# 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
and expanding the solution in series in \(a\) yields that to the accuracy of \(O\big{(}a^{3}\big{)}\), the ratio \(\frac{P}{T}\) is approximated by the following second-order polynomial:  
\[y(0)=0 \tag{24}\]4  
The derivation assumes that the normalized penetration \(y\) or/and the normalized speed \(V\) are sufficiently small compared to unity. For the Nolte-Smith slope parameter \(b_{P}\), the approximation (31) yields the following:  
\[y(0)=0 \tag{24}\]5  
The maximal value of the slope parameter corresponds to an impenetrable barrier and equals 1.0. This implies that the nominator on the right side of (32) cannot be negative. Hence, the approximate Equation (31) and its corollary (32) are applicable only in the interval \(0\leq a\leq 5/12\). In this range, the slope parameter changes from its maximal value \(b_{P}=1.0\) when \(a=0\), through the minimum \(b_{P}=0.845\) reached at \(a=3.0-\sqrt{7.75}=0.2161\), back to the maximum \(b_{P}=1.0\) for \(a=5/12\). Therefore, in the whole range considered, the slope parameter is close to its maximal value \(b_{P}=1.0\), corresponding to an impenetrable barrier. By the definition (30), the corresponding interval of the normalized time \(t\prime\) is \(0\leq t\prime\leq t^{\prime}{}_{A}\), where  
\[t\prime_{A}= \begin{cases}\hskip 14.226378 pt0&\text{when }0<R_{0}\leq 1.724\\ 0.581R_{0}-1.0&\text{when }R_{0}>1.724\end{cases}  \tag{33}\]  
The time \(t\prime_{A}\) is the maximal normalized time during which the barrier may be considered practically impenetrable (\(b_{P}\approx 1.0\)). This time may be associated with the _arrest time_ caused by the barrier for the fracture propagation. The value \(R_{0}=1.724\) is found in (30) for the maximal acceptable value of \(a\) (\(a=5/12\)) and the minimal possible value of time (\(t\prime=0\)). From (33), it follows that for \(R_{0}>1.724\), the arrest time \(t^{\prime}{}_{A}\) grows linearly with the increase in the starting barrier intensity \(R_{0}\). With these prerequisites, we may compare the exact solution for pressure history \(P(t\prime)\), defined by (29), with the approximation (31). The comparison is given in Figure 4. The arrest times \(t\prime_{A}\), defined by Equation (33), are shown by crosses for each of the starting barrier intensities \(R_{0}\). It can be seen that the approximation (31) is quite accurate in the entire range of the normalized time \(0\leq t^{\prime}\leq t^{\prime}{}_{A}\), from reaching the barrier until the arrest time. After that, the approximation drastically deteriorates. As mentioned, the arrest time \(t\prime_{A}=0.58R_{0}-1.0\) grows linearly with the barrier intensity from zero, when the barrier is weak (\(R_{0}<1.72\)), so that it does not influence the fracture propagation, to values exceeding 1.0 for a sufficiently strong barrier (\(R_{0}>3.5\)). As appears in Figure 3a, the normalized penetration \(y\) during the arrest time is quite small, being less than 0.1 for \(R_{0}=2\) and decreasing to less than 0.01 for \(R_{0}=7\). Consequently, \(\varsigma=1/(1+y)\approx 1\), and by the mass conservation law, the normalized opening, defined in (27), grows during the arrest time proportionally to the normalized time: \(W=T=1+t\prime\). Since \(w=Ww_{B}\) and \(w=Tt_{B}\), this implies that for strong barriers, the cubed opening grows rapidly in time; within the limit, it goes to infinity with the increase in time. Then, as evidenced by Equations (9), (12), (14), and (15), the stability and efficiency of any scheme using spatial discretization would deteriorate. Below, we shall illustrate this by considering the results of the paper [44] obtained for the plane-strain problem solved with spatial discretization. Within the arrest interval \(0\leq t^{\prime}\leq t^{\prime}{}_{A}\), the Nolte-Smith slope parameter is close to its maximal value \(b_{P}=1.0\), corresponding to an impenetrable barrier. When the normalized time exceeds the arrest time, the slope parameter \(b_{P}\) rapidly decreases, which indicates overcoming the barrier.