# Numerical Investigation of Slurry Fracturing during Shield Tunneling under a Reservoir
## 2 Numerical Method
### Unit Cracking Principle  
To represent the discontinuity of the cracked element, the elements of the enriched region introduce virtual nodes superimposed on the original real nodes. When the element under load reaches the threshold set for element failure, the element node and the virtual node are separated to produce cracks, and the real nodes on one side of the crack surface on the penetrating cracked element are combined with the virtual nodes on the other side to form a \"unit\". Figure 1 illustrates the crack propagation. a', b', c', and d' are virtual nodes superimposed with conventional finite element nodes a, b, c, and d, respectively. After the unit fails to form a crack surface that penetrates the unit, a new \"unit\" is formed as follows: a, b, c', and d' and a', b', c, and d. Of course, the failed real elements in the finite element model are not deleted, but the failed elements do not participate in the rest of the numerical calculation process.  
To simulate water pressure, additional virtual nodes with pore pressure degrees of freedom were introduced at the edge nodes of each enriched element to simulate fluid flow within the surface of the cracked element. Further, additional virtual nodes are combined with virtual nodes superimposed on the original real nodes to represent the discontinuities in displacement and fluid pressure in the rupture element. A virtual node on an enrichment cell is only activated when a fracture intersects the enrichment cell's boundary. The flow pattern of fracture fluid in the fracture unit is illustrated in Figure 2.  
Assuming that the fluid is incompressible, in order to maintain continuity of the fluid flow, the constitutive response of the fluid flow on the crack surface of the damaged element  
Figure 1: Virtual node and formation of new elements as crack propagates.  
Figure 2: Flow pattern of fluid in fracture.  
includes the tangential flow of fluid on the crack surface and normal fluid loss through the crack surface. The fluid pressure between the fissures acts on the fissure surface, causing the element to have traction separation behavior, thus realizing the simulation of slurry fracturing propagation [36].  
The tangential flow at the fracture surface can be modeled using either Newtonian or power models. When the Newton model is used, the calculation formula of the flow density vector of the fluid is as follows:  
\[Q=-k_{t}\
abla p \tag{4}\]  
where \(Q\) is the fluid displacement in the fracture, \(k_{t}\) is the tangential flow coefficient, which is used to describe the frictional resistance of fluid flow, and \(\
abla p\) is the pressure gradient on the fracture surface.  
The normal flow of fluid on the crack surface can be achieved by defining the fluid loss coefficient of the material in the XFEM. This coefficient describes the relationship between the pressure difference and the flow velocity between the virtual nodes of the fracture element and the fracture surface; that is, the flow velocity of the fracture surface after the action of the permeable layer is considered, as illustrated in Figure 3.  
The normal flow velocity of the fluid can be calculated from Equations (5) and (6):  
\[q_{t}=c_{t}(p_{i}-p_{t}) \tag{5}\]  
\[q_{b}=c_{b}(p_{i}-p_{b}) \tag{6}\]  
where \(q_{t}\) and \(q_{b}\) are the flow rates of the interstitial fluid above and below the fracture, respectively, \(p_{i}\) is the pressure at the virtual node at the boundary of the ruptured element, and \(p_{t}\) and \(p_{b}\) are the pore water pressures at the upper and lower surfaces of the ruptured element, respectively.