Fractal and geostatistical methods for modeling of a fracture network

The modeling of fracture networks is useful for fluid flow and rock mechanics studies. About 6600 fracture traces were recorded on drifts of a uranium mine in a granite massif. The traces have an extension of 0.20–20 m. The network was studied by fractal and by geostatistical methods but can be considered neither as a fractal with a constant dimension nor a set of purely randomly located fractures. Two kinds of generalization of conventional models can still provide more flexibility for the characterization of the network: (a) a nonscaling fractal model with variable similarity dimension (for a 2-D network of traces, the dimension varying from 2 for the 10-m scale to 1 for the centimeter scale, (b) a parent-daughter model with a regionalized density; the geostatistical study allows a 3-D model to be established where: fractures are assumed to be discs; fractures are grouped in clusters or swarms; and fracturation density is regionalized (with two ranges at about 30 and 300 m). The fractal model is easy to fit and to simulate along a line, but 2-D and 3-D simulations are more difficult. The geostatistical model is more complex, but easy to simulate, even in 3-D.This paper was presented at Emerging Concepts, MGUS-87 Conference, Redwood City, California, 13–15 April 1987.     

Modelling Spatial Variability Along Drainage Networks with Geostatistics

Local characteristics of drainage networks such as cross-section geometry and hydraulic roughness coefficient, influence surface water transfers and must be taken into account when assessing the impact of human activities on hydrological risks. However, as these characteristics have not been available till now through remote sensing or hydrological modelling, the only available methods are interpolation or simulation based on scarce data. In this paper we propose a statistical model based on geostatistics that allows us to take account of both the spatial distribution and spatial uncertainties. To do this, we modify the geostatistical framework to suit directed tree supports corresponding to drainage network structures. The stationarity concept is specified assuming conditional independence between parts of the network; variogram fitting and modelling are then modified accordingly. A sequential multi Gaussian simulation procedure going upstream along the network is proposed. We illustrate this approach by studying the width of an 11-km long artificial drainage network in the south of France.     

Comparison of three geostatistical methods for hydrofacies simulation: a test on alluvial sediments

The hydrodispersive properties of porous sediments are strongly influenced by the heterogeneity at fine scales, which can be modeled by geostatistical simulations. In order to improve the assessment of the properties of three different geostatistical simulation methods (Sequential indicator simulation, SISIM; Transition probability geostatistical simulation, T-PROGS; Multiple point simulation, MPS) a comparison test at different scales was performed for a well-exposed aquifer analogue. In the analysed volume (approximately 30,000?m³) four operative hydrofacies have been recognised: very fine sand and silt, sand, gravelly sand and open framework gravel. Several equiprobable realizations were computed with SISIM, MPS and T-PROGS for a test volume of approximately 400?m³ and for the entire volume, and the different outcomes were compared with visual inspection and connectivity analysis of the very or poorly permeable structures. The comparison of the different simulations shows that the geological model is best reproduced when the simulations are realised separately for each highest rank depositional element and subsequently merged. Moreover, the three methods yield different images of the volume; in particular MPS is efficient in mapping the geometries of the most represented hydrofacies, whereas SISIM and T-PROGS can account for the distribution of the less represented facies.     

Geostatistical modelling of spatial and depth variability of SPT data for Bangalore

The purpose of this study is to develop a geostatistical model to evaluate the spatial and depth variability of Standard Penetration Test (SPT) data from Bangalore, India. The database consists of 766 boreholes spread over a 220 km² area, with several SPT values (N) in each of them. The geostatistical analysis is done for N corrected (N _corrected) values. The N _corrected value has been corrected for different parameters such as overburden stress, size of the bore hole, type of sampler, hammer energy and length of the connecting rod. The knowledge of the semivariogram of the SPT data is used with kriging theory to estimate the values at points in the subsurface of Bangalore where field measurements are not available. The model is used to compute the variance of estimated data. The model predicts reasonably well the SPT data. The geostatistical model provides valuable results that can be used for seismic hazard analysis, site response and liquefaction studies for the development of microzonation maps. The predicted N values from the developed model can also be used to estimate the subsurface information, allowable bearing pressure of soils and elastic modulus of soils.     

Challenges in reservoir forecasting

The combination of geostatistics-based numerical geological models and finite difference flow simulation has improved our ability to predict reservoir performance. The main contribution of geostatistical modeling has been more realistic representations of reservoir heterogeneity. Our understanding of the physics of fluid flow in porous media is reasonably captured by flow simulators in common usage. Notwithstanding the increasing application and success of geostatistics and flow simulation there remain many important challenges in reservoir forecasting. This application has alerted geoscientists and physicists that geostatistical/flow models in many respects, are, engineering approximations to thereal spatial distribution andreal flow processes. This paper reviews current research directions and presents some new ideas of where reserach could be focused to improve our ability to model geological features, model flow processes, and, ultimately, improve reservoir performance predictions.     

Conditional Latin Hypercube Simulation of (Log)Gaussian Random Fields

In earth and environmental sciences applications, uncertainty analysis regarding the outputs of models whose parameters are spatially varying (or spatially distributed) is often performed in a Monte Carlo framework. In this context, alternative realizations of the spatial distribution of model inputs, typically conditioned to reproduce attribute values at locations where measurements are obtained, are generated via geostatistical simulation using simple random (SR) sampling. The environmental model under consideration is then evaluated using each of these realizations as a plausible input, in order to construct a distribution of plausible model outputs for uncertainty analysis purposes. In hydrogeological investigations, for example, conditional simulations of saturated hydraulic conductivity are used as input to physically-based simulators of flow and transport to evaluate the associated uncertainty in the spatial distribution of solute concentration. Realistic uncertainty analysis via SR sampling, however, requires a large number of simulated attribute realizations for the model inputs in order to yield a representative distribution of model outputs; this often hinders the application of uncertainty analysis due to the computational expense of evaluating complex environmental models. Stratified sampling methods, including variants of Latin hypercube sampling, constitute more efficient sampling aternatives, often resulting in a more representative distribution of model outputs (e.g., solute concentration) with fewer model input realizations (e.g., hydraulic conductivity), thus reducing the computational cost of uncertainty analysis. The application of stratified and Latin hypercube sampling in a geostatistical simulation context, however, is not widespread, and, apart from a few exceptions, has been limited to the unconditional simulation case. This paper proposes methodological modifications for adopting existing methods for stratified sampling (including Latin hypercube sampling), employed to date in an unconditional geostatistical simulation context, for the purpose of efficient conditional simulation of Gaussian random fields. The proposed conditional simulation methods are compared to traditional geostatistical simulation, based on SR sampling, in the context of a hydrogeological flow and transport model via a synthetic case study. The results indicate that stratified sampling methods (including Latin hypercube sampling) are more efficient than SR, overall reproducing to a similar extent statistics of the conductivity (and subsequently concentration) fields, yet with smaller sampling variability. These findings suggest that the proposed efficient conditional sampling methods could contribute to the wider application of uncertainty analysis in spatially distributed environmental models using geostatistical simulation.     

Scales of Reservoir Heterogeneities and Impact of Seismic Resolution on Geostatistical Integration

Seismic measurements may be used in geostatistical techniques for estimation and simulation of petrophysical properties such as porosity. The good correlation between seismic and rock properties provides a basis for these techniques. Seismic data have a wide spatial coverage not available in log or core data. However, each seismic measurement has a characteristic response function determined by the source-receiver geometry and signal bandwidth. The image response of the seismic measurement gives a filtered version of the true velocity image. Therefore the seismic image cannot reflect exactly the true seismic velocity at all scales of spatial heterogeneities present in the Earth. The seismic response function can be approximated conveniently in the spatial spectral domain using the Born approximation. How the seismic image response affects the estimation of variogram. and spatial scales and its impact on geostatistical results is the focus of this paper. Limitations of view angles and signal bandwidth not only smooth the seismic image, increasing the variogram range, but also can introduce anisotropic spatial structures into the image. The seismic data are enhanced by better characterizing and quantifying these attributes. As an exercise, examples of seismically assisted cokriging and cosimulation of porosity between wells are presented.     

The Importance of Time Synchronization in the Local Networks of the Science and Application Center for Lunar and Deep-space Exploration

The data acquisition stations and the data processing center of the Science and Application Center for Lunar and Deep-space Exploration (SACLuDE) are located at different geographical sites. They respectively have their own local networks and interconnect with each other through access to the core data network. This paper describes the clock drift in the computer and other networked devices building up the infrastructure of the above local networks. The network time variance of the stochastic model is also estimated. The poor precision of network synchronization will bring about potential hazards to the network operation and application running in the networks, which is clarified in the present paper. At the end of the paper, a cost-effective and feasible solution is proposed based on the Global Position System (GPS) and the Network Time Protocol (NTP).     

Modelling a real-world buried valley system with vertical non-stationarity using multiple-point statistics

Stationarity has traditionally been a requirement of geostatistical simulations. A common way to deal with non-stationarity is to divide the system into stationary sub-regions and subsequently merge the realizations for each region. Recently, the so-called partition approach that has the flexibility to model non-stationary systems directly was developed for multiple-point statistics simulation (MPS). The objective of this study is to apply the MPS partition method with conventional borehole logs and high-resolution airborne electromagnetic (AEM) data, for simulation of a real-world non-stationary geological system characterized by a network of connected buried valleys that incise deeply into layered Miocene sediments (case study in Denmark). The results show that, based on fragmented information of the formation boundaries, the MPS partition method is able to simulate a non-stationary system including valley structures embedded in a layered Miocene sequence in a single run. Besides, statistical information retrieved from the AEM data improved the simulation of the geology significantly, especially for the deep-seated buried valley sediments where borehole information is sparse.     

Neural networks and their derivatives for history matching and reservoir optimization problems

In geosciences, complex forward problems met in geophysics, petroleum system analysis, and reservoir engineering problems often require replacing these forward problems by proxies, and these proxies are used for optimizations problems. For instance, history matching of observed field data requires a so large number of reservoir simulation runs (especially when using geostatistical geological models) that it is often impossible to use the full reservoir simulator. Therefore, several techniques have been proposed to mimic the reservoir simulations using proxies. Due to the use of experimental approach, most authors propose to use second-order polynomials. In this paper, we demonstrate that (1) neural networks can also be second-order polynomials. Therefore, the use of a neural network as a proxy is much more flexible and adaptable to the nonlinearity of the problem to be solved; (2) first-order and second-order derivatives of the neural network can be obtained providing gradients and Hessian for optimizers. For inverse problems met in seismic inversion, well by well production data, optimal well locations, source rock generation, etc., most of the time, gradient methods are used for finding an optimal solution. The paper will describe how to calculate these gradients from a neural network built as a proxy. When needed, the Hessian can also be obtained from the neural network approach. On a real case study, the ability of neural networks to reproduce complex phenomena (water cuts, production rates, etc.) is shown. Comparisons with second polynomials (and kriging methods) will be done demonstrating the superiority of the neural network approach as soon as nonlinearity behaviors are present in the responses of the simulator. The gradients and the Hessian of the neural network will be compared to those of the real response function.     

A Comparison of Methods for the Stochastic Simulation of Rock Fractures

Methods reported in the literature for rock fracture simulations include approaches based on stochastic geometry, multiple-point statistics and a combination of geostatistics for fracture density and object-based modelling for fracture geometries. The advantages and disadvantages of each of these approaches are discussed with examples. By way of review, the authors begin with the geostatistical indicator simulation method, based on the truncated–Gaussian algorithm; this is followed by multiple-point statistical simulation and then the stochastic geometry approach, which is based on marked point process simulation. A new approach, based on pluriGaussian structural simulation, is then introduced. The new approach incorporates in the simulation the spatial correlation between different sets of fractures, which in general, is very difficult, if not impossible, to accomplish in the three methods reviewed. Each simulation method is summarised together with detailed simulation procedures for each. A published two-dimensional fracture dataset is used as a means of assessing the performance of each simulation method and of demonstrating the concepts discussed in the text.     

An intelligent neural network model for evaluating performance of immobilized cell biofilter treating hydrogen sulphide vapors

Biofiltration has shown to be a promising technique for handling malodours arising from process industries. The present investigation pertains to the removal of hydrogen sulphide in a lab scale biofilter packed with biomedia, encapsulated by sodium alginate and poly vinyl alcohol. The experimental data obtained under both steady state and shock loaded conditions were modelled using the basic principles of artificial neural networks. Artificial neural networks are powerful data driven modelling tools which has the potential to approximate and interpret complex input/output relationships based on the given sets of data matrix. A predictive computerised approach has been proposed to predict the performance parameters namely, removal efficiency and elimination capacity using inlet concentration, loading rate, flow rate and pressure drop as the input parameters to the artificial neural network model. Earlier, experiments from continuous operation in the biofilter showed removal efficiencies from 50 to 100 % at inlet loading rates varying up to 13 g H₂S/m³h. The internal network parameter of the artificial neural network model during simulation was selected using the 2^k factorial design and the best network topology for the model was thus estimated. The results showed that a multilayer network (4-4-2) with a back propagation algorithm was able to predict biofilter performance effectively with R² values of 0.9157 and 0.9965 for removal efficiency and elimination capacity in the test data. The proposed artificial neural network model for biofilter operation could be used as a potential alternative for knowledge based models through proper training and testing of the state variables.     

Probability of plume interception using conditional simulation of hydraulic head and inverse modeling

A procedure to estimate the probability of intercepting a contaminant groundwater plume for monitoring network design has been developed and demonstrated. The objective of the procedure is to use all available information in a method that accounts for the heterogeneity of the aquifer and the paucity of data. The major components of the procedure are geostatistical conditional simulation and parameter estimation that are used sequentially to generate flow paths from a suspected contaminant source location to a designated monitoring transect. From the flow paths, a histogram is constructed that represents the spatial probability distribution of plume centerlines. With an independent estimate of the plume width, a relationship between the total cost and the probability of detecting a plume can be made. The method uses geostatistical information from hydraulic head measurements and is conditioned by the data and the physics of groundwater flow. This procedure was developed specifically for the design of monitoring systems at sites where very few, if any, hydraulic conductivity data are available.     

Characterization and reconstruction of a rock fracture surface by geostatistics

It is well understood that, in studying the mechanical and hydromechanical behaviour of rock joints, their morphology must be taken into account. A geostatistical approach has been developed for characterizing the morphology of fracture surfaces at a decimetre scale. This allows the analysis of the spatial variability of elevations, and their first and second derivatives, with the intention of producing a model that gives a numerical three‐dimensional (3D) representation of the lower and upper surfaces of the fracture. Two samples (I and II) located close together were cored across a natural fracture. The experimental data are the elevations recorded along profiles (using recording steps of 0.5 and 0.02 mm, respectively, for the samples I and II). The goal of this study is to model the surface topography of sample I, so getting estimates for elevations at each node of a square grid whose mesh size will be, for mechanical purposes, no larger than the recording step. Since the fracture surface within the sample core is not strictly horizontal, geostatistical methods are applied to residuals of elevations of sample I. Further, since structural information is necessary at very low scale, theoretical models of variograms of elevations, first and second derivatives are fitted using data of both that sample I and sample II. The geostatistical reconstructions are computed using kriging and conditional simulation methods. In order to validate these reconstructions, variograms and distributions of experimental data are compared with variograms and distributions of the fitted data. Copyright © 2002 John Wiley & Sons, Ltd.     

Calculation of Uncertainty in the Variogram

There are often limited data available in early stages of geostatistical modeling. This leads to considerable uncertainty in statistical parameters including the variogram. This article presents an approach to calculate the uncertainty in the variogram. A methodology to transfer this uncertainty through geostatistical simulation and decision making is also presented.The experimental variogram value for a separation lag vector h is a mean of squared differences. The variance of a mean can be calculated with a model of the correlation between the pairs of data used in the calculation. The data here are squared differences; therefore, we need a measure of a 4-point correlation. A theoretical multi-Gaussian approach is presented for this uncertainty assessment together with a number of examples. The theoretical results are validated by numerical simulation. The simulation approach permits generalization to non-Gaussian situations.Multiple plausible variograms may be fit knowing the uncertainty at each variogram point, . Multiple geostatistical realizations may then be constructed and subjected to process assessment to measure the impact of this uncertainty.     

Geostatistical Seismic Inversion with Direct Sequential Simulation and Co-simulation with Multi-local Distribution Functions

Stochastic sequential simulation is a common modelling technique used in Earth sciences and an integral part of iterative geostatistical seismic inversion methodologies. Traditional stochastic sequential simulation techniques based on bi-point statistics assume, for the entire study area, stationarity of the spatial continuity pattern and a single probability distribution function, as revealed by a single variogram model and inferred from the available experimental data, respectively. In this paper, the traditional direct sequential simulation algorithm is extended to handle non-stationary natural phenomena. The proposed stochastic sequential simulation algorithm can take into consideration multiple regionalized spatial continuity patterns and probability distribution functions, depending on the spatial location of the grid node to be simulated. This work shows the application and discusses the benefits of the proposed stochastic sequential simulation as part of an iterative geostatistical seismic inversion methodology in two distinct geological environments in which non-stationarity behaviour can be assessed by the simultaneous interpretation of the available well-log and seismic reflection data. The results show that the elastic models generated by the proposed stochastic sequential simulation are able to reproduce simultaneously the regional and global variogram models and target distribution functions relative to the average volume of each sub-region. When used as part of a geostatistical seismic inversion procedure, the retrieved inverse models are more geologically realistic, since they incorporate the knowledge of the subsurface geology as provided, for example, by seismic and well-log data interpretation.     

Improving Sequential Simulation with a Structured Path Guided by Information Content

Multiple-point simulation is a newly developed geostatistical method that aims at combining the strengths of two mainstream geostatistical methods: object-based and pixel-based methods. It maintains the flexibility of pixel-based algorithms in data conditioning, while enhancing its capability of reproducing realistic geological shapes, which is traditionally reserved to object-based algorithms. However, the current snesim program for multiple-point simulation has difficulty in reproducing large-scale structures, which have a significant impact on the flow response. To address this problem, we propose to simulate along a structured path based on an information content measure. This structured path accounts for not only the information from the data, but also some prior structural information provided by geological knowledge. Various case studies show a better reproduction of large-scale structures. This concept of simulating along a structured path guided by information content can be applied to any sequential simulation algorithms, including traditional variogram-based two-point geostatistical algorithms.     

Prediction of creep characteristic of rock under varying environment

The strain developed due to creep is mainly proportional to the logarithm of the time under load, and is mostly proportional to the stress and temperature. At higher temperature the creep rate falls slowly with respect to time, and the creep strain is proportional to a fractional power of time, with the exponent increasing as the temperature increases and reaching a value approximately one-third at temperatures of about 0.5°C. At these temperatures, the creep increases with stress according to a power greater than unity and possibly exponentially. It increases with temperature as (−U/kT), where U is an activation energy and k is Boltzman’s constant. There are different methods to determine the creep strain and the energy of Jog (B) including experimental methods, multivariate regression analysis, and by numerical simulation. These methods are less cumbersome and time consuming. In the present investigation, artificial neural network technique has been used for prediction of the creep strain and energy of Jog (B). Two different networks have been tested and validated. Both the networks have four input neurons and one hidden layer with five neurons, and one output neuron. The data for different rocks at temperatures up to 750°C under conditions of compressive or tortional stress are taken from the literatures. The training and testing data sets used were 163 and 14, respectively. To deal with the problem of overfitting of data, Bayesian regulation has been used and network is trained with suitable training epochs. The coefficients of correlation among the predicted and observed values are found high and they improve the confidence of the users. The mean absolute percentage error obtained are also very low.     

An adaptive grid technique for minimizing heterogeneity of cells or elements

Geostatistical techniques allow simulation of properties such as porosity or conductivity on a fine scale. Typically, porous media flow modeling is performed at a coarser scale. Upscaling properties from the fine scale to the coarser scale introduces potential errors which are constrained by the degree of homogeneity of the cell or element. Adaptive grid techniques can be used to minimize the heterogeneity in the individual cells or elements, thus minimizing potential upscaling errors. A geostatistical adaptive grid (GAG) algorithm based on local minimization of heterogeneity is introduced. Local minimization allows greater control over the type of distortion permitted. Comparisons are made with a general elastic grid adjustment (GEGA) algorithm based on global minimization of heterogeneity. Several sample problems are used to test and demonstrate the two approaches.     

Conditional simulation with data subject to measurement error: Post-simulation filtering with modified factorial kriging

Conditional simulation with data subject to measurement error has received little attention in the geostatistical literature. The treatment of measurement error in simulation must be different from its treatment in estimation. Two approaches are examined: pre- and post-simulation filtering of data measurement error. The pre-simulation filtering is shown to be inefficient. The post-simulation filtering performs best. It is done by factorial kriging and a modified version of factorial kriging which ensures predetermined theoretical variance for the filtered data. It also is shown that the theoretical variogram of the filtered data reproduces the underlying variogram (i.e., without noise) almost perfectly. A simulation with a high level of correlated noise is used for validation and comparison. The post-simulation filtered values show an experimental variogram in agreement with the previously identified underlying variogram. Moreover, the filtered image compares well with the true image. The theoretical variogram corresponding to the post-simulation filter can be computed beforehand. Thus, the size of the simulation grid and of the filter neighborhood can be adjusted to ensure good reproduction of the underlying variogram.