larger times simulated, most fluid just passes through this highly conductive channel to reach the exit zone. Thus, the localised flow channel provides the lowest resistance pathway for fluid flow. Here, we note the contribution of residual hydraulic aperture on the fluid movement and storage. As stated above, each natural fracture has a pre-existing aperture of 0.01 mm in the

computations. Thus, some fluid enters and is stored in this pre-existing aperture.

It is found that fractures with wider opening have had their opening enhanced by the large slip along the longest oblique natural fracture, which is oriented at 45 degrees to the x-axis, as shown in Figure 4(b). The kinematic transfer between slip and opening assists the fracture

outlet 1 for Type II

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149

outlet 2 for Type I

outlet 1 for Type I

Q out /Q

opening.

0.0

0.1

0.2

0.3

0.4

breakthrough

top to the bottom. For case (b) there is only one outlet at the top.

0.5

0.6

0

**Figure 2.** Fracture pathways at the time of breakthrough for two different injection conditions: (a) Type I and (b) Type II. The fracture system is subject to the applied stresses σ*xx* <sup>∞</sup>=6 MPa, σ*yy* <sup>∞</sup>=4 MPa and σ*xy* <sup>∞</sup>=0. The initial hydraulic aperture <sup>0</sup> for all natural fractures is 0.01 mm. The fluid comes into the area from four entry fractures at the left (x=-1 m) and its pressure drives some fractures to propagate and connect the separated fractures as the hydraulic fractures grow to‐ ward the right outlet zone (x=2 m). The initial fracture configuration is shown in green and the generated fracture segments are in red.

(a)

**y(m)**



0

0.5

1

1.5

148 Effective and Sustainable Hydraulic Fracturing

**y(m)**

**x (m)**

**x (m)**

<sup>∞</sup>=6 MPa, σ*yy*

<sup>∞</sup>=4 MPa and σ*xy*

<sup>∞</sup>=0. The initial hydraulic aperture


**Figure 2.** Fracture pathways at the time of breakthrough for two different injection conditions: (a) Type I and (b) Type

<sup>0</sup> for all natural fractures is 0.01 mm. The fluid comes into the area from four entry fractures at the left (x=-1 m) and its pressure drives some fractures to propagate and connect the separated fractures as the hydraulic fractures grow to‐ ward the right outlet zone (x=2 m). The initial fracture configuration is shown in green and the generated fracture


(b)

segments are in red.


II. The fracture system is subject to the applied stresses σ*xx*


0

0.5

1

1.5

**Figure 3.** Evolution of normalised outlet flux that is defined by the entrance flow rate of the outlet fractures by the injection rates for two cases provided in Figure 2. There are two outlets for case (a) and they are numbered from the top to the bottom. For case (b) there is only one outlet at the top.

The rapidly increasing trend in outlet flux is also reflected in the opening profiles as shown in Figure 4(a) with the wider open fracture path corresponding to the path that carries the most fluid. The wider fracture channels are localised along the preferential pathway along the upper fracture set to the right-top fracture outlet under type II injection. It is found that at the given time instant, up to 85 percent of the injected fluid is pumped into this preferential path and the fractures not included in this path are all static at late times in the simulation. At the larger times simulated, most fluid just passes through this highly conductive channel to reach the exit zone. Thus, the localised flow channel provides the lowest resistance pathway for fluid flow. Here, we note the contribution of residual hydraulic aperture on the fluid movement and storage. As stated above, each natural fracture has a pre-existing aperture of 0.01 mm in the computations. Thus, some fluid enters and is stored in this pre-existing aperture.

It is found that fractures with wider opening have had their opening enhanced by the large slip along the longest oblique natural fracture, which is oriented at 45 degrees to the x-axis, as shown in Figure 4(b). The kinematic transfer between slip and opening assists the fracture opening.

**Figure 5.** Evolution of injection pressure at the left entry zone for the type II injection method.

indicated in Figure 3.

Also, it is found that some fracture connections are made even after the preferential fluid pathway is established, since fracture growth can still be generated in the higher pressure region near the entry. After the breakthrough, the injection pressure has to be retained at a higher level, to force fluid through width restrictions that occur at intersections and offsets, as shown in Figure 5. The continuing fracture growth in the network can alter the earlier opening distributions. Comparing Figure 4 and Figure 2(b), it is clear that fracture growth and fracture interconnection can occur inside the network region after the flow breakthrough. Of course, one reason for pressure increasing after breakthrough is attributed to the strong resistance to fluid flow at the last vertical natural fracture on the connected flow path for this specific geometry. This vertical fracture provides the strongest barrier to fluid flow as its opening is highly constrained by the geometry, with the approaching hydraulic fracture making a right angle to the natural fracture. A new fracture would possibly be nucleated at some location along this vertical natural fracture which would reduce this restriction. Although Figure 5 shows the increasing trend of the injection pressure, the pressure will eventually level off or even decrease as the outflux from the network increases to 100 percent of the input rate, as

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151

**Figure 4.** Fracture trajectories and opening (a) and slip (b) profiles, which are represented by the blue bars perpendic‐ ular to the fractures, under type II injection at a specific time.

**Figure 5.** Evolution of injection pressure at the left entry zone for the type II injection method.

(a)

**y (m)** -1


0

0.5

1

150 Effective and Sustainable Hydraulic Fracturing

**y (m)**

**x (m)**

**x (m)**

**Figure 4.** Fracture trajectories and opening (a) and slip (b) profiles, which are represented by the blue bars perpendic‐



**Time = 95.53 secomds**

**Time = 95.53 secomds**

**0.5 mm**

**0.5 mm**

(b)


ular to the fractures, under type II injection at a specific time.


0

0.5

1

Also, it is found that some fracture connections are made even after the preferential fluid pathway is established, since fracture growth can still be generated in the higher pressure region near the entry. After the breakthrough, the injection pressure has to be retained at a higher level, to force fluid through width restrictions that occur at intersections and offsets, as shown in Figure 5. The continuing fracture growth in the network can alter the earlier opening distributions. Comparing Figure 4 and Figure 2(b), it is clear that fracture growth and fracture interconnection can occur inside the network region after the flow breakthrough. Of course, one reason for pressure increasing after breakthrough is attributed to the strong resistance to fluid flow at the last vertical natural fracture on the connected flow path for this specific geometry. This vertical fracture provides the strongest barrier to fluid flow as its opening is highly constrained by the geometry, with the approaching hydraulic fracture making a right angle to the natural fracture. A new fracture would possibly be nucleated at some location along this vertical natural fracture which would reduce this restriction. Although Figure 5 shows the increasing trend of the injection pressure, the pressure will eventually level off or even decrease as the outflux from the network increases to 100 percent of the input rate, as indicated in Figure 3.
