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Adaptive Fup multi-resolution approach to flow and advective transport in highly heterogeneous porous media: Methodology, accuracy and convergence
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, Water Resources Engineering.
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.

Place, publisher, year, edition, pages
2009. Vol. 32, no 6, 885-905 p.
Keyword [en]
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
URN: urn:nbn:se:kth:diva-18506DOI: 10.1016/j.advwatres.2009.02.013ISI: 000266895000009ScopusID: 2-s2.0-67349160471OAI: diva2:336553
QC 20100525Available from: 2010-08-05 Created: 2010-08-05 Last updated: 2011-08-30Bibliographically 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.
Trita-LWR. PHD, ISSN 1650-8602 ; 1051
Multi-resolution adaptive approach; Atomic and Fup basis functions; Monte- Carlo method; Heterogeneous porous media; Flow; Transport; Travel time
National Category
Water Engineering
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)
QC 20100714Available from: 2009-06-09 Created: 2009-06-09 Last updated: 2011-08-30Bibliographically approved

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