Session H05: Porous Media Flows: Convection and Heat Transfer I

Buoyant convection in a porous medium with an inclined permeability jump: An experimental investigation of filling box-type flows

Employing the concept of a ‘filling-box’ model, we report upon experiments featuring a dense, Boussinesq plume falling through a two-layered porous medium characterized by a high permeability upper layer and a low permeability lower layer. Upon striking the permeability jump, the plume fluid bifurcates leading to the formation of a pair of (up- and downdip) ‘primary’ gravity currents. The associated early time spreading behavior has been investigated in our previous work (Bharath et.al, JFM, vol. 902, 2020). It was shown that the primary gravity currents reach a state of runout wherein the inflow from the plume is precisely matched by the outflow due to draining. As an extension of our previous work, we herein examine the flow evolution over longer times and thereby study the impact of the lower layer depth. To this end, we characterize the formation of `secondary’ gravity currents, which form when the fluid draining from the underside of the gravity currents strikes the bottom (impermeable) boundary. As a consequence of the propagation of these secondary gravity currents, the primary gravity currents may become remobilized. Later, the gravity currents flow become impeded by the vertical sidewall boundaries, allowing us to categorize two qualitatively different filling regimes.   

Influence of non-Darcy effects on scaling laws of convection in Hele-Shaw flows

In this work, we examined the influence of non-Darcy terms on convective dissolution in confined porous media. We performed experiments in Hele-Shaw configuration and we focused on buoyancy-driven convection, where the flow is controlled by the Rayleigh-Darcy number, Ra, which measures the strength of convection compared to diffusion. The Hele-Shaw cell is suitable to mimic Darcy flows only under certain geometrical constraints, and precise limits exist for Ra and cell geometry beyond which the flow exhibits non-Darcy effects. We performed experiments in Rayleigh-Bénard-like configuration and we clearly identify the application limits of Darcy flow assumptions. Besides confirming previous theoretical predictions, current results are of relevance in the context of porous media flows - which are often studied experimentally with Hele-Shaw setups. Using our original datasets, we have been able to explain and reconcile the discrepancies observed between scaling laws previously proposed for Rayleigh-Bénard-like experiments and simulations in similar contexts. Specifically, we discuss the conditions beyond which Hele-Shaw experiment results are influenced by three-dimensional effects.   

Effective solvent-solute transport across micro-structured thin membranes

A multi-scale homogenization based model to simulate the transport of a passive solute across a membrane and the hydrodynamic of the surrounding fluid solvent is presented.

It provides a description of the micro- and macroscopic solvent behavior and solute convection and consists of some constraints to be satisfied by the solvent velocity $\mathbf{u}$ and solute concentration $c$, imposed within the fluid domain, over a virtual smooth surface passing through the center of each membrane pore

$$\mathbf{u}=-{\bf M}:{\bf{\Sigma}}^--{\bf N}:{\bf{\Sigma}}^+, \quad {c}=-{\bf X}\cdot{\bf{F}}^--{\bf Y}\cdot {\bf{F}}^+,$$ 

where $\mathbf{\Sigma}^{\pm}$ and $\mathbf{F}^\pm$ denotes the upward and downward solvent stresses and solute fluxes and ${\bf M}$, $\bf N$, $\bf X$ and $\bf Y$ represent the microscopic feedback of the solid skeleton on the macroscopic fields and can be computed once and for all at the pore-scale. 

The model shows that the membrane produces a jump in solvent stresses  and solute fluxes whose intensity and direction,

evaluated solving problems at the microscale, depend on the external transport phenomenon and on the pore geometry. To assert the validity of the macroscopic model developed, its solution is compared with the solution of the full-scale problem.   

Experimental investigation on Rayleigh-Taylor instability in confined porous media

Experiments in Hele-Shaw cells are used to mimic the problem of Rayleigh-Taylor instability in confined porous media. Rayleigh-Taylor instability phenomena arise when a layer of heavy fluid sits on top of a layer of a lighter fluid. The flow is initially controlled by diffusion, but rather quickly the action of gravity produces efficient fluid mixing in the entire domain. Small fluctuations of the interface separating the two fluid layers produce larger structures that eventually drive the flow into a nonlinear convective stage. The competition between buoyancy and diffusion is measured by the Rayleigh-Darcy number, the value of which controls the entire dynamics of the flow. We employ an optical method to obtain accurate measurements of the solute concentration and we perform experiments for a wide range of Rayleigh-Darcy numbers. The fluids consist of water and an aqueous solution of potassium permanganate. We characterize the flow evolution with the mixing length (a suitably defined extension of the mixing region) and with the degree of mixing of the two fluids. In this work, mixing is quantified by the mean scalar dissipation rate. The results obtained are compared against previous experimental and numerical works.   

Acoustic flow in porous media

We calculate the acoustic flow – the steady drift of fluid mass to appear due to the convection of momentum along the path of an acoustic wave – in a porous medium. In particular, we suggest a mechanism to explain observations of acoustic contributions to mass transport in porous media at geological, unit operation, and lab on a chip length scales. We study three limits for the case of an acoustic planar wave (that is, sound or ultrasound waves) whose wavelength is large compared to the pore size. We commence our analysis at the ideal limit of similar acoustic properties in the solid and fluid. The acoustic flow may then be treated according with the Darcy theory for flow through porous media in addition to a correction for the average azimuth of the pores compared to the acoustic path. The two other limits are taken within the framework of a rigid porous frame. The presence of a flow forcing mechanism to result from the viscous dissipation of the acoustic wave at the solid surface of the pores in this case hinders the direct use of the Darcy theory. The analysis is conducted by a detailed calculation of the convective transport of mass through cylindrical pores of similar size but arbitrary inclination, where we consider large and medium to small pore diameter limits.   

Towards the ultimate regime in Rayleigh-Darcy convection

Numerical simulations are used to probe Rayleigh-Darcy convection in fluid-saturated porous media towards the ultimate regime. The present three-dimensional dataset, up to Rayleigh-Darcy number Ra=80 000,  suggests that the appropriate scaling of the Nusselt number is  Nu=0.0081Ra+0.067Ra^0.61, fitting the computed data for Ra>1000. Extrapolation of current predictions to the ultimate linear regime yields the asymptotic law Nu=0.0081Ra, about 16% less than indicated in previous studies. Upon examination of the flow structures near the boundaries, we confirm previous indications of small flow cells hierarchically nesting into supercells, and we show evidence that the supercells at the boundary are the footprints of the megaplumes that dominate the interior part of the flow. The present findings pave the way for more accurate modeling of geophysical systems, with special reference to geological CO₂ sequestration.   

Brinkman number effect and Conjugate heat transfer in a porous microchannel

The problem of conjugate heat transfer in a fully saturated porous medium confined in a microchannel between parallel plates is analyzed. Two conjugate heat transfer parameters were defined in the microchannel wall and in the fluid. It was found that the values of these parameters have certain limits under which, longitudinal temperature gradients can occur both in the wall and in the porous medium. Additionally, we studied the effects of the Darcy number Da and the Brinkman number Br, on fluid and wall temperature profiles for different values of the conjugate heat transfer parameters. Finally, several values of the Peclet number are analyzed in order to find out which of them promotes longitudinal (axial) heat conduction in the fluid and also in the wall of the microchannel. The main objective of this work is to determine for which Brinkman values the viscous dissipation effects are more important than the longitudinal heat conduction effects in the microchannel wall.   

Modelling the Navier-Stokes-Darcy-Boussinesq system

To investigate convection in coupled fluid-porous media systems, we analyze the Navier-Stokes-Darcy Heat model with numerical simulations conducted via a finite element method. To help validate numerical results, we compare our simulations against stability thresholds for the system.

Additionally, we investigate various parameter regimes of the system and present a notable case with altering the depth ratio of the two regions. In these superposed fluid-porous media systems, the ratio of the fluid height to the porous medium height exerts a significant influence on the behavior of the coupled system with its impact on resulting convection cells. Altering the depth ratio slightly can trigger a transition from  deep convection where convection cells encapsulate the entire domain to shallow convection where cells occupy only the fluid region. With current interest surrounding superposed fluid-porous medium systems in numerous projects of industrial, environmental, and geophysical importance (oil recovery, carbon dioxide sequestration, contamination in sub-soil reservoirs, etc.), being able to predict the critical depth ratio where this convection shift occurs is particularly timely.