Heterogeneous Media Research Articles

Cobalt Tetracationic 3,4‐Pyridinoporphyrazine for Direct CO2 to Methanol Conversion Escaping the CO Intermediate Pathway

AbstractMolecular catalysts offer a unique opportunity to implement different chemical functionalities to steer the efficiency and selectivity for the CO2 reduction for instance. Metalloporphyrins and metallophthalocyanines are under high scrutiny since their most classic derivatives the tetraphenylporphyrin (TPP) and parent phthalocyanine (Pc), have been used as the molecular platform to install, hydrogen bonds donors, proton relays, cationic fragments, incorporation in MOFs and COFs, to enhance the catalytic power of these catalysts. Herein, we examine the electrocatalytic properties of the tetramethyl cobalt (II) tetrapyridinoporphyrazine (CoTmTPyPz) for the reduction of CO2 in heterogeneous medium when adsorbed on carbon nanotubes (CNT) at a carbon paper (CP) electrode. Unlike reported electrocatalysis with cobalt based phthalocyanine where CO was advocated as the two electron and two protons reduced intermediate on the way to the formation of methanol, we found here that CoTmTPyPz does not reduce CO to methanol. Henceforth, ruling out a mechanistic pathway where CO is a reaction intermediate.

Strong effect of often-overlooked initial spilled oil distribution on subsequent soil remediation: A pore-scale perspective

The removal of spilled oil within soil by water injection is essentially an immiscible fluid displacement process. Numerous studies have been conducted to investigate the way to achieve a higher displacement efficiency. However, these studies often assumed that the pore space was initially saturated by oil only, ignoring the effect of initial oil distribution on the remediation, which led to incorrectly computed flow field and biased analysis. This work aimed to reveal the effect of initial oil distribution on flow physics during the remediation. Based on a pore-scale model and the phase-field method, the spilled oil invasion and remediation in a heterogeneous porous media were investigated, and the dynamic evolution of the phase interface was examined in particular. Four different distribution patterns of spilled oil were obtained through various velocities during the oil invasion stage, which were considered as initial states for the subsequent remediation stage. The results showed that ignoring the oil invasion process led to a more than 10% reduction in the ultimate remediation efficiency. However, the maximum interfacial area of fluids was larger and the obtained relative permeability curve was much smoother for the scenario considering the oil invasion. Besides, a higher initial water saturation led to a lower starting pressure. This work provided new pore-scale insights into the often-overlooked roles of initial phase distribution in fluid dynamics behind the remediation of oil-contaminated soil by water injection, which was of fundamental importance to the development of effective response strategy for soil contamination.

Effective diffusion along the backbone of combs with finite-span 1D and 2D fingers.

Diffusion in complex heterogeneous media, such as biological tissues or porous materials, typically involves constrained displacements in tortuous structures and sticky environments. Therefore, diffusing particles experience both entropic (excluded-volume) forces and the presence of complex energy landscapes. In this situation, one may describe transport through an effective diffusion coefficient. In this paper, we examine comb structures with finite-length 1D and finite-area 2D fingers, which act as purely diffusive traps. We find that there exists a critical width of 2D fingers, above which the effective diffusion along the backbone is faster than for an equivalent arrangement of 1D fingers. Moreover, we show that the effective diffusion coefficient is described by a general analytical form for both 1D and 2D fingers, provided the correct scaling variable is identified as a function of the structural parameters. Interestingly, this formula corresponds to the well-known general situation of diffusion in a medium with fast reversible adsorption. Finally, we show that the same formula describes diffusion in the presence of dilute potential energy traps, e.g., through a landscape of square wells. While diffusion is ultimately always the result of microscopic interactions (with particles in the fluid, other solutes, and the environment), effective representations are often of great practical use. The results reported in this paper help clarify the microscopic origins and the applicability of global, integrated descriptions of diffusion in complex media.

Thermovoltaic response in two-layered thin-film zinc oxide structures

A method of measuring the thermovoltaic effect in heterogeneous media with gradient doping impurity distributions producing gradient carrier distributions has been proposed. Iron doped zinc oxide specimens have been produced using ion beam sputtering on thin foil tantalum substrates for thermovoltaic effect measurements, glass-ceramic substrates for Hall measurements and silicon substrates for structural study. The doping impurity concentration хFe in the specimens has been varied from 0.34 to 4.18 at.%. X-ray phase analysis has shown that all the specimens have a hexagonal zinc oxide crystal structure. The films have preferential [002] orientation. The carrier concentration in the experimental specimen layers according Hall data obtained on an ECOPIA 5500 measurement system in a 0.5 T DC magnetic field has varied in the 1016–1020 cm-3 range. The specimens have an n-type conductivity. Thermovoltaic measurements have been carried out for two-layered iron doped zinc oxide specimens with different carrier and iron doping impurity concentrations using the method proposed. The maximum thermovoltaic response (U ~ 80 μV) has been observed in the two-layered thin-film specimen with the carrier concentration difference between the layers (Δn ≈ 2∙103 cm-3). The observed saturation of the thermovoltaic response has been attributed to the establishment of dynamic equilibrium between carrier diffusion from the high carrier concentration layer to the low carrier concentration layer and carrier drift due to internal electric field.

Role of capillary pressure on carbon dioxide sequestration: mathematical analysis

ABSTRACT Geological disposal of CO2 after capturing the gas from large emitting sources is one of the safest and most prominent methods to eliminate CO2 from the atmosphere and mitigate climate change. The heterogeneities in porosity, permeability and capillary pressure in various underground formations impact gas distribution, affecting their storage capacity. Therefore, a numerical model is developed to study the plume migration under various physical heterogeneities by upscaling the variation in capillary pressure. Accordingly, homogeneous, heterogeneous and layered formations have been considered. The influence of capillary pressure and field-scale heterogeneities, such as porosity and permeability, is incorporated using the Leverett J function. The impact of variations in capillary pressure on gas migration is calibrated in different formations for ten years of injection. The analysis project that plume evolution in different formations highly deviates and a plume length of 3410 m is obtained for layered formation. A sensitivity analysis was conducted to understand the influence of interfacial tension in formations with differing heterogeneity. The change in interfacial tension impacts the average saturation of CO2, ranging from 0.287 to 0.155 in heterogeneous aquifers. The plume migration decreases in layered and increases in homogenous and heterogeneous media with decreased interfacial tension.

Generalized Transition Waves for Neural Field Equations in Time Heterogeneous Media

Generalized Transition Waves for Neural Field Equations in Time Heterogeneous Media

Evaluation of a practical approach for field scale moisture flow modeling in heterogeneous media at a semiarid site

AbstractA practical approach for modeling field‐scale moisture flow in a highly heterogeneous unsaturated medium is described in this study. The validity of this approach is demonstrated through comparison of the numerical simulations with field observations at a semiarid site located in southcentral Washington State. The methodology is based on upscaling the core scale hydraulic properties and combining power‐law and tensorial connectivity‐tortuosity (PA‐TCT) approaches to derive macroscopic anisotropy parameters for each hydrostratigraphic unit (HSU) identified in the field. Each heterogeneous HSU is approximated by an equivalent homogeneous medium (EHM) model for which PA‐TCT parameters are used in the flow simulations. The available field data on moisture content and matric potential are compared with steady‐state flow simulations based on the mean form of Richards' equation. While the homogenization or averaging of heterogeneities, embedded in the EHM modeling approximation, cannot capture all of the field‐scale variability, the simulated steady‐state moisture and matric potential profiles capture well the central tendency of the field data. This approach is deemed practical for assessing the fate and transport of contaminants in highly heterogeneous unsaturated media at the transport scale of hundreds of meters.

Highly-enhanced propagation of long-range kinks in heterogeneous media

We investigate a field-theoretical model that describes the interaction between kinks and antikinks and between kinks and other heterogeneous fields and impurities. We show that the long-range kink can tunnel through a barrier created by heterogeneous fields and impurities even when the energy of the center of mass of the kink is less than the height of the energy barrier. We also study the conditions under which the kink can pass freely through a disordered medium. We introduce the concept of “effective translational symmetry”. We compare our results with those from recent papers published in this journal, where the Bogomol’nyi-Prasad-Sommerfield property is discussed.

Non-Intrusive Reduced Basis two-grid method for flow and transport problems in heterogeneous porous media

Due to its non-intrusive nature and ease of implementation, the Non-Intrusive Reduced Basis (NIRB) two-grid method has gained significant popularity in numerical computational fluid dynamics simulations. The efficiency of the NIRB method hinges on separating the procedure into offline and online stages. In the offline stage, a set of high-fidelity computations is performed to construct the reduced basis functions, which is time-consuming but is only executed once. In contrast, the online stage adapts a coarse-grid model to retrieve the expansion coefficients of the reduced basis functions. Thus it is much less costly than directly solving a high-fidelity model. However, coarse grids in heterogeneous porous media of flow models are often accompanied by upscaled hydraulic parameters (e.g. hydraulic conductivity), thus introducing upscaling errors. In this work, we introduce the two-scale idea to the existing NIRB two-grid method: when dealing with coarse-grid models, we also employ upscaled model parameters. Both the discretization and upscaling errors are compensated by the rectification post-processing. The numerical examples involve flow and heat transport problems in heterogeneous hydraulic conductivity fields, which are generated by self-affine random fields. Our research findings indicate that the modified NIRB method can effectively capture the large-scale features of numerical solutions, including pressure, velocity, and temperature. However, accurately retrieving velocity fields with small-scale features remains highly challenging.

Sensitivity of mass flux reduction and mass removal of perfluoroalkyl substances to groundwater flow and transport parameter variability and heterogeneity

The heterogeneity of soil hydraulic (e.g., hydraulic conductivity (KS), porosity (θS)) and chemical (e.g., solid-phase adsorption (Kd)) properties complicate contaminant transport by creating spatial variability in sources of contaminant leaching. There is a knowledge gap on the effect of the interplay between these properties on the retardation and transport of per- and polyfluoroalkyl substances (PFAS) with different properties including carbon–fluorine chain-length and functional groups even in water-saturated conditions. Breakthrough curves (BTCs) have been used to evaluate PFAS transport behavior through heterogeneous media, including arrival time, maximum concentration, and tailing behavior. Contaminant mass flux reduction (MFR) and mass removal (MR) correlations are also compared using numerical modeling herein to characterize PFAS transport through different source zones within a two-domain, heterogeneous system with comparison to homogeneous scenarios under water-saturated conditions. With heterogeneous properties, model sensitivity to KS was the highest among the other parameters and was controlled by the KS ratio between the different soils. The PFAS models in the homogeneous and heterogeneous scenarios were both sensitive to θS depending on PFAS chain length. However, long-chain PFAS were less sensitive to θS variability compared to short-chain PFAS, due to their higher Kd. The homogeneous and heterogeneous scenarios were equally sensitive to Kd variability, which was dependent on PFAS chain length.

A Domain-Adaptive Physics-Informed Neural Network for Inverse Problems of Maxwell's Equations in Heterogeneous Media

A Domain-Adaptive Physics-Informed Neural Network for Inverse Problems of Maxwell's Equations in Heterogeneous Media

Stochastic analysis of the interaction between excess fluid flow and soil deformation in heterogeneous deformable porous media

Stochastic analysis of the interaction between excess fluid flow and soil deformation in heterogeneous deformable porous media

Comprehensive analysis of wave interactions and resulting spall damage in layered heterogeneous medium under impact

The effect of dynamic spall damage on the response of layered medium subjected to one-dimensional impact involving an impactor and a multi-layered target is investigated. The study examines the damage caused by elastic waves below the Hugoniot Elastic Limit (HEL) during impact events. Analytical expressions of stress and particle velocity derived from mass and momentum conservation principles are utilized to solve the wave interactions within the impactor-target system, considering reflections, transmissions, and wave interactions between layers. These analytical expressions, integrated into a computational framework, facilitate the monitoring of material responses following each wave interaction, thereby analysing the impact response of the layered medium. Wave interactions, resulting in tensile stresses, that can cause damage to the target have been identified. A strain-based damage initiation and evolution law is incorporated into the developed computer program to predict damage in each layer of a medium. Damage alters the medium’s impact behaviour by decreasing stress magnitude and inducing temporal delays in the response due to attenuated wave propagation speed. The occurrences of tensile stress are explored, analyzing spatial and temporal variations of multiple damage incidents within the layered medium. The tension at the interface between two layers can induce damage in one or both adjacent layers, leading to debonding between layers. The effect of damage on the impact response of the medium is analysed by comparing the results for the damaged and undamaged target scenarios. The impact behaviour of a single-layer and a multi-layer target obtained from the present model, in terms of stress and particle velocity, is verified through Finite Element simulations of the identical impact problems, where the damage evolution criterion is incorporated using a VUMAT subroutine. The outcomes derived from this study align closely with Finite Element (FE) results.

Characterization of weakly nonlinear effects in relationship to transducer parameters in focused ultrasound therapy.

Focused ultrasound therapy has been widely used for the treatment of various diseases, employing different types of transducers. The focused ultrasound pressure fields inevitably exhibit nonlinear effects, which can influence the ablation region. However, the nonlinear effects exhibit noticeable variations across different applications. The characterization of the nonlinear pressure fields of ultrasound is important for the effective implementation of focused ultrasound therapy. The traditional angular spectrum method (ASM) was extended to accurately and efficiently simulate the propagation of weakly nonlinear ultrasound in heterogeneous mediums of clinical model. The nonlinear effects were further analyzed in relationship to the transducer parameters that are different in various applications. The pressure fields were simulated using the extended ASM, incorporating calculations for phase aberration in the frequency domain and magnitude compensation in the spatial domain to account for heterogeneous acoustic impedance mismatch. Validation was performed by comparison to k-Wave simulation results using two simplified clinical models, an abdominal soft tissue and a transcranial skull model. The nonlinear effects were then analyzed in relation to the transducer parameters of f-number and effective source area based on the same acoustic output power. The analysis of nonlinear effects was conducted under both homogeneous medium and the clinical models. The simulation results demonstrated a maximum error of 3.93% in the calculated harmonic pressure of the abdominal model, and a maximum error of 4.89% within the transcranial model when comparing the extended ASM simulation results to those obtained from k-Wave simulations. The characterization of the nonlinear effects reveals a strong correlation with the transducer parameters. Specifically, the results indicate that the nonlinear effects intensify with an increase in the effective source area and f-number, under the same acoustic output power of the transducer. However, the clinical model also showed an influence on the nonlinear effects in relation to the f-number. The extended ASM was demonstrated as an accurate and efficient simulation tool, and the simulation results provide a reference for evaluating the intensity of nonlinear effects in various transducer designs.

MultIHeaTS: A Fast and Stable Thermal Solver for Multilayered Planetary Surfaces

A fully implicit scheme is proposed for solving the heat equation in 1D heterogeneous media, available as a computationally efficient open-source Python code. The algorithm uses finite differences on an irregular grid and is unconditionally stable due to the implicit formulation. The thermal solver is validated against a stiff analytical solution, demonstrating its robustness in handling stiff initial conditions. Its general applicability for heterogeneous cases is demonstrated through its use in a planetary surface scenario with nonlinear boundary conditions induced by blackbody thermal emission. MultIHeaTS's advantageous stability allows for computation times up to 100 times faster than Spencer’s explicit solver, making it ideal for simulating processes on large timescales. This solver is used to compare the thermal signatures of homogeneous and bilayer profiles on Europa. Results show that homogeneous materials cannot reproduce the thermal signature observed in bilayer profiles, emphasizing the need for multilayer solvers. In order to optimize the scientific return of a space mission, we propose a strategy made of three local time observations that is enough to identify bilayer media, for instance, for the next missions to the Jovian system. A second application of the solver is the estimation of the temperature profile of Europa’s near surface (first 10s m) throughout a 1 million yr simulation with varying orbital parameters. The probability distribution of temperature through depth is obtained. Among its various applications, MultIHeaTS serves as the core thermal solver in a multiphysics simulation model detailed in the companion article by C. Mergny & F. Schmidt.

Numerical Investigation of Heterogeneous Calcite Distributions in MICP Processes

Microbially induced calcite precipitation (MICP) is a sustainable and environmentally friendly technology with applications in soil stabilization, concrete crack repair, and wastewater treatment. This study presents an improved Darcy-scale numerical model to simulate the MICP processes in heterogeneous porous media. It focuses on the effects of porosity heterogeneity, characterized by average porosity and correlation length, as well as injection strategies. Both average porosity and correlation length are critical factors influencing mass transport and calcite distribution during MICP treatment. An increase in average porosity leads to significant reductions in transport distance and total calcite mass. Notably, in the case of low averaged porosity, a larger correlation length results in more heterogeneous calcite distributions. However, there exists an upper threshold value of the initial averaged porosity (ϕ0=0.45) above which the heterogeneity of the calcite does not present clear dependence on the correlation length. Additionally, injection strategies significantly impact the consolidation effects. Compared to continuous injection, using the phased injection strategy can greatly improve the precipitated calcite area and mass due to its high utility and the efficiency of reactants.

Two-Phase Incompressible Flow with Dynamic Capillary Pressure in a Heterogeneous Porous Media

We prove the existence of weak solutions of a two-incompressible immiscible wetting and non-wetting fluids phase flow model in porous media with dynamic capillary pressure. This model is a coupled system which includes a nonlinear parabolic saturation equation and an elliptic pressure–velocity equation. In the regularized case, the existence and uniqueness of the weak solution are obtained. We let the regularization parameter be η→0 to prove the existence of weak solutions.

A combined Markov Chain Monte Carlo and Levenberg–Marquardt inversion method for heterogeneous subsurface reservoir modeling

In this study, we systematically investigate an inverse method for heterogeneous porous media to obtain porosity and permeability fields considering only production well data. The forward modeling of the input data, namely pressure and saturation fields, is based on the motion equations of a coupled Darcy flow system involving two phases of isothermal fluid flow. We discretize these equations using a multiscale finite volume simulation technique in the spatial domain, and the backward Euler method in time domain. In the inversion procedure, we combine a global optimization method, the Markov Chain Monte Carlo (MCMC) method, with a local optimization method, the Levenberg–Marquardt (LM) method. The MCMC was implemented as the Random Walk algorithm, and to generate the samples of the porosity and permeability fields, we employed Karhunen–Loève (KL) expansion of second-order stationary fields with Gaussian covariance. The coefficients of the KL expansion are estimated by minimizing the norm of the residual dependent on these fields. At the end of the MCMC iterations, we refine the KL coefficients associated with the porosity and permeability fields using the LM method. We verify in the numerical experiments that the accuracy of the porosity and permeability fields is improved by the LM refinement step.

CFD analysis of pore morphology, gravity, and fluid characteristics influences on water flooding process

In this research, computational fluid dynamics (CFD) was employed to examine the two-phase flow of water and oil in a porous medium. For this purpose, the Navier-Stockes equations, which describe fluid motion, and the Cahn-Hillard phase field, which defines the interface between two phases, are coupled. The numerical discretization system of equations was solved using the finite element method (FEM) with the COMSOL software. To validate the results and the phase-field model (PFM), the Lucas and Washburn equation was used together with the experimental data. Through this study, the effective parameters were evaluated in two parts. In the first part, seven models with different pore morphologies were designed, and the impact of petrophysical parameters of the reservoir, including the shape of pores, connectivity of pores with or without throat lines, porous media heterogeneity, and relative permeability, on oil recovery was investigated. The second part was devoted to performing sensitivity analysis on the effect of fluid properties, including interfacial tension, wettability, viscosity ratio, injected fluid flow, and gravity, upon enhanced oil recovery (EOR) in different morphologies. Owing to the uniform distribution of capillary pressure in the patterns with the throat lines, the sweep efficiency of the injected fluid was found to be better, and thereby oil production increased. The results of the present work proved the significant influence of gravity on EOR, so that by applying gravity to the solution domain, the breakthrough time and oil recovery factor increased by 20 minutes and 28.6 %, respectively. Moreover, the velocity of the injected fluid as a representative of the flow rate had the greatest effect on EOR, with a 35 % increase in production.

Velocity of ultrasound diffusion in heterogeneous media and its applications

This paper investigates the velocity of ultrasonic diffusion in heterogeneous media under the conditions where the diffuse wave approximation is valid (λ≤a). The diffusion velocity is defined as the moving speed of the peak of the energy evolution curve. The peak arrival time as a function of transport distance is calculated for infinite spaces and bounded domains. The results show that the peak arrival time is independent of the domain’s geometry, i.e. the size, shape, and boundary conditions. This is confirmed by comparing the arrival times in an infinite space and a bounded domain with a geometric feature – a surface-breaking crack of varying depth. Therefore, the velocity of ultrasonic diffusion is an intrinsic property of a medium, and the formula for the infinite three-dimensional space is sufficient to calculate the arrival time–transport distance relationship, given the diffuse properties of the medium. These findings eliminate the needs for the finite element numerical simulations in the applications to determine geometric parameters such as the crack depth using diffuse ultrasound. The diffusion velocity is a function of transport distance in general, while its far-field asymptotic value is constant regardless of the dimensionality and geometry of the domain.