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Significance of higher order moments to the completecharacterization of the travel time pdf in heterogeneous porous media using the maximum entropyprinciple
KTH, School of Architecture and the Built Environment (ABE), Land and Water Resources Engineering.
KTH, School of Architecture and the Built Environment (ABE), Land and Water Resources Engineering.
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.

Place, publisher, year, edition, pages
2010. Vol. 46, no W05502
Keyword [en]
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: urn:nbn:se:kth:diva-10651DOI: 10.1029/2009WR008220ISI: 000277266300007Scopus ID: 2-s2.0-77952264451OAI: oai:DiVA.org:kth-10651DiVA: diva2:222646
Note
QC 20100714Available from: 2009-06-09 Created: 2009-06-09 Last updated: 2011-01-03Bibliographically approved
In thesis
1. A multi-resolution approach for modeling flow and solute transport in heterogeneous porous media
Open this publication in new window or tab >>A multi-resolution approach for modeling flow and solute transport in heterogeneous porous media
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
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:nbn:se:kth:diva-10655 (URN)978-91-7415-377-4 (ISBN)
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

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