Example Problem 2. Representation of Aquitards

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Example Problem 3, Head-Dependent Boundary Conditions

Introduction

A fully worked-out version of Example 2 can be downloaded from Argus. The files you can download include a fully worked-out versions of this example named "aquitard1.mmb" to "aquitard5.mmb" and an Excel spreadsheet with the model results named aquitard.xls.

This example is based on problem 11 of Andersen (1993). It includes several different models but they will all be similar to one another. They are all one-dimensional vertical models. Thus, they will only have one row and one column in each layer but they will have multiple layers. Many of the aquifer characteristics will be the same in all the models too. All the models begin with an initial head in the bottom aquifer of 1000 ft. The other model layers will have a higher initial head. We will run a series of models in which we vary the way the aquitard is modeled.

Aquifer Characteristics

The aquifer characteristics for the first model are as follows:

Areal dimensions = x= y = 10 ft.
(A convenient drawing size for this model is a horizontal extent of 30 and a vertical extent of 20.5 with the horizontal and vertical origins both at -1. You learned how to set these in the last example.)

Aquifers

Aquitard

You don't assign aquifer thickness directly. Instead, you assign the top and bottom elevations of the aquifer. I used an elevation of 200 for the top of the upper most unit.

The model duration is one year (31,536,000 seconds) with 25 time steps and a time-step multiplier of 1.3.

I suggest that you again use the PCG2 solver with the two convergence criteria (HCLOSE and RCLOSE) both set to 0.01.

A good name for this model might be aquitard1. However, you can't use that for the root on the output files tab too. The version of MODFLOW-96 distributed by the USGS does not support long file names so the root must be 8 characters or less. Again, you should print the water budget on every time step. The first model will have three layers. All three layers will be simulated. On the geology tab of the Project Info dialog, be sure that you change the aquitard from a nonsimulated to a simulated layer.

In the first model you can set the following characteristics

Layer 1 (upper aquifer)
Elev Top Unit1 200
Elev Bottom Unit1 150
Initial Head Unit1 1010
Prescribed Head Unit1 1010

 

Layer 2 (aquitard)
Elev Top Unit2 150
Elev Bottom Unit2 50
Initial Head Unit2 1010
Hydr Cond. Unit2 Kx 1x10-8 ft/s
Hydr Cond. Unit2 Kz 1x10-8 ft/s
Specific Storage Unit2 = 5xl0-6/ft.

Layer 3 (lower aquifer)
Elev Top Unit3 50
Elev Bottom Unit3 0
Initial Head Unit3 1000
Prescribed Head Unit3 1000

You can assign these values using contours. However, in many cases, it is easier to assign values with the expression editor.


Using the Expression Editor to set Default Aquifer Properties and to Link Layer Properties

You can set all these characteristics by using the "Layers Dialog box".

Call up the Layers Dialog box by selecting "Layers..." from the "View" menu. or pressing the "Layers Dialog" button on the toolbar. You can tell which that is by putting the cursor over the buttons and waiting for a "hint" to appear. (Be sure you select the "Layers Dialog" button and not the "Layers" button.)

Now select "Elevation Top Unit1" in the top window. In the bottom window, click and hold on the arrow in the "Value" Column. A pop-up menu with a single choice "Expression" will appear. Select it and the Expression editor will appear. Type 200 in the upper edit box.

In some cases, instead of entering a value in the expression editor, we will enter a more complex expression. For example, instead of entering the top of Unit2 as a number, we can link it to the bottom of Unit1. Then any changes we make to the bottom of Unit1 will automatically apply to the top of Unit2 as well. To do this call up the Layers dialog box again and select Elev Top Unit2 in the upper half of the dialog box. In the lower half of the dialog box, select the one and only parameter. In the bottom window, click and hold on the arrow in the "Value" Column. Then click on "Expression" in the popup menu to call up the Expression Editor. This time, instead of entering a number in the upper half of the dialog box, select Elevation Bottom Unit1 from the menu in the lower half of the dialog box. A list of parameters for that layer will appear on the right. In this case there is only one parameter; "Elevation Bottom Unit1". Double click on it. It will be transferred to the upper half of the dialog box. Click the OK button to close the Expression Editor. "Elevation Bottom Unit1" will now appear as the parameter value in the Layers Dialog box.

I suggest you use this procedure to link to top of each layer to the bottom of the layer above it. You can also link the initial head of Unit2 (the aquitard) to the initial head of Unit1. Those two units will have the same initial head in all variations of the model but in aquitard3.mmp, we will increase the head in both to a higher value. By linking the starting heads in the two layers, you won't have to remember to change the starting heads in both layers.

This is a very simple use of the Expression Editor. You can also use more complex expressions. The Expression Editor is one of the most powerful features of Argus ONE. If you are curious about some more complicated expressions, try looking at some of the parameters in the MODFLOW FD Grid layer. (Those parameter expressions are locked. You will have to unlock them to look at them. Consult the Argus ONE documentation on how to unlock expressions. The reason they are locked is that if you change those expressions, you may mess up your model.)


The Uses of Prescribed Heads (Constant Heads)

The one thing that is a bit different about this model from the previous example is that we now have prescribed heads. Prescribed heads remain the same throughout the execution of the model. A real-world example where you might want to use a prescribed head would be a large lake that always was filled to the same level. You might also want to use prescribed heads for a river with relatively low fluctuations in water level. In the MODFLOW documentation, prescribed heads are referred to as constant heads.


Extracting Heads and Discharges from the Output Files

When you are finished running the model, open up aquitard.lst with Budgeteer. With Budgeteer, save the budget data to a text file that can be readily imported into a spreadsheet program. Next start HydrographExtractor.exe and select column, row and layer to the aquitard cell (1,1,2). Then read drawdown file (aquitard.ddn) with HydrographExtractor.exe and save the results to a text file. We will graph the results from these two text files and compare the results with ressults from modified versions of the model. We will plot drawdown versus log(time) and the log(rate of change in storage) versus log(time)

There is an MS Excel spreadsheet with the data already prepared if you don't want to extract all the data from the listing file yourself (aquitard.xls). However, be sure to look at the listing file anyway so you can become more familiar with it. You can also look at the formatted head file and extract the head in the aquitard for each time step.


Modifying the Model to Show the Effects of Representing the Aquitard with More Layers

Save this model with the name Aquitard1.mmb. Then save it again with the name Aquitard2.mmb. We will be modifying this model further and you may wish to go back and compare the results of the two versions of the model. In Aquitard2 we will subdivide the aquifers and aquitard to see how that affects the results of the model. The aquifers will both be split into two layers each of which is 25 ft. thick. The aquitard will be split into five layers each 20 ft. thick.

In Aquitard2 open up the Project Info dialog box (PIEs|Edit Project Info...) and select the Geology tab. Now select the top layer. Enter 2 for the "Vertical discretization". Do the same with the bottom aquifer and the aquitard enter a value of 5 instead of 2 for the aquitard.

When exporting to MODFLOW, the upper and lower aquifers will now be split into two layers and the aquitard will be split into five layers. You can do this regardless of the complexity of the topography of the upper and lower surfaces of the geologic unit. We could have also created new layers by pushing the insert button on the geology tab. You might want to create new layers if you wanted to have properties in the new layer different from those of its neighbors.


Comparing the Results of the 3-Layer and 9-Layer Models

Now save the model with the name aquitard2.mmb. Be sure that the "root" in the Project Info dialog box is set to Aquitar2 and run the model.

After you run the model, plot the head in layer 5 and rate of change in storage just as you did before. Because we haven't really changed any of the aquifer characteristics, you might expect that the results would be the same as before but they aren't. The nine-layer model reaches equilibrium sooner due to the higher flux into storage.

  Rate of Change in Storage, Layer 5 Head, Layer 5
cumulative time (days) Nine-Layer Model Three-Layer Model Nine-Layer Model Three-Layer Model
0.155 9.5388E-07 2.1281E-07 1010 1009.9
0.357 8.2863E-07 2.0974E-07 1010 1009.9
0.620 7.0446E-07 2.0587E-07 1009.9 1009.8
0.961 5.8786E-07 2.0117E-07 1009.8 1009.7
1.405 4.8392E-07 1.953E-07 1009.7 1009.5
1.982 3.9524E-07 1.878E-07 1009.4 1009.4
2.732 3.2146E-07 1.7853E-07 1009 1009.2
3.707 2.5971E-07 1.6723E-07 1008.5 1008.9
4.975 2.0637E-07 1.5374E-07 1008 1008.6
6.623 1.5877E-07 1.3805E-07 1007.3 1008.2
8.765 1.161E-07 1.2025E-07 1006.7 1007.8
11.550 7.9212E-08 1.0084E-07 1006.2 1007.3
15.171 4.9512E-08 8.0665E-08 1005.7 1006.9
19.877 2.7855E-08 6.0864E-08 1005.4 1006.4
25.996 1.3861E-08 4.2773E-08 1005.2 1006
33.950 5.9952E-09 2.7604E-08 1005.1 1005.6
44.291 2.2148E-09 1.6101E-08 1005 1005.4
57.733 6.8859E-10 8.3461E-09 1005 1005.2
75.209 1.7729E-10 3.7804E-09 1005 1005.1
97.927 3.7102E-11 1.4707E-09 1005 1005
127.460 6.4279E-12 4.8466E-10 1005 1005
165.853 9.889E-13 1.3251E-10 1005 1005
215.765   2.9667E-11 1005 1005
280.650   5.2663E-12 1005 1005
365.000   9.0023E-13 1005 1005

ChartObject Chart 2

ChartObject Chart 1

These two examples illustrate how changing the way in which aquitards are represented in a model can change the results. You can usually increase the accuracy of a model by increasing the number of layers. Of course, when you do that, the model will take longer to run. You will have to make a decision about which is more important; increased accuracy or a shorter model run-time.

Another test you can try is to increase the difference in head between the two aquifers to 100 feet. To do this, raise the initial heads and prescribed heads in the upper aquifer and aquitard to 1100 feet in the nine-layer model. You could lower the initial heads and prescribed heads in the lower aquifer to 910 feet instead to get the same effect. In either case, the response will be 10 times greater than when the difference in head was 10 feet but otherwise the results will be nearly identical. If you linked the initial heads in the aquitard to the initial head in Unit1, all you have to do will be to change the initial head in Unit1.

Finally, you can try either doubling the hydraulic conductivity of the aquitard or dividing its specific storage by two. Either will have the same effect on head in the aquitard; the time required to reach equilibrium will be cut in half. To understand this consider this equation by Bredehoeft and Pinder (1970).

tD = K't/(Ss' b2)

where

tD is dimensionless time

K' is aquitard hydraulic conductivity

t is time

Ss' is aquitard specific storage

b is aquitard thickness

When tD is less than 0.1 all the interactions of the aquifer with the aquitard are due to changes in storage in the aquitard. When tD is greater than 0.5, the aquitard will have equilibrated with the adjoining aquifers. For this problem, a dimensionless time of 0.1 corresponds to 5.8 days and a dimensionless time of 0.5 corresponds to 29 days. If tD is kept constant, any increase in the hydraulic conductivity of the aquitard or decrease in the specific storage will result in the time required to reach equilibrium be decreased by a corresponding factor. These last two models are examples of how you can get exactly the same distribution of head for models that have different parameters.

Summary

Argus ONE allows you to link the properties of different layers using the Expression Editor. One useful application of this is to link the tops of layers to the bottoms of overlying layers. If you do this, you don't need to worry about editing the top elevations of any layers except the top layer.

MODFLOW requires that the user specify the vertical conductance. (MODFLOW calculates the horizontal conductance.) The Argus ONE PIE calculates the vertical conductance for you based on the vertical hydraulic conductivity and layer thicknesses. (This is a big time-saver because you are unlikely to know vertical conductance but you should know the vertical hydraulic conductivity and layer geometry.)

You can increase the accuracy of a model by including additional layers. However, including more layers may make the model take longer to execute. Thus, you must make a trade-off between the accuracy and speed.

Models with different parameters may give exactly the same head distribution. Thus, calibrating a model to match an observed head distribution does not guarantee that the model is correct.

Example Problem 3, Head-Dependent Boundary Conditions
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