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Simulation of Ground-Water Discharge to Biscayne Bay

Governing Equations

Simulation of ground-water flow is performed by numerically solving the ground-water flow and solute-transport equations. The ground-water flow equation can take many forms depending on the assumptions that are valid for the problem of interest. In most cases, it is assumed that the density of ground water is spatially and temporally constant. To simulate ground-water flow in coastal environments, the assumption of constant density is not valid because seawater contains a higher concentration of dissolved salts than rainfall, which is the primary source for aquifer recharge. As previously discussed, fluid density is a function of dissolved salt. Kohout (1960a) and many other investigators have shown that the density difference between fresh ground water and seawater can greatly affect ground-water flow patterns. Accordingly, the ground-water flow equation used in the present study does not assume that ground-water density is constant. A general form of the equation that describes the flow of variable-density ground water is:

,

(2)      (D)

where:

	is the spatial gradient operator,
	is fluid density [M / L3],
q	is specific discharge [L / T],
	is source/sink fluid density [M / L3],
	is a source/sink rate term [T -1],
SP	is specific storage in terms of pressure [LT2 / M],
P	is pressure [M / LT2],
t	is time [ T ],
n	is porosity [dimensionless], and
C	is the concentration of the dissolved constituent that affects fluid density [M / L3].

Equation 2 is valid when the density variations are caused by solute concentration rather than temperature. For this study, temperature is assumed to be spatially uniform and temporally constant.

To solve the ground-water flow equation, the solute-transport equation also must be solved because concentration is a function of time and location. For dissolved constituents that are conservative, such as those found in seawater, the solute-transport equation is:

,

(3)      (D)

where:

D	is the dispersion coefficient [L2 / T],
	is velocity [L / T],
qS	is the volumetric flux of a source or sink [T -1 ], and
CS	is the concentration of the source or sink [M / L3],

The ground-water flow and transport equations are coupled through the velocity term , which is the specific discharge divided by porosity, and . Exact solutions to equation 2 and equation 3, therefore, require that both mathematical expressions be simultaneously solved. In this study, however, the equations are not simultaneously solved, but rather, a one timestep lag is used.

In the form presented above, equation 2 uses pressure as an independent variable. Guo and Bennet (1998) use an alternative form of the ground-water flow equation written in terms of equivalent fresh-water head. The following discussion presents the mathematical derivation of the ground-water flow equation in terms of freshwater head. This derivation is presented because the variable-density flow code, SEAWAT, uses freshwater head rather than pressure.

Darcy's law for ground water of variable density is written as:

,

(4)      (D)

where is the gravity vector. For flow in the horizontal plane (denoted by x), equation 4 is simplified as:

,

(5)      (D)

For flow in the vertical plane (denoted by z), Darcy's law is written as:

,

(6)      (D)

In equation 6, the gravity vector , which is directed downward, has been replaced by a scalar quantity, -g. For aquifers with constant fluid density, head is generally used as the independent variable because it is easily measured. Head is defined as the elevation to which ground water will rise in a cased well. Mathematically, head (h) is expressed by the following equation:

,

(7)      (D)

For variable-density systems, the flow of ground water cannot be described solely by head, and thus, equations are written in terms of pressure. Rather than use pressure as an independent variable, Lusczynski (1961) first suggested the use of equivalent freshwater head. Freshwater head is defined as the elevation to which freshwater will rise in a cased well. The following equation converts from head at a specific density to freshwater head h_f:

,

(8)      (D)

where _f is the density of freshwater. The freshwater head is also described by the following equation:

,

(9)      (D)

By solving for P in equation 9, substituting for P in equation 5 and equation 6, and simplifying, Darcy's law for horizontal and vertical flow is:

,

(10)      (D)

and

,

(11)      (D)

respectively.

The coefficients on the right-hand side of equation 10 and equation 11, , are collectively referred to as equivalent freshwater hydraulic conductivity (K_f), which is the hydraulic conductivity of a porous media that is saturated with freshwater. The storage term in equation 2 can also be rewritten in terms of freshwater head using the relation,
S_f = g _f S_P. The equivalent freshwater storage term, S_f, is defined as the volumetric release of freshwater from storage per unit area per unit decline in freshwater head.

By substituting equivalent freshwater hydraulic conductivity and the equivalent freshwater storage term, the ground-water flow equation becomes:

,

(12)      (D)

Included in this equation is the assumption that the dynamic viscosity of ground water with dissolved solids is equal to the dynamic viscosity of freshwater.

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