# Modeling Hydraulic Fracture Entering Stress Barrier: Theory and Practical Recommendations
## 4 Practical Recommendations
### Numerical Indication of Desirable Changes in Computational Scheme  
In the case when the rough estimations of the previous subsection show that at the moment of reaching a barrier its intensity \(R_{0}\) exceeds 3.5, the dominant propagation near the barrier may be along it. At this moment, much depends on the conditions in other parts of the fracture front. For instance, if the high intensity \(R_{0}\) occurs only at the upper boundary of a pay layer, while there is no stress contrast on its lower boundary, then with the growing distance from the source, the solution tends to that corresponding to the solution for contrast-free propagation (\(R_{0}=0\)) of the fracture driven by the source with two-fold lower pumping rate [31]. Clearly, _no changes_ in the computational scheme are needed. In this case, the net pressure decreases in time as \(t^{bp}\) where \(b_{p}=-n/(n+2)\). In the opposite case, when strong stress contrasts occur at both boundaries of a pay layer, the propagation becomes entirely channelized within the layer. The net pressure, as mentioned, after reaching the barriers becomes nearly uniform across the pay layer. This meets the key suggestion of a P3D model and justifies its use. The maximal and average openings in a cross-section become proportional to the net pressure, and the pressure and opening change in time with the same exponent \(b_{P}\) in their monomial dependence \(t^{bp}\)[31]. For a strong barrier, \(b_{P}\) is positive. In the case of a Newtonian fluid, \(b_{P}=0.2\) for an impermeable and \(b_{P}=0.125\) for a highly permeable pay layer. The _pressure gradient decreases_ as \(1/t^{1-bp}\), and the Poiseuille-type equation implies that the flux is defined by the product of two terms, one of which grows in time as \(t^{3 bp}\), while the other decreases as \(1/t^{1-bp}\). Since \(b_{P}\leq 0.2\), the maximal product is of the form \(t^{0.6}\cdot\left(\frac{1}{t^{3b}}\right)\). It decreases at the rate of \(t^{-0.2}\) in time. Therefore, no computational problems, caused by uncertainty of the type \(\infty\)-\(0\), arise. Still, problems may arise due to another reason, discussed in Section 2.2. Specifically, when modeling a channelized propagation by using a spatial grid, the number of the unknowns grows in time proportionally to the fracture length; for a Newtonian fluid, it is proportional to \(t^{0.8}\). Then, in the time interval from seconds to first hours, the number of unknowns grows nearly three orders. This drastically increases the number of work units, the cost of a single work unit (single matrix-to-vector multiplication) and also the condition number of the elasticity matrix. The practical implication is as follows: in the case when the propagation tends to become channelized (\(R_{0}>3.5\)), it is reasonable to change a truly 3D computational scheme to a much more robust, stable, and accurate 1D scheme of the _P3D model_ in its improved form [15]. The latter may be notably simplified by using the asymptotic approach in the form (21) to simplify tracing the fracture height, since the improved P3D model suffices using simple analytical formulae to calculate SIFs. At last, in the case, when a strong barrier has a closed contour surrounding the injection source, both the average net pressure and average opening grow linearly in time. As noted, this implies that with the increase in time, computations, even performed with double precision, would deteriorate. The implication is as follows: it is reasonable to change the computational method when modeling a time interval of practical interest. In this case, the problem is promptly removed by employing the _asymptotic approach_. It is sufficient to use (21) and to solve a linear elasticity problem for a planar crack under uniform pressure.