Fast Diffusion Limit Research Articles

Surface relaxivity estimation of coals using the cutting grain packing method for coalbed methane reservoirs

Low-field nuclear magnetic resonance (NMR) has been widely used to express the pore-size distributions of conventional and unconventional reservoirs. Based on the equation 1T2=ρFs1r, the determination of the surface relaxivity ρ can help to quantitatively convert the transverse relaxation time T2 to pore radius (r). The existing calculation of ρ using NMR requires supplementary and expensive destructive or non-destructive methods, and consolidated cylinders are required as samples to be well tested. However, the operation of multiple drilling projects and the production of many drilling cuttings are barriers to making rapid measurement of the ρ value of the unconsolidated coal cuttings challenging. In this study, an NMR grain packing method is proposed using different coal grains. The surface-to-volume ratio was transformed into the ratio of the derivative of an arbitrary pore volume to its volume, and the governing NMR equation was established by considering only the NMR porosity and T2 distribution. Sphere and cubic-like models were also constructed. The results showed that the coal with different grain sizes had different relaxation responses, and the different assemblies with the same wgrain/wgrain,c also exhibited some relaxation differences, which might be affected by the diversity in grain packing patterns and grain morphology, resulting in the variation in porosity. In addition, the water film distribution on the grain surface and the soluble minerals in the coal contribute to the variation in ρ values by influencing the relaxation properties of the voids between grains and the pores in the grains. The T2 spectrum indicated that the macropores contributed the most to the porosity, owing to the loose grain packing, and the larger grains might not strictly obey the fast diffusion limit. For the sphere-like model, the ρ values of the Ji15 and Ji17 coals were in the range of 19.47–44.04 μm/s and 36.50–46.66 μm/s, respectively; whereas for the cubic-like model, the corresponding ρ values of these two coals ranged from 24.16 μm/s to 54.64 μm/s and from 45.29 μm/s to 57.90 μm/s, respectively. Finally, different methods of ρ calculation are compared, and its potential field application is discussed.

Numerical study of NMR relaxation responses in synthetic clayey sandstone by dual-scale modeling

Nuclear Magnetic Resonance (NMR) relaxometry is a common technique for petrophysical characterization of sedimentary rocks. The standard interpretation of NMR relaxation response assumes that the fast diffusion limit is valid for the whole pore space, allowing to translate transverse relaxation components into pore apertures. However, porous media naturally exhibit multiple length scales. The diffusion between different sized pores may modify the transverse relaxation rate, weakening the relationship with corresponding pore size populations. Focusing on sandstones, we investigate the impact of diffusion coupling on transverse relaxation depending on kaolinite amount, spatial distribution and temperature. A series of synthetic clayey sandstone models with different clay amounts and morphological distributions (pore-lining, pore-filling and laminated) are generated based on a micro-CT image of an actual Bentheimer sandstone. A dual-scale random walk NMR relaxation simulation with resolved multi-porosity kaolinite models is utilized to avoid problems in near to interface exchange regions typical for effective medium representations. Simulations provide spatially resolved dynamics of magnetization exchange between different porosity populations. The results indicate that increased temperature and kaolinite clay amount with lower micro-porosity allows higher magnetization exchange between micro- and macro- porous regions. Pore-lining clay demonstrates stronger diffusional coupling effects, leading to an overestimation of micro-porosity. We further discuss the impact of diffusion coupling on NMR-estimated permeability via SDR and Coates models.

Equilibrium return times of small fluctuating clusters and vacancies.

The expected return time of a fluctuating two-dimensional cluster or vacancy to a given configuration is studied in thermodynamic equilibrium. We define a family of bond-breaking models that preserve the number of particles. This family includes edge diffusion and surface diffusion inside vacancies in the limit of fast particle diffusion and slow attachment-detachment kinetics. Within the frame of these bond-breaking models, the expected return time is found to depend on the energies of the configurations and on the energies of the excited states formed by removing a single particle from the cluster. High- and low-temperature regimes are studied. We clarify the conditions under which the return time is a nonmonotonous function of temperature: a minimum is found when the energy obtained by the average over the excited states of the configuration weighted by their attachment probabilities is lower than the energy averaged over all states. In addition, we show that the optimal temperature at which the return time is minimum is shifted to a higher temperature as compared to the temperature at which the equilibrium probability is maximum. This shift is influenced by the average curvature of the cluster edge, and is therefore larger for vacancies.

Finite-density-induced motility and turbulence of chimera solitons

We consider a one-dimensional oscillatory medium with a coupling through a diffusive linear field. In the limit of fast diffusion this setup reduces to the classical Kuramoto–Battogtokh model. We demonstrate that for a finite diffusion stable chimera solitons, namely localized synchronous domain in an infinite asynchronous environment, are possible. The solitons are stable also for finite density of oscillators, but in this case they sway with a nearly constant speed. This finite-density-induced motility disappears in the continuum limit, as the velocity of the solitons is inverse proportional to the density. A long-wave instability of the homogeneous asynchronous state causes soliton turbulence, which appears as a sequence of soliton mergings and creations. As the instability of the asynchronous state becomes stronger, this turbulence develops into a spatio-temporal intermittency.

The stability analysis of a 2D Keller–Segel–Navier–Stokes system in fast signal diffusion

This paper investigates the stability of a fully parabolic–parabolic-fluid (PP-fluid) system of the Keller–Segel–Navier–Stokes type in a bounded planar domain under the natural volume-filling hypothesis. In the limit of fast signal diffusion, we first show that the global classical solutions of the PP-fluid system will converge to the solution of the corresponding parabolic–elliptic-fluid (PE-fluid) system. As a by-product, we obtain the global well-posedness of the PE-fluid system for general large initial data. We also establish some new exponential time decay estimates for suitable small initial cell mass, which in particular ensure an improvement of convergence rate on time. To further explore the stability property, we carry out three numerical examples of different types: the nontrivial and trivial equilibriums, and the rotating aggregation. The simulation results illustrate the possibility to achieve the optimal convergence and show the vanishment of the deviation between the PP-fluid system and PE-fluid system for the equilibriums and the drastic fluctuation of error for the rotating solution.

Characterization of porous media by nuclear magnetic resonance beyond fast diffusion limit with numerical simulation

Nuclear magnetic resonance logging is an important formation detection technique in the petroleum industry, which is always used to detect formation pore structure and fluid identification. However, conventional nuclear magnetic resonance characterization methods assume that the formation pore medium are in fast diffusion regime. In complex formations, non-fast diffusion regimes (intermediate diffusion and slow diffusion) are also present. In the previous paper [13], a new method for characterizing porous media with non-fast diffusion regime and uniform pore size distribution by T2-T2 pulse sequence is proposed. In this research, the T2-T2 signals of periodic stacked pore models beyond fast diffusion regime are numerically simulated by the random walk algorithm, and the pore size - surface relaxivity maps are respectively inverted, the relationship between pore structure and correlation maps are discussed. The results provide theoretical foundation for characterizing pore structure beyond fast diffusion limit.

Machine-Learned Corrections to Mean-Field Microkinetic Models at the Fast Diffusion Limit

We present a machine learning-based formalism to correct the mean-field assumption in microkinetic models to incorporate adsorbate interactions and surface inhomogeneity at the fast diffusion limit. Lattice Monte Carlo simulations are used to compute the macroscopic reaction rate in the presence of adsorbate interaction at different values of surface coverage. This dataset is then used to train an artificial neural network to compute precise reaction rates as a function of surface coverage of intermediates, and the underlying microkinetic model of the reaction system is modified by correcting the typical mean-field rate terms with these data-driven functions. An example of CO oxidation on the square ordered lattice is used to illustrate the speed, accuracy, and robustness of this approach, vis-à-vis a full-fledged kinetic Monte Carlo simulation. In particular, we show that while the traditional mean-field model completely misses the bistability of this system under certain conditions, the neural network-modified formalism correctly captures this phenomenon. We posit that this method scales well to larger reaction systems and is a cost-effective means to improve the accuracy of differential equation-based microkinetic models.

Approximate dynamics of stochastic partial differential equations under fast dynamical boundary conditions

This work is concerned with a class of stochastic partial differential equations with a fast random dynamical boundary condition. In the limit of fast diffusion, it derives an effective stochastic partial differential equation to describe the evolution of the dominant pattern. Using the multiscale analysis and the averaging principle, it then establishes deviation estimates of the original stochastic system towards the effective approximating system. A concrete example further illustrates the result on a large time scale.

A numerical study of field strength and clay morphology impact on NMR transverse relaxation in sandstones

Nuclear magnetic resonance (NMR) relaxometry is a common technique for the petrophysical characterization of sedimentary rocks. The standard interpretation of NMR transverse relaxation responses is based on the assumptions of the fast diffusion limit, dominant surface relaxation, and weak coupling between pores, allowing an interpretation of the NMR response as pore size distribution. In general, three asymptotic relaxation regimes can be defined by comparing dephasing, structural and diffusion length scales. The shortest length scale defines the dominating relaxation regime, which is either the free diffusion, motional averaging, or localization regime. Complex mineralogy and morphology of rocks, especially due to clay minerals, may lead to the co-existence and coupling of multiple relaxation regimes in the presence of surface relaxivity heterogeneity and internal gradient effects.In this work the relationship between morphology of rocks, strength of applied magnetic field, resulting relaxation regimes and observed NMR relaxation responses is studied numerically. Limitations in the discretization of clays typical for micro-CT images are addressed by introducing a fine-scale kaolinite model based on SEM images. The model is utilized to evaluate the field-dependent mean transverse relaxation time and effective diffusion coefficient of clay regions. These effective parameters are incorporated into random-walk simulations on micro-CT images. Relaxation regimes are analysed spatially for three different types of sandstone as function of external magnetic field strength. Increase of paramagnetic clay content, field strength and echo-time increases the fraction of spins relaxing in the localization regime within macro-pores, typically being localised in the regions neighbouring clay pockets and grain surfaces. We demonstrate that a significant contribution of the localization regime to relaxation may lead to incorrect predictions of pore-size distributions and fluid typing. The prefactor of permeability correlations varies significantly due to clay effects. Accounting for these effects should enable more robust petrophysical interpretations.

Statistics of Reaction Flux and Dynamical Activity Associated with a Diffusion-Influenced Ligand-Binding Reaction.

In this paper, we consider a macromolecule with two competitive binding sites where a ligand can bind to and gives rise to a unicyclic reaction network consisting of four states-(i) a single state with both binding sites vacant, (ii) two states with one bound site and one free binding site, and (iii) an another single state with both sites occupied. We obtain probability densities of the time-integrated current along the clockwise direction and the dynamical activity or mean number of jumps between different states for finite times at a fast diffusion limit. On the other hand, in the diffusion-limited case, ligand diffusion between the two binding sites directly connects the mono-ligated states-changing the reaction scheme. Addition of the new reaction channel alters the precision of ligand occupancy to a single site, the mean dynamical activity, and the mean entropy production rate. All of these quantities are calculated with varying degrees of competition between the two sites for ligands, and we find that increase in the competition between the two sites decreases all above-mentioned additive functionals. The upper bound of precision associated with a single-site ligand occupancy for a diffusion-influenced reaction network is set by the mean dynamical activity (mean entropy production rate) at small (large) ligand concentrations at the steady-state limit.

The convergence rate of the fast signal diffusion limit for a Keller–Segel–Stokes system with large initial data

In this paper, we investigate the fast signal diffusion limit of solutions of the fully parabolic Keller–Segel–Stokes system to solution of the parabolic–elliptic-fluid counterpart in a two-dimensional or three-dimensional bounded domain with smooth boundary. Under the natural volume-filling assumption, we establish an algebraic convergence rate of the fast signal diffusion limit for general large initial data by developing a series of subtle bootstrap arguments for combinational functionals and using some maximal regularities. In our current setting, in particular, we can remove the restriction to asserting convergence only along some subsequence in Wang–Winkler and the second author (Cal. Var., 2019).

Fast-diffusion limit for reaction-diffusion equations with multiplicative noise

In this paper we consider a class of a stochastic partial differential equations with multiplicative noise and consider the limit of fast diffusion or other fast elliptic operator. We show that one can approximate solutions of the stochastic partial differential equations by the solution of a suitable stochastic ordinary differential equations on the kernel of the linear operator. We focus on equations with polynomial nonlinearities and give applications to Fisher–Kolmogorov–Petrovsky–Piscounov equation and the Ginzburg–Landau equation.

Pattern formation in a coupled membrane-bulk reaction-diffusion model for intracellular polarization and oscillations

Reaction-diffusion systems have been widely used to study spatio-temporal phenomena in cell biology, such as cell polarization. Coupled bulk-surface models naturally include compartmentalization of cytosolic and membrane-bound polarity molecules. Here we study the distribution of the polarity protein Cdc42 in a mass-conserved membrane-bulk model, and explore the effects of diffusion and spatial dimensionality on spatio-temporal pattern formation. We first analyze a one-dimensional (1-D) model for Cdc42 oscillations in fission yeast, consisting of two diffusion equations in the bulk domain coupled to nonlinear ODEs for binding kinetics at each end of the cell. In 1-D, our analysis reveals the existence of symmetric and asymmetric steady states, as well as anti-phase relaxation oscillations typical of slow-fast systems. We then extend our analysis to a two-dimensional (2-D) model with circular bulk geometry, for which species can either diffuse inside the cell or become bound to the membrane and undergo a nonlinear reaction-diffusion process. We also consider a nonlocal system of PDEs approximating the dynamics of the 2-D membrane-bulk model in the limit of fast bulk diffusion. In all three model variants we find that mass conservation selects perturbations of spatial modes that simply redistribute mass. In 1-D, only anti-phase oscillations between the two ends of the cell can occur, and in-phase oscillations are excluded. In higher dimensions, no radially symmetric oscillations are observed. Instead, the only instabilities are symmetry-breaking, either corresponding to stationary Turing instabilities, leading to the formation of stationary patterns, or to oscillatory Turing instabilities, leading to traveling and standing waves. Codimension-two Bogdanov–Takens bifurcations occur when the two distinct instabilities coincide, causing traveling waves to slow down and to eventually become stationary patterns. Our work clarifies the effect of geometry and dimensionality on behaviors observed in mass-conserved cell polarity models.

Fast-Diffusion Limit for Reaction–Diffusion Equations with Degenerate Multiplicative and Additive Noise

In the present work we consider a quite general class of reaction–diffusion equations forced by additive and multiplicative noise. When the diffusion is large, one can approximate the solutions of the stochastic reaction–diffusion equations with polynomial term by the solutions of a stochastic ordinary equations with additive noise. We illustrate our results by applying it to logistic equation and nonlinear heat equation.

Large-time asymptotics of a public goods game model with diffusion

We consider a spatially inhomogeneous public goods game model with diffusion. By utilising a generalised Hamiltonian structure of the model we study the existence of global classical solutions as well as the large time behaviour: first, the asymptotic convergence of the PDE to the corresponding ODE system is proven. This result entails also the periodic behaviour of PDE solutions in the large time limit. Secondly, a shadow system approximation is considered and the convergence of the PDE to the shadow system in the associated fast-diffusion limit is shown. Finally, the asymptotic convergence of the shadow to the ODE system is proven.

Erratum to the paper “The fast signal diffusion limit in a chemotaxis system with strong signal sensitivity”

Theorem 1.3.Let Ω be a bounded domain in with smooth boundary, and assume that χ satisfies (1.3) and (1.6). Then for all satisfying (1.5), there exist unique functions Proof.Combination of the estimate (1) and arguments similar to those in the proof of [2, Theorem 1.1], along with the uniqueness of classical solutions to the parabolic–elliptic chemotaxis system (1.2) (cf. Lemma 4.4) derives Theorem 1.3. The author is partially supported by JSPS Research Fellowships for Young Scientists (No. 17J00101).

Characterization of porous media by T2-T2 correlation beyond fast diffusion limit

Pore size distribution and surface relaxivity are two important properties of porous media such as rock samples and can be obtained by NMR methods. However, it is difficult to obtain these information beyond the fast diffusion limit. Here we present a new method to directly characterize the averaged pore size of a porous sample with a narrow pore size distribution. This method is based on the parallel plates pore model and the T2-T2 correlation sequence. The pore size (a) - surface relaxivity (ρ) correlation maps were obtained using the non-negative least squares method. Three kinds of glass bead samples were measured and the averaged pore size and surface relaxivity were extracted.

Well-posedness and fast-diffusion limit for a bulk–surface reaction–diffusion system

We analyze a certain class of coupled bulk–surface reaction–drift–diffusion systems arising in the modeling of signalling networks in biological cells. The coupling is by a nonlinear Robin-type boundary condition for the bulk variable and a corresponding source term on the cell boundary. For reaction terms with at most linear growth and under different regularity assumptions on the data we prove the existence of weak and classical solutions. In particular, we show that solutions grow at most exponentially with time. Furthermore, we rigorously derive an asymptotic reduction to a non-local reaction–drift–diffusion system on the membrane in the fast-diffusion limit.

The fast signal diffusion limit in a chemotaxis system with strong signal sensitivity

AbstractThis paper gives an insight into making a mathematical bridge between the parabolic‐parabolic signal‐dependent chemotaxis system and its parabolic‐elliptic version. To be more precise, this paper deals with convergence of a solution for the parabolic‐parabolic chemotaxis system with strong signal sensitivity urn:x-wiley:0025584X:media:mana201700270:mana201700270-math-0001to that for the parabolic‐elliptic chemotaxis system urn:x-wiley:0025584X:media:mana201700270:mana201700270-math-0002where Ω is a bounded domain in () with smooth boundary, is a constant and χ is a function generalizing urn:x-wiley:0025584X:media:mana201700270:mana201700270-math-0006In chemotaxis systems parabolic‐elliptic systems often gave some guide to methods and results for parabolic‐parabolic systems. However, the relation between parabolic‐elliptic systems and parabolic‐parabolic systems has not been studied except for the case that . Namely, in the case that Ω is a bounded domain, it still remains to analyze on the following question: Does a solution of the parabolic‐parabolic system converge to that of the parabolic‐elliptic system as ? This paper gives some positive answer in the chemotaxis system with strong signal sensitivity.

Spectral diffusion and dynamic nuclear polarization: Beyond the high temperature approximation

Dynamic Nuclear Polarization (DNP) has proven itself most powerful for the orientation of nuclear spins in polarized targets and for hyperpolarization in magnetic resonance imaging (MRI). Unfortunately, the theoretical description of some of the processes involved in DNP invokes the high temperature approximation, in which Boltzmann factors are expanded up to first order, while the high electron and nuclear spin polarization required for many applications do not justify such an approximation. A previous article extended the description of one of the mechanisms of DNP—thermal mixing—beyond the high temperature approximation (Wenckebach, 2017). But that extension is still limited: it assumes that fast spectral diffusion creates a local equilibrium in the electron spin system.Provotorov’s theory of cross-relaxation enables a consistent further extension to slower spectral diffusion, but also invokes the high temperature approximation. The present article extends the theory of cross-relaxation to low temperature and applies it to spectral diffusion in glasses doped with paramagnetic centres with anisotropic g-tensors. The formalism is used to describe DNP via the mechanism of the cross effect. In the limit of fast spectral diffusion the results converge to those obtained in Wenckebach (2017) for thermal mixing. In the limit of slow spectral diffusion a hole is burnt in the electron spin resonance (ESR) signal, just as predicted by more simple models. The theory is applied to DNP of proton and 13C spins in samples doped with the radical TEMPO.