Biomedical Engineering Reference
In-Depth Information
equilibrium with the solid,
k
is the rate constant, and
β
is the order of the
heterogeneous reactions, is
dC
i
dt
1
m
i
(
D
i
n
i
m
i
+
D
j
n
j
m
j
)(
C
i
−
C
j
)
=
(
r
i
−
·∇
i
W
(
r
i
−
r
j
,h
)
r
j
)
n
i
n
j
(
r
i
−
r
j
)
2
j
∈
fluid
C
eq
)
β
k∈
solid
W
(
r
i
−
r
k
,h
)
n
k
l∈
solid
−
k
(
C
i
−
(7.33)
n
l
W
(
r
k
−
r
l
,h
)
Here,
γ
=2
/
3 for three-dimensional simulations,
γ
=
1
/
2
for two-dimensional
simulations, and
D
is the diffusion coecient. In equation (7.33) the symbol
j∈
fluid
indicates summation over all fluid particles and
j∈
solid
indicates
summation over all solid particles. The last term in equation (7.33) is pro-
portional to the rate of the mass loss/gain (per unit mass of solution) due
to the precipitation/dissolution. Mass conservation requires the rate of solid
mass gain/loss due to precipitation or dissolution to be
r
i
,h
)
i∈
fluid
dm
k
dt
k
C
eq
)
β
W
(
r
i
−
n
k
i∈
fluid
=
(
C
i
−
r
k
,h
)
(7.34)
n
l
W
(
r
k
−
Homogeneous reactions can be included by adding terms to the right-hand
side of equation (7.33). For example, if the solute is unstable or reacts with
itself, the term
rC
i
, where
α
is the order of the reaction, can be added to
the right-hand side of equation (7.33).
The masses,
m
i
, of the solid particles are tracked, and once the mass of a
solid particle,
m
k
, exceeds 2
m
k
, where
m
k
is the mass of the mineral phase
within a volume of 1/
n
k
(where
n
k
is the SPH particle number density of
the solid particle
i
), the nearest fluid particle “precipitates,” becoming a new
solid particle, and the mass of the new solid particle is set to
m
i
−
−
m
k
while
the mass of the old solid particle becomes
m
k
. Similarly, if the mass of a solid
particle reaches zero, the solid particle becomes a new fluid particle. Since the
fluid velocity adjacent to a solid surface is very small, the velocity of the new
fluid particle is set to zero and the concentration of the fluid particle is set to
the equilibrium concentration.
Figure 7.7 illustrates an SPH simulation of the injection of a supersaturated
solution into a fractured porous medium that was initially saturated with
a solution at equilibrium with the solid. In the simulation, the precipitate
eventually seals the fracture walls completely.
Figure 7.8 illustrates simulations of multicomponent reactive transport in a
two-dimensional porous medium. Solutions of
A
and
B
were injected into two
different halves of a porous medium, and the product,
C
, of the homogeneous
Search WWH ::
Custom Search