1. Aspects of Pure Mathematics, Experimental Design and Mathematical Modelling DIANE DONOVAN STEVE TYSON BEVAN THOMPSON LIAM O’SULLIVAN MARVIN TAS

2. Groundwater  Water table: underground water seeping slowly through aquifers.  Depth: varying from meters to hundreds of meters below the surface.  Aquifers: permeable material such as gravel, sand, sandstone, or fractured rock.  Flow speed: dependant on the size of the aquifers and how well they are connected. Agency for Toxic Substances and Disease Registry

3. Groundwater contamination  May be introduced through  Erosion  Industrial discharge  Agricultural discharge  Household discharge  Hazardous waste sites  Landfills  Road salts or chemicals.  Primary concerns: synthetic compounds, including solvents, pesticides, paints, varnishes, gasoline and nitrate. Agency for Toxic Substances and Disease Registry

4. Erosion  Soil erosion significantly impacts on our environment:  Capadocia, Turkey  Burdekin Basin (Aust) and Great Barrier Reef  The Australian Research Organization (CSİRO) is studying the effect of soil erosion contamination of the Great Barrier Reef. Toprak Erozyonu/Soil Erosion

5. Erosion and the Great Barrier Reef

6. Latin Hypercube Sampling

7. 3-D Latin Hypercube Sampling

8.  Advantages  Propagation of uncertainty through models.  Good coverage of parameter space.  Easy updating, given new data.  Each parameter is fully stratified and each sub- division is sampled with the same density.  Variance reduction when compared with random sampling.  Fast implementation. Latin Hypercube Sampling

9. 3-D Latin Hypercube Sampling

10.  A LHT is an Orthogonal sample (OS) if the n = p d sample points are distributed evenly across all sub-blocks.  Advantages:  Uniformity of small dimensional margins.  Improved representation of the underlying variability.  A form of variance reduction.  Better screening for effective parameters.  Equally fast implementation. Orthogonal Sampling

11. Coverage of Parameter Space  Theoretical & computational arguments show that as the number of trials increases the size of the un-sampled space decreases exponentially.  LHS & OS (with n = p d ) the expected percentage coverage of parameter space is given by

12. Transportation of Chemical Contaminants in Groundwater

13. Chemical Contaminants in Groundwater Retardation factor with uncertainty in organic carbon content Longitudinal dispersion coefficient with uncertainty in organic carbon content and hydraulic conductivity Pore water velocity with uncertainty in hydraulic conductivity ParameterDistributionLower LimitUpper Limit K(cm/s) hydraulic conductivity Uniform1.0E-71.0E-3 Uniform20500

14. Chemical Contaminants in Groundwater ParameterDistributionLower LimitUpper Limit K(cm/s)Uniform1.0E-71.0E-3 Uniform20500

15. Polynomial Chaos Expansion

16. Finite Differences

17. Polynomial Chaos Expansion

18.  Advantages  Fast and efficient.  Different probability distributions can be assigned to input parameters.  Simplifying implementation using spectral representation & orthogonal bases.  Reduced computational costs.  Easy post processing statistics, including moments and the probability density function--zero-index term contains the solution mean.  Sensitivity to underlying probability distribution, propagating uncertainty & variability through the simulation. Polynomial Chaos Expansion

19.  Disadvantages  Non-normal random input distributions must be treated with care.  Convergence domains must be studied with care for both smooth and non-smooth outputs.  PC does not quantify the approximation error as a component of uncertainty.  Changes to the input distribution may require output values, the convergence (of the approximation) and truncation parameters to be recomputed. Polynomial Chaos Expansion

20. References  A O'Hagan, Polynomial Chaos: A Tutorial and Critique from a Statistician's Perspective, 2013  D Datta & S Kushwaha, Uncertainty Quantification Using Stochastic Response Surface Method Case Study-Transport of Chemical Contaminants through Groundwater, International Journal of Energy, Information and Communication, 2011, 2(3), 49-58  DL Parkhurst & CAJ Appelo, User's Guide to PHREEQC (Version 2)--A Computer Program for Speciation, Batch-Reaction, One-Dimensional Transport, and Inverse Geochemical Calculations, accessed at http://wwwbrr.cr.usgs.gov/projects/GWC_coupled/phreeqc/html/final.html on 15/9/2015http://wwwbrr.cr.usgs.gov/projects/GWC_coupled/phreeqc/html/final.html  K Burrage, PM Burrage, D Donovan, T McCourt & HB Thompson, Estimates on the coverage of parameter space using populations of model, Modelling and Simulation, IASTED, ACTA Press, 2014  K Burrage, PM Burrage, D Donovan & HB Thompson, Populations of Models, Experimental Designs and Coverage of Parameter Space by Latin Hypercube and Orthogonal Sampling, Procedia Computer Science, 2015, 51, 1762-1771  S Tyson, D Donovan, B Thompson, S Lynch & M Tas, Uncertainty Modelling with Polynomial Chaos, Report to the Centre for Coal Seam Gas, University of Queensland, August 2015.

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