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A multi-resolution approach for modeling flow and solute transport in heterogeneous porous media
KTH, School of Architecture and the Built Environment (ABE), Land and Water Resources Engineering.
2009 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Subsurface processes are usually characterized by rare field experiments, sparse measurements,multi-resolution interpretations, stochastic description, related uncertainties and computational complexity. Over the last few decades, different computational techniques and strategies have become indispensable tools for flow and solute transport prediction in heterogeneous porousmedia. This thesis develops a multi-resolution approach based on Fup basis functions with compactsupport, enabling the use of an efficient and adaptive procedure, closely related to currentunderstood physical interpretation. All flow and transport variables, as well as intrinsic heterogeneity,are described in a multi-resolution representation, in the form of a linear combination ofFup basis functions. Each variable is represented on a particular adaptive grid with a prescribedaccuracy. The methodology is applied to solving problems with sharp fronts, and to solving flowand advective transport in highly heterogeneous porous media, under mean uniform flow conditions.The adaptive Fup collocation method, through the well known method of lines, efficientlytracks solutions with sharp fronts, resolving locations and frequencies at all spatial and/or temporalscales. The methodology yields continuous velocity fields and fluxes, enabling accurate andreliable transport analysis. Analysis of the advective transport proves the robustness of the firstordertheory for low and mild heterogeneity. Moreover, due to the accuracy of the improved Monte-Carlo methodology, this thesis presents the effects of high heterogeneity on ensembleflow and travel time statistics. The difference between Eulerian and Lagrangian velocity statisticsand the importance of higher travel time moments are indicative of high heterogeneity. The thirdtravel time moment mostly describes a peak and late arrivals, while higher moments are requiredfor early arrivals which are linked with the largest uncertainty. A particular finding is the linearityof all travel time moments, which implies that in the limit an advective transport in multi-Gaussian field becomes Fickian. By comparison, the transverse displacement pdf converges to aGaussian distribution around 20 integral scales after injection, even for high heterogeneity. Thecapabilities of the presented multi-resolution approach, and the quality of the obtained results,open new areas for further research.

Abstract [sv]

Markprocesser karakteriseras ofta av fåtaliga fältexperiment, glesa mätningar, heterogenitet påolika skalor, slumpmässighet och relaterade osäkerheter, samt beräkningsmässiga svårigheter.Under de senaste årtiondena har olika beräkningstekniker och strategier blivit ovärderliga verktygför att förutspå vattenflöde och ämnestransport i heterogena porösa medier. Denna doktorsavhandling utvecklar ett angreppssätt med flerskaliga upplösningar baserat på Fup basis funktionermed kompakt stöd, som möjliggör en effektiv och anpassningsbar procedur, nära relaterad tillrådande fysiska tolkningar. Alla flödes- och transportvariabler, så väl som heterogeniteten, beskrivsav en flerskaligt upplöst representation, i form av linjära kombinationer av Fup basis funktioner.Varje variabel representeras på ett speciellt anpassningsbar gridnät med given noggrannhet.Metoden appliceras för att lösa problem med skarpa fronter, samt vattenflöde och advektivämnestransport i starkt heterogena porösa medier. Adaptive Fup collocation metoden tillsammansmed den välkända Method of lines, spårar effektivt lösningar med skarpa fronter och löserupp positioner och frekvenser på alla rums- och/eller tidsskalor. Metoden ger kontinuerliga hastighetsfältoch flöden, och möjliggör noggrann och tillförlitlig transportanalys. Analys av advektivtransport understöder stabiliteten i första-ordningens transport teori för låg och mild heterogenitet.Utöver detta, som resultat av noggrannheten i den förbättrade Monte-Carlo metodiken, visardenna avhandling effekten av hög heterogenitet på ensemble statistiken för flöden och transporttider.Skillnaden mellan Eulerisk och Lagrangian hastighetsstatistik och betydelsen av högrestatistiska moment för transporttider, indikerar hög heterogenitet. Det tredje transporttidsmomentetbeskriver huvudsakligen sannolikhetspiken och de långa transporttiderna, medan högremoment behövs för de korta transporttiderna, som har den största osäkerheten. En speciell upptäcktär linjäariteten i transporttidsmoment, som indikerar att advektiv transport i multi-Gaussiska fält blir Gaussisk i gränsen. Som jämförelse konvergerar sannolikhetsfunktioner förden transversella transportförflyttningen mot en Gaussisk fördelning vid runt 20 korrelationslängder efter injektion, även för hög heterogenitet. Förmågan i det presenterade angreppssättet med flerskalig upplösning, och resultatens noggrannhet, öppnar nya områden för fortsatt forskning.

Place, publisher, year, edition, pages
Stockholm: KTH , 2009. , xiv, 51 p.
Series
Trita-LWR. PHD, ISSN 1650-8602 ; 1051
Keyword [en]
Multi-resolution adaptive approach; Atomic and Fup basis functions; Monte- Carlo method; Heterogeneous porous media; Flow; Transport; Travel time
National Category
Water Engineering
Identifiers
URN: urn:nbn:se:kth:diva-10655ISBN: 978-91-7415-377-4 (print)OAI: oai:DiVA.org:kth-10655DiVA: diva2:222662
Public defence
2009-06-18, M3, Entreplan, KTH, Brinellvägen 64, Stockholm, 10:00 (English)
Opponent
Supervisors
Note
QC 20100714Available from: 2009-06-09 Created: 2009-06-09 Last updated: 2011-08-30Bibliographically approved
List of papers
1. Multi-resolution adaptive modeling of groundwater flow and transport problems
Open this publication in new window or tab >>Multi-resolution adaptive modeling of groundwater flow and transport problems
2007 (English)In: Advances in Water Resources, ISSN 0309-1708, E-ISSN 1872-9657, Vol. 30, 1105-1126 p.Article in journal (Refereed) Published
Abstract [en]

Many groundwater flow and transport problems, especially those with sharp fronts, narrow transition zones, layers and fingers, require extensive computational resources. In this paper, we present a novel multi-resolution adaptive Fup approach to solve the above mentioned problems. Our numerical procedure is the Adaptive Fup Collocation Method (AFCM), based on Fup basis functions and designed through a method of lines (MOL). Fup basis functions are localized and infinitely differentiable functions with compact support and are related to more standard choices such as splines or wavelets. This method enables the adaptive multi-reso In tion approach to solve problems with different spatial and temporal scales with a desired level of accuracy using the entire family of Fup basis functions. In addition, the utilized collocation algorithm enables the mesh free approach with consistent velocity approximation and flux continuity due to properties of the Fup basis functions. The introduced numerical procedure was tested and verified by a few characteristic groundwater flow and transport problems, the Buckley-Leverett multiphase flow problem, the 1-D vertical density driven problem and the standard 2-D seawater intrusion benchmark-Henry problem. The results demonstrate that the method is robust and efficient particularly when describing sharp fronts and narrow transition zones changing in space and time.

Keyword
Fup basis functions, compact support, method of lines, adaptive Fup collocation method, multi-resolution approach, numerical dispersion, groundwater flow and transport problems
National Category
Civil Engineering
Identifiers
urn:nbn:se:kth:diva-10644 (URN)10.1016/j.advwatres.2006.10.007 (DOI)000246092800006 ()2-s2.0-33847309618 (Scopus ID)
Note
QC 20100714Available from: 2009-06-09 Created: 2009-06-09 Last updated: 2017-12-13Bibliographically approved
2. Adaptive Fup multi-resolution approach to flow and advective transport in highly heterogeneous porous media: Methodology, accuracy and convergence
Open this publication in new window or tab >>Adaptive Fup multi-resolution approach to flow and advective transport in highly heterogeneous porous media: Methodology, accuracy and convergence
2009 (English)In: Advances in Water Resources, ISSN 0309-1708, E-ISSN 1872-9657, Vol. 32, no 6, 885-905 p.Article in journal (Refereed) Published
Abstract [en]

In this paper, we present a new Monte-Carlo methodology referred to as Adaptive Fup Monte-Carlo Method (AFMCM) based on compactly supported Fup basis functions and a multi-resolution approach. We consider for illustration 2-D steady, linear and unidirectional flow and advective transport defined on a domain of size 64I(Y) * 32I(Y) with isotropic exponential correlation heterogeneity structure and sigma(2)(Y) up to 8. Accuracy and convergence issues are rigorously analyzed for each realization as well as for the ensemble. Log-conductivity is presented by continuous function at high resolution level (n(Y) = 4-32 points per integral scale) reproducing very accurately prescribed statistics. The flow problem is the most demanding Monte-Carlo step due to satisfying detailed log-conductivity properties. Presented methodology inherently gives continuous and mesh-free velocity fields, which enables the construction of a new efficient and accurate particle tracking algorithm. Results indicate that resolutions n(Y) = 8 and n(h) = 32 enable very accurate flow solutions in each realization with mass balance error less than 3% and accurate ensemble velocity statistics. Results show that the proposed AFMCM enables tracking of an unlimited number of injected particles and calculates required transport variables as continuous functions with desired relative accuracy (0.1%) in each realization. Furthermore, we show that the resolution n(Y) = 8 yields a quite accurate pdf of the transverse displacement and travel time. All required flow and transport variables require 500 Monte-Carlo realizations in order to stabilize fluctuations of the higher-order moments and the probability density functions.

Keyword
Adaptive multi-resolution approach, Flow and advective transport, Highly heterogeneous porous media, Fup basis functions, Monte-Carlo, method, Particle tracking, wavelet collocation method, solute flux approach, groundwater-flow, atomic functions, 1st-order approximations, stochastic-analysis, elliptic problems, log-conductivity, finite-elements, path lines
National Category
Water Engineering
Identifiers
urn:nbn:se:kth:diva-18506 (URN)10.1016/j.advwatres.2009.02.013 (DOI)000266895000009 ()2-s2.0-67349160471 (Scopus ID)
Note
QC 20100525Available from: 2010-08-05 Created: 2010-08-05 Last updated: 2017-12-12Bibliographically approved
3. Flow and travel time statistics in highly heterogeneous porous media
Open this publication in new window or tab >>Flow and travel time statistics in highly heterogeneous porous media
2009 (English)In: Water resources research, ISSN 0043-1397, E-ISSN 1944-7973, Vol. 45, no W07402Article in journal (Refereed) Published
Abstract [en]

In this paper we present flow and travel time ensemble statistics based on a new simulation methodology, the adaptive Fup Monte Carlo method (AFMCM). As a benchmark case, we considered two-dimensional steady flow in a rectangular domain characterized by multi-Gaussian heterogeneity structure with an isotropic exponential correlation and lnK variance sigma(2)(Y) up to 8. Advective transport is investigated using the travel time framework where Lagrangian variables (e. g., velocity, transverse displacement, or travel time) depend on space rather than on time. We find that Eulerian and Lagrangian velocity distributions diverge for increasing lnK variance due to enhanced channeling. Transverse displacement is a nonnormal for all sigma(2)(Y) and control planes close to the injection area, but after xI(Y) = 20 was found to be nearly normal even for high sigma(2)(Y). Travel time distribution deviates from the Fickian model for large lnK variance and exhibits increasing skewness and a power law tail for large lnK variance, the slope of which decreases for increasing distance from the source; no anomalous features are found. Second moment of advective transport is analyzed with respect to the covariance of two Lagrangian velocity variables: slowness and slope which are directly related to the travel time and transverse displacement variance, which are subsequently related to the longitudinal and transverse dispersion. We provide simple estimators for the Eulerian velocity variance, travel time variance, slowness, and longitudinal dispersivity as a practical contribution of this analysis. Both two-parameter models considered (the advection-dispersion equation and the lognormal model) provide relatively poor representations of the initial part of the travel time probability density function in highly heterogeneous porous media. We identify the need for further theoretical and experimental scrutiny of early arrival times, and the need for computing higher-order moments for a more accurate characterization of the travel time probability density function. A brief discussion is presented on the challenges and extensions for which AFMCM is suggested as a suitable approach.

Keyword
HEAD SPATIAL VARIABILITY; SOLUTE FLUX APPROACH; MEAN UNIFORM-FLOW; STOCHASTIC-ANALYSIS; GROUNDWATER-FLOW; NUMERICAL SIMULATIONS; CONSERVATIVE SOLUTES; VELOCITY COVARIANCE; EVOLVING SCALES; CAPE-COD
National Category
Water Engineering
Identifiers
urn:nbn:se:kth:diva-10650 (URN)10.1029/2008WR007168 (DOI)000267648200001 ()2-s2.0-69249101625 (Scopus ID)
Note
QC 20100714Available from: 2009-06-09 Created: 2009-06-09 Last updated: 2017-12-13Bibliographically approved
4. Significance of higher order moments to the completecharacterization of the travel time pdf in heterogeneous porous media using the maximum entropyprinciple
Open this publication in new window or tab >>Significance of higher order moments to the completecharacterization of the travel time pdf in heterogeneous porous media using the maximum entropyprinciple
2010 (English)In: Water resources research, ISSN 0043-1397, E-ISSN 1944-7973, Vol. 46, no W05502Article in journal (Refereed) Published
Abstract [en]

The travel time formulation of advective transport in heterogeneous porous media is of interest both conceptually, e. g., for incorporating retention processes, and in applications where typically the travel time peak, early, and late arrivals of contaminants are of major concern in a regulatory or remediation context. Furthermore, the travel time moments are of interest for quantifying uncertainty in advective transport of tracers released from point sources in heterogeneous aquifers. In view of this interest, the travel time distribution has been studied in the literature; however, the link to the hydraulic conductivity statistics has been typically restricted to the first two moments. Here we investigate the influence of higher travel time moments on the travel time probability density function (pdf) in heterogeneous porous media combining Monte Carlo simulations with the maximum entropy principle. The Monte Carlo experimental pdf is obtained by the adaptive Fup Monte Carlo method (AFMCM) for advective transport characterized by a multi-Gaussian structure with exponential covariance considering two injection modes (in-flux and resident) and lnK variance up to 8. A maximum entropy (MaxEnt) algorithm based on Fup basis functions is used for the complete characterization of the travel time pdf. All travel time moments become linear with distance. Initial nonlinearity is found mainly for the resident injection mode, which exhibits a strong nonlinearity within first 5I(Y) for high heterogeneity. For the resident injection mode, the form of variance and all higher moments changes from the familiar concave form predicted by the first-order theory to a convex form; for the in-flux mode, linearity is preserved even for high heterogeneity. The number of moments sufficient for a complete characterization of the travel time pdf mainly depends on the heterogeneity level. Mean and variance completely describe travel time pdf for low and mild heterogeneity, skewness is dominant for lnK variance around 4, while kurtosis and fifth moment are required for lnK variance higher than 4. Including skewness seems sufficient for describing the peak and late arrivals. Linearity of travel time moments enables the prediction of asymptotic behavior of the travel time pdf which in the limit converges to a symmetric distribution and Fickian transport. However, higher-order travel time moments may be important for most practical purposes and in particular for advective transport in highly heterogeneous porous media for a long distance from the source.

Keyword
FULL-BAYESIAN APPROACH; SOLUTE FLUX APPROACH; MEAN UNIFORM-FLOW; GROUNDWATER-FLOW; INVERSE PROBLEM; CONSERVATIVE SOLUTES; MASS-TRANSFER; TRANSPORT; INFORMATION; STATISTICS
National Category
Water Engineering
Identifiers
urn:nbn:se:kth:diva-10651 (URN)10.1029/2009WR008220 (DOI)000277266300007 ()2-s2.0-77952264451 (Scopus ID)
Note
QC 20100714Available from: 2009-06-09 Created: 2009-06-09 Last updated: 2017-12-13Bibliographically approved
5. Maximum entropy algorithm with inexact upper entropy bound based on Fup basis functions with compact support
Open this publication in new window or tab >>Maximum entropy algorithm with inexact upper entropy bound based on Fup basis functions with compact support
2009 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 228, no 24, 9079-9091 p.Article in journal (Refereed) Published
Abstract [en]

The maximum entropy (MaxEnt) principle is a versatile tool for statistical inference of the probability density function (pdf) from its moments as a least-biased estimation among all other possible pdfs. It maximizes Shannon entropy, satisfying the moment constraints. Thus, the MaxEnt algorithm transforms the original constrained optimization problem to the unconstrained dual optimization problem using Lagrangian multipliers. The Classic Moment Problem (CMP) uses algebraic power moments, causing typical conventional numerical methods to fail for higher-order moments (m > 5-10) due to different sensitivities of Lagrangian multipliers and unbalanced nonlinearities. Classic MaxEnt algorithms overcome these difficulties by using orthogonal polynomials, which enable roughly the same sensitivity for all Lagrangian multipliers. In this paper, we employ an idea based on different principles, using Fup(n) basis functions with compact support, which can exactly describe algebraic polynomials, but only if the Fup order-n is greater than or equal to the polynomial's order. Our algorithm solves the CMP with respect to the moments of only low order Fup(2) basis functions, finding a Fup(2) optimal pdf with better balanced Lagrangian multipliers. The algorithm is numerically very efficient due to localized properties of Fup(2) basis functions implying a weaker dependence between Lagrangian multipliers and faster convergence. Only consequences are an iterative scheme of the algorithm where power moments are a sum of Fup(2) and residual moments and an inexact entropy upper bound. However, due to small residual moments, the algorithm converges very quickly as demonstrated on two continuous pdf examples - the beta distribution and a bi-modal pdf, and two discontinuous pdf examples - the step and double Dirac pdf. Finally, these pdf examples present that Fup MaxEnt algorithm yields smaller entropy value than classic MaxEnt algorithm, but differences are very small for all practical engineering purposes.

Keyword
Maximum entropy algorithm; Classic Moment Problem; Fup basis functions; Compact support
National Category
Water Engineering
Identifiers
urn:nbn:se:kth:diva-10653 (URN)10.1016/j.jcp.2009.09.011 (DOI)000271790600008 ()2-s2.0-70350026655 (Scopus ID)
Note
Tidigare titel: Inexact Maximum Entropy algorithm based on Fup basis functions with compact support. QC 20100714. Updated from submitted to published 20120326Available from: 2009-06-09 Created: 2009-06-09 Last updated: 2017-12-13Bibliographically approved

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  • apa
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  • Other locale
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Output format
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