Use of hierarchical cluster analysis to assess the representativeness of a baseline groundwater quality monitoring network: comparison of New Zealand’s national and regional groundwater monitoring programs

Baseline monitoring of groundwater quality aims to characterize the ambient condition of the resource and identify spatial or temporal trends. Sites comprising any baseline monitoring network must be selected to provide a representative perspective of groundwater quality across the aquifer(s) of interest. Hierarchical cluster analysis (HCA) has been used as a means of assessing the representativeness of a groundwater quality monitoring network, using example datasets from New Zealand. HCA allows New Zealand's national and regional monitoring networks to be compared in terms of the number of water-quality categories identified in each network, the hydrochemistry at the centroids of these water-quality categories, the proportions of monitoring sites assigned to each water-quality category, and the range of concentrations for each analyte within each water-quality category. Through the HCA approach, the National Groundwater Monitoring Programme (117 sites) is shown to provide a highly representative perspective of groundwater quality across New Zealand, relative to the amalgamated regional monitoring networks operated by 15 different regional authorities (680 sites have sufficient data for inclusion in HCA). This methodology can be applied to evaluate the representativeness of any subset of monitoring sites taken from a larger network.

A new implicit numerical method for the space and time fractional Bloch-Torrey equation

In recent years, it has been found that many phenomena in engineering, physics, chemistry and other sciences can be described very successfully by models using mathematical tools from fractional calculus. Recently, noted a new space and time fractional Bloch-Torrey equation (ST-FBTE) has been proposed (see Magin et al. (2008)), and successfully applied to analyse diffusion images of human brain tissues to provide new insights for further investigations of tissue structures. In this paper, we consider the ST-FBTE on a finite domain. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Firstly, we propose a new effective implicit numerical method (INM) for the STFBTE whereby we discretize the Riesz fractional derivative using a fractional centered difference. Secondly, we prove that the implicit numerical method for the ST-FBTE is unconditionally stable and convergent, and the order of convergence of the implicit numerical method is ( T2 - α + h2 x + h2 y + h2 z ). Finally, some numerical results are presented to support our theoretical analysis.

Numerical simulation of a new two-dimensional variable-order fractional percolation equation in non-homogeneous porous media

Percolation flow problems are discussed in many research fields, such as seepage hydraulics, groundwater hydraulics, groundwater dynamics and fluid dynamics in porous media. Many physical processes appear to exhibit fractional-order behavior that may vary with time, or space, or space and time. The theory of pseudodifferential operators and equations has been used to deal with this situation. In this paper we use a fractional Darcys law with variable order Riemann-Liouville fractional derivatives, this leads to a new variable-order fractional percolation equation. In this paper, a new two-dimensional variable-order fractional percolation equation is considered. A new implicit numerical method and an alternating direct method for the two-dimensional variable-order fractional model is proposed. Consistency, stability and convergence of the implicit finite difference method are established. Finally, some numerical examples are given. The numerical results demonstrate the effectiveness of the methods. This technique can be used to simulate a three-dimensional variable-order fractional percolation equation.