Subsurface Solute Transport Modelsand Case HistoriesTheory and Applications of Transport in Porous MediaSeries Editor:Jacob Bear, Department of Civil and Environmental Engineering,Technion Israel Institute of Technology, Haifa, and School of Engineering,Kinneret College on the Sea of Galilee, IsraelVolume 25For further volumes:http://www.springer.com/series/6612Vyacheslav G. RumyninSubsurface Solute TransportModels and Case HistoriesWith Applications to Radionuclide MigrationABCVyacheslav G. RumyninThe Russian Academy of SciencesInstitute of Environmental GeologySaint Petersburg DivisionSaint Petersburg State UniversityGeological DepartmentSredniy Ave., 41, of 519199004 St. PetersburgRussian Federationrumynin@hge.pu.ruThisbookisarevisedandupdatedversionofthebookinRussianGeomigracionnyemodeli v gidrogeologii(GeomigrationModels withApplicationtoGroundwaterHydrology), byVyacheslavG. Rumynin, publishedbyNauka (Science) Publisher,St. Petersburg, 2009, ISBN 978-5-02-025140-3.ISBN 978-94-007-1305-5 e-ISBN 978-94-007-1306-2DOI 10.1007/978-94-007-1306-2Springer Dordrecht Heidelberg London New YorkLibrary of Congress Control Number: 2011930258cSpringer Science+Business Media B.V. 2011No part of this work may be reproduced, stored in a retrievalsystem, or transmitted in any form or byany means, electronic, mechanical, photocopying, microlming, recording or otherwise, without writtenpermission from the Publisher, with the exception of any material supplied specically for the purposeof being entered and executed on a computer system, for exclusive use by the purchaser of the work.Cover design: deblikPrinted on acid-free paperSpringer is part of Springer Science+Business Media (www.springer.com)PrefaceStudies of solute fate and transport in the subsurface environment have been playinga signicant role in hydrogeology over the past half century. The problem directlyrelatestothequalityofnaturalwaterresources,whichareessentialtoallkindsof life, and are a basic element in many sectors of human society. Most migrationstudies of both natural and anthropogenically derived species have considered themotion of a uid (groundwater) accompanied by diffusiondispersion phenomena,physicochemical interactions, as well as microbiological transformations, known tobethedominant factors providing theimpact ofcontaminants upon groundwatersupplies.Over the last decades, essential progress in the migration process description hasbeen achieved due to the development of mathematical background and numericalmethods and laboratory and eld investigations of particular transport mechanismsand physicochemical interactions. However, in many real situations, the subsurfacematerial heterogeneity and variations in uid properties, resulting in nonlinear con-taminant plumebehavior, maketheprediction accuracyofthetransferprocessestoo low to satisfy the practical needs. The lack of comprehensive eld studies of so-lute movement is often cited as a major impediment to our understanding of solutetransport in such systems.Therefore, this work is aimed at the development of the basic knowledge of thesubsurface solutetransferwithaparticular emphasis onelddatacollectionandanalysiscoupledwithmodeling(analyticalandnumerical)toolapplication. Thebookisbasedmostlyoneldmaterialsfromauthorslong-standing, recent,andcurrent experience inthestudyofgroundwater qualityrelatedproblems. Thedi-versityoftheseproblemsisconcernedwiththevarietyofgeological settingsaswellastheanthropogenic effects andprocessescausedbyhuman activity.Someproblems encountered in practice looked as challenge-like and, thus, the author wasencouraged tosearchfornewsolutionsandapproaches.Therelevant theoreticaldevelopments are concerned mainly with the formulation and solution of determin-istic mass-transport equations for a wide range of engineering issues in groundwaterquality assessment and forecasting that can be of some interest for bridging the gapsstill existing in our knowledge of contaminant hydrogeology.The book gives many computation examples and casestudies drawn from theconducted eld investigations. Those examples show the applicability of the theoryvvi Prefaceand methods for solving various practical problems and making decisions in con-taminant hydrology to explain the observed and to forecast the future groundwaterquality. The analyzed problems are as follows:(1) investigation andprediction ofgroundwater contamination byindustrial con-taminants and solutions (radionuclides, chloride and nitrate brine) with specialfocus on the effect of (a) aquifer heterogeneity, anisotropy, and dual porosity,(b)densitycontrast betweenindustrial wasteandgroundwater, (c)physico-chemical interactions that playamajorroleinretarding (e.g., adsorption) orenhancing(e.g.,interactionsbetweendissolvedspeciesandmobilecolloids)contaminant transport;(2) predictionof theeffects of pumpingongroundwater qualityat wellelds:(a)thedisplacement ofstratiedinitialconcentration inartesianandcoastal(off-shore) groundwater systems due to water pumping, (b) downward move-ment of mineral-weathering products in the vadoze zone (above the loweringwater table) with water recharge to the producing aquifers;(3) groundwaterdatingusingstableandradioactiveisotopesforpredictionandassessmentofcontamination potentialandthetimethatwouldbeneededtodisplace contaminants from the groundwater system;(4) eld and laboratory tests design and analysis, and monitoring data interpreta-tion;(5) partitioning of surface and subsurface ows using isotope technique;(6) formation of evaporated salt deposits in closed surface water reservoirs havinga hydraulic connection with the surrounding groundwater systems.Several parts of the book demonstrate the potential for using numerical ground-waterowandtransportmodelsinenvironmental riskassessmentofsubsurfacecontamination by dense or light miscible liquid waste. Environmental isotope datawereutilizedfordeningthegroundwatersystemsandmodelingdataanalysis.However, numerical modeling emerged in the book mostly as one of the primarytools used to understand the most important physical and physicochemical processesthat occur in groundwater systems, as well as for getting analytical approximationsfor some coupled problems, which do not necessarily have exact solutions in closedanalytical forms or cannot be treated with the classical methods.One of the most essential topics addressed in the book is the migration and fate ofradionuclides. Model development is motivated by eld data analysis froma numberof radioactively contaminated sites in the Russian Federation: near-surface radioac-tive waste (RW) disposal sites in northwestern Russia and the Southern Urals, andtwo deep-well RW injection sites in Western Siberia. These sites are part of hugenuclear industry enterprises licensed to possess radioactive materials and also in-volved in hazardous-waste operations, which are supervised by RosAtom, the StateNuclear Energy Corporation, Russian Federation.The total activity of radionuclides that were released (accidentally or intention-ally) in aquifers at many sites reaches hundred thousands to hundred millions Ci.Any of the three RW disposal sites out of the four mentioned here (located in South-ern Urals and Western Siberia) probably contains more radioactive contaminationPreface viiin the subsurface than any other site in the world. Additionally, detailed informationon physical, mechanical, and solute transfer properties of clay formation (which isconsidered as a host medium for the engineered underground RW repository in thenorthwestern part of the Russian Federation) is also analyzed.Those sites play a unique role in the advancement of knowledge of the subsurfacebehavior and fate of many hazardous radionuclides and can be considered as eld-scale laboratories. The book is focused on the modeling and analytical assessmentsof a range of physical and chemical processes and interactions of concern. Some ofthe key issues needed to be addressed included:(1) study of the behavior of a broad spectrum of radionuclides (ssion products andactinides) in waste (with low content of dissolved solids and brine) based onlong-term(up to 50 years) monitoring data in shallow and deep aquifer systems;(2) study of the spatial variability of migration properties of aquifer materials andclayey semipervious formations;(3) assessment of the role of brine-induced advection in redistribution of radioactivecomponents at waste disposal sites;(4) study of adsorption hysteresis implying isotherm nonsingularity and other non-idealsorptionphenomena,aswellastheassessment oftheirroleinnaturalattenuation of radioactively contaminated sites;(5) analysis of transient hydrogeochemical-barrier effects, facilitating radionuclidetransport, and some other mechanisms responsible for fast radionuclide trans-port in aquifers;(6) experimental evidence for colloid-facilitated radionuclide (actinide) transport,and mathematical description of the phenomena.The model developments were accompanied by laboratory studies into naturalattenuation,radionuclideadsorptionanddesorptionkineticsandequilibrium(in-cludingwhencolloidal particlesareinvolved).Batchtestswereconductedwithdifferent radioactive solutions under different temperature and pressure conditions.Anomalous behavior of radionuclides was observed and modeled.This study can be regarded as the continuation of a series of works started by theauthor in the 1970s in cooperation with the outstanding Russian scientist, hydroge-ologist, V.A. Mironenko, whose contribution to the development of several lines ofstudies in hydrogeology and hydrogeomechanics is difcult to overestimate. At thesame time, this book could not appeared were it not for the all-round support fromcolleagues researchers fromE.M. Sergeev Institute of Environmental Geology, St.Petersburg Division, RAS, and St. Petersburg State University, who rendered assis-tanceinthepreparation ofpartsofthebook. Inthisconnection, theauthor verymuch appreciates the help of Leonid Sindalovsky in implementation of many nu-merical algorithms considered in the book, the contribution of Pavel Konosavsky tothe joint studies of adsorption hysteresis and the development of some models ofsolute transfer in the porous media under disturbed ow conditions. The author alsoappreciates Igor Tokarevs willingness to share his data on regional isotope study ofa groundwater system in the area of RW disposal at Tomsk-7 site.viii PrefaceThestudydiscussesexperimentscarriedoutinlaboratoriesofA.N. FrumkinInstitute ofPhysical Chemistry and Electrochemistry, RAS,andA.P. AlexandrovTechnical Institute under supervision of Drs. Elena Zakharova, Elena Kaimin, andElena Pankina. The author expresses his sincere gratitude to these groups for coop-eration that have yielded new results.The author appreciates the cooperation of Aretech Solutions and TIHGSA En-terprises allowing him to learn new hydrogeological aspects related to the formationof groundwater resources and quality in arid regions.Theauthoralsomuchappreciatestheattentiontohisworkandfruitfulldis-cussionswithProfs. VsevolodShestakovandSergeyPozdniakov,MoscowStateUniversity, and Dr. Andrei Zubkov, the head of the Environmental Protection Di-vision (Siberian Chemical Plant), and many other brilliant expertshydrogeologists,whose talent and enthusiasm in scientic and production work allows the author tobelieve in the future of the Russian hydrogeological school.Many efforts were made by Dr. Chin-Fu Tsang and Prof. Jacob Bear to organizethis work in a proper way in order to prepare the book in a format acceptable for theinternational publishing company, Springer. Discussions and exchange of informa-tion, ideas, and opinions with them was a great support to this work.Finally, the author very much appreciates the help of Dr. Gennady Krichevets inprofessional translation of the book and many useful comments from him allowingthe author to make certain improvements to the book. The author would also like toacknowledge the help of Ekaterina Kaplan for her editorial assistance and technicalsupport of the work.Thus, the book, along with theoretical ndings, contains eld information, whichwill facilitate the understanding of subsurface solute transport and the developmentofamethodology forpracticalapplicationtogroundwater hydrology. Thisbookaddresses scientists and engineers who are interested in the quantitative approach tostudying groundwater migration processes. The book can also be protably read bystudents.December 28, 2010 Vyacheslav G. RumyninContentsPart I The Essentials of Dissolved Species Transportin the Subsurface Environment: Basic Denitions, FundamentalMechanisms and Mathematical Formulation1 Advection and Dispersion of Dissolved Species in Aquifers. . . . . . . . . . . . . 31.1 Governing Equations and Solute Transport Parameters . . . . . . . . . . . . . . 31.1.1 Advection of Conservative Componentsin Porous and Fractured Media. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.2 Molecular Diffusion and HydrodynamicDispersion (Microdispersion) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1.3 Initial and Boundary Conditions; Denitionsof Concentration Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.2 Models for Advective Transport in HomogeneousIsotropic Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.2.1 A Characteristics-Based Method for Solvingthe Transport Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.2.2 SoluteTransport ProcessAnalysisin Curvilinear Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.3 A One-Dimensional Model of Microdispersion . . . . . . . . . . . . . . . . . . . . . . 331.3.1 Solutions for Innite Porous Domain. . . . . . . . . . . . . . . . . . . . . . . . 341.3.2 ABasic(Fundamental)Solutionfor Semi-Innite Porous Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361.3.3 On the Solution and Analysis of SoluteTransport Problems by Applying the Laplace Transform. . 381.3.4 Quasi-One-Dimensional Solutionof Microdispersion Problems in DeformedFlows in Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451.4 Spatial (2D and 3D) Models of Microdispersionin Unidirectional Steady-State Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461.4.1 Basic Solutions for a Point Source. . . . . . . . . . . . . . . . . . . . . . . . . . . 471.4.2 Approximate Solutions for 2D and 3D SoluteTransport Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501.4.3 Steady-State Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52ixx Contents1.4.4 Approximate Solutions for a Finite-Size Source . . . . . . . . . . . . 541.4.5 Exact Solutions for 3D Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571.4.6 The Inuence of Geological Boundaries . . . . . . . . . . . . . . . . . . . . 581.5 Equations for Simplest Chemical Reactions and Transformations. . 601.5.1 Sorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601.5.2 Decay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702 Water Movement and Solute Transport in UnsaturatedPorous Media. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772.1 Basic Soil-Water Movement and Inltration Models . . . . . . . . . . . . . . . . . 782.1.1 Governing Functions and Parameters. . . . . . . . . . . . . . . . . . . . . . . . 792.1.2 Continuity Equation and its Major Representations . . . . . . . . 852.1.3 Particular Solutions for Moisture Migrationand Their Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 882.2 On Models Coupling Water Inltration and Solute Transport . . . . . . .1062.2.1 Advection: A Characteristic Solution. . . . . . . . . . . . . . . . . . . . . . . .1072.2.2 Dispersion During Adsorption of Water by Soil . . . . . . . . . . . .1112.2.3 AdvectionDispersion Transport. . . . . . . . . . . . . . . . . . . . . . . . . . . .114References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .116Part II Conceptual Models for Regional Assessment of Solute Transport(Under Homogeneous Liquid Flow Conditions)3 One-Dimensional Hydrodynamic Mixing Models forRegional Flow Systems Under Areal Recharge Conditionsand Their Application to the Interpretation of Isotopic Data . . . . . . . . . . .1233.1 Stable Component Migration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1243.1.1 Flow and Mass Balance Under Conned FlowConditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1243.1.2 Basic Analytical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1253.1.3 Correspondence with a Reservoir Model:Transit Time and Transit Time Distribution . . . . . . . . . . . . . . . . .1283.2 Transport of a Solute Subject to First-Order Single-Stage Decay. . .1313.2.1 Basic Analytical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1313.2.2 Variable Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1323.3 Migration of a Solute Subject to Chain Decay. . . . . . . . . . . . . . . . . . . . . . . .1353.3.1 Two-Stage Chain Decay of an UnstableIsotope Coming into an Aquifer withInltration Recharge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1363.3.2 Two-Stage Chain Decay in Aquifer witha Radioactive Element in Solids as the OnlySource of Radioactivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1393.3.3 Two-Member Chain Decay in Aquifer SolidsContaining Several Radioactive Elements. . . . . . . . . . . . . . . . . . .141Contents xi3.3.4 Basic Concept and Model Developmentfor 4He Groundwater Dating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1433.3.5 Converting Physical Units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1463.4 Hydrodynamic Interpretation of Isotopic GroundwaterMonitoring Data: Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1493.4.1 On Groundwater Dating Using Global Isotopes . . . . . . . . . . . .1493.4.2 Calculated Distributions of Atmospheric 3Hand Its Decay Product3He in Groundwater(Typical Curves) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1523.4.3 ACaseHistoryof3H3HeGroundwaterAnalysisandDataInterpretation (Izhora Plateau, LeningradRegion, Russia) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1573.4.4 Hydrodynamic Interpretation of GroundwaterIsotopic Data from a Site of Deep LiquidRadioactive Waste Disposal, Siberia ChemicalCombine, Russian Federation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .159References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1684 Prole (Two-Dimensional in Vertical Cross-Section)Models for Solute Transport in Regional Flow Systems. . . . . . . . . . . . . . . . . .1734.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1734.2 Homogeneous Conned Aquifer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1754.2.1 Flow Velocity Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1754.2.2 Flow Kinematic Equations and ConcentrationDistributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1774.2.3 Semi-Analytical Solution for the Distributionof Global Tritium over the Aquifer Depth(Typical Curves) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1814.3 Two-Layer Conned Aquifer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1834.3.1 Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1834.3.2 An Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1864.4 Multi-Layer (Stratied) Aquifer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1874.4.1 Hydrodynamic Features of Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . .1874.4.2 Characteristic-Based Relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .192References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1965 Models for Assessment of Transverse Diffusive andAdvective Transfer in Regional Two-Layer Systems. . . . . . . . . . . . . . . . . . . . .1995.1 Diffusion-Dispersion Interlayer Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . .2005.1.1 Balance Estimation for Layer-by-Layer MassTransport Scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2015.1.2 A Case of Two-Layer Stratum with SharpPermeability Contrast Between Layers . . . . . . . . . . . . . . . . . . . . . .202xii Contents5.1.3 The Case of a Reservoir Consisting of TwoPermeable Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2075.2 Combined Inuence of Vertical Advection andDiffusion in a Two-Layer Leaky System on Solute Transport . . . . . . .2085.2.1 Derivation of Analytical Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . .2085.2.2 ACaseStudy: TheFormation andDegradation of a Subsurface Iodine-WaterDeposit (Paleohydrogeology Reconstruction). . . . . . . . . . . . . . .213References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2176 Analytical Models for Solute Transport in SaturatedFractured-Porous Media. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2196.1 Governing Parameters and Conceptual Model Formulation . . . . . . . . .2206.1.1 Parameters and Topological Presentationof Fractured Rock Continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2206.1.2 Mass Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2226.1.3 Basic Analytical Solutions (for Asymptotic Models) . . . . . . .2286.2 Generalized Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2376.2.1 A Streamline-Based Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2376.2.2 Application of the Convolution Propertyof the Laplace Transform for Solving theProblem of Solute Advective Dispersionin Dual Porosity Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2386.3 Solute Transport in Heterogeneous Dual PorosityMedia (Qualitative Analysis) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2436.4 Adsorption and Decay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2456.4.1 Adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2456.4.2 Decay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2466.4.3 Migration of Unstable Components UnderAreal Recharge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .253References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2557 Flow and Transport Through UnsaturatedFractured-Porous Rocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2597.1 Problem Conceptualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2597.2 Saturation Prole at Steady-State. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2627.3 Solute Transport Under Steady-State MoistureDistribution Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2647.4 Nonequilibrium Flow and Transport Processes . . . . . . . . . . . . . . . . . . . . . . .2667.4.1 Model-Based Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2677.4.2 A Solution Describing the Early Stageof Wetting Front Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2697.4.3 The Integral Mass Balance Approach . . . . . . . . . . . . . . . . . . . . . . .272Contents xiii7.4.4 A Solution for Leading Front PropagationUnder Exponentially Damped Regimeof Water Imbibition into a Gas-SaturatedMatrix Block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2737.4.5 A Generalized Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2747.4.6 Kinematic Wave Approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . .2787.4.7 Solute Transport Problem Formulation . . . . . . . . . . . . . . . . . . . . . .282References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .282Part III Solute Transport Processes Induced by Recharge and DischargeWells8 Models for Tracer Test Analysis and Interpretation. . . . . . . . . . . . . . . . . . . . . .2878.1 Tracer Migration in a Radially Divergent Flow Field. . . . . . . . . . . . . . . .2888.1.1 Mathematical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2888.1.2 Microdispersion: A Full Analytical Solution. . . . . . . . . . . . . . . .2938.1.3 Approximate Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2958.1.4 Tracer Tests in Fractured-Porous Aquifers . . . . . . . . . . . . . . . . . .2998.2 Tracer Migration in a Radially Convergent Flow Field. . . . . . . . . . . . . .3028.2.1 On the Application of Approximated Modelswith Linear Geometry and the Assessmentof Distorting Factors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3038.2.2 Microdispersion of Tracer in a HomogeneousSingle Porosity Aquifer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3068.2.3 Tracer Transport in a Fractured-Porous Aquifer . . . . . . . . . . . .3098.3 The Time Lag for Breakthrough Curves and TracerDilution in a Source Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3118.3.1 The Time Lag for Breakthrough CurvesDetected in an Observation Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3118.3.2 Effect of Tracer Dilution in the Source Well . . . . . . . . . . . . . . . .3158.4 Analytical Models for Doublet Tracer Testing. . . . . . . . . . . . . . . . . . . . . . . .3168.4.1 Flow Field and Travel Time BetweenRecharge and Discharge Wells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3178.4.2 Piston-LikeTracerDisplacementin a Homogeneous Single Porosity Aquifer . . . . . . . . . . . . . . . . .3188.4.3 An Approximate Solution for Microdispersionin a Homogeneous Aquifer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3218.4.4 Solutions for Mass Transfer in a Fractured-Porous Aquifer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3228.5 Problems Related to the Subvertical Migrationof Tracers in a Field of Recharge and Discharge Wells. . . . . . . . . . . . . .3238.5.1 Problem Conceptualization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3238.5.2 Partially-Penetrating Well OperationUnder the Condition of Nonuniform InitialConcentration Prole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .325xiv Contents8.5.3 Plots and Formulas for the Analysisof Vertical Dipole Tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .329References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3359 ModelsforPredictionofEffectsofPumpingon Groundwater Quality at Well-Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3399.1 Change in the Groundwater Quality in Leaky Aquifer Systems . . . . .3399.1.1 Flow and Mass Balance Equations. . . . . . . . . . . . . . . . . . . . . . . . . .3409.1.2 Solutions of Radial Flow Problems . . . . . . . . . . . . . . . . . . . . . . . . . .3429.1.3 Solution of Solute Transport Problems . . . . . . . . . . . . . . . . . . . . . .3449.2 Change in the Water Quality of Unconned ProducingAquifer Under the Inuence of Weathering SulphideMineral Products in Vadoze Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3519.2.1 Governing Factors and the Scale of the Process. . . . . . . . . . . .3519.2.2 Thermodynamics of Chemical Weathering Process. . . . . . . .3549.2.3 Sulde Oxidation Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3559.2.4 Distribution of Oxygen and Sulfatesin the Vadoze Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3569.2.5 Sulfate Migration in an Aquifer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .359References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .365Part IV Lumped-Parameter Models for Flow and Solute Balance in CoupledSurface-Water/Groundwater Systems10 Conceptual Lumped-Parameter Models for CoupledTransient Flow and Solute Transport in Catchments. . . . . . . . . . . . . . . . . . . .36910.1 Basic Concepts and Denitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36910.2 A Two-Layer Model with Lumped Parametersfor Lateral Subsurface Flow and Base Flow . . . . . . . . . . . . . . . . . . . . . . . . . .37210.3 Basic Analytical Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37510.3.1 Steady-State Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37510.3.2 Unsteady-State Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37510.4 Time-Varying Inltration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37710.4.1 Computation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37710.4.2 An Illustrative (Synthesized) Example . . . . . . . . . . . . . . . . . . . . . .37810.5 A Coupled Solution of Fluid Flow and SoluteTransport Equations for Time-Independent Boundary Conditions. .37910.5.1 Steady-State Flow Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37910.5.2 Transient Flow Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38010.6 A Coupled Solution of Fluid Flow and SoluteTransport Equations for Time-Variable Input Functions . . . . . . . . . . . . .38310.6.1 NumericalAnalytical Solution Algorithm. . . . . . . . . . . . . . . . . .38310.6.2 An Illustrative (Synthesized) Example . . . . . . . . . . . . . . . . . . . . . .38410.7 Runoff, Inltration, and Groundwater Recharge. . . . . . . . . . . . . . . . . . . . . .38510.7.1 Water Budget. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .386Contents xv10.7.2 Inltration Models and Conceptual Scenariosfor Runoff Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38810.8 A Modied SCS-CN Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39010.8.1 A Basic Semi-Empirical Formula for Runoff Calculation. .39010.8.2 Basic Relationships for Flow Characteristics . . . . . . . . . . . . . . .39210.8.3 Concentration Response Function . . . . . . . . . . . . . . . . . . . . . . . . . . .39310.8.4 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .394References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40211 Unsteady-State Hydrogeological Model of Evaporation-Induced Sedimentation in a Surface Reservoir. . . . . . . . . . . . . . . . . . . . . . . . . . .40511.1 Problem Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40511.2 Basic Balance Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40711.2.1 The Case of C1 < C1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40711.2.2 The Case of C1C1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40911.3 Numerical Solutions of the Problem and Their Analysis. . . . . . . . . . . . .410References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .413Part V Variable-Density Flow and Solute Transport: Physical Phenomenaand Mathematical Formulation12 Dynamic Equilibrium of FreshwaterSaltwater Interface . . . . . . . . . . . . . . .41712.1 Basic Steady-State Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41712.1.1 Interface Between Two Immiscible Liquidsin Equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41712.1.2 GhybenHerzberg Relation (Approximation) .. . . . . . . . . . . . . .41912.2 Approximate Solutions of the Problem of the Shapeof the SeawaterFresh Groundwater Interface . . . . . . . . . . . . . . . . . . . . . . . .42112.2.1 A Conned Coastal Aquifer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .42112.2.2 A Leaky Conned Coastal Aquifer . . . . . . . . . . . . . . . . . . . . . . . . . .42312.2.3 A Phreatic Coastal Aquifer Under Recharge Conditions . . .42612.2.4 Freshwater Lens on an Elongated Oceanic Island. . . . . . . . . .42712.3 Equilibrium for Saltwater Upconing Beneatha Partially Penetrating Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .42912.3.1 Problem Setting and Analysis of ExistingApproaches and Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43012.3.2 Analytical Solutions for the Critical PumpingRate and the Critical Interface Rise. . . . . . . . . . . . . . . . . . . . . . . . . .432References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43613 Dynamics of SaltwaterFreshwater Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43913.1 Two-Dimensional Prole Models for ImmiscibleFluids Interface Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43913.1.1 Linear Displacement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44013.1.2 Radial Displacement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .447xvi Contents13.2 Application of Two-Phase Flow Approach for BrineTransport in Porous Media Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45313.2.1 Physical and Mathematical Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . .45313.2.2 Properties of Particular Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . .455References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46014 Studying Subsurface Density-Induced Phenomena UsingNumerical Modeling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46314.1 OnPhysicalApproaches toMathematicalProgramming Formalism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46414.2 Brine Migration in Idealized Aquifer Systems. . . . . . . . . . . . . . . . . . . . . . . .46814.2.1 Numerical Simulators PerformanceCapabilities and Their Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46814.2.2 Physical Phenomena Analysis for Migrationof a Brine Released from a Surface Reservoir. . . . . . . . . . . . . .47314.2.3 Solute Concentration in a Pumping WellAffected by SaltwaterFreshwater Interface Upconing. . . . .481References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .489Part VI Case Histories of Subsurface Contamination by Industrialand Environmental Brines: Field Data Analysis and Modeling of MigrationProcesses15 Radioactive Brine Migration at the Lake Karachai Site(South Urals, Russian Federation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .49515.1 Introduction Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .49515.2 Hydrogeological Setting and General Descriptionof the Migration Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .49815.3 Groundwater Contamination Plume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .50015.3.1 A Structure of Groundwater Flow at the Site. . . . . . . . . . . . . . .50015.3.2 The Distribution of the Radionuclides andPrincipal Ions Within Contamination Plume . . . . . . . . . . . . . . . .50115.4 Overview of Modeling Analysis Approach . . . . . . . . . . . . . . . . . . . . . . . . . . .50815.5 Model Setup and Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .50915.5.1 Model Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .51015.5.2 Sharp-Interface Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .51115.5.3 Fully Miscible Transport Modeling Approach . . . . . . . . . . . . . .51315.5.4 Brine Plume Simulation and Prediction . . . . . . . . . . . . . . . . . . . . .515References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .51716 Modeling of Seawater Intrusion in Coastal Area of RiverAndarax Delta (Almeria, Spain) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .51916.1 Study Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .51916.1.1 Brief Geological Description of the Site. . . . . . . . . . . . . . . . . . . .52016.1.2 Hydrogeological Setting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .521Contents xvii16.2 Groundwater Salinization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52316.2.1 Spatial and Temporal Changes in Groundwater Quality. . . .52316.2.2 Major Results of Vertical Electrical Soundings . . . . . . . . . . . . .52616.3 Conceptualization and Model Design of SeawaterIntrusion Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52716.4 Modeling Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .530References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .53417 Studying and Modeling of Uncontrolled Discharge of DeepBrine into Mine Drainage Systems at the KorshunovskyIron Ore Mine (Eastern Siberia, Russian Federation). . . . . . . . . . . . . . . . . . .53517.1 ABriefDescriptionoftheGeological andHydrological Structure of the Site, Drainage Measuresand Groundwater Regime Disturbed by Mining Operations . . . . . . . . .53617.1.1 Hydrogeological Units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .53717.1.2 Drainage of the Open Pit Mine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .53717.1.3 Vertical Hydrogeochemical Straticationof the Groundwater System and TemporalChanges in Groundwater Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . .53817.2 Analytical Assessments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .53917.3 Numerical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .54117.3.1 Process Conceptualization and Model Design . . . . . . . . . . . . . .54217.3.2 Modeling Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .54217.3.3 Experimental Verication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .543References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .54418 Light Wastewater Injection into a Deep GeologicalFormation Containing Brine (Volzhsky OrgsintezDeep-Well Disposal Site, Central Russia Region) . . . . . . . . . . . . . . . . . . . . . . . . .54518.1 Hydrogeological Characteristics and Settingof the Geological Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .54518.1.1 Available Field Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .54518.1.2 Conceptualization of the HydrogeologicalSetting and Model Description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55118.2 The Concept and Technique of Numerical Analysis. . . . . . . . . . . . . . . . .55218.3 Numerical Solution of a Groundwater Transport Problem . . . . . . . . . . .554References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .556Part VII Physicochemical Description and Mathematical Formulationof Sorption Processes19 Conceptual Models for Sorption Under Batch Conditions. . . . . . . . . . . . . . .56119.1 Sorption Equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .56119.1.1 Principal Sorption Isotherms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .561xviii Contents19.1.2 Principal Factors Affecting the ExperimentalIsotherm Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .56419.1.3 Hysteresis Phenomenon in Sorption . . . . . . . . . . . . . . . . . . . . . . . . .56619.2 Models of Sorption/Desorption Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .57119.2.1 Sorption Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .57219.2.2 A Generalized Nonequilibrium Sorption Model . . . . . . . . . . . .57219.3 Models for Static (Batch) Sorption Experiments . . . . . . . . . . . . . . . . . . . . .57319.3.1 Mass Balance in a Batch Experiment . . . . . . . . . . . . . . . . . . . . . . . .57319.3.2 One-Site Kinetic Model of Sorptionwith Concomitant Mineral Dissolution . . . . . . . . . . . . . . . . . . . . . .574References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58020 Conceptual Transport Models for Adsorbable Solutes . . . . . . . . . . . . . . . . . . .58520.1 Equilibrium Sorption in Groundwater Flow . . . . . . . . . . . . . . . . . . . . . . . . . .58520.1.1 Effective Transfer Parameters for EquilibriumReversible Sorption. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58520.1.2 The Inuence of Nonlinear Sorptionon AdvectiveDispersive Solute Transport . . . . . . . . . . . . . . . . . .59020.1.3 The Inuence of Nonlinear Sorptionon Advective Transport of a Decayed Component . . . . . . . . . .59420.1.4 The Inuence of Sorption Hysteresison Concentration Front Displacement . . . . . . . . . . . . . . . . . . . . . . .59820.1.5 OnIncorporation of aGeochemicalPhenomenon into a Radionuclide Transport Model. . . . . . . . .60220.2 Nonequilibrium Sorption. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .60720.2.1 A Fundamental Solution for Linear Sorption . . . . . . . . . . . . . . .60720.2.2 Asymptotic Solution of the Problemof Nonlinear Sorption Kinetics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61020.2.3 A Numerical Model of an AdsorbableComponent Transport in Porous Mediawith Discrete Sorption Sites. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .612References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .613Part VIII Experimental and Modeling Study of SorptionDesorptionProcesses21 Radon Site for Near-Surface Disposal of Solid RW. . . . . . . . . . . . . . . . . . . . . . .61721.1 AGeneral DescriptionoftheGroundwaterContamination Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61721.1.1 Hydrogeological Setting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61721.1.2 The Causes and Extent of Groundwater Contamination . . . .61821.1.3 Dynamics of Groundwater Contamination . . . . . . . . . . . . . . . . . .62021.2 Variation in Sorption Coefcients and Controlling Factors . . . . . . . . . .62221.2.1 Distribution Coefcients (Linear Model) . . . . . . . . . . . . . . . . . . . .62221.2.2 Nonlinear Freundlich Sorption of Co-60 . . . . . . . . . . . . . . . . . . . .626Contents xix21.3 Hysteresis in Sorption. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62721.3.1 Experiments with Reference Samplesof Cambrian Sands. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62721.3.2 Model EstimatesoftheFormationof Concentration Fronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63321.4 Spatial Variability of Sorption Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . .63421.4.1 A Review of Published Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . .63421.4.2 Spatial Variability of Nonlinear SorptionParameters for Sr-90. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .636References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64422 Study of Physical, Mechanical, Flow, and Solute TransferProperties of Clay Formations with Respect to the Designof Underground Storage Facilities for RW Disposal . . . . . . . . . . . . . . . . . . . . . .64722.1 Introduction Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64722.2 The Structure, Chemical and Mineral Composition,and the Physical Properties of the Clays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64922.3 Rock Mechanical and Hydraulic Properties. . . . . . . . . . . . . . . . . . . . . . . . . . .65222.4 Variations in Physical and Mechanical Properties over Depth . . . . . . .65322.5 A Comparative Analysis of the Clay Formations . . . . . . . . . . . . . . . . . . . . .65722.6 SorptionDesorption Experiments (Vkt Clay) . . . . . . . . . . . . . . . . . . . . . . . .65822.6.1 Sorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65822.6.2 Desorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .66022.7 Diffusion Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .66222.7.1 Single-Chamber Diffusion Cell(In/Out-Diffusion) Tests with Packing Cambrian Clay . . . . .66222.7.2 Single-Chamber Diffusion Cell Testswith Undisturbed Vendian Clay Samples. . . . . . . . . . . . . . . . . . . .66922.7.3 A 3D Diffusion Test with a Cambrian ClaySample of Natural Structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .67222.7.4 A Comparative Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .677References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .67923 Tomsk-7 and Krasnoyarsk-26 Sites for Deep-WellInjection Radioactive Waste Disposal, and Lake KarachaiSite of Near-Surface Disposal of Radioactive Brine. . . . . . . . . . . . . . . . . . . . . .68123.1 Nonideal Behavior of Sorption Curves Observedin Batch Tests with Core Material from the Tomsk-7 Site. . . . . . . . . . .68123.1.1 Materials, Methods, and Experimental Series . . . . . . . . . . . . . . .68223.1.2 Results: Qualitative Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .68423.1.3 Modeling Results and Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . .69423.1.4 On the Direction of Processes UnderExtremely High Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .698xx Contents23.2 The Inuence of Mineral Transformation of AquiferMatrix on Radionuclide Sorption in Batch Testswith Core Material from the Krasnoyarsk-26 Site . . . . . . . . . . . . . . . . . . . .70123.2.1 General Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .70123.2.2 Experimental Setup and Analysis of the Major Results . . . .70223.3 Radionuclide Sorption ontoFreshFracturesof Volcanogenic Metamorphized Rocks from the LakeKarachai Site. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .70623.3.1 Samples and Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . .70723.3.2 Sorption Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .70823.3.3 Hysteresis in Sorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .709References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .710Part IX Colloid-Facilitated Solute Transport in Aquifers24 Colloidal Systems and Equilibrium in Such Systems . . . . . . . . . . . . . . . . . . . . .71524.1 General Views on Colloids and Their Genesis . . . . . . . . . . . . . . . . . . . . . . . .71524.2 Properties of Colloidal Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71924.2.1 Stability of Colloidal System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71924.2.2 Mobility and Accumulation of Colloidsin the Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72224.3 Sorption of Chemical Species onto Colloids(Under Batch Conditions). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72424.3.1 Basic Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72424.3.2 Governing Equations for Sorption Equilibrium . . . . . . . . . . . . .72624.3.3 Sorption Kinetics and Some Experimental Data . . . . . . . . . . . .72724.4 Subsurface Behavior of Actinides at Existingand Proposed RW Disposal Sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73124.4.1 Sites Where Pu and Some Other ActinidesHave Been Detected in the Environment . . . . . . . . . . . . . . . . . . . .73124.4.2 Designed and Engineered Repositoriesfor RW Disposal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .733References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73525 Experimental Study of Radionuclide Interactionwith Colloids with Respect to Tomsk-7 Deep-Well RWDisposal in a Geological Formation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73925.1 A Study of Sorption of Plutonium on Colloidsin Ultraltration Experiments with Synthesized Solutions . . . . . . . . . . .73925.1.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73925.1.2 Interpretation of Experimental Results . . . . . . . . . . . . . . . . . . . . . .74325.2 A Study of Colloidal Forms of Radionuclide Migrationat a Radioactive Waste Disposal Site . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75025.2.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75025.2.2 Calculation Algorithm and ObtainedParameter Values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .752Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .754Contents xxi26 Models of Sorption Type for Colloid-Facilitated Transportin Aquifers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75526.1 The Governing Equations for Migration of Colloidal Solutions . . . . .75526.1.1 A Dual-Species Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75526.1.2 Transport of a Polydisperse Colloidal Solution . . . . . . . . . . . . .75726.2 A Model with Effective Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75826.2.1 Equilibrium Reversible Sorption. . . . . . . . . . . . . . . . . . . . . . . . . . . . .75826.2.2 Irreversible Sorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .76026.3 Numerical Modeling and Illustrative Examples . . . . . . . . . . . . . . . . . . . . . .76126.3.1 Introduction Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .76126.3.2 Equilibrium Sorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .76326.3.3 The Inuence of Sorption Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . .767References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .76927 A Thermodynamics-Based Conceptual Model forColloid-Facilitated Solute Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .77127.1 Surface Complexation Models (Static Formulation) . . . . . . . . . . . . . . . . .77127.2 On Modeling Approach for Multicomponent Solute Transport . . . . . .77827.2.1 Tests and Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .77927.2.2 Sample Problems and Analysis of Migration Process . . . . . .78127.3 A Conceptual Model for the Subsurface Transportof Plutonium on Colloidal Particles Involving SurfaceComplexation Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .78427.3.1 The Basic Chemistry of Plutonium . . . . . . . . . . . . . . . . . . . . . . . . . .78427.3.2 Examples of Modeling Assessmentsfor Migration of the Sodium Nitrate SolutionContaining Pu(IV) and Colloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . .793References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .797Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .801Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .803Part IThe Essentials of Dissolved SpeciesTransport in the Subsurface Environment:Basic Denitions, FundamentalMechanisms and MathematicalFormulationThe theory of solute migration in groundwater originates from studies devoted tothe description of elementary pore-scale mechanisms (processes) of the movementof dissolved species in a single uid phase, including advection, molecular diffusion,and hydrodynamic dispersion, which are accompanied by acts of simplest sorption-type physicochemical interactions and solute decay (destruction) reactions. Thosestudies were mostly based on the classical theory of uid motion in idealized porousmedia (Muskat 1937; Scheidegger 1957), as well as on chemical kinetics and reac-tion engineering. However, it has become clear that there exist some specic featuresin the application of conventional hydrodynamics methods to the formulation, so-lution,andanalysisofmanypracticallysignicanthydrogeological problems.Inparticular, the relative signicance of those mechanisms and interactions in the gen-eral migration process was soon found to depend on the spatial and temporal scale oftheir analysis, the lithological and genetic type of geological sections, and the spa-tial correlation structure of their physical parameters, the structure of water ows,and the conditions on their inner and outer boundaries. This, as well as the specicfeatures of the application of physico-mathematical apparatus to the solution of ap-propriate boundary problems regarding dissolved species transport in single-phaseconstant-density groundwater ows, will be the focus of the rst part of this book.Mathematical models used to describe solute transport in the unsaturated zoneofthesubsurfacearealsoincludedinthispart ofthebook. Aswill beshown,for accurate prediction of contaminant transport through the unsaturated zone, eldequations for transport of moisture and chemicals must be coupled.Theequations givenhererepresent adeterministicapproach todescribing thesubsurfacetransport phenomena, andhavebeenassembledfromaconsiderablecollectionof previousworksandinvestigationsconductedbymanyrecognized2 I The Essentials of Dissolved Species Transport in the Subsurface Environmentauthorities intheeldofsubsurface uid dynamics. More generalized uidowand transport models, accounting for the stochastic nature of aquifers and soil ma-terials are subject of high prole, well-publicized special investigations.The proposed material forms a bridge to the understanding of solute transport un-der near-natural conditions and the analysis of migration of complex-compositionsolutions (liquids) whose properties differ from those of formation waters. Besides,the approaches developed here will be used to assess the contributions of variousphysicochemical processes,whichinmanycasescontrol thepotential ofanthro-pogenic impact on groundwater quality under natural conditions.Chapter 1Advection and Dispersion of Dissolved Speciesin AquifersThetransferofchemicalcomponentsthat, wheninsolutions, havenoeffectonthephysical properties ofaquifer materials and groundwater, isinseparable fromthe groundwater ow. Their advective transport involves micro- and macrodisper-sion processes, which control the extent of solute dispersion in homogeneous andheterogeneousaquifers. Inthischapter, wewill consider themigrationmodelsthat describe the motion of solutions miscible with groundwater in homogeneousaquifers. The solute migration processes in heterogeneous (stratied and fractured-porous) systems will be discussed in separate chapters.1.1 Governing Equations and Solute Transport ParametersThe traditional description of ow and solute transport in natural porous and frac-turedmedia(soils, sediments, rocks), aswell asthesolutionofthemajorityofsubsurface hydrology problems, are based on the continuum mechanics approach(Bear 1972; Shestakov 1995; Bearand Cheng 2010). The continuum assumptionconsiders the uid and solid phase as acontinuous medium withow and solute(mass)transportparameterstakentobewell-denedattheREV(representativeelementary volume) scale. The appropriate level of statistical averaging of mediumproperties is a priori attained in complexes of weakly lithied porous sediments. Theconstruction of solute migration models in fractured rocks requires the validity ofmediums continuity assumption, REV, to be analyzed more thoroughly and the pos-sible scale effects associated with the structuring of groundwater ows in fracturespace to be taken into account in the models (Schwartz et al. 1983; Berkowitz 2002;Kosakowski 2004; Neuman 2005; Reeves et al. 2008a, b). Effective (or equivalent)continuum models utilizing REV approach may not be applicable for real fracturedrocks (see below).Themathematical formalizationof thesubsurfacemigrationisbasedonthegroundwater ow continuity equation (which is equation of conservation of mass),Darcyslawandthesolutetransferequationtogetherwithanappropriateset ofboundaryand/or initial conditions. This sectionis devotedtoanalysis of suchmodels.V.G. Rumynin, Subsurface Solute Transport Models and Case Histories,Theory and Applications of Transport in Porous Media 25,DOI 10.1007/978-94-007-1306-2 1, cSpringer Science+Business Media B.V. 201134 1 Advection and Dispersion of Dissolved Species in Aquifers1.1.1 Advection of Conservative Components in Porousand Fractured MediaAdvection of particles or heat, from the classical uid-dynamical point of view, isthetransfer ofmatter orheat bytheowof auid. Advection duetotheuidsbulk motion in pores or fractures is among the major mechanisms governing solutetransport in aquifers lying in the hydrodynamic zone of active water exchange. Thedriving force for advection is the gradient in the hydraulic head.Inunconsolidated granularporousmedia,thisprocessproceedsinpracticallynonstructured void space; whereas liquid motion in fractured rocks, consisting ofan assemblage of intact rock blocks (matrix) separated by intersecting sets of joints,proceeds in the space which generally has a distinct structure. In some cases (pri-marily, when the process is considered at a local scale), these distinctions requiredifferentiation of the computation schemes (models) used to describe solute migra-tion in two types of rock formations with different nature of void space.1.1.1.1 Flow Field and Actual Fluid VelocityIn subsurface uid dynamics, ow velocity eld,creating potential for advectionof dissolved solutes is a vector eld. This eld can be mathematically described bya continuity equation written in the most general form for compressible pore-uidmixtures as followst+ (q) = 0, (1.1)where q is the specic discharge or Darcy velocity (a vector with 3 components)[LT1],q =K (Pg); (1.2)istheliquiddensity[ML3]; istheporositydenedasthevoidspacebe-tween grains (in porous-type formations) or fracture walls (in consolidated rocks)lled with water [L3L3]; P is the hydraulic pressure; K is the permeability [L2](second-order tensor with 9 components, three of which are Kx, Ky, Kz); is thedynamic viscosity [ML1T1]; g is the gravity vector [LT2]. The term (q) iscalled the divergence of uid ux, representing the net uid inux/efux throughthe element and sometimes is written as div(q). Equation 1.1 does not include theinow/outow source-terms.The rst chapters of this book deal with calculating the motion of componentswhose concentration C has no effect on the density ( = const) and viscosity ( =const)ofliquid inthepores (fractures). Thepossible initial(t = 0) variations ofgroundwater densityarealsoneglected.Theporesareassumedtobelledwith1.1 Governing Equations and Solute Transport Parameters 5water alone: no other liquids or gas phase are present. In such case, Eqs. 1.1 and 1.2can be rewritten in terms of hydraulic head, h (Bear 1972, p. 207),Ssht+ q = 0, (1.3)q =kh, (1.4)where Ss is the specic storage of the porous medium [L1], which is the volume ofwater, dVw, that a volume of an aquifer, dWa, releases from storage under a unit de-cline in hydraulic head, dh; k is the hydraulic conductivity [LT1]. Hydraulic headsprovide a measure of the total mechanical uid potential, and Eq. 1.3, formulated onthe principle of conservation of uid volume, conserves uid mass.To determine the specic discharge the gradient-based Darcy law can be writtenin the more convenient, indicial notation, form which is valid for the general case ofgroundwater ow in an anisotropic mediumqi =kijh xj, (1.4a)where qiare the components of specic discharge [LT1], h is the hydraulic head[L],kij(i = 1, 2, 3, j = 1, 2, 3) arethecomponents ofsymmetrical matrix(tensor)of hydraulic conductivity [LT1]; coefcient kij is connected with the permeability,Kij, by the relationship kij = Kijg/. Formula (1.4a) implies the assumption thatsummation over the sameindices is carried out. In the general three-dimensionalcase /xj = /x +/y +/z is the Cartesian coordinate system, isgradient operator (grad), sometimes referred to as Hamiltonian operator.Now dissolved passive species will move with the same velocity as water parti-cles (average water velocity) ui (u)xi tC= ui = qi, orx tC= u =q. (1.5)Equation (1.5) specifyrelationship between thespecicdischarge, which isusedtodetermine thevolumes ofuidpassing through given surfaces,and theactual(advective)uidvelocitycontrollingthefrontofsolutemovement intheporousspace. Actual uid velocity varies over the pore space, due to the connectivity andgeometric complexity of that space. This variable velocity can be characterized byits mean or average value. The average uid velocity depends on what part of thecross-section area is made up of pores, and to what extent the pore space is con-nected. Therefore is the effective porosity (fracture porosity in fractured rocks)also called kinematic, advective and open porosity.Taking C as the volumetric concentration of a chemical component (ML3, M isthe amount of the species), the advective ux (ML2T1) can be expressed in termsof specic discharge (qi):Jai= qiC = uiC,or Ja= qC =uC. (1.6)6 1 Advection and Dispersion of Dissolved Species in AquifersHere, Jai(Ja) is the mass of a component carried across a unit area, oriented normaltoi direction, perunit time. DirectionCqcoincideswiththedirectionofuidmotion.In the cases where solute transport causes the appearance of density gradients orwhere such gradients originally exist in groundwater systems, the use of hydraulicheadasonlydependent variableintheanalysisneglectingbuoyancycomponentoftheow-driving forceisnotacceptable(BachuandMichael2002; Postetal.2007). Darcys specic discharge in such systems should be expressed in terms ofa pressure function with allowance made for the space and time variations in thephysical characteristics and (1.2). Thus, a uid pressure-based formulation isgenerally preferable in modeling variable density problems. Such problems, whichbelong to the class of coupled problems, are considered in the following parts of thisbook (Chaps. 1218). The coupling of ow and transport phenomena is caused bythe dependence of the water density on the salt concentration.Strictly speaking, the use of relationship (1.5) implies that the scales and dimen-sions of the ow and solute migration problems are consistent. Thus, the effectivevalue of the hydraulic conductivity, derived from pumping tests of heterogeneousaquifers, reects the three-dimensional ow conditions. This value is always greaterthan the hydraulic conductivity, which governs the migration of components underconstrained conditions of a one-dimensional or two-dimensional groundwater ow(Rovey and Niemann 2005). This fact follows from the analysis of basic stochasticmodels (Gelhar 1993; Neuman 1994), demonstrating the effect of the groundwaterow dimensions on the effective hydraulic conductivity. The ratio of calculated toactual migration velocity values can be as large as two or three, meaning that therate of aquifer pollution will be considerably overestimated.1.1.1.2 Effective Porosity (Fracture Porosity)The characteristic values ofactive porosity for loose (not cemented, sandy) sedi-ments commonly varies within a relatively narrow range ( 0.2 0.4). In sandtype of sediments that have not experienced cementation, the value of is com-monly near the total porosity value 0. Silt, loam and clay types of sediments alsofeature sufciently high values (0.30.45). However, in argillite-like clays,where molecular diffusion dominates, a considerable portion of voids (0) isinaccessible for dissolved species (Sect. 1.1.2.1). This is due to the presence of ce-ment walls in the pore space. The conrmation is the radical difference betweenmolecular diffusion coefcients obtained in experiments with undisturbed rock sam-ples andwithpackingclaypreparedfromthesamesamples (Garca-Guti errezet al. 2006). Moreover, the diffusion-accessible porosity depends on the type of themigrating ion (Huysmans and Dassargues 2006).Infracturedcrystallinerocksandhardsedimentaryrockssuchassandstone,limestone and chalk, conceptualized as nonuniform continua with bulk properties,the scatter of parameter is much wider, while its absolute values are much less1.1 Governing Equations and Solute Transport Parameters 7(the averageinterval is 0.0050.03). Whence, it follows that the macroscopictransport in fractured rocks, all other conditions being the same, should be muchfaster than in porous ones.Clearly, the errors in advection velocity estimates in porous sedimentary depositsare primarily determined by errors in the description of permeability eld and thestructure of groundwater ows. Variations in the porosity, , are less signicant thanthe space variations in sediment permeability (hydraulic conductivity), so expert es-timates of porosity can be used in some cases, while the hydraulic conductivity nodoubt requires detailed experimental studies. Conversely, as it can be seen later, thevalues of in fractured type of rock formations are hardly predictable at the intuitivelevel. Therefore, wehave toaccept the fact that the results of predicting ground-waterpollutioninfracturedrockcomplexes, becauseoftheirheterogeneityandanisotropism in terms of permeability and advective porosity, are often unreliable.1.1.1.3 Anisotropy of Sediment and Rock PropertiesThe form of Darcys law (1.4a) corresponds to the general case of uid ow in ananisotropic medium. The hydraulic anisotropy of a bed is the governing factor inthe advection in heavily deformed groundwater ow that form, for example, due toconcentrated water withdrawal or when density advection develops in the aquifer.Of major importance in sedimentary (porous type) complexes is the anisotropy ofpermeability in the prole, which is due to the lithologic and facies variability. Theanisotropy in fractured-rock complexes is mostly due to the presence of several sys-tems of subvertical fractures (planar anisotropy) and the existence of bedding joints(prole anisotropy). Large tectonic fractures with distinct unidirectional orientationin a mediumwith primary lithogenetic jointing are most often responsible for planaranisotropy. Therefore, one of the principal anisotropy directions commonly lies inthe plane of the water-bearing bed (aquifer), while the other is perpendicular to it.1.1.1.4 On the Microstructure of Flows in Porous and Fractured RocksActiveporosity, , forunconsolidatedorweaklyconsolidated(sandclay)sedi-mentsisaconventional characteristic,since, inadditiontoow-through (active)zones, there always exist stagnant zones not involved in the ow but still playingaconsiderableroleintheformationofthegeneralmassow(CoatsandSmith1964; van Genuchten and Wierenga 1976; Rose 1977; Golubev 1981): by accumu-lating the dissolved species via molecular diffusion, such dead-end zones enhancethe overall salt-related capacity of the systemas compared to the active poros-ity. Therefore, more appropriate characteristic for long-term forecasts would be thevalue of the total connected porosity of rocks (0).Taking into account the interaction between individual elements of ow-bearingmediais of fundamental importancefor fracturedrocks (Tsanget al.1991; Gelhar 1993; Berkowitz 2002; Park et al. 2003; Kosakowski 2004;8 1 Advection and Dispersion of Dissolved Species in AquifersPozdniakov and Tsang 2004). The heterogeneity of the fracture void space is due totectonic movements, weathering processes, secondary mineral formation, fractureslling by ne material delivered by inltration waters from the surface, and karstprocesses.Themajor(upto7090%) discharge ofauidmoving insuchrockscan be concentrated in essentially disjunct channels, which account for as little as520% ofthetotalnumberoffractures (RasmusonandNeretnieks 1986; Dahanet al. 1999; Salve 2005). Based on this statistics one may suggest the ow systemfollows the Paretos power law related to the class of heavy-tailed distributions. Theow-focusinginindividualfracturesphenomenon, especiallywhenfracturesareintersectedbyshortscreensofwaterextraction/injection wells, canbeattributedto their poor hydraulic interaction, which is often the case in the practice, even inheavily fractured rock formations (Berkowitz 2002).In accordance with the current theoretical views, a continuous (connected) owthrough a fracture system can form only when the density of fractures with speciedgeometric characteristics and statistical distribution exceeds some threshold referredtoaspercolation threshold. Inthecaseofidealized model fracture systemswithisometric conguration, the critical percolation criteria can be derived analyticallyor by numerically modeling the ow processes.For example, in the case of fractures with a length L [L], randomly oriented ina two-dimensional space, theoretical analysis (Pozdniakov and Tsang 2004) yieldsthe following percolation criterionN2L22D =8(1 +2+2), (1.7)whichcontrols theonset oftheowviainteracting fractures; hereN2isfracturedensity (the number of fractures per unit area) [L2], is the ratio of the fracturewidth (in horizontal plane) to its length [LL1]. This relationship was obtained bythe self-consistence method for asystem withbinary distribution of permeabilityand is valid for the conditions where the permeability of the matrix (the rock massbetween fractures) can be neglected, and the fracture-related porosity of the system 1. Thus, the percolation threshold for long thin fractures ( 1) is2D 8/ = 2.55. In the case of three-dimensional groundwater ow in idealized modelsystemrepresented by fractures in the formof discs, which have diameter Dand alsoare randomly distributed in the space (Pozdniakov and Tsang 2004), the criterionbecomes:N3D33D = 94, (1.7a)where N3is thevolumetric density offractures (the number of fractures perunitvolume) [L3].One should generally expect three-dimensional systems to yield less strict perco-lation criteria since the probability for fractures to intersect in a three-dimensionalspaceislargerthaninatwo-dimensionalone(Gelhar1993). Insuchcase, the1.1 Governing Equations and Solute Transport Parameters 9existence of individual fractures with anomalously large aperture in a statisticallyaveraged system is not a governing factor in terms of their inuence on the effectivepermeability of the medium: fracture intersections may sometimes exert a greaterinuence on the overall hydraulic conductivity than do the fracture planes (Neuman2005).For qualitative estimates, one can assume that a connected groundwater ow isvery likely to form in a fractured rock when each fracture is crossed on the averageby 35 other fractures. However, when met, such criteria by no means imply thattheowisperfectly similartoaowinacontinuous porous medium, whereanoverwhelmingly major partofthewater-lledspaceisinvolved intheformationof ow.The main factors determining the channel-type ow mechanism are high vari-ations in the permeability (hydraulic conductivity) of fractures and poor hydraulicinteraction betweenthem.Channeling inthefracture systemhasaninuenceonmatrix-diffusion mechanisms (Chaps. 6 and 7) because it tends to reduce the effec-tive contact surface between the solute and the matrix (Moreno et al. 1988; Bodinet al. 2003). Thus, this mechanism reduces the damping effect of the porous matrixspace (Kurtzman et al. 2005), as well as the role of sorption on fracture surface byreducing the travel times of pollutants (tracers) and increasing the peak concen-tration values, primarily, in processes at relatively small scales. Moreover, when thenumber of intersections of fractures and channels is small, individual ows throughthem do not reach complete mixing, as is tacitly recognized by all models (Kupperet al. 1995). This circumstance is one more factor that limits the use of models ofequivalent porous media for the description of water and solute transfer in fracturedsystems.This implies the need to signicantly shift (toward greater values) the custom-ary estimates of elementary representative volume that are based on characteristicdistances between fractures: for the continuity condition to be met for a medium,the water-bearing channels in the identied volume should be intersected by a largenumber of transverse fractures, which ensure the hydraulic interaction between thechannels. Thechannel-basedtransport mechanismshouldbetakenintoaccountprimarilywhenestimationismadeofasituationintheimmediatevicinityofapollution source, as well as when indicator tests of beds are carried out.Overall, when describing ows in porous media, one often faces situations withhigh uncertainty, resulting from the lack of reliable information about the location,orientation, length, width, hydraulic interaction, and density of individual fracturesand their systems. The uncertainty in forecasts of ow and solute transport in frac-ture systems has always been considerable (Schwartz et al. 1983; Nordqvist et al.1996; Tsang et al. 1996; Masciopinto 2005; Neuman 2005).Having in mind the restrictions that can be imposed on the description of waterowandsolutetransfer infractured rocks,letus consideraveraged models withbulk rock properties whose development supposes that the medium is continuouswithin elementary representative volume.10 1 Advection and Dispersion of Dissolved Species in Aquifers1.1.1.5 Continuity EquationsIn the overwhelming majority of practically signicant situations, the progress insolutetransfer ismuchslower thanthepropagationof hydrodynamicperturba-tions; therefore, Eq. 1.3canbetakeninthesteady-stateformulation, assumingh/t = 0.1Now the system of equations describing advection in aquifers, in par-ticular,theadvection-induced shiftofinterfacesbetweenliquids(solutions)withidentical physical properties with no regard to diffusiondispersion phenomena intheir contact zone can be written in the following generalized form:Ct+ (Cq) +Ws = 0, (1.8) q+Wf = 0, (1.9)where Cisthevolumetricconcentrationofacomponent intheowingsolution[ML3]; Ws is the rate at which the mass of the component is produced or disappearsper unit volume of the system [ML3T1], e.g., mass-exchange rate between layerswith different permeability in stratied systems or fractures and porous matrix infractured-porous groundwater reservoirs (Chaps. 5 and 6); Wfis the rate at whichvolume of the uid is produced or disappears per unit volume of the system [T1],e.g., areal recharge, interlayer leakage (Chaps. 35 and 9).Equations (1.8) and (1.9) represent a differential form of the mass conservationlawforthedissolvedsubstanceandowinguid.Equation1.8isthecontinuityequationfor themassux. Itssolutiondescribestheconcentrationdistributionforming under theeffect of advection in thegiven domain under given boundaryconditions for the solute migration problem. The continuity equation for water ow(1.9) allows one to describe the head eld and evaluate the specic discharges asrequired for solving the former problem.These equations admit additional transformations. Thus, the well-known formulaof vector algebra (Cq) = q C+C q allows the system of Eqs. 1.8, 1.9 to bewritten as C t+qC+C q+Ws = 0, (1.10) q+Wf = 0, (1.11)where q = divq = qx/x +qy/y +qz/z is the divergence of the vectoreld q (the scalar product of two vectors, and q).1However, there exists a class of problems where the unsteady-state phase of the ow process maydetermine behavior of the concentration function (Chap. 9).1.1 Governing Equations and Solute Transport Parameters 11Substituting (1.11) into (1.10) yields the following generalized equation C t+uCWf C+ Ws= 0. (1.12)Such formalization of the process, which preserves the linear form of the trans-port equation, is possible only in the case where solute transport has no effect on thestructure of the ow eld.1.1.2 Molecular Diffusion and Hydrodynamic Dispersion(Microdispersion)Strictly speaking, the abrupt concentration fronts, as predicted by the model of ad-vection transport, are nonexistent in nature: molecular diffusion and hydrodynamicdispersion result in the formation of a transient (mixing) zone between two liquids,which were initially separated by an abrupt interface.1.1.2.1 Molecular-Diffusion TransportThe signicance of this process increases when the specic discharge is low, pri-marily, in the ow through low-permeability formations (separation layers, porousblocks of fractured rocks).DiffusionisaprocessinducedbytherandomBrownian(thermal)motionofmolecules and ions. The diffusive ux through a unit cross-sectional area of a porousisotropic medium can be described by Ficks rst law:Jm=DeC =DmC, (1.13)where Jmis the diffusion ux [ML2T1], C is the concentration [ML3], De andDm=De/arethe effective diffusionandpore diffusioncoefcients [L2T1](Ruthven1984). Deis a coefcient characterizingsteady-state diffusionux.Coefcient Dmdescribestransient diffusionandisaparameterincludedintheFicks second law (see Sect. 1.1.2.5). Along with De and Dm, an apparent diffusioncoefcient, Da, can be used to characterize migration of sorbed species. It takes intoaccount not only the geometry of the porous medium but also the retardation factor,R (Sect. 1.5.1): Da = Dm/R.Thus, the relationship between the different dispersion coefcients is therefore:Dm = DaR = De/. (1.13a)The driving force for the one-dimensional (along the x direction) diffusion is thequantityC_x, thus in a simple one-dimensional system at uniform temperatureand pressure under steady state conditionsJmx=DedCdx . (1.14)12 1 Advection and Dispersion of Dissolved Species in AquifersResults of laboratory and eld tests showthat the diffusion coefcient depends onthe magnitude of the porosity and on its spatial distribution. The basic equation is:De = D0, (1.15)whereD0isthediffusioncoefcient inbulkwater[L2T1], isthediffusion-accessible porosity [L3L3], is the tortuosity [L2L2] which accounts for the poregeometry (Grathwohl 1998) and ranges between 0.01 and 0.5 for most geologicalmaterials.AnalternativepresentationforDefollowsfromtheformal analogybetweendiffusion and electrical conductivity inporous sediments (Boving and Grathwohl2001):De = D0m(1.16)(Archieslaw), wheremisanempiricalexponent (cementationorshapefactor).The value of m lies between 1.3 and 5.4 (Polak et al. 2002), depending upon rockconsolidation and some other factors. Studies focused ondifferent limestone andsandstone rocks, argillaceous rocks and chalk showed that the data for various inor-ganic and organic compounds resulted in an exponent between 1.5 and 2.5 (Bovingand Grathwohl 2001; Polak et al. 2002; van Loon et al. 2003; Blum et al. 2007). Itis very likely that m=2 corresponds to the upper limit for a rock matrix with poros-ity 100 m)thickness, which are ofinterest inthe context of their possibleuse for long-termisolation of radioactive wastes (RW). Molecular diffusion is regarded as the mainradionuclidetransport mechanismintheestimationof theriskassociatedwithRWburial.Thus,anumberoflabandinsitutestshasbeenperformed tostudydiffusioninAalenianOpalinusclay(Switzerland), RupelianBoomclayat Mol(Belgium), Callovo-Oxfordianclayeysiltstone/siltyclayat Bure(France), Ven-dian/Kothlin, Cambrian/Blue clays (the north-western region of Russian Federation)(see Chap. 22).The studies show all these strata to feature anisotropic diffusion properties: theeffective-diffusion coefcient along the beds, DeL, is many times greater than thesamecoefcient DeTgoverningthesolutetransferacrossthebeds(Palut et al.2003; Yllera et al. 2004; Clay Club Catalogue. . . 2005; Garca-Guti errez et al. 2006;Samper et al. 2006; Samper et al. 2008; Soler et al. 2008; Cormenzana et al. 2008;Rumynin et al. 2009; see alsoSects. 22.7.2 and 22.7.3 in this book). This differ-ence isdue tothe microstratication ofclay strata,i.e., alternation of beds (withthickness varying from several millimeters to a few centimeters) with different con-centrations of clay minerals. Such effect can be caused, for example, by intercalationof argillite and aleurolite. The anisotropy of the effective diffusion coefcient is dueto stratication, the preferred orientation of the microlayers at the mm-cmscale. Thetortuosity of the diffusion pathways is larger when the direction is normal rather thanparallel to the preferred orientation of the layers (Suzuki et al. 2004). So, diffusionanisotropy is a consequence of rock heterogeneity and bedding.Tracer transport in an anisotropic clay domain is described by the following dif-fusion equation (Bear 1972; Crank 1975; Palut et al. 2003; Samper et al. 2006, 2008;Soler et al. 2008):Ct (Dm C) = 0, (1.17)14 1 Advection and Dispersion of Dissolved Species in Aquiferswhere Dm is the orthotropic pore diffusion tensor. The expression for the latter is asfollows:Dm =__DmL0 00 DmL00 0 DmT__, (1.18)where DmL and DmTare the principal components of the tensor. In this study, DmL isthe effective pore diffusion coefcient parallel to the horizontal bedding while DmTis the effective pore diffusion coefcient perpendicular to the bedding.The values of the molecular diffusioncoefcient commonlycorrelate onlyslightly with aquifer material permeability, thoughthere are some exceptions(Boving and Grathwohl 2001; Reimus et al. 2007). Some researchers found labora-tory estimates of Deto be less than the values determined in the eld (Zhou et al.2007).Suchdifferenceisprimarilyduetotheexistenceofaweathering(degra-dation) zone at the contact between a porous block and afracture, as well as thepresence of stagnant water in dead-end microfractures.1.1.2.2 Mechanical Dispersion (Hydrodynamic Dispersion) of SolutesThis phenomenon is due to two effects: (1) a dynamic effect, i.e., local (at the porescale)variabilityofowvelocityeld;and(2)kinematic,i.e., thebranchingoftrajectories of motion in the pore space. Their combination looks like a diffusionprocess,facilitatingtheformation ofatransient (intermsofconcentration) zonebetween the solutions replacing and being replaced.Hydrodynamic dispersioncanbeconsideredjointlywithmoleculardiffusion,and is in many cases taken to be a Fickian process, whose transport law takes theform of Ficks law of molecular diffusion (1.13)Jd=DC, (1.19)where Jdis the vector of total dispersion ux [ML2T1], C is the concentration[ML3],Disthetotaldispersioncoefcient(secondordertensor)[L2T1]. Thedispersion ux (1.19) can be presented in the indicial notation formJdi=DijC xj. (1.19a)Similar to hydraulic conductivity, kij, in the Darcys law (1.4a), equality (1.19a) forFickian dispersion is valid for the general case of an anisotropic porous medium.Despitethemathematical similarityofthetwofundamentallaws, (1.4a)and(1.19a), there isabasicdifference between thecoefcients kijandDij(BearandCheng 2010): the former is a function of the microscopic geometry of the void space,while the latter depends also on the macroscopic velocity eld. The properties of thedispersion coefcient for isotropic and anisotropic models of porous media are thor-oughly investigated by Bear and Cheng (2010).1.1 Governing Equations and Solute Transport Parameters 15In the general case of isotropic porous medium, this total dispersion coefcientdepends onthe specic dischargeandcomprises a hydrodynamic, Dhij, andamolecular-diffusion, Dm, components (Bear 1972):Dij = Dhij + Dmij, (1.20)Dhij =_(LT)uiuj[u[2+Tij_[u[ , ij_= 1for i =j ,= 0for i ,=j,(1.21)whereLandTarethelongitudinal andtransversedispersivityof theporousmedium (factors inthelinear relationship between components of hydrodynamicdispersion Dhijand uid actual velocity ui/uj) [L], Dmis the pore diffusion coef-cient [L2T1], [u[ = u = (u2x +u2y +u2z)1/2,ijis Kroneckers delta function [], i,j = 1, 2, 3.For example, in a unidirectional (along the x axis) horizontal ow:DxxDL =Lux +Dm, Dyy = DzzDT =Tux +Dm. (1.22)The longitudinal dispersivity, L, reects the variability of pores and fractures ge-ometry. This proportionality constant is accepted to be a characteristic length scaleof the medium. Having been measured for a relatively identical spatial scale, disper-sivity of fractured rocks usually exceeds dispersivity of granular type of formations.Dispersivity values measured at typical interwell distances (1001,000 m) are abouttwo to four orders of magnitude larger than those measured i