# Groundwater/Base Flow

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):

&#x20;         $$Q_{gw}=\frac{8000*K_{sat}}{L_{gw}^2}*h_{wtbl}$$                                                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).

&#x20;       Water table fluctuations due to non-steady-state response of groundwater flow to periodic recharge is calculated (Smedema and Rycroft, 1983):

&#x20;     $$\frac{dh_{wtbl}}{dt}=\frac{w_{rchrg,sh}-Q_{gw}}{800*\mu}$$                                                      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).

&#x20;   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:

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

&#x20;                                                                                           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:    &#x20;

$$Q_{gw,i}=Q_{gw,i-1}*exp\lfloor-\alpha_{gw}*\Delta t\rfloor+w_{rchrg,sh}*(1-exp\lfloor-\alpha_{gw}*\Delta t\rfloor)$$&#x20;

&#x20;                                                      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).&#x20;

&#x20;    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.

&#x20;   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

&#x20;    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).

&#x20;         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.
