download pdf version of PhD book - Universiteit Utrecht

More documents

Recommendations

Info

7. Adsorption under Partially-Saturated Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . where, kd,ij,k sw and kaw d,ij,k [L] are pore scale adsorption distribution coefficients at SW and AW interfaces, respectively. The specific surface area, a αw ij,k , is defines as a αw ij,k = Aαw ij,k V ij,k where α = s, a (7.6) and A αw ij,k is the total area of the appropriate αw interface within the kth edge of pore throats ij. Combining Equations (7.1) through (7.5) results in a linear set of equations to be solved for c ij,k and c CU,i . Since we discretize pore bodies and pore throats on the basis of their saturation state (i.e., depending on that a pore is saturated or drained by the non-wetting phase), the number of unknowns are different for different simulations. In the case of a fully saturated domain, the number of unknowns is equal to N tube + N node (N tube is the total number of pore throats and N node is the total number of pore bodies). In general, the number of pore throats is larger than the number of pore bodies in pore network models. In order to increase numerical efficiency, first, applying a fully implicit scheme, we discretized Equation (7.5) and determined c ij,k in terms of c CU,i . This was then substituted into the discretized form of Equation (7.1), resulting in a set of equations for c CU,i . This method considerably reduced the number of unknowns, and thus the computational times. The details of the method are given in Chapter 8, Section 8.3.2. For the accuracy of the scheme, the minimum time step was chosen on the basis of residence times [Suchomel et al., 1998a, Sun, 1996] ∆t ≤ min {T ij , T ij,k , T CU,i , T i } (7.7) where T denotes the residence time pertaining to different elements within the pore network. After obtaining the solution at any given time, BTCs at a given longitudinal position were found by averaging the concentrations of pore bodies that possess the same longitudinal coordinate. In calculating BTCs, the concentrations of pore bodies were weighted by their volumetric flow rate, resulting in a fluxaveraged concentration. That is, the normalized average concentration, c(x, t), given by c(x, t) = [ ∑N x t i ∑ N x t ] c i (x, t)Q i i Q i where c 0 is inlet solute concentration, and N x t 168 1 c 0 i = 1, 2, 3, . . . , N t (7.8) denotes the total number of
7.5 Macro-scale formulations of solute transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pore body elements that are located at the longitudinal coordinate x. The longitudinal coordinate can be written as multiples of lattice size l, i.e. x = 1l, 2l, . . . , L. where l is the horizontal distance between centers of two adjacent pore bodies. The breakthrough curve at the outlet is obtained by plotting c(x = L, t). 7.5 Macro-scale formulations of solute transport At the macro scale we consider two models: Advection-Dispersion Equation (ADE) with adsorption, and a two-site kinetic model. Indeed, the nonequilibrium transport model could be in terms of alternative physical or chemical nonequilibrium models. For the ADE model, the dispersion coefficient, D, and retardation factor, R, are the only parameters to be estimated since average flow velocity is obtained directly from the corresponding pore network model simulations. For the nonequilibrium model, in addition to the dispersion coefficient, D, and the retardation factor, R, the coefficient of partitioning between the equilibrium and nonequilibrium sites, f, and the mass transfer coefficient, ω, for transfer between the mobile and immobile zones need to be determined. 7.5.1 Advection-Dispersion Equation (ADE) The transport of a adsorptive solute through a porous medium may be described by the hydrodynamic dispersion theory [Bear, 1972]. The one-dimensional transport equation for a adsorptive solute is: R ∂θC ∂t = ∂ ∂z ( θD ∂C ) − ∂qC ∂z ∂z (7.9) where C is the solute concentration, θ is the water content of the porous medium, q is the water flux, D is the hydrodynamic longitudinal dispersion coefficient, R is the retardation factor, and z is the distance. Dispersion coefficient (and thus dispersivity) at a given saturation could be determined by fitting the analytical solution of ADE to the BTCs of average concentration at the outlet of the network at that saturation. 169
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 24 and 25:
1. Introduction . . . . . . . . . .
Page 26 and 27:
1. Introduction . . . . . . . . . .
Page 28 and 29:
1. Introduction . . . . . . . . . .
Page 30 and 31:
1. Introduction . . . . . . . . . .
Page 32 and 33:
1. Introduction . . . . . . . . . .
Page 34 and 35:
1. Introduction . . . . . . . . . .
Page 37:
Part I Generation of Multi-Directio
Page 40 and 41:
2. Generating MultiDirectional Pore
Page 42 and 43:
Page 44 and 45:
Page 46 and 47:
Page 48 and 49:
Page 50 and 51:
Page 52 and 53:
Page 54 and 55:
Page 56 and 57:
Page 58 and 59:
Page 61 and 62:
Published as: Raoof, A. and Hassani
Page 63 and 64:
3.1 Introduction . . . . . . . . .
Page 65 and 66:
3.1 Introduction . . . . . . . . .
Page 67 and 68:
3.2 Theoretical upscaling of adsorp
Page 69 and 70:
Page 71 and 72:
Page 73 and 74:
3.3 Numerical upscaling of adsorbin
Page 75 and 76:
Page 77 and 78:
Page 79 and 80:
Page 81 and 82:
3.4 Discussion of results . . . . .
Page 83 and 84:
3.5 Conclusion . . . . . . . . . .
Page 85 and 86:
Published as: Raoof, A. and Hassani
Page 87 and 88:
4.1 Introduction . . . . . . . . .
Page 89 and 90:
4.1 Introduction . . . . . . . . .
Page 91 and 92:
4.2 Description of the Pore-Network
Page 93 and 94:
4.3 Simulating flow and transport w
Page 95 and 96:
4.3 Simulating flow and transport w
Page 97 and 98:
4.4 Macro-scale adsorption coeffici
Page 99 and 100:
Page 101 and 102:
Page 103 and 104:
4.5 Discussion . . . . . . . . . .
Page 105 and 106:
4.5 Discussion . . . . . . . . . .
Page 107:
4.6 Conclusions . . . . . . . . . .
Page 111 and 112:
Raoof, A. and Hassanizadeh S. M.,
Page 113 and 114:
5.1 Introduction . . . . . . . . .
Page 115 and 116:
5.1 Introduction . . . . . . . . .
Page 117 and 118:
5.2 Network Generation . . . . . .
Page 119 and 120:
5.2 Network Generation . . . . . .
Page 121 and 122:
5.3 Modeling flow in the network .
Page 123 and 124:
Page 125 and 126:
Page 127 and 128:
Page 129 and 130:
Page 131 and 132:
5.4 Results . . . . . . . . . . . .
Page 133 and 134:
5.4 Results . . . . . . . . . . . .
Page 135 and 136:
5.4 Results . . . . . . . . . . . .
Page 137 and 138: 5.4 Results . . . . . . . . . . . .
Page 139 and 140: 5.4 Results . . . . . . . . . . . .
Page 141 and 142: 5.4 Results . . . . . . . . . . . .
Page 143: 5.5 Conclusion Our approach is also
Page 146 and 147: 6. Dispersivity under Partially-Sat
Page 178 and 179: 7. Adsorption under Partially-Satur
Page 198 and 199: 8. Numerical scheme . . . . . . . .
Page 214 and 215: 8. Numerical scheme substituting fo
Page 216 and 217: 9. Summary and Conclusions . . . .
Page 222 and 223: Appendix A . . . . . . . . . . . .
Page 224 and 225: Appendix A . . . . . . . . . . . .
Page 226 and 227: Appendix B . . . . . . . . . . . .
Page 228 and 229: Appendix C average velocity: ṽ =
Page 230 and 231: REFERENCES . . . . . . . . . . . .
Page 238 and 239:
REFERENCES . . . . . . . . . . . .
Page 240 and 241:
REFERENCES . . . . . . . . . . . .
Page 242 and 243:
REFERENCES . . . . . . . . . . . .
Page 244 and 245:
REFERENCES . . . . . . . . . . . .
Page 246 and 247:
REFERENCES . . . . . . . . . . . .
Page 248 and 249:
REFERENCES . . . . . . . . . . . .
Page 250 and 251:
REFERENCES . . . . . . . . . . . .
Page 252 and 253:
REFERENCES . . . . . . . . . . . .
Page 254 and 255:
REFERENCES . . . . . . . . . . . .
Page 256 and 257:
REFERENCES D.F. Yule and W.R. Gardn
Page 258:
و ‏ه ‏د ‏ر ش ل ات
show all

download pdf version of PhD book - Universiteit Utrecht

Create successful ePaper yourself

Delete template?

Save as template?