Buckley–Leverett Solution for Two-Phase Displacement in a Composite Porous–Cavernous–Porous System

Fluid flow in fractured-vuggy carbonate reservoirs is characterized by extreme multiscale heterogeneity, where the coexistence of tight matrix rock and macroscopic cave challenges traditional Darcy-based continuum models. This paper presents a semi-analytical solution for two-phase immiscible displacement in a one-dimensional composite porous–cavernous–porous (PCP) system. The main feature of the model is that the cave region is treated separately from the porous domains: classical Darcy flow is used in the surrounding matrix, whereas an idealized free-flow representation is introduced for open caves based on a simplified one-dimensional treatment of the cave momentum balance. To elucidate the impact of distinct flow regimes on displacement dynamics, three physical models are compared for the cave region: (1) an open-cave model represented by a simplified free-flow formulation; (2) a filled-cave non-Darcy model governed by the Forchheimer equation using the Ergun correlation; and (3) a creeping-flow model governed by Darcy’s law. A piecewise semi-analytical solution procedure is established to enforce flux continuity, characterize interfacial state remapping, and determine the downstream front under global water-balance closure. The results show that both cave geometry and internal cave-flow mechanism critically control water-front advancement. While the open-cave model exhibits piston-like displacement behavior with high local displacement efficiency but stronger preferential flow, the Forchheimer model shows that inertial resistance can modify the saturation profile and delay breakthrough relative to the Darcy prediction. The proposed framework provides an idealized theoretical reference for benchmarking numerical simulators and for interpreting waterflooding behavior in complex vuggy reservoirs under one-dimensional, incompressible, gravity-free, and capillarity-free conditions.