Pore-scale dispersion: Bridging the gap between microscopic pore structure and the emerging macroscopic transport behavior

We devise an efficient methodology to provide a universal statistical description of advection-dominated dispersion ($\text{Péclet}\to \infty$) in natural porous media including carbonates. First, we investigate the dispersion of tracer particles by direct numerical simulation (DNS). The transverse dispersion is found to be essentially determined by the tortuosity and it approaches a Fickian limit within a dozen characteristic scales. Longitudinal dispersion was found to be Fickian in the limit for bead packs and superdiffusive for all other natural media inspected. We demonstrate that the Lagrangian velocity correlation length is a quantity that characterizes the spatial variability for transport. Finally, a statistical transport model is presented that sheds light on the connection between pore-scale characteristics and the resulting macroscopic transport behavior. Our computationally efficient model accurately reproduces the transport behavior in longitudinal direction and approaches the Fickian limit in transverse direction.

Figure 1

Convergence of particle plumes in terms of sample size. Depicted are (left) longitudinal and (right) transverse particle position histograms after nondimensional time $t/T={10}^4$.

Figure 2

Spreading of particle plumes in (top half) the DNS and (bottom half) the model computations. Depicted are the particle position variances in (a) longitudinal $x_1$ and (b) transverse $x_2$ direction, respectively. The thin straight lines at the left and right ends of the curves correspond to quadratic and linear scalings with $t$, respectively. Superdiffusive spreading exceeds the linear scaling.

Figure 3

Comparison of longitudinal particle plumes resulting from (solid red) DNS and (dashed blue) the statistical model. Plotted are different snapshot times. Data corresponding to the bead pack 500 sample are depicted.

Figure 4

See caption to Fig. 3. Data corresponding to the Ketton carbonate sample are depicted.

Figure 5

See caption to Fig. 3. Data corresponding to the Bentheimer sandstone 1000 sample are depicted.

Figure 6

See caption to Fig. 3. Data corresponding to the Estaillades carbonate sample are depicted.

Figure 7

(a) Local coordinate system spanned by the mean velocity $\mathbf{U}={(U,0,0)}^T$ and the instantaneous particle velocity $\mathbf{u}\left(t\right)$. Positive angles are depicted. (b) Gaussian model for flow-direction vectors $\mathbf{u}\left(l\right)/u\left(l\right)$.

Figure 8

Exemplary velocity-magnitude time series $u\left(t\right)$ resulting from randomly placed particles in different samples. (a) Bead pack 500, (b) Ketton, (c) Bentheimer 1000, (d) Estaillades.

Figure 9

Autocorrelation function ${\rho }_u\left(t\right)$ of the Lagrangian particle velocity $u\left(t\right)$ and histogram $p\left(v\right)$ of the log velocity-magnitude $v$. Dots represent the skew-normal model for $p\left(v\right)$.

Figure 10

Drift and diffusion coefficients $a\left(v\right)$ and $d\left(v\right)$, respectively, for the log velocity-magnitude process $v\left(t\right)$ of the Estaillades carbonate sample. Depicted are the parametric fits (lines) and the transition moments (dots).

Figure 11

Autocorrelation function ${\rho }_{\theta }\left(l\right)$ of the Lagrangian velocity direction-angle $\theta \left(l\right)$ and histogram $p\left(\right|\theta \left|\right)$. Dots represent the analytical model given by Eq. (5) for $p\left(\right|\theta \left|\right)$.

Figure 12

Histogram of the travel-distance $L_{\theta }$ with no sign change in $\theta \left(l\right)$. Dots represent the exponential PDF model.

Figure 13

Compensated variance evolution $\langle {[\beta (l+s)-\beta \left(l\right)]}^2\rangle /s$ as a function of separation distance $s$. Dots represent the compensated variance resulting from the integrated Wiener process model.

Figure 14

List of parameters of our stochastic transport model. The rows correspond to different parameters and the columns to different samples. The gray level encodes the parameter value.

Figure 15

Exemplary particle path lines tracked in the (1) monodisperse bead pack and (2) Estaillades carbonate. Panels correspond to path lines from (a) DNS and (b) the statistical model. The color represents $ln\left[u\right(t)/U]$.