Viscous fingering analysis for water-drive oil in the inclined plane

Viscous fingering is a common instability event that occurs during the process of water-drive oil for oil recovery, significantly limiting the efficiency of oil extraction. In this study, we propose a film flow model that accounts for the variation in height at the water-oil two phase interface, enabling the calculation and analysis of the triggering mechanism and flow evolution process of this unstable phenomenon. We theoretically derive the equation of water film flow, which can be used to explore the flow evolution of the two-phase interface in the process of oil displacement. By numerically solving the two-dimensional flow equation, we obtain the traveling wave profile and find that the morphology of the two-phase interface is significantly affected by the plane's inclined angle, capillary number and density ratio of the two-phase liquid. Furthermore, we perform linear stability analysis and finite element numerical simulation considering small initial disturbances to explore the triggering conditions of viscous fingering phenomenon and the full time from gentle displacement to unstable flow. The results reveal that the moving contact line of the driven liquid front is more stable when the viscosity of the oil is less different from the driven liquid and has a smaller density, thereby improving of the driving efficiency in the water-driven oil process. These insights have significant implications for guiding efforts to enhance oil recovery efficiency, and we provide concrete engineering suggestions to achieve this aim.