Uncertainty Quantification Algorithms Research Articles

Uncertainty quantification in hybrid dynamical systems

Uncertainty quantification (UQ) techniques are frequently used to ascertain output variability in systems with parametric uncertainty. Traditional algorithms for UQ are either system-agnostic and slow (such as Monte Carlo) or fast with stringent assumptions on smoothness (such as polynomial chaos and Quasi-Monte Carlo). In this work, we develop a fast UQ approach for hybrid dynamical systems by extending the polynomial chaos methodology to these systems. To capture discontinuities, we use a wavelet-based Wiener–Haar expansion. We develop a boundary layer approach to propagate uncertainty through separable reset conditions. We also introduce a transport theory based approach for propagating uncertainty through hybrid dynamical systems. Here the expansion yields a set of hyperbolic equations that are solved by integrating along characteristics. The solution of the partial differential equation along the characteristics allows one to quantify uncertainty in hybrid or switching dynamical systems. The above methods are demonstrated on example problems.

Chemical kinetic uncertainty quantification for Large Eddy Simulation of turbulent nonpremixed combustion

While the accuracy of chemical kinetic mechanisms continues to improve, these mechanisms are still models with, sometimes considerable, uncertainty. In order to rigorously validate turbulent combustion simulations against experimental data, this uncertainty must be separated from deficiencies in the turbulent combustion model itself. In this work, a method is developed for quantifying the uncertainty in turbulent flame simulations due to input uncertainty in the chemical mechanism. Here the method is developed for Large Eddy Simulation (LES) combined with a steady flamelet model. Rather than a brute force probabilistic approach in which hundreds or thousands of LES runs are required to compute statistics of outputs of interest, the method takes advantage of the actual algorithm employed with the steady flamelet model. First, the high-dimensional uncertainty in the chemical kinetics is propagated through the flamelet equations, and the resulting lower-dimensional joint distribution of the temperature, species mass fractions, and other derived quantities is used as a stochastic equation of state in the LES. Since only a few “active” quantities are needed to evolve the LES governing equations, efficient non-intrusive stochastic collocation is used to propagate the uncertainty in the density, requiring only a few LES runs. This process captures the uncertainty in the flow field induced by the uncertainty in the chemical kinetic rates. The remaining uncertainty in “passive” quantities, that is, quantities needed only for post-processing such as the temperature and species mass fractions, is computed with random sampling during the LES runs. The uncertainty quantification algorithm is demonstrated with Sandia flame D, and it is shown that the uncertainty in the simulation results caused by uncertainties in the kinetic rates is sufficiently large to account for the discrepancies with the experimental measurements. The implication is that the turbulent combustion model cannot be fairly assessed with such a large uncertainty.

Decision support system for water distribution systems based on neural networks and graphs theory for leakage detection

This paper presents an efficient and effective decision support system (DSS) for operational monitoring and control of water distribution systems based on a three layer General Fuzzy Min–Max Neural Network (GFMMNN) and graph theory. The operational monitoring and control involves detection of pipe leakages. The training data for the GFMMNN is obtained through simulation of leakages in a water network for a 24h operational period. The training data generation scheme includes a simulator algorithm based on loop corrective flows equations, a Least Squares (LS) loop flows state estimator and a Confidence Limit Analysis (CLA) algorithm for uncertainty quantification entitled Error Maximization (EM) algorithm. These three numerical algorithms for modeling and simulation of water networks are based on loop corrective flows equations and graph theory. It is shown that the detection of leakages based on the training and testing of the GFMMNN with patterns of variation of nodal consumptions with or without confidence limits produces better recognition rates in comparison to the training based on patterns of nodal heads and pipe flows state estimates with or without confidence limits. It produces also comparable recognition rates to the original recognition system trained with patterns of data obtained with the LS nodal heads state estimator while being computationally superior by requiring a single architecture of the GFMMNN type and using a small number of pattern recognition hyperbox fuzzy sets built by the same GFMMNN architecture. In this case the GFMMNN relies on the ability of the LS loop flows state estimator of making full use of the pressure/nodal heads measurements existent in a water network.

Population-Based Optimization Algorithms for History Matching and Uncertainty Quantification: PUNQ-S3

This article, written by Senior Technology Editor Dennis Denney, contains highlights of paper SPE 136861, ’Comparative Study of Novel Population-Based Op timization Algorithms for History Matching and Uncertainty Quantification: PUNQ-S3 Revisited,’ by Yasin Hajizadeh, SPE, Mike Christie, SPE, and Vasily Demyanov, Institute of Petroleum En gineering, Heriot-Watt University, prepared for the 2010 Abu Dhabi International Petroleum Exhibition & Conference, Abu Dhabi, UAE, 1-4 November. The paper has not been peer reviewed. Multiple history-matched models are essential for a realistic uncertainty estimate of the field’s future behavior. This study compared application of ant-colony optimization (ACO), differential evolution (DE), and neighborhood algorithms (NAs) for history matching and uncertainty quantification of the PUNQ-S3 reservoir, a synthetic benchmark case with challenging parameterization, history-matching, and uncertainty-quantification steps. Introduction History matching of reservoir models and uncertainty quantification of the predictions are two important steps in any field-management process. In the history-matching phase, simulation models are calibrated on the basis of the observed production history of the reservoir. Parameters of the reservoir model are perturbed, and the model’s output is compared with available fluid-production rates or pressure measurements. This procedure is repeated with new parameter values until good agreement is obtained between output of the simulation and observations. History matching is an inverse problem with nonunique solutions. Multiple combinations of reservoir properties can provide a good match to observed field behavior. To quantify the uncertainty of predictions realistically, multiple history-matched reservoir models are required. Diverse models are likely to show different production behavior in the future because of different reservoir parameters. Recently, ACO and DE algorithms have been applied to history matching of a simple reservoir model with a small number of unknowns (eight). The subject methods for history matching were applied to a more-complex reservoir with 45 uncertain parameters, and the results were compared to those from NA. Originally, ACO was proposed for problems with discrete parameter space, and it later was adopted to handle continuous-optimization problems. DE is a powerful global-optimization algorithm that is a parallel-population-based search algorithm. The NA is a recent stochastic-optimization method that uses simple geometrical constructs called Voronoi cells to find good-fitting regions of the search space. Voronoi cell is a method to decompose the search space into n cells around n points by centering on the generated points. Each of these cells is the nearest neighborhood region of the points. Case Study The PUNQ-S3 reservoir-simulation model is a five-layer model. The top depth of PUNQ-S3 reservoir is 2430 m. It has a 1.5° dip angle and is bounded by faults to the east and south. A relatively strong aquifer on the north and west provides pressure support. Because of this pressure support, no injection wells have been drilled in this reservoir. There also is a small gas cap in the reservoir in Layer 1, and no wells are completed in this layer because of the effect of free-gas production on recovery from the reservoir. There are six production wells, all near the initial gas/oil contact. Three producers are perforated in Layers 4 and 5. Two producers are completed in Layers 3 and 4, and one producer is perforated only in Layer 4. One of the producers is completed near the aquifer, and water breakthrough was observed in the seventh year. Free-gas production starts in Years 4 and 5 in two of the producers. There also are positions for five infill wells.

Fast Algorithms for Bayesian Uncertainty Quantification in Large-Scale Linear Inverse Problems Based on Low-Rank Partial Hessian Approximations

We consider the problem of estimating the uncertainty in large-scale linear statistical inverse problems with high-dimensional parameter spaces within the framework of Bayesian inference. When the noise and prior probability densities are Gaussian, the solution to the inverse problem is also Gaussian and is thus characterized by the mean and covariance matrix of the posterior probability density. Unfortunately, explicitly computing the posterior covariance matrix requires as many forward solutions as there are parameters and is thus prohibitive when the forward problem is expensive and the parameter dimension is large. However, for many ill-posed inverse problems, the Hessian matrix of the data misfit term has a spectrum that collapses rapidly to zero. We present a fast method for computation of an approximation to the posterior covariance that exploits the low-rank structure of the preconditioned (by the prior covariance) Hessian of the data misfit. Analysis of an infinite-dimensional model convection-diffusion problem, and numerical experiments on large-scale three-dimensional convection-diffusion inverse problems with up to 1.5 million parameters, demonstrate that the number of forward PDE solves required for an accurate low-rank approximation is independent of the problem dimension. This permits scalable estimation of the uncertainty in large-scale ill-posed linear inverse problems at a small multiple (independent of the problem dimension) of the cost of solving the forward problem.

Comparison of Stochastic Sampling Algorithms for Uncertainty Quantification

Summary History matching and uncertainty quantification are two important research topics in reservoir simulation currently. In the Bayesian approach, we start with prior information about a reservoir (e.g., from analog outcrop data) and update our reservoir models with observations (e.g., from production data or time-lapse seismic). The goal of this activity is often to generate multiple models that match the history and use the models to quantify uncertainties in predictions of reservoir performance. A critical aspect of generating multiple history-matched models is the sampling algorithm used to generate the models. Algorithms that have been studied include gradient methods, genetic algorithms, and the ensemble Kalman filter (EnKF). This paper investigates the efficiency of three stochastic sampling algorithms: Hamiltonian Monte Carlo (HMC) algorithm, Particle Swarm Optimization (PSO) algorithm, and the Neighbourhood Algorithm (NA). HMC is a Markov chain Monte Carlo (MCMC) technique that uses Hamiltonian dynamics to achieve larger jumps than are possible with other MCMC techniques. PSO is a swarm intelligence algorithm that uses similar dynamics to HMC to guide the search but incorporates acceleration and damping parameters to provide rapid convergence to possible multiple minima. NA is a sampling technique that uses the properties of Voronoi cells in high dimensions to achieve multiple history-matched models. The algorithms are compared by generating multiple history- matched reservoir models and comparing the Bayesian credible intervals (p10-p50-p90) produced by each algorithm. We show that all the algorithms are able to find equivalent match qualities for this example but that some algorithms are able to find good fitting models quickly, whereas others are able to find a more diverse set of models in parameter space. The effects of the different sampling of model parameter space are compared in terms of the p10-p50-p90 uncertainty envelopes in forecast oil rate. These results show that algorithms based on Hamiltonian dynamics and swarm intelligence concepts have the potential to be effective tools in uncertainty quantification in the oil industry.

Technology Focus: Reservoir Simulation and Visualization (July 2009)

Technology Focus For many years, the title of this feature has been "Reservoir Simulation and Visualization," but year by year fewer papers are submitted on visualization. This clearly reflects that 3D visualization is no longer a novelty—it is a standard part of our workflow. This trend ties with another I noted last year—greater inclusion of geological modeling in history-matching workflows. Visualization is an essential part of the geological-modeling workflow, one of the geologist's key quality-checking steps, and only with good 3D-graphics workstations did geological modeling become mainstream. However, it does not seem likely that history matching will die out as a topic for papers any time soon! A great variety of techniques are being deployed to automate the process of history matching (improving the engineer's productivity) and to improve the quality of the result. Several experimental-design techniques are reported, as researchers continue attempts to extract the maximum benefit out of the fewest runs. It will be interesting to see if developments in hardware, leading to ever-reducing computing costs, result in a change of emphasis. Although history matching remains a key focus, the broader application of mathematical-optimization techniques to prediction is emerging as a key topic of interest. This application goes to the heart of our role as reservoir engineers, and it is good to see history matching treated as merely a means to an end, rather than an end in itself. The variety of reservoirs being developed and studied with simulation continues to grow, and the boundaries of technology continue to stretch. Naturally fractured carbonates, heavy oil under thermal recovery, tight gas, shale gas, coalbed methane, and other projects all feature more-complex and sophisticated wells. The 2009 SPE Reservoir Simulation Symposium produced many excellent papers on the underlying technology and algorithms of simulation and its application: It is good to see that innovation continues. Sadly, many of these papers do not lend themselves well to the synopsis format of this feature, so I strongly urge you to track down the Proceedings from a colleague if you were unable to attend. Reservoir Simulation and Visualization additional reading available at OnePetro: www.onepetro.org SPE 118178 • "An Integrated Case Study From Seismic to Simulation Through Geostatistical Inversion" by S. Soni, SPE, Fugro-Jason, et al. SPE 119139 • "Comparison of Stochastic Sampling Algorithms for Uncertainty Quantification" by L. Mohamed, SPE, Heriot-Watt University, et al. SPE 122934 • "A Workflow for Integrated Barnett Shale Gas Reservoir Modeling and Simulation" by C. Du, SPE, Schlumberger, et al.

Investigation of reliability method formulations in DAKOTA/UQ

Reliability methods are probabilistic algorithms for quantifying the effect of simulation input uncertainties on response metrics of interest. In particular, they compute approximate response function distribution statistics (probability, reliability and response levels) based on specified input random variable probability distributions. In this paper, a number of algorithmic variations are explored for both the forward reliability analysis of computing probabilities for specified response levels (the reliability index approach (RIA)) and the inverse reliability analysis of computing response levels for specified probabilities (the performance measure approach (PMA)). These variations include limit state linearizations, probability integrations, warm starting and optimization algorithm selections. The resulting RIA/PMA reliability algorithms for uncertainty quantification are then employed within bi-level and sequential reliability-based design optimization approaches. Relative performance of these uncertainty quantification and reliability-based design optimization algorithms are presented for a number of computational experiments performed using the DAKOTA/UQ software.

Coarse-gradient Langevin algorithms for dynamic data integration and uncertainty quantification

The main goal of this paper is to design an efficient sampling technique for dynamic data integration using the Langevin algorithms. Based on a coarse-scale model of the problem, we compute the proposals of the Langevin algorithms using the coarse-scale gradient of the target distribution. To guarantee a correct and efficient sampling, each proposal is first tested by a Metropolis acceptance–rejection step with a coarse-scale distribution. If the proposal is accepted in the first stage, then a fine-scale simulation is performed at the second stage to determine the acceptance probability. Comparing with the direct Langevin algorithm, the new method generates a modified Markov chain by incorporating the coarse-scale information of the problem. Under some mild technical conditions we prove that the modified Markov chain converges to the correct posterior distribution. We would like to note that the coarse-scale models used in the simulations need to be inexpensive, but not necessarily very accurate, as our analysis and numerical simulations demonstrate. We present numerical examples for sampling permeability fields using two-point geostatistics. Karhunen–Loève expansion is used to represent the realizations of the permeability field conditioned to the dynamic data, such as the production data, as well as the static data. The numerical examples show that the coarse-gradient Langevin algorithms are much faster than the direct Langevin algorithms but have similar acceptance rates.