Mixing In Porous Media Research Articles

Hydro-dynamic Solute Transport under Two-Phase Flow Conditions

There are abundant examples of natural, engineering and industrial applications, in which “solute transport” and “mixing” in porous media occur under multiphase flow conditions. Current state-of-the-art understanding and modelling of such processes are established based on flawed and non-representative models. Moreover, there is no direct experimental result to show the true hydrodynamics of transport and mixing under multiphase flow conditions while the saturation topology is being kept constant for a number of flow rates. With the use of a custom-made microscope, and under well-controlled flow boundary conditions, we visualized directly the transport of a tracer in a Reservoir-on-Chip (RoC) micromodel filled with two immiscible fluids. This study provides novel insights into the saturation-dependency of transport and mixing in porous media. To our knowledge, this is the first reported pore-scale experiment in which the saturation topology, relative permeability, and tortuosity were kept constant and transport was studied under different dynamic conditions in a wide range of saturation. The critical role of two-phase hydrodynamic properties on non-Fickian transport and saturation-dependency of dispersion are discussed, which highlight the major flaws in parametrization of existing models.

Impact of saturation on dispersion and mixing in porous media: Photobleaching pulse injection experiments and shear‐enhanced mixing model

AbstractThe dynamics of solute dispersion and mixing in unsaturated flows is analyzed from photobleaching experiments in two‐dimensional porous micromodels. This technique allows producing pulse line (delta‐Dirac) injections of a conservative tracer by bleaching a finite volume of fluorescent without disturbing the flow field. The temporal evolution of the concentration field and the spatial distribution of the air and water phases can be monitored at pore scale. We study the dispersion and mixing of a line of tracer under different water saturations. While dispersion in saturated porous media follows an approximately Fickian scaling, a shift to ballistic scaling is observed as soon as saturation is lowered. Hence, at the time scale of observation, dispersion in our unsaturated flows is dominated by the ballistic separation of tracer blobs within the water phase, between trapped clusters and preferential flow paths. While diffusion plays a minor role in the longitudinal dispersion during the time scale of the experiments, its interplay with fluid deformation is apparent in the dynamics of mixing. The scalar dissipation rates show an initial stretching regime, during which mixing is enhanced by fluid deformation, followed by a dissipation regime, during which diffusion overcomes compression induced by stretching. The transition between these two regimes occurs at the mixing time, when concentration gradients are maximum. We propose a predictive analytical model, based on shear‐enhanced diffusion, that captures the dynamics of mixing from basic unsaturated porous media parameters, suggesting that this type of model may be a useful framework at larger scales.

Impact of Mineral Precipitation on Flow and Mixing in Porous Media Determined by Microcomputed Tomography and MRI.

Precipitation reactions influence transport properties in porous media and can be coupled to advective and dispersive transport. For example, in subsurface environments, mixing of groundwater and injected solutions can induce mineral supersaturation of constituents and drive precipitation reactions. Magnetic resonance imaging (MRI) and microcomputed tomography (μ-CT) were employed as complementary techniques to evaluate advection, dispersion, and formation of precipitate in a 3D porous media flow cell. Two parallel fluids were flowed concentrically through packed glass beads under two relative flow rates with Na2CO3 and CaCl2 in the inner and outer fluids, respectively. CaCO3 became supersaturated and formed a precipitate at the mixing interface between the two solutions. Spatial maps of changing local velocity fields and dispersion in the flow cell were generated from MRI, while high resolution μ-CT imaging visualized the precipitate formed in the porous media. Formation of a precipitate minimized dispersive and advective transport between the two fluids and the shape of the precipitation front was influenced by the relative flow rates. This work demonstrates that the combined use of MRI and μ-CT can be highly complementary in the study of reactive transport processes in porous media.

Experimental investigation of transverse mixing in porous media under helical flow conditions.

Plume dilution and transverse mixing can be considerably enhanced by helical flow occurring in three-dimensional heterogeneous anisotropic porous media. In this study, we perform tracer experiments in a fully three-dimensional flow-through chamber to investigate the effects of helical flow on plume spiraling and deformation, as well as on its dilution. Porous media were packed in angled stripes of materials with different grain sizes to create blocks with macroscopically anisotropic hydraulic conductivity, which caused helical flows. Steady-state transport experiments were carried out by continuously injecting dye tracers at different inlet ports. High-resolution measurements of concentration and flow rates were performed at 49 outlet ports. These measurements allowed quantifying the spreading and dilution of the solute plumes at the outlet cross section. Direct evidence of plume spiraling and visual proof of helical flow was obtained by freezing and slicing the porous media at different cross sections and observing the dye-tracer distribution. We simulated flow and transport to interpret our experimental observations and investigate the effects of helical flow on mixing-controlled reactive transport. The simulation results were evaluated using metrics of reactive mixing such as the critical dilution index and the length of continuously injected steady-state plumes. The results show considerable reaction enhancement, quantified by the remarkable decrease of reactive plume lengths (up to four times) in helical flows compared to analogous scenarios in uniform flows.

Experimental Evidence of Helical Flow in Porous Media

Helical flow leads to deformation of solute plumes and enhances transverse mixing in porous media. We present experiments in which macroscopic helical flow is created by arranging different materials to obtain an anisotropic macroscopic permeability tensor with spatially variable orientation. The resulting helical flow entails twisting streamlines which cause a significant increase in lateral mass exchange and thus a large enhancement of plume dilution (up to 235%) compared to transport in homogenous media. The setup may be used to effectively mix solutes in parallel streams similarly to static mixers, but in porous media.

Do we really need a large number of particles to simulate bimolecular reactive transport with random walk methods? A kernel density estimation approach

Random walk particle tracking methods are a computationally efficient family of methods to solve reactive transport problems. While the number of particles in most realistic applications is in the order of 106–109, the number of reactive molecules even in diluted systems might be in the order of fractions of the Avogadro number. Thus, each particle actually represents a group of potentially reactive molecules. The use of a low number of particles may result not only in loss of accuracy, but also may lead to an improper reproduction of the mixing process, limited by diffusion. Recent works have used this effect as a proxy to model incomplete mixing in porous media. In this work, we propose using a Kernel Density Estimation (KDE) of the concentrations that allows getting the expected results for a well-mixed solution with a limited number of particles. The idea consists of treating each particle as a sample drawn from the pool of molecules that it represents; this way, the actual location of a tracked particle is seen as a sample drawn from the density function of the location of molecules represented by that given particle, rigorously represented by a kernel density function. The probability of reaction can be obtained by combining the kernels associated to two potentially reactive particles. We demonstrate that the observed deviation in the reaction vs time curves in numerical experiments reported in the literature could be attributed to the statistical method used to reconstruct concentrations (fixed particle support) from discrete particle distributions, and not to the occurrence of true incomplete mixing. We further explore the evolution of the kernel size with time, linking it to the diffusion process. Our results show that KDEs are powerful tools to improve computational efficiency and robustness in reactive transport simulations, and indicates that incomplete mixing in diluted systems should be modeled based on alternative mechanistic models and not on a limited number of particles.

The lamellar description of mixing in porous media

We develop a general framework for modelling mixing in porous media flows, in which the scalar mixture is represented as an ensemble of lamellae evolving through stretching, diffusion and coalescence. Detailed numerical simulations in Darcy scale heterogeneous permeability fields are used to analyse the lamella deformation process, which controls the local concentration gradients and thus the evolution of the concentration mixture through stretching enhanced diffusion. The corresponding Lagrangian deformation process is shown to be well modelled by a Langevin equation with multiplicative noise, which can be coupled with diffusion to predict the temporal evolution of the concentration probability density function (PDF). At late times, lamella interaction is enforced by confinement of the mixture within the dispersion area. This process is shown to be well represented by a random aggregation model, which quantifies the frequency of lamella coalescence and allows us to predict the temporal evolution of the concentration PDF in this regime. The proposed theoretical framework provides an accurate prediction of the concentration PDFs at all investigated times, heterogeneity levels and Péclet numbers. In particular, it relates the temporal behaviour of mixing, as quantified by concentration moments, scalar dissipation rate or spatial increments of concentration, to the degree of structural heterogeneity.

Impact of viscous fingering and permeability heterogeneity on fluid mixing in porous media

AbstractFluid mixing plays a fundamental role in many natural and engineered processes, including groundwater flows in porous media, enhanced oil recovery, and microfluidic lab‐on‐a‐chip systems. Recent developments have explored the effect of viscosity contrast on mixing, suggesting that the unstable displacement of fluids with different viscosities, or viscous fingering, provides a powerful mechanism to increase fluid‐fluid interfacial area and enhance mixing. However, existing studies have not incorporated the effect of medium heterogeneity on the mixing rate. Here, we characterize the evolution of mixing between two fluids of different viscosity in heterogeneous porous media. We focus on a practical scenario of divergent‐convergent flow in a quarter five spot geometry prototypical of well‐driven groundwater flows. We study by means of numerical simulations the impact of permeability heterogeneity and viscosity contrast on the breakthrough curves and mixing efficiency, and we rationalize the nontrivial mixing behavior that emerges from the competition between the creation of fluid‐fluid interfacial area and channeling.

Scaling Analysis of the Convective Mixing in Porous Media for Geological Storage of CO2: An Experimental Approach

Prediction of the behavior of convective mixing and the effectiveness of this mechanism is essential for permanent sequestration of CO2 in deep saline aquifers. Simulation of the diffusion–convection mechanism at a large scale is very expensive and time-consuming; therefore, scaling relationships can be used to find suitable candidates for storage sites. In this study, scaling analysis is performed for the convective mixing of CO2 in saline aquifers based on experimental results. The scaling relationships are presented for the prediction of convective dissolution behavior. In the presented scaling analysis, different systems with a wide range of Rayleigh numbers were used. All experiments were conducted in a dissolution cell with different ranges of grain sizes. The pressure decay data are used to determine the dissolution rate of CO2, Sherwood number, and convective flux. In addition, the fraction of ultimate dissolution is calculated for each experiment to investigate the mixing regimes (convective mixing and diffusive mixing). The results indicate that the mixing of CO2 in water can be approximated by a scaling relationship for the Sherwood number and convective flux. These relations can be used in the implementation of large-scale CO2 storage in deep saline aquifers.

Rock dissolution patterns and geochemical shutdown of –brine–carbonate reactions during convective mixing in porous media

AbstractMotivated by the process of $\text{CO}_{2}$ convective mixing in porous media, here we study the formation of rock-dissolution patterns that arise from geochemical reactions during Rayleigh–Bénard–Darcy convection. Under the assumption of instantaneous chemical equilibrium, we adopt a formulation of the local reaction rate as a function of scalar dissipation rate, a measure that depends solely on flow and transport, and chemical speciation, which is a measure that depends only on the equilibrium thermodynamics of the chemical system. We use high-resolution simulations to examine the interplay between the density-driven hydrodynamic instability and the rock dissolution reactions, and analyse the impact of geochemical reactions on the macroscopic mass exchange rate. We find that dissolution of carbonate rock initiates in regions of locally high mixing, but that the geochemical reaction shuts down significantly earlier than shutdown of convective mixing. This early shutdown feature reflects the important role that chemical speciation plays in this hydrodynamics–reaction coupled process. Finally, we extend our analysis to three dimensions and explore the morphology of dissolution patterns in three dimensions.

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

Geological carbon dioxide (CO2) sequestration entails capturing and injecting CO2 into deep saline aquifers for long-term storage. The injected CO2 partially dissolves in groundwater to form a mixture that is denser than the initial groundwater. The local increase in density triggers a gravitational instability at the boundary layer that further develops into columnar plumes of CO2-rich brine, a process that greatly accelerates solubility trapping of the CO2. Here, we investigate the pattern-formation aspects of convective mixing during geological CO2 sequestration by means of high-resolution three-dimensional simulation. We find that the CO2 concentration field self-organizes as a cellular network structure in the diffusive boundary layer at the top boundary. By studying the statistics of the cellular network, we identify various regimes of finger coarsening over time, the existence of a non-equilibrium stationary state, and a universal scaling of three-dimensional convective mixing.

Nonlinear simulation of transverse flow interactions with chemically driven convective mixing in porous media

Buoyancy‐driven hydrodynamic instabilities of a miscible reactive interface in a homogeneous porous medium is examined. A bimolecular chemical reaction (A+B→C) is triggered at the interface between two reactant solutions A and B resulting in a chemical product solution C with different density and the viscosity from those of the reactants. The effects of the chemical reaction and a transverse flow parallel to the initial interface between the reactants are numerically analyzed. It was found that as a result of the transverse flow, fingers with sharp concentration gradients tend to develop and advance fast downward leading to higher rates of chemical production. Furthermore, a detailed analysis of the finger growth and the effects of buoyancy, transverse flow and chemical reaction allowed to reach a physical interpretation of the trends observed. Finally, a special tuning of the transverse velocity is proposed to ensure maximum or minimum chemical production applicable to subsurface flows.

Evaluation of the Role of Heterogeneities on Transverse Mixing in Bench‐Scale Tank Experiments by Numerical Modeling

In this work, numerical modeling is used to evaluate and interpret a series of detailed and well-controlled two-dimensional bench-scale conservative tracer tank experiments performed to investigate transverse mixing in porous media. The porous medium used consists of a fine matrix and a more permeable lens vertically aligned with the tracer source and the flow direction. A sensitivity analysis shows that the tracer distribution after passing the lens is only slightly sensitive to variations in transverse dispersivity, but strongly sensitive to the contrast of hydraulic conductivities. A unique parameter set could be calibrated to closely fit the experimental observations. On the basis of calibrated and validated model, synthetic experiments with different contrasts in hydraulic conductivity and more complex setups were performed and the efficiency of mixing evaluated. Flux-related dilution indices derived from these simulations show that the contrasts in hydraulic conductivity between matrix and high-permeable lenses as well as the spatial configuration of tracer plumes and lenses dominate mixing, rather than the actual pore scale dispersivities. These results indicate that local material distributions, the magnitude of permeability contrasts, and their spatial and scale relation to solute plumes are more important for macro-scale transverse dispersion than the micro-scale dispersivities of individual materials. Local material characterization by thorough site investigation hence is of utmost importance for the evaluation of mixing-influenced or -governed problems in groundwater, such as tracer test evaluation or an assessment of contaminant natural attenuation.

Stretching, coalescence, and mixing in porous media.

We study scalar mixing in heterogeneous conductivity fields, whose structural disorder varies from weak to strong. A range of stretching regimes is observed, depending on the level of structural heterogeneity, measured by the log-conductivity field variance. We propose a unified framework to quantify the overall concentration distribution predicting its shape and rate of deformation as it progresses toward uniformity in the medium. The scalar mixture is represented by a set of stretched lamellae whose rate of diffusive smoothing is locally enhanced by kinematic stretching. Overlap between the lamellae is enforced by confinement of the scalar line support within the dispersion area. Based on these elementary processes, we derive analytical expressions for the concentration distribution, resulting from the interplay between stretching, diffusion, and random overlaps, holding for all field heterogeneities, residence times, and Péclet numbers.

Scaling of Convective Mixing in Porous Media

Convective mixing in porous media is triggered by a Rayleigh-Bénard-type hydrodynamic instability as a result of an unstable density stratification of fluids. While convective mixing has been studied extensively, the fundamental behavior of the dissolution flux and its dependence on the system parameters are not yet well understood. Here, we show that the dissolution flux and the rate of fluid mixing are determined by the mean scalar dissipation rate. We use this theoretical result to provide computational evidence that the classical model of convective mixing in porous media exhibits, in the regime of high Rayleigh number, a dissolution flux that is constant and independent of the Rayleigh number. Our findings support the universal character of convective mixing and point to the need for alternative explanations for nonlinear scalings of the dissolution flux with the Rayleigh number, recently observed experimentally.

Effect of particle diameter of porous media on flow and heat transfer in a mixing tee

Numerical simulations have been carried out to investigate flow and heat transfer in a mixing tee filled with periodic sintered copper spheres. Three particle diameter cases of 28mm, 14mm and 7mm with the array of 4×4, 8×8 and 16×16 at the same porosity of 0.3 have been calculated using large-eddy simulations and the Smagorinsky–Lilly sub-grid scale model. With the particle diameter decreasing, the mixture scale of hot and cold fluid tends to decrease in the mixing tee; the pressure drop of fluid flow through porous media increases. When the particle diameter ratios are 4:2:1 and the specific surface ratios are 1:2:4, the pressure drop ratios are 1:2:4; the thermal mixing in porous media is weakened because the temperature fluctuation decreases and the stratification of hot and cold fluids is observed.

Evaluation of transverse dispersion effects in tank experiments by numerical modeling: Parameter estimation, sensitivity analysis and revision of experimental design

Transverse dispersion represents an important mixing process for transport of contaminants in groundwater and constitutes an essential prerequisite for geochemical and biodegradation reactions. Within this context, this work describes the detailed numerical simulation of highly controlled laboratory experiments using uranine, bromide and oxygen depleted water as conservative tracers for the quantification of transverse mixing in porous media. Synthetic numerical experiments reproducing an existing laboratory experimental set-up of quasi two-dimensional flow through tank were performed to assess the applicability of an analytical solution of the 2D advection-dispersion equation for the estimation of transverse dispersivity as fitting parameter. The fitted dispersivities were compared to the “true” values introduced in the numerical simulations and the associated error could be precisely estimated. A sensitivity analysis was performed on the experimental set-up in order to evaluate the sensitivities of the measurements taken at the tank experiment on the individual hydraulic and transport parameters. From the results, an improved experimental set-up as well as a numerical evaluation procedure could be developed, which allow for a precise and reliable determination of dispersivities. The improved tank set-up was used for new laboratory experiments, performed at advective velocities of 4.9m d−1 and 10.5m d−1. Numerical evaluation of these experiments yielded a unique and reliable parameter set, which closely fits the measured tracer concentration data. For the porous medium with a grain size of 0.25–0.30mm, the fitted longitudinal and transverse dispersivities were 3.49×10−4m and 1.48×10−5m, respectively. The procedures developed in this paper for the synthetic and rigorous design and evaluation of the experiments can be generalized and transferred to comparable applications.

Experimental Investigation and Pore-Scale Modeling Interpretation of Compound-Specific Transverse Dispersion in Porous Media

In this study, we performed multitracer laboratory bench-scale experiments and pore-scale simulations in different homogeneous saturated porous media (i.e., different grain sizes) with the objective of (i) obtaining a generalized parameterization of transverse hydrodynamic dispersion at the continuum Darcy scale; (ii) gaining an improved understanding of the role of basic transport processes (i.e., advection and molecular diffusion) at the subcontinuum scale and their effect on the macroscopic description of transverse mixing in porous media; (iii) quantifying the importance of compound-specific properties such as aqueous diffusivities for transport of different solutes. The results show that a non-linear compound-dependent parameterization of transverse hydrodynamic dispersion is required to capture the observed lateral displacement over a wide range of seepage velocities (0.1–35 m/day). With pore-scale simulations, we can prove the hypothesis that the interplay between advective and diffusive mass transfer results in vertical concentration gradients leading to incomplete mixing in the pore channels. We quantify mixing in the pore throats using the concept of flux-related dilution index and show that different solutes undergoing transport in a flow-through system with a given average velocity can show different degrees of incomplete mixing. Furthermore, it is this compound-specific incomplete mixing within pores that causes different local transverse (mechanical) dispersion to result at the Darcy scale for high flow velocities. We conclude that physical processes at the microscopic level significantly determine the observed macroscopic behavior and, therefore, should be properly reflected in up-scaled parameterizations of transport processes such as local hydrodynamic dispersion coefficients.

Transverse Mixing Enhancement due to Bacterial Random Motility in Porous Microfluidic Devices

Bacterial swimming in groundwater may create flow disturbances in the surrounding microenvironment thereby enhancing contaminant mixing. Porous microfluidic devices (MFDs) were fabricated in three different pore geometry designs: uniform grain size with large pore throats (MFD-I), nonuniform grain size with restricted pore space (MFD-II), and uniform grain size with small pore throats (MFD-III). Escherichia coli HCB33 was used to assess the effect of bacterial random motility on transverse mixing of a tracer, fluorescent labeled dextran, under three experimental conditions in which motile bacteria, nonmotile bacteria, and plain buffer suspensions were flown through the MFDs at four different flow rates. Mixing was quantified in terms of the best-fit effective transverse dispersion coefficient ((D(cy))(eff)). A mixing enhancement index (MEI) was defined as the ratio of the (D(cy))(eff) of tracer in experiments with motile bacteria and without bacteria. Motile bacteria caused a maximum 5-6 fold increase in MEI in MFD-II, a nearly 4-fold increase in MFD-I, and very little observed change in MFD-III. The apparent transverse dispersivities (α(app)) of MFD-II and MFD-I increased by 3 and 2.3 times, respectively, with no change in MFD-III. These observations indicate that both pore throat size and pore arrangement are critical factors for contaminant mixing in porous media.

Enhancement of dilution and transverse reactive mixing in porous media: Experiments and model-based interpretation

Transport and natural attenuation of contaminant plumes in groundwater are often controlled by transverse dispersion. The extent of mixing between dissolved reaction partners at the fringe of a plume determines its length and depends strongly on the groundwater flow field. Transient flow conditions as well as the focusing of the flow in high-permeability zones may enhance transverse mixing of dissolved species and, therefore, create favorable conditions for the natural attenuation of contaminant plumes. The aim of the present study is to experimentally test the influence of these processes on solute mixing and to directly compare the results with those under analogous homogeneous and steady-state conditions. We have performed conservative and reactive tracer experiments in a quasi two-dimensional tank filled with glass beads of different sizes. The experiments have been carried out in both homogeneous and heterogeneous porous media under steady-state and transient (i.e. oscillating) flow fields. We used fluorescein as conservative tracer; whereas an alkaline solution (NaOH) was injected into ambient acidic water (HCl) in the reactive experiments. A pH indicator was added to the reacting solutions in order to visualize the emerging plume. We simulated the laboratory experiments with a numerical model and compared the outcomes of the model with the measured concentrations at the outlet of the tank and with the observed tracer plumes. Spatial moments, a newly defined flux-related dilution index, the product mass fluxes and the reaction enhancement factors were calculated to quantify the differences in mixing and reaction extent under various experimental conditions. The results show that flow focusing in heterogeneous porous media significantly enhances transverse mixing and mixing-controlled reactions, whereas temporally changing flow fields appear to be of minor importance.