Pore-scale statistics of flow and transport through porous media

Flow in porous media is known to be largely affected by pore morphology. In this work, we investigate the effects of pore geometry on the transport and spatial correlations of flow through porous media in two distinct pore structures arising from three-dimensional assemblies of overlapping and nonoverlapping spheres. Using high-resolution direct numerical simulations (DNS), we perform Eulerian and Lagrangian analysis of the flow and transport characteristics in porous media. We show that the Eulerian velocity distributions change from nearly exponential to Gaussian distributions as porosity increases. A stretched exponential distribution can be used to represent this behavior for a wide range of porosities. Evolution of Lagrangian velocities is studied for the uniform injection rule. Evaluation of tortuosity and trajectory length distributions of each porous medium shows that the model of overlapping spheres results in higher tortuosity and more skewed trajectory length distributions compared to the model of nonoverlapping spheres. Wider velocity distribution and higher tortuosity for overlapping spheres model give rise to non-Fickian transport while transport in nonoverlapping spheres model is found to be Fickian. Particularly, for overlapping spheres model our analysis of first-passage time distribution shows that the transport is very similar to those observed for sandstone. Finally, using three-dimensional (3D) velocity field obtained by DNS at the pore-scale, we quantitatively show that despite the randomness of pore-space, the spatially fluctuating velocity field and the 3D pore-space distribution are strongly correlated for a range of porous media from relatively homogeneous monodisperse sphere packs to Castlegate sandstone.

Figure 1

(a) Comparison of interstitial velocity profiles at $\mathrm{Re}=105.57$ with experimental data of Suekane et al. [49] $(◯)$ and numerical results of the same experiment by Finn and Apte [50] $\left(--\right)$. $-$ simulation data with grid spacing of $\mathrm{\Delta }/\mathrm{D}=56$; $·-·$ simulation data with grid spacing of $\mathrm{\Delta }/\mathrm{D}=28$. (b) Schematic of the experimental setup of Suekane et al. Simulation is performed with four lateral faces as wall and inflow and outflow boundary conditions for bottom and top faces, respectively, according to the experiment of Suekane et al. Normalized velocity is measured along the centerline of the plane shown in white (along $x$ axis). Solids are colored gray.

Figure 2

Three-dimensional periodic assemblies of spheres and cross sectional views of indicator function (solid or pore-space) for (a), (b) monodisperse hard-spheres, (c), (d) polydisperse hard-spheres, and (e), (f) (monodisperse) overlapping spheres. Pore-space is colored black.

Figure 3

Different models of porous media and their two-point correlation functions. Significant difference in the structure of $S_2\left(r\right)$ is an indication of different morphology of porous media models. In hard-sphere systems (a), oscillations in $S_2\left(r\right)$ is due to exclusion volume effect. These oscillations becomes weaker in the case of polydisperse hard-spheres (b), and completely vanishes for overlapping spheres (c). Different statistics could be obtained simply by calculating $S_2\left(r\right)$, as shown in the figure.

Figure 4

Representative elementary volume calculation: (a) variation of permeability with respect to domain size; (b) variogram of porosity and velocity in the direction of flow normalized by their theoretical asymptotic values ${\gamma }_{\infty }=\varphi (1-\varphi )$ for porosity and ${\gamma }_{\infty }=\langle u^2\rangle -{\langle u\rangle }^2$ for velocity variograms. Variograms are plotted as functions of distance from the inlet face. Data is for the monodisperse hard-sphere model at $\varphi =0.40$.

Figure 5

Variation of trajectory length distribution with respect to porosity for (a)–(c) monodisperse hard-sphere, (d)–(f) polydisperse hard-sphere, and (g)–(i) overlapping sphere models of porous media.

Figure 6

Variation of tortuosity with different models of porous media and porosity. Error bars for the case of hard-sphere model are the size of symbols and thus not plotted.

Figure 7

First-passage time distribution for (a) monodisperse hard-spheres and (b) overlapping spheres for high ($\square$) and low ($◯$) porosity media.

Figure 8

Evolution of the mean velocity sampled equidistantly in space for hard-sphere ($◯$) and overlapping sphere ($\square$) models. Due to uniform injection of particles, mean velocity sampled equidistantly in space evolves from its initial value which coincides with mean Eulerian velocity ($·-·$) and reaches its steady-state value calculated from Eq. (21) ($--$). Inset: Mean velocity sampled isochronally remains constant until breakthrough and is equal to mean Eulerian velocity as expected. $n_t$ is the number of time steps.

Figure 9

Comparison of stationary $s$-Lagrangian velocity PDF as given by Eq. (21) and numerical simulation.

Figure 10

Top row: PDF of streamwise velocity. Bottom row: PDF of velocity perpendicular to mean flow direction (spanwise). All velocities are normalized by mean interstitial velocity $\left(q/\varphi \right)$ for (a), (b) monodisperse hard-sphere, (c), (d) polydisperse hard-sphere, and (e), (f) overlapping sphere models of porous media.

Figure 11

PDFs of velocity norm for monodisperse hard-sphere ($◯$) and overlapping sphere ($\square$) models for $\varphi =0.36$.

Figure 12

Dependence of model parameters in Eq. (23) on porosity for two models of porous media (hard-sphere and overlapping). $\eta$ strongly depends on the porosity and is consistently larger for hard-spheres (a). However, the dependence of $\alpha$ on the porosity is not very strong (b). This figure clearly quantifies the transition in velocity distribution shape as a function of porosity.

Figure 13

Streamwise velocity distributions for (monodisperse) hard-sphere and overlapping sphere models of porous media with their corresponding fits obtained by Eq. (23) (solid line). Porosity is $0.45$.

Figure 14

Spatial correlation in velocity fluctuations for monodisperse hard-sphere model along with their two-point correlation function calculated on pore-space. Strong similarities suggests spatial correlations of velocity and pore-space are almost identical in the creeping flow regime.

Figure 15

Spatial correlation in velocity fluctuations for (a) polydisperse hard-sphere and (b) overlapping sphere models along with their two-point correlation function calculated on pore-space.

Figure 16

Cross-sectional view of the indicator function (solid or pore-space) and normalized velocity magnitude ($\left|\mathrm{U}\right|/{\left|\mathrm{U}\right|}_{\max }$) for (a), (b) the unconsolidated sandpack and (c), (d) Castlegate sandstone with porosity of $36\%$ and $20.6\%$, respectively. Pore-space is colored black.

Figure 17

Spatial correlation in velocity fluctuations for different models of porous media along with their two-point correlation function calculated on pore-space. Strong similarities suggest spatial correlations of velocity and pore-space are almost identical in the creeping flow regime.