Diffusion Coefficients In Porous Media Research Articles

Mass transfer kinetics inside bio-(aero)gels during solvent exchange and supercritical drying: On the relevance of advection, gel-porosity and a peculiarity regarding the tortuosity

Spatially and temporally resolved composition profiles were measured inside biopolymer gels during solvent exchange and supercritical (sc) drying using one-dimensional Raman spectroscopy. The effective diffusion coefficients of solvent exchange and sc drying were obtained by fitting modelled composition profiles to the experimentally obtained ones. We considered one purely diffusional model, which excludes advection, and one convective model, which regards the superposition of diffusion and advection inside the pore system. The tortuosity was obtained by relating the obtained effective diffusion coefficients in porous media with the mutual diffusion coefficients. When considering the solvent exchange experiments the derived tortuosities for agar gels are about 1.1 and are in good agreement with literature data. When considering the sc drying experiments, the derived tortuosities are about 3.2, which is again in good agreement with literature data, but in contradiction to the tortuosities derived for the same agar gels from the solvent exchange experiments before.

Homogenized model for diffusion and heterogeneous reaction in porous media: Numerical study and validation.

In our previous work, macroscopic models for multi-species diffusion/heterogeneous reaction with different diffusion coefficients in porous media were derived from the homogenization technique based on a spectral approach (Le et al., Applied Mathematical Modelling, 104, 666–681, 2022). This work aims to analyze numerically the properties of the eigenvalues and eigenfunctions of the spectral problem, the solution of the closure problems and the effective coefficients as a function of key parameters such as the Damköhler number, the ratios of the reaction rates and diffusion coefficients. This provides in-depth understanding of the role of the coefficients involved in the macroscopic models in the whole range of the Damköhler number. Simulation results of the macroscopic equations are compared with direct numerical simulations in unsteady regime, showing a very satisfactory agreement and thus numerically validating the proposed models.

A Probabilistic Formulation of the Diffusion Coefficient in Porous Media as Function of Porosity

We investigate the upscaling of diffusive transport parameters using a stochastic framework. At sub-REV (representative elementary volume) scale, the complexity of the pore space geometry leads to a significant scatter of the observed diffusive transport. We study a large set of volumes reconstructed from focused ion beam-scanning electron microscopy data. Each individual volume provides us sub-REV measurements on porosity and the so-called transport-ability, being a dimensionless parameter representing the ratio of diffusive flux through the porous volume to that through an empty volume. The detected scatter of the transport-ability is mathematically characterized through a probability distribution function (PDF) with a mean and variance as function of porosity, which includes implicitly the effect of pore structure differences among sub-REV volumes. We then investigate domain size effects and predict when REV scale is reached. While the scatter in porosity observations decreases linearly with increasing sample size as expected, the observed scatter in transport-ability does not converge to zero. Our results confirm that differences in pore structure impact transport parameters at all scales. Consequently, the use of PDFs to describe the relationship of effective transport coefficients to porosity is advantageous to deterministic semiempirical functions. We discuss the consequences and advocate the use of PDFs for effective parameters in both continuum equations and data interpretation of experimental or computational work. The presented statistics-based upscaling technique of sub-REV microscopy data provides a new tool in understanding, describing and predicting macroscopic transport behavior of microporous media.

A new prediction model of CO2 diffusion coefficient in crude oil under reservoir conditions based on BP neural network

Diffusion is the key mechanism of enhanced oil recovery (EOR) by CO2 injection in unconventional oil reservoirs. The accurate measurement of the diffusion coefficient in porous media is essential for forecasting and optimizing CO2 injection. The pressure decay technique is the most commonly used method for measuring the diffusion coefficient, which is well acknowledged. However, it has a long experimental period with higher requirements on the equipment and operation. This paper firstly proposed a quick and simple prediction methods of diffusion coefficient for both CO2-oil systems within/without porous media based on back propagation (BP) neural network. The average errors are 18.73% and 18.80%, respectively. With the continuous supplement of the data, models can be continuously updated to provide more accurate estimates of the supercritical CO2-oil system without/with porous media conditions. Temperature, pressure, permeability, porosity and surface area positively correlate with the diffusion coefficient. Oil viscosity, oil density, and volume of porous media have a negative correlation with the diffusion coefficient. It is worth noting that for rocks with certain volume, the increase of surface area can significantly increase the diffusion coefficient, which implies that direct upscale of the measured CO2 diffusion coefficient in the lab is totally unreasonable.

Structural dependence of effective mass transport properties in lithium battery electrodes

Scanning electrochemical microscopy (SECM) has recently been reported as a convenient technique to characterize the apparent diffusion coefficients in porous media (D⋆) such as lithium-ion battery electrodes. The technique relies on hemispherical diffusion at the probe in the presence and absence of porous media. However, transition towards cylindrical diffusion follows as the diffusion layer becomes comparable to the media thickness, gradually limiting direct correlation between D⋆ and the SECM probe steady state current. In this work, we examine the experimental parameters that define this transitional frontier using simulations. A best practice thickness criterium is derived that ensures that the measured steady state current is ≳ 90% of that found in the ideal hemispherical case. Below this thickness limit the response was found to follow an empirical formulation with only two variables. Fundamentally, this quantifies diffusion in the transition regime between hemispherical and cylindrical transport. Practically, this correction formulation may be included into standard steady-state current expression to account for the deviation from hemispherical transport in “thin” electrodes. The analysis consolidates SECM as a reliable tool to characterize liquid phase mass transport within porous films over wide range of thicknesses, including films supported on blocking substrates, like lithium-ion battery electrodes.

Mathematical Model for Effective Gas Diffusion Coefficient in Multi-scale Fractal Porous Media

Based on the capillary hypothesis and fractal theory, a mathematical model for calculating the effective gas diffusion coefficient in porous media is established. By using fractal geometry theory, pore area fractal dimension, tortuosity fractal dimension and pore connectivity are introduced to quantitatively characterize the real internal structure in the porous media. An effective gas diffusion coefficient model for the fractal porous media is derived, and the influence of multi-scale porous media microstructure parameters on the effective gas diffusion coefficient is discussed. The results show that effective gas diffusion coefficient approximates to linearly increase with the increase of porosity, the pore area fractal dimension and the effective gas diffusion coefficient is positive correlation, but the tortuosity fractal dimension is negatively related to it. In the case of different porosities, the gas effective diffusion coefficient varies with the change of the pore diameter ratio, the effective gas diffusion coefficient increases with the increase of pore connectivity.

Estimating diffusion coefficients of shale oil, gas, and condensate with nano-confinement effect

Molecular diffusion plays an important role in oil and gas migration in tight shale formations. However, there are insufficient reference data in the literature to specify the diffusion coefficients within porous media. This study aims at calculating diffusion coefficients of shale gas, shale condensate, and shale oil at reservoir conditions with CO2 injection for EOR/EGR. The large nano-confinement effects including large gas-oil capillary pressure and critical property shifts could alter the phase behaviors. This paper estimates the diffusivities of shale fluids in nanometer-scale shale rock from two perspectives: 1) examining the shift of diffusivity caused by nanopore confinement effects from phase change (phase composition and fluid property) perspective, and 2) calculating the effective diffusion coefficient in porous media by incorporating rock intrinsic properties (porosity and tortuosity factor). The tortuosity is obtained by using tortuosity-porosity relations as well as the measured tortuosity of shale from 3D imaging techniques. The results indicated that nano-confinement effects could affect the diffusion coefficient through altering the phase properties, such as phase compositions and densities. Compared to bulk phase diffusivity, the effective diffusion coefficient in porous shale rock is reduced by 102 to 104 times as porosity decreases from 0.1 to 0.03.

Monte Carlo method for determining radon diffusion coefficients in porous media

Radon diffusivity is a key parameter for estimating radon emanation from porous media. We have developed a Monte Carlo method for determining radon diffusion coefficients in porous media. In the proposed method, the probability model for pore diameter derived from fractal theory is used to describe the microscopic pore structure of porous media. The Monte Carlo technique is utilized to search for the pore size that satisfies the probability model for porous media. The convergence rate of the proposed is fast (∼3 min). Radon diffusion coefficients measured by the closed chamber method are used to validate the accuracy of the proposed method.

Experimental study of the supercritical CO2 diffusion coefficient in porous media under reservoir conditions.

Reliable measurement of the CO2 diffusion coefficient in consolidated oil-saturated porous media is critical for the design and performance of CO2-enhanced oil recovery (EOR) and carbon capture and storage (CCS) projects. A thorough experimental investigation of the supercritical CO2 diffusion in n-decane-saturated Berea cores with permeabilities of 50 and 100 mD was conducted in this study at elevated pressure (10–25 MPa) and temperature (333.15–373.15 K), which simulated actual reservoir conditions. The supercritical CO2 diffusion coefficients in the Berea cores were calculated by a model appropriate for diffusion in porous media based on Fick's Law. The results show that the supercritical CO2 diffusion coefficient increases as the pressure, temperature and permeability increase. The supercritical CO2 diffusion coefficient first increases slowly at 10 MPa and then grows significantly with increasing pressure. The impact of the pressure decreases at elevated temperature. The effect of permeability remains steady despite the temperature change during the experiments. The effect of gas state and porous media on the supercritical CO2 diffusion coefficient was further discussed by comparing the results of this study with previous study. Based on the experimental results, an empirical correlation for supercritical CO2 diffusion coefficient in n-decane-saturated porous media was developed. The experimental results contribute to the study of supercritical CO2 diffusion in compact porous media.

The effect of permeability on supercritical CO2 diffusion coefficient and determination of diffusive tortuosity of porous media under reservoir conditions

A general method for determining the diffusivity coefficient of supercritical CO2 in cores saturated with oil is presented in this paper. Theoretically, a mathematical model including Fick’s diffusion equation and Peng–Robinson Equation of State (PR EOS) is proposed to evaluate the mass transfer of CO2 in the cores with different permeabilities. Experimentally, the pressure-decay method is employed by monitoring the CO2 pressure in the diffusion cell during diffusion experiments. The CO2 diffusion coefficients in the cores with different permeabilities are determined when the discrepancy of the calculated and measured pressure-decay curves has been minimized. The diffusion coefficient of supercritical CO2 increases gradually with rising permeability at the range of 0.1–10 mD, and then it reaches a plateau at the permeability range of 10–300 mD. The impact of the permeability on the CO2 diffusion coefficient is attributed to the pore radius and pore structure of the cores. The Pore radius of the core whose permeability is less than 10 mD is less than 1 μm, and pore walls restrict CO2 mass transfer under such condition, which accounts for the relative low diffusion coefficient. When the permeability exceeds 10 mD, the pore radius is larger than 1 μm, and the influence of the solid boundary is negligible. Moreover, the textural coefficients of cores decrease with the permeability, which shows that the less tortuous pore structure facilitates the mass transfer process. Diffusive tortuosities for the cores with different permeabilities are determined by using the CO2 diffusion coefficients in porous media and bulk phase, which show the same trend with the pore radius distribution.

Experimental investigation of gas mass transport and diffusion coefficients in porous media with nanopores

Understanding gas mass transport and determining diffusion coefficients are essential for investigating the gas flow mechanisms and evaluating porous media with nanopores. Multiple gas transport mechanisms coexist in porous media with complex pore size distribution, including viscous flow, Knudsen diffusion and surface diffusion. During pressure depletion of a reservoir, the adsorbed gas desorbs into pore space asadditional ‘free gas’, and meanwhile, diffuses along the surface of nanopores in porous media. The surface diffusion itself increases the total gas transport capacity in pores and its effect cannot be neglected. The bulk gas transport (non-surface diffusion) data was excluded experimentally to intensively investigate the surface diffusion during gas mass transport based on the gas storage and flow mechanisms. Accordingly, a mathematical model is developed by incorporating the surface diffusion. The results show that the equilibrium time for gas transport process decreases quickly with temperature. Higher saturation pressure could accelerate the process and increase the amount of produced gas. Besides, the two-stage process of the gas mass transport can be identified by recording the decay of gas pressure, which implies that the surface diffusion dominates the late stage of the gas mass transport. The surface diffusion coefficient for shale is between 10−18 and 10−16m2/s. This study provides a straightforward method to describe the gas mass transport in shale, simple but information–rich for the assessment of shale gas targets.

Predicted Dependence of Gas—Liquid Diffusion Coefficient on Capillary Pressure in Porous Media

Theoretical and experimental studies of the change of diffusion coefficient in porous media are reported. Mathematical equations were derived for calculating the diffusion coefficient in oil and gas phases. Experiments were conducted by measuring the pressure decline in both a PVT-cell and a sample of actual core. The medium porosity was found to have a significant effect on the system pressure decline. The diffusion coefficient in the PVT-cell was two orders of magnitude greater than that in the core. The diffusion coefficient was studied as a function of rock permeability and porosity. It was found that the porosity was the main factor affecting the diffusion coefficient whereas the permeability could be neglected.

Moisture transport in cementitious materials. Periodic homogenization and numerical analysis

This article presents a new approach based on periodic homogenisation for the computation of moisture diffusion coefficients in porous media. Experimental data allow fixing the parameters in the homogenised model. A series of parametric simulations are performed in order to compute the homogenised moisture diffusion coefficient through a gradual complexity of elementary cells approaching the microstructure of the cementitious materials tested.

Treatment of boundary conditions in through-diffusion: A case study of 85Sr2 + diffusion in compacted illite

Valuable techniques to measure effective diffusion coefficients in porous media are an indispensable prerequisite for a proper understanding of the migration of chemical-toxic and radioactive micropollutants in the subsurface and geosphere. The present article discusses possible pitfalls and difficulties in the classical through-diffusion technique applied to situations where large diffusive fluxes of cations in compacted clay minerals or clay rocks occur. The results obtained from a benchmark study, in which the diffusion of 85Sr2+ tracer in compacted illite has been studied using different experimental techniques, are presented. It is shown that these techniques may yield valuable results provided that an appropriate model is used for numerical simulations. It is further shown that effective diffusion coefficients may be systematically underestimated when the concentration at the downstream boundary is not taken adequately into account in modelling, even for very low concentrations. A criterion is derived for quasi steady-state situations, by which it can be decided whether the simplifying assumption of a zero-concentration at the downstream boundary in through-diffusion is justified or not. The application of the criterion requires, however, knowledge of the effective diffusion coefficient of the clay sample. Such knowledge is often absent or only approximately available during the planning phase of a diffusion experiment.

A Study on CO2 Diffusion Coefficient in n-decane Saturated Porous Media by MRI

The measurement of CO2 diffusion coefficient in porous media is considered to have great significance for enhancing oil recovery rate in oil industry. In the study, MRI technology was used to observe the CO2 diffusion process in porous media.The measurement is conducted at 24°C under four different pressures. This paper conducted in-depth analysis on the related parameters which affect the diffusion process of CO2 in n-decanesaturated porous media. And the results showed that the diffusion coefficient increased as the pressure increased, also the diffusion coefficient was influenced by parameters including pore-throatstructure, diameter of glass sands and so on.

Determination of spatially-resolved porosity, tracer distributions and diffusion coefficients in porous media using MRI measurements and numerical simulations

This work is focused on measuring the concentration distribution of a conservative tracer in a homogeneous synthetic porous material and in heterogeneous natural sandstone using MRI techniques, and on the use of spatially resolved porosity data to define spatially variable diffusion coefficients in heterogeneous media. The measurements are made by employing SPRITE, a fast MRI method that yields quantitative, spatially-resolved tracer concentrations in porous media. Diffusion experiments involving the migration of H(2)O into D(2)O-saturated porous media are conducted. One-dimensional spatial distributions of H(2)O-tracer concentrations acquired from experiments with the homogeneous synthetic calcium silicate are fitted with the one-dimensional analytical solution of Fick's second law to confirm that the experimental method provides results that are consistent with expectations for Fickian diffusion in porous media. The MRI-measured concentration profiles match well with the solution for Fick's second law and provide a pore-water diffusion coefficient of 1.75×10(-9)m(2)s(-1). The experimental approach was then extended to evaluate diffusion in a heterogeneous natural sandstone in three dimensions. The relatively high hydraulic conductivity of the sandstone, and the contrast in fluid density between the H(2)O tracer and the D(2)O pore fluid, lead to solute transport by a combination of diffusion and density-driven advection. The MRI measurements of spatially distributed tracer concentration, combined with numerical simulations allow for the identification of the respective influences of advection and diffusion. The experimental data are interpreted with the aid of MIN3P-D - a multicomponent reactive transport code that includes the coupled processes of diffusion and density-driven advection. The model defines local diffusion coefficients as a function of spatially resolved porosity measurements. The D(e) values calculated for the heterogeneous sandstone and used to simulate diffusive and advective transport range from 5.4×10(-12) to 1.0×10(-10)m(2)s(-1). These methods have broad applicability to studies of contaminant migration in geological materials.

Experimental Study of Diffusive Tortuosity of Liquid-Saturated Consolidated Porous Media

Diffusive tortuosity factor is one of the key parameters in modeling solute diffusion in liquid-saturated porous media. However, the determination of diffusive tortuosity factor has to involve a diffusion process in liquid-saturated porous media, which was usually found complicated in laboratories. The incorrect use of diffusive tortuosity factor may cause significant errors in certain circumstances. This paper presents a method to evaluate the diffusive tortuosity factors for liquid-saturated consolidated porous media, i.e., sandstones, which are the typical porous media commonly encountered in contaminant transport in underground water and gas migration in liquid-saturated reservoirs. The proposed method applies two specific experiments to determine the diffusion coefficient in bulk liquid phase and the effective diffusion coefficient in liquid-saturated porous media, respectively. Diffusive tortuosity factor of the porous media is obtained by comparing the effective diffusion coefficient in porous media to the diffusion coefficient in bulk liquid. This study provides a procedure to evaluate the diffusive tortuosity factor for consolidated porous media and also the measured values of diffusive tortuosity factors for selected sandstone samples which can be used as input data for further studies. Another application of the proposed method is to determine the diffusion coefficient in bulk liquid phase for CO2 from the measured effective diffusion coefficients in porous media.

Unified Multilayer Diffusion Model and Application to Diffusion Experiment in Porous Media by Method of Chambers

Diffusion coefficient is an important parameter for examining contaminant transport in the environment. Chamber methods (with or without external mixing devices) are the most popular methods for measuring effective diffusion coefficients in porous media (Deff) through air or water. The objectives of this paper were to apply simplified and unified analytical methods for both perfectly mixed and nonmixed (one- or two-) chamber systems and to examine how mixing affects the estimation of Deff. An analytical solution for a multilayer transient diffusion model was applied to the chamber methods without external mixing. By increasing the diffusion coefficient in reservoirs (D1 and D3) more than 10 times from the value for air or water (D0), the model was sufficient to approximate the well-mixed condition and, consequently, can be used to model transient diffusion in chamber systems with external mixing devices. We demonstrated that at long time Deff was related to the first eigenvalue (beta1) of a transcendental equation, which provided a quick method for determining Deff accurately from experimental data. The error caused by using the well-mixed approximation can be significant for a single-chamber system when there are no external mixing devices. This error increased rapidly with decreases in the experimental duration. A good fit for the concentration versus time curve could not be obtained forthe well-mixed solution, especially when sampling ports were near the boundary (x=0) and interface (x = l1). The proposed solutions are useful when the reservoir or chamber methods are used for measuring Deff and have wide applications in predicting contaminates transport in porous media and groundwater.

Determination of the effective diffusion coefficient in porous media including Knudsen effects

A 3-D bond pore network model is presented and used to evaluate the effect of pore size and connectivity on the effective diffusion coefficient in random porous media. The control equations of the system are set up and the simulation method is discussed. The simulation results show that when the average pore size dm < 1 μm, the effective diffusion coefficient is strongly dependent on pore size. The analysis shows that Knudsen and bulk diffusion effects can be decoupled, and for any given diffusion conditions, the effect coefficient accounting for Knudsen diffusion can be obtained. Thus, the effect of pore size can been readily accounted for by correcting the bulk effective diffusivity with the Knudsen effect coefficient. The simulations also show that the percolation threshold in random porous materials decreases with increased pore network connectivity.

Analytical solution and analysis of solute transport in rivers affected by diffusive transfer in the hyporheic zone

Summary A model is presented for solute transport in rivers including transient storage in hyporheic zones. The model consists of an advection–dispersion equation for transport in the main channel with a sink term describing diffusive solute transfer to the hyporheic zone. This system of equations is solved analytically for instantaneous injection of a conservative tracer in an infinite uniform river reach with steady flow. The solution enables to estimate the temporal and spatial evolution of tracer concentrations downstream of the injection point with fewer parameters than any other model before. The solution is linked to a non-linear least squares optimisation algorithm to analyse breakthrough curves and estimate solute transport parameters. The model is applied to tracer experiments conducted in the Chillan River, Chile, which were previously analysed with a model including mass exchange between the river and a stagnant storage zone. The fit between observations and model results is good, except for some experiments where the tailing of the fitted curves is more pronounced than observed. Estimates of the water flow velocity are practically identical with previous findings, but the estimates of the cross-sectional area and the dispersion coefficient are markedly different. Estimated values for the diffusion coefficient in the hyporheic zone agree with values cited in literature and with the magnitude of chemical diffusion coefficients in porous media.