(1)
The saturated zone (SZ) component in SHE calculates the saturated subsurface flow in the catchment. The spatial and temporal variations of the hydraulic head is described mathematically by the non-linear Boussinesq equation and solved numerically by an iterative implicit finite difference technique.
The SZ component interacts with the other components of SHE mainly by using the boundary flow from other components as sources/sinks terms in the Governing Equation; that is:
(1)
Where h is the piezometric head; Q is the volumetric flux per unit volume representing source/sink terms; S is the specific storage coefficient defined as the volume of water released from storage per unit change in head per unit volume of porous media.
The river-aquifer exchange is calculated for all grid squares adjacent to river link. The exchange flow between the saturated zone component and the river component is described as a conductance multiplied by a head difference; that is:
(2)
where the head difference is calculated as:
(3)
when groundwater level hSZ is higher than river flow level hriver. If the groundwater level drops below the river bed level the head difference is calculated as:
(4)
and the flow is from river to aquifer.
The conductance between saturated zone and river takes the “soil-flow resistance in the saturated zone” and “the flow resistance in the river lining” into account, and is given as:
(5)
where
CSZ-river: conductance between aquifer and river (m2/s )
CSZ: hydraulic conductivity in the saturated zone (m/s)
Criver: leakage coefficient of river lining (1/s)
da: saturated zone thickness (m)
dx: grid size (m)
ds: average flow length (i.e., discharge from center of grid to half-width of river bank ) (m)
w: assumed wetted perimeter in river (for a river grid) (m)
The flow exchange to the river system is included in the source/sink term of the governing equation (i.e., groundwater flow equation (1)) and can be regarded as a boundary condition of the third type (i.e., Fourier’s condition, where the head dependent flux is prescribed on the boundary) for all grid squares with “contact” to te river system.. In summary, the flow exchange rate is a function of the (1) groundwater level, (2) aquifer properties in the concerned grid squares, (3) river water level, (4) river width, (5) the elevation of river bed, and (6) hydraulic properties at the river bed.
The overland flow and channel flow modules in SHE are rather similar. Both of them are based on a diffusive wave approximation of the Saint Venant equations. The only difference is that the overland flow equation is two-dimensional, while the channel flow has only one dimension.
For the 2-D overland flow, the conservation of mass states:
(6)
where h(x,y) is the flow depth above the ground surface; u(x,y) and v(x,y) are the flow velocities in x- and y- directions, respectively; i(x,y) is the net input flux into overland flow (i.e., effective rainfall less infiltration).
By using the diffusion wave approximation at the St. Venant equation, (6) can be simplified as:
(7a)
(7b)
where Zg is the ground surface level; Sfx and Sfy are the friction slopes in the x- and y- directions, respectively. They are related to the Strickler/Manning coefficients Kx and Ky in the two directions as following:
(8a)
(8b)
From Eqs. (7) and (8) the relationship between the flow velocities and flow depth can be written as:
(9a)
(9b)
where z = Zg + h is the water surface elevation referred to datum. Notice that the left hand sides of Eqs. (9a) and (9b) represents the discharge per unit area in the x- and y- directions, respectively.
Similar to Eq. (6) but for the 1-D channel flow, the conservation of mass states:
(10)
where A(x) is the cross-section area of the flow in the channel; qL is the lateral inflow volume rate per unit of length into the channel.
The relationship between the rate of channel flow volume Au and the flow depth in the channel is:
(11)
here z is the water surface level referred to datum.
2003 Pat Yeh and Lincoln Fok