## 1. Introduction

[2] Pore-scale modeling of subsurface flow applications has the potential to provide links between fundamental pore-scale physics and macroscopic phenomena that are observed at the continuum scale. Qualitative modeling techniques have been used for years for phenomenological research. In recent years, emphasis has shifted toward quantitative and even predictive modeling. This change is driven largely by improvements in high-resolution 3-D imaging, which have allowed rapid advances in image-based modeling techniques (i.e., algorithms that operate directly on digital images of real materials).

[3] The evolution toward predictive pore-scale modeling brings questions about how to integrate pore-level results into reservoir-scale models using upscaling and/or numerical coupling. The most direct approach is simple numerical upscaling: pore-scale simulations are performed on a sufficiently large domain to extract continuum-scale parameters; the predicted parameters are then used in larger-scale models. This approach mirrors what is done with physical laboratory experiments, but does not use multiscale modeling to its full potential. A more integrated approach is to allow communication in both directions: continuum-scale parameters are obtained from fundamental pore-scale models and passed up to the continuum scale; simultaneously, the continuum scale model passes relevant conditions back down to the pore-scale models so that these simulations are performed using updated conditions.

[4] This latter approach is more complicated, but it is difficult to overstate its potential impact if it can be made practical. Consider reservoir-scale modeling of multiphase flow, which is vital to both subsurface hydrology and oil and gas production. Relative permeability is the main continuum-scale parameter used to characterize the dynamic component of continuum-scale multiphase models. For practical reasons, it is usually tabulated as a nonlinear function of phase saturation. However, this basic parameterization significantly oversimplifies the reality, which is that relative permeability is usually hysteretic and depends on numerous other parameters including wettability, capillary number (*Ca =µv/σ*), viscosity ratio, absolute permeability, and saturation history. True multiscale modeling represents a fundamental change to the traditional approach to reservoir-scale modeling of multiphase flow and has a number of practical implications. First, communication in both directions would allow the continuum model to operate using current relative permeability values from the pore-scale model, thus eliminating the need to tabulate or fit a relative permeability curve. (Here the important benefit is not convenience as much as it is the fact that relative permeability no longer needs to operate as a function of saturation only.) Second, provided that the pore-scale model captures relevant first-principles behavior, relative permeability would respond to unanticipated changes to the system that either are revealed by the continuum-scale model or are forced via changes in applied boundary conditions at the continuum scale. Examples include changes in injection rate, wettability alteration, formation damage, stimulation, or subsidence. Third, a pore-scale model that is coupled in simulated time with the continuum-scale model provides a mechanism to implicitly store saturation history (or other relevant history such as wettability change). This is not possible using traditional techniques.

[5] The general concept of exchanging information between a practical continuum simulation and a first-principles pore-scale simulation is not new. However, the hurdles between concept and implementation are large enough to have prevented a fully coupled approach from being developed until now. This paper describes the first multiscale algorithm that allows a pore-scale model to be embedded inside continuum gridblocks such that relative permeability is passed from the pore-scale model to the continuum models as a current gridblock parameter, while fractional flows are passed from the continuum model to the pore-scale model as current boundary conditions. The two models march forward in time together such that saturation evolves naturally, which in turn allows saturation history to be retained implicitly via the pore-scale model.