A multi-phase upscaling framework for relative permeability of representative elementary volumes

Simulating multi-phase flow across multiple scales remains a central challenge in subsurface modeling. This study addresses numerical difficulties associated with proper selection of boundary conditions in upscaling effective relative permeability and capillary pressure (saturation functions) for heterogeneous systems at bedform scale. We introduce an upscaling framework that employs mirror-periodic boundary conditions within a steady-state workflow. The approach applies a pre-processing mirror reflection to produce an extended computational domain out of original domain for the sake of preserving continuity of high permeability paths. This modification effectively overcomes saturation and capillary discontinuities inherent in conventional periodic boundaries, providing a more physically consistent representation of flow across a Representative Elementary Volume (REV). Our method seamlessly enables a continuous transition from capillary limit (CL) to viscous limit (VL). The proposed approach produces a rate-dependent relative permeability tensor, which captures directional flow effects in complex geometries. The framework is demonstrated by estimating effective multi-phase parameters for 2D and 3D models of bioturbated strata. The upscaled results exhibit rate-dependent anisotropy, with the critical non-wetting phase saturation ranging from 0 to 0.2 depending on flow direction relative to the burrow network. Results further show that the mirror-periodic approach eliminates the artifacts of conventional methods while retaining their advantage of allowing a natural pressure gradient, unlike constant fractional flow approaches. We also discuss the computational costs and implementation considerations for systems with preferentially oriented laminations.