Apparent Toughness Anisotropy Induced by Roughness of in Situ Stress: A Mechanism that Hinders Vertical Growth of Hydraulic Fractures and Its Simplified Modeling

A hydraulic fracture's height growth is known to be affected by many factors that are related to the layered structure of sedimentary rocks. While these factors are often used to qualitatively explain why hydraulic fractures usually have well-bounded height growth, most of them cannot be directly and quantitatively characterized for a given reservoir to enable a priori prediction of fracture height growth. In this work, we study the role of the "roughness" of in situ stress profiles, namely alternating low- and high-stress among rock layers, in determining the tendency of a hydraulic fracture to propagate horizontally versus vertically. We found that a hydraulic fracture propagates horizontally in low-stress layers ahead of neighboring high-stress layers. Under such a configuration, a fracture mechanics principle dictates that the net pressure required for horizontal growth of high-stress layers within the current fracture height is significantly lower than that required for additional vertical growth across rock layers. Without explicit consideration of the rough stress profile, the system behaves as if the rock is tougher against vertical propagation than it is against horizontal fracture propagation. We developed a simple relationship between the apparent differential rock toughness and characteristics of the stress roughness that induce equivalent overall fracture shapes. This relationship enables existing hydraulic fracture models to represent the effects of rough in situ stress on fracture growth without directly representing the fine-resolution rough stress profiles.