Immobile Fluid Research Articles

Darcy–Forchheimer gravity currents in porous media

We theoretically and experimentally study gravity currents of a Newtonian fluid advancing in a two-dimensional, infinite and saturated porous domain over a horizontal impermeable bed. The driving force is due to the density difference between the denser flowing fluid and the lighter, immobile ambient fluid. The current is taken to be in the Darcy–Forchheimer regime, where a term quadratic in the seepage velocity accounts for inertial contributions to the resistance. The volume of fluid of the current varies as a function of time as $\sim T^{\gamma }$ , where the exponent parameterizes the case of constant volume subject to dam break ( $\gamma =0$ ), of constant ( $\gamma =1$ ), waning ( $\gamma <1$ ) and waxing inflow rate ( $\gamma >1$ ). The nonlinear governing equations, developed within the lubrication theory, admit self-similar solutions for some combinations of the parameters involved and for two limiting conditions of low and high local Forchheimer number, a dimensionless quantity involving the local slope of the current profile. Another parameter $N$ expresses the relative importance of the nonlinear term in Darcy–Forchheimer's law; values of $N$ in practical applications may vary in a large interval around unity, e.g. $N\in [10^{-5},10^{2}]$ ; in our experiments, $N\in [2.8,64]$ . Sixteen experiments with three different grain sizes of the porous medium and different inflow rates corroborate the theory: the experimental nose speed and current profiles are in good agreement with the theory. Moreover, the asymptotic behaviour of the self-similar solutions is in excellent agreement with the numerical results of the direct integration of the full problem, confirming the validity of a relatively simple one-dimensional model.

Time-lapse 3D imaging and modelling of acid-rock interactions within a mudstone: Implications for cap-rock behaviour during carbon storage

Assessing caprock integrity is crucial for safe geological carbon storage. This study addresses this by utilising time-lapse micro-CT imaging to capture real-time, dynamic changes in mudstone during acid injection, providing a detailed analysis of the 3D geometry and evolution of minerals and fractures. Additionally, the integration of reactive transport modelling offers insights into the potential changes occurring immediately after acid injection, enhancing our understanding of fluid-rock interactions and their impact on caprock stability. Different acid concentrations were used to mimic a range of possible acid concentrations during CO2 injection and storage. Changes in mudstones due to acid interaction include initial pre-existing fracture closure, followed by fracture growth and sample swelling. Acid predominantly followed pathways like fractures or laminations. Reactive transport models showed most dissolution occurring near the inlet, driven by a decrease in H+ concentration along fluid pathways. Carbonate distribution controlled dissolved areas, with calcite dissolution and acid migration rates decreasing over time. Simulation indicated reduced shear stress after acid injection, especially with low-pH acids. This correlates to decreased fluid velocity, resulting in slower fluid flow. Slower acid movement and prolonged presence in immobile fluid flow zones led to more extensive calcite dissolution compared to mobile zones. This study provides valuable insights into microstructural changes in caprock following CO2 injection, emphasizing the potential risk of fracture development and leakage.

Solute Transport in a Multi-Channel Karst System with Immobile Zones: An Example of Downtown Salado Spring Complex, Salado, Texas

To investigate the influence of flow rate increment on the solute transport parameter of immobile zones in a karst system, a dye tracer test was conducted in the Downtown Salado Spring Complex (DSSC) comprising three springs: Big Boiling, Anderson, and Doc Benedict springs. The Multiflow two-region nonequilibrium model (2RNE) was used to simulate the breakthrough curve (BTC) of the springs, and changes in the solute transport parameters in response to flow rate increment were observed. The simulation result showed that the 2RNE model was capable of reproducing the BTC of all the DSSC springs, with an R-squared value greater than 0.9 in all flow rate increment scenarios. The research demonstrates that a positive correlation will exist between the flow rate and solute transport parameter of the immobile zones if the tracer transport to the spring is truly influenced by immobile zones. In contrast, a negative correlation will exist between the flow rate and mass transfer coefficient if the immobile zone has less influence. Overall, the research provides insights into contaminant movement in karst by documenting how tracers are retained in the immobile fluid zone.

Effect of charge inversion on the electrokinetic transport of nanoconfined multivalent ionic solutions

Understanding the effects of phenomena occurring at electrically charged interfaces, such as charge inversion (CI), is crucial for enabling electroosmosis as an efficient transport mechanism in nanodevices. Here, we employ molecular dynamics (MD) simulations to systematically analyze the effect of CI on the electrokinetic transport of multivalent ionic solutions confined in amorphous silica nanochannels. We employ mixtures of monovalent and multivalent counterions while fixing the total ionic concentration to establish correlations between observed phenomena and the amount of multivalent ionic species in the electrolyte solution. The results show that the development of CI is related to a decrease in the mobility of the fluid layers adjacent to the charged surface. In addition, we observe that interfacial overcharging disrupts the water molecular orientation in the fluid layers adjacent to the channel walls. From the non-equilibrium MD simulations of electro-osmotic flow, we disclose the influence of phenomena related to the presence of CI. In particular, flow reversal occurs in scenarios involving CI due to increased local viscosity and a higher concentration of coions within the hydrodynamically mobile and electrokinetically active region of the charged interface. We also find that the magnitude of the wall zeta (ζ) potential displays a monotonic increase with the development of CI in the system. Moreover, we explain why positioning the wall ζ potential at an imaginary (slip) plane, which separates the hydrodynamically mobile and immobile fluid, is misleading.

Thermosolutal Convection of Natural and Anti-Natural Solutions Through an Angled Cavity Under Cross Gradients in Temperature and Concentration

In the current study, cross-temperature and concentration gradients are used to model the in a binary fluid contained in an angled square cavity. Using a method, the , , and conservation equations were numerically solved. The inclined cavity under equal solutal buoyancy and thermal forces was the subject of the study . Since the horizontal components of the thermal and singular volume forces were equal but opposed to one another, an equilibrium solution for this situation that corresponds to the rest state of the immobile fluid is feasible. However, this equilibrium solution becomes unstable above a specific critical value of the , leading to vertical density stratification inside the enclosure. The results are shown using the and as well as the and for the flow intensity. The existence of the commencement of convection is demonstrated in this work, and both natural and anti-natural flow solutions are obtained. Subcritical convection has also been seen for the natural solution when the is more or less than unity. For the start of supercritical and subcritical convection, the number's critical values are identified. As the climbed, so did the flow's intensity and the rates at which heat and mass were transferred. Reducing flow intensity and accelerating mass transfer are the results of raising the . Different flow patterns are shown for an aspect ratio of 4, and the existence interval of the oscillatory solutions is calculated.

Control of Chemoconvection in a Rectangular Slot by Changing Its Spatial Orientation

Recently, we found that a two-layer miscible system placed in a vertical slab reactor shows an occurrence of a density shock-wave-like pattern. This wave resembles a turbulent bore separating immobile fluid and an area of intense mixing. It travels away from the convective core of the system and is highly dependent on the intensity of a gravity-dependent chemoconvection in the cocurrent flow. The novelty of this work is that we demonstrate that the change in angle between gravity and wave direction allows controlling the chemoconvection intensity and, consequently, the rate of a spatially-extended reaction. We study both experimentally and numerically the effect of the spatial orientation of a slab reactor to a gravity field on a flow structure induced by a neutralization reaction. In experiments, we use aqueous mixtures of nitric acid and sodium hydroxide. We apply the Fizeau interferometry to visualize the flow and use the PIV method to measure the fluid velocity. The mathematical model includes reaction–diffusion–convection equations that describe 3D flows. We study the flow modifications with a change in the inclination angle from 0 to 90 degrees. At small angles (up to 30), the cocurrent flow becomes spatially heterogeneous, and the fields of salt and acid are separated. If the inclination exceeds 50 degrees, the wavefront is deformed, and the wave breaks up, resulting in a sharp decrease in the reaction rate.

Advancing the knowledge of solute transport in the presence of bound water in mixed porous media based on low-field nuclear magnetic resonance

Bound water (BW) has significant effects on the physical and chemical properties of soils and other geologic formations. However, little is known about its effect on the non-Fickian transport of solute in porous media. This paper investigated the effect of BW on the non-Fickian solute transport characteristics in mixed porous media. Different porous media of quartz sand with variable contents of powdered minerals were prepared. The low-field nuclear magnetic resonance (LF-NMR) was used to quantitatively characterise BW and its spatial distribution. Then solute transport experiments were conducted in different prepared porous media and the measured breakthrough curves (BTCs) were analysed using the advection–dispersion equation (ADE), mobile-immobile (MIM) and continuous time random walk (CTRW) models. The results revealed that all measured BTCs exhibited the characteristics of early arrival and late-time tailing. However, stronger non-Fickian characteristics were observed as BW increased in content, indicating that BW influenced the non-Fickian transport of the solute. The optimised immobile water fraction of the MIM model consistently increased with increasing estimated BW fraction. Moreover, the coefficient of mass transfer rate (α) of MIM decreased as BW increased, meaning that the solute exchange between the mobile and immobile fluids became slower and consequently yielded strongly tailed BTCs. From the CTRW model, the solute transport velocity (vCTRW) increased with increasing BW, leading to a reduction in the breakthrough time of the solute. An increase in BW yielded a decreasing power law exponent (βCTRW) of the CTRW model, implying an enhancement of the non-Fickian transport characteristics. Furthermore, the porous media with BW<16.4% showed moderately non-Fickian solute transport whereas those with BW≥16.4% exhibited highly non-Fickian solute transport.

A Semi‐Analytical Solution for Heat Transport in Rock With Parallel Fractures and a Heat Source in Both Fracture and Matrix

AbstractIn recent decades, numerous analytical solutions have been developed to quantify temperature perturbations in fractured rock having mobile and immobile fluid phases. The assumption of one‐dimensional heat conduction in the matrix, or neglecting heat dispersion and storage in fractures, however, are typical simplifications adopted to overcome the difficulties in mathematically representing the problem. In this study, we propose a two‐dimensional semi‐analytical solution framework based on Green's function approach for a flexible heat source definition, including source dimensions, energy delivery strength and duration, and the presence of a heat source in the matrix and/or fracture. The solution fully accounts for heat conduction, advection, dispersion, and transient heat exchange between the fracture fluid and rock matrix in a system of parallel fractures. The solution having a strip heat source extending from a fracture into the matrix indicates that one‐dimensional heat conduction in the matrix underestimates and overestimates temperature responses at early and later times, respectively. Additionally, the temperature peak arrival time is also substantially delayed by simplification. The fracture temperature grows slower near the heat source area as the fracture aperture increases. The fracture temperature growth is enhanced via the overlapped heating areas between the parallel fractures. The transient temperature analyses imply that the spatial temperature variation is strongly associated with heat delivery strength. The early time temperature variances are closely related to the heat source configurations, and the later time temperatures in the domain are mainly determined by the total energy being delivered into the domain.

Numerical Model for Heat Transfer in Layered Soil with Local Thermal Nonequilibrium

A numerical model, called HT2, is presented for one-dimensional heat transfer in saturated incompressible layered soil with steady fluid flow, effective porosity, and possible local thermal nonequilibrium (LTNE) between solid and fluid phases. The model uses a series-parallel approach and accounts for advection, conduction, linear and nonlinear thermal dispersion, and interstitial heat transfer. The key to HT2 is the definition of separate columns for the solid matrix (SM) and mobile pore fluid (MPF). The SM column includes the solid phase and immobile pore fluid and consists of fixed elements. The MPF column consists of moving elements and uses Lagrangian element-tracking to follow the fluid motion, which reduces numerical dispersion and simplifies heat transfer within the column to that of dispersive flux between contiguous elements. The model uses an interstitial heat transfer coefficient to calculate kinetic heat transfer between SM and MPF elements. The development of HT2 is first presented, followed by verification checks and a limited parametric study. Numerical simulations indicate that larger particle size and higher fluid discharge velocity yields greater local temperature disequilibrium between solid and fluid phases during heat transfer through saturated soil.

Extended classification of the buoyancy-driven flows induced by a neutralization reaction in miscible fluids. Part 1. Experimental study

Abstract

Numerical Model for Heat Transfer in Saturated Layered Soil with Effective Porosity

A numerical model, called HT1, is presented for one-dimensional (1D) heat transfer in saturated incompressible layered soil with effective porosity and steady fluid flow. The model uses a series-parallel approach for heat transfer and accounts for advection, conduction, and thermal mechanical dispersion assuming local thermal equilibrium between solid and fluid phases. The key to HT1 is the definition of separate columns for the solid matrix and mobile pore fluid. The solid matrix column includes the solid phase and immobile pore fluid and consists of fixed elements. The mobile pore fluid column uses Lagrangian element-tracking to follow the fluid motion, which reduces numerical dispersion and simplifies heat transfer to that of dispersive flux between contiguous elements. Development of the HT1 model is first presented, followed by verification checks using available analytical and numerical solutions. Simulation results for several numeric examples are used to demonstrate model performance and the effects of various parameters, including effective porosity and multiple soil layers, on heat transfer behavior for saturated incompressible soil.

Free-Rising Bubbles Bounce More Strongly from Mobile than from Immobile Water-Air Interfaces.

Recently it was reported that the interface mobility of bubbles and emulsion droplets can have a dramatic effect not only on the characteristic coalescence times but also on the way that bubbles and droplets bounce back after collision (Vakarelski, I. U.; Yang, F.; Tian, Y. S.; Li, E. Q.; Chan D. Y. C.; Thoroddsen, S. T. Sci. Adv.2019, 5, eaaw4292). Experiments with free-rising bubbles in a pure perfluorocarbon liquid showed that collisions involving mobile interfaces result in a stronger series of rebounds before the eventual rapid coalescence. Here we examine this effect for the case of pure water. We compare the bounce of millimeter-sized free-rising bubbles from a pure water–air interface with the bounce from a water–air interface on which a Langmuir monolayer of arachidic acid molecules has been deposited. The Langmuir monolayer surface concentration is kept low enough not to affect the water surface tension but high enough to fully immobilize the interface due to Marangoni stress effects. Bubbles were found to bounce much stronger (up to a factor of 1.8 increase in the rebounding distance) from the clean water interface compared to the water interface with the Langmuir monolayer. These experiments confirm that mobile surfaces enhance bouncing and at the same time demonstrate that the pure water–air interfaces behave as mobile fluid interfaces in our system. A complementary finding in our study is that the ethanol–air interface behaves as a robust mobile liquid interface. The experimental findings are supported by numerical simulations of the bubble bouncing from both mobile and immobile fluid interfaces.

Relative Permeability Model Taking the Roughness and Actual Fluid Distributions into Consideration for Water Flooding Reservoirs

Reservoir relative permeability is greatly important to the development of water flooding reservoirs. Currently, most researches on relative permeability have not taken the roughness of pore surface and actual fluid distributions into consideration. In this paper, a novel relative permeability model for water flooding reservoirs taking the roughness and actual fluid distributions into consideration has been proposed by using the fractal theory. The novel model contains some key parameters, all of which have clear physical meanings, such as the immobile liquid film thickness, relative roughness, tortuosity fractal dimension $$ D_{\text{T}} $$ and pore fractal dimension $$ D_{\text{f}} $$ . The predicted results of the novel fractal relative permeability model are consistent with published experimental data. That verifies the correctness of the novel fractal relative permeability model. Finally, sensitive factor analysis of novel relative permeability model is conducted. We can find that the wetting fluid relative permeability decreases as the immobile wetting fluid film thickness or relative roughness increases. When the tortuosity fractal dimension or pore fractal dimension increases, the wetting relative permeability and non-wetting relative permeability will both decrease. An increase in maximum pore diameter or the decreasing of minimum pore diameter results in the reduction in fractal dimension of flow channel and discontinuous saturation. The increasing of maximum pore diameter results in an increase in the relative permeability of wetting fluid. The minimum pore diameter has tiny effect on the relative permeability.

Spatial resolution of transport parameters in a subtropical karst conduit system during dry and wet seasons

Karst aquifers are characterized by a high degree of hydrologic variability and spatial heterogeneity of transport parameters. Tracer tests allow the quantification of these parameters, but conventional point-to-point experiments fail to capture spatiotemporal variations of flow and transport. The goal of this study was to elucidate the spatial distribution of transport parameters in a karst conduit system at different flow conditions. Therefore, six tracer tests were conducted in an active and accessible cave system in Vietnam during dry and wet seasons. Injections and monitoring were done at five sites along the flow system: a swallow hole, two sites inside the cave, and two springs draining the system. Breakthrough curves (BTCs) were modeled with CXTFIT software using the one-dimensional advection-dispersion model and the two-region nonequilibrium model. In order to obtain transport parameters in the individual sections of the system, a multi-pulse injection approach was used, which was realized by using the BTCs from one section as input functions for the next section. Major findings include: (1) In the entire system, mean flow velocities increase from 183 to 1,043 m/h with increasing discharge, while (2) the proportion of immobile fluid regions decrease; (3) the lowest dispersivity was found at intermediate discharge; (4) in the individual cave sections, flow velocities decrease along the flow direction, related to decreasing gradients, while (5) dispersivity is highest in the middle section of the cave. The obtained results provide a valuable basis for the development of an adapted water management strategy for a projected water-supply system.

Predicting stagnant pore volume in porous media using temporal moments of tracer breakthrough curves

The heterogeneity in complex porous media like oil reservoirs and aquifers many times renders a fraction of fluid in the pores immobile. The tracer breakthrough curves that are used to characterize the porous media are impacted by the immobile volume fraction and show a tailing effect. It is important to first recognize the presence of the immobile volume and quantify it based on a single tracer breakthrough curve in complex geological environments. In this work, we use the method of moments on the tracer breakthrough curve to solve the inverse problem of porous media characterization with stagnant volume fractions. In particular, we use a three parameter model describing the tracer transport in a porous medium with an immobile fluid fraction and solve the forward problem analytically. We derive mean, variance, skewness, and kurtosis of the analytical solution for the exit age density of the injected tracer in the porous medium. We then describe the behavior of the moments in the parameter space of the transport model and relate it with the tracer transport behavior in the porous medium. We then develop the solution for the inverse problem of using these higher order temporal moments of the effluent tracer concentration to estimate the immobile fluid volume in the porous medium.

Coalescence Dynamics of Mobile and Immobile Fluid Interfaces.

Coalescence dynamics between deformable bubbles and droplets can be dramatically affected by the mobility of the interfaces with fully tangentially mobile bubble-liquid or droplet-liquid interfaces expected to accelerate the coalescence by orders of magnitude. However, there is a lack of systematic experimental investigations that quantify this effect. By using high speed camera imaging we examine the free rise and coalescence of small air-bubbles (100 to 1300 μm in diameter) with a liquid interface. A perfluorocarbon liquid, PP11, is used as a model liquid to investigate coalescence dynamics between fully mobile and immobile deformable interfaces. The mobility of the bubble surface was determined by measuring the terminal rise velocity of small bubbles rising at Reynolds numbers, Re, less than 0.1 and the mobility of free PP11 surface by measuring the deceleration kinetics of the small bubble toward the interface. Induction or film drainage times of a bubble at the mobile PP11-air surface were found to be more than 2 orders of magnitude shorter compared to the case of bubble and an immobile PP11-water interface. A theoretical model is used to illustrate the effect of hydrodynamics and interfacial mobility on the induction time or film drainage time. The results of this study are expected to stimulate the development of a comprehensive theoretical model for coalescence dynamics between two fully or partially mobile fluid interfaces.

流路として機能しない空隙を導入した貯留層シミュレーションモデルによる 圧力・トレーサー濃度変化の理論的検討

流路として機能しない空隙を導入した貯留層シミュレーションモデルによる 圧力・トレーサー濃度変化の理論的検討

Regional-scale analysis of karst underground flow deduced from tracing experiments: examples from carbonate aquifers in Malaga province, southern Spain

Tracer concentration data from field experiments conducted in several carbonate aquifers (Malaga province, southern Spain) were analyzed following a dual approach based on the graphical evaluation method (GEM) and solute transport modeling to decipher flow mechanisms in karst systems at regional scale. The results show that conduit system geometry and flow conditions are the principal factors influencing tracer migration through the examined karst flow routes. Solute transport is mainly controlled by longitudinal advection and dispersion throughout the conduit length, but also by flow partitioning between mobile and immobile fluid phases, while the matrix diffusion process appears to be less relevant. The simulation of tracer breakthrough curves (BTCs) suggests that diffuse and concentrated flow through the unsaturated zone can have equivalent transport properties under extreme recharge, with high flow velocities and efficient mixing due to the high hydraulic gradients generated. Tracer mobilization within the saturated zone under low flow conditions mainly depends on the hydrodynamics (rather than on the karst conduit development), which promote a lower longitudinal advection and retardation in the tracer migration, resulting in a marked tailing effect of BTCs. The analytical advection-dispersion equation better approximates the effective flow velocity and longitudinal dispersion estimations provided by the GEM, while the non-equilibrium transport model achieves a better adjustment of most asymmetric and long-tailed BTCs. The assessment of karst underground flow properties from tracing tests at regional scale can aid design of groundwater management and protection strategies, particularly in large hydrogeological systems (i.e. transboundary carbonate aquifers) and/or in poorly investigated ones.

Convective instability in a two-layer system of reacting fluids with concentration-dependent diffusion

The development of convective instability in a two-layer system of miscible fluids placed in a narrow vertical gap has been studied theoretically and experimentally. The upper and lower layers are formed with aqueous solutions of acid and base, respectively. When the layers are brought into contact, the frontal neutralization reaction begins. We have found experimentally a new type of convective instability, which is characterized by the spatial localization and the periodicity of the structure observed for the first time in the miscible systems. We have tested a number of different acid–base systems and have found a similar patterning there. In our opinion, it may indicate that the discovered effect is of a general nature and should be taken into account in reaction–diffusion–convection problems as another tool with which the reaction can govern the movement of the reacting fluids. We have shown that, at least in one case (aqueous solutions of nitric acid and sodium hydroxide), a new type of instability called as the concentration-dependent diffusion convection is responsible for the onset of the fluid flow. It arises when the diffusion coefficients of species are different and depend on their concentrations. This type of instability can be attributed to a variety of double-diffusion convection. A mathematical model of the new phenomenon has been developed using the system of reaction–diffusion–convection equations written in the Hele–Shaw approximation. It is shown that the instability can be reproduced in the numerical experiment if only one takes into account the concentration dependence of the diffusion coefficients of the reagents. The dynamics of the base state, its linear stability and nonlinear development of the instability are presented. It is also shown that by varying the concentration of acid in the upper layer one can achieve the occurrence of chemo-convective solitary cell in the bulk of an almost immobile fluid. Good agreement between the experimental data and the results of numerical simulations is observed.

Continuum‐scale characterization of solute transport based on pore‐scale velocity distributions

AbstractWe present a methodology to characterize a continuum‐scale model of transport in porous media on the basis of pore‐scale distributions of velocities computed in three‐dimensional pore‐space images. The methodology is tested against pore‐scale simulations of flow and transport for a bead pack and a sandstone sample. We employ a double‐continuum approach to describe transport in mobile and immobile regions. Model parameters are characterized through inputs resulting from the micron‐scale reconstruction of the pore space geometry and the related velocity field. We employ the outputs of pore‐scale analysis to (i) quantify the proportion of mobile and immobile fluid regions and (ii) assign the velocity distribution in an effective representation of the medium internal structure. Our results (1) show that this simple conceptual model reproduces the spatial profiles of solute concentration rendered by pore‐scale simulation without resorting to model calibration and (2) highlight the critical role of pore‐scale velocities in the characterization of the model parameters.