Volume averaging: Local and nonlocal closures using a Green’s function approach

Modeling transport phenomena in discretely hierarchical systems can be carried out using any number of upscaling techniques. In this paper, we revisit the method of volume averaging as a technique to pass from a microscopic level of description to a macroscopic one. Our focus is primarily on developing a more consistent and rigorous foundation for the relation between the microscale and averaged levels of description. We have put a particular focus on (1) carefully establishing statistical representations of the length scales used in volume averaging, (2) developing a time–space nonlocal closure scheme with as few assumptions and constraints as are possible, and (3) carefully identifying a sequence of simplifications (in terms of scaling postulates) that explain the conditions for which various upscaled models are valid. Although the approach is general for linear differential equations, we upscale the problem of linear convective diffusion as an example to help keep the discussion from becoming overly abstract. In our efforts, we have also revisited the concept of a closure variable, and explain how closure variables can be based on an integral formulation in terms of Green’s functions. In such a framework, a closure variable then represents the integration (in time and space) of the associated Green’s functions that describe the influence of the average sources over the spatial deviations. The approach using Green’s functions has utility not only in formalizing the method of volume averaging, but by clearly identifying how the method can be extended to transient and time or space nonlocal formulations. In addition to formalizing the upscaling process using Green’s functions, we also discuss the upscaling process itself in some detail to help foster improved understanding of how the process works. Discussion about the role of scaling postulates in the upscaling process is provided, and poised, whenever possible, in terms of measurable properties of (1) the parameter fields (including the indicator fields describing the medium geometry) associated with the transport phenomenon of interest, and (2) measurable properties of the independent variable itself. To highlight the relevance of this interpretation we study the benchmark problem of linear nonlocal diffusion in porous media.