Impact of porous media heterogeneity on convective mixing in a Rayleigh-Bénard instability

TL;DR

This study addresses how permeability heterogeneity in porous media modifies Rayleigh-Bénard convection and convective mixing by simulating the Horton-Rogers-Lapwood problem with a multi-Gaussian log-normal permeability field. A parametric suite over $Ra$, $\sigma^2_{\log k}$, and horizontal correlation length $\lambda_x$ reveals that heterogeneity increases heat flux, promotes dispersive fingering aligned with high-permeability pathways, and intensifies mixing segregation, while also shifting stagnation-point dynamics toward boundary-permeability features. The work links interface compression, velocity structure, and mixing state, showing altered scaling of interface width $s_B$ and flux $F$ with $Ra$ and heterogeneity, and provides practical scalings to estimate heat transport in geothermal and CCS contexts. Overall, the results emphasize that accounting for permeability structure is essential for accurate prediction of heat transfer and mixing efficiency in natural and engineered porous systems.

Abstract

This work studies the effect of the heterogeneity of a porous medium on convective mixing. We consider a system where a Rayleigh-Bénard instability is triggered by a temperature difference between the top and bottom boundaries. Heterogeneity is represented by multi-Gaussian log-normally distributed permeability fields. We explore the effect of the Rayleigh number, the variance and correlation length of the log-permeability field on the fingering patterns, heat flux, mixing state and flow structure. Heat flux increases for all heterogeneous cases compared to the homogeneous ones. When heterogeneity is weak and the horizontal correlation length small, flux exhibits minimal sensitivity to the variance of the log-permeability. When the correlation length increases, flux increases proportionally to the log-permeability variance. The mixing state is evaluated through the temperature variance and the intensity of segregation. Both take higher values, compared to their homogeneous analogues, when the correlation length and the variance of the permeability are increased. This indicates that even if heat flux increases, the system is less well mixed. The flow structure shows that in homogeneous and weakly heterogeneous cases there is a relation between the location of high strain rate and stagnation point, while for strongly heterogeneous cases, high strain rate zones are linked to high permeability areas near the boundaries, where temperature plume originate. The interface width tends to decrease as the variance and the correlation length of the permeability field are augmented, suggesting that the interface undergoes a higher stretching in heterogeneous porous media.

Figures (10)

• Figure 1: Log-permeability field (leftmost column) and time-averaged temperature maps at convective regime ($60<t<250$) for the homogeneous (a) and $\sigma_{\log k}^2=1$ cases (b--e) with varying $Ra, \lambda_{x}$, and $\lambda_{z}$. Increasing $Ra$ increases the number of patterns in the homogeneous (a) and heterogeneous case (b) thus increasing the interfacial area between the two fluids in contact, unlike in the other heterogeneous cases where merging occurs faster and fingers become distorted and dispersive (particularly in cases (c) and (d)). Instantaneous temperature snapshots are shown in figures 1--4 of the supplementary material for times $t=20,60,100,140$
• Figure 2: Time-averaged temperature maps at convective regime ($60<t<250$) for $Ra=1e3$, different $\sigma_{\log k}^{2}, \lambda_{x}$ and $\lambda_{z}$. Increasing $\sigma_{\log k}^{2}$ and $\lambda_x$ (especially with a smaller anisotropy ratio), promotes faster merging of the fingers. Instantaneous temperature snapshots are shown in the supplementary material for times $t=20,60,100,140$
• Figure 3: Heat flux through the boundary versus $Ra$ for the homogeneous case. A transition in the scaling is observed for $Ra> 10^{3}$ corresponding to the transition from organised patterns to chaotic patterns. Figures of heat fluxes versus time are shown in figures 5 and 6 of the supplementary material
• Figure 4: Flux versus $Ra$ for different correlation lengths and different variances of the permeability field. Flux is computed as the average of five realisations. Error bars show the standard deviation of the ensemble average. A higher $\sigma^2_{\log k}$ increases the error and causes greater separation between cases with different $\lambda_x$. The fluxes are also sensitive to $\lambda_x$: the larger it is, the higher the fluxes. For the same $\lambda_x$, cases with a smaller anisotropy ratio have bigger fluxes than those with a larger anisotropy ratio. Figures of heat flux versus time fro the heterogeneous cases are shown in figures 5 and 6 of the supplementary material
• Figure 5: Log-permeability ($\log k$), temperature $T$ and logarithm of the determinant of the strain tensor $\mathbf{E}$ normalised by its mean for $Ra=1e2$, $\sigma_{\log k}^{2}=1,3$ and different $\lambda_{x}$, $\lambda_{z}$. Values are time averaged in the convective regime ($60 < t <250$)