Solute transport in porous media studied by lattice Boltzmann simulations at pore scale and x-ray tomography experiments

With the aid of nondestructive microfocus x-ray computed tomography (CT), we performed three-dimensional (3D) tracer dispersion experiments on randomly unconsolidated packed beds. Plumes of nonreactive sodium iodide solution were point injected into a sodium chloride solvent as a tracer for the evaluation of the dispersion process. The asymptotic dispersion coefficient was obtainable within the experimental scale and was summarized over Péclet numbers from 11.7 to ∼860. Then, the lattice Boltzmann method and moment propagation method were used to elucidate the mechanisms embedded in the dispersion phenomenon. The methods were rigorously verified against the classical Taylor dispersion problem and extended to simulate fluid flow and tracer dispersion in high-resolution 3D digital porous structures from CT. The method of moments, Lagrangian velocity correction function, and dilution index were thoroughly analyzed to evaluate the dispersion behaviors. Numerical simulations revealed ballistic and superdiffusive regimes at the transient times, whereas asymptotic dispersion behaviors appear at longer characteristic times. Besides, the observed transient times unanimously persist over convective length scales of around 12 particles transversely and 16 particles longitudinally. The estimated dispersion coefficients from simulation are in consistence with the experimental result. Furthermore, the simulation also enabled the identification of regimes, including diffusive, power law, and mechanical dispersion. Thus, the proposed experimental and computational schemes are of practical means to study dispersion behaviors by direct pore scale imaging and modeling.

Figure 1

Schematic view of the dispersion experiment apparatus with x-ray CT. The illustrated 3D configuration of the NaI tracer plume is visualized using the vgstudio/max2.1 software.

Figure 2

Concentration profile along the flow direction averaged at the $y$ plane of the tube. The isovolume plots (a)–(f) illustrate the concentration profile at times $t=60000$, $t=120000$, $t=180000$, $t=240000$, $t=300,000$, and $t=360000\mathrm{l}.\mathrm{u}.$, respectively. The tracer dispersion after point (e) has passed the transient period and becomes stabilized.

Figure 3

Evolution of the nondimensional dispersion coefficients after injection of an immiscible passive tracer plume with a diameter of ten lattices into a fully developed pressure-driven Hagen-Poiseuille tube flow with a pipe diameter of 40 lattices. The inset depicts the temporal evolution of the MSD.

Figure 4

Porous media showing pore size distributions and steady-state flow fields. (a) The left half represents the melamine resin particles (red represents particles and blue denotes pore spaces), and the right half represents the streamlines at $\mathrm{Pe}=350.4\mathrm{and}{d}_p=338\mu \mathrm{m}$. (b) Probability density function of the interstitial velocity (PDF). (c) Cumulative density function (CDF) for the pore size distribution, representing the probability of the occurrence of a pore size larger than $d_{\mathrm{pore}}$. (d) CDF plotted as a function of the rescaled particle diameter.

Figure 5

Visualization of the local NaI concentration distributions over time projected in the $xz$ plane. The NaCl solution flow is in the $z$ direction. The concentration of the NaI solution at the initial condition was nearly 10 wt %, and it decayed to approximately 2–3 wt % after injection of 0.5 PV NaCl solution $\left(\mathrm{Pe}=350.4,d_p=338\mu \mathrm{m}\right)$.

Figure 6

Temporal evolution of the transverse (a) and longitudinal (b) tracer plume configurations projected in the $x$ and $y$ planes ($\mathrm{Pe}=350.4,d_p=338\mu \mathrm{m}$).

Figure 7

Temporal evolution of the longitudinal and transverse spatial variances. The time intervals are 129.2, 64.6, 28.5, 8.6, and 4.3 s for volume flow rates of $Q=100$, 200, 500, 1500, and 3000 ml/h, respectively; six continuous x-ray scans were taken for each experiment. The dashed lines denote the linearly fitted lines of the data acquired at each time step, with half of the slope value representing the dispersion coefficients.

Figure 8

Temporal evolution of the NaI tracer plume concentration profiles from LBM simulations. The time shown at the bottom of each profile is the transformed physical time. The initial shape of the tracer plume is spherical and occupies the space of around four particles. The colors denote isosurfaces of the solute concentration. The outer surface threshold is 6.25% of the initial concentration ($\mathrm{Pe}=350.4,d_p=338\mu \mathrm{m}$).

Figure 9

Temporal evolution of the transverse (a) and longitudinal (b) tracer concentration. ($\mathrm{Pe}=350.4,d_p=338\mu \mathrm{m}$).

Figure 10

Average velocity of the tracer particles and NaCl solution over the $x$ and $y$ planes. ($\mathrm{Pe}=350.4,d_p=338\mu \mathrm{m},\mathrm{t}=2.856\mathrm{s}$). The dotted lines with open circles represent the average velocity of the tracer particles: (a) refers to the transverse direction and (b) refers to the longitudinal direction. The solid lines denote the average flow velocity with its statistical average expressed as the black dashed lines.

Figure 11

MSD for LBM simulations. The total simulation times for $\mathrm{Pe}=11.7$, 23.4, 58.4, 175.2, and 350.4 are 214.2, 107.1, 42.84, 14.28, and 7.14 s, respectively. The closed-symbol dotted lines show the MSD in the transverse direction (a), whereas the open-symbol dotted lines represent that in the longitudinal direction (b). (c),(d) plot MSD with the rescaled time $\mathrm{t}/\tau$, where $\tau =d_p/u_{\mathrm{av}}$ is the characteristic convective time required for the tracer particle to transport a characteristic length $L$ (in our study $L=d_p$).

Figure 12

LVCF ($\mathrm{Pe}=350.4,d_p=338\mu \mathrm{m}$).$\tau$ is characteristic time as previously specified. The filled squares represent the LVCF in the transverse direction, and the open squares denote the longitudinal direction.

Figure 13

Evolution of the dilution index $E\left(t\right)$ and temporal derivative of $\ln E\left(t\right)$.

Figure 14

Dependence of the normalized dispersion coefficients ($D_{T/L}/D_m$) on Pe. The filled symbols represent the current experimental results (red circles, blue triangles, and brown squares are for the mean particle diameters $d_p$ of 338, 513, and $780\mu \mathrm{m}$, respectively). The dashed lines correspond to the fitted curves from the present LBM and MPM simulations. The open symbols denote the dispersion data from previous studies. In (a), the open circles represent the results for the Berea sandstones [25], whereas the open triangles represent the same porous media but with larger sizes [40]. In (b), the open squares are associated with the CFD simulation results for confined sand packing [67], whereas the open circles and triangles are for the experimental results of the Ketton limestone and packed beads [24].