Ground flow formula (Water Overlay)

Underground flow is different from surface flow, since it has to account for the slowdown and porousness of the medium. In general, horizontal underground flow is calculated using formulas described in Harbaugh 2005^[1]^[2]. However, when an aquifer is present, the aquifer variant is applied.

Hydraulic Conductivity without Thickness

Two adjacent cells, where underground water level of cell 1 is larger than cell 2.

The flow between the two cells is calculated as:

$\Delta w_{t}=w_{1,t}-w_{2,t}$

$B_{ex}=max(B_{1},B_{2})-d_{ground}$

$A_{c,t}=\Delta x\cdot (w_{avg,t}-B_{ex})$

$K=min(K_{1},K_{2})$

$V_{t}=\Delta w\cdot K\cdot A_{c}/\Delta x\cdot \Delta t$

where:

w_{n}

= The underground water level of cell

n

at time

t

.

B_{n}

= The surface height of cell n.

K_{n}

= The hydraulic conductivity of the cell, defined in HYDRAULIC_CONDUCTIVITY_MD of the underground terrain.

d_{ground}

= The configured ground bottom distance, defined in GROUND_BOTTOM_DISTANCE_M of the Water Overlay.

A_{c,t}

= Area of conductance at time

t

..

\Delta w

= Underground water level difference at time

t

.

\Delta t

= Computational timestep.

\Delta x

= Size of grid cell.

w_{avg,t}

= Averaged underground water level at time

t

, based on water levels in underground, WATER_STORAGE_PERCENTAGE and potentially the surface water level, when the underground is filled to the top.

V_{t}

= The amount of water to be transported at time

t

between one cell and the other

Hydraulic Conductivity with Thickness

The flow between the two cells is calculated as:

$\Delta w_{t}=w_{1,t}-w_{2,t}$

$KD=min(KD_{1},KD_{2})$

$V_{t}={\frac {\Delta w\cdot KD\cdot \Delta x}{\Delta x}}\cdot \Delta t$

where:

w_{n,t}

= The underground water level of cell

n

at time

t

.

B_{n}

= The surface height of cell n.

KD_{n}

= The hydraulic conductivity of the cell, multiplied with the thickness of the layer, defined in HYDRAULIC_CONDUCTIVITY_WITH_THICKNESS_MD of the underground terrain.

\Delta w

= Underground water level difference at time

t

.

\Delta t

= Computational timestep.

\Delta x

= Size of grid cell.

V_{t}

= The amount of water to be transported at time

t

between one cell and the other.

Aquifer formula

When an aquifer is present, its hydraulic diffusivity is used to calculate the water flow.

First the following condition is checked in order to allow water movement through the aquifer:

V_{t}=w_{g,t}>z_{a}

Based on this condition being true, the transported volume V is calculated:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V = \frac{\Delta w_{g,t} \cdot {KD}_a \codt \Delta x}{\Delta x} \cdot \Delta t}

Where:

w_{g,t}

= Ground water level in the cell at time

t

;

z_{a}

= the datum height of the aquifer at the cell.

V_{t}

= Volume in

m^{3}

that flows between the two adjacent cells due to the aquifer at time

t

.

\Delta w_{g,t}

= Ground water level difference between the two adjacent cells at time

t

;

\Delta t

= Computational timestep.

{KD}_{a}

= The AQUIFER_KD attribute of aquifer.

References

↑ Harbaugh, A.W., 2005, MODFLOW-2005, the U.S. Geological Survey modular ground-water model-the Ground-Water Flow Process: U.S. Geological Survey Techniques and Methods 6-A16, variously paginated.
↑ 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 ∙ p 31 ∙ found at: https://doi.org/10.3133/tm6A55 (last visited 2019-02-04)