A note on volume-averaging

In conceptualizing and quantifying the physical world of the porous medium, we are led to the consideration of properties averaged over finite subdomains. These averages are evaluated through some meaningful procedure of integration and normalization. In order that the physical significance of the averaging is assured, the procedure of integration must be consistent with the axiomatic foundations of integration and its relation to the nature of physical quantities. This paper points out the difference between extensive quantities (measures) and intensive quantities (potentials) and shows how the latter are to be first converted to the former before physically meaningful integration can be performed. It then proceeds to show how the averaging procedure can be simplified in special circumstances. It concludes with a consideration of how the volume-average of the conceptual-computational world can meaningfully simulate the physical world perceived through measurements.