# Modeling Hydraulic Fracture Entering Stress Barrier: Theory and Practical Recommendations
## 3 Accuracy and Range of Applicability of Asymptotic Approach
### Accuracy of Asymptotic Approach in General Case  
_Errors of asymptotic approach caused by rough evaluation of driving SIF_: The benchmark solutions are found by using in (16) the exact formulae for the SIFs \(K_{I}\), \(K_{I\Delta\sigma}\), and the average opening \(w_{av}\). The solutions may serve to estimate the error of the penetration \(\Delta z\) in a general case. The most unfavorable estimation (17) of the average opening \(w_{av}\) corresponds to the maximal value of the distance \(r=d\), which is the fracture half-length \(x_{*}\) in the plane-strain and the radius \(r_{*}\) in the axisymmetric case. In the plane-strain case, the worst estimation of the average opening is \(w_{av}=\pi x_{*}\frac{p_{tot}}{E^{\prime}}\). Its substitution into (22) gives \(\frac{\Delta z}{x_{*}}=\left(\frac{3\pi^{2}}{3E}\right)^{2}\left(\frac{p_{tot} }{\Delta\sigma}\right)^{2}\). Comparing the factor \(\left(\frac{3\pi^{2}}{32}\right)^{2}\) with the factor \(\frac{\pi^{2}}{8}\) in Equation (21), obtained from the analysis of the exact solution, shows that the approximate numerical factor is 30.6% lower. A similar result follows also for the axisymmetric problem. The given estimation refers to the upper bound of the errors because we have used the most unfavorable value of the distance \(d\). Having in mind that, in practical modeling of hydraulic fractures, the driving SIF is found quite inaccurately, the accuracy provided by using (22) appears acceptable. _Accuracy of asymptotic solution as compared with results of truly 3D modeling_: The conclusion that, in the case of a closed impenetrable contour, the net pressure becomes nearly constant suggests that, in general, when the stress barrier is high (\(R_{0}>3.5\)), the net pressure at a plane-strain rectangular zone, discussed in Section 3.3, is nearly constant. Hence, when solving numerically the elasticity equation, the entire rectangle may be taken as a single grid cell. For a barrier with a straight boundary, a set of such rectangles presents a strip, in which the pressure changes in the direction parallel to the boundary. In the cases when there are two high barriers with parallel boundaries, the rectangles with constant pressure elongate up to merger. Then, we have merely a 1D grid along barriers, and the scheme becomes that of the P3D model. Detailed truly 3D calculations, performed in [30] and reproduced in [31] for a fracture between parallel stress barriers with various intensities, have shown that the net pressure rapidly becomes constant over a cross-section. The propagation becomes channelized with the length-to-height ratio about 30 when \(R_{0}\approx 3.5\). For the greatest of the modeled intensities, \(R_{0}=4.66\), the ratio reaches 50. Thus, the numerical results of the papers cited may serve to conclude on the accuracy of the approximate Equation (22) when the stress contrast is not extremely high. We shall use _our_ results, obtained for the same input data as those in [30]: the pay layer height is \(H=20\) m; the pumping rate is \(Q=0.083\) m\({}^{3}\)/s; the fluid is Newtonian (\(n=1\)) with the dynamic viscosity \(\mu=1\) mPa-\(s\); and the plane-strain elasticity modulus is \(E^{\prime}=3\cdot 10^{4}\) MPa. We consider two stress contrasts in the embedding layers, \(\Delta\sigma=2\) MPa (\(R=2.34\)) and \(\Delta\sigma=3\) MPa (\(R=3.5\)). For these intensities, the barrier is in the range between quite week (\(\Delta\sigma=1\) MPa, \(R=1.17\)), for which the propagation is nearly contrast-free, and quite high (\(\Delta\sigma=4\) MPa, \(R=4.66\)), for which the propagation is entirely channelized. The fracture toughness is set equal to zero (\(K_{IC}=0\)). Similar to [30], the time stepping is performed by the _explicit_ stabilized Runge-Kutta-Legendre (RKL) method of the second order (RKL2) [51]. In contrast to [30], the system of ODE includes speed equations in accordance with [34]. Similarly to [30], the final time of modeling is about 10 min.