Biomedical Engineering Reference
In-Depth Information
7.3.3 Implementation of the SPH Model
At each time step in a simulation, the particle number densities,
n
i,
at each
of the particles are calculated using equation (7.25) and the pressure at each
particle is obtained using the equation of state,
P
i
=
P
eq
n
i
/
n
eq
. In Model 1,
particles with nonzero biomass,
M
i
>
0 are assumed to be immobile. The
positions of fluid particles with
M
i
= 0 and new concentrations
A
,
B,
and
M
are found from (7.26) and (7.28-7.30) using the explicit “velocity Verlet”
algorithm (Allen and Tildesley 2001). In Model 2, all fluid particles with or
without biomass are considered to be mobile, and the new positions of the
fluid particles and new concentrations are found by explicit time integration
of equations (7.32) and (7.28-7.30). An
M
6
spline function (Schoenberg 1946)
was used for the SPH weighting function.
7.3.4 Numerical Results
To initialize the simulations, particles were placed randomly into a 16
32
box (in units of
h
), and the SPH equation (7.27), with
g
= 0, and periodic
boundary conditions in all directions were used to bring the system into an
equilibrium state. The equilibrium particle density was
n
eq
=19
h
−
2
(19 parti-
cles in an area of
h
2
). The coecient
P
eq
in the equation of state was
P
eq
=20
and the viscosity was
µ
i
= 1. Model units of time, length, and mass are used
in the descriptions of the simulations presented later. After equilibrium was
reached, the particles at positions
r
i
covered by the discs representing soil
grains were “frozen” to form impermeable boundaries to the flow. A fractured
porous medium was generated by randomly inserting nonoverlapping discs
with random radii on either side of the gap between two self-ane fractal
curves representing a microfracture. The initial concentrations of
A
and
B
were set to zero. A body force was then applied in the
y
direction. No-flow
boundary conditions were imposed at the boundaries of the computational
domain in the
x
direction, and periodic flow boundary conditions were used
in the
y
direction. Particles exiting the flow domain at
y
= 0 were returned
into the flow domain at
y
= 32.
The injection of electron donors and acceptors in different halves of the
computational domain was simulated by assigning concentrations
A
=1,
B
=0,
M
= 0 to the particles entering at
y
= 32 in the left part of domain
and
A
=0,
B
=1,
M
= 0 in the right part of the domain. The parameters
D
A
=
D
B
=0
.
5,
g
=[0
,
×
0
.
01],
K
A
=0
.
2,
K
B
=0
.
1,
Y
=0
.
07,
k
S
=0
.
1, and
k
d
=0
.
001 were used in the simulations. Figure 7.5 shows the steady-state dis-
tribution of biomass resulting from continuous injection of solutions containing
electron donors and acceptors for two different values of the critical stress,
τ
cr,
when Model 1 was used. For the smaller value of
τ
cr
the biomass grew pref-
erentially near the entrance of the fracture where the solutes were injected.
From the profiles of the product of concentrations
A
and
B
,
AB
, depicted in
Figure 7.5, it can be seen that the nutrient substrate becomes rapidly depleted
−
Search WWH ::
Custom Search