Stochastic modeling of solute transport in 3-D heterogeneous porous media with random source condition

During probabilistic analysis of flow and transport in porous media, the uncertainty due to spatial heterogeneity of governing parameters are often taken into account. The randomness in the source conditions also play a major role on the stochastic behavior in distribution of the dependent variable. The present paper is focused on studying the effect of both uncertainty in the governing system parameters as well as the input source conditions. Under such circumstances, a method is proposed which combines with stochastic finite element method (SFEM) and is illustrated for probabilistic analysis of concentration distribution in a 3-D heterogeneous porous media under the influence of random source condition. In the first step SFEM used for probabilistic solution due to spatial heterogeneity of governing parameters for a unit source pulse. Further, the results from the unit source pulse case have been used for the analysis of multiple pulse case using the numerical convolution when the source condition is a random process. The source condition is modeled as a discrete release of random amount of masses at fixed intervals of time. The mean and standard deviation of concentration is compared for the deterministic and the stochastic system scenarios as well as for different values of system parameters. The effect of uncertainty of source condition is also demonstrated in terms of mean and standard deviation of concentration at various locations in the domain.     

Effective dispersion in a chemically heterogeneous medium under temporally fluctuating flow conditions

We investigate effective solute transport in a chemically heterogeneous medium subject to temporal fluctuations of the flow conditions. Focusing on spatial variations in the equilibrium adsorption properties, the corresponding fluctuating retardation factor is modeled as a stationary random space function. The temporal variability of the flow is represented by a stationary temporal random process. Solute spreading is quantified by effective dispersion coefficients, which are derived from the ensemble average of the second centered moments of the normalized solute distribution in a single disorder realization. Using first-order expansions in the variances of the respective random fields, we derive explicit compact expressions for the time behavior of the disorder induced contributions to the effective dispersion coefficients. Focusing on the contributions due to chemical heterogeneity and temporal fluctuations, we find enhanced transverse spreading characterized by a transverse effective dispersion coefficient that, in contrast to transport in steady flow fields, evolves to a disorder-induced macroscopic value (i.e., independent of local dispersion). At the same time, the asymptotic longitudinal dispersion coefficient can decrease. Under certain conditions the contribution to the longitudinal effective dispersion coefficient shows superdiffusive behavior, similar to that observed for transport in s stratified porous medium, before it decreases to its asymptotic value. The presented compact and easy to use expressions for the longitudinal and transverse effective dispersion coefficients can be used for the quantification of effective spreading and mixing in the context of the groundwater remediation based on hydraulic manipulation and for the effective modeling of reactive transport in heterogeneous media in general.     

Block-effective macrodispersion for numerical simulations of sorbing solute transport in heterogeneous porous formations

Heterogeneity is prevalent in aquifers and has an enormous impact on contaminant transport in groundwater. Numerical simulations are an effective way to deal with heterogeneity directly by assigning different hydraulic property values to each numerical grid block. Because hydraulic properties vary on different scales, but they cannot be sampled exhaustively and the number of numerical grid blocks is limited by computational considerations, the dispersive effects of unmodeled heterogeneity need to be accounted for. Dispersion tensors can be used to model the dispersion caused by unmodeled heterogeneity. The concept of block-effective macrodispersion tensors for modeling the effects of small-scale variability on solute transport introduced by Rubin et al. [Rubin Y, Sun A, Maxwell R, Bellin A. The concept of block-effective macrodispersivity and a unified approach for grid-scale- and plume-scale-dependent transport. J Fluid Mech 1999;395:161–80] is extended in this paper for use with reactive solutes. The tensors are derived for reactive solutes with spatially variable retardation factors and for solutes experiencing spatially uniform rate-limited sorption. The longitudinal block-effective macrodispersion coefficient is largest for perfect negative correlation between the log-hydraulic conductivity and the retardation factor. Because dispersion tensors, as they are usually implemented in numerical simulations, produce symmetric spreading, the applicability of the concept depends on the portion of the plume asymmetry caused by small-scale variability. The presented results show that the concept is applicable for rate-limited sorption for block sizes of one and two integral scales.     

Stochastic modeling of reactive transport in wetlands

This study describes the development of a general model for reaction in and performance of spatially heterogeneous bioreactors such as treatment wetlands. The modeled domain possesses local-scale velocities, reaction rates and transverse dispersion coefficients that are functions of an underlying heterogeneity variate representing one or more controlling biophysical attributes, for example, reactive surface area (submerged plant) density. Reaction rate coefficients are treated as related to local velocities in an inverse square fashion via their mutual dependence upon the variate. The study focuses on the solution for the steady-state case with constant inlet concentration. Results compare well with exact solutions developed for laterally-bounded systems in which the heterogeneity is represented explicitly. Employing the bicontinuum analogue of a second-order model, an expression for an effective longitudinal dispersion coefficient as a function of travel distance is developed using the method of moments. The result provides insights into the behavior of concentration as transverse mixing drives the system asymptotically toward Fickian longitudinal dispersion. The model may represent an improvement over other approaches for characterizing treatment wetland performance because it accounts for evolving shear flow dispersion, and because parameters are few in number, physically based, and invariant with mean velocity.     

新疆和田台阵PSD与PDF分析

和田台阵是我国第一个自主建成并运行至今的小孔径台阵,承担着监测印巴地区核试验以及我国西部地震活动的重要使命.台阵波形数据中充斥着背景噪声,直接影响着数据质量.为了评估台阵噪声水平,本文利用Welch平均周期法对9个子台记录的数十万条噪声样本进行功率谱估计,对出现的谱异常进行了总结归纳,通过绘制概率密度函数图以及单频曲线来研究背景噪声变化范围和规律,最后针对台阵降噪提出了建设性意见.研究结果表明,中心台长周期噪声功率谱密度随季节变化显著,具有周期性;受温度和气压影响,水平分量长周期噪声变化幅度较大,局部频段超出新高噪声模型,建议改善仪器安装条件,或者利用数学方法进行校正.所有子台短周期噪声变化规律与长周期相反,受到采石场影响,谱密度曲线在4~8 Hz之间出现形态规则的高频尖刺,A1、B3、B4子台最为明显,可以通过窄带滤波或者聚束予以压制.本文取得的研究成果为台阵运维提供重要依据,除此之外,总结出的不同地震频谱特征也为地震解释工作提供重要参考.

     

A practical reliability-based method for assessing soil liquefaction potential

The current simplified methods for assessing soil liquefaction potential use a deterministic safety factor in order to judge whether liquefaction will occur or not. However, these methods are unable to determine the liquefaction probability related to a safety factor. An answer to this problem can be found by reliability analysis. This paper presents a reliability analysis method based on the popular Seed'85 liquefaction analysis method. This reliability method uses the empirical acceleration attenuation law in the Taiwan area to derive the probability density distribution function (PDF) and the statistics for the earthquake-induced cyclic shear stress ratio (CSR). The PDF and the statistics for the cyclic resistance ratio (CRR) can be deduced from some probabilistic cyclic resistance curves. These curves are produced by the regression of the liquefaction and non-liquefaction data from the Chi-Chi earthquake and others around the world, using, with minor modifications, the logistic model proposed by Liao [J. Geotech. Eng. 114 (1988) 389]. The CSR and CRR statistics are used in conjunction with the first order and second moment method, to calculate the relation between the liquefaction probability, the safety factor and the reliability index. Based on the proposed method, the liquefaction probability related to a safety factor can be easily calculated. The influence of some of the soil parameters on the liquefaction probability can be quantitatively evaluated.     

A method for incorporating equilibrium chemical reactions into multiphase flow models for CO2 storage

CO₂ injection and storage in deep saline aquifers involves many coupled processes, including multiphase flow, heat and mass transport, rock deformation and mineral precipitation and dissolution. Coupling is especially critical in carbonate aquifers, where minerals will tend to dissolve in response to the dissolution of CO₂ into the brine. The resulting neutralization will drive further dissolution of both CO₂ and calcite. This suggests that large cavities may be formed and that proper simulation may require full coupling of reactive transport and multiphase flow. We show that solving the latter may suffice whenever two requirements are met: (1) all reactions can be assumed to occur in equilibrium and (2) the chemical system can be calculated as a function of the state variables of the multiphase flow model (i.e., liquid and gas pressure, and temperature). We redefine the components of multiphase flow codes (traditionally, water and CO₂), so that they are conservative for all reactions of the chemical system. This requires modifying the traditional constitutive relationships of the multiphase flow codes, but yields the concentrations of all species and all reaction rates by simply performing speciation and mass balance calculations at the end of each time step. We applied this method to the H₂O–CO₂–Na–Cl–CaCO₃ system, so as to model CO₂ injection into a carbonate aquifer containing brine. Results were very similar to those obtained with traditional formulations, which implies that full coupling of reactive transport and multi-phase flow is not really needed for this kind of systems, but the resulting simplifications may make it advisable even for cases where the above requirements are not met. Regarding the behavior of carbonate rocks, we find that porosity development near the injection well is small because of the low solubility of calcite. Moreover, dissolution concentrates at the front of the advancing CO₂ plume because the brine below the plume tends to reach high CO₂ concentrations quite rapidly. We conclude that carbonate dissolution needs not to be feared.     

Stochastic analysis of flood series

A stochastic model based on the renewal process was developed and used to analyse the characteristics of floods: the volume exceedence, the duration of the flood and the maximum annual flow. The model contains a method for determination of total annual volume exceedence and total annual duration of floods, as well as a method for calculation of maximum annual exceedence, maximum flood duration and maximum flow. The subset of the flood occurrence number in a given time interval is common for all analysed phenomena (volume exceedence, flood duration, maximum flow). The subset of given exceedences is common for total annual volume exceedence, as well as for maximum annual volume exceedence. The same holds for durations of individual floods. The model was then applied to analyse the floods on the Drina River at the Paunci hydrological station and on the Danube River at the Bezdan station.     

A multidimensional streamline-based method to simulate reactive solute transport in heterogeneous porous media

We present a new streamline-based numerical method for simulating reactive solute transport in porous media. The key innovation of the method is that both longitudinal and transverse dispersion are incorporated accurately without numerical dispersion. Dispersion is approximated in a flow-oriented grid using a combination of a one-dimensional finite difference scheme and a meshless approximation. In contrast to previous hybrid alternatives to incorporate dispersion in streamline-based simulations, the proposed scheme does not require a grid and, hence, it does not introduce numerical dispersion. In addition, the proposed scheme eliminates numerical oscillations and negative concentration values even when the dispersion tensor includes the off-diagonal coefficients and the flow field is non-uniform. We demonstrate that for a set of two- and three-dimensional benchmark problems, the new proposed streamline-based formulation compares favorably to two state of the art finite volume and hybrid Eulerian–Lagrangian solvers.     

A cross entropy method for flood frequency analysis

The cross-entropy method with fractile constraints has been developed to estimate a random variable when the data are a set of independent observations of the variable. The method can claim several advantages over existing methods. It uses a reference distribution like the prior distribution in Bayesian analysis and likewise generates a posterior distribution.The method is of interest, in particular, because it satisfies two fundamental requirements for selfconsistency in the analysis of a probabilistic system based on data: a principle of invariance and a principle of data monotonicity.The method is applied to flood analysis. Robustness of the minimum cross-entropy method is compared with other methods: the methods of moments and the maximum likehood.     

Stochastic analysis of biodegradation fronts in one-dimensional heterogeneous porous media

We consider a one-dimensional model biodegradation system consisting of two reaction–advection equations for nutrient and pollutant concentrations and a rate equation for biomass. The hydrodynamic dispersion is ignored. Under an explicit condition on the decay and growth rates of biomass, the system can be approximated by two component models by setting biomass kinetics to equilibrium. We derive closed form solutions for constant speed traveling fronts for the reduced two component models and compare their profiles in homogeneous media. For a spatially random velocity field, we introduce travel time and study statistics of degradation fronts via representations in terms of the travel time probability density function (pdf) and the traveling front profiles. The travel time pdf does not vary with the nutrient and pollutant concentrations and only depends on the random water velocity. The traveling front profiles are expressed analytically or semi-analytically as functions of the travel time. The problem of nonlinear transport by a random velocity reduces to two subproblems: one being nonlinear transport by a known (unit) velocity, and the other being linear (advective) transport by a random velocity. The approach is illustrated through some examples where the randomness in velocity stems from the spatial variability of porosity.     

A mathematical model for steady‐state regolith production at constant erosion rate

It has been hypothesized that many soil profiles reach a steady‐state thickness. In this work, such profiles were simulated using a one‐dimensional model of reaction with advective and diffusive solute transport. A model ‘rock’ is considered, consisting of albite that weathers to kaolinite in the presence of chemically inert quartz. The model yields three different steady‐state regimes of weathering. At the lowest erosion rates, a local‐equilibrium regime is established where albite is completely depleted in the weathering zone. This regime is equivalent to the transport‐limited regime described in the literature. With an increase in erosion rate, transition and kinetic regimes are established. In the transition regime, both albite and kaolinite are present in the weathering zone, but albite does not persist to the soil–air interface. In the weathering‐limited regime, here called the kinetic regime, albite persists to the soil–air interface. The steady‐state thickness of regolith decreases with increasing erosion rate in the local equilibrium and transition regimes, but in the kinetic regime, this thickness is independent of erosion rate. Analytical expressions derived from the model are used to show that regolith production rates decrease exponentially with regolith thickness. The steady‐state regolith thickness increases with the Darcy velocity of the pore fluid, and in the local equilibrium regime may vary markedly with small variations in this velocity and erosion rate. In the weathering‐limited regime, the temperature dependences for chemical weathering rates are related to the activation energy for the rate constant for mineral reaction and to the ΔH of dissolution, while for local equilibrium regimes they are related to the ΔH only. The model illustrates how geochemical and geomorphological observations are related for a simple compositional system. The insights provided will be useful in interpreting natural regolith profiles. Copyright © 2010 John Wiley & Sons, Ltd.     

Stochastic analysis of a field-scale unsaturated transport experiment

Modelling of field-scale transport of chemicals is of deep interest to public as well as private sectors, and it represents an area of active theoretical research in many environmentally-based disciplines. However, the experimental data needed to validate field-scale transport models are very limited due to the numerous logistic difficulties that one faces out.     

砂岩储层AVO特征影响因素的不确定性研究

传统的地震AVO正演研究多采用参数固定的岩石物理模型,而实际地层属性参数在勘探范围内具有不确定性.本研究以目标地层岩芯样品的实验室测试数据为基础,通过样品孔隙度和干燥状态下纵、横波阻抗的高度线性关系对岩石物理模型进行了简化,并结合实验测量和测井解释建立了主要模型参数的概率密度函数.采用Monte-Carlo随机正演和Gassmann流体替换技术,对模型参数不确定性与地震AVO响应关系的研究表明,模型饱和不同流体时的AVO截距-斜率点束的分布形态和偏离饱水背景趋势线的程度受盖层泥岩速度的不确定性影响非常大,而储层砂岩孔隙度的不确定性也有明显影响.正确解释实际地震AVO异常的关键,是将岩石物理模型的参数合理地概率化,通过正演模拟来获得储层物性和流体信息的最大可能性.     

Stochastic analysis of velocity spatial variability in bounded rectangular heterogeneous aquifers

Transport of inert solutes in two-dimensional bounded heterogeneous porous media is investigated in a stochastic framework. After adopting a first-order approximation of the flow equations, analytical expressions are derived for the velocity covariances. Effects of the boundary conditions and aquifer size upon the statistical moments are analyzed. While the size of the domain is shown to have small influence on the covariances in most cases, the solutions are considerably modified by the boundaries. The results are compared with analytical solutions on infinite domains, and several discrepancies are demonstrated. For example, while the velocity variances on infinite domains are homogeneous, the present results are strongly non-stationary. Finally, the problem is solved numerically by the Monte Carlo simulation method. The results, including the behavior near the boundaries, are shown to be in close agreement with analytical solutions.     

Stochastic analysis of flux and head moments in a heterogeneous aquifer system

This study investigates the behavior of flux and head in a strongly heterogeneous three-dimensional aquifer system. The analyses relied on data from 520 slug tests together with 38,000 one-foot core intervals lithological data from the site of the General Separations Area in central Savannah River Site, South Carolina, USA. The skewness in the hydraulic conductivity histograms supported the geologic information for the top two aquifers, but revealed stronger clay content, than was reported for the bottom aquifer. The log-normal distribution model described adequately the hydraulic conductivity measurements for all three aquifers although, other distributions described equally well the bottom aquifer measurements. No apparent anisotropy on the horizontal plane was found for the three aquifers, but ratios of horizontal to vertical correlation lengths between 33 and 75 indicated a strong stratification at the site. Three-dimensional Monte Carlo stochastic simulations utilized a grid with larger elements than the support volume of measurements, but of sub-REV (representative elementary volume) dimensions. This necessitated, on one hand, the use of upscaled hydraulic conductivity expressions, but on the other hand did not allow for the use of anisotropic effective hydraulic conductivity expressions (Sarris and Paleologos in J Stoch Environ Res Risk Assess 18: 188–197, 2004). Flux mean and standard deviations components were evaluated on three vertical cross-sections. The mean and variance of the horizontal flux component normal to a no-flow boundary tended to zero at approximately two to three integral scales from that boundary. Close to a prescribed head boundary both the mean and variance of the horizontal flux component normal to the boundary increased from a stable value attained at a distance of about five integral scales from that boundary. The velocity field 〈q_x〉 was found to be mildly anisotropic in the top two aquifers, becoming highly anisotropic in the bottom aquifer; 〈q_y〉 was anisotropic in all three aquifers with directions of high continuity normal to those of the 〈q_x〉 field; finally, 〈q_z〉 was highly anisotropic in all three aquifers, with higher continuity along the east–west direction. The mean head field was found to be continuous, despite the high heterogeneity of the underlying hydraulic conductivity field. Directions of high continuity were in alignment with field boundaries and mean flow direction. Conditioning did not influence significantly the expected value of the flux terms, with more pronounced being the effect on the standard deviation of the flux vector components. Conditioning reduced the standard deviations of the horizontal flux components by as much as 50% in the bottom aquifer. Variability in the head cross-sections was affected only marginally, with an average 10% reduction in the respective standard deviation. Finally, the location of the conditioning data did not appear to have a significant effect on the surrounding area, with uniform reduction in standard deviations.     

Stochastic analysis of unsaturated transport in soils with fractal log-conductivity distribution

Within the framework of stochastic theory and the spectral perturbation techniques, three-dimensional dispersion in partially saturated soils with fractal log hydraulic conductivity distribution is analyzed. Our analysis is focused on the impact of fractal dimension of log hydraulic conductivity distribution, local dispersivity, and unsaturated flow parameters, such as the soil poresize distribution parameter and the moisture distribution parameter, on the spreading behavior of solute plume and the concentration variance. Approximate analytical solutions to the stochastic partial differential equations are derived for the variance of asymptotic solute concentration and asymptotic macrodispersivities.