[8] In equation (3), the most problematic coefficient is *φ*, which has rather transient characteristics. In the present study, to address the temporally varying recharge rate, the concept of transient *φ* will be introduced. The *φ* is given by *V*/*Δ**h*, where *V* is the required volume of water for the *Δh* [L] water table rise per unit area [*Kayane*, 1983; *Sophocleous*, 1991; *Park and Parker*, 2008]. In fact, *V* is a nonlinear function of *Δh* and unsaturated hydraulics. Figure 1 shows the concept of transient *φ*. In the figure, the solid line at *t*_{∞} is the equilibrium water content profile due to a long period of time without precipitation, and the dashed line at *t* is a profile, which has not yet reached the equilibrium state. In terms of residual moisture content at a reference elevation, the solid line reaches its irreducible condition, *θ*_{r}, while the dashed line is at interim water content, *θ*_{tr}, because the moisture from the preceding precipitation in the unsaturated zone has not yet drained or stabilized. If there is another precipitation event and the change of the water table is *Δ**h*, the engaged water volume corresponding to *t* and *t*_{∞} curves are II and I + II, respectively. The discrepancies in the water volumes originate from the temporal change of the water content profile. From this reasoning, it can be inferred that a recharge rate needs to be decided based both on the water table fluctuation of a saturated zone and the transient water content profile of an unsaturated zone. The temporal change of unstable soil moisture can be described by the kinematic wave model [*Charbeneau*, 2000; *Nachabe*, 2002] as,

where the effective saturation Θ [–] is equal to (*θ* – *θ*_{r})/(*θ*_{s} – *θ*_{r}), *θ* is the volumetric water content of porous media [–], *θ*_{s} is saturated water content equivalent to effective porosity [–], *θ*_{r} is the irreducible or residual water content [–], *K* is the unsaturated hydraulic conductivity [LT^{−1}] given as a function of *θ*, and *z* is the distance [L] along the vertical direction toward the ground surface, of which the origin is at the water table. By applying the van Genuchten-Mualem unsaturated hydraulic conductivity model [*van Genuchten*, 1980] to equation (4), the rate of effective saturation change with time, *ρ* [T^{−1}], becomes

where *m* = 1 *–* 1/*n*, *Γ* = 1 *–* Θ^{1/m} = (*αz*)^{n}/{1 + (*αz*)^{n}}, and *α*, *m*, and *n* are the van Genuchten model parameters. Further detail on the derivation of equation (5) is summarized in Appendix A. Therefore, Θ(*z*, *t*) is given by,

where Θ_{0} is the initial effective saturation. Immediately after a water table rises because of recharge, the pores right above the water table are close to saturation. The remaining excessive moisture in this zone is drained by gravitational force with a diffusive pattern and this diffusive component of recharge does not involve a water table rise, except for the case when the aquifer hydraulic conductivity is considerably low. In this study, the diffusive recharge flux is not counted as the recharge rate. Thus, the estimation of recharge rate by the developed methodology could be slightly biased. As previously mentioned, the water content profile before the cessation of drainage (Figure 1) generally deviates from its steady state curve. The excessive soil moisture at this stage is drained relatively quickly by gravitational force until the capillary pressure of the pore reaches the entry threshold, which may be represented by the inflection height of the water content profile. In the present study, it is assumed that the pattern of the transient curve may be approximated by manipulating the transient effective residual water content (*θ*_{tr}). *θ*_{tr} can be delineated from equation (6) with equation (5) by substituting *z* with a distance from the water table to a reference point which may be set as a height above maximum water table fluctuation. Also, Θ_{0} for this case is assumed to be that of the inflection height, *z**, of the moisture retention curve (i.e., *z** = 1/*α*), and equal 1/2^{m}.

[9] As an exercise of the derived equations, *θ*_{tr} at 1 m above water table is computed with equations (5) and (6) and the unsaturated hydraulic parameters of sand (*θ*_{s} = 0.43; *θ*_{r} = 0.045; *α* = 14.5 m^{−1}; *n* = 2.68; *K*_{s} = 7.128 m d^{−1}) [*Carsel and Parrish*, 1988]. In the computation, it is assumed that the starting value is equal to the entry effective saturation of Θ_{0} = 0.6476. In 1 d, the effective saturation of sandy loam drops to Θ(*z* = 1 m, *t* = 1 d) = 0.2413, which is 37% of its starting value. Based on these observations, one can set *θ*_{tr} for sand in 1 d since a recharge event is *θ*_{tr} = Θ_{t}θ_{s} + (1 – Θ_{t})*θ*_{r} = 0.2413*θ*_{s} + (1 – 0.2413)*θ*_{r} = 0.1379. From the computed *θ*_{tr}, the fillable porosity corresponds to *Δ**h* = 0.1 m water table rise can be computed by the definition, the volume of water for unit rise of water table per unit area (Figure 1) as,

and the value is 0.0661. This value is only 15% of effective porosity of sand.