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A sound knowledge of the transmissivity of rock fractures is of special importance to the fields of mining (Li et al. 2014), CO2 sequestration (Macminn et al. 2010; Li et al. 2016), underground oil storage (Wang et al. 2015), groundwater use (Juanes and Tchelepi 2008), and contaminant containment (Zhao et al. 2011; Perera et al. 2011). For fluid flow in fractured rock masses, it is commonly assumed that the transmissivity of fractures is much larger than that of the rock matrix (Cai et al. 2010, 2015; Zhou et al. 2015) and that flow in fractures follows the cubic law, in which the flow rate is linearly correlated with the pressure drop (Liu et al. 2015, 2016a). However, in situ, high-pressure packer tests have shown that the flow rate is nonlinearly correlated with the pressure drop (Chen et al. 2015; Wang et al. 2016a). Thus, the linear cubic law is not suitable. Moreover, the fluid flow through fractured rock masses is influenced by factors such as fracture length, orientation, aperture, and surface roughness. Therefore, it is difficult to quantitatively predict the magnitude of transmissivity in complex fracture networks (Leung and Zimmerman 2012; Liu et al. 2016b), and quantitative estimates of transmissivity and the onset of nonlinear flow of fluid through single fractures, which are fundamental elements in fracture networks, should be fully understood. …