# 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
Actually, in the graphical representation, it is indistinguishable from the exact solution to ODE (23). Similarly, the dependence of the net pressure on time also practically coincides with the approximation (31) in the whole range, from reaching the barrier at \(t_{B}=203.904\) s to the arrest time \(t_{A}=9127\) s.  
We should now compare the solution to (23) and (24) with the solution [44, 61], obtained by _spatial discretization_. In the paper cited, the problem was solved for grids having from 20 to 200 nodes at a fracture half-length and for stress contrasts varying from \(R_{0}=0\) (classic KGD case), through moderate (\(R_{0}\) from 2 to 3.5) and strong (\(R_{0}=10\)) barriers, up to an extremely high barrier with the mentioned intensity \(R_{0}=77.04\). For the finest grid \(\Delta x=1.25\cdot 10^{-2}\) m, the Courant time (9) when reaching the barriers was \(cou=1.1\cdot 10^{-8}\) s. Since for an explicit scheme of integration, a time step \(\Delta t\) cannot exceed \(cou\), and the step  
Figure 8: Dependence of average opening on time after reaching the barrier. \(\Delta t\) used in the paper cited was quite small (\(\Delta t<1.1\cdot 10^{-8}\) s). Using such a small time step was possible because in the problem, which is one-dimensional, the order of the matrix resulting from spatial discretization was relatively small. Conventional _explicit_ (forward Euler and Runge-Kutta of the fourth order) and a number of _implicit_ schemes (backward Euler, Petzold-Gear backward differential formula (BDF) and Brayton-Gustavson-Hachtel BDF [62]) were used for time stepping. It was established that even for rather high stress contrasts of the intensity \(R_{0}=10\), all the methods, both explicit and implicit, were stable and provided the same (within the tolerance accepted) numerical results. However, for _extremely high barrier_ (\(R_{0}=77.04\)), studied above with the asymptotic approach, with the increase in time, the opening becomes 30-fold greater. Thus, the Courant time (9) becomes four orders smaller. As a consequence, the stability of calculations drastically decreases. As reported in the paper cited, the explicit as well as the implicit backward Euler and Petzold-Geer BDF methods failed. In this case, only the Brayton-Gustavson-Hachtel BDF method provided stable, physically consistent results. As mentioned above, it provided results indistinguishable from those of the asymptotic approach. Still, as it is clear from the specific difficulty discussed in Section 2.3 that even this extremely stable method would unavoidably fail due to further growth of the opening when considering barriers of higher intensity and greater time intervals. It is also valuable to compare details of the solutions revealed for the extremely high barrier. Note first that, as mentioned, the error of finding the front position was of the grid size order. Apart from that, _immediately after_ reaching the barrier, the calculated _particle velocity_ had high-frequency oscillations with the length of the mesh size \(\Delta x\) along the feature (Figure 4 of the paper [61]). The oscillations rapidly (during 10 s) damped practically to zero. Evidently, this was a grid-dependent effect. Hence, the solution, obtained with spatial discretization cannot accurately reproduce the abrupt drop in the propagation speed after reaching the high barrier. _In contrast_, the asymptotic approach, avoiding spatial discretization, provides physically consistent results on the speed drop immediately after reaching the barriers. Figure 5 above illustrates this. It shows that the speed drops by an order during a very short interval of 0.04 s after reaching the barriers. In the scale of minutes (Figure 6), the drop looks instant. Another noteworthy feature refers to the change in the propagation speed _well after_ reaching the barrier. As appears from the paper [61], for 50 min after reaching the barrier, the _propagation speed_ has oscillations.