Miscible Fluids Research Articles

Quantitative Tortuosity Measurements of Carbonate Rocks Using Pulsed Field Gradient NMR

Tortuosity is an important physical characteristic of porous materials; for example, it is a critical parameter determining the effective diffusion coefficient dictating mixing between miscible fluids in porous rock structures as is relevant to enhanced gas recovery processes. Accurate measurement of tortuosity remains challenging, resulting in various definitions dictated largely by the measurement protocol applied. Here, we focus primarily on ‘diffusive’ tortuosity (τd), which is defined as the ratio of the bulk fluid diffusion coefficient to the restricted diffusion coefficient applicable to the porous media under study. Specifically, we consider carbonate rock cores ranging in permeability from 2 to 5300 mD and adapt pulsed field gradient (PFG) NMR methodology such that accurate measurements of tortuosity are obtained over a sufficiently representative length scale of the porous media. To this end, we deploy supercritical methane as a probe molecule exploiting both its high mobility and proton density. Tortuosity measurements are shown to be independent of both pressure and diffusion observation time, conclusively proving that our measurements are in the asymptotic regime in which all of the pore space is adequately sampled by the diffusing methane molecules. The resultant ‘diffusive’ tortuosity measurements (which ranged from 3.1 to 5.6) are then compared against independent electrical conductivity measurements of tortuosity using a two-electrode impedance technique applied to the carbonate samples saturated with brine solution. Agreement between the ‘diffusive tortuosity,’ as measured by PFG NMR, and ‘electrical’ tortuosity was remarkably good (within 10%), given the very different measurements techniques used, for most of the carbonate rock samples considered.

Mixing intensification under turbulent conditions in a high pressure microreactor

The turbulent mixing of two miscible fluids is investigated in a high pressure (HP) coflow microreactor operated at 100 bar. Ethanol and CO2 are selected as model solvents to mimic the final targeted application, i.e.: a supercritical antisolvent process at microscale (μSAS). We first demonstrate experimentally that turbulent mixing can be reached in a microchannel using HP microfluidics. A computational fluid dynamic (CFD) model, performed using direct numerical simulation (DNS) down to the Kolmogorov scale has been applied for the turbulent mixing simulations. The effects of the main operating parameters on the final mixing efficiency have been studied, namely: the temperature, the fluid flowrates, the microchannel dimensions and the capillary inner and outer diameters. According to a predefined intensity of segregation, the characteristic mixing times are determined and used for determining mixing efficiency. The ratio of the total mixing time to the diffusion time depends on the ratio of the kinetic energies (the outer fluid to the inner one). The obtained micromixing times have been related to the turbulent energy dissipation rate ∊, calculated directly from the velocity fluctuations. The mixing intensification is obtained with much lower characteristic mixing times in the microreactor (one order of magnitude) than previously reported. This fundamental study is an indispensable guidance for several processes, including the μSAS applications.

Spatial analog of the two-frequency torus breakup in a nonlinear system of reactive miscible fluids.

We present a theoretical study on pattern formation occurring in miscible fluids reacting by a second-order reaction A+B→C in a vertical Hele-Shaw cell under constant gravity. We have recently reported that the concentration-dependent diffusion of species coupled with a frontal neutralization reaction can produce a multilayer system where low-density depleted zones could be embedded between the denser layers. This leads to the excitation of chemoconvective modes spatially separated from each other by a motionless fluid. In this Rapid Communication, we show that the layers can interact via a diffusion mechanism. Since diffusively coupled instabilities initially have different wavelengths, this causes a long-wave modulation of one pattern by another. We have developed a mathematical model which includes a system of reaction-diffusion-convection equations. The linear stability of a transient base state is studied by calculating the growth rate of the Lyapunov exponent for each unstable layer. Numerical simulations supported by phase portrait reconstruction and Fourier spectra calculation have revealed that nonlinear dynamics consistently passes through (i) a perfect spatially periodic system of chemoconvective cells, (ii) a quasiperiodic system of the same cells, and (iii) a disordered fingering structure. We show that in this system, the coordinate codirected to the reaction front paradoxically plays the role of time, time itself acts as a bifurcation parameter, and a complete spatial analog of the two-frequency torus breakup is observed.

Pressure‐driven displacement flows of yield stress fluids: Viscosity ratio effects

AbstractWe numerically investigate pressure‐driven, density‐unstable displacement flows of two miscible fluids along a near‐horizontal 2D channel. The displacing fluid is a Newtonian fluid, slightly heavier than the displaced yield stress (Bingham) fluid. The imposed displacement flow is laminar. We show that the displacement flow is mainly governed by five dimensionless numbers, and their combinations, including the Reynolds number (Re), the Bingham number (Bn), the densimetric Froude number (Fr), the viscosity ratio (m), and the channel inclination angle (β). In this work, we primarily focus on the viscosity ratio and provide a detailed understanding of the flow behaviours via studying the effects of m on displacement flow patterns, regime classifications based on slump‐type and centre‐type displacement flow regimes, leading and trailing displacement front features, and finally the effects of m at different inclination angles.

Front dynamics in exchange flow of two immiscible iso-viscous fluids in two-dimensional straight and curved plane channels

Exchange flow of two immiscible fluids at a low Atwood number in a straight and curved plane channel is considered in this analytical study. The fluids are considered immiscible, but practically, the results can be applied to miscible fluids for short times and in nearly horizontal channels where mixing is negligible due to strong segregation. The exchange flow and displacement flow in pipes at different inclinations with respect to vertical have been extensively studied and have many applications in industry or environmental settings. For the case of plane two-dimensional channels, however, because of the simpler geometry, it is more convenient to understand the physics of the problem and formulate the physical phenomena mathematically. An equation has been derived that describes the transient front velocity in exchange flow in a straight plane channel. The steady state front velocity in straight channels is estimated. The exchange flow in curved channels demonstrates an unstable front or a separated trail because of the curvature of the path. In the case of curved channels, some of the general behavior of the interface is predicted and validated against some experimental observations in curved pipes but quantitative analysis of the interface and the flow requires more advanced mathematical formulation and more detailed experiments for validation.

Optimization of a static mixing device using the continuous adjoint to a two-phase mixing model

The continuous adjoint method is formulated and utilized for the optimization of a static mixing device. The CFD tool used for the simulations is based on a two-phase model governing flows of two miscible fluids. The formulation of the corresponding continuous adjoint problem is presented and the computed gradients are utilized in an optimization loop. In specific, a multi-objective optimization problem is formulated and solved for maximum mixture uniformity at the outlet and minimum total pressure losses inside a static mixing device. The weighted sum of these two quantities of interest is the objective function to be minimized by solving a single-objective problem. Through the solution of a number of optimization problems, with different weights each, the Pareto front of optimal solutions is computed. Two optimization approaches are employed taking the manufacturability of the final shape into consideration, giving rise to different optimal designs to be discussed and compared. Differences in the efficiency and the optimal shapes between the two approaches are thoroughly discussed and compared.

A Novel Numerical Model for Simulating the Quantity of Tailing Oil in the Mixed Segment between Two Batches in Product Pipelines

Cutting mixed oil in product pipelines has a great influence on the economy of the pipeline operation processes. The reasonable prediction of CDMS (the concentration distribution in the mixed segment) is important for cutting mixed segments. The classical model cannot explain the tailing phenomenon well which should not be neglected during operation processes. Based on Fick’s diffusion law, a new model for calculating the diffusion coefficient is proposed in this article, which originates from the essence of the diffusion phenomenon and considers the effects of both physical properties of oil products and the turbulence. At the same time, the dynamic fluid equilibrium model of CDMS near the pipe wall is given which has considered the adsorption effect of wall roughness. Based on these two factors, a novel numerical model for simulating the quantity of tailing oil is proposed, which is solved via the characteristic method and the finite difference method. The effects of different physical properties, as well as the adsorption, on both LFMS (the length of the front of the mixed segment) and LTMS (the length of the tail of the mixed segment), are analyzed. The comparison between the simulation results and the experimental data is utilized to validate the proposed numerical model. The simulation results show that the novel model can well describe the mixed segment tailing phenomenon and also explain the mixing essence of two miscible but dissimilar fluids in the pipeline more clearly. To sum up, this model can provide theoretical guidance for the prediction of CDMS and cutting process in practical operation processes; therefore, more economic benefit can be obtained.

Effective interfacial tension in flow-focusing of colloidal dispersions: 3-D numerical simulations and experiments

An interface between two miscible fluids is transient, existing as a non-equilibrium state before complete molecular mixing is reached. However, during the existence of such an interface, which typically occurs at relatively short time scales, composition gradients at the boundary between the two liquids cause stresses effectively mimicking an interfacial tension. Here, we combine numerical modelling and experiments to study the influence of an effective interfacial tension between a colloidal fibre dispersion and its own solvent on the flow in a microfluidic system. In a flow-focusing channel, the dispersion is injected as core flow that is hydrodynamically focused by its solvent as sheath flows. This leads to the formation of a long fluid thread, which is characterized in three dimensions using optical coherence tomography and simulated using a volume of fluid method. The simulated flow and thread geometries very closely reproduce the experimental results in terms of thread topology and velocity flow fields. By varying the interfacial tension numerically, we show that it controls the thread development, which can be described by an effective capillary number. Furthermore, we demonstrate that the applied methodology provide the means to measure the ultra-low but dynamically highly significant effective interfacial tension.

Spinning Drop Dynamics in Miscible and Immiscible Environments.

We report on the extensional dynamics of spinning drops in miscible and immiscible background fluids following a rotational speed jump. Two radically different behaviors are observed. Drops in immiscible environments relax exponentially to their equilibrium shape, with a relaxation time that does not depend on the centrifugal force. We find an excellent quantitative agreement with the relaxation time predicted for quasi-spherical drops by Stone and Bush (Q. Appl. Math. 1996, 54, 551), while other models proposed in the literature fail to capture our data. By contrast, drops immersed in a miscible background fluid do not relax to a steady shape: they elongate indefinitely, their length following a power-law in very good agreement with the dynamics predicted by Lister and Stone (J. Fluid Mech. 1996, 317, 275) for inviscid drops. Our results strongly suggest that low compositional gradients in miscible fluids do not give rise to an effective interfacial tension measurable by spinning drop tensiometry.

The structure of a turbulent line fountain

Line fountains form when heavy miscible fluid is ejected steadily upwards as a jet from a high-aspect-ratio rectangular slot, of length $L$ and half-width $b_{0}$, into lighter quiescent surroundings. Viewed along the slot from one end, previous observations reveal that the ejected fluid mixes with the environment and reaches a peak height before partially collapsing back downward under gravity to form a fountain whose top thereafter fluctuates vertically about a mean height. While the motion as perceived from this single view has provided insights that have successfully guided theoretical predictions for the initial rise height, until now a wider understanding of line fountains, and corresponding predictive capability, has been limited to this single prediction due to a lack of any other observational data. Indeed, the general behaviour of line fountains, including the structure internally and along the spanwise length $L$ of the slot, has not been reported previously. To address this, flow visualisations and comprehensive measurements of saline fountains in an aqueous environment are presented here that reveal their complex overall structure and behaviours. After establishing the uniformity of the source conditions from slots of aspect ratio $600:1$ and $300:1$, we first show that double-averaged (spanwise and time) rise heights $\overline{\overline{z}}_{v}/b_{0}$ scale on $Fr_{0}^{4/3}$, $Fr_{0}$ being the source Froude number, with vertical fluctuations being circa 20 % of these heights. Then, simultaneously interrogating the flow as viewed from above and from the side onto the spanwise dimension, we identify three distinct patterns of behaviour. Instrumental to distinguishing these behaviours were the contrasting signatures we observed in the time series of rise height departures from the mean which led us to the following classification: (i) non-uniform flapping for $0.05\lesssim \overline{\overline{z}}_{v}/L\lesssim 0.30$, in which the lateral motion of the fountain takes the form of an oscillatory wave with a wavelength of $2L/3$ (approx.); (ii) uniform flapping for $0.30\lesssim \overline{\overline{z}}_{v}/L\lesssim 0.45$, in which the entire fountain sways to the left and then to the right side of the slot; and (iii) disorganised flapping for $\overline{\overline{z}}_{v}/L\gtrsim 0.45$. Regarding the internal structure, we show that unlike a classic round fountain, eddying structures comparable in scale with the rise height form towards the top of the fountain, and the counterflow forms predominantly to one side of the jet. We then identify the single dominant mechanism driving the flapping motions, successfully linking the wave-like behaviour observed along the span to the internal structure and vertical oscillations. Quantifying the oscillatory motions, both the vertical and flapping frequencies scale as $Fr_{0}^{-2}$, and we demonstrate and explain a robust coupling between these frequencies that follows a ratio of 2:1.

Strict Equivalence between Maxwell–Stefan and Fast-Mode Theory for Multicomponent Polymer Mixtures

The applicability of theories describing the kinetic evolution of fluid mixtures depends on the underlying physical assumptions. The Maxwell-Stefan equations, widely used for miscible fluids, express forces depending on coupled fluxes. They need to be inverted to recover a Fickian form which is generally impossible analytically. Moreover, the concentration dependence of the diffusivities has to be modelled, e.g. by the multicomponent Darken equation. Cahn-Hilliard type equations are preferred for immiscible mixtures, whereby different assumptions on the coupling of fluxes lead to the slow-mode and fast-mode theories. For two components, these were derived from the Maxwell-Stefan theory in the past. Here, we prove that the fast-mode theory and the generalized Maxwell-Stefan theory together with the multicomponent Darken equation are strictly equivalent even for multicomponent systems with very different molecular sizes. Our findings allow to reduce the choice of a suitable theory to the most efficient algorithm for solving the underlying equations.

Cylindrical flowing-junction cell for the investigation of fluctuations and pattern-formation in miscible fluids.

We describe a flowing-junction cell with cylindrical symmetry suitable to investigate fluctuations and pattern formation at the diffusing interface between two miscible phases of a liquid mixture. The continuous outflow of the remixed fluid through a thin slit located at the midheight of the sample allows the preparation of an initially sharp interface. The system can be used in both gravity-stable and unstable conditions. In the stable case, the denser liquid is on the bottom of the cell and mass diffusion is the only active process for remixing the two liquids. Once the flow is stopped, one can investigate nonequilibrium fluctuations during free-diffusion in a binary mixture or double diffusive instabilities in multicomponent mixtures. Two horizontal transparent windows allow vertical mapping of the fluid flow by using shadowgraphy. In the unstable condition, with the denser fluid on top, stopping the radial flow at the interface gives rise to a Rayleigh-Taylor instability, which drives the denser liquid toward the bottom of the cell. The fact that the cell can maintain the system in the unstable condition shows that it is suitable to perform experiments under microgravity conditions. With respect to other free-diffusion cells, the proposed configuration has the advantage that the interface is extremely stable and flat, and that the experiments can be repeated by just flowing the cell with fresh liquids.

Analysis of the 3D non-linear Stokes problem coupled to transport-diffusion for shear-thinning heterogeneous microscale flows, applications to digital rock physics and mucociliary clearance

This study provides the analysis of the generalized 3D Stokes problem in a time dependent domain, modeling a solid in motion. The fluid viscosity is a non-linear function of the shear-rate and depends on a transported and diffused quantity. This is a natural model of flow at very low Reynolds numbers, typically at the microscale, involving a miscible, heterogeneous and shear-thinning incompressible fluid filling a complex geometry in motion. This one-way coupling is meaningful when the action produced by a solid in motion has a dominant effect on the fluid. Several mathematical aspects are developed. The penalized version of this problem is introduced, involving the penalization of the solid in a deformable motion but defined in a simple geometry (a periodic domain and/or between planes), which is of crucial interest for many numerical methods. All the equations of this partial differential system are analyzed separately, and then the coupled model is shown to be well-posed and to converge toward the solution of the initial problem. In order to illustrate the pertinence of such models, two meaningful micrometer scale real-life problems are presented: on the one hand, the dynamics of a polymer percolating the pores of a real rock and miscible in water; on the other hand, the dynamics of the strongly heterogeneous mucus bio-film, covering the human lungs surface, propelled by the vibrating ciliated cells. For both these examples the mathematical hypothesis are satisfied.

Spatiotemporal electrochemistry on flexible microelectrode arrays: Progress towards smart contact lens integration

We demonstrate a real-speed spatiotemporal electrochemical map showing both time- and position-varying concentration of an analyte in contact with a flexible microelectrode array. A polymer-based device of 11 μm in thickness comprising patterned gold metallisation on a polyimide substrate was fabricated, with eight individually addressable working electrodes (diameter 30 μm) and an integrated counter electrode. We performed a repeated sequence of high-speed chronoamperometric measurements at each electrode and processed the data to generate a spatiotemporal concentration map, in which a number of fluid effects, including bulk flow, diffusive mixing and homogenisation of two miscible fluids of different concentration were observed. This device was fabricated using processes compatible with an existing smart contact lens platform, with a view to develop integrated sensors in future work. We believe this technique has significant potential in the field of electrochemical smart contact lenses, both in introducing new functionality and in improving our ability to draw accurate and clinically-relevant conclusions from measurements made in the tear film.

Simultaneous Occurrence of Miscible and Immiscible Displacement Processes During Solvent Displacement of Heavy Oil: A Parametric Analysis Using Visual Capillary-Tube Experiments

Summary Oil/solvent mixing is essential during solvent-injection applications to reduce the viscosity of oil, but mass transfer by diffusion becomes slower because the oil becomes heavier. Thus, an interface exists between the oil and solvent at certain times, being stronger in the beginning of the process. This results in an immiscible displacement controlled by the capillary forces while mixing is in progress. It is of practical and fundamental importance to determine the mechanisms responsible for the displacement of heavy oil and the behavior of solvents (acting as both immiscible and miscible displacement agents) because it could be advantageous to accelerate the dilution of heavy oil in many circumstances, including heterogeneous (fractured, layered, wormholed) systems. This is a complex process consisting of multiple pore phases (oil, solvent, their mixtures, and aqueous and vapor phases) at the same time, while different mechanisms such as capillary imbibition, miscible interaction (diffusion and convection), and gravity also act simultaneously. To investigate this complex phenomenon for different oil/solvent systems, a novel experimental method was used. The underlying mechanisms that dominate the solvent displacement process were comprehensively identified. The movement and evolution of interfaces among different fluid phases in glass capillary tubes were observed and recorded. Oil samples with different viscosities were used to examine the effects of oil viscosity on the mass transfer accelerated by imbibition transfer. The effects of temperature, wettability, and boundary conditions on the interaction of miscible fluid pairs were also studied. Pentane, heptane, and decane were used as the solvent phases. Advanced photographic techniques using ultraviolet (UV) light and dyed fluids were applied to better track the flow of different phases in the mixing zone. The experiments demonstrated a slowly smearing interface between the solvent and viscous oil. A unique natural convection was induced, with the combined effect of gravity, diffusion (mixing), and capillarity all contributing to the recovery of heavy oil. On the basis of the saturation method, boundary condition, and the Bond number, four different motion modes of mixing zone and interfaces of miscible fluids in the capillary tube were revealed and categorized to identify the degree of interface development (immiscible flooding). Also, the mixing zone, mass flux, and flow behavior were quantified using dimensionless parameters. The results indicate that priority may be given to a solvent with a high interfacial tension (IFT) for the solvent-based oil-recovery technique because of a strong imbibition and further enhancement of the dilution and displacement processes under conditions of a similar viscosity ratio. The data provided will be useful for the accuracy of modeling studies, especially for complex geologies where oil/solvent interaction is critically difficult to develop in order for mixing to occur.

Frozen waves in turbulent mixing layers

The horizontal vibrations of an interface between miscible fluids trigger a parametric instability, leading to frozen wave patterns. Theory and numerical simulations reveal that the mixing layer size varies as the square of the forcing amplitude in the turbulent regime.

Mesoscale modelling of miscible and immiscible multicomponent fluids

A mesoscopic simulation method based on the integration of dissipative particle dynamics (DPD), smoothed particle hydrodynamics (SPH) and computational thermodynamics (CT) has been developed. The kinetic behaviours of miscible and immiscible fluids were investigated. The interaction force between multicomponent mesoscopic particles is derived from the system free energy. The diffusivity of the components in non-ideal solution is determined by the chemical potential. The proposed method provides convincing predictions to the effects of convection, diffusion and microscopic interaction on the non-equilibrium evolution of engineering fluids, and demonstrates a potential to simulate more complicated phenomena in materials processing.

The effect of density difference on the development of density-driven convection under large Rayleigh number

In this study, we visualized density-driven convection in an improved Hele-Shaw cell at 25 °C and 0.1 MPa by examining the convection mixing process of analogous fluids and total dissolution rates in the high Rayleigh number (Ra) region. The high Ra number was inspired by data referring to the geological reservoir at the Sleipner site. We investigated the effect of the density difference between different fluid pairs on the initial moment and development of convective mixing. Though direct imaging and processing, we examined the convective mixing development regime, finger number density, onset time and movement speed of fingers. The fingering instability and mixing time of miscible fluids were also analysed and show that as the density difference increases, the density of the finger number increases and the fingers quickly merge. The onset time decreases sharply and then becomes stable and is inversely proportional to Ra2. The convection mixing time has the same tendency but is inversely proportional to Ra. Moreover, the finger speed linearly increases and is directly proportional to the characteristic velocity. The dissolution rate was investigated with the dimensionless convection flux Sherwood number (Sh) and shows a power law relationship with Ra. The density difference accelerates the dissolution rate. The experimental conditions were different from those of real CO2 geological storage, but this study has implications for predicting the CO2 dissolution trapping time and accumulation for geological storage of carbon dioxide.

Three-dimensional visualization of viscous fingering for non-Newtonian fluids with chemical reactions that change viscosity

Three-dimensional viscous fingering for miscible non-Newtonian fluids with and without chemical reactions were studied with a microfocus X-ray computed tomography scanner. We find that the area fraction of injected fluid in the reactive cases was lower than that in the nonreactive cases.

Thermal mixing, cooling and entropy generation in a micromixer with a porous zone by the lattice Boltzmann method

Short transport paths with high surface to volume ratios in microfluidic devices have the potential to increase the heat transfer significantly. However, because of the very low Reynolds numbers reached in microchannel flows, it is difficult to achieve effective thermal mixing. Microfluidic mixing requires fast mixing process of low diffusivity fluids. To obtain a rapid thermal mixing in passive micromixers, a porous block may be inserted to enhance the thermal performance. A 2D lattice Boltzmann thermal model is utilized here to investigate the thermal performance of a Y-micromixer with a porous block. Different parameters of porous block including its aspect ratio, its position, its porosity and its effective thermal conductivity are considered. The thermal mixing and cooling of two miscible fluids at 50 and 25 °C entering the micromixer are investigated in detail. The results show that the porous block significantly improves the thermal mixing of both streams. Increasing the porous block aspect ratio leads to better cooling, low entropy generation and high dimensionless entropy generation. It is also observed that thermal mixing and cooling performance increase by decreasing the effective thermal conductivity and porosity of the block, which causes a low entropy generation.