2.1. Boundary Coupling of Pore-Scale and Continuum-Scale Models
 For the sake of discussion, we divide pore-to-continuum multiscale techniques into two categories: boundary coupling and hierarchical coupling. Boundary coupling links pore-scale models with adjacent continuum regions at an interface between the two. Balhoff et al.  coupled a pore-network model to a lower-permeability continuum region by matching the flux at a shared interface. They demonstrated that both pore-scale heterogeneity and the resistance in the low-permeability region affect the near-interface flow distribution in the continuum region, making it essential to match flux and pressure at the coupling boundary. Subsequently, Balhoff et al.  developed a more general approach that couples pore-scale network models to other adjacent pore-scale or continuum-scale models using a mortar method. This approach demands a weaker flux match at the interface than the earlier technique but has a more flexible formulation. Boundary coupling techniques work well for joining adjacent regions, but they are not designed to upscale information across multiple length scales at the same location.
2.2. Sequential Coupling of Pore-Scale and Continuum-Scale Models
 A second class of multiscale modeling is hierarchical coupling, which links pore-scale and continuum-scale models in the same region (e.g., from an embedded network model to the continuum gridblock in which it resides). This technique must address large disparities in both length and time scales [e.g., Patzek, 2001], and can require both numerical coupling and upscaling. A simple but limited hierarchical approach is sequential coupling, where rock properties or transport parameters such as permeability, relative permeability, or capillary pressure are simulated in a preprocessing step and then used in a reservoir simulation in much the same way as if they had been obtained from a physical laboratory experiment. Blunt et al.  showed that the assignment of relative permeability based on network modeling gives significantly different predictions of oil recovery compared to using traditional empirical relative permeability models, which implies sequential coupling could improve the predictive power of reservoir models if pore-scale prediction of relative permeability can be made more accurate than general empirical methods. Jackson et al.  used the same coupled model to predict oil recovery for different wettabilities. White et al.  developed a multiscale model for simulating single-phase flow through porous media. The pore space was discretized into smaller domains. LBM was used to calculate microscale velocity distributions and permeability at the pore scale for each subdomain, and the results were then transmitted to a continuum model via tabular or empirical permeability relationships as one would do with laboratory data. Lichtner and Kang  developed a multiscale model to study reactive flow. Pore-scale transport equations were solved using LBM at the pore scale, and the results were upscaled through simple volume averaging at slow reaction rate and fit into a multiscale continuum model. Rhodes et al.  studied particle transport at four different scales: pore, core, gridblock, and field. At the pore scale, transport was simulated using a network model. Transport properties were then upscaled to the larger scales using transit time distribution functions. They argued that pore-scale or gridblock-scale heterogeneity delayed transport and increased breakthrough times by up to an order of magnitude compared to those predicted using a traditional advection and dispersion model. Chen et al.  developed a multiscale LBM simulation: pore-scale LBM was used to estimate permeability after colloid deposition, and this estimated that permeability distribution was used in a second LBM model to simulate flow at the continuum scale. Tsakiroglou  developed a multiscale network type algorithm. They estimated capillary pressure and relative permeability functions using a quasi-static network model and fed the information as the input to a macroscopic simulator as part of an analysis of the axial distribution of water saturation and pressure across a column-scale soil sample. Sun et al.  studied the effect of subdomain size on the accuracy of a multiscale method, which couples LBM with a continuum model.
2.3. Concurrent Coupling of Pore-Scale and Continuum-Scale Models
 The aforementioned sequential coupling techniques offer the potential to replace or augment coreflood experiments with numerical simulations. However, they do not exploit the most powerful aspect of pore-scale to continuum-scale coupling, which is the ability of the pore-scale model to march forward in time with the reservoir simulator, providing “just-in-time” continuum-scale parameters needed at the larger scale. This approach, which we will refer to as concurrent coupling, has the advantages described earlier.
 To date, few studies have been published in the area of concurrent coupling. Heiba et al.  coupled a statistical network model for two-phase flow with a reservoir-scale finite element solver. Relative permeability was calculated using percolation theory. They argued that this approach provides more accurate and reliable prediction of relative permeability and capillary pressure in two-phase displacements compared to empirical correlations, but with a factor of 100 increase in computational cost compared to conventional reservoir simulation because of the capillary pressure and relative permeability simulations. Celia et al.  proposed a concurrent multiscale simulation framework to couple a quasi-static network model with a continuum model. The idea consisted of a two-way exchange process: network simulators located in the center of selected blocks in the continuum model providing bulk properties to these blocks. In turn, the continuum simulator provides boundary conditions (in the form of fluid pressures) for the network model. The focus of their approach was to allow material heterogeneities to be properly accounted for in the continuum model. A concurrent multiscale simulation strategy using LBM was also proposed by Van den Akker . Neither of these latter two models was implemented at the time that they were proposed because of numerical limitations. Battiato et al.  developed a multiscale approach that allows a pore-scale model to operate at locations where flow or transport violates assumptions inherent in the continuum equations. The averaging process in the pore-scale region produces integral source terms in the continuum model, and multiscale coupling allows these terms to be evaluated. Chu et al.  developed a single-phase multiscale algorithm that couples network models to continuum-scale models. Network properties such as throat conductances are nonlinear functions of the macroscopic properties at a much larger scale, which is designed to allow multiscale simulation of non-Darcy flow. Chu et al.  integrated a dynamic pore-scale model for immiscible displacement with a reservoir model for multiphase flow. They decoupled the pressure and saturation computations by using two separate sets of networks: networks placed at gridblock boundaries were used for pressure calculation, while separate networks were placed along the displacement front (where the saturation gradient is larger) to be used for saturation updates.
 For the type of multiscale simulation we are interested in (i.e., embedding a network model into a much larger gridblock for the purpose of predicting continuum parameters in that gridblock), a number of simplifications can be made to allow multiscale coupling and/or to improve numerical efficiency. However, these simplifications have corresponding restrictions associated with the physics of flow, including the following:
 1. Quasi-static models are limited to zero-capillary-number behavior, meaning not only that viscous effects are negligible but also that time is not a parameter in the model. Giving up the ability to model viscous behavior or any time-dependent effects eliminates many reservoir processes that would benefit most from multiscale, multiphase modeling.
 2. Most dynamic network algorithms model immiscible displacement (injection of only one phase) rather than more general multiphase flow processes. These simulations force a large saturation change to occur over very small (order-mm) linear dimensions, which translates to an essentially infinite saturation gradient in the direction of flow at the reservoir scale. Consequently, both the magnitude of the saturation gradient and the spatial distribution of phases (which dictates relative permeability) are inconsistent with what would be expected in a reservoir.
 3. For finite-difference or finite-volume algorithms (at the continuum scale), two possible mappings of pore-scale models onto the continuum grid are to (i) associate a pore-scale model with the interior of a gridblock or (ii) place pore-scale models at gridblock boundaries. The latter allows for a simpler numerical approach if it is run such that a pore-scale model on an yz face of a gridblock (for instance) is operated using only a dP/dx pressure gradient. However, this approach implies that the continuum grid is aligned with the principal directions for the permeability tensor, which restricts generality of the algorithm.
 4. As discussed below, no explicit equation links phase saturation in a pore-scale model with the saturation in its home gridblock, at least in a natural formulation derived directly from the conservation equations. If these values are forced to be equal (for instance, by a forced update to the pore-scale saturation at each time step so that it matches the current gridblock saturation), this procedure in effect erases saturation history.
 The issue of viscous versus quasi-static multiphase modeling is not unique to the multiscale framework, and it has a fairly long history in network modeling. Multiphase network models that account for viscous effects (termed dynamic algorithms) remain tricky to implement even after 25 years of research, and research remains active in this area. Lenormand et al.  developed a dynamic two-phase network model that couples viscous force and capillary force together using nonlinear Poiseuille equations. Constantinides and Payatakes  developed a dynamic two-phase model by incorporating Washburn-type equations (for immiscible displacement in a tube) into the material balance. Using their model, they simulated steady state flow in a cubic lattice network with geometrically identical inlet and outlet zones and calculated steady state relative permeability by averaging transient values over a sufficiently long time. Mogensen and Stenby  developed a dynamic two-phase network model and studied the competition between frontal pore-filling and snap-off. Thompson  developed a model in which the two-phase conservation equations were solved simultaneously within each pore, helping to reduce rule-based decisions in the algorithm. Nguyen et al.  developed a two-phase network model for dynamic imbibition; the thickness of wetting films was allowed to vary as a function of local capillary pressure, and pore invasion was controlled by the competition between frontal displacement and snap-off. Joekar-Niasar et al.  used the aforementioned approach in which the phase conservation equations were solved simultaneously, and studied nonequilibrium capillarity effects and the dynamics of two-phase flow. This latter model was also used for the coupled model by Chu et al.  mentioned above.
 Using the lattice Boltzmann modeling for multiphase flow simulation automatically addresses the quasi-static problem. However, while LBM models are more rigorous than network models, they are less appealing for this type of multiscale simulation because of their additional computational overhead. Additionally, most LBM models employ periodic or mirror boundary conditions, which prevent them from being used to monitor the natural evolution of saturation history (because overall saturation cannot change due to differences in inflow/outflow).
 As with the quasi-static issue, the problem of length scales is not unique to coupled models. In fact, it is a significant concern for nearly all modern image-based modeling applications because of the small size of the samples that can be imaged at high resolutions. However, in the context of multiscale modeling, the problem is particularly striking because pore-scale models derived from high-resolution imaging are usually order-mm in size, whereas continuum-scale gridblocks are typically three orders of magnitude greater. Similarly, timescales for immiscible displacement in a mm-sized pore-scale model are order seconds or even less than one second, while numerical time steps at the continuum scale are orders of magnitude larger. At this point, we should clarify two issues which are not the focus of work in this paper. The first is the traditional role of upscaling over a hierarchy of scales. While we do not discount the importance of this problem for modeling heterogeneous, multiscale geologic media, the focus in this work is on a single transition in scales: from pore-scale (where fundamental equations of motion can be applied) to the continuum scale (where Darcy-type equations are used, and which in turn requires spatially averaged parameters such as permeability). The second issue not addressed here is the data-poor conditions that exist in the context of pore-to-continuum modeling, which will always prevent us from having sufficient pore-scale data to accurately populate a reservoir model. We again acknowledge the significance of this problem, but argue that it should not slow the development of these powerful multiscale techniques. In fact, we point out the similarity to the situation that the subsurface simulation community has dealt with for decades: experimental tests are performed on a limited set of cm-scale core samples or sand columns, and these results are used to populate much larger continuum grids, which in turn describe heterogeneous subsurface structures.
 Discussion in the remainder of the paper assumes the following. First, separate tests have been performed to ensure that the length scales used in the pore-scale modeling are sufficient to extract spatially averaged continuum scale parameters such as permeability, relative permeability, and capillary pressure. Second, continuum properties within a gridblock are reasonably uniform and flow conditions are such that the original continuum-scale equation remains valid everywhere.
 In addition to the issues of scale discussed in the previous two paragraphs, there are time-scale and length-scale considerations that are more critical to the coupled, multiscale problem than to general pore-scale modeling. Consider first that the most common methods for pore-scale modeling of multiphase flow use immiscible displacement algorithms. In these simulations, both saturation and fractional flow exhibit order-unity changes over very small length scales, thus creating essentially infinite gradients in these parameters at the continuum scale. This behavior is not consistent with the magnitude of continuum-scale saturation gradients, nor is it consistent with the physics of flow at either scale. The problem is of particular concern if relative permeabilities are extracted from these unrealistically sharp saturation fronts because the pore-scale distribution of fluid phases is the dominant factor in dictating relative permeability.
 The problems associated with time scales are analogous. The flat front that passes through a pore-scale region during an immiscible displacement simulation represents an order-unity saturation change in approximately one pore volume, which in turn implies a very small time scale for the saturation change to occur (order milliseconds or seconds depending on the capillary number). In contrast, an Eulerian view of a mm-scale section of a reservoir would exhibit a much slower saturation change during most reservoir-scale flow scenarios, during which time an enormous number of pore volumes would pass through the small domain.
 Given the above considerations, we argue that the more proper pore-scale simulation to couple to a continuum-scale time step is steady-state multiphase flow over a short period of time. Generally, this would be at a new saturation that has evolved to match the current continuum-scale time step.