  Reactive transport in saturated/unsaturated porous media is numerically
upscaled to the space-time scale of a hypothetical measurement through
coarse-grained space-time (CGST) averages. The reactive transport is modeled at
the fine-grained Darcy scale by the actual number of molecules involved in
reactions which undergo advective and diffusive movements described by global
random walk (GRW) simulations. The CGST averages verify identities similar to a
local balance equation which allow us to derive expressions for the flow
velocity and the intrinsic diffusion coefficient in terms of averaged
microscopic quantities. The latter are further used to verify the CGST-GRW
numerical approach. The upscaling approach is applied to biodegradation
processes in saturated aquifers and variably saturated soils and the CGST
averages are compared to classical volume averages. One finds that if the
process is characterized by slow variations in time, as in homogeneous reaction
systems, the differences between the two averages are negligible. Instead, the
differences are significant and can be extremely large in simulations of
time-dependent biodegradation processes in both soils and saturated aquifers.
