Download or read book Stochastic Models for Nonlinear Transport in Multiphase and Multiscale Heterogeneous Media written by Farzaneh Rajabi and published by . This book was released on 2021 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Elucidating multiscale, multiphase and multiphysics phenomena of flow and transport processes in porous media is the cornerstone of numerous environmental and engineering applications. Several factors including spatial and temporal heterogeneity on a continuity of scales, the strong coupling of processes at such different scales at least at a localized region within the domain, combined with the nonlinearity of processes calls for a new modeling paradigm called multiscale models, which are able to properly address all such issues while presenting an accurate descriptive model for processes occurring at field scale applications. Furthermore, the typical temporal resolution used in modern simulations significantly exceeds characteristic time scales at which the system is driven and a solution is sought. This is especially so when systems are simulated over time scales that are much longer than the typical temporal scales of forcing factors. In addition to spatial and temporal heterogeneity, mixing and spreading of contaminants in the subsurface is remarkably influenced by oscillatory forcing factors. While the pore-scale models are able to handle the experimentally-observed phenomena, they are not always the best choice due to the high computational burden. Although handling across-scale coupling in environments with several simultaneous physical mechanisms such as advection, diffusion, reaction, and fluctuating boundary forcing factors complicates the theoretical and numerical modeling capabilities at high resolutions, multiscale models come to rescue. To this end, we investigate the impact of space-time upscaling on reactive transport in porous media driven by time-dependent boundary conditions whose characteristic time scale is much smaller than that at which transport is studied or observed at the macroscopic level. We first introduce the concept of spatiotemporal upscaling in the context of homogenization by multiple-scale expansions, and demonstrate the impact of time-dependent forcings and boundary conditions on macroscopic reactive transport. Proposing such a framework, we scrutinize the behavior of porous media for ``quasisteady stage time'' (thousands of years), where there is an interplay between signal frequency and the three physical underlying mechanisms; advection, molecular diffusion and heterogeneous reaction. To this end, we demonstrate that the transient forcing factors augment the solute mixing as they are combined with diffusion at the pore-scale. We then derive the macroscopic equation as well as the corresponding applicability criteria based on the order of magnitude of the dimensionless Peclet and Damkohler numbers. Also, we demonstrate that the dynamics at the continuum scale is strongly influenced by the interplay between signal frequency at the boundary and transport processes at the pore level. We validate such a framework for reactive transport in a planar fracture in which the single-component solute particle is undergoing nonlinear first-order heterogeneous reaction at the solid-liquid interface, while the medium is episodically influenced by time-dependent boundary conditions at the inlet. We also present the alternative effective transport model at a much lower cost, albeit at the regions where the corresponding applicability criteria are satisfied. We perform direct numerical simulations to study several test cases with different controlling parameters i.e. Peclet and Damkohler numbers and the space/time scale separation parameters; the ratio of characteristic transversal and longitudinal lengths $\varepsilon$ and $\omega$; the ratio of period of time-fluctuating boundary conditions to the observation time scale. A rigorous justification of the effective transport model for the given applicability conditions is demonstrated, essentially by comparing the local vertically averaged microscopic simulations with their corresponding macroscopic counterparts. Moreover, as a special case, we employ a singular perturbation technique to look at the effective model for vertical mixing through a narrow and long two-dimensional pore. We obtain explicit expressions for dispersion tensor as well as the other effective coefficients in the coarse-scale homogenized equation. Our analysis manifests robustness of the sufficient and necessary applicability constraints which validate the upscaled model as a solid replacement of the pore-scale one within the accuracy prescribed by homogenization theory. While a deterministic model is sufficiently robust for a plethora of subsurface applications, a more realistic setting is often required when dealing with other scopes of engineering applications, e.g. reservoir engineering and enhanced oil recovery. Rigorous modeling of these systems calls for sophisticated strategies for uncertainty quantification and stochastic treatment of the system under study. Such an uncertainty is inherent to, and critical for any physical modeling, essentially due to the incomplete knowledge of state of the world, noisy observations, and limitations in systematically recasting physical processes in a suitable mathematical framework. To this end, accurate predictions of outputs (e.g. saturation fields) from reservoir simulations guarantee precise oil recovery forecasts. These quantitative predictions rely on the quality of the input measurements/data, such as the reservoir permeability and porosity fields as well as forcings, such as initial and boundary conditions. However, the available information about a particular geologic formation, e.g. from well logs and seismic data of an outcrop, is usually sparse and inaccurate compared to the size of the natural system and the complexity arising from multiscale heterogeneity of the underlying system. Eventually, the uncertainty in the flow prediction can have a huge impact on the oil recovery. Consequently, we also develop a probabilistic approach to map the parametric uncertainty to the output state uncertainty in first-order hyperbolic conservation laws. We analyze this problem for nonlinear immiscible two-phase transport (Buckley-Leverett displacement) in heterogeneous porous media in the presence of a stochastic velocity field, where the uncertainty in the velocity field can arise from the incomplete description of either porosity field, injection flux, or both. Such uncertainty leads to the spatiotemporal uncertainty in the outputs of the problem. Given information about the spatial/temporal statistics of the correlated heterogeneity, we leverage method of distributions (MD) to derive deterministic equations that govern the evolution of single-point CDF of saturation in the form of linear hyperbolic conservation laws. We first derive the semi-analytical solution of the raw CDF of saturation at a given point, for the cases in which two shocks are present due to the gravitational forces. Then, we describe development of the partial differential equation that governs the evolution of the raw CDF of saturation, subject to uniquely specified boundary conditions in the phase space, wherein no closure approximations are required. Hereby, we give routes to circumventing the computational cost of Monte Carlo scheme while obtaining the full statistical description of saturation. This derivation is followed by conducting a set of numerical experiments for horizontal reservoirs and more complex scenarios in which gravity segregation takes place. We then compare the CDFs as well as the first two moments of saturation computed with the method of distributions, against those obtained using the statistical moment equations (SME) approach and kernel density estimation post-processing of exhaustive high-resolution Monte Carlo simulations (MCS). This comparison demonstrates that the CDF equations remain accurate over a wide range of statistical properties, i.e. standard deviation and correlation length of the underlying random fields, while the corresponding low-order statistical moment equations significantly deviate from Monte Carlo results, unless for very small values of standard deviation and correlation length.

Download or read book Dispersive Transport Equations and Multiscale Models written by Ben Abdallah Naoufel and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 297 pages. Available in PDF, EPUB and Kindle. Book excerpt: IMA Volumes 135: Transport in Transition Regimes and 136: Dispersive Transport Equations and Multiscale Models focus on the modeling of processes for which transport is one of the most complicated components. This includes processes that involve a wdie range of length scales over different spatio-temporal regions of the problem, ranging from the order of mean-free paths to many times this scale. Consequently, effective modeling techniques require different transport models in each region. The first issue is that of finding efficient simulations techniques, since a fully resolved kinetic simulation is often impractical. One therefore develops homogenization, stochastic, or moment based subgrid models. Another issue is to quantify the discrepancy between macroscopic models and the underlying kinetic description, especially when dispersive effects become macroscopic, for example due to quantum effects in semiconductors and superfluids. These two volumes address these questions in relation to a wide variety of application areas, such as semiconductors, plasmas, fluids, chemically reactive gases, etc.

Download or read book Multiscale Finite Element Methods written by Yalchin Efendiev and published by Springer Science & Business Media. This book was released on 2009-01-10 with total page 242 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this monograph is to describe the main concepts and recent - vances in multiscale ?nite element methods. This monograph is intended for thebroaderaudienceincludingengineers,appliedscientists,andforthosewho are interested in multiscale simulations. The book is intended for graduate students in applied mathematics and those interested in multiscale compu- tions. It combines a practical introduction, numerical results, and analysis of multiscale ?nite element methods. Due to the page limitation, the material has been condensed. Each chapter of the book starts with an introduction and description of the proposed methods and motivating examples. Some new techniques are introduced using formal arguments that are justi?ed later in the last chapter. Numerical examples demonstrating the signi?cance of the proposed methods are presented in each chapter following the description of the methods. In the last chapter, we analyze a few representative cases with the objective of demonstrating the main error sources and the convergence of the proposed methods. A brief outline of the book is as follows. The ?rst chapter gives a general introductiontomultiscalemethodsandanoutlineofeachchapter.Thesecond chapter discusses the main idea of the multiscale ?nite element method and its extensions. This chapter also gives an overview of multiscale ?nite element methods and other related methods. The third chapter discusses the ext- sion of multiscale ?nite element methods to nonlinear problems. The fourth chapter focuses on multiscale methods that use limited global information.

Download or read book Stochastic Models of Solute Transport in Highly Heterogeneous Geologic Media written by and published by . This book was released on 2009 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: A stochastic model of anomalous diffusion was developed in which transport occurs by random motion of Brownian particles, described by distribution functions of random displacements with heavy (power-law) tails. One variant of an effective algorithm for random function generation with a power-law asymptotic and arbitrary factor of asymmetry is proposed that is based on the Gnedenko-Levy limit theorem and makes it possible to reproduce all known Levy [alpha]-stable fractal processes. A two-dimensional stochastic random walk algorithm has been developed that approximates anomalous diffusion with streamline-dependent and space-dependent parameters. The motivation for introducing such a type of dispersion model is the observed fact that tracers in natural aquifers spread at different super-Fickian rates in different directions. For this and other important cases, stochastic random walk models are the only known way to solve the so-called multiscaling fractional order diffusion equation with space-dependent parameters. Some comparisons of model results and field experiments are presented.

Download or read book Quantifying Non Fickian Transport in Porous and Fractured Media Using Fractional calculus Based Stochastic Models written by Bingqing Lu and published by . This book was released on 2019 with total page 218 pages. Available in PDF, EPUB and Kindle. Book excerpt: Non-Fickian or "anomalous" transport, where the target's spatial variance grows nonlinearly in time, describes the pollutant dynamics widely observed in heterogeneous geological media deviating significantly from that described by the classical advection dispersion equation (ADE). The ADE describes the Fickian-type of transport, with symmetric snapshots like the Gaussian distribution in space (Berkowitz et al., 2006). Non-Fickian transport can be observed at all scales. Non-Fickian transport is typically characterized by apparent (as heavy as power-law) early arrivals and late time tailing behaviors in the tracer breakthrough curves (BTCs). Non-Fickian transport is well known to be affected by medium heterogeneity. Heterogeneity can refer to variations in the distribution of geometrical properties, as well as variations in the biogeochemical properties of the medium, which cannot be mapped exhaustively at all relevant scales. Complex geometric structures and intrinsic heterogeneity in geological formations affect predictions of tracer transport and further challenge remediation analyses. Hence, efficient quantification of non-Fickian transport requires parsimonious models such as the fractional engine based physical models. In this dissertation, I first compared three types of time non-local transport models, which include the multi-rate mass transfer (MRMT) model, the continuous time random walk (CTRW) framework, and the tempered time fractional advection dispersion equation (tt-fADE) model. I then found that tt-fADE can model the rate-limited diffusion and sorption-desorption of Arsenic in soil. Additionally, non-Fickian dynamics for pollutant transport in field-scale discrete fracture networks (DFNs) were explored. Monte Carlo simulations of water flow were then conducted through field-scale DFNs to identify non-Darcian flow and non-Fickian pressure propagation. Finally, to address non-Fickian transport for reactive pollutants, I proposed a time fractional derivative model with the reaction term. Findings of this dissertation improve our understanding of the nature of water flow and pollutant transport in porous and fractured media at different scales. The correlated parameters and relationships between media properties and parameters can enhance the applicability of fractional partial differential equations that can be parameterized using the measurable media characteristics. This provides one of the most likely ways to improve the model predictability, which remained the most challenge for stochastic hydrologic models

Download or read book Fractals written by Behzad Ghanbarian and published by CRC Press. This book was released on 2017-11-23 with total page 289 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides theoretical concepts and applications of fractals and multifractals to a broad range of audiences from various scientific communities, such as petroleum, chemical, civil and environmental engineering, atmospheric research, and hydrology. In the first chapter, we introduce fractals and multifractals from physics and math viewpoints. We then discuss theory and practical applications in detail. In what follows, in chapter 2, fragmentation process is modeled using fractals. Fragmentation is the breaking of aggregates into smaller pieces or fragments, a typical phenomenon in nature. In chapter 3, the advantages and disadvantages of two- and three-phase fractal models are discussed in detail. These two kinds of approach have been widely applied in the literature to model different characteristics of natural phenomena. In chapter 4, two- and three-phase fractal techniques are used to develop capillary pressure curve models, which characterize pore-size distribution of porous media. Percolation theory provides a theoretical framework to model flow and transport in disordered networks and systems. Therefore, following chapter 4, in chapter 5 the fractal basis of percolation theory and its applications in surface and subsurface hydrology are discussed. In chapter 6, fracture networks are shown to be modeled using fractal approaches. Chapter 7 provides different applications of fractals and multifractals to petrophysics and relevant area in petroleum engineering. In chapter 8, we introduce the practical advantages of fractals and multifractals in geostatistics at large scales, which have broad applications in stochastic hydrology and hydrogeology. Multifractals have been also widely applied to model atmospheric characteristics, such as precipitation, temperature, and cloud shape. In chapter 9, these kinds of properties are addressed using multifractals. At watershed scales, river networks have been shown to follow fractal behavior. Therefore, the applications of fractals are addressed in chapter 10. Time series analysis has been under investigations for several decades in physics, hydrology, atmospheric research, civil engineering, and water resources. In chapter 11, we therefore, provide fractal, multifractal, multifractal detrended fluctuation analyses, which can be used to study temporal characterization of a phenomenon, such as flow discharge at a specific location of a river. Chapter 12 addresses signals and again time series using a novel fractal Fourier analysis. In chapter 13, we discuss constructal theory, which has a perspective opposite to fractal theories, and is based on optimizationof diffusive exchange. In the case of river drainages, for example, the constructal approach begins at the divide and generates headwater streams first, rather than starting from the fundamental drainage pattern.

Download or read book Mathematical Reviews written by and published by . This book was released on 2005 with total page 1884 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Rock Fractures and Fluid Flow written by National Research Council and published by National Academies Press. This book was released on 1996-08-27 with total page 568 pages. Available in PDF, EPUB and Kindle. Book excerpt: Scientific understanding of fluid flow in rock fracturesâ€"a process underlying contemporary earth science problems from the search for petroleum to the controversy over nuclear waste storageâ€"has grown significantly in the past 20 years. This volume presents a comprehensive report on the state of the field, with an interdisciplinary viewpoint, case studies of fracture sites, illustrations, conclusions, and research recommendations. The book addresses these questions: How can fractures that are significant hydraulic conductors be identified, located, and characterized? How do flow and transport occur in fracture systems? How can changes in fracture systems be predicted and controlled? Among other topics, the committee provides a geomechanical understanding of fracture formation, reviews methods for detecting subsurface fractures, and looks at the use of hydraulic and tracer tests to investigate fluid flow. The volume examines the state of conceptual and mathematical modeling, and it provides a useful framework for understanding the complexity of fracture changes that occur during fluid pumping and other engineering practices. With a practical and multidisciplinary outlook, this volume will be welcomed by geologists, petroleum geologists, geoengineers, geophysicists, hydrologists, researchers, educators and students in these fields, and public officials involved in geological projects.

Download or read book Multiscale Model Reduction written by Eric Chung and published by Springer Nature. This book was released on 2023-06-07 with total page 499 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph is devoted to the study of multiscale model reduction methods from the point of view of multiscale finite element methods. Multiscale numerical methods have become popular tools for modeling processes with multiple scales. These methods allow reducing the degrees of freedom based on local offline computations. Moreover, these methods allow deriving rigorous macroscopic equations for multiscale problems without scale separation and high contrast. Multiscale methods are also used to design efficient solvers. This book offers a combination of analytical and numerical methods designed for solving multiscale problems. The book mostly focuses on methods that are based on multiscale finite element methods. Both applications and theoretical developments in this field are presented. The book is suitable for graduate students and researchers, who are interested in this topic.

Download or read book Physics of Fluid Flow and Transport in Unconventional Reservoir Rocks written by Behzad Ghanbarian and published by John Wiley & Sons. This book was released on 2023-04-13 with total page 388 pages. Available in PDF, EPUB and Kindle. Book excerpt: Physics of Fluid Flow and Transport in Unconventional Reservoir Rocks Understanding and predicting fluid flow in hydrocarbon shale and other non-conventional reservoir rocks Oil and natural gas reservoirs found in shale and other tight and ultra-tight porous rocks have become increasingly important sources of energy in both North America and East Asia. As a result, extensive research in recent decades has focused on the mechanisms of fluid transfer within these reservoirs, which have complex pore networks at multiple scales. Continued research into these important energy sources requires detailed knowledge of the emerging theoretical and computational developments in this field. Following a multidisciplinary approach that combines engineering, geosciences and rock physics, Physics of Fluid Flow and Transport in Unconventional Reservoir Rocks provides both academic and industrial readers with a thorough grounding in this cutting-edge area of rock geology, combining an explanation of the underlying theories and models with practical applications in the field. Readers will also find: An introduction to the digital modeling of rocks Detailed treatment of digital rock physics, including decline curve analysis and non-Darcy flow Solutions for difficult-to-acquire measurements of key petrophysical characteristics such as shale wettability, effective permeability, stress sensitivity, and sweet spots Physics of Fluid Flow and Transport in Unconventional Reservoir Rocks is a fundamental resource for academic and industrial researchers in hydrocarbon exploration, fluid flow, and rock physics, as well as professionals in related fields.

Download or read book Pore Scale Geochemical Processes written by Carl Steefel and published by Walter de Gruyter GmbH & Co KG. This book was released on 2015-09-25 with total page 496 pages. Available in PDF, EPUB and Kindle. Book excerpt: This RiMG (Reviews in Mineralogy & Geochemistry) volume includes contributions that review experimental, characterization, and modeling advances in our understanding of pore-scale geochemical processes. The volume had its origins in a special theme session at the 2015 Goldschmidt Conference in Prague. From a diversity of pore-scale topics that ranged from multi-scale characterization to modeling, this work summarizes the state-of-the-science in this subject. Topics include: modification of thermodynamics and kinetics in small pores. chemo-mechanical processes and how they affect porosity evolution in geological media. small angle neutron scattering (SANS) techniques. how isotopic gradients across fluid–mineral boundaries can develop and how these provide insight into pore-scale processes. Information on an important class of models referred to as "pore network" and much more. The material in this book is accessible for graduate students, researchers, and professionals in the earth, material, environmental, hydrological, and biological sciences. The pore scale is readily recognizable to geochemists, and yet in the past it has not received a great deal of attention as a distinct scale or environment that is associated with its own set of questions and challenges. Is the pore scale merely an environment in which smaller scale (molecular) processes aggregate, or are there emergent phenomena unique to this scale? Is it simply a finer-grained version of the "continuum" scale that is addressed in larger-scale models and interpretations? The scale is important because it accounts for the pore architecture within which such diverse processes as multi-mineral reaction networks, microbial community interaction, and transport play out, giving rise to new geochemical behavior that might not be understood or predicted by considering smaller or larger scales alone.

Download or read book The Engineering Index Annual written by and published by . This book was released on 1992 with total page 2264 pages. Available in PDF, EPUB and Kindle. Book excerpt: Since its creation in 1884, Engineering Index has covered virtually every major engineering innovation from around the world. It serves as the historical record of virtually every major engineering innovation of the 20th century. Recent content is a vital resource for current awareness, new production information, technological forecasting and competitive intelligence. The world?s most comprehensive interdisciplinary engineering database, Engineering Index contains over 10.7 million records. Each year, over 500,000 new abstracts are added from over 5,000 scholarly journals, trade magazines, and conference proceedings. Coverage spans over 175 engineering disciplines from over 80 countries. Updated weekly.

Download or read book Simulation of Flow in Porous Media written by Peter Bastian and published by Walter de Gruyter. This book was released on 2013-07-31 with total page 224 pages. Available in PDF, EPUB and Kindle. Book excerpt: Subsurface flow problems are inherently multiscale in space due to the large variability of material properties and in time due to the coupling of many different physical processes, such as advection, diffusion, reaction and phase exchange. Subsurface flow models still need considerable development. For example, nonequilibrium effects, entrapped air, anomalous dispersion and hysteresis effects can still not be adequately described. Moreover, parameters of the models are diffcult to access and often uncertain. Computational issues in subsurface flows include the treatment of strong heterogeneities and anisotropies in the models, the effcient solution of transport-reaction problems with many species, treatment of multiphase-multicomponent flows and the coupling of subsurface flow models to surface flow models given by shallow water or Stokes equations. With respect to energy and the environment, in particular the modelling and simulation of radioactive waste management and sequestration of CO2 underground have gained high interest in the community in recent years. Both applications provide unique challenges ranging from modelling of clay materials to treating very large scale models with high-performance computing. This book brings together key numerical mathematicians whose interest is in the analysis and computation of multiscale subsurface flow and practitioners from engineering and industry whose interest is in the applications of these core problems.

Download or read book Multiscale Cancer Modeling written by Thomas S. Deisboeck and published by CRC Press. This book was released on 2010-12-08 with total page 492 pages. Available in PDF, EPUB and Kindle. Book excerpt: Cancer is a complex disease process that spans multiple scales in space and time. Driven by cutting-edge mathematical and computational techniques, in silico biology provides powerful tools to investigate the mechanistic relationships of genes, cells, and tissues. It enables the creation of experimentally testable hypotheses, the integration of dat

Download or read book Flowing Matter written by Federico Toschi and published by Springer Nature. This book was released on 2019-09-25 with total page 309 pages. Available in PDF, EPUB and Kindle. Book excerpt: This open access book, published in the Soft and Biological Matter series, presents an introduction to selected research topics in the broad field of flowing matter, including the dynamics of fluids with a complex internal structure -from nematic fluids to soft glasses- as well as active matter and turbulent phenomena. Flowing matter is a subject at the crossroads between physics, mathematics, chemistry, engineering, biology and earth sciences, and relies on a multidisciplinary approach to describe the emergence of the macroscopic behaviours in a system from the coordinated dynamics of its microscopic constituents. Depending on the microscopic interactions, an assembly of molecules or of mesoscopic particles can flow like a simple Newtonian fluid, deform elastically like a solid or behave in a complex manner. When the internal constituents are active, as for biological entities, one generally observes complex large-scale collective motions. Phenomenology is further complicated by the invariable tendency of fluids to display chaos at the large scales or when stirred strongly enough. This volume presents several research topics that address these phenomena encompassing the traditional micro-, meso-, and macro-scales descriptions, and contributes to our understanding of the fundamentals of flowing matter. This book is the legacy of the COST Action MP1305 “Flowing Matter”.

Download or read book Uncertainty Quantification in Multiscale Materials Modeling written by Yan Wang and published by Woodhead Publishing Limited. This book was released on 2020-03-12 with total page 604 pages. Available in PDF, EPUB and Kindle. Book excerpt: Uncertainty Quantification in Multiscale Materials Modeling provides a complete overview of uncertainty quantification (UQ) in computational materials science. It provides practical tools and methods along with examples of their application to problems in materials modeling. UQ methods are applied to various multiscale models ranging from the nanoscale to macroscale. This book presents a thorough synthesis of the state-of-the-art in UQ methods for materials modeling, including Bayesian inference, surrogate modeling, random fields, interval analysis, and sensitivity analysis, providing insight into the unique characteristics of models framed at each scale, as well as common issues in modeling across scales.

Download or read book Homogenization Methods For Multiscale Mechanics written by Chiang C Mei and published by World Scientific. This book was released on 2010-09-23 with total page 349 pages. Available in PDF, EPUB and Kindle. Book excerpt: In many physical problems several scales are present in space or time, caused by inhomogeneity of the medium or complexity of the mechanical process. A fundamental approach is to first construct micro-scale models, and then deduce the macro-scale laws and the constitutive relations by properly averaging over the micro-scale. The perturbation method of multiple scales can be used to derive averaged equations for a much larger scale from considerations of the small scales. In the mechanics of multiscale media, the analytical scheme of upscaling is known as the Theory of Homogenization.The authors share the view that the general methods of homogenization should be more widely understood and practiced by applied scientists and engineers. Hence this book is aimed at providing a less abstract treatment of the theory of homogenization for treating inhomogeneous media, and at illustrating its broad range of applications. Each chapter deals with a different class of physical problems. To tackle a new problem, the approach of first discussing the physically relevant scales, then identifying the small parameters and their roles in the normalized governing equations is adopted. The details of asymptotic analysis are only explained afterwards.