Physical and numerical experiments of flow and transport in heterogeneous fractured media : single fracture flow at high Reynolds numbers, and reactive particle transport

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(1)N° d’ordre : 2143. THESE présentée pour obtenir LE TITRE DE DOCTEUR DE L’INSTITUT NATIONAL POLYTECHNIQUE DE TOULOUSE. École doctorale : Sciences de l’Univers de l’Environnement et de l’Espace Spécialité : Sciences de la Terre et de l’Environnement, Mécanique des Fluides. Par M. Martin Werner SPILLER. Titre de la thèse MODELISATION PHYSIQUE ET NUMERIQUE D'ECOULEMENTS ET TRANSPORTS EN MILIEUX HETEROGENES FRACTURES (ECOULEMENT A HAUT REYNOLDS ET TRANSPORT PARTICULAIRE REACTIF). Soutenue le 25/10/2004. devant le jury composé de :. M. Prof. Dr. Rachid ABABOU. Codirecteur de thèse. M. Univ.-Prof. Dr.-Ing. Jürgen KÖNGETER. Codirecteur de thèse. M. Prof. Dr.-Ing. Rainer HELMIG. Rapporteur. M. Univ.-Prof. Dr.-Ing. Konstantin MESKOURIS. Rapporteur. M. Dr. Franck PLOURABOUE. Membre. M. Dr. Michel QUINTARD. Membre.

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(5) III. Vorwort Die Frage der Wassergewinnung aus Festgesteinsaquiferen gewinnt zunehmend an Bedeutung, da Grundwasser in Lockergesteinsaquiferen in vielen Gebieten nur noch begrenzt nutzbar sind. Für den Transport des Wassers sind die Risse im Festgestein, die sogenannten Klüfte, maßgeblich. Das Ziel der hier vorgestellten Dissertation ist, einen Beitrag zur Verbesserung des Prozessverständnisses bei reaktiven Transportprozessen in Einzelklüften mit angrenzender permeabler Matrix zu leisten. Dazu sind umfangreiche experimentelle und numerische Untersuchungen durchgeführt worden. Die vorliegende Arbeit ist im Rahmen eines Deutsch-Französischen Promotionsvorhabens (Thèse en Cotutelle) gemeinsam von Prof. Dr.-Ing. J. Köngeter am Institut für Wasserbau und Wasserwirtschaft der RWTH Aachen (IWW) und Prof. R. Ababou vom Institut de Mécanique des Fluides de Toulouse (IMFT) betreut worden. Herrn Prof. Dr.-Ing. J. Köngeter danke ich an dieser Stelle ganz herzlich für die fortwährende große Unterstützung bei der Durchführung der Promotion. Des Weiteren bedanke ich mich für den Freiraum, den ich während meiner Zeit am IWW hatte. Ich danke Herrn Prof. R. Ababou für die tatkräftige Unterstützung, dessen Gastfreundschaft und seine Anregungen, die mir immer eine große Hilfe gewesen sind. Meine Arbeit hat sehr profitiert von den vielen fruchtbaren Diskussionen, auch mit Herrn Dr. F. Plouraboué, dem ich an dieser Stelle ebenfalls herzlich danke. Herrn Prof. Dr.-Ing. K. Meskouris, Herrn Prof. Dr.-Ing. R. Helmig und Herrn Dr. M. Quintard danke ich für ihre Bereitschaft, die Aufgabe als Berichter für meine Promotion übernommen zu haben. Herzlicher Dank gilt den Kollegen, insbesondere Herrn Dipl.-Ing. T. Becker und Herrn Dipl.-Ing. W. Hamelmann, sowie den studentischen Mitarbeitern für die wichtigen Diskussionen, die tatkräftige Mithilfe und für ein hervorragendes Arbeitsumfeld. Abschließend möchte ich mich bei der Deutsch-Französischen Hochschule bedanken, die mit Mitteln der Robert-Bosch-Stiftung meine Aufenthalte in Toulouse unterstützt hat. Aachen, im Januar 2004. Martin Spiller.

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(7) V. Abstract The first part of the thesis is focusing on an experimental study of single phase flow processes in a single granite fracture. In order to obtain the aperture field, two different measurement methods were developed and evaluated. The effect of heterogeneities of the aperture field was shown by a visualization of streamlines. The flow experiments revealed that inertial effects lead to a correction of the effective permeability which is of second order in the mean flux. The comparison of results of the experiment with numerical simulations performed on the basis of the local Cubic Law is discussed. The second part of the thesis is about numerical modeling of reactive transport processes. Two different models have been developed. The first one is based on a Lagrangian approach, whereas the second one is based on a microscopic description of reactions and transport using a Master equation approach. A comparison of the two algorithms is discussed based on three selected application examples..

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(9) VII. Zusammenfassung Das Ziel der hier vorgestellten Dissertation ist, einen Beitrag zur Verbesserung des Prozessverständnisses. bei. reaktiven. Transportprozessen. in. Einzelklüften. mit. angrenzender permeabler Matrix zu leisten. Dazu sind umfangreiche experimentelle und numerische Untersuchungen durchgeführt worden. Die experimentellen Untersuchungen konzentrierten sich auf die Identifikation von Strömungsprozessen in einer heterogenen Einzelkluft. Es wurde ein transparenter Epoxydharzabguss einer realen Granit-Einzelkluft verwendet, wobei die Ober- und Unterseite der Kluft in Strömungsrichtung gegeneinander verschoben werden können und. sich. somit. verschiedene. Scherversatzbeträge. einstellen. lassen.. Das. Öffnungsweitenfeld ist mit zwei unabhängigen Messmethoden (Absorptionsmethode und 3D-Scan Methode) vermessen worden, womit ein Vergleich zwischen experimentell gemessenen Strömungsverlusten und Stromlinien mit Ergebnissen numerischer Simulationen möglich ist. Während die Stromlinien eine sehr gute Übereinstimmung zeigen, gibt es signifikante Abweichungen bei der Bestimmung der effektiven hydraulischen Durchlässigkeit der Einzelkluft: Die experimentell gemessene Permeabilität ist ca. 25% geringer als die mit dem Ansatz eines lokalen "Cubic Laws" simulierte Permeabilität. Diese Abweichungen stimmen mit aus der Literatur bekannten Ergebnissen experimenteller und numerischer Studien überein. Es konnte in numerischen und experimentellen Untersuchungen gezeigt werden, dass in dem Bereich der untersuchten Reynoldszahlen zwischen Re = 5 und Re = 180 der Einfluss inertieller Strömungseffekte mit dem Fließgesetz von Darcy-WardForchheimer beschrieben werden kann. Die zusätzlichen hydraulischen Verluste reduzieren die hydraulische Heterogenität der Einzelkluft, was zu dem makroskopisch beobachteten Effekt einer mit zunehmender Reynoldszahl abnehmenden Krümmung (Tortuosität) der Stromlinien führt. Ferner wurde experimentell gezeigt, dass die dimensionslose Ward-Konstante, welche in der Darcy-Ward-Forchheimer-Gleichung die sekundären Strömungseffekte charakterisiert, mit steigender Temperatur zunimmt. Dieses Verhalten ist bislang nur für Strömungen in porösen Medien bekannt..

(10) VIII. Auf Grundlage einer auf den Navier-Stokes-Gleichungen basierenden Skalenanalyse und den experimentell ermittelten Werten der Ward-Konstante kann ein einfacher Zusammenhang zwischen statistischen Parametern des Öffnungsweitenfeldes und der dimensionslosen Ward-Konstante abgeleitet werden. Im Bereich der Numerik ist die Entwicklung eines Algorithmus gelungen, der effizient die unentkoppelten Advektions-Diffusions-Reaktions-Gleichungen simuliert. Die Kopplung von kinetischen Reaktionen beliebiger Ordnung mit Transportprozessen geschieht durch die Einführung verallgemeinerter Reaktionen, die Transportprozesse durch den Austausch von Partikeln mit benachbarten Zellen abbilden. Die zeitliche Entwicklung. des. Systems. wird. durch. die. stochastische. Folge. einzelner. Reaktionsschritte simuliert, wobei der Zeitraum zwischen zwei aufeinander folgenden Reaktionen durch exponentiell verteilte Wartezeiten gegeben ist, welche im Mittel den Kehrwert der Gesamtreaktivität des Reaktionssystems betragen. Der Vorteil dieser Vorgehensweise ist, dass kein externer Zeitschritt vorgegeben werden muss, sondern sich das System auf der Zeitskala der beteiligten Prozesse entwickelt. Anhand von drei ausgewählten Beispielen ist die prinzipielle Anwendbarkeit des entwickelten Algorithmus auf komplexe Anwendungen gezeigt worden. Die gewonnenen Erkenntnisse zu Strömungsprozessen in Einzelklüften zusammen mit den im Rahmen dieser Dissertation entwickelten numerischen Methoden können in Zukunft auf reaktive Stofftransportprozesse in Kluft-Matrix-Systemen angewendet werden mit dem Ziel, die bisher an stark vereinfachten numerischen Modellen abgeleiteten nicht-linearen Zusammenhänge zwischen dimensionslosen Parametern der diskreten Modellierung und denen der Kontinuum Modellierung zu überprüfen..

(11) IX. Résume Ce travail est consacré à l'analyse des écoulements et du transport en milieux poreux géologiques (roches), naturellement hétérogènes et / ou fracturés. On s'intéresse en particulier à deux questions importantes pour les problèmes de migration de polluants toxiques (chimiques ou radioactifs): 1) Les écoulements rapides à haute Reynolds dans les fractures elles-mêmes, et 2) le transport des polluants réactifs à travers des milieux à la fois poreux et fracturés. Plusieurs types de méthodes sont employées : analyses théoriques. (Navier. Stokes,. lubrification,. champs. aléatoires) ;. expérimentales. (écoulement et traçage sur modèle physique de laboratoire) ; et numériques (transport des particules). La première partie de cette thèse étudie la configuration des écoulements monophoniques à travers une fracture rugueuse unique de granite en régime nonDarcien. (haute. Reynolds. 0 < Re < 180).. Cette. étude. est. menée. à. l'aide. d'expérimentations physiques en laboratoire. Différentes méthodes de mesure ont été développées pour déterminer à la fois le champ des hauteurs des surfaces en regard et le champ d'ouverture entre les deux faces de la fracture. L'étude expérimentale est complétée par des simulations numériques des écoulements non-Darciens à perte de charge quadratique, et par une analyse stochastique de rugosité de parois de la fracture à l'aide de la théorie des champs aléatoires. On s'intéresse plus particulièrement à la nonlinéarité de la loi de perte de charge globale à travers de la fracture et à la structure de l'écoulement (ligne de courant du traceur expérimental). Des visualisations directes ont montré l'influence de l'hétérogénéité du champ d'ouverture sur l'écoulement. Des mesures hydrauliques ont ensuite permis la mesure de la conductance hydraulique en faisant varier les effets inertiels. La comparaison de cette mesure avec les calculs obtenus à partir d'une loi cubique locale est discutée. Elle conduit à proposer une loi simple de perte de charge, de type quadratique, qui tient compte des effets inertiels et de la géométrie de champs d'ouverture . La seconde partie de cette thèse est consacrée au développement d'outils de modélisation numérique performants pour simuler le transport de polluants réactifs à travers divers types de milieux poreux hétérogènes ou fracturées (bloc fracturé). Les méthodes utilisées sont toutes basées sur la discrétisation des champs de concentration.

(12) X. en particules. Nous utilisons une approche Lagrangienne, qui traite des problèmes de transport advective-diffusive-reactive par décomposition d'opérateur et marche aléatoire sans grille pour la diffusion. Cette approche Lagrangienne est ensuite comparé à une méthode novatrice dite automate de Markov (Markov automaton), qui utilise une description microscopique du transport associée à une équation maîtresse. L'équation d'évolution en temps est alors décrite à l'aide d'un processus de Markov utilisant l'algorithme de Gillespie pour lequel tous les termes de réaction-advection-diffusion sont traités de la même manière. Dans cette méthode de type "particle in cell", l'espace est discrétisé, les pas de temps sont stochastiques, et les "réactions" généralisées sont traitées comme des réactions entre particules ("birth-death processes", processus de naissance et de mort, processus de temps d'attente de type Poisson, processus Markovien à accroissement indépendant). La comparaison entre les deux méthodes (Lagrangienne et automate de Markov) est menée et discutée sur trois exemples: 1. Le transport diffusif avec réaction cinétique linéaire de premier ordre (réaction hétérogène de type adsorption-desorption) 2. Diffusion pure bidimensionnelle dans un milieux poreux fracturé (système matricejoints) 3. Problème de réaction-diffusion non-linéaire (Fisher). Le comparaison entre les deux méthodes a montré que pour les problèmes fortement diffusives, l'automate de Markov est numériquement plus performant. Cette méthode semble la plus prometteuse grâce à sa flexibilité dans le cas de réactions complexes (hétérogènes, non-linéaires) et de géométries complexes (systèmes matrice / joints irréguliers). Cependant, pour les problèmes fortement advectifs, la méthode Lagrangienne reste une alternative très intéressante..

(13) Table of contents. XI. Table of contents Table of contents ........................................................................................................................ XI Figures......................................................................................................................................XVII Tables ......................................................................................................................................XXIII Notations ................................................................................................................................ XXIV 1. 2. Introduction .......................................................................................................................... 1 1.1. Context ........................................................................................................................ 1. 1.2. Definition of the problems under study......................................................................... 4. 1.3. Outline of the thesis ..................................................................................................... 6. 1.4. Structure of the thesis .................................................................................................. 6. Geometrical characterization and hydraulic models for rough single fractures: theory .................................................................................................................................... 9 2.1. Introduction and outline of chapter 2............................................................................ 9. 2.2. Elements of stochastic calculus applied to 2D random fields..................................... 10 2.2.1. 2.3. 2.4. Spatial-statistical averages, moments and structure functions...................... 10 2.2.1.1. One-point statistics, moments, and power averages ..................... 10. 2.2.1.2. Two-point statistics and structure functions ................................... 11. 2.2.2. Ensemble averages, spatial averages, and ergodicity .................................. 12. 2.2.3. Spatial-statistical averages, moments, and structure functions revisited ...... 13. 2.2.4. Estimation of the fluctuation scale................................................................. 14. 2.2.5. Preliminary considerations on structure functions versus real data............... 16. 2.2.6. Random fields: Gaussian and lognormal distributions .................................. 18. Geometrical and statistical characterization of single fractures.................................. 19 2.3.1. Statistical methods for the characterization of fracture surfaces ................... 19. 2.3.2. Statistical methods for the characterization of aperture fields ....................... 24. 2.3.3. Statistical relationships between surface and aperture field.......................... 28. From Navier-Stokes to Darcy-Ward-Forchheimer in rough fractures ......................... 32 2.4.1. 3D Navier-Stokes equations ......................................................................... 32. 2.4.2. 3D Stokes equations ..................................................................................... 33. 2.4.3. Lubrication approximation and pseudo-3D Reynolds equations ................... 34.

(14) XII. 2.5. 3. 2.4.4. Vertically integrated lubrication equations: 2D Reynolds-Darcy equations or local 'Cubic Law'....................................................................... 36. 2.4.5. From 2D Reynolds-Darcy to 2D Darcy-Ward-Forchheimer........................... 38 The Darcy-Ward-Forchheimer model for 3D isotropic media......... 38. 2.4.5.2. A 2D Darcy-Ward-Forchheimer model for rough fractures ............ 40. 2.4.5.3. Scale analysis of Navier-Stokes equations: From NavierStokes to Ward's law ..................................................................... 43. Darcian flow statistics for a random fracture in 2D..................................................... 45 2.5.1. Effective conductivity for 2D Darcian flows ................................................... 46. 2.5.2. Spectral perturbation solutions for 2D isotropic Darcian flows ...................... 48. Single fracture geometry: measurements and statistics................................................ 49 3.1. Introduction and outline of the structure of chapter 3 ................................................. 49. 3.2. Choice of fracture type and discussion of requirements for experimental study ........ 49. 3.3. Measurement of topology and aperture field.............................................................. 50 3.3.1. 3.3.2. 3.3.3. 3.4. 4. 2.4.5.1. Literature review ........................................................................................... 51 3.3.1.1. Measurement of surface elevation................................................. 51. 3.3.1.2. Measurement of aperture field....................................................... 52. The dye absorption method .......................................................................... 55 3.3.2.1. The measurement of a normalized aperture field .......................... 55. 3.3.2.2. From the normalized to the actual aperture field............................ 57. 3.3.2.3. Discussion of accuracy and possible error sources ....................... 57. The 3D-scan method..................................................................................... 61 3.3.3.1. Numerical assembly: from point measurements of surface elevation to aperture fields............................................................. 62. 3.3.3.2. Discussion of the accuracy and possible error sources ................. 64. Statistical analysis of discrete topology and aperture data......................................... 65 3.4.1. Statistical analysis of surface topology.......................................................... 65. 3.4.2. Statistical analysis of aperture fields ............................................................. 73. 3.5. Interrelation of aperture field measurement and flow model ...................................... 91. 3.6. Summary and evaluation of measurement techniques .............................................. 93. Single fracture flow and transport experiment: set up and measurements.................. 94 4.1. Introduction and outline of the structure of chapter 4 ................................................. 94. 4.2. The experimental setup ............................................................................................. 94.

(15) Table of contents. 4.3. 4.2.1. The measurement of hydraulic head-loss ..................................................... 96. 4.2.2. The tracer injection........................................................................................ 97. Flow experiment: measurement of transmissivity for different Reynolds numbers..................................................................................................................... 97 4.3.1. 4.3.2 4.4 5. Influence of the Reynolds number on fracture transmissivity ........................ 99 4.3.1.1. Application of the Ergun approach to flow through a single fracture .......................................................................................... 99. 4.3.1.2. Measurement errors .................................................................... 100. 4.3.1.3. Results of the flow experiments................................................... 102. Influence of temperature on the dimensionless Ward constant................... 104. Transport experiment: alteration of streamlines with different Reynolds numbers ... 105. Rough fracture flow: analyses and comparisons of experimental results and numerical models............................................................................................................. 108 5.1. Introduction and outline of the structure of chapter 5 ............................................... 108. 5.2. Simulating the Darcy-Ward-Forchheimer equations using the BIGFLOW code....... 108. 5.3. Numerical experiment with random aperture fields .................................................. 109. 5.4. Comparison of numerical and experimental results ................................................. 111. 5.5. 5.6 6. XIII. 5.4.1. Comparison of numerically and experimentally obtained streamlines......... 112. 5.4.2. Comparison of numerically and experimentally obtained hydraulic aperture....................................................................................................... 117. 5.4.3. Comparison of numerical and experimental results for larger Reynolds numbers ...................................................................................................... 122. Prediction of the Ward constant for single fractures................................................. 129 5.5.1. Estimation of the Ward constant using the single-step model of BODARWÉ (1999).......................................................................................... 129. 5.5.2. Estimation of the Ward constant based on experimental results and scaling analysis........................................................................................... 131. Summary ................................................................................................................. 133. Advective-diffusive-reactive transport with particle methods: modeling and analysis............................................................................................................................. 136 6.1. Introduction and outline of the structure of chapter 6 ............................................... 136. 6.2. Overview of simulation of reactive transport ............................................................ 136. 6.3. Review and outline of numerical approaches to reactive transport .......................... 137 6.3.1. The local equilibrium assumption and a heuristic criterion for its applicability ................................................................................................. 139.

(16) XIV. 6.3.2 6.4. Lagrangian particle tracking transport with random walk diffusion (LPT3D code).... 142 6.4.1. 6.5 7. Kinetic reactions: two stochastic approaches for reactive transport developed and applied in this work ............................................................. 141 Mass transport with heterogeneous diffusion: interpolation schemes for random walks.............................................................................................. 144 6.4.1.1. Particle reflection techniques....................................................... 145. 6.4.1.2. Interpolation techniques............................................................... 148. 6.4.2. Proposed quadratic interpolation scheme for random walk in fields of heterogeneous diffusion.............................................................................. 148. 6.4.3. Results and comparison: two blocks with a sharp D(x) interface ................ 152. 6.4.4. Summary of Lagrangian random walk methods with discontinuous diffusion coefficients.................................................................................... 154. Coupling of Lagrangian transport and reactions ...................................................... 155. The Gillespie algorithm for the simulation of reactions: Markov processes and Master equation................................................................................................................ 158 7.1. Introduction and outline of the structure of chapter 7 ............................................... 158. 7.2. General remarks on the modeling of reaction systems: introduction to the theory of Markov processes and to the Master equation.......................................... 158. 7.3. 7.4. 7.5. 7.2.1. The interrelationship between Markov processes and the Poisson point process ....................................................................................................... 159. 7.2.2. The Master equation ................................................................................... 161. A mathematical derivation of the Gillespie algorithm ............................................... 163 7.3.1. Relationship between collision parameter and reaction rate constant......... 164. 7.3.2. Formulation of the Master equation............................................................. 167. 7.3.3. Derivation of the reaction probability density function ................................. 168. Outline of the implementation of the Gillespie algorithm .......................................... 170 7.4.1. Generation of pseudo-random numbers ..................................................... 170. 7.4.2. Summary of the characteristics of the Gillespie algorithm........................... 172. The simulation of chemical reactions using the example of radioactive decay and kinetic sorption .................................................................................................. 173 7.5.1. Radioactive decay and the Poisson point process...................................... 173. 7.5.2. Kinetic sorption ........................................................................................... 175 7.5.2.1. Description of the problem........................................................... 176. 7.5.2.2. Sorption kinetics: a stochastic model........................................... 177.

(17) Table of contents. 8. The Markov automaton SMART2D: reactive transport modeling based on the Master equation................................................................................................................ 181 8.1. Introduction and outline of the structure of chapter 8 ............................................... 181. 8.2. Modeling of reactive transport with the Markov automaton: a mathematical derivation ................................................................................................................. 181. 8.3. 8.4. 9. XV. 8.2.1. A Master equation formulation of one-dimensional A-D-R processes ......... 182. 8.2.2. Equivalence of the macroscopic A-D-R equation and the Master equation ...................................................................................................... 184. Implementation of stochastic transitions .................................................................. 188 8.3.1. Progression of time ..................................................................................... 189. 8.3.2. The selection of the next reaction ............................................................... 189 8.3.2.1. The inverse CDF algorithm and linear selection .......................... 190. 8.3.2.2. The Von Neumann rejection algorithm ........................................ 192. 8.3.2.3. The logarithmic class algorithm ................................................... 193. Extensions of the concept of the Markov automaton ............................................... 196 8.4.1. Extension of the Markov automaton to heterogeneous two-dimensional domains ...................................................................................................... 196. 8.4.2. Extension of the Markov automaton for coping with grid Peclet numbers greater than two .......................................................................................... 200. 8.4.3. Verification tests.......................................................................................... 203. 8.4.4. Summary..................................................................................................... 205. Simulations of A-D-R transport: applications, analyses and comparisons ................ 206 9.1. Introduction and outline of the structure of chapter 9 ............................................... 206. 9.2. Application 1: modeling of diffusive transport with linear adsorption and desorption ................................................................................................................ 206. 9.3. Application 2: 2D simulation of highly heterogeneous joint / matrix diffusion ........... 210. 9.4. 9.5. 9.3.1. Formulation of the problem and simulations ............................................... 212. 9.3.2. Comparison of simulation results and evaluation of numerical methods..... 214. Application 3: simulation of the Fisher’s equations: a nonlinear reactiondiffusion problem...................................................................................................... 218 9.4.1. Simulation of Fisher’s equation with the Lagrangian approach using an operator-splitting technique for the reaction (GILLPT3D)............................ 218. 9.4.2. Simulation of Fisher’s equation via stochastic Markov processes (SMART2D) ................................................................................................ 221. Discussion and comparison ..................................................................................... 222.

(18) XVI. 10 Conclusion and outlook .................................................................................................. 225 10.1 Accurate measurement of aperture fields ................................................................ 225 10.2 Physical and numerical flow experiments ................................................................ 226 10.3 Modeling of reactive transport.................................................................................. 228 10.4 Outlook .................................................................................................................... 230 References ............................................................................................................................... 233 A1 Manufacturing of transparent plexiglass replica of a real granite fracture................. 249 A1.1 The making of a mould for casting the fracture surface ........................................... 249 A1.2 The casting of the rock fracture using silicone rubber.............................................. 250 A1.3 The reproduction of the mating fracture surface casting the silicone rubber mould ....................................................................................................................... 251 A1.4 Casting the silicone rubber moulds with epoxy resin ............................................... 252.

(19) Figures. XVII. Figures Fig. 1.1: Fig. 2.1:. Structure of the thesis. ................................................................................................. 7 Distributions with significant deviations from the Gaussian distribution characterized by the parameters skewness (left) and kurtosis (right) (PRESS et al., 1994)................................................................................................................ 11 Fig. 2.2: Characterization of the fracture surfaces. .................................................................. 20 Fig. 2.3 Characterization of the aperture field. ........................................................................ 24 Fig. 2.4: Outline of different methods for the definition of a local 'Cubic Law' aperture. The different definitions are plotted as follows: the vertical aperture: long dashed lines, the aperture defined by MOURZENKO et al. (1995): straight line, the aperture defined by GE (1997): dots, and the method by ORON & BERKOWITZ (1998): bold dashes. Figure modified from ORON & BERKOWITZ (1998). ....................................................................................................................... 26 Fig. 2.5: Plot of normalized covariance of aperture field as a function of normalized lag and normalized shear displacement based on the assumption of identical stationary and isotropic fracture surfaces having a Gaussian correlation function..................................................................................................... 31 Fig. 2.6: Calculation of the bounds of CARDWELL & PARSONS (1945) for flow along the x-axes. ....................................................................................................................... 47 Fig. 3.1: Image of the observation field with the stepped density tablet attached to the fracture cell (on the right hand side) and the fracture filled with a dye solution (HAMELMANN, 2001)....................................................................................... 55 Fig. 3.2: Error sources of the measurement system. Picture modified from HAMELMANN (2001)..................................................................................................... 58 Fig. 3.3: Possible overestimation of aperture due to the geometry of the measurement system (HAMELMANN, 2001)................................................................. 60 Fig. 3.4: Sketch of the setup of the fringe-projection sensor (BERGMANN et al., 1997) The sensor head basically consists of a projection unit and two digital, high-resolution CCD cameras (12bit, 1280 u 1024 pixels) whose optical axes are in the same plane (left). Periodically projected equidistant fringes on a car model (GOM, right). Pictures taken from HAMELMANN (2001)....................... 62 Fig. 3.5: Elevation of the surface for both fracture surfaces..................................................... 65 Fig. 3.6: Histogram of the surface elevation for both fracture surfaces and Gaussian fit................................................................................................................................ 66 Fig. 3.7: Covariogram of the surface elevation sampled in the direction of the possible translation (x-direction) based on the whole domain and a subdomain of size 15 cm u 12 cm. ............................................................................ 67 Fig. 3.8: Covariogram of the surface elevation sampled perpendicular to the direction of the possible translation (y-direction) based on the whole domain and a subdomain of size 15 cm u 12 cm. ............................................................................ 67 Fig. 3.9: Scale of fluctuation of the fracture surfaces in x-direction (left) and ydirection (right) based on the approach of MESA & POVEDA (1993). The scale of fluctuation is plotted on a logarithmic scale............................................................ 68 Fig. 3.10: Averaged power spectra of elevation profiles of the two fracture surfaces. ............... 69 Fig. 3.11: Results obtained with the windowing method: root mean square roughness (left) and maximum elevation difference (right) vs. normalized windowing size. ........................................................................................................................... 69.

(20) XVIII. Fig. 3.12: Histogram of the gradients of surface elevation for both surfaces, where x denotes the direction of possible translation and y is perpendicular to this direction. .................................................................................................................... 71 Fig. 3.13: Aperture fields for different displacements obtained by the dye method. ................... 73 Fig. 3.14: Aperture fields for different displacemets obtained by the 3D-scan method. ............. 74 Fig. 3.15: Bias of aperture obtained with the 3D-scan method: sketch of the problem. Circles and squares mark location of exactly measured surface coordinates. ........... 75 Fig. 3.16: Comparison of a cloud of tracer injected into the fracture aperture for 7.2 mm displacement (top) and the aperture field obtained with the 3D-scan method (bottom). Regions where apertures are artificially set to a small value (i.e. 0.05 mm) are framed by black lines........................................................... 76 Fig. 3.17: Comparison of the aperture histogram and the fitted Gaussian for four different displacements and both aperture measurement techniques (3Dscan on the left and dye method on the right) based on the whole aperture field. ........................................................................................................................... 78 Fig. 3.18: Standard deviation of the aperture versus the displacement of the fracture surfaces, obtained by the 3D-scan method and by the dye method. In addition, results of theoretical estimates of standard deviation of the aperture are shown. ................................................................................................... 80 Fig. 3.19: Covariogram sampled in x-direction based on the whole aperture field..................... 81 Fig. 3.20: Covariogram sampled in y-direction based on the whole aperture field..................... 82 Fig. 3.21: Covariogram sampled in x-direction based on a 8 cm u 8 cm subdomain of the aperture field. ....................................................................................................... 83 Fig. 3.22: Covariogram sampled in y-direction based on a 8 cm u 8 cm subdomain of the aperture field. ....................................................................................................... 83 Fig. 3.23: Scale of fluctuation in x-direction (left) and y-direction (right) based on the approach of MESA & POVEDA (1993). The scale of fluctuation is plotted on a logarithmic scale. ....................................................................................................... 86 Fig. 3.24: Averaged power spectra of the aperture fields for the four displacement configurations studied in a log-log plot. The average was taken over 433 slices of length 512'x, where the discretization length was 'x = 0.3853 mm. ....................................................................................................... 87 Fig. 3.25: Results obtained with the windowing method: root mean square roughness (left) and maximum elevation difference (right) vs. normalized windowing size. ........................................................................................................................... 88 Fig. 3.26: Comparison of the correlation scales obtained by the covariogram analysis, the estimator of MESA & POVEDA (1993), and the fluctuation scale obtained by fractal analysis. ..................................................................................................... 89 Fig. 3.27: Histogram of gradients of aperture field obtained by the 3D-scan method (top) and the dye method (bottom) for the 7.2 mm displacement in the direction of shear displacement (x-direction, left) and perpendicular to the direction of shear displacement (y-direction, right)..................................................... 91 Fig. 4.1: The fracture-flow apparatus operates in conjunction with a re-circulating water supply system. It is composed of an upstream and a downstream area with an intermediate observation field........................................................................ 95 Fig. 4.2: The principle of hydraulic head measurement. .......................................................... 96 Fig. 4.3: The principle of tracer injection. ................................................................................. 97.

(21) Figures. Fig. 4.4:. XIX. Effective hydraulic aperture versus Reynolds number for zero displacement (T | 21 °C). .............................................................................................................. 102 Fig. 4.5: Effective hydraulic aperture versus Reynolds number for 3.5 mm shear displacement (T | 21 °C). ........................................................................................ 103 Fig. 4.6: Effective hydraulic aperture versus Reynolds number for 7.2 mm shear displacement (T | 21 °C). ........................................................................................ 103 Fig. 4.7: Effective hydraulic aperture versus Reynolds number for 7.2 mm shear displacement and three different water temperatures. ............................................. 104 Fig. 4.8: Change of streamlines with different Reynolds numbers. The direction of flow is from right to left; the section shown has a length of about 30 cm.................. 106 Fig. 4.9: Multiple crossing of streamlines at Re = 80. The direction of flow is from right to left; the section shown has a length of about 30 cm..................................... 107 Fig. 5.1: Sketch of the boundary conditions applied in the numerical simulations.................. 111 Fig. 5.2: Simulated streamlines in the fracture obtained by 0 mm shear displacement (top) and 3.5 mm shear displacement (bottom). Flow is from right to left................. 113 Fig. 5.3: Simulated streamlines in the fracture obtained by 7.2 mm shear displacement (top) and 10.8 mm shear displacement (bottom). Flow is from right to left. ............................................................................................................... 114 Fig. 5.4: Normalized distribution of Reynolds number in the fracture plane for different shear displacements. ................................................................................. 115 Fig. 5.5: Streamlines in the fracture obtained by 7.2 mm shear displacement: comparison of numerically obtained streamlines for Darcy-flow and streamlines obtained experimentally at Re = 20. Flow is from right to left. The length of the section of the flow field shown in this figure is about 20 cm in the direction of flow. ............................................................................................. 116 Fig. 5.6: Dimensionless transmissivities for three different dimensionless correlation lengths O/Vh (O/Vh = 1, 2, and 3) following MOURZENKO et al. (1995). In this case, the quotient of the transmissivity BS obtained by a simulation of the Stokes equation over the 'Cubic Law' transmissivity Bp1 based on the arithmetic mean aperture is denoted by dashed lines. Dotted lines show this quotient for the case of simulating the Reynolds equations. The experimentally obtained values are added in this figure as follows: a) for 0 mm displacement () O/Vh = 2.2, bm/Vh = 0.35 and a quotient of the transmissivities BExp/Bp1 = 0.51, b) for 3.5 mm displacement (▲) O/Vh = 1.8, bm/Vh = 0.99, and BExp/Bp1 = 0.31, and c) for 7.2 mm displacement (▼) O/Vh = 1.8, bm/Vh = 1.16, and BExp/Bp1 = 0.4. ............................................................. 119 Fig. 5.7: Streamlines for the 0 mm displacement configuration obtained by simulations based on the Darcy approach (gray) and Darcy-WardForchheimer (black) for Re = 149. ........................................................................... 122 Fig. 5.8: Streamlines for the 3.5 mm displacement configuration obtained by simulations based on the Darcy approach (gray) and Darcy-WardForchheimer (black) for Re = 135. ........................................................................... 123 Fig. 5.9: Streamlines for the 7.2 mm displacement configuration obtained by simulations based on the Darcy approach (gray) and Darcy-WardForchheimer (black) for Re = 92. ............................................................................. 123 Fig. 5.10: Comparison of the alteration of experimentally and numerically obtained streamlines with Re using the example of the 7.2 mm displacement configuration. ........................................................................................................... 124.

(22) XX. Fig. 5.11: Effect of Re on tortuosity of the numerically obtained velocity field for the three different configurations studied in the experiment........................................... 125 Fig. 5.12: Effect of Re on statistical properties of the velocity field for 0 mm displacement configuration. ..................................................................................... 126 Fig. 5.13: Effect of Re on statistical properties of the velocity field with Re for 3.5 mm displacement configuration. ..................................................................................... 126 Fig. 5.14: Effect of Re on statistical properties of the velocity field with Re for 7.2 mm displacement configuration. ..................................................................................... 126 Fig. 5.15: Effect of Re on the effective hydraulic aperture for the three different configurations studied in the experiment (1/(aeff)³ vs. Re): comparison between experiment and simulation......................................................................... 127 Fig. 5.16: Effect of Re on the effective hydraulic aperture for the three different configurations studied in the experiment (aeff vs. Re): comparison between experiment and simulation. ...................................................................................... 128 Fig. 5.17: Single-step model of BODARWÉ (1999). Picture modified from BODARWÉ et al. (1997). ..................................................................................................................... 129 Fig. 5.18: Prediction of the effective Ward coefficient using the single-step model of BODARWÉ (1999). ..................................................................................................... 131 Fig. 5.19: Scaling of the experimentally obtained Ward constant with mean geometric aperture over correlation length O. Results are shown for the three configurations with the granite fracture presented in this work and one flow experiment with a schist fracture sample studied in KÖNGETER et al. (2000a). ........ 132 Fig. 6.1: Schematic of fracture-matrix interaction. Picture modified from XU & WÖRMAN (1999)........................................................................................................ 137 Fig. 6.2: Classification of chemical reactions in solute transport systems. Picture modified from RUBIN et al. (1983)............................................................................. 138 Fig. 6.3: Stratified system with discontinuity of velocity and dispersion at x = 0. Picture modified from LABOLLE et al. (1996)............................................................. 145 Fig. 6.4: Quadratic interpolation of D(x) in case of an abrupt change of diffusion coefficient (two domains with diffusion coefficients D1 and D2). The quadratic interpolation (straight line) between D1 and D2 , which is performed in the transition zone, leads to a piecewise-linear change of the drift velocity (dashed line). Picture modified from SPILLER et al. (2002). ...................................... 149 Fig. 6.5a,b: 1D Concentration distributions for a) correct choice of time step and b) too large time step. Here, the discretization length is chosen as 'x = 2 (SPILLER et al., 2002).............................................................................................................. 153 Fig. 6.6a,b: Concentration distribution for D1=4 [L2/T], D2=1 [L2/T] and 'x0=0.02 [L] at 5 different times: ■ : 0.05 sec, ▲ : 0.1 sec, ▼ : 0.25 sec, ◆ : 0.5 sec, and ● : 1 sec. (SPILLER et al., 2002). ............................................................................. 154 Fig. 6.7: Different operator-splitting schemes (SPILLER et al., 1998). ..................................... 156 Fig. 7.1: Radioactive decay with O = 1 [1/T]: normalized concentration vs. time. .............. 174 Fig. 7.2: Radioactive decay with O = 1 [1/T]: deviation from analytical solution vs. time.......................................................................................................................... 174 Fig. 7.3: Scheme of the phase transition within the time interval 't (SPILLER et al., 1998)........................................................................................................................ 178 Fig. 8.1: Possible transitions of state in an advection-diffusion-adsorption / desorption system. .................................................................................................................... 183 Fig. 8.2: Possible transitions of the chosen A-D-R process. .................................................. 184.

(23) Figures. Fig. 8.3:. Fig. 8.4 Fig. 8.5 Fig. 8.6 Fig. 8.7: Fig. 8.8: Fig. 8.9: Fig. 8.10: Fig. 8.11: Fig. 9.1: Fig. 9.2: Fig. 9.3: Fig. 9.4: Fig. 9.5: Fig. 9.6: Fig. 9.7: Fig. 9.8: Fig. 9.9: Fig. 9.10: Fig. 9.11: Fig. 9.12:. XXI. Flow chart for the simulation of one reaction step with the Gillespie algorithm. The two processes, i.e., progression in time and change of the system’s configuration space are depicted using the example of a onedimensional A-D-R process. .................................................................................... 188 Sketch of the linear selection of a cell D. Picture modified from FRICKE & WENDT (1995). ......................................................................................................... 192 Sketch of the Von Neumann rejection method for the selection of a cell. Picture modified from FRICKE & WENDT (1995)......................................................... 193 Sketch of the method of logarithmic classes for the selection of a cell. Picture modified from FRICKE & WENDT (1995)......................................................... 194 Two-dimensional finite volume grid for homogeneously discretized domain (BECKER, 2003). ....................................................................................................... 196 Splitting of the grid into several subcells (gray background). Here the number of subcells n is n = 4. Picture modified from BECKER (2002)........................ 201 Grid types and subgrids: grid with centered nodes (left) and centered interfaces (right). Subcells are represented by dashed lines. Picture modified from BECKER (2002)................................................................................... 202 Simulation of one-dimensional advective-diffusive transport with a grid Peclet number of 1 (top) and 100 (bottom). Picture modified from BECKER (2002). ..................................................................................................................... 204 Simulation of one-dimensional diffusive transport with SMART2D using the same parameters as for the verification test of the code GILLPT3D. Picture modified from BECKER (2002)................................................................................... 204 Simulated residence time distribution and analytic marginal distribution for ¢Npt('t)² = 2. ............................................................................................................ 207 Simulated residence time distribution and analytic marginal distribution for ¢Npt('t)² = 20. .......................................................................................................... 208 Time evolution of the retardation factor for different Damköhler 2 numbers............. 209 Comparison of the time evolution of the retardation factor obtained with GILLPT3D and SMART2D....................................................................................... 210 Microstructures of the NiO layer generated by oxidation of nickel at 700 ° C (from left upper corner to right lower corner: layer thickness is 1 µm, 3 µm, 6 µm, 10 µm). The pictures were provided by D. MONCEAU. ................................... 211 Schematic outline of the nickel oxide layer. Picture modified from MONCEAU et al. (2000).............................................................................................................. 212 Two-dimensional finite-differences net: channel-like structures represent grain boundaries and matrix-like structures represent the grains (SPILLER et al., 2002).................................................................................................................. 213 Concentration distribution for t = 5.3 [h] obtained with GILLPT3D. .......................... 213 Concentration distribution for t = 5.3 [h] obtained with SMART2D. .......................... 214 Normalized concentration versus time obtained with the two codes GILLPT3D and SMART2D. This figure shows a zoom on the first four hours of the simulation....................................................................................................... 215 Normalized concentration versus time obtained with the two codes GILLPT3D and SMART2D. This figure shows the evolution of the concentration until steady-state is reached.............................................................. 216 Concentration distribution for ten different times obtained with SMART2D.............. 217.

(24) XXII. Fig. 9.13: Advance of the concentration wave for four different times without a mass refinement at the concentration front. Picture modified from SPILLER et al. (1998). ..................................................................................................................... 219 Fig. 9.14: Wave speed without mass refinement. Picture modified from SPILLER et al. (1998). ..................................................................................................................... 219 Fig. 9.15: Advance of the concentration wave for four different times with a mass refinement at the concentration front. Picture modified from SPILLER et al. (1998). ..................................................................................................................... 220 Fig. 9.16: Wave speed for three different time discretizations with mass refinement. Picture modified from SPILLER et al. (1998).............................................................. 220 Fig. 9.17: Possible transitions of state in a reaction-diffusion system described by Fisher’s equation (SPILLER et al., 2000). .................................................................. 221 Fig. 9.18: Velocity of the concentration wave for different mass resolutions obtained with the Markov automaton code SMART2D (Spiller et al., 2000). .......................... 222 Fig. A1.1: Sketch (left) and photograph (right) of the wooden mould with the embedded granite fracture surface (HAMELMANN, 2001). ........................................................... 250 Fig. A1.2: Images of the transparent fracture cell (disassembled and reassembled), with elongated holes and locating holes in the lateral borders and the roughwalled fracture field in the center. (HAMELMANN, 2001)............................................. 233.

(25) Tables. XXIII. Tables Table 2.1: Normalized covariance functions and their correlation scale or fluctuation scale. ..................................................................................................................... 15 Table 2.2: Different combinations of the macroscopic statistical parameters mean aperture a , standard deviation of aperture V and correlation length scale O............ 25 Table 3.1: Overview of statistical parameters of the fracture surfaces........................................ 70 Table 3.2: Overview of statistical parameters of the gradient of the fracture surfaces. The values in brackets give the result of the statistical analysis of the slopes. .......... 72 Table 3.3: Overview of statistical parameters of the aperture fields obtained by the 3D-scan method and by the dye method. .................................................................. 79 Table 3.4: Overview of results of the covariogram analysis for zero displacement..................... 84 Table 3.5: Overview of results of the covariogram analysis for 3.5 mm displacement ................ 84 Table 3.6: Overview of results of the covariogram analysis for 7.2 mm displacement ................ 85 Table 3.7: Overview of results of the covariogram analysis for 10.8 mm displacement. ............. 85 Table 3.8: Overview of the identified correlation scales and fluctuation scales of the aperture fields. The values in brackets denote ambiguous estimates........................ 86 Table 3.9: Roughness exponents for the four displacement configurations studied, based on the results obtained with the windowing method. ....................................... 88 Table 3.10: Overview of statistical parameters of the gradient of the aperture fields obtained by the 3D-scan method and by the dye method. The values in brackets give the result of the statistical analysis of the slopes.............................. 90 Table 3.11: Comparison of the two measurement techniques for the determination of the aperture field applied in this thesis............................................................................. 93 Table 4.1: Overview of the results obtained from the flow experiments. ................................... 104 Table 4.2: Overview of the results obtained for the 7.2 mm displacement at three different temperatures.............................................................................................. 105 Table 5.1: Numerical study on the influence of spatial discretization on the flow field. ............. 110 Table 5.2: Summary of simulation results obtained with BIGFLOW.......................................... 112 Table 5.3: Comparison of different estimates for the effective hydraulic aperture based on different types of averaging of the discrete heterogeneous aperture field obtained with the two different methods (3D-scan and dye method). ...................... 117 Table 5.4: Summary of global statistical parameters for the fracture’s aperture field. The different configurations are marked with different background colors. .............. 120 Table 5.5: Comparison of different empirical estimates for the effective hydraulic aperture. The same colors as in Table 5.4 are used to distinguish between the different configurations, i.e., dark gray for 0 mm displacement, light gray for 3.5 mm displacement, and white for 7.2 mm displacement. ............................... 121 Table 5.6: Values for the proportional factor F(E,Re) obtained in BODARWÉ (1999).................. 130 Table 5.7: Comparison of the mean value of the Ward constant c obtained with the single-step model of BODARWÉ (1999), the proposed scaling approach and the experimentally obtained Ward constant. ............................................................ 132 Table 6.1: Selection of reflection approaches for D1 > D2. ........................................................ 146 Table 7.1: Selected publications of application examples of the Gillespie algorithm................. 162 Table 7.2: Different combinations of a particle’s state at the beginning and the end of a time step. ................................................................................................................. 178.

(26) XXIV. Notations Coordinates, indices and operators. x, y, z t i, j P, I D U, D, L, R I Ig. [L] [T] [-] [-] [-] [-]. global coordinates time indices indices of cells cell index labels upper, lower, left and right neighbor cell arithmetic mean geometric mean. Ih Ip. harmonic mean power average. G ' w / wx i D/Dt 2. small displacement difference between two scalars gradient partial differentiation total differentiation Laplace operator. Scalars (Latin letters). A ADWF ADWF,Flux AFluid AR a ac,min ac,max aeff ah BDWF BDWF,Flux b C Cdye c cP D Da1 Da2 dh g. [-] [L-1T] [L-2T] [L2] [M-1T-1] [L] [L] [L] [L] [L] [L-2T2] [L-4T2] [L] [-] [ML-3] [-] [T-1] [L2T-1] [-] [-] [L] [LT-2]. absorbance of the solute coefficient of linear term in Darcy-Ward-Forchheimer law coefficient of linear term in Darcy-Ward-Forchheimer law area occupied by fluid transition rate coefficients arithmetic mean of the aperture field Cardwell bounds of the aperture field Cardwell bounds of the aperture field effective mean aperture of the fracture harmonic mean of the aperture field coeff. of quadratic term in Darcy-Ward-Forchheimer law coeff. of quadratic term in Darcy-Ward-Forchheimer law fracture width constant of Ergun concentration of the dye Ward constant collision parameter diffusion coefficient Damköhler 1 number Damköhler 2 number hydraulic diameter gravitational constant.

(27) Notations. H h hP I I0 J K Keff k k kP kd ks ks L Li Lz lQ M N Ndim Ni ni Pe p ~ p Q R R RpP R1 R(D) RA(D) RC(D) RD(D) RR(D) RP Re r s s i( P ) T T*a T*d ta td texit. XXV. [-] [L] [-] [MT-3] [MT-3] [-] [LT-1] [LT-1] [L-1] [L2] [T-1] [-] [T-1] [T-1] [L] [L] [-] [L] [M] [-] [-] [-] [M-1] [-] [ML-1T-2] [-] [L3T-1] [T-1] [-] [-] [-] [T-1] [T-1] [T-1] [T-1] [T-1] [T-1] [-] [-] [ML-3] [-] [L2T-1] [-] [-] [T] [T] [T]. roughness exponent hydraulic head different arrangements of reaction partners light intensity initial light intensity hydraulic gradient hydraulic conductivity effective hydraulic conductivity wave number Darcy permeability reaction rate constant equilibrium distribution coefficient adsorption rate desorption rate length scale spatial extension of the surface in i-direction logarithmic class characteristic length scale of the viscous forces mass number of events dimension number of particles of species Xi molecular concentration Peclet number pressure normalized pressure discharge rate total reaction rate retardation coefficient reaction product of the reaction P reflection coefficient total reaction rate in cell D advection rate of cell D cumulative transition rate of cell 1 to D diffusion rate of cell D reaction rate of cell D transition or reaction rate Reynolds number random number concentration of the immobile phase stoichiometric coefficient of species i in reaction P transmissivity dimensionless transition time for dominant advection dimensionless transition time for diffusion dominant characteristic time for advection characteristic time for diffusion time at which particle leaves transition zone.

(28) XXVI. U0 u, v, w ~ ~ u, ~ v, w ud w 2 (/). [LT-1] [LT-1] [-] [L] [L]. characteristic velocity velocities in x, y, z direction normalized velocities in x, y, z direction displacement length standard deviation of the elevations in a window of size /. w f (/) Xi. [L] [-]. difference between the maximum and minimum elevation species “i”. Scalars (Greek letters). E 'x 't G* H N O OV OM P P Pdye Q T U Vqx W : Z [. [-] [L] [T] [L] [-] [-] [L] [L] [L] [ML-1T-1] [-] [M-1L2] [L2T-1] [-] [ML-3] [LT-1] [T] [ LN dim ] [ LN dim ] [L]. factor of aperture change discretization length time step thickness of the viscous layer small parameter dimensionality correlation length or scale of fluctuation scale of fluctuation unbiased estimator of fluctuation scale dynamic viscosity reaction absorptivity of dye kinematic viscosity porosity density standard deviation of the velocity in x direction time interval or waiting time total volume in Ndim dimensions subvolume in Ndim dimensions lag. Tensors & N & q & v & v drift & [ Dij Kij Tij. [-] [LT-1] [LT-1] [LT-1] [L] [L2T-1] [LT-1] [L2T-1]. different states of the system flux or filter velocity velocity drift velocity lag dispersion tensor effective hydraulic conductivity effective hydraulic transmissivity.

(29) Notations. XXVII. Functions. a(x,y) & C( [ ) & c( x ) & D( x ) & D*( x ) E{.} floor(arg) F(E,Re) H(.) h 0r ( x, y) h±(x,y) hd(ud) K(x,y) & R( [ ) T(x,y) z±(x,y) & GD( x ) )(.) I(z)Binomial & * [

(30) <.> Abbreviations ADR PDF PDE RMS SDE. [L] [ML-3] [L2T-1] [L2T-1] [-] [-] [-] [L] [L] [L] [L2] [-] [L2T-1] [L] [L2T-1] [-] [-] [-]. local aperture field covariance function concentration diffusion coefficient interpolated diffusion coefficient expectation value largest integer that is not greater than the function argument proportional factor Heaviside step function surface mean plane fluctuations around the mean surface vertical displacement of upper wall local permeability normalized covariance function local fracture transmissivity surface elevation of upper (+) and lower surface (-) discrepancy between real diffusion and interpolated one distribution function probability generating function of Binomial distribution variance function spatial average. advection-diffusion-reaction probability density function partial differential equation root mean square stochastic differential equation.

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