Analytical solution of oil and water two-phase Buckley-Leverett equation in inclined stratified heterogeneous reservoirs

Abstract

Water flooding technology is widely applied in medium-to high-permeability reservoirs and plays a crucial role in global oilfield development. The traditional Buckley-Leverett theory provides a theoretical foundation for water flooding. However, its assumptions, such as reservoir homogeneity and the neglect of gravity effects, present limitations in practical applications. In inclined reservoirs, gravity segregation causes the injected water to preferentially advance along the lower part of the reservoir, resulting in the formation of a water cone or water ridge. This reduces vertical sweep efficiency and limits the effective displacement of crude oil. In high-permeability layers, the advance speed of the water front is much faster than in low-permeability layers, leading to a premature breakthrough of the injected water, which reduces the plane sweep efficiency and weakens the water drive effect. Therefore, addressing the issues of gravity effects and reservoir heterogeneity is key to achieving efficient water flooding development. This paper establishes the Buckley-Leverett equation for water flooding in tilted, layered, heterogeneous reservoirs. The implicit solution is derived using fractional flow theory, and the explicit analytical solution is estimated using the Lambert W function. The analytical solution provides guidance for addressing gravity effects and heterogeneity in two-phase flow. A comparison between the analytical results and numerical simulation results for different permeability values and varying reservoir dip angles reveals a good match between the two. The analysis shows that as the reservoir inclination angle and permeability heterogeneity increase, the water drive breakthrough time lengthens, the front advance distance decreases, and micro-displacement efficiency diminishes. The analytical model presented in this paper accurately describes the two-phase flow behavior in reservoirs, considering both gravity differentiation and heterogeneity, and offers valuable insights for optimizing reservoir development strategies. Using this model, the position changes of the water front can be predicted, facilitating the formulation of more effective development plans that enhance sweep and recovery efficiencies.