Begell House Inc.
International Journal for Uncertainty Quantification
IJUQ
2152-5080
2
4
2012
GRID-BASED INVERSION OF PRESSURE TRANSIENT TEST DATA WITH STOCHASTIC GRADIENT TECHNIQUES
323-339
10.1615/Int.J.UncertaintyQuantification.2012003480
Richard
Booth
Schlumberger BRGC, Rio de Janeiro, Brazil
Kirsty
Morton
Schlumberger BRGC, Rio de Janeiro, Brazil
Mustafa
Onur
Technical University of Istanbul, Istanbul, Turkey
Fikri
Kuchuk
Schlumberger Ribould Product Centre, Clamart, France
Bayesian inference
geological applications
transient well test interpretation
reservoir characterization
In any subsurface exploration and development, indirect information and measurements, such as detailed geological description, outcrop studies, and direct measurements (such as seismic, cores, logs, and fluid samples), provide useful data and information for static reservoir characterization, simulation, and forecasting. However, core and log data delineate rock properties only in the vicinity of the wellbore, while geological and seismic data are usually not directly related to formation permeability. Pressure transient test (PTT) data provide dynamic information about the reservoir and can be used to estimate rock properties, fluid samples for well productivity, and dynamic reservoir description. Therefore, PTT data are essential in the industry for the general purposes of production and reservoir engineering as well as commonly used for exploration environments. With the need for improved spatial resolution of the reservoir parameters, grid-based techniques have been developed in which the reservoir properties are discretized over a fine grid and characterization of the probable state of the reservoir is sought using the Bayesian framework. Unfortunately, for the exploration of hydrocarbon-bearing formations, the available prior information is often limited: in particular, unexpected geological features, such as fracture and faults, may be present. There are two groups of recent methods for dynamic characterization of the reservoir: (i) data assimilation techniques, e.g., ensemble Kalman filter (EnKF) and (ii) maximum-likelihood techniques, such as gradient-based methods. The EnKF is designed to produce a set of realizations of the reservoir properties that fit the PTT data; however, the method often fails to honor the data when unexpected features are not captured by the prior model. The alternative gradient-based methods do provide a good fit to the PTT data. They can also be made efficient for high-dimensional problems by using an adjoint scheme for determining the gradient of the log-likelihood function. However, as a maximum likelihood technique, this method only yields a single realization of the reservoir. It is important to maintain a model of the uncertainty of the reservoir characterization after PTT data assimilation, so that the risk associated with future decisions is understood. We therefore present and investigate a stochastic, gradient-based method that allows for proper sampling of realizations of the reservoir parameters that preserve the fit with the PTT data. The results indicate that our proposed method is quite encouraging for efficiently generating realizations of rock property distributions conditioned to PTT data sets and a given prior geostatistical model.
REDUCED ORDER MODELING FOR NONLINEAR MULTI-COMPONENT MODELS
341-361
10.1615/Int.J.UncertaintyQuantification.2012003523
Hany S.
Abdel-Khalik
Department of Nuclear Engineering, North Carolina State University, Raleigh, North Carolina 27695-7909, USA
Youngsuk
Bang
Department of Nuclear Engineering, North Carolina State University, Raleigh, North Carolina 27695-7909, USA
Christopher
Kennedy
Department of Nuclear Engineering, North Carolina State University, Raleigh, North Carolina 27695-7909, USA
Jason
Hite
Department of Nuclear Engineering, North Carolina State University, Raleigh, North Carolina 27695-7909, USA
nonlinear sensitivity analysis
reduced order modeling
subspace methods
Reduced order modeling plays an indispensible role in most real-world complex models. A hybrid application of order reduction methods, introduced previously, has been shown to effectively reduce the computational cost required to find a reduced order model with quantifiable bounds on the reduction errors, which is achieved by hybridizing the application of local variational and global sampling methods for order reduction. The method requires the evaluation of first-order derivatives of pseudo-responses with respect to input parameters and the ability to perturb input parameters within their user-specified ranges of variations. The derivatives are employed to find a subspace that captures all possible response variations resulting from all possible parameter variations with quantifiable accuracy. This paper extends the applicability of this methodology to multi-component models. This is achieved by employing a hybrid methodology to enable the transfer of sensitivity information between the various components in an efficient manner precluding the need for a global sensitivity analysis procedure, which is often envisaged to be computationally intractable. Finally, we introduce a new measure of conditioning for the subspace employed for order reduction. Although, the developments are general, they are applied here to smoothly behaving functions only. Extension to non-smooth functions will be addressed in a future article. In addition to introducing these new developments, this manuscript is intended to provide a pedagogical overview of our current developments in the area of reduced order modeling to real-world engineering models.
A HOLISTIC APPROACH TO UNCERTAINTY QUANTIFICATION WITH APPLICATION TO SUPERSONIC NOZZLE THRUST
363-381
10.1615/Int.J.UncertaintyQuantification.2012003562
Christopher J.
Roy
Aerospace and Ocean Engineering Department, Virginia Tech, Blacksburg, Virginia 24061, USA
Michael S.
Balch
Applied Biomathematics, Setauket, New York 11733, USA
uncertainty quantification
verification
validation
modeling
simulation
In modeling and simulation (M&S), we seek to predict the state of a system using a computer-based simulation of a differential equation-based model. In general, the inputs to the model may contain uncertainty due to inherent randomness (aleatory uncertainty), a lack of knowledge (epistemic uncertainty), or a combination of the two. In many practical cases, there is so little knowledge of a model input that it should be characterized as an interval, the weakest statement of knowledge. When some model inputs are probabilistic and others are intervals, segregated uncertainty propagation should be used. The resulting uncertainty structure on the M&S output can take the form of a cumulative distribution function with a finite width; i.e., a p-box. Implications of sampling over interval versus probabilistic uncertainties in the outer loop are discussed and examples are given showing the effects of the choice of uncertainty propagation and characterization methods. In addition to the uncertainties in model inputs, uncertainties also arise due to modeling deficiencies and numerical approximations. Modeling uncertainties can be reduced by performing additional experiments and numerical uncertainties can be reduced by using additional computational resources; thus, both sources of uncertainty can be modeled as epistemic and can be characterized as intervals and included in the total predictive uncertainty by appropriately broadening the prediction p-box. A simple example is given for the M&S predictions of supersonic nozzle thrust that incorporates and quantifies all three sources of uncertainty.
CONVOLVED ORTHOGONAL EXPANSIONS FOR UNCERTAINTY PROPAGATION: APPLICATION TO RANDOM VIBRATION PROBLEMS
383-395
10.1615/Int.J.UncertaintyQuantification.2012004041
X. Frank
Xu
Department of Civil, Environmental and Ocean Engineering, Stevens Institute of Technology, Hoboken, New Jersey 07030, USA
George
Stefanou
Aristotle University of Thessaloniki
orthogonal expansion
non-Gaussian
random process
nonlinear dynamics
Physical nonlinear systems are typically characterized with n-fold convolution of the Green′s function, e.g., nonlinear
oscillators, inhomogeneous media, and scattering theory in continuum and quantum mechanics. A novel stochastic
computation method based on orthogonal expansions of random fields has been recently proposed [1]. In this study, the
idea of orthogonal expansion is formalized as the so-called nth-order convolved orthogonal expansion (COE) method,
especially in dealing with random processes in time. Although the paper is focused on presentation of the properties
of the convolved random basis processes, examples are also provided to demonstrate application of the COE method to
random vibration problems. In addition, the relation to the classical Volterra-type expansions is discussed.
INTERACTIVE VISUALIZATION OF PROBABILITY AND CUMULATIVE DENSITY FUNCTIONS
397-412
10.1615/Int.J.UncertaintyQuantification.2012004074
Kristin
Potter
NREL
Mike
Kirby
Scientific Computing and Imaging Institute, University of Utah, Salt Lake City, Utah, 84112, USA
Dongbin
Xiu
Ohio Eminent Scholar Department of Mathematics The Ohio State University Columbus, Ohio 43210, USA
Chris R.
Johnson
Scientific Computing and Imaging Institute, School of Computing
University of Utah
Salt Lake City, Utah 84112, USA
visualization
probability density function
cumulative density function
generalized polynomial chaos
stochastic Galerkin methods
stochastic collocation methods
The probability density function (PDF), and its corresponding cumulative density function (CDF), provide direct statistical insight into the characterization of a random process or field. Typically displayed as a histogram, one can infer probabilities of the occurrence of particular events. When examining a field over some two-dimensional domain in which at each point a PDF of the function values is available, it is challenging to assess the global (stochastic) features present within the field. In this paper, we present a visualization system that allows the user to examine two-dimensional data sets in which PDF (or CDF) information is available at any position within the domain. The tool provides a contour display showing the normed difference between the PDFs and an ansatz PDF selected by the user and, furthermore, allows the user to interactively examine the PDF at any particular position. Canonical examples of the tool are provided to help guide the reader into the mapping of stochastic information to visual cues along with a description of the use of the tool for examining data generated from an uncertainty quantification exercise accomplished within the field of electrophysiology.
ITERATIVE METHODS FOR SCALABLE UNCERTAINTY QUANTIFICATION IN COMPLEX NETWORKS
413-439
10.1615/Int.J.UncertaintyQuantification.2012004138
Amit
Surana
United Technologies Research Center, East Hartford, Connecticut 06118, USA
Tuhin
Sahai
United Technologies Research Center, East Hartford, Connecticut 06118, USA
Andrzej
Banaszuk
United Technologies Research Center, East Hartford, Connecticut 06118, USA
collocation
polynomial chaos
graph decomposition
waveform relaxation
In this paper we address the problem of uncertainty management for robust design, and verification of large dynamic networks whose performance is affected by an equally large number of uncertain parameters. Many such networks (e.g., power, thermal, and communication networks) are often composed of weakly interacting subnetworks. We propose intrusive and nonintrusive iterative schemes that exploit such weak interconnections to overcome the dimensionality curse associated with traditional uncertainty quantification methods (e.g., generalized polynomial chaos, probabilistic collocation) and accelerate uncertainty propagation in systems with a large number of uncertain parameters. This approach relies on integrating graph theoretic methods and waveform relaxation with generalized polynomial chaos, and probabilistic collocation, rendering these techniques scalable. We introduce an approximate Galerkin projection that based on the results of graph decomposition computes "strong" and "weak" influence of parameters on states. An appropriate order of expansion, in terms of the parameters, is then selected for the various states. We analyze convergence properties of this scheme and illustrate it in several examples.