A model was created to help understand the dynamics of the water table. Data from well 1 was used as a boundary condition on one end of the system. At the distance of well 2, the modeled response was compared to the observed water table. The system was run numerous times until a diffusivity was found for which the two sets of data matched.
These tests served two purposes. They provided an in situ measurement of the diffusivity which could be compared to the value obtained by lab tests on the sand. More importantly, the model gave us an understanding of what the water table was likely doing between the wells. Specifically, the model confirmed that the water table in well 2 responds to tidal changes within a couple hours. There had been concern that these flucuations could, in fact, be due to previous tidal cycles which took tens of hours to reach well 2. This is not the case.
mechanics of model
Data from well 1 served as the right hand boundary condition on one end of the system. The influence of tides on the water table decreases away from the shore until a point where the variation is negligible. The model box was made long enough that the left hand boundary condition could be set to a constant value. Several model parameters could be varied including the node spacing, time interval spacing and the diffusivity of the sand. The quality of the diffusivity value was determined by how well the modeled water table fit the observed data at the distance of well 2.
The evolution of hydraulic head (water table height) over a given area satisfies the diffusion equation. Based on observations in the wells, we take the top few meters of sand to be relatively homogeneous. Thus our 2D system can be modeled as diffusion in one dimension.
Sensitivity of the model
In addition to the physical parameters (boundary conditions and diffusivity), there were a number of model parameters. They include the node spacing, the time interval, the length of the 1D system and the degree of implicit vs. explicit integration. Several tests were performed to ensure these parameters were chosen to accurately represent the real system.
Boundary conditions
| node n BC=Constant value |
node n-1 | node n-2 | node n-3 | node ... | node ... | node 4 | node 3 | node 2 | node 1 BC=Specfied by well 1 data |
| Right hand (well 1) boundary condition |
|---|
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Initially, the water table height in all nodes was set the constant value used at the left side of the model. Only the right hand boundary condition was different. The minimize the effects of this unrealistic initial condition, the model needed to run for several cycles before observations were made. It was quickly clear that, for reasonable values of diffusivity, the "memory" of this system was less than one cycle. So the same 24.8 hour signal was fed into the right hand boundary condition multiple times until the system had reached equilibrium.
figure The evolution of the water table surface over three interatinos of the well 1 boundry condition. Note that at a distance of 50 meters the height variations with time are neglible. This implies that our100 m box is sufficiently large. The signal is decaying on its own and is not being dampened by the constant value boundary condition at 100 m. Notice also, that the constant-valued initial conditions adjust quickly to the time variant signal.
figureThe modeled water table height (blue) for a diffusivity of 120 compared to the observed water table height (red) projected onto three cycles.
Conclusions from model study