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3.2 Theoretical upscaling of adsorption in porous media
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∂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)
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