Sergey Oladyshkin

Data-driven uncertainty quantification using the arbitrary polynomial chaos expansion

We discuss the arbitrary polynomial chaos (aPC), which has been subject of research in a few recent theoretical papers. Like all polynomial chaos expansion techniques, aPC approximates the dependence of simulation model output on model parameters by expansion in an orthogonal polynomial basis. The aPC generalizes chaos expansion techniques towards arbitrary distributions with arbitrary probability measures, which can be either discrete, continuous, or discretized continuous and can be specified either analytically (as probability density/cumulative distribution functions), numerically as histogram or as raw data sets. We show that the aPC at finite expansion order only demands the existence of a finite number of moments and does not require the complete knowledge or even existence of a probability density function. This avoids the necessity to assign parametric probability distributions that are not sufficiently supported by limited available data. Alternatively, it allows modellers to choose freely of technical constraints the shapes of their statistical assumptions. Our key idea is to align the complexity level and order of analysis with the reliability and detail level of statistical information on the input parameters. We provide conditions for existence and clarify the relation of the aPC to statistical moments of model parameters. We test the performance of the aPC with diverse statistical distributions and with raw data. In these exemplary test cases, we illustrate the convergence with increasing expansion order and, for the first time, with increasing reliability level of statistical input information. Our results indicate that the aPC shows an exponential convergence rate and converges faster than classical polynomial chaos expansion techniques.

Bayesian updating via bootstrap filtering combined with data-driven polynomial chaos expansions: methodology and application to history matching for carbon dioxide storage in geological formations

Model calibration and history matching are important techniques to adapt simulation tools to real-world systems. When prediction uncertainty needs to be quantified, one has to use the respective statistical counterparts, e.g., Bayesian updating of model parameters and data assimilation. For complex and large-scale systems, however, even single forward deterministic simulations may require parallel high-performance computing. This often makes accurate brute-force and nonlinear statistical approaches infeasible. We propose an advanced framework for parameter inference or history matching based on the arbitrary polynomial chaos expansion (aPC) and strict Bayesian principles. Our framework consists of two main steps. In step 1, the original model is projected onto a mathematically optimal response surface via the aPC technique. The resulting response surface can be viewed as a reduced (surrogate) model. It captures the model’s dependence on all parameters relevant for history matching at high-order accuracy. Step 2 consists of matching the reduced model from step 1 to observation data via bootstrap filtering. Bootstrap filtering is a fully nonlinear and Bayesian statistical approach to the inverse problem in history matching. It allows to quantify post-calibration parameter and prediction uncertainty and is more accurate than ensemble Kalman filtering or linearized methods. Through this combination, we obtain a statistical method for history matching that is accurate, yet has a computational speed that is more than sufficient to be developed towards real-time application. We motivate and demonstrate our method on the problem of CO2 storage in geological formations, using a low-parametric homogeneous 3D benchmark problem. In a synthetic case study, we update the parameters of a CO2/brine multiphase model on monitored pressure data during CO2 injection.

Polynomial Response Surfaces for Probabilistic Risk Assessment and Risk Control via Robust Design

Many engineering systems represent challenging classes of complex dynamic systems. Lacking information about their system properties leads to model uncertainties up to a level where quantification of uncertainties may become the dominant question in modeling, simulation and application tasks. Uncertainty quantification is the prerequisite for probabilistic risk assessment and related tasks. Current numerical simulationmodels are often too expensive for advanced application tasks that involve accurate uncertainty quantification, risk assessment and robust design. This Chapter will present recent approaches for these challenges based on polynomial response surface techniques, which reduce massively the initial complex model at surprising accuracy. The reduction is achieved via projections on orthonormal polynomial bases, which form a so-called response surface. This way, the model response to changes in uncertain parameters and design or control variables is represented by multi-variate polynomials for each output quantity of interest. This technique is known as polynomial chaos expansion (PCE) in the field of stochastic PDE solutions. The reduced model represented by the response surface is vastly faster than the original complex one, and thus provides a promising starting point for follow-up tasks: global sensitivity analysis, uncertainty quantification and probabilistic risk assessment and as well as robust design and control under uncertainty. Often, the fact that the response surface has known polynomial properties can further simplify these tasks. We will emphasize a more engineering-like language as compared to otherwise intense mathematical derivations found in the literature on PCE. Also we will make use of most recent developments in the theory of stochastic PDE solutions for engineering applications. The current Chapter provides tools based on PCE for global sensitivity analysis, uncertainty quantification and risk analysis as well as design under uncertainty (robust design).

Application of the polynomial chaos expansion to approximate the homogenised response of the intervertebral disc

A possibility to simulate the mechanical behaviour of the human spine is given by modelling the stiffer structures, i.e. the vertebrae, as a discrete multi-body system (MBS), whereas the softer connecting tissue, i.e. the softer intervertebral discs (IVD), is represented in a continuum-mechanical sense using the finite-element method (FEM). From a modelling point of view, the mechanical behaviour of the IVD can be included into the MBS in two different ways. They can either be computed online in a so-called co-simulation of a MBS and a FEM or offline in a pre-computation step, where a representation of the discrete mechanical response of the IVD needs to be defined in terms of the applied degrees of freedom (DOF) of the MBS. For both methods, an appropriate homogenisation step needs to be applied to obtain the discrete mechanical response of the IVD, i.e. the resulting forces and moments. The goal of this paper was to present an efficient method to approximate the mechanical response of an IVD in an offline computation. In a previous paper (Karajan et al. in Biomech Model Mechanobiol 12(3):453–466, 2012), it was proven that a cubic polynomial for the homogenised forces and moments of the FE model is a suitable choice to approximate the purely elastic response as a coupled function of the DOF of the MBS. In this contribution, the polynomial chaos expansion (PCE) is applied to generate these high-dimensional polynomials. Following this, the main challenge is to determine suitable deformation states of the IVD for pre-computation, such that the polynomials can be constructed with high accuracy and low numerical cost. For the sake of a simple verification, the coupling method and the PCE are applied to the same simplified motion segment of the spine as was used in the previous paper, i.e. two cylindrical vertebrae and a cylindrical IVD in between. In a next step, the loading rates are included as variables in the polynomial response functions to account for a more realistic response of the overall viscoelastic intervertebral disc. Herein, an additive split into elastic and inelastic contributions to the homogenised forces and moments is applied.

Probabilistic Assessment of Above Zone Pressure Predictions at a Geologic Carbon Storage Site

Carbon dioxide (CO2) storage into geological formations is regarded as an important mitigation strategy for anthropogenic CO2 emissions to the atmosphere. This study first simulates the leakage of CO2 and brine from a storage reservoir through the caprock. Then, we estimate the resulting pressure changes at the zone overlying the caprock also known as Above Zone Monitoring Interval (AZMI). A data-driven approach of arbitrary Polynomial Chaos (aPC) Expansion is then used to quantify the uncertainty in the above zone pressure prediction based on the uncertainties in different geologic parameters. Finally, a global sensitivity analysis is performed with Sobol indices based on the aPC technique to determine the relative importance of different parameters on pressure prediction. The results indicate that there can be uncertainty in pressure prediction locally around the leakage zones. The degree of such uncertainty in prediction depends on the quality of site specific information available for analysis. The scientific results from this study provide substantial insight that there is a need for site-specific data for efficient predictions of risks associated with storage activities. The presented approach can provide a basis of optimized pressure based monitoring network design at carbon storage sites.

Incomplete statistical information limits the utility of high-order polynomial chaos expansions

Polynomial chaos expansion (PCE) is a well-established massive stochastic model reduction technique that approximates the dependence of model output on uncertain input parameters. In many practical situations, only incomplete and inaccurate statistical knowledge on uncertain input parameters are available. Fortunately, to construct a finite-order expansion, only some partial information on the probability measure is required that can be simply represented by a finite number of statistical moments. Such situations, however, trigger the question to what degree higher-order statistical moments of input data are increasingly uncertain. On the one hand, increasing uncertainty in higher moments will lead to increasing inaccuracy in the corresponding chaos expansion and its result. On the other hand, the degree of expansion should adequately reflect the non-linearity of the analyzed model to minimize the approximation error of the expansion. Observation of the PCE convergence when statistical input information is incomplete demonstrates that higher-order PCE expansions without adequate data support are useless. Moreover, it makes apparent that PCE of a certain order is adequate just for a corresponding amount of available input data. The key idea of the current work is to align the order of expansion with a compromise between the degree of non-linearity of the model and the reliability of statistical information on the input parameters. To assure an optimal choice of the expansion order, we offer a simple relation that helps to align available input statistical data with an adequate expansion order. As fundamental steps into this direction, we propose overall error estimates for the statistical type of error that results from inaccurate statistical information plus the error that results from truncating the expansion of a non-linear model. Our key message is that any order of expansion is only justified if accompanied by reliable statistical information on input moments of a certain higher order.

Data-driven surrogates for rapid simulation and optimization of WAG injection in fractured carbonate reservoirs

Conventional simulation of fractured carbonate reservoirs is computationally expensive because of the multiscale heterogeneities and fracture–matrix transfer mechanisms that must be taken into account using numerical transfer functions and/or detailed models with a large number of simulation grid cells. The computational requirement increases significantly when multiple simulation runs are required for sensitivity analysis, uncertainty quantification and optimization. This can be prohibitive, especially for giant carbonate reservoirs. Yet, sensitivity analysis, uncertainty quantification and optimization are particularly important to analyse, determine and rank the impact of geological and engineering parameters on the economics and sustainability of different Enhanced Oil Recovery (EOR) techniques. We use experimental design to set up multiple simulations of a high-resolution model of a Jurassic carbonate ramp, which is an analogue for the highly prolific reservoirs of the Arab D Formation in Qatar. We consider CO2 water-alternating-gas (WAG) injection, which is a successful EOR method for carbonate reservoirs. The simulations are employed as a basis for generating data-driven surrogate models using polynomial regression and polynomial chaos expansion. Furthermore, the surrogates are validated by comparing surrogate predictions with results from numerical simulation and estimating goodness-of-fit measures. In the current work, parameter uncertainties affecting WAG modelling in fractured carbonates are evaluated, including fracture network properties, wettability and fault transmissibility. The results enable us to adequately explore the parameter space, and to quantify and rank the interrelated effect of uncertain model parameters on CO2 WAG efficiency. The results highlight the first-order impact of the fracture network properties and wettability on hydrocarbon recovery and CO2 utilization during WAG injection. In addition, the surrogate models enable us to calculate quick estimates of probabilistic uncertainty and to rapidly optimize WAG injection, while achieving significant computational speed-up compared with the conventional simulation framework.

Estimating the Probability of CO2 Leakage Using Rare Event Simulation

Estimating the probability of rare events is an extremely challenging task. For example, estimating the probability of leakage of CO2 from a saline aquifer using a direct Monte-Carlo approach would in general require a number of simulations proportional to the inverse of the rare event probability. Since it is likely that any action will require a probability of failure of less than

Non-Equlibrium Two-Velocity Effects in Gas-Condensate Flow through Porous Media

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A concept for data-driven uncertainty quantification and its application to carbon dioxide storage in geological formations

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