Time-dependent Diffusion Model Research Articles

Time-dependent gas dynamic diffusion process in shale matrix: model development and numerical analysis

Gas diffusion is a pivotal process during shale gas recovery, which is determined by diffusion coefficient to a large extent. In previous studies, the gas diffusion coefficient is generally assumed as a constant. However, increasing experiments prove that the diffusion coefficient of shale gas is strongly time-dependent. Therefore, to perfect the theory of shale gas diffusion, this paper proposes a time-dependent diffusion model for shale gas, which incorporates time-dependent gas diffusion coefficient, composing of the bulk diffusion coefficient for free gas in organic and inorganic pores, as well as the surface diffusion coefficient for adsorbed gas in organic pores. To validate the accuracy of the new theory, we calibrate the theoretical results against experimental data, and the results show that they have strong correlation, and the time-dependent diffusion model is superior to classical model. Finally, the numerical analysis of gas dynamic diffusion process in shale matrix is conducted. The results show that at the end of diffusion, a large amounts of shale gas remain trapped in the matrix core due to the attenuation of gas diffusion coefficient. In addition, neglecting the time-dependent nature of gas diffusion in shale matrix leads to a significant overestimation of gas production.

Time-dependent concrete chloride diffusion coupled with nonlinear chloride binding: An analytical study

Both the temporal dependence of chloride diffusion coefficient and the effect of nonlinear chloride binding substantially affect the behavior of concrete chloride diffusion. This study develops an exact chloride diffusion solution that accommodates the above two factors by introducing a simplified but reasonably realistic nonlinear chloride binding isotherm. The introduction of this isotherm makes it possible to achieve an analytical solution to a highly nonlinear chloride diffusion equation that involves a free chloride concentration dependent dimensionless factor which reflects the effect of chloride binding. First, a time-dependent chloride diffusion model incorporating the introduced isotherm is established with the corresponding exact solution derived. The proposed solution is successfully validated against two existing analytical solutions and a numerical solution. Parametric study is conducted using the proposed solution to investigate how the parameters regarding chloride ingress influence the corrosion initiation time, and results show that the increase of the initial effective chloride diffusivity or the before-ingress age decreases the corrosion initiation time bringing an acceleration of a different level to chloride diffusion while the increase of the age factor prolongs the corrosion initiation time decelerating chloride diffusion to some extent, and as each of the three nonlinear chloride binding associated parameters (i.e., the deflection-point concentration, the former slope and the latter slope) increases, the corrosion initiation time rises and the retardation effect on chloride diffusion is enhanced to a various extent. Lastly, to demonstrate the practical validity of the proposed solution, the solution is applied to reproducing a reported chloride diffusion test, and it turns out that our solution reproduces the diffusion test reasonably well, justifying its practical validity.

Modelling and verification for dual time-dependent chloride diffusion of circular concrete columns in marine environment

A dual time-dependent chloride diffusion model (DTC) for circular concrete columns in a marine environment was proposed by considering the time dependency of chloride diffusion coefficient (D) and surface chloride concentration (Cs). The partial differential equation (PDF) describing the time-dependent chloride diffusion in circular concrete columns was established in the polar coordinate system firstly. Then Duhamel's theorem, variable separation method and the properties of the Bessel function were adopted to solve the PDF and to develop the DTC for circular concrete columns. Finally, the accuracy and applicability of the DTC for circular concrete columns were verified by natural diffusion tests and finite element method. The results indicated that the larger the radius of the circular concrete columns is, the lower the chloride concentration will be, and the difference in chloride concentration distribution in circular concrete columns caused by the change of radius becomes more significant with the exposure time. The chloride concentration would be overestimated if the time-dependent behavior of D or Cs were ignored. Furthermore, the effect of the time dependency of D on the chloride diffusion is more significant than that of Cs. The proposed model is applicable for various time-dependent function forms (e.g., reciprocal, exponential, power and logarithmic functions) of Cs with good applicability.

Probabilistic analysis of corrosion initiation in existing reinforced concrete structures with imprecise random field

Reasonable assessment of corrosion initiation probability in existing reinforced concrete structures is an important premise for ensuring the safety and durability of such structures. In this paper, a simplified dual time-dependent chloride diffusion model considering the load effect is established and verified by comparing its results with the experimental data. Then, a modelling method of imprecise random fields is developed. In addition, a novel probabilistic analysis method of corrosion initiation in existing RC structures with imprecise random fields is proposed. The case analysis results show that the corrosion resistance of existing structures will be overestimated without considering the effect of imprecise random fields. In comparison, the upper bound of cumulative probability of corrosion initiation is 2.29 times of that using the existing deterministic method in this case. Parametric analyses show that the imprecise random field of concrete cover (Cd) has the greatest effect on the cumulative probability of corrosion initiation, followed by initial surface chloride concentration (Cs) and chloride diffusion coefficient (Da). Compared with the interval coefficient of variation and correlation length, the cumulative probability of corrosion initiation is more sensitive to the interval mean, especially the interval mean of Cd. Additionally, the effect of interval correlation length of Cs is sequentially greater than that of Cd and Da.

Healthful choices depend on the latency and rate of information accumulation.

The drift diffusion model provides a parsimonious explanation of decisions across neurobiological, psychological and behavioural levels of analysis. Although most drift diffusion model implementations assume that only a single value guides decisions, choices often involve multiple attributes that could make separable contributions to choice. Here we fit incentive-compatible dietary choices to a multi-attribute, time-dependent drift diffusion model, in which taste and health could differentially influence the evidence accumulation process. We find that these attributes shaped both the relative value signal and the latency of evidence accumulation in a manner consistent with participants' idiosyncratic preferences. Moreover, by using a dietary prime, we showed how a healthy choice intervention alters multi-attribute, time-dependent drift diffusion model parameters that in turn predict prime-dependent choices. Our results reveal that different decision attributes make separable contributions to the strength and timing of evidence accumulation, providing new insights into the construction of interventions to alter the processes of choice.

A time-dependent chloride diffusion model for predicting initial corrosion time of reinforced concrete with slag addition

The effect of granulated blast furnace slag (GBFS) addition on the threshold chloride concentration (TCC) for rebar corrosion was investigated. A modified diffusion model, coupled with a time-dependent effective diffusion coefficient and surface chloride concentration, was proposed to predict the chloride profile. The corrosion initiation time was estimated based on the model predictions and the measured TCC. The results indicate that adding GBFS decreases the TCC by lowering the pH value of the pore solution. The evolution of corrosion potential and current density is found to obey a 3-parameter Weibull distribution. MIP tests show that adding GBFS contributes to refinement of pore structure by decreasing the fraction of large capillary pores. The time-dependent model exhibits good predictive strength and helps understand how GBFS addition delays the corrosion initiation by retarding the chloride diffusion, though a lower TCC is obtained.

A sequential seed scheduling heuristic based on determinate and latent margin for influence maximization problem with limited budget

The influence maximization problem in social networks aims to select a subset of most influential nodes, denoted as seed set, to maximize the influence diffusion of the seed nodes. The majority of existing works on this problem would ignite all the seed nodes simultaneously at the beginning of the diffusion process and let the influence diffuses passively in the network. However, it cannot depict the practical dynamics exactly of viral marketing campaigns in reality and fails to provide driving policies to control over the diffusion. In this paper, we focus on the dynamic influence maximization problem with limited budget to study the scheduling strategies including which influential node is to be seeded during the diffusion process and when to seed it at the right time. A time-dependent seed activating feedback scheme is modeled firstly by considering the time factor and its impact on the influence obligation in diffusion process. Then a scheduling heuristic based on determinate and latent margin is proposed to evaluate the marginal return of candidate nodes and activate the right seed node to promote the viral marketing. Extensive experiments on four social networks show that the proposed algorithm achieves significantly better results than a typical static influence maximization algorithm based on swarm intelligence and can improve the influence propagation under the time-dependent diffusion model comparing with the centrality-based scheduling heuristics.

TIME-DEPENDENT CHLORIDE DIFFUSION MODEL BASED ON LUMPED CONCENTRATION MATRIX AND PRECISE TIME-INTEGRATION METHOD

In order to overcome the disadvantages of traditional models which cannot balance the computational accuracy, efficiency and numerical stability by adopting the coordinated concentration matrix and the Taylor series expansion technology, a time-dependent chloride diffusion model was proposed based on the lumped concentration matrix and the precise time-integration method: The governing equation of time-varying chloride diffusion was transformed into that of equivalent constant chloride diffusion by introducing the equivalent diffusion time. a finite element model for time-varying chloride diffusion was established based on the lumped concentration matrix and the Galerkin weighted residual method. a time-dependent chloride diffusion model which is based on the lumped concentration matrix and the precise time-integration method was proposed by adopting the Pade series expansion technology. The applicability of the proposed model was verified through a comparison among the traditional finite element model, analytical model and natural exposure field data. The analysis shows that: it is more accurate to adopt the lumped concentration matrix than to adopt the traditional coordinated concentration matrix since the former can solve the numerical stability problems such as oscillations and negative values. The computational accuracy and efficiency can be improved obviously by adopting the Pade series expansion technology which requires only a small scale factor to guarantee the computational accuracy, compared with the traditional Taylor series expansion technology. The proposed model can not only balance the computational accuracy, efficiency and numerical stability, but also be insensitive to the meshed sizes of spatial grids and time steps.

Corrosion initiation life prediction of RC square piles based on a dual time-dependent bidirectional chloride diffusion model

An analytical solution for predicting the bidirectional chloride penetration in reinforced concrete (RC) square piles is proposed by considering the four sides of piles exposed to marine environments. This proposed solution considers the dual time-dependence of the surface chloride concentration and the chloride diffusion coefficient based on the Green function method. The distributions of the surface chloride concentrations for different marine zones along the altitude of square piles are introduced. The versatility of the proposed analytical solution is validated by comparison with the traditional solution considering different conditions and values of the surface chloride concentration and the chloride diffusion coefficient. Furthermore, the proposed solution is applied to predict the corrosion initiation life of RC square piles, and the main factors influencing the corrosion initiation life of piles are also analysed. The results show that the corrosion initiation life of piles in the tidal and splash zones is shortest, while that in the submerged and atmospheric zones becomes longer in turn. The study also indicates that the corrosion initiation life of RC square piles is sensitive to both the material and environmental parameters such as the concrete cover depth and the relative humidity.

Molecular dynamics simulation study on the isomerization reaction in a solvent with slow structural relaxation

• Isomerization reaction in viscous solvent was studied by MD simulation. • Slow structural relaxation of solvent is coupled to reaction dynamics. • Retardation of reaction in viscous solvent was explained by two-step model. The trans–gauche isomerization reaction of model 1,2-dichloroethane in water and ethylene glycol (EG) was studied by molecular dynamics (MD) simulation. With low barrier height, the reaction in EG was slower than that in water, and their difference decreased with increasing barrier height. Compared with the time-dependent diffusion model, in water, the effective diffusion coefficient was almost independent of time, whereas it decreased with time in EG. The trends were reproduced by Langevin dynamics simulation with a time-dependent friction coefficient from MD simulation. The effective diffusion coefficient in water agreed well with the prediction of the Grote–Hynes (GH) theory, whereas for EG, the GH theory overestimated the effective diffusion coefficient. It was suggested that the coupling with the slow structural relaxation of EG slows down the dynamics far from the transition state, which may slow down the overall reaction dynamics when the activation barrier is not high.

Locating small inclusions in diffuse optical tomography by a direct imaging method

Abstract Optical tomography is a typical non-invasive medical imaging technique, which aims to reconstruct geometric and physical properties of tissues by passing near infrared light through tissues for obtaining the intensity measurements. Other than optical properties of tissues, we are interested in finding locations of small inclusions inside the object from boundary measurements, based on the time-dependent diffusion model. First, we analyze the asymptotic behavior of the boundary measurements weighted by the fundamental solution of a backward diffusion equation as the diameters of inclusions go to zero. Then, we derive an efficient algorithm for locating small inclusions by finite boundary measurements. This algorithm is direct, simple and easy to be implemented numerically, since it only involves matrix operations and has no iteration process. Finally, some numerical results are presented to illustrate the feasibility and robustness of the algorithm. A new observation of the algorithm is that we can take the source points and test points independently and increase the resolution of numerical results by taking more test points.

Experimental investigation of the constant and time-dependent dynamic diffusion coefficient: Implication for CO2 injection method

Gas diffusion in coal is an important transport mechanism that plays a crucial role in carbon dioxide (CO2) storage in and methane (CH4) extraction from coal seams. Studies on coal diffusion are largely based on experimental work on different coal particles and the establishment of uni-pore or bidisphere theoretical models. However, problems remain in both experimental and theoretical analyses. Therefore, the initial desorption time of adsorbed gas, something that is not done using conventional experimental devices, was revealed in this study. The initial desorption time was obtained by calculating the relationship between the amount of gas collected and the theoretical amount of free gas. An improved diffusion model considering lost gas was established based on the initial desorption time of adsorbed gas. The mpdel provides diffusion coefficients for different equilibrium pressures and particle sizes on the basis of a unipore diffusion model. By correlating diffusion coefficient and time, an empirical time-dependent diffusion model was established to solve the dynamic diffusion coefficient under different conditions. Finally, in order to substantiate that the variation in diffusion length is the primary cause of the variation of diffusion coefficients with time, the variation of diffusion length with desorption time was obtained. As a result, a stepwise pressure-rasing method was proposed. This new injection method can effectively improve CO2 storage and CH4 recovery from coal seams. The study has a certain guiding significance in engineering practice.

Rheological quantification of the extent of dissolution of ultrahigh molecular weight polyethylene in melt-compounded blends with high density polyethylene

Melt compounding of ultrahigh molecular weight polyethylene (UHMWPE) with high density polyethylene (HDPE) promises to be an alternative route to prepare bimodal polyethylene grades. However, complete dissolution of UHMWPE in HDPE cannot be guaranteed during melt compounding. Indeed, in an earlier work [K. Chaudhuri et al., Polym. Eng. Sci. 59, 821–829 (2019)], it was shown that a fully entangled UHMWPE did not mix well with commercial HDPE. However, a disentangled UHMWPE (dPE) could be melt-mixed in the same HDPE as evidenced qualitatively by rheological measurements. The present work is focused on quantifying the extent of dissolution of dPE in HDPE. The proposed method involves fitting rheological models for linear viscoelasticity of entangled bimodal blends of polydisperse polymers to dynamic oscillatory shear data and extracting information on the extent of dissolved species. The time-dependent diffusion model of des Cloizeaux is used along with the theory of double reptation (DR) to describe the dynamics of polydisperse homopolymers and also to extract the molecular weight distribution of the UHMWPE sample. A quadratic mixing rule, consistent with the DR model, is used to describe the dynamics of bimodal blends. Melt-mixed dPE/HDPE blends were prepared and characterized for their linear viscoelastic response by frequency sweep tests. The blends showed complex behavior with multiple crossover points, especially for the higher content of dPE. The bimodal model was then fit to the experimental frequency sweep data to determine the only unknown parameter, namely, the extent of dissolved dPE. It was found that a considerable fraction of dPE is dissolved in HDPE during melt compounding.

Time-dependent dynamic diffusion processes in coal: Model development and analysis

Gas diffusion, a critical process during underground gas drainage and coalbed methane recovery, is viewed as the bridge connecting the coal matrix and fractures. Study on gas diffusion is of great significance to the understanding of gas migration in coal seams and the prediction of gas production. Gas diffusivity is one of the most important parameters characterizing the diffusion process. In previous studies, diffusivity was generally viewed as a constant, but the experimental results show that it is strongly dependent on diffusion time. In order to fill the gap between the existing theories and experimental results, this paper proposes a physical model for gas diffusion in coal matrix, based on which a time-dependent diffusivity model is developed from the perspective of fluid dynamics. In this model, the diffusivity of coal particles is divided into two parts, namely time-dependent diffusivity and residual diffusivity. With the continuation of the diffusion process, the diffusivity exponentially decreases with the time. Based on the time-dependent diffusivity and the classical unipore diffusion model, a time-dependent diffusion model is established. To indicate the rationality and superiority of the newly developed model, both the unipore diffusion model and time-dependent diffusion model are used to fit the experimental data, and the results show that the fitting effect of time-dependent diffusion model is obviously better than that of the unipore diffusion model. Finally, the physical nature of the time dependence of the diffusivity is discussed, and it is believed that the change of the pore pressure and its resultant alteration of the pore structure are the main reasons for the variation of diffusivity with time.

Time Dependent Analytical Diffusion Model of Air Pollutant in a Protected Zone due to Point Source with Wet Deposition

Time Dependent Analytical Diffusion Model of Air Pollutant in a Protected Zone due to Point Source with Wet Deposition

An Interface Equilibria-Triggered Time-Dependent Diffusion Model of the Boundary Potential and Its Application for the Numerical Simulation of the Ion-Selective Electrode Response in Real Systems.

A simple dynamic model of the phase boundary potential of ion-selective electrodes is presented. The model is based on the calculations of the concentration profiles of the components in membrane and sample solution phases by means of the finite difference method. The fundamental idea behind the discussed model is that the concentration gradients in both membrane and sample solution phases determine only the diffusion of the components inside the corresponding phases but not the transfer across the interface. The transfer of the components across the interface at any time is determined by the corresponding local interphase equilibria. According to the presented model, each new calculation cycle begins with the correction of the components' concentrations in the near-boundary (first) layers of the membrane and solution, based on the constants of the interphase equilibria and the concentrations established at a given time as a result of diffusion. The corrected concentrations of the components in the boundary layers indicate the start of a new cycle every time with respect to the calculations of diffusion processes inside each phase from the first layer to the second one, and so on. In contrast to the well-known Morf's model, the above-mentioned layers do not comprise an imaginary part and are entirely localized in the corresponding phases, and this allows performing the calculations of the equilibrium concentrations by taking into account material balance for each component. The model remains operational for any realistic scenarios of the electrode functioning. The efficiency and predictive ability of the proposed model are confirmed by comparing the results of calculations with the experimental data on the dynamics of the potential change of a picrate-selective electrode in nitrate solutions when determining the selectivity coefficients using the methods recommended by IUPAC.

Putting atomic diffusion theory of magnetic ApBp stars to the test: evaluation of the predictions of time-dependent diffusion models

A series of recent theoretical atomic diffusion studies has address the challenging problem of predicting inhomogeneous vertical and horizontal chemical element distributions in the atmospheres of magnetic ApBp stars. Here we critically assess the most sophisticated of such diffusion models - based on a time-dependent treatment of the atomic diffusion in a magnetised stellar atmosphere - by direct comparison with observations as well by testing the widely used surface mapping tools with the spectral line profiles predicted by this theory. We show that the mean abundances of Fe and Cr are grossly underestimated by the time-dependent theoretical diffusion model, with discrepancies reaching a factor of 1000 for Cr. We also demonstrate that Doppler imaging inversion codes, based either on modelling of individual metal lines or line-averaged profiles simulated according to theoretical three-dimensional abundance distribution, are able to reconstruct correct horizontal chemical spot maps despite ignoring the vertical abundance variation. These numerical experiments justify a direct comparison of the empirical two-dimensional Doppler maps with theoretical diffusion calculations. This comparison is generally unfavourable for the current diffusion theory, as very few chemical elements are observed to form overabundance rings in the horizontal field regions as predicted by the theory and there are numerous examples of element accumulations in the vicinity of radial field zones, which cannot be explained by diffusion calculations.

Time-Dependent Methane Diffusion Behavior in Coal: Measurement and Modeling

In-depth understanding of how methane diffuses in porous media like coal is critical for enhanced coalbed methane recovery and coal seam methane drainage. However, the classic unipore gas diffusion model can only describe methane diffusion behavior in coal at the early time diffusion stage. Describing the whole methane diffusion behavior in coal has not been reasonably addressed in the coalbed methane field. Considering the large surface-to-volume ratio of coal, the authors proposed a time-dependent gas diffusion model to describe the whole diffusion process. This work first proposed a power–law relationship between diffusion coefficient and time according to previous studies of the time-dependent gas diffusion process in porous media. Then, both the analytical solution and its approximation solution for the time-dependent gas diffusion model were derived. Six methane diffusion tests in crushed coal were conducted to validate the proposed model under different pressures (0.5, 1.5, 2.5 MPa) and temperatures (20 and \(30\,{^{\circ }}\hbox {C}\)). Modeling results show that the time-dependent gas diffusion model is superior to the classic unipore gas diffusion model for describing the whole methane diffusion process in coal. Increasing temperature always accelerates methane diffusion in coal; the higher the temperature the larger the diffusion coefficient.

Estimation and Inference of a Time-dependent Second-order Diffusion Model with Application to Financial Time Series

A specific second-order diffusion model which has been found to be promising in analyzing financial time series is further extended to incorporate the time factor into the coefficients in the drift and diffusion functions. A two-step estimation approach is proposed to estimate the time-dependent coefficients in the drift and diffusion functions. A generalized likelihood ratio test is formulated to identify whether the coefficients vary actually with time. Some simulations are conducted to assess the performance of the estimation and testing approaches. Two stock price series are analyzed to demonstrate the applications of the proposed estimation and testing methods.

The Effect of Concrete Cracks on Chloride Erosion

To explore the effect of concrete cracks on chloride erosion, different dry-wet cycle immersion experiment for bending cracking component is designed. The chloride content in concrete is tested by using the method of RCT in simulated marine environment. It is showed that the chloride convection diffusion zone depth in the cracking component is between 15mm and 25mm. The chloride erosion process is accelerated by macro-cracks. According to experiment result, a new chloride time-dependent diffusion model is put forward, considering crack width and effective diffusion coefficient. It is found that the chloride concentration is increased with the increase of crack width, approaching to the saturated concentration.