Author: Wenyue Sun
Publisher:
ISBN:
Category :
Languages : en
Pages :
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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.
Author: Jiachuan He
Publisher:
ISBN:
Category :
Languages : en
Pages : 190
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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
Author: Markus Wahlsten
Publisher: Linköping University Electronic Press
ISBN: 917685339X
Category :
Languages : en
Pages : 45
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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.
Author: Philippe Renard
Publisher: Frontiers Media SA
ISBN: 2889636747
Category :
Languages : en
Pages : 177
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Book Description
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.
Author: Jihoon Park
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Book Description
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.
Author: Yorum Rubin
Publisher: Springer Science & Business Media
ISBN: 1402031025
Category : Science
Languages : en
Pages : 518
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Book Description
This ground-breaking work is the first to cover the fundamentals of hydrogeophysics from both the hydrogeological and geophysical perspectives. Authored by leading experts and expert groups, the book starts out by explaining the fundamentals of hydrological characterization, with focus on hydrological data acquisition and measurement analysis as well as geostatistical approaches. The fundamentals of geophysical characterization are then at length, including the geophysical techniques that are often used for hydrogeological characterization. Unlike other books, the geophysical methods and petrophysical discussions presented here emphasize the theory, assumptions, approaches, and interpretations that are particularly important for hydrogeological applications. A series of hydrogeophysical case studies illustrate hydrogeophysical approaches for mapping hydrological units, estimation of hydrogeological parameters, and monitoring of hydrogeological processes. Finally, the book concludes with hydrogeophysical frontiers, i.e. on emerging technologies and stochastic hydrogeophysical inversion approaches.
Author: Marta D'Elia
Publisher: Springer
ISBN: 9783030487232
Category : Mathematics
Languages : en
Pages : 282
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Book Description
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.
Author: Hans-Jörg G. Diersch
Publisher: Springer Science & Business Media
ISBN: 364238739X
Category : Science
Languages : en
Pages : 1018
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Book Description
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).
Author: Xiang Ma
Publisher:
ISBN:
Category :
Languages : en
Pages : 224
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Book Description
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.
Author: Hyung Jun Yang
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Book Description
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.
