**3. Verification test: Penny-shaped crack propagation in medium with zero toughness**

The non-steady response of rock to injection of fluid depends on fracture toughness, the viscosity of the fluid and the rate of leak-off. In the case of zero fracture toughness and no leakoff, the response is viscosity-dominated, which corresponds to the "M-asymptote" identified by [5]. This condition is used for verification of *HF Simulator*.

also can simulate the porous medium flow through unfractured blocks as a way to represent the leakoff. This capability is not discussed further in this paper.) The flow pipe network is dynamic and automatically updated by connecting newly formed microcracks to the existing flow network. The model uses the lubrication equation to approximate the flow within a fracture as a function of aperture. The flow rate along a pipe, from fluid node "A" to node "B,"

( ) <sup>3</sup>

= -+ - é ù

 r

where *a* is hydraulic aperture, *μ* is viscosity of the fluid, *p <sup>A</sup>* and *p <sup>B</sup>* are fluid pressures at nodes "A" and "B", respectively, *z <sup>A</sup>* and *z <sup>B</sup>* are elevations of nodes "A" and "B", respectively, and

Clearly, when the pipe is saturated, *s* =1 and the relative permeability is 1. The dimensionless number *β* is a calibration parameter, a function of resolution, used to match conductivity of a pipe network to the conductivity of a joint represented by parallel plates with aperture *a*. The

**1.** Fracture permeability depends on aperture, or on the deformation of the solid model. **2.** Fluid pressure affects both deformation and the strength of the solid model. The effective

**3.** The deformation of the solid model affects the fluid pressures. In particular, the code can

A new coupling scheme, in which the relaxation parameter is proportional to *KRa* / *R*, where *KR* is rock bulk modulus and *R* is the lattice resolution, is implemented in *HF Simulator*, allowing larger explicit time steps and faster simulation times compared to conventional

**3. Verification test: Penny-shaped crack propagation in medium with zero**

The non-steady response of rock to injection of fluid depends on fracture toughness, the viscosity of the fluid and the rate of leak-off. In the case of zero fracture toughness and no leak-

*AB AB*

ë û (3)

(3 2 ) *<sup>r</sup> ks s* = - (4)

12

calibrated relation between *β* and the resolution is built into the code.

In *HF Simulator*, the mechanical and flow models are fully coupled.

predict changes in fluid pressure under undrained conditions.

methods that use fluid bulk modulus as a relaxation parameter.

m

b

*r w <sup>a</sup> q k p p gz z*

*ρw* is fluid density. The relative permeability, *kr*, is a function of saturation, *s*:

2

is calculated based on the following relation:

824 Effective and Sustainable Hydraulic Fracturing

**2.4. Hydro-mechanical coupling**

**toughness**

stress calculations are carried out.

In the simulated example, fluid is injected at a constant rate into a penny-shaped crack of low initial aperture (10−<sup>5</sup> m). The crack has zero normal strength, and the in-situ stresses are also zero. Thus, the test conditions approximate those of the analytical solution for the no-lag case (i.e., no fluid pressure tension cut-off) provided by [5]. The injection rate is 0.01 m3 /s; the dynamic viscosity is 0.001 Pa×s. The mechanical properties of the rock are characterized by Young's modulus of 7×1010Pa and Poisson's ratio of 0.22. Figure 3 provides a visualization of the state of the model at 10 s of elapsed time. Note that pressures are negative in the outer annulus of the flow disk.

**Figure 3.** View of pressure (Pa) field (icons, colored according to magnitude) and cross-section of displacement (m) field (vectors, colored according to magnitude)

Figure 4 shows the aperture profiles at three times during the simulation — averaged numer‐ ical results (for 30 radial distances), together with asymptotic solutions (derived from the equations of [5]). Figure 5 shows the pressure profile at 10 s, together with the asymptotic solution. Note that there is a lack of match at small and large radial distances: at small distances, the numerical source is a finite volume, rather than a point source (which is assumed in the exact solution); at large distances, the finite initial aperture allows seepage (compared to zero seepage in the exact solution, which assumes zero initial aperture).

### **4. Example application**

Two example problems are discussed in this section. Fracture propagation in a homogeneous (unfractured) and fractured media is analyzed. These two problems involve a horizontal borehole segment with two injection clusters with centers at 4.8 m distance (Figure 6). The model domain is 18 m × 18 m × 18 m, and the lattice resolution was set to 0.5 m. Fluid is injected into the clusters at rate of 0.01 m3 /s. The assumed stress state is anisotropic with *σxx* =1 MPa, *σyy* =12 MPa and *σzz* =10 MPa. The least principal stress is aligned with the horizontal section of the borehole. This stress state favors crack propagation in the direction normal to the horizontal section of the borehole. In order to initiate the fluid calculation, fluid-filled joints have been placed at the center of each cluster; these joints are slightly larger than the cluster size. The initial apertures in these joints have been set to 0.1 mm. Both example problems use this model configuration. The example shown on the left in Figure 6 simulates the response of an unfractured medium to fluid injection. Three discrete joints that interact with the induced fractures are introduced in the example on the right in Figure 6.

depending on a number of parameters, including stress state, strength and permeability of the pre-existing joint.) The propagation continues by reinitiation along the edges of pre-existing

Three-Dimensional Numerical Model of Hydraulic Fracturing in Fractured Rock Masses

http://dx.doi.org/10.5772/56313

827

joints.

**Figure 5.** Pressure profile at 10 s

**Figure 6.** Geometry of two example problems

**Figure 4.** Aperture profiles for three times

The induced microcracks in the homogeneous model after 15 s of injection are shown in Figure 7. The microcracks form two roughly circular (penny-shape) hydraulic fractures. In this example, the fractures are not parallel. There is a slight trend of fractures curving away from each other as a result of stress interaction.

In the second example, the HF propagation is clearly affected by the pre-existing joints, as shown in Figure 8. When the HF intersects the pre-existing joint, the fluid is diverted into the pre-existing joints. (In general case, the HF can cross or be diverted into the pre-existing joint, depending on a number of parameters, including stress state, strength and permeability of the pre-existing joint.) The propagation continues by reinitiation along the edges of pre-existing joints.

**Figure 5.** Pressure profile at 10 s

**4. Example application**

826 Effective and Sustainable Hydraulic Fracturing

into the clusters at rate of 0.01 m3

**Figure 4.** Aperture profiles for three times

each other as a result of stress interaction.

Two example problems are discussed in this section. Fracture propagation in a homogeneous (unfractured) and fractured media is analyzed. These two problems involve a horizontal borehole segment with two injection clusters with centers at 4.8 m distance (Figure 6). The model domain is 18 m × 18 m × 18 m, and the lattice resolution was set to 0.5 m. Fluid is injected

*σyy* =12 MPa and *σzz* =10 MPa. The least principal stress is aligned with the horizontal section of the borehole. This stress state favors crack propagation in the direction normal to the horizontal section of the borehole. In order to initiate the fluid calculation, fluid-filled joints have been placed at the center of each cluster; these joints are slightly larger than the cluster size. The initial apertures in these joints have been set to 0.1 mm. Both example problems use this model configuration. The example shown on the left in Figure 6 simulates the response of an unfractured medium to fluid injection. Three discrete joints that interact with the induced

The induced microcracks in the homogeneous model after 15 s of injection are shown in Figure 7. The microcracks form two roughly circular (penny-shape) hydraulic fractures. In this example, the fractures are not parallel. There is a slight trend of fractures curving away from

In the second example, the HF propagation is clearly affected by the pre-existing joints, as shown in Figure 8. When the HF intersects the pre-existing joint, the fluid is diverted into the pre-existing joints. (In general case, the HF can cross or be diverted into the pre-existing joint,

fractures are introduced in the example on the right in Figure 6.

/s. The assumed stress state is anisotropic with *σxx* =1 MPa,

**Figure 6.** Geometry of two example problems
