**2. Procedure**

In this section, steps to measure embedment and fracture conductivity of fractured shale have been presented. Governing equations of numerical simulation to study the impact of embedment on the reduction of fracture conductivity have been presented in Section 2.3. The conductivity reduction due to embedment was modeled with a computational fluid dynamics (CFD) approach. We used the CFD software package CFX (ANSYS Inc.) in this work and applied the boundary conditions as shown in **Figure 3** and **Table 1**. Initially, geotechnical characteristics of shale formation were calculated to define these properties in the software.

#### **2.1 Elastic geo-mechanical properties**

Dynamic elastic properties of the shale lithofacies, that is, Young's modulus and Poisson's ratio were calculated using compressional and shear velocities measured on shale core samples. The following equations [32] are used to obtain the respective values.

$$\text{Dynamic Young Modulus } (\nu) = \frac{P\_b. V\_s^2 \left(3V\_p^2 - 4V\_s^2\right)}{V\_p^2 - V\_s^2} \tag{1}$$

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

$$\text{Dynamic Poisson's Ratio } (\mathbf{E}) = \frac{\mathbf{V}\_{\text{p}}^2 - 2\mathbf{V}\_{\text{s}}^2}{2\left(\mathbf{V}\_{\text{p}}^2 - \mathbf{V}\_{\text{s}}^2\right)}\tag{2}$$

where *Vs* and *Vp* represent the shear and dynamic wave velocities in Km/s and *Pb* is the bulk density of the shale in gm/cc. The results (**Table 2**) show that massive siliceous shale has a high value of Young's modulus and a low Poisson's ratio in

#### **Figure 3.**

*The direction of flow at inlet and outlet across the proppant.*


#### **Table 1.**

*Basic input parameters, conditions, and assumptions.*


#### **Table 2.**

*Elastic and strength properties of the lithofacies identified in this study.*

**Figure 4.** *Reduction in the fracture conductivity by embedment in the fracture surfaces, modified from Zhang et al. [29].*

comparison with massive argillaceous shale facie. Static Young's modulus and static Poisson's ratio were measured on cylindrical core samples of 2.0 in diameter and 4.0 in length. The specimens were failed in a triaxial setup and the deformations (axial and radial strains) were measured on the installed strain gauges. The static elastic parameters were calculated using the following equations.

$$\text{Static Young Modulus } (E) = \frac{\sigma\_a}{\varepsilon\_a} \tag{3}$$

$$\text{Static Poisson's Ratio } (\nu) = -\frac{\varepsilon\_r}{\varepsilon\_d} \tag{4}$$

where *σ<sup>a</sup>* represents axial stress, while *ε<sup>a</sup>* and *ε<sup>r</sup>* represent axial and radial strains measured on the samples under deformation. Like dynamic parameters, the static Young's modulus of the massive siliceous shale is also greater than the static Young's modulus of argillaceous shale facie. Overall, values obtained for the static moduli are less than the dynamic moduli values measured on the same samples (**Table 2**). This is per previous findings done on producing shale formations. The correlation between dynamic and static moduli is shown in **Figure 4**. Strength parameters of the lithofacies are calculated using the equations below.

$$\text{Shear Modulus } (G) = \frac{E}{2(1+v)}\tag{5}$$

$$\text{Plane Strain Modulus } (E') = \frac{E}{1 + v^2} \tag{6}$$
