Biomedical Engineering Reference
In-Depth Information
area (length in two-dimensional simulations) associated with solid particle
b
.
The third term on the right-hand side of (7.41) represents the heterogeneous
reaction term obtained by discretization of the integral in equation (7.37),
where
W
(
x,h
r
) was used to approximate the Dirac delta function. In general,
the support lengths,
h
r
and
h
, of the smoothing function,
W
, in the reactions
and diffusion terms are not the same, and there is often good reason why they
should be different. Criteria that could be used to select the problem-specific
values for
h
r
and
h
are provided in the following section. The normalization
factor,
W
b
, guarantees that the total change of concentration due to the het-
erogeneous reaction is equal to
k
F
(
C
C
C
eq
)
da
.
The SPH particles representing solids are frozen in space. We neglect the
movement of fluid particles due to mineral precipitation, so that the hydro-
static conditions assumed in this study render fluid particles immobile (oth-
erwise, their dynamics can be described by an SPH discretization of the NS
equations).
To describe the evolution of the fluid-solid interfaces due to precipitation
and dissolution, we introduce “ghost” particles whose initial mass is zero and
whose initial locations coincide with those of fluid particles. The masses of
the ghost particles,
m
a
, change because of the heterogeneous precipitation
reaction according to (7.41) so that
−
=
k
(
C
A,
0
+
C
B,
0
)
b
dm
a
dt
V
b
W
−
1
∆
b
(
C
a
−
C
eq
)
W
(
x
ab
,h
r
)
(7.42)
b
∈
Ω
s
Once
m
a
reaches the prescribed solid particle mass,
m
0
, the ghost parti-
cle is converted into a solid particle and the corresponding fluid particle is
removed. Dissolution is modeled in a similar fashion by tracking the mass of
solid particles according to (7.42). When the mass of a solid particle reaches
zero, the solid particle is reclassified as a new fluid particle.
7.5.4 Darcy-Scale (Continuum) Description
On the Darcy scale, the porous medium is treated as a continuum, and equa-
tions (7.36-7.39) are replaced by a system of averaged coupled RDEs
∂φC
I
∂t
(
φD
I
C
I
)
φk
AB
eff
C
A
C
B
,I
=
A, B,
=
∇·
∇
−
x
∈
Ω
c
(7.43)
∂φC
C
∂t
(
φD
C
C
C
)+
φk
AB
eff
C
A
C
B
φk
eff
(
C
C
=
∇·
∇
−
−
C
eq
)
,
x
∈
Ω
c
(7.44)
where
D
I
(
I
=
A, B,C
) are the effective diffusion coecients;
φ
is the porosity
of the porous medium; and
k
AB
eff
and
k
eff
are the effective rate coecients for
the homogeneous and heterogeneous reactions, respectively. In principle, the
parameters
φ
,
D
I
,
k
AB
eff
, and
k
eff
are measurable quantities, which are assumed
to be known as long as the internal structure and topology of a porous medium
are not significantly affected by precipitation and/or dissolution.
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