Lagrangian Particle Methods for Biological Systems - Porous Media: Applications in Biological Systems and Biotechnology - page 276

defined over Ω p and these equations are subject to boundary conditions on
its boundary, ∂ Ω p . Darcy-scale models treat the porous medium as a contin-
uum, that is, the corresponding RDEs are defined at every point x

Ω T .In
a hybrid formulation, pore-scale simulations are carried out in Ω d , a (small)
portion of the porous medium, Ω T , while the Darcy-scale model is solved in
the remainder of the porous medium, Ω c (Ω T

∈

Ω d ). While the hybrid
algorithm developed in this study is applicable to a large class of multi-
component reaction-diffusion systems in porous media, we formulate it in
terms of mixing-induced heterogeneous precipitation. Here, we assume that
mixing-induced heterogeneous precipitation involves a homogeneous reaction
between two mixing solutes A and B , which forms a reaction product C
that grows heterogeneously from a supersaturated solution. For simplicity,
we disregard both the reverse homogeneous reaction and homogeneous nucle-
ation. (The latter approximation implies that the supersaturation index is
not large enough to support precipitation in the liquid phase.) Furthermore,
we assume that precipitation occurs only as a dense overgrowth on solid sur-
faces. In practice, the chemistry of precipitation and dissolution can be quite
complex. A high supersaturation may be required to initiate heterogeneous
precipitation, and a number of surface and solution species may play signif-
icant roles in the precipitation process. A simple alternative to the forma-
tion of intermediate C , which precipitates heterogeneously on the surface,
would be the separate incorporation of A and B , in equal amounts, into the
solid at the interface. The nonequilibrium thermodynamics approach to inter-
facial chemistry (Onsager 1931a,b; Prigogine 1947) indicates that the rate
of precipitation will depend on the supersaturation index, and both sim-
ple models for the precipitation chemistry can be expected to give similar
results.

=Ω c ∪

7.5.2 Pore-Scale Description and Its SPH Formulation
The subdomain, Ω d =Ω p ( t ) Ω s ( t ), in which the pore-scale simulations are

conducted, consists of the fluid-filled pore space, Ω p ( t ), and the solid matrix,
Ω s ( t ), with F ( t )=Ω s ( t )

∪

Ω p ( t ) denoting the corresponding (multiconnected)
fluid-solid interface. Precipitation/dissolution causes the pore geometry, that
is, Ω p ( t ), Ω s ( t ), and F ( t ), to change with time t .
Let C A ( x ,t ), C B ( x ,t ), and C C ( x ,t ) denote the concentrations in the sol-
vent of solutes A and B and the reaction product, C . The concentrations are
defined as the mass dissolved in a unit volume of fluid; the concentrations,
C A and C B , are normalized with the corresponding initial concentrations C A, 0
and C B, 0 ; and the concentration, C C , is normalized with C A, 0 + C B, 0 . The
precipitation process can be described by a system of coupled RDEs:

∩

∂C I
∂t

( D I C I ) k AB C A C B ,I = A, B, x

= ∇· ∇ − ∈ Ω p (7.36)