Technical note: Revisiting the geometric theorems for volume averaging

• Re-examination of the geometrical theorems presented by Quintard and Whitaker [5].
• Illustrates how the geometrical theorems are related to the moments of the porous medium geometry.
• Discussion indicating a revision of the concept of disorder as presented by Quintard and Whitaker [5].
• Direct computation of the geometrical theorems supplied as a pedagogical aid.

The geometric theorems reported by Quintard and Whitaker [5, Appendix B] are re-examined. We show (1) The geometrical theorems can be interpreted in terms of the raw spatial moments of the pore structure within the averaging volume. (2) For the case where the first spatial moment is aligned with the center of mass of the averaging volume, the geometric theorems can be expressed in terms of the central moments of the porous medium. (3) When the spatial moments of the pore structure are spatially stationary, the geometrical theorems allow substantial simplification of nonlocal terms arising in the averaged equations. (4) In the context of volume averaging, the geometric theorems of Quintard and Whitaker [5, Appendix B] are better interpreted as statements regarding the spatial stationarity of specific volume averaged quantities rather than an explicit statement about the media disorder.