Geostatistical modeling of clay spatial distribution in siliciclastic rock samples using the plurigaussian simulation method

In order to implement secondary and enhanced oil recovery processes in complex terrigenous formations as is usual in turbidite deposits, a precise knowledge of the spatial distribution of shale grains is a crucial element for the fluid flow prediction. The reason of this is that the interaction of water with shale grains can significantly modify their size and/or shape, which in turn would cause porous space sealing with the subsequent impact in the flow. In this work, a methodology for stochastic simulations of spatial grains distributions obtained from scanning electron microscopy images of siliciclastic rock samples is proposed. The aim of the methodology is to obtain stochastic models would let us investigate the shale grain behavior under various physico-chemical interactions and flux regimes, which in turn, will help us get effective petrophysical properties (porosity and permeability) at core scale. For stochastic spatial grains simulations a plurigaussian method is applied, which is based on the truncation of several standard Gaussian random functions. This approach is very flexible, since it allows to simultaneously manage the proportions of each grain category in a very general manner and to rigorously handle their spatial dependency relationships in the case of two or more grain categories. The obtained results show that the stochastically simulated porous media using the plurigaussian method adequately reproduces the proportions, basic statistics and sizes of the pore structures present in the studied reference images.     

Mapping optimization based on sampling size in earth related and environmental phenomena

A critical sampling grid can be defined for an earth related natural variable distributed in space, according to established theoretical results and under certain mathematical conditions. Sampling above this critical limit does not substantially improve mapping results, while based on this limit the ideal process of reproducing the original phenomenon is theoretically defined. The aim of the present paper is, by using an innovative approach; to investigate the validity of commonly used interpolation algorithms, both stochastic and deterministic, below and above this critical sampling limit. When sampling is dense, application to a simulated spatial random field shows that the results are equally accurate with those derived with more sophisticated stochastic methods. On the other hand, when the sampling grid is sparse, deterministic methods produce less accurate results, therefore stochastic algorithms with minimum estimation error are a much better option. To further demonstrate these points, the interpolation algorithms were applied in three different sampling grid densities in a contaminated waste disposal site in Russia.     

Spatial prediction model and its application to chemistry of atmospheric precipitation in Jordan

This study is concerned with the spatial variability of some wet atmospheric precipitation parameters such as; pH, conductivity (EC). The study also depicts the spatial variability of some ions (cations and anions) of atmospheric precipitation in Jordan such as, Ca²⁺, Mg²⁺, Na⁺ and K⁺, HCO₃⁻, Cl⁻, NO₃⁻ and SO₄²⁻. The basis of the work is to establish a relationship through the cumulative semivariogram technique between the distance ratios and the spatial dependence structure of the chemical composition of atmospheric precipitation. All semivariogram models are constructed in this study in order to understand the behavior of the spatial distribution. The spatial distributions of rainwater parameters show differences from station to station which is expressed in terms of angle, where the larger the angle the weaker the correlation. The semivariogram (SV) models are constructed to show the variation of the rainfall chemistry in Jordan. The SV models show weak correlation between mountain and leeside mountain stations, i.e. mountain and desert stations. On the other hand, good correlations are observed when transferring from south to north of the country. The larger is the found angle, the weaker is the correlation. For most of the SV model the correlation is found to be very weak between desert and mountainous locality. The Standard Regional Dependence Factor (SRDF) is used for prediction of the distribution of rain fall parameters. It shows the relative error between observed and predicted values of rainwater parameters. The overall regional relative error between the observed and estimated concentrations remains less than 15%.     

Increasing the accuracy of nitrogen dioxide (NO2) pollution mapping using geographically weighted regression (GWR) and geostatistics

Nitrogen dioxide (NO₂) is known to act as an environmental trigger for many respiratory illnesses. As a pollutant it is difficult to map accurately, as concentrations can vary greatly over small distances. In this study three geostatistical techniques were compared, producing maps of NO₂ concentrations in the United Kingdom (UK). The primary data source for each technique was NO₂ point data, generated from background automatic monitoring and background diffusion tubes, which are analysed by different laboratories on behalf of local councils and authorities in the UK. The techniques used were simple kriging (SK), ordinary kriging (OK) and simple kriging with a locally varying mean (SKlm). SK and OK make use of the primary variable only. SKlm differs in that it utilises additional data to inform prediction, and hence potentially reduces uncertainty. The secondary data source was oxides of nitrogen (NO_x) derived from dispersion modelling outputs, at 1 km × 1 km resolution for the UK. These data were used to define the locally varying mean in SKlm, using two regression approaches: (i) global regression (GR) and (ii) geographically weighted regression (GWR). Based upon summary statistics and cross-validation prediction errors, SKlm using GWR derived local means produced the most accurate predictions. Therefore, using GWR to inform SKlm was beneficial in this study.     

Geostatistical analysis of arsenic concentration in groundwater in Bangladesh using disjunctive kriging

The National Hydrochemical Survey of Bangladesh sampled the water from 3,534 tube wells for arsenic throughout most of Bangladesh. It showed that 27% of the shallow tube wells (less than 150 m deep) and 1% of the deep tube wells (more than 150 m deep) exceeded the Bangladesh standard for arsenic in drinking water (50 µg L^–1). Statistical analyses revealed the main characteristics of the arsenic distribution. Concentrations ranged from less than the detection limit (0.5 µg L^–1), to as much as 1,600 µg L^–1, though with a very skewed distribution, and with spatial dependence extending to some 180 km. Disjunctive kriging was used to estimate concentrations of arsenic in the shallow ground water and to map the probability that the national limit for arsenic in drinking water was exceeded for most of the country (the Chittagong Hill Tracts and the southern coastal region were excluded). A clear regional pattern was identified, with large probabilities in the south of the country and small probabilities in much of the north including the Pleistocene Tracts. Using these probabilities, it was estimated that approximately 35 million people are exposed to arsenic concentrations in groundwater exceeding 50 µg L^–1 and 57 million people are exposed to concentrations exceeding 10 µg L^–1 (the WHO guideline value).     

Conditional simulation of intrinsic random functions of order k

Conditional simulation of intrinsic random functions of orderk is a stochastic method that generates realizations which mimic the spatial fluctuation of nonstationary phenomena, reproduce their generalized covariance and honor the available data at sampled locations. The technique proposed here requires the following steps: (i) on-line simulation of Wiener-Levy processes and of their integrations; (ii) use of the turning-bands method to generate realizations in Rⁿ; (iii) conditioning to available data; and (iv) verification of the reproduced generalized covariance using generalized variograms. The applicational aspects of the technique are demonstrated in two and three dimensions. Examples include the conditional simulation of geological variates of the Crystal Viking petroleum reservoir, Alberta, Canada.     

一种新的储量估算法—杨赤中滤波与推估法及其应用

杨赤中滤波与推估法是一种对矿床地质变量统计的新方法。为使这一新理论用于勘探网规则或不规则情况下及在矿体小而勘探抽样点比较少,产状复杂的铀矿体的储量估算,我们用此法对2个矿床的5个矿体进行了储量估算,取得了较满意的效果。实践证明,杨赤中滤波与推估法与传统的块段法比较,具有公式简明、使用方便、易掌握、计算速度快、结果准确,花费少、经济效益好等优点。本文主要介绍了此种方法实际应用的步骤,应用的体会以及效果的初步评价。     

地质统计学新进展

地质统计学(空间信息统计学)是数学地质领域中一门发展迅速且有着广泛应用前景的新兴科学。结合地质统计学发展现状,对地质统计学的新进展进行了研究,从地质统计学理论体系、应用及软件开发等方面探讨了地质统计学的发展前缘。并指出时空多元技术、条件模拟、非参数和非线性将是地质统计学今后的发展趋势。     

Spatial and temporal study of nitrate concentration in groundwater by means of coregionalization

 Spatial and temporal behavior of hydrochemical parameters in groundwater can be studied using tools provided by geostatistics. The cross-variogram can be used to measure the spatial increments between observations at two given times as a function of distance (spatial structure). Taking into account the existence of such a spatial structure, two different data sets (sampled at two different times), representing concentrations of the same hydrochemical parameter, can be analyzed by cokriging in order to reduce the uncertainty of the estimation. In particular, if one of the two data sets is a subset of the other (that is, an undersampled set), cokriging allows us to study the spatial distribution of the hydrochemical parameter at that time, while also considering the statistical characteristics of the full data set established at a different time. This paper presents an application of cokriging by using temporal subsets to study the spatial distribution of nitrate concentration in the aquifer of the Lucca Plain, central Italy. Three data sets of nitrate concentration in groundwater were collected during three different periods in 1991. The first set was from 47 wells, but the second and the third are undersampled and represent 28 and 27 wells, respectively. Comparing the result of cokriging with ordinary kriging showed an improvement of the uncertainty in terms of reducing the estimation variance. The application of cokriging to the undersampled data sets reduced the uncertainty in estimating nitrate concentration and at the same time decreased the cost of the field sampling and laboratory analysis. Received: 23 July 1997 · Accepted: 31 March 1998     

A Distance-based Prior Model Parameterization for Constraining Solutions of Spatial Inverse Problems

Spatial inverse problems in the Earth Sciences are often ill-posed, requiring the specification of a prior model to constrain the nature of the inverse solutions. Otherwise, inverted model realizations lack geological realism. In spatial modeling, such prior model determines the spatial variability of the inverse solution, for example as constrained by a variogram, a Boolean model, or a training image-based model. In many cases, particularly in subsurface modeling, one lacks the amount of data to fully determine the nature of the spatial variability. For example, many different training images could be proposed for a given study area. Such alternative training images or scenarios relate to the different possible geological concepts each exhibiting a distinctive geological architecture. Many inverse methods rely on priors that represent a single subjectively chosen geological concept (a single variogram within a multi-Gaussian model or a single training image). This paper proposes a novel and practical parameterization of the prior model allowing several discrete choices of geological architectures within the prior. This method does not attempt to parameterize the possibly complex architectures by a set of model parameters. Instead, a large set of prior model realizations is provided in advance, by means of Monte Carlo simulation, where the training image is randomized. The parameterization is achieved by defining a metric space which accommodates this large set of model realizations. This metric space is equipped with a “similarity distance” function or a distance function that measures the similarity of geometry between any two model realizations relevant to the problem at hand. Through examples, inverse solutions can be efficiently found in this metric space using a simple stochastic search method.