Sensitivity Analysis and Quantification of the Role of Governing Transport Mechanisms and Parameters in a Gas Flow Model for Low-Permeability Porous Media

Recent models represent gas (methane) migration in low-permeability media as a weighted sum of various contributions, each associated with a given flow regime. These models typically embed numerous chemical/physical parameters that cannot be easily and unambiguously evaluated via experimental investigations. In this context, modern sensitivity analysis techniques enable us to diagnose the behavior of a given model through the quantification of the importance and role of model input uncertainties with respect to a target model output. Here, we rely on two global sensitivity analysis approaches and metrics (i.e., variance-based Sobol’ indices and moment-based AMA indices) to assess the behavior of a recent interpretive model that conceptualizes gas migration as the sum of a surface diffusion mechanism and two weighted bulk flow components. We quantitatively investigate the impact of (i) each uncertain model parameter and (ii) the type of their associated probability distribution on the evaluation of methane flow. We then derive the structure of an effective diffusion coefficient embedding all complex mechanisms of the model considered and allowing quantification of the relative contribution of each flow mechanism to the overall gas flow.

     

Uncertainty quantification for dynamics of geometrically nonlinear structures coupled with internal acoustic fluids in presence of sloshing and capillarity

In this paper, we propose an uncertainty quantification analysis, which is the continuation of a recent work performed in a deterministic framework. The fluid–structure system under consideration is the one experimentally studied in the sixties by Abramson, Kana, and Lindholm from the Southwest Research Institute under NASA contract. This coupled system is constituted of a linear acoustic liquid contained in an elastic tank that undergoes finite dynamical displacements, inducing geometrical nonlinear effects in the structure. The liquid has a free surface on which sloshing and capillarity effects are taken into account. The problem is expressed in terms of the acoustic pressure field in the fluid, of the displacement field of the elastic structure, and of the normal elevation field of the free surface. The nonlinear reduced-order model constructed in the recent work evoked above is reused for implementing the nonparametric probabilistic approach of uncertainties. The objective of this paper is to present a sensitivity analysis of this coupled fluid–structure system with respect to uncertainties and to use a classical statistical inverse problem for carrying out the experimental identification of the hyperparameter of the stochastic model. The analysis show a significant sensitivity of the displacement of the structure, of the acoustic pressure in the liquid, and of the free-surface elevation to uncertainties in both linear and geometrically nonlinear simulations.     

New hybrid Cartesian cut cell/enriched multipoint flux approximation approach for modelling and quantification of structural uncertainties in petroleum reservoirs

Efficient and profitable oil production is subject to make reliable predictions about reservoir performance. However, restricted knowledge about reservoir rock and fluid properties and its geometrical structure calls for history matching in which the reservoir model is calibrated to emulate the field observed history. Such an inverse problem yields multiple history‐matched models, which might result in different predictions of reservoir performance. Uncertainty quantification narrows down the model uncertainties and boosts the model reliability for the forecasts of future reservoir behaviour. Conventional approaches of uncertainty quantification ignore large‐scale uncertainties related to reservoir structure, while structural uncertainties can influence the reservoir forecasts more significantly compared with petrophysical uncertainty. Quantification of structural uncertainty has been usually considered impracticable because of the need for global regridding at each step of history matching process. To resolve this obstacle, we develop an efficient methodology based on Cartesian cut cell method that decouples the model from its representation onto the grid and allows uncertain structures to be varied as a part of history matching process. Reduced numerical accuracy due to cell degeneracies in the vicinity of geological structures is adequately compensated with an enhanced scheme of a class of locally conservative flux continuous methods (extended enriched multipoint flux approximation method or extended EMPFA). The robustness and consistency of the proposed hybrid Cartesian cut cell/extended EMPFA approach are demonstrated in terms of true representation of geological structures influence on flow behaviour. Significant improvements in the quality of reservoir recovery forecasts and reservoir volume estimation are presented for synthetic model of uncertain structures. Copyright © 2015 John Wiley & Sons, Ltd.     

Dynamic stability of a pipe conveying fluid with an uncertain computational model

This paper deals with the problem of a pipe conveying fluid of interest in several engineering applications, such as micro-systems or drill-string dynamics. The deterministic stability analysis developed by Paidoussis and Issid (1974) is extended to the case for which there are model uncertainties induced by modeling errors in the computational model. The aim of this work is twofold: (1) to propose a probabilistic model for the fluid–structure interaction considering modeling errors and (2) to analyze the stability and reliability of the stochastic system. The Euler–Bernoulli beam model is used to model the pipe and the plug flow model is used to take into account the internal flow in the pipe. The resulting differential equation is discretized by means of the finite element method and a reduced-order model is constructed from some eigenmodes of the beam. A probabilistic approach is used to model uncertainties in the fluid–structure interaction. The proposed strategy takes into account global uncertainties related to the noninertial coupled fluid forces (related to damping and stiffness). The resulting random eigenvalue problem is used to analyze flutter and divergence unstable modes of the system for different values of the dimensionless flow speed. The numerical results show the random response of the system for different levels of uncertainty, and the reliability of the system for different dimensionless speeds and levels of uncertainty.     

Analysis of a benchmark solution for non-Newtonian radial displacement in porous media

We present an analytical formulation useful to interpret the key phenomena involved in non-Newtonian displacement in porous media and an analysis of the results obtained by considering the uncertainty associated with relevant problem parameters. To derive a benchmark solution, we consider the radial dynamics of a moving stable interface in a porous domain saturated by two fluids, displacing and displaced, both non-Newtonian of shear-thinning power-law behavior, assuming the pressure and velocity to be continuous at the interface, and constant initial pressure. The flow law for both fluids is a modified Darcy’s law. Coupling the nonlinear flow law with the continuity equation, and taking into account compressibility effects, yields a set of nonlinear second-order partial differential equations. Considering two fluids with the same flow behavior index n allows transformation of the latter equations via a self-similar variable; further transformation of the equations incorporating the conditions at the interface shows for n<1 the existence of a compression front ahead of the moving interface. Solving the resulting set of nonlinear equations yields the positions of the moving interface and compression front, and the pressure distributions; the latter are derived in closed form for any value of n. A sensitivity analysis of the model responses is conducted both in a deterministic and a stochastic framework. In the latter case, Global Sensitivity Analysis (GSA) of the benchmark analytical model is adopted to study how the effects of uncertainty affecting selected parameters: (a) the fluids flow behavior index, (b) the relative total compressibility and mobility in the displaced and displacing fluid domains, and (c) the domain permeability and porosity, propagate to state variables. The relative influence of input parameters on model outputs is evaluated by means of associated Sobol indices, calculated via the Polynomial Chaos Expansion (PCE) technique. The goodness of the results obtained by the PCE is assessed by comparison against a traditional Monte Carlo (MC) approach.     

Surrogate model‐based strategy for cryogenic cavitation model validation and sensitivity evaluation

The study of cavitation dynamics in cryogenic environment has critical implications for the performance and safety of liquid rocket engines, but there is no established method to estimate cavitation‐induced loads. To help develop such a computational capability, we employ a multiple‐surrogate model‐based approach to aid in the model validation and calibration process of a transport‐based, homogeneous cryogenic cavitation model. We assess the role of empirical parameters in the cavitation model and uncertainties in material properties via global sensitivity analysis coupled with multiple surrogates including polynomial response surface, radial basis neural network, kriging, and a predicted residual sum of squares‐based weighted average surrogate model. The global sensitivity analysis results indicate that the performance of cavitation model is more sensitive to the changes in model parameters than to uncertainties in material properties. Although the impact of uncertainty in temperature‐dependent vapor pressure on the predictions seems significant, uncertainty in latent heat influences only temperature field. The influence of wall heat transfer on pressure load is insignificant. We find that slower onset of vapor condensation leads to deviation of the predictions from the experiments. The recalibrated model parameters rectify the importance of evaporation source terms, resulting in significant improvements in pressure predictions. The model parameters need to be adjusted for different fluids, but for a given fluid, they help capture the essential fluid physics with different geometry and operating conditions. Copyright © 2008 John Wiley & Sons, Ltd.     

翼型跨声速气动特性的不确定性及全局灵敏度分析

针对马赫数和仰角的随机不确定性会导致气动性能波动的现象, 采用非嵌入式的混沌多项式方法对绕NACA0012 翼型跨声速随机气动特性进行不确定性及全局灵敏度分析. 具体分析了飞行状态的不确定性对气动载荷分布、流场及气动力系数的影响并通过全局灵敏度分析找出重要因素. 不确定性分析结果表明翼型上表面的激波以及激波后分离泡是造成气动性能剧烈波动的主要原因. 灵敏度分析结果表明在跨声速区域马赫数对激波处气动性能影响最大, 此外, 虽然马赫数和仰角相互耦合作用对气动力系数贡献比较小, 但对于激波位置处的流场, 这种互耦合作用不可忽略.      

Sensitivity analysis and Kriging based models for robust stability analysis of brake systems

This paper presents a global strategy for the prediction of brake squeal. This approach is based on the global sensitivity analysis combined with Kriging modeling. The main aim is to assess the pertinence of using this strategy to build suitable and efficient instability predictors that can be potentially associated with numerical optimization schemes and/or robustness analysis algorithms for a robust design of brake systems. Through the use of a simplified brake system, the global sensitivity analysis is, firstly, shown to be essential for obtaining great insight on how design parameters influence individually and/or collectively the stability behavior. The latter is characterized by the distance of all the systems eigenvalues from the imaginary axis. It is shown that the so-called Sobol indices help for an objective quantification of the importance of taking parameter uncertainty into account in the whole design process. Based on these conclusions, Kriging modeling is, secondly, proposed to robustly predict friction induced instabilities. Its efficiency is then demonstrated. Consequently, the global sensitivity analysis and Kriging modeling give a very promising strategy helping for squeal prediction and, more generally, for optimal and robust design of brake systems.     

A Bayesian Calibration–Prediction Method for Reducing Model-Form Uncertainties with Application in RANS Simulations

Model-form uncertainties in complex mechanics systems are a major obstacle for predictive simulations. Reducing these uncertainties is critical for stake-holders to make risk-informed decisions based on numerical simulations. For example, Reynolds-Averaged Navier-Stokes (RANS) simulations are increasingly used in the design, analysis, and safety assessment of mission-critical systems involving turbulent flows. However, for many practical flows the RANS predictions have large model-form uncertainties originating from the uncertainty in the modeled Reynolds stresses. Recently, a physics-informed Bayesian framework has been proposed to quantify and reduce model-form uncertainties in RANS simulations for flows by utilizing sparse observation data. However, in the design stage of engineering systems, when the system or device has not been built yet, measurement data are usually not available. In the present work we extend the original framework to scenarios where there are no available data on the flow to be predicted. In the proposed method, we first calibrate the model discrepancy on a related flow with available data, leading to a statistical model for the uncertainty distribution of the Reynolds stress discrepancy. The obtained distribution is then sampled to correct the RANS-modeled Reynolds stresses for the flow to be predicted. The extended framework is a Bayesian calibration–prediction method for reducing model-form uncertainties. The merits of the proposed method are demonstrated on two flows that are challenging to standard RANS models. By not requiring observation data on the flow to be predicted, the present calibration–prediction method will gain wider acceptance in practical engineering design and analysis compared to the original framework. While RANS modeling is chosen to demonstrate the merits of the proposed framework, the methodology is generally applicable to other complex mechanics models involving solids, fluids flows, or the coupling between the two (e.g., mechanics models for the cardiovascular systems), where model-form uncertainties are present in the constitutive relations.     

Modeling of Solute Transport in a 3D Rough-Walled Fracture–Matrix System

Fluid flow and solute transport in a 3D rough-walled fracture–matrix system were simulated by directly solving the Navier–Stokes equations for fracture flow and solving the transport equation for the whole domain of fracture and matrix with considering matrix diffusion. The rough-walled fracture–matrix model was built from laser-scanned surface tomography of a real rock sample, by considering realistic features of surfaces roughness and asperity contacts. The numerical modeling results were compared with both analytical solutions based on simplified fracture surface geometry and numerical results by particle tracking based on the Reynolds equation. The aim is to investigate impacts of surface roughness on solute transport in natural fracture–matrix systems and to quantify the uncertainties in application of simplified models. The results show that fracture surface roughness significantly increases heterogeneity of velocity field in the rough-walled fractures, which consequently cause complex transport behavior, especially the dispersive distributions of solute concentration in the fracture and complex concentration profiles in the matrix. Such complex transport behaviors caused by surface roughness are important sources of uncertainty that needs to be considered for modeling of solute transport processes in fractured rocks. The presented direct numerical simulations of fluid flow and solute transport serve as efficient numerical experiments that provide reliable results for the analysis of effective transmissivity as well as effective dispersion coefficient in rough-walled fracture–matrix systems. Such analysis is helpful in model verifications, uncertainty quantifications and design of laboratorial experiments.     

One-dimensional model of two-phase fluid displacement in a slot with permeable walls

A one-dimensional model is proposed for transportation of a two-phase fluid (sandcontaining fluid and pure fluid) in the Hele-Shaw cell with permeable walls through which the pure fluid can leak off, causing the growth of the sand concentration. The model describes the process of pure fluid displacement with the emergence of the Saffman–Taylor instability and extends Koval’s model to the case of sand concentration variation owing to pure fluid outflow through the cell walls. The Riemann problem is analyzed. New flow configurations, which are not predicted by Koval’s model, are discovered.     

混合不确定度量化方法及其在计算流体动力学迎风格式中的应用

针对流体力学数值求解间断问题时,初始状态含有偶然和认知混合型的不确定性,将认知不确定度作为外层,偶然不确定度作为内层,分别使用非嵌入多项式混沌方法(non-intrusive polynomial chaos, NIPC)和概率盒(P-box)理论处理偶然不确定度和认知不确定度,发展了流体力学数值求解过程中,初始状态含有混合不确定度传播量化的一种方法。以迎风格式和黎曼解法器求解Sod问题为例,评估了左状态密度(偶然不确定度)和理想气体多方指数(认知不确定度)对模型输出结果的影响,有效验证了该方法的可行性。     

Horizontal Redistribution of Two Fluid Phases in a Porous Medium: Experimental Investigations

Classical models for flow and transport processes in porous media employ the so-called extended Darcy’s Law. Originally, it was proposed empirically for one-dimensional isothermal flow of an incompressible fluid in a rigid, homogeneous, and isotropic porous medium. Nowadays, the extended Darcy’s Law is used for highly complex situations like non-isothermal, multi-phase and multi-component flow and transport, without introducing any additional driving forces. In this work, an alternative approach by Hassanizadeh and Gray identifying additional driving forces were tested in an experimental setup for horizontal redistribution of two fluid phases with an initial saturation discontinuity. Analytical and numerical solutions based on traditional models predict that the saturation discontinuity will persist, but a uniform saturation distribution will be established in each subdomain after an infinite amount of time. The pressure field, however, is predicted to be continuous throughout the domain at all times and is expected to become uniform when there is no flow. In our experiments, we also find that the saturation discontinuity persists. But, gradients in both saturation and pressure remain in both subdomains even when the flow of fluids stops. This indicates that the identified additional driving forces present in the truly extended Darcy’s Law are potentially significant.     

Dynamic response of a single-mesh gear system with periodic mesh stiffness and backlash nonlinearity under uncertainty

Parametric uncertainties play a critical role in the response predictions of a gear system. However, accurately determining the effects of the uncertainty propagation in nonlinear time-varying models of gear systems is awkward and difficult. This paper improves the interval harmonic balance method (IHBM) to solve the dynamic problems of gear systems with backlash nonlinearity and time-varying mesh stiffness under uncertainties. To deal with the nonlinear problem including the fold points and uncertainties, the IHBM is improved by introducing the pseudo-arc length method in combination with the Chebyshev inclusion function. The proposed approach is demonstrated using a single-mesh gear system model, including the parametrically varying mesh stiffness and the gear backlash nonlinearity, excited by the transmission error. The results of the improved IHBM are compared with those obtained from the scanning method. Effects of parameter uncertainties on its dynamic behavior are also discussed in detail. From various numerical examples, it is shown that the results are consistent meanwhile the computational cost is significantly reduced. Furthermore, the proposed approach could be effectively applied for sensitivity analysis of the system response to parameter variations.     

On the uncertainty quantification of the unsteady aerodynamics of 2D free falling plates

The aim of this paper is to conduct a statistical analysis of the effects of the fillet radii on the dynamics of the falling plate using the nonintrusive spectral projection (NISP) method. The free fall of two-dimensional cards immersed in a fluid was studied using a deterministic and stochastic numerical approach. The motion is characterized by the fluid-body interaction described by coupling the Navier–Stokes and rigid body dynamic equations. The model’s predictions have been validated using both experimental and numerical data available in the literature. In the stochastic simulations, the fillet radius of the plate was considered a random variable characterized by a uniform probability density function introducing, in this way, some uncertainties in the plate’s trajectory. To take into account the uncertainties, we employed the NISP method based on polynomial chaos expansion. The analysis was focused on finding the ensemble mean trajectory and error bar for a confidence interval of 95 % for both tumbling and fluttering regimes.     

Gas/shear-thinning liquid flows through pipes: Modeling and experiments

In chemical and oil industry gas/shear-thinning liquid two-phase flows are frequently encountered. In this work, we investigate experimentally the flow characteristics of air/shear-thinning liquid systems in horizontal and slightly inclined smooth pipes. The experiments are performed in a 9-m-long glass pipe using air and three different carboxymethyl cellulose (CMC) solutions as test fluids. Flow pattern maps are built by visual observation using a high-speed camera. The observed flow patterns are stratified, plug, and slug flow. The effects of the pipe inclination and the rheology of the shear-thinning fluid in terms of flow pattern maps are presented. The predicted existence region of the stratified flow regime is compared with the experimental observation showing a good agreement. A mechanistic model valid for air/power-law slug flow is proposed and model predictions are compared to the experimental data showing a good agreement. Slug flow characteristics are investigated by the analysis of the signals of a capacitance probe: slug velocity, slug frequency, and slug lengths are measured. A new correlation for the slug frequency is proposed and the results are promising.     

Global sensitivity analysis based on high-dimensional sparse surrogate construction

Surrogate models are usually used to perform global sensitivity analysis(GSA) by avoiding a large ensemble of deterministic simulations of the Monte Carlo method to provide a reliable estimate of GSA indices. However, most surrogate models such as polynomial chaos(PC) expansions suffer from the curse of dimensionality due to the high-dimensional input space. Thus, sparse surrogate models have been proposed to alleviate the curse of dimensionality. In this paper, three techniques of sparse reconstruction are used to construct sparse PC expansions that are easily applicable to computing variance-based sensitivity indices(Sobol indices). These are orthogonal matching pursuit(OMP), spectral projected gradient for L_1 minimization(SPGL1), and Bayesian compressive sensing with Laplace priors. By computing Sobol indices for several benchmark response models including the Sobol function, the Morris function, and the Sod shock tube problem, effective implementations of high-dimensional sparse surrogate construction are exhibited for GSA.     

A robust ‘time-marching’ solver for one-dimensional nucleating steam flows

A mixed Lagrangian/Eulerian ‘time-marching’ solver capable of predicting one-dimensional nucleating steam flows is described. Simple nucleation and droplet growth models are employed which avoid the use of variable empirical factors and which have been validated using existing experimental data from nozzle experiments performed in the steam tunnel of the Central Electricity Research Laboratories. Theoretical predictions are compared against experimental results encompassing all flow regimes likely to be encountered in a one-dimensional analysis of flow in a low pressure steam turbine. These include supercritical heat addition cases which display both steady and unsteady shock wave formation.     

A sensitivity equation method for fast evaluation of nearby flows and uncertainty analysis for shape parameters

This paper presents the application of the continuous sensitivity equation method (CSEM) to fast evaluation of nearby flows and to uncertainty analysis for shape parameters. The flow and sensitivity fields are solved using an adaptive finite-element method. A new approach is presented to extract accurate flow derivatives at the boundary, which are needed in the shape sensitivity boundary conditions. Boundary derivatives are evaluated via high order Taylor series expansions used in a constrained least-squares procedure. The proposed method is first applied to fast evaluation of nearby flows: the baseline flow and sensitivity fields around a NACA 0012 airfoil are used to predict the flow around airfoils with nearby shapes obtained by modifications of the thickness (NACA 0015), the angle of attack and the camber (NACA 4512). The method is then applied to evaluate the influence of geometrical uncertainties on the flow around a NACA 0012 airfoil.     

Modeling study of three-phase low liquid loading flow in horizontal pipes

A theoretical study is conducted to model the flow characteristics of three-phase stratified wavy flow in horizontal pipelines with a focus on the low liquid loading condition, which is commonly observed in wet gas pipelines. The model predictions are compared to the experimental data of Karami et al. (2016a, b). These experiments were conducted with water or 51 wt% of MEG in the aqueous phase, and inlet aqueous phase fraction values from 0 to100%.Modeling of three-phase flow can be described as a combination of two-phase gas-liquid flow modeling, and a liquid phase oil-water mixing modeling. A mechanistic model is proposed to predict flow characteristics of three-phase stratified wavy flow in pipeline. For the gas-liquid interactions, Watson's (1989) combined momentum balance equation derivation was applied. However, the calculation procedure was reversed, and the wave celerity was assumed as an input, while interfacial friction factor was one of the model's outputs. The liquid-liquid interactions were modeled using a simple energy balance equation and shift in liquid phase center of gravity calculations. The liquid phases can be separated, partially mixed, or fully mixed. The bottom aqueous film velocity was calculated using the law of the wall formulation, and was used to calculate the flowing aqueous phase fraction.The model predictions of different flow characteristics for two and/or three-phase flows were compared with available experimental data. The pressure gradient, wave amplitude, and aqueous phase fraction predictions were in good agreement with the experimental data. However, the liquid holdup predictions were slightly under-predicted by the model. Overall, an acceptable agreement was observed for all cases.Most of the common multiphase stratified flow models are developed with the assumption of steady-state conditions and with constant interfacial friction factor value. This study proposes a novel method to model stratified flow. The predictions are in acceptable agreement with experimental data conducted under stratified wavy flow pattern conditions.