Henning Omre

Bayesian kriging—Merging observations and qualified guesses in kriging

Frequently a user wants to merge general knowledge of the regionalized variable under study with available observations. Introduction of fake observations is the usual way of doing this. Bayesian kriging allows the user to specify a qualified guess, associated with uncertainty, for the expected surface. The method will provide predictions which are based on both observations and this qualified guess.

The Bayesian bridge between simple and universal kriging

Kriging techniques are suited well for evaluation of continuous, spatial phenomena. Bayesian statistics are characterized by using prior qualified guesses on the model parameters. By merging kriging techniques and Bayesian theory, prior guesses may be used in a spatial setting. Partial knowledge of model parameters defines a continuum of models between what is named simple and universal kriging in geostatistical terminology. The Bayesian approach to kriging is developed and discussed, and a case study concerning depth conversion of seismic reflection times is presented.

A Two-Stage Stochastic Model Applied to a North Sea Reservoir

This paper presents a two-stage stochastic model that caters to the large-scale geological heterogeneities resulting from different rock types and the inherent spatial variability of rock properties. The suggested approach combines several elements from a variety of models, methods, and algorithms that have emerged during the last few years. This two-stage procedure can be used to generate several geologically sound realizations of an oil or gas reservoir in an efficient manner. Stage 1 preserves the important geological architecture, while Stage 2 provides small-scale variability in the rock properties. At both stages, the stochastic models are conditional on the actual values observed in wells. Hence, every realization honors the observations. An example from a highly heterogeneous North Sea reservoir, deposited in an upper shore-face environment illustrates application of the model.

Spatial Interpolation Errors for Monitoring Data

Abstract Separate modeling of the spatial mean field, the spatial variance field, and the space-time residual fields can give a more detailed and possibly more accurate representation of spatial interpolation errors when we have repeated observations on a fixed monitoring network. This article gives expressions for the spatial interpolation errors in terms of the statistics of the component fields, which enable us to assess the relative importance of different kinds of uncertainty. This modeling approach is applied to data of sulfur dioxide concentrations in Europe, and a comparison with neighborhood kriging is done by means of cross-validation.

Uncertainties in Reservoir Production Forecasts

Being aware of uncertainties in forecasts of production characteristics is important for reservoir management. Decisions concerning further appraisal drilling, flexibility in development plans, and selection among reservoir prospects all require that uncertainties are taken into account. This study addresses the problem of quantifying uncertainties due to incomplete knowledge of the initial reservoir characteristics, with emphasis on the difference between heterogeneity modeling and assessment of uncertainty. In this study we outline a formalism for uncertainty modeling in a Bayesian framework and present an extensive case study of a North Sea Brent Group reservoir. The results are obtained by computer software implemented such that no human interference is required once the stochastic reservoir model is established. Once the stochastic reservoir model is established, multiple realizations can be generated, rescaled, and fed into a fluid flow simulator to forecast production characteristics and quantify uncertainty associated with response parameters such as cumulative oil production, recovery factors, and water cuts. The results demonstrate that uncertainty in model parameters contributes significantly to the overall uncertainty. The most influential parameters in our case study include the sealing capacity of major faults, seismic velocities used in depth conversion, and average porosities and shale continuity within the main reservoir sandstone.

Combining Fibre Processes and Gaussian Random Functions for Modelling Fluvial Reservoirs

A combination of fibre processes, a special version of marked point processes, and Gaussian random functions is applied to build a 3D model for fluvial reservoir. A fluvial reservoir consists of a set of channel-belts, each containing several river-beds.

Stochastic Simulation and Conditioning by Annealing in Reservoir Description

Simulation of realizations of high dimensional probability distributions is often complicated. In the continuous case under Gaussian assumptions, several well understood algorithms are available. In the general case the picture seems to be fairly confusing.

Sampling from Bayesian Models in Reservoir Characterization

In the evaluation of reservoir characteristics of a petroleum reservoir both observations from the reservoir under study and general geologic knowledge should be taken into account. In this paper, the observations from the reservoir under study and the sampling procedure used to collect them are modeled by a probability distribution in which the reservoir characteristics are considered as model parameters. The general geologic knowledge is modeled by a prior distribution for the reservoir characteristics, and it is argued that it must be doubly stochastic in order to represent the prior uncertainty realistically. Bayes formula is used to obtain the corresponding posterior distribution for the reservoir characteristics. The posterior distribution will usually be too complex to make any analytical calculations possible. Hence the properties of the posterior distribution must be assessed through sampling. In the construction of algorithms sampling from the posterior distribution sequential simulation and Markov chain Monte Carlo simulation appear as especially flexible. Two examples of Bayesian models with doubly stochastic prior distributions are presented and two possible simulation algorithms are specified.

A Bayesian Approach to Surface Estimation

The motive for working with Bayesian approaches to surface estimation is the lack of flexibility in choosing expectation functions in the kriging methods most frequently used. In the method presented, the user can provide qualified guesses for the expectation function and specify the uncertainty associated with this guess. Predictions will be based on the qualified guess and the available observations. A couple of examples are presented.

Random Functions and Geological Subsurfaces

The objective of the presentation is to show how the theory of Gaussian random functions (fields) can be used for describing geological structures. It will be demonstrated how Gaussian random functions can be used to obtain the most probable description and to model variability. Depth conversion of seismic travel time maps to depth maps will be used as an illustration. The ability for Gaussian random field models to integrate such diverse information as depth, clip and velocity information in wells, seismic travel time and velocity maps, and even subjective knowledge on velocity fields, will be outlined. Properties of Gaussian random functions will be presented. Some underlying theoretical properties will be given, but emphasis is made on the practical side. Especially the use of spatial prediction and spatial simulation will be considered in some detail.