Multiscale estimation of the Freundlich adsorption isotherm

Adsorption plays an important role in water and wastewater treatment. The analysis and design of processes that involve adsorption rely on the availability of isotherms that describe these adsorption processes. Adsorption isotherms are usually estimated empirically from measurements of the adsorption process variables. Unfortunately, these measurements are usually contaminated with errors that degrade the accuracy of estimated isotherms. Therefore, these errors need to be filtered for improved isotherm estimation accuracy. Multiscale wavelet based filtering has been shown to be a powerful filtering tool. In this work, multiscale filtering is utilized to improve the estimation accuracy of the Freundlich adsorption isotherm in the presence of measurement noise in the data by developing a multiscale algorithm for the estimation of Freundlich isotherm parameters. The idea behind the algorithm is to use multiscale filtering to filter the data at different scales, use the filtered data from all scales to construct multiple isotherms and then select among all scales the isotherm that best represents the data based on a cross validation mean squares error criterion. The developed multiscale isotherm estimation algorithm is shown to outperform the conventional time-domain estimation method through a simulated example.     

Statistical properties of sediment bed profiles in alluvial channels

The objective of this study is to investigate in detail the statistical properties of series of bed elevations measured on gravel-bed and sand-bed alluvial channels, in order to identify means of quantifying bed roughness effects on streamflow. The semivariogram is used as the basic statistical method for investigating roughness properties of bed profiles obtained from field work and laboratory experiments. For sand bedforms, the semivariograms include exponential and periodic components from which can be obtained reliable measures of bedform spacing and height, as well as information on the degree of regularity of bedform arrangement. Because of the irregular nature of gravel-bed profiles, the approach in this case uses the semivariogram to investigate fractal properties of series of bed elevations to determine scales of bed roughness associated with grain sizes and small-scale bedforms and to estimate the Hausdorff dimension corresponding to each scale. These superimposed scales of roughness may be responsible for the greater flow resistance generally observed in gravel-bed rivers rather than predicted from the theoretical friction equation.     

测井多尺度分析方法用于层序地层划分研究

测井数据包含了丰富的地质信息,是研究地层多尺度沉积旋回的主要资料.本文阐述了小波变换及多尺度分析方法,探讨了测井多尺度分析方法在层序地层划分中的应用.以东营凹陷某井为例,选取Morlet小波基函数对GR测井曲线进行连续小波变换,将测井信号与深度的关系转换为与深度和尺度域的变化关系.通过研究多种伸缩尺度下小波系数曲线表现出的周期性振荡特征,并结合不同测井曲线多尺度分解后的高频信号特征,划分出各级层序界面,与传统方法所划分的界面基本一致.     

The fractal properties of geochemical landscapes as an indicator of weathering and transport processes within the Eastern Alps

Variations in surface morphology and lithology provide an opportunity to study lithologic and morphologic influences on the spatial pattern of stream-sediment geochemistry within two contrasting environments of the Eastern Alps (Hohe Tauern Range and Gurktaler Alpen Range). The fractal dimension, a measure of surface roughness over a variety of scales, is used to model the dissipation of erosive products due to climatic controlled denudation and fluvial mass transport. Based on a spatial correlation analysis, specific elemental concentrations are used as indicators for a dominant lithotype. Fractal geometry of these elements has been estimated by sequential Gaussian simulation of the area/perimeter relationship (Dal) and by the estimation of multifractal spectra. It is shown that within a 510–780 km² survey area the spatial variations of Al, Ga, Ni and Ca can be approximated by single fractals but for those of Ag and Sn multifractal models must be used. Fractal properties derived from simulated surfaces are explainable by the process controlling the spatial structure of the data. Climatic and tectonic parameters apparently influences Dal at large scales. At smaller scales rock-type variation exert an additional influence on Dal.     

Determination of Joint Roughness Coefficients Using Roughness Parameters

This study used precisely digitized standard roughness profiles to determine roughness parameters such as statistical and 2D discontinuity roughness, and fractal dimensions. Our methods were based on the relationship between the joint roughness coefficient (JRC) values and roughness parameters calculated using power law equations. Statistical and 2D roughness parameters, and fractal dimensions correlated well with JRC values, and had correlation coefficients of over 0.96. However, all of these relationships have a 4th profile (JRC 6–8) that deviates by more than ±5 % from the JRC values given in the standard roughness profiles. This indicates that this profile is statistically different than the others. We suggest that fractal dimensions should be measured within the entire range of the divider, instead of merely measuring values within a suitable range. Normalized intercept values also correlated with the JRC values, similarly to the fractal dimension values discussed above. The root mean square first derivative values, roughness profile indexes, 2D roughness parameter, and fractal dimension values decreased as the sampling interval increased. However, the structure function values increased very rapidly with increasing sampling intervals. This indicates that the roughness parameters are not independent of the sampling interval, and that the different relationships between the JRC values and these roughness parameters are dependent on the sampling interval.     

Changes in velocity profiles at roughness transitions in coarse grained channels

Heterogeneous coarse grained channels are often characterized by local transitions in bed surface roughness. Distinct spatial zones in terms of grain size have been reported, for example sand ribbons and bedload sheets. The transition from areas of finer to coarser grained surface sediment is often abrupt. However, the effects of these transitions on the shape of the velocity profile and associated shear velocity and roughness length estimates have not been investigated in detail in coarse grained channels. This paper therefore examines the combined effects of a sudden change in surface roughness and of superimposed scales of resistancé on the structure of the turbulent boundary layer. Measurements along roughness transitions from smooth to rough beds were conducted in a flume using artificial roughness features and in a natural gravel bed river. Immediately at the transition from a zone of close packed roughness to a rougher section dominated by obstacles superimposed on the more or less uniform roughness surface, boundary shear stress and roughness length increase considerably. Downstream from this transition, velocity profiles become concave upwards. Downstream and upstream sections show significant differences in terms of near bed velocities (deceleration downstream of the transition), velocity gradient and turbulence intensity of the streamwise velocity component. Comparing the mean velocity profiles corresponding to these two different roughness surfaces gives some indication of the proportion of total shear velocity (or shear stress) associated with the pressure drag produced by large and isolated obstacles.     

Investigation of natural rock joint roughness

The paper provides a comprehensive review on rock joint roughness measurement and quantification procedures. Superiority of fractal based methods over JRC, statistical parameters and statistical functions in quantifying roughness is discussed in the paper. Two of the best fractal based methodologies available in the literature, the modified 2-D divider and variogram methods, are used to quantify natural rock joint roughness in 3-D and 2-D, respectively. The capability of these two methods in accurate quantification of natural rock joint roughness is shown in the paper by applying the procedures to four natural rock joints. A good comparison has been obtained from the values obtained through the two methods. Both these methodologies have two parameters to capture the stationary roughness. The fractal dimension captures the spatial auto correlation of roughness; the other parameter captures the amplitude of roughness. Anisotropic roughness has been studied by applying two other methodologies: (a) a triangular plate methodology and (b) a light source methodology to the same four natural rock joints. A reasonably good comparison has been obtained through the results of these two methodologies. All four roughness quantification methodologies can be applied to any size of sample covering from laboratory to field scales. The results of the triangular plate and light source methodologies provided possible sliding direction values (under the gravitational loading) close to that reported in the literature for the rough discontinuity planes used in the study.     

Fractal properties of simulated bed profiles in coarse-grained channels

Bed roughness characteristics in coarse-grained channels are fairly complex. A hierarchy of roughness elements can be observed, ranging from variable particle sizes and shapes and small-scale sedimentary structures, to large-scale bedforms such as riffle-pool sequences. The effects of these scales of roughness on the flow geometry still remain to be thoroughly investigated. The semivariogram has been suggested in the past as a means of quantifying bed roughness effects on streamflow, as well as for distinguishing between scales of roughness. However, field measurements are rather time-consuming. The low number of bed profiles measured in the field precludes the identification of generally applicable relationships between the statistical properties derived from the semivariograms (such as the Hausdorff dimensions and the scale of autocorrelation corresponding to each fractal band) and the bed configuration itself (geometrical and sedimentological properties). Simulation results of gravel-bed profiles are, therefore, presented in order to complement the original investigation of Robert (1988a). The simulation experiments, based on grain characteristics of sizes and shapes and on morphological properties of small-scale bedforms, yield very significant information on boundary roughness at the microscale and give insight into the interpretation of empirical semivariograms (derived from field measurements). Bed-material sorting, variable grain shapes, and height and spacing of cluster bedforms control the fractal dimensions obtained from the semivariograms, as well as the location of the break of slope and the range of the process.     

基于粗糙度的雨海地形及撞击坑特征分析

以月球雨海地区为研究区,利用Matlab编程语言,结合分辨率为118 m的DEM数据,得出了该地区的均方根高程及均方根偏差分布图,并在研究区内选取18条剖面线,对其进行多种粗糙度参数计算。结果表明雨海地区粗糙度较大的区域主要分布在环形山和撞击坑等区域,粗糙度较小值则主要分布在被火山熔岩流填充的平原等区域,可以确定在百米尺度上环形山和撞击坑是影响雨海地形粗糙度的重要因素。对单个撞击坑而言,坑壁粗糙度最大,坑缘次之,坑底较为平坦,粗糙度最小。坑底粗糙度增大的现象可初步推断该撞击坑存在中央峰。     

Modified governing equation and numerical simulation of seepage flow in a single fracture with three-dimensional roughness

Roughness and tortuosity influence groundwater flow through a fracture. Steady flow through a single fracture can be described primitively by the well-known Cubic Law and Reynolds equation with the assumption that the fracture is made of smooth parallel plates. However, ignoring the roughness and tortuosity of the fracture will lead to inaccurate estimations of the flow rate. To obtain a more accurate flow rate through a rough fracture, this paper has derived a modified governing equation, taking into account the three-dimensional effect of the roughness. The equation modifies the Reynolds equation by adding correction coefficients to the terms of the flow rates, which are relative to the roughness angles in both the longitudinal and transverse directions. Experiments of steady seepage flow through sawtooth fractures were conducted. The accuracy of the modified equation has been verified by comparing the experimental data and the theoretical computational data. Furthermore, three-dimensional numerical models were established to simulate the steady flow in rough fractures with the triangular, sinusoidal surfaces and the typical joint roughness coefficient (JRC) profiles. The simulation results were compared with the calculation results of the modified equation and the current equations. The comparison indicates that the flow rate calculated by the modified equation is the closest to the numerical result.     

Quantification of Natural Fracture Surfaces Using Fractal Geometry

The purpose of this paper is to present an extensive evaluation of the methods to calculate the fractal dimension of natural fracture surfaces. Three methods; variogram analysis (VA), power spectral density (PSD), and roughness-length method (RMS) are applied to 2-D surface data (PSD) and 1-D profiles (VA and RMS) extracted from the surface data of 54 mm diameter crystallized limestone samples. Surface topography of the samples is quantified through a newly designed fully automated device. Before the application, self-affinity of the surface roughness and the applicability of these methods are validated using synthetically generated fractal surfaces. Fractal dimension values of the profiles are obtained as between 1 and 1.5 with a few exceptions. VA and RMS methods yield consistent fractal dimensions while the PSD values are lower than those of the other two methods. In terms of practical applicability, the VA is found more convenient than other two methods because there still exists shortcomings with the PSD and RMS methods due to difficulties in the mathematical analysis of the plots whose slopes are used in the computation of fractal dimension. However, it is observed that the data of limited size fracture surfaces are convenient for fractal analysis and the results are promising for further applications if the fracture surface size is restricted like cores recovered from deep boreholes.     

Parametric study of factors affecting fluid flow through a fracture

Understanding the flow behavior through fractures is critically important in a wide variety of applications. In many situations, the fluid flow can be highly irregular and non-linear in nature. Numerical simulation can be employed to simulate such conditions which are difficult to replicate in laboratory experiments. Therefore, a parametric study has been conducted on the fluid flow through micro-fracture over a large range of inlet pressure, fluid density, fluid viscosity, temperature, joint roughness coefficient (JRC), and fracture using finite element analysis. Irregular fracture profiles were created using Barton’s joint roughness coefficient. The Navier-Stokes (NS) equation was used to simulate the flow of water in those micro-fractures. The result showed that the fracture, fluid, and ambient conditions have a wide and varied effect on the fluid flow behavior. The interrelationship between these parameters was also studied. The model simulation provided result in the form of velocity and pressure drop profile, which can be used to determine the behavior of flow under different condition. The volumetric flow was calculated for each condition and has been plotted against the corresponding parameter to study the interrelationship.     

Correlations Developed for Estimation of Hydraulic Parameters of Rough Fractures Through the Simulation of JRC Flow Channels

The hydro-mechanical response of fractured rock masses is complex, due partly to the presence of fractures at different scales. Surface morphology has a significant influence on fluid flow behaviour of a fracture. Different empirical correlations and statistical models have been proposed to estimate the equivalent hydraulic aperture and determine the pressure drop along a fracture. However, the existing models suffer from not being adequately generalised to be applicable to a wide range of real fracture surfaces. To incorporate the effect of profile roughness in the hydro-mechanical behaviour of fractured rock masses, the joint roughness coefficient (JRC) is the most widely used empirical approach. However, the average JRC of two fracture walls in fluid flow analysis, as is a common practice, appears to be inappropriate. It will be shown how different combinations of pairs of JRCs could lead to a similar JRC value. Also, changing the position of the top and bottom walls of a fracture can significantly change the hydraulic response of the fracture while the average JRC is identical in both cases. In this paper, correlations are developed which are based on the simulation of JRCs using estimated fluid flow parameters of 2D fractures can be estimated. In order to widen the application range of the correlations, JRC flow channels were generated: these are 2D channels with their top and bottom walls being made from two of the JRC profiles. To estimate the JRC of linear profiles, a correlation developed between JRC and a newly developed Riemannian roughness parameter, D _R1, is proposed. Considering ten JRC profiles, a total of 100 JRC flow channels were generated. In order to only investigate the effect of surface roughness on fluid flow, the minimum closure between the top and bottom walls of JRC flow channels were considered to be constant. Three cases with minimum closures of 0.01, 0.05 and 0.10 cm were considered in this study. All JRC flow channels were subjected to fluid analysis using FLUENT software. Based on these results, correlations were developed between the geometrical and hydraulic properties of flow channels. Analysis of several real fractures demonstrated the applicability of these correlations.     

Underestimation of roughness in rough rock joints

Numerous studies have been made to improve Barton's shear strength model for the quantification of rock joints. However, in these previous studies, the roughness and shear strength of the rock joint have been underestimated especially for relatively high undulated profiles (joint roughness coefficient (JRC) >14). The main factors of roughness underestimation in rough rock joints are investigated for the proper quantification of rock joint roughness. The aliasing effect and the roughness characteristics are analyzed by using artificial joint profiles and natural rock joint profiles. A 3D camera scanner is adopted to verify the main source of underestimation when using conventional measurement methods. Shear strength tests are carried out by using two types of shear apparatus to study the roughness mobilization characteristics, which may also affect the roughness underestimation. The results of joint roughness assessment, such as aliasing and undulation of waviness, show that the roughness can be underestimated in relatively rough joint profiles (JRC>14). At least two components of roughness parameters are needed to properly represent the joint roughness, for example, the amplitude and the inclination angle of joint asperity. Roughness mobilization is affected by both the normal stress and the asperity scale. Copyright © 2008 John Wiley & Sons, Ltd.     

汶川地震地表破裂面形貌特征

准确描述破裂面形貌对于我们理解地震断层作用是非常重要的,破裂面的形貌特征包含许多关于地震和断层机制的有用信息。在Mw7.9 2008汶川地震中断层活动产生了两个新鲜的破裂面,八角庙破裂面和沙坝破裂面。我们使用3D便携式激光扫描仪(Tri mble GX)对两个破裂表面进行测量,在野外微观尺度上研究了破裂面形貌特征。通过能谱密度和均方值两个方法分析破裂面形貌,新鲜的破裂面表现为自相仿性,能谱密度和均方值均与剖面长度存在幂律关系。在能谱密度与空间频率的对数图中,能谱密度曲线存在明显的拐点,该拐点所对应的波长称为"特征波长",表明单一分形不能准确描述破裂面形貌。八角庙破裂面在平行滑动方向上的特征波长为7 mm,在垂直方向上特征波长略大一些(区域Ⅰ为10 mm,区域Ⅱ为9 mm);沙坝破裂面在平行滑动方向上的特征波长为8 mm,但垂直方向上特征波长略小(6 mm)。均方值曲线的最小二乘拟合直线的斜率为Hurst指数,该指数依赖于剖面线方向并描述破裂面形貌的各向异性,H指数的最小值和最大值分别与平行擦痕和垂直擦痕方向对应,这与野外断层面擦痕测量结果一致。沙坝破裂面的H指数极坐标图中存在次级H指数峰值(对应剖面线方向为85°和160°),这揭示破裂面上存在一组隐匿擦痕。该组隐匿擦痕为汶川地震之前断层活动中形成的,但这还不足以推测上一次断层活动的时间和规模。另外,通过比较新鲜节理面和破裂面表明H指数是否大于0.8反映了断层类型。在整个空间频率域上,能谱密度曲线斜率(-α)和均方值曲线斜率(H)的线性拟合关系为α=1.22+1.72H,并不严格满足两者之间理论关系式,α=1+2H。这个差异是由于测量信号噪音、破裂面的多分形性和分析方法的差异造成的。     

柴西南地区古近系-新近系井-震统一层序地层格架的建立

柴西南地区古近系—新近系是柴达木盆地最主要的油气勘探区和开发区,通过对５４口钻井和录井资料的详细层序地层学的研究,对研究区古近系－新近系的干柴沟组—狮子沟组地层中识别出了一个初次湖泛面和六个最大湖泛面,将其划分为６个三级层序和１７个准层序组,并对十条连井剖面进行了层序对比。在此基础上,对柴西南三维地震区切出了与前述的钻井连井剖面相同的连井地震剖面,并通过对地震剖面层序界限识别和划分,以及将钻井剖 面上的层序划分结果投影到连井地震剖面中等方法,对钻井剖面和地震剖面的层序划分结果进行了对比和调整,使两者完全统一,建立了柴西南区古近系－新近系井—震统一的高精度层序地层格架,这为该区开展岩性油气藏勘探奠定了基础。     

A simple kriging method incorporating multiscale measurements in geochemical survey

In this study, we propose a kriging algorithm, multiscale kriging model, to incorporate geochemical data observed at multiscales (multiresolutions). We assume that there are a number of measurements at different scales, and that the target scale at which the parameter values are needed may be different from the measurement scales. Several synthetic examples and the vanadium geochemical data from 8402 stream sediment samples in Zhejiang Province, China, have been used to illustrate the method. These examples demonstrate that, by incorporating measurements from all scales, the estimated field is better than the field estimated using measurements from any individual scale. This method also allows us to estimate a parameter field at the scale that does not have any measurements.     

An Investigation of Discontinuity Roughness Scale Dependency Using High-Resolution Surface Measurements

The influence of roughness on the hydro-mechanical behavior of rock discontinuities has long been recognized. As a result, several definitions and measures of roughness have been developed. According to the ISRM (Int J Rock Mech Min Sci Geomech Abstr 15(6):319–368, 1978), discontinuity roughness comprises large-scale (waviness) and small-scale (unevenness) components. However, the division between these scales is not clear and most investigations of surface roughness have been restricted to small fracture surfaces (<1 m²). Hence, the large-scale components of roughness are often neglected. Furthermore, these investigations typically define roughness using two-dimensional profiles rather than three-dimensional surfaces, which can lead to biased estimates of roughness. These limitations have led to some contradictory findings regarding roughness scale dependency (scale effects). This paper aims to provide some explanation of these contradictory findings. Through the in situ digitization and analysis of two adjacent large-scale (~2 × 3 m² and ~2 × 2 m²) migmatitic-gneiss fracture surfaces, the influence of sample size on roughness estimates are investigated. In addition, the influence of measurement resolution on roughness estimates is investigated by digitizing small-scale (100 × 100 mm²) samples from the same fracture with varying resolution. The findings show roughness to increase as a function of the sampling window size, in contrast to what is commonly assumed. That is, the combined waviness and unevenness of a discontinuity relative to its mean plane increases with scale. Compared to the sampling window size, the resolution of surface measurements is shown to have a far greater influence on roughness estimates. This influence of measurement resolution may explain some of the contradictory roughness scale relationships that have been published previously. It is important to note that the observed decrease in shear strength with increasing scale, as observed in many prior studies, is not being questioned; rather, a clarification of the role of roughness in this phenomenon is sought.     

Linear flow behaviour of matched joints – case study on standard JRC profiles

Roughness on rock joints produces a variable aperture across the joints and increases the flow path length. These conditions should be taken into account for a good approximation from cubic law. In this paper, the concept of local true aperture and tortuosity is applied to assumed joints where surfaces are matched to each other and correspond with standard Joint Roughness Coefficient (JRC) profiles. Furthermore, the hydraulic behaviour of JRC profiles is studied by a new laboratory experiment setup. The analytical approach provides new insights into the effects of roughness on hydraulic properties of rock joints. The results indicate that for a constant mechanical aperture, both the minimum local aperture and hydraulic aperture decrease with increasing JRC. Furthermore, tortuosity and standard deviation of local true aperture increase with JRC increment. The trend obtained between different parameters and JRC shows an obvious fluctuation for JRC lower than 10. On one hand, the results of this study along with a critical review of previous studies demonstrate that JRC profiles cannot present a precise roughness increment when JRC is less than 10. A new laboratory setup was designed to study the flow behaviour of JRC profiles. The results obtained from laboratory experiments under linear flow conditions validate the accuracy of the applied analytical method.