Markov chain Monte Carlo algorithms generate samples from a target distribution by simulating a Markov chain. Large¯exibility exists in speci®cation of transition matrix of the chain. In practice, however, most algorithms used only allow small changes in the state vector in each iteration. This choice typically causes problems for multimodal distributions as moves between modes become rare and, in turn, results in slow convergence to the target distribution. In this paper we consider continuous distributions on R n and specify how optimization for local maxima of the target distribution can be incorporated in the speci®cation of the Markov chain. Thereby, we obtain a chain with frequent jumps between modes. We demonstrate the effectiveness of the approach in three examples. The ®rst considers a simple mixture of bivariate normal distributions, whereas the two last examples consider sampling from posterior distributions based on previously analysed data sets.

Summary A stochastic model in a Bayesian setting, conditioned on well observations, seismic amplitude data and production history, is defined. Samples of reservoir characteristics and production forecasts from the posterior model are used to evaluate the impact of various observation types. Well observations are found to be important to production forecasts due to near-well conditioning, while seismic data impact facies geometries but not the production forecasts. Production history contributes significantly only if certain events, such as gas-breakthrough time, are observed in wells. A brute-force rejection sampling approach may work well if proper conditioning on well observations and seismic data is done. Introduction The objective of reservoir evaluation is to forecast production characteristics under various recovery strategies and eventually decide on a management strategy. The forecasts should be as accurate as possible; therefore, both general reservoir knowledge and reservoir-specific observations should be used in the evaluation. General reservoir knowledge includes geologic understanding, physically based models for fluid flow, and insight into the data acquisition procedures. The reservoir-specific observations include well observations, seismic amplitude data, and production history collected from the reservoir under study. In Omre and Tjelmeland,1 a Bayesian approach to integrated reservoir evaluation is presented. Stochastic reservoir modeling based on geologic knowledge and well observations only has evolved over the last two decades. Inclusion of seismic amplitude data has been an active field of research the last few years (see Bortoli et al.2 and Eide et al.3). In order to represent uncertainty, seismic inversion must be a part of the model. History matching of production data has only recently been phrased in a stochastic setting (see Oliver,4 Wen et al.,5 and Hegstad and Omre6). This requires the use of a fluid-flow simulator, which normally needs considerable computer resources to run. The current paper integrates all these sources of information in a framework defined along the lines of Omre and Tjelmeland.1 A graphical model is introduced to communicate the model assumptions more easily to the user. Models and algorithms developed in Eide et al.3 and Hegstad and Omre6 are integrated and applied to a case study inspired by the Troll field in the North Sea. The focus of the paper is to evaluate the contribution of the well observations, seismic-amplitude data, and production history to the reduction of uncertainty in the production forecasts. Conditioning to the production history constitutes a major challenge due to the nonlinearity of the fluid-flow model. The study also sheds some light on which sampling algorithms are suitable for this conditioning. The study is thoroughly documented in Hegstad and Omre,7 in which further details can be found. Reference Reservoir and Production The reference reservoir and the recovery strategy are related to the Troll field in the North Sea offshore Norway (see Eide et al.3). This is a multilayer reservoir with alternating C-sands and M-sands having high- and low-permeability properties, respectively. The reservoir is represented in a 3D domain being 104×104×102 feet3 then discretized into a (50×50×15) - grid termed LD. The primary interest of the study is the future production, being crucially dependent on the reservoir variables of porosity and absolute log-permeability. The acoustic impedances and seismic-reflection coefficients are introduced as support variables. This multivariate spatial reservoir variable, rt, is defined as a high-dimensional vector on the grid LD, with indexes (f,k, z, c) indicating porosity, absolute log-permeability, acoustic impedance, and seismic reflection coefficients respectively. Index t indicates that it is the reference reservoir. The lateral-vertical anisotropy in permeability is 1:600. A more detailed description of the construction can be found in Hegstad and Omre.7 The cross section of the reference reservoir, displayed in Fig. 1, shows distinct layering. The histograms and cross plots of the reservoir variables in Fig. 2 display definite bimodality and clustering. The reference reservoir variables have characteristics far from Gaussian. Initially, oil is assigned every where, and the reservoir is in pressure equilibrium before start of production (i.e., no fluid movements in the reservoir prior to production start). For further details and other fluid characteristics, see Hegstad and Omre. 7 The recovery strategy is based on the injection of gas in one vertical well perforated in the upper layers and the production through two horizontal wells (see Fig. 3.) The reference production is generated by the fluid-flow simulator Eclipse100 (see GeoQuest8) on a (10×10×15) laterally upscaled grid. The permeability characteristics are upscaled by harmonic averaging, while all other variables are upscaled by arithmetic averaging. The oil production-rate and gas-oil ratio in the two producing wells and the bottomhole pressure (BHP) in the injection well are monitored monthly for 16 years, or 5,840 days. This is represented by the time vector on L T (see Fig. 4). Note that gas breakthrough appears after about 3 years and that the plateau production is reached in about 5 years. Stochastic Model and Simulation The stochastic model consists of two components: a prior model for the reservoir and production variables and a likelihood function model for the reservoir-specific observations available. The stochastic model is graphically displayed in Fig. 5. The variables and the relations in this graph will be defined in this section.

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The joint probability density function of the state space vector of a white noise excited van der Pol oscillator satisfies a Fokker-Planck-Kolmogorov (FPK) equation. The paper describes a numerical procedure for solving the transient FPK equation based on the path integral solution (PIS) technique. It is shown that by combining the PIS with a cubic B-spline interpolation method, numerical solution algorithms can be implemented giving solutions of the FPK equation that can be made accurate down to very low probability levels. The method is illustrated by application to two specific examples of a van der Pol oscillator.

TX 75083-3836, U.S.A., fax 01-972-952-9435. AbstractIncreasingly, the industry is aware of the need to improve field development planning decisions with more rigorous risk analysis. A key is to have technology and work processes supporting the complex evaluation of projects as a whole, i.e. from subsurface to processing with economics, while preserving physical fidelity and interdependent uncertainties. This paper will illustrate how an integrated stochastic approach with scenario analysis, sensitivity analysis, and Monte Carlo simulation assisted an asset team to understand overall project uncertainties and sensitivities for a large gas project. This understanding gave a better basis for the planning decisions.The richness and speed of the integrated stochastic approach is compared with more conventional case study analysis. The latter requires manual iteration to investigate how variations of input variables as e.g. reservoir properties impact reserves and production. It is followed by manual input to the economic model and manual analysis.Outputs presented from hundreds of simulations from the integrated stoachastic approach include distributions and histograms of original fluids in place, cumulative production, plateau period, net present value, production rate, discounted cash flows, and rates of return etc. Correlation coefficients between input uncertainties and output uncertainites indicate which input uncertainties give the major contribution to the output uncertainties. This helps the asset team to focus on the important factors for major decisions, and use less time on the less important issues.

This paper describes the workflow for an uncertainty study for a field development case offshore Norway. It is a workflow where uncertainties in seismic time interpretation, depth conversion, contacts, fault scenarios, alternative conceptual models, facies models, relative permeability, schedule etc. are included in an automated way to generate multiple realizations (hundreds) of the geomodel and reservoir model. Hence all uncertainties in the geomodel are also included in the reservoir model explicitly, and are captured by the ensemble of realizations. Since structure, contacts and facies distribution are different from realization to realization the workflow ensures that the planned well trajectories are automatically adjusted accordingly. Having such a workflow which can generate and manage multiple realizations makes it straight forward to get robust and realistic estimates of uncertainties in in-place volumes and produced volumes. It is also used for risk mitigation and decision support as e.g. to evaluated robustness of well placement, well number, top side capacities etc. This automatic workflow made it easy to rerun the uncertainty study when new well-data arrived. It made it also easy to run sensitivities on any part of the reservoir modelling workflow to gain valuable insight. Furthermore, having such a workflow made it possible to do quick and simple soft conditioning on dynamic data (such as drill stem test data) or alternatively use the dynamic data as direct input to the geomodel. Few real data from the field under study will be included. Those who are included are anonymized and scales on axes are masked.

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