# Modeling Hydraulic Fracture Entering Stress Barrier: Theory and Practical Recommendations
## 1 Introduction  
Since the early studies of hydraulic fracturing (HF) (e.g., [1, 2]), the impact of stress inhomogeneity on fracture propagation has been taken into account. Specifically, channelized propagation along a pay layer between impenetrable neighbors was simulated by the classical Perkins-Kern-Nordgren (PKN) model. However, with increasing pumping pressure, a fracture also extends into neighbor layers, which affects final footprints and openings, and thus, the efficiency of an HF treatment. This stimulated efforts to account for fracture growth beyond a pay layer, which resulted in the successive development ofpseudo-three-dimensional (P3D) models [3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15]. They complemented the PKN model with equations for out-of-layer growth. All these models presume that the variation in in situ stresses, caused by the difference in elastic modules of rock layers, impacts fracture growth much more significantly than the difference in the modules itself. Then, rock mass may be assumed homogeneous, while the assigned initial stresses change. Numerical results, obtained by solving problems for crack intersecting layers with different elastic modules and different assigned stresses [16; 17], support the suggestion of the predominant impact of stress contrasts. Other solutions for 3D layered structures with planar cracks [17; 18; 19; 20; 21] agree with it as well. This assumption notably simplifies computations by avoiding the need to build, store, and repeat using a special green function for layered media. It suggests an extension to general problems when a crack arises and propagates in an inhomogeneous medium with initial (in particular, residual) stresses, where the sizes of inhomogeneities are comparable or exceed those of a considered crack. Concerning inhomogeneities in sizes less than those of a crack considered, they may be accounted for through effective modules by well-developed methods (e.g., [22; 23]). Thus, the model of a homogenous medium with non-uniform initial stresses seems acceptable for solving practical problems, especially those like modeling HF, for which some important input parameters (including rock structure, properties, and in situ stresses) are often uncertain or unknown and estimated quite roughly. For HF problems, the assumption of a homogeneous medium is additionally supported by a specific physical feature. The driving factor of hydraulic fracture propagation is the _net pressure_, which is the _difference_ between the actual fluid pressure and a typical in situ stress at a treatment depth. The net pressure is much less than typical in situ stresses, and the spatial changes in the latter are of dominant significance. Thus, in a further discussion, we focus on jump-like changes in in situ confining stresses, while rock properties are assumed homogeneous. Furthermore, the case when the fracture enters the surface from the area with lower confining stresses will be solely discussed. In this case, the surface of the jump presents a stress _barrier_, which hampers the propagation and complicates numerical modeling. Importantly, in this review and in the following exposition, we consider only _stress_ barriers. Other types of barriers on the path of fracture propagation caused by heterogeneous conditions, such as sudden changes in the fracture toughness of layers (e.g., [24]), multiple weak interfaces [25], or changes in material properties on a propagation path (e.g., [26]), are beyond our study. It focuses on the _stress barriers_ arising from _jumps in in situ stresses_, which are assumed known from special experiments or assigned from theoretical considerations. Furthermore, this study focuses primarily on computational aspects of accounting for stress barriers under the assumption that the conventional model of a fracture as a surface of displacement discontinuity (crack) is applicable. In particular, the model may refer to low-permeability reservoirs, and, in some cases, to shale reservoirs.