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6. Dispersivity under Partially-Saturated Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.0 var L var M var H vity [cm] Dispersiv 2.0 1.0 0.0 0.0 0.2 0.4 S 0.6 0.8 1.0 w Figure 6.9: The relation between dispersivity of the MIM model and the wetting phase saturation (α − S w) for three networks shown in Figure (6.1) The dispersivity values obtained using MIM (Figure 6.9) are lower than those obtained using the ADE model (Figure 6.6). This result has been observed by others [Toride et al., 2003]. The reason for lower dispersivities using MIM is that the contribution of mass transfer between mobile and immobile domains causes extra mixing. Compared to the ADE model, the MIM model fits the BTCs better under unsaturated conditions (results not shown). Functionally, the ADE model provides mean transport characteristics, e.g. average velocity, necessary to predict solute transport. ADE results will be appropriate provided it is possible to estimate effective (or average) values for the transport characteristics. However, under intermediate saturations, due to the high velocity variations, one single average velocity (such as the one used in the ADE model) may not sufficiently describe the mean advective flux within the system. Under such a condition, the MIM model gives better results, since it divides the pore-scale velocities into two groups, one with a non-zero average velocity and the other one with a velocity of zero. At saturated conditions the velocity field variation is narrower (i.e., with an smaller value for c.v.), and transport may be described well using the ADE. 6.5.3 Case study In this section, we use our pore-network model to simulate solute transport within a real porous medium. We have chosen data reported by Toride et al. [2003], who studied the hydrodynamic dispersion coefficient under unsaturated 150
6.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . conditions. In their experiment, they used non-aggregated dune sand to minimize the effect of immobile water. Unit-gradient flow experiments (i.e., the water flux being equal to the unsaturated conductivity) were conducted to measure solute BTCs under uniform saturations and steady-state flow conditions. Transport parameters for the ADE and MIM models were determined by fitting analytical solutions to the observed BTCs. They found a maximum dispersivity, α max , of 0.97 cm corresponding to S cr = 0.40, whereas for saturated flow dispersivity was equal to 0.1 cm, irrespective of pore-water velocity. The maximum value of dispersivity was almost ten times greater than it’s value under saturated conditions. The BTCs for unsaturated flow tended to be less symmetrical with considerable tailing compared with the BTCs for saturated flow. The ADE model described the observed data better for saturated than for unsaturated conditions. To model the experimental results, first, we have constructed a MDPN model, based on the information reported by Toride et al. [2003]. They have used a well-sorted dune sand, which was uniformly packed. The dune sand had an average particle size of 0.28 mm. The average coordination number for a sand packing is about 4.5 [Talabi and Blunt, 2010, Talabi et al., 2008]. We have used the same procedure mentioned in earlier sections to generate a MDPN model to match an average coordination number of 4.5 and calculate the BTCs under different saturations, and to plot the dispersivity-saturation relationship (α − S w curve) shown in Figure (6.10). Figure 6.10: Comparison between dispersivities calculated using the MDPN model and results based on experiments [Toride et al., 2003] for various degrees of saturation. Considering the complications of flow and solute transport under unsaturated 151
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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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Published as: Raoof, A. and Hassani
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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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4.3 Simulating flow and transport w
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4.4 Macro-scale adsorption coeffici
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4.5 Discussion . . . . . . . . . .
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4.5 Discussion . . . . . . . . . .
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4.6 Conclusions . . . . . . . . . .
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Raoof, A. and Hassanizadeh S. M.,
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5.1 Introduction . . . . . . . . .
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5.1 Introduction . . . . . . . . .
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5.2 Network Generation . . . . . .
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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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