A fractal model for relative permeability in fractures under stress dependence

Relative permeability (RP) plays a critical role in fluid flow and transport in fracture systems, with considerable advances in theoretical and computational methods over the last few decades. Nevertheless, there is no standard procedure for measuring RP in fractures, and the essential controls on RP in rough fractures are not yet definitive. In this study, we developed a theoretical model to investigate the RP in inclined rough fractures under stress dependence based on fractal theory. The topography of the rough fracture surfaces is well addressed by fractal theory and lubrication theory, and thin bending theory is used to characterize permeability evolution characteristics of fractures under stress conditions. The model accounts for multiple key factors, including the microstructure of the rough fractures (e.g. the fractal dimension, the area ratio, the length ratio, and the maximum and minimum base radii), rock lithology (e.g. elastic modulus and Poisson's ratio), gravity and effective stress. The predicted results agree well with the available experimental data. It is inferred that the effect of gravity on RP decreases as pressure gradient increases. In general, the theoretical fractal model reveals the coupled flow-deformation mechanisms in fractures, and tends to improve the efficacy of reservoir development strategies. With this new analytical solution, it can help to reduce the uncertainty in flow through fractures and obtain data with high accuracy.