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8. Numerical scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ever, under partially-saturated conditions, the choice of network elements is different. Upon drained of a pore body, the non-wetting phase occupies the bulk space of the pore. We consider each corner of the pore body as a separate network element (i.e., with its own pressure and concentration). Thus, for a cubic pore body, 8 different corner elements exist. Fluid flow and solute mass fluxes between these elements occurs through the 12 edges of the pore body (see Figure 6.3(a)). In the same manner, after invasion of the pore throat by the non-wetting, each edge of the pore throat will be a separate network element (i.e., with separate flow rate and concentration assigned to it). For example a pore throat with triangular cross section, upon drainage will break into three network elements, one for each edge flow. We applied a fully implicit numerical scheme for transport of adsorptive solute under (partially-) saturated conditions, undergoing both/either equilibrium and/or kinetic types of adsorption. Although defining edges of drained pores as separate network elements will increase accuracy of modeling, it will make the computational process heavier, since each network element will be an unknown variable during the simulation. Under such a conditions, applying an efficient numerical algorithm becomes very important. We have developed, and did modeling using efficient numerical algorithms which, though substitution of unknown variables, reduced the size of system of linear equations by a factor of at least three. This significantly decreased the computational times. In this chapter we first present numerical scheme for kinetic adsorption under saturated conditions and then we proceed with transport of non-adsorptive solutes well as adsorptive solutes under partially saturated conditions. 180
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 8.1: Nomenclature Abbr. Description Dimension number of pore throats flowing into the corner unit — i N ij edge number of edges within angular pore throat ij — (e.g., for triangular cross section, N ij edge = 3) N CU,i in,edge number of pore body edges (within the same pore — body) flowing into corner unit i ∆t discretized time step [T] c ij concentration of saturated pore throat ij [ML −3 ] c ij,k concentration of k th edge of drained angular pore [ML −3 ] throat ij V ij volume of saturated pore throat ij [L 3 ] V ij,k volume of k th edge of drained angular pore throat [L 3 ] ij q ij volumetric flow of saturated pore throat ij [L −3 T −1 ] q ij,k volumetric flow of k th edge of drained angular pore [L −3 T −1 ] throat ij s ij adsorbed mass concentration of saturated pore [ML −3 ] throat ij s sw ij,k adsorbed mass concentration at SW interface in [ML −3 ] k th edge of pore throat ij s aw ij,k adsorbed mass concentration at AW interface in [ML −3 ] k th edge of pore throat ij KD,ij sw upscaled distribution coefficient of SW interface in [-] pore throat ij KD,ij aw upscaled distribution coefficient of AW interface in [-] pore throat ij kd,ij,k sw pore-scale distribution coefficient of SW interface [L] in pore throat ij kd,ij,k aw pore-scale distribution coefficient of AW interface [L] in pore throat ij a sw ij,k specific surface area of SW interface in k th edge of [L −1 ] pore throat ij a aw ij,k specific surface area of AW interface in k th edge of [L −1 ] pore throat ij N tube in 181
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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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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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6. Dispersivity under Partially-Sat
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Page 228 and 229: Appendix C average velocity: ṽ =
Page 230 and 231: REFERENCES . . . . . . . . . . . .
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REFERENCES . . . . . . . . . . . .
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REFERENCES . . . . . . . . . . . .
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REFERENCES . . . . . . . . . . . .
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REFERENCES D.F. Yule and W.R. Gardn
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