# 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
The results of truly 3D modeling are presented in Figure 9. Of special interest are the final values of the displayed quantities. For the final time, it can be seen that the length-to-height ratios are about 15 and 26 for the stress contrasts \(\Delta\sigma=2\) MPa and \(\Delta\sigma=3\) MPa, respectively. Thus, in both cases, the influence of the barrier is significant, while it is notably greater for the higher barrier, so that the final penetrations into the barrier decrease by nearly two-fold, from \(\Delta z_{F}=19.4\) m for \(\Delta\sigma=2\) MPa to \(\Delta z_{F}=10.6\) m when \(\Delta\sigma=3\) MPa. In contrast to the footprints, the final values of the pressure and opening stay at the same level in both cases. Specifically, the final pressures are, respectively, 21.6 MPa and 21.9 MPa; the final openings are, respectively, \(w_{F}(0)=3.2\) mm and \(w_{F}(0)=3.3\) mm. The latter values may be used in (22) to see to which extent this rough estimation of the penetration agrees with the penetration obtained for a truly 3D model. When using in (22) the most unfavorable \(d=\frac{H}{2}=10\) m, the corresponding penetrations are \(\Delta z_{F}=20.0\) m and \(\Delta z_{F}=9.4\) m for, respectively, \(\Delta\sigma=2\) MPa and \(\Delta\sigma=3\) MPa. The agreement with the final penetrations \(\Delta z_{F}=19.4\) m and \(\Delta z_{F}=10.6\) m, discussed above, is satisfactory. Thus, despite the fact that estimation (22) is rough, it may be used for practical calculations performed in a conventional way. Figure 9: \(Cont.\)  
Truly 3D modeling tends to take into account the major factors influencing hydraulic fracture propagation. Still, it cannot reliably reproduce field conditions because of uncertain, incomplete, and unavailable input data and also due to specific local factors that are not present in the model. The final footprints and openings are unknown as well. The lack of complete field data for case studies is compensated by employing data of specially designed laboratory experiments performed under assigned input and controlled output data. The successful validation of truly 3D models against available laboratory experiments (e.g., [14; 29; 30]) justifies using truly 3D modeling as an important method to generate _pseudo-experimental_ dependencies of output quantities on the time and to produce \"real case\" examples. Recently, it was employed in our work [31] to generate pseudo-experimental pressure histories for various stress contrasts. These histories were considered _experimental_. They served for the identification of actual stress contrasts by minimizing the misfit between them and _theoretical_ pressure histories. Since the problem is non-linear, the minimization required iterations with updated stress contrasts for generating a current _theoretical_ history. To avoid enormous time expense for repeated iterations, the theoretical histories were generated by using the _key Equation (16) of the asymptotic approach_ at each of the iterations. Its application made the time expense for a single iteration not exceed 3 s. The iterations employing (16) were normally converged. In many examples, given in the cited paper, they converged to sound estimations of the identified stress contrasts. These examples, as well as rapid intermediate calculations in each of the iterations, may serve as \"real case\" examples of positive validation of the approach efficiency.