Anisotropic stress-induced permeability change in cracked rock during the crack closure stage: A tensorial approach incorporating tortuosity and bottlenecks

Understanding stress-dependent permeability in cracked rocks is crucial for predicting fluid migration in the subsurface subjected to stress and in controlled laboratory experiments. In Oda's permeability tensor, the parameter related to connectivity is represented solely by the fraction of geometrically connected cracks, and neither tortuosity nor bottleneck effects, which are strongly influenced by crack closure, particularly under anisotropic stress, are taken into account. We propose a new crack network parameter, T ij , in tensor form, which is defined to incorporate tortuosity and bottleneck effects. By integrating this parameter into the permeability tensor, fluid flow through crack networks under anisotropic stress conditions can be evaluated. Numerical simulations were performed using idealised three mutually orthogonal crack-set models under hydrostatic, triaxial, and polyaxial stress conditions, with varying crack densities and degrees of anisotropy. In addition, we then compared and validated the equivalent permeability obtained in this study with the results of flow simulations based on a DFN model. The results show that under low crack density, the closure of cracks normal to the loading axis significantly increases tortuosity and forms bottlenecks. It was also shown that permeability decreases in the direction of the loading axis, even within the elastic regime. Moreover, as failure progresses and crack density increases due to crack development, the flow paths tend to become more linear, thereby reducing the influence of tortuosity on permeability. The proposed tensor T ij successfully reproduces this reduction in permeability along the loading direction, which conventional models cannot explain—for instance, the decrease in permeability observed during the crack closure stage of triaxial compression tests. Furthermore, this study presents a practical procedure for estimating T ij from experimental and observational data, including crack orientation, aperture, and elastic wave velocity measurements. These data are incorporated into T ij , which is formulated as a correction factor within the conventional permeability tensor framework.