## 1. Introduction

[2] In many low-lying, humid regions where shallow water tables (WTs) are common, vegetation growth is largely driven by the water table dynamics [*Bierkens*, 1998; *Nachabe*, 2002]. It is therefore important to understand WT dynamics for efficient management of crops as well as natural vegetation. Shallow WT fluctuations in unconfined aquifers are generally studied by means of a storage parameter called the drainable porosity (or specific yield). Drainable porosity (*λ _{d}*) represents the amount of water released by the aquifer when the water table drops by a unit distance [

*Bouwer*, 1978;

*Freeze and Cherry*, 1979;

*Neuman*1987]. It can be estimated as the change in total water storage of the aquifer,

*S*, per unit drop in WT elevation [

*Hilberts et al.*, 2005]:

where *h* is height of the phreatic surface above a reference datum. Analogous to the concept of *λ _{d}* is the fillable porosity (

*λ*), which can be defined as the amount of water that is imbibed or absorbed by an aquifer per unit rise in the water table [

_{f}*Bouwer*, 1978;

*Sophocleous*, 1991;

*Park and Parker*, 2008]. It can be estimated, in a similar manner to

*λ*, as the change in total storage deficit of the aquifer,

_{d}*D*, per unit rise in WT elevation:

_{s}Drainable and fillable porosity play key roles in WT fluctuations due to groundwater flow, evapotranspiration (ET), rainfall, and are integral parameters of Boussinesq-type groundwater flow models [e.g.,*Chapman and Dressler*, 1984; *Brutsaert*, 1994; *Hilberts et al.*, 2005]. These parameters are also required when ET is estimated from groundwater fluctuations [*Loheide et al.*, 2005; *Gribovszki et al.*, 2007].

[3] In shallow phreatic aquifers, WT fluctuations are significantly influenced by the soil capillary retention [*Childs*, 1960; *Gillham*, 1984; *Nielsen and Perrochet*, 2000; *Healy and Cook*, 2002]. Hence the magnitudes of *λ _{d}* and

*λ*depend on the position of the WT and soil moisture status in the unsaturated zone [

_{f}*Hillel*, 1998], rather than being unique for a given soil type [

*Hilberts et al.*, 2005]. While existing analytical expressions for

*λ*do account for the effect of unsaturated zone soil moisture [e.g.,

_{d}*Duke*, 1972;

*Bierkens*, 1998;

*Nachabe*, 2002;

*Hilberts et al.*, 2005], these methods assume hydrostatic conditions in the soil profile.

[4] Although *λ _{d}* expressions derived using static moisture profiles have been successfully used in several studies, it is well known that the storage parameters of unconfined aquifers are also significantly affected by moisture flux in the unsaturated zone [

*Childs*, 1960;

*Chapman and Dressler*, 1984;

*Chapman*, 1995;

*Tritscher et al.*, 2000;

*Brutsaert*, 2005].

*Tritscher et al.*[2000] used a two-dimensional flow model to show that the error associated with estimation of

*λ*from static moisture profile may be as high as 35% when the infiltration rates are high.

_{d}*Chapman and Dressler*[1984] showed that fillable porosity was directly affected by the vertical recharge rate (

*R*) and approached zero as

_{e}*R*increased. Thus the application of

_{e}*λ*expressions based on static moisture profile should ideally be limited to hydrostatic conditions. However, this condition rarely holds in most field situations, unless the soil is very coarse [

_{d}*Bear and Cheng*, 2008]. The assumption of hydrostatic pressure distribution is especially inappropriate in shallow WT environments such as wetlands, riparian areas and crop fields with controlled WTs where a significant portion of the ET demand is directly fulfilled by the water table [

*Loheide et al.*, 2005;

*Nachabe et al.*, 2005].

[5] Another important limitation of *λ _{d}* expressions based on the hydrostatic assumption is that they result in the same value for both

*λ*and

_{d}*λ*for a given WT depth. Normally

_{f}*λ*is regarded as smaller than

_{f}*λ*due to hysteresis in moisture retention [

_{d}*Bouwer*, 1978;

*Sophocleous*, 1991;

*Stauffer et al.*, 1992], or air compression or encapsulation in the profile [

*Nachabe et al.*, 2004]. However, in many practical situations it may not be possible to incorporate these factors while estimating

*λ*and

_{d}*λ*. Therefore, most previous studies have implemented only a nonhysteretic

_{f}*λ*based on the assumption of a hydrostatic soil moisture profile [e.g.,

_{d}*Bierkens*, 1998;

*Hilberts et al.*, 2005, 2007;

*Gribovszki et al.*, 2007]. However, as we show below, as the moisture profile deviates from a hydrostatic pressure distribution, different

*λ*and

_{d}*λ*values result for a given WT depth, even when hysteresis and air encapsulation are neglected. Yet, separate

_{f}*λ*and

_{d}*λ*parameters for water table decline and rise have not previously been implemented in unconfined groundwater flow studies. These parameters have also been treated as common in all of the hydraulic groundwater models and their derivatives [e.g.,

_{f}*Troch et al.*, 2003;

*Hilberts et al.*, 2005, 2007;

*Laio et al.*, 2009;

*Pumo et al.*, 2010] which may potentially lead to erroneous results especially during water table rise scenarios.

[6] *Laio et al.* [2009] recently developed an approach for incorporating dynamic moisture profiles explicitly into *λ _{d}* estimations. These authors used the soil moisture retention model of

*Brooks and Corey*[1964] to develop an approximate analytical expression for

*λ*accounting for both

_{d}*R*and root uptake. Their relation was expressed in terms of a critical WT depth term, representing the point of transition from a shallow to deep WT. However, this critical WT depth lacked a closed-form expression and was approximated by a hypergeometric function. Also, these authors did not separately consider both

_{e}*λ*and

_{d}*λ*.

_{f}[7] The main objective of this study was to develop approximate closed-form expressions for*λ _{d}* and

*λ*as separate aquifer parameters that account for vertical soil moisture flow in the unsaturated zone. First, a simple mass balance approach is used to illustrate that when hydrodynamic conditions prevail in the soil profile,

_{f}*λ*and

_{d}*λ*are unequal (but they are equal under hydrostatic conditions). This simple analysis also establishes the theoretical basis for the development of separate analytical expressions for the two parameters. Assuming no hysteresis, the soil moisture retention curve was first integrated to obtain

_{f}*S*and

*D*under hydrostatic conditions. Here we use a variant of the

_{s}*van Genuchten*[1980] retention model [

*Troch*, 1992;

*Troch et al.*, 1993] which enables analytical solutions for both

*S*and

*D*. These solutions are expressed in terms of matric suction at the soil surface ( ) which is a linear function of WT under a hydrostatic pressure distribution above the water table. The hydrostatic expressions were then modified to approximate

_{s}*S*and

*D*under dynamic soil moisture profiles by substituting the linear function by a flow-dependent, nonlinear relationship estimated under successive steady state ET and/or recharge (

_{s}*R*) from/to a shallow water table. Differentiating these modified expressions according to equations (1) and (2) results in generalized, approximate expressions for

_{e}*λ*and

_{d}*λ*in terms of and soil hydraulic parameters. The effects of unsaturated zone soil moisture flux to and from the WT in the

_{f}*λ*and

_{d}*λ*expressions are therefore accounted for by the steady state , which was estimated using the hydraulic conductivity model of

_{f}*Gardner*[1958] at successive time steps. The final

*λ*and

_{d}*λ*expressions are flexible enough to incorporate any predefined ET and

_{f}*R*functions. Finally, the analytical expressions are implemented in a one-dimensional hydraulic groundwater model to simulate shallow WT dynamics in a 15 ha potato field in northeast Florida, managed under a WT control system. Simulated hourly WT dynamics are compared for models implemented with only a single, hydrostatic profile-based expression for

_{e}*λ*, and with distinct

_{d}*λ*and

_{d}*λ*expressions developed in this study based on a dynamic moisture profile.

_{f}