Biomedical Engineering Reference
In-Depth Information
The left column in Figure 7.9 shows three temporal snapshots of the pore-
scale SPH simulations of this reaction-diffusion system for
L
= 31,
B
=8,
D
A
=
D
B
=
D
C
=0
.
5,
k
AB
=1
.
5, and
k
= 20. The precipitation of the reaction
product,
C
, (the black particles in Figure 7.9) modifies the structure of the
porous medium, thus leading to changes in its effective properties, including
porosity, effective diffusion coecient, and effective reaction rates. As should
be expected, these processes occur only in the narrow reaction zone separating
the solutions of the two reactants,
A
and
B
.
7.5.13 Hybrid Simulations
The presence of a small region within a computational domain in which a
coarse-grain description breaks down is, of course, the
raison d'etre
for a
hybrid algorithm. Since evolution of the pore geometry cannot be accurately
described on the Darcy (continuum) scale, we employed the hybrid pore-
scale/Darcy-scale algorithm. It combines a pore-scale simulation in the cen-
tral part of the computational domain with a continuum reaction-diffusion
description elsewhere (the right column in Figure 7.9).
To parameterize the Darcy-scale model, we obtained a steady-state finite-
element solution of the one-dimensional version of the Darcy-scale diffu-
sion equation subject to the boundary conditions
C
A
(0)=1,
C
A
(
L
)=0,
C
B
(0)=0,
C
B
(
L
)=1, and
C
C
(0) =
C
C
(
L
) = 0. This solution was fitted
to the
y
-averaged steady-state SPH solution of the corresponding pore-scale
model with
L
= 31,
B
=8,
D
=0
.
5,
k
AB
=1
.
5, and
k
= 20 to yield the
effective transport parameters
D
I
=0
.
31 (
I
=
A, B, C
),
k
AB
eff
=0
.
98, and
k
eff
=3
.
3.
The right column in Figure 7.9 presents the results of the hybrid pore-
scale/Darcy-scale simulations. Figure 7.10 shows the temporal evolution of
the porosity of the portion of the porous medium affected by precipitation,
x
[8
,
24], obtained with the pore-scale and hybrid simulations. It can be seen
that the two are in a close agreement.
In the examples presented here, the hybrid algorithm allowed the number
of SPH particles to be reduced from 4,096 in the pore-scale simulations over
the whole computational domain to 2,416 in the hybrid pore-scale/Darcy-scale
simulations. In these simulations, the domain of the pore-scale simulations
occupied half of the total computational domain. In practical applications of
the hybrid model, this ratio is expected to be orders of magnitude smaller,
so that the savings in the number of particles is expected to be substan-
tially larger. The savings can be increased further by reducing the number of
particles used to discretize the continuum (Darcy-scale) subdomains as the
distance from the nearest pore-scale domain increases. Since the number of
operations in SPH simulations increases linearly with the number of parti-
cles, the hybrid algorithm allows for a significant reduction in computational
time.
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