Watertable dynamics under capillary fringes : experiments and modelling

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 <i>h(t)</i> responded very closely to the driving head compared with the predictions of previous analytical and numerical models. The total moisture quantified as an equivalent, saturated height <i>h</i>_tot<i>(t)</i> varied very little compared with the watertable height. Neither <i>h(t)</i> nor <i>h</i>_tot<i>(t)</i> deviated significantly from simple harmonics when the driving head was simple harmonic. This indicates that non-linear effects are weak and hence that analysis based on linear solutions have fairly broad applicability. When <i>h(t)</i> and <i>h</i>_tot<i>(t)</i> are simple harmonic, the ratio <i>n</i>_d=[d<i>h</i>_tot/d<i>t</i>]/[d<i>h</i>/d<i>t</i>] is a constant in the complex formalism. Its magnitude

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is the usual effective porosity while its argument accounts for the phase shift which is always observed between <i>h(t)</i> and <i>h</i>_tot<i>(t)</i>. Within the current range of experiments this dynamic, effective porosity <i>n</i>_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 <i>n</i>_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 Richard’s equation with van Genuchten parameters corresponding to the measured water retention curve. The complex <i>n</i>_d can be immediately adopted into linear watertable problems in 1 or 2 horizontal dimensions. Compared with “no fringe solutions”, this leads to modification of the watertable behaviour which is in agreement with experiments and previous models. The use of a complex <i>n</i>_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 flow of groundwater. Water Resour Res 1987; 23(5):805–8.] that the order of the differential equations is lower. For example, the linearised Boussinesq equation with complex <i>n</i>_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 insignificant compared to dealing with higher order equations. On the basis of the presently available data it also seems that the “complex <i>n</i>_d approach” is more accurate. This is to be expected since the complex <i>n</i>_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 <i>n</i>_d is a very simple extension since the determination of wave numbers already involves complex numbers.