Longitudinal and Transverse Dispersion in Porous Media by Pore-scale Modeling and Continuous Time Random Walk (CTRW) Theory

We provide a physically based explanation for the complex macroscopic behavior of longitudinal and transverse dispersion in porous media as a function of Peclet number, Pe. A Lagrangian pore-scale transport model incorporating flow and molecular diffusion is applied in 2D and 3D lattices of throats with square cross-section whose characteristics are representative of Berea sandstone. The model accurately predicts NMR, laser fluorescence and the classic breakthrough data for the experimental dependence of the longitudinal dispersion coefficient, DL, on Pe. We compute the probability, y(t)dt that a particle moves from one throat junction to a nearest neighbor junction in the time interval t to t+dt and fit it to the simple analytical expression, that depends on the mean advective transit time, a late-time diffusive cut-off and a parameter b characterizing the distribution of transit times between pores. Then, interpreting transport as a continuous time random walk, we show: (1) that the power-law dispersion regime is controlled by the variation in average velocity between throats (the distribution of local Pe rather than by diffusion from boundary layers within throats, giving with d = 3-b »1.2; (2) the cross-over to a linear regime for for Pe > Pecrit ≈ 400 is due to a transition from a diffusion-controlled late-time cut-off, to transport governed by advective movement; and (3) that the transverse dispersion coefficient for all Pe >>1. We provide a quantitative description of both asymptotic and pre-asymptotic dispersion using CTRW with a physical interpretation of the parameters. Hence, through an analysis of experiment, numerical modeling and theory we provide a physical explanation of the subtle and surprising dependence of dispersion coefficients in porous media on Pe. Our demonstration that the power-law dependence of dispersion coefficient on Pe is due to the distribution of flow speeds in individual pores contrasts with the traditional theories of Saffman and Koch and Brady that emphasize the contribution to dispersion within pores or at pore junctions.Presenters: name: Bijeljic, Branko affiliation: Imperial Colleg