Diffusion In Porous Media Research Articles

Critical exponent to a cancer invasion model with nonlinear diffusion

This paper is concerned with a cancer invasion model that incorporates porous medium diffusion (Δum) and extracellular matrix remodeling effects [ηω(1 − u − ω)] in a bounded domain of RN (N ≥ 2). Rich achievements have been achieved for the case η = 0 in the past ten years for the nonlinear diffusion case, but there is no any progress for η > 0. In this paper, we pay our attention to the global existence of solutions of the case η > 0, and establish the critical exponent m*=2N−2N of global solvability. More precisely, if m > m*, the solution will always exist globally, while if m < m*, there exist blow-up solutions. In this system, the remodeling effect of extracellular matrix [ηω(1 − u − ω)] bring some essential difficulties to the estimation of the haptotactic term, so the main technique we used is completely different from the case of η = 0.

Influences of thermal and mass stratification on unsteady magnetohydrodynamics parabolic flow along an infinite vertical plate with periodic temperature variation and exponential mass diffusion in porous medium

AbstractThis study explores the dynamics of unsteady magnetohydrodynamics (MHD) parabolic flow along an infinite vertical plate, emphasizing the effects of thermal and mass stratification in a porous medium subjected to periodic temperature variation and exponential mass diffusion. Utilizing the Laplace transform technique to obtain precise solutions, this study effectively integrates the impacts of both thermal and mass stratification without dependence on approximations. The main goal is to assess how thermal and mass stratification impact MHD flow dynamics, temperature, and concentration profiles under varying conditions. The study provides a thorough comparison of these findings with traditional nonstratified scenarios, presenting a comprehensive analysis of fluid behavior under diverse conditions. The conclusions reveal that thermal and mass stratifications considerably diminish velocity and stabilize temperature distributions, which suggests a damping influence on fluid movement and improved management of diffusion processes. Enhanced Grashof numbers improve heat and mass transfer efficiency, while magnetic and Darcy parameters significantly influence flow resistance and heat transfer characteristics. These conditions also result in higher Nusselt and Sherwood numbers, indicating increased efficiency in heat and mass transfer. In contrast, scenarios without stratification display higher velocities and more unstable temperature and concentration profiles. The findings highlight the critical role of stratification in improving fluid dynamics and increasing the efficiency of heat and mass transfer processes, offering valuable insights for engineering and environmental applications in similar conditions. The main novelty of the research is being the first to use the Laplace transform for exact solutions on combined thermal and mass stratification in MHD flows, enhancing prediction accuracy and process control.

Enhance Ethanol Sensing Performance of Fe-Doped Tetragonal SnO2 Films on Glass Substrate with a Proposed Mathematical Model for Diffusion in Porous Media.

This research enhances ethanol sensing with Fe-doped tetragonal SnO2 films on glass, improving gas sensor reliability and sensitivity. The primary objective was to improve the sensitivity and operational efficiency of SnO2 sensors through Fe doping. The SnO2 sensors were synthesized using a flexible and adaptable method that allows for precise doping control, with energy-dispersive X-ray spectroscopy (EDX) confirming homogeneous Fe distribution within the SnO2 matrix. A morphological analysis showed a surface structure ideal for gas sensing. The results demonstrated significant improvement in ethanol response (1 to 20 ppm) and lower temperatures compared to undoped SnO2 sensors. The Fe-doped sensors exhibited higher sensitivity, enabling the detection of low ethanol concentrations and showing rapid response and recovery times. These findings suggest that Fe doping enhances the interaction between ethanol molecules and the sensor surface, improving performance. A mathematical model based on diffusion in porous media was employed to further analyze and optimize sensor performance. The model considers the diffusion of ethanol molecules through the porous SnO2 matrix, considering factors such as surface morphology and doping concentration. Additionally, the choice of electrode material plays a crucial role in extending the sensor's lifespan, highlighting the importance of material selection in sensor design.

Global solvability in a three-dimensional self-consistent chemotaxis-Navier-Stokes system with porous medium diffusion

Global solvability in a three-dimensional self-consistent chemotaxis-Navier-Stokes system with porous medium diffusion

Time-fractional fabric to quantify non-Fickian diffusion in porous media: New vision from previous studies

The diffusion kinetics of different substances in porous media is a priori described by the second Fick's law of diffusion. Herein, we demonstrate that such a description sometimes yields erratic results. In contrast, the time-fractional diffusion equation is found to reflect the diffusion kinetics in the case of Fick's law failure. The analytic solutions of the time-fractional diffusion equation are based on the Mittag–Leffler function. At large times, it is shown that the utilization of the approximation of the Mittag–Leffler function for the large values of its argument for the description of the experimental data gives almost the same results as those obtained using the Mittag–Leffler function itself. The time-fractional diffusion is applied on a phenomenological basis. Nevertheless, we speculate that the deviations from Fick's law may be governed by strong adsorption of the diffusing species inside the small micropores of a porous material.

On a nonlinear diffussive model for the evolution of cells within a moving domain

We investigate the dynamics of a nonlinear model describing the motion of cells under the effect of porous-medium diffusion and transport and in the presence of nutrient and drug application. The momentum equation for the evolution of the velocity field is governed by Darcy’s law, while the evolution of the chemical attractant (nutrient or drug) is governed by a diffusion equation. The system evolves within a moving domain in R 3 \mathbb {R}^3 accounting for the expansion or shrinkage of the tumor. The global existence of weak solutions is established with the aid of a regularized approximating scheme and an Arbitrary Lagrangian-Eulerian (ALE) mapping for the motion of the tumor. This work provides a variational framework suitable for both analysis and simulations.

Thermal and mass stratification effects on unsteady parabolic flow past an infinite vertical plate with exponential decaying temperature and variable mass diffusion in porous medium

AbstractThis study looks at how thermal and mass stratification affect the unsteady parabolic flow past an infinite vertical plate with exponentially decaying temperature and variable mass diffusion in a porous medium. The Laplace transform method is applied to solve the governing equations for flow, energy, and mass diffusion. The effects of the both stratification on the flow field, temperature, and mass diffusion are then investigated. The research suggests that the existence of thermal and mass stratification has a considerable influence on the profiles of velocity, temperature, and mass diffusion. It is also found that a stable state is reached for the velocity when both stratification parameters are increased, while stable states are reached for the temperature and concentration when the mass stratification is increased but the thermal stratification is decreased.

Ultrafast Diffusion Modeling via the Riemann–Liouville Nonlocal Structural Derivative and Its Application in Porous Media

Ultrafast diffusion disperses faster than super-diffusion, and this has been proven by several theoretical and experimental investigations. The mean square displacement of ultrafast diffusion grows exponentially, which provides a significant challenge for modeling. Due to the inhomogeneity, nonlinear interactions, and high porosity of cement materials, the motion of particles on their surfaces satisfies the conditions for ultrafast diffusion. The investigation of the diffusion behavior in cementitious materials is crucial for predicting the mechanical properties of cement. In this study, we first attempted to investigate the dynamic of ultrafast diffusion in cementitious materials underlying the Riemann–Liouville nonlocal structural derivative. We constructed a Riemann–Liouville nonlocal structural derivative ultrafast diffusion model with an exponential function and then extended the modeling strategy using the Mittag–Leffler function. The mean square displacement is analogous to the integral of the corresponding structural derivative, providing a reference standard for the selection of structural functions in practical applications. Based on experimental data on cement mortar, the accuracy of the Riemann–Liouville nonlocal structural derivative ultrafast diffusion model was verified. Compared to the power law diffusion and the exponential law diffusion, the mean square displacement with respect to the Mittag–Leffler law is closely tied to the actual data. The modeling approach based on the Riemann–Liouville nonlocal structural derivative provides an efficient tool for depicting ultrafast diffusion in porous media.

Global boundedness and eventual regularity of chemotaxis-fluid model driven by porous medium diffusion

Global boundedness and eventual regularity of chemotaxis-fluid model driven by porous medium diffusion

Thermal and Mass Stratification Effects on Unsteady Flow Past an Accelerated Infinite Vertical Plate with Variable Temperature and Exponential Mass Diffusion in Porous Medium

This study looks at how thermal and mass stratification affect the unsteady flow past an infinitely fast-moving vertical plate when the temperature is changing and there is exponential mass diffusion in a porous medium. By applying the Laplace transformation method, we determine the solutions to the equations that govern the system for the case of unitary Prandtl and Schmidt numbers. Graphical representations of the concentration, temperature, and velocity profiles, as well as the Nusselt Number, Sherwood number, and the Skin friction are provided to facilitate discussion of the cause of the different variables. To see the effects of thermal and mass stratification on the fluid flow, we compare the classical solution (Fluid with out stratification) with the primary solution (Fluid with the stratification) by using graph. The combined effects of the two stratification lead to a quicker approach to steady states. The outcomes can be helpful for heat exchange design and other engineering applications.

Empirical and Numerical Modelling of Gas–Gas Diffusion for Binary Hydrogen–Methane Systems at Underground Gas Storage Conditions

The physical process in which a substance moves from a location with a higher concentration to a location with a lower concentration is known as molecular diffusion. It plays a crucial role during the mixing process between different gases in porous media. Due to the petrophysical properties of the porous medium, the diffusion process occurs slower than in bulk, and the overall process is also affected by thermodynamic conditions. The complexity of measuring gas–gas diffusion in porous media at increased pressure and temperature resulted in significant gaps in data availability for modelling this process. Therefore, correlations for ambient conditions and simplified diffusivity models have been used for modelling purposes. In this study, correlations in dependency of petrophysical and thermodynamic properties were developed based on more than 30 measurements of the molecular diffusion of the binary system hydrogen–methane in gas storage rock samples at typical subsurface conditions. It allows reproducing the laboratory observations by evaluating the bulk diffusion coefficient and the tortuosity factor with relative errors of less than 50 % with minor exceptions, leading to a strong improvement compared to existing correlations. The developed correlation was implemented in the open-source simulator DuMux and the implementation was validated by reproducing the measurement results. The validated implementation in DuMux allows to model scenarios such as Underground Hydrogen Storage (UHS) on a field-scale and, as a result, can be used to estimate the temporary loss of hydrogen into the cushion gas and the purity of withdrawn gas due to the gas–gas mixing process.

H2-gas diffusion in porous media as observed by NMR

Not only is hydrogen (H2) the simplest and most abundant gas in our universe, it also plays a pivotal role in the decarbonization of our energy. As such hydrogen will be one of the most important critical enablers for industry in the global energy transition. Hydrogen’s low density hinders the wide-scale deployment and therefore, it is important to come to large scale low-cost H2 storage capacity as a source of hydrogen gas. In order, to store the energy in the TW h/GW h-range subsurface storage of hydrogen in depleted hydrocarbon reservoir and/or deep saline aquifers is needed. Concerning the hydrogen storage in depleted gas/oil reservoirs there are still many scientific questions open, which have to be addressed in order to come to reliable, efficient and cost-effective storage [1, 2]. For example, the injected hydrogen will displace the pore residual oil/gas and spread out in a reservoir; a trap structure is needed to prevent the hydrogen from escaping and allowing reproduction of the hydrogen, i.e., leakage is a major concern. Accurate predictions are needed of multi-phase fluid displacement in porous media, e.g., the interaction/displacement of cushion gas and the interactions with the solid matrix.
 In this study we have used NMR to image the diffusion and interaction in porous media. NMR is one the few methods by which H2-gas can be directly imaged in porous media. In order to do so, we have constructed a special NMR setup to measure the diffusion in porous media up to 60 bar. As the relaxation times measured for hydrogen are too short for 3D imaging this setup is limited to 1D measurement [3]. As, to be able to measure also on real porous media containing magnetic impurities, it was chosen to keep the main field at 0.7 T. The present setup is made up of an electromagnet generating a main magnetic field of 0.7 T with a gap of 70 mm and a picture is given in Fig 1. The NMR setup itself is equipped with Anderson gradient coils which can generate up to 0.3 T/m. An RF insert was made which is equipped with a special Faraday-shield, to be able to perform quantitative measurements and as such this setup is directed towards quantitative H2 profile measurements [4].
 For the NMR measurements a special PEEK reactor was purchased, which can be operated on up to 60 bars. The PEEK reactor has an outer diameter of 50 mm and inner diameter of 30 mm with a length of 500 mm. The reactor can be moved through the RF-coil with the help of a stepper motor, hence giving the possibility to measure the H2 content profile over the length of the reactor.
 In the initial tests we looked at the quantitative measurement of H2 as a function of the gas pressure. These initial measurements show that the H2 gas has a T1 in the order of 0.5 ms. Hence, as the signal of the H2 gas is quite low in comparison to the background noise, it was decided to measure the H2 gas at a high repetition rate, i.e., TR=2 ms. The results as function of the gas pressure up in the initial test upto 15 bars are given in Fig 2. As can be seen a clear linear behavior is found and only a minor contribution of the background is observed. In the next experiment the reactor was filled with 0.06 mm glass beads to mimic an ideal porous material and the results are also given in Fig 2. Again, a clear linear behavior can be observed, indicating measurements can also be performed in porous media.
 After this preliminary study in which we have tested the setup, we now intend to conduct measurements on porous glass beads up to 2 mm under various prewetting conditions, e.g., water, NaCl solution, oil and at various pressures. Here we want to measure the exchange of H2 gas by other gases such as N2 up to 60 bar and see the influence of the various prewetting conditions.

Heat pump efficiency with energy geostructure: Numerical long term modelling

Heat pump efficiency with energy geostructure: Numerical long term modelling

Determination of Concentration-Dependent Effective Diffusivity of Each Gas Component of a Binary Mixture in Porous Media Saturated with Heavy Oil under Reservoir Conditions

Summary One frequently used enhanced heavy oil recovery technique is gas injection, during which heavy oil viscosity is reduced due to diffusion of gaseous components and heavy oil swelling in porous media. Effective diffusivities of gas components are generally assumed to be constants, while no attempts have been made to determine both the concentration-dependent effective diffusivity in porous media saturated with heavy oil and the preferential contribution of each component in a binary/ternary gas mixture. In this study, a pragmatic and robust technique has been proposed to determine the concentration-dependent effective diffusivity of each gas component by reproducing the experimental measurements during pressure decay tests for CO2-C3H8-heavy oil systems in porous media. Experimentally, CO2 and C3H8 are utilized to diffuse into sandpacks fully saturated with heavy oil. Under a constant temperature within a thermostatic chamber, the pressures of the aforementioned gas(es)-heavy oil systems are consistently tracked and saved while gas samples are taken at the start and end of the diffusion tests for gas chromatography analyses. Theoretically, a mass transfer model is formulated to determine effective gas diffusivity in heavy oil as a concentration-dependent function by incorporating Fick’s second law and the modified Peng-Robinson equation of state (PR EOS). The concentration-dependent effective diffusivity for each gas component is ascertained when the measured pressure profiles and gas compositions are matched well to their correspondingly calculated values with minimum deviations. Compared to either a constant assumption or a linear concentration-dependent relation with respect to diffusivity, an exponential concentration-dependent relation leads to more accurately reproducing the measured pressure profiles. Compared with pure CO2, its effective diffusivity in a binary (i.e., CO2 and C3H8) gas system is found to be larger, indicating that C3H8 accelerates the CO2 mass transfer into heavy oil under the same circumstances. Furthermore, this study confirms that a larger tortuosity of a porous medium leads to a longer diffusion path with less contact between gas and liquid phases and that a lower concentration of a gaseous component yields a lower effective diffusivity.

Global Boundedness to a 3D Chemotaxis–Stokes System with Porous Medium Cell Diffusion and General Sensitivity Under Dirichlet Signal Boundary Condition

Global Boundedness to a 3D Chemotaxis–Stokes System with Porous Medium Cell Diffusion and General Sensitivity Under Dirichlet Signal Boundary Condition

Dynamic monitoring of leaking oil diffusion in porous media: An improved method assisting buried oil pipeline condition assessment

The environmental pollution caused by buried oil pipeline leakage is a typical non-aqueous phase liquid (NAPL) pollution accident. Therefore, it seems more reasonable to assess the condition of buried oil pipelines considering the potential contamination range. The law of oil diffusion in porous soil is very important for the safety management of buried pipelines and the remedy of leakage accidents. This paper proposes a dynamic observation technique of NAPL in porous media for buried oil pipeline leakage. A quantitative relationship between the diesel saturation and color components is established based on the theoretical analysis combined with 21 groups of dyed diesel experimental results. The accuracy of the observation technique is verified by a quantitative analysis of a series of discrete point samplings and a comparison of diesel volume in a diesel diffusion experiment. The results show that the combination of a specified saturation boundary and the color components S and I can effectively track the diesel diffusion fronts. The relative error between the calculated cumulative volume and the injected cumulative volume fluctuates around 5% at different times. This improved technology can effectively make up for the limitation of the traditional light transmission visualization (LTV) under low-intensity light and promote the study of NAPLs diffusion characteristics. Furthermore, the dynamic observation technique of NAPL in porous media for buried oil pipeline leakage monitoring can make a significant contribution to the improvement of the level of buried pipeline leakage accidents and remediation.

Deep learning for diffusion in porous media

We adopt convolutional neural networks (CNN) to predict the basic properties of the porous media. Two different media types are considered: one mimics the sand packings, and the other mimics the systems derived from the extracellular space of biological tissues. The Lattice Boltzmann Method is used to obtain the labeled data necessary for performing supervised learning. We distinguish two tasks. In the first, networks based on the analysis of the system’s geometry predict porosity and effective diffusion coefficient. In the second, networks reconstruct the concentration map. In the first task, we propose two types of CNN models: the C-Net and the encoder part of the U-Net. Both networks are modified by adding a self-normalization module [Graczyk et al. in Sci Rep 12, 10583 (2022)]. The models predict with reasonable accuracy but only within the data type, they are trained on. For instance, the model trained on sand packings-like samples overshoots or undershoots for biological-like samples. In the second task, we propose the usage of the U-Net architecture. It accurately reconstructs the concentration fields. In contrast to the first task, the network trained on one data type works well for the other. For instance, the model trained on sand packings-like samples works perfectly on biological-like samples. Eventually, for both types of the data, we fit exponents in the Archie’s law to find tortuosity that is used to describe the dependence of the effective diffusion on porosity.

Justification of a New Original Homogenized Model for Ionic Diffusion in Porous Media

Abstract In this work, a new original justification of an homogenized model for ionic diffusion in porous media is proposed. The approach used enables to specify clearly the domain of validity of this homogenized model, involving a source term characterizing the electrical double layer effect at the macroscale. This homogenized model is obtained from the formal periodic homogenization of a Nernst–Planck–Poisson system at the pore scale accounting for conductivity of the solid phase which is generally neglected. The Poisson equation is defined in both fluid and solid phases and the discontinuity of fluxes at the solid–fluid interface is modeled by a jump of the electrical field, linked to the surface electrical charge of the solid interface. Numerical simulations are carried out at the scale of the unit cell to underscore the influence of the contrast on the electrical permittivity between fluid and solid phases. The comparison of the concentrations and the electrical potential given at the macroscale by the homogenized model and by a direct pore scale model reveals the accuracy of the homogenized model which is very simple to use.

Global Solvability, Pattern Formation and Stability to a Chemotaxis-haptotaxis Model with Porous Medium Diffusion

Global Solvability, Pattern Formation and Stability to a Chemotaxis-haptotaxis Model with Porous Medium Diffusion

Advances in Permeation of Solutes into Hair: Influencing Factors and Theoretical Models

The permeation and absorption of solutes into human hair are highly relevant to various applications, including the formulation of hair-care products, the development of water pollution control and remediation, and the risk assessment of environmental exposure. Based on a detailed introduction of the structure and composition of hair, the effects of the properties of hair (structure, composition, and charge properties), the physicochemical properties of solutes (molecular size, shape, and hydrophobicity), and the conditions of the surrounding medium (solvent composition, temperature, and pH) on the permeation and absorption were comprehensively analyzed. Several theoretical models were reviewed, including two-part/two-state, porous media diffusion, homogeneous medium diffusion, heterogeneous medium partition, and diffusion models. Finally, future research directions for the permeation and absorption of solutes in hair were proposed to provide a foundation for the further optimization and application of permeation models.