外文数值解论文A comparison of numerical solution

errors, conflicting data and site heterogeneities all contribute to the complexities in site characterization. Detailed site information needed to fully characterize a site are usually lacking as large number of samples are required to completely capture a site's descriptions and its variabilities. Substantial data requirement means high costs and usually the site assessor makes decisions with much less data. Apart from issues related to site variabilities, multiple sources of information have to be compiled such as historical, geological and hydrogeologic information which may be in “soft” (descriptive) or “hard” (numeric) form. There may be data from laboratory studies, field investigations as well as from expert opinions. All these information are required to determine soil characteristics and site parameters such as hydraulic conductivity, storage coefficient, and porosity which
Kejiang Zhang a, Gopal Achari a,⁎, Hua Li b
a b
Department of Civil Engineering, University of Calgary, Calgary, AB, Canada T2N 1N4 Department of Mathematics, Zhengzhou University, Zhengzhou, Henan, PR China
Journal of Contaminant Hydrology 110 (2009) 45–59
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Journal of Contaminant Hydrology
j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / j c o n h yd
A comparison of numerical solutions of partial differential equations with probabilistic and possibilistic parameters for the quantification of uncertainty in subsurface solute transport
1. Introduction Modeling contaminant fate and transport is an integral part of exposure assessment, a necessary step in environmental risk assessment. The various physical phenomena in groundwater flow and contaminant transport are represented by partial differential equations (PDEs) in space and time. The geologic and hydrogeologic site characteristics are included as parameters in the governing PDEs. To determine these parameters is a challenging part of modeling. Scarcity of information, uncertainty in the data collected, mea. / Journal of Contaminant Hydrology 110 (2009) 45–59
are then integrated into a groundwater flow and contaminant transport model. For contaminant fate and transport, additionally, the biogeochemical information about the target contaminant such as precipitation–dissolution, redox reaction, and biodegradation are needed. Uncertain, variable and multi-sourced “soft” and “hard” information not only require proper representations of different uncertainties but also their integration (Porter et al., 2000; Dubois et al., 2000; Sentz and Ferson, 2002; Dubois et al., 2004), and incorporation of the integrated information into the groundwater flow and contaminant transport modeling. Three types of uncertainty (Möller and Beer, 2004), stochastic, informal and lexical, are present in site characterization. Stochastic uncertainty is best described by classical probability theory. Informal uncertainty results from an information deficit, such as when only a small number of observations are available. Lexical uncertainty occurs when instead of numbers, words such as “high”, “medium”, and “low” are used. Opinions of experts are usually linguistic and will have this type of uncertainty. Probability theory is suitable for representation of probabilistic information. Fuzzy set theory (Zadeh, 1965), and possibility theory (Zadeh, 1978; Dubois and Prade, 1988), evidence theory (Shafer, 1976), and random sets (Matheron, 1975; Molchanov, 2005; Nguyen, 2006) are employed to represent informal and lexical uncertain information or possibilistic information (Liu and Peng, 2005). Evidence theory is more suitable when a combination of different types of uncertain information exist (Ross, 2004). This may occur when one theory best represents one parameter whereas another theory may be more suitable for another parameter. Both types of parameters are however required in one governing equation. This leads to the interesting question of hybrid uncertainties and their propagation. Recent research by Guyonnet et al. (2003) and Baudrit et al. (2004, 2007) identified a method of hybrid uncertainty propagation through the conventional risk assessment process. Chen et al. (2003), Li et al. (2007) and Li et al. (2008) use the hybrid fuzzy-stochastic modeling approach to evaluate risk posed by contaminated groundwater and air pollution. However, different types of uncertainties associated with contaminant fate and transport were all captured using probability theory and fuzzy sets were only used to quantify uncertainties inherent in environmental quality guidelines and health evaluation criteria. Baraldi and Zio (2008) used a combined Monte Carlo and possibilistic approach to propagate uncertainty in event tree analysis. Conventionally, uncertainties associated with different parameters in a PDE are modeled using stochastic partial differential equations (SPDEs) (Dagan and Neuman, 1997; Glimm and Sharp, 1998). There are two limitations of stochastic modeling: (1) it needs sufficient data to justify the assumed statistical distributions. Given sufficient statistical data, the probability theory can better capture the random uncertainties; (2) different types of uncertainty that may be associated with the parameters cannot all be properly represented by probability distributions. Chen et al. (2003), Li et al. (2007), and Maxwell et al. (1998) used stochastic methods to simulate the uncertainties in groundwater flow and contaminant transport. Both used the Monte Carlo method for obtaining the simulation results.