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6. Dispersivity under Partially-Saturated Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The numerically-computed dispersivities are compared with the results from reported experimental studies. The agreement between the results demonstrates the capability of this formulation to properly produce the effect of saturation on solute dispersion. While these computations have been restricted to the flow and transport of inert solutes and determination of their dispersivities, they demonstrate the significant potential of this formulation of pore-network modeling for predicting other transport properties, such as mass transfer coefficients under reactive/adsorptive solute transport, through including limited mixing within pores. 6.1 Introduction Mechanical dispersion in porous media occurs because water flow velocity varies in magnitude and direction as a result of meandering through the complex pore structure [Perfect and Sukop, 2001]. The degree of spreading is related to: distribution of the water velocity within the pores; the degree of solute mixing because of convergence and divergence of flow paths; and molecular diffusion [Bolt, 1979, Leij and van Genuchten, 2002]. 6.1.1 Dispersion under unsaturated conditions The most established model for describing solute transport in porous media is the Advection-Dispersion Equation (ADE), which can be used to model the porous medium as a single-porosity domain. However, when there is preferential transport by a secondary pore system, another theoretical description must be used. For the latter case, several models exist, including a two-domain approach for both water and solute transport [Gerke and van Genuchten, 1993], and the mobile-immobile model for solute transport [Smet et al., 1981]. When observed breakthrough curves (BTCs) in unsaturated porous media show a tailing effect, a mobile-immobile model can be applied successfully [Gaudet et al., 1977]. In saturated porous media, the longitudinal dispersion coefficient has often been expressed [Bear, 1972, Scheidegger, 1961, Freeze and Cherry, 1979] D = D e + αv n (6.1) where the first term, D e [L 2 T −1 ], is the effective diffusion coefficient, while the second term describes the coefficient of hydrodynamic dispersion, where α is 126
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the (longitudinal) dispersivity, v [LT −1 ], denotes the pore-water velocity, and n is an empirical constant. Often, D exhibits an almost linear dependence with pore-water velocity, (i.e., n = 1), especially in the case of non-aggregated soil or glass beads [Bear, 1972, Bolt, 1979]. The dispersivity, α, is assumed to be an intrinsic soil property for saturated flow. Equation (6.1) has been also used to describe the dispersion coefficient for solute transport under unsaturated conditions, (e.g., Kirda et al. [1973], Yule and Gardner [1978] and De Smedt et al. [1986]) in a modified form. A commonly employed relationship is D(θ, v) = D e (θ) + α(θ)v n (6.2) Hydrodynamic dispersion in unsaturated soils is more complicated than in saturated soils. The resulting BTCs from experiments show greater solute spreading and longer tailing at lower water contents compared to the saturated conditions [Gupta et al., 1973, Krupp and Elrick, 1968]. High values of the dispersion coefficient, which have been observed in unsaturated experiments [Kirda et al., 1973, De Smedt and Wierenga, 1984, Maraqa et al., 1997, Matsubayashi et al., 1997, Padilla et al., 1999], have been attributed to the presence of immobile water [Gaudet et al., 1977, De Smedt and Wierenga, 1979, 1984]. However, dispersion phenomenon of solute movement through unsaturated porous medium has not been fully studied, and there is no single theory which can unify dispersion under both saturated and unsaturated conditions. Although dispersion is known to be strongly dependent on both flow velocity and water content in soil columns [Maciejewski, 1993], there is not much information on the nature and functional form of such dependency [Bear and Alexander, 2008]. This need prompts systematic investigations of the relationships between saturation, and water velocity fluctuations and dispersivity in unsaturated porous media. The fraction of immobile water (or mobile water with a very low velocity compared to the average pore-water velocity) depends on the pore structure as well as the saturation. When saturation decreases, flow paths will be longer and the arrival time distribution will be more diverse, which causes higher dispersion. Consequently, the variability of microscopic velocity and its direction can be (much) larger than in saturated porous media. These effects, observed in soil column experiments, have been subjected to different interpretations, e.g. by describing molecular diffusion as a function of water content [Nielsen and Biggar, 1960, Bresler, 1973, Kemper, 1966]. As water content 127
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Reactive/Adsorptive Transport in (P
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text Reactive/Adsorptive Transport
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Promoter: Prof.dr.ir. S.M. Hassaniz
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My colleagues and other staff in th
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CONTENTS 2.2.3 Connection Matrix .
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CONTENTS 5.3.3 Regular hyperbolic p
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CONTENTS 8.3.1 Non-adsorptive solut
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LIST OF FIGURES 3.6 The relation be
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LIST OF FIGURES 6.5 Breakthrough cu
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LIST OF TABLES 2.1 Expressions for
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1. Introduction . . . . . . . . . .
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1. Introduction . . . . . . . . . .
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1. Introduction . . . . . . . . . .
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1. Introduction . . . . . . . . . .
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1. Introduction . . . . . . . . . .
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1. Introduction . . . . . . . . . .
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1. Introduction . . . . . . . . . .
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Part I Generation of Multi-Directio
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2. Generating MultiDirectional Pore
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Published as: Raoof, A. and Hassani
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3.1 Introduction . . . . . . . . .
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3.1 Introduction . . . . . . . . .
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3.2 Theoretical upscaling of adsorp
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3.3 Numerical upscaling of adsorbin
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3.4 Discussion of results . . . . .
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3.5 Conclusion . . . . . . . . . .
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4.1 Introduction . . . . . . . . .
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4.1 Introduction . . . . . . . . .
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4.2 Description of the Pore-Network
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4.3 Simulating flow and transport w
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7. Adsorption under Partially-Satur
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8. Numerical scheme . . . . . . . .
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8. Numerical scheme substituting fo
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9. Summary and Conclusions . . . .
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Appendix A . . . . . . . . . . . .
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Appendix A . . . . . . . . . . . .
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Appendix B . . . . . . . . . . . .
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Appendix C average velocity: ṽ =
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REFERENCES D.F. Yule and W.R. Gardn
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