Stochastic analysis of immiscible displacement of the fluids with arbitrary viscosities and its dependence on support scale of hydrological data

Stochastic analysis is commonly used to address uncertainty in the modeling of flow and transport in porous media. In the stochastic approach, the properties of porous media are treated as random functions with statistics obtained from field measurements. Several studies indicate that hydrological properties depend on the scale of measurements or support scales, but most stochastic analysis does not address the effects of support scale on stochastic predictions of subsurface processes. In this work we propose a new approach to study the scale dependence of stochastic predictions. We present a stochastic analysis of immiscible fluid–fluid displacement in randomly heterogeneous porous media. While existing solutions are applicable only to systems in which the viscosity of one phase is negligible compare with the viscosity of the other (water–air systems for example), our solutions can be applied to the immiscible displacement of fluids having arbitrarily viscosities such as NAPL–water and water–oil. Treating intrinsic permeability as a random field with statistics dependant on the permeability support scale (scale of measurements) we obtained, for one-dimensional systems, analytical solutions for the first moments characterizing unbiased predictions (estimates) of system variables, such as the pressure and fluid–fluid interface position, and we also obtained second moments, which characterize the uncertainties associated with such predictions. Next we obtained empirically scale dependent exponential correlation function of the intrinsic permeability that allowed us to study solutions of stochastic equations as a function of the support scale. We found that the first and second moments converge to asymptotic values as the support scale decreases. In our examples, the statistical moments reached asymptotic values for support scale that were approximately 1/10000 of the flow domain size. We show that analytical moment solutions compare well with the results of Monte Carlo simulations for moderately heterogeneous porous media, and that they can be used to study the effects of heterogeneity on the dynamics and stability of immiscible flow.     

骨架导电的混合泥质砂岩通用孔隙结合电阻率模型研究

在水介质中顺序添加分散粘土颗粒、油珠、导电骨架颗粒、层状泥质,并对每一种成分进行连续积分,建立了一种适用骨架导电及含有分散粘土和层状泥质的泥质砂岩通用电阻率模型.通过对该模型的影响因素分析,发现泥质分布形式对模型计算的含水饱和度有很大影响;对应两个不同粘土颗粒电阻率或骨架颗粒电阻率的地层电导率之差,几乎与总含水饱和度无关,而对应两个不同层状泥质电阻率的地层电导率之差,随总含水饱和度增大而增大;骨架胶结指数变化对地层电导率与总含水饱和度关系曲线的影响最大,而粘土胶结指数变化对地层电导率与总含水饱和度关系曲线的影响最小;饱和度指数对地层电导率与总含水饱和度关系曲线的影响随总含水饱和度的增大而减小.通过一组骨架导电的人造岩样的试验,表明当地层水电阻率.小于颗粒电阻率时,该模型可以用于不含粘土的骨架导电的岩石.通过两组分散泥质砂岩岩样实验测量数据和一组层状泥质砂岩测井资料及实际测井资料的计算,表明本文给出的电阻率模型既适用于分散泥质砂岩地层解释又适用于层状泥质砂岩地层解释,同时,还适用于含有分散粘土和层状泥质的混合泥质砂岩地层解释.     

基于沉积过程的数字岩石建模方法研究

本文提出模拟地层沉积及成岩过程的矿物沉积算法,建立数字岩石模型,并通过对比Micro-CT扫描图像和数值模型的局部孔隙度及平均渗流概率函数分布特征,评价建模的准确性.结果表明,由二维扫描提取的粒径信息作为输入参数,模拟矿物沉积过程建模得到的三维数字岩石模型,能够准确重构原始岩心的非均质性及渗流特性,成功应用于泥质砂岩、碳酸盐岩、页岩等存在多矿物或多尺度孔隙的数字岩石建模中.数字岩石物理是正在兴起的重要技术.数字岩石采用超高分辨率先进成像装备,采集和表征微纳尺度岩石结构,在岩石弹性、电性、核磁、渗流特性等数值计算中发挥重要作用.但是,由于三维直接成像在有限视域内难以表征足够的岩石非均质性,提取二维结构统计特征,利用统计或地质过程法重构具有代表性的三维岩石结构成为十分有价值的研究课题,而且,对业界大量存在的岩石薄片及电镜高清二维图像的深度开发应用也具有重要的现实意义.本文发展的新方法,复原沉积过程,较好地解决了孔隙尺度岩石物理定量研究中数值建模与理论计算的技术瓶颈.     

Multiscale study for stochastic characterization of shale samples

Characterization of shale reservoirs, which are typically of low permeability, is very difficult because of the presence of multiscale structures. While three-dimensional (3D) imaging can be an ultimate solution for revealing important complexities of such reservoirs, acquiring such images is costly and time consuming. On the other hand, high-quality 2D images, which are widely available, also reveal useful information about shales’ pore connectivity and size. Most of the current modeling methods that are based on 2D images use limited and insufficient extracted information. One remedy to the shortcoming is direct use of qualitative images, a concept that we introduce in this paper. We demonstrate that higher-order statistics (as opposed to the traditional two-point statistics, such as variograms) are necessary for developing an accurate model of shales, and describe an efficient method for using 2D images that is capable of utilizing qualitative and physical information within an image and generating stochastic realizations of shales. We then further refine the model by describing and utilizing several techniques, including an iterative framework, for removing some possible artifacts and better pattern reproduction. Next, we introduce a new histogram-matching algorithm that accounts for concealed nanostructures in shale samples. We also present two new multiresolution and multiscale approaches for dealing with distinct pore structures that are common in shale reservoirs. In the multiresolution method, the original high-quality image is upscaled in a pyramid-like manner in order to achieve more accurate global and long-range structures. The multiscale approach integrates two images, each containing diverse pore networks – the nano- and microscale pores – using a high-resolution image representing small-scale pores and, at the same time, reconstructing large pores using a low-quality image. Eventually, the results are integrated to generate a 3D model. The methods are tested on two shale samples for which full 3D samples are available. The quantitative accuracy of the models is demonstrated by computing their morphological and flow properties and comparing them with those of the actual 3D images. The success of the method hinges upon the use of very different low- and high-resolution images.     

A stochastic model for initial movement of sand grains by wind

A stochastic model for entrainment of sand grains by wind is presented through analysis of the forces exerted on a single spherical grain, coupled with fluctuations of wind velocity and the change in grain position on the surface. The structure of the stochastic model is consistent with experimental data in the literature. The probability of initial motion increases first, and then decreases, with grain size. It reaches a maximum at diameters of about 0·9 mm. Some sand grains are still in motion at less than the conventional threshold velocity, even at very low velocities. The probability of sand grain movement reaches unity at twice the conventional threshold velocity. Considerable discrepancies amongst conventional threshold formulae may result from the different probabilities of initial movement implied in these formulae. Copyright © 2008 John Wiley & Sons, Ltd.     

Interface dynamics in randomly heterogeneous porous media

We consider the dynamics of a fluid interface in heterogeneous porous media, whose hydraulic properties are uncertain. Modeling hydraulic conductivity as a random field of given statistics allows us to predict the interface dynamics and to estimate the corresponding predictive uncertainty by means of statistical moments. The novelty of our approach to obtaining the interface statistics consists of dynamically mapping the Cartesian coordinate system onto a coordinate system associated with the moving front. This transforms a difficult problem of deriving closure relationships for highly nonlinear stochastic flows with free surfaces into a relatively simple problem of deriving stochastic closures for linear flows in domains with fixed boundaries. We derive a set of deterministic equations for the statistical moments of the interfacial dynamics, which hold in one and two spatial dimensions, and analyze their solutions for one-dimensional flow.     

Fundamental Changes in In Situ Air Sparging How Patterns

Two types of gas-phase flow patterns have been discussed and observed in the in situ air sparging (ISAS) literature: bubble flow and air channels. A critical factor affecting the flow pattern at a given location is the grain size of the porous medium. Visualization experiments reported in the literature indicate that a change in the flow pattern occurs around 1 to 2 mm grain diameters, with air channels occurring below the transition size and bubbles above. Analysis of capillary and buoyancy forces suggests that for a given gas-liquid-solid system, there is a critical size that dictates the dominant force, and the dominant force will in turn dictate the flow pattern. The dominant forces, and consequently the two-phase flow patterns, were characterized using a Bond number modified with the porous media aspect ratio (pore throat to pore body ratio). Laboratory experiments were conducted to observe flow patterns as a function of porous media size and air flow rate. The experimental results and the modified Bond number analysis support the relationship of flow patterns to grain size reported in the literature.     

Real gas flow through heterogeneous porous media: theoretical aspects of upscaling

We consider the problem of upscaling transient real gas flow through heterogeneous bounded reservoirs. One of the commonly used methods for deriving effective permeabilities is based on stochastic averaging of nonlinear flow equations. Such an approach, however, would require rather restrictive assumptions about pressure-dependent coefficients. Instead, we use Kirchhoff transformation to linearize the governing stochastic equations prior to their averaging. The linearized problem is similar to that used in stochastic analysis of groundwater flow. We discuss the effects of temporal localization of the nonlocal averaged Darcy's law, as well as boundary effects, on the upscaled gas permeability. Extension of the results obtained by means of small perturbation analysis to highly heterogeneous porous formations is also discussed.     

An erosion criterion for gravel-bed rivers

A theoretically-based erosion criterion is developed for gravel-bed rivers which incorporates the effect of both grain geometry and turbulent velocity fluctuations. It is derived from a balance of instantaneous drag, lift, and gravity forces operating on individual grains and is calculated for spherical grains arranged in three distinct geometries. To accommodate the temporal variation in bed shear stress, the model includes a stochastic element based on the characteristics of turbulence derived from the flume evidence of McQuivey (1973a, b). In terms of the Shields parameter, results show reasonable agreement with the range of observations quoted from the field and with the experimental data of Fenton and Abbott (1977). Finally, the argument is generalized to cover applications in the wider context of field conditions including a range of grain sizes and flow conditions.     

Truncated plurigaussian simulations to characterize aquifer heterogeneity

Integrating geological concepts, such as relative positions and proportions of the different lithofacies, is of highest importance in order to render realistic geological patterns. The truncated plurigaussian simulation method provides a way of using both local and conceptual geological information to infer the distributions of the facies and then those of hydraulic parameters. The method ( Le Loc'h and Galli 1994 ) is based on the idea of truncating at least two underlying multi-Gaussian simulations in order to create maps of categorical variable. In this article, we show how this technique can be used to assess contaminant migration in highly heterogeneous media. We illustrate its application on the biggest contaminated site of Switzerland. It consists of a contaminant plume located in the lower fresh water Molasse on the western Swiss Plateau. The highly heterogeneous character of this formation calls for efficient stochastic methods in order to characterize transport processes.     

A multiple-point statistics algorithm for 3D pore space reconstruction from 2D images

Fluid flow behavior in a porous medium is a function of the geometry and topology of its pore space. The construction of a three dimensional pore space model of a porous medium is therefore an important first step in characterizing the medium and predicting its flow properties. A stochastic technique for reconstruction of the 3D pore structure of unstructured random porous media from a 2D thin section training image is presented. The proposed technique relies on successive 2D multiple point statistics simulations coupled to a multi-scale conditioning data extraction procedure. The Single Normal Equation Simulation Algorithm (SNESIM), originally developed as a tool for reproduction of long-range, curvilinear features of geological structures, serves as the simulation engine. Various validating criteria such as marginal distributions of pore and grain, directional variograms, multiple-point connectivity curves, single phase effective permeability and two phase relative permeability calculations are used to analyze the results. The method is tested on a sample of Berea sandstone for which a 3D micro-CT scanning image is available. The results confirm that the equi-probable 3D realizations obtained preserve the typical patterns of the pore space that exist in thin sections, reproduce the long-range connectivities, capture the characteristics of anisotropy in both horizontal and vertical directions and have single and two phase flow characteristics consistent with those of the measured 3D micro-CT image.     

Relating electrical conduction of alluvial sediments to textural properties and pore‐fluid conductivity

Electrical conductivity of alluvial sediments depends on litho‐textural properties, fluid saturation and porewater conductivity. Therefore, for hydrostratigraphic applications of direct current resistivity methods in porous sedimentary aquifers, it can be useful to characterize the prevailing mechanisms of electrical conduction (electrolytic or shale conduction) according to the litho‐textural properties and to the porewater characteristics. An experimental device and a measurement protocol were developed and applied to collect data on eight samples of alluvial sediments from the Po plain (Northern Italy), characterized by different grain‐size distribution, and fully saturated with porewater of variable conductivity. The bulk electrical conductivities obtained with the laboratory tests were interpreted with a classical two‐component model, which requires the identification of the intrinsic conductivity of clay particles and the effective porosity for each sample, and with a three‐component model. The latter is based on the two endmember mechanisms, surface and electrolytic conduction, but takes into account also the interaction between dissolved ions in the pores and the fluid‐grain interface. The experimental data and their interpretation with the phenomenological models show that the volumetric ratio between coarse and fine grains is a simple but effective parameter to determine the electrical behaviour of clastic hydrofacies at the scale of the representative elementary volume.     

Constraining complex aquifer geometry with geophysics (2-D ERT and MRS measurements) for stochastic modelling of groundwater flow

Stochastic modelling is a useful way of simulating complex hard-rock aquifers as hydrological properties (permeability, porosity etc.) can be described using random variables with known statistics. However, very few studies have assessed the influence of topological uncertainty (i.e. the variability of thickness of conductive zones in the aquifer), probably because it is not easy to retrieve accurate statistics of the aquifer geometry, especially in hard rock context. In this paper, we assessed the potential of using geophysical surveys to describe the geometry of a hard rock-aquifer in a stochastic modelling framework.The study site was a small experimental watershed in South India, where the aquifer consisted of a clayey to loamy–sandy zone (regolith) underlain by a conductive fissured rock layer (protolith) and the unweathered gneiss (bedrock) at the bottom. The spatial variability of the thickness of the regolith and fissured layers was estimated by electrical resistivity tomography (ERT) profiles, which were performed along a few cross sections in the watershed. For stochastic analysis using Monte Carlo simulation, the generated random layer thickness was made conditional to the available data from the geophysics. In order to simulate steady state flow in the irregular domain with variable geometry, we used an isoparametric finite element method to discretize the flow equation over an unstructured grid with irregular hexahedral elements.The results indicated that the spatial variability of the layer thickness had a significant effect on reducing the simulated effective steady seepage flux and that using the conditional simulations reduced the uncertainty of the simulated seepage flux.As a conclusion, combining information on the aquifer geometry obtained from geophysical surveys with stochastic modelling is a promising methodology to improve the simulation of groundwater flow in complex hard-rock aquifers.     

Geostatistical simulation of spatial variability of convective storms in Mexico City Valley

Precipitation is one of the main components of the hydrological cycle and knowledge of its spatial distribution is fundamental for the prediction of other closely related environmental variables, for example, runoff, flooding and aquifer recharge. Most of the precipitation in Mexico City is due to convective storms characterized by a high spatial variability, implying that modeling its behavior is very complex. In this work stochastic simulation techniques with a geostatistical approach were applied to model the spatial variability of the rainfall of three convective storms. The analysis of the results shows that using the proposed methodology spatial distributions of rain are obtained that reproduce the statistical characteristics present in the available information.     

Stochastic analysis of multiphase flow in porous media: 1. Spectral/perturbation approach

Stochastic analysis of steady-state multiphase (water, oil, and air) flow in heterogeneous porous media was performed using the perturbation theory and spectral representation techniques. The gas phase is assumed to have constant pressure. The governing equations describing the flow of oil and water are coupled and nonlinear. The key stochastic input variables are intrinsic permeability,k, and the soil grain size distribution index, . Three different stochastic combinations of these two input parameters were considered. The perturbation/spectral analysis was used to develop closed-form expressions that describe stochastic variability of key output processes, such as capillary and individual phase pressures and specific discharges. The analysis also included the derivation of the mean flow equations and estimation of the effective flow properties. The impact of the spatial variability ofk and on the effective conductivities and the variances of pressures and specific discharges was examined.     

利用普及型CT机获得HRCT图像的方法及临床应用

ＨＲＣＴ（高分辨ＣＴ）图像是较常规ＣＴ具有更清晰的空间分辨率的图像。可明显提高图像质量，从而提高病灶细节以及正常和异常细微结构的显示能力，对配合临床诊断及指导临床治疗有着很高的实用价值。如何利用普及型ＣＴ机获得ＨＲＣＴ图像。我们做了技术上的改进，并在不增加硬件设备的基础上利用软件功能，提高一些技术参数，主要是扫描层厚，管电流及扫描时间，高分辨薄层重建，尽可能的缩小显示野，加大矩阵等。所得到的图像分辨率明显高于常规ＣＴ，如肺小结节的显示，腔隙性梗塞的检出率，可疑小腔隙与大的像素颗粒的辨别等。我们认为，进一步开发软件功能。会获得更加理想的ＨＲＣＴ图像     

Induced Groundwater Flux by Increases in the Aquifer's Total Stress

Fluid‐filled granular soils experience changes in total stress because of earth and oceanic tides, earthquakes, erosion, sedimentation, and changes in atmospheric pressure. The pore volume may deform in response to the changes in stress and this may lead to changes in pore fluid pressure. The transient fluid flow can therefore be induced by the gradient in excess pressure in a fluid‐saturated porous medium. This work demonstrates the use of stochastic methodology in prediction of induced one‐dimensional field‐scale groundwater flow through a heterogeneous aquifer. A closed‐form of mean groundwater flux is developed to quantify the induced field‐scale mean behavior of groundwater flow and analyze the impacts of the spatial correlation length scale of log hydraulic conductivity and the pore compressibility. The findings provided here could be useful for the rational planning and management of groundwater resources in aquifers that contain lenses with large vertical aquifer matrix compressibility values.     

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.     

Plurigaussian modeling of geological domains based on the truncation of non-stationary Gaussian random fields

The plurigaussian model is used in mining engineering, oil reservoir characterization, hydrology and environmental sciences to simulate the layout of geological domains in the subsurface, while reproducing their spatial continuity and dependence relationships. However, this model is well-established only in the stationary case, when the spatial distribution of the domains is homogeneous in space, and suffers from theoretical and practical impediments in the non-stationary case. To overcome these limitations, this paper proposes extending the model to the truncation of intrinsic random fields of order k with Gaussian generalized increments, which allows reproducing spatial trends in the distribution of the geological domains. Methodological tools and algorithms are presented to infer the model parameters and to construct realizations of the geological domains conditioned to existing data. The proposal is illustrated with the simulation of rock type domains in an ore deposit in order to demonstrate its applicability. Despite the limited number of conditioning data, the results show a remarkable agreement between the simulated domains and the lithological model interpreted by geologists, while the conventional stationary plurigaussian model turns out to be unsuccessful.     

Continuous TDEM for monitoring shale hydraulic fracturing

Monitoring and delineating the spatial distribution of shale fracturing is fundamentally important to shale gas production. Standard monitoring methods, such as time-lapse seismic, cross-well seismic and micro-seismic methods, are expensive, timeconsuming, and do not show the changes in the formation with time. The resistivities of hydraulic fracturing fluid and reservoir rocks were measured. The results suggest that the injection fluid and consequently the injected reservoir are characterized by very low resistivity and high chargeability. This allows using of the controlled-source electromagnetic method (CSEM) to monitor shale gas hydraulic fracturing. Based on the geoelectrical model which was proposed according to the well-log and seismic data in the test area the change rule of the reacted electrical field was studied to account for the change of shale resistivity, and then the normalized residual resistivity method for time lapse processing was given. The time-domain electromagnetic method (TDEM) was used to continuously monitor the shale gas fracturing at the Fulin shale gas field in southern China. A high-power transmitter and multi-channel transient electromagnetic receiver array were adopted. 9 h time series of Ex component of 224 sites which were laid out on the surface and over three fracturing stages of a horizontal well at 2800 m depth was recorded. After data processing and calculation of the normalized resistivity residuals, the changes in the Ex signal were determined and a dynamic 3D image of the change in resistivity was constructed. This allows modeling the spatial distribution of the fracturing fluid. The model results suggest that TDEM is promising for monitoring hydraulic fracturing of shale.