**3.3 Impact of embedment on fracture conductivity**

In this section, the change in pressure and velocity of backflow of fluid across the proppant has been presented. Embedment has a profound impact on the pressure drop as well as velocity profile as shown in **Figure 10**. In this study, three different embedment cases have been considered (0, 60, and 80%). The percentage of embedment is defined as the proportion of the total proppant that is embedded through the fracture surface. Without embedment, a slight difference between inlet and outlet pressure has been recorded; however, a significant difference

*Hydraulic Fracture Conductivity in Shale Reservoirs DOI: http://dx.doi.org/10.5772/intechopen.100473*

#### **Figure 10.**

*Velocity and pressure profiles in the fracture zone around the single proppant with and without embedment. (a) Pressure profile with no embedment. (b) Velocity streamlines with no embedment. (c) Pressure profile with 60% embedment. (d) Velocity streamlines (a streamline is a line that is tangential to the instantaneous velocity direction (velocity is a vector, and it has a magnitude and a direction. Color represents velocity magnitude) with 60% embedment). (e) Pressure profile with 80% embedment. (f) Velocity streamlines with 80% embedment.*

between inlet and outlet pressures can be seen at 60 and 80% embedment as shown in **Figure 10(b)** and **(c)**. Inlet velocity in all cases is 0.5 m/s but around the proppant, the flow velocity is recorded around 2 m/s and at outlet, the velocity is achieved 1.5 m/s.

Based on the different embedment depths, the velocity of the injected fluid varies significantly as shown in **Figure 11**. In all cases, injection velocity is constant, that is, 0.5 m/s. A sudden increase in the fluid velocity is recorded around the proppant and a decrease in velocities is presented at the end of the proppant. The results show a significant decline in the velocity at 80% embedment; therefore, fracture conductivity is recorded significantly low at high embedment (see **Figure 12**). As fluid flowing continues around the exit sides of the proppant, it begins to slow down due to eddy generated at the outlet/backside of the proppant.

#### *Emerging Technologies in Hydraulic Fracturing and Gas Flow Modelling*

**Figure 11.** *Velocity profile of injection fluid around the proppant under different embedment percentages.*

#### **Figure 12.**

*Fracture conductivity obtained with finite volume method based on experimental and numerical measured embedment depths with finite element method.*

The Lagrangian analysis is capable of revealing the underlying structure and complex phenomena in unsteady flows [41].

Distance is measured from one side (fracture inlet) of the proppant until the other side (fracture outlet) of the proppant. All the three positions of proppants have been presented in **Figure 10(a), (c)**, and **(e)**. Numerical analysis is conducted by finite volume method to obtain pressure drop across the proppant and resultant fracture permeability. **Table 4** shows the fracture conductivity based on embedment percentage.

Finally, fracture conductivity is achieved based on fracture permeability and fracture width. **Figure 12** shows that fracture conductivity was measured based on *Hydraulic Fracture Conductivity in Shale Reservoirs DOI: http://dx.doi.org/10.5772/intechopen.100473*


**Table 4.**

*Embedment and resultant fracture conductivity.*

embedment depth obtained with experimental and finite element methods. A dramatic decrease in fracture conductivity has been obtained with the increase of embedment depth. The reason for a significant decrease in the conductivity is the significant pressure drop across the embedment. The results show that pressure loss at 60 and 80% embedment is 29,716 and 64,721 pa, respectively.
