Instability mechanism and a fractal theory-based relative permeability model of immiscible two-phase displacement in rough fractures

The two-phase displacement in rock fractures is a critical scientific challenge in deep energy development projects, characterized by the intricate and dynamic flow patterns resulting from the complex fracture geometry and mutual interference between the two-phase fluids. The present study involves the construction of a three-dimensional rough fracture fine model using the fast Fourier transform and Gaussian distribution method. The phase field-finite element method is employed to simulate the dynamic displacement of water-oil flow and the evolution of the water-oil interface, aiming to investigate the impact of roughness, aperture, and wettability on the viscous fingering pattern observed in rough fractures. The numerical results indicate that the immiscible two-phase displacement presents a viscous fingering pattern at a high flow rate. Moreover, the greater the roughness is, the higher the number of fingering is, and the more volatile the change in displacement velocity is. The width of the wet-phase flow channel increases proportionally with the fracture aperture increasing. Furthermore, the morphology of immiscible two-phase interface within fractures is also influenced by the wetting angle, and the fingering, bifurcation, and crushing phenomena appear in the oil-wet state with the strong displacement interface instability. Finally, a fractal model of relative permeability for waterflooding in rough fractures is proposed based on the cubic law of flow and fractal dimension of aperture distribution, which has been validated through comparison with numerical results.