# Modeling Hydraulic Fracture Entering Stress Barrier: Theory and Practical Recommendations
## 2 Theory of Fracture Penetration into Stress Barrier
### Specific Computational Difficulty for Modeling High Barriers  
The core physical reason of the difficulties is revealed by considering the extreme case of when at the moment \(t_{B}\), the _entire_ fracture contour reaches a very high, in limit impenetrable, stress barrier. Then, at \(t=t_{B}\), the propagation speed \(v_{*}(t_{B}-0)=v_{*B}\
eq 0\) instantly drops to zero (\(v_{*}(t_{B}+0)=0\)). This drastically changes the asymptotics of fields near the fracture contour. Now, the speed is zero (\(v_{*}(t)\equiv 0\)) for any \(t>t_{B}\). Hence, the fluid particle velocity is also zero at the barrier. Then, in a vicinity of the barrier, the key suggestion, accepted when deriving the Poiseuille-type equation, that in-plane component of the particle velocity is much greater than the component normal to the channel walls, is invalid. The Poiseuille-type equation becomes inapplicable. Physically, a reverse flow, caused by reflection of fluid from the impenetrable barrier, arises. Its detailed description requires inclusion in the analysis of fluid compressibility, the corresponding inertial term, and, at least near the barrier, the cross-sectional component of the particle velocity. This tremendously complicates the numerical solution of the problem. However, with the increase in time, an approximate limiting picture for a viscous fluid tends to become quite simple because of the damping influence of viscous losses. As usual, due to viscosity, the flow \"forgets\" about a state fairly before the current time (cf., e.g., [53; 54]). With the increase in time, the pressure between impenetrable barriers tends to become uniform. For a uniform net pressure and unchanged fracture contour, from the elasticity equation, it follows that the opening between the barriers changes proportionally to the net pressure, with an unchanged form of the distribution. Hence, with the increase in time, the average opening and the uniform net pressure become connected by a factor depending merely on the time. The global mass conservation implies that the average opening and, consequently, the net pressure grows proportionally to the fluid volume between the barriers. For a constant pumping rate, the growth is linear. These specific features of the case considered have unfavorable computational consequences for conventional evaluation of the fluid flux \(\mathbf{q}\) by using the Poiseuille-type Equation (2). The latter implies that \(|\mathbf{q}|^{n}=\frac{w^{2n+1}}{\mu^{\prime}}|\text{grad}p|\), so that for a finite flux, the pressure gradient decreases with time, at least as \(1/w^{2n+1}\). Hence, the conventional calculation of the flux involves the product of the term, which grows in time as \(w^{2n+1}\), by the term, which decreases as \(1/w^{2n+1}\). With the increase in time, conventional calculations unavoidably involve the uncertainty of the type \(\infty\)-\(0\). In particular, for a Newtonian fluid (\(n=1\)) and constant pumping rate (\(Q(t)=Q_{0}\)), commonly used in calculations, the terms are of the orders of \(t^{3}\) and \(1/t^{3}\), respectively. When the time changes three orders from an initial to a final value, each of the factors changes 9 orders with the second of them going to zero. Clearly, practical computations, even performed with arithmetic of double precision, would deteriorate when using the values of openings in grid cells. The deterioration is due to uncertainty. We see that the deterioration of a conventional scheme will occur for any grid size and for any time step. It is caused by the specific features of the particular problem. The error blows up due to the uncertainty \(\infty\)-\(0\) when calculating the flux by using the Poiseuille-type equation. It is noteworthy, however, that in essence the difficulties discussed, both general and specific, arise as a consequence of _spatial discretization_.