Joint Inversion of Fracture Model Properties for CO2 Storage Monitoring or Oil Recovery History Matching

International audienceFor oil recovery or CO2 storage, "reservoirs" are commonly used to designate geological structures where oil can be found or CO2 can be stored. All reservoirs present a heterogeneity in terms of rock type and properties (such as porosity and permeability). In addition, some of these reservoirs present fractures and faults. Fractured reservoirs are an important part of the oil reserves in the world (Middle East, Gulf of Mexico, etc.) and some of them are important reservoirs in terms of oil volume and productivity in spite of the fractures. In addition, studies of reservoirs for geologic storage of CO2 have shown the existence of diffuse fractures and faults and their strong impacts on flow. A key point in fractured reservoirs is to understand the geometry and hydraulic conductivity of the network formed by the fractures. This requires the construction of a reservoir model that integrates all available conceptual knowledge and quantitative data. The topic of the present paper deals with a new methodology able to perform the history matching of a fractured reservoir model by adapting the sub-seismic fault properties and positions. The main difficulty of this work is to generate a sub-seismic fault network whose fault positions can be easily modified while respecting the statistical fault model. The sub-seismic fault model we have chosen allows us to obtain a sub-seismic fault network that is consistent with the seismic fault network and that succeeds in capturing the specific spatial organization of the faults. In a first step, the geometry of the seismic fault network is characterized using fractal methods. Sub-seismic faults are then generated according to a stochastic algorithm. Finally, the geometry of this discrete fracture network is optimized in order to match the hydrodynamic data about the reservoir. The optimization algorithm modifies the sub-seismic fault positions, leading to the history matching of the reservoir model. Fractal properties are preserved during the deformation process. These different steps are demonstrated on a realistic synthetic case

GEOSTATISTICAL ANALYSIS OF VALIDATION DATA OF AN AIR POLLUTION SIMULATOR

International audienceChemistry-transport models for air quality forecasting are affected by the uncertainty on the input data (emissions of pollutants and meteorological conditions), the approximations in the modelling of the physicochemical reactions, and numerical approximations (space and time discretization). The validation of the accuracy of these simulators can be done by comparing predictions with actual measurements. This exercise has been carried out for a model for daily forecasting at the scale of Europe, with reference to daily measurements at about one hundred stations over one year. A thorough variographic analysis shows that the error field cannot be characterized independently of the predicted and observed fields. Indeed the forecasts usually display space and time variations similar to those of the measurement data, up to a multiplicative factor, but are often poorly correlated with the reality. These results can be used to define priorities in the improvement of the chemistrytransport model. The presentation is focused on sulphates and nitrogen dioxide

Pointwise consistency of the kriging predictor with known mean and covariance functions

This paper deals with several issues related to the pointwise consistency of the kriging predictor when the mean and the covariance functions are known. These questions are of general importance in the context of computer experiments. The analysis is based on the properties of approximations in reproducing kernel Hilbert spaces. We fix an erroneous claim of Yakowitz and Szidarovszky (J. Multivariate Analysis, 1985) that the kriging predictor is pointwise consistent for all continuous sample paths under some assumptions.Comment: Submitted to mODa9 (the Model-Oriented Data Analysis and Optimum Design Conference), 14th-19th June 2010, Bertinoro, Ital

Density of States for a Specified Correlation Function and the Energy Landscape

The degeneracy of two-phase disordered microstructures consistent with a specified correlation function is analyzed by mapping it to a ground-state degeneracy. We determine for the first time the associated density of states via a Monte Carlo algorithm. Our results are described in terms of the roughness of the energy landscape, defined on a hypercubic configuration space. The use of a Hamming distance in this space enables us to define a roughness metric, which is calculated from the correlation function alone and related quantitatively to the structural degeneracy. This relation is validated for a wide variety of disordered systems.Comment: Accepted for publication in Physical Review Letter

Means and covariance functions for geostatistical compositional data: an axiomatic approach

This work focuses on the characterization of the central tendency of a sample of compositional data. It provides new results about theoretical properties of means and covariance functions for compositional data, with an axiomatic perspective. Original results that shed new light on the geostatistical modeling of compositional data are presented. As a first result, it is shown that the weighted arithmetic mean is the only central tendency characteristic satisfying a small set of axioms, namely continuity, reflexivity and marginal stability. Moreover, this set of axioms also implies that the weights must be identical for all parts of the composition. This result has deep consequences on the spatial multivariate covariance modeling of compositional data. In a geostatistical setting, it is shown as a second result that the proportional model of covariance functions (i.e., the product of a covariance matrix and a single correlation function) is the only model that provides identical kriging weights for all components of the compositional data. As a consequence of these two results, the proportional model of covariance function is the only covariance model compatible with reflexivity and marginal stability

Nested sampling and spatial analysis for reconnaissance investigations of soil: an example from agricultural land near mine tailings in Zambia

A reconnaissance survey was undertaken on soil near mine tailings to investigate variation in the content of copper, chromium and uranium. A nested sampling design was used. The data showed significant relations between the content of copper and uranium in the soil and its organic matter content, and a significant spatial trend in uranium content with distance from the tailings. Soil pH was not significantly related to any of the metals. The variance components associated with different scales of the sample design had large confidence intervals, but it was possible to show that the random variation was spatially dependent for all spatial models, whether for variation around a constant mean, or with a mean given by a linear effect of organic matter or distance to the tailings. For copper, we showed that a fractal or multifractal random model, with equal variance components for scales in a logarithmic progression, could be rejected for the model of variation around the fixed mean. The inclusion of organic matter as an explanatory factor meant that the fractal model could no longer be rejected, suggesting that the effect of organic matter results in spatial variation that is not scale invariant. It was shown, taking uranium as a case study, that further spatially nested sampling to estimate scale-dependent variance components, or to test a non-fractal model with adequate power, would require in the order of 200–250 samples in total

Sequential design of computer experiments for the estimation of a probability of failure

This paper deals with the problem of estimating the volume of the excursion set of a function $f:\mathbb{R}^d \to \mathbb{R}$ above a given threshold, under a probability measure on $\mathbb{R}^d$ that is assumed to be known. In the industrial world, this corresponds to the problem of estimating a probability of failure of a system. When only an expensive-to-simulate model of the system is available, the budget for simulations is usually severely limited and therefore classical Monte Carlo methods ought to be avoided. One of the main contributions of this article is to derive SUR (stepwise uncertainty reduction) strategies from a Bayesian-theoretic formulation of the problem of estimating a probability of failure. These sequential strategies use a Gaussian process model of $f$ and aim at performing evaluations of $f$ as efficiently as possible to infer the value of the probability of failure. We compare these strategies to other strategies also based on a Gaussian process model for estimating a probability of failure.Comment: This is an author-generated postprint version. The published version is available at http://www.springerlink.co

Optimal spatial design for air quality measurement surveys

International audienceMeasurement surveys using passive diffusion tubes are regularly car-ried out to elaborate atmospheric concentration maps over various areas. Sampling schemes must be designed to characterize both contaminant concentrations (of benzene or nitrogen dioxide for example) and their re-lations to environmental variables so as to obtain pollution maps as precise as possible. Here, a spatial statistical methodology to design benzene air concentration measurement surveys on the urban scale is exposed. In a first step, an a priori modeling is conducted that is based on the analysis of data coming from previous campaigns on two different agglomerations. More precisely, we retain a modeling with an external drift which consists of a drift plus a spatially correlated residual. The statistical analysis per-formed on available data leads to choose the most relevant auxiliary vari-ables and to determine an a priori variogram model for the residual. An a priori distribution is also defined for the variogram parameters, whose values appear to vary from a campaign to another. In a second step, we optimize the positioning of the measuring devices on a third agglomera-tion according to a Bayesian criterion. Practically, we aim at finding the design that minimizes the mean over the urban domain of the universal kriging variance, whose parameters are based on the a priori modeling, while accounting for the prior distribution over the variogram parameters. Two global optimization algorithms are compared: simulated annealing and a particle filter based algorithm

Analysis of multispecies point patterns by usingmultivariate log-Gaussian Cox processes

Multivariate log-Gaussian Cox processes are flexible models for multivariate point patterns. However, they have so far been applied in bivariate cases only. We move beyond the bivariate case to model multispecies point patterns of tree locations. In particular we address the problems of identifying parsimonious models and of extracting biologically relevant information from the models fitted. The latent multivariate Gaussian field is decomposed into components given in terms of random fields common to all species and components which are species specific. This allows a decomposition of variance that can be used to quantify to what extent the spatial variation of a species is governed by common or species-specific factors. Cross-validation is used to select the number of common latent fields to obtain a suitable trade-off between parsimony and fit of the data. The selected number of common latent fields provides an index of complexity of the multivariate covariance structure. Hierarchical clustering is used to identify groups of species with similar patterns of dependence on the common latent fields.We thank the Joint Editor, the Associate Editor and the two referees for constructive comments that helped to improve both content and exposition of this paper. Abdollah Jalilian and Rasmus Waagepetersen's research was supported by the Danish Natural Science Research Council, grant 09-072331 ‘Point process modelling and statistical inference’, Danish Council for Independent Research—Natural Sciences, grant 12-124675, ‘Mathematical and statistical analysis of spatial data’, and by Centre for Stochastic Geometry and Advanced Bioimaging, funded by a grant from the Villum Foundation. Yongtao Guan's research was supported by National Science Foundation grant DMS-0845368, by National Institutes of Health grant 1R01CA169043 and by the VELUX Visiting Professor programme. Jorge Mateu's research was supported by grants P1-1B2012-52 and MTM2013-43917-P. The BCI forest dynamics research project was made possible by National Science Foundation grants to Stephen P. Hubbell: DEB-0640386, DEB-0425651, DEB-0346488, DEB-0129874, DEB-00753102, DEB-9909347, DEB-9615226, DEB-9405933, DEB-9221033, DEB-9100058, DEB-8906869, DEB-8605042, DEB-8206992 and DEB-7922197, support from the Center for Tropical Forest Science, the Smithsonian Tropical Research Institute, the John D. and Catherine T. MacArthur Foundation, the Mellon Foundation, the Celera Foundation and numerous private individuals, and through the hard work of over 100 people from 10 countries over the past two decades. The plot project is part of the Center for Tropical Forest Science, a global network of large-scale demographic tree plots. The BCI soils data set was collected and analysed by J. Dalling, R. John, K. Harms, R. Stallard and J. Yavitt with support from National Science Foudation grants DEB021104, DEB021115, DEB0212284, DEB0212818 and Office of International Science and Engineering grant 0314581, Smithsonian Tropical Research Institute and Center for Tropical Forest Science. Paolo Segre and Juan Di Trani provided assistance in the field. The covariates dem, grad, mrvbf, solar and twi were computed in SAGA GIS by Tomislav Hengl (http://spatial-analyst.net/). We thank Dr Joseph Wright for sharing data on dispersal modes and life forms for the BCI tree specie