# Groundwater flow

Single layer groundwater flow requires the four following components, and each is described in more detail below:

- aquifer
- connectivity
- constanthead
- boundaries

## Aquifer types

Groundwater flow can occur either in a confined or unconfined aquifer. Confined aquifers are overlain by a poorly permeable confining layer (e.g. clay). No air can get in to fill the pore space so that the aquifer always remains fully saturated. For a confined aquifer, water will always flow along the complete height $H$ [m] over the aquifer and transmissivity $kH$ [m$^2$ d$^{-1}$] is a constant ($k$ [m d$^{-1}$] is the horizontal hydraulic conductivity). Specific storage is the amount of water an aquifer releases per unit change in hydraulic head, per unit volume of aquifer, as the aquifer and the groundwater itself is compressed. Its value is much smaller than specific yield, between 1e-5 (stiff) and 0.01 (weak).

The upper boundary of an unconfined aquifer is the water table (the phreatic surface). Specific yield (or drainable porosity) represents the volumetric fraction the aquifer will yield when all water drains and the pore volume is filled by air instead. Specific yield will vary roughly between 0.05 (clay) and 0.45 (peat) (Johnson, 1967).

Groundwater flow is solved forward in time and central in space. The vertically averaged governing equation for an inhomogeneous and isotropic aquifer in one dimension can be written as:

\[ S \frac{\phi}{\delta t} = \frac{\delta}{\delta x} (kH \frac{\phi}{\delta x}) + Q\]

where $S$ [m m$^{-1}$] is storativity (or specific yield), $\phi$ [m] is hydraulic head, $t$ is time, $k$ [m t$^{-1}$] is horizontal hydraulic conductivity, $H$ [m] is the (saturated) aquifer height: groundwater level - aquifer bottom elevation and $Q$ [m t$^{-1}$] represents fluxes from boundary conditions (e.g. recharge or abstraction), see also Aquifer boundary conditions.

The simplest finite difference formulation is forward in time, central in space, and can be written as:

\[ S_i \frac{(\phi_{i}^{t+1} - \phi_i^{t})}{\Delta t} = -C_{i-1} (\phi_{i-1} - \phi_i) - C_i (\phi_{i+1} - \phi_i) + Q_ᵢ\]

where $_i$ is the cell index, $^t$ is time, $\Delta t$ is the step size, $C_{i-1}$ is the the intercell conductance between cell $i-1$ and $i$ and $C_i$ is the intercell conductance between cell $i$ and $i+1$. The connection data between cells is stored as part of the `Connectivity`

struct, see also Connectivity for more information.

Conductance $C$ is defined as:

\[ C = \frac{kH w}{l}\]

where $w$ [m] is the width of the cell to cell connection, and $l$ [m] is the length of the cell to cell connection. $k$ and $H$ may both vary in space; intercell conductance is therefore an average using the properties of two cells. For the calculation of the intercell conductance $C$ the harmonic mean is used (see also Goode and Appel, 1992), here between cell index $i$ and cell index $i+1$, in the $x$ direction:

\[ C_i = w \frac{(k_iH_i\cdot k_{i+1}H_{i+1})}{(k_iH_i \cdot l_{i+1} + k_{i+1}H_{i+1} \cdot l_i)}\]

where $H$ [m] is the aquifer top - aquifer bottom, and $k$, $l_i$ is the length in cell $i$ ($0.5 \Delta x_i$), $l_{i+1}$ is the length in cell $i+1$ ($0.5 \Delta x_{i+1}$) and $w$ as previously defined. For an unconfined aquifer the intercell conductance is scaled by using the "upstream saturated fraction" as the MODFLOW documentation calls it. In this approach, the saturated thickness of a cell-to-cell is approximated using the cell with the highest head. This results in a consistent overestimation of the saturated thickness, but it avoids complexities related with cell drying and rewetting, such as having to define a "wetting threshold" or a "wetting factor". See also the documentation for MODFLOW-NWT (Niswonger et al., 2011) or MODFLOW6 (Langevin et al., 2017) for more background information. For more background on drying and rewetting, see for example McDonald et al. (1991).

For the finite difference formulation, there is only one unknown, $\phi_i^{t+1}$. Reshuffling terms:

\[\phi_i^{t+1} = \phi_i^t + (C_{i-1} (\phi_i - \phi_{i-1}) + C_i (\phi_{i+1} - \phi_i) + Q_i) \frac{Δt}{S_i}\]

This can be generalized to two dimensions, for both regular and irregular cell connectivity. Finally, a stable time step size can be computed given the forward-in-time, central in space scheme, based on the following criterion from Chu and Willis (1984):

\[\frac{\Delta t k H}{(\Delta x \Delta y S)} \le \frac{1}{4}\]

where $\Delta t$ [d] is the stable time step size, $\Delta x$ [m] is the cell length in the $x$ direction and $\Delta y$ [m] is the cell length in the $y$ direction, $k$ is the horizontal hydraulic conductivity [m$^2$ d$^{-1}$] and $H$ [m] is the saturated thickness of the aquifer. For each cell $\frac{(\Delta x \Delta y S)}{k H}$ is calculated, the minimum of these values is determined, and multiplied by $\frac{1}{4}$, to get the stable time step size.

For more details about the finite difference formulation and the stable time step size criterion we refer to the paper of Chu and Willis (1984).

Boundary conditions can be classified into three categories:

- specified head (Dirichlet)
- specified flux (Neumann)
- head-dependent flux (Robin)

Neumann and Robin conditions are implemented by adding to or subtracting from a net (lumped) cell flux. Dirichlet conditions are special cased, since they cannot (easily) be implemented via the flux, but the head is set directly instead.

## Connectivity

The connectivity between cells is defined as follows.

`Wflow.Connectivity`

— Type`Connectivity{T}`

Stores connection data between cells. Connections are stored in a compressed sparse column (CSC) adjacency matrix: only non-zero values are stored. Primarily, this consist of two vectors:

- the row value vector holds the cell indices of neighbors
- the column pointers marks the start and end index of the row value vector

This matrix is square: n = m = ncell. nconnection is equal to nnz (the number of non-zero values).

- ncell: the number of (active) cells in the simulation
- nconnection: the number of connections between cells
- length1: for every connection, the length in the first cell, size nconnection
- length2: for every connection, the length in the second cell, size nconnection
- width: width for every connection, (approx.) perpendicular to length1 and length2, size nconnection
- colptr: CSC column pointer (size ncell + 1)
- rowval: CSC row value (size nconnection)

## Constant head

Dirichlet boundary conditions can be specified through the field `constanthead`

(type `ConstantHead`

) of the `GroundwaterFlow`

struct.

```
@get_units struct ConstantHead{T}
head::Vector{T} | "m"
index::Vector{Int} | "-"
end
```

For the model `SBM + Groundwater flow`

this boundary condition is optional, and if used should be specified in the TOML file as follows (see also sbm_gwf_config.toml):

```
[model]
constanthead = true
```

## Aquifer boundary conditions

### River

The flux between river and aquifer is calculated using Darcy's law following the approach in MODFLOW:

\[ Q_{riv} = \Bigg\lbrace{C_{i} \,\text{min}(h_{riv} - B_{riv}, h_{riv} - \phi), \,h_{riv} > \phi \atop C_{e} (h_{riv} - \phi) , \,h_{riv} \leq \phi}\]

where $Q_{riv}$ is the exchange flux from river to aquifer [L$^3$ T$^{-1}$], $C_i$ [L$^2$ T$^{-1}$] is the river bed infiltration conductance, $C_e$ [L$^2$ T$^{-1}$] is the river bed exfiltration conductance, $B_{riv}$ the bottom of the river bed [L], $h_{riv}$ is the river stage [L] and $\phi$ is the hydraulic head in the river cell [L].

The Table in the Groundwater flow river boundary condition section of the Model parameters provides the parameters of the struct `River`

. Parameters that can be set directly from the static input data (netCDF) are marked in this Table.

The exchange flux (river to aquifer) $Q_{riv}$ is an output variable (field `flux`

of the `River`

struct), and is used to update the total flux in a river cell. For the model `SBM + Groundwater flow`

, the water level `h`

[m] of the river kinematic wave in combination with the river `bottom`

is used to update the `stage`

field of the `River`

struct each time step.

### Drainage

The flux from drains to the aquifer is calculated as follows:

\[Q_{drain} = C_{drain} \text{min}(0, h_{drain} - \phi)\]

where $Q_{drain}$ is the exchange flux from drains to aquifer [L$^3$ T$^{-1}$], $C_{drain}$ [L$^2$ T$^{-1}$] is the drain conductance, $h_{drain}$ is the drain elevation [L] and $\phi$ is the hydraulic head in the cell with drainage [L].

The Table in the Groundwater flow drainage boundary condition section of the Model parameters provides the parameters of the struct `Drainage`

. Parameters that can be set directly from the static input data (netCDF) are marked in this Table.

The exchange flux (drains to aquifer) $Q_{drain}$ is an output variable (field `flux`

of struct `Drainage`

), and is used to update the total flux in a cell with drains. For the model `SBM + Groundwater flow`

this boundary condition is optional, and if used should be specified in the TOML file as follows (see also sbm_gwf_config.toml):

```
[model]
drains = true
```

### Recharge

The recharge flux $Q_{r}$ to the aquifer is calculated as follows:

\[Q_{r} = R \, A\]

with $R$ the recharge rate [L T$^{-1}$] and $A$ the area [L$^2$ ] of the aquifer cell.

The Table in the Groundwater flow recharge boundary condition section of the Model parameters section provides the parameters of the struct `Recharge`

. Parameters that can be set directly from the static input data (netCDF) are marked in this Table.

The recharge flux $Q_r$ is an output variable (field `flux`

of struct `Recharge`

), and is used to update the total flux in a cell where recharge occurs. For the model `SBM + Groundwater flow`

, the recharge rate from the vertical SBM concept `recharge`

[mm] is used to update the `rate`

field of the `Recharge`

struct each time step. The `rate`

field is multiplied by the `area`

field of the aquifer.

### Head boundary

This boundary is a fixed head with time (not affected by the model stresses over time)) outside of the model domain, and is generally used to avoid an unnecessary extension of the model domain to the location of the fixed boundary (for example a large lake). The flux from the boundary $Q_{hb}$ [L$^3$ T$^{-1}$] is calculated as follows:

\[Q_{hb} = C_{hb} (\phi_{hb} - \phi)\]

with $C_{hb}$ the conductance of the head boundary [L$^2$ T$^{-1}$], $\phi_{hb}$ the head [L] of the head boundary and $\phi$ the head of the aquifer cell.

The Table in the Groundwater flow head boundary condition section of the Model parameters provides the parameters of the struct `HeadBoundary`

.

The head boundary flux $Q_{hb}$ is an output variable (field `flux`

of struct `HeadBoundary`

), and is used to update the total flux in a cell where this type of boundary occurs. The parameter Head $\phi_{hb}$ can be specified as a fixed or time dependent value.

This boundary is not (yet) part of the model `SBM + Groundwater flow`

.

### Well boundary

A volumetric well rate [L$^3$ T$^{-1}$] can be specified as a boundary condition.

The Table in the well boundary condition section of the Model parameters provides the parameters of the struct `Well`

.

The volumetric well rate $Q_{well}$ can be can be specified as a fixed or time dependent value. If a cell is dry, the actual well flux `flux`

is set to zero (see also the last note on this page).

This boundary is not (yet) part of the model `SBM + Groundwater flow`

.

For an unconfined aquifer the boundary fluxes are checked, in case of a dry aquifer cell a negative flux is not allowed.

## References

- Chu, W. S., & Willis, R. (1984). An explicit finite difference model for unconfined aquifers. Groundwater, 22(6), 728-734.
- Goode, D. J., & Appel, C. A. (1992). Finite-Difference Interblock Transmissivity for Unconﬁned Aquifers and for Aquifers having Smoothly Varying Transmissivity Water-resources investigations report, 92, 4124.
- Johnson, A. I. (1967), Specific yield: compilation of specific yields for various materials, Water Supply Paper 1662-D, Washington, D.C.: U.S. Government Printing Office, p. 74, doi:10.3133/wsp1662D.
- Langevin, C.D., Hughes, J.D., Banta, E.R., Niswonger, R.G., Panday, Sorab, and Provost, A.M., 2017, Documentation for the MODFLOW 6 Groundwater Flow Model: U.S. Geological Survey Techniques and Methods, book 6, chap. A55, 197 p., https://doi.org/10.3133/tm6A55.
- McDonald, M.G., Harbaugh, A.W., Orr, B.R., and Ackerman, D.J., 1991, A method of converting no-flow cells to variable-head cells for the U.S. Geological Survey modular finite-difference groundwater flow model: U.S. Geological Survey Open-File Report 91-536, 99 p.
- Niswonger, R.G., Panday, Sorab, and Ibaraki, Motomu, 2011, MODFLOW-NWT, A Newton formulation for MODFLOW-2005: U.S. Geological Survey Techniques and Methods 6-A37, 44 p.