Immiscible Fluid Phases Research Articles

Chemical Programming of Solubilizing, Nonequilibrium Active Droplets.

ConspectusThe multifunctionality and resilience of living systems has inspired an explosion of interest in creating materials with life-like properties. Just as life persists out-of-equilibrium, we too should try to design materials that are thermodynamically unstable but can be harnessed to achieve desirable, adaptive behaviors. Studying minimalistic chemical systems that exhibit relatively simple emergent behaviors, such as motility, communication, or self-organization, can provide insight into fundamental principles which may enable the design of more complex and life-like synthetic materials in the future.Emulsions, which are composed of liquid droplets dispersed in another immiscible fluid phase, have emerged as fascinating chemically minimal materials in which to study nonequilibrium, life-like properties. As covered in this Account, our group has focused on studying oil-in-water emulsions, specifically those which destabilize by solubilization, a process wherein oil is released into the continuous phase over time to create gradients of oil-filled micelles. These chemical gradients can create interfacial tension gradients that lead to droplet self-propulsion as well as mediate communication between neighboring oil droplets. As such, oil-in-water emulsions present an interesting platform for studying active matter. However, despite being chemically minimal with sometimes as few as three chemicals (oil, water, and a surfactant), emulsions present surprising complexity across the molecular to macroscale. Fundamental processes governing their active behavior, such as micelle-mediated interfacial transport, are still not well understood. This complexity is compounded by the challenges of studying systems out-of-equilibrium which typically require new analytical methods and may break our intuition derived from equilibrium thermodynamics.In this Account, we highlight our group's efforts toward developing chemical frameworks for understanding active and interactive oil-in-water emulsions. How do the chemical properties and physical spatial organization of the oil, water, and surfactant combine to yield colloidal-scale active properties? Our group tackles this question by employing systematic studies of active behavior working across the chemical space of oils and surfactants to link molecular structure to active behavior. The Account begins with an introduction to the self-propulsion of single, isolated droplets and how by applying biases, such as with a gravitational field or interfacially adsorbed particles, drop speeds can be manipulated. Next, we illustrate that some droplets can be attractive, as well as self-propulsive/repulsive, which does not fall in line with the current understanding of the impact of oil-filled micelle gradients on interfacial tensions. The mechanisms by which oil-filled micelles influence interfacial tensions of nonequilibrium interfaces is poorly understood and requires deeper molecular understanding. Regardless, we extend our knowledge of droplet motility to design emulsions with nonreciprocal predator-prey interactions and describe the dynamic self-organization that arises from the combination of reciprocal and nonreciprocal interactions between droplets. Finally, we highlight our group's progress toward answering key chemical questions surrounding nonequilibrium processes in emulsions that remain to be answered. We hope that our progress in understanding the chemical principles governing the dynamic nonequilibrium properties of oil-in-water droplets can help inform research in tangential research areas such as cell biology and origins of life.

Chaotic Transport of Solutes in Unsaturated Porous Media.

Unsaturated porous media, characterized by the combined presence of several immiscible fluid phases in the pore space, are highly relevant systems in nature, because they control the fate of contaminants and the availability of nutrients in the subsoil. However, a full understanding of the mechanisms controlling solute mixing in such systems is still missing. In particular, the role of saturation in the development of chaotic solute mixing has remained unexplored. Using three-dimensional numerical simulations of flow and transport at the pore scale, built upon X-ray tomograms of a porous medium at different degrees of liquid (wetting)-phase saturation, we show the occurrence of chaotic dynamics in both the deformation of the solute plume, as characterized by computed chaos metrics (Lyapunov exponents), and the mixing of the injected solute. Our results show an enhancement of these chaotic dynamics at lower saturation and their occurrence even under diffusion-relevant conditions over the medium's length, also being strengthened by larger flow velocities. These findings highlight the dominant role of the pore-scale spatial heterogeneity of the system, enhanced by the presence of an immiscible phase (e.g., air), on the mixing efficiency. This represents a stepping stone for the assessment of mixing and reactions in unsaturated porous media.

Mass Transfer Performance of a Rotating Spiral: Comparison with Conventional Contacting Methods

Previous work by the authors develops a novel technology for counter-current contacting of immiscible fluid phases based on a rotating spiral channel. The work has established theoretical modelling allowing prediction of mass transfer rates for physical absorption and desorption of dilute solute species and has verified the main hydrodynamic elements of the theory, with experimental measurements over a wide range of flow rates and liquid viscosity. A key feature of rotating spiral contacting is that the relative thicknesses of the phase layers is constant throughout the contacting process and, along with the relative phase flow rates, can be varied regardless of phase and solute properties to produce optimum conditions for the contacting. It has been shown that there are two criteria that determine the optimum. First, the flow rate ratio of the phases is matched to the equilibrium distribution of the solute between the two phases, so that sufficient solvent phase flows to allow full removal of solute from the processed stream. Second, the relative layer thicknesses should maximise specific throughput, i.e. the flow rate of the processed stream per device volume. These ideas of optimum contacting guide a comparison here of recent mass transfer data for the spiral with corresponding literature data for the packed column and rotating packed bed.

Dynamics of a viscoelastic droplet migrating in a ratchet microchannel under AC electric field

Droplet-based microfluidic devices can be powered or manipulated by applying an external electric field, and the ability to precisely control the flow in such devices is essential for various engineering and biomedical applications. In this numerical study, we investigate the deformation dynamics of a viscoelastic droplet in a ratchet microchannel under the influence of an AC electric field. We employ the leaky-dielectric electrohydrodynamic model for both the immiscible fluid phases coupled with the Oldroyd-B model for the droplet fluid. The effect of geometrical parameters such as the type of ratchet and the wavenumber of the ratchets along with the flow parameters such as the electrocapillary number, Weissenberg number and the capillary number significantly affect the droplet shape dynamics and the polymer chain extension. For the parameters considered in this work, the electric force tends to stretch the droplet in the streamwise direction and enhances the droplet deformation and polymer extension. Several interesting effects arise as a result of the coupling of the periodic hydrodynamic forcing of the ratchet walls and the electric field. Specifically, an exponential rise in the polymer chain extension for higher ratchet wavenumbers is observed, along with the cross-stream migration of the droplet for higher electrocapillary numbers when it reaches the outlet of the ratchet constriction.

Oscillations of a spherical particle in the presence of a flat interface separating two fluid phases

An analytical–numerical approach based on collocation techniques is introduced to examine the challenge of characterizing Stokes flow induced by a spherical particle in perpendicular motion to a plane interface that divides two semi-infinite, immiscible viscous fluid phases. The movement is examined under the conditions of low Reynolds and capillary numbers, where the interface undergoes negligible deformation. We contemplate both the rectilinear oscillations along and the rotational oscillation around an axis that is perpendicular to the interface. A comprehensive solution is formed by combining basic solutions in both cylindrical and spherical coordinate systems through superposition. Firstly, the boundary conditions at the fluid–fluid interface are fulfilled through Fourier–Bessel transforms, followed by addressing the conditions at the particle’s surface using a boundary collocation scheme. The drag force and torque coefficients, both in-phase and out-of-phase, acting on the particle are determined with satisfactory convergence across different geometric and physical parameter values. These results are illustrated through graphs and tables. We compare our results of drag force and torque coefficients with existing data in the literature, specifically for the limiting cases. The outcomes of this study highlight the substantial impact of the interface on both drag force and torque coefficients. This inquiry is driven by the necessity to enhance our comprehension of the fluid tapping mode in atomic force microscope devices, particularly when this mode involves an interface substrate with another fluid.

Impact of a spherical interface on a concentrical spherical droplet

<p>In this paper, an analytical and numerical technique are examined in order to analyse the Stokes flow determination problem due to a viscous sphere droplet moving at a concentric instantaneous position inside a spherical interface separating finite and semi-infinite immiscible fluid phases. Here, when only one of the three phases of the fluid (micropolar fluid) has a microstructure, attention is focused on this case. The motion is considered when Reynolds- and capillary-numbers are low, and the droplet surface and the fluid-fluid interface have insignificant deformation. A general solution is obtained in a spherical coordinate system based on a concentric position to analyse the slow axisymmetric movement of the micropolar fluid, considering microrotation and velocity components. Boundary conditions are initially fulfilled at the fluid-fluid interface and subsequently at the droplet surface. The normalised hydrodynamic drag force applying to a moving viscous droplet appears to be a function of the droplet-to-interface radius ratio, which increases monotonically and becomes unbounded when the droplet surface touches the fluid-fluid interface. The numerical outcomes of the normalised drag force acting on the viscous droplet are derived for different values of the parameters, and are presented in a tabular and graphical framework. A comparison was made between our numerical outcomes for the drag force and the pertinent data for the special cases found in the literature.</p>

Magmatic–hydrothermal evolution of the Koktokay No. 3 pegmatite, Altai, northwestern China: Constraints from in situ boron isotope and chemical compositions of tourmaline

Lithium–Cs–Ta-enriched (LCT-type) pegmatites are characterized by zoning texture and extreme enrichment of rare-metals. The role of fluids in zonation development and rare-metal mineralization remains a topic of debate. The Koktokay No. 3 pegmatite in northwestern China represents a typical LCT-type pegmatite characterized by economically significant Li–Be–Nb–Ta–Cs mineralization. This pegmatite exhibits a complete evolutionary sequence from outer to inner zones, with diverse rare-metal mineralization. Tourmaline is ubiquitously present in all textural zones of LCT-type pegmatites. In the Koktokay No. 3 pegmatite, the tourmalines exhibit a range of compositions from dravite and schorl to elbaite and rossmanite. The tourmalines in this pegmatite are characterized by significantly enriched light B isotope compositions (δ11B = −14.8‰ to −8.1‰), similar to those typically observed in most pegmatites derived from continental crust sources. The tourmalines in the outer zones have less variable δ11B values (−13.8‰ to −11.5‰) and are interpreted to have crystallized in a viscous magma. The tourmalines in the inner zones exhibit larger variations in δ11B values (−14.8‰ to −8.1‰) than those from the outer zones. The decreasing δ11B values of tourmalines from the outer to inner zones are attributed to the separation of an immiscible fluid phase that preferentially incorporated heavy B isotopes. The occurrence of hydrothermal tourmaline as veinlets indicates the exsolution of fluids in inner zones, with high δ11B values ranging from −10.2‰ to −9.3‰. Therefore, the hydrothermal fluid can be saturated when the magma has been strongly evolved. In addition to fractional crystallization, exsolved fluids play a significant role in the formation of internal zones and rare-metal mineralization in LCT-type pegmatites. The rare-metals are gradually enriched through fractional crystallization of magma, and the fluid contains mobile elements such as Cs that facilitate precipitation of rare-metal minerals.

Flow-Dependent Relative Permeability Scaling for Steady-State Two-Phase Flow in Porous Media: Laboratory Validation on a Microfluidic Network

Conventionally, the relative permeabilities of two immiscible fluid phases flowing in porous media are considered and expressed as functions of saturation. Yet, this has been put into challenge by theoretical, numerical, and laboratory studies of flow in artificial pore network models and real porous media. These works have revealed a significant dependency of the relative permeabilities on the flow rates, especially when the flow regime is capillary to capillary-viscous dominated, and part of the disconnected nonwetting phase remains mobile. These studies suggest that relative permeability models should include the functional dependence on flow intensities. However, revealing the explicit form of such dependence remains a persistent problem. Just recently, a general form of dependence was inferred based on extensive simulations with the DeProF model for steady-state two-phase flows in pore networks. The simulations revealed a systematic dependence of the relative permeabilities on the local flow rate intensities. This dependence can be described analytically by a universal scaling functional form of the actual independent variables of the process, namely, the capillary number, Ca, and the flow rate ratio, r. The proposed scaling incorporated a kernel function, the intrinsic dynamic capillary pressure (IDCP) function, describing the transition between capillarity- and viscosity-dominated flow phenomena. In a parallel laboratory study, SCAL measurements provided a preliminary proof-of-concept on the applicability of the model. In the laboratory study presented here, we examine the applicability of the scaling model by taking extensive, ex-core measurements of relative permeabilities for steady-state co-injections of two immiscible fluids within an artificial microfluidic pore network, across different flow regimes in Ca and r. From these measurements, we calculate the values of the mobility ratio, and we compare these to the corresponding values of the flow rate ratio. We also extract the IDCP curve, the locus of critical flow conditions, whereby the process is more efficient in terms of energy utilization – accounted by the nonwetting phase flow rate per unit of total power provided to the process, as well as the locus of flow conditions of equal relative permeabilities. We show that the degree of consistency between flow rate ratio and mobility ratio values, the IDCP curve, the locus of critical flow conditions, and the locus of equal relative permeabilities, as well as some associated invariant characteristic values, can be used for assessing the extent of end effects and for characterizing the flow as capillary- or viscous-dominated. The proposed scaling introduces new opportunities for enhancing SCAL protocols and their associated applications. These include the characterization of systems and flow conditions, dynamic rock typing, evaluation of capillary end effects, as well as the advancement of more efficient field-scale simulators. Additionally, it paves the way in designing more energy-efficient EOR interventions.

Dynamic Relative Permeabilities for Partially Saturated Porous Media Accounting for Viscous Coupling Effects: An Analytical Solution

We present an analytical model to compute frequency-dependent relative permeability functions for partially saturated porous media accounting for viscous coupling effects. For this, we consider the oscillatory motion of two immiscible fluid phases and solve the Navier–Stokes equations at the pore scale using suitable interface conditions between fluids. These calculations are combined with the generalized two-phase flow Darcy equations to obtain the corresponding upscaled macroscopic fluxes. By means of an analog pore model consisting of a bundle of cylindrical capillaries in which pore fluids are distributed in a concentric manner, we find closed analytical expressions for the complex-valued and frequency- and saturation-dependent relative permeability functions. These expressions allow for a direct assessment of viscous coupling effects on oscillatory flow for all frequencies and saturations. Our results show that viscous coupling effects significantly affect flow characteristics in the viscous and inertial regimes. Dynamic relative permeabilities are affected by the pore fluid densities and viscosities. Moreover, viscous coupling effects may induce two critical frequencies in the dynamic relative permeability curves, a characteristic that cannot be addressed by extending the classic dynamic permeability definition to partially saturated scenarios using effective fluids. The theoretical derivations and results presented in this work have implications for the estimation and interpretation of seismic and seismoelectric responses of partially saturated porous media.

A lattice Boltzmann exploration of two-phase displacement in 2D porous media under various pressure boundary conditions

While experimental designs developed in recent decades have contributed to research on dynamic nonequilibrium effects in transient two-phase flow in porous media, this problem has been seldom investigated using direct numerical simulation (DNS). Only a few studies have sought to numerically solve Navier–Stokes equations with level-set (LS) or volume-of-fluid (VoF) methods, each of which has constraints in terms of meniscus dynamics for various flow velocities in the control volume (CV) domain. The Shan–Chen multiphase multicomponent lattice Boltzmann method (SC-LBM) has a fundamental mechanism to separate immiscible fluid phases in the density domain without these limitations. Therefore, this study applied it to explore two-phase displacement in a single representative elementary volume (REV) of two-dimensional (2D) porous media. As a continuation of a previous investigation into one-step inflow/outflow in 2D porous media, this work seeks to identify dynamic nonequilibrium effects on capillary pressure–saturation relationship ( P c – S ) for quasi-steady-state flow and multistep inflow/outflow under various pressure boundary conditions. The simulation outcomes show that P c , S and specific interfacial area ( a nw ) had multistep-wise dynamic effects corresponding to the multistep-wise pressure boundary conditions. With finer adjustments to the increase in pressure over more steps, dynamic nonequilibrium effects were significantly alleviated and even finally disappeared to achieve quasi-steady-state inflow/outflow conditions. Furthermore, triangular wave-formed pressure boundary conditions were applied in different periods to investigate dynamic nonequilibrium effects for hysteretical P c – S . The results showed overshoot and undershoot of P c to S in loops of the nonequilibrium hysteresis. In addition, the flow regimes of multistep-wise dynamic effects were analyzed in terms of Reynolds and capillary numbers ( Re and Ca ). The analysis of REV-scale flow regimes showed higher Re (1 < Re < 10) for more significant dynamic nonequilibrium effects. This indicates that inertia is critical for transient two-phase flow in porous media under dynamic nonequilibrium conditions.

A new potential interface tension model for MPS method avoiding unphysical particle cohesion

Improvement of the capability to accurately track the interface between immiscible phases/components is of great importance to the numerical analysis with complex multi-component and multi-phase flow dynamics, which might be involved in severe accidents of light water reactors, such as Molten Core-Concrete Interaction (MCCI). In order to model the interface tension between immiscible phases/components, Zhu et, al., precedingly developed a potential interface tension model in MPS method based on Kondo et al.‘s earlier model for surface tension and succeeded to stably model the interface tension qualitatively by introducing different attractive forces for different pairs of particle types. However, it has been known that pressure is overestimated in Zhu et, al.‘s potential interface tension model. Moreover, this study has shown that the fluid flow can be numerically hindered with Zhu et al.‘s potential interface tension model. These issues of Zhu et al.‘s potential interface tension model is caused by unphysical particle cohesion due to the inter-particle attractive forces determined by the surface tension coefficients, which are required even when the free surface is not considered. Meanwhile, in the framework of SPH method, Zhou et, al., proposed to model interface tension by introducing the repulsive force between different phases. Hence, this study aimed to develop a new potential interface tension model for MPS method which can overcome the conventional drawbacks of pressure overestimation or unphysical flow hindrance, by incorporating the repulsive force between immiscible fluid phases referring to Zhou et, al.‘s idea within SPH method.The interface tension in the present method is modeled as the repulsive force derived from the “repulsive potential” and is considered only near the interface to avoid the unphysical particle cohesion due to the inter-particle attractive forces. The amplitude coefficient in the repulsive potential is determined by the methodology similar to that proposed by Kondo, et, al.With the newly developed interface tension model, the common issue of numerical flow hindrance that occurred in the previous interface tension models in MPS method has been resolved. The validation of the newly developed interface tension model shows good agreement with theoretical evaluations of Laplace pressure and the droplet deformation ratio in a shear flow. Moreover, the new interface tension model has been validated against a 3-dimensional immiscible fluid stratification test, in which the new interface tension model shows good agreement with the test results and improvement of the pressure profile evaluations compared to that by Zhu et al.‘s model. Further extension of the present model so as to incorporate surface tension and wall wetting is one of the most important issues which require future development.

A phase field model for partially saturated geomaterials describing fluid–fluid displacements. Part I: The model and one-dimensional analysis

A model describing immiscible fluid–fluid displacements in partially saturated porous media is presented. This is based on a phase field approach that interprets the mixture of wetting (liquid water) and non-wetting (air) fluids within the pore space as a single saturating non-uniform pore fluid characterized by a phase field parameter, which is considered to be the saturation degree of the wetting fluid. While the standard retention curve provides for the retention properties of the pore walls, a Cahn–Hilliard like double-well energy is employed to describe the possible co-existence of the immiscible fluid phases. An enhanced description of the macroscopic surface tension between the fluid phases is obtained naturally within the phase field framework due to a regularization that depends on the spatial gradient of the water content. A generalized Darcy’s law is used to describe dissipation due to fluid flow driven by the gradient of a generalized chemical potential. Thus, in the context of soil hydrology this model is interpreted as an extension to the classical Richards equation governing the spatio-temporal evolution of the phase field parameter. Employing a convex–concave flux function it is shown, using one-dimensional analysis, that both imbibition and drainage fronts can be modeled in this phase field framework. The non-monotonicities observed in the resolved solutions are explained using a combination of asymptotic matching techniques and dynamical systems analysis.

Upscaling a Navier-Stokes-Cahn-Hilliard model for two-phase porous-media flow with solute-dependent surface tension effects

We consider a model for the flow of two immiscible and incompressible fluid phases in a porous medium. A surfactant is dissolved in one of the fluid phases, and its concentration at the interface separating the two fluids can change the surface tension. At the scale of pores, we assume that the flow is governed by the Navier-Stokes equations, while for the phase separation, a Cahn-Hilliard phase-field model is adopted. Using formal homogenization, we derive a two-scale model describing the averaged behaviour of the system at the larger Darcy scale, where effective quantities are found through local (cell) problems at the smaller pore scale. For this two-scale model, we formulate a numerical scheme and present numerical results highlighting the influence of the solute-dependent surface tension.

A consistent and conservative Phase-Field method for multiphase incompressible flows

In the present study, a consistent and conservative Phase-Field method, including both the model and scheme, is developed for multiphase flows with an arbitrary number of immiscible and incompressible fluid phases. The consistency of mass conservation and the consistency of mass and momentum transport are implemented to address the issue of physically coupling the Phase-Field equation, which locates different phases, to the hydrodynamics. These two consistency conditions, as illustrated, provide the “optimal” coupling because (i) the new momentum equation resulting from them is Galilean invariant and implies the kinetic energy conservation, regardless of the details of the Phase-Field equation, and (ii) failures of satisfying the second law of thermodynamics or the consistency of reduction of the multiphase flow model only result from the same failures of the Phase-Field equation but are not due to the new momentum equation. Physical interpretation of the consistency conditions and their formulations are first provided, and general formulations that are obtained from the consistency conditions and independent of the interpretation of the velocity are summarized. Then, the present consistent and conservative multiphase flow model is completed by selecting a reduction consistent Phase-Field equation. Several novel techniques are developed to inherit the physical properties of the multiphase flows after discretization, including the gradient-based phase selection procedure, the momentum conservative method for the surface force, and the general theorems to preserve the consistency conditions on the discrete level. Equipped with those novel techniques, a consistent and conservative scheme for the present multiphase flow model is developed and analyzed. The scheme satisfies the consistency conditions, conserves the mass and momentum, and assures the summation of the volume fractions to be unity, on the fully discrete level and for an arbitrary number of phases. All those properties are numerically validated. Numerical applications demonstrate that the present model and scheme are robust and effective in studying complicated multiphase dynamics, especially for those with large-density ratios.

Motion of a slip spherical particle near a planar micropolar-viscous interface

An analytical–numerical method based on collocation techniques is presented to investigate the problem of determining Stokes flow caused by a spherical particle moving perpendicularly to a plane interface separating two semi-infinite immiscible fluid phases. Attention is focused on the case when one of the two fluid phases is of a microstructure nature (micropolar fluid). A linear slip boundary conditions, of Basset type, is used on the surface of the particle. The motion is considered in the limit of small Reynolds and capillary numbers where in this case the interface is of insignificant deformation. To solve the axisymmetric creeping motion of the micropolar fluid for velocity and microrotation components, a general solution is constructed from the superposition of some basic solutions in both cylindrical and spherical coordinate systems. Boundary conditions are satisfied first at the fluid–fluid interface using the Fourier Bessel transforms and then on the surface of the particle by the collocation scheme. Numerical results of the normalized drag force acting on the particle are obtained with good convergence for various values of the relevant parameters and presented both in graphical and tabular. Our results of the normalized drag force are compared with the available data in the literature for the limiting cases.

On upscaling pore-scale models for two-phase flow with evolving interfaces

The modelling and simulation of the unsaturated flow or the flow of two immiscible fluid phases in a porous medium is challenging as this flow takes place through the pores of the medium, which form a highly complex domain. Next to the complexity of the domain, a major challenge is to account for the interface separating the fluids, or the unsaturated fluid from the inert filling part, as the location of this interface is not known a-priori. The evolution of this interface depends on the flow of both fluids and of the surface tension. Moreover, the surface tension may depend on the concentration of a surfactant dissolved in one fluid phase. In this work, such aspects are taken into account, and effective, Darcy-scale models are derived based on the known physics at the pore scale. In this sense a thin strip is used as the representation of a single pore in the porous medium. The Darcy-scale models are derived for various regimes, accounting for different pore-scale processes. Numerical examples show that the upscaled models are a good approximation of the transversal average of the solution to the pore-scale models, as the ratio of the width and the length of the pore approaches zero.

Modeling of capillary-driven microfluidic networks using electric circuit analogy

Capillary-driven flow in complex microfluidic devices is increasingly encountered in life science applications, and powerful modeling tools are necessary to assist the design of these devices. In this work, we present a modeling framework for capillary-driven flow in closed complex microfluidic networks using electric circuit analogy. The model handles two immiscible fluid phases, including the capillary pressure jump across the interface(s) between the phases, in a large variety of fluidic structures. As outputs, the position and velocity of each interface in the analyzed microfluidic network are provided as a function of time. Single channels, successive channels of different dimensions, tapered channels, micropillar arrays, flow splitters, mixing of two liquids, and the presence of vents can be modeled and combined in different complex structures. Static and dynamic contact angles can be employed. Advancing and receding interfaces can both be modeled. The model was validated against both experimental and analytical results for specific microfluidic structures. Furthermore, the capabilities of the model are demonstrated using a complex microfluidic network.

Dynamic permeability functions for partially saturated porous media

SUMMARY While the frequency-dependence of permeability under fully saturated conditions has been studied for decades, the corresponding characteristics of partially saturated porous media remain unexplored. Notably, it is not clear whether the use of effective pore fluid approaches under such conditions is valid. To address this issue, we propose a method that allows us to obtain dynamic permeability functions for partially saturated porous media. To this end, we conceptualize the considered pore space as a bundle of capillary tubes of different radii saturated by two immiscible fluid phases. We then solve the Navier–Stokes equations within the pore space and define a capillary pressure–saturation relationship, which permits to obtain saturation- and frequency-dependent effective permeability estimates. The application of this method to a realistic model of an unconsolidated granular sediment demonstrates that dynamic effective permeability functions for wetting and non-wetting fluid phases exhibit distinct characteristics, thus rendering effective pore fluid approaches inadequate. Finally, we explore the capability of the seminal dynamic permeability model developed by Johnson et al.[J. Fluid Mech. 176, 379 (1987)] to account for the effects of partial saturation. We find that the frequency scaling proposed by Johnson et al. prevails in partially saturated scenarios. However, the parameters associated with this model need to be redefined to account for saturation-dependent effects.

Fluids in High-Pressure Granulites

Properties of fluids in high-pressure granulites were studied in HP granulites (~8.7–11 kbar, ~800–900°С) and syngranulite fluid infiltration-driven HP metasomatites (~11–9 kbar, ~920–850°С) from the Lapland granulite belt of the Fennoscandian Shield. The study involved large-scale mapping of the rocks, microthermometry of mineral-hosted fluid inclusions, multiequilibrium mineral thermobarometry, and calculations of H2O activity based on mineral equilibria. The mafic pyroxene granulites and syngranulite metasomatites (quartz blastomylonites with orthopyroxene, sillimanite, and garnet; veins and vein-like bodies of orthopyroxene–garnet and diopside–scapolite rocks) contain similar assemblages of syngenetic fluid inclusions (which are hosted mostly in quartz and also in garnet, orthopyroxene, and scapolite) of contrasting chemical composition: nearly pure СО2 (distinctly predominant), brines (the dominant salts are CaCl2 and NaCl), and N2 ± H2O. These three types of inclusions coexist in the same generations of early inclusions: rarer primary (p) and predominant primary–secondary (ps). The CO2 inclusions have either high or low densities, and the N2 inclusions are of low density. The brine inclusions show a wide range of total salt contents (up to 30–35 wt%) and variable concentration proportions of the dominant salts: p-inclusions with a salinity of 20 wt% CaCl2 + 10 wt% NaCl; ps-inclusions with a salinity of 5 wt% CaCl2 + 20 wt% NaCl; p- and ps-inclusions with a salinity of 5–23 wt% NaCl eq; and p-inclusions with halite (up to 35 wt% NaCl). In general, CaCl2 is the predominant salt component in the early p- and ps-inclusions of the rocks. Considered together, currently available data (including Sr, Nd, and O isotope systems) on these rocks indicate that the external fluid flow during the origin of granulites was evidently of mantle origin. At the peak P–T parameters, the inclusions were entrapped from a heterogeneous fluid in which immiscible water–salt and CO2-rich fluids, which initially contained N2, coexisted. Data on the chemical composition and salt concentrations of the fluids, $${{a}_{{{{{\text{H}}}_{2}}{\text{O}}}}}$$ = 0.40–0.51, are compared with the theoretically predicted phase state of the fluids and the properties of the coexisting immiscible fluid phases at the estimated P–T parameters of granulite petrogenesis on the basis of numerical models in the H2O–CO2–NaCl and H2O–CO2–CaCl2 ternary systems. The location of the tie-lines and solvus were calculated to subsequently use for the thermodynamic prediction. The paper discusses similarity and the reasons for the difference between the theoretical compositions of the generated fluid phases and the composition of fluid inclusions, geochemical consequences of the heterogenization of granulite fluids (the formation of concentrated alkaline brines and a potentially acidic CO2-rich fluid phase, the values of the mass and volume fractions of these phases depending on variations in the composition of the initial homogeneous fluid, etc.). It follows that an extensive region of the compositions of aqueous fluids with different concentrations of CO2 and Na and Ca chlorides exists at the P–T parameters of HP granulites in which originally homogeneous fluid splits into compositionally contrasting fluid phases with different properties. This region of coexisting immiscible fluids significantly expands with increasing CaCl2 concentration. Hence, the lower crust at the level of the HP granulite facies may be the region where high-temperature immiscible fluids are generated. One of these fluids is a denser phase of alkaline brines, and the other is a less dense potentially acidic phase of H2O–CO2 fluids rich in CO2. Ascending along regional permeable zones, these fluid phases of deep origin can play an important role in magmatic, metamorphic, metasomatic, and ore-forming processes in the middle and upper crust.

Tracing fluid saturation during pegmatite differentiation by studying the fluid inclusion evolution and multiphase cassiterite mineralisation of the Gatumba pegmatite dyke system (NW Rwanda)

Aqueous fluid saturation during the internal evolution of phosphorus- and boron-rich pegmatites in NW Rwanda has been investigated using petrography, fluid inclusion microthermometry, and microanalysis of minerals and fluid inclusions (Raman microspectroscopy and LA-ICP-MS). The Gatumba dyke system (GDS) in Rwanda has been selected for this study because it hosts a multiphase cassiterite mineralisation, in wall, core and internal replacement zones, suggesting an advancing influence of magmatic-hydrothermal conditions during dyke solidification. As such, multiphase cassiterite precipitation is applied as a tracer for magmatic to hydrothermal crystallisation processes in granitic pegmatites. The GDS consists of a group of six major LCT (lithium-caesium-tantalum) family dykes, which all belong to the most differentiated rare-element pegmatites in the Gatumba-Gitarama field. Petrography identified a well-developed internal anatomy consisting of a border, wall, intermediate and a quartz core zone. In addition, replacement zones developed as cleavelandite after perthitic microcline units in the intermediate zone and as muscovite-quartz pockets in the intermediate and wall zones. The latter replacement zone display muscovitisation reactions with the formation of greisens. Texturally, the greisen pockets replace the secondary cleavelandite units. During internal differentiation, multiphase cassiterite mineralisation formed in 1) large microcline crystals of the wall zone (Cst1), 2) assemblage with quartz, F-poor montebrasite, and carbonate- and boron-enriched Mn-fluorapatite in the core zone (Cst2), and 3) the greisen replacement units in the wall and intermediate zones (Cst3). Fluid inclusion micro-analyses demonstrate that a saline aqueous H2O-NaCl-KCl-(CO2,N2) L1-type fluid (∼20 wt.% NaCl and ∼3 wt.% KCl) was saturated during onset of crystallisation of the wall and intermediate zones and precipitation of the disseminated, magmatic-hydrothermal Cst1 mineralisation in and around large perthitic K-feldspars. Isochore reconstructions of this L1 fluid indicate crystallisation conditions of 535–560 °C and 5.1–5.6 kbar for the GDS. Extensive microcline fractionation and subsequent replacement reactions of cleavelandite after microcline in the intermediate zone were induced by a Li-enriched and more Na-depleted H2O-NaCl-LiCl-(CO2,N2) L2-type fluid, which was present during solidification of the core zone and present during precipitation of Cst2 phase mineralisation (6–12 wt.% NaCl, 1–10 wt.% LiCl). Late-stage, hydrothermal Cst3 mineralisation in the greisen pockets precipitated from a Cs-enriched H2O-NaCl-(KCl, CO2,N2) L3-type fluid (∼15 wt.% NaCl) distinctly after the formation of the cleavelandite replacement units. Subsolidus Cst3 precipitation is dominantly driven by metasomatic, hydrolytic fluid-rock reactions. Combining textural, microthermometric and fluid compositional data with reported Sn melt-fluid partitioning and solubility data demonstrate that primary cassiterite (i.e. Cst1 and Cst2) in the wall and core zone of the GDS precipitated from the exsolved saline aqueous L1-and L2-fluid present at the interface of the highly-fractionated melt phase. Fast disequilibrium growth of (near)-anhydrous mineral assemblages likely caused the formation of flux-rich and, at least locally, water-saturated melt compositions at the crystallisation front of the large crystals. This study emphasises the importance of local magmatic-hydrothermal conditions and the presence of an immiscible aqueous fluid phase during the internal crystallisation and cassiterite mineralisation of rare-element pegmatites.