# Modeling Hydraulic Fracture Entering Stress Barrier: Theory and Practical Recommendations
## 2 Theory of Fracture Penetration into Stress Barrier
### General Restrictions of Time Stepping Schemes aggravated by Stress Barrier
Notably, the condition (14), like (9) and (12), distinctly shows the rapid growth of computational complications with the growth of the opening \(w\) at the near-barrier zone and/or with decreasing grid size \(\Delta x\). _Comment on implicit schemes_: Employing _implicit_ schemes is another option for overcoming limitations of the explicit Euler integration. For these schemes, high stiffness of the system appears through the high condition number (_cond_) of the matrix. Since implicit integration allows for a time step \(\Delta t\) much greater than that of the Euler step (11), we may employ the analysis of the spectrum given in [47, 48] to conclude on the typical condition number. Omitting details, the result is as follows:  
\[cond\approx 2\frac{\Delta t}{cou} \tag{15}\]  
From (15), it appears that again the Courant time _cou_ is of prime significance for both the stability and efficiency of computations. Using (9) in (15) shows that the condition number grows rapidly, as \(\left(w/\Delta x\right)^{3}\), with the decreasing grid size \(\Delta x\) and growing average opening \(w\). This implies [47, 48] that to avoid an enormous number of iterations in the internal cycle, it is desirable to develop a proper _preconditioner_. The need for an effective preconditioner is further increased when there are _jumps in_ in situ confining _stress_. For such problems, two efficient preconditioners were specially designed in [47, 48]. Both of them distinguish the highly oscillating part of the matrix spectrum by \"discarding all but nearest-neighbor influences\" ([47], p. 1807). Furthermore, the first of the preconditioners employs the _multigrid_ approach (e.g., [52]) to detect the input of high frequencies, generated by the area near a stress jump. The authors used two grids, so that the fine grid was twice smaller in size than the rough one. Thus, the fine grid could catch perturbations caused by the jump. The second of the preconditioners [48] did not directly account for the local perturbation; rather, to decrease the number of iterations, it employed approximate inversion of the distinguished part of the matrix. Their superiority over conventional preconditioners was demonstrated in the cited papers, for example, when a fracture enters a layer with _accelerating_ (negative) stress contrast. However, one should recall that an implicit method requires inversions of the matrix and repeated matrix-to-vector multiplications within a time step. This burden is quite unfavorable for computational complexity and consequently for the time expense. As established in the paper [30] (p. 370), modeling HF by the _stabilized_ explicit RKL2 method \"can be up to 30 times faster\" as compared to the _implicit_ integration. Hence, the limitations on the computational efficiency, revealed for the RKL2 method, certainly refer to explicit methods as well. In particular, excessive time expense, indicated for the channelized propagation between _barriers of high intensity_, is even greater when employing implicit methods. The considerations above show that the options for modeling an HF with spatial discretization are restricted even in the case of _parallel barriers_, when there is a passageway along which a fluid may flow and, consequently, the fracture may propagate. The _general_ computational difficulties discussed further increase when the passageway itself is blocked by high stress barriers. Moreover, there is a _specific_ difficulty, exposed in the next subsection, due to which any method of modeling with spatial discretization may become inapplicable.