# 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
Therefore, there arises the need for using either _implicit_ (e.g., [50]) or _stabilized explicit_ (e.g., [49; 51]) schemes. Let us consider the limitations of these methods when applied to penetration into a barrier. We start from the stabilized explicit schemes as a natural improvement in the classical explicit Euler stepping. _Stabilized explicit Runge-Kutta schemes_: These schemes were developed specially to smoothen the quite unfavorable limitation of the Courant-Friedrich-Levi (CFL) condition of instability [49; 51]. The explicit stabilized Runge-Kutta-Legendre 2 nd order (RKL2) method, suggested and studied in [51], appears to be highly stable, accurate, and efficient. Its first applications to hydraulic fracture modeling [30] have evidently demonstrated its advantages over other presently used methods employing spatial discretization. The RKL2 method enables the easy estimation of the corresponding number of work units \(NWU_{RKL2}\). \(N_{S}\) is denoted as the (odd) number of stages, employed in a super step of the RKL2 method [51]. The required number of work units performed in a super step is as follows:  
\[NWU_{RKL2}\approx\frac{4}{N_{S}}NWU_{E} \tag{13}\]  
Equation (13) shows that the efficiency of the RKL2 method is \(N_{S}/4\)-fold greater than the efficiency (12) of the explicit Euler scheme. In practical calculations for HF modeling by the RKL2 method, the number of stages \(N_{S}\) does not exceed 399 (in average, it is at the level of 111). Further increasing \(N_{S}\) leads either to inacceptable growth of the time step, so that the front propagates several grid cells, or what is worse, to the notable influence of rounding errors. With the maximal \(N_{S}=399\), the efficiency (13) of the RKL2 method is 100-fold greater than that of the explicit Euler. Thus, the number of work units \(NWU_{RKL2}\), computational complexity \(C_{RKL2}\), and time cost \(T_{CRKL2}\) of the most advanced modern integration scheme for tracing HF are 100-fold lower than those given above for the explicit Euler scheme. When employing the RKL2 method, the critical value \(cou_{RKL2}\) of the Courant time (9), acceptable for practical calculations, may be taken 100-fold lower than its value acceptable when using explicit Euler. We see that the Courant time \(cou\) is also of prime significance for the _time expense_ of an HF modeling by the _explicit stabilized_ Runge-Kutta (RK) methods. The numerical data, obtained with the RKL2 method for the HF problem with parallel barriers, give an estimation of the minimal \(cou\), which is close to the bound, beyond which the time expense of the modeling becomes enormous. From the results of modeling channelized propagation of a fracture driven by a fluid with the dynamic viscosity \(\mu~{}=~{}1\) mPa\(\cdot\)s [30; 31], it appears that tracing the propagation time of the duration of 10 min required the computation time expense of 3 h. The corresponding Courant time, calculated by using (9), is \(cou=1.75{\cdot}10^{-4}\) s. To model further propagation up to the duration of 1.5 h, it would require much greater time expense. Roughly, the less restrictive critical \(cou_{cr}=1.5{\cdot}10^{-4}\) s may be taken as an acceptable bound. Thus, to model 3D propagation of an HF on a given spatial grid, the following condition of computational efficiency is to be met:  
\[t_{n}\bigg{(}\frac{\Delta x}{w}\bigg{)}^{3}>cou_{cr} \tag{14}\]  
where \(cou_{cr}\approx 1.5{\cdot}10^{-4}\) s. The condition (14) may serve to control if numerical modeling with spatial discretization is possible using the quite advanced explicit stabilized RKL2 method. Our calculations with this method for channelized fracture propagation between _high barriers_ (\(R_{0}>10\)) show that, when the driving fluid is thin, in particular \(\mu=1\) mPa\(\cdot\)s, the modeling is nearly at the limit of computational potential.