A generalization of averaging theorems for porous medium analysis

•An averaging theorem for common points is provided.•Averaging theorems are extended to include singularities on the entity boundaries.•Unified notation reveals similarities among the forms and types of averaging theorems.

The contributions of Stephen Whitaker to the rigorous analysis of porous medium flow and transport are built on the use of temporal and spatial averaging theorems applied to phases in representative elementary volumes. Here, these theorems are revisited, common point theorems are considered, extensions of existing theorems are developed to include the effects of lower dimensional entities represented as singularities, and a unified form of the theorems for phases, interfaces, common curves, and common points is established for both macroscale and mixed macroscale–megascale systems. The availability of the full set of theorems facilitates detailed analysis of a variety of porous medium systems. Explicit modeling of the physical processes associated with interfaces, common curves, and common points, as well as the kinematics of these entities, can be undertaken at both the macroscale and megascale based on these theorems.