Spatial Correlation Length Research Articles

A Stochastic Method for Modelling the Geometry of a Single Fracture: Spatially Controlled Distributions of Aperture, Roughness and Anisotropy

We describe a simple but effective stochastic method to model the void structure of a single fracture in a form of voxel representation. A fracture void is delineated by two bounding wall surfaces that are separated by some distance (i.e. the local aperture) at each location on the medial surface that lies within the fracture void and serves as a model reference frame. The three surface height fields are generated based on four parameters (mean, standard deviation and two spatial correlation lengths) for each field and two parameters (coefficient and synergistic length) for the spatial correlation between the fracture walls. Testing of generated models demonstrates that not only are the model fracture apertures spatially correlated and characterized as a Gaussian field, but also the two fracture walls are closely correlated, with a similar shape and/or height. With respect to fracture apertures, three quantities, i.e. the mean aperture, roughness and anisotropy, can be derived from the fracture models to describe fracture morphology. The effect of model fractures on fluid flow is investigated in order to establish the relationship between fracture permeability and the three morphological quantities, revealing a way to avoid the significant estimation error associated with the use of the cubic law.

Redistribution of local fracture aperture and flow patterns by acidizing

Statistical and spatial distributions of apertures may be changed due to chemical dissolution. The chemical dissolution in rocks may result from the passage of acidic water through fractures or acid fracturing treatment. In this study, an acid injection cell was developed to simulate acidizing stimulation into fractures and investigate fractures surfaces etching and aperture evolution due to acidizing. Results showed that initial hydraulic and mechanical apertures and permeability (under zero stress) were increased due to fracture acidizing and a varying range of 2%–12% contact area was observed. Apertures before and after fracture acidizing followed a log-normal distribution and all of statistical moments were increased by acid injection while the calculated coefficient of variance in both arithmetic and logarithmic scales was decreased after acidizing. Correlation length increased in the direction of acid flow (x-direction) after acidizing. Results of fluid flow modeling using a developed finite element code showed a channeling path in the x-direction. Furthermore, using fast sequential simulation algorithm, apertures with various spatial correlation length in x and y directions were generated. Numerical modeling results of these generated aperture patterns showed that flow channelizing was increased by increasing of correlation length in the x-direction. Fluid flow rate was increased with increasing of correlation length in x-direction, while it decreased with increasing correlation length in y-direction.

The effects of spatially correlated heterogeneities on acidising wormhole development

A commonly used rock stimulation technique in subsurface geoenergy technologies is matrix acidisation. During this process, an acidic solution is injected into the hydrocarbon or geothermal reservoirs to dissolve certain minerals and enhance the injectivity and productivity of the hydrocarbon or heat recovery operation. This study aims to investigate the influence of three-dimensional (3D) lithological heterogeneity of the rock on reactive transport of acid in the porous domain. Through a development and application of a reactive transport model with restricting assumptions (linear calcite dissolution kinetics with hydrochloric acid (HCl)) and simplifications (mineralogical homogeneity) on the geochemistry embedded into the model, we present the results of a series of 3D simulation of carbonate acidisation in presence of varying spatial correlation lengths of petrophysical properties of the domain. The results are compared with the case where petrophysical properties are distributed randomly throughout the domain. The study provides new insights into the impact of increasing correlation lengths that lack in the existing literature. Under the conditions of case studies, the pore volume of the acid injected reduces 10·5 and 12·2% for Damköhler number of 100, when correlation lengths of only 0·0667 and 0·1667 cm are considered in core of length 5 cm.

Inverse percolation by removing straight rigid rods from square lattices in the presence of impurities

Numerical simulations and finite-size scaling analysis have been carried out to study the problem of inverse percolation by removing straight rigid rods from square lattices contaminated with non-conducting impurities. The presence of impurities provides a more realistic approach to the deposited monolayer, which usually presents inhomogeneities due to the irregular arrangement of surface and bulk atoms, the presence of various chemical species, etc. The process starts with an initial configuration, where all lattice sites are occupied by an impurity (with a concentration ) or a conducting particle (with a concentration ). Then, the system is diluted by randomly removing linear k-mers (linear clusters of k consecutive conducting particles) from the lattice. The impurities remain fixed in its position and cannot be removed. The central idea of this paper is based on finding the maximum concentration of conducting sites (minimum concentration of empty sites) for which the connectivity disappears. This particular value of the concentration is called inverse percolation threshold, and determines a well-defined geometrical phase transition in the system. On the other hand, the inverse jamming coverage is the coverage of the limit state, in which no more objects can be removed from the lattice due to the absence of linear clusters of nearest-neighbour sites of appropriate size. The dependence of percolation and jamming thresholds on the concentration of defects was investigated for different values of k, ranging from 1 to 120. The obtained results show that the behaviour of the system is significantly affected by the presence of impurities. In addition, the nature of the jamming and percolation transitions was studied. In the first case, the corresponding spatial correlation length critical exponent was measured, being . This value coincides with previous calculations of this exponent for the standard random sequential adsorption of linear k-mers on square lattices. Critical exponents were also calculated for the percolation phase transition, showing that the universality class corresponding to ordinary percolation is preserved regardless the values of k and considered.

Spontaneous Spatial Correlation of Elastic Modulus in Jammed Epithelial Monolayers Observed by AFM

For isolated single cells on a substrate, the intracellular stiffness, which is often measured as the Young’s modulus, E, by atomic force microscopy (AFM), depends on the substrate rigidity. However, little is known about how the E of cells is influenced by the surrounding cells in a cell population system in which cells physically and tightly contact adjacent cells. In this study, we investigated the spatial heterogeneities of E in a jammed epithelial monolayer in which cell migration was highly inhibited, allowing us to precisely measure the spatial distribution of E in large-scale regions by AFM. The AFM measurements showed that E can be characterized using two spatial correlation lengths: the shorter correlation length, lS, is within the single cell size, whereas the longer correlation length, lL, is longer than the distance between adjacent cells and corresponds to the intercellular correlation of E. We found that lL decreased significantly when the actin filaments were disrupted or calcium ions were chelated using chemical treatments, and the decreased lL recovered to the value in the control condition after the treatments were washed out. Moreover, we found that lL decreased significantly when E-cadherin was knocked down. These results indicate that the observed long-range correlation of E is not fixed within the jammed state but inherently arises from the formation of a large-scale actin filament structure via E-cadherin-dependent cell-cell junctions.

Revisiting the structure–property relationship of metallic glasses: Common spatial correlation revealed as a hidden rule

What determines the glass property remains one of the major unsolved problems in both condensed matter physics and materials science. Despite extensive research attempting to identify possible structural features as property signatures in glasses obeying the conventional philosophy of ``structure determines property'' in matter, the hidden rule about why some proposed structures predict properties effectively but others do not is still poorly understood. Here we revisit some earlier proposed successful ``structural descriptors'' in glasses, e.g., vibrational mean-squared displacement, flexibility volume, participation fraction of low-frequency vibrational modes, and two-body excess entropy, correlating them with the long-time dynamic property of model glass probed via the activation energy for local structural excitation. We find that all four structural descriptors correlate strongly with the activation energy, presenting large Pearson's correlation coefficients. By examining the spatial nature of the activation energy and the structural descriptors, a common rule for the robustness of structure--property relationships in glasses is established, according to which there exists a critical characteristic correlation length. We further demonstrate the concept that complex structures determine the glass property by manipulating the cutoff distance used to define the two-body excess entropy. Only if this structural descriptor is defined involving atoms beyond the first nearest neighbor does it reproduce the feature of common spatial correlation range in glass. The presence of a common spatial correlation length strongly indicates that it is necessary to include the spatial correlation of a complex structure accounting for the dynamic property of metallic glasses.

Effects of Understory Shrub Biomass on Variation of Soil Respiration in a Temperate-Subtropical Transitional Oak Forest

Quantification of the temporal and spatial variations of soil respiration is an essential step in modeling soil carbon (C) emission associated with the spatial distribution of plants. To examine the temporal and spatial variations of soil respiration and its driving factors, we investigated soil respiration, microclimate, and understory vegetation in a 50 m × 70 m plot in a climatic transitional zone oak forest in Central China. The temporal variation of soil respiration based on the 21 measurements ranged from 15.01% to 30.21% across the 48 subplots. Structural equation modeling showed that soil temperature and understory shrub biomass had greater positive effects on the seasonal variability of soil respiration. The spatial variation of soil respiration of the 48 subplots varied from 3.61% to 6.99% during the 21 measurement campaigns. Understory shrub biomass and belowground fine root biomass positively regulated the spatial variation of soil respiration. Soil respiration displayed strong spatial autocorrelation, with an average spatial correlation length of 20.1 m. The findings highlight the importance of understory shrub and belowground biomass in regulating the temporal and spatial heterogeneity of soil respiration in forest ecosystems, and the need to carefully address it to robustly estimate the contribution of soil C emission in terrestrial C cycling.

Worst-case spatial correlation length in probabilistic slope stability analysis

This note investigates the reliability of undrained slopes by the random finite-element method. The focus of this note is on the worst-case spatial correlation length, namely the correlation length at which the probability of slope failure reaches a maximum. It is shown that the worst-case phenomenon is most pronounced when the mean factor of safety is relatively low or the coefficient of variation is relatively high. Knowledge of the worst-case spatial correlation length is valuable, as it can be employed for conservative design in the absence of substantial soil field data.

Statistical parameters of seismicity induced by the impoundment of the Three Gorges Reservoir, Central China

Induced seismicity has been observed in the Three Gorges Reservoir (TGR) area, central China, since its impoundment in May, 2003. Data from a local seismic network in the TGR area and a temporal network allow us to obtain a detailed picture of the spatio-temporal distribution of earthquakes in the Badong area, in which earthquakes are clustered. Statistical parameters, namely the seismic b-value, spatial correlation length, and fractal dimension of the hypocentre distribution, as an integrated set, are investigated and taken as indicators to divide the seismicity in the study area into typical stages. Epidemic-type aftershock sequence modelling, which is useful for extracting possible fluid signals from earthquake occurrences, is applied to the data set. The temporal variations in the parameters indicate a long-term increasing and diffusing stress until the present. Thus, much more attention should be given to seismic activity in the TGR area. The largest earthquake (M5.1), which was followed by aftershocks, a large fraction of which were Omori-type, primarily resulted from a triggered fault movement. In the M4.3 earthquake sequence, both Omori-type seismic triggering and reservoir triggering are statistically important. After the Omori-type aftershocks (which accounted for a small fraction (<20% in total) of earthquakes) are removed, the forced seismic rate is correlated with the water level of the reservoir, indicating that most of the small earthquakes were induced by the reservoir.

Iterative filter based estimation of fully 3D heterogeneous fields of permeability and Mualem-van Genuchten parameters

The accurate modeling of flow and transport in the vadose zone for agricultural and environmental applications requires knowledge about soil parameters. Soil parameters vary in space depending on soil texture and structure. In the present synthetic study we considered spatial variation of permeability (k), inverse of capillary entry pressure head (αVG) and exponent (n) of the Mualem-van Genuchten model. The iterative Ensemble Kalman filter (IEnKF) can estimate the spatially variable soil parameters if measurements of water saturation at different locations and times are available. We used as input daily precipitation data from the Berambadi catchment (southern India). We first considered that the parameters vary horizontally but are constant in the vertical direction. In this case log (k) and log (αVG) can be estimated satisfactorily with 30%–40% reduction of RMSE (compared to open loop runs), if the initial guess of the spatial correlation lengths of the heterogeneous fields is equal to or larger than the unknown, true values. The estimation of exponent n is poorer as the reduction of RMSE is just 20%. If vertical heterogeneity of the parameters is considered the estimation of log (k) and log (αVG) is only improved for the upper 1.5 m and estimation of n is not improved. We also demonstrate that the estimation problem can be simplified when flow in the unsaturated zone is predominantly vertical. If in this case soil hydraulic parameters are estimated with IEnKF at measurement locations and afterwards interpolated with kriging, results are produced with a similar quality as with 3D-IEnKF.

Detection of colon cancer stages via fractal dimension analysis of optical transmission imaging of tissue microarrays (TMA)

At epidemic proportions worldwide, one in four persons now has cancer, and this statistic will change to one in two persons in near future. An important step in the fight against cancer is its early and accurate detection. This calls for affordable, quick and easy diagnostic methods. Standard pathological detection of cancer involves microscopic examination of morphological changes using stained biopsy samples, but this method is prone to human error and misdiagnosis. A tissue is a spatially heterogeneous medium with fractal properties owing to self-similarity in mass distribution. With the progress of cancer, this tissue heterogeneity changes owing to more mass accumulation and rearrangement of intracellular macromolecules like DNA, RNA, and lipids. Recently, commercially available tissue microarray (TMA) samples have gained significant attention in research studies, as the array of numerous tissue samples of different cases of a disease on a glass slide allows ease of conducting comprehensive studies. The present study uses optical transmission imaging to analyze the fractal dimension of colon TMA samples of 5 μm thickness and 1.5 mm diameter to correctly distinguish different stages of colon cancer. Results of this specialized analysis are also supported by entropy and spatial correlation length analysis. The application of this method of cancer diagnostics is discussed.

Characterisation of the spatial variability of material properties of Gilsocarbon and NBG-18 using random fields

Graphite is a candidate material for Generation IV concepts and is used as a moderator in Advanced Gas-cooled Reactors (AGR) in the UK. Spatial material variability is present within billets causing different material property values between different components. Variations in material properties and irradiation effects can produce stress concentrations and diverse mechanical responses in a nuclear reactor graphite core. In order to characterise the material variability, geostatistical techniques called variography and random field theory were adapted for studying the density and Young's modulus of a billet of Gilsocarbon and NBG-18 graphite grades. Variography is a technique for estimating the distance over which material property values have significant spatial correlation, known as the scale of fluctuation or spatial correlation length. The paper uses random field theory to create models that mimic the original spatial and statistical distributions of the original data set. This study found different values of correlation length for density and Young's modulus around the edges of a Gilsocarbon billet, while in the case of NBG-18, similar correlation lengths where found across the billet. Examples of several random fields are given to reproduce the spatial patterns and values found in the original data.

Extensions to the Groningen ground-motion model for seismic risk calculations: component-to-component variability and spatial correlation

A bespoke ground-motion model has been developed for the prediction of response spectral accelerations, peak ground velocity and significant duration due to induced earthquakes in the Groningen gas field in the Netherlands. For applications to the calculation of risk to the exposed building stock, extensions to the model are required. The use of the geometric mean horizontal component in the ground-motion predictions and the arbitrary horizontal component for the building fragility functions requires the addition of component-to-component variability. A model for this variability has been developed that both reflects the strong horizontal polarisation of motions observed in many Groningen records obtained at short distances and the fact that the strong polarisation is unlikely to persist at larger magnitudes. The other extension of the model is the spatial correlation of ground motions for the calculation of aggregated risk, which can be approximated through simple rules for sampling the variance within site response zones. Making use of ground-motion recordings from several networks in the field and the results of finite difference waveform simulations, a Groningen-specific spatial correlation model has been developed. The new model also combines results from traditional variogram fitting approaches with a new method to infer spatial correlation lengths from observed variance reduction. The development of the new spatial correlation model relaxes the need to approximate spatial correlation through the sampling of site response, although the results obtained herein suggest that similar results could be obtained using either approach. The preliminary consideration of the numerical waveform modelling results in this study paves the way for significant extensions to be made for the modelling of spatial correlations and the decomposition of apparent spatial variability into systematic and random components within a fully non-ergodic framework .

A Geometric Framework for Stochastic Shape Analysis

We introduce a stochastic model of diffeomorphisms, whose action on a variety of data types descends to stochastic evolution of shapes, images and landmarks. The stochasticity is introduced in the vector field which transports the data in the large deformation diffeomorphic metric mapping framework for shape analysis and image registration. The stochasticity thereby models errors or uncertainties of the flow in following the prescribed deformation velocity. The approach is illustrated in the example of finite-dimensional landmark manifolds, whose stochastic evolution is studied both via the Fokker–Planck equation and by numerical simulations. We derive two approaches for inferring parameters of the stochastic model from landmark configurations observed at discrete time points. The first of the two approaches matches moments of the Fokker–Planck equation to sample moments of the data, while the second approach employs an expectation-maximization based algorithm using a Monte Carlo bridge sampling scheme to optimise the data likelihood. We derive and numerically test the ability of the two approaches to infer the spatial correlation length of the underlying noise.

Probabilistic analysis of the bearing capacity of inclined loaded strip footings near cohesive slopes

ABSTRACT The bearing capacity of strip footings has been widely investigated using deterministic methods . In practise, soil properties are spatially variable, which makes the use of the probabilistic approach seem more reliable. In this article, a probabilistic study using finite element limit analysis, combined with random field theory, through OptumG2 code is carried out to evaluate the undrained bearing capacity of an inclined loaded strip footing adjacent to a slope. This article presents a new estimation of the probabilistic parameters effect on the mean bearing capacity factors μNc , and the mean failure envelopes, by taking into account the combination of load inclination and sloping ground. It is found that the coefficient of variation CovSu and the spatial correlation length Θ of soil shear strength have a significant effect on the undrained bearing capacity of the strip footing, in which the probabilistic analysis gives conservative values compared to the deterministic one.

Analysis of Rock Mass Stability Based on Mining-Induced Seismicity: A Case Study at the Hongtoushan Copper Mine in China

In deep metal mines, seismic hazards (for example, rock bursts and roof collapses) occur easily due to the influence of complicated geological conditions, high-stress environments and strong blast disturbances. A microseismic (MS) monitoring technique was used to evaluate rock mass stability in the Hongtoushan copper mine in China. The changes in the multiple MS parameters, including the apparent volume, energy index, spatial correlation length, fractal dimension and b value, during the mining process were presented. The results showed that the MS sequences decayed following Omori’s power-law and that the rock mass returned to a relatively stable state after approximately 20 days of blasting. After each stoping, the proportion of large-scale fractures increased, the MS events become more concentrated, and the long-range correlations of the rock mass weakened. These changes caused b value, fractal characteristics and spatial correlation length to decrease significantly. Although the risk of a large-scale rock mass failure was reduced, the risk of a local failure increased. As the mining process continued, stress and deformation gradually increased, and the areas with concentrated stress and large deformation were not consistent. Control measures for hazards should be implemented according to the corresponding occurrence mechanisms as determined from the differences in the rock mass physics mechanics.

Numerical Study of Spatial Behavior of Solute Particle Transport in Single Fracture with Variable Apertures

The aim of this study is to numerically analyze spatial behaviors of solute particle transport in a single fracture with spatially correlated variable apertures under application of effective normal stress conditions. The numerical results show that solute particle transport in a single fracture is strongly affected by spatial correlation length of variable apertures and applied effective normal stress. As spatial correlation length increases, mean residence time of solute particles decreases and tortuosity and Peclet number (a dimensionless number representing the relationship between the rate of advection of solute particles by the flow and the rate of diffusion of solute particles) also decreases. These results indicate that the geometry of the aperture distribution is favorable to solute particle transport when the spatial correlation length is increased. However, as effective normal stress increases, the mean residence time and tortuosity tend to increase but the Peclet number decreases. The main reason for a decreasing Peclet number is that the solute particle is transported by one or two channels with relatively higher localized flow rates owing to increase in contact areas resulting from increasing effective normal stress. Based on the numerical results of the solute particle transport produced in this study, an exponential-type correlation formula between the mean residence time and the effective normal stress is proposed.

Customizing speckle intensity statistics

We develop a general method for customizing the intensity statistics of speckle patterns on a target plane. By judiciously modulating the phase-front of a monochromatic laser beam, we experimentally generate speckle patterns with arbitrarily-tailored intensity probability-density functions. Relative to Rayleigh speckles, our customized speckles exhibit radically different topologies yet maintain the same spatial correlation length. The customized speckles are fully developed, ergodic, and stationary: with circular non-Gaussian statistics for the complex field. Propagating away from the target plane, the customized speckles revert back to Rayleigh speckles. This work provides a versatile framework for tailoring speckle patterns with varied applications in microscopy, imaging and optical manipulation.

Vertical spatial correlation length based on standard penetration tests

ABSTRACTThe spatial correlation length (SCL), or the scale of fluctuation, is a parameter for describing the spatial variability of soil and one of the important parameters used in random field theory. Studies reporting the spatial correlation length based on real field data of offshore/nearshore sea bottom soils are rather limited in the literature, so in this study, the vertical spatial correlation length is determined using site investigation data from two sites of the southern coast of Turkey. Based on quite extensive data, the vertical spatial correlation length is estimated using four different autocovariance functions. The values are within typical ranges reported in the literature for similar soil groups, both onshore and offshore. It is also noted that the widely-used exponential function almost always gives the lowest value of spatial correlation length. The results of this study add to the database of spatial correlation lengths based on real data and could be useful for future studies on reliability assessment of offshore foundations using random finite element method.

Spatial correlation of elastic heterogeneity tunes the deformation behavior of metallic glasses

Metallic glasses (MGs) possess remarkably high strength but often display only minimal tensile ductility due to the formation of catastrophic shear bands. Purposely enhancing the inherent heterogeneity to promote distributed flow offers new possibilities in improving the ductility of monolithic MGs. Here, we report the effect of the spatial heterogeneity of elasticity, resulting from the inherently inhomogeneous amorphous structures, on the deformation behavior of MGs, specifically focusing on the ductility using multiscale modeling methods. A highly heterogeneous, Gaussian-type shear modulus distribution at the nanoscale is revealed by atomistic simulations in Cu64Zr36 MGs, in which the soft population of the distribution exhibits a marked propensity to undergo the inelastic shear transformation. By employing a mesoscale shear transformation zone dynamics model, we find that the organization of such nanometer-scale shear transformation events into shear-band patterns is dependent on the spatial heterogeneity of the local shear moduli. A critical spatial correlation length of elastic heterogeneity is identified for the simulated MGs to achieve the best tensile ductility, which is associated with a transition of shear-band formation mechanisms, from stress-dictated nucleation and growth to structure-dictated strain percolation, as well as a saturation of elastically soft sites participating in the plastic flow. This discovery is important for the fundamental understanding of the role of spatial heterogeneity in influencing the deformation behavior of MGs. We believe that this can facilitate the design and development of new ductile monolithic MGs by a process of tuning the inherent heterogeneity to achieve enhanced ductility in these high-strength metallic alloys.