A method to quantify the exchange of water between surface-water channels and the ground-water aquifer based on the concept of reach transmissivity was evaluated for use in numerical models. Linking ground-water and surface-water models to each other is frequently problematic because the two models use different sets of governing equations; additionally, the time-scale of interest is often significantly longer for ground-water modeling than for surface-water modeling.

Currently, the formulation of the most common method used for leakage calculations assumes vertical flow of water through a low permeability layer at the bottom of the surface water channel (fig. 1A). The mathematical formulation of this relation is based on Darcy’s law and may be expressed as follows:

where

q    is leakage to the aquifer from the channel (volume of water per unit channel length per unit time).
k'    is hydraulic conductivity of low permeability layer of the channel bottom.
b'    is thickness of the low permeability layer of the channel bottom.
B    is top width of the channel.
Z    is surface-water elevation in the channel.
h    is piezometric head directly beneath the channel bed.

In the reach transmissivity leakage relation, the flow resistance is based on the transmissivity of the aquifer, and the reference ground-water head is measured at points located a distance L from the center of the channel (fig. 1B). The mathematical formulation of the reach transmissivity relation is

where T is the transmissivity of the aquifer and the subscripts L and R designate the left and right sides of the channel, respectively.

Differences between the vertical flow and reach transmissivity relations were examined. The input parameters required for the reach transmissivity relation are easier to obtain from published sources than those required for the vertical flow relation, which must be established through model calibration or site-specific sampling. The reach transmissivity relation also calculates leakage to each side of the channel separately, and its parameters are less dependent on model grid spacing.

The derivation of a form of the reach transmissivity relation that is suitable for use in numerical modeling relies on the following assumptions: steady state conditions, full penetration of the aquifer by the channel, and the Dupuit-Forcheimer assumption. These assumptions may preclude use of the reach transmissivity relation in certain conditions, such as when very short time steps are used in a numerical model. Furthermore, the validity of the Dupuit-Forcheimer assumption requires that ground-water heads used as input to leakage calculations be obtained from locations a significant distance from the edge of the surface-water channel. Despite these restrictions, the reach transmissivity leakage relation is applicable to a wide range of conditions.

Methods were developed to quantify the error associated with use of the reach transmissivity relation to simulate both periodic and aperiodic transient conditions; these methods can be used to evaluate the suitability of the reach transmissivity relation for a particular application. The differences in leakage between fully and partially penetrating channels were examined using a simple MODFLOW model; the results suggest that the assumption of full penetration does not usually introduce significant error. Leakage calculations based on the reach transmissivity relation were compared to measured leakage on the L-31N Canal; differences between the calculated and measured data had approximately the same magnitude as measurement errors in the gaging data. In addition, leakage calculated using the reach transmissivity relation matched the measured data better than leakage calculated using the vertical flow relation.

The equations associated with the reach transmissivity relation were developed in finite-difference form and incorporated into a modified version of MODBRANCH, a numerical flow model that couples a ground-water model (MODFLOW) and surface-water model (BRANCH). The modified program was then tested for three problems with analytical solutions and one problem that had previously been solved with the original version of MODBRANCH. The reach transmissivity relation was judged to have functioned satisfactorily in these tests. Additionally, the modified model required only about 60 percent as many iterations as the original model.

Figure 2. Location of Levee 31N study area. Click for larger image.

A model using both the vertical flow and reach transmissivity versions of MODBRANCH was developed for a region centered on Levee 31N that includes wetland areas of the Everglades and nonwetland areas of western Miami (fig. 2). The model grid consisted of 49 rows, 58 columns, and 6 layers. Row and column spacing was 500 ft near the center of the study area and 1,000 ft elsewhere. The top layer was assigned a hydraulic conductivity of 3,000,000 ft/d to simulate the wetlands environment; the hydraulic conductivities of the other layers were based on hydrogeologic properties of the surficial aquifer, which is exceptionally transmissive. Each MODFLOW stress period was 1 day, and each BRANCH stress period was 1 hour. The model was run to simulate transient conditions throughout a calendar year. Ground-water boundary conditions consisted of interpolated heads obtained from monitoring wells and canal gaging stations around the perimeter of the study area; initial conditions were obtained by interpolation, using an inverse distance method, of measured heads at the beginning of the simulation. Evapotranspiration and recharge were obtained from measured data. Potential evapotranspiration was constant throughout the model area, and the extinction depth was specified as 20 ft, based on previous research. Recharge was obtained from three rain gages within the study site and was spatially variable. A 6-mi reach of the L-31N Canal was simulated by BRANCH; boundary conditions consisted of specified head and discharge at the upstream and downstream ends of the channel, respectively.

The computer model of the Levee 31N area was first run using the existing version of MODBRANCH (with the vertical flow relation) and calibrated by varying aquifer hydraulic conductivity and the vertical flow leakage coefficient. Calibration was based on data from the 1996 calendar year. The model was found to be more sensitive to changes in the aquifer hydraulic conductivity than in the leakage coefficient. The overall transmissivity of the surficial aquifer was calibrated to 1.4 x 10⁶ ft/d, and the vertical flow leakage coefficient was established as 0.0009 s^-1 ; both results were similar to those of previous studies.

The version of MODBRANCH modified to use the reach transmissivity relation was then calibrated with the aquifer hydraulic conductivities previously obtained using the vertical flow relation. When aquifer transmissivity is fixed, the reach transmissivity leakage coefficients are only a function of the distance from the channel at which the reference ground-water head is obtained. The best results were obtained when this distance was such that the ground-water head was obtained from the model cells immediately adjacent to the channel, but results were similar for varying distances. Comparisons of modeled and measured ground-water head are presented in figure 3. There were no large differences between results modeled using the vertical flow and reach transmissivity leakage relations. The mean annual modeled ground-water heads differed by only 0.02 ft, and the mean yearly modeled canal discharges varied less than 1.0 ft³/s. The vertical flow and reach transmissivity models’ output provided coefficients of determination (R²) values within 0.03 of one another for mean ground-water head, canal stage, and canal discharge within the study area. The reach transmissivity version of MODBRANCH, however, reached a solution in about 40 percent fewer iterations.

An estimation of seepage beneath Levee 31N was obtained by summing the MODFLOW cell-by-cell flow terms for all layers of model cells directly west of the levee. These values were converted to a seepage rate per foot of distance along the levee, yielding mean values of 198.9 ft³/s and 179.1 ft³/s per foot of levee for 1996 and 1997, respectively.

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(This abstract was taken from the Greater Everglades Ecosystem Restoration (GEER) Open File Report (PDF, 8.7 MB))

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