# Modeling Hydraulic Fracture Entering Stress Barrier: Theory and Practical Recommendations
## 3 Accuracy and Range of Applicability of Asymptotic Approach
### Comparison with Plane-Strain Solution Obtained by Spatial Discretization  
We consider the example of very high stress barrier studied in the paper [44] by using spatial discretization. All the physical input parameters, except for the intensity of the stress contrast, were the same as in the paper [14]: \(n=1\), \(\mu^{\prime}=1.2\) Pa\(\cdot\) s, \(E^{\prime}=2.5\cdot 10^{4}\) MPa, \(K_{IC}=0\), \(x_{B}=25\) m, and \(Q_{0}=5\cdot 10^{-4}\) m\({}^{2}/\)s. For them, the self-similar solution in [35] yields \(t_{n}=4.8\cdot 10^{-11}\) s, \(t_{B}=203.904\) s, \(w_{B}=2.039\cdot 10^{-3}\) m, \(p_{B}=0.649\) MPa, \(K_{IB}=5.752\) MPa\(\cdot\)\(\sqrt{\text{s}}\), and \(v_{B}=0.0817\) m\(/\)s. The results below refer to the case of the highest stress contrast \(\Delta\sigma=50\) MPa studied in [44] using rather a fine grid with 200 cells along a half-length. The corresponding intensity of the barrier is \(R_{0}=77.04\). Let us first present the results obtained by the _asymptotic approach_ without computational difficulties. They are given in Figures 5-8 with regard to physical time, penetration, propagation speed, and average opening. For the considered example, the stress barrier, while being very high, is _finite_. Thus, the propagation speed stays continuous. The calculations show that it drastically drops from the initial value \(v_{B}=0.0817\) m\(/\)s to \(v_{smin}=0.600\cdot 10^{-4}\) m\(/\)s during the time interval of 12 s. Figure 5 illustrates how fast the drop occurs immediately after reaching the barrier (during the first second). Figure 5: Rapid drop of the propagation speed immediately after reaching the barrier at \(t_{B}=203.904\) s.  
Figure 6: Change in the propagation speed in the entire interval of calculations. Figure 7: Dependence of penetration on time after reaching the barrier. The change in the propagation speed in the whole interval of calculations from \(t_{B}=203.904\) s to \(t=11,000\) s is shown in Figure 6. It clearly demonstrates that the speed decrease happens practically as a jump. After that, the speed starts to grow. The growth is slow and almost linear in the major part of the further penetration of the tip in the barrier. The tip position is presented in Figure 7. For the almost linear change in speed, the penetration \(\Delta z\) grows approximately proportionally to the squared time elapsed after reaching the barrier. From Figure 7, it can be seen that a 5% increase in the fracture length occurs only after some 3000 s. Its length \(2x_{*}\) is nearly the same as the distance between barriers \(2x_{B}\). Then, by the global mass balance \(w_{av}(t)x_{*}(t)=\frac{1}{2}\mathbb{Q}_{0}t\), the growth of the average opening is approximately linear \(w_{av}(t)=w_{B}+\frac{\mathbb{Q}_{0}}{2x_{B}}(t-t_{B})\). The graph of the calculated opening in Figure 8 evidently agrees with such growth. At the time \(t=3203\) s, the opening is 15-fold greater than its starting value \(w_{B}=2.039\cdot 10^{-3}\) m. At the arrest time, which is \(t_{A}=9127\) s, the opening becomes 32-fold greater. In view of (9), (12), and (14), these estimations indicate that there may be difficulties when using spatial discretization. At a large time of \(t=11,000\) s (more than three hours), the fracture length \(2x_{*}\) becomes 1.5-fold greater than the distance \(2x_{B}\) between the barriers. The average opening is \(w_{av}=7.47\cdot 10^{-2}\) m; it is 37-fold greater than the starting value. The results presented in Figures 6-8 evidently demonstrate that beyond a very short interval immediately after reaching the barrier (Figure 5), the input of the viscosity term is negligible, and the penetration occurs in the storage-dominated regime. Mathematically, this means that, in the major part of the penetration into high barrier, the viscosity resistance term \(K_{IA}\) may be neglected in the key Equation (16). Then, the ODE (23) becomes an algebraic equation with the solution \(y=\sqrt{1+2a^{2}}-1\), where \(a\) is defined by (30). The corresponding dimensional penetration is shown in Figure 7 by white circles.