# Modeling Hydraulic Fracture Entering Stress Barrier: Theory and Practical Recommendations
## 2 Theory of Fracture Penetration into Stress Barrier
### Asymptotic Approach
#### 2.4.2 Benchmark Problems
Integration of the ODE (23) under the initial condition (24) was performed by using the standard Fortran subroutine IVPRK. It solves the Cauchy problem by the Runge-Kutta-Verner fifth-order method to an assigned tolerance. The tolerance was set as \(TOL=5\cdot 10^{-4}\). The calculations covered six orders of time intervals. The corresponding normalized time \(t^{\prime}=\frac{t-t_{B}}{t_{B}}\) changed from 0.5-10\({}^{-3}\) to 0.5-10\({}^{3}\). To preserve the accuracy on the tolerance level, the calculations were performed in a number of stages by using an output of a finer time scale, to assign the initial condition for the next rougher scale. When having normalized penetration \(y(t^{\prime})\), the normalized average opening \(W\) and net pressure \(P\) are found by means of (27). In the case of the axisymmetric problem, they are given in the remark above. The calculated dependencies of the normalized penetration \(y\), penetration speed \(V=dy/dt^{\prime}\), opening \(W\), and net pressure \(P\) on the normalized time \(t^{\prime}\) are presented in Figure 3. The normalized time, normalized penetration, and speed, which change by some orders, are given in logarithmic scales. \[y(0)=0 \tag{24}\]0 _Cont._  
The graphs quantitatively illustrate the expected physical features of the penetration into the barrier. The curves for the speed (Figure 3b) are especially instructive. They show that there are three distinct stages of penetration. Firstly, the speed decreases due to hampering influence of a barrier (stage I). In the second stage (II), the speed increases due to the growth of the average net pressure (Figure 3d) caused by the barrier. In the third stage (III), the speed again decreases. The decrease starts when the net pressure increases to the level of the stress contrast, so that the _current_ barrier intensity \(R(t)=\Delta\sigma/p_{net}(t)\) amounts about 1.4. The decrease in speed occurs due to turning to the _asymptotic regime_, corresponding to the propagation under the dominating influence of the new _nearly uniform_ in situ stress. Recall that, while the starting in situ stress between the barriers is \(\sigma_{0}\), the in situ stress behind the barriers is \(\sigma_{0}+\Delta\sigma\). Thus, with the growing length of the fracture, the net pressure is to be defined with respect to the latter value. In limit, the propagation corresponds to that in a medium with uniform in situ stresses \(\sigma_{0}+\Delta\sigma\). Consequently, the solution tends to the dependence corresponding to the solution for the case \(R_{0}=0\), studied in [35]. In particular, for a Newtonian fluid, this solution yields that the speed decreases as \(t^{-1/3}\) with the increase in time. Such limiting behavior clearly appears in Figure 3a-c, which show that the solutions for \(R_{0}=0\) envelope all the curves for various starting intensities \(R_{0}\), when the time increases. _Nolte-Smith parameter and arrest time_: Recall that the change in the pressure with time is of special value for conclusions on the fracture propagation (e.g., [31; 58; 59]). Thus, it is reasonable to study the dependence \(P(t^{\prime})\) in more detail. From the second dependency in (27), it may be inferred that, in the extreme cases of zero (\(R_{0}=0\)) and infinite (\(R_{0}=\infty\)) intensities of the barrier, the dependences of the pressure on the time become monomial \(P=T^{b_{P}}\) with the Nolte-Smith [58] slope parameter \(b_{P}=\frac{d\log P}{d\log P}\) equal to \(-1/3\) for \(R_{0}=0\) and \(1.0\) for \(R_{0}=\infty\). Furthermore, the second dependency in (27) is written as follows:  
\[y(0)=0 \tag{24}\]1  
with  
\[y(0)=0 \tag{24}\]2  
\[y(0)=0 \tag{24}\]3 Dependencies of (**a**) the normalized penetration \(y\), (**b**) speed \(V\), (**c**) opening \(W\), and (**d**) pressure \(P\) on the normalized time \(t^{\prime}\) for various values of the barrier intensity.