Time–space fractional governing equations of transient groundwater flow in confined aquifers: Numerical investigation

In order to model non‐Fickian transport behaviour in groundwater aquifers, various forms of the time–space fractional advection–dispersion equation have been developed and used by several researchers in the last decade. The solute transport in groundwater aquifers in fractional time–space takes place by means of an underlying groundwater flow field. However, the governing equations for such groundwater flow in fractional time–space are yet to be developed in a comprehensive framework. In this study, a finite difference numerical scheme based on Caputo fractional derivative is proposed to investigate the properties of a newly developed time–space fractional governing equations of transient groundwater flow in confined aquifers in terms of the time–space fractional mass conservation equation and the time–space fractional water flux equation. Here, we apply these time–space fractional governing equations numerically to transient groundwater flow in a confined aquifer for different boundary conditions to explore their behaviour in modelling groundwater flow in fractional time–space. The numerical results demonstrate that the proposed time–space fractional governing equation for groundwater flow in confined aquifers may provide a new perspective on modelling groundwater flow and on interpreting the dynamics of groundwater level fluctuations. Additionally, the numerical results may imply that the newly derived fractional groundwater governing equation may help explain the observed heavy‐tailed solute transport behaviour in groundwater flow by incorporating nonlocal or long‐range dependence of the underlying groundwater flow field.     

A critical state model for sands dependent on stress and density

An elastoplastic model for sands is presented in this paper, which can describe stress–strain behaviour dependent on mean effective stress level and void ratio. The main features of the proposed model are: (a) a new state parameter, which is dependent on the initial void ratio and initial mean stress, is proposed and applied to the yield function in order to predict the plastic deformation for very loose sands; and (b) another new state parameter, which is used to determine the peak strength and describe the critical state behaviour of sands during shearing, is proposed in order to predict simply negative/positive dilatancy and the hardening/softening behaviour of medium or dense sands. In addition, the proposed model can also predict the stress–strain behaviour of sands under three-dimensional stress conditions by using a transformed stress tensor instead of ordinary stress tensor. Copyright © 2004 John Wiley & Sons, Ltd.     

On the practice of estimating fractal dimension

Coastlines epitomize deterministic fractals and fractal (Hausdorff-Besicovitch) dimensions; a divider [compass] method can be used to calculate fractal dimensions for these features. Noise models are used to develop another notion of fractals, a stochastic one. Spectral and variogram methods are used to estimate fractal dimensions for stochastic fractals. When estimating fractal dimension, the objective of the analysis must be consistent with the method chosen for fractal dimension calculation. Spectal and variogram methods yield fractal dimensions which indicate the similarity of the feature under study to noise (e.g., Brownian noise). A divider measurement method yields a fractal dimension which is a measure of complexity of shape.     

Influence of Structural Non-Stationarity of Surface Roughness on Morphological Characterization and Mechanical Deformation of Rock Joints

Summary Structural non-stationarity of surface roughness affects accurate morphological characterization as well as mechanical behaviour of rock joints at the laboratory scale using samples with a size below the stationarity threshold. In this paper, the effect of structural non-stationarity of surface roughness is investigated by studying the scale dependence of surface roughness and mechanical behaviour of rock joints. The results show that the structural non-stationarity mainly affects the accurate characterization of the surface roughness of the fracture samples. It also controls the amount and location of the contact areas during shear tests, which in turn affects the mechanical properties and asperity degradation of the samples. It is concluded that for accurate determination of the morphological and mechanical properties of rock joints at laboratory and field scales, samples with size equal to or larger than the stationarity threshold are required. Author’s address: Nader Fardin, Rock Mechanics Group, Department of Mining Engineering, Faculty of Engineering, University of Tehran, P.O. Box: 11365/4563, Tehran, Iran     

Fractal Determination of Coastal Types Based on GIS--A Case Study of China

The probability of fractal determination of coastal types based on GIS is preliminarily discussed with China as an example. Finally, some significant conclusions are drawn: (1) The fractal dimension of coastline of the bedrock coast is larger than that of the plain coast on the same scale map; (2) As far as the bedrock coast is concerned, the larger fractal dimension of coastline of the bedrock coast on the same scale map indicates that the bedrock coast is probably not typical; (3) As far as the plain coast is concerned, the smaller fractal dimension of coastline of the plain coast on the same scale map indicates that it is probably the silt plain coast; (4) The different substantial compositions affect the fractal dimensions of coastlines of different coastal types. In general, the coast which lies in the north of the Hangzhou Bay consists of sand mainly, its surface is flat, and it is connected with the coastal plain, its landform is broad shoal, its total change is comparatively homogenous in the tidal dynamic process, and thus, the relatively smaller fractal dimension of coastline results from this. For the bedrock coast, there is more bedrock, the coastline is comparatively smooth and straight, being affected by the faults and ocean dynamic process, which result in the larger fractal dimension.     

富钴结壳分布的分形特征

利用我国在西太平洋国际海底两个海山获得的富钴结壳资料进行分形研究后发现，富钴结壳的分布具有变维分形分布的特点，结壳丰度呈二阶累计和分形分布，结壳厚度呈多阶不定累计和分形分布，通过计算得到了它们各自的分维数，进而对两个海山的结壳资源进行了初步评价。     

Fractal Dimension and Fractals in Ocean Engineering

- This paper discusses the application of fractal dimension and fractals in ocean engineering. To handle some ocean environment problems, the existing fractal method, in which the fractal dimension is a constant, can be used. For some complicated problems in ocean engineering, this paper presents the concept of the variable dimension fractals (D = f(r)), i. e., the fractal dimension D is the function of characteristic scale r instead of a constant. By using variable dimension fractals, several deformation and stress states of offshore structures are described.     

Optimizing interpolation of shoot density data from a Posidonia oceanica seagrass bed

A case study on the optimization of Posidonia oceanica density interpolation, using a data set from a large meadow at Porto Conte Bay (NW Sardinia, Italy), is presented. Ordinary point kriging, cokriging and a weighted average based on inverse square distance were used to interpolate density data measured in 36 sampling stations. The results obtained from different methods were then compared by means of a leave‐one‐out cross‐validation procedure. The scale at which interpolation was carried out was defined on the basis of the Hausdorff dimension of the variogram. Optimizing spatial scale and data points search strategy allowed obtaining more accurate density estimates independently of the interpolation method.     

相空间反演方法在表层水温预报中的应用

利用相空间理论及方法对渤、黄、东海共4个站位近十几年的旬平均SST进行分析。结果表明：表层水温具有混饨特性，其吸引子关联维数平均约为1．23、嵌入相空间维数为6（渤、黄海）和7（东海178号站位）、二阶Renyi熵平均约为3．7×10-4（1／d）及平均可预报时间尺度平均为27个点；基于以上分析结果运用相空间反演方法建立了旬平均SST的反演模型，并且在试预报的前5旬的最大相对误差约为4．2％。     

广西沿岸海平面变化的关联维数研究

采用相空间重构和关联维数方法，研究了广西沿岸月平均海平面变化的分形特征，结果表明，相空间重构的延迟时间τ为3△t，当关联维数趋于饱和时，石头埠，北海和龙尾的相空间维数为7，而涠洲岛为9，它们的关联维数平均值平均值分别为6.6507,6.5414,6.7059.