# Modeling Hydraulic Fracture Entering Stress Barrier: Theory and Practical Recommendations
## 2 Theory of Fracture Penetration into Stress Barrier
### Concept and Range of Barrier Intensity  
In order to develop a theory, it is necessary, first of all, to distinctly define what the barrier _strength_ is. In the Introduction Section, it is mentioned that barrier is a surface of sudden increase in the in situ stress hampering fracture propagation. Thus, the positive stress contrast (6) across the propagation path presents a _dimensional_ characteristic of a stress barrier. Clearly, with a growing stress contrast, the strength of barrier increases. However, it remains unclear which values of this _external_ parameter refer to low, intermediate, or high barriers. Much depends also on the internal parameters driving the fracture growth. This may be seen in Figure 1, which illustrates the asymptotic fields near a fracture tip entering a barrier. It shows that the internal quantity with the stress dimension is the average net pressure \(p_{av}\) near the crack tip. Thus, an appropriate _non-dimensional_ parameter, characterizing hydrofracture penetration, is the following ratio:  
\[R=\frac{\Delta\sigma}{p_{av}} \tag{7}\]  
This parameter defines the actual strength of a stress contrast in reference to the net pressure activating the penetration. We shall call this important parameter barrier _intensity_. Since net pressure changes in time, depends on the pumping rate, fluid viscosity and compliance of rock, the intensity R strongly depends on these factors. Figure 1: Asymptotic scheme for a fracture entering stress barrier. Of special significance is the _starting_ value \(R_{0}\) of intensity (7) at the moment, when the fracture front reaches the barrier. It may be evaluated by using the self-similar solutions to plane-strain (Figure 2a) and axisymmetric (Figure 2b) problems [35; 36; 43]. By using these solutions, it may be inferred that at the moment of reaching a barrier at the distance of \(H/2\) from the inlet, the barrier intensity for a Newtonian fluid is as follows:  
\[R_{0}=k_{0}\frac{\Delta\sigma}{E^{\prime}}\bigg{(}\frac{H^{m}E^{\prime}}{Q_{0} \mu}\bigg{)}^{\frac{1}{4}} \tag{8}\]  
where \(k_{0}=2.36\), \(m=2\), and \(H=x_{B}\) for the plain-strain problem [44], and \(k_{0}=1.63\), \(m=3\), and \(H=r_{B}\) for the axisymmetric problem [31]. In (8), it may be seen that barrier intensity may change over a very wide range, from zero for contrast-free propagation to large values, characterizing practically impenetrable barriers, for which \(R_{0}=\infty\). In particular, with the decreasing pumping rate, the intensity \(R_{0}\) goes to infinity, and a barrier with whatever _small_ stress contrast becomes _impenetrable_. In the opposite case, with the increasing pumping rate, a barrier with whatever _large_ stress contrast becomes \"_invisible_\" for fracture propagation. From the results of [31], it may be concluded that a barrier becomes practically impenetrable when \(R_{0}\) exceeds 7; it is practically invisible when \(R_{0}\) is less than 1. Hereafter, these estimations are confirmed by a special analysis. With the intensity defined, the following questions arise: if, when, why, and how a barrier of a given intensity may cause computational complications? The fact that complications sometimes arise, results from the failure of both explicit and implicit algorithms discussed in the paper [44] for Khristianovich-Geertsma-de-Clerk (KGD) problem [45; 46] with stress contrasts. Evident signs of instability and extreme growth of time expense for calculations have also been observed in truly 3D modeling of fracture propagation within and out of a pay layer between half-spaces, in cases when the fluid viscosity was small, while the stress contrasts grew [30; 31]. As it turns out from the results of these papers, it happens for \(R_{0}>3.5\), when the propagation becomes practically channelized.