Download or read book Exact Averaging of Stochastic Equations for Flow in Porous Media written by and published by . This book was released on 2008 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: It is well known that at present, exact averaging of the equations for flow and transport in random porous media have been proposed for limited special fields. Moreover, approximate averaging methods--for example, the convergence behavior and the accuracy of truncated perturbation series--are not well studied, and in addition, calculation of high-order perturbations is very complicated. These problems have for a long time stimulated attempts to find the answer to the question: Are there in existence some, exact, and sufficiently general forms of averaged equations? Here, we present an approach for finding the general exactly averaged system of basic equations for steady flow with sources in unbounded stochastically homogeneous fields. We do this by using (1) the existence and some general properties of Green's functions for the appropriate stochastic problem, and (2) some information about the random field of conductivity. This approach enables us to find the form of the averaged equations without directly solving the stochastic equations or using the usual assumption regarding any small parameters. In the common case of a stochastically homogeneous conductivity field we present the exactly averaged new basic nonlocal equation with a unique kernel-vector. We show that in the case of some type of global symmetry (isotropy, transversal isotropy, or orthotropy), we can for three-dimensional and two-dimensional flow in the same way derive the exact averaged nonlocal equations with a unique kernel-tensor. When global symmetry does not exist, the nonlocal equation with a kernel-tensor involves complications and leads to an ill-posed problem.

Download or read book Averaging of Stochastic Equations for Flow and Transport in PorousMedia written by and published by . This book was released on 2005 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: It is well known that at present exact averaging of theequations of flow and transport in random porous media have been realizedfor only a small number of special fields. Moreover, the approximateaveraging methods are not yet fully understood. For example, theconvergence behavior and the accuracy of truncated perturbation seriesare not well known; and in addition, the calculation of the high-orderperturbations is very complicated. These problems for a long time havestimulated attempts to find the answer for the question: Are there inexistence some exact general and sufficiently universal forms of averagedequations? If the answer is positive, there arises the problem of theconstruction of these equations and analyzing them. There are manypublications on different applications of this problem to various fields, including: Hydrodynamics, flow and transport in porous media, theory ofelasticity, acoustic and electromagnetic waves in random fields, etc. Here, we present a method of finding some general form of exactlyaveraged equations for flow and transport in random fields by using (1)some general properties of the Green s functions for appropriatestochastic problems, and (2) some basic information about the randomfields of the conductivity, porosity and flow velocity. We presentgeneral forms of exactly averaged non-local equations for the followingcases: (1) steady-state flow with sources in porous media with randomconductivity, (2) transient flow with sources in compressible media withrandom conductivity and porosity; and (3) Nonreactive solute transport inrandom porous media. We discuss the problem of uniqueness and theproperties of the non-local averaged equations for cases with some typeof symmetry (isotropic, transversal isotropic and orthotropic), and weanalyze the structure of the nonlocal equations in the general case ofstochastically homogeneous fields.

Download or read book Exactly Averaged Stochastic Equations for Flow and Transport in Random Media written by and published by . This book was released on 2001 with total page 5 pages. Available in PDF, EPUB and Kindle. Book excerpt: It is well known that exact averaging of the equations of flow and transport in random porous media are at present realized only for a small number of special, occasionally exotic, fields. On the other hand, the properties of approximate averaging methods are not yet fully understood. For example, the convergence behavior and the accuracy of truncated perturbation series are not well known. Furthermore, the calculation of the high-order perturbations is very complicated. These problems for a long time have stimulated attempts to find the answer for the question: Are there in existence some exact general and sufficiently universal forms of averaged equations? If the answer is positive, there arises the problem of the construction of these equations and analyzing them. There exist many publications related to these problems and oriented on different applications: hydrodynamics, flow and transport in porous media, theory of elasticity, acoustic and electromagnetic waves in random fields, etc. We present a method of finding some general forms of exactly averaged equations for flow and transport in random fields by using (1) an assumption of the existence of Green's functions for appropriate stochastic problems, (2) some general properties of the Green's functions, and (3) the some basic information about the random fields of the conductivity, porosity and flow velocity. We present some general forms of the exactly averaged non-local equations for the following cases. 1. Steady-state flow with sources in porous media with random conductivity. 2. Transient flow with sources in compressible media with random conductivity and porosity. 3. Non-reactive solute transport in random porous media. We discuss the problem of uniqueness and the properties of the non-local averaged equations, for the cases with some types of symmetry (isotropic, transversal isotropic, orthotropic) and we analyze the hypothesis of the structure of non-local equations in a general case of stochastical ly homogeneous fields.

Download or read book Stochastic Methods for Flow in Porous Media written by Dongxiao Zhang and published by Elsevier. This book was released on 2001-10-11 with total page 371 pages. Available in PDF, EPUB and Kindle. Book excerpt: Stochastic Methods for Flow in Porous Media: Coping with Uncertainties explores fluid flow in complex geologic environments. The parameterization of uncertainty into flow models is important for managing water resources, preserving subsurface water quality, storing energy and wastes, and improving the safety and economics of extracting subsurface mineral and energy resources. This volume systematically introduces a number of stochastic methods used by researchers in the community in a tutorial way and presents methodologies for spatially and temporally stationary as well as nonstationary flows. The author compiles a number of well-known results and useful formulae and includes exercises at the end of each chapter. Balanced viewpoint of several stochastic methods, including Greens' function, perturbative expansion, spectral, Feynman diagram, adjoint state, Monte Carlo simulation, and renormalization group methods Tutorial style of presentation will facilitate use by readers without a prior in-depth knowledge of Stochastic processes Practical examples throughout the text Exercises at the end of each chapter reinforce specific concepts and techniques For the reader who is interested in hands-on experience, a number of computer codes are included and discussed

Download or read book A Note on the Averaging Approach for the Random Linear Transport Equation written by Fábio A. Dorini and published by . This book was released on 2007 with total page 18 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Averaged Description of Flow Steady and Transient and Nonreactive Solute Transport in Random Porous Media written by and published by . This book was released on 2011 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: In previous papers (Shvidler and Karasaki, 1999, 2001, 2005, and 2008) we presented and analyzed an approach for finding the general forms of exactly averaged equations of flow and transport in porous media. We studied systems of basic equations for steady flow with sources in unbounded domains with stochastically homogeneous conductivity fields. A brief analysis of exactly averaged equations of nonsteady flow and nonreactive solute transport was also presented. At the core of this approach is the existence of appropriate random Green's functions. For example, we showed that in the case of a 3-dimensional unbounded domain the existence of appropriate random Green's functions is sufficient for finding the exact nonlocal averaged equations for flow velocity using the operator with a unique kernel-vector. Examination of random fields with global symmetry (isotropy, transversal isotropy and orthotropy) makes it possible to describe significantly different types of averaged equations with nonlocal unique operators. It is evident that the existence of random Green's functions for physical linear processes is equivalent to assuming the existence of some linear random operators for appropriate stochastic equations. If we restricted ourselves to this assumption only, as we have done in this paper, we can study the processes in any dimensional bounded or unbounded fields and in addition, cases in which the random fields of conductivity and porosity are stochastically nonhomogeneous, nonglobally symmetrical, etc. It is clear that examining more general cases involves significant difficulty and constricts the analysis of structural types for the processes being studied. Nevertheless, we show that we obtain the essential information regarding averaged equations for steady and transient flow, as well as for solute transport.

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 Convective Heat Transfer in Porous Media written by Yasser Mahmoudi and published by CRC Press. This book was released on 2019-11-06 with total page 381 pages. Available in PDF, EPUB and Kindle. Book excerpt: Focusing on heat transfer in porous media, this book covers recent advances in nano and macro’ scales. Apart from introducing heat flux bifurcation and splitting within porous media, it highlights two-phase flow, nanofluids, wicking, and convection in bi-disperse porous media. New methods in modeling heat and transport in porous media, such as pore-scale analysis and Lattice–Boltzmann methods, are introduced. The book covers related engineering applications, such as enhanced geothermal systems, porous burners, solar systems, transpiration cooling in aerospace, heat transfer enhancement and electronic cooling, drying and soil evaporation, foam heat exchangers, and polymer-electrolyte fuel cells.

Download or read book A Stochastic Differential Equation Approach to Multiphase Flow in Porous Media written by David W. Dean and published by . This book was released on 2001 with total page 24 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Probabilistic Solution to Stochastic Overland Flow Equation Using the Cumulant Expansion Method written by Jaeyoung Yoon and published by . This book was released on 2001 with total page 230 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book The Probability Density Function to the Random Linear Transport Equation written by Lúcio T. Santos and published by . This book was released on 2008 with total page 20 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Stochastic Modelling of Fluid Flow in Porous Media written by Tom Lindstrøm and published by . This book was released on 1991 with total page 34 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Flow and Transport in Porous Formations written by Gedeon Dagan and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 472 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the mid-seventies, a new area of research has emerged in subsurface hydrology, namely sto chastic modeling of flow and transport. This development has been motivated by the recognition of the ubiquitous presence of heterogeneities in natural formations and of their effect upon transport and flow, on the one hand, and by the vast expansion of computational capability provided by elec tronic machines, on the other. Apart from this, one of the areas in which spatial variability of for mation properties plays a cardinal role is of contaminant transport, a subject of growing interest and concern. I have been quite fortunate to be engaged in research in this area from its inception and to wit ness the rapid growth of the community and of the literature on spatial variability and its impact upon subsurface hydrology. In view of this increasing interest, I decided a few years ago that it would be useful to present the subject in a systematic and comprehensive manner in order to help those who wish to engage themselves in research or application of this new field. I viewed as my primary task to analyze the large scale heterogeneity of aquifers and its effect, presuming that the reader already possesses a background in traditional hydrology. This is achieved in Parts 3, 4 and 5 of the text which incorporate the pertinent material.

Download or read book Macroscale Models of Flow Through Highly Heterogeneous Porous Media written by M. Panfilov and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 384 pages. Available in PDF, EPUB and Kindle. Book excerpt: The The book book was was planned planned in in such such a a manner manner that that two two basic basic goals goals would would be be reached. reached. On On the the one one hand, hand, the the goal goal was was to to show show some some new new results results in in the the field field of of modeling modeling transport transport through through highly highly heterogeneous heterogeneous media, media, based based on on the the homogenization homogenization theory. theory. Multiple Multiple new new mathematical mathematical models models of of transport transport are are presented presented herein, herein, studying studying their their properties, properties, developing developing methods methods to to compute compute effective effective parameters parameters of of the the averaged averaged media, media, simulation simulation of of cell cell problems, problems, using using new new models models to to simulate simulate some some practical practical problems. problems. High High heterogeneity heterogeneity being being subjected subjected to to the the homogenization homogenization procedure, procedure, generates generates non-local non-local phenomena phenomena and and then then gives gives a a possibility possibility to to develop develop a a new, new, non-local non-local (or (or "dynamic"), "dynamic"), theory theory of of transport transport in in porous porous media. media.

Download or read book Computation and Applied Mathematics written by and published by . This book was released on 1998 with total page 124 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Averaging of Flows with Capillary Hysteresis in Stochastic Porous Media written by Ben Schweizer and published by . This book was released on 2004 with total page 30 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Numerical Models in Geomechanics written by G.N. Pande and published by CRC Press. This book was released on 2004-08-15 with total page 760 pages. Available in PDF, EPUB and Kindle. Book excerpt: Reflecting the current research and advances made in the application of numerical methods in geotechnical engineering, this volume details proceedings of the Ninth International Symposium on 'Numerical Models in Geomechanics - NUMOG IX' held in Ottawa, Canada, 25-27 August 2004. Highlighting a number of new developments in the area, papers concentrate upon the following four main areas: * constitutive relations for geomaterials * numerical algorithms: formulation and performance * modelling of transient, coupled and dynamic problems * application of numerical techniques to practical problems. Representing the most advanced, modern findings in the field, Numerical Models in Geomechanics is a comprehensive and impeccably-researched text, ideal for students and researchers as well as practising engineers.