On the definition of turbulent kinetic energy for flow in porous media

In the literature, there are two distinct approaches for developing turbulent models for flow in a porous medium. The first one starts with the macroscopic equations using the extended Darcy-Forchheimer model. The second method considers first the microscopic balance equations. In both cases, time and volume averaging operators are applied in a different order. The turbulence kinetic energy equation resulting from application of the two averaging operators, following both orders of integration, are different. In this work, a new double-decomposition (time and volume) methodology is suggested and the differences between those two mathematical treatments are highlighted.