Universal transport laws in buoyancy-driven porous mixing

Abstract

Buoyancy-driven convection in porous media governs heat and mass transport in a wide range of natural and engineered systems, from groundwater aquifers and geothermal reservoirs to carbon storage in geological formations and flows through planetary interiors. Yet the transient regime, in which fingering flows emerge and transport is strongly enhanced, is still described largely through empirical scaling laws, limiting predictive capability across conditions. Here we show that transient porous buoyancy-driven mixing obeys exact time-dependent balances that couple transport, flow intensity, and scalar dissipation. These balances remain accurate when restricted to the actively mixing layer, revealing that the essential dynamics are localized within a finite region. Leveraging these results, we derive a minimal one-parameter closure for the mean scalar field. The theory we propose predicts self-similar mean profiles, universal second-order statistics, and a linear transport law without case-by-case tuning. Direct numerical simulations up to spatial resolution of $2048\times2048\times16384$ points validate these predictions. Our results place transient porous mixing on a predictive footing, showing how macroscopic transport laws emerge from exact balances and self-similar dynamics, and provide a general framework for buoyancy-driven transport in porous media.

Figures (5)

• Figure 1: (a) Conceptualized hydrogeology of the River Murray basin area (New South Wales, Australia), adapted from Narayan et al. narayan1995simulation. Inset: modelling of the saline seepage through the bottom of Lake Ranfurly West. The high-permeability sands aquifer is confined by two low-permeability layers. Lake Ranfurly West supplies high-salt-concentration (high-density) water from the top, while low-salinity (low-density) groundwater is present in the aquifer. (b) Evolution of the density distribution in time ($t$) for simulation S1 ($\mathop{\mathrm{\mathit{Ra}}}\nolimits=3.2\times10^4$, quantities shown in dimensionless units). (b-i) Initial flow configuration with indication of gravity ($\mathbf{g}$), of the reference frame ($x,y,z$) and of the domain extension ($L_x,L_y,L_z$). (b-ii) A developed field during the convective phase, with indication of the extension of the mixing region, $h$. (b-iii) Field after the fingers reached the boundaries.
• Figure 2: Verification of the thermal fluctuations relations against numerical simulations ($\mathop{\mathrm{\mathit{Ra}}}\nolimits=3.2\times10^4$, corresponding to simulation S1, see Tab. \ref{['tab:numdet_text']} for further details). The validity of Eqs. \ref{['eq:peb2']} and \ref{['eq:ref32']} and the validity of Eqs. \ref{['eq:ref32cd']} is verified in (a) and (b), respectively. The validity of Eqs. \ref{['eq:peb2b']} and \ref{['eq:ref32b']} and the validity of Eqs. \ref{['eq:budgetb2']} is verified in (c) and (d), respectively.
• Figure 3: Profiles obtained at $\mathop{\mathrm{\mathit{Ra}}}\nolimits=3.2\times 10^4$ (simulation S1, panels a and b) and $\mathop{\mathrm{\mathit{Ra}}}\nolimits=2.56\times 10^5$ (simulation S4, panels c and d) for $0.5\le t \le 2.0$. Original profiles (panels a,c) and profiles rescaled using the self-similar coordinate $\xi$ (panels b,d) are shown. The blue dashed line represents the self-similar solution derived in Eq. \ref{['eq:cases']}.
• Figure 4: Verification of Eq. \ref{['eq:w2bis']}, with the blue dashed line given by $9a/16(1-\xi^2)^2$. Simulations S1 ($\mathop{\mathrm{\mathit{Ra}}}\nolimits=3.2\times10^4$) and S4 ($\mathop{\mathrm{\mathit{Ra}}}\nolimits=2.56\times10^5$) are considered in the top and bottom panels, respectively. Additional cases are reported in Supplementary information.
• Figure 5: Evolution of the mixing-layer Nusselt number $\mathop{\mathrm{\mathit{Nu}}}\nolimits_m$ in time (a) and as a function of the instantaneous Rayleigh number $\mathop{\mathrm{\mathit{Ra}}}\nolimits_m$ (b) for the simulations performed (symbols). Equation (\ref{['eq:numml2']}) (dashed line in panel b) provides an excellent description of the collapsed transport data, using a single coefficient, $a$ (details in Supplementary information).