node spacing=5 meters
node spacing=1 meter
node spacing=0.2 meters
node spacing=5 meters |
node spacing=1 meter |
node spacing=0.2 meters |
Observed water table height in well 2: red circles
Modeled water table height in well 2:blue lines
The three tests shown at right were run for a 150 meter wide system with a diffusivity of 120. The time step used in the integration was 0.124 hours. The 150 meter region was divided into 5, 1 and .2 meter blocks. Smaller blocks allow the model to represent the water table without significant "steps" from one block into the next. Alternatively, the model will ignore all signals with a wavelength less than two times the block width. However, small spacings also require more time to run.
The required spacing is therefore a function of the topography of the water table. The purpose of this test was to ensure that the spacing used in the model was adequately small to capture the full signal.
The modeled water table height in well 2 was calculated for three node spacings. If the spacing is decreased from 5 to 1 meter, there is a noticeable shift inthe result. Decreasing the block size by another factor of 5 however, does not alter the signal significantly. Thus for a diffusivity of 120, a block spacing of 1 meter is sufficient.
 dt = 0.620 hours |
 dt = 0.124 hours |
 dt = 0.031 hours |
Observed water table height in well 2: red circles
Modeled water table height in well 2:blue lines
The time interval refers to the jump in time between successive interations of the diffusion equation. Like the block spacing above, the short time intervals imply longer run times for the model while long time intervals run the risk of inaccuracy.
All of the tests shown here used a diffusivity of 120 and a node spacing of 0.1 meters. The anticipated signal in well 2 is shown. As the time interval, dt, is decreased from .62 to 0.124 to .031 hours, the phase of the signal shifts slightly to the right. The change from 0.124 to 0.031 is minimal and we conclude that 0.031 is a sufficently small time interval for these tests.
 diffusivity=15 |
 diffusivity=30 |
 diffusivity=60 |
 diffusivity=120 |
 diffusivity=240 |
Observed water table height in well 2: red circles
Modeled water table height in well 2:blue lines
Diffusivity is the one parameter which real world implications. It is not a function of the model but rather a property of the soil. It is the scale factor between the partial derivatives in the expression:
dH/dt = Dd2H/dx2
The concept of diffusivity allows hydraulic conductivity and storativity to be condensed into one term by dividing them. Loosely speaking it is a measure of how easily water diffuses through the medium.
It's influence on the model is intuitive. There are three significant ways an increase in diffusivity impacts the system.
If our model accurately describes the system then BOTH of the amplitude and phase of the signal should be matched - a daunting task since there is only one physical parameter to vary. Examination of these plots however shows that a diffusivity of 120 fits both the amplitude and phase relatively well. We take this as confirmation that our simplistic model portrays the beach system reasonably well.
The response of the water table to changing conditions depends on two soil parameters. Hydraulic conductivity describes the..... Storativity describes
 implicit/explicit weight=0.47 |
 implicit/explicit weight=0.50 |
 implicit/explicit weight=0.75 |
 implicit/explicit weight=1.00 |
 implicit/explicit weight=5.00 |
Observed water table height in well 2: red circles
Modeled water table height in well 2:blue lines
The model could be run via using an explicit or implicit scheme to obtain water table heights given the distribution one time step earlier. A mix of these approaches offers a good stability without oversmoothing the results.
In this model, an implicit/explicit weight of 0 is fully explicit while a weight of 1 is fully implicit. Below 0.5, the model was found to be unstable with behavior shown in the top diagram. Above 0.5 the results appear similar regardless of the weight. Tests in this experiment were run with a weight of 0.5, otherwise known as a Crank-Nicolson scheme.