Download Tortuosity-porosity relation in the porous media flow

Tortuosity of Sediments:
A Mathematical Model
Maciej Matyka
Arzhang Khalili
Institute of Theoretical Physics
University of Wrocław
Poland
Max Planck Institute
for Marine Microbiology,
Germany
Zbigniew Koza
Institute of Theoretical Physics
University of Wrocław
Poland
Mathematical Modeling Group
2. Methodology
1. Introduction
Permeability k and porosity

are two important physical properties of marine sediments.
Another physical characteristics is tortuosity, which affects all diffusive and dynamic processes
involved in marine sediments, or generally speaking, in porous media. From hydrodynamic
We use D2Q9 Lattice Boltzmann BGK model
for the creeping flow problem:
 
t 
t 
n (r  ci )  ni (r )   i (r )  Fi
t 1
i
Transport equation:
point of view, tortuosity T in a fluid medium would be unity if a particle could travel a distance
on a stricktly horizontal pathline. However, due to the existence of solid matrices in a porous
Equilibrium density function:

 
n (u ,  )  i [1  3(u  ci )]
medium, the pathlines become tortuous, which leads to T>1. The more `wavy' the pathlines
eq
i
become, the larger the tortousity. As one can not easily measure tortousity directly, and
because, it plays an important role in almost all geophysical and geochemical transport
processes, it has been the subject of intensive research. The question is whether or not
ni - distribution function (DF)
ti - collision operator
nieq - equilibrium DF
 i - lattice weights

u - velocity (macro)
Fi - body force (i.e. gravity)

ci - lattice vectors (i=0..8)
tortuosity can be described as a function of porosity.
 - density (macro)
Weighted average of pathline lengths:
3. Results
1
T
L
 u ( x) ( x)dx
 u ( x)dx
A
y
A
y
Simplifies for flux averaged spreading:
k
1 1 N
T
(x j )

L N j 1
4. Conclusion
As shown here, tortuosity can be obtained
mathematically, as a function of the medium
porosity by calculating the average of all
pathlines
of
problem,
and
the
corresponding
that
it
follows
flow
the
experimental relation given by Comiti &
Renaud:
  0.9
Fig 1: Velocity magnitudes squared
  0.65
  0.8
(u2+v2)
and streamlines generated for three different porous media porosities.
T  1  p log(  ) .
Matyka, M., Khalili, A. & Koza, Z. (2008)
Tortuosity-porosity relation in the porous
media flow (submitted to Phys. Rev. E.)
5. Perspectives
- Experimental pathline visualizations for
calculation of tortuosity in a micro-channel
device are planned, and will be made to
verify the generality of the model results.
Fig 2: (left) Dependency of tortuosity T on the system size for two different porosities

.
Fig 3: (right) The relation between tortuosity and porosity for the system of overlapping rectangles. Our calculation (symbols)
with a fit to empirical relation T  1  p log(  ) obtained from experimental measurements (solid line) [J. Comiti and M. Renaud,
Chem. Eng. Sci. 44, 1539 (1989)]. Calculation of Koponen et al, Phys. Rev. E 56, 3319 (1997) (dashed line) without finite size
scaling analysis, leading to a bigger deviation from experiments.
- Also planning a joint EU project on
tortuosity between MPI Bremen and IFT
Wrocław.