## 3 Numerical Examples
### Edge Discontinuities in Unsaturated Porous Media  
This problem is designed to investigate the isothermal behavior of an unsaturated porous plate for verification purposes of the proposed model. The problem is solved in a single edge crack by [20] and two discontinuities are considered in this study. A square plate in the plane strain conditions is given, where two existing cracks with an initial length of 0.05 m are aligned, as in Figure 3. The top and bottom surfaces of the domain are stretched out with the constant displacement rate of \(2.37\times 10^{-7}\)\(\frac{\text{m}}{\text{s}}\). All boundaries are assumed to be impermeable to the moisture. In order to detect the presence of discontinuities via the XFEM, the level-set method is implemented. To define the geometry of each crack independently, three separate functions are required; one represents the crack body, and the other two show crack tips. For a large number of fractures, the calculation of Level Set functions for each crack requires high computational cost. To resolve this problem, the Level Set functions are only calculated for a narrow band of nodes around each crack [25]. Moreover, the leakage flux is calculated for each crack based on Equation (9), and then is added to general equilibrium equations. The material properties are presented in Table 1.  
Due to the geometrical symmetry of the problem, two tensile cracks are initiated and propagate toward each other along the horizontal direction. Since the stress significantly varies around the crack tip, the nonlocal stress may better capture the onset of crack propagation. For stress regularization around a specified radius in the neighborhood of the crack tip, the averaged stress values are calculated instead of the real crack tip stresses. When the value of this nonlocal stress exceeds the material's cohesive strength, the crack starts to propagate. The length of the crack extension is determined based on the stress value at the Gauss points on the potential crack trajectory. If this stress on both Gauss points of an element surpasses the material's tensile strength, the crack bisects that element. In this manner, it is possible that the crack can pass several elements in one step. The crack tip may exist inside an element. To satisfy the zero openings for crack tip displacement, \begin{table}
\begin{tabular}{c c} Young’s Modulus (GPa) & 25.85 \\ Poisson’s ratio & 0.2 \\ Tensile strength (MPa) & 2.7 \\ Fracture energy (N/m) & 95 \\ Initial saturation degree & 1 \\ Residual saturation degree & 0 \\ \(\alpha_{VG}\left(\frac{1}{\text{m}}\right)\) & \(1\times 10^{-6}\) \\ \(n_{VG}\) & 2.3 \\ \(k_{\text{up}}\left(\frac{\text{m}}{\text{m}}\right)\) & \(1.2\times 10^{-11}\) \\ Porosity & 0.2 \\ \end{tabular}
\end{table} 
Table 1: Material Properties of Problem 1. Figure 3: Geometry and loading of Problem 1 (two existing cracks with the initial length of 0.05 m along the middle of a square domain). however, it must be located on the edge of the element where the connected nodes are not enriched. The normal gradient of the suction presented in Figure 4d indicates moisture leakage flux. The normal gradient of suction has opposite values on the two sides of discontinuities, and the difference between normal gradients of suction on two sides of fractures activates the leakage flux term. As a consequence of moisture leakage flux, moisture is absorbed to the discontinuity, and it ends in the change of water content around the cracks. Thus, as depicted in Figure 4c, the degree of saturation decreases around the cracks. Figure 4: _Cont._  
To better analyze the distribution of the cohesive traction along the crack, the normal stress and crack opening at different times of crack propagation are presented in Figure 5a,b, respectively. As the crack propagates, traction-separation force is applied to the crack body. It is clear that the stress value at the crack tip reaches the cohesive strength of the material where the crack has zero openings.