3.4. Recharge and Evapotranspiration¶
The recharge and evapotranspiration boundary conditions simulate flux across the ground surface to or from the groundwater system. Users may directly input a recharge rate that varies spatially and/or temporally, or input precipitation rates into MINEDW to calculate a recharge rate. Recharge and evapotranspiration are both simulated using boundary fluxes—distributed source-sink fluxes that occur on the boundary surface of the flow domain. Recharge and evapotranspiration boundary fluxes both apply to the upper boundary surface of the model domain. In the 3D finite-element grid, the boundary surface consists of a collection of individual triangular surfaces assembled contiguously in 3D space. The finite-element grid forms from an assembly of contiguous triangular prisms, and selected faces of these prisms form the collection of triangular surfaces comprising the flow domain boundary.
3.4.1. Recharge¶
The recharge boundary condition simulates precipitation infiltrating into the groundwater system. Users can:
Enter recharge information directly as a recharge rate,
Vary the recharge rate spatially by specifying zones, in which different recharge characteristics are specified for different areas of the model,
Specify recharge rates that are constant in time or vary annually, and
Calculate recharge rate from precipitation information.
3.4.1.1. Calculating Recharge Rate from Precipitation Rate¶
To calculate the recharge rate from precipitation information, users may directly input precipitation rates that vary temporally or spatially. Alternatively, users may fit an empirical curve of the form shown below to observed precipitation at a range of ground-surface elevations. MINEDW then calculates the precipitation rate at each surface node in the model based on its elevation.
Where:
\(P\) = precipitation rate [LT⁻¹],
\(A\) [LT⁻¹], \(B [L^{-C}]\) and \(C\) [-] are curve-fitting parameters, and
\(z\) = ground-surface elevation [L].
If precipitation rates are entered or calculated by MINEDW using the equation, the water from precipitation entering the groundwater system as recharge can be defined using elevation factors and temporal factors. Elevation factors represent the percentage of precipitation applied to the groundwater system in a specified range of ground-surface elevations. Users can specify elevation factors for multiple ground-surface elevation ranges. This method is adapted from Maxey and Eakin (1949) [2]. Temporal factors can also be applied to precipitation rate so the percentage of precipitation applied as recharge varies with time, using either annual temporal factors or temporal factors that vary as prescribed by the user. Elevation factors and temporal factors define the recharge rate as shown below.
Where:
R = recharge rate [LT⁻¹],
TF = temporal factor [%], and
EF= elevation factor [%].
The empirical relationship shown in the equation and the temporal factors formulation shown in the equation can both be varied by zone in order to represent spatial variation in recharge.
3.4.1.2. Applying Recharge to Model¶
Once determined, the recharge rate applies to the model using the following equation.
Where:
\(q_r\) = recharge discharge [L³T⁻¹],
n = number of elements [L⁰],
\(A_e\) = surface area associated with element e [L²], and
\(R_e\) = recharge rate per unit area at element e [LT⁻¹].
Elements with user-input recharge rates have non-zero recharge rates (\(R_e\)); all others equal zero.
3.4.1.3. Surface Recharge Application¶
Recharge applies either at the surface (default) or to the uppermost saturated model node. Surface-applied recharge infiltrates downward through the unsaturated zone between the ground surface and phreatic surface following Darcian flow with an assumed unit vertical hydraulic gradient. Therefore, the maximum infiltration rate (\(R_e\)) equals the hydraulic conductivity of the hydrogeologic unit through which it flows (the relative hydraulic conductivity coefficient (\(k_R\)) does not apply to vertical hydraulic conductivity).
If the recharge rate (user-defined or calculated by MINEDW) exceeds the hydraulic conductivity of the uppermost hydrogeologic unit, recharge above the maximum infiltration rate does not enter the groundwater system and counts as surface runoff in the MINEDW water budget. This represents a case where user-specified recharge becomes surface runoff.
If a high hydraulic conductivity unit on the surface overlies a low hydraulic conductivity unit, recharge may not infiltrate to the phreatic surface at the applied rate. In this case, the computed hydraulic head at the surface may exceed the ground-surface elevation, causing water to discharge from the groundwater system at drain boundary conditions applied to surface nodes. This discharge also counts as surface runoff in the water balance.
3.4.1.4. Phreatic Surface Recharge Application¶
If recharge is applied to the model at the top of the phreatic surface, the maximum recharge rate is equal to the hydraulic conductivity of the hydrogeologic unit which contains the phreatic surface. If the user-defined recharge rate is greater than the hydraulic conductivity of the hydrogeologic unit, in which the phreatic surface is located, a recharge rate equal to the hydraulic conductivity of that unit is applied. In this case, the difference between the user-defined recharge rate and the recharge rate applied to the groundwater system is accounted for as surface runoff in the water budget. Note that if the phreatic surface rises or falls to a different unit, the maximum recharge rate will change with the hydraulic conductivity of the unit which contains the phreatic surface.
3.4.2. Evapotranspiration¶
If the phreatic surface is near the ground surface elevation, evapotranspiration can be used to simulate water discharged by vegetation (transpiration) and evaporation. The simulation is performed by assuming that groundwater discharge related to evapotranspiration is linearly related to the depth from the land surface to the phreatic surface. The linear relationship holds until a maximum depth is reached, which is the extinction depth. At the extinction depth, evapotranspiration ceases. The evapotranspiration rate depends on:
Local depth to the phreatic surface,
Extinction depth,
Potential (maximum possible) evapotranspiration rate, and
Size of the groundwater discharge area.
The discharge of groundwater from a shallow phreatic surface has the general form shown below:
Where:
\(q_{ET}\) = evapotranspiration discharge [L³T⁻¹],
\(C_{ETe}\) = coefficient representing the evapotranspiration at element e [L²T⁻¹],
\(H_{ETe}\) = extinction elevation at element e [L],
\(H_e\) = computed hydraulic head at element e for the current time step [L],
\(H_{Le}\) = ground-surface elevation at element e [L],
\(A_e\) = discharge area of element e [L²],
\(ET_{max}\) = potential evapotranspiration rate [LT⁻¹], and
n = number of elements [L°].
The discharge (\(q_{ET}\)) has a non-zero value for nodes where evapotranspiration occurs, otherwise it has a zero value. The coefficient \(C_{ETe}\) in the equation depends on a number of factors as indicated in the equation and the equation.
Where:
\(d_o\) = extinction depth, specified by the user [L].
A number of different cases can occur based on the computed hydraulic head for an element e (\(H_e\)) as shown in the equation. First, if \(H_e\) is below the extinction depth, then the contribution to \(q_{ET}\) from element e equals zero, and no groundwater discharge related to evapotranspiration occurs. Second, if \(H_e\) is above the extinction elevation (\(H_{ETe}\)) and below the ground-surface elevation, the contribution to \(q_{ET}\) depends on the difference between \(H_e\) and \(H_{ETe}\), and the value of \(C_{ETe}\). Third, if \(H_e\) is at or above the land surface, the contribution to \(q_{ET}\) is proportional to \(A_e\) and \(ET_{max}\).
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