Upscaling at the MADE site: When, Why, How, and What Next? (Invited)

Abstract:

It has long been known that averaging of heterogeneous velocities, through sampling or discretization, leads to spatial and temporal non-locality in the governing equation of the upscaled dependent variable. It is important to recognize that the variables in the small-scale and up-scaled equations are not the same, so it is improper to ask which equation is ``correct" without also asking which variable is being studied. Here we review the appearance of non-locality and efforts to simulate the MADE plume with either small-scale heterogeneity explicitly represented or averaged into nonlocal operators. The spatial and temporal non-locality can have similar effects (e.g., imparting skewness and rapid spreading), but are distinct in other ways, making their effects observably separable. But new questions arise, including ``What are the scales at which one or both cannot be ignored?'' and ``What are the data requirements to characterize (especially in a predictive sense) large-scale transport?'' Some newer studies, including high-resolution $K$ measurements (not without their own ambiguities) shed some light on these questions. In short, recognizing the different facies allows fairly simple representation of the ``fast movers'' that generate spatial non-locality, but mass transfer between relatively mobile and immobile phases tends to occur at a very small scale and presently cannot be represented by fine-scale heterogeneous velocities.