**2.2 Calculation of embedment depth**

The factors responsible for the change in fracture width and conductivity after hydraulic fracturing in shale reservoirs include proppant embedment and proppant deformation. The embedment of the proppant involves the penetration of the proppant inside the fracture surface, while proppant deformation is directly related to the strength of the proppant [33]. The deformation of the proppant can be described by Eq. (7), as the proppant can be assumed as an elastic body while the penetration of proppant into a body can be solved using contact mechanics [34]. The contact problem can be formulated as a constrained minimization problem, where the objective function to be minimized is the total potential energy (*Π*) of the bodies in contact. The energy for this system can be written as

$$H(\mathbf{u}) = \frac{1}{2}\mathbf{k}\mathbf{u}^2 - \mathbf{f}\mathbf{u} \tag{7}$$

where *k* is the stiffness matrix, *u* is the displacement field, and *f* is the external force. Several constrained minimization algorithms can be used to solve the problem of the equation, such as the penalty method, the Lagrange multipliers method, and the augmented Lagrangian method. The results presented in this paper are based on the augmented Lagrangian method according to the ANSYS implementation. Augmented Lagrangian methods are a certain class of algorithms for solving c onstrained optimization problems. They have similarities to penalty methods in that they replace a constrained optimization problem with a series of unconstrained problems and add a penalty term to the objective; the difference is that the augmented Lagrangian method adds yet another term, designed to mimic a Lagrange multiplier. The augmented Lagrangian is related to, but not identical with the meth od of Lagrange multipliers. A general discussion of these techniques can be found in the literature on contact mechanics [35–37]. In this study, a numerical model is developed based on the experimental investigation carried out on the embedment of 20/40 mesh proppant (size between 20 and 40 μm) on shale samples from Sungai Perlis beds, Terengganu, Malaysia. The core samples were subjected to uni-axial stress of 20 MPa to find the proppant embedment in the formation, as shown in **Figure 4**. Such a high compression force was applied to investigate the embedment under reservoir conditions.

The embedment cell consists of a transparent cylindrical tube where a shale sample is placed. A metal loading ram is used to load the shale-proppant stack and deformation is measured as the axial load is increased. The deformation in shale, assuming elastic behavior, is quantified using Young's modulus and the applied load. **Figure 5** shows the universal testing machine (UTM) used for measuring

**Figure 6.** *Illustration of embedment of the surface due to external load.*

compression and tensile strengths of materials and samples used in the proppant embedment tests.

In this study, the contact behavior of proppant and rock was carried out using structural mechanics. Experimental boundary conditions are implemented to find the impact of load on proppant penetration in the rock surface. The resulting properties from experiments are introduced in a finite element model to find the effect of fracture surface on the embedment of proppant, as shown in **Figure 6**. A finite element study with ANSYS Workbench has been performed for the computational contact analysis [38]. In a subsurface reservoir, proppants experiences compression from both sides in the formation; therefore, biaxial test should be carried for precise estimation of embedment. Proppants inside the fracture also contact each other due to subsurface stresses. Due to external force, the stressdisplacement relationship is as follows:

$$
\rho\_n \frac{\partial^2 u\_n}{\partial t^2} = \nabla. \sigma\_n + F\_{\rm vn} \tag{8}
$$

where *ρ* is the density, kg/m<sup>3</sup> ; u is the displacement, mm; *σ* is the stress tensor; *Fv* is the external force per unit volume; *n* = 1, indicating the proppant, *n* = 2, indicating the rock matrix. The stress–strain relationship and strain–displacement relationship are shown in Eqs. (9) and (10):

$$
\sigma = \mathbb{C}.\varepsilon \tag{9}
$$

$$\varepsilon = \frac{1}{2} \left[ (\nabla u)^T + \nabla u \right] \tag{10}$$

where *ε* is the strain tensor; *E* is Young's modulus, GPa; *υ* is the Poisson's ratio. As we assume that Young's modulus and Poisson's ratio of the reservoir rock would change from the outer surface of the fracture toward the inside as shown in **Figure 7** and the specific correlation is expressed as follows:

$$E = f(l) \tag{11}$$

$$
v = f(l) \tag{12}$$

where *l* is the distance from the surface of the fracture toward the inside as shown in **Figure 7**. The closure pressure causes the proppant to embed into the fracture surface and the porosity of the propped fracture to change. The embedment of the proppant (*h*em) can be expressed as follows:

$$h\_{\rm em} = u\_m - u\_p \tag{13}$$

where *um* is the displacement of the no-contact part between the fracture and the proppant under closure pressure; *up* is the displacement of the contact part between the fracture and the proppant.

**Figure 7.** *Change in Young's modulus due to fracturing fluid interaction with the shale.*

As the focus is proppant embedment into the shale formation, therefore, proppant is assumed as a rigid body. Thus, no deformation occurs in the proppant. However, the deformation of the shale surface is simulated with the penetration of proppant under uni-axial and bi-axial loads.
