Two-branch retention behavior in unsaturated fractured rock driven by fracture-matrix flow partitioning

Upscaling unsaturated flow in fractured rock remains challenging because fractures and matrix often exhibit sharply contrasting hydraulic behaviors across saturation states. Here, we demonstrate that unsaturated flow undergoes a transition between matrix- and fracture-dominated regimes. Three-dimensional direct numerical simulations reveal that both relative permeability and capillary pressure curves display a robust two-branch structure. We analytically derive a generalized retention formulation that identifies a critical saturation marking the transition between the two distinct retention regimes and reproduces the two-branch behavior captured in the numerical simulations. An analytical expression for the critical pressure head is further derived to represent the limiting case of fully connected fracture networks, providing a physical explanation for the retention regime shift and showing good agreement with the numerical results for systems above the percolation threshold. Our results provide a mechanistic framework for understanding and upscaling unsaturated flow in fractured rock, with broad implications for hydrology and geophysics.