Aspects of Pure Mathematics, Experimental Design and Mathematical Modelling

DIANE DONOVAN STEVE TYSON BEVAN THOMPSON LIAM O’SULLIVAN MARVIN TAS

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

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

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

Regular/systematic sweeps, sampling erosion.

Sampling at distinct times varying by longitude and latitude

Erosion and the Great Barrier Reef

         

         

         

         

         

A Latin hypercube trial, LHT, (d-trial):

n sample points in d dimensional sample space where each the n sub-divisions for each of the d parameters appears in precisely one sample point.

d=2 take a square grid, Latin Hypercube trial has 1 sample per row and 1 sample per column.

Repeated applications of the LHT gives a Latin Hypercube Sample, LHS.

Latin Hypercube Sampling

         

         

         

         

         

3-D Latin Hypercube Sampling

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

         

         

         

         

         

3-D Latin Hypercube Sampling

A LHT is an Orthogonal sample (OS) if the

n = pd 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

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 = pd ) the expected percentage coverage of parameter space is given by

Transportation of Chemical Contaminants in Groundwater

1-D steady state flow chemical transportation with advection, dispersion, retardation.

Governing equation 𝑑𝑥𝑑𝑦

𝑑𝑧−𝐷𝐿

𝜕𝐶𝜕𝑥

𝑣𝐶

−𝐷𝐿 (𝜕𝐶𝜕𝑥 + 𝜕2𝐶𝜕𝑥2

𝑑𝑥)

𝑣 (𝐶+𝜕𝐶𝜕𝑥

𝑑𝑥 )

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

Uncertainty in and due to uncertainty in hydraulic conductivity K and organic carbon partition coefficient .

Parameter Distribution Lower Limit Upper Limit

K(cm/s) hydraulic conductivity

Uniform 1.0E-7 1.0E-3

(cc/g) organic carbon content

Uniform 20 500

Chemical Contaminants in Groundwater

Monte Carlo techniques: modelling uncertainty in hydraulic conductivity K and organic carbon partition coefficient .

Populations of Models: different sets of admissible parameter values (models) capable of reproducing the observed output within given tolerances.

Latin Hypercube & Orthogonal sampling: efficient techniques especially useful for POM propagating the uncertainty through the simulations.

Parameter Distribution Lower Limit Upper Limit

K(cm/s) Uniform 1.0E-7 1.0E-3

(cc/g) Uniform 20 500

Polynomial Chaos Expansion

distributed uniformly with mean & variance /3.

organic carbon is distributed uniformly with mean & variance /3.

(.

Orthogonal Legendre Polynomials

where

Finite Differences An a approximations for , .

𝜕𝐶 (𝑥𝑖 ,𝑡 𝑗)𝜕𝑡

𝐶 (𝑥𝑖 ,𝑡 𝑗+1 )−𝐶 (𝑥 𝑖 ,𝑡 𝑗)∆ 𝑡

𝜕2𝐶𝜕𝑥2

𝐶 (𝑥 𝑖+1, 𝑡 𝑗 )−2𝐶 (𝑥 𝑖 ,𝑡 𝑗 )+𝐶 (𝑥𝑖 −1 ,𝑡 𝑗)

∆ 𝑥2

𝜕𝐶𝜕 𝑥

𝐶 (𝑥 𝑖+1 , 𝑡 𝑗 )−𝐶 (𝑥 𝑖− 1, 𝑡 𝑗)2∆ 𝑥

𝑡 𝑗+1=𝑡 𝑗+∆𝑡 ,𝑥 𝑖+1=𝑥𝑖+∆ 𝑥

Polynomial Chaos Expansion

.

.

(.

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

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

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 15/9/2015

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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