Ground flow formula (Water Overlay)

Underground flow is calculated differently from surface flow, to account for the slowdown and porousness of the medium.

In general, Darcy's law is used. When an aquifer is present a variant is applied.

Darcy's law

Underground flow between cells is calculated using Darcy's law.

C_d = Δt * I_und * width * A * ( (WL_source - WL_target) / distance )

Since both width and distance are directly related to the cell size, and the result should be in water height rather than volume, the formula can be rewritten as follows:

C_d = Δt * I_und * A * (W_source - W_target) / cell

Because the underground may have a different porousness, the maximum amount of water that can flow to another cell has to take into account the relative water storage capacities of the underground.

C_wsp = (WL_source - WL_target) * (WSP_source / (WSP_source + WSP_target) )

The amount of water which flows from the source to the target cell is calculated as follows:

Δw = max( 0 , min( C_d , C_wsp ) )

Where:

• Δw = The underground flow which takes place.
• Δt = Computational timestep.
• cell = Cell size.

• C_d = The capacity for water flow possible based on the relative water heights.
• C_wsp = The capacity for water flow possible based on the relative water storage percentages.

• WL_source = The amount of water in the saturated zone of the source cell. The height of the water column if the equivalent amount of water was placed on the surface.
• WL_target = The amount of water in the saturated zone of the target cell. The height of the water column if the equivalent amount of water was placed on the surface.
• A = Contact area of the underground cells
• I_und = The GROUND_INFILTRATION_MD of the underground terrain type of the origin cell.
• WSP_source = The WATER_STORAGE_PERCENTAGE of the underground terrain type of the origin cell.
• WSP_target = The WATER_STORAGE_PERCENTAGE of the underground terrain type of the target cell.

Aquifer formula

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

First, the hydraulic diffusivity dictates the fraction of the water height difference which will flow.

F = 2 * sqrt( KD / WSP_source ) * sqrt( Δt ) * ( 1 / cell )

Based on this fraction, the actual amount of water flow is calculated.

Δw = ( (WL_source/WSP_source) - (WL_target/WSP_source) ) * F

Where:

• Δw = The underground flow which takes place.
• Δt = Computational timestep.
• cell = Cell size.

• F = Fraction of water which flows between cells

• KD = The AQUIFER_KD attribute of aquifer.
• WSP_source = The WATER_STORAGE_PERCENTAGE attribute of the underground terrain type of the origin cell.
• WSP_target = The WATER_STORAGE_PERCENTAGE attribute of the underground terrain type of the target cell.
• WL_source = The amount of water in the saturated zone of the source cell. The height of the water column if the equivalent amount of water was placed on the surface.
• WL_target = The amount of water in the saturated zone of the target cell. The height of the water column if the equivalent amount of water was placed on the surface.