Interpretation of macroscale variables in Darcy's law

The pursuit of a theoretical foundation of Darcy's law based on volume averaging of equations at the scale of flow in pores has a long history. While theories are well established for homogeneous systems, more complex systems exhibit inconsistencies in the resulting equations. The difficulties often lie in the treatment of surface integral terms arising from the classical averaging theorems used to transform averages of derivatives into derivatives of averages. In this work we extend the intrinsic phase average as a macroscale variable to a family of more general macroscale variables, which take into account systematic dependencies of averaging volume size on the macroscale. Comparison to Darcy's law gives new insight into the relationship between variables at the microscale and macroscale.