Uncertainty In Reservoir Performance Predictions Research Articles

MPS realization selection with an innovative LSTM tool

Multiple point statistics (MPS) methods are good tools to recreate the spatial heterogeneity of permeability in porous media. On the other hand, a reliable forecast of reservoir performance necessitates a large number of realistic permeability realizations which could be achieved by MPS techniques. However, the cost of numerical simulation on a large number of realizations, especially for screening, optimization or history matching purposes, is completely prohibitive. Emerging techniques in deep learning such as Long Short-Term Memory (LSTM) networks can provide robust proxy models replacing the numerical simulation with reasonable run-time and acceptable predictability. We train LSTM networks using some of realizations as input with a novel implementation. The LSTM network accepts the permeability values in a state of art manner, i.e. accepting the permeability values row by row. The network is trained with reservoir simulation outputs of corresponding MPS-generated realizations. In fact, the output values for training (the labels) are the reservoir simulation outputs for each realization which are obtained by an MPS method. In this way, MPS can be coupled with deep learning to find the realizations having the best match with a reference case.The architecture of the trained LSTM network is illustrated with details. The results show that the trained network is able to find realizations resulting in an adequate match with the reference case. These realizations not only exhibit fairly good prediction in future times of simulation due to MPS-derived permeability heterogeneity within but have an acceptable match with the historical reservoir performance. Our hybrid MPS-LSTM approach allows reducing the uncertainty in reservoir performance prediction. The reason is both meticulous heterogeneity modeling and deep-learning-assisted approach of realization screening. The proposed workflow could be a promising basis for history matching and uncertainty quantification calculation of reservoir.

Improving Forecast Uncertainty Quantification by Incorporating Production History and Using a Modified Ranking Method of Geostatistical Realizations

Abstract The majority of the geostatistical realizations ranking methods disregard the production history in selection of realizations, due to its requirement of high simulation run time. They also ignore to consider the degree of linear relationship between the “ranks based on the ranking measure” and “ranks based on the performance parameter” in choosing the employed ranking measure. To address these concerns, we propose an uncertainty quantification workflow, which includes two sequential stages of history matching and realization selection. In the first stage, production data are incorporated in the uncertainty quantification procedure through a history matching process. A fast simulator is employed to find the realizations with consistent flow behavior with the production history data in shorter time, compared to a comprehensive simulator. The selected realizations are the input of the second stage of the workflow, which can be any type of the realization selection method. In this study, we used the most convenient realization selection method, i.e., ranking of the realizations. To select the most efficient ranking measure, we investigated the degree of the linear correlation between the ranks based on the several ranking measures and the performance parameter. In addition, due to the shortcomings of the traditional ranking methods in uncertainty quantiles identification, a modified ranking method is introduced. This modification increases the certainty in the probability of the selected realizations. The obtained results on 3D close-to-real synthetic reservoir models revealed the capability of the modified ranking method in more accurate quantification of the uncertainty in reservoir performance prediction.

Improved History Matching and Uncertainty Characterization

This article, written by Senior Technology Editor Dennis Denney, contains highlights of paper SPE 141336, ’Combining the Ensemble Kalman Filter With Markov-Chain Monte Carlo for Improved History Matching and Uncertainty Characterization,’ by Alexandre A. Emerick, SPE, Petrobras, and Albert C. Reynolds, SPE, University of Tulsa, prepared for the 2011 SPE Reservoir Simulation Symposium, The Woodlands, Texas, 21-23 February. The paper has not been peer reviewed. With the use of the ensemble Kalman filter (EnKF) for assisted history matching, the approximation of covariance matrices from a finite ensemble of states and a finite ensemble of predicted-data vectors can lead to a large underestimation of uncertainty in the posterior probability-density function (PDF) for the state vector. Regardless of whether covariance localization is used, EnKF can lead to an overestimation of the uncertainty in predictions of reservoir performance. Under reasonable assumptions, a Markov-chain Monte Carlo (MCMC) algorithm will, theoretically, generate a correct sampling of the posterior PDF conditional to dynamic data. Introduction In a reservoir-simulation study, history matching is essential to establish a measure of legitimacy of reservoir-performance predictions. Over the last decade, increased importance has been attached to quantifying uncertainty in reservoir-performance predictions and in reservoir description to manage risk. Because of this interest in characterizing uncertainty, it is common to generate multiple history-matched models. However, generating multiple history-matched models does not necessarily lead to a correct assessment of uncertainty. Uncertainty has no scientific meaning outside the realm of statistics and probability. The MCMC process provides a theoretically attractive method for sampling the posterior PDF for reservoir-model parameters. MCMC does not require knowledge of the normalizing constant in the target PDF to be sampled. A properly designed MCMC method will sample the target PDF correctly in the limit as the number of states in the chain approaches infinity. However, for high-dimension problems, MCMC typically requires a large number of iterations to sample the target PDF correctly. When MCMC is used to estimate the posterior PDF of reservoir-model parameters conditional to observed production data, calculation of the probability of accepting the transition from the current state to the proposed state requires a run of the reservoir simulator to evaluate the likelihood portion of the posterior PDF. Therefore, the direct application of MCMC can be prohibitively expensive computationally for realistic reservoir cases.

Markov Chain Monte Carlo for Modelling Permeability Variations in Reservoirs

Abstract Accurate representation of reservoir heterogeneity using stochastic modelling techniques requires careful synthesis of the multivariate probability distribution characterizing the spatial variations of petrophysical properties. A well designed scheme for sampling from that distribution is necessary in order to accurately portray the uncertainty in identifying reservoir properties arising from our imperfect knowledge of the reservoir under study. That uncertainty is data-dependent and most importantly, model-dependent. The paper presents a Markov Chain Monte Carlo (MCMC) methodology for sampling from the invariant or stationary probability distribution characterizing the reservoir permeability field. An initial random field is perturbed successively following a Gibbs sampling procedure. The updated values at the perturbed node are obtained by sampling from local probability distributions obtained by kriging. This iterative updating procedure is continued until a prescribed large number of iterations are performed. Hard data, histogram, and variogram models are honoured, as expected. The multiple point histogram (MPH) and entropy of MPH are used to assess the joint spatial uncertainty exhibited by the MCMC model. The paper also discusses and demonstrates a new approach for integrating secondary information within the MCMC framework. The reduction in joint spatial uncertainty subsequent to integrating the secondary information is demonstrated. Introduction The assessment of uncertainty is a crucial step for decisionmakers to reduce the financial risks during the development of a reservoir. The uncertainty in reservoir performance predictions is usually reflected by a number of parameters of interest, such as OIP, production profiles, ultimate recovery, and any forecast figure, etc. Reservoir complexity and the lack of data available are two main reasons that lead to uncertainty in reservoir modeling(1). Several papers(1–4) have addressed the issue of uncertainty assessment in detail. However, all efforts for quantifying uncertainty are model and data specific. The true reservoir permeability field is entirely characterized by a multivariate probability distribution describing the joint uncertainty in permeability at all locations in the reservoir. That multivariate distribution is usually unknown and hence the objective of stochastic simulation is to obtain the requisite multivariate distribution using the available data and the prior model for spatial continuity (or variability). Ultimately, this posterior probability distribution is sampled in order to obtain realizations of the permeability field. In this paper, a Markov Chain Monte Carlo (MCMC) procedure is used to update and sample from the posterior probability density function. The simulation commences from an initial kriged map. The local conditional probability distribution at each location is constructed using the values at neighbouring locations and subsequently, a value is sampled from that distribution. A Gibbs sampler is utilized implying that the sampled value is retained regardless of whether the mismatch against target statistics increases. The procedure is continued at all locations by following a random path. This iterative procedure is continued until a final realization is obtained that identifies target statistics, such as the prior variogram and/or the prior histogram of permeability values. The details of the MCMC algorithm are described next.

Conditioning fractal (fBm/fGn) porosity and permeability fields to multiwell pressure data

Stochastic fractal (fGn and fBm) porosity and permeability fields are conditioned to given variogram, static (or hard), and multiwell pressure data within a Bayesian estimation framework. Because fGn distributions are normal/second-order stationary, it is shown that the Bayesian estimation methods based on the assumption of normal/second-order stationary distributions can be directly used to generate fGn porosity/permeability fields conditional to pressure data. However, because fBm is not second-order stationary, it is shown that such Bayesian estimation methods can be used with implementation of a pseudocovariance approach to generate fBm porosity/permeability fields conditional to multiwell pressure data. In addition, we provide methods to generate unconditional realizations of fBm/fGn fields honoring all variogram parameters. These unconditional realizations can then be conditioned to hard and pressure data observed at wells by using the randomized maximum likelihood method. Synthetic examples generated from one-, two-, and three-dimensional single-phase flow simulators are used to show the applicability of our methodology for generating realizations of fBm/fGn porosity and permeability fields conditioned to well-test pressure data and evaluating the uncertainty in reservoir performance predictions appropriately using these history-matched realizations.

A Hybrid Markov Chain Monte Carlo Method for Generating Permeability Fields Conditioned to Multiwell Pressure Data and Prior Information

Summary It is necessary to construct and sample the a posteriori probability density functions (pdf's) for the rock property fields to properly evaluate the uncertainty in reservoir performance predictions. In this work, the a posteriori pdf is constructed based on prior means and variograms (covariance function) for log-permeability and multiwell pressure data. Within the context of sampling the probability density function, we argue that the notion of equally probable realizations is the wrong paradigm for reservoir characterization. If the simulation of Gaussian random fields with a known variogram is the objective, it is shown that the variogram should not be incorporated directly into the objective function if simulated annealing (SA) is applied either to sample the a posteriori pdf or to estimate a global minimum of the associated objective function. It is shown that the hybrid Markov chain Monte Carlo (MCMC) method provides a way to explore more fully the set of plausible log-permeability fields and does not suffer from the high rejection rates of more standard MCMC methods.