If the shallow aquifer is specified as the source of irrigation water or water removed for use outside the watershed, the model will allow an amount of water up to the total volume of the shallow aquifer to be removed on any given day. Detailed information on water management may be found in Chapter 6:2.

The water balance for the shallow aquifer is:

$aq_{sh,i}=aq_{sh,i-1}+w_{rchrg,sh}-Q_{gw}-w_{revap}-w_{seep}-w_{pump,sh}$ 2:4.2.1

where $aq_{sh,i}$ is the amount of water stored in the shallow aquifer on day $i$ (mm H$_2$O), $aq_{sh,i-1}$ is the amount of water stored in the shallow aquifer on day $i-1$ (mm H$_2$O), $w_{rchrg,sh}$ is the amount of recharge entering the shallow aquifer on day $i$ (mm H$_2$O), $Q_{gw}$ is the groundwater flow, or base flow, into the main channel on day $i$ (mm H$_2$O), $w_{revap}$ is the amount of water moving into the soil zone in response to water deficiencies on day $i$ (mm H$_2$O), $w_{seep}$ is seepage from the shallow aquifer to the deep aquifer on day $i$ (mm H$_2$O) and $w_{pump,sh}$ is the amount of water removed from the shallow aquifer by pumping on day $i$ (mm H$_2$O).

Water that moves past the lowest depth of the soil profile by percolation or bypass flow enters and flows through the vadose zone before becoming shallow and/or deep aquifer recharge. The lag between the time that water exits the soil profile and enters the shallow aquifer will depend on the depth to the water table and the hydraulic properties of the geologic formations in the vadose and groundwater zones.

An exponential decay weighting function proposed by Venetis (1969) and used by Sangrey et al. (1984) in a precipitation/groundwater response model is utilized in SWAT+ to account for the time delay in aquifer recharge once the water exits the soil profile. The delay function accommodates situations where the recharge from the soil zone to the aquifer is not instantaneous, i.e. 1 day or less.

The recharge to both aquifers on a given day is calculated:

$w_{rchrg,i}=(1-exp\lfloor-1/\delta_{gw}\rfloor)*w_{seep}+exp\lfloor-1/\delta_{gw}\rfloor*w_{rchrg,i-1}$

2:4.2.2

where $w_{rchrg,i}$ is the amount of recharge entering the aquifers on day $i$ (mm H$_2$O),$\delta_{gw}$ is the delay time or drainage time of the overlying geologic formations (days), $w_{seep}$ is the total amount of water exiting the bottom of the soil profile on day $i$ (mm H$_2$O), and $w_{rchrg,i-1}$ is the amount of recharge entering the aquifers on day $i-1$ (mm H$_2$O). The total amount of water exiting the bottom of the soil profile on day $i$ is calculated:

$w_{seep}=w_{perc,ly=n}+w_{crk,btm}$ 2:4.2.3

where $w_{seep}$ is the total amount of water exiting the bottom of the soil profile on day $i$ (mm H$_2$O), $w_{perc,ly=n}$ is the amount of water percolating out of the lowest layer, $n$, in the soil profile on day $i$ (mm H$_2$O), and $w_{crk,btm}$ is the amount of water flow past the lower boundary of the soil profile due to bypass flow on day $i$ (mm H$_2$O).

The delay time, $\delta_{gw}$, cannot be directly measured. It can be estimated by simulating aquifer recharge using different values for $\delta_{gw}$ and comparing the simulated variations in water table level with observed values. Johnson (1977) developed a simple program to iteratively test and statistically evaluate different delay times for a watershed. Sangrey et al. (1984) noted that monitoring wells in the same area had similar values for $\delta_{gw}$, so once a delay time value for a geomorphic area is defined, similar delay times can be used in adjoining watersheds within the same geomorphic province.

Water may move from the shallow aquifer into the overlying unsaturated zone. In periods when the material overlying the aquifer is dry, water in the capillary fringe that separates the saturated and unsaturated zones will evaporate and diffuse upward. As water is removed from the capillary fringe by evaporation, it is replaced by water from the underlying aquifer. Water may also be removed from the aquifer by deep-rooted plants which are able to uptake water directly from the aquifer.

SWAT+ models the movement of water into overlying unsaturated layers as a function of water demand for evapotranspiration. To avoid confusion with soil evaporation and transpiration, this process has been termed ‘revap’. This process is significant in watersheds where the saturated zone is not very far below the surface or where deep-rooted plants are growing. Because the type of plant cover will affect the importance of revap in the water balance, the parameters governing revap are usually varied by land use. Revap is allowed to occur only if the amount of water stored in the shallow aquifer exceeds a threshold value specified by the user, $aq_{shthr,rvp}$.

The maximum amount of water than will be removed from the aquifer via ‘revap’ on a given day is:

$w_{revap,mx}=\beta_{rev} *E_o$ 2:4.2.15

where $w_{revap,mx}$ is the maximum amount of water moving into the soil zone in response to water deficiencies (mm H$_2$O), $\beta_{rev}$ is the revap coefficient, and $E_o$ is the potential evapotranspiration for the day (mm H$_2$O). The actual amount of revap that will occur on a given day is calculated:

$w_{revap}=0$ if $aq_{sh} \le aq_{shthr,rvp}$ 2:4.2.16

$w_{revap}=w_{revap,mx} - aq_{shthr,rvp}$ if $aq_{shthr,rvp}<aq_{sh}<(aq_{shthr,rvp}+w_{revap,mx})$ 2:4.2.17

$w_{revap}=w_{revap,mx}$ if $aq_{sh} \ge(aq_{shthr,rvp}+w_{revap,mx})$ 2:4.2.18

where $w_{revap}$ is the actual amount of water moving into the soil zone in response to water deficiencies (mm H$_2$O), $w_{revap,mx}$ is the maximum amount of water moving into the soil zone in response to water deficiencies (mm H$_2$O), $aq_{sh}$ is the amount of water stored in the shallow aquifer at the beginning of day $i$ (mm H$_2$O) and $aq_{shthr,rvp}$ is the threshold water level in the shallow aquifer for revap to occur (mm H$_2$O).

A fraction of the total daily recharge can be routed to the deep aquifer. The amount of water than will be diverted from the shallow aquifer due to percolation to the deep aquifer on a given day is:

$w_{deep}=\beta_{deep}*w_{rchrg}$ 2:4.2.4

where $w_{deep}$ is the amount of water moving into the deep aquifer on day $i$ (mm H$_2$O), $\beta_{deep}$ is the aquifer percolation coefficient, and $w_{rchrg}$ is the amount of recharge entering both aquifers on day $i$ (mm H$_2$O). The amount of recharge to the shallow aquifer is:

$w_{rchrg,sh}=.w_{rchrg}-w_{deep}$ 2:4.2.5

where $w_{rchrg,sh}$ is the amount of recharge entering the shallow aquifer on day $i$ (mm H$_2$O).

The shallow aquifer contributes base flow to the main channel or reach within the subbasin. Base flow is allowed to enter the reach only if the amount of water stored in the shallow aquifer exceeds a threshold value specified by the user, $aq_{shthr,q}$.

The steady-state response of groundwater flow to recharge is (Hooghoudt, 1940):

$Q_{gw}=\frac{8000*K_{sat}}{L_{gw}^2}*h_{wtbl}$ 2:4.2.6

where $Q_{gw}$ is the groundwater flow, or base flow, into the main channel on day $i$ (mm H$_2$O), $K_{sat}$ is the hydraulic conductivity of the aquifer (mm/day), $L_{gw}$ is the distance from the ridge or subbasin divide for the groundwater system to the main channel (m), and $h_{wtbl}$ is the water table height (m).

Water table fluctuations due to non-steady-state response of groundwater flow to periodic recharge is calculated (Smedema and Rycroft, 1983):

$\frac{dh_{wtbl}}{dt}=\frac{w_{rchrg,sh}-Q_{gw}}{800*\mu}$ 2:4.2.7

where $\frac{dh_{wtbl}}{dt}$ is the change in water table height with time (mm/day), $w_{rchrg,sh}$ is the amount of recharge entering the shallow aquifer on day $i$ (mm H$_2$O), $Q_{gw}$ is the groundwater flow into the main channel on day $i$ (mm H$_2$O), and $\mu$ is the specific yield of the shallow aquifer (m/m).

Assuming that variation in groundwater flow is linearly related to the rate of change in water table height, equations 2:4.2.7 and 2:4.2.6 can be combined to obtain:

$\frac{dQ_{gw}}{dt}=10*\frac{K_{sat}}{\mu *L^2_{gw}}*(w_{rchrg,sh}-Q_{gw})=\alpha_{gw}*(w_{rchrg,sh}-Q_{gw})$

2:4.2.8

where $Q_{gw}$ is the groundwater flow into the main channel on day $i$ (mm H$_2$O), $K_{sat}$ is the hydraulic conductivity of the aquifer (mm/day),$\mu$ is the specific yield of the shallow aquifer (m/m), $L_{gw}$ is the distance from the ridge or subbasin divide for the groundwater system to the main channel (m), $w_{rchrg,sh}$ is the amount of recharge entering the shallow aquifer on day $i$ (mm H$_2$O) and $\alpha_{gw}$ is the baseflow recession constant or constant of proportionality. Integration of equation 2:4.2.8 and rearranging to solve for $Q_{gw}$ yields:

$Q_{gw,i}=Q_{gw,i-1}*exp\lfloor-\alpha_{gw}*\Delta t\rfloor+w_{rchrg,sh}*(1-exp\lfloor-\alpha_{gw}*\Delta t\rfloor)$

if $aq_{sh}>aq_{shthr,q}$ 2:4.2.9

$Q_{gw,i}=0$ if $aq_{sh} \le aq_{shthr,q}$ 2:4.2.10

where $Q_{gw,i}$ is the groundwater flow into the main channel on day $i$ (mm H$_2$O), $Q_{gw,i-1}$ is the groundwater flow into the main channel on day $i-1$ (mm H$_2$O), $\alpha_{gw}$ is the baseflow recession constant, $\Delta t$ is the time step (1 day), $w_{rchrg,sh}$ is the amount of recharge entering the shallow aquifer on day $i$ (mm H$_2$O), $aq_{sh}$ is the amount of water stored in the shallow aquifer at the beginning of day $i$ (mm H$_2$O) and $aq_{shthr,q}$ is the threshold water level in the shallow aquifer for groundwater contribution to the main channel to occur (mm H$_2$O).

The baseflow recession constant, $\alpha_{gw}$, is a direct index of groundwater flow response to changes in recharge (Smedema and Rycroft, 1983). Values vary from 0.1-0.3 for land with slow response to recharge to 0.9-1.0 for land with a rapid response. Although the baseflow recession constant may be calculated, the best estimates are obtained by analyzing measured streamflow during periods of no recharge in the watershed.

When the shallow aquifer receives no recharge, equation 2:4.2.9 simplifies to:

$Q_{gw}=Q_{gw,0}*exp\lfloor-\alpha_{gw}*t\rfloor$ if $aq_{sh}>aq_{shthr,q}$ 2:4.2.11

$Q_{gw,i}=0$ if $aq_{sh} \le aq_{shthr,q}$ 2:4.2.12

where $Q_{gw}$ is the groundwater flow into the main channel at time $t$ (mm H$_2$O), $Q_{gw,0}$ is the groundwater flow into the main channel at the beginning of the recession (time $t$=0) (mm H$_2$O), $\alpha_{gw}$ is the baseflow recession constant, and t is the time lapsed since the beginning of the recession (days), $aq_{sh}$ is the amount of water stored in the shallow aquifer at the beginning of day $i$ (mm H$_2$O) and $aq_{shthr,q}$ is the threshold water level in the shallow aquifer for groundwater contribution to the main channel to occur (mm H$_2$O). The baseflow recession constant is measured by rearranging equation 2:4.2.11.

$\alpha_{gw}=\frac{1}{N}*ln[\frac{Q_{gw,N}}{Q_{gw,0}}]$ 2:4.2.13

where $\alpha_{gw}$ is the baseflow recession constant, $N$ is the time lapsed since the start of the recession (days), $Q_{gw,N}$ is the groundwater flow on day $N$ (mm H$_2$O), $Q_{gw,0}$ is the groundwater flow at the start of the recession (mm H$_2$O).

It is common to find the baseflow days reported for a stream gage or watershed. This is the number of days for base flow recession to decline through one log cycle. When baseflow days are used, equation 2:4.2.13 can be further simplified:

$\alpha_{gw}=\frac{1}{N}*ln[\frac{Q_{gw,N}}{Q_{gw,0}}]=\frac{1}{BFD}*ln[10]=\frac{2.3}{BFD}$ 2:4.2.14

where $\alpha_{gw}$ is the baseflow recession constant, and $BFD$ is the number of baseflow days for the watershed.

Although SWAT+ does not currently print groundwater height in the output files, the water table height is updated daily by the model. Groundwater height is related to groundwater flow by equation 2:4.2.6.

$Q_{gw}=\frac{8000*K_{sat}}{L^2_{gw}}*h_{wtbl}=\frac{8000*\mu}{10}*\frac{10*K_{sat}}{\mu * L^2_{gw}}*h_{wtbl}=800*\mu*\alpha_{gw}*h_{wtbl}$

2:4.2.19

where $Q_{gw}$ is the groundwater flow into the main channel on day $i$ (mm H$_2$O), $K_{sat}$ is the hydraulic conductivity of the aquifer (mm/day), $L_{gw}$ is the distance from the ridge or subbasin divide for the groundwater system to the main channel (m), $h_{wtbl}$ is the water table height (m), $\mu$ is the specific yield of the shallow aquifer (m/m), and $\alpha_{gw}$ is the baseflow recession constant. Substituting this definition for $Q_{gw}$ into equation 2:4.2.9 gives

$h_{wtbl,i}=h_{wtbl,i-1}*exp[-\alpha_{gw}*\Delta t]+\frac{w_{rchrg}*(1-exp\lfloor-\alpha_{gw} *\Delta t\rfloor)}{800*\mu*\alpha_{gw}}$ 2:4.2.20

where $h_{wtbl,i}$ is the water table height on day $i$ (m), $h_{wtbl,i-1}$ is the water table height on day $i-1$ (m), $\alpha_{gw}$ is the baseflow recession constant, $\Delta t$ is the time step (1 day), $w_{rchrg}$ is the amount of recharge entering the aquifer on day $i$ (mm H$_2$O), and $\mu$ is the specific yield of the shallow aquifer (m/m).

Table 2:4-1: SWAT+ input variables used in shallow aquifer calculations.