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ASCEM : Uncertainty Quantification (UQ)

  1. ASCEM
  2. Tutorials

ASCEM : Uncertainty Quantification (UQ)

Introduction

Objective

The objective of this tutorial is to demonstrate an uncertainty quantification (UQ) analysis using Monte Carlo simulation.

Toolsets/modules exercised

The following Akuna modules are being exercised:

  1. UQ using Monte Carlo method with Latin Hypercube sampling
    1. Input parameter selection
    2. Observation definition
  2. Job submission and monitoring
  3. Visualization of UQ output

The following Amanzi modules are being exercised:
  1. Variably saturated flow using Richards equation
    1. Steady-state
    2. Transient
  2. Transient transport

For this tutorial, it is assumed that the 1D-richard model described

in Single-Run (SR) Tutorial has been generated. If this simulation has not yet been generated, create a new model by copying the Richard-1D-transport_exa model from tutorial folder.

  1. Right click on projects > tutorial > Richard-1D-transport_exa and choose copy.
  2. Right-click on your user folder and choose paste.
  3. The following window will appear:

image0
Figure 1. Select tools to import when copying an existing model
  1. Click the radio button for single-run to import the single-run example. Press OK.
  2. A new model will appear under the user’s working folder.

Problem Description

The tutorial problem is a one-dimensional, 3 layer, unsaturated flow and transport problem, where

  1. steady-state unsaturated flow is calculated with a constant infiltration rate, providing the initial conditions for year 1950;
  2. discharge of water and 99Tc occurs at the top of the ground surface between 1950.1-1950.26 years; and
  3. continuing infiltration drives the 99Tc plume downward, to the water table, which is represented by the bottom boundary of the domain.

Uncertainty in the concentration predictions of 99Tc at different well locations after 2200 years of simulation will be analyzed with respect to 1) uncertainties in permeabilities for the three different geologic layers; and 2) the uncertainty in the rate of discharge for the source released between 1950.1-1950.26 years.

Initialize UQ Toolset

To start the UQ Toolset:

  1. On the main window (Figure 2), click on the Richard-1D-transport conceptual model |image1| icon representing the conceptual model that was built in SR Tutorial; it will be highlighted as shown in the figure below.
  2. Click image2 to launch the Uncertainty Quantification Toolset. The item UQ is not clickable unless a conceptual model is selected.

image3
Figure 2. Create a new UQ Analysis

3.  A window (Figure 3) that prompts for a UQ Name will appear. Name the UQ and click OK.


image4
Figure 3. Provide a name for the UQ Analysis
This action launches the Uncertainty Quantification Toolset window

(Figure 4). The tabs are the same as those in the SR Toolset and presented in the SR Tutorial, except for the Analysis Options tab. | image5 |

Figure 4. UQ Toolset window

Setup UQ Toolset

Akuna simplifies the setup by providing the user with the ability to import existing information from another analysis (alternatively, all fields can be populated manually). Here it is demonstrated how to import the data from the SR Tutorial into the UQ Toolset.

  1. Click image6 on the upper left corner of the UQ Toolset window.
  2. Select all sections to be imported by checking all the radio buttons, as shown in Figure 5. Press Ok.

image7
Figure 5. Select all radio buttons to import all the information.

3. Select the single-run created in the SR tutorial as shown in Figure 6. Press OK and all the tabs will be filled with data used in the SR (Figure 7).


image8
Figure 6. Import from a SR run that was previously created.
image9
Figure 7. Data imported from an existing SR run into UQ analytis.

Parameter Setup

The parameters that are considered uncertain in the UQ analysis are setup as follows:

  1. Click the Parameters section and then click image10 to add a parameter, as shown in Figure 8.

image11
Figure 8. Parameters section.

2.  The dialog box shown in Figure 9 will appear. Click on image12 to expand the list of parameters that can be chosen.


image13

Figure 9. Select Parameters From Model Setup window.

3.  For the 1st parameter, choose:  Material Properties > Facies1 > Permeability > x; Material Properties > Facies1 > Permeability > y; Material Properties > Facies1 > Permeability > z   simultaneously while holding down the Shift key These three variables will be treated as a single parameter. Click Add.

 4.  For the 2nd and 3rd parameters, repeat the previous step with Facies2 and Facies3

 5.  For the 4th parameters, choose Boundary Conditions > B17 > 1950.1 y > Liquid Phase > Inward mass flux. Click Add again.

 6.  Close the window.

 7.  Double click on any of the entries in the table to define properties related to each parameter. Change the entries to those shown in Figure 10.  The first 3 parameters are examined in the logarithm space and thus the Transformation Type (column 5) is changed to Logarithm. (After selecting the transformation type, the user will be prompted for converting the initial and default values before the transformation is applied.) As a result the distributions of the permeabilities are log-normal when the Distribution Type (column 6) is Normal.  The 4th parameter is examined in the absolute space and the transformation type is uniform.

image14
Figure 10. Edit the entries of the Parameter table based on entries

shown above.

Define Observations

Observations are defined under the Observations tab. Observations have been prefilled since they were defined during model setup and in the SR Tutorial. Observations can be added or deleted in this tab, if desired.

Specify Analysis options

The analysis options are set in the Analysis Options tab:

  1. Choose Probabilistic predictions for Type.
  2. Choose Monte Carlo for Method.
  3. Choose Latin Hypercube for Sampling Method.
  4. Set the Number of Forward Runs to 100 and press ENTER; the plots on the right will automatically show the distribution of the samples generated.  The user needs to ensure that the generated histograms are consistent with the defined distributions for the parameters, as shown in Figure 11.
  5. Change the Random Seed or click image15 found on the upper right corner to generate a different set of parameters.

image16
Figure 11. Generate samples for UQ analysis.

Job Configuration

The job configuration process is similar to job configuration demonstrated in the SR Tutorial (Figure 12). The only additional step involves specifying the number of Processors and Processors per Tasks. In the SR Tutorial, only a single simulation is executed, the number of Processors and Processors per Tasks are the same. In the UQ, 100 simulations will be executed, with each simulation using 24 processors. Set Processors to 72, Processors per Tasks to 2 and Max Simultaneous Simulations to 3.

Save the model by clicking image17 at the top left corner. The  |image18|  button will turn green, indicating that all information needed to run the analysis has been provided.


image19
Figure 12. Job Configuration in the UQ analysis.

Submit UQ Analysis

Click image20 to submit the job (credential required). The remaining steps for submission and monitoring of jobs follows the SR Tutorial.

Analyze UQ Results

When UQ simulations are completed, the Plot button is activated, and is accessible in two places: 1) in the Summary Tab in the right panel; and 2) above the data browser in the left panel. Note that even if all simulations do not run successfully, results from the completed simulations can still be viewed.

Concentration Histogram

  1. Click on the Plot button, as shown in Figure 13.

image21
Figure 13. Location of the Plot button.

2.  The Analysis windows (Figure 14) will appear. Under the Guided Control in the left column, 3 plot types are available: histogram, scatter plot and line plot. In this exercise, a histogram of aqueous concentrations for all realizations at Well1 will be created. The entries in the Guided Control are changed to those shown in Figure 14.


image22
Figure 14. Entries under Guided Controls for generating a

concentration histogram.

3.  Press Generate. Because only two times (year 2180 and 2200) and multiple plots are selected, then two plots will be generated in the right plotting area, as shown in Figure 15.


image23
Figure 15. Histogram of the results for Well 1 at year 2180 and 2200.
  1.  Click image24 in the upper left corner to save the results.

Breakthrough Curves

In this exercise, breakthrough curves for all of the simulations will be generated at a single well location.

  1. Under the Guided Control in the left column, select the entries as shown in the Guided Control in Figure 16.
  2. Press Generate. A single plot showing the aqueous concentration vs. time for all 100 simulations is shown in Figure 17.
  3. To plot only the mean and 95% confidence intervals for all 100 simulations, use the Filters menu in the upper right corner of the plotting panel to select
  4. Filters – Add - Mean to add a line that plots the mean concentration at each point in time.
  5. Filters – Add - 95% to add the lines plotting the upper and lower 95% confidence bounds.
  6. Filters – Remove – Unfiltered data, will turn off the lines for each of the 100 simulations, leaving only the mean and confidence interval lines as shown in Figure 18.
  7. Options – Show Legend/Hide Legend to toggle the legend.

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Figure 16. Entries under Guided Controls for the generating breakthrough curves at Well1.

image26

Figure 17. Breakthrough curves for all simulations at Well1.
image27

Figure 18. Mean and 95% confidence intervals for concentration vs. time.

Number of Years to Reach a Minimum or Maximum Concentration

Uncertainty analyses often use the drinking water standard maximum contaminant level (MCL) to determine risk. In this section, a histogram is generated that use a threshold metric (analogous to an MCL) to examine UQ results. The final plot shows the number of years required to reach the peak concentration for the100 simulations.

  1. Using the Advanced Controls tab, click on the blue arrow shown in Figure 19. This creates drop-down menus for plotting.

image28
Figure 19. Startup screen for Advanced Controls tab.

2.  Within the Advanced Controls tab, select the entries as shown in the Guided Control in Figure 20.


image29

Figure 20. Entries under Advanced Controls for generating peak concentration histograms.

  1. Once the plots showing the aqueous concentration for all 100 simulations are generated for both locations (Well1 and Well2), use the dropdown menus at the right to perform the following steps:
    1. Transform – Set lower threshold and check the y-axis check-box, and type 3.5e-7 in the blank next to the y-axis label. Only concentrations above this threshold value will be plotted.
    2. Filter – Use Filtered Data. This step allows all subsequent operations to be performed on the modified data set (i.e., all concentrations above the threshold).
    3. Extract – Time (years) to y-axis. This step plots the time in years along the y-axis at which concentrations are above the threshold occur.
    4. Transform – Shift, check the y-axis check-box, and enter the value -1950.1, the year the source was released. This will calculate the number of years from the time of release that the concentration was above the threshold.
    5. Filters – Add – Min. This action will extract the year on the y-axis in which the concentration first exceeds the threshold concentration.
    6. Filter – Remove – Unfiltered Data. Removing the unfiltered data will permit only the minimum to be plotted.
    7. Options – Render as – Histogram – Use 10 bins to change the chart type to a histogram. This will yield the histogram shown in Figure 21, which shows the number of years it takes to reach the threshold concentration at Well1 and Well2.

image30

Figure 21. Number of years to reach the threshold concentration at locations Well1 and Well2.

To plot the number of years required to reach the peak concentration, perform the following steps:

  1. Using the Advanced Controls tab, click on the blue arrow shown in Figure 19, if the Plot Controls menu is not already showing. Then create the same entries as shown in Figure 20.
  2. Once the plots showing the aqueous concentration for all 100 simulations are generated for both locations (Well1 and Well2), use the menus at the top right of the plotting window:

a.  Filters – Add - Max. This action will plot the maximum concentration for each run as a separate data set.

b.  Filter – Remove Unfiltered Data. This step allows all subsequent operations to only be performed on the modified (filtered) data set (i.e., the maximum concentration for each run).

c.  Extract – Time (years) to y-axis. This step plots the time in years along the y-axis at which the maximum concentration occurs.

d.  Transform – Shift, check the y-axis check-box, and enter the value -1950.1, the year the source was released. This will calculate the number of years from the time of release to the time the peak concentration occurred.

e.  Options – Render as – Histogram – Use 10 bins to change the chart type to a histogram. This will yield the histogram shown in Figure 22, which shows the number of years it takes to reach the peak concentration at Well1 and Well2.


image31

Figure 22. Number of years to reach the peak concentration at locations Well1 and Well2.

Number of Years Above a Threshold Concentration

Uncertainty analyses often use the drinking water standard maximum contaminant level (MCL) to determine risk. In this section, a histogram is generated that use a threshold metric (analogous to an MCL) to examine UQ results. The final plot shows the number of years that the concentration is above the threshold concentration (i.e., MCL).

  1. Using the Advanced Controls tab, click on the blue arrow shown in Figure 19, if the Plot Controls menu is not already showing. Then create the same entries shown in Figure 20 that were used in the previous section.
  2. Once the plots showing the aqueous concentration for all 100 simulations are generated for both locations (Well1 and Well2), use the menus at the top right of the plotting window to perform the following steps:

a.  Transform – Set lower threshold and check the y-axis check-box, and type 3.5e-7 in the blank next to the y-axis label. Only concentrations above this threshold value will be plotted.

b.  Filter – Use Filtered Data. This step allows the following operations to be performed on the modified data set (i.e., all concentrations above the threshold).

c.  Extract – Time (years) to y-axis. This step plots the time in years along the y-axis at which concentrations are above the threshold occur.

d.  Transform – Shift, check the y-axis check-box, and enter the value -1950.1, the year the source was released. This will calculate the number of years from the time of release that the concentration was above the threshold.

e.  Filters – Add - Range. This action will extract the range of years on the y-axis, for which the concentration was above the threshold.

f.  Filter – Remove – Unfiltered Data. Removing the unfiltered data will permit only the range to be plotted.

g.  Options – Render as – Histogram – Use 10 bins to change the chart type to a histogram. This will yield the histogram shown in Figure 23, which shows the number of years the threshold concentration is exceeded at Well1 and Well2.


image32
Figure 23. Number of years the threshold concentration is exceeded at

Well1 and Well2.