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Whitaker
6.2 VOLUME AVERAGED CONSERVATION EQUATIONS
To capture the essential features of mass transfer in porous media, we develop the
local volume averaged forms of the governing equations. Volume averaging can occur
at many levels and some of those levels associated with a study of biofilm growth
in porous media are illustrated in Figure 6.2. In that figure, Level I represents an
heterogeneous aquifer in which mass transport and biological reactions occur. The
averaging volume used to develop transport equations in the aquifer is illustrated as
Level II. There it is apparent that a two-region model is being used to model the
aquifer. In order to describe mass transport and reaction in one of those two regions,
we make use of the averaging volume illustrated as Level III in Figure 6.2. There we
have shown a fluid phase, a solid phase, and the biofilm phase. Continuing on to Level
IV we indicate that the solid phase at Level III is actually a micro-porous medium
in which the diffusive transport may play an important role in the biofilm growth.
Although not shown, one must make use of another averaging volume to analysis the
process of diffusion and reaction in the biofilms (Wood and Whitaker, 1998). In this
work, we consider volume averaging at Level III, which is often referred to as the

I

II

fluid
III
solid biofilm

IV

Figure 6.2. Hierarchical porous medium containing biofilms