Tortuosity of Sediments: A Mathematical Model

Maciej Matyka Arzhang Khalili Zbigniew Koza Institute of Theoretical Physics Max Planck Institute Institute of Theoretical Physics

University of Wrocław for Marine Microbiology, University of Wrocław Poland Germany Poland

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

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

medium, the pathlines become tortuous, which leads to T>1. The more `wavy' the pathlines

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

tortuosity can be described as a function of porosity.

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.

- Also planning a joint EU project on tortuosity between MPI Bremen and IFT Wrocław.

Mathematical Modeling Group

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 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.

3. Results

2. Methodology

Fig 1: Velocity magnitudes squared (u2+v2) and streamlines generated for three different porous media porosities.

kk

A y

A y

dxxu

dxxxu

LT

)(

)()(1

Weighted average of pathline lengths:

Simplifies for flux averaged spreading:

N

jjxNL

T1

)(11

iti

tii

ti Frrncrn )()()(1

)](31[),( iieqi cuun

We use D2Q9 Lattice Boltzmann BGK model

for the creeping flow problem:

eqin

iF

ic

i

u

in ti

- distribution function (DF)

- equilibrium DF

- body force (i.e. gravity)

- lattice vectors (i=0..8)

- collision operator

- lattice weights

- velocity (macro)

- density (macro)

Transport equation:

Equilibrium density function:

)log(1 pT

9.0 8.0 65.0

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 the corresponding flow

problem, and that it follows the

experimental relation given by Comiti &

Renaud: .

Matyka, M., Khalili, A. & Koza, Z. (2008)

Tortuosity-porosity relation in the porous

media flow (submitted to Phys. Rev. E.)

)log(1 pT