| Periodic transient flow of groundwater due to time dependent boundary conditions occurs in a
few circumstances (e.g.rainfall seasonal, tidal diurnal and high frequency oscillatory well
tomography, variations). Solving the flow equations (Darcy’s Law, mass conservation)
requires knowledge of hydraulic conductivity K and storativity s. In case of heterogeneous
aquifers of random spatial variations of K and s, it is common to derive the mean flow
variables (heads, fluxes) by adopting constant effective properties in the flow equations. The
usual approach is the quasi-steady one, which adopts the well known steady state
Kefst and the arithmetic mean sA values, supposedly applying to sufficiently low
frequency Ï. We derive the effective conductivity Kef of heterogeneous aquifers of
lognormal conductivity distribution for arbitrary Ï. It leads for the first time to the
new effect of emergence of dynamic Kef, which is complex. This implies a phase
difference between the average flux and head gradient in the upscaled Darcy’s Law,
in contrast with the local Darcy’s Law for which Kef is real. The dependence of
both the amplitude and phase of Kef upon Ï and the logconductivity variance are
illustrated graphically, with delimitation of the parameters values for which the
quasi-steady approximation applies. The results are illustrated for a few realistic values of
aquifer properties, for which the quasi-steady approximation generally applies,
with the exception of formations of very low mean conductivity. The results can be
used for more complex time variations by superimposing harmonics of different Ï. |