We investigate the heat transport in porous media convection over a wide Rayleigh--Darcy number range of $26.8\leq Ra\leq 2.62\times 10^5$, and a Darcy number range of $6.18\times10^{-7}\leq Da\leq 1.21\times 10^{-5}$. In the experiments, we employ 3D-printed lattice structures as the solid porous matrix and water as the working fluid. Quantitative analyses of the porous medium Nusselt number $Nu_m$ and local temperature statistics reveal that the present system undergoes a transition through five distinct regimes: I. Conduction, II. Convection, III. Oscillation, IV. Transition, V. Classical Rayleigh--Bénard convection. This transitional process bridges the gap between Rayleigh--Darcy-like behaviour and Rayleigh--Bénard-like behaviour in porous media convection. By varying the permeability of the matrix, we further examine the role of the Darcy number $Da$, which turns out to have a profound impact on the transitional processes across different regimes. Flow field measurements reveal that the flow structures within Regime IV and Regime V evolve from several horizontally stacked convection rolls to a single-roll structure, and the pore-scale Reynolds number both exceeds unity in these two regimes. Finally, we report the corresponding phase diagram in the $Ra$-$Da$ space.
Recent planetary data and geophysical modelling suggest that hydrothermal activity is ongoing under the ice crust of Enceladus, one of Saturn's moons. According to these models, hydrothermal flow in the porous, rocky core of the satellite is driven by tidal deformation that induces dissipation and volumetric internal heating. Despite the effort in the modelling of Enceladus' interior, systematic understanding---and even basic scaling laws---of internally-heated porous convection and hydrothermal activity are still lacking. In this article, using an idealised model of an internally-heated porou…
In the above paper the authors treat the natural convection boundary layer flow in a Darcy-Brinkman-Forchheimer porous medium. In the energy equation the radiation effect has been taken into account. A two-parameter perturbation method is used for the solution of the equations. The first order results are presented in tables and figures. This is an interesting work but there are some weak points which are presented below:
In a range of physical systems, the first instability in Rayleigh-Bernard convection between nearly thermally insulating horizontal plates is large scale. This holds for thermal convection of fluids saturating porous media. Large-scale thermal convection in a horizontal layer is governed by remarkably similar equations both in the presence of a porous matrix and without it, with only one additional term for the latter case, which, however, vanishes under certain conditions (e.g., two-dimensional flows or infinite Prandtl number). We provide a rigorous derivation of long-wavelength equations fo…
We present a numerical study of convection in a horizontal layer comprising a fluid-saturated porous bed overlain by an unconfined fluid layer. Convection is driven by a vertical, destabilising temperature difference applied across the whole system, as in the canonical Rayleigh-Bénard problem. Numerical simulations are carried out using a single-domain formulation of the two-layer problem based on the Darcy-Brinkman equations. We explore the dynamics and heat flux through the system in the limit of large Rayleigh number, but small Darcy number, such that the flow exhibits vigorous convection i…
The problem of steady mixed convection boundary-layer flow on a cooled vertical permeable circular cylinder embedded in a fluid-saturated porous medium is studied. Here, we evaluate the flow and heat transfer characteristics numerically for various values of the governing parameters and demonstrate the existence of dual solutions beyond a critical point.
Convection heat transfer in porous media is a universal phenomenon in nature, and it is also frequently encountered in scientific and engineering fields. An in-depth understanding of the fundamental mechanism of convection heat transfer in porous media requires efficient and powerful numerical tools. In this paper, a cascaded lattice Boltzmann (CLB) method for convection heat transfer in porous media at the representative elementary volume (REV) scale is presented. In the CLB method, the flow field is solved by an isothermal CLB model with the D2Q9 lattice based on the generalized non-Darcy mo…
We report a numerical study of Rayleigh--Bénard convection through random porous media using pore-scale modelling, focusing on the Lagrangian dynamics of fluid particles and heat transfer for varied porosities $ϕ$. Due to the interaction between the porous medium and the coherent flow structures, the flow is found to be highly heterogeneous, consisting of convection channels with strong flow strength and stagnant regions with low velocities. The modifications of flow field due to porous structure have a significant influence on the dynamics of fluid particles. Evaluation of the particle displa…
A ubiquitous arrangement in nature is a free-flowing fluid coupled to a porous medium, for example a river or lake lying above a porous bed. Depending on the environmental conditions, thermal convection can occur and may be confined to the clear fluid region, forming shallow convection cells, or it can penetrate into the porous medium, forming deep cells. Here, we combine three complementary approaches -- linear stability analysis, fully nonlinear numerical simulations, and a coarse-grained model -- to determine the circumstances that lead to each configuration. The coarse-grained model yields…
We perform a numerical study of the heat transfer and flow structure of Rayleigh-Bénard (RB) convection in (in most cases regular) porous media, which are comprised of circular, solid obstacles located on a square lattice. This study is focused on the role of porosity $ϕ$ in the flow properties during the transition process from the traditional RB convection with $ϕ=1$ (so no obstacles included) to Darcy-type porous-media convection with $ϕ$ approaching 0. Simulations are carried out in a cell with unity aspect ratio, for the Rayleigh number $Ra$ from $10^5$ to $10^{10}$ and varying porosities…
The paper presents a theoretical study using the Karman-Pohlhausen method for describing the transient heat exchange between the boundary-layer free convection and a vertical flat plate embedded in a porous medium. The unsteady behavior is developed after the generation of an impulsive heat flux step at the right-hand side of the plate. Two cases are considered according to whether the plate has a finite thickness or no thickness. The time and space evolution of the interface temperature is evidenced.
In the above paper the author treats the boundary layer flow along a vertical flat plate, immersed in a Darcy Brinkman Forchheimer porous medium. The porosity and the permeability of the porous medium are variable across the boundary layer. In addition a magnetic field with constant strength is applied normal to the plate. The fluid temperature at the plate is constant and different from that of the ambient fluid. This temperature difference creates a buoyancy force and the flow is characterized as mixed convection. The partial differential equations of the boundary layer flow are transformed …
In this paper, a double multiple-relaxation-time lattice Boltzmann model is developed for simulating transient solid-liquid phase change problems in porous media at the representative elementary volume scale. The model uses two different multiple-relaxation-time lattice Boltzmann equations, one for the flow field and the other for the temperature field with nonlinear latent heat source term. The model is based on the generalized non-Darcy formulation, and the solid-liquid phase change interface is traced through the liquid fraction which is determined by the enthalpy method. The model is valid…
In this paper we propose a continuous data assimilation (downscaling) algorithm for the Bénard convection in porous media using only coarse mesh measurements of the temperature. In this algorithm, we incorporate the observables as a feedback (nudging) term in the evolution equation of the temperature. We show that under an appropriate choice of the nudging parameter and the size of the mesh, and under the assumption that the observed data is error free, the solution of the proposed algorithm converges at an exponential rate, asymptotically in time, to the unique exact unknown reference solutio…
We perform linear and nonlinear stability analysis for thermal convection in a fluid overlying a saturated porous medium. We use a coupled system, with the Navier-Stokes equations and Darcy's equation governing the free-flow and the porous regions respectively. Incorporating a dynamic pressure term in the Lions interface condition (which specifies the normal force balance across the fluid-medium interface) permits an energy bound on the typically uncooperative nonlinear advection term, enabling new nonlinear stability results. Within certain regimes, the nonlinear stability thresholds agree cl…
The study investigates the problem of magnetohydrodynamic (MHD) free convection over a non-isothermal axisymmetric body under the action of transverse magnetic field. The body is embedded in a porous medium saturated with electrically conducting non-Newtonian power law fluid. In order to obtain similarity solution, it is assumed that the viscosity of the fluid decays exponentially with temperature. The qualitative results are illustrated for a vertical flat plate, horizontal cylinder and sphere.
This investigation deals with the analysis of stagnation point heat transfer and corresponding flow features of hydromagnetic viscous incompressible fluid over a vertical shrinking sheet. The considered sheet is assumed to be permeable and subject to addition of stagnation point to control the generated vorticity in the boundary layer. The sheet is placed on the right side of the fluid saturated porous medium which is having permeability of specified form. Nonlinear convection waves in the flow field are realized due to the envisaged nonlinear relation between density and temperature. The equa…
Natural convection in porous media is a fundamental process for the long-term storage of CO2 in deep saline aquifers. Typically, details of mass transfer in porous media are inferred from the numerical solution of the volume-averaged Darcy-Oberbeck-Boussinesq (DOB) equations, even though these equations do not account for the microscopic properties of a porous medium. According to the DOB equations, natural convection in a porous medium is uniquely determined by the Rayleigh number. However, in contrast with experiments, DOB simulations yield a linear scaling of the Sherwood number with the Ra…
Double-diffusive convection in porous media is a common phenomenon in nature, and has received considerable attention in a wide variety of engineering applications. In this paper, a multiple-relaxation-time (MRT) lattice Boltzmann (LB) model is developed for simulating double-diffusive convection in porous media at the representative elementary volume scale. The MRT-LB model is constructed in the framework of the triple-distribution-function approach: the velocity field, the temperature and concentration fields are solved separately by three different MRT-LB equations. The present model has tw…
We study the fractionation of two components between a well-mixed gas and a saturated convecting porous layer. Motivated by geological carbon dioxide (CO$_2$) storage we assume that convection is driven only by the dissolved concentration of the first component, while the second acts as a tracer with increased diffusivity. Direct numerical simulations for convection at high Rayleigh numbers reveal that the partitioning of the components, in general, does not follow a Rayleigh fractionation trend, as commonly assumed. Initially, increases in tracer diffusivity also increase its flux, because th…
A bidisperse porous medium is one with two porosity scales. There are the usual pores known as macro pores but also cracks or fissures in the skeleton which give rise to micro pores. In this article we develop and analyse a model for thermal convection where a layer of viscous incompressible fluid overlies a layer of bidisperse porous medium. Care has to be taken with the boundary conditions at the interface of the fluid and the porous material and this aspect is investigated. The situation is one in a layer which is heated from below and under appropriate conditions bimodal neutral curves are…
New insight into the contribution of the microscale vortex evolution towards convection heat transfer in porous media is presented in this paper. The objective is to determine how the microscale vortices influence convection heat transfer in turbulent flow inside porous media. The microscale temperature distribution is analyzed using flow visualization in 2D using streamlines and in 3D using Q structures. The pertinent observations are supplemented with the comparison of surface skin friction and heat transfer using: (1) surface skin friction lines and (2) joint PDF of pressure and skin fricti…
We investigate theoretically the nonlinear state of ideal straight rolls in the Rayleigh-Bénard system of a fluid layer heated from below with a porous medium using a Galerkin method. Applying the Oberbeck-Boussinesq approximation, binary mixtures with positive separation ratio are studied and compared to one-component fluids. Our results for the structural properties of roll convection resemble qualitatively the situation in the Rayleigh--Bénard system without porous medium except for the fact that the streamlines of binary mixtures are deformed in the so-called Soret regime. The deformation …
We numerically examine solutal convection in porous media, driven by the dissolution of carbon dioxide (CO2) into water---an effective mechanism for CO2 storage in saline aquifers. Dissolution is associated with slow diffusion of free-phase CO2 into the underlying aqueous phase followed by density-driven convective mixing of CO2 throughout the water-saturated layer. We study the fluid dynamics of CO2 convection in the single aqueous-phase region. A comparison is made between two different boundary conditions in the top of the formation: (i) a constant, maximum aqueous-phase concentration of CO…
Gravity-driven convection of fluid at parameters near its thermodynamic critical point inside a porous layer heated from below (Rayleigh-Darcy convection) is studied. The fluid having the temperature slightly above the critical one is one-component. The hydrodynamic model describing a high compressible fluid phase at variable physical properties inside a solid matrix at a uniform porosity is analyzed. A near-critical fluid is assumed to be the van der Waals gas. In the limit of small variations in the density and thermodynamic coefficients where the Oberbeck-Boussinesq approximation is applica…
Fully developed forced convective flow in a channel filled with porous material bounded by two impermeable walls subject to constant heat flux is considered. We consider Brinkman-Forchhimer equation to govern the flow inside the porous medium which accounts for the presence of inertial term. We assume that the porous medium is anisotropic in nature and permeability is varying along all the directions, correspondingly, permeability will be a matrix which appear as a positive semi-definite matrix in the momentum equation. The anisotropic nature of the porous medium is characterized by anisotropi…
In this paper, global sensitivity analysis (GSA) and uncertainty quantification (UQ) have been applied to the problem of natural convection (NC) in a porous square cavity. This problem is widely used to provide physical insights into the processes of fluid flow and heat transfer in porous media. It introduces however several parameters whose values are usually uncertain. We herein explore the effect of the imperfect knowledge of the system parameters and their variability on the model quantities of interest (QoIs) characterizing the NC mechanisms. To this end, we use GSA in conjunction with th…
In this paper, a multiple-relaxation-time (MRT) lattice Boltzmann (LB) model is proposed for convection heat transfer in porous media under local thermal non-equilibrium (LTNE) condition. The model is constructed within the framework of the three-distribution-function approach: two temperature-based MRT-LB equations are proposed for the temperature fields of fluid and solid phases in addition to the MRT-LB equation of a density distribution function for the velocity field described by the generalized non-Darcy model. The thermal non-equilibrium effects are incorporated into the model by adding…
Immiscible fluid-fluid displacement in porous media is of great importance in many engineering applications, such as enhanced oil recovery, agricultural irrigation, and geologic CO2 storage. Fingering phenomena, induced by the interface instability, are commonly encountered during displacement processes and somehow detrimental since such hydrodynamic instabilities can significantly reduce displacement efficiency. In this study, we report a possible adjustment in pore geometry which aims to suppress the capillary fingering in porous media with hierarchical structures. Through pore-scale simulat…
Direct numerical simulations (DNS) indicate that at large values of the Rayleigh number ($Ra$) convection in porous media self-organizes into narrowly-spaced columnar flows, with more complex spatiotemporal features being confined to boundary layers near the top and bottom walls. In this investigation of high-$Ra$ porous media convection in a minimal flow unit, two reduced modeling strategies are proposed that exploit these specific flow characteristics. Both approaches utilize the idea of decomposition since the flow exhibits different dynamics in different regions of the domain: small-scale …
We investigate the effect of dispersion on convection in porous media by performing direct numerical simulations (DNS) in a two-dimensional Rayleigh-Darcy domain. Scaling analysis of the governing equations shows that the dynamics of this system are not only controlled by the classical Rayleigh-Darcy number based on molecular diffusion, $Ra_m$, and the domain aspect ratio, but also controlled by two other dimensionless parameters: the dispersive Rayleigh number $Ra_d = H/α_t$ and the dispersivity ratio $r = α_l/α_t$, where $H$ is the domain height, $α_t$ and $α_l$ are the transverse and longit…
Perfectly incompressible materials do not exist in nature but are a useful approximation of several media which can be deformed in non-isothermal processes but undergo very small volume variation. In this paper the linear analysis of the Darcy-Bénard problem is performed in the class of extended-quasi-thermal-incompressible fluids, introducing a factor $β$ which describes the compressibility of the fluid and plays an essential role in the instability results. In particular, in the Oberbeck-Boussinesq approximation, a more realistic constitutive equation for the fluid density is employed in ord…
Porous materials are used in many fields, including energy industry, agriculture, medical industry, etc. The generation of digital porous media facilitates the fabrication of real porous media and the analysis of their properties. The past random digital porous media generation methods are unable to generate a porous medium with a specific permeability. A new method is proposed in the present study, which can generate the random packed isotropic porous media with specific porosity and permeability. Firstly, the process of generating the random packed isotropic porous media is detailed. Secondl…
Deep saline aquifers used for CO$_2$ sequestration are commonly made of sedimentary formations consisting of several layers of distinguishable permeability. In this work, the effect of a non-monotonic, vertically varying permeability profile on the onset of convective instability is studied theoretically using linear stability analyses. The onset time depends on the interaction between the permeability profile and the location of the concentration perturbation peak beyond which the concentration of CO$_2$ decays. A thin low-permeability layer can either accelerate or delay the onset time of th…
Motivated by geological carbon dioxide (CO$_2$) storage, many recent studies have investigated the fluid dynamics of solutal convection in porous media. Here we study the convective dissolution of CO$_2$ in a closed system, where the pressure in the gas declines as convection proceeds. This introduces a negative feedback that reduces the convective dissolution rate even before the brine becomes saturated. We analyse the case of an ideal gas with a solubility given by Henry's law, in the limits of very low and very high Rayleigh numbers. The equilibrium state in this system is determined by the…
Comment on "Scalings of Rayleigh-Taylor Instability at Large Viscosity Contrasts in Porous Media" (N. Sabet, H. Hassanzadeh, A. De Wit, and J. Abedi (2021) Phys. Rev. Lett. 126:094501).
The flow of a gas through porous medium is considered in the case of pressure dependent permeability. Approximate self-similar solutions of the boundary-value problems are found.
We study the mixing in the presence of convective flow in a porous medium. Convection is characterized by the formation of vortices and stagnation points, where the fluid interface is stretched and compressed enhancing mixing. We analyze the behavior of the mixing dynamics in different scenarios using an interface deformation model. We show that the scalar dissipation rate, which is related to the dissolution fluxes, is controlled by interfacial processes, specifically the equilibrium between interface compression and diffusion, which depends on the flow field configuration. We consider differ…
We compute the fluid flow time-correlation functions of incompressible, immiscible two-phase flow in porous media using a 2D network model. Given a properly chosen representative elementary volume, the flow rate distributions are Gaussian and the integrals of time correlation functions of the flows are found to converge to a finite value. The integrated cross-correlations become symmetric, obeying Onsager's reciprocal relations. These findings support the proposal of a non-equilibrium thermodynamic description for two-phase flow in porous media.
In this paper we show the global well-posedness of solutions to a three-dimensional magnetohydrodynamical (MHD) model in porous media. Compared to the classical MHD equations, our system involves a nonlinear damping term in the momentum equations due to the "Brinkman-Forcheimer-extended-Darcy" law of flow in porous media.
The coefficients of dispersion, D_{ij}, and the dispersivity, a_{ijkl}, appear in the expression for the flux of a solute in saturated flow through porous media. We present a detailed analysis of these tensors in an axially symmetric porous medium and show that in such a medium, the dispersivity is governed by 6 independent moduli. We present also the constraints that have to be satisfied by these moduli. We also show that at least two independent experiments are required in order to obtain the values of these coefficients for any three-dimensional porous medium domain.
Steady laminar flows through porous media spontaneously generate Lagrangian chaos at pore scale, with qualitative implications for a range of transport, reactive and biological processes. The characterization and understanding of mixing dynamics in these opaque environments is an outstanding challenge. We address this issue by developing a novel technique based upon high-resolution imaging of the scalar signature produced by push-pull flows through porous media samples. Owing to the rapid decorrelation of particle trajectories in chaotic flows, the scalar image measured outside the porous mate…
We present the visual analysis of our novel parameter study of porous media experiments, focusing on gaining a better understanding of drainage processes on the micro-scale. We analyze the temporal evolution of extracted characteristic values, and discuss how to directly compare experiments that exhibit processes at different temporal scales due to varying boundary and physical conditions. To enable spatio-temporal analysis, we introduce a new abstract visual representation showing which paths through the porous media were occupied to what extent, e.g., allowing for classification into viscous…
