Effect of Scale - Transport and Fate of Chemicals in Soils: Principles and Applications - page 136

D D − 1

B ′′ =− +β− 1 v av

fr

( j + 1)

fr

γ (C5)

3

1
2

D D − 1

Ca vj ′′ =

fr

( ) +

fr

γ− β v

(C6)

3

1
2

( )

DD − 1

j j j j

Ec ′′ =− − av

fr

j

fr

γ −+ c 2 cc

(C7) i 3 i + 1 i i − 1

Obviously, Equation C3 reduces to Equation A5 for D fr = .
For this case, the upper boundary conditions after discretization are
given by:

D − 1 D − 1

t

fr

t

fr

D − 1 D − 1 D − 1 D − 1

j + 1 j + 1

1 + av

fr

( j + 1)

fr

c − a v

fr

( ) j +

fr

c = c ,

3 1 3 2 0

x x

(C8)

0( 1) ≤+ ≤ j t T

D − 1 D − 1

t

fr

t

fr

D − 1 D − 1 D − 1 D − 1

j + 1 j + 1

1 + av

fr

( j + 1)

fr

c − a v

fr

( ) j +

fr

c = 0,

3 1 3 2

x x

(C9)

( ) j +> t T

where c 0 is the solute concentration in input pulse, and T is the pulse duration.

4.10 Appendix D: Derivations of Finite-Difference
Equations for CDE with a Nonlinearly
Distance-Dependent Dispersivity
For a parabolic dispersivity-distance model, the governing equation is
(Equation 4.28):

2

∂
∂

c
t

∂
∂ −− −

c
x

∂
∂

c
x

D − 1 D − 1

= avx

fr

1( aD 1) x

fr

v

(D1) 4 4 fr

2

For convenience, we let

t
x

t

β= , ξ=

3 − D fr

x