# 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  
_Problem formulation in terms of SIFs_: The conclusion on the jump to uniform pressure has far-reaching implications for our topic. It suggests using the approximation of _uniform pressure_ in the cross-section orthogonal to a barrier. Then, as follows from the elasticity theory (e.g., [55]), the dominating asymptotics near the barrier are of the square-root type. The factor defining their intensity is the classical tensile stress intensity factor (SIF) (e.g., [42]). Then, employing SIFs becomes the natural way to study penetration into a barrier. For years, the suggestion on the uniform cross-sectional net pressure between plane parallel barriers and the option to use SIFs have been employed in pseudo-three-dimensional (P3D) modeling (e.g., [3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15]). The differences between various P3D models arise merely when accounting for viscous losses by means of the apparent viscous SIF \(K_{IA}\). Actually, the proper way to define \(K_{IA}\) for _symmetric_ stress contrasts was found in [14]. Its general form for _arbitrary_ contrasts is expressed by the correspondence principle [15]. The applicability of this principle in a wide range of stress contrasts has been confirmed by the good agreement of the footprints obtained in comparative P3D and truly 3D calculations [14; 15; 31]. Thus, it is reasonable to follow this path to model stress barriers of arbitrary forms. Let us return to the scheme of Figure 1 for a fracture tip starting penetration into a barrier. The penetration is hampered by the stress contrast \(\Delta\sigma\), which, for a high barrier, is much greater than the tensile strength of rock. In terms of the linear fracture mechanics (e.g., [42]), this means that the absolute value of the (negative) SIF \(K_{I\Delta\sigma}\) may dominate in the combined resistance \(K_{IR}=(-K_{I\Delta\sigma})+K_{IC}+K_{IA}\) as compared with the inputs of the fracture toughness \(K_{IC}\) and the apparent viscous SIF \(K_{IA}\). \(K_{I}\) is denoted as the SIF generated by the driving force, which is the net pressure \(p_{net}\). The fracture propagation into a barrier is possible when \(K_{I}\) is equal (for stable fracture growth) or exceeds (for unstable, jump-like growth) the combined resistance \(K_{IR}\). The expected fracture growth is stable due to the need to adjust the distance \(\Delta z\) of the penetration to a current driving SIF \(K_{I}\). Thus, the fracture condition, defining penetration, is as follows:  
\[(-K_{I\Delta\sigma})+K_{IA}+K_{IC}=K_{I} \tag{16}\]  
This is the _key equation_ for tracing penetration into a barrier. It is formulated in terms of asymptotic characteristics of fields near the fracture tip. Thus, using (16) will be called the _asymptotic approach_. _Asymptotic solution for penetrable high barriers_: The terms substituted in (16) for a general case of stress barrier are specified. _Driving SIF \(K_{I}\):_ According to the general theory (e.g., [56]), Equation (16) implies that, when neglecting the toughness \(K_{IC}\) and apparent viscous \(K_{IA}\) SIFs, the normal stress ahead of a crack tip is finite, while the opening goes to zero as \(w=O\left(r^{3/2}\right)\), where \(r\) is the distance from the tip. However, this asymptotics is valid merely at a very small distance from a tip. Beyond it, the next (square root) term of the asymptotics becomes leading. This asymptotics, conventional in the linear fracture mechanics (e.g., [42; 57]), holds at the distance \(r\) much greater than an initial penetration \(\Delta z\). The distance may be large enough to include a number of grid cells, even for quite rough discretization normally used for HF modeling.