**4.1. Dimensionless toughness** *κIC* **vs. angle** *β* **diagram**

Fig. 4 shows the results of a large number of numerical computations performed with various independent parameters of the problem. They are projected on the diagram of dimensionless toughness versus interaction angle. These results represent the final outcome of the HF-NF interaction in terms of either fracture crossing or arresting at the NF. We observe that the HF crosses the NF predominantly at large angles of intersection (i.e., close to 90°) and smaller values of dimensionless toughness as expected. Note that the dimensionless toughness *κIC* we used for plotting depends not only on the Mode I toughness but also on the injection rate and fluid viscosity, as seen from Eqs. (22-23). If the fluid viscosity *μ* or the flow rate at the fracture inlet *Q* increases, the value of the dimensionless toughness decreases and the crossing behavior becomes more favorable.

The computations have been performed for several values of in-situ rock stress difference *ΔΣ*. To denote this important parameter in the plotted results, we have used different colors for different values of *ΔΣ*. To separate the crossing or arrest tendency at small stress contrast from the similar tendency at large stress contrast, we have presented the results (Fig. 4) in two diagrams for the small and large values of *ΔΣ* respectively. After using such a representation of the results, one can still observe that some points remained nonseparated and overlapped. For example, in the bottom diagram, the results from the computations with different stress contrast coincide at the same point in the angle-toughness diagram. They can have opposite outcome as shown by arrows at the bottom of Fig. 4.

#### **4.2.** *κIC* **vs.***ΔΣ* **diagram**

After the scaling we obtain nine dimensionless parameters

also note the parameters have nonequal sensitivity to the result of fracture interaction. For example, all previous theoretical and experimental studies have shown that the significant effect of the relative difference of the applied stresses can be written as *ΔΣ* =(*Σ*<sup>1</sup> −*Σ*2)/ *Σ*2, rather than as the mean stress. The effect of NF compliance and the difference between Mode I and Mode II NF toughness can be neglected in the first attempt of parametric study, so we assume

This study is mainly intended for the oil and gas industry, so the comprehensive numerical study in infinite limits of these parameters in the present study is not necessary. In what follows we limit the range of the dimensionless parameter values to the practical range by use of the compilation of dimensional parameters of the problem shown in Table 1. Using the scaling introduced previously it is possible to calculate the corresponding range of the dimensionless

> **Parameter Minimum Value Maximum Value** κIC 0.002 11 κIC (NF)/ κIC (HF) 0 0.5 ΔΣ 0 2 Ω<sup>h</sup> 0.0001 0.37 β 30° 90° λ 0.2 1

The following parametric study is restricted to the range of values of dimensionless parameters

We performed numerical simulations of the HF propagation and interaction with the preex‐ isting NF by use of the modified CSIRO code (with stress-and-energy initiation criterion) [12, 20]. Dimensionless parameters of the simulations have been selected within the range specified in Table 2. Systematic analysis of the obtained results of HF-NF interaction allowed us to project them onto the specific parametric diagrams containing dimensionless toughness of the rock, fracture intersection angle, stress contrast, frictional coefficient. and relative toughness of the NF. For better representation they are divided into several cross sections of one global

, *Σ*1, *Σ*2, *Ω<sup>h</sup>* , *β*, *λ*, *χ*) that characterize our dimensionless problem. We

. Consequently, we restrict our parametric analysis with the following six

(*κIC*, *κIC*

here *κIC* (*NF* ) =*κIIC* (*NF* )

(NF) , *κIIC* (NF)

parameters { *κIC*, *ΔΣ*, *β*, *λ*, *κIC*

166 Effective and Sustainable Hydraulic Fracturing

parameters (Table 2).

(*NF* )

, *Ω<sup>h</sup>* }.

**Table 2.** Range of dimensionless parameters calculated from Table 1 using introduced scaling

shown in Table 2, as practically required.

**4. Results of parametric study**

multiparametrical cube.

The next series of parametric diagrams represent the results of numerical experiments in the dimensional toughness vs. stress contrast cross section. This representation better emphasizes the role of the dimensionless relative stress and with the flow rate and viscosity that are inversely proportional to the dimensionless toughness. Plots in Fig. 5 clearly show that the higher relative stress difference *ΔΣ*, the larger the region of the dimensionless toughness values where fracture crossing occurs. The highest threshold value of the dimensionless toughness for crossing behavior increases with the stress contrast; i.e., in the formations with high stress contrast the crossing behavior will most probably happen even with low viscosity and/or low pumping rate during fracturing jobs. At the same time, it is unlikely that the rock with relatively small stress difference *ΔΣ* <1 will support fracture crossing unless the dimensionless tough‐ ness value is sufficiently decreased by stronger injection power (larger pumping rates or higher fluid viscosity). This is especially the case if the preferential inclination of the NFs with respect to the orientation of the HF is far from 90°.

#### **4.3.** *λ* **vs.** *κIC***(NF) /** *κIC* **diagram**

Next we analyzed the influence of the friction coefficient *λ* and the relative NF Mode I toughness *κIC*(NF) /*κIC* on the result of HF-NF interaction (Fig. 6). In this series of numerical tests we observe a sharp boundary for the friction coefficient values where the crossing behavior

1

2 3

ceases. We also observed that there is no influence of the relative NF toughness, compared to that of the friction, although this parameter deserves more attention in the future.

10 Book Title

0.002

0.1 1.0 10.0

**Figure 5.** Crossing vs. arresting diagram in (κ*IC*, ΔΣ) for different angles of fracture interaction. κ*IC* (NF) = 0.

**Relative stress contrast,** '**�=(�1��2)/�2**

**��= 90**

crossing, k = 2e�10,���= 90 crossing, k =8e�8,���= 90 crossing, k = 4e�7,���= 90 crossing, k =1e�6,���= 90 crossing, k = 2e�6,���= 90 crossing, k = 4e�6,���= 90 crossing, k = 1e�5,���= 90 crossing, k = 3e�5,���= 90 crossing, k = 8e�5,���= 90 crossing, k = 1e�4,���= 90 crossing, k = 2e�4,���= 90 crossing, k = 3e�4,���= 90 crossing, k = 4e�4,���= 90 crossing, k = 5e�4,���= 90 no crossing, k = 2e�6,���= 90 no crossing, k = 4e�6,���= 90 no crossing, k = 1e�4,���= 90 no crossing, k = 2e�4,���= 90 no crossing, k = 3e�4,���= 90 no crossing, k = 5e�4,���= 90

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169

Hydraulic Fracture Propagation Across a Weak Discontinuity Controlled by Fluid Injection

0.02

**Dimensionless**

0.2

**toughness,** N**IC** 2

**Figure 4.** Crossing vs. arresting diagram in (κ*IC*, β) for different range of relative stress contrast. κ*IC* (NF) = 0.

Hydraulic Fracture Propagation Across a Weak Discontinuity Controlled by Fluid Injection http://dx.doi.org/10.5772/55941 169

ceases. We also observed that there is no influence of the relative NF toughness, compared to

10 Book Title

**0<=ΔΣ<1** crossing,ΔΣ=0.6, k=1e‐4

crossing,ΔΣ=0.6, k=2e‐4 crossing,ΔΣ=0.6, k=3e‐4 crossing,ΔΣ=0.6, k=4e‐4 crossing,ΔΣ=0.2, k=6e‐7 crossing,ΔΣ=0.2, k=2e‐6 crossing,ΔΣ=0, k=2e‐6 no crossing,ΔΣ=0.9, k=4e‐7 no crossing,ΔΣ=0.9, k=1e‐6 no crossing, ΔΣ=0.6,k=1e‐4 no crossing, ΔΣ=0.6,k=2e‐4 no crossing, ΔΣ=0.6,k=3e‐4 no crossing, ΔΣ=0.2,k=3e‐6 no crossing, ΔΣ=0.2,k=2e‐5 no crossing, ΔΣ=0.2,k=2e‐4 no crossing, ΔΣ=0.2,k=5e‐4 no crossing, ΔΣ=0.17,k=2e‐6 no crossing, ΔΣ=0.1,k=2e‐4 no crossing, ΔΣ=0.1,k=5e‐4

crossing,ΔΣ=1, k=2e‐6 crossing,ΔΣ=1.5, k=2e‐4 crossing,ΔΣ=1.5, k=5e‐4 crossing,ΔΣ=1.6, k=2e‐4 crossing,ΔΣ=1.8, k=4e‐7 crossing,ΔΣ=1.8, k=1e‐6 crossing,ΔΣ=3.0, k=2e‐4 no crossing, ΔΣ=1,k=2e‐6 no crossing, ΔΣ=1.5,k=2e‐4 no crossing, ΔΣ=1.5,k=5e‐4 no crossing, ΔΣ=1.6,k=2e‐4 no crossing, ΔΣ=1.8,k=4e‐7 no crossing, ΔΣ=1.8,k=1e‐6 no crossing,ΔΣ=3.0, k=2e‐4

(NF) = 0.

that of the friction, although this parameter deserves more attention in the future.

30 40 50 60 70 80 90

**β**

**ΔΣ=1‐3**

30 40 50 60 70 80 90

**β**

**Figure 4.** Crossing vs. arresting diagram in (κ*IC*, β) for different range of relative stress contrast. κ*IC*

1

0.002

0.002

0.02

**Dimensionless**

**toughness,**

**IC**

0.2

2

0.02

**Dimensionless**

**toughness,**

**IC**

168 Effective and Sustainable Hydraulic Fracturing

0.2

2

2 3

**Figure 5.** Crossing vs. arresting diagram in (κ*IC*, ΔΣ) for different angles of fracture interaction. κ*IC* (NF) = 0.

Figure 6. Crossing vs. arresting diagram in (*λ, κIC*(NF)/*κIC*). **Figure 6.** Crossing vs. arresting diagram in (λ, κ*IC*(NF)/κ*IC*).

**5. Effect of NF permeability** 

#### **5.1. Analytical model 5. Effect of NF permeability**

#### of the fracture interaction outcome on the permeability of the NF. To accomplish the goal of the parametric study, we built an analytical model of the T-shape contact between the HF and the permeable NF with the following assumptions (Fig. 7). **5.1. Analytical model**

The numerical investigation of the HF-NF interaction with the help of MineHF2D code did not allow us to extract the dependency of the fracture interaction outcome on the permeability of the NF. To accomplish the goal of the parametric study, we built an analytical model of the Tshape contact between the HF and the permeable NF with the following assumptions (Fig. 7).

The numerical investigation of the HF-NF interaction with the help of MineHF2D code did not allow us to extract the dependency

Consider the HF in a T-shape contact with a permeable NF as schematically shown in Fig. 7. Both fractures have uniformly distributed but different hydraulic openings. After the contact, the fracturing fluid penetrates the NF from the tip of the HF. Representation of the HF with blunted tip of width *w* is realistic when the HF contacts a frictionally weak NF [25]. The NF remains mechanically closed with finite hydraulic conductivity described by the residual hydraulic opening *wh* . It is assumed that *wh* does change with time and the NF remains mechanically closed all the time. We also neglect fluid lag in the HF and solve the problem assuming that the injected fluid entirely fills the HF right after contact. The fracturing fluid penetrates the NF from the junction point symmetrically on both sides along the NF (Fig. 7).

Hydraulic Fracture Propagation Across a Weak Discontinuity Controlled by Fluid Injection

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171

First, we prescribe the uniform distribution of the fluid pressure along the HF from the inlet

where *p*<sup>0</sup> is the fluid pressure at the inlet. The approximate solution of the elasticity equation

*L wt p t <sup>E</sup>* = -

Eqns. (6–7). The fluid flow rate at the junction point is denoted as

0 2 <sup>4</sup> () [ () ] '

Poiseuille's law and continuity equation for the fluid flow *q*(*x,t*) along the HF are described by

As long as *w* =*wh* =const along the NF, the Poiseuille's law and continuity equation can be

<sup>2</sup> 3

12 ' *<sup>f</sup> <sup>h</sup> <sup>w</sup> dp <sup>q</sup> dx dq dx*

where *q*<sup>2</sup> is the fluid flow rate along the NF on one side of the junction point (see Fig. 7), and *pf*2 (*x',t*) is the distribution of fluid pressure in the NF. As the fluid penetrates the NF symmet‐

m

2 '

2

ì <sup>ï</sup> = - <sup>ï</sup> í ï ï î

rically from the junction point, using (26) we can write

s

<sup>0</sup> ( ,) () *<sup>f</sup> p xt p t* = (24)

*qx L q* ( )\* = = (26)

(25)

(27)

point to the contact point:

for the opening of the HF *w* can be written as

where *σ*2 is the minimum far-field stress.

written for the fluid flow along the NF as

First, we prescribe the uniform distribution of the fluid pressure along the HF from the inlet point to the contact point:

where *p*0 is the fluid pressure at the inlet. The approximate solution of the elasticity equation for the opening of the HF *w* can be

<sup>0</sup> ( ,) () *<sup>f</sup> p xt p t* (24) **Figure 7.** Illustration of the HF-NF interaction scheme in analytical model

0 2 <sup>4</sup> () [ () ] '

(25)

written as

*<sup>L</sup> wt p t <sup>E</sup>*

Consider the HF in a T-shape contact with a permeable NF as schematically shown in Fig. 7. Both fractures have uniformly distributed but different hydraulic openings. After the contact, the fracturing fluid penetrates the NF from the tip of the HF. Representation of the HF with blunted tip of width *w* is realistic when the HF contacts a frictionally weak NF [25]. The NF remains mechanically closed with finite hydraulic conductivity described by the residual hydraulic opening *wh* . It is assumed that *wh* does change with time and the NF remains mechanically closed all the time. We also neglect fluid lag in the HF and solve the problem assuming that the injected fluid entirely fills the HF right after contact. The fracturing fluid penetrates the NF from the junction point symmetrically on both sides along the NF (Fig. 7).

First, we prescribe the uniform distribution of the fluid pressure along the HF from the inlet point to the contact point:

$$p\_f(\mathbf{x}, t) = p\_0(t) \tag{24}$$

where *p*<sup>0</sup> is the fluid pressure at the inlet. The approximate solution of the elasticity equation for the opening of the HF *w* can be written as

$$w(t) = \frac{4L}{E} \left[ p\_0(t) - \sigma\_2 \right] \tag{25}$$

where *σ*2 is the minimum far-field stress.

Figure 6. Crossing vs. arresting diagram in (*λ, κIC*(NF)/*κIC*).

0 0.2 0.4 0.6 0.8 1 1.2

**β = 90 IC =0.2**

**IC(NF) /IC**

The numerical investigation of the HF-NF interaction with the help of MineHF2D code did not allow us to extract the dependency of the fracture interaction outcome on the permeability of the NF. To accomplish the goal of the parametric study, we built an analytical model of the Tshape contact between the HF and the permeable NF with the following assumptions (Fig. 7).

Figure 7. Illustration of the HF-NF interaction scheme in analytical model

point symmetrically on both sides along the NF (Fig. 7).

<sup>0</sup> ( ,) () *<sup>f</sup> p xt p t* (24)

*<sup>L</sup> wt p t <sup>E</sup>*

0 2 <sup>4</sup> () [ () ] '

**Figure 7.** Illustration of the HF-NF interaction scheme in analytical model

(25)

written as

The numerical investigation of the HF-NF interaction with the help of MineHF2D code did not allow us to extract the dependency of the fracture interaction outcome on the permeability of the NF. To accomplish the goal of the parametric study, we built an

crossing no crossing

Consider the HF in a T-shape contact with a permeable NF as schematically shown in Fig. 7. Both fractures have uniformly distributed but different hydraulic openings. After the contact, the fracturing fluid penetrates the NF from the tip of the HF. Representation of the HF with blunted tip of width ݓ is realistic when the HF contacts a frictionally weak NF [25]. The NF remains mechanically closed with finite hydraulic conductivity described by the residual hydraulic opening ݓ. It is assumed that ݓ does change with time and the NF remains mechanically closed all the time. We also neglect fluid lag in the HF and solve the problem assuming that the injected fluid entirely fills the HF right after contact. The fracturing fluid penetrates the NF from the junction

where *p*0 is the fluid pressure at the inlet. The approximate solution of the elasticity equation for the opening of the HF *w* can be

First, we prescribe the uniform distribution of the fluid pressure along the HF from the inlet point to the contact point:

analytical model of the T-shape contact between the HF and the permeable NF with the following assumptions (Fig. 7).

**5. Effect of NF permeability** 

**5.1. Analytical model** 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

**Figure 6.** Crossing vs. arresting diagram in (λ, κ*IC*(NF)/κ*IC*).

**5. Effect of NF permeability**

**5.1. Analytical model**

**Friction**

**coefficient**

170 Effective and Sustainable Hydraulic Fracturing

Poiseuille's law and continuity equation for the fluid flow *q*(*x,t*) along the HF are described by Eqns. (6–7). The fluid flow rate at the junction point is denoted as

$$
\eta(\mathbf{x} = L) = \mathbf{q}^\* \tag{26}
$$

As long as *w* =*wh* =const along the NF, the Poiseuille's law and continuity equation can be written for the fluid flow along the NF as

$$\begin{cases} q\_2 = -\frac{\pi w\_h^3}{12\mu} \frac{dp\_{f^2}}{d\mathbf{x}^\cdot} \\ \qquad \frac{dq\_2}{d\mathbf{x}^\cdot} \end{cases} \tag{27}$$

where *q*<sup>2</sup> is the fluid flow rate along the NF on one side of the junction point (see Fig. 7), and *pf*2 (*x',t*) is the distribution of fluid pressure in the NF. As the fluid penetrates the NF symmet‐ rically from the junction point, using (26) we can write

$$
\eta\_2(t) = \frac{1}{2} \eta^\*(t) \tag{28}
$$

Rewriting the equations (32) and initial conditions (33) in terms of dimensionless parameters already introduced, and using substitutions *L <sup>f</sup>* = *L γ<sup>f</sup>* , *wh* =*W Ψ<sup>h</sup>* , *p*<sup>0</sup> =*PΠ*0, *t* =*Tτ*, we get

2 0

g

*f h f*

Numerical solution of the problem (34) is plotted in Fig. 8. Fluid penetration length (Fig. 8, left) grows fast at the very beginning (τ << 1) and turns to nearly linear dependence on time when τ > 1. The pressure at the junction point drops right after the HF-NF contact (Fig. 8, right). The velocity of pressure drop rapidly slows, and after the certain inflection point it starts to grow linearly. As it will be shown later, the inflection point depends on the NF permeability

The numerical solution helps in the general solution of the problem (34) but hides the para‐ metrical dependency of the fluid penetration and pressure behavior shown in Fig. 8. To evaluate the pressure inflection point *τ\** and reveal the parametric sensitivity of the velocity of fluid penetration into the NF and the HF pressure dynamics during the contact, we further investigate the asymptotical behavior of the fluid length *γf* and the fluid pressure *Π*0 at the

Running Title <sup>17</sup>

**Figure 8.** Analytical solution of the dimensionless problem for fluid penetration into NF (34) for dimensionless perme‐ ability Ψh = 1, and initial pressure Π00 = 1.5. Left plot is the dimensionless fluid front propagation along the NF in time.

Let us assume the following law for the fluid pressure and penetration length at the beginning

0 00

12 4 (0) 0 (0)

í -Y =P

g

*f h*

<sup>ì</sup> <sup>P</sup> <sup>ï</sup> = Y

*f*

<sup>ï</sup> <sup>=</sup> <sup>ï</sup> ï

îP =P

g

very beginning of fracture contact and at large time after contact.

Right plot is the change of dimensionless pressure at the HF-NF intersection point.

*Early-time (left) asymptote: 0 < τ << τ\*.*

and initial fluid pressure at fracture contact.

1

*f*

2 3

of fracture contact,

g

&

ï ï

0

Hydraulic Fracture Propagation Across a Weak Discontinuity Controlled by Fluid Injection

& & (34)

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173

For steady flow the fluid pressure at the fracture junction point must be the same at the HF and NF sides, so using assumption (24) we write

$$p\_{f2}(\mathbf{x}'=\mathbf{0}) = p\_f(\mathbf{x}=L) \approx p\_0(t) \tag{29}$$

The total fluid mass balance in the system of the HF and the NF can be written by making use of continuity equations (6) and (27), inlet condition (10), and elasticity equation (25) as follows:

$$Q = \frac{d}{dt}(\int\_0^L wdx + 2\int\_0^{L\_f} w\_h dx') = \frac{4L^2}{E'}\frac{dp\_0}{dt} + 2w\_h\dot{L}\_f\tag{30}$$

where *Lf* is the length of fluid penetration into the NF, and the upper point denotes the differentiation with respect to time. Using now the fluid flow equations (27) and the pressure relationship at the junction point (29), we can write

$$w\_2 = w\_h \dot{L}\_f = \frac{\pi v\_h^3}{12\,\mu} \frac{p\_0}{L\_f} \tag{31}$$

Thus, from (30) and (31) we obtain the following system of ordinary differential equations for the fluid penetration length *L <sup>f</sup>* and fluid pressure at the inlet *p*<sup>0</sup>

$$\begin{cases} \dot{L}\_f = \frac{w\_h^2}{12\mu} \frac{p\_0}{L\_f} \\\\ Q - 2w\_h \dot{L}\_f = \frac{4L^2}{E^\circ} \dot{p}\_0 \end{cases} \tag{32}$$

The solution of these equations can be found after setting up the initial condition at the time of fracture contact, *t =* 0. We presume the fluid pressure at the HF tip is prescribed and there is not yet fracturing fluid penetration into the NF. The initial conditions are thus written as

$$\begin{cases} p\_0(0) = p\_{00} \\ L\_f(0) = 0 \end{cases} \tag{33}$$

Rewriting the equations (32) and initial conditions (33) in terms of dimensionless parameters already introduced, and using substitutions *L <sup>f</sup>* = *L γ<sup>f</sup>* , *wh* =*W Ψ<sup>h</sup>* , *p*<sup>0</sup> =*PΠ*0, *t* =*Tτ*, we get

$$\begin{cases} \dot{\boldsymbol{\gamma}}\_f = \boldsymbol{\Psi}\_h^2 \frac{\boldsymbol{\Pi}\_0}{\boldsymbol{\mathcal{Y}}\_f} \\ \mathbf{1} - 2\boldsymbol{\Psi}\_h \dot{\boldsymbol{\mathcal{Y}}}\_f = \mathbf{4} \dot{\boldsymbol{\Pi}}\_0 \\ \boldsymbol{\gamma}\_f(0) = 0 \\ \boldsymbol{\Pi}\_0(0) = \boldsymbol{\Pi}\_{00} \end{cases} \tag{34}$$

Numerical solution of the problem (34) is plotted in Fig. 8. Fluid penetration length (Fig. 8, left) grows fast at the very beginning (τ << 1) and turns to nearly linear dependence on time when τ > 1. The pressure at the junction point drops right after the HF-NF contact (Fig. 8, right). The velocity of pressure drop rapidly slows, and after the certain inflection point it starts to grow linearly. As it will be shown later, the inflection point depends on the NF permeability and initial fluid pressure at fracture contact.

The numerical solution helps in the general solution of the problem (34) but hides the para‐ metrical dependency of the fluid penetration and pressure behavior shown in Fig. 8. To evaluate the pressure inflection point *τ\** and reveal the parametric sensitivity of the velocity of fluid penetration into the NF and the HF pressure dynamics during the contact, we further investigate the asymptotical behavior of the fluid length *γf* and the fluid pressure *Π*0 at the very beginning of fracture contact and at large time after contact.

Running Title <sup>17</sup>

3 **Figure 8.** Analytical solution of the dimensionless problem for fluid penetration into NF (34) for dimensionless perme‐ ability Ψh = 1, and initial pressure Π00 = 1.5. Left plot is the dimensionless fluid front propagation along the NF in time. Right plot is the change of dimensionless pressure at the HF-NF intersection point.

*Early-time (left) asymptote: 0 < τ << τ\*.*

1

2

2

0 0

the fluid penetration length *L <sup>f</sup>* and fluid pressure at the inlet *p*<sup>0</sup>

*<sup>f</sup> L L*

and NF sides, so using assumption (24) we write

172 Effective and Sustainable Hydraulic Fracturing

relationship at the junction point (29), we can write

where *Lf*

For steady flow the fluid pressure at the fracture junction point must be the same at the HF

The total fluid mass balance in the system of the HF and the NF can be written by making use of continuity equations (6) and (27), inlet condition (10), and elasticity equation (25) as follows:

<sup>4</sup> ( 2 ') <sup>2</sup> '

*d L dp Q wdx w dx w L dt E dt*

<sup>2</sup> 12

2 0

*h*

m

<sup>4</sup> <sup>2</sup> '

0 00 (0) (0) 0 *<sup>f</sup> p p*

*L* ìï = <sup>í</sup> <sup>=</sup> ïî

*h f*

*<sup>L</sup> Q wL p <sup>E</sup>*

*f*

& &

The solution of these equations can be found after setting up the initial condition at the time of fracture contact, *t =* 0. We presume the fluid pressure at the HF tip is prescribed and there is not yet fracturing fluid penetration into the NF. The initial conditions are thus written as

*L*

12

*w p <sup>L</sup>*

<sup>ï</sup> - = <sup>î</sup>

*f*

&

ì <sup>ï</sup> <sup>=</sup> <sup>ï</sup> í ï

*h f*

*w p q wL*

2 0

is the length of fluid penetration into the NF, and the upper point denotes the

differentiation with respect to time. Using now the fluid flow equations (27) and the pressure

3 0

m

2 0

*f*

*h*

Thus, from (30) and (31) we obtain the following system of ordinary differential equations for

*h h f*

= + =+ ò ò & (30)

*<sup>L</sup>* = = & (31)

(32)

(33)

<sup>1</sup> ( ) \*( ) <sup>2</sup> *qt q t* <sup>=</sup> (28)

2 0 ( ' 0) ( ) ( ) *f f p x p x L pt* = = = » (29)

Let us assume the following law for the fluid pressure and penetration length at the beginning of fracture contact,

$$
\begin{aligned}
\Delta\Pi\_0 &= \Pi\_{00} - \Pi\_1 \sqrt{\tau} \\
\mathcal{V}\_f &= \upsilon\_f \sqrt{\tau}
\end{aligned}
\tag{35}
$$

0 1 *f f v*

Õ =Õ

g

where *Π*1 and *vf* are new unknown positive constants. From (34) we then obtain

2 2

ï =Y P

*v*

ì

í

Solution of (39) for the unknown *Π*1 and *vf* gives the expressions

1

n

excellent agreement. Fig. 10 demonstrates this comparison.

fluid pressure at junction point at large time of fracture contact

*The inflection point: τ = τ\*, Π˙ <sup>0</sup> <sup>=</sup> <sup>0</sup>*

t

<sup>=</sup> (38)

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Hydraulic Fracture Propagation Across a Weak Discontinuity Controlled by Fluid Injection

(39)

175

(40)

 t

1 <sup>1</sup> 12 4 *f h*

4 2

*h h*

4 2

4

Comparison of the asymptotes (38) with numerical solution of the equations (34) gives

**Figure 10.** Asymptotic (red) and numerical (blue) solutions for the dimensionless fluid penetration length (left) and

To find the parametric expression for the inflection point in time scale, where the fluid pressure

1 2 *<sup>f</sup>*

*h*

<sup>=</sup> <sup>Y</sup> & (41)

starts to increase, we substitute *Π*˙ <sup>0</sup> =0 into the second equation of (34) and have

g

*h f*

<sup>ï</sup> -Y =P <sup>î</sup>

2

*h*

*f hh*

<sup>ï</sup> <sup>Y</sup> é ù <sup>ï</sup> = Y + -Y ê ú <sup>î</sup> ë û

4 8

4

*h*

<sup>1</sup> <sup>4</sup>

<sup>ì</sup> <sup>Y</sup> é ù ïP = - Y + -Y <sup>ï</sup> ê ú ë û <sup>í</sup>

*v*

where *Π*1 and *vf* are unknown positive constants that must be determined. After substitution of (35) in (34), in the limit *τ*≪1 we arrive at the following relations for *Π*1 and *vf* :

$$\begin{cases} \upsilon\_f^2 = 2\Psi\_h^2 \Pi\_{00} \\ 2\Pi\_1 = \Psi\_h \upsilon\_f \end{cases} \tag{36}$$

The unknown constants *Π*1 and *vf* are thus found as

$$\begin{cases} \Pi\_1 = \Psi\_h^2 \sqrt{\Pi\_{00}/2} \\ \nu\_f = \Psi\_h \sqrt{2\Pi\_{00}} \end{cases} \tag{37}$$

From this analysis it is obvious that the rate of fluid pressure drop at the beginning of fracture contact as well as the velocity of fluid penetration into the NF are affected more strongly by the permeability of the NF *Ψh* than by the initial fluid pressure in the HF *Π*00.

The asymptotes (35) can be plotted with the accurate numerical solution of the equations (34). Fig. 9 shows the comparison of the numerical solution of the problem and asymptotes (35) with coefficients (37). 18 Book Title

2 3 **Figure 9.** Asymptotic (red) and numerical (blue) solutions for the dimensionless fluid penetration length (left) and flu‐ id pressure in junction point (right) at the early time of fracture contact

*Large-time (right) asymptote: τ >> τ\**

1

Now consider the large-time behavior of the system of contacted fractures. In that limit let us employ the linear asymptotes for the pressure and fluid penetration length

Hydraulic Fracture Propagation Across a Weak Discontinuity Controlled by Fluid Injection http://dx.doi.org/10.5772/55941 175

$$\begin{aligned} \Pi\_0 &= \Pi\_1 \,\,\pi \\ \mathcal{V}\_f &= \mathcal{v}\_f \,\,\pi \end{aligned} \tag{38}$$

where *Π*1 and *vf* are new unknown positive constants. From (34) we then obtain

$$\begin{cases} \upsilon\_f^2 = \Psi\_h^2 \,\Pi\_1\\ \mathbf{1} - 2\Psi\_h \upsilon\_f = \mathbf{4}\Pi\_1 \end{cases} \tag{39}$$

Solution of (39) for the unknown *Π*1 and *vf* gives the expressions

$$\begin{cases} \Pi\_1 = \frac{1}{4} - \frac{\Psi\_h^2}{8} \left[ \sqrt{\Psi\_h^4 + 4} - \Psi\_h^2 \right] \\\\ \nu\_f = \frac{\Psi\_h}{4} \left[ \sqrt{\Psi\_h^4 + 4} - \Psi\_h^2 \right] \end{cases} \tag{40}$$

Comparison of the asymptotes (38) with numerical solution of the equations (34) gives excellent agreement. Fig. 10 demonstrates this comparison.

**Figure 10.** Asymptotic (red) and numerical (blue) solutions for the dimensionless fluid penetration length (left) and fluid pressure at junction point at large time of fracture contact

*The inflection point: τ = τ\*, Π˙ <sup>0</sup> <sup>=</sup> <sup>0</sup>*

0 00 1

P =P -P

of (35) in (34), in the limit *τ*≪1 we arrive at the following relations for *Π*1 and *vf* :

2 2

ï = YP

2

2 1 00

*f h* n

the permeability of the NF *Ψh* than by the initial fluid pressure in the HF *Π*00.

<sup>ï</sup> =Y P <sup>î</sup>

ïP =Y P

2 *h*

From this analysis it is obvious that the rate of fluid pressure drop at the beginning of fracture contact as well as the velocity of fluid penetration into the NF are affected more strongly by

The asymptotes (35) can be plotted with the accurate numerical solution of the equations (34). Fig. 9 shows the comparison of the numerical solution of the problem and asymptotes (35)

**Figure 9.** Asymptotic (red) and numerical (blue) solutions for the dimensionless fluid penetration length (left) and flu‐

Now consider the large-time behavior of the system of contacted fractures. In that limit let us

employ the linear asymptotes for the pressure and fluid penetration length

with coefficients (37). 18 Book Title

*f h*

1

2

*v*

ì

í <sup>ï</sup> P =Y <sup>î</sup>

ì

í

The unknown constants *Π*1 and *vf* are thus found as

174 Effective and Sustainable Hydraulic Fracturing

id pressure in junction point (right) at the early time of fracture contact

*Large-time (right) asymptote: τ >> τ\**

1

*f*

2 3  t

where *Π*1 and *vf* are unknown positive constants that must be determined. After substitution

00

00

/ 2

*h f*

*v*

t

<sup>=</sup> (35)

(36)

(37)

*f f v*

g

To find the parametric expression for the inflection point in time scale, where the fluid pressure starts to increase, we substitute *Π*˙ <sup>0</sup> =0 into the second equation of (34) and have

$$
\dot{\mathcal{V}}\_f = \frac{1}{2\Psi\_h} \tag{41}
$$

If *τ\** is close to zero, the velocity of fluid penetration can be taken from the early-time solution (35) with (37):

$$
\dot{\boldsymbol{\gamma}}\_f = \boldsymbol{\Psi}\_h \sqrt{\frac{\prod\_{\alpha 0}}{2\pi^\*}} \tag{42}
$$

Careful investigation of the complex dynamics of fluid-coupled fracture interaction deserves

Hydraulic Fracture Propagation Across a Weak Discontinuity Controlled by Fluid Injection

http://dx.doi.org/10.5772/55941

177

1 mm, MineHF2D 0.1 mm, MineHF2D 0.01 mm, MineHF2D 1 mm, Analytical solution 0.1 mm, Analytical solution 0.01 mm, Analytical solution

MineHF2D 0.1 mm, MineHF2D 0.01 mm, MineHF2D 1 mm, Analytical solution 0.1 mm, Analytical solution 0.01 mm, Analytical solution

**Figure 11.** Results of numerical experiments with κ*IC* =0.32, Σ<sup>2</sup> =0.8, ΔΣ =1, β =90, λ =0.2 and three values of residu‐ al hydraulic opening at the NF permeability (blue: 1 mm; red: 0.1 mm; green: 0.01 mm). Top left: fluid pressure changes at the inlet. Top right: opening of the HF at junction. Bottom left: fluid penetration length along the NF. Bot‐

In conclusion, we conducted an extensive parametric study of the problem of HF-NF interac‐ tion by means of numerical simulator MineHF2D developed by CSIRO. This research was mainly focused on the result of fracture interaction in terms of crossing or arresting of the HF at the NF as a function of the most sensitive parameters, such as fracture approach angle β, friction coefficient λ, dimensionless toughness κIC, inversely proportional to the injection rate

In a system of interacting hydraulic and NFs, the number of physical parameters that can affect the result of interaction is large. The proper scaling of the problem allowed us to decrease the number of independent parameters from 14 dimensional parameters to 9 dimensionless parameters, and effectively perform the parametric study in the space of 6 of the most critical dimensionless parameters. We summarized the results of the large number of numerical

and fluid viscosity, relative stress contrast*Δ Σ,* and the NF permeability *k*.

a separate work.


**6. Conclusions**

**L, m**

**P, MPa**

0 0.2 0.4 0.6

**Time, sec**

**Fluid penetration into NF** 1 mm,

0 0.2 0.4 0.6 0.8

**Time, sec**

tom right: total length of the mechanically open zone at the NF in meters.

**Pressure records at the injection point**

Comparing now (41) with (42), one obtains the following estimation for the time of pressure drop-growth inflection:

$$
\tau^\* = 2\,\Pi\_{00}\Psi\_h^4 \tag{43}
$$

This simple expression tells again that practically it is the magnitude of NF permeability that plays a key role in the transient pressure behavior and intensity of NF infiltration by the fracturing fluid.

#### **5.2. Numerical simulations**

To validate the analytical model predictions we performed several numerical simulations by MineHF2D code with *κIC* =0.32, *Σ*<sup>2</sup> =0.8, *ΔΣ* =1, *β* =90, *λ* =0.2 and various permeabilities of the NF. The residual hydraulic opening of the NF Ω*h* in these numerical runs was intentionally chosen to be a magnitude close to the average opening of the HF at the point of intersection.

Fig. 11 shows the results of simulations for three different values of the residual hydraulic opening prescribed at the NF Ω*<sup>h</sup>* : 1 mm, 0.1 mm, and 0.01 mm. In all these test cases the average opening of the HF close to the fracture intersection was 3 mm. Pressure records at the injection point and the length of fluid penetration into the NF after the fracture contact are compared with predictions of the analytical model and shown in Fig. 11 (left top and left bottom).

For two cases with lower NF permeability (green-, red-dashed line in Fig. 11) the pressure starts to grow immediately after the fracture intersection. As the fluid weakly propagates into the NF in these cases, the net pressure and HF opening almost equally quickly grow with time. In the second example, for Ω*h* = 0.1 mm at *t* = 0.55 s, the NF opens mechanical‐ ly as a result of increased net pressure of the penetrated fluid and the penetration rate increases (red line, Fig. 11); the pressure and opening of the HF decrease because of the enhanced leakoff along the NF.

In the third case, when the residual hydraulic opening at the NF is of the same order of magnitude as the average opening of the HF (3 times less), we observe comparably fast fluid leakoff into the NF after the contact (blue line, Fig. 11), which given the fixed injection rate into the HF does not allow the net pressure in the HF to grow. Very soon after the fluid reaches the tips of the NF prescribed in the numerical model (1.5 m in this example), the NF rapidly opens mechanically and the HF opening starts to grow.

Careful investigation of the complex dynamics of fluid-coupled fracture interaction deserves a separate work.

**Figure 11.** Results of numerical experiments with κ*IC* =0.32, Σ<sup>2</sup> =0.8, ΔΣ =1, β =90, λ =0.2 and three values of residu‐ al hydraulic opening at the NF permeability (blue: 1 mm; red: 0.1 mm; green: 0.01 mm). Top left: fluid pressure changes at the inlet. Top right: opening of the HF at junction. Bottom left: fluid penetration length along the NF. Bot‐ tom right: total length of the mechanically open zone at the NF in meters.

#### **6. Conclusions**

If *τ\** is close to zero, the velocity of fluid penetration can be taken from the early-time solution

00

<sup>P</sup> & = Y (42)

=PY (43)

t

Comparing now (41) with (42), one obtains the following estimation for the time of pressure

4

This simple expression tells again that practically it is the magnitude of NF permeability that plays a key role in the transient pressure behavior and intensity of NF infiltration by the

To validate the analytical model predictions we performed several numerical simulations by MineHF2D code with *κIC* =0.32, *Σ*<sup>2</sup> =0.8, *ΔΣ* =1, *β* =90, *λ* =0.2 and various permeabilities of the NF. The residual hydraulic opening of the NF Ω*h* in these numerical runs was intentionally chosen to be a magnitude close to the average opening of the HF at the point of intersection.

Fig. 11 shows the results of simulations for three different values of the residual hydraulic opening prescribed at the NF Ω*<sup>h</sup>* : 1 mm, 0.1 mm, and 0.01 mm. In all these test cases the average opening of the HF close to the fracture intersection was 3 mm. Pressure records at the injection point and the length of fluid penetration into the NF after the fracture contact are compared with predictions of the analytical model and shown in Fig. 11 (left top and left bottom).

For two cases with lower NF permeability (green-, red-dashed line in Fig. 11) the pressure starts to grow immediately after the fracture intersection. As the fluid weakly propagates into the NF in these cases, the net pressure and HF opening almost equally quickly grow with time. In the second example, for Ω*h* = 0.1 mm at *t* = 0.55 s, the NF opens mechanical‐ ly as a result of increased net pressure of the penetrated fluid and the penetration rate increases (red line, Fig. 11); the pressure and opening of the HF decrease because of the

In the third case, when the residual hydraulic opening at the NF is of the same order of magnitude as the average opening of the HF (3 times less), we observe comparably fast fluid leakoff into the NF after the contact (blue line, Fig. 11), which given the fixed injection rate into the HF does not allow the net pressure in the HF to grow. Very soon after the fluid reaches the tips of the NF prescribed in the numerical model (1.5 m in this example), the NF rapidly opens

2 \* *f h*

<sup>00</sup> \* 2 *<sup>h</sup>*

g

t

(35) with (37):

drop-growth inflection:

176 Effective and Sustainable Hydraulic Fracturing

fracturing fluid.

**5.2. Numerical simulations**

enhanced leakoff along the NF.

mechanically and the HF opening starts to grow.

In conclusion, we conducted an extensive parametric study of the problem of HF-NF interac‐ tion by means of numerical simulator MineHF2D developed by CSIRO. This research was mainly focused on the result of fracture interaction in terms of crossing or arresting of the HF at the NF as a function of the most sensitive parameters, such as fracture approach angle β, friction coefficient λ, dimensionless toughness κIC, inversely proportional to the injection rate and fluid viscosity, relative stress contrast*Δ Σ,* and the NF permeability *k*.

In a system of interacting hydraulic and NFs, the number of physical parameters that can affect the result of interaction is large. The proper scaling of the problem allowed us to decrease the number of independent parameters from 14 dimensional parameters to 9 dimensionless parameters, and effectively perform the parametric study in the space of 6 of the most critical dimensionless parameters. We summarized the results of the large number of numerical simulations performed in the range of parameters values relevant to fracturing field opera‐ tions. The resultant solid picture of parametric sensitivity to the arresting versus crossing behavior helps provide a better understanding of the relative role of each parameter. In particular, aside from the well-known effect of the fracture approach angle and stress contrast, we revealed the influence of the injection rate and fluid viscosity on reducing the angle and stress threshold for HF-NF crossing.

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The presented analytical model of the infiltration of the permeable NF by the contact with the HF allowed us to understand a parametric dependency of the HF pressure response and fracturing fluid penetration at early and large time after the HF-NF contact. We have seen a predominant role of the hydraulic permeability of the NF in the evaluation of the pressure decay curve after the fracture contact. It appears that at T-shape fracture contact, initially the pressure quickly drops and after some saturation it rebounds to grow. The rebound time of pressure response *τ \** separating the early and large time regimes is strongly dependent on the permeability of the NF (as its forth power) and to a much lesser extent by fluid pressure at the HF-NF junction (linearly). Such fast pressure decay means that during *τ* <*τ \** the fluid pene‐ tration supports temporal arrest of the HF by the NF. Independent numerical computations with large residual aperture of the NF led us to the conclusion that the result of HF-NF interaction is affected by permeable properties of the NF only when the residual opening of the NF is comparable in magnitude with the opening of the HF at the contact.

### **Acknowledgements**

The authors are grateful to Schlumberger for permission to publish this paper. Special thanks go to Xi Zhang (CSIRO) for his constant attention and enormous help with improvement of the numerical code MineHF2D during the study, and to Xiaowei Weng (Schlumberger) for constructive discussions of the obtained results and valuable recommendations.
