Recent studies reveal the central role of chaotic advection in controlling pore-scale processes including solute mixing and dispersion, chemical reactions, and biological activity. These dynamics have been observed in porous media (PM) with a continuous solid phase (such as porous networks) and PM comprising discrete elements (such as granular matter). However, a unified theory of chaotic advection across these continuous and discrete classes of PM is lacking. Key outstanding questions include: (i) topological unification of discrete and continuous PM; (ii) the impact of the non-smooth geometry of discrete PM; (iii) how exponential stretching arises at contact points in discrete PM; (iv) how fluid folding arises in continuous PM; (v) the impact of discontinuous mixing in continuous PM; and (vi) generalised models for the Lyapunov exponent in both PM classes. We address these questions via a unified theory of pore-scale chaotic advection. We show that fluid stretching and folding (SF) in discrete and continuous PM arise via the topological complexity of the medium. Mixing in continuous PM manifests as discontinuous mixing through a combination of SF and cutting and shuffling (CS) actions, but the rate of mixing is governed by SF only. Conversely, discrete PM involves SF motions only. These mechanisms are unified by showing that continuous PM is analogous to discrete PM with smooth, finite contacts. This unified theory provides insights into the pore-scale chaotic advection across a broad class of porous materials and points to design of novel porous architectures with tuneable mixing and transport properties.

The evaporation of liquid from within a porous medium is a complicated process involving coupled capillary flow, vapour diffusion and phase change. Different drying behaviour is observed at different stages during the process. Initially, liquid is drawn to the surface by capillary forces, where it evaporates at a near constant rate; thereafter, a drying front recedes into the material, with a slower net evaporation rate. Modelling drying porous media accurately is challenging due to the multitude of relevant spatial and temporal scales, and the large number of constitutive laws required for model closure. Key aspects of the drying process, including the net evaporation rate and the time of the sudden transition between stages, are not well understood or reliably predicted. We derive simplified mathematical models for both stages of this drying process by systematically reducing an averaged continuum multi-phase flow model, using the method of matched asymptotic expansions, in the physically relevant limit of slow vapour diffusion relative to the local evaporation rate (the large-Péclet-number limit). By solving our reduced models, we compute the evolving net evaporation rate, fluid fluxes and saturation profiles, and estimate the transition time to be when the initial constant-rate-period model ceases to be valid. We additionally characterise properties of the constitutive laws that affect the qualitative drying behaviour: the model is shown to exhibit a receding-front period only if the relative permeability for the liquid phase decays sufficiently quickly relative to the blow up in the capillary pressure as the liquid saturation decreases.

An experimental study was conducted to investigate the impingement of a vortex ring onto a porous wall by laser-induced fluorescence and particle image velocimetry. The effects of different Reynolds numbers (${{Re}}_{\it\Gamma } = 700$ and $1800$) and hole diameters ($d_{h}^{*} = 0.067$, $0.10$, $0.133$ and $0.20$) on the flow characteristics were examined at a constant porosity ($\phi = 0.75$). To characterise fluid transport through a porous wall, we recall the model proposed by Naaktgeboren, Krueger & Lage (2012, J. Fluid Mech., vol. 707, 260–286), which shows rough agreement with the experimental results due to the absence of vortex ring characteristics. This highlights the need for a more accurate model to correlate the losses in kinetic energy ($\Delta E^{*}$) and impulse ($\Delta I^{*}$) resulting from the vortex ring–porous wall interaction. Starting from Lamb’s vortex ring model and considering the flow transition from the upstream laminar state to the downstream turbulent state caused by the porous wall disturbance, a new model is derived theoretically: $\Delta E^{*} = 1 - k(1 - \Delta I^{*})^2$, where $k$ is a parameter dependent on the dimensionless core radius $\varepsilon$, with $k = 1$ when no flow state change occurs. This new model effectively correlates $\Delta E^{*}$ and $\Delta I^{*}$ across more than 70 cases from current and previous experiments, capturing the dominant flow physics of the vortex ring–porous wall interaction.

We present a theoretical framework for porous media gravity currents propagating over rigid curvilinear surfaces. By reducing the flow dynamics to low-dimensional models applicable on surfaces where curvature effects are negligible, we demonstrate that, for finite-volume releases, the flow behaviour in both two-dimensional and axisymmetric configurations is primarily governed by the ratio of the released viscous fluid volume to the characteristic volume of the curvilinear surface. Our theoretical predictions are validated using computational fluid dynamics simulations based on a sharp-interface model for macroscopic flow in porous media. In the context of carbon dioxide geological sequestration, our findings suggest that wavy cap rock geometries can enhance trapping capacity compared with traditional flat-surface assumptions, highlighting the importance of incorporating realistic topographic features into subsurface flow models.

Assemblies of slender structures forming brushes are common in daily life from sweepers to pastry brushes and paintbrushes. These types of porous objects can easily trap liquid in their interstices when removed from a liquid bath. This property is exploited to transport liquids in many applications, ranging from painting, dip-coating and brush-coating to the capture of nectar by bees, bats and honeyeaters. Rationalising the viscous entrainment flow beyond simple scaling laws is complex due to the multiscale structure and the multidirectional flow. Here, we provide an analytical model, together with precision experiments with ideal rigid brushes, to fully characterise the flow through this anisotropic porous medium as it is withdrawn from a liquid bath. We show that the amount of liquid entrained by a brush varies non-monotonically during the withdrawal at low speed, is highly sensitive to the different parameters at play and is very well described by the model without any fitting parameter. Finally, an optimal brush geometry maximising the amount of liquid captured at a given retraction speed is derived from the model and experimentally validated. These optimal designs open routes towards efficient liquid-manipulating devices.

Based on data from pore-resolved direct numerical simulation of turbulent flow over mono-disperse random sphere packs, we evaluate the budgets of the double-averaged turbulent kinetic energy (TKE) and the wake kinetic energy (WKE). While TKE results from temporal velocity fluctuations, WKE describes the kinetic energy in spatial variations of the time-averaged flow field. We analyse eight cases which represent sampling points within a parameter space spanned by friction Reynolds numbers $Re_\tau \in [150, 500]$ and permeability Reynolds numbers $Re_K \in [0.4, 2.8]$. A systematic exploration of the parameter space is possible by varying the ratio between flow depth and sphere diameter $h/D \in \{ 3, 5, 10 \}$. With roughness Reynolds numbers of $k_s^+ \in [20,200]$, the simulated cases lie within the transitionally or fully rough regime. Revisiting the budget equations, we identify a WKE production mechanism via viscous interaction of the flow field with solid surfaces. The scaling behaviour of different processes over $Re_K$ and $Re_\tau$ suggests that this previously unexplored mechanism has a non-negligible contribution to the WKE production. With increasing $Re_K$, progressively more WKE is transferred into TKE by wake production. A near-interface peak in the TKE production, however, primarily results from shear production and scales with interface-related scales. Conversely, further above the sediment bed, the TKE budget terms of cases with comparable $Re_\tau$ show similarity under outer-scaling. Most transport processes relocate energy in the near-interface region, whereas pressure diffusion propagates TKE and WKE into deeper regions of the sphere pack.

Previous studies claimed that the non-monotonic effects of wettability came mainly from the heterogeneity of geometries or flow conditions on multiphase displacements in porous media. For macroscopic homogeneous porous media, without permeability contrast or obvious preferential flow pathways, most pore-scale evidence showed a monotonic trend of the wettability effect. However, this work reports transitions from monotonic to non-monotonic wettability effects when the dimension of the model system rises from two-dimensional (2-D) to three-dimensional (3-D), validated by both the network modelling and the microfluidic experiments. The mechanisms linking the pore-scale events to macroscopic displacement patterns have been analysed through direct simulations. For 2-D porous media, the monotonic effect of wettability comes from the consistent transition pattern for the full range of capillary numbers $Ca$, where the capillary fingering mode transitions to the compact displacement mode as the contact angle $\theta$ decreases. Yet, it is indicated that the 3-D porous geometries, even though homogeneous without permeability contrast or obvious preferential flow pathways, introduce a different $Ca$–$\theta$ phase diagram with new pore-scale events, such as the coupling of capillary fingering with snap-off during strong drainage, and frequent snap-off events during strong imbibition. These events depend strongly on geometric confinements and capillary numbers, leading to the non-monotonicity of wettability effects. Our findings provide new insights into the multiphase displacement dependent on wettability in various natural porous media and offer design principles for engineering artificial porous media to achieve desired immiscible displacement behaviours.

The interaction between the dynamics of a flame front and the acoustic field within a combustion chamber represents an aerothermochemical problem with the potential to generate hazardous instabilities, which limit burner performance by constraining design and operational parameters. The experimental configuration described here involves a laminar premixed flame burning in an open–closed slender tube, which can also be studied through simplified modelling. The constructive coupling of the chamber acoustic modes with the flame front can be affected via strategic placement of porous plugs, which serve to dissipate thermoacoustic instabilities. These plugs are lattice-based, 3-D-printed using low-force stereolithography, allowing for complex geometries and optimal material properties. A series of porous plugs was tested, with variations in their porous density and location, in order to assess the effects of these variables on viscous dissipation and acoustic eigenmode variation. Pressure transducers and high-speed cameras are used to measure oscillations of a stoichiometric methane–air flame ignited at the tube’s open end. The findings indicate that the porous medium is effective in dissipating both pressure amplitude and flame-front oscillations, contingent on the position of the plug. Specifically, the theoretical fluid mechanics model is developed to calculate frequency shifts and energy dissipation as a function of plug properties and positioning. The theoretical predictions show a high degree of agreement with the experimental results, thereby indicating the potential of the model for the design of dissipators of this nature and highlighting the first-order interactions of acoustics, viscous flow in porous media and heat transfer processes.

Mixing-induced reactions play an important role in a wide range of porous media processes. Recent advances have shown that fluid flow through porous media leads to chaotic advection at the pore scale. However, how this impacts Darcy-scale reaction rates is unknown. Here, we measure the reaction rates in steady mixing fronts using a chemiluminescence reaction in index-matched three-dimensional porous media. We consider two common mixing scenarios for reacting species, flowing either in parallel in a uniform flow or towards each other in a converging flow. We study the reactive properties of these fronts for a range of Péclet numbers. In both scenarios, we find that the reaction rates significantly depart from the prediction of hydrodynamic dispersion models, which obey different scaling laws. We attribute this departure to incomplete mixing effects at the pore scale, and propose a mechanistic model describing the pore-scale deformations of the front triggered by chaotic advection and their impact on the reaction kinetics. The model shows good agreement with the effective Darcy-scale reaction kinetics observed in both uniform and converging flows, opening new perspectives for upscaling reactive transport in porous media.

Viscous flow through high-permeability channels occurs in many environmental and industrial applications, including carbon sequestration, groundwater flow and enhanced oil recovery. In this work, we study the displacement of a less-viscous fluid by a more-viscous fluid in a layered porous medium in a rectilinear configuration, where two low-permeability layers sandwich a higher-permeability layer. We derive a theoretical model that is validated using corroborative laboratory experiments, when the influence of the density difference is negligible. We find that the location of the propagating front increases with time according to a power-law form $x_f \propto t^{1/2}$, while the fluid–fluid interface exhibits a self-similar shape, when the motion of the displaced fluid is negligible in an unconfined porous medium. In the experimental set-up, distinct permeability layers were constructed using various sizes of spherical glass beads. The working fluids comprised fresh water as the less-viscous ambient fluid, and a glycerine–water mixture as the more-viscous injecting fluid. Our experimental measurement show a better match with the theory for the experiments performed at low Reynolds numbers and with permeable boundaries in the far field.

In this work we focus on expected flow in porous formations with highly conductive isolated fractures, which are of non-negligible length compared with the scales of interest. Accordingly, the definition of a representative elementary volume (REV) for flow and transport predictions may not be possible. Recently, a non-local kernel-based theory for flow in such formations has been proposed. There, fracture properties like their expected pressure are represented as field quantities. Unlike existing models, where fractures are assumed to be small compared with the scale of interest, a non-local kernel function is used to quantify the expected flow transfer between a point in the fracture domain and a potentially distant point in the matrix continuum. The transfer coefficient implied by the kernel is a function of the fracture characteristics that are in turn captured statistically. So far the model has successfully been applied for statistically homogeneous cases. In the present work we demonstrate the applicability for heterogeneous cases with spatially varying fracture statistics. Moreover, a scaling law is presented that relates the transfer coefficient to the fracture characteristics. Test cases involving discontinuously and continuously varying fracture statistics are presented, and the validity of the scaling law is demonstrated.

We derive boundary conditions for two-dimensional parallel and non-parallel flows at the interface of a homogeneous and isotropic porous medium and an overlying fluid layer by solving a macroscopic closure problem based on the asymptotic solution to the generalised transport equations (GTE) in the interfacial region. We obtained jump boundary conditions at the effective sharp surface dividing the homogeneous fluid and porous layers for either the Darcy or the Darcy–Brinkman equations. We discuss the choice of the location of the dividing surface and propose choices which reduce the distance with the GTE solutions. We propose an ad hoc expression of the permeability distribution within the interfacial region which enables us to preserve the invariance of the fluid-side-averaged velocity profile with respect to the radius $r_0$ of the averaging volume. Solutions to the GTE, equipped with the proposed permeability distribution, compare favourably with the averaged solutions to the pore-scale simulations when the interfacial thickness $\delta$ is adjusted to $r_0$. Numerical tests for parallel and non-parallel flows using the obtained jump boundary conditions or the generalised transport equations show quantitative agreement with the GTE solutions, with experiments and pore-scale simulations. The proposed model of mass and momentum transport is predictive, requiring solely information on the bulk porosity and permeability and the location of the solid matrix of the porous medium. Our results suggest that the Brinkman corrections may be avoided if the ratio $a=\delta /\delta _B$ of the thickness $\delta$ of the interfacial region to the Brinkman penetration depth $\delta _B$ is large enough, as the Brinkman sub-layer is entirely contained within the interfacial region in that case. Our formulation has been extended to anisotropic porous media and can be easily dealt with for three-dimensional configurations.

Porous membranes, like nets or filters, are thin structures that allow fluid to flow through their pores. Homogenisation can be used to rigorously link the flow velocity with the stresses on the membrane via several coefficients (e.g. permeability and slip) stemming from the solution of Stokes problems at the pore level. For a periodic microstructure, the geometry of a single pore determines these coefficients for the whole membrane. However, many biological membranes are not periodic, and the porous microstructure of industrial membranes can be modified to address specific needs, resulting in non-periodic patterns of solid inclusions and pores. In this case, multiple microscopic calculations are needed to retrieve the local non-periodic membrane properties, negatively affecting the efficiency of the homogenised model. In this paper, we introduce an adjoint-based procedure that drastically reduces the computational cost of these operations by computing the pore-scale solution’s first- and second-order sensitivities to geometric modifications. This adjoint-based technique only requires the solution of a few problems on a reference geometry and allows us to find the homogenised solution on any number of modified geometries. This new adjoint-based homogenisation procedure predicts the macroscopic flow around a thin aperiodic porous membrane at a fraction of the computational cost of classical approaches while maintaining comparable accuracy.

A new lattice Boltzmann model (LBM) is presented to describe chemically reacting multicomponent fluid flow in homogenised porous media. In this work, towards further generalising the multicomponent reactive lattice Boltzmann model, we propose a formulation which is capable of performing reactive multicomponent flow computation in porous media at the representative elementary volume (REV) scale. To that end, the submodel responsible for interspecies diffusion has been upgraded to include Knudsen diffusion, whereas the kinetic equations for the species, the momentum and the energy have been rewritten to accommodate the effects of volume fraction of porous media through careful choice of the equilibrium distribution functions. Verification of the mesoscale kinetic system of equations by a Chapman–Enskog analysis reveals that at the macroscopic scale, the homogenised Navier–Stokes equations for compressible multicomponent reactive flows are recovered. The dusty gas model (DGM) capability hence formulated is validated over a wide pressure range by comparison of experimental flow rates of component species counter diffusing through capillary tubes. Next, for developing a capability to compute heterogeneous reactions, source terms for maintaining energy and mass balance across the fluid phase species and the surface adsorbed phase species are proposed. The complete model is then used to perform detailed chemistry simulations in porous electrodes of a solid oxide fuel cell (SOFC), thereby predicting polarisation curves which are of practical interest.

Thermo-responsive hydrogels are smart materials that rapidly switch between hydrophilic (swollen) and hydrophobic (shrunken) states when heated past a threshold temperature, resulting in order-of-magnitude changes in gel volume. Modelling the dynamics of this switch is notoriously difficult and typically involves fitting a large number of microscopic material parameters to experimental data. In this paper, we present and validate an intuitive, macroscopic description of responsive gel dynamics and use it to explore the shrinking, swelling and pumping of responsive hydrogel displacement pumps for microfluidic devices. We finish with a discussion on how such tubular structures may be used to speed up the response times of larger hydrogel smart actuators and unlock new possibilities for dynamic shape change.

Dispersion in spatio-temporal random flows is dominated by the competition between spatial and temporal velocity resets along particle paths. This competition admits a range of normal and anomalous dispersion behaviours characterised by the Kubo number, which compares the relative strength of spatial and temporal velocity resets. To shed light on these behaviours, we develop a Lagrangian stochastic approach for particle motion in spatio-temporally fluctuating flow fields. For space–time separable flows, particle motion is mapped onto a continuous time random walk (CTRW) for steady flow in warped time, which enables the upscaling and prediction of the large-scale dispersion behaviour. For non-separable flows, we measure Lagrangian velocities in terms of a new sampling variable, the average number of velocity transitions (both temporal and spatial) along pathlines, which renders the velocity series Markovian. Based on this, we derive a Lagrangian stochastic model that represents particle motion as a coupled space–time random walk, that is, a CTRW for which the space and time increments are intrinsically coupled. This approach sheds light on the fundamental mechanisms of particle motion in space–time variable flows, and allows for its systematic quantification. Furthermore, these results indicate that alternative strategies for the analysis of Lagrangian velocity data using new sampling variables may facilitate the identification of (hidden) Markov models, and enable the development of reduced-order models for otherwise complex particle dynamics.

In soft porous media, deformation drives solute transport via the intrinsic coupling between flow of the fluid and rearrangement of the pore structure. Solute transport driven by periodic loading, in particular, can be of great relevance in applications including the geomechanics of contaminants in the subsurface and the biomechanics of nutrient transport in living tissues and scaffolds for tissue engineering. However, the basic features of this process have not previously been systematically investigated. Here, we fill this hole in the context of a one-dimensional model problem. We do so by expanding the results from a companion study, in which we explored the poromechanics of periodic deformations, by introducing and analysing the impact of the resulting fluid and solid motion on solute transport. We first characterise the independent roles of the three main mechanisms of solute transport in porous media – advection, molecular diffusion and hydrodynamic dispersion – by examining their impacts on the solute concentration profile during one loading cycle. We next explore the impact of the transport parameters, showing how these alter the relative importance of diffusion and dispersion. We then explore the loading parameters by considering a range of loading periods – from slow to fast, relative to the poroelastic time scale – and amplitudes – from infinitesimal to large. We show that solute spreading over several loading cycles increases monotonically with amplitude, but is maximised for intermediate periods because of the increasing poromechanical localisation of the flow and deformation near the permeable boundary as the period decreases.

We report an anomalous capillary phenomenon that reverses typical capillary trapping via nanoparticle suspension and leads to a counterintuitive self-removal of non-aqueous fluid from dead-end structures under weakly hydrophilic conditions. Fluid interfacial energy drives the trapped liquid out by multiscale surfaces: the nanoscopic structure formed by nanoparticle adsorption transfers the molecular-level adsorption film to hydrodynamic film by capillary condensation, and maintains its robust connectivity, then the capillary pressure gradient in the dead-end structures drives trapped fluid motion out of the pore continuously. The developed mathematical models agree well with the measured evolution dynamics of the released fluid. This reversing capillary trapping phenomenon via nanoparticle suspension can be a general event in a random porous medium and could dramatically increase displacement efficiency. Our findings have implications for manipulating capillary pressure gradient direction via nanoparticle suspensions to trap or release the trapped fluid from complex geometries, especially for site-specific delivery, self-cleaning, or self-recover systems.

A point force acting on a Brinkman fluid in confinement is always counterbalanced by the force on the porous medium, the force on the walls and the stress at open boundaries. We discuss the distribution of those forces in different geometries: a long pipe, a medium with a single no-slip planar boundary, a porous sphere with an open boundary and a porous sphere with a no-slip wall. We determine the forces using the Lorentz reciprocal theorem and additionally validate the results with explicit analytical flow solutions. We discuss the relevance of our findings for cellular processes such as cytoplasmic streaming and centrosome positioning.

During a rainfall event, water infiltrates into the ground where it accumulates in porous rocks. This accumulation pushes the underlying groundwater towards neighbouring streams, where it runs to the sea. After the rain has stopped, the aquifer gradually releases its excess water, as the water table relaxes, until the next rain. In the absence of recharge, the water table would eventually reach its horizontal equilibrium position. The volume of groundwater stored above this level, which we call the active volume, sustains the river between two rainfall events. In this article, we use an experimental aquifer recharged by artificial rain to investigate how this active volume depends on the rainfall rate. Restricting our analysis to the steady-state regime, wherein the discharge into the stream balances rainfall, we explore a broad range of rainfall rates, for which the water table deforms significantly. We find that the active volume of water stored in the aquifer decreases with its depth. Using conformal mapping, we derive the flow equations and develop a numerical procedure that accounts for the active volume of groundwater in our experiments. In the case of an infinitely deep aquifer, the problem admits a closed-form solution, which provides a satisfying estimate of the active volume when the aquifer's depth is at least half its width. In the general case, a rougher estimate results from the energy balance of the dissipative groundwater flow.