Watertable dynamics under capillary fringes: experiments and modelling

Abstract

Watertable heights and total moisture content were measured in a sand column where the piezometric head at the base (``the driving head'') varied as a simple harmonic with periods in the range from 14.5 min to 6.5 h. The watertable height ht† responded very closely to the driving head compared with the predictions of previous analytical and numerical models. The total moisture quanti®ed as an equivalent, saturated height h tot t† varied very little compared with the watertable height. Neither ht† nor h tot t† deviated signi®cantly from simple harmonics when the driving head was simple harmonic. This indicates that non-linear e€ects are weak and hence that analysis based on linear solutions have fairly broad applicability. When ht† and h tot t† are simple harmonic, the ratio n d ˆ ‰dh tot adtŠa‰dhadtŠ is a constant in the complex formalism. Its magnitude jn d j is the usual e€ective porosity while its argument accounts for the phase shift which is always observed between ht† and h tot t†. Within the current range of experiments this dynamic, e€ective porosity n d appears to be almost independent of the forcing frequency, i.e., it is a function of the soil and its compaction only. Introducing the complex n d enables analytical solution for the watertable height in the column which is simpler and more consistently accurate over a range of frequencies than previous models including RichardÕs equation with van Genuchten parameters corresponding to the measured water retention curve. The complex n d can be immediately adopted into linear watertable problems in 1 or 2 horizontal dimensions. Compared with``with``no fringe solutions'', this leads to modi®cation of the watertable behaviour which is in agreement with experiments and previous models. The use of a complex n d to account for the capillary fringe in watertable models has the advantage, compared with previous models, e.g., [Parlange J-Y, Brutsaert W. A capillary correction for free surface ¯ow of groundwater. Water Resour Res 1987; 23(5):805±8.] that the order of the di€erential equations is lower. For example, the linearised Boussinesq equation with complex n d is still of second order while the Parlange and Brutsaert equation is of third order. The extra work of calculating the imaginary part of the initially complex solution is insigni®cant compared to dealing with higher order equations. On the basis of the presently available data it also seems that thè`complex n d approach'' is more accurate. This is to be expected since the complex n d accounts implicitly for hysteresis while the Green±Ampt model does not. With respect to linear watertable waves, accounting for the capillary fringe through the complex n d is a very simple extension since the determination of wave numbers already involves complex numbers.

Influence of capillarity on a simple harmonic oscillating water table: Sand column experiments and modeling

“…Therefore the complex nature of n w was adopted to account for the damping through its magnitude jn w j and the phase lag through its argument Arg{n w }. Note that we have elected to use the subscript w as opposed to Nielsen and Perrochet's [2000] n d to denote the dependence of the complex effective porosity on the oscillation frequency w. This will be clearly evidenced in section 4.…”

“…[4] The influence of the capillary fringe on water table oscillations has received much attention in the literature from experimental [e.g., Lehman et al, 1998;Nielsen and Perrochet, 2000;Stauffer and Kinzelbach, 2001], theoretical [e.g., Parlange and Brutsaert, 1987;Barry et al, 1996;Nielsen and Perrochet, 2000;Li et al, 2000] and numerical [e.g., Li et al, 1997;Stauffer and Kinzelbach, 2001;Werner and Lockington, 2003] perspectives.…”

“…[7] Nielsen and Perrochet [2000] introduced the concept of a dynamic complex effective porosity n w in order to account implicitly for hysteresis effects on an oscillating water table. The complex nature of n w was required to account for the observation that fluctuations in the equivalent saturated height of the total moisture dh tot were both damped and lagged relative to those in water table dh.…”

“…[9] From their experiments over the range of oscillation periods, 14 min < T < 6.5 hours, Nielsen and Perrochet [2000] concluded that the complex effective porosity was a constant for the sand they tested. In this paper, further sand column experiments are conducted over a wider range of oscillation frequencies and for two additional porous materials with the results indicating a strong dependence of capillarity effects (n w ) on oscillation frequency.…”

Water table waves in an unconfined aquifer: Experiments and modeling

“…[27] At higher frequencies, water table oscillations will be influenced by capillarity [e.g., Li et al, 1997;Nielsen and Perrochet, 2000]. Under the shallow aquifer assumption (neglecting vertical flow effects), Barry et al [1996] solved a modified Boussinesq equation with the capillarity correction of Parlange and Brutsaert [1987] for water table waves described by (2) and obtained the following dispersion relation, where B = nH ψ is the capillary fringe thickness.…”