Novel analytical and numerical methods for solving fractional dynamical systems

During the past three decades, the subject of fractional calculus (that is, calculus of integrals and derivatives of arbitrary order) has gained considerable popularity and importance, mainly due to its demonstrated applications in numerous diverse and widespread fields in science and engineering. For example, fractional calculus has been successfully applied to problems in system biology, physics, chemistry and biochemistry, hydrology, medicine, and finance. In many cases these new fractional-order models are more adequate than the previously used integer-order models, because fractional derivatives and integrals enable the description of the memory and hereditary properties inherent in various materials and processes that are governed by anomalous diffusion. Hence, there is a growing need to find the solution behaviour of these fractional differential equations. However, the analytic solutions of most fractional differential equations generally cannot be obtained. As a consequence, approximate and numerical techniques are playing an important role in identifying the solution behaviour of such fractional equations and exploring their applications. The main objective of this thesis is to develop new effective numerical methods and supporting analysis, based on the finite difference and finite element methods, for solving time, space and time-space fractional dynamical systems involving fractional derivatives in one and two spatial dimensions. A series of five published papers and one manuscript in preparation will be presented on the solution of the space fractional diffusion equation, space fractional advectiondispersion equation, time and space fractional diffusion equation, time and space fractional Fokker-Planck equation with a linear or non-linear source term, and fractional cable equation involving two time fractional derivatives, respectively. One important contribution of this thesis is the demonstration of how to choose different approximation techniques for different fractional derivatives. Special attention has been paid to the Riesz space fractional derivative, due to its important application in the field of groundwater flow, system biology and finance. We present three numerical methods to approximate the Riesz space fractional derivative, namely the L1/ L2-approximation method, the standard/shifted Gr¨unwald method, and the matrix transform method (MTM). The first two methods are based on the finite difference method, while the MTM allows discretisation in space using either the finite difference or finite element methods. Furthermore, we prove the equivalence of the Riesz fractional derivative and the fractional Laplacian operator under homogeneous Dirichlet boundary conditions – a result that had not previously been established. This result justifies the aforementioned use of the MTM to approximate the Riesz fractional derivative. After spatial discretisation, the time-space fractional partial differential equation is transformed into a system of fractional-in-time differential equations. We then investigate numerical methods to handle time fractional derivatives, be they Caputo type or Riemann-Liouville type. This leads to new methods utilising either finite difference strategies or the Laplace transform method for advancing the solution in time. The stability and convergence of our proposed numerical methods are also investigated. Numerical experiments are carried out in support of our theoretical analysis. We also emphasise that the numerical methods we develop are applicable for many other types of fractional partial differential equations.

Coal Seam Gas Water from Maramarua, New Zealand: Characterisation and Comparison to United States Analogues

Groundwater from Maramarua has been identified as coal seam gas (CSG) water by studying its composition, and comparing it against the geochemical signature from other CSG basins. CSG is natural gas that has been produced through thermogenic and biogenic processes in underground coal seams; CSG extraction requires the abstraction of significant amounts of CSG water. To date, no international literature has described coal seam gas water in New Zealand, however recent CSG exploration work has resulted in CSG water quality data from a coal seam in Maramarua, New Zealand. Water quality from this site closely follows the geochemical signature associated with United States CSG waters, and this has helped to characterise the type of water being abstracted. CSG water from this part of Maramarua has low calcium, magnesium, and sulphate concentrations but high sodium (334 mg/l), chloride (146 mg/l) and bicarbonate (435 mg/l) concentrations. In addition, this water has high pH (7.8) and alkalinity (360 mg/l as CaCO3), which is a direct consequence of carbonate dissolution and biogenic processes. Different analyte ratios ('source-rock deduction' method) have helped to identify the different formation processes responsible in shaping Maramarua CSG water

A comprehensive investigation into the hydrodynamic and capture/retention performance of a gross pollutant trap

A novel and comprehensive testing approach to examine the performance of gross pollutant traps (GPTs) was developed. A proprietary GPT with internal screens for capturing gross pollutants—organic matter and anthropogenic litter—was used as a case study. This work is the first investigation of its kind and provides valuable practical information for the design, selection and operation of GPTs and also the management of street waste in an urban environment. It used a combination of physical and theoretical models to examine in detail the hydrodynamic and capture/retention characteristics of the GPT. The results showed that the GPT operated efficiently until at least 68% of the screens were blocked, particularly at high flow rates. At lower flow rates, the high capture/retention performance trend was reversed. It was also found that a raised inlet GPT offered a better capture/retention performance. This finding indicates that cleaning operations could be more effectively planned in conjunction with the deterioration in GPT’s capture/retention performance.

Atmospheric deposition as a source of stormwater pollution in Gold Coast, Australia

Atmospheric deposition is one of the most important pollutant pathways for urban stormwater pollution. Atmospheric deposition can be in the form of dry and wet depositions which have distinct characteristics in terms of pollutant types, pollutant sources and influential parameters. This paper discusses the outcomes of a comprehensive study undertaken to identify the characteristics of wet and dry deposition of pollutants. Sample collection was undertaken at eight study sites with distinct characteristics. Four sites were close to road sites with varying traffic characteristics, whilst the other four sites had different land use characteristics. Dry deposition samples were collected for different antecedent dry days and wet deposition samples were collected immediately after rainfall events. The dry deposition was found to increase with the antecedent dry days and consisted of relatively coarser particles (greater than 1 µm) when compared to wet deposition. The wet deposition showed a strong affinity to rainfall depth, but was not related to the antecedent dry period. It was also found that smaller size particles (less than 1 µm) travel much longer distances from the source and deposit mainly with the wet deposition

Use of 3D visualisation to enhance groundwater stable isotope data and its interpretation, Lockyer Valley, southeast Queensland

The use of stable isotope ratios δ18O and δ2H are well established in assessment of groundwater systems and their hydrology. The conventional approach is based on x/y plots and relation to various MWL’s, and plots of either ratio against parameters such as Clor EC. An extension of interpretation is the use of 2D maps and contour plots, and 2D hydrogeological vertical sections. An enhancement of presentation and interpretation is the production of “isoscapes”, usually as 2.5D surface projections. We have applied groundwater isotopic data to a 3D visualisation, using the alluvial aquifer system of the Lockyer Valley. The 3D framework is produced in GVS (Groundwater Visualisation System). This format enables enhanced presentation by displaying the spatial relationships and allowing interpolation between “data points” i.e. borehole screened zones where groundwater enters. The relative variations in the δ18O and δ2H values are similar in these ambient temperature systems. However, δ2H better reflects hydrological processes, whereas δ18O also reflects aquifer/groundwater exchange reactions. The 3D model has the advantage that it displays borehole relations to spatial features, enabling isotopic ratios and their values to be associated with, for example, bedrock groundwater mixing, interaction between aquifers, relation to stream recharge, and to near-surface and return irrigation water evaporation. Some specific features are also shown, such as zones of leakage of deeper groundwater (in this case with a GAB signature). Variations in source of recharging water at a catchment scale can be displayed. Interpolation between bores is not always possible depending on numbers and spacing, and by elongate configuration of the alluvium. In these cases, the visualisation uses discs around the screens that can be manually expanded to test extent or intersections. Separate displays are used for each of δ18O and δ2H and colour coding for isotope values.

Relationships between rainfall intensity, duration and suspended particle washoff from an urban road surface

A basic understanding of the relationships between rainfall intensity, duration of rainfall and the amount of suspended particles in stormwater runoff generated from road surfaces has been gained mainly from past washoff experiments using rainfall simulators. Simulated rainfall was generally applied at constant intensities, whereas rainfall temporal patterns during actual storms are typically highly variable. This paper discusses a rationale for the application of the constant-intensity washoff concepts to actual storm event runoff. The rationale is tested using suspended particle load data collected at a road site located in Toowoomba, Australia. Agreement between the washoff concepts and measured data is most consistent for intermediate-duration storms (duration <5 h and >1 h). Particle loads resulting from these storm events increase linearly with average rainfall intensity. Above a threshold intensity, there is evidence to suggest a constant or plateau particle load is reached. The inclusion of a peak discharge factor (maximum 6 min rainfall intensity) enhances the ability to predict particle loads.

Development of a sewerage infrastructure buffer assessment tool for local authorities

Urban expansion continues to encroach on existing or newly implemented sewerage infrastructure. In this context, legislation and guidelines, both national and international, provide limited direction to the amenity allocation of appropriate buffering distances for land use planners and infrastructure providers. A review of published literature suggests the dominant influences include topography, wind speed and direction, temperature, humidity, existing land uses and vegetation profiles. A statistical criteria review of these factors against six years of sewerage odour complaint data was undertaken to ascertain their influence and a complaint severity hierarchy was established. These hierarchical results suggested the main criteria were: topographical location, elevation relative to the odour source and wind speed. Establishing a justifiable criterion for buffer zone allocations will assist in analytically determining a basis for buffer separations and will assist planners and infrastructure designers in assessing lower impact sewerage infrastructure locations.

Three-dimensional geological modelling and multivariate statistical analysis of water chemistry data to analyse and visualise aquifer structure and groundwater composition in the Wairau Plain, Marlborough District, New Zealand

Concerns regarding groundwater contamination with nitrate and the long-term sustainability of groundwater resources have prompted the development of a multi-layered three dimensional (3D) geological model to characterise the aquifer geometry of the Wairau Plain, Marlborough District, New Zealand. The 3D geological model which consists of eight litho-stratigraphic units has been subsequently used to synthesise hydrogeological and hydrogeochemical data for different aquifers in an approach that aims to demonstrate how integration of water chemistry data within the physical framework of a 3D geological model can help to better understand and conceptualise groundwater systems in complex geological settings. Multivariate statistical techniques(e.g. Principal Component Analysis and Hierarchical Cluster Analysis) were applied to groundwater chemistry data to identify hydrochemical facies which are characteristic of distinct evolutionary pathways and a common hydrologic history of groundwaters. Principal Component Analysis on hydrochemical data demonstrated that natural water-rock interactions, redox potential and human agricultural impact are the key controls of groundwater quality in the Wairau Plain. Hierarchical Cluster Analysis revealed distinct hydrochemical water quality groups in the Wairau Plain groundwater system. Visualisation of the results of the multivariate statistical analyses and distribution of groundwater nitrate concentrations in the context of aquifer lithology highlighted the link between groundwater chemistry and the lithology of host aquifers. The methodology followed in this study can be applied in a variety of hydrogeological settings to synthesise geological, hydrogeological and hydrochemical data and present them in a format readily understood by a wide range of stakeholders. This enables a more efficient communication of the results of scientific studies to the wider community.

Inactivation and structural changes of E. Cloacae and B. Subtilis Endospores during IR laser water treatment

IR radiation has been studied for micro-organism inactivation of bacterial spores on metal substrates [1] and on metal and paper substrates [2]. A near-point near infrared laser water treatment apparatus for use in dental hand-pieces was also developed [3]. To date water sterilisation research using a mid-IR laser technique is very rare. According to the World Health Organisation [4], examinations for faecal indicator bacteria remain the most sensitive and specific way of assessing the hygienic quality of water. Bacteria that fall into this group are E. coli, other coliform bacteria (including E. cloacae) and to a lesser extent, faecal streptococci [5]. Protozoan cysts from organisms which cause giardiasis are the most frequently identified cause of waterborne diseases in developed countries [6,7]. The use of aerobic bacterial endospores to monitor the efficiency of various water treatments has been shown to provide a reliable and simple indicator of overall performance of water treatment[8,9].The efficacy of IR radiation for water disinfection compared to UV treatment has been further investigated in the present study. In addition FTIR spectroscopy in conjunction with Principle Component Analysis was used to characterise structural changes within the bacterial cells and endospores following IR laser treatment. Changes in carbohydrate content of E. cloacae following IR laser treatment were observed.

Understanding treatment characteristics of constructed stormwater wetlands

Constructed wetlands are a common structural treatment measure employed to remove stormwater pollutants and forms an important part of the Water Sensitive Urban Design (WSUD) treatment suite. In a constructed wetland, a range of processes such as settling, filtration, adsorption, and biological uptake play a role in stormwater treatment. Occurrence and effectiveness of these processes are variable and influenced by hydraulic, chemical and biological factors. The influence of hydraulic factors on treatment processes are of particular concern. This paper presents outcomes of a comprehensive study undertaken to define the treatment performance of a constructed wetland highlighting the influence of hydraulic factors. The study included field monitoring of a well established constructed wetland for quantity and quality factors, development of a conceptual hydraulic model to simulate water movement within the wetland and multivariate analysis of quantity and quality data to investigate correlations and to define linkages between treatment performance and influential hydraulic factors. Total Suspended Solids (TSS), Total Nitrogen (TN) and Total Phosphorus (TP) concentrations formed the primary pollutant parameters investigated in the data analysis. The outcomes of the analysis revealed significant reduction in event mean concentrations of all three pollutants species. Treatment performance of the wetland was significantly different for storm events above and below the prescribed design event. For events below design event, TSS and TN load reduction was comparatively high and strongly influenced by high retention time. For events above design event, TP load reduction was comparatively high and was found to be influenced by the characteristics of TP wash-off from catchment surfaces.