Combining stochastic facies and fractal models for representing natural heterogeneity

Sedimentary deposits are often characterized by various distinct facies, with facies structure relating to the depositional and post-depositional environments. Permeability (k) varies within each facies, and mean values in one facies may be several orders of magnitude larger or smaller than those in another facies. Empirical probability density functions (PDFs) of log(k) increments from multi-facies structures often exhibit properties well modeled by the Levy PDF, which appears unrealistic physically. It is probable that the statistical properties of log(k) variations within a facies are very different from those between facies. Thus, it may not make sense to perform a single statistical analysis on permeability values taken from a mix of distinct facies. As an alternative, we employed an indicator simulation approach to generate large-scale facies distributions, and a mono-fractal model, fractional Brownian motion (fBm), to generate the log(k) increments within facies. Analyses show that the simulated log(k) distributions for the entire multi-facies domain produce apparent non-Gaussian log(k) increment distributions similar to those observed in field measurements. An important implication is that Levy-like behavior is not real in a statistical sense and that rigorous statistical measures of the log(k) increments will have to be extracted from within each individual facies. Electronic Publication     

近50年济南岩溶泉域地下水位对降水响应的时滞差异

依据近50年降水及年均水位等的差异,将1959-2011年济南岩溶泉域地下水位与降水观测数据划分为若干时段。采用小波变换方法定量分析了泉域地下水位对降水的响应时滞,并采用相关分析讨论了时滞与地下水开采量等的关系。结果表明,(1)泉域地下水位与降水的主周期均为1 a。(2)两种划分方式下,泉域排泄区水位时滞分别为83.44~161.24 d、88.62~150.56 d,径流区水位时滞分别为67.87~81.66 d、76.58~82.21 d,径流区水位时滞明显小于排泄区。(3)依据降水划分的时段,排泄区与径流区均有随年降水减少,水位时滞增加的趋势。(4)依据水位划分的时段,排泄区有随水位降低,时滞增加的趋势,而径流区不明显。(5)泉域水位时滞与泉域地下水开采量、泉群流量等存在相关关系,泉域开采量越大,泉群流量越小,则水位时滞相对越大。上述时滞差异分析,有助于增加地下水位与降水之间的非线性耦合过程的认识,可为地下水位的预报预警提供帮助。     

Stochastic Modeling of Variably Saturated Transient Flow in Fractal Porous Media

This work presents the application of a Monte Carlo simulation method to perform an statistical analysis of transient variably saturated flow in an hypothetical random porous media. For each realization of the stochastic soil parameters entering as coefficients in Richards' flow equation, the pressure head and the flow field are computed using a mixed finite element procedure for the spatial discretization combined with a backward Euler and a modified Picard iteration in time. The hybridization of the mixed method provides a novel way for evaluating hydraulic conductivity on interelement boundaries. The proposed methodology can handle both large variability and fractal structure in the hydraulic parameters. The saturated conductivity K _s and the shape parameter _vg in the van Genuchten model are treated as stochastic fractal functions known as fractional Brownian motion (fBm) or fractional Gaussian noise (fGn). The statistical moments of the pressure head, water content, and flow components are obtained by averaging realizations of the fractal parameters in Monte Carlo fashion. A numerical example showing the application of the proposed methodology to characterize groundwater flow in highly heterogeneous soils is presented.     

剪切带倾角尺度律与局部化启动跳跃稳定研究

研究了剪切带倾角是如何依赖于岩样高度以及剪切带的不稳定性。建立了准脆性材料试件剪切带倾角尺度效应模型,得到了剪切带倾角尺度效应的解析解,且与实验结果比较相符。研究结果表明:剪切带倾角随着试件高度的增加而增加,但其增加幅逐渐减缓,最终趋于稳定值。笔者还对剪应变局部化启动、跳跃和稳定进行了理论分析,解释了实验所观测到的剪切带跳跃现象和砂岩岩样应变局部化较煤样滞后的原因。剪应变局部化是否发生跳跃,关键取决于全程应力-应变曲线软化段是否存在拐点。对于没有拐点的情形,宏观剪切带图案不跳跃。局部化是导致准脆性材料试件剪切带倾角尺度效应的原因。     

http://www.sciencedirect.com/science/article/pii/S1674987110000320

The self-similar is a common phenomena arising in the field of geology.It has been shown that geochemical element data,mineral deposits,and spacial distribution conform to a fractal structure.A fractal distribution requires that the number of objects larger than a specified size have a power-law dependence on size.This paper shows that a number of distributions,including power-function,Pareto, lognormal,and Zipf,display fractal properties under certain conditions and that this may be used as the mathemat...     

分形不变分布及其在山东地区金矿床中的应用

自相似性(标度不变性)是地学中的一个普遍现象。研究表明,地球化学元素含量、矿床储量规模及其空间分布具有分形结构。分形不变分布的特点要求大于某一尺度物体的数目与物体大小之间存在幂指数关系。论证了幂函数分布、帕累托分布、对数正态分布和齐波夫定律在一定条件下具有分形不变性质,它们是分形模型的数学基础。基于分形模型,用求和方法确定中国山东省金地球化学元素异常值范围。等值线大于或等于金地球化学元素临界值(200×10-9)围成的异常面积包含了已知的大型、超大型金矿床。     

Improved barometric and loading efficiency estimates using packers in monitoring wells

Measurement of barometric efficiency (BE) from open monitoring wells or loading efficiency (LE) from formation pore pressures provides valuable information about the hydraulic properties and confinement of a formation. Drained compressibility (α) can be calculated from LE (or BE) in confined and semi-confined formations and used to calculate specific storage (S _s). S _s and α are important for predicting the effects of groundwater extraction and therefore for sustainable extraction management. However, in low hydraulic conductivity (K) formations or large diameter monitoring wells, time lags caused by well storage may be so long that BE cannot be properly assessed in open monitoring wells in confined or unconfined settings. This study demonstrates the use of packers to reduce monitoring-well time lags and enable reliable assessments of LE. In one example from a confined, high-K formation, estimates of BE in the open monitoring well were in good agreement with shut-in LE estimates. In a second example, from a low-K confining clay layer, BE could not be adequately assessed in the open monitoring well due to time lag. Sealing the monitoring well with a packer reduced the time lag sufficiently that a reliable assessment of LE could be made from a 24-day monitoring period. The shut-in response confirmed confined conditions at the well screen and provided confidence in the assessment of hydraulic parameters. A short (time-lag-dependent) period of high-frequency shut-in monitoring can therefore enhance understanding of hydrogeological systems and potentially provide hydraulic parameters to improve conceptual/numerical groundwater models.     

On the Use of Multivariate Lévy-Stable Random Field Models for Geological Heterogeneity

Increasing attention has been paid to the use of non-Gaussian distributions as models of heterogeneity in sedimentary formations in recent years. In particular, the Lévy-stable distribution has been shown to be a useful model of the distribution of the increments of data measured in well logs. Frequently, the width of this distribution follows a power–law type scaling with increment lag, thus suggesting a nonstationary, fractal, multivariate Lévy distribution as a useful random field model. However, in this paper we show that it is very difficult to formulate a multivariate Lévy distribution with any nontrivial spatial correlations that can be sampled from rigorously in large models. Conventional sequential simulation techniques require two properties to hold of a multivariate distribution in order to work: (1) the marginal distributions must be of relatively simple form, and (2) in the uncorrelated limit, the multivariate distribution must factor into a product of independent distributions. At least one of these properties will break down in a multivariate Lévy distribution, depending on how it is formulated. This makes a rigorous derivation of a sequential simulation algorithm impossible. Nonetheless, many of the original observations that spurred the original interest in multivariate Lévy distributions can be reproduced with a conventional normal scoring procedure. Secondly, an approximate formulation of a sequential simulation algorithm can adequately reproduce the Lévy distributions of increments and fractal scaling frequently seen in real data.     

Study on fractal dimensions of the silty soil around the tunnel under the subway loading in Shanghai

Based on the global digital systems, by fully considering the factors of the accumulated deformation, such as the frequency, the dynamic stress amplitude as well as the consolidation ratio, the mercury intrusion test was conducted to quantitatively study the microstructures of the silty soil around the tunnel near the Guoquan station of line No. 10 Subway in Shanghai. The thermodynamic model was used to calculate fractal dimensions in the analysis of the mercury intrusion data. The results show that the relationship of ln(W _n /r _n ² ) and lnQ _n was a significant linear correlation and the correlation coefficient is over 0.98. The relationship between the fractal dimensions and the frequency, the dynamic stress amplitude as well as the consolidation ratio was analyzed; the variation law of the residual strain with the fractal dimensions was obtained. The study offers the theoretical support for damage mechanics, the subway axial deformation and the ground settlement under the subway vibration loading.     

基于土壤水分动态随机模型的土壤湿度概率密度函数研究进展

简要回顾和评述了几个典型土壤水分动态随机模型和相应的土壤湿度概率密度函数,并基于这些模型的模拟结果就系统随机波动、气候因素、土壤特性、植被条件、地形特征对稳定状态土壤湿度概率密度函数的影响分别进行了分析和讨论,同时还引用相关的实验结果对这些影响进行佐证.结果表明,土壤湿度概率密度函数对以上因子的响应非常敏感,对揭示土壤水分动态和诠释生态水文现象有很好的指导意义.指出未来土壤湿度概率密度函数的研究方向:开发更加逼近真实情况的随机模型;用土壤湿度概率密度函数的思想量化植被水分胁迫、耦合土壤水分动态与养分动态;对某一地区建立其土壤湿度概率密度函数,并用该函数指导其他相关的生态建设和研究.     

Anatomy of a transgressive lag: Panther Tongue Sandstone, Star Point Formation, central Utah

Abstract The Panther Tongue of the Star Point Formation in central Utah contains a variety of transgressive lag deposits that, when mapped regionally, show a sensitive dependence upon pre‐existing topography of the palaeoshoreline. The Panther Tongue consists of a coarsening‐upward sandstone wedge that prograded into the Western Interior Seaway during Late Cretaceous (Santonian) time. High‐resolution sequence stratigraphic analysis revealed that this member was deposited during the long‐distance (>50 km) regression and transgression of a delta into shallow‐marine environments, containing basal highstand, forced regression, lowstand and transgressive systems tracts. Based on grain size, clast composition, lateral extent and stratigraphic position, the coarse sandstones on top of the Panther Tongue were classified into four types: (1) simple; (2) dispersed; (3) oxidized; and (4) local lags. The simple lag is composed of dark grey coarse sandstone with oyster fragments and shark teeth. This lag is typically extensively bioturbated and massive. Laminated and cross‐bedded units are also common. This type of coarse sandstone is interpreted as a typical transgressive lag. The dispersed lag differs in that it contains abundant mud and commonly occurs as multiple beds in thick intervals of muddy sandstone. Mixing of bay/estuarine and shallow ‐ marine mud with simple lag sand may be responsible for deposition of this type of coarse sandstone. The oxidized lag is distinctive in its reddish colour with extensive bioturbation and is commonly overlain by a simple lag. The local lag is composed of thin‐bedded, dark grey, coarse sandstone, occurring locally between the mouth bar and distributary channel. The variation in types, grain size and bed thickness of the coarse‐grained lags was mainly controlled by antecedent topography as suggested by immediately underlying lithofacies. Relatively thick (≈30 cm) simple lags are present on top of mouth‐bar sandstones, whereas dispersed lags are common on top of the distributary channel sandstone and in bay/estuarine and shallow‐marine mudstones. Erosion of topographic highs (mouth bar) resulted in relatively thick accumulation of simple lags. In topographic low areas such as distributary channel, estuary, bay and shallow‐marine environments, fine‐grained muddy sands that were eroded from the nearby topographic highs were redeposited. Intermittent storm waves transported coarse sands both landward and seaward, forming a dispersed lag. The net effect was reworking of local topographic relief during overall transgression, forming an apparently planar transgressive surface of erosion.     

Kinetic rate laws as derived from order parameter theory II: Interpretation of experimental data by laplace-transformation,the relaxation spectrum,and kinetic gradient coupling between two order parameters

A unifying theory of kinetic rate laws, based on order parameter theory, is presented. The time evolution of the average order parameter is described by $$\langle Q\rangle \propto \smallint P(x)e^{^{^{^{^{^{^{ - xt} } } } } } } dx = L(P)$$ where t is the time, x is the effective inverse susceptibility, and L indicates the Laplace transformation. The probability function P(x) can be determined from experimental data by inverse Laplace transformation. Five models are presented:

1. Polynomial distributions of P(x) lead to Taylor expansions of 〈Q〉 as $$\langle Q\rangle = \frac{{\rho _1 }}{t} + \frac{{\rho _2 }}{{t^2 }} + ...$$
2. Gaussian distributions (e.g. due to defects) lead to a rate law $$\langle Q\rangle = e^{ - x_0 t} e^{^{^{^{^{\frac{1}{2}\Gamma t^2 } } } } } erfc\left( {\sqrt {\frac{\Gamma }{2}} t} \right)$$ where x ₀ is the most probable inverse time constant, Γ is the Gaussian line width and erfc is the complement error integral.
3. Maxwell distributions of P are equivalent to the rate law 〈Q〉∝e^?k√t .
4. Pseudo spin glasses possess a logarithmic rate law 〈Q〉∝lnt.
5. Power laws with P(x)=x ^a lead to a rate law: ln〈Q〉=-(α + 1) ln t.

The power spectra of Q are shown for Gaussian distributions and pseudo spin glasses. The mechanism of kinetic gradient coupling between two order parameters is evaluated.     

Fractal Analysis of <Emphasis Type="Italic">In Situ</Emphasis> Host Rock Nepheline Syenite Xenoliths in a Micro-Shonkinite Dyke (The Elchuru Alkaline Complex,SE India)

Formation of the fragments of the wall-rock during dyking is one of the important manifestations of instantaneous magmatic events. This process is well documented at shallower depths of Earth’s crust but not at deeper levels. In this paper the in situ xenoliths of host rock nepheline syenite within a micro-shonkinite dyke emplaced at mid-crustal depths is described and the fractal theory applied to evaluate origin of the xenoliths. The nepheline syenite xenoliths are angular to oval shaped and sub-millimetre to ~50 cm long. The xenoliths are matrix supported with clasts and matrix being in equal proportions. Partly detached wall-rock fragments indicate incipient xenolith formation, which suggested that the model fragmentation processes is solely due to dyke emplacement. Fractal analytical techniques including clast size distribution, boundary roughness fractal dimension and clast circularity was carried out. The fractal data suggests that hydraulic (tensile) fracturing is the main process of host rock brecciation. However, the clast size and shape are further affected by postfragmentation processes including shear and thermal fracturing, and chemical erosion. The study demonstrates that dyking in an isotropic medium produces fractal size distributions of host rock xenoliths; however, post-fragmentation processes modify original fractal size distributions.     

A thermodynamic theory of the Grüneisen ratio at extreme conditions: MgO as an example

The Grüneisen ratio, γ, is defined as γy=αK _TV/C_v. The volume dependence of γ(V) is solved for a wide range in temperature. The volume dependence of αK _T is solved from the identity (? ln(αK _T)/? ln V)T ≡ δ _T-K′. α is the thermal expansivity; K _T is the bulk modulus; C _V is specific heat; and δ _Tand K′ are dimensionless thermoelastic constants. The approach is to find values of δ _T and K′, each as functions of T and V. We also solve for q=(? ln γ/? ln V) where q=δ _T -K′+ 1-(? ln C _V/? ln V)T. Calculations are taken down to a compression of 0.6, thus covering all possible values pertaining to the earth's mantle, q=? ln γ/? ln V; δ _T=? ln α/? ln V; and K′= (?K _T/?P)T. New experimental information related to the volume dependence of δ _T, q, K′ and C _V was used. For MgO, as the compression, η=V/V ₀, drops from 1.0 to 0.7 at 2000 K, the results show that q drops from 1.2 to about 0.8; δ _T drops from 5.0 to 3.2; δ _T becomes slightly less than K′; ? ln C _V/? In V→0; and γ drops from 1.5 to about 1. These observations are all in accord with recent laboratory data, seismic observations, and theoretical results.     

Distribution of time lags between successive petroleum exploration activities

Four time lags are studied: (1) drilling activity behind seismic activity; (2) oil and gas reserves discovered behind drilling activity; (3) oil and gas production behind reserves discovered; and (4) oil and gas production behind drilling activity. A ridge regression method is used to estimate the distributions of these lags for 50 countries and regions active in petroleum exploration. A complete review of obtained results is given as well as an interpretation of estimated models and failed ridge regressions. Five types of distributions have been found: decreasing monotonically, unimodal, bimodal, uniform, and increasing monotonically. The distributions allow estimating the capital freezing in the exploration process and assessing the resulting policy made by all companies active in a country. For example, the actual exploration history in Poland is analyzed with the use of distributed time-lag approach and net present-value concept. An overinvestment in drilling in comparison with petroleum infrastructure has been stated on the basis of the excessive lagging of production behind drilling activity. A direction of further lag studies is also suggested.     

Bias in estimating fractal dimension with the rescaled-range (R/S) technique

Fractal geostatistics are being applied to subsurface geological data as a way of predicting the spatial distribution of hydrocarbon reservoir properties. The fractal dimension is the controlling parameter in stochastic methods to produce random fields of porosity and permeability. Rescaled range (R/S)analysis has become a popular way of estimating the fractal dimension, via determination of the Hurst exponent (H). A systematic investigation has been undertaken of the bias to be expected due to a range of factors commonly inherent in borehole data, particularly downhole wireline logs. The results are integrated with a review of previous work in this area. Small datasets. overlapping samples, drift and nonstationariry of means can produce a very large bias, and convergence of estimates of H around 0.85–0.90 regardless of original fractal dimension. Nonstationarity can also account for H>1, which has been reported in the literature but which is theoretically impossible for fractal time series. These results call into question the validity of fractal stochastic models built using fractal dimensions estimated with the R/Smethod.     

Thermal properties of forsterite at mantle pressures derived from vibrational spectroscopy

The pressure dependence of the Raman spectrum of forsterite was measured over its entire frequency range to over 200 kbar. The shifts of the Raman modes were used to calculate the pressure dependence of the heat capacity, C _v, and entropy, S, by using statistical thermodynamics of the lattice vibrations. Using the pressure dependence of C _v and other previously measured thermodynamic parameters, the thermal expansion coefficient, α, at room temperature was calculated from α = K _S (?T/?P) _S C _V/TVK _T, which yields a constant value of (? ln α/? ln V)_T= 6.1(5) for forsterite to 10% compression. This value is in agreement with (? ln α/? ln V)_T for a large variety of materials. At 91 kbar, the compression mechanism of the forsterite lattice abruptly changes causing a strong decrease of the pressure derivative of 6 Raman modes accompanied by large reductions in the intensities of all of the modes. This observation is in agreement with single crystal x-ray diffraction studies to 150 kbar and is interpreted as a second order phase transition.     

A Corrected and Generalized Successive Random Additions Algorithm for Simulating Fractional Levy Motions

Simulation of subsurface heterogeneity is important for modeling subsurface flow and transport processes. Previous studies have indicated that subsurface property variations can often be characterized by fractional Brownian motion (fBm) or (truncated) fractional Levy motion (fLm). Because Levy-stable distributions have many novel and often unfamiliar properties, studies on generating fLm distributions are rare in the literature. In this study, we generalize a relatively simple and computationally efficient successive random additions (SRA) algorithm, originally developed for generating Gaussian fractals, to simulate fLm distributions. We also propose an additional important step in response to continued observations that the traditional SRA algorithm often generates fractal distributions having poor scaling and correlation properties. Finally, the generalized and modified SRA algorithm is validated through numerical tests.     

A stochastic model of fractional crystallization and its application to the Japanese geosynclinal basalt

A stochastic model of the magmatic differentiation by fractional crystallization is given. The probability distribution of the quantity of crystallized solid removed from silicate melt was assumed to be binomial at any stage of the differentiation. According to this model, when concentrations of an element are transformed into their powers by using the reciprocal of the difference between bulk-partition coefficient for the element and unity as an exponent, the resultant frequency-distribution pattern becomes absolutely normal. The patterns of concentrations, however, cannot be expressed by a particular type of function but are variable according to the values of bulk-partition coefficients. Frequency distributions of element concentrations were computed under selected conditions on the basis of the model. The result shows that the skewed pattern often observed in frequency distributions of minor-element concentrations is explained by this fractional crystallization process. The frequency distributions of Ni and Cr concentrations in the geosynclinal basalt of southwestern Japan were examined in terms of this model. It was concluded that the model can be applied to the formation of the basalt concerning at least these two elements.     

Dependence of flotation rate on particle size and fractional mineral content

Specific rates of flotation, K, as a function of particle size, d, and fractional mineral content, W, were measured for a quartz-hematite ore in a continuous flotation system using hydroxamate collector. The resulting K-d-W surface is explained in terms of flotation mechanisms and results of previous investigators. A K-d-W relationship will be of value to flotation models.