The Effect of varying correlation lengths on Anomalous Transport

Abstract

Conventional concepts for transport in porous media assume that the heterogeneous distribution of hydraulic conductivities is the source for the contaminant temporal and spatial heavy tail. This tailing, known as anomalous or non-Fickian transport, can be captured by the β parameter in the continuous-time random walk framework. This study shows that with the increase in spatial correlation length between these heterogeneous distributions of hydraulic conductivities, the transport’s anomaly reduces; yet, the β value is unchanged, suggesting a topological component of the conductivity field, captured by the β. This finding is verified by an analysis of the solute transport, showing that the changing conductivity values have a moderate effect on the transport shape.

Appendix

As stated in Method section, a parallel simulation is presented for Δ = 0.02 so to verify that a change of Δ will have no effect on the transport. As previously, we use a field of 3000 × 1200 cells from which a 300 × 120 field is taken. Unlike the previous simulation, the absolute field size is now 6 × 1.2, an order of magnitude smaller. Thus, the head difference was reduced from 100 to 10 so to maintain the same mean velocity. The dimensionless correlation length (normalized by the field length) ranges from 1 × 10^–5 to 1 × 10^–4, yet there are 15 cells for the low correlation length and 150 cells for the high correlation length.

The BTC fit in Fig. 5a–c and the corresponding juxtaposition in Fig. 5d–f show that the β is still persistent for the variations in correlation length, while the t₂ is reducing due to the correlation length increase, as before. Moreover, the same dispersion increase can be found with an increase in correlation length. This shows the consistency of these results, mainly since the juxtaposition in Fig. 5d–f still reproduces the same slope associated with the β. At the same time, the t₂ cutoff is apparent in the PDF tail dispersion. The PPF’s analysis, as the correlation length changes, shows even higher values of PPF’s spatial occurrence for varying correlation lengths of more than 60% to 85% (Fig. 6). As for the larger field, the weighted conductivity distribution for the PPFs has the same tailing as shown in Fig. 3a–c, where the backward tail follows the weighted conductivity, and the forward tail follows the unweighted distribution (Figs. 6a–c, 7)

Fig. 5

a–c Ensemble BTCs for correlation lengths of 1 × 10^–5, 5 × 10^–5 and 1 × 10^–4 with CTRW-TPL fits for each. As can be seen, the tail decreases, yet it is the t₂ and D that displays the biggest changes. For correlation length 1 × 10^–5, the CTRW-TPL parameters are v = 4.4, D = 0.086, β = 1.63, t₁ = 0.013 and t₂ = 10^2.1. For correlation length 5 × 10^–5, the CTRW-TPL parameters are v = 4.2, D = 0.45, β = 1.63, t₁ = 0.01 and t₂ = 10^1.8. And for correlation length 1 × 10^–4, the CTRW-TPL parameters are v = 4.4, D = 1.55, β = 1.63, t₁ = 0.0061 and t₂ = 10^1.6. As can be seen, the tail decreases, yet it is the t₂ that displays the most significant changes, while β is unchanged. d–f The PDF of 1 × 10^–5, 5 × 10^–5 and 1 × 10^–4, respectively, for the ensemble, weighted by the particle visitation with the juxtaposition of the CTRW PDF with the same parameters found for a–c. As can be seen, the logarithmic slope, which corresponds to β, is identical, while the t₂ tailing and the t₁ mean reduce with the increase in correlation length. All values are in consistent, arbitrary length and time units; the R² difference is bigger than 0.95 for all figures

Fig. 6

a–c Blue circles are the conductivity distribution for correlation lengths of 1 × 10^–5, 5 × 10^–5 and 1 × 10^–4, respectively, averaged for 1000 realizations, with average conductivity of 0.15, 0.24 and 0.36, respectively, and conductivity variance of 4.9, 4.8 and 4.5, respectively. Red squares are the same conductivity distributions only weighted by the particle’s visitations at each cell, with average conductivity of 0.67, 0.9 and 1.2, respectively, and conductivity variance of 4.9, 4.8 and 4.4, respectively; green asterisks are the permeabilities weighted by the particle visitations only on the PPF, with average conductivity of 0.54, 0.7 and 0.7, respectively, and conductivity variance of 4.9, 4.6 and 4.3, respectively. d–f Particle visitations on the PPF for correlation lengths of 1 × 10^–5, 5 × 10^–5 and 1 × 10^–4, respectively

Fig. 7

Ensemble BTCs at a x = 30 and b x = 120 for correlation lengths of 5 × 10^–3, 1.6 × 10^–2 and 5 × 10^–2 (red, green and blue circles, respectively) with CTRW-TPL fits for each. a For x = 30 correlation length 5 × 10^–3, the CTRW-TPL parameters are v = 4.8, D = 1, β = 1.63, t₁ = 0.1 and t₂ = 10^2.4. For x = 30 correlation length 1.6 × 10^–2, the CTRW-TPL parameters are v = 4.8, D = 2.25, β = 1.63, t₁ = 0.18 and t₂ = 10^1.9. And for x = 30 correlation length 5 × 10^–2, the CTRW-TPL parameters are v = 5.2, D = 8.46, β = 1.63, t₁ = 0.4 and t₂ = 10^1.6. b For x = 120 correlation length 5 × 10^–3, the CTRW-TPL parameters are v = 4.16, D = 1, β = 1.63, t₁ = 0.24 and t₂ = 10^2.6. For x = 120 correlation length 1.6 × 10^–2, the CTRW-TPL parameters are v = 4.17, D = 2.3, β = 1.63, t₁ = 0.39 and t₂ = 10^2.2. And for x = 120 correlation length 5 × 10^–2, the CTRW-TPL parameters are v = 4.5, D = 9.5, β = 1.63, t₁ = 0.86 and t₂ = 10^1.9