Microscale modeling of nondilute flow and transport in porous medium systems

Nondilute transport occurs routinely in porous medium systems. Experimental observations have revealed effects that seemingly depend upon density, viscosity, velocity, and chemical activity. Macroscale models based upon averaged behavior over many pores have been relied upon to describe such systems to date, which require parametrization of important physical phenomena in material coefficients. To advance fundamental understanding of these complex systems, we examine nondilute transport from a fundamental microscale, or pore-scale, continuum modeling perspective. We approximate the solution of a model based upon the variable-density Navier-Stokes equations and a nondilute species transport equation. Known dependencies of the densities, viscosities, chemical activity, and diffusion for a salt solution on chemical composition are included in the model. Microscale model solutions are averaged to the macroscale and compared with extant experimental observations. Investigation of the effects of various physical phenomena on the microscale velocity distribution and the observed macroscale dispersion are considered using dimensional analysis and constrained simulations. Simulation results are used to explain observed experimental results in light of underlying mechanisms. Conditions under which the various physicochemical effects investigated are important are revealed.

Figure 1

REV scale domain: (a) sphere pack; and (b) grain diameter distribution with a sample mean of 0.11 cm and standard deviation of 0.02 cm. The direction of flow for all simulations was upwards and the blue box represents the domain that was simulated.

Figure 2

Sub-REV domain: (a) sphere pack; and (b) grain diameter distribution with a sample mean of 0.12 cm and standard deviation of 0.019 cm. The blue box represents the domain that was simulated.

Figure 3

Results of fitting the dilute macroscale model (lines) to REV-scale cross-section averaged microscale data (points) for Pe = 0.026 (a), 2.6 (b), and 260 (c).

Figure 4

Dilute microscale sub-REV velocity distributions for ${\mathrm{Re}}_0=3.1\times {10}^{-5}$ (a) and ${\mathrm{Re}}_0=0.31$ (b). The means and standard deviations of the velocity are included. The distributions were sampled at a cross-section orthogonal to the mean direction of flow.

Figure 5

Nondilute species breakthrough curves: (a) laboratory experiments from [6]; and (b) averaged REV-scale microscale simulations.

Figure 6

Averaged REV-scale nondilute breakthrough curves for Re $\approx {10}^{-3}$.

Figure 7

Molecular diffusion sensitivity to density, viscosity, and activity [see Eq. (4)].

Figure 8

Normalized microscale mass fraction simulations at $\mathrm{Re}\approx {10}^{-3}$ for an upward flow displacement pattern: (a) dilute case; and (b) ${\omega }_{\mathrm{in}}=0.4$ and ${\omega }_{\mathrm{res}}=0$.

Figure 9

Leading edge (dotted line) and trailing edge (dashed line) of the mixing zone, and the mixing zone thickness (solid line) for the dilute (blue) and ${\omega }_{\mathrm{in}}=0.4$ and ${\omega }_{\mathrm{res}}=0$ (yellow) simulations at $\mathrm{Re}\approx {10}^{-3}$.

Figure 10

Normalized mass fraction variance averaged over a plane located at $z=0.513$ for dilute and nondilute simulations at $\mathrm{Re}\approx {10}^{-3}$.

Figure 11

Distribution of velocity components and mass fraction sampled for a cross-section at $z=0.513$ for simulations at $\mathrm{Re}\approx {10}^{-3}$: (a) ${\omega }_{\mathrm{in}}=0.1$; and (b) ${\omega }_{\mathrm{in}}=0.4$.

Figure 12

Normalized microscale mass fraction simulations at $\mathrm{Re}\approx {10}^{-1}$ for an upward flow displacement pattern: (a) dilute case; and (b) ${\omega }_{\mathrm{in}}=0.4$ and ${\omega }_{\mathrm{res}}=0$.

Figure 13

Leading edge (dotted line) and trailing edge (dashed line) of the mixing zone, and the mixing zone thickness (solid line) for the dilute (blue) and ${\omega }_{\mathrm{in}}=0.4$ and ${\omega }_{\mathrm{res}}=0$ (yellow) simulations at $\mathrm{Re}\approx {10}^{-1}$.

Figure 14

Normalized mass fraction variance averaged over a plane located at $z=0.513$ for dilute and nondilute simulations at Re $\approx {10}^{-1}$.

Figure 15

Distribution of velocity components and mass fraction sampled for a cross-section at $z=0.513$ for simulations at $\mathrm{Re}\approx {10}^{-1}$ and ${\omega }_{\mathrm{in}}=0.4$.

Figure 16

Sub-REV-scale breakthrough curve sensitivity at $\mathrm{Re}\approx {10}^{-3}$: (a) ${\omega }_{\mathrm{in}}=0.4$ and ${\omega }_{\mathrm{res}}=0$; and (b) ${\omega }_{\mathrm{in}}=0.5$ and ${\omega }_{\mathrm{res}}=0.4$.

Figure 17

Sub-REV-scale breakthrough curve sensitivity at $\mathrm{Re}\approx {10}^{-1}$: (a) ${\omega }_{\mathrm{in}}=0.4$ and ${\omega }_{\mathrm{res}}=0$; and (b) ${\omega }_{\mathrm{in}}=0.5$ and ${\omega }_{\mathrm{res}}=0.4$.