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

#### **Figure 9.**

*Proppant under vertical principal stress: (a) total deformation and penetration due to an external load, (b) directional deformation along the Z axis, one surface under load while another surface in static, (c) plot of directional deformation profile, (d) normal stress profile along with the formation and proppant, and (e) plot of normal stress profile along with the formation and proppant under uni-axial load.*

applied at the top surface, while the lower surface has been kept static to validate the results with a uni-axial compression test. In the case of the uniaxial load test, the resulted embedment depth in this study is found different for the top and bottom of the fracture surface, similar to an earlier experimental study performed using different types of proppants [39]. The significant difference between embedment profiles is the result of different proppant types being tested. 20/40 Ceramic proppants are rounder, more spherical, uniform, and stronger than 20/40 Ottawa proppants. This difference in proppants makes 20/40 Ottawa sand more prone to proppant embedment as well as any other damage mechanism caused by mechanical and chemical factors in fractures [39]. The present study shows that even though the proppant is strong enough to get deformed, and there is a penetration of proppant in the rock surface due to fracking fluid flow that has reduced Young's modulus of the fractured surface. Initially, ceramic proppant and rock properties are introduced based on the shale formation. A pair of contact between surfaces and proppant is defined. Then, the external loads are applied to the rock surfaces to obtain the stress transfer across the proppant of contact surfaces. The penalty-based method is used to simulate the contact behavior [27, 40]. The finite element method is a numerical method that can be successfully used to generate solutions for problems belonging to a vast array of engineering fields: stationary, transitory, linear, or nonlinear problems. For the linear case, computing the solution to the given problem is a straightforward process, and the displacements are obtained in a single step and all the other quantities are evaluated afterward. When faced with a nonlinear problem, in this case with a contact nonlinearity, one needs to account for the fact that the stiffness matrix of the systems varies with the loading, the force vs. stiffness relation being unknown before the beginning of the analysis. Modern software using the finite element method to solve contact problems usually approaches such problems *via* two basic theories that, although different in their approaches, lead to the desired solutions. One of the theories is known as the penalty function method [40]. The penalty method is simple to implement in practice. The penalty is a sort of friction coefficient, and one can specify a friction model that defines the force resisting the relative tangential motion of the surfaces in a mechanical contact analysis. By selecting a penalty, one can use a stiffness (penalty) method that permits some relative motion of the surfaces when they should be sticking. By applying the penalty method, the penetration of the proppant has been achieved higher at the top which is 75 μm, whereas the penetration at the bottom surface is recorded 60 μm. The penetration of proppant in numerical model has been achieved almost the same as recorded in the experiments, which was 76 μm at the top and 64 μm at the bottom surface. The similarity of the results shows that the developed model has satisfactory results and parametric study can be carried out for further analysis. Once the model has been validated with the experimental results, then the external force was applied on both sides of the proppant to represent the actual condition in the fracture formation.
