# Modeling Hydraulic Fracture Entering Stress Barrier: Theory and Practical Recommendations
## 2 Theory of Fracture Penetration into Stress Barrier  
For completeness and convenience, the conventional equations for planar propagation of a fracture in homogeneous rock are reproduced (e.g., [12; 27; 30; 32; 33; 34]). These include the following equations. The continuity equation:  
\[\frac{\partial w}{\partial t}=-div\mathbf{q}-q_{l}-Q=0 \tag{1}\]  
The Poiseuille-type equation:  
\[\mathbf{q}=-\bigg{(}\frac{w^{2n+1}}{\mu t}\bigg{)}^{1/n}\Big{|}\
abla p_{f}\Big{|} ^{1/n-1}\
abla p_{f} \tag{2}\]  
The elasticity equation:  
\[-\frac{E^{\prime}}{8\pi}\int_{\rm S}\frac{w(\xi)}{r^{3}}dS_{\xi}=p_{f}(\mathbf{x}) -\sigma_{0n}(\mathbf{x}) \tag{3}\]  
The units of quantities entering these and the forthcoming equations are those of the SI system. Herein, \(w\) is the opening, \(t\) is the time, \(\mathbf{q}\) is the fluid flux in the propagation plane, \(q_{l}\geq 0\), and \(Q\) are the terms accounting for, respectively, fluid leak-off and fluid influx into the fracture, \(\mu^{\prime}=2\left[2(2n+1)\,/n\right]^{n}\!M\), \(n\) is the fluid behavior index, \(M\) is its consistency index (for a Newtonian fluid, \(n=1\), \(\mu^{\prime}=12\mu\) with \(M=\mu\) being the dynamic viscosity), \(p_{f}\) is the fluid pressure, \(E^{\prime}=E/\left(1-v^{2}\right)\) is the plane-strain elasticity modulus, \(E\) is the [PERSON]'s modulus, \(\
u\) is the [PERSON]'s ratio, \(r=\sqrt{(x_{1}-\xi_{1})^{2}+(x_{2}-\xi_{2})^{2}}\), \(\sigma_{0n}\) is in situ traction normal to the fracture surface, the coordinates \(x_{1}\) and \(x_{2}\) of the right Cartesian system are located in the fracture plane, and the coordinate \(x_{3}\) is orthogonal to them; the normal to the plane has the direction of the \(x_{3}\) axis. To simplify the notation, compressive stresses and tractions, as well as the fluid pressure, are assumed positive (thus, \(\sigma_{0n}>0\)). The simplified forms of (1)-(3) for plain-strain and axisymmetric problems may be found in the papers [35; 36], respectively. The forementioned equations are complemented with initial, boundary, fluid front propagation and fracture conditions. The _initial_ condition commonly presumes zero opening along any perspective fracture surface before the start of fluid pumping. When neglecting the lag between the fluid and fracture fronts, the condition of zero opening at each of front points \(\mathbf{x_{\star}}\) serves as the _boundary_ condition \(w(\mathbf{x_{\star}},t)=0\). This condition is met identically when looking for the solution of the hypersingular Equation (3) in the class of functions equal to zero at the fracture front. The fluid front _propagation_ condition is expressed by the speed equation (e.g., [33; 37]). Its formulation makes use of the asymptotic behavior of a solution, which, depending on a particular problem, may correspond to various propagation regimes (see, e.g., [33; 38; 39; 40; 41]). The _fracture_ conditions define the very possibility and the direction of fracture propagation. Commonly, they are formulated in terms of linear elasticity fracture mechanics (LEFM), and the tensile mode (see, e.g., [42]) is assumed. Then, the fracture conditions are as follows:  
\[K_{I}=K_{IC},\ K_{II}=0,\ \ K_{III}=0 \tag{4}\]  
where \(K_{I}\), \(K_{II}\), and \(K_{III}\) are, respectively, normal, shear plane-strain, and shear anti-plane stress intensity factors (SIFs); \(K_{IC}\) is the critical SIF, defined by the strength of a material. The first SIF in (4) defines the very possibility of a fracture growth, and the two remaining SIFs define the direction of the propagation. For a planar fracture, it is assumed that the third equation of (4) is met in the identical way, while the second of (4) conventionally drops out from considerations, when the direction of propagation is assigned along the in-plane normal to the front curve. The discussion of a very complicated issue of the criteria defining the possibility and direction of _out-of-plane_ propagation (e.g., [16]) is not in the scope of the present paper.