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6. Dispersivity under Partially-Saturated Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . performed to determine pore-level distribution of each phase. Then, steadystate flow is established and equations of mass balance are solved to calculate transport properties of such a distribution and to obtain BTCs of solute concentration. The results of the modeling are also compared with experimental observations to show the capability of this formulation. The following morphological and computational features are introduced in the present study (1) The topology of the pore space is modeled using a MDPN which allows a distribution of coordination numbers ranging between one and 26. (2) To take into account the angularity of pores in natural porous media, pore throats with various cross sections, with a wide range of shape factor values and pore sizes, are used in the network. This includes rectangular, circular, and various irregular triangular cross sections. (3) The pore body size distributions are assumed to follow a truncated lognormal distribution, without any correlation. The pore-throat size distributions are related to the pore body size distributions. (4) Both pore bodies and pore throats are considered to have volume. This means we solve mass balance equations and calculate solute concentrations and mass fluxes within both pore bodies and pore throats. (5) As soon as a pore body is (partially) saturated, it will be discretized further into smaller regions occupied by water, each with its own flow rate and concentration, in order to capture the effect of limited mixing due to the partial filling of the pore. (6) Upon invasion of a pore throat by the non-wetting phase, each edge of the pore throat will be considered as a separate domain with its own flow rate and concentration. (7) Employing a fully implicit numerical scheme to calculate the unknown connections, a substitution method is introduced which considerably reduces the computational time. (8) Various parameters and relations, including coefficient of variation of the velocities field, relative permeability-saturation (k r − S w ) curves, capillary pressure-saturation (P c − S w ) curves, and fraction of percolating saturated pores are also computed. 132
6.2 Network Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Since we are employing angular pores, the wetting phase is always connected through the edges. This, together with the fact that we are ignoring the effect of diffusion, means we do not have any tailing effect, as there is no diffusive mass flux from zones of immobile water to the domain of mobile water. However, there are many pores with velocities much smaller than average velocity, in which the fluid is practically immobile. It has been shown that the effect of diffusion under unsaturated condition is mostly negligible. For example, Maraqa et al. [1997] found the relative contribution of molecular diffusion to the value of dispersion coefficient to be less than 1.0%. This is of course not the case for the saturated experiments with low pore-water velocities (e.g., velocities around 1 cm/h or less), where there is a potential contribution of molecular diffusion to the value of the hydrodynamic dispersion coefficient [Maraqa et al., 1997]. 6.2 Network Generation 6.2.1 Pore size distributions In the present study, the pore structure is represented using a 3D MDPN model. Because natural porous media can be mostly described by a lognormal distribution [Bear, 1988], the pore-body radii are assigned from such a distribution, with no spatial correlation, expressed by: f (R i , σ) = √ 2 exp [ [ ( √ πσ2 R i erf − 1 2 ln Rmax Rm √ 2σ 2 ( ) ] ln R i 2 Rm σ ( ) − erf ln R min Rm √ 2σ 2 )] (6.6) where R min , R max , and R m , are the minimum, maximum, and mean of the distribution, respectively; and σ 2 is the variance. The pore structure is constrained to be isotropic, in the sense that the same values for the R parameters σ 2 are specified for all pores oriented along all network directions. In this study, four different networks were constructed: three generic networks, and one network which represented a specific porous medium. The three generic networks had different distributions of pore body and pore throat sizes, shown in Figure (6.1), but the same coordination number distribution (figure 6.2). Pore body radii are taken from uncorrelated truncated lognormal distributions, 133
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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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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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4.4 Macro-scale adsorption coeffici
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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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