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1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . our formulation in a MDPN model. Resulting saturation-relative permeability relationships show very good agreement with measured curves. Chapter 6 is devoted to the study of the dispersion coefficients under partially saturated conditions using a new formulation. It is known that in unsaturated porous media, the dispersion coefficient depends on the Darcy velocity as well as saturation. The dependency of dispersion on velocity is fairly studied, however, there is not much known about its dependence on saturation and the underlying process. The purpose of this chapter is to investigate how the longitudinal dispersivity varies with saturation. We have represented the porous medium with our MDPN. Both pore bodies and pore throats have volumes and we assign separate concentrations to each of them. Further, since pore geometry and corner flows greatly influence transport properties, efforts are made to include different angular cross sectional shapes for the pore throats. This includes circular, rectangular, and irregular triangular cross sections, which are important especially under unsaturated/two-phase flow and reactive transport. After the construction of the pore network, dispersivity was calculated by solving the mass balance equations for solute concentration in all network elements and averaging the concentrations over a large number of pores. We have introduced a new formulation of solute transport within a pore network which helps to capture the effect of limited mixing under partially-saturated conditions. In this formulation we refine the discretization on the basis of the saturation state of pores. We assign separate concentrations to different corners of a given drained pore body and also assign different concentrations to different corners of a drained angular pore throat. This formulation allows a very detailed description of pore-scale solute transport processes by accounting for limitations in mixing as a result of reduced water content. The numerically computed dispersivities successfully explain the results obtained through experimental studies, and show the underlaying pore scale processes contributing to dispersion under unsaturated conditions. Chapter 7 deals with transport of adsorptive solute under partially saturated conditions. All of the modeling capabilities developed in previous chapters are included in this chapter. Compared to column scale experimental studies on adsorptive transport, there is lack of pore scale modeling studies, especially under unsaturated porous media. Under unsaturated conditions, the system contains three phases: air, water, and solid. The principal interactions usually occur at the solid-water interfaces (sw) and air-water interfaces (aw) and are thus greatly influenced by water content. In this chapter, we have for- 12
1.6 Programming issue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . mulated various types of adsorption: i) two site (sw and aw interfaces) kinetic; ii) two site equilibrium; and iii) one site (sw or aw interfaces) kinetic and one site equilibrium. To numerically solve the mass balance equations, we have applied a fully implicit numerical scheme for transport of adsorptive solute under unsaturated conditions. Through applying an efficient numerical algorithm, we have reduced the size of system of linear equations by a factor of at least three, which significantly decreases computational time. 1.6 Programming issue CPNS: Complex Pore Network Simulator Through this study to generate pore networks and accurately simulate fluid flow and transport of reactive/adsorptive solute, we have developed an advanced FORTRAN 90 modular package. All the above mentioned formulations and capabilities are included in CPNS, which enables one to simulate fluid flow and transport of adsorptive/reactive solutes under both saturated and partially saturated conditions to upscale from pore scale to the core scale. For this study, three programming languages are used: Visual Basic Application (VBA) in Excel, MATLAB programming, and FORTRAN 90 programming. The inputs for the model are inserted into Excel sheets which are then recorded as input files using Excel VBA to be used by the FORTRAN simulator. Throughout the execution of the model, data and results are transferred to MATLAB using MAT- LAB Engine for possible analysis and post processing. Since various physical (saturated and unsaturated) and chemical (tracer, equilibrium adsorption, kinetic adsorption, system of reactions) conditions should be simulated, to keep numerical commutations efficient, we designed CPNS as a modular package composed of separate interchangeable components, each of which accomplishes one mode of simulation. Figure (1.2) shows the flowchart of CPNS. The last version of CPNS (not included in this thesis) is capable of simulating transport of multi-competent chemical species undergoing equilibrium and/or kinetic reactions [Raoof et al., 2011]. Both advective and diffusive transport processes are included within the pore spaces. To simulate chemical reactions, MDPN is coupled with the Biogeochemical Reaction Simulator (BRNS) [Regnier et al., 2002, 2003], which performs the reaction part. This gives major advantages for simulation of complicated system of reactions. The coupling between transport and reaction parts is done though a non-iterative sequen- 13
Page 1 and 2: Reactive/Adsorptive Transport in (P
Page 3: text Reactive/Adsorptive Transport
Page 6 and 7: Promoter: Prof.dr.ir. S.M. Hassaniz
Page 8 and 9: My colleagues and other staff in th
Page 10 and 11: CONTENTS 2.2.3 Connection Matrix .
Page 12 and 13: CONTENTS 5.3.3 Regular hyperbolic p
Page 14 and 15: CONTENTS 8.3.1 Non-adsorptive solut
Page 16 and 17: LIST OF FIGURES 3.6 The relation be
Page 18 and 19: LIST OF FIGURES 6.5 Breakthrough cu
Page 20 and 21: LIST OF TABLES 2.1 Expressions for
Page 22 and 23: 1. Introduction . . . . . . . . . .
Page 37: Part I Generation of Multi-Directio
Page 40 and 41: 2. Generating MultiDirectional Pore
Page 61 and 62: Published as: Raoof, A. and Hassani
Page 63 and 64: 3.1 Introduction . . . . . . . . .
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Page 67 and 68: 3.2 Theoretical upscaling of adsorp
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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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5.2 Network Generation . . . . . .
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5.3 Modeling flow in the network .
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5.4 Results . . . . . . . . . . . .
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5.4 Results . . . . . . . . . . . .
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5.4 Results . . . . . . . . . . . .
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5.4 Results . . . . . . . . . . . .
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5.4 Results . . . . . . . . . . . .
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5.4 Results . . . . . . . . . . . .
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5.5 Conclusion Our approach is also
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6. Dispersivity under Partially-Sat
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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 . . . . . . . . . . . .
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REFERENCES D.F. Yule and W.R. Gardn
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