Hydrodynamic dispersion in porous media is important in various engineering fields. In environmental engineering, it plays a key role in explaining how the disposed pollutants penetrate the soil, spread in the groundwater and contaminate aquifers. Moreover, in petroleum engineering, it shows how the injected chemicals travel in a hydrocarbon reservoir and enhance the recovery of oil. Therefore, hydrodynamic dispersion has drawn much research attention. As a result, most of scientists agree that convection together effective molecular diffusion in porous media determine the effectiveness of dispersion. In addition, its coefficient in different porous medium structures is found as different functions of Eulerian Peclect number, which is based on a length scale characteristic of the porous medium geometry. In this study, we examine hydrodynamic diserpersion but from a Lagrangian approach, in which molecular diffusion of nano-particles is taken into account for the time and space scales in addition to dispersion due to convection. In our work, Lattice Boltzman method (LBM) is employed to simulate flow in various porous media configurations, having pore velocity in the range of 0 to 0.4 cm/s. The porous media are packed beds with different sphere packing types. The Lagrangian particle tracking method (LPT) is then applied to keep track of the position and velocity of nanoparticles dispersing in the porous medium as a function of time. The Schmidt number of the nanoparticles is in the range from 100 to 10,000. The simulation codes have been validated carefully with various known analytical solutions [1]–[3]. While the trajectory data of particles allow the direct calculation of hydrodynamic dispersion coefficient, their velocity with time leads to the determination of Lagrangian length and time scales by calculating the Lagrangian velocity autocorrelation coefficient. We also probe the correlation between Lagrangian and Eulerian timescales, as well as their effects on dispersivity. Results of the LBM/LPT method provide direct calculations of the effective diffusivity (De) within the porous medium at zero flow velocity. The ratio of the hydrodynamic dispersivity (Dh) over the effective diffusivity, given as Dh/De, can be correlated to the Peclet number. The results show that there is no single correlation between Dh/De and the effective Eulerian Peclet number. In contrast, that ratio is linearly dependent on the effective Lagrangian Peclet number for different sphere packing types. The slope of the line varies with the porous medium type and depends on the porous medium structure (such as its porosity, the Darcy number, etc.). The effective Lagrangian Peclet number, which is defined with the Lagrangian length scale and the effective diffusivity, De., should therefore be the appropriate dimensionless number for dispersion in porous media.
Literature Cited:
[1] R. Voronov, S. VanGordon, V. I. Sikavitsas, and D. V. Papavassiliou, “Computational modeling of flow-induced shear stresses within 3D salt-leached porous scaffolds imaged via micro-CT,†J. Biomech., vol. 43, no. 7, pp. 1279–1286, 2010, doi: 10.1016/j.jbiomech.2010.01.007.
[2] N. H. Pham, R. S. Voronov, N. R. Tummala, and D. V. Papavassiliou, “Bulk stress distributions in the pore space of sphere-packed beds under Darcy flow conditions,†Phys. Rev. E - Stat. Nonlinear, Soft Matter Phys., vol. 89, no. 3, pp. 1–13, 2014, doi: 10.1103/PhysRevE.89.033016.
[3] R. S. Voronov, S. B. VanGordon, V. I. Sikavitsas, and D. V. Papavassiliou, “Efficient Lagrangian scalar tracking method for reactive local mass transport simulation through porous media,†Int. J. Numer. Methods Fluids, vol. 67, no. 4, pp. 501–517, Oct. 2011, doi: 10.1002/fld.2369.
