In this work, the Renshaw and Pollard criterion for orthogonal intersections is extended to a fracture crossing frictional interfaces at non-orthogonal angles. A similar approach as in Renshaw and Pollard is used, and the stress field near the fracture tip and the interface is analyzed. The extended criterion could not be expressed explicitly in a simple formula, but crossing or no-crossing can be determined from a procedure involving a quadratic equation that can be easily solved using a simple computer routine. The results of crossing at all possible angles, i.e., angles fall between 0 and 90°, are obtained and explained. The smaller the angle between the fracture and the interface is, the more difficultly crossing occurs. In other words, the fracture is more likely to turn and propagate along the interface than to cross it as the angle is less than 90°. The quantitative results of crossing and no-crossing are also expressed graphically as functions of stress ratio, coefficient of friction, and intersection angle. The criterion can be applied in general analysis of fracture network complexities and in numerical fracture network simulators.
Hydraulic fracturing treatments are necessary in shale gas field development. Shale formations often contain natural fractures, and hydraulic fracture propagation is affected by the natural fractures with the result that complex fracture networks may form. The complexity of a hydraulic fracture network depends on the behavior of a hydraulic fracture when it encounters natural fractures. One of the mechanisms causing fracture complexity is a hydraulic fracture crossing natural fractures. If the hydraulic fracture propagates across a natural fracture without changing direction, the hydraulic fracture remains essentially a planar fracture; if the hydraulic fracture does not cross the natural fracture, but instead dilates and propagates along the natural fracture, a complex fracture network may result. It is important to determine whether a hydraulic fracture crosses natural fractures under particular field conditions of in-situ stresses, rock and natural fracture properties, fracturing fluid properties, and pumping conditions. Natural fractures can be considered as frictional interfaces when the intersection of hydraulic fractures and natural fractures are considered. Extensive theoretical, numerical and experimental works have been conducted on the fracture propagation across interfaces [1-7]. Renshaw and Pollard  developed a simple criterion for predicting whether a fracture will propagate across a frictional interface orthogonal to the approaching fracture. The criterion is based on the linear elastic fracture mechanics solution for the stresses near the fracture tip to determine the stresses required to prevent slip along the interface at the moment when the stress on the opposite side of the interface is sufficient to reinitiate a fracture. The criterion was validated by laboratory experiments. Natural fractures are often not aligned with the contemporary principal in-situ stress directions in the rock formation, in which case the intersection angle of a hydraulic fracture approaching a natural fracture is between 0 and 90°. The results from the extended criterion are presented, with particular emphasis on the effect of intersection angle.