Petter Abrahamsen

Kriging with inequality constraints

A Gaussian random field with an unknown linear trend for the mean is considered. Methods for obtaining the distribution of the trend coefficients given exact data and inequality constraints are established. Moreover, the conditional distribution for the random field at any location is calculated so that predictions using e.g. the expectation, the mode, or the median can be evaluated and prediction error estimates using quantiles or variance can be obtained. Conditional simulation techniques are also provided.

Bayesian Kriging for Seismic Depth Conversion of a Multi-Layer Reservoir

A stochastic model for a petroleum reservoir with L seismic subsurfaces is presented. The seismic velocities within each layer is described by linear regression models and Gaussian random fields. Seismic interpretation errors are modeled as Gaussian random fields. Intercorrelations between all subsurfaces and all velocity fields are taken into consideration. This simplifies the handling of deviating wells and ensures consistent prediction and prediction variances for all L subsurfaces and L velocity fields. Bayesian kriging is used for prediction of subsurfaces and velocity fields.

Geostatistics : Oslo 2012

Foreword Preface Acknowledgements Part I Theory Sequential Simulation with Iterative Methods Applications of Randomized Methods for Decomposing and Simulating from Large Covariance Matrices Event-based Geostatistical Modeling: Description and Applications A Plurigaussian Model for Simulating Regionalized Compositions New Flexible Non-parametric Data Transformation for Trans-Gaussian Kriging Revisiting the Linear Model of Coregionalisation Modeling Nonlinear Beta Probability Fields Approximations of High-Order Spatial Statistics through Decomposition Multiple-Point Geostatistical Simulation Based on Genetic Algorithms Implemented in a Shared-Memory Supercomputer The Edge Effect in Geostatistical Simulations Extensions of the Parametric Inference of Spatial Covariances by Maximum Likelihood Part II Petroleum Constraining a Heavy Oil Reservoir to Temperature and Time Lapse Seismic Data Using the EnKF Micro-Modeling for Enhanced Small Scale Porosity-Permeability Relationships Applications of Data Coherency for Data Analysis and Geological Zonation Multiscale Modeling of Fracture Network in a Carbonate Reservoir Uncertainty Quantification and Feedback Control Using a Model Selection Approach Applied to a Polymer Flooding Process Efficient Conditional Simulation of Spatial Patterns Using a Pattern-Growth Algorithm Multiple-Point Statistics in a Non-Gridded Domain: Application to Karst/Fracture Network Modeling Sequential Simulations of Mixed Discrete-Continuous Properties: Sequential Gaussian Mixture Simulation Accounting for Seismic Trends in Stochastic Well Correlation Some Newer Algorithms in Joint Categorical and Continuous Inversion Problems around Seismic Data Non-Random Discrete Fracture Network Modelling Part III Mining Kriging and Simulation in Presence of Stationary Domains: Developments in Boundary Modeling Assessing Uncertainty in Recovery Functions: A Practical Approach Comparative Study of Localized Block Simulations and Localized Uniform Conditioning in the Multivariate Case Application of Stochastic Simulations and Quantifying Uncertainties in the Drilling of Roll Front Uranium Deposits Multivariate Estimation using Log Ratios: A Worked Alternative Measuring the Impact of the Change of Support and Information Effect at Olympic Dam Comparative Study of Two Gaussian Simulation Algorithms, Boddington Gold Deposit Non-multi-Gaussian Multivariate Simulations with Guaranteed Reproduction of Inter-Variable Correlations Field Parametric Geostatistics-a Rigorous Theory to Solve Problems of Highly Skewed Distributions Multiple-Point Geostatistics for Modeling Lithological Domains at a Brazilian Iron Ore Deposit Using the Single Normal Equations Simulation Algorithm Practical Implementation of Non-Linear Transforms for Modeling Geometallurgical Variables Combined Use of Lithological and Grade Simulations for Risk Analysis in Iron Ore, Brazil The Use of Geostatistical Simulation to Optimize the Homogenization and Blending Strategies Plurigaussian Simulations Used to Analyze the Uncertainty in Resources Estimation from a Lateritic Nickel Deposit Domaining by Clustering Multivariate Geostatistical Data The Influence of Geostatistical Techniques on Geological Uncertainty Part IV Environmental, Climate and Hydrology A Study on How Top-Surface Morphology Influences the Storage Capacity of CO2 in Saline Aquifers Modeling and Analysis of Daily Rainfall Data A Stochastic Model in Space and Time for Monthly Maximum Significant Wave Height Interpolation of Concentration Measurements by Kriging Using Flow Coordinates A Comparison of Methods for Solving the Sensor Location Problem Comparing Geostatistical Models for River Networks Author Index Subject Index

An Uncertainty Model for Fault Shape and Location

Fault models are often based on interpretations of seismic data that are constrained by observations of faults and associated strata in wells. Because of uncertainties in depth migration, seismic interpretations and well data, there often is significant uncertainty in the geometry and position of the faults. Fault uncertainty impacts determinations of reservoir volume, flow properties and well planning. Stochastic simulation of the faults is important for quantifying the uncertainties and minimizing the impacts. In this paper, a framework for representing and modeling uncertainty in fault location and geometry is presented. This framework can be used for prediction and stochastic simulation of fault surfaces, visualization of fault location uncertainty, and assessments of the sensitivity of fault location on reservoir performance. The uncertainty in fault location is represented by a fault uncertainty envelope and a marginal probability distribution. To be able to use standard geostatistical methods, quantile mapping is employed to construct a transformation from the fault surface domain to a transformed domain. Well conditioning is undertaken in the transformed domain using kriging or conditional simulations. The final fault surface is obtained by transforming back to the fault surface domain. Fault location uncertainty can be visualized by transforming the surfaces associated with a given quantile back to the fault surface domain.

Using Multiple Grids in Markov Mesh Facies Modeling

A multigrid Markov mesh model for geological facies is formulated by defining a hierarchy of nested grids and defining a Markov mesh model for each of these grids. The facies probabilities in the Markov mesh models are formulated as generalized linear models that combine functions of the grid values in a sequential neighborhood. The parameters in the generalized linear model for each grid are estimated from the training image. During simulation, the coarse patterns are first laid out, and by simulating increasingly finer grids we are able to recreate patterns at different scales. The method is applied to several tests cases and results are compared to the training image and the results of a commercially available snesim algorithm. In each test case, simulation results are compared qualitatively by visual inspection, and quantitatively by using volume fractions, and an upscaled permeability tensor. When compared to the training image, the method produces results that only have a few percent deviation from the values of the training image. When compared with the snesim algorithm the results in general have the same quality. The largest computational cost in the multigrid Markov mesh is the estimation of model parameters from the training image. This is of comparable CPU time to that of creating one snesim realization. The simulation of one realization is typically ten times faster than the estimation.

Modification of the Snesim Algorithm

Snesim is an algorithm for simulating complex structures. It is a pixel based method that sequentially simulates the nodes in a random order, while considering multipoint statistics. The conditional probabilities are extracted directly from a training image. If a configuration of the set of conditioned nodes is not found in the training image, the set is decreased node by node until the configuration of the final set is found. We have modified the Snesim algorithm by enabling it to also delete nodes, i.e. instead of just removing nodes from the set of conditioning nodes we let the set remain intact but delete the values of the relevant nodes. When all nodes are visited, the deleted nodes are revisited in a second iteration, and this is repeated until all nodes are sampled. For a training image consisting of channels, the modified algorithm has shown improved results. The original Snesim algorithm had problems with loose end channels, and with the modified algorithm the number of loose ends has decreased significantly. The number of iterations required for the simulations of the modified algorithm is insignificant, and only an increase of 23% of the total number of nodes had to be re-sampled.

Estimation of Gross Rock Volume of Filled Geological Structures With Uncertainty Measures

The gross rock volume of a filled structure is uncertain because of uncertainty in the determination of caprock depth and the uncertainty in depth to the hydrocarbon contact determined by the spill point of the caprock. Ignoring this uncertainty might lead to biased volume estimates. This paper reports two procedures to assist with assessing this uncertainty to obtain better estimates. The first is to use conditional simulation techniques to generate realizations of the depth to the caprock. The second procedure is a new fast algorithm that determines the location of the spill point and trapped area of each caprock realization. Taken together, the two procedures determine the thickness and lateral extension of each reservoir realization. Finally, gross rock volume for each realization can be calculated and the volumetric uncertainty can be quantified in terms of expectation, histograms, percentiles, etc., for the whole set of realizations. A synthetic example and an example from the North Sea illustrate the use of these procedures. A method for including knowledge of the spill-point depth for improving depth maps is also presented.

A Study on How Top-Surface Morphology Influences the Storage Capacity of CO2 in Saline Aquifers

The primary trapping mechanism in CO2 storage is structural trapping, which means accumulation of a CO2 column under a deformation in the caprock. We present a study on how different top-seal morphologies will influence the CO2 storage capacity and migration patterns. Alternative top-surface morphologies are created stochastically by combining different stratigraphic scenarios with different structural scenarios. Stratigraphic surfaces are generated by Gaussian random fields, while faults are generated by marked point processes. The storage capacity is calculated by a simple and fast spill-point analysis, and by a more extensive method including fluid flow simulation in which parameters such as pressure and injection rate are taken into account. Results from the two approaches are compared. Moreover, by generating multiple realizations, we quantify how uncertainty in the top-surface morphology impacts the primary storage capacity. The study shows that the morphology of the top seal is of great importance both for the primary storage capacity and for migration patterns.

COMBINING METHODS FOR SUBSURFACE PREDICTION

The depth to subsurfaces in a multi-layer model is obtained by adding the thickness of layers. However, the choice of layering is not unique so there will often be alternative ways of obtaining the depth to a particular subsurface. Each layer thickness can be described by a stochastic model accounting for uncertainties in the thickness. Stochastic models for the depth to subsurfaces are obtained from these. Alternative layer models will give alternative stochastic models and thus alternative depth predictions for the same subsurface. Two approaches to resolve this ambiguity is proposed. The first uses an established method of unbiased linear combination of predictors. The second and new approach combines the alterna- tive stochastic models into a single stochastic model giving a single predictor for subsurface depth. This predictor performs similarly to the approach combining several predictors while drastically reducing computational costs. The proposed method applies to layered geological structures using a combination of universal or Bayesian kriging and cokriging.

Efficient Neighborhoods for Kriging with Numerous Data

Kriging is a data interpolation method that can be used to populate regular grids from data scattered in space, and requires the solution of a linear equation system the size of the number of data. When the data is numerous the speed of the calculation is slow. In this paper we propose to divide the regular grid into rectangular sub-segments and let all the grid cells in each sub-segment share a common data neighborhood. The advantage of this approach is that the number of data in the neighborhoods can be small compared to the complete dataset and it is possible to reuse some of the computations for all grid cells in each sub-segment. We show that the precision can be controlled through selection of neighbourhood size, and that the speed of the calculations can be optimized through selection of sub-segment size. We show that this is an efficient method for kriging when number of data is huge, giving a significant speed-up even for high data densities and precisions.