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3. Upscaling of Adsorbing Solutes; Pore Scale
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average flow velocity:
v f ≡ 1
V f ∫
V f vdV ; (3.6)
Note that this volume-averaged definition of the flow velocity is admissible only
if the fluid mass density variations are small (see Hassanizadeh and Gray, 1979,
for discussion);
average diffusion flux
average chemical reaction rate
and average adsorbed mass fraction
j i ≡ 1
V f ∫
r i ≡ 1
V f ∫
s i ≡ 1
ρ s V s
V f j i dV ; (3.7)
V f r i dV ; (3.8)
∫
A fs s i dA, (3.9)
where ρ s [ML −3 ] denotes the solid mass density.
Note that the average mass density of the sorbed solute is now defined in the
form of mass fraction (mass of solute per unit mass of grains), s i [MM −1 ], as
is common in solute transport.
To upscale Equation (3.1), we need to integrate it over V f . To do so, we
need averaging theorems which relate the average of a derivative to the derivative
of the average [Whitaker, 1969, 1986, Hassanizadeh et al., 1986, Gray
and Hassanizadeh, 1998]. Averaging of Equation (3.1) over V f results in the
(macro-scale) Equation (3.10). Details of averaging are given in Appendix C.
∂nc i
∂t
+ ∇ · (nc i v ) + ∇ · (nJ i) = nr i − Û i , (3.10)
where v is the average flow velocity, and r i is the reaction rate; the macro-scale
adsorption term, Û i [ML −3 T −1 ], is defined by
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