# 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  
_Formulation of benchmark problems for HF penetration into stress barrier._ The approximate solution (22) is derived by using, for driving and resisting SIFs, their asymptotic forms (18) and (19). In cases of plane-strain and axisymmetric problems, its accuracy may be checked by comparing with the results obtained using exact equations for these SIFs. In general, we shall not neglect the inputs of the toughness \(K_{IC}\) and viscosity \(K_{IA}\) terms in the key Equation (16). Including them into the analysis will help us to clarify details of the rapid (nearly instant) speed drop immediately after reaching a high barrier, and to thoroughly compare the asymptotic solution with the solution obtained by spatial discretization. Consider the symmetric scheme (Figure 2a) for a straight fracture of the length of \(2x_{*}(t)\), with tips penetrating into barriers with stress contrast \(\Delta\sigma\). The boundaries of the barrier are at the distance \(x_{B}\) from the origin, where a pointed source with the pumping rate \(Q_{0}\) is located. The penetration \(\Delta z(t)\) into the barrier is \(\Delta z(t)=x_{*}(t)-x_{B}\). Because of symmetry, it is sufficient to consider the right part of the picture, for which the fluid influx is \(Q_{0}/2\). The exact equations for SIFs entering (16) are given explicitly in the paper [15]. There is no need to reproduce them because only the resulting ODE is of interest for our topic. Omitting technical details, the resulting ODE is as follows:  
\[\frac{dy}{dt^{\prime}}=\gamma_{x}\left[\left(\frac{1}{1+y}\right)^{\alpha_{x} }\left(1+t\prime-\frac{2\sqrt{2}}{\pi}R_{0}\sqrt{y+y^{2}/2}\right)-\left( \frac{1}{1+y}\right)^{\beta}\frac{K_{IC}}{K_{IB}}\right]^{\omega} \tag{23}\]  
where \(y=\frac{\Delta z}{x_{B}}\) is the normalized penetration; \(t^{\prime}=\frac{t-t_{B}}{t_{B}}\) is the normalized time, counted from the time of reaching the barrier; and \(R_{0}\) is the _starting intensity_ of the barrier at the moment. The exponents entering (23) are \(\gamma_{x}=\frac{n+1}{n+2}\), \(\alpha_{x}=\frac{n+4}{n+2}\), \(\beta=\frac{1-0.5n}{n+2}\), and \(\omega=\frac{n+2}{n}\); they depend merely on the fluid behavior index \(n\). The ODE (23) is to be solved under the initial condition of zero penetration at \(t^{\prime}=0\):  
\[y(0)=0 \tag{24}\]  
The normalized variables in (23) and (24) are defined as the following ratios:  
\[y(t)=\frac{\Delta z(t)}{x_{B}}=\frac{x_{*}(t)-x_{B}}{x_{B}},\ \ t^{\prime}=\frac{ \Delta t}{t_{B}}=\frac{t-t_{B}}{t_{B}} \tag{25}\]  
Recall that \(x_{B}\) is the half-distance between the barriers, and \(t_{B}\) is the time of reaching the barriers. Apart from that, the derivation employs the average opening \(w_{B}\), net pressure \(p_{B}\), and stress intensity factor (SIF) \(K_{IB}=p_{B}\sqrt{\pi x_{B}}\) at the instant \(t_{B}\). With \(x_{B}\) being assigned, other normalizing quantities are known from the self-similar solution [35] to the classical KGD problem. They are as follows:  
\[t_{B}=t_{n}\left(\frac{x_{B}{}^{2}}{\xi_{*n}^{2}Q_{0}t_{n}}\right)^{1/(2\gamma _{x})},\ \ w_{B}=\frac{Q_{0}t_{B}}{2x_{B}},\ p_{B}=\frac{E^{\prime}}{\pi}\frac{w_{B}}{x_ {B}},\ K_{IB}=p_{B}\sqrt{\pi x_{B}},\ v_{B}=\gamma_{x}\frac{x_{B}}{t_{B}} \tag{26}\]  
where \(\xi_{*n}\) is the self-similar fracture half-length, depending solely on the fluid behavior index \(n\); it is tabulated in the cited paper and its values slowly increase with decreasing \(n\) from \(\xi_{*n}=0.615\) for a Newtonian fluid (\(n=1\)) to \(\xi_{*n}=0.654\) for a perfectly plastic fluid (\(n=0\)). The normalized start velocity is \(\left.\frac{dy}{dt^{\prime}}\right|_{t^{\prime}=0}=\gamma_{x}\) to the accuracy of the ratio \(\frac{K_{IC}}{K_{IB}}\), which is neglected at \(t\leq t_{B}\)_before_ reaching the barrier in the viscosity dominated regime. The ratio may become of essence only well _after_ reaching the barrier (at \(t\gg t_{B}\)), when the regime turns to the toughness dominated due to significant drop of the propagation speed.