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3. Upscaling of Adsorbing Solutes; Pore Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . is formulated at the pore scale utilizing equilibrium adsorption. Then the equations are averaged, and upscaled equations are obtained. As a result, explicit expressions are derived for the mass exchange between fluid and solid phases in terms of average concentrations. Next, steady state flow and transient adsorption in a single tube are simulated numerically. The resulting breakthrough curves from this single-tube model are compared to the solution of the 1-D, continuum scale, transport equation to estimate upscaled adsorption parameters. 3.2 Theoretical upscaling of adsorption in porous media 3.2.1 Formulation of the pore-scale transport problem Consider the transport of a solute in the pore space of a granular soil. Processes affecting transport are considered to be advection, diffusion, and chemical reaction within the water phase, plus adsorption to the solid phase at the pore boundaries. The general form of the equation of mass balance for the solute is: ∂c i ∂t + ∇ · (c i v ) + ∇ · j i = ̂r i (3.1) where: c i (ML −3 ) is the mass concentration of solute i in a pore; v (LT −1 ) is the interstitial water velocity; ̂r i (ML −3 T −1 ) is the rate of chemical reaction with other solutes, and j i (ML −2 T −1 ) is the diffusive flux of the solute. The diffusive flux, j i , is given by Fick’s first law: j i = −D i 0∇c i (3.2) where D0 i [L 2 T −1 ] is the molecular diffusion coefficient of solute i in water. In principle, one should solve this equation within the pore space of the soil subject to boundary conditions. At a point on the pore boundary, adsorption causes a flux of solute from the fluid to the solid phase; this gives rise to an increase in the mass density of adsorbed solutes. The rate of adsorption is equal to the solute mass flux normal to the pore boundary, ( c i v + j i)·n. Thus, the following condition at the pore boundary holds 46
3.2 Theoretical upscaling of adsorption in porous media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∂s i ∂t = (ci v + j i ).n| s (3.3) where: s i [ML −2 ] is the mass of adsorbed solute per unit area of the solid grains, n is the unit vector normal to the pore wall; and | s denotes evaluation of the preceding quantity within the pore but at the solid surface. Because s i is unknown, an additional equation is needed in order to have a determinate system. That extra equation comes from the continuity of chemical potential at the grain surface. This condition leads to an equilibrium relationship, a linear approximation of which yields s i = k i D c i∣ ∣ s (3.4) where c i∣ ∣ s is the solute concentration of fluid at the pore wall and kD i [L] is an equilibrium, pore-scale distribution coefficient. The set of Equations 3.1 through 3.4, together with conditions at the outside boundaries of the porous medium and an appropriate set of initial conditions, completely specify the solute transport problem at the pore scale. 3.2.2 Averaging of pore-scale equations We would like to upscale Equations (3.1)-(3.4) to the macro scale. To do so, we need to define an averaging volume, commonly denoted as the representative elementary volume (REV) [Bachmat and Bear, 1987]. This is a well known concept and has been extensively discussed in the porous medium literature (see, e.g., Bear [1988], Bachmat and Bear [1986], Hassanizadeh and Gray [1979]). Denote the volume occupied an REV by V . Volume V is, in turn, composed of two subvolumes: V f occupied by the fluid phase and V s occupied by the solid grains. The boundary of solid grains is denoted A fs ; the superscripts, f and s, are used to denote the fluid and solid phases, respectively. Note that the averaging volume V is taken to be invariant in time and space, whereas the subvolumes V f and V s may vary both in time and space. We shall integrate the fluid Equation (3.1) over V f , and the interface condition (Equation 3.3) will be integrated over A fs . First, we need to define the following average properties: average mass concentration c i ≡ 1 V f ∫ V f c i dV ; (3.5) 47
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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
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
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Page 63 and 64: 3.1 Introduction . . . . . . . . .
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Page 69 and 70: 3.2 Theoretical upscaling of adsorp
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Page 81 and 82: 3.4 Discussion of results . . . . .
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Page 91 and 92: 4.2 Description of the Pore-Network
Page 93 and 94: 4.3 Simulating flow and transport w
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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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