Modeling Hydraulic Fracture Entering Stress Barrier: Theory and Practical Recommendations

Numerical modeling of hydraulic fracturing is complicated when a fracture reaches a stress barrier. For high barriers, it may require changing the computational scheme. In view of the strong influence of stress barriers on the final footprint and opening of a hydraulic fracture, for decades, their modeling has been the subject of special investigations. Actually, classical models of propagation within a pay layer with impenetrable boundaries referred to the case of extremely high-stress contrast in neighbor layers. Further improvements tended to account for the fracture growth in these layers by including stress contrasts as input parameters of a model. This tendency resulted in the suggestion and successive enhancements of pseudo-three-dimensional models. All of them have used stress intensity factors (SIFs) to characterize the combined resistance caused by stress contrasts, material strength, and fluid viscosity. Specifically, the SIFs served to formulate the conditions that control the front penetration into a neighbor layer. This key concept presents the background of our research. Despite examples of modeling propagation through barriers, there is no general theory clarifying when and why conventional schemes may become inefficient and how to overcome computational difficulties. This paper presents the theory and practical recommendations following it. We start with the definition of the barrier intensity, which exposes that the barrier strength may change from zero for contrast-free propagation to infinity for channelized propagation. The analysis reveals two types of computational difficulties caused by spatial discretization as follows: (i) general, arising for fine grids and aggravated by a barrier, and (ii) specific, caused entirely by a strong barrier. The asymptotic approach, which avoids spatial discretization, is suggested. It is illustrated by solving benchmark problems for barriers of arbitrary intensity. The analysis distinguishes three typical stages of the fracture penetration into a barrier and provides theoretical values of the Nolte–Smith slope parameter and arrest time as functions of the barrier intensity. Special analysis establishes the accuracy and bounds of the asymptotic approach. It appears that the approach provides physically significant and accurate results for fracture penetration into high, intermediate, and even weak stress barriers. On this basis, simple practical recommendations are given for modeling hydraulic fractures in rocks with stress barriers. The recommendations may be promptly implemented in any program using spatial discretization to model fracture propagation.