Efficient incorporation of time-dependent global effects in two-phase upscaling of flow and transport in heterogeneous formations

In subsurface reservoir simulation, upscaling techniques are often applied to coarsen highly detailed geological descriptions. Upscaled two-phase functions are time-dependent functions and are more challenging to compute than upscaled single-phase flow parameters. In two-phase upscaling, local boundary conditions for both pressure and saturation are needed. Due to the hyperbolic feature of the saturation equation, its nonlocal effects evolve in both space and time. In this work, we present a TOF (time-of-flight)-based two-phase upscaling approach, in which local saturation boundary conditions are time-dependent and determined from single-phase time-of-flight. Based on an asymptotic analysis, we propose an approximation to represent the local saturation boundary conditions as a function of time and TOF. The single-phase TOF can represent local saturation variations (due to permeability heterogeneity), as well as the spatial trend of global flow. A pre-determined parameter associated with time is used to represent the temporal effects of global flow. We apply the TOF-based two-phase upscaling to permeability fields with various correlation lengths and fluid mobility ratios. It is shown that the use of the TOF-based two-phase upscaling can considerably improve upon existing two-phase upscaling methods (e.g., standard local boundary conditions and effective flux boundary conditions), and provide accurate coarse-scale predictions for both flow and transport.