Polynomial Chaos Based Solution to Inverse Problems in Petroleum Reservoir Engineering

2019 ◽  
Author(s):  
Sufia Khatoon ◽  
Jyoti Phirani ◽  
Supreet Singh Bahga
Keyword(s):  
Bayesian Inference ◽  
History Matching ◽  
Polynomial Chaos ◽  
Reservoir Engineering ◽  
Model Parameters ◽  
Petroleum Reservoir ◽  
Model Predictions ◽  
Production History ◽  
Porosity And Permeability ◽  
Computationally Expensive

Abstract In reservoir simulations, model parameters such as porosity and permeability are often uncertain and therefore better estimates of these parameters are obtained by matching the simulation predictions with the production history. Bayesian inference provides a convenient way of estimating parameters of a mathematical model, starting from a probable range of parameter values and knowing the production history. Bayesian inference techniques for history matching require computationally expensive Monte Carlo simulations, which limit their use in petroleum reservoir engineering. To overcome this limitation, we perform accelerated Bayesian inference based history matching by employing polynomial chaos (PC) expansions to represent random variables and stochastic processes. As a substitute to computationally expensive Monte Carlo simulations, we use a stochastic technique based on PC expansions for propagation of uncertainty from model parameters to model predictions. The PC expansions of the stochastic variables are obtained using relatively few deterministic simulations, which are then used to calculate the probability density of the model predictions. These results are used along with the measured data to obtain a better estimate (posterior distribution) of the model parameters using the Bayes rule. We demonstrate this method for history matching using an example case of SPE1CASE2 problem of SPEs Comparative Solution Projects. We estimate the porosity and permeability of the reservoir from limited and noisy production data.

Integrated Characterization of Hydraulically Fractured Shale-Gas Reservoirs—Production History Matching

2015 ◽  
Vol 18 (04) ◽  
pp. 481-494 ◽  
Author(s):  
Siavash Nejadi ◽  
Juliana Y. Leung ◽  
Japan J. Trivedi ◽  
Claudio Virues
Keyword(s):  
Shale Gas ◽  
Hydraulic Fracture ◽  
History Matching ◽  
Flow Simulation ◽  
Simulation Models ◽  
Model Parameters ◽  
Fracture Parameters ◽  
Production History ◽  
Dfn Model

Summary Advancements in horizontal-well drilling and multistage hydraulic fracturing have enabled economically viable gas production from tight formations. Reservoir-simulation models play an important role in the production forecasting and field-development planning. To enhance their predictive capabilities and to capture the uncertainties in model parameters, one should calibrate stochastic reservoir models to both geologic and flow observations. In this paper, a novel approach to characterization and history matching of hydrocarbon production from a hydraulic-fractured shale is presented. This new methodology includes generating multiple discrete-fracture-network (DFN) models, upscaling the models for numerical multiphase-flow simulation, and updating the DFN-model parameters with dynamic-flow responses. First, measurements from hydraulic-fracture treatment, petrophysical interpretation, and in-situ stress data are used to estimate the initial probability distribution of hydraulic-fracture and induced-microfracture parameters, and multiple initial DFN models are generated. Next, the DFN models are upscaled into an equivalent continuum dual-porosity model with analytical techniques. The upscaled models are subjected to the flow simulation, and their production performances are compared with the actual responses. Finally, an assisted-history-matching algorithm is implemented to assess the uncertainties of the DFN-model parameters. Hydraulic-fracture parameters including half-length and transmissivity are updated, and the length, transmissivity, intensity, and spatial distribution of the induced fractures are also estimated. The proposed methodology is applied to facilitate characterization of fracture parameters of a multifractured shale-gas well in the Horn River basin. Fracture parameters and stimulated reservoir volume (SRV) derived from the updated DFN models are in agreement with estimates from microseismic interpretation and rate-transient analysis. The key advantage of this integrated assisted-history-matching approach is that uncertainties in fracture parameters are represented by the multiple equally probable DFN models and their upscaled flow-simulation models, which honor the hard data and match the dynamic production history. This work highlights the significance of uncertainties in SRV and hydraulic-fracture parameters. It also provides insight into the value of microseismic data when integrated into a rigorous production-history-matching work flow.

scholarly journals Estimating parameters from multiple time series of population dynamics using Bayesian inference

2018 ◽  
Author(s):  
Benjamin Rosenbaum ◽  
Michael Raatz ◽  
Guntram Weithoff ◽  
Gregor F. Fussmann ◽  
Ursula Gaedke
Keyword(s):  
Time Series ◽  
Population Dynamics ◽  
Bayesian Inference ◽  
Steady States ◽  
Series Data ◽  
Model Parameters ◽  
List Type ◽  
Model Predictions ◽  
Cyclic Dynamics ◽  
Parameter Values

AbstractEmpirical time series of interacting entities, e.g. species abundances, are highly useful to study ecological mechanisms. Mathematical models are valuable tools to further elucidate those mechanisms and underlying processes. However, obtaining an agreement between model predictions and experimental observations remains a demanding task. As models always abstract from reality one parameter often summarizes several properties. Parameter measurements are performed in additional experiments independent of the ones delivering the time series. Transferring these parameter values to different settings may result in incorrect parametrizations. On top of that, the properties of organisms and thus the respective parameter values may vary considerably. These issues limit the use of a priori model parametrizations.In this study, we present a method suited for a direct estimation of model parameters and their variability from experimental time series data. We combine numerical simulations of a continuous-time dynamical population model with Bayesian inference, using a hierarchical framework that allows for variability of individual parameters. The method is applied to a comprehensive set of time series from a laboratory predator-prey system that features both steady states and cyclic population dynamics.Our model predictions are able to reproduce both steady states and cyclic dynamics of the data. Additionally to the direct estimates of the parameter values, the Bayesian approach also provides their uncertainties. We found that fitting cyclic population dynamics, which contain more information on the process rates than steady states, yields more precise parameter estimates. We detected significant variability among parameters of different time series and identified the variation in the maximum growth rate of the prey as a source for the transition from steady states to cyclic dynamics.By lending more flexibility to the model, our approach facilitates parametrizations and shows more easily which patterns in time series can be explained also by simple models. Applying Bayesian inference and dynamical population models in conjunction may help to quantify the profound variability in organismal properties in nature.

Reservoir Modeling and Numerical Simulation Controlled by Flow Units

2013 ◽  
Vol 690-693 ◽  
pp. 3190-3193
Author(s):  
Yong Chao Xue ◽  
Lin Song Cheng
Keyword(s):  
Numerical Simulation ◽  
Reservoir Simulation ◽  
History Matching ◽  
Oil Field ◽  
Flow Unit ◽  
Reservoir Engineering ◽  
Flow Units ◽  
Study Results ◽  
Porosity And Permeability ◽  
Static Flow

Geological model controlled by sedimentary microfacies can not able to accurately reflect the actual seepage characteristic, How to apply the static flow units to the dynamic reservoir numerical simulation is the advancing edge of petroleum industry. The method of reservoir geological modeling controlled by flow units is proposed. Firstly, the 3D models of flow unit should be build, secondly, the 3D porosity and permeability model are established controlled by the model of flow unit, thirdly, the 3D fluids saturation model is calculated by Leverett J function based on porosity and permeability mode. Selecting different relative permeability curves according to different flow units in history matching. which realizes the dynamic (development geological) and static (reservoir engineering) combination. The oilfield examples shown that the velocity and precision of history matching can be significantly improved by using the method mentioned. Flow units were proposed by Hearn in 1984[. Study on flow units, which could not only deepen the understanding of reservoir geology, make reservoir evaluation more reasonable and reduce the heterogeneity impact on oil development, but also could be of great significance to improve oil field development effect especially for tertiary oil recovery[. Previous studies mainly focused on defining the concept of flow units, divide method of flow units and so on, but few studies on how to apply the study results of static flow units to the dynamic reservoir engineering and reservoir simulation[3-. Our research aimed at how to connect the static flow units and the dynamic reservoir simulation closely so as to achieve "dynamic and static combination".

scholarly journals Comparing two sequential Monte Carlo samplers for exact and approximate Bayesian inference on biological models

2017 ◽  
Vol 14 (134) ◽  
pp. 20170340 ◽  
Author(s):  
Aidan C. Daly ◽  
Jonathan Cooper ◽  
David J. Gavaghan ◽  
Chris Holmes
Keyword(s):  
Bayesian Inference ◽  
Bayesian Methods ◽  
Sequential Monte Carlo ◽  
Model Parameters ◽  
Biological Models ◽  
Exact Inference ◽  
Modelling Studies ◽  
Approximate Bayesian ◽  
Approximate Bayesian Inference ◽  
Abc Methods

Bayesian methods are advantageous for biological modelling studies due to their ability to quantify and characterize posterior variability in model parameters. When Bayesian methods cannot be applied, due either to non-determinism in the model or limitations on system observability, approximate Bayesian computation (ABC) methods can be used to similar effect, despite producing inflated estimates of the true posterior variance. Owing to generally differing application domains, there are few studies comparing Bayesian and ABC methods, and thus there is little understanding of the properties and magnitude of this uncertainty inflation. To address this problem, we present two popular strategies for ABC sampling that we have adapted to perform exact Bayesian inference, and compare them on several model problems. We find that one sampler was impractical for exact inference due to its sensitivity to a key normalizing constant, and additionally highlight sensitivities of both samplers to various algorithmic parameters and model conditions. We conclude with a study of the O'Hara–Rudy cardiac action potential model to quantify the uncertainty amplification resulting from employing ABC using a set of clinically relevant biomarkers. We hope that this work serves to guide the implementation and comparative assessment of Bayesian and ABC sampling techniques in biological models.

scholarly journals Gaussian Processes Proxy Model with Latent Variable Models and Variogram-Based Sensitivity Analysis for Assisted History Matching

Energies ◽  
2020 ◽  
Vol 13 (17) ◽  
pp. 4290
Author(s):  
Dongmei Zhang ◽  
Yuyang Zhang ◽  
Bohou Jiang ◽  
Xinwei Jiang ◽  
Zhijiang Kang
Keyword(s):  
Sensitivity Analysis ◽  
Gaussian Processes ◽  
Latent Variable ◽  
History Matching ◽  
Latent Variable Models ◽  
High Dimensional ◽  
Model Parameters ◽  
Variable Model ◽  
Assisted History Matching ◽  
Proxy Models

Reservoir history matching is a well-known inverse problem for production prediction where enormous uncertain reservoir parameters of a reservoir numerical model are optimized by minimizing the misfit between the simulated and history production data. Gaussian Process (GP) has shown promising performance for assisted history matching due to the efficient nonparametric and nonlinear model with few model parameters to be tuned automatically. Recently introduced Gaussian Processes proxy models and Variogram Analysis of Response Surface-based sensitivity analysis (GP-VARS) uses forward and inverse Gaussian Processes (GP) based proxy models with the VARS-based sensitivity analysis to optimize the high-dimensional reservoir parameters. However, the inverse GP solution (GPIS) in GP-VARS are unsatisfactory especially for enormous reservoir parameters where the mapping from low-dimensional misfits to high-dimensional uncertain reservoir parameters could be poorly modeled by GP. To improve the performance of GP-VARS, in this paper we propose the Gaussian Processes proxy models with Latent Variable Models and VARS-based sensitivity analysis (GPLVM-VARS) where Gaussian Processes Latent Variable Model (GPLVM)-based inverse solution (GPLVMIS) instead of GP-based GPIS is provided with the inputs and outputs of GPIS reversed. The experimental results demonstrate the effectiveness of the proposed GPLVM-VARS in terms of accuracy and complexity. The source code of the proposed GPLVM-VARS is available at https://github.com/XinweiJiang/GPLVM-VARS.

scholarly journals Formulating the history matching problem with consistent error statistics

2021 ◽  
Author(s):  
Geir Evensen
Keyword(s):  
Simulation Model ◽  
Measurement Errors ◽  
History Matching ◽  
Critical Role ◽  
Bayes Theorem ◽  
Conditional Probability Density ◽  
Model Parameters ◽  
Rate Data ◽  
Matching Problem ◽  
Reservoir Prediction

AbstractIt is common to formulate the history-matching problem using Bayes’ theorem. From Bayes’, the conditional probability density function (pdf) of the uncertain model parameters is proportional to the prior pdf of the model parameters, multiplied by the likelihood of the measurements. The static model parameters are random variables characterizing the reservoir model while the observations include, e.g., historical rates of oil, gas, and water produced from the wells. The reservoir prediction model is assumed perfect, and there are no errors besides those in the static parameters. However, this formulation is flawed. The historical rate data only approximately represent the real production of the reservoir and contain errors. History-matching methods usually take these errors into account in the conditioning but neglect them when forcing the simulation model by the observed rates during the historical integration. Thus, the model prediction depends on some of the same data used in the conditioning. The paper presents a formulation of Bayes’ theorem that considers the data dependency of the simulation model. In the new formulation, one must update both the poorly known model parameters and the rate-data errors. The result is an improved posterior ensemble of prediction models that better cover the observations with more substantial and realistic uncertainty. The implementation accounts correctly for correlated measurement errors and demonstrates the critical role of these correlations in reducing the update’s magnitude. The paper also shows the consistency of the subspace inversion scheme by Evensen (Ocean Dyn. 54, 539–560 2004) in the case with correlated measurement errors and demonstrates its accuracy when using a “larger” ensemble of perturbations to represent the measurement error covariance matrix.

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