Download or read book Data space Approaches for Efficient Uncertainty Quantification in Subsurface Flow Problems written by Wenyue Sun and published by . This book was released on 2018 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Uncertainty quantification for subsurface flow problems is typically accomplished through the use of model inversion procedures in which multiple posterior (history-matched) geological models are generated and used for flow predictions. These procedures can be demanding computationally, and it is not always straightforward to maintain geological realism in the resulting history-matched models. In some applications, it is the flow predictions themselves (and the uncertainty associated with these predictions), rather than the posterior geological models, that are of primary interest. This is the motivation for the data-space inversion (DSI) procedures developed in this work. In the DSI framework, an ensemble of prior model realizations, honoring prior geostatistical information and hard data at wells, are generated and then (flow) simulated. The resulting reservoir responses (e.g., time-series of flow rate data at wells, and/or limited spatial saturation fields) are assembled into data vectors that represent prior `realizations' in the data space. The conditional distribution of data variables given observed data is then constructed within a Bayesian framework. This distribution is directly sampled using a data-space randomized maximum likelihood method. Due to the non-Gaussian characteristics of the data variables, we introduce pattern-based mapping operations, or histogram transformation, along with principal component analysis. These treatments allow us to represent the data variables using a set of low-dimensional variables that are closer to multivariate Gaussian, which is shown to improve the performance of DSI. We present extensive numerical results for two example cases involving oil-water flow in a bimodal channelized system and oil-water-gas flow in a Gaussian permeability system, in which the quantities of interest (QoI) are time-series data at wells. DSI results, with pattern-based mapping operations, for uncertainty quantification (e.g., P10, P50, P90 posterior predictions) are compared with those obtained from a strict rejection sampling (RS) procedure. Reasonable agreement between the DSI and RS results is consistently achieved, even when the (synthetic) true data to be matched fall near the edge of the prior distribution. Computational savings using DSI are very substantial in that RS requires O(10^5--10^6) flow simulations, in contrast to 500 for DSI, for the cases considered. We then apply the DSI procedure, with the histogram transformation treatment for data reparameterization, for naturally fractured reservoirs (NFRs), represented as general discrete-fracture-matrix (DFM) models. This DSI procedure is first tested on two-dimensional DFM systems involving multiple fracture scenarios. Comparison with an approximate rejection sampling procedure for this case indicates the DSI results for the P10, P50 and P90 responses are again consistent with RS results. The DSI method is then applied to a realistic NFR that has undergone 15 years of primary production and is under consideration for waterflooding. To construct the DSI representation, around 400 prior DFM models, which correspond to different geologic concepts and properties, are simulated. Two different reference `true' models, along with different data-assimilation durations, are considered. In all cases, the DSI predictions are shown to be consistent with the forecasts from the `true' model, and to provide reasonable quantification of forecast uncertainty. Finally, we investigate the application of DSI to quantify the uncertainty associated with carbon storage operations, in which the QoI is the spatial distribution of CO2 saturation in the top layer of a storage aquifer, and the observed data are pressure and CO2 saturation measurements from a few monitoring wells. We also introduce a procedure to optimize the locations of monitoring wells using only prior-model simulation results. This approach is based on analytical DSI results, and determines monitoring well locations such that the reduction in expected posterior variance of a relevant quantity is maximized. The new DSI procedure is applied to three-dimensional heterogeneous aquifer models involving uncertainties in a wide range of geological parameters, including variogram orientation, porosity and permeability fields, and regional pressure gradient. Multiple monitoring scenarios, involving four to eight monitoring wells, are considered in this evaluation. Application of DSI with optimal monitoring wells is shown to consistently reduce the posterior variance in predictions of the average CO2 saturation in the top layer, and to provide detailed saturation fields in reasonable correspondence with the `true' saturation distribution.

Download or read book Data driven Uncertainty Quantification for Predictive Subsurface Flow and Transport Modeling written by Jiachuan He and published by . This book was released on 2018 with total page 190 pages. Available in PDF, EPUB and Kindle. Book excerpt: Specification of hydraulic conductivity as a model parameter in groundwater flow and transport equations is an essential step in predictive simulations. It is often infeasible in practice to characterize this model parameter at all points in space due to complex hydrogeological environments leading to significant parameter uncertainties. Quantifying these uncertainties requires the formulation and solution of an inverse problem using data corresponding to observable model responses. Several types of inverse problems may be formulated under various physical and statistical assumptions on the model parameters, model response, and the data. Solutions to most types of inverse problems require large numbers of model evaluations. In this study, we incorporate the use of surrogate models based on support vector machines to increase the number of samples used in approximating a solution to an inverse problem at a relatively low computational cost. To test the global capabilities of this type of surrogate model for quantifying uncertainties, we use a framework for constructing pullback and push-forward probability measures to study the data-to-parameter-to-prediction propagation of uncertainties under minimal statistical assumptions. Additionally, we demonstrate that it is possible to build a support vector machine using relatively low-dimensional representations of the hydraulic conductivity to propagate distributions. The numerical examples further demonstrate that we can make reliable probabilistic predictions of contaminant concentration at spatial locations corresponding to data not used in the solution to the inverse problem. This dissertation is based on the article entitled Data-driven uncertainty quantification for predictive flow and transport modeling using support vector machines by Jiachuan He, Steven Mattis, Troy Butler and Clint Dawson [32]. This material is based upon work supported by the U.S. Department of Energy Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under Award Number DE-SC0009286 as part of the DiaMonD Multifaceted Mathematics Integrated Capability Center

Download or read book Uncertainty quantification for wave propagation and flow problems with random data written by Markus Wahlsten and published by Linköping University Electronic Press. This book was released on 2018-04-09 with total page 45 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this thesis we study partial differential equations with random inputs. The effects that different boundary conditions with random data and uncertain geometries have on the solution are analyzed. Further, comparisons and couplings between different uncertainty quantification methods are performed. The numerical simulations are based on provably strongly stable finite difference formulations based on summation-by-parts operators and a weak implementation of boundary and interface conditions. The first part of this thesis treats the construction of variance reducing boundary conditions. It is shown how the variance of the solution can be manipulated by the choice of boundary conditions, and a close relation between the variance of the solution and the energy estimate is established. The technique is studied on both a purely hyperbolic system as well as an incompletely parabolic system of equations. The applications considered are the Euler, Maxwell's, and Navier--Stokes equations. The second part focuses on the effect of uncertain geometry on the solution. We consider a two-dimensional advection-diffusion equation with a stochastically varying boundary. We transform the problem to a fixed domain where comparisons can be made. Numerical results are performed on a problem in heat transfer, where the frequency and amplitude of the prescribed uncertainty are varied. The final part of the thesis is devoted to the comparison and coupling of different uncertainty quantification methods. An efficiency analysis is performed using the intrusive polynomial chaos expansion with stochastic Galerkin projection, and nonintrusive numerical integration. The techniques are compared using the non-linear viscous Burgers' equation. A provably stable coupling procedure for the two methods is also constructed. The general coupling procedure is exemplified using a hyperbolic system of equations.

Download or read book Parameter Estimation and Uncertainty Quantification in Water Resources Modeling written by Philippe Renard and published by Frontiers Media SA. This book was released on 2020-04-22 with total page 177 pages. Available in PDF, EPUB and Kindle. Book excerpt: Numerical models of flow and transport processes are heavily employed in the fields of surface, soil, and groundwater hydrology. They are used to interpret field observations, analyze complex and coupled processes, or to support decision making related to large societal issues such as the water-energy nexus or sustainable water management and food production. Parameter estimation and uncertainty quantification are two key features of modern science-based predictions. When applied to water resources, these tasks must cope with many degrees of freedom and large datasets. Both are challenging and require novel theoretical and computational approaches to handle complex models with large number of unknown parameters.

Download or read book Uncertainty Quantification and Sensitivity Analysis of Geoscientific Predictions with Data driven Approaches written by Jihoon Park and published by . This book was released on 2019 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Uncertainty quantification in the Earth Sciences forms an integral component in decision making. Such decision has different objectives depending on the subsurface system. For example, the goals include maximizing profits in exploitation of resources or minimizing the effects on the environment. It is often the case that the decision has to balance between multiple conflicting objectives. Because the decision is made on prediction uncertainty, it is crucial to quantify realistic uncertainty which necessitates identification of a variety of sources of model uncertainty. The sources of model uncertainty include different interpretations on subsurface structures and depositional scenarios, unknown spatial distributions of properties, uncertainty in boundary conditions, hydrological/hydraulic properties and errors in measurements. The subsurface system is parameterized to represent model uncertainty. The model variable can be either global (takes scalar value) or spatially distributed. With limited available data, a large number of uncertain model variables exists. One of key tasks is to quantify how each model variable contribute to response uncertainty, which can be achieved by means of sensitivity analysis. Sensitivity analysis plays an important role in geoscientific computer experiments, whether for forecasting, data assimilation or model calibration. Some methods of sensitivity analysis have been used in Earth Sciences but they have clear limitations -- they cannot efficiently deal with multivariate responses, excessive calculations are required, and it is hard to take into account categorical input uncertainty. Overcoming these limitations, we revisit the idea of regionalized sensitivity analysis. In particular, we focus on distance-based global sensitivity analysis to estimate sensitivities of multivariate responses with limited number of samples. We demonstrate how the results from sensitivity analysis can be utilized to reduce model uncertainty with minimal impact on response uncertainty. The results can be used to design second Monte Carlo or building a surrogate model. Uncertainty needs to be updated as more data are required from different sources. In a Bayesian framework, this requires sampling from a posterior density of model and prediction variables. The key components of the workflow are dimensionality reduction of data variables and building of a statistical surrogate model to replace full forward models. It is demonstrated that the methodology successfully performs model inversions with limited number of full forward model runs. In many geoscientific applications, both global and spatial variables are uncertain. For convenience in computations, spatial variables are often converted to a few global variables. Even if the approach is efficient, the inversion results may not be consistent with the stated geological prior which leads to unrealistic uncertainty. In this dissertation, we propose to extend direct forecasting to predict model variables themselves. It is shown that successful inversion can be performed with both global and spatial variables characterizing a field-scale subsurface system. All the methodologies are demonstrated with the case studies. The first case deals with an oil reservoir in Libya. The case is used to study the proposed methods for global sensitivity analysis and approaches for model inversions to integrate dynamic data. The second case deals with the groundwater reservoir in Denmark. The case is used to integrate different sources of data to offer the inputs of decision models for groundwater management.

Download or read book Quantification of Uncertainty Improving Efficiency and Technology written by Marta D'Elia and published by Springer. This book was released on 2021-08-01 with total page 282 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book explores four guiding themes – reduced order modelling, high dimensional problems, efficient algorithms, and applications – by reviewing recent algorithmic and mathematical advances and the development of new research directions for uncertainty quantification in the context of partial differential equations with random inputs. Highlighting the most promising approaches for (near-) future improvements in the way uncertainty quantification problems in the partial differential equation setting are solved, and gathering contributions by leading international experts, the book’s content will impact the scientific, engineering, financial, economic, environmental, social, and commercial sectors.

Download or read book An Efficient Computational Framework for Uncertainty Quantification in Multiscale Systems written by Xiang Ma and published by . This book was released on 2011 with total page 224 pages. Available in PDF, EPUB and Kindle. Book excerpt: To accurately predict the performance of physical systems, it becomes essential for one to include the effects of input uncertainties into the model system and understand how they propagate and alter the final solution. The presence of uncertainties can be modeled in the system through reformulation of the governing equations as stochastic partial differential equations (SPDEs). The spectral stochastic finite element method (SSFEM) and stochastic collocation methods are the most popular simulation methods for SPDEs. However, both methods utilize global polynomials in the stochastic space. Thus when there are steep gradients or finite discontinuities in the stochastic space, these methods converge slowly or even fail to converge. In order to resolve the above mentioned issues, an adaptive sparse grid collocation (ASGC) strategy is developed using piecewise multi-linear hierarchical basis functions. Hierarchical surplus is used as an error indicator to automatically detect the discontinuity region in the stochastic space and adaptively refine the collocation points in this region. However, this method is limited to a moderate number of random variables. To address the solution of high-dimensional stochastic problems, a computational methodology is further introduced that utilizes the High Dimensional Model Representation (HDMR) technique in the stochastic space to represent the model output as a finite hierarchical correlated function expansion in terms of the stochastic inputs starting from lower-order to higher-order component functions. An adaptive version of HDMR is also developed to automatically detect the important dimensions and construct higherorder terms using only the important dimensions. The ASGC is integrated with HDMR to solve the resulting sub-problems. Uncertainty quantification for fluid transport in porous media in the presence of both stochastic permeability and multiple scales is addressed using the developed HDMR framework. In order to capture the small scale heterogeneity, a new mixed multiscale finite element method is developed within the framework of the heterogeneous multiscale method in the spatial domain. Several numerical examples are considered to examine the accuracy of the multiscale and stochastic frameworks developed. A summary of suggestions for future research in the area of stochastic multiscale modeling are given at the end of the thesis.

Download or read book Distribution based Framework for Uncertainty Quantification of Flow in Porous Media written by Hyung Jun Yang and published by . This book was released on 2022 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Quantitative predictions of fluid flow and transport in porous media are often compromised by multi-scale heterogeneity and insufficient site characterization. These factors introduce uncertainty on the input and output of physical systems which are generally expressed as partial differential equations (PDEs). The characterization of this predictive uncertainty is typically done with forward propagation of input uncertainty as well as inverse modeling for the dynamic data integration. The main challenges of forward uncertainty propagation arise from the slow convergence of Monte Carlo Simulations (MCS) especially when the goal is to compute the probability distribution which is necessary for risk assessment and decision making under uncertainty. On the other hand, reliable inverse modeling is often hampered by the ill-posedness of the problem, thus the incorporation of geological constraints becomes increasingly important. In the thesis, four significant contributions are made to alleviate these outstanding issues on forward and inverse problems. First, the method of distributions for the steady-state flow problem is developed to yield a full probabilistic description of outputs via probability distribution function (PDF) or cumulative distribution (CDF). The derivation of deterministic equation for CDF relies on stochastic averaging techniques and self-consistent closure approximation which ensures the resulting CDF has the same mean and variance as those computed with moment equations or MCS. We conduct a series of numerical experiments dealing with steady-state two-dimensional flow driven by either a natural hydraulic head gradient or a pumping well. These experiments reveal that the proposed method remains accurate and robust for highly heterogeneous formations with the variance of log conductivity as large as five. For the same accuracy, it is also up to four orders of magnitude faster than MCS with a required degree of confidence. The second contribution of this work is the extension of the distribution-based method to account for uncertainty in the geologic makeup of a subsurface environment and non-stationary cases. Our CDF-RDD framework provides a probabilistic assessment of uncertainty in highly heterogeneous subsurface formations by combining the method of distributions and the random domain decomposition (RDD). Our numerical experiments reveal that the CDF-RDD remains accurate for two-dimensional flow in a porous material composed of two heterogeneous geo-facies, a setting in which the original distribution method fails. For the same accuracy, the CDF-RDD is an order of magnitude faster than MCS. Next, we develop a complete distribution-based method for the probabilistic forecast of two-phase flow in porous media. The CDF equation for travel time is derived within the efficient streamline-based framework to replace the MCS in the previous FROST method. For getting fast and stable results, we employ numerical techniques including pseudo-time integration, flux-limited scheme, and exponential grid spacing. Our CDF-FROST framework uses the results of the method of distributions for travel time as an input of FROST method. The proposed method provides a probability distribution of saturation without using any sampling-based methods. The numerical tests demonstrate that the CDF-FROST shows good accuracy in estimating the probability distributions of both saturation and travel time. For the same accuracy, it is about 5 and 10 times faster than the previous FROST method and naive MCS, respectively. Lastly, we propose a consensus equilibrium (CE) framework to reconstruct the realistic geological model by the inverse modeling of sparse dynamic data. The optimization-based inversion techniques are integrated with recent machine learning-based methods (e.g., variational auto-encoder and convolutional neural network) by the proposed CE algorithm to capture the complicated geological features. The numerical examples verify that the proposed method well preserves the geological realism, and it efficiently quantifies the uncertainty conditioned on dynamic information.

Download or read book Geothermal Power Generation written by Ronald DiPippo and published by Elsevier. This book was released on 2024-10-11 with total page 977 pages. Available in PDF, EPUB and Kindle. Book excerpt: Geothermal Power Generation, New Developments and Innovations, Second Edition provides an update to the advanced energy technologies that are urgently required to meet the challenges of economic development, climate change mitigation, and energy security. Edited by respected and leading experts in the field, this book provides a comprehensive overview of the major aspects of geothermal power production. Chapters cover resource discovery, resource characterization, energy conversion systems, design, economic considerations, and a range of fascinating and updated case studies from across the world.Geothermal resources are considered renewable and are currently the only renewable source able to generate baseload electricity while producing very low levels of greenhouse gas emissions, thus playing a key role in future energy needs. - Provides readers with a comprehensive and systematic overview of geothermal power generation - Presents an update to advanced energy technologies that are urgently required to meet the challenges of economic development, climate change mitigation, and energy security - Edited by authorities in the field and contributed to by global experts in their areas - Supports sustainability and the United Nations Sustainable Development Goals (UN SDGs) 7, 9, 11 and 13

Download or read book Numerical Methods for Porous Media Flow written by Bradley W. McCaskill and published by . This book was released on 2018 with total page 139 pages. Available in PDF, EPUB and Kindle. Book excerpt: The way in which we manage subsurface resources is directly determined by the availability and quality of information we have about the dynamical systems that govern them. Typically this information is obtained by solving mathematical models that are posed on the domain of interest. Unfortunately, both the construction and process of solving these models can be a nontrivial task. In this dissertation we explore solutions to several problems related to modeling fluid flow through porous media. One aspect of this dissertation is the development of a nonstandard multiscale finite element method for solving elliptic boundary value problems. The so-called multiscale Robin method can be viewed as a merger of traditional domain decomposition methods with the framework of multiscale finite element methods. The novelty of this method is that its efficiency and accuracy are governed by a geometric enrichment of the solution space. An application of the multiscale Robin method to uncertainty quantification through the use of a stochastic representation method is considered. To this end, the multiscale Robin methodology is adapted to the framework of coupled elliptic boundary value problems. A computationally cheap and efficient method for the simulation of two-phase flow through poroelastic media is also proposed. Specifically, through the use of a artificial stabilization term and an element based post processing reasonable estimates of solutions to the associated geomechanic subsystem can be obtained. Finally, we adapt a continuous data assimilation algorithm to a model for miscible flow through porous media. The existence of weak solutions and convergence properties of the resulting model solution are studied. In all chapters a variety of numerical examples are used to evaluate the performance of each proposed solution methodology.

Download or read book Hessian based Response Surface Approximations for Uncertainty Quantification in Large scale Statistical Inverse Problems with Applications to Groundwater Flow written by Hannah Pearl Flath and published by . This book was released on 2013 with total page 316 pages. Available in PDF, EPUB and Kindle. Book excerpt: Subsurface flow phenomena characterize many important societal issues in energy and the environment. A key feature of these problems is that subsurface properties are uncertain, due to the sparsity of direct observations of the subsurface. The Bayesian formulation of this inverse problem provides a systematic framework for inferring uncertainty in the properties given uncertainties in the data, the forward model, and prior knowledge of the properties. We address the problem: given noisy measurements of the head, the pdf describing the noise, prior information in the form of a pdf of the hydraulic conductivity, and a groundwater flow model relating the head to the hydraulic conductivity, find the posterior probability density function (pdf) of the parameters describing the hydraulic conductivity field. Unfortunately, conventional sampling of this pdf to compute statistical moments is intractable for problems governed by large-scale forward models and high-dimensional parameter spaces. We construct a Gaussian process surrogate of the posterior pdf based on Bayesian interpolation between a set of "training" points. We employ a greedy algorithm to find the training points by solving a sequence of optimization problems where each new training point is placed at the maximizer of the error in the approximation. Scalable Newton optimization methods solve this "optimal" training point problem. We tailor the Gaussian process surrogate to the curvature of the underlying posterior pdf according to the Hessian of the log posterior at a subset of training points, made computationally tractable by a low-rank approximation of the data misfit Hessian. A Gaussian mixture approximation of the posterior is extracted from the Gaussian process surrogate, and used as a proposal in a Markov chain Monte Carlo method for sampling both the surrogate as well as the true posterior. The Gaussian process surrogate is used as a first stage approximation in a two-stage delayed acceptance MCMC method. We provide evidence for the viability of the low-rank approximation of the Hessian through numerical experiments on a large scale atmospheric contaminant transport problem and analysis of an infinite dimensional model problem. We provide similar results for our groundwater problem. We then present results from the proposed MCMC algorithms.

Download or read book FEFLOW written by Hans-Jörg G. Diersch and published by Springer Science & Business Media. This book was released on 2013-11-22 with total page 1018 pages. Available in PDF, EPUB and Kindle. Book excerpt: FEFLOW is an acronym of Finite Element subsurface FLOW simulation system and solves the governing flow, mass and heat transport equations in porous and fractured media by a multidimensional finite element method for complex geometric and parametric situations including variable fluid density, variable saturation, free surface(s), multispecies reaction kinetics, non-isothermal flow and multidiffusive effects. FEFLOW comprises theoretical work, modeling experiences and simulation practice from a period of about 40 years. In this light, the main objective of the present book is to share this achieved level of modeling with all required details of the physical and numerical background with the reader. The book is intended to put advanced theoretical and numerical methods into the hands of modeling practitioners and scientists. It starts with a more general theory for all relevant flow and transport phenomena on the basis of the continuum approach, systematically develops the basic framework for important classes of problems (e.g., multiphase/multispecies non-isothermal flow and transport phenomena, discrete features, aquifer-averaged equations, geothermal processes), introduces finite-element techniques for solving the basic balance equations, in detail discusses advanced numerical algorithms for the resulting nonlinear and linear problems and completes with a number of benchmarks, applications and exercises to illustrate the different types of problems and ways to tackle them successfully (e.g., flow and seepage problems, unsaturated-saturated flow, advective-diffusion transport, saltwater intrusion, geothermal and thermohaline flow).

Download or read book Artificial Intelligence and Data Analytics for Energy Exploration and Production written by Fred Aminzadeh and published by John Wiley & Sons. This book was released on 2022-08-26 with total page 613 pages. Available in PDF, EPUB and Kindle. Book excerpt: ARTIFICAL INTELLIGENCE AND DATA ANALYTICS FOR ENERGY EXPLORATION AND PRODUCTION This groundbreaking new book is written by some of the foremost authorities on the application of data science and artificial intelligence techniques in exploration and production in the energy industry, covering the most comprehensive and updated new processes, concepts, and practical applications in the field. The book provides an in-depth treatment of the foundations of Artificial Intelligence (AI) Machine Learning, and Data Analytics (DA). It also includes many of AI-DA applications in oil and gas reservoirs exploration, development, and production. The book covers the basic technical details on many tools used in “smart oil fields”. This includes topics such as pattern recognition, neural networks, fuzzy logic, evolutionary computing, expert systems, artificial intelligence machine learning, human-computer interface, natural language processing, data analytics and next-generation visualization. While theoretical details will be kept to the minimum, these topics are introduced from oil and gas applications viewpoints. In this volume, many case histories from the recent applications of intelligent data to a number of different oil and gas problems are highlighted. The applications cover a wide spectrum of practical problems from exploration to drilling and field development to production optimization, artificial lift, and secondary recovery. Also, the authors demonstrate the effectiveness of intelligent data analysis methods in dealing with many oil and gas problems requiring combining machine and human intelligence as well as dealing with linguistic and imprecise data and rules.

Download or read book Machine Learning Methods for Uncertainty Quantification in Subsurface Reservoirs written by Shing Cheng Chan Chang and published by . This book was released on 2018 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book History Matching and Uncertainty Quantification Using Sampling Method written by Xianlin Ma and published by . This book was released on 2010 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Uncertainty quantification involves sampling the reservoir parameters correctly from a posterior probability function that is conditioned to both static and dynamic data. Rigorous sampling methods like Markov Chain Monte Carlo (MCMC) are known to sample from the distribution but can be computationally prohibitive for high resolution reservoir models. Approximate sampling methods are more efficient but less rigorous for nonlinear inverse problems. There is a need for an efficient and rigorous approach to uncertainty quantification for the nonlinear inverse problems. First, we propose a two-stage MCMC approach using sensitivities for quantifying uncertainty in history matching geological models. In the first stage, we compute the acceptance probability for a proposed change in reservoir parameters based on a linearized approximation to flow simulation in a small neighborhood of the previously computed dynamic data. In the second stage, those proposals that passed a selected criterion of the first stage are assessed by running full flow simulations to assure the rigorousness. Second, we propose a two-stage MCMC approach using response surface models for quantifying uncertainty. The formulation allows us to history match three-phase flow simultaneously. The built response exists independently of expensive flow simulation, and provides efficient samples for the reservoir simulation and MCMC in the second stage. Third, we propose a two-stage MCMC approach using upscaling and non-parametric regressions for quantifying uncertainty. A coarse grid model acts as a surrogate for the fine grid model by flow-based upscaling. The response correction of the coarse-scale model is performed by error modeling via the non-parametric regression to approximate the response of the computationally expensive fine-scale model. Our proposed two-stage sampling approaches are computationally efficient and rigorous with a significantly higher acceptance rate compared to traditional MCMC algorithms. Finally, we developed a coarsening algorithm to determine an optimal reservoir simulation grid by grouping fine scale layers in such a way that the heterogeneity measure of a defined static property is minimized within the layers. The optimal number of layers is then selected based on a statistical analysis. The power and utility of our approaches have been demonstrated using both synthetic and field examples.

Download or read book Reduced Order Model and Uncertainty Quantification for Stochastic Porous Media Flows written by Jia Wei and published by . This book was released on 2012 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: In this dissertation, we focus on the uncertainty quantification problems where the goal is to sample the porous media properties given integrated responses. We first introduce a reduced order model using the level set method to characterize the channelized features of permeability fields. The sampling process is completed under Bayesian framework. We hence study the regularity of posterior distributions with respect to the prior measures. The stochastic flow equations that contain both spatial and random components must be resolved in order to sample the porous media properties. Some type of upscaling or multiscale technique is needed when solving the flow and transport through heterogeneous porous media. We propose ensemble-level multiscale finite element method and ensemble-level preconditioner technique for solving the stochastic flow equations, when the permeability fields have certain topology features. These methods can be used to accelerate the forward computations in the sampling processes. Additionally, we develop analysis-of-variance-based mixed multiscale finite element method as well as a novel adaptive version. These methods are used to study the forward uncertainty propagation of input random fields. The computational cost is saved since the high dimensional problem is decomposed into lower dimensional problems. We also work on developing efficient advanced Markov Chain Monte Carlo methods. Algorithms are proposed based on the multi-stage Markov Chain Monte Carlo and Stochastic Approximation Monte Carlo methods. The new methods have the ability to search the whole sample space for optimizations. Analysis and detailed numerical results are presented for applications of all the above methods.

Download or read book Analyticity and Sparsity in Uncertainty Quantification for PDEs with Gaussian Random Field Inputs written by Dinh Dũng and published by Springer Nature. This book was released on 2023-11-16 with total page 216 pages. Available in PDF, EPUB and Kindle. Book excerpt: The present book develops the mathematical and numerical analysis of linear, elliptic and parabolic partial differential equations (PDEs) with coefficients whose logarithms are modelled as Gaussian random fields (GRFs), in polygonal and polyhedral physical domains. Both, forward and Bayesian inverse PDE problems subject to GRF priors are considered. Adopting a pathwise, affine-parametric representation of the GRFs, turns the random PDEs into equivalent, countably-parametric, deterministic PDEs, with nonuniform ellipticity constants. A detailed sparsity analysis of Wiener-Hermite polynomial chaos expansions of the corresponding parametric PDE solution families by analytic continuation into the complex domain is developed, in corner- and edge-weighted function spaces on the physical domain. The presented Algorithms and results are relevant for the mathematical analysis of many approximation methods for PDEs with GRF inputs, such as model order reduction, neural network and tensor-formatted surrogates of parametric solution families. They are expected to impact computational uncertainty quantification subject to GRF models of uncertainty in PDEs, and are of interest for researchers and graduate students in both, applied and computational mathematics, as well as in computational science and engineering.