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Hydraulic fracturing technology is being widely used within the oil and gas industry for both waste injection and unconventional gas production wells. It is essential to predict the behavior of hydraulic fractures accurately based on understanding the fundamental mechanism(s). The prevailing approach for hydraulic fracture modeling continues to rely on computational methods based on Linear Elastic Fracture Mechanics (LEFM). Generally, these methods give reasonable predictions for hard rock hydraulic fracture processes, but still have inherent limitations, especially when fluid injection is performed in soft rock/sand or other non-conventional formations. These methods typically give very conservative predictions on fracture geometry and inaccurate estimation of required fracture pressure. One of the reasons the LEFM-based methods fail to give accurate predictions for these materials is that the fracture process zone ahead of the crack tip and softening effect should not be neglected in ductile rock fracture analysis. A 3D pore pressure cohesive zone model has been developed and applied to predict hydraulic fracturing under fluid injection. The cohesive zone method is a numerical tool developed to model crack initiation and growth in quasi-brittle materials considering the material softening effect. The pore pressure cohesive zone model has been applied to investigate the hydraulic fracture with different rock properties. The hydraulic fracture predictions of a three-layer water injection case have been compared using the pore pressure cohesive zone model with revised parameters, LEFM-based pseudo 3D model, a Perkins-Kern-Nordgren (PKN) model, and an analytical solution. Based on the size of the fracture process zone and its effect on crack extension in ductile rock, the fundamental mechanical difference of LEFM and cohesive fracture mechanics-based methods is discussed. An effective fracture toughness method has been proposed to consider the fracture process zone effect on the ductile rock fracture.

Liquid–liquid phase separation, driven by multivalent macromolecular interactions, causes formation of membraneless compartments, which are biomolecular condensates containing concentrated macromolecules. These condensates are essential in diverse cellular processes. Formation and dynamics of micrometer-scale phase-separated condensates are examined routinely. However, limited by commonly used methods which cannot capture small-sized free-diffusing condensates, the transition process from miscible individual molecules to micrometer-scale condensates is mostly unknown. Herein, with a dual-color fluorescence cross-correlation spectroscopy (dcFCCS) method, we captured formation of nanoscale condensates beyond the detection limit of conventional fluorescence microscopy. In addition, dcFCCS is able to quantify size and growth rate of condensates as well as molecular stoichiometry and binding affinity of client molecules within condensates. The critical concentration to form nanoscale condensates, identified by our experimental measurements and Monte Carlo simulations, is at least several fold lower than the detection limit of conventional fluorescence microscopy. Our results emphasize that, in addition to micrometer-scale condensates, nanoscale condensates are likely to play important roles in various cellular processes and dcFCCS is a simple and powerful quantitative tool to examine them in detail.