1. Introduction

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 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 porousmedia flow(submitted to Phys. Rev. E.) Tortuosity of Sediments: A Mathematical ModelMaciej Matyka Arzhang Khalili Zbigniew Koza Institute of Theoretical Physics Max Planck Institute Institute of Theoretical PhysicsUniversity of Wrocław for Marine Microbiology,University of Wrocław Poland Germany Poland Mathematical Modeling Group 1. Introduction Permeability kand porosityare two important physical properties ofmarine sediments. Another physical characteristics is tortuosity, which affects alldiffusive and dynamic processes involved in marine sediments, or generallyspeaking, in porous media.From hydrodynamic point of view, tortuosity T in a fluid medium would beunity if a particle could travel a distance on a stricktly horizontalpathline.However, due to the existence of solid matrices in a porous medium, the pathlinesbecome tortuous, which leads to T>1. The more `wavy' the pathlines become, thelarger the tortousity.As one can not easily measure tortousity directly, andbecause, it plays an important role in almost all geophysical and geochemicaltransport processes, it has beenthe subject of intensive research.The question is whether or not tortuosity can be described as a function of porosity. 2. Methodology We use D2Q9 Lattice Boltzmann BGK model for the creeping flow problem: Transport equation: Equilibrium density function: - distribution function (DF) • collision operator - equilibrium DF • lattice weights • velocity (macro) - body force (i.e. gravity) • density (macro) - lattice vectors (i=0..8) Weighted average of pathline lengths: 3. Results Simplifies for flux averaged spreading: Fig 1: Velocity magnitudes squared (u2+v2) and streamlines generated for three different porous media porosities. 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. 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.