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

Abstract

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.

Difficulties arising from different definitions of tortuosity for wave propagation in saturated poroelastic media models

“…Figure 2a shows the viscous skin depth, χ −1 , as a function of the frequency, from Eq. (20). Figure 2b shows the permeability, K(ω), as a function of frequency along the three principal axes of material symmetry, obtained from Eq.…”

“…While Kμ −1 is called the permeability coefficient, K is called the intrinsic permeability. Gray et al [20] review the averaging processes over different structural scales that may be used to deal with the inhomogeneous material version of Darcy's law. In particular, Gray et al [20] consider problems with the proper interpretation of pressures appearing in Darcy's law.…”

“…Gray et al [20] review the averaging processes over different structural scales that may be used to deal with the inhomogeneous material version of Darcy's law. In particular, Gray et al [20] consider problems with the proper interpretation of pressures appearing in Darcy's law. Coussy [14] and Cowin and Cardoso [17] employ the Darcy definition (1) of permeability.…”

The role of cell lysis and matrix deposition in tumor growth modeling

“…It is recalled that tumors are modeled at the macroscopic scale, being the domain of interest too large and the phase distributions too complex for modeling at the microscale. The TCAT approach [24][25][26] is used to transform known microscale relations to mathematically and physically consistent macroscale relations. These macroscale relations are adequate for describing the tumor behavior while filtering out high frequency spatial variability.…”

A monolithic multiphase porous medium framework for (a-)vascular tumor growth

“…Consequently, a more detailed and natural description with at least one solid and one liquid phase [4] characterizes these approaches. In [4,8] they are roughly categorized into models based on mixture theory or Thermodynamically Constrained Averaging Theory (TCAT) [9][10][11]. Mixture theory models [12,13] lack a stringent connection between microscale pore mechanics and macroscale closure relations [9].…”