Averaging theory for description of environmental problems: What have we learned?

Advances in Water Resources has been a prime archival source for implementation of averaging theories in changing the scale at which processes of importance in environmental modeling are described. Thus in celebration of the 35th year of this journal, it seems appropriate to assess what has been learned about these theories and about their utility in describing systems of interest. We review advances in understanding and use of averaging theories to describe porous medium flow and transport at the macroscale, an averaged scale that models spatial variability, and at the megascale, an integral scale that only considers time variation of system properties. We detail physical insights gained from the development and application of averaging theory for flow through porous medium systems and for the behavior of solids at the macroscale. We show the relationship between standard models that are typically applied and more rigorous models that are derived using modern averaging theory. We discuss how the results derived from averaging theory that are available can be built upon and applied broadly within the community. We highlight opportunities and needs that exist for collaborations among theorists, numerical analysts, and experimentalists to advance the new classes of models that have been derived. Lastly, we comment on averaging developments for rivers, estuaries, and watersheds.