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Maucec, Marko (Shell International Ltd.) | Douma, Sippe G. (Shell Intl. E&P Co.) | Hohl, Detlef (Shell International Exploration and Production Inc.) | Leguijt, Jaap (SIEP-RTS) | Jimenez, Eduardo (Texas A&M University) | Datta-Gupta, Akhil
Streamline models have shown great potential for the inversion of dynamic reservoir models using well production data. The solution of "history matching?? inverse problem is greatly simplified when sensitivities are known. This applies to both single-model optimization and multi-realization stochastic sampling of the objective function with uncertainty evaluation.
Bayesian modeling with sequential Markov chain Monte Carlo (MCMC) algorithms provides a rigorous framework for stochastic sampling. However, a naive approach of calculating the likelihood term in Bayes' formula using a dynamic reservoir simulator incurs prohibitively high computational cost. Each MCMC proposal requires a time-consuming reservoir simulation, and when no Jacobian information is available acceptance ratios are very low.
To reduce these high computational costs, approximate models have been introduced. In this contribution we present a Bayesian multi-stage MCMC approach, based on an approximation with a linear expansion around the current model state using the semi-analytically computed streamline sensitivities. The algorithm is benchmarked to a simple synthetic permeability model with one injecting well and four producing wells to study efficiency.
Another crucial ingredient in our inversion and sampling scheme is a fast method to generate new realizations of reservoir models (here: permeability fields) from the prior probability density function that obey known geostatistics (variograms) and well constraints. This is accomplished using a novel Fourier-space filter method that can be used with very large numbers of variables (~106) without the computational and memory cost of traditional algorithms like the Cholesky decomposition.
We ultimately demonstrate the application of the multi-stage MCMC algorithm for an efficient dynamic inversion of water-cut data in structurally complex and faulted offshore turbidite oil reservoir. Timing studies, validation and implementation of MCMC algorithm convergence criteria as well as full parallelization of the computer code, rendering substantial reduction of computing time, are described in detail.
Gao, Guohua (Shell Global Solutions US, Inc.) | Vink, Jeroen C. (Shell Global Solutions International B.V.) | Chen, Chaohui (Shell International Exploration & Production Inc.) | Tarrahi, Mohammadali (Shell Global Solutions US, Inc.) | El Khamra, Yaakoub (Shell Global Solutions US, Inc.)
It is crucially important for decision making and an extremely challenging task to properly quantify uncertainty of model parameters and production forecasts after conditioning to production data. A novel approach is proposed to generate approximate conditional realizations using the distributed Gauss-Newton (DGN) method together with a multiple local Gaussian approximation technique. Results are compared with those obtained from other approaches such as Randomized Maximum Likelihood (RML), Ensemble-Kalman-Filter (EnKF), and Markov-Chain-Monte-Carlo (MCMC) simulation.
The DGN method is developed to find multiple local minima of the objective function in parallel, by collecting and sharing information from dispersed regions in parameter space dynamically. Around each local minimum, the estimated Hessian obtained from the Gauss-Newton approximation along with the prior inverse covariance matrix is used as a local approximation of the posterior inverse covariance matrix. The posterior joint PDF can then be approximated as a weighted linear superposition of multiple local Gaussian distributions, which can be sampled very efficiently without having to resort to expensive MCMC methods.
The proposed approach is first validated using a nonlinear history matching toy problem with multiple modes. In terms of efficiency, the new approach can significantly reduce the computational cost and accelerate the uncertainty quantification process, when compared to the traditional RML method or traditional MCMC approaches. In terms of accuracy, uncertainty characteristics obtained from the proposed approach are comparable to those generated from the MCMC simulation, and they are much better than those obtained from EnKF or RML. The approach is then also applied to a real field history matching problem, where the dynamic system of multi-phase flow in the reservoir exhibits very strong nonlinear behavior, and the objective function has multiple local minima. Uncertainty ranges of production forecasts for the real field case are quantified by generating an ensemble of conditional realizations. The production forecasts for all conditional realizations are consistent with the production data observed after the history matching period, which further validates the applicability of the proposed method to real field problems.
Its high efficiency makes the new approach practical for large-scale problems, for which methods based on Design of Experiment break down. Furthermore, as was argued by
Friedel, Torsten (Schlumberger Data & Consulting Services) | Tewari, Raj Deo (Schlumberger Data & Consulting Services) | Flew, Steve (Schlumberger Logelco, Inc) | Strasser, Ralf (Schlumberger) | Trebolle, Ramiro (Schlumberger Data & Consulting Services) | Belfield, William (Schlumberger) | Othman, Tg Rasidi B. Tg. | Caretta, Fausto
Copyright 2009, Society of Petroleum Engineers This paper was prepared for presentation at the 2009 SPE/EAGE Reservoir Characterization and Simulation Conference held in Abu Dhabi, UAE, 19-21 October 2009. This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Abstract Brownfields are often characterized by a varying degree of maturity, both within the field and within individual reservoir units. A novel workflow solves the complex problem of uncertainty assessment and risk management in a brownfield redevelopment. Traditionally, a single deterministic reservoir model is built, matched, and used for predictions and infill planning. The availability of sophisticated simulation workflow tools enable the team now to explore the practical aspects of performing sophisticated reservoir description, static model construction, history matching, and forecast uncertainty analysis. History matching is conducted for all realizations, and the match quality is assessed by means of statistical analysis. The workflow facilitates generating hydrocarbon thickness maps by using the average column thickness of many simulation models instead of a dedicated single one. Target selection also accounts for possible sweep and sand risks by means of maps showing the standard deviation of the column thickness. The new framework is applied to a conceptual redevelopment of a brownfield. It increases the understanding of fluid flow processes in the reservoirs and is a vital component of the decision and risk analysis for the concept selection stage. Introduction History matching is traditionally conducted as a deterministic process with a single realization considered representative. Many researchers have dealt with the problem of history matching and uncertainty assessment in reservoirs.
In this work we discuss the successful application of our previously developed automated scenario reduction approach applied to life-cycle optimization of a real field case. The inherent uncertainty present in the description of reservoir properties motivates the use of an ensemble of model scenarios to achieve an optimized robust reservoir development strategy. In order to accurately span the range of uncertainties it is imperative to build a relatively large ensemble of model scenarios. The size of the ensemble is directly proportional to the computational effort required in robust optimization. For high-dimensional, complex field case models this implies that a large ensemble of model scenarios which albeit accurately captures the inherent uncertainties would be computationally infeasible to be utilized for robust optimization. One of the ways to circumvent this problem is to work with a reduced subset of model scenarios. Methods based on heuristics and ad-hoc rules exist to select this reduced subset. However, in most of the cases, the optimal number of model realizations must be known upfront. Excessively small number of realizations may result in a subset that does not always capture the span of uncertainties present, leading to sub-optimal optimization results. This raises the question on how to effectively select a subset that contains an optimal number of realizations which both is able to capture the uncertainties present and allow for a computationally efficient robust optimization. To answer this question we have developed an automated framework to select the reduced ensemble which has been applied to an original ensemble of 300 equiprobable model scenarios of a real field case. The methodology relies on the fact that, ideally, the distance between the cumulative distribution functions (CDF) of the objective function (OF) of the full and reduced ensembles should be minimal. This allows the method to determine the smallest subset of realizations that both spans the range of uncertainties and provides an OF CDF that is representative of the full ensemble based on a statistical metric. In this real field case application we optimize the injection rates throughout the assets life-cycle with expected cumulative oil production as the OF. The newly developed framework selected a small subset of 17 model scenarios out of the original ensemble which was used for robust optimization. The optimal injection strategy achieved an average increase of 6% in cumulative oil production with a significant reduction, approximately 90%, in the computational effort. Validation of this optimal strategy over the original ensemble lead to very similar improvements in cumulative oil production, highlighting the reliability and accuracy of our framework.
Chen, Chaohui (Shell International Exploration & Production Inc.) | Li, Ruijian (Shell Exploration & Production Co.) | Gao, Guohua (Shell Global Solutions (US) Inc.) | Vink, Jeroen C. (Shell Global Solutions International B.V.) | Cao, Richard (Shell Exploration & Production Co.)
For unconventional reservoirs, it is very difficult to determine the values of key parameters or properties that govern fluid flow in the subsurface due to unknown fracture growth and rock properties. These parameters generally have quite large uncertainty ranges and need to be calibrated by available production data. Using an ensemble of history matched reservoir models to predict the Estimated Ultimate Recovery (EUR) is one of popular approaches when parallelized computing facilities become cheaper and cheaper to customers. The Randomized Maximum Likelihood (RML) method has been proved quite robust for generating multiple realizations by conditioning to production data. However, it is still expensive to apply traditional optimization algorithms to find a conditional realization by minimizing the objective function defined within a Bayesian framework, especially when adjoint-derivatives are unavailable. How to generate multiple conditional realizations efficiently is critically important but still a very challenging task for proper uncertainty quantification.
In this paper, a novel approach that hybrids the direct-pattern-search and the Gauss-Newton algorithm is developed to generate multiple conditional realizations simultaneously. The proposed method is applied to history match a real unconventional Liquid Rich Shale reservoir. The reservoir is stimulated by multiple stage hydraulic fractures. In this example, uncertainty parameters include those characterizing uncertainties of reservoir properties (including matrix permeability, permeability reduction coefficient, porosity, initial water saturation and pressure) and those for hydraulic fractures (height, width, length, and effective permeability of SRV zone). Uncertainty of production forecasts are quantified with both unconditional and conditional realizations.
The case study indicates that the new method is very efficient and robust. Uncertainty ranges of parameters and production forecasts before and after conditioning to production data are quantified and compared. The new approach enhances the EUR assessment confidence level and therefore significantly reduces risks for unconventional assets development.
For unconventional reservoirs, the key reservoir properties, such as effective flowing fracture length (Xf), effective fracture height (Hf), permeability and permeability reduction coefficient, fracture conductivity (FCD), drainage area (A) etc., that govern fluid flow in subsurface are very difficult to obtain due to unknown fracture growth in tight rock. Uncertainties associated with these parameters are usually quite large. Understanding the uncertainty of the subsurface model is helpful to define how to drill wells and determine fracture stages spacing or the number of wells. There are mainly three categories of Estimated Ultimate Recovery (EUR) prediction methodologies for unconventionals: