**3. A simple test problem**

( , )/ ( , ) *ff f l ql t wl t* <sup>=</sup> & (7)

Q Q - Q= (8)

The fracture growth is based on using the maximum hoop stress criterion, with the maximum

*K KK <sup>I</sup> II Ic*

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

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

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

/s

<sup>2</sup> <sup>3</sup> cos ( cos sin ) 2 22

mixed-mode stress intensity factor reaching a critical value [14]

144 Effective and Sustainable Hydraulic Fracturing

are approximated by the analytical LEFM solutions [14].

is assumed to initially be stationary and unsaturated.

**Property Value** Young's modulus E 50 GPa Poisson's ratio 0.22

Coefficient of friction along fractures 0.8

Mode I Toughness 1.0 MPa⋅m0.5 Fluid dynamic viscosity 0.01 Pa⋅ s Injection rate 0.00002 m<sup>2</sup>

of newly-created fractures.

**Table 1.** Material properties

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 specified.

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 Figure 1(d).
