Nonlinear upscaling of two-phase flow using non-local multi-continuum approach

The objective of this work is to design upscaled model concepts for multi-phase flow and transport. Our approaches are based on recent developments in multiscale simulations and their relations to upscaling.
We propose a novel multi-phase upscaling technique, which employs rigorous multiscale concepts based on the Constraint Energy Minimization (CEM-GMsFEM). CEM-GMsFEM concepts utilize local spectral problems and an energy minimization principle to design multiscale basis functions, which are supported in oversampled regions. A coarse-grid solution defined by these basis functions provides first-order accuracy with respect to the coarse-mesh size and is independent of high contrast of the permeability. The degrees of freedom in multiscale methods represent the coordinates of the solution in the multiscale space. To design an upscaled model, we modify these basis functions such that the degrees of freedom have physical meanings, in particular, the averages of the solution in each continuum. This allows deriving rigorous upscaled models and account for both local and non-local on the effects. The transmissibilities in our upscaled models are non-local and take into account non-neighboring connections.
To extend this approach to nonlinear problems in the context of two-phase flow, we develop nonlinear upscaling, where the pressures and saturations are interpolated within an oversampled region based on average values of these quantities. Multicontinua concepts are used to localize the problem to the oversampled regions. Our upscaled model shares some similarities with the pseudo-relative permeability approach with the following differences: (1) the upscaled relative permeabilities depend non-locally on the saturations; and (2) local problems, formulated in oversampled regions, involve constraints and require multi-contiuum concepts.
The numerical results will utilize upscaled methods to predict the solution of single-and two-phase flow dynamics. We will describe upscaled equations, which include the non-local neighborhood connections. Our results demonstrate that the proposed approaches provide a good accuracy and robustness. We consider various types of heterogeneities. The proposed concepts will benefit developing coarse-grid and upscaled models for many applications involving multi-phase flow and transport.