Pattern formation and coarsening dynamics in three-dimensional convective mixing in porous media

Convective mixing in porous media: a review of Darcy, pore-scale and Hele-Shaw studies

Convection-driven porous media flows are common in industrial processes and in nature. The multiscale and multiphase character of these systems and the inherent nonlinear flow dynamics make convection in porous media a complex phenomenon. As a result, a combination of different complementary approaches, namely theory, simulations and experiments, have been deployed to elucidate the intricate physics of convection in porous media. In this work, we review recent findings on mixing in fluid-saturated porous media convection. We focus on the dissolution of a heavy fluid layer into a lighter one, and we consider different flow configurations. We present Darcy, pore-scale and Hele-Shaw investigations inspired by geophysical processes. While the results obtained for Darcy flows match the dissolution behaviour predicted theoretically, Hele-Shaw and pore-scale investigations reveal a different and tangled scenario in which finite-size effects play a key role. Finally, we present recent numerical and experimental developments and we highlight possible future research directions. The findings reviewed in this work will be crucial to make reliable predictions about the long-term behaviour of dissolution and mixing in engineering and natural processes, which are required to tackle societal challenges such as climate change mitigation and energy transition.

Geometry-Driven Detection, Tracking and Visual Analysis of Viscous and Gravitational Fingers

Viscous and gravitational flow instabilities cause a displacement front to break up into finger-like fluids. The detection and evolutionary analysis of these fingering instabilities are critical in multiple scientific disciplines such as fluid mechanics and hydrogeology. However, previous detection methods of the viscous and gravitational fingers are based on density thresholding, which provides limited geometric information of the fingers. The geometric structures of fingers and their evolution are important yet little studied in the literature. In this article, we explore the geometric detection and evolution of the fingers in detail to elucidate the dynamics of the instability. We propose a ridge voxel detection method to guide the extraction of finger cores from three-dimensional (3D) scalar fields. After skeletonizing finger cores into skeletons, we design a spanning tree based approach to capture how fingers branch spatially from the finger skeletons. Finally, we devise a novel geometric-glyph augmented tracking graph to study how the fingers and their branches grow, merge, and split over time. Feedback from earth scientists demonstrates the usefulness of our approach to performing spatio-temporal geometric analyses of fingers.

Vigorous convection in porous media

The problem of convection in a fluid-saturated porous medium is reviewed with a focus on 'vigorous' convective flow, when the driving buoyancy forces are large relative to any dissipative forces in the system. This limit of strong convection is applicable in numerous settings in geophysics and beyond, including geothermal circulation, thermohaline mixing in the subsurface and heat transport through the lithosphere. Its manifestations range from 'black smoker' chimneys at mid-ocean ridges to salt-desert patterns to astrological plumes, and it has received a great deal of recent attention because of its important role in the long-term stability of geologically sequestered CO₂. In this review, the basic mathematical framework for convection in porous media governed by Darcy's Law is outlined, and its validity and limitations discussed. The main focus of the review is split between 'two-sided' and 'one-sided' systems: the former mimics the classical Rayleigh-Bénard set-up of a cell heated from below and cooled from above, allowing for detailed examination of convective dynamics and fluxes; the latter involves convection from one boundary only, which evolves in time through a series of regimes. Both set-ups are reviewed, accounting for theoretical, numerical and experimental studies in each case, and studies that incorporate additional physical effects are discussed. Future research in this area and various associated modelling challenges are also discussed.

Multidimensional Observations of Dissolution-Driven Convection in Simple Porous Media Using X-ray CT Scanning

We present an experimental study of dissolution-driven convection in a three-dimensional porous medium formed from a dense random packing of glass beads. Measurements are conducted using the model fluid system MEG/water in the regime of Rayleigh numbers, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Ra=2000{-}5000$$\end{document}Ra=2000-5000. X-ray computed tomography is applied to image the spatial and temporal evolution of the solute plume non-invasively. The tomograms are used to compute macroscopic quantities including the rate of dissolution and horizontally averaged concentration profiles, and enable the visualisation of the flow patterns that arise upon mixing at a spatial resolution of about (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\times 2\times 2)\,\hbox {mm}^3$$\end{document}2×2×2)mm3. The latter highlights that under this Ra regime convection becomes truly three-dimensional with the emergence of characteristic patterns that closely resemble the dynamical flow structures produced by high-resolution numerical simulations reported in the literature. We observe that the mixing process evolves systematically through three stages, starting from pure diffusion, followed by convection-dominated and shutdown. A modified diffusion equation is applied to model the convective process with an onset time of convection that compares favourably with the literature data and an effective diffusion coefficient that is almost two orders of magnitude larger than the molecular diffusivity of the solute. The comparison of the experimental observations of convective mixing against their numerical counterparts of the purely diffusive scenario enables the estimation of a non-dimensional convective mass flux in terms of the Sherwood number, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Sh=0.025Ra$$\end{document}Sh=0.025Ra. We observe that the latter scales linearly with Ra, in agreement with both experimental and numerical studies on thermal convection over the same Ra regime.

Steady Flux Regime During Convective Mixing in Three-Dimensional Heterogeneous Porous Media

Density-driven convective mixing in porous media can be influenced by the spatial heterogeneity of the medium. Previous studies using two-dimensional models have shown that while the initial flow regimes are sensitive to local permeability variation, the later steady flux regime (where the dissolution flux is relatively constant) can be approximated with an equivalent anisotropic porous media, suggesting that it is the average properties of the porous media that affect this regime. This work extends the previous results for two-dimensional porous media to consider convection in three-dimensional porous media. Through the use of massively parallel numerical simulations, we verify that the steady dissolution rate in the models of heterogeneity considered also scales as k v k h in three dimensions, where k v and k h are the vertical and horizontal permeabilities, respectively, providing further evidence that convective mixing in heterogeneous models can be approximated with equivalent anisotropic models.

Pattern formation in the geosciences

Pattern formation is a natural property of nonlinear and non-equilibrium dynamical systems. Geophysical examples of such systems span practically all observable length scales, from rhythmic banding of chemical species within a single mineral crystal, to the morphology of cusps and spits along hundreds of kilometres of coastlines. This article briefly introduces the general principles of pattern formation and argues how they can be applied to open problems in the Earth sciences. Particular examples are then discussed, which summarize the contents of the rest of this Theme Issue.