Analytical expressions for drainable and fillable porosity of phreatic aquifers under vertical fluxes from evapotranspiration and recharge


Corresponding author: S. Acharya, Soil and Water Science Department, University of Florida, Gainesville, FL 32611, USA. (


[1] In shallow unconfined aquifers, the response of the water table (WT) to input and output water fluxes is controlled by two distinct storage parameters, drainable and fillable porosity, which are applicable for WT drawdown and rise, respectively. However, only the drainable porosity estimated from the hydrostatic soil moisture profile is in common use. In this study, we show that under conditions of evapotranspiration and/or recharge from or to a shallow water table, drainable and fillable porosity have different values. Separate analytical expressions are developed for drainable and fillable porosity accounting for dynamic soil moisture conditions through the assumption of successive steady state fluxes in the unsaturated zone. The equations are expressed in terms of soil hydraulic parameters and matric suction at the soil surface. Parametric evapotranspiration and recharge functions are used to estimate the suction at the soil surface. The final expressions are independent of evapotranspiration or recharge function, thus allowing the use of any appropriate function to estimate the storage parameters. It is shown that the occurrence of unsaturated zone fluxes can result in significantly different values of drainable and fillable porosity, even when hysteresis is neglected. Application of the two parameters in a Boussinesq-type groundwater model resulted in significantly improved estimates of field-measured water table dynamics compared to the hydrostatic, single-parameter model.

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 (λd) 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]:

display math

where h is height of the phreatic surface above a reference datum. Analogous to the concept of λd is the fillable porosity (λf), which can be defined as the amount of water that is imbibed or absorbed by an aquifer per unit rise in the water table [Bouwer, 1978; Sophocleous, 1991; Park and Parker, 2008]. It can be estimated, in a similar manner to λd, as the change in total storage deficit of the aquifer, Ds, per unit rise in WT elevation:

display math

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 λd and λf depend on the position of the WT and soil moisture status in the unsaturated zone [Hillel, 1998], rather than being unique for a given soil type [Hilberts et al., 2005]. While existing analytical expressions for λd do account for the effect of unsaturated zone soil moisture [e.g., Duke, 1972; Bierkens, 1998; Nachabe, 2002; Hilberts et al., 2005], these methods assume hydrostatic conditions in the soil profile.

[4] Although λd 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 ofλd from static moisture profile may be as high as 35% when the infiltration rates are high. Chapman and Dressler [1984] showed that fillable porosity was directly affected by the vertical recharge rate (Re) and approached zero as Re increased. Thus the application of λd 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 [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 λd expressions based on the hydrostatic assumption is that they result in the same value for both λd and λf for a given WT depth. Normally λf is regarded as smaller than λd due to hysteresis in moisture retention [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 λd and λf. Therefore, most previous studies have implemented only a nonhysteretic λd based on the assumption of a hydrostatic soil moisture profile [e.g., 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 λd and λf values result for a given WT depth, even when hysteresis and air encapsulation are neglected. Yet, separate λd and λf 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., 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 λd estimations. These authors used the soil moisture retention model of Brooks and Corey [1964] to develop an approximate analytical expression for λd accounting for both Reand 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 bothλd and λf.

[7] The main objective of this study was to develop approximate closed-form expressions forλd and λf 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, λd and λf 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 S and Ds under hydrostatic conditions. Here we use a variant of the van Genuchten [1980] retention model [Troch, 1992; Troch et al., 1993] which enables analytical solutions for both S and Ds. These solutions are expressed in terms of matric suction at the soil surface ( math formula) 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 and Ds under dynamic soil moisture profiles by substituting the linear math formulafunction by a flow-dependent, nonlinear relationship estimated under successive steady state ET and/or recharge (Re) from/to a shallow water table. Differentiating these modified expressions according to equations (1) and (2) results in generalized, approximate expressions for λd and λf in terms of math formula and soil hydraulic parameters. The effects of unsaturated zone soil moisture flux to and from the WT in the λd and λf expressions are therefore accounted for by the steady state math formula, which was estimated using the hydraulic conductivity model of Gardner [1958] at successive time steps. The final λd and λf expressions are flexible enough to incorporate any predefined ET and Refunctions. 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 forλd, and with distinct λd and λf expressions developed in this study based on a dynamic moisture profile.

2. Theory

2.1. Inequality in λd and λf Under ET and/or Recharge From/to a Shallow Water Table

[8] We consider an unconfined aquifer with a shallow WT and plants growing at the soil surface. The aquifer is assumed to be incompressible with no hysteresis in the soil moisture retention behavior. The water balance components of the cross section of such an aquifer with a shallow WT bounded by a ditch or stream are illustrated in Figure 1. A conceptual representation of saturated (Ss) and unsaturated storage (Us), storage deficit (Ds), and the changes in these quantities due to WT decline or rise is presented in Figure 2.

Figure 1.

Conceptual representation of the water table profile and water balance components in a typical unconfined aquifer with a shallow phreatic surface, bounded by a ditch or stream.

Figure 2.

Conceptual representations of (left) change storage deficit (Ds) due to a unit rise in WT elevation and (middle and right) changes in unsaturated storage (Us) and saturated storage (Ss) due to a unit decline in WT in an unconfined aquifer system. Although both S and Ds change with any change in WT elevation, only the change in S during WT drawdown and Ds during WT rise are illustrated.

[9] Following the definitions of drainable and fillable porosity in equations (1) and (2), it can be shown, using the conservation of mass principle, that the two parameters have different values under conditions of ET and Re from/to the shallow water table (dynamic soil moisture conditions) while they converge to one single value as the soil profile moves toward a hydrostatic pressure distribution. Initially, the water table is at an elevation h from the reference datum at which it has the potential to move in either upward or downward direction depending on the boundary conditions. Considering the drainage scenario, if the WT moves down by a small distance math formula to a new position, the total storage of the profile (Us + Ss) will also change by math formula to a new value. Under shallow WT environments, it is normally true that changes in WT elevation can occur in direct response to surface ET [Loheide et al., 2005; Gribovszki et al., 2007] or recharge fluxes. Therefore, this drawdown in WT is the combined result of drainage, ET and Re. The volume of water drained ( math formula) from the profile may then be expressed as

display math

From equation (3), it is apparent that less drainage is necessary to effect a unit drop in WT elevation when ET > 0 than when ET = 0 and Re = 0 while the opposite is true when Re > 0 and ET = 0.

[10] Similarly, if the WT instead rises from the initial position by the same distance math formula, the aquifer S and Ds will also change to new values. This rise in WT is the result of the combined effects of water imbibition, ET and Re. The volume of water imbibed ( math formula) during this WT rise can be expressed as

display math

Equation (4) implies that more water is necessary to effect a unit rise in WT when ET > 0 than when ET = 0 and Re = 0 while less water is needed when Re > 0 and ET = 0. Note that equations (3) and (4) may be expressed in terms of either math formula or math formula as every change in the WT position is accompanied by simultaneous changes in both S and Ds. The forms used here were chosen to be consistent with the respective definitions of λd and λf in equations (1) and (2).

[11] From equations (3) and (4), it is clear that math formula when water is either lost from (ET) or accreted to (Re) the soil profile. Equations (3) and (4) also indicate that when ET > 0, λd < λf and vice versa, when Re> 0. Under hydrostatic condition, the math formula due to a unit drop in WT from initial position h is equal to the math formula due to an equivalent rise from h, since the shape of the soil moisture distribution curve under this assumption remains the same during WT movement. In the following sections, the inequality in λd and λf is investigated under specific flow conditions for different soil types by developing separate approximate analytical expressions for λd and λf.

2.2. Soil Water Storage and Storage Deficit

[12] The unsaturated soil water storage is normally calculated by integrating the moisture profile from the WT to the soil surface with respect to depth (z), expressed as

display math

where, H is the distance of soil surface from the datum, θ(ψ) is the water content as function of soil matric suction (ψ), and θr is the residual water content. Changing the variable of integration in equation (5) yields

display math

where math formula is the matric suction at the soil surface. From Figure 2, the total soil water storage (S) of the profile can now be estimated as the sum of the area math formula and area under ABDE given by equation (6), which yields

display math

where math formula is the saturated water content. Integration of the second term on the right side of equation (7) requires a constitutive math formula relationship. In this study we used a modified van Genuchten [1980] retention model introduced by Troch [1992], which has been widely used [e.g., Bierkens, 1998; Hilberts et al., 2005, 2007]:

display math

where α and nare soil-specific parameters of the modified model. Substituting(8) in (7) yields

display math

Similarly the storage deficit (Ds) of the profile is given by the area under BCD in Figure 2, which can be expressed as

display math

Analytical solution of equations (9) and (10) is possible only if math formula = 1; i.e., when the pressure distribution in the unsaturated zone is hydrostatic [e.g., Bierkens 1998; Hilberts et al., 2005], which yields

display math
display math

It also follows from equations (11) and (12) that math formula, which gives the total area of the profile above the reference datum.

[13] Note that in equations (11) and (12), the critical variable is math formula, the suction at the soil surface, given as math formula for a hydrostatic pressure distribution. However, hydrostatic conditions are rarely reached under shallow water table environments because of the effects of ET and recharge. Therefore, use of equations (11) and (12), which assume a hydrostatic moisture profile, may introduce errors in the estimates of S and Ds when dynamic moisture conditions prevail.

[14] Under dynamic soil moisture conditions, math formula ≠ 1 and its value depends on the magnitude of the flux, precluding analytical integration of equations (9) and (10). Therefore, in this study we adopt an approximate approach to estimate S and Ds under conditions of ET and/or Re from/to shallow WTs. We modified equation (11) and (12) by replacing the hydrostatic math formulawith a flow-dependent math formula estimated as a function of the WT elevation ( math formula).This approach assumes that (11) and (12) hold nearly true even under dynamic conditions, if math formula is estimated accounting for the flow situation in the unsaturated zone. As we describe below, math formula under ET and Re can be estimated by assuming successive steady state fluxes to and from the WT respectively, which can then be used in equations (11) and (12) to estimate S and Ds. Comparison of the S and Dsapproximations with values obtained from HYDRUS-1D [Simunek et al., 1998] numerical solution of Richards equation under respective draining and rising WTs showed reasonably good agreement. Figure 3 shows S and Dsas functions of WT under ET and recharge for the Ellzey fine sand series found at our field sites (described in detail below) as computed from HYDRUS-1D,equations (11) and (12) modified with math formula, and the original hydrostatic forms of equations (11) and (12) (i.e., with math formula). For the Ellzey series, the root mean square errors (RMSEs) between the numerical solution for S and Ds and the estimates from equations (11) and (12) modified with math formula were as much as 50% lower than the RMSE based on the hydrostatic estimate (equations (11) and (12) with math formula). Thus, our approximation of a flow-dependent math formula in equations (11) and (12) was able to reasonably estimate storage and storage deficit under dynamic moisture profiles and also showed a significant improvement compared to simply applying the hydrostatic expressions.

Figure 3.

(left) Water table depth versus storage during ET and WT drawdown and (right) WT depth versus storage deficit during WT rise due to rainfall for the Ellzey fine sand series. The black lines are from HYDRUS-1D, the blue lines are fromequations (11) and (12) with math formula (i.e., hydrostatic pressure distribution), and the red lines are from the modification of (11) and (12) with the nonlinear math formula function estimated under steady state moisture fluxes.

2.3. General Expressions for Drainable and Fillable Porosity

[15] From equations (11) and (12), λd and λf can be obtained by taking the derivative of S and Ds with respect to h:

display math
display math

It also follows from equations (13) and (14) that, for a given water table depth, λd and λf can be related as

display math

Equations (13) and (14) are generalized expressions for λd and λf as functions of the WT elevation. Note that both of these equations can be defined for every WT elevation because at every point below the surface, the phreatic surface has a potential to move either upward or downward based on the combined effect of the water balance components (see Figure 1). Both (13) and (14) suggest that λd and λf depend on math formula [Brutsaert, 2005]. Under hydrostatic conditions, both equations (13) and (14) yield the drainable porosity expression derived by Bierkens [1998] and Hilberts et al. [2005]:

display math

Under nonhydrostatic conditions, on the other hand, the behavior of math formula becomes much more complex. Due to highly nonlinear nature of soil water movement, normally it is not possible to obtain an explicit expression for math formula under transient flow conditions. The simplified approach used here is to assume steady state flux above the water table, which enables direct calculation of math formula

2.4. Steady Flow From/to the Water Table: Evaporation, Root Water Uptake, and Infiltration

[16] In this section, expressions for math formula are derived considering steady state evaporation or infiltration under shallow water table conditions.

2.4.1. Evaporation and Root Water Uptake

[17] Steady state soil moisture flux from the water table, q0, in the absence of plant root uptake, is given by the Darcy equation as

display math

where K(ψ) is the unsaturated hydraulic conductivity function and z is the vertical coordinate (Figure 1). Equation (17) can be integrated analytically for certain K(ψ) functions to obtain a closed-form expression forψ. One such solution was presented by Gardner [1958] using an exponential K(ψ) function:

display math

where Ks is the saturated hydraulic conductivity and math formula is an empirical parameter. Owing to its simplicity, many have used this hydraulic conductivity model to obtain analytical solutions for nonlinear moisture flow [Philip, 1969; Raats, 1974; Zhu and Mohanty, 2002]. Combining equations (17) and (18) and integrating within the limits of h and H yields the following expression for math formula

display math

[18] When plant roots are present, a significant portion of the soil moisture is taken up by the roots to meet the transpiration demand. Under steady state, applying the principle of mass conservation yields [Raats, 1974]

display math

where math formula is the flux at the soil surface, U is the total root water uptake, and hr is the height of bottom of the root zone from the datum (Figure 1). The magnitude of U in equation (20) can be estimated by defining an appropriate root distribution function [e.g., Raats, 1974; Jackson, 1996; Lai and Katul, 2000] and transpiration (T). Transpiration can be calculated using the potential ET (PET) and the leaf area index [Ritchie, 1972]. Once these fluxes are determined, math formula can be calculated simply by substituting math formula for math formula in equation (19). Evaporation and root water uptake, however, are highly dynamic processes that are dictated by moisture status of the soil profile [Feddes et al., 1976; Lai and Katul, 2000] and hence by the position of the phreatic surface. This is especially true for many crops. It is difficult to incorporate these dynamic phenomena in the analytical solutions and therefore require numerical solution methods [e.g., Simunek et al., 1998]. In this paper we use an approximate method to estimate ET from WT depth based on the soil texture and vegetation type, as discussed in section 2.4.2.

2.4.2. Evapotranspiration Approximation Based on Water Table Elevation

[19] In the case of natural shallow WT environments such as wetlands and riparian zones, most of the vegetation is well adapted to perpetual saturated or near-saturated conditions, thus showing little or no reduction in root water uptake due to high soil moisture [Laio et al., 2009; Pumo et al., 2010; Tamea et al., 2010]. In such environments, it is reasonable to assume that math formula which, from equation (19), yields

display math

Both linear [Harbaugh, 2005] and exponential [Shah et al., 2007] relations have been developed to estimate ET from water table elevation. In this study we used the exponential function given by Shah et al. [2007] since it considers both soil and vegetation type:

display math
display math

where math formula and β are parameters that respectively describe ET “transition depth” and the rate of reduction in ET as the WT falls below d′ (Figure 1). It was also shown by Shah et al. [2007] that the magnitude of β and math formula depend both on the type of soil and vegetation. Smaller β and larger d′ values correspond to finer soils and deeper rooted vegetation and vice versa. Figure 4 illustrates the relationship in (22) and (23)for clay, loam, and sand soils under typical grass-type vegetation based on the parameter values reported byShah et al. [2007]. Note that for clay soil the reduction in the rate of ET with decreasing WT is much smaller followed by loam and sand. Estimation of the ET parameters for our field site is discussed in section 3.3. The parameter values for the ET versus WT relationship for major soil textural classes under different vegetation types are given by Shah et al. [2007]. Once ET is estimated from equations (22) and (23), math formula can be easily calculated from equation (21).

Figure 4.

Relationship between ET rate (PET = 0.02 cm h−1) and WT depth for clay, loam, and sand soils under grass-type vegetation based onequations (22) and (23). The parameter values for the soils were d′ = 95 cm, β = 0.011 for clay, d′ = 85 cm, β = 0.028 for loam, and d′ = 30 cm, β = 0.043 for sand [Shah et al., 2007].

2.4.3. Water Table Recharge During Rainfall

[20] Water table recharge during rainfall begins once the wetting front reaches the phreatic surface [Woods et al., 1997] resulting in its rise. Depending on the depth to WT and the soil hydraulic properties, it can be assumed that a fraction of total rainfall during a time step (e.g., daily, hourly) contributes to recharge [Park and Parker, 2008] and subsequent water table rise. Under shallow WT conditions, almost all of the infiltrated water contributes to recharge instantaneously [Novakowski and Gillham, 1988; Hilberts et al., 2007] because the capillary fringe is usually close to the surface [Laio et al., 2009]. As the WT depth increases, more infiltrated water is held in the unsaturated zone, thus reducing the fraction of recharge during the same interval of time. A simplified approximation method would be to estimate recharge as a function of precipitation and WT depth without having to explicitly model the unsaturated flow processes. For example, Woods et al. [1997] estimated spatially variable recharge as a function of effective precipitation and WT depth in a hillslope. However, their recharge function was solely dependent on the WT elevation regardless of the soil type. Here, we assume that the recharge is spatially uniform throughout the field site and depends on the soil type, in addition to the position of the phreatic surface. In this study, we propose that the recharge fraction of rainfall at a given time step can be estimated by an exponential function expressed as

display math
display math

where Re is the recharge rate, P is the rainfall amount during the given time step, and βr and dr are the parameters that determine what fraction of rainfall is converted to recharge depending on the soil type. Note that unsaturated flow processes are not explicitly considered when equation (24) is used to estimate WT recharge, so it cannot be applied at very short time steps [Woods et al., 1997]. However, if the WT is close to the surface, hourly time steps may be adequate since instantaneous recharge can be assumed. Now, if we treat Re (−ve) as the steady flux in successive times steps, math formula is given as

display math

Note that because we make the simplifying assumption of constant flux, Re, throughout the unsaturated zone, math formula given by equation (26) may not correspond to values measured during rainfall infiltration. Under field conditions, math formula is likely to reach zero (or positive) before the water table experiences any recharge. Positive math formula in equations (13) and (14) would produce physically impossible negative λd or λf.

2.5. Parameters λd and λf During ET or Recharge From the Water Table

[21] Differentiating equations (21) and (26) with respect h yields

display math

where μ represents either ET (+ve) or recharge, Re (−ve). Combining equation (27) with equations (13) and (14) gives the final expressions for λd and λf under successive steady state fluxes from the water table

display math
display math

It can be seen from equations (28) and (29) that λdλf. The disparity between the two parameters is determined by the soil hydraulic parameters, and the unsaturated zone flux. The direction of the flux determines which storage parameter attains the larger value at a given water table depth while the hydraulic parameters determine how different the two parameters will be. Note that under hydrostatic condition where μ = 0, math formula, both equations (28) and (29) reduce to (15), which is the expression developed by Hilberts et al. [2005] and Bierkens [1998]. The behavior of λd and λf for different soil textures under ET and Re is discussed further in section 4.1.

2.6. Implementation of λd and λfin a 1-D Groundwater Model

[22] Water movement through soils and other porous media is most accurately represented by the Richards equation. However, due to high nonlinearity and associated difficulties of solving this equation, simpler approaches are usually sought [Pikul et al., 1974]. The overall water balance per unit area of an aquifer can be expressed as

display math

Note that in equation (30) only a single porosity parameter λ is implemented, which is the conventional approach. Substituting Darcy's law for the groundwater flux, assuming horizontal flow and neglecting vertical flow, yields the one dimensional Boussinesq equation

display math

where, the first term on the right gives lateral the groundwater flux. However, using only a single λ in equation (31) to express the effects of both ET and Re on WT elevation is theoretically inconsistent. Therefore, implementation of separate λd and λf developed in this study requires reformulation of (31), depending on the direction of groundwater flow:

display math

during groundwater inflow toward the aquifer from the lateral boundaries, and

display math

during groundwater outflow from the aquifer toward the boundaries. Note that only one or both of the storage parameters may appear in the governing equation depending on which unsaturated fluxes are active. For example, when groundwater is flowing in to the system and ET = 0, then the governing relation (equation (32)) will include only fillable porosity. But for ET > 0, both math formula and math formula appear in the governing equation because groundwater inflow raises the WT (controlled by math formula) while ET tends to lower the WT (controlled by math formula). Hence, the response of the phreatic surface will depend on the net difference between the magnitude of groundwater inflow and the ET loss. A common example of this condition is the diurnal fluctuations normally observed in shallow WT environments [e.g., Loheide et al., 2005; Gribovszki et al., 2007] where the WT normally drops during the day (ET > groundwater inflow) and recovers during the night (ET ≈ 0).

[23] In equation (33), note that the recharge component is linked with both math formula and math formula. An example scenario governed by equation (33) occurs in artificially drained fields where the water table is usually lowered before anticipated rainfall events in order to avoid the inundation of the field. While the groundwater flow direction at the lateral boundaries is outward, the response of the water table (rise or drop) will depend on the magnitude of the recharge at the phreatic surface. When Re > 0, the magnitude of λd increases because at each time step there will be some accretion of moisture [Chapman and Dressler, 1984] which needs to be removed from the aquifer to bring the same decline in the water table as compared to when Re = 0. Therefore, at each time step with Re > 0, the total amount of moisture added by recharge to the system will be partitioned into two fractions: (a) the fraction which contributes to WT rise and is controlled by λf, and (b) the fraction which is drained from the profile as the WT drops due to groundwater outflow, and is controlled by λd. Therefore, the net effect of this Re is determined by the difference between the reciprocals of math formula and math formula.

3. Methods

3.1. Study Site

[24] The study field chosen to test the expressions for λd and λf was located in northeast Florida (29.694′N, 81.446′W; Figure 5). This area is a low (<10 m above sea level), flat landscape with a naturally shallow water table (<2 m from surface). A locally important crop is potato grown under a WT management system known as seepage irrigation [Smajstrla et al., 2000; Munoz-Arboleda et al., 2008; Acharya and Mylavarapu, 2011] wherein the WT is maintained close to the crop root zone by applying water from shallow furrows spaced uniformly at 18–20 m between main lateral ditches (1.5–2.0 m deep) at the edges of fields on the order of 3–15 ha. The flow system in these agricultural fields therefore can be appropriately represented by the Boussinesq equation (i.e., equations (27) and (28)). The soil in the field site is highly sandy classified as Ellzey fine sand series (sandy, siliceous, hyperthermic, Arenic Endoaqualf) [Natural Resources Conservation Service, 1999] with some clay content at depths 75–120 cm [Acharya and Mylavarapu, 2011].

Figure 5.

Study site in northeast Florida showing agricultural land use and typical water table management system with ditches and irrigation furrows.

3.2. Water Table and Weather Data

[25] Three shallow (1.2 m deep) wells were installed in a 15 ha field (Figure 5) prior to the Spring 2010 potato season (February to June). Hourly WT data were collected from each well using data logging pressure transducers. In the 2011 potato season, hourly WT data were collected from 3 new wells installed in the same field at approximately the same locations. In both years, hourly PET estimates were calculated using the Penman-Monteith [Monteith, 1965] equation based on the weather data collected from a Florida Automated Weather Network (FAWN; weather station located adjacent to the field site. Hourly PET estimates were converted to the actual potato ET estimates using available potato crop coefficient values [Allen et al., 1998].

3.3. Parameter Estimation

[26] Moisture retention parameters of Ellzey fine sand were estimated by fitting soil moisture retention data to the modified van Genuchten function (equation (4)) which showed a very good fit for both the original and modified equations (Figure 6). Acharya and Mylavarapu [2011]reported a lab-determined averageKs = 10.0 cm/h for Ellzey fine sand, while Rosa [2000] reported Ks = 7.0 cm h−1 below the WT of the same soil using a slug test method. In this study we used the latter value since in situ observations are more likely to represent field conditions. Table 1 shows the values of the hydraulic parameters for Ellzey series.

Figure 6.

Original (m = 1 − 1/n) and modified (m = 1 + 1/n) van Genuchten moisture retention models (equation (8)) fitted to the measured date for Ellzey fine sand.

Table 1. Original and Modified van Genuchten (VG) Soil Moisture Retention Model Parameters and Gardner's Alpha (aG) for Clay, Loam, and Sand and Ellzey Seriesa
α (1 cm−1)0.0080.0000.0360.0010.1450.0790.0190.011
aG0.001 0.0335 0.25 0.068 
Ks (cm h−1)0.2 1.04 29.7 7.0 

[27] The most rigorous method for estimating Gardner's parameter math formula would be to fit the model to experimental ψ-K data. However, due to the constraints in collecting such experimental data, it is often necessary to revert to other methods. One such method is estimating math formula by establishing correspondence with the parameters of the soil moisture retention model [e.g., Zhu et al., 2004; Rucker et al., 2005; Ghezzehei et al., 2007; Warrick, 1995]. Ghezzehei et al. [2007] developed a simple formula to calculate math formula from the original van Genuchten model parameters α and n by defining two ψ values at which α and math formula match. They reported that, for soils with n > 2, math formula can be estimated reliably by

display math

Equation (34) was therefore used to estimate math formula for Ellzey sand since the value of n for Ellzey series was 2.62 (Table 1).

[28] Parameters β and d′ of the ET-WT relationship in 22 and 23 were adapted from Shah et al. [2007], who determined the parameter values for the major soil textural classes under bare soil and grass- and forest-type vegetation. The soil in our field site was dominated by fine sand with clay content up to 11% at 70–120 cm depth [Acharya and Mylavarapu, 2011]. Based on these results, the soil was considered close to the loamy sand texture. The parameter values were then determined by taking the average of the values for bare soil and grass since the root system of potato is normally shallow. The estimated values of β and d′ for Ellzey sand were 0.08 and 45 cm respectively. In order to estimate the parameters of the recharge function given by equation (24), several hypothetical simulations of rainfall infiltration with different initial WT depths and rainfall rates were conducted using HYDRUS-1D. The magnitudes of the soil moisture flux at the saturated-unsaturated zone interface (i.e., the phreatic surface) were recorded at each time step which were then then fitted with the recharge function in(24), which provided the estimates of βr = 0.15 and dr = 60 cm for Ellzey fine sand.

3.4. Boundary Conditions and Numerical Simulation

[29] The governing equations (32) and (33)are nonlinear partial differential equations whose analytical solutions are not always readily available. For our study, the equations were solved numerically by implicit finite difference method for one-dimensional lateral flow. In the field, ditches and water furrows are used to control the WT during irrigation and drainage. The furrows thus act as internal boundaries in addition to the ditches during irrigation and the early stage of drainage. Therefore, in the numerical solution, the locations of the water furrows were treated as internal boundaries in addition to the external boundary conditions imposed at the main ditches. The numerical solution was obtained using the tridiagonal matrix algorithm in the statistical programming language R [R Development Core Team, 2011]. The simulated WT dynamics were then compared with the field data collected from the study site during the spring of 2010 and 2011.The Nash and Sutcliffe [1970] efficiency coefficient, Ceff, and RMSE were used as goodness of fit measures between the model implemented only with λd and the model with both λd and λf.

4. Results

4.1. Drainable Porosity and Fillable Porosity Under ET and Recharge

[30] Figure 7 shows λd and λfas functions of WT for clay, loam, and sand with typical grass-type vegetation at different atmospheric ET demands. Increasing ET demand reducesλd to values smaller than hydrostatic (HS-λ), while at the same WT depths and ET demands, λf values are larger than HS-λ. This deviation from the HS-λ is different for different soils and is maximum for WT depths near the ET “transition depth” (see Figure 4). For example, in loam soil at ET = 0.02 cm/h, when the WT is near 80 cm from the surface, λd is 22% less than HS-λ while at the same WT depth λf is 51% more than HS-λ. For clay soil, this difference is not as extreme as in loam while for sandy soils the difference seems negligible even at ET rate of 0.02 cm h−1. As the WT depth increases beyond the ET transition depth, both λd and λf converge toward HS-λ since the pressure distribution in the soil profile moves toward hydrostatic. When the WT is very close to the surface, both λd and λf are close to zero irrespective of the ET demand. For sandy soils, which normally have low capillary retention, λd and λf hardly differ from HS-λ even when the ET demand is high. On the other hand, clay and loam soils show significant differences from HS-λ even under small ET demand because of their high moisture retention capacity.

Figure 7.

(left) Drainable and (right) fillable porosity as a function of WT depth for clay, sand, and loam soils under grass-type vegetation during hydrostatic and different steady ET rates from WT table based onequations (28) and (29). The ET parameters were adopted from Shah et al. [2007].

[31] The soil in our study site also did not show as large a shift in λd and λf from HS-λ as would clay or loam soils under ET (Figure 8). This can be attributed to the sandy nature of the profile and shallow root system of potato plants. Nonetheless, there is still a discernible downward shift in λd as ET demand increases. Figure 8 also suggests that the difference between λd and λf reaches as much as 18% of the effective porosity (θsθr) at 45 cm WT depth for ET = 0.03cm/h. Hourly ET rates in the study area can reach as high as 0.07 cm/h during midday (FAWN; which would result in significantly lower λd or higher λf values.

Figure 8.

(left) Drainable and (right) fillable porosity of the field site Ellzey series soil under a potato field at different ET demands.

[32] The shift in the values of λd and λf under shallow water table conditions was more pronounced for recharge (Figure 9). Under recharge, for the same WT depth, λf becomes smaller than HS-λ as Re increases while the opposite is true for λd. Since Re tends to saturate the soil profile, extra water needs to be drained in order to achieve the same drop in WT that would have occurred under hydrostatic conditions, resulting in larger λd. On the other hand, λf decreases sharply since recharge reduces the storage deficit [Chapman and Dressler, 1984; Chapman, 1995] consequently reducing the amount of water required to achieve a unit rise in the water table. It should be noted that using different ET or Re functions may result in slightly different relations between λd and λf WT for the same soil and vegetation types.

Figure 9.

(left) Drainable and (right) fillable porosity of the field site Ellzey series soil under a potato field for different rainfall (P) rates.

4.2. Water Table Dynamics

[33] The observed and simulated hourly WT dynamics in Well 1 are shown in Figure 10for 50 day periods in spring 2010 and 2011. Similar patterns were also observed in the other two wells (data not shown). Model results are compared for the single-parameter model (SPM) withHS-λ (equation (31)) and the dual-parameter models (DPM) with separate steady stateλd and λf expressions developed in this study (equations (32) and (33)). Note that the ET and Re values required for the estimation of λd and λf from equations (28) and (29) were determined a priori, which enabled analytical calculation of these storage parameters as functions of the predicted WT elevations during each simulation. In both years, the WT data clearly show the timing of irrigation and drainage in the field. During irrigation periods, the WT was consistently maintained around 40–60 cm depth from the surface by raising water levels in the lateral ditches and irrigation furrows accordingly. Diurnal fluctuations in the WT are also clearly discernible during the days with irrigation. This diurnal fluctuation, despite the continuous subsurface lateral inflow of water, indicates that the WT supports most of the ET demand of potato crop under this water management system. The observed WT dynamics were matched better by the DPM simulations than the SPM. Note that during the early part of the 2011 simulations, the observed diurnal WT fluctuations were larger than predicted for both SPM and DPM. This indicates that ET for the potato crop during this period was underestimated, likely because the crop coefficient for the midgrowth stage of potato in this area may be higher than the general Food and Agricultural Organization value [Allen et al., 1998] value. The average DPM Ceff values in both 2010 and 2011 (0.56, 0.55) were higher compared to those for SPM (0.34, −0.18), indicating improved performance for the DPM, with an especially marked difference in 2011 (Table 2). The average RMSE (cm) values in both years were correspondingly smaller for the DPM (0.34, 0.85) than the SPM (3.14, 0.86). Note that the Ceff values for Well 3 were significantly smaller than the values for Well 1 and Well 2. This was likely due to the proximity of this well to the ditch (12 m) where the assumption of purely horizontal flow may not have been supported due to flow convergence.

Figure 10.

Simulated and observed hourly water table dynamics in Well 1 located at the center of the study field in the (top) 2010 and (bottom) 2011 spring potato seasons.

Table 2. Nash-Sutcliffe Coefficient of Efficiency (Ceff) and Root-Mean-Square Error (RMSE) of the Single-Parameter Model Implemented Only WithHS-λand the Dual-Parameter Model (Separateλd and λf) Expressions
 Single-Parameter Model (HS-λ)Dual-Parameter Model (λd and λf)
CeffRMSE (cm)CeffRMSE (cm)
Well 10.283.300.800.39
Well 20.373.140.750.22
Well 3−0.022.990.120.41
Well 10.210.590.752.06
Well 2−0.101.510.630.03
Well 3−0.652.500.280.46

[34] The SPM-simulated WT fluctuated at higher elevations than the observed WT with narrower amplitudes both in 2010 and 2011. The observed diurnal amplitudes of WT depth were well matched by the DPM. This discrepancy between DPM and SPM was more pronounced toward the end of the simulation period both in 2010 and 2011. This period coincided with later stages of the potato crop when the water level in the ditches and furrows were brought to the shallowest elevation (approximately 30–35 cm below ground surface). Rising WT during rainfall events was also predicted by DPM better than SPM, which underestimated WT rise during smaller events. These results highlight the contribution of the recharge-dependentλf parameter in improving the simulations.

5. Discussion

[35] When flux-dependentλd and λf are considered in groundwater studies, it is also important to understand the relationship between these parameters and the magnitude of flux. Of particular interest is the relationship between Re and λf because it determines the rate of WT rise during rainfall events. The relationship between λf (equation (29)) and Re (scaled to Ks) under Ellzey fine sand for four different WT depths is shown in Figure 11. These results suggest that when the WT is deep, even a small Re corresponds to a λf value very close to zero. That is, by the time the wetting front has reached a deep WT, most of the soil profile is already saturated. Whereas under the same Re in a shallow WT, some portion of the soil pores may still be empty (i.e., λf > 0). However, for larger recharge rates λf approaches zero irrespective of the WT depth. Chapman and Dressler [1984] showed a similar relationship between Re and an effective storage coefficient.

Figure 11.

Recharge rate (Re/Ks) and fillable porosity for Ellzey fine sand. The relationship is presented in terms log(λf) against (Re/Ks) to enhance the visual pattern in the curves.

[36] An important advantage of a method to estimate λf in the presence of vertical fluxes is that it enables simple estimation of WT rise during rainfall events. In shallow phreatic aquifers, the WT normally shows a quick response to the rainfall events. Using λf and neglecting the contribution of groundwater flow and ET in equation (32), the water table position at time t during rainfall be estimated simply as

display math
display math

where Δt is the time step increment, and P(tt, t) is the rainfall during the period of each time step. Note that Re is estimated from P using equations (24) and (25), which is then used to estimate λf using equation (29). The WT position is then estimated by simple Euler integration of equation (35) as shown in equation (36). Application of equation (35) to the field site using λf is compared in Figure 12 to the estimates based on HS-λ. The WT rise is significantly underestimated by HS-λ as compared to our λfexpression, especially when the rainfall rates are smaller. During high-intensity rainfall events, however, the error of estimation is relatively large for both expressions. Comparison ofequation (34)with HYDRUS-1D for an initial WT depth of 120 cm (Figure 13) shows good agreement for all rainfall rates (0.05–2.0 cm h−1), although equation (34) estimates faster WT rise during the initial phase. This is an anticipated pattern because in deeper water tables Re will not occur instantaneously, since the unsaturated zone captures most of the infiltrated moisture [Hilberts et al., 2007]. Application of λf (equation (29)) in deeper initial WT conditions therefore may require additional modification of the recharge function to account for the lag between rainfall and recharge.

Figure 12.

Observed water table rise at the study site during four rainfall events (28 March, 25 April, 4 May, and 17 May) in 2010 compared to predictions from equations (35) and (36), neglecting groundwater outflow to the ditches.

Figure 13.

Water table rise predictions during five hypothetical rainfall events in Ellzey soil with initial WT = 120 cm. Lines are from equations (35) and (36)using recharge-dependentλf; solid circles are from the numerical simulation of the one-dimensional Richards equation using HYDRUS-1D.

6. Conclusions

[37] Drainable and fillable porosity parameters are critical in modeling groundwater flow in unconfined aquifers. They not only determine the water table dynamics but also influence the subsurface discharge at the aquifer outlet. In this paper we have presented a method to incorporate effects of unsaturated zone moisture fluxes while estimating these two parameters. By assuming successive steady state ET and Refluxes, closed-form expressions forλd and λf were developed. The main advantage of these expressions is that the dynamic behavior of soil water is accounted for in the estimation of storage parameters. These expressions therefore relax the limitations of previously available methods that relied on a hydrostatic assumption to incorporate the effect of soil moisture retention. The strongest merit of these expressions, however, is the ability to estimate and λd and λf separately for WT drawdown and rise scenarios. This study shows that depending on the magnitude of vertical flux and soil hydraulic properties, the difference between λf and λdcan be substantial. With the help of a hydraulic groundwater model implemented with flux-dependentλd and λf, it is shown that the simulation of WT dynamics can be strongly improved.

[38] The values of drainable and fillable porosity are different under hydrodynamic conditions, but they collapse into a single equation under a static moisture profile. The analytical expressions revealed that under the influence of ET, λd and λf of sandy soils hardly differ from HS-λ whereas they can be significantly different in clay and loam. In general, the results suggest that at a given WT depth, the finer the soil, the more pronounced the difference between HS-λ and λd or λf will be for the same ET rate. Recharge also had strong effects on the magnitudes of both λd and λf. It was found that the same recharge rates correspond to a much smaller λf values (very close to zero) under deeper WT than under shallower WT. This study highlights the importance of using both λd and λf, instead of only λd, for modeling water table dynamics. The inequality between λd and λf that develops during recharge or ET in shallow WT environments is significant and should not be neglected in most soil types. Fillable porosity is normally regarded to be smaller than λd due to hysteresis in soil moisture retention [Bouwer, 1978; Sophocleous, 1991]. We found differences between λd and λf even when hysteresis is not considered. Extending these analyses to incorporate hysteretic effects precludes analytical solutions and requires numerical methods [e.g., Stauffer et al., 1992].

[39] Note that we examined shallow water table dynamics, but the expressions developed in this study may be effectively used in modeling shallow subsurface flow in hillslopes [e.g., Hilberts et al., 2005, 2007], land drainage [e.g., Skaggs, 1980], and estimating ET from diurnal water table fluctuations [Loheide et al., 2005; Nachabe et al., 2005; Gribovszki et al., 2007].