# 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
In terms of physical quantities, the equality \(\left.\frac{dy}{dt^{\prime}}\right|_{t^{\prime}=0}=\gamma_{x}\) expresses that, as it should be, the propagation speed \(v_{*}=\frac{d\Delta z}{dt}\) at the moment of reaching the barrier equals to the speed \(v_{B}=\gamma_{x}\frac{x_{B}}{t_{B}}\), defined by the self-similar solution. Thus, the problem is reduced to the Cauchy problem (23) and (24). It contains solely three external parameters; these are the starting barrier intensity \(R_{0}=\frac{\Delta\sigma}{p_{B}}\), the fluid behavior index \(n\), and the ratio \(\frac{K_{IC}}{K_{IB}}\). The solution \(y(t^{\prime})\) to the Cauchy problem (23) and (24) is promptly found numerically by using a standard subroutine. When having \(y(t^{\prime})\), the normalized average opening \(W\) and normalized net pressure \(P\) become known as well. Omitting again technical details, they are as follows:  
\[W=T_{\xi},\;\;P=T_{\xi}{}^{2}+\frac{2}{\pi}R_{0}\bigg{(}\arccos\varsigma- \varsigma\sqrt{1-\varsigma^{2}}\bigg{)} \tag{27}\]  
where  
\[W=\frac{w_{av}}{w_{B}},\;P=\frac{p_{net}}{p_{B}},\;T=\frac{t}{t_{B}}=1+t^{ \prime},\;\varsigma=\frac{1}{X},\;\;X=\frac{x_{*}}{x_{B}}=1+y \tag{28}\]  
Note that the first part of (28) actually expresses the mass conservation law written in the normalized variables. It shows that, for small penetration (\(y\ll 1\)), the average opening is proportional to time. _Remark on axisymmetric problem_: Similar to the modified KGD scheme (Figure 2a), the modified axisymmetric scheme (Figure 2b) is one-dimensional with \(x\), \(x_{*}(t)\), and \(x_{B}\) replaced now with polar radii \(r\), \(r_{*}(t)\), and \(r_{B}\). As a result, we arrive at the same Cauchy problem (23) and (24), with changes in the constants entering the normalizing quantities (26) and in numerical coefficients. Specifically, they are changed to the following:  
\[t_{B}=t_{n}\bigg{(}\frac{r_{B}^{3}}{\zeta_{n}^{2}\lambda_{0} \lambda_{n}}\bigg{)}^{1/(3\gamma_{r})},\;\;w_{B}=\frac{Q_{0}t_{B}}{\pi r_{B}}, \;\;p_{B}=\frac{E^{\prime}}{\pi}\frac{w_{B}}{r_{B}},\;K_{IB}=\frac{2}{\sqrt{ \pi}}p_{B}\sqrt{r_{B}},\;v_{B}=\gamma_{r}\frac{r_{B}}{t_{B}},\] \[\gamma_{r}=\frac{2}{3}\frac{n+1}{n+2},\;\alpha_{r}=2\frac{n+3}{n+2 },\;\beta=\frac{1-0.5n}{n+2},\omega=\frac{n+2}{n}\]  
where \(\zeta_{*n}\) is tabulated in the paper [43]; its values slowly increase with decreasing \(n\) from \(\xi_{*n}=0.698\) for a Newtonian fluid (\(n=1\)) to \(\zeta_{*n}=0.733\) for a perfectly plastic fluid (\(n=0\)). The variables \(y=\frac{\Delta z}{r_{B}}\), \(t^{\prime}=\frac{t-t_{B}}{t_{B}}\) are defined as in (23) by (25). In view of the entire analogy with the plain-strain case, it is sufficient to consider the latter. The normalized average opening \(W=\frac{w_{av}}{w_{B}}\) and the normalized net pressure \(P=\frac{p_{net}}{p_{B}}\) for the axisymmetric problem are as follows:  
\[W=T_{\xi}{}^{2},\;\;P=T_{\xi}{}^{3}+R_{0}\Big{(}1-\varsigma^{2}\Big{)}^{3/2}\]  
with \(T\) and \(\varsigma\) defined in (28). _Numerical results and their interpretation_: Both for plane-strain and axisymmetric problems, the Cauchy problem (23) and (24) contains merely three external parameters; these are the fluid behavior index \(n\), the ratio \(\frac{K_{IC}}{K_{IB}}\), and the starting intensity of the stress contrast \(R_{0}=\frac{\Delta\sigma}{p_{B}}\). It is promptly solved by a standard solver when the behavior index \(n\) is non-zero. If \(n=0\), the problem may be reformulated to account for the degeneration of the Poiseuille-type equation in this case. For certainty, we focus on the plane-strain problem and on a commonly considered case of a Newtonian fluid (\(n=1\)). Then, \(\gamma_{x}=2/3\), \(\alpha=5/3\), \(\beta=1/6\), \(\omega=3\), \(t_{n}=u^{\prime}/E^{\prime}\), and \(\xi_{*n}=0.615\). When neglecting the toughness \(K_{IC}\), the ODE (23) contains only one parameter, the starting intensity of stress contrast \(R_{0}\). We present its benchmark solutions for various \(R_{0}\).