Mixing In Porous Media Research Articles

Rayleigh–Taylor mixing in porous media at an extreme viscosity contrast

We present experimental findings of Rayleigh–Taylor (RT) instability within porous materials, with a significant viscosity contrast of M ≈ 106, where M represents the ratio of the dynamic viscosity of heavy fluid to that of light fluid, M = μH/μL, and Ra = 6.62 × 104–6.67 × 105, and Rayleigh number (Ra) quantifies the relative significance of buoyancy forces compared to viscous forces. We observe that the lighter fluid diffuses into the denser one, creating a transient diffusive boundary layer that rapidly becomes unstable, transitioning into a convection-dominated regime. Initially, the instability manifests as small fingers protruding upward. However, these fingers coalesce and form fewer major fingers. Convection persists until fingers reach the upper boundary, transitioning into a shutdown regime. During the convection-dominated phase, the extracted solute concentration exhibits a linear relationship with time on a log –log scale, suggesting a constant mass flux. However, this flux diminishes upon entering the shutdown regime. The steady flux, quantified by the Sherwood number, correlates with the Rayleigh number as Sh = 0.046Ra, indicating independence from the height of the porous medium. We have also developed a simple conceptual model that effectively captures the dynamics of RT mixing.

Phase Saturation Control on Vorticity Enhances Mixing in Porous Media

AbstractMixing controls the fate of any solute entering porous media. Hence, an understanding of the involved processes is essential for assessing subsurface contamination and planning for its protection. However, the three‐dimensional mechanisms dominating solute mixing in the presence of several fluid phases in the pore space, and their dependency on phase saturation degree (fraction of the pore volume occupied by a phase) are unknown. Here, we analyze solute mixing in unsaturated porous media at the pore scale using X‐ray micro‐tomography performed with synchrotron radiation at unprecedented temporal and spatial resolutions for such an investigation. Transport experiments through a synthetic, sand‐like porous medium, followed in 4D using a contrast solution, are performed at different liquid phase saturation degrees. The results reveal larger solute's front deformation at lower saturation, which translates into an enhanced mixing with time. Using different topological indexes, defined based on a description of the liquid phase geometry and of the resulting hydrodynamics, we show an increase in the spatial convergence of flow streamlines at lower saturation, which, in turn, leads to a strengthened helical flow inside the liquid phase. Consequently, this increases the number of shear‐ and vorticity‐dominated deformation regions, as characterized by larger negative and positive Q‐criterion values, respectively. These findings represent a major step toward understanding the control of both saturation and the system's heterogeneity on solute mixing, essential, among others, to assess reactivity in porous media.

Evolution of plume geometry, dilution and reactive mixing in porous media under highly transient flow fields at the surface water-groundwater interface

Highly transient boundary conditions affect mixing of dissolved solutes in groundwater. An example of these transient boundary conditions occurs at the surface water-groundwater interface, where the water level in rivers can change rapidly due to the operation of hydropower plants, leading to a regime known as hydropeaking. Inspired by this phenomenon, this work studies at laboratory scale the effects of fluctuating surface water bodies on solute transport in aquifers. We performed flow-through experiments at two different flow velocities and under steady and transient flow conditions where a conservative tracer was injected in the system and its concentration measured with optical imaging methods. The experimental results were quantitatively interpreted with numerical simulations implementing a non-linear velocity-dependent dispersive transport model. We estimated plume dilution by computing the dilution index and its evolution as well as two key geometrical metrics of the transient plumes: the perimeter and the area. We further investigated reactive mixing and mixing enhancement considering mixing-controlled bimolecular reactions using different critical mixing ratios. In general, highly transient boundary conditions lead to a larger area, perimeter and plume dilution and the results show greater relative enhancement for the scenarios with low groundwater flow velocity. A linear relationship was observed between the evolution of the area and the dilution index of the plumes for the transient flow scenarios investigated. Considering reactive transport and mixing-limited reactions at the surface water-groundwater interface, we identified different dilution and reaction dominated regimes, characterized, respectively, by increasing and decreasing plume entropies at different mixing ratios of the reactants. Furthermore, reactive mixing was enhanced by transient flows leading to a faster degradation of contaminant plumes compared to corresponding steady flow conditions.

Nondilute effects reshape miscible displacement in porous media

We provide an experimental benchmark for modeling nondilute miscible mixing in porous media, which emerges in CO${}_{2}$ sequestration and solvent-based enhanced oil recovery. In a near-fracture porous medium, we show that conventional Darcy-Fickian coupling fails to predict nondilute fluid-fluid displacement, although it works perfectly for dilute systems. Nondilute fluid systems show ``diffusive'' mixing kinetics, wedge-like mixing front, and significant convection at the front that is perpendicular to the concentration gradient. These above abnormal phenomena are rationalized by Korteweg stress, which is still challenging in modeling.

Impact of variable density on electrokinetic transport and mixing in porous media

Electrokinetic (EK) techniques can effectively mobilize amendments and contaminants in porous media and therefore have a great potential for in situ subsurface remediation of a wide variety of organic and inorganic contaminants. However, the fundamental mechanisms governing EK applications in porous media are complex and entail coupling between different flow, transport, electrostatic and geochemical processes. In this study we investigate the interactions between density driven effects and electrokinetic transport in permeable porous media and we compare the observed dynamics with the outcomes of analogous scenarios in which advection and dispersion are the only transport mechanisms. We perform a series of multidimensional experiments and we demonstrate the strong interplay between Coulombic interactions among charged species in pore water and density-driven effects during electrokinetics. Our study illuminates non-trivial interactions between the pore water composition and the macroscopic behavior of variable density dye tracer plumes, showing different sinking and floating patterns, distinct plume shapes and spreading, as well as different extent of mixing depending on the chemical composition of the pore water. A numerical model, based on the Nernst-Planck-Poisson formulation and on the coupling with the geochemical code PHREEQC, was developed to simulate variable density flow and transport and to update the density of the aqueous solution resulting from mixing and electrostatic interactions between all chemical species in the injected plume and in the surrounding pore water electrolyte. The model allowed us to interpret the experiments and to visualize the impact of small density gradients and Coulombic effects on the macroscopic transport of the dye tracer. The outcomes of this study highlight the importance of density-driven flow also during EK transport, where the electrostatic interactions not only influence the concentration of charged species but can also affect the density and result in complex flow patterns, plume shapes and mixing behavior, radically different from the ones generated during advective-dispersive transport.

Editorial to the Special Issue: Mixing in Porous Media

Editorial to the Special Issue: Mixing in Porous Media

Mixing in Porous Media: Concepts and Approaches Across Scales

This review provides an overview of concepts and approaches for the quantification of passive, non-reactive solute mixing in steady uniform porous media flows across scales. Mixing in porous media is the result of the interaction of spatial velocity fluctuations and diffusion or local-scale dispersion, which may lead to the homogenization of an initially segregated system. Velocity fluctuations are induced by spatial medium heterogeneities at the pore, Darcy or regional scales. Thus, mixing in porous media is a multiscale process, which depends on the medium structure and flow conditions. In the first part of the review, we discuss the interrelated processes of stirring, dispersion and mixing, and review approaches to quantify them that apply across scales. This implies concepts of hydrodynamic dispersion, approaches to quantify mixing state and mixing dynamics in terms of concentration statistics, and approaches to quantify the mechanisms of mixing. We review the characterization of stirring in terms of fluid deformation and folding and its relation with hydrodynamic dispersion. The integration of these dynamics to quantify the mechanisms of mixing is discussed in terms of lamellar mixing models. In the second part of this review, we discuss these concepts and approaches for the characterization of mixing in Poiseuille flow, and in porous media flows at the pore, Darcy and regional scales. Due to the fundamental nature of the mechanisms and processes of mixing, the concepts and approaches discussed in this review underpin the quantitative analysis of mixing phenomena in porous media flow systems in general.

Enhancing drop mixing in powder bed by alternative particle arrangements with contradictory hydrophilicity

BackgroundDrop mixing in porous media is critical to printing quality of inkjet printing and binder jetting 3D printing, however, the mixing is poor due to restricted pore space. In this study, an alternative hydrophobic/hydrophilic particle arrangement was proposed to enhance dual droplets mixing in a powder bed. MethodsA computational fluid dynamic (CFD) model was first developed to simulate the fluid flow for drops falling on a powder bed by using volume of fluid (VOF) technique. Two drops containing separate solutes sequentially impact an assembly of particles. An index of mixing was proposed to quantitatively describe the degree of mixing by calculating the standard deviation of species concentrations. Significant FindingsTo enhance mixing, a new approach was developed to improve liquid mixing by arranging hydrophobic/hydrophilic particles alternatively. Furthermore, drops falling with a horizontal offset was shown to prolong the vorticity for convective mass transportation. With an optimal offset distance of 15μm, the mixing degree can be enhanced from 34.2% to 56.2%. The printing strategy developed from this study could apparently enhance drop mixing in porous media and provided potential developments for multiple-component mixing in binder jetting 3D printing.

Impact of solute charge and diffusion coefficient on electromigration and mixing in porous media

The application of electrokinetic techniques in porous media has great potential to enhance mass transfer rates and, thus, to mobilize contaminants and effectively deliver reactants and amendments. However, the transport mechanisms induced by the application of an external electric field are complex and entail the coupling of physical, chemical and electrostatic processes. In this study we focus on electromigration and we provide experimental evidence of the impact of compound-specific properties, such as the aqueous diffusivity and the valence of charged species, on the macroscopic electrokinetic transport. We performed a series of multidimensional experiments considering the displacement of three different tracer plumes (i.e., permanganate, allura red and new coccine) in different background electrolyte solutions. The outcomes of the experiments clearly show that both the compound-specific diffusivity and the charge of the injected and resident ions impact the transport of the selected color tracer plumes, whose evolution was monitored with image analysis. The investigated experimental scenarios led to distinct plume behavior characterized by different mass distribution, average displacement velocities, longitudinal and lateral plume spreading, shape of the invading and receding fronts, as well as dilution of the injected solutes. A numerical simulator, based on the Nernst-Planck-Poisson equations and on aqueous speciation reactions in the pore water, allowed us to quantitatively interpret the experimental results, to capture the observed patterns of plume evolution, and to illuminate the coupling between the governing physico-chemical mechanisms and the controlling role of small scale compound-specific and electrostatic properties. Finally, the model was also extended to a typical configuration of in situ electrokinetic remediation of contaminated groundwater to show the impact of such mechanisms at larger scale.

Modeling solute transport and mixing in heterogeneous porous media under turbulent flow conditions

We develop and test a modeling approach to quantify turbulence-driven solute transport and mixing in porous media. Our approach addresses two key elements: (a) the spatial variability of the effective diffusion coefficient which is typically documented in the presence of a sediment–fluid interface and (b) the need to provide a model that can yield the complete distribution of the concentration probability density function, not being limited only to the mean concentration value and thus fully addressing solute mixing. Our work is motivated by the importance of solute transport processes in the hyporheic zone, which can have strong implications in natural attenuation of pollutants. Our approach combines Lagrangian schemes to address transport and mixing in the presence of spatial variability of effective diffusion. An exemplary scenario we consider targets a setup constituted by a homogeneous (fully saturated) porous medium underlying a clear water column where turbulent flow is generated. Solute concentration histories obtained through a model based solely on diffusive transport are benchmarked against an analytical solution. These are then compared against the results obtained by modeling the combined effects of diffusion and mixing. A rigorous sensitivity analysis is performed to evaluate the influence of model parameters on solute concentrations and mixing, the latter being quantified in terms of the scalar dissipation rate.

Dimensional transition in Darcy-Rayleigh-Taylor mixing

The effect of geometrical confinement on Rayleigh-Taylor mixing in porous media is investigated by means of numerical simulations. We find a dimensional transition from three-dimensional to two-dimensional phenomenology when the density plumes become larger than the confining scale. In the two-dimensional regime we observe a faster mixing process and a larger value of the density Nusselt number.

Relative impacts of permeability heterogeneity and viscosity contrast on solute mixing

Solute transport and mixing in porous media are affected by the heterogeneous structure of the permeability field and the viscosity contrast between the dissolved solute and the ambient fluid. We provide a quantitative separation of the impacts of fingering and heterogeneity and we parameterize the concentration probability distribution function.

Scalar Signatures of Chaotic Mixing in Porous Media.

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 material is representative of insitu mixing dynamics. We present a theoretical framework for estimation of the Lyapunov exponent based on extension of Lagrangian stretching theories to correlated aggregation. This method provides a full characterization of chaotic mixing dynamics in a large class of porous materials.

Scaling of Rayleigh-Taylor mixing in porous media

Pushing two fluids with different density one against the other causes the development of the Rayleigh-Taylor instability at their interface, which further evolves in a complex mixing layer. In porous media, this process is influenced by the viscous resistance experienced while flowing through the pores, which is described by the Darcy's law. Here, we investigate the mixing properties of the Darcy-Rayleigh-Taylor system in the limit of large P\'eclet number by means of direct numerical simulations in three and two dimensions. In the mixing zone, the balance between gravity and viscous forces results in a non-self-similar growth of elongated plumes, whose length increases linearly in time while their width follows a diffusive growth. The mass-transfer Nusselt number is found to increase linearly with the Darcy-Rayleigh number supporting a universal scaling in porous convection at high Ra numbers. Finally, we find that the mixing process displays important quantitative differences between two and three dimensions.

Velocity distributions, dispersion and stretching in three-dimensional porous media

Using index matching and particle tracking, we measure the three-dimensional velocity field in an isotropic porous medium composed of randomly packed solid spheres. This high-resolution experimental dataset provides new insights into the dynamics of dispersion and stretching in porous media. Dynamic-range velocity measurements indicate that the distribution of the velocity magnitude, , is flat at low velocity (probability density function ). While such a distribution should lead to a persistent anomalous dispersion process for advected non-diffusive point particles, we show that the dispersion of non-diffusive tracers nonetheless becomes Fickian beyond a time set by the smallest effective velocity of the tracers. We derive expressions for the onset time of the Fickian regime and the longitudinal and transverse dispersion coefficients as a function of the velocity field properties. The experimental velocity field is also used to study, by numerical advection, the stretching histories of fluid material lines. The mean and the variance of the line elongations are found to grow exponentially in time and the distribution of elongation is log-normal. These results confirm the chaotic nature of advection within three-dimensional porous media. By providing the laws of dispersion and stretching, the present study opens the way to a complete description of mixing in porous media.

Incomplete mixing in porous media: Todd-Longstaff upscaling approach versus a dynamic local grid refinement method

Field-scale simulation of flow in porous media in presence of incomplete mixing demands for high-resolution computational grids, much beyond the scope of state-of-the-art simulators. Hence, the upscaling-based Todd and Longstaff (TL) approach is typically used, where coarse grid cells are employed with effective mixing fluid properties and parameters found by matching results obtained with fully resolved reference simulations. Dynamic local grid refinement (DLGR) techniques, on the other hand, only employ fine-scale grid resolution where the fully mixed assumption is not valid. The rest of the domain is then solved at coarser resolutions, where the fully mixed assumption is valid. Here, we assess the accuracy and the robustness of DLGR- and TL-based simulations of miscible displacements in homogeneous and heterogeneous porous media. Due to the intrinsic uncertainty within the unstable displacement nature of the studied incomplete mixing processes, the performance of the methods is also investigated based on a range of acceptable solutions rather than relying only on a single reference one. Systematic numerical results illustrate that the DLGR method is much more robust and accurate than the upscaling-based TL approach, and employs only a small fraction of fine-scale reference grids. Especially, the TL upscaling results (though history matched with computationally expensive fine-scale results) are very sensitive to the change of the simulation parameters. Based on this study, we propose a dynamic multilevel simulation strategy for efficient and reliable large-scale simulation of the complex incomplete mixing processes.

Numerical modelling of solute flow dispersion in porous media using simulator MUFITS

An extension of the MUFITS reservoir simulator for modelling solute transport in a porous medium is presented. The new option concerns modelling of the longitudinal and transverse mechanical dispersion of the solute flow. The numerical dispersion coefficients for the solute transport are derived. The sufficient condition for the time step and the grid resolution that ensure accurate modelling of mixing in porous media is presented. Two benchmark problems which allow exact analytical solution are described and are used for validation of the developed simulation option. The solute distributions calculated with MUFITS are in a good agreement with the exact solutions.

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.

Space-Group Symmetries Generate Chaotic Fluid Advection in Crystalline Granular Media.

The classical connection between symmetry breaking and the onset of chaos in dynamical systems harks back to the seminal theory of Noether [Transp. Theory Statist. Phys. 1, 186 (1918)10.1080/00411457108231446]. We study the Lagrangian kinematics of steady 3D Stokes flow through simple cubic and body-centered cubic (bcc) crystalline lattices of close-packed spheres, and uncover an important exception. While breaking of point-group symmetries is a necessary condition for chaotic mixing in both lattices, a further space-group (glide) symmetry of the bcc lattice generates a transition from globally regular to globally chaotic dynamics. This finding provides new insights into chaotic mixing in porous media and has significant implications for understanding the impact of symmetries upon generic dynamical systems.

Refractive‐Light‐Transmission Technique Applied to Density‐Driven Convective Mixing in Porous Media With Implications for Geological CO2 Storage

AbstractDensity‐driven convection has been identified to accelerate the rate of CO2 solubility trapping during geological CO2 storage in deep saline aquifers. In this paper, we present an experimental method using the refractive properties of fluids (their impact on light transmission), and an analogous system design, which enables the study of transport mechanisms in saturated porous media. The method is used to investigate solutally induced density‐driven convective mixing under conditions relevant to geological CO2 storage. The analogous system design allows us by choice of initial solute concentration and bead size to duplicate a wide range of conditions ( ‐values), making it possible to study the convective process in general, and as a laboratory analogue for systems found in the field. We show that the method accurately determines the solute concentration in the system with high spatial and temporal resolution. The onset time of convection ( ), mass flux ( ), and flow dynamics are quantified and compared with experimental and numerical findings in the literature. Our data yield a scaling law for which gives new insight into its dependence on , indicating to be more sensitive to large than previously thought. Furthermore, our data show and explain why is described equally well by a ‐dependent or a ‐independent scaling law. These findings improve the understanding of the physical process of convective mixing in saturated porous media in general and help to assess the CO2 solubility trapping rate under certain field conditions.