# Modeling Hydraulic Fracture Entering Stress Barrier: Theory and Practical Recommendations
## 4 Practical Recommendations  
### Rough Numerical Estimations of Barrier Intensity  
Conventional calculations for problems involving stress contrasts (e.g., [14; 15; 30; 31; 61]) show that fracture growth tends to become channelized when \(R_{0}>3.5\). In general, using the barrier intensity (7) may serve to indicate when this occurs. Then, as discussed above, conventional tracing of further propagation may be significantly complicated.  
There are two options to use the intensity (7) of a stress barrier, (i) either directly by means of the calculated net pressure, or (ii) through the average opening \(w_{av}\) in the plane-strain zone near the barrier (in particular, in ribbon elements [32]). The first option involves the net pressure, which for low stress contrasts is found quite inaccurately, especially in ribbon elements [34]. The second option employs the opening, calculated much more accurately. It requires to express the intensity \(R_{0}\) via the opening. From equation \(K_{IB}=p_{B}\sqrt{\pi x_{*}}\) for a straight fracture of the half-length \(x_{*}\) (e.g., [42]), it follows that  
Figure 9: Graphs of time dependencies, calculated for a truly 3D model, (**a**) pressure \(p\), (**b**) opening \(w_{F}\), (**c**) penetration \(\Delta z_{F}\), and (**d**) fracture length \(x_{*}\).  
\(p_{B}=K_{IB}/\sqrt{\pi x_{*}}\). Then, using Equation (18) for the driving SIF and substitution of the result into (7) gives the following:  
\[R_{0}=\frac{8\sqrt{2}}{3}\frac{\Delta\sigma}{E^{\prime}}\frac{\sqrt{x_{*}d}}{w_{ B}} \tag{36}\]  
Let us illustrate the two options with the data of the examples, given in [30] for a fracture propagating between parallel barriers. To use (7), it is crucial that for a fluid with the viscosity \(\mu=1\) mPa\(\cdot\)s, roughly \(p_{B}=1\) MPa, \(w_{B}=1\) mm. For a fluid with the viscosity \(\mu=100\) mPa\(\cdot\)s, these values are three-fold greater. Moreover, four values of stress contrasts used in the cited paper were \(\Delta\sigma=1.0,\ 2.0,\ 3.0,\ 4.0\) MPa. The corresponding _theoretical_ values of the intensities for \(\mu=1\) mPa\(\cdot\)s were \(R_{0}=1.17,\ 2.34,\ 3.50,\ 4.68\); for \(\mu=100\) mPa\(\cdot\)s, they are \(R_{0}=0.37,\ 0.74,\ 1.11,\ 1.48\).  
For \(\mu=1\) mPa\(\cdot\)s, using \(p_{B}=1\) MPa in (7) yields \(R_{0}=1.0,\ 2.0,\ 3.0,\ 4.0\) when \(\Delta\sigma=1.0,\ 2.0,\ 3.0,\ 4.0\) MPa, respectively. These values are 17% lower than the theoretical values \(R_{0}=1.17,\ 2.34,\ 3.50,\ 4.68\). The agreement looks strikingly good. It may be a consequence of using a quite coarse mesh (four elements along the half-height). By using \(w_{B}=1\) mm in (36) and the input parameters \(E^{\prime}=3\cdot 10^{4}\) MPa, \(x_{*}=d=10\) m in the example, it gives \(R_{0}=1.26,\ 2.51,\ 3.77,\ 5.03\). These intensities are 8% greater than the theoretical values given above. Again, the agreement is unexpectedly good having in mind that the estimation (36) is quite rough.  
For \(\mu=100\) mPa\(\cdot\)s, the numerical results in [30] show that both the net pressure \(p\) and the opening \(w_{B}\) are three-fold greater than for a fracture driven by a fluid with \(\mu=1\) mPa\(\cdot\)s. Hence, the estimations of intensities \(R_{0}\) by using (7) and (36) are now three-fold lower. By (7), they are \(R_{0}=0.33,\ 0.66,\ 0.99,\ 1.32\). By (36), \(R_{0}=0.42,\ 0.84,\ 1.26,\ 1.68\). Again, they agree with the theoretical values \(R_{0}=0.37,\ 0.74,\ 1.11,\ 1.48\). Much greater errors of up to 50-70% would be acceptable to distinguish a case, when a barrier is certainly weak (\(R_{0}<1\)) and the propagation is nearly contrast-free, from a case when a barrier is certainly high enough (\(R_{0}>10\)) to make the propagation channelized.  
When an estimation shows that the stress contrast is weak, the conventional calculations do not require caution. Otherwise, it is reasonable to trace the net pressure or/and opening histories to conclude if the changes in the computational scheme are desirable.