Lagrangian Particle Methods for Biological Systems - Porous Media: Applications in Biological Systems and Biotechnology - page 281

Here
d a = D I ,a = k AB ,a

∈ Ω p ∈ Ω p

AB
a

,

I ,a

k AB
eff

D ∈ Ω c ,a ∈ Ω c

k a = 0 , ,k a = k, ,ω a = 1 ,a

a ∈ Ω p a ∈ Ω p ∈ Ω p

(7.50)

k eff ,a ∈ Ω c 0 ,a ∈ Ω c φ, a ∈ Ω c

The last term in (7.49) vanishes in the Darcy-scale domain, since the soil
grains and mineral surfaces are not explicitly represented there.
The symmetric form of the interactions between “pore scale” and “Darcy
scale” particles in equations (7.48-7.49) ensure the continuity of mass fluxes
across the interface between the Darcy-scale and pore-scale domains and
implicitly imposes the boundary condition (7.47). The parameters in the
hybrid model, defined by (7.50), characterize the continuum properties of the
porous media and/or properties of the solutions. These parameters can be
measured by standard laboratory methods or field experiments, and they are
tabulated for the wide class of soils and chemical compounds. The hybrid
model does not require any additional parameters beyond those used in the
Darcy-scale or pore-scale models and, hence, parameterization of the hybrid
model can be easily achieved.

7.5.9 Multiresolution Implementation of the Hybrid
Algorithm
To increase the computational eciency of the pore-scale/Darcy-scale hybrid
algorithm, a lower spatial resolution (increased spacing between adjacent par-
ticles in SPH models) was used in the continuum domain relative to its coun-
terpart in the pore-scale domain. Particles in both domains were placed on the
nodes of square lattices. In the discrete domain, Ω d , particles were placed on a
square lattice with a size ∆ x =0 . 25 (in model length units). In the continuum
domain, Ω c , the spacing between particles was increased from ∆ x =0 . 25, near
the boundary, Γ, to ∆ x = 1, away from Γ. The support length for each particle
was set at

h i =4∆ x i (7.51)

where ∆ x i is the size of the lattice on which particle i is located. This defines
a smoothing length over which each particle interacts with approximately 50
neighbors. Finally, h r was set to

h r = √ 2∆ x +0 . 01∆ x

(7.52)

so that only one layer of fluid particles can react with the solid particles
and vice versa. Upon setting h ab =( h a + h b ) / 2, this leads to the following