# Modeling Hydraulic Fracture Entering Stress Barrier: Theory and Practical Recommendations
## 2 Theory of Fracture Penetration into Stress Barrier
### Asymptotic Approach
#### Asymptotic Solution in General Case
Thus, for a small penetration, the square-root asymptotics connect the opening with the driving SIF by the classical dependence commonly used in papers on HF modeling:  
\[w(r)=\sqrt{\frac{32}{\pi}}\frac{K_{I}}{E^{\prime}}\sqrt{r} \tag{17}\]  
By using (17), the driving SIF \(K_{I}\) may be evaluated via a fracture opening \(w(d)\) at a distance \(d\) from the barrier as \(K_{I}=\sqrt{\frac{\pi}{32}}E^{\prime}\frac{w(d)}{\sqrt{d}}\). This equation is used in computational and experimental fracture mechanics to find the SIF and/or its critical value \(K_{IC}\). For our topic, it is reasonable to use its integral form, adjusted to accounting for the mass conservation law, which is exactly met by most schemes of HF modeling. To this end, the driving SIF is expressed in terms of the volume \(V_{\Delta s}\) of incompressible fluid, filling a near-barrier rectangular zone. The rectangle has the longer side \(r=d\) orthogonal to the barrier and the side \(\Delta s\) along it. Thus, \(V_{\Delta s}=w_{av}\Delta sd\), where \(w_{av}\) is the average opening over the rectangle. A set of such rectangles may be considered as a grid along the barrier. Actually, this means applying a P3D model to a near-barrier strip. Integration (17) over the rectangle and solving the result in \(K_{I}\) gives the following equation:  
\[K_{I}=\frac{3}{2}\sqrt{\frac{\pi}{32}}E^{\prime}\frac{w_{av}}{\sqrt{d}} \tag{18}\]  
With a growing distance \(d\), the accuracy of (18) is notably better than that of the starting equation. This becomes evident when employing the most unfavorable (maximal) values of the distance \(d\) for a straight or penny-shaped crack under uniform pressure \(p_{net}\). By using analytical solutions for SIFs and openings in these cases, it may be seen that, for the most unfavorable distance \(d=x_{*}\), the error of (18) is about two-fold lower than the error of the starting Equation (17). Thus, in further discussion, we shall use \(K_{I}\) defined by (18). SIF of stress contrast \(K_{I\Delta\sigma}\): For a small penetration \(\Delta z\), the asymptotic picture corresponds to a plane-strain problem for a semi-infinite crack. Then, the asymptotic expression for \(K_{I\Delta\sigma}\) is (e.g., [57]) as follows:  
\[K_{I\Delta\sigma}=-\sqrt{\frac{8}{\pi}}\Delta\sigma\sqrt{\Delta z} \tag{19}\]  
It can be seen, that with a growing stress contrast \(\Delta\sigma\), the term \(K_{I\Delta\sigma}\) in (16) becomes dominating. Tensile critical SIF: The tensile critical SIF \(K_{IC}\) is an assigned characteristic of a material. It is scale dependent, and in field conditions, its value is quite uncertain. For rocks, it normally does not exceed 1 MPa \(\sqrt{m}\). For a high barrier, its input into the condition (16) is much smaller than that of the SIF of stress contrast (19). For instance, in the example of high barrier, given in [30], the stress contrast was high when \(\Delta\sigma=4\) MPa; the grid size was \(\Delta x=2.5\) m. Hence, when modeling the penetration to the accuracy of the mesh size (\(\Delta z=\Delta x=2.5\) m), the resisting SIF (19) was \(-K_{I\Delta\sigma}=10.1\) MPa \(\sqrt{m}\). Therefore, in the problem considered, the input of material resistance \(K_{IC}\) may be neglected. SIF of viscous resistance \(K_{IA}\): As mentioned before, the apparent viscous SIF \(K_{IA}\) is defined by the correspondence principle [15]. The latter connects the SIF \(K_{IA}\) with the propagation speed \(v_{*}\). Since \(v_{*}=\frac{d\Delta z}{dt}\), using the correspondence principle to find \(K_{IA}\) makes the condition (16) an ordinary differential equation (ODE), defining the penetration \(\Delta z(t)\) and, consequently \(v_{*}(t)\), as functions of time. At a high barrier, the penetration speed \(v_{*}\) drastically drops, which results in the drop in viscous resistance. By the correspondence principle, for the zero-penetration speed \(v_{*}\), the apparent SIF \(K_{IA}\) is zero as well. Hence, with growing stress contrast, when penetration speed reaches zero, the viscous term in (16) becomes negligible.