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 formula is applied.

It depends on the configuration of the HYDRAULIC_CONDUCTIVITY_WITH_THICKNESS attribute value in the Water Overlay what Hydraulic Conductivity formula is used:

A value $<=0$ means Hydraulic Conductivity without Thickness formula will be used.
A value $>0$ means Hydraulic Conductivity with Thickness formula will be used.

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:

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

$B_{im}=max(B_{1}-d_{b,1},B_{2}-d_{b,2})$

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

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

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

where:

w_{n,t}

= The underground water level of cell

n

at time

t

.

B_{c}

= the datum height of the surface of cell c, set by the elevation or a Terrain elevation prequel.

B_{im}

= the calculated datum height of the impenetrable layer.

K_{n}

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

d_{b,c}

= The ground bottom distance of the cell c, defined by a Bottom distance prequel or a general 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.

{\bar {w_{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_{K,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:

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

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

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

where:

w_{n,t}

= The underground water level of cell

n

at time

t

.

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_{KD,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.

Based on conditions being true, the calculated volume of water that is transported through the aquifer is calculated as:

V_{a,t}={\begin{cases}{\dfrac {\Delta w_{n,t}\cdot {KD}_{a}\cdot \Delta x}{\Delta x}}\cdot \Delta t&{\text{if }}w_{n,t}>z_{a}{\text{ and }}KD_{a}>0\\V_{K(D),t}&{\text{otherwise}}\end{cases}}

Where:

w_{n,t}

= Ground water level in cell

n

at time

t

;

z_{a}

= the datum height of the aquifer at the cell.

V_{a,t}

= Volume in

m^{3}

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

t

.

\Delta w_{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.

V_{K(D),t}

= The calculated amount of water to be transported at time

t

between one cell and the other.

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)