PARTICLE AND STREAMLINE NUMERICAL METHODS FOR CONSERVATIVEAND REACTIVE TRANSPORT SIMULATIONS IN POROUS MEDIAbyPAULO ANDRES HERRERA RICCIM.Sc., University of Illinois at Urbana-Champaign, 2003Ingeniero Civil, Universidad de Chile, 2001A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIES(Geological Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)November 2009cPaulo Andres Herrera Ricci, 2009AbstractReactive transport modeling has become an important tool to study and understandthe transport and fate of solutes in the subsurface. However, the accurate simulationof reactive transport represents a formidable challenge because of the characteristics offlow, transport and chemical reactions that govern the migration of solutes in geologicalformations.In particular, solute transport in natural porous media is advection-controlled and disper-sion is higher in the direction of flow than in the transverse direction. Both characteristicscreate diﬃculties for traditional numerical schemes that result in numerical dispersionand/or spurious oscillations. While these errors can often be tolerated in conservativetransport simulations, they can be amplified in presence of chemical reactions resultingin much larger errors or unstable solutions.In this thesis, new Lagrangian based methods to simulate conservative and reactivetransport in porous media are investigated. First, the derivation of a new meshlessapproximation based on smoothed particle hydrodynamics (SPH) to simulate conserva-tive multidimensional solute transport, including advection and anisotropic dispersion, ispresented. Second, a hybrid scheme that combines some of the advantages of streamline-based simulations and meshless methods and that allows simulating longitudinal andtransverse dispersion without requiring a background grid is also derived. The numer-ical properties of both methods are analyzed analytical and numerically. Furthermore,both formulations are compared with existing numerical techniques in a set of two- andthree-dimensional benchmark problems.It is demonstrated that the proposed schemes provide accurate and eﬃcient solutionsof physical transport processes in heterogeneous porous media and overcome most ofthe issues in existing numerical formulations. The new methods have the potential toremove or minimize numerical dispersion and grid orientation eﬀects and, in the caseiiof the hybrid streamline method, also eliminate spurious oscillations even in presence oflarge longitudinal to transverse dispersivity ratios.Therefore, the results presented in this thesis confirm that the Lagrangian formulationsof solute transport investigated here are viable and compelling alternatives to simulatereactive transport versus more standard numerical techniques.iiiTable of ContentsAbstract ....................................... iiTable of Contents ................................. ivList of Tables ................................... ixList of Figures ................................... xivAcknowledgments ................................. xvDedication ..................................... xviCo-authorship Statement ............................ xvii1 Introduction .................................. 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Reactive Transport Modeling . . . . . . . . . . . . . . . . . . . . . . . 31.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.2 Conceptual Model . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.3 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.1 Particularities of Flow and Transport in Porous Media . . . . . 61.3.2 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . 101.3.2.1 Mesh-based numerical methods . . . . . . . . . . . . . 101.3.2.2 Hybrid Eulerian-Lagrangian methods . . . . . . . . . . 121.3.2.3 Random walk particle tracking methods . . . . . . . . 121.3.3 Limitations of Current Numerical Methods . . . . . . . . . . . . 131.3.3.1 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . 131.3.3.2 Monotonicity . . . . . . . . . . . . . . . . . . . . . . . 151.3.3.3 Performance . . . . . . . . . . . . . . . . . . . . . . . . 161.4 Lagrangian Numerical Methods . . . . . . . . . . . . . . . . . . . . . . 171.4.1 Meshless Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 171.4.2 Streamline-Based Simulations . . . . . . . . . . . . . . . . . . . 181.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18iv1.6 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.7 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 A Meshless Method to Simulate Solute Transport in HeterogeneousPorous Media ................................. 292.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.1.2 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . 302.2 Monte Carlo SPH method . . . . . . . . . . . . . . . . . . . . . . . . . 332.2.1 Time Integration . . . . . . . . . . . . . . . . . . . . . . . . . . 372.2.2 Accuracy and Spatial Resolution . . . . . . . . . . . . . . . . . 392.2.3 Mass Conservation . . . . . . . . . . . . . . . . . . . . . . . . . 402.3 Numerical evaluation of the MC-SPH method . . . . . . . . . . . . . . 412.3.1 One-Dimensional Dispersion . . . . . . . . . . . . . . . . . . . . 412.3.2 Two-Dimensional Dispersion . . . . . . . . . . . . . . . . . . . . 422.3.2.1 Initial particle and concentration distribution . . . . . 462.3.2.2 Performance . . . . . . . . . . . . . . . . . . . . . . . . 482.3.2.3 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . 512.3.3 Advection–Dispersion in Heterogeneous Porous Media . . . . . . 542.3.3.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 542.3.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 562.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673 Evaluation of Particle Approximations to Simulate Anisotropic Dis-persion ...................................... 723.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.2 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 743.3 Smoothed Particle Hydrodynamics (SPH) Approximation . . . . . . . . 753.3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.3.2 SPH Approximation for Tensorial Dispersion . . . . . . . . . . . 763.3.3 Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.4 Particle Strength Exchange (PSE) Approximation . . . . . . . . . . . . 803.5 Numerical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813.5.1 Simulation Cases . . . . . . . . . . . . . . . . . . . . . . . . . . 833.5.2 Equispaced Particles . . . . . . . . . . . . . . . . . . . . . . . . 86v3.5.2.1 Eﬀect of particle spacing . . . . . . . . . . . . . . . . . 863.5.2.2 Maximum concentration . . . . . . . . . . . . . . . . . 893.5.2.3 Negative concentrations . . . . . . . . . . . . . . . . . 893.5.2.4 Eﬀect of ratio between smoothing length and particlespacing . . . . . . . . . . . . . . . . . . . . . . . . . . 923.5.2.5 Eﬀect of anisotropy ratio . . . . . . . . . . . . . . . . . 933.5.2.6 Eﬀect of kernel function . . . . . . . . . . . . . . . . . 933.5.2.7 Eﬀect of velocity orientation . . . . . . . . . . . . . . . 963.5.3 Irregularly Spaced Particles . . . . . . . . . . . . . . . . . . . . 973.5.3.1 Isotropic case . . . . . . . . . . . . . . . . . . . . . . . 993.5.3.2 Anisotropic case . . . . . . . . . . . . . . . . . . . . . 1043.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084 A Multidimensional Streamline-Based Method to Simulate ReactiveSolute Transport in Heterogeneous Porous Media ........... 1114.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.2 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 1164.2.1 Governing Equation . . . . . . . . . . . . . . . . . . . . . . . . 1164.2.2 Streamline Formulation . . . . . . . . . . . . . . . . . . . . . . . 1174.2.3 Numerical Approximation . . . . . . . . . . . . . . . . . . . . . 1184.2.3.1 Advection along streamlines . . . . . . . . . . . . . . . 1194.2.3.2 Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . 1204.3 Implementation Details . . . . . . . . . . . . . . . . . . . . . . . . . . . 1224.3.1 Streamline Tracing . . . . . . . . . . . . . . . . . . . . . . . . . 1224.3.2 Time Integration . . . . . . . . . . . . . . . . . . . . . . . . . . 1244.3.3 Advection Solution . . . . . . . . . . . . . . . . . . . . . . . . . 1244.3.4 MC-SPH Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.3.4.1 SPH kernel . . . . . . . . . . . . . . . . . . . . . . . . 1254.3.4.2 Neighbor search . . . . . . . . . . . . . . . . . . . . . . 1254.3.4.3 Time integration . . . . . . . . . . . . . . . . . . . . . 1264.3.5 Longitudinal Dispersion . . . . . . . . . . . . . . . . . . . . . . 1274.3.5.1 Interface coeﬃcients . . . . . . . . . . . . . . . . . . . 1274.3.5.2 Time integration . . . . . . . . . . . . . . . . . . . . . 1274.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128vi4.4.1 Example 1: Continuous Solute Release in Uniform Flow . . . . . 1304.4.2 Example 2: Quarter Five-Spot in Heterogeneous Medium . . . . 1324.4.2.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 1324.4.2.2 Simulated concentrations . . . . . . . . . . . . . . . . 1344.4.2.3 Numerical oscillations . . . . . . . . . . . . . . . . . . 1404.4.2.4 Performance comparison . . . . . . . . . . . . . . . . . 1424.4.3 Example 3: Quarter Five-Spot in Heterogeneous Medium withRate-Limited Sorption . . . . . . . . . . . . . . . . . . . . . . . 1434.4.4 Example 4: Natural Biodegradation in Three-dimensional Hetero-geneous Porous Media . . . . . . . . . . . . . . . . . . . . . . . 1464.4.4.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 1464.4.4.2 Simulated concentrations . . . . . . . . . . . . . . . . 1494.4.4.3 Breakthrough curves . . . . . . . . . . . . . . . . . . . 1564.4.4.4 Performance . . . . . . . . . . . . . . . . . . . . . . . . 1594.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1664.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1675 Conclusions ................................... 1725.1 Limitations of Proposed Numerical Schemes . . . . . . . . . . . . . . . 1735.2 General Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1755.3 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1765.4 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1775.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178Appendix A Derivation of SPH Approximation for Isotropic Dispersion 181AppendixB DerivationofSPHApproximationforSecondOrderDeriva-tives ....................................... 183Appendix C Random Walk Particle Method ............... 188Appendix D Streamline Tracing ....................... 191viiList of Tables1.1 Estimated dispersivity values from field- and laboratory-scale experiments. 92.1 Comparison of MC-SPH and RWPT methods. . . . . . . . . . . . . . 442.2 Parameters used in RWPT simulations. . . . . . . . . . . . . . . . . . . 452.3 Parameters used in SPH simulations. . . . . . . . . . . . . . . . . . . . 452.4 Parameter and results of flow model. . . . . . . . . . . . . . . . . . . . 542.5 Parameter values used in transport model. . . . . . . . . . . . . . . . . 553.1 Parameters used in all simulations. . . . . . . . . . . . . . . . . . . . . 823.2 Parameters used to define diﬀerent simulation scenarios to evaluate ap-proximations for anisotropic dispersion. . . . . . . . . . . . . . . . . . 853.3 Definition of diﬀerent runs used to study convergence properties. . . . . 853.4 Normalized error for diﬀerent SPH kernels and PSE cutoﬀ functions. . 963.5 Error versus flow velocity direction. . . . . . . . . . . . . . . . . . . . . 974.1 Dispersivity and equivalent longitudinal (PeL) and transverse (PeT) gridPéclet values used in Example 1. . . . . . . . . . . . . . . . . . . . . . 1314.2 Parameters used in MOC simulations. . . . . . . . . . . . . . . . . . . . 1334.3 Number of nodes or cells, time step size and number of time steps used insimulations of Example 2. . . . . . . . . . . . . . . . . . . . . . . . . . 1344.4 Dispersivity and equivalent longitudinal (PeL) and transverse (PeT) gridPéclet values used in Example 2. . . . . . . . . . . . . . . . . . . . . . 1344.5 Normalized minimum simulated concentration values for Example 2. . . 1424.6 Normalized maximum simulated concentration values for Example 2. . 1424.7 Normalized CPU time required to simulate Example 2 for diﬀerent sce-narios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1434.8 Parameters of the rate-limited sorption model used in Example 3. . . . 1444.9 Definition of three scenarios simulated in Example 4. . . . . . . . . . . 1474.10 Spatial and temporal discretizations used to simulate Example 4. . . . 149viii4.11 Normalized CPU time required to simulate Example 4 for the two scenar-ios than include biodegradation. . . . . . . . . . . . . . . . . . . . . . . 162D.1 Comparison of the performance of Pollock’s and explicit adaptive algo-rithm to trace streamlines in heterogeneous quarter five-spot problem. . 204D.2 Comparison of alternative seed distributions to trace streamlines in theheterogeneous quarter five-spot problem. . . . . . . . . . . . . . . . . 209ixList of Figures1.1 Groundwater pollution due to tailings infiltration. . . . . . . . . . . . 21.2 Vertical cross-section of a synthetically generated aquifer using the esti-mated statistics of the sandy aquifer at Canadian Forces Base, Borden,Ontario. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Solute plume migration in vertical cross-section. . . . . . . . . . . . . 81.4 Eﬀect of advection and local-scale dispersion on solute concentration. 111.5 Subgrid-scale segregation and cell averaged concentration values. . . . 142.1 SPH kernels and derivatives. . . . . . . . . . . . . . . . . . . . . . . . 342.2 Error versus smoothing length. . . . . . . . . . . . . . . . . . . . . . . 432.3 Initial distribution of particles and solute concentration in RWPT sim-ulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.4 Initial solute concentration distribution in MC-SPH simulations. . . . 472.5 Normalized CPU time versus number of particles in RWPT simulations. 492.6 Normalized CPU time versus total number of particles and average num-ber of particles per kernel support volume in MC-SOH simulations. . 502.7 Maximum concentration versus time in MC-SPH and RWPT simulations. 522.8 Normalized global error versus total number of particles and normalizedCPU time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.9 Domain dimensions, square initial plume, and breakthrough observationpoints P1 and P2 along centerline. . . . . . . . . . . . . . . . . . . . . 552.10 Spatial concentration distribution for TVD, HMOC and SPH simula-tions at τ = Ut/IY=62for Pe= ∞................... 582.11 Concentration versus accumulated distance along centerline at dimen-sionless time τ = Ut/IY=62....................... 592.12 Breakthrough curve at point P1 located 26IYdownstream from initialplume center. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61x2.13 Breakthrough curve at point P2 located 42IYdownstream from initialplume center. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622.14 Dimensionless maximum concentration versus dimensionless time. . . 632.15 Mean concentration versus dimensionless time. . . . . . . . . . . . . . 653.1 Sum of dispersion components as function of flow velocity direction. . 793.2 SPH kernels and PSE cutoﬀ functions. . . . . . . . . . . . . . . . . . 843.3 Error E2as function of particle or grid spacing for equispaced particles. 873.4 Normalized error E∞as function of particle or grid spacing for equis-paced particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883.5 Diﬀerence between maximum concentration values of numerical and an-alytical solutions as function of time for equispaced particles. . . . . . 903.6 Concentration distribution for equispaced particles. . . . . . . . . . . 913.7 Diﬀerence between analytical and numerical solutions for equispacedparticles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 923.8 Error E2versus the ratio between smoothing length or core size andparticle spacing, γ = h/∆x........................ 943.9 Error as function of the anisotropy ratio αT/αL............. 953.10 Particle distortion due to flow velocity. . . . . . . . . . . . . . . . . . 983.11 Particle locations considering random and quasi-random distributions. 993.12 E2error versus average particle spacing using equispaced, random, andquasi-random particle distributions. . . . . . . . . . . . . . . . . . . . 1003.13 Normalized E∞error versus average particle spacing using equispaced,random, and quasi-random particle distributions. . . . . . . . . . . . 1023.14 Diﬀerence between maximum concentration values of analytical and nu-merical solutions as function of time for random and quasi-random par-ticle distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033.15 Error as function of average particle spacing for αT/αL=0.01 usingrandomly and quasi-randomly distributed particles. . . . . . . . . . . 1053.16 Concentration distribution after 300 time steps for αT/αL=0.01 andquasi-randomly distributed particles. . . . . . . . . . . . . . . . . . . 1064.1 Meshless MC-SPH method. . . . . . . . . . . . . . . . . . . . . . . . . 1134.2 Hybrid streamline-SPH method. . . . . . . . . . . . . . . . . . . . . . 115xi4.3 Flow oriented coordinate system. . . . . . . . . . . . . . . . . . . . . 1184.4 Overall solution approach implemented in streamline-based simulator. 1294.5 Comparison of simulated concentrations for Example 1. . . . . . . . . 1314.6 Spatial distribution of the natural logarithm of the hydraulic conductiv-ity and streamlines in Example 2. . . . . . . . . . . . . . . . . . . . . 1324.7 Simulated concentration values after injection of 0.4 pore volume of con-taminated fluid for Example 2. . . . . . . . . . . . . . . . . . . . . . . 1354.8 Breakthrough curves at observation point P1 in Example 4. . . . . . . 1374.9 Breakthrough curves at observation point P2 in Example 4. . . . . . . 1384.10 Comparison of simulated breakthrough curves at observation point P1for Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1394.11 Comparison of simulated breakthrough curves at observation point P2for Example 2, (a) streamline simulator and (b) MOC solver using finegrid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1404.12 Numerical oscillations in simulated concentrations for Example 2. . . 1414.13 Breakthrough at observation point P2 for diﬀerent mass transfer (β) andpartition (Kd) coeﬃcients considered in Example 3 . . . . . . . . . . 1454.14 Spatial distribution of natural logarithm of hydraulic conductivity andflow velocity magnitude used in Example 4 . . . . . . . . . . . . . . . 1484.15 Simulated concentrations at nodes along streamlines after 10,000 dayssince the initial release of BTEX for the scenario that includes advectivetransport with biodegradation in Example 4. . . . . . . . . . . . . . . 1514.16 Simulated concentrations at nodes along streamlines after 10,000 dayssince the initial release for the scenario that includes advection, disper-sion and biodegradation in Example 4. . . . . . . . . . . . . . . . . . 1524.17 imulated concentration values for Example 4 at vertical plane defined byy=11.5 m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1534.18 Simulated concentration values for Example 4 at horizontal plane definedby z=2.5 m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1544.19 Comparison of simulated concentration values for Example 4 at horizon-tal plane defined by z=2.5 m. . . . . . . . . . . . . . . . . . . . . . . 1554.20 Simulated concentration values for Example 4 along the profile parallelto y direction at coordinates x=35 m and z=2.5 m. . . . . . . . . . . 157xii4.21 Location of contaminant source and the two observation wells in Exam-ple 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1584.22 Simulated BTEX concentration versus time for Example 4 assumingadvective transport only. . . . . . . . . . . . . . . . . . . . . . . . . . 1604.23 Simulated BTEX concentration versus time for Example 4 assumingadvective transport and biodegradation. . . . . . . . . . . . . . . . . . 1614.24 Distribution of cells according to the flow velocity magnitude. . . . . 1634.25 Distribution of (a) nodes along streamlines according to the flow veloc-ity magnitude and (b) number of streamlines based on the maximumvelocity magnitude along individual streamlines. . . . . . . . . . . . . 1644.26 Spatial distribution of nodes and number of nodes according to the num-ber of neighboring nodes that contribute to the SPH summation to ap-proximate dispersion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 165D.1 Scehamatic of Pollock’s particle tracking method. . . . . . . . . . . . 193D.2 Schematic of explicit particle tracking method. . . . . . . . . . . . . . 198D.3 Comparison of Pollock’s and explicit integration methods for the homo-geneous quarter five-spot problem. . . . . . . . . . . . . . . . . . . . 200D.4 Comparison of time of flight and arc length computed with Pollock’s andexplicit integration methods for homogeneous quarter five-spot problem........................................ 201D.5 Spatialdistributionofhydraulicconductivityusedinheterogeneousquaterfive-spot problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201D.6 Comparison of Pollock’s and explicit integration methods for the het-erogeneous quarter five-spot problem. . . . . . . . . . . . . . . . . . 202D.7 Comparison of time of flight and arc length computed with Pollock’s andexplicit integration methods for heterogeneous quarter five-spot problem. 203D.8 Streamline distribution in heterogeneous quater five-spot problem. . . 207D.9 Comparison of two alternative streamline distributions for the homoge-neous quater five-spot problem. . . . . . . . . . . . . . . . . . . . . . 207D.10 Comparison of four alternative streamline distributions for the hetero-geneous quater five-spot problem. . . . . . . . . . . . . . . . . . . . . 208D.11 Schematic of flow problem considering low permeability inclusion. . . 210xiiiD.12 Comparison of Pollock’s and time of flight streamline discretization ap-proaches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211D.13 Comparison of Pollock’s and arc length streamline discretization ap-proaches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212xivAcknowledgmentsI especially want to thank my advisor, Prof. Roger Beckie, who gave me freedom andconstant encouragement to pursue my research interests. He has been an excellent mentornot only on the technical aspects of this research, but also on the many other aspects ofthe academic world. I feel fortunate I have had the opportunity to collaborate with suchgreat person.I also appreciate the interesting conversations I had with the others members of my thesiscommittee, Prof. Leslie Smith and Prof. Ulrich Mayer. Prof. Albert Valocchi and Dr.Marco Massabó also provided comments that improved the text and help to clarify someof the points presented in this thesis.I am grateful to all the people that directly or indirectly have funded my educationand research through the years. My PhD education and research have been foundedthrough a University of British Columbia Graduate Fellowship and a NSERC DiscoveryGrant awarded to Prof. R. Beckie. I am also grateful to have received the Thomas andMarguerite MacKay Memorial Scholarship and the Hugh Nasmith Graduate Scholarshipwhile I was a graduate student at UBC.I am deeply indebted with thousands of programmers who have generously donated theirwork to create free software. This thesis would not have been possible without such greattools.Finally, I wish to thank my family who has been a constant source of support andencouragement during these long years. To Françoise, Clara and Matias, I will neverforget all the sacrifices you have made to allow me to pursue my dreams.xvTo Françoise, Clara and MatiasxviCo-authorship StatementThis thesis has been prepared as a collection of manuscripts, either published, submittedor in preparation, that have been co-authored with individuals other than myself. Ineach case I am the first author and have conducted all the research work and manuscriptpreparation.The specific objectives of each chapter and research approach are based on my initiative inconsultation with my thesis supervisor Prof. Roger Beckie. The research work, includingderivation of numerical approximations, code development and numerical simulations,with the exception of the one-dimensional simulations presented in Chapter 2, have beenentirely done by myself with support from the co-authors as outlined below.Dr. Marco Massabó from CIMA Research Foundation in Savona, Italy; provided insight-ful comments and collaboration on the interpretation of the smoothed particle hydrody-namics (SPH) method in the context of simulations of subsurface solute transport. Inaddition, Dr. Massabó provided the results of the one-dimensional SPH simulations in-cluded in Chapter 2. Prof. Beckie, Dr. Massabó, and Prof. Valocchi from the Universityof Illinois at Urbana-Champaign, USA, also assisted with corrections and suggestions toimprove the editing of the chapters which they were involved with.xviiChapter 1Introduction1.1 MotivationGroundwater pollution has become a serious problem during recent decades. The releaseand infiltration into aquifers of pesticides, organic compounds, contaminants of biolog-ical origin and nuclear waste, among others, pose a risk for the human health and theenvironment.In 1996, the U.S. Environmental Protection Agency (EPA) identified 217,000 contam-inated sites in the U.S. that required mitigation action. Another 300,000 sites werereported cleaned up or were found to no require mitigation. It was estimated that sev-eral thousands of those sites were polluted with highly radioactive nuclear waste andwould require coordinated mitigation actions for several decades before been declaredcleaned up. In addition, it was estimated that there were between 130,000 to 450,000additional sites that could potentially require some mitigation action (EPA, 1996).In Canada, the Federal Contaminated Site Inventory (FCS, retrieved on August 10,2009) lists 3,208 sites that have at least one substance in the groundwater that occurs atconcentrations above natural levels and that pose an immediate or long-term hazard tohuman health or the environment. Those sites includes only the small proportion of casesfor which the Government of Canada has accepted some or all financial responsibility.Figure 1.1 shows a common example of groundwater contamination due to tailings infil-tration from a tailings impoundment. Once the tailings plume reaches the water tableit migrates carried by the regional groundwater flow and it can eventually impact the1Figure 1.1: Groundwater pollution due to tailings infiltration. Once the tailingsplume reaches the water table it can migrate up to several kilometersdownstream from the contaminant source carried by the regional ground-water flow.water quality of wells located up to several kilometers downstream from the contaminantsource. This example demonstrates the potential large spatial scale of problems relatedto the pollution of natural aquifers.Because of the scale of the problem, it is essential to find the most eﬀective and financiallysound mitigation actions. Possible mitigation actions include: natural attenuation (ei-ther due to dilution or natural biodegradation), passive containment, and active cleanupmeasures such as pump and treat and enhanced bioremediation (EPA, 1996). The selec-tion of the most eﬀective action requires a good understanding of the physical, chemicaland biological processes that govern the migration and transformation of contaminantsin the subsurface. At the same time, scientists and engineers who are involved in theremediation of contaminated sites are interested in finding answers to questions like:• How long will the contaminant plume take to reach a well or a river?• What will be the contaminant concentration at a given location and time?• Will dilution due to the advection and dispersion of the contaminant plume beenough to decrease the contaminant concentration to acceptable levels within areasonable time frame?2• Will the contaminant of concern be retarded with respect to the groundwater flow?• Will biodegradation be an eﬀective process to transform and remove the contami-nant from the groundwater?Reactive transport modeling has emerged during the last decades as an instrument toanswer these practical questions and as a tool to integrate fundamental knowledge of thecomplex processes that control flow, transport and chemical reactions in porous media(Steefel et al.,2005).In this thesis, new methods for the simulation of conservative and reactive transportare investigated. The methods focus upon accurate and eﬃcient solution of physicaltransport processes in heterogeneous porous media. This research addresses problemsin existing formulations that lead to, for example, negative concentrations, which areparticularly problematic because of non-linear chemical reaction rates that are commonin reactive transport simulations.1.2 Reactive Transport Modeling1.2.1 DefinitionReactive transport in natural porous media is a broad term that is used to refer tocomplex physical and chemical processes that occur at disparate spatial and temporalscales and that involve fluid flow, mass transport and chemical reactions in the subsurface(Steefel et al.,2005).The interaction between transport, and reactions is complex. On one hand, reactionssuch as mineral precipitation and dissolution can change the porosity and permeabilityof a porous medium, hence, aﬀect the fluid flow and transport properties of the medium.On the other hand, mass transport plays a key role in enabling reactions because itprovides the driving force to perturb a chemical system out of equilibrium by transport-ing and mixing reactants and, because it sets a characteristic time scale during whichreactions can take place (Valocchi, 1985). The most important transport processes forenabling reactions are advection, molecular diﬀusion, and mechanical dispersion (Steefeland MacQuarrie,1996;De Simoni et al.,2005;Steefel and Maher,2009).31.2.2 Conceptual ModelThe migration and transformation of contaminants in the subsurface is the result offluid and transport processes that occur at length scales of a single or few pores andof chemical reactions that happen at even smaller scales (molecular scale). Although,pore-scale modeling has gained considerable attraction during recent years, their use islimited to very small scale problems where they have provided insightful understanding ofbasic mechanisms. However, pore-scale modeling is not appropriate for studying practicalfield-scale problems that occur at much larger scales (Steefel et al.,2005).Therefore, most current conservative and reactive transport models used to simulatefield-scale problems are based on a continuum representation of porous media, such thatthe system properties are averaged over a representative elementary volume (REV) withlength scale equal to many pore lengths (Bear,1988;Steefel et al., 2005). Thus, theREV scale, often called Darcy’s scale or local-scale, defines the spatial scale or volumesize where fluid velocity, transport properties, concentrations and reactions rates arecomputed.In what follows we will concentrate our discussion on the use of reactive transport model-ing in hydrogeology. Furthermore, we will assume isothermal saturated groundwater flowwith constant density and negligible eﬀect of reactions on flow and transport properties.1.2.3 Mathematical ModelThe groundwater specific discharge, q, can be calculated using Darcy’s law,q = −K∇φ (1.1)where K [L/T] is the hydraulic conductivity tensor and φ [L] is the hydraulic head.The two main transport mechanisms at the REV scale are: advection, which involvesthe movement of the solute with the flow; and hydrodynamic or local-scale dispersionthat includes molecular diﬀusion and mechanical dispersion due to variations of the flowvelocity at the pore-scale (Bear, 1988). Thus, reactive solute transport at the local-scaleis modeled by a system of partial diﬀerential equations, which is given as follows for thecase of constant porosity (Bear,1988;Steefel and MacQuarrie,1996):4∂Ck∂t= ∇·(D∇Ck)−∇·(vCk) + Rk(c) k =1,...,m∂Ck∂t= Rk(c) k = m +1,...M (1.2)where Ck[M/L3] is the solute concentration of species or component k, D [L2/T] isthe hydrodynamic dispersion tensor, v = q/η [L/T] is the pore water velocity, η is theporosity of the medium, Rk[M/L3/T] is the total reaction rate for species or componentk,c = (C1,...,CM) is the concentration vector, m is the number of species or componentsin the aqueous (mobile) phase, and M is the total number of species or components.The most common expression to compute the coeﬃcients of the dispersion tensor D foran isotropic porous medium considering a Cartesian coordinate system is (Bear,1988)Dij= (αTq + Dm)δij+ (αL−αT)vivjq(1.3)where Dm[L2/T] is the molecular diﬀusivity, δijis Kronecker’s delta, q = |v| [L/T]is the magnitude of the pore water velocity, and αLand αT[L] are the longitudinaland transverse dispersivity of the medium, respectively. Alternative expressions for thedispersion tensor components include diﬀerent transverse dispersivities for the horizontaland vertical directions (Burnett and Frind,1987;Lichtner et al., 2002), however, thosemodels are less commonly used.The reaction term in (1.2) may include homogeneous reactions that occur in a singlephase or heterogeneous reactions that include constituents in more than one phase, e.g.sorption which includes the solid and aqueous phases (Rubin,1983;Mayer et al.,2002).From a practical point of view, sorption and biodegradation are the two most relevantreactions in groundwater, because of their role in retarding the migration of heavy metalsand in the transformation of hydrocarbons, respectively; which are the two most commonsubstances in contaminated aquifers (EPA, 1996).51.3 Numerical SolutionThe system of equations in (1.2) corresponds to a set of non-linear partial diﬀerentialequations (PDE) and, in general, must be integrated numerically. There are two diﬀerentnumerical approaches to integrate the system of PDEs: a fully-implicit approximation oran operator splitting approach (Yeh and Tripathi,1989;Steefel and MacQuarrie,1996).The simplest one is based on an operator splitting formulation that allows decouplingof the transport and reaction terms. Then, equation (1.2) is split into two terms rep-resenting transport and chemical reactions. Usually, the transport component, whichcorresponds to a linear PDE, is solved first, then the concentrations computed as resultof the transport step are used as initial conditions to compute the solution of the non-linear set of ordinary diﬀerential equations that represent chemical reactions. Variationsof the operator splitting approach include schemes that iterate between the solutions ofthe transport and reaction terms or that switch the order of the evaluation (Steefel andMacQuarrie,1996).In the discussion that follows we will assume that an operator splitting approach is usedto evaluate (1.2) and we will focus our analysis on the numerical solution of the transportterm. However, we must emphasize that the analysis presented below is also valid forfully implicit implementations.1.3.1 Particularities of Flow and Transport in Porous MediaThe numerical integration of the advection-dispersion-reaction (ADR) equation, (1.2),presents some unique challenges.First, because of the geological origin of aquifers, hydraulic conductivity varies by sev-eral orders of magnitudes within relatively short distances. For example, Sudicky (1986)reports diﬀerences of more than thirty times between hydraulic conductivity values sepa-rated by few centimeters in a shallow sandy aquifer at Canadian Forces Base in Borden,Ontario, Canada. Similar variations were also observed at the Macrodispersion Experi-ment (MADE) site in Mississippi, USA, where local-scale hydraulic conductivity valuesvaried by more than four orders of magnitude within an area of approximately 250 x 300 m(Boggs et al.,1992;Zheng and Gorelick,2003).Large variations in hydraulic conductivity produce not only important variations in flowvelocity magnitude, but also in the direction of the flow (Sudicky, 1986). For example,6Figure 1.2: Vertical cross-section of a synthetically generated aquifer using the esti-mated statistics of the sandy aquifer at Canadian Forces Base, Borden,Ontario (Mackay et al., 1986; Sudicky, 1986; Freyberg, 1986). Naturallogarithm of the hydraulic conductivity (top) and simulated groundwatervelocity (bottom). The cross-section is 32 m long and 8 m high. Largevariations of hydraulic conductivity within short distances result in im-portant variation of the magnitude and direction (arrows) of the flowvelocity.Figure 1.2 shows a vertical cross-section of a synthetically generated aquifer using theestimated statistics of the sandy aquifer at the Borden site (Mackay et al.,1986;Sudicky,1986; Freyberg, 1986). The figure also shows simulated groundwater velocities, thatdemonstrate that variations of hydraulic conductivity within short distances result inimportant variations of the magnitude and direction of the flow velocity.Because of the heterogeneity of the flow velocity, adjacent fluid parcels may travel at verydiﬀerent velocities. Such variations in travel time result in stretching and spreading ofthe contaminant plume, which manifests as large variations of concentration within shortdistances even at the order of few centimeters (Mackay et al.,1986;Molz and Widdowson,71988; Benson and Meerschaert, 2008). Figure 1.3 shows how variations in velocity canproduce dramatic changes in the shape of a initially regular plume within short traveldistances.Figure 1.3: Solute plume migration in vertical cross-section shown in Figure 1.2. Anidealized initial rectangular solute plume (red rectangle on top figure)migrates carried by the flow velocity. Because of the heterogeneity of theflow field, the shape of the plume becomes very irregular after the centerof mass of the plume has travelled approximately 10 m (bottom).Second, local-scale dispersion is anisotropic and it is much more important in the directionof the flow than in the transverse directions. Table 1.1 lists estimated dispersivity valuesbased on data collected in field- and laboratory-scale experiments. While longitudinaldispersivity ranges between 3 · 10−2and 5 · 10−1m, typical transverse dispersivity is ofthe order of 10−3m, which is equivalent to transverse dispersion coeﬃcients of the orderof molecular diﬀusion for typical pore water velocities (Cirpka et al., 2006). Note thatexperimental measurements of local-scale dispersion are diﬃcult, thus most estimateddispersivity values are based on spatial moment analysis, which measures the spreading ofthe plume, or eﬀective parameters that measure the combined eﬀect of flow heterogeneityand local-scale dispersion (Jose et al., 2004). Therefore, most of the values presented in8Table 1.1 represent upper bounds for local-scale longitudinal and transverse dispersivities.αL(m) αT(m) Comments3 · 10−2−5 · 10−15 · 10−4−1 · 10−3Based on data collected duringfield-scale tracer test in a shallowunconfined sand and gravel aquifer onCape Cod, Massachusetts, USA (Hesset al.,2002)4 · 10−14 · 10−2Based on spatial moment analysis ofdata collected during the large-scaleexperiment conducted at the Bordensite in Ontario, Canada (Freyberg,1986)– 1 · 10−6−2 · 10−4Calculated from reactive plume lengthsin laboratory-scale experiments withhomogeneous materials (Cirpka et al.,2006)4 · 10−12 · 10−3Eﬀective dispersivities obtained fromanalysis of breakthrough curves in a14 m long sandbox filled with fourdiﬀerent types of silica sand (Joseet al.,2004)Table 1.1: Estimated dispersivity values from field- and laboratory-scale experiments.The heterogeneity of the flow velocity field together with the small magnitude of local-scale dispersion coeﬃcients in natural porous media, have important consequences forkey transport processes such as spreading, dilution and mixing.Kitanidis (1994) discusses the diﬀerence between the spreading and dilution of a soluteplume. Spreading is defined as the stretching of the plume and can be measured as therate of change of the second central spatial moment. Dilution is the process by whichthe initial solute mass is distributed in an increasing volume. Mixing is the result ofthe combined action of the stretching and folding of material lines of the plume, and themass exchange due to local-scale dispersion (Weeks and Sposito, 1998). While spreading isproduced by the spatial velocity variability, dilution and mixing are due to the combinedaction of the heterogeneity of the flow field and most importantly local-scale dispersion.Because of the relatively small magnitude of the local-scale dispersion and in absenceof sorption, mixing between a contaminant and other chemical species present in the9natural groundwater occurs in a narrow zone located along the irregular edges of thecontaminant plume (Oya and Valocchi,1998).Figure 1.4 shows the diﬀerence between spreading and mixing using results of simulationsincluded in Section 2.3.3. The figures show simulated concentrations for two conservativetransport scenarios with diﬀerent Péclet number (Pe): advection-only (Pe= ∞) and foradvection and dispersion (Pe=200). Under a purely-advective scenario the solute plumebecomes irregular due to variations in velocity, however initial concentration values do notchange and there is a sharp interface between fluid zones with and without solute. Whenlocal-dispersion is included, there is solute mass transfer in areas of high concentrationsnear the plume edges that results in a thin area, relative to the typical length of thelocal-scale heterogeneity, with concentrations lower than the initial value. In many fieldsituations the zone with lower concentrations would correspond to a mixing area wherethe solute and the natural groundwater mix enabling chemical reactions. For example,natural and enhanced attenuation of organic contaminants occur in a narrow zone closeto the plume boundaries where the contaminant (substrate) and the electron acceptor(e.g. oxygen) mix (Oya and Valocchi,1998;Cirpka et al., 1999b; Ham et al.,2004).In the next section, we will argue that the characteristics of the flow and solute transportprocess described above must be considered in the selection of numerical methods tosimulate conservative and reactive solute transport in groundwater.1.3.2 Numerical MethodsWe start this section by reviewing some of the most common numerical schemes that areused to simulate conservative and reactive solute transport.1.3.2.1 Mesh-based numerical methodsThis category includes finite diﬀerence, finite volume or finite element methods. In finitediﬀerence and finite volume schemes each grid cell defines a new control volume withinwhich parameters and variables are considered constant (Steefel and MacQuarrie,1996).Mesh-based methods are relatively easy to implement, have convergence, stability andaccuracy properties that are well understood, and it is possible to develop formulationsthat are mass conservative. Because of those characteristics, mesh-based approximations10Figure 1.4: Eﬀect of advection and local-scale dispersion on solute concentration. Fig-ure shows a small part of the simulated solute plume in Section 2.3.3 forPéclet number, Pe= ∞ (top) and Pe= 200 (bottom). Advection onlyaﬀects the shape of a contaminant plume (top). If local-dispersion is in-cluded, mass transfer occurs in areas of high concentration gradients anda mixing zone develops around the plume edges (bottom). The mixingzone is critical to enable some chemical reactions such as biodegradation.11are used in most reactive transport packages (e.g. Pruess,1991;White et al.,1995;Mayeret al.,2002;Mills et al.,2007).Low-order mesh-based schemes to approximate advection, e.g. upstream finite diﬀer-ence, introduce large amount of artificial diﬀusion and mixing (Steefel and MacQuarrie,1996). Therefore, alternative schemes based on high-order approximations that were firstdeveloped to simulate problems in computational fluid dynamics, have been adopted byreactive transport modelers. Examples of high-order schemes are flux-corrected transport(FCT) methods (Boris and Book,1973;Zalesak, 1979), which combine high- and low-order schemes; and total variation diminishing (TVD) schemes (Harten and Lax,1984;Yee et al.,1985;Cox and Nishikawa,1991).1.3.2.2 Hybrid Eulerian-Lagrangian methodsHybrid schemes are based on the same general concept, the use of particles to handleadvection and a grid-based method to handle dispersion. Each time step is split into twosub-steps. First, changes in concentrations due to advection are computed by forwardor backward particle tracking. Then, concentration values are interpolated onto a grid.Next, grid concentration values are used to solve for dispersion, and eventually reactions,using some traditional mesh-based solver. Multiple variations of this approach existdepending on the interpolation methods and tracking algorithm. Examples of these kindsof methods are: hybrid Eulerian-Lagrangian methods (Neuman, 1981, 1984), method ofcharacteristics (MOC) and hybrid method of characteristics (HMOC) in the MT3DMSpackage (Zheng and Wang, 1999) and MOC3D (Konikow et al., 1996), and Eulerian-Lagrangian localized adjoint methods (ELLAM) (Celia et al.,1990;Russell and Celia,2002).1.3.2.3 Random walk particle tracking methodsRandom walk particle-tracking methods (RWPT) have long been used to simulate con-servative solute transport in porous media (Ahlstrom et al.,1977;Pickens and Grisak,1981; Tompson and Gelhar,1990;Tompson, 1993). In this type of model, solute massis distributed among a set of particles that move carried by the flow velocity and by arandom drift that models dispersive transport (Delay et al.,2005;Salamon et al.,2006).Solute concentrations are estimated by averaging the mass contained in the particles12found in some specified volume. Therefore, concentration values depend upon the totalnumber of particles, size of the averaging volume, and spatial particle distribution.The popularity of RWPT methods is due to its natural capacity to accurately simulateadvection and ease of implementation. Because of its advantages RWPT has becomethe de facto standard method in numerical studies of plume spreading and dilution (e.g.see Delay et al.,2005;Salamon et al., 2006, and references therein). However, RWPTmethods are less attractive for the simulation of reactive transport because: (i) it isdiﬃcult to simulate general heterogeneous reactions that include the solid phase, (ii)a very large number of particles is required to obtain an accurate estimation of lowconcentration values that is crucial to approximate reactions that occur in the mixingzone along the plume edges, and (iii) simulations that include multiple species need alarge number of particles to track individual species.1.3.3 Limitations of Current Numerical MethodsAccording to Steefel and MacQuarrie (1996) there are three main properties that a nu-merical method must satisfy to be used in reactive transport simulations: (i) accuracyin space and time, which includes minimizing numerical diﬀusion and mass conserva-tion errors, (ii) monotonicity, which means avoiding spurious oscillations (e.g., negativeconcentrations); and (iii) computational eﬃciency. Next, we evaluate current numericalmethods based on those three criteria.1.3.3.1 AccuracyMesh-based numerical methods, including high resolution methods, have problems to ac-curately simulate multidimensional advection-dominated transport because of numericaldispersion that results in excessive artificial mixing, dilution, and overestimation of reac-tions rates (Steefel and MacQuarrie,1996;Cirpka et al.,1999;Zheng and Wang,1999).Numerical dispersion is more important when the grid is not aligned with the directionof the flow (Frind et al., 1987), which is always the case in non-uniform flows as foundin heterogeneous porous media.On the other hand, hybrid schemes that require interpolating concentrations to a back-ground grid also introduce numerical dispersion even if they provide a very accuratesolution for advection. In the case of RWPT methods, concentration values can only13A!B!A + B!Figure 1.5: Subgrid-scale segregation and cell averaged concentration values. SpeciesA and B are physically segregated at the subgrid-scale (left). However,they would appear well-mixed at the cell scale (right). If A and B aretwo reactants in a chemical reaction, then numerical simulations basedon cell averaged concentrations would overestimate the reaction rate.be obtained after averaging the mass of particles over some control volume, which alsoresults in numerical mixing.The use of cell average concentration values can introduce large errors in the estimationof dilution and reaction rates. For example, Figure 1.5 shows two species, A and B,which are physically segregated, however, they would appear well-mixed in numericalsimulations that use cell averaged concentration values. If A and B are reactants in achemical reaction, then the simulated reaction rate would overestimate the real reactionrate (which is zero, in this case).The errors due to numerical dispersion and cell averaging are smaller for larger values oflocal-scale dispersion, because concentration values within the cell volume are smoothedout by dispersion (Steefel and MacQuarrie, 1996). However, as discussed earlier, local-scale dispersion in porous media, particularly in the transverse direction, is small andits eﬀect to smooth out concentration fluctuations is limited. Therefore, sub-grid scaleconcentration fluctuations are important and the use of cell averaged concentration val-ues is an important source of error in conservative and reactive transport simulations(Frind and Germain,1986;Frind et al.,1987;Molz and Widdowson,1988;Benson andMeerschaert,2008).In evaluating the accuracy of a numerical method for simulating reactive transport, it14is also important to keep in mind that any error in the solution of the transport com-ponent can be greatly amplified by non-linear reactions. Thus, methods that performacceptably well to simulate conservative solute transport, can produce large errors whenchemical reactions are included (Steefel and MacQuarrie, 1996). For example, Cirpkaet al. (1999b) demonstrated that small amounts of numerical dispersion in simulations ofbiodegradation controlled by transverse mixing simulated using a high-order finite volumemethod, can result in larger errors in the estimation of reaction rates and contaminantmass removal.1.3.3.2 MonotonicitySpurious oscillations in simulations of conservative solute transport arise due to the useof non-linear high-order methods to control numerical dispersion (Steefel and MacQuar-rie,1996;Cirpka et al., 1999) and numerical approximations of the oﬀ-diagonal entries(“cross-terms”) in the local-dispersion tensor (Herrera and Valocchi, 2006). Multidimen-sional high-order mesh-based solvers for advection based on the FCT and TVD schemes,which are supposed to suppress numerical oscillations, often result in small oscillations(Steefel and MacQuarrie,1996;Herrera and Valocchi, 2006). Numerical oscillations thatarise from the solution of parabolic or elliptic PDEs that include mixed derivatives or“cross-terms” are a well known problem and have been the subject of many researcheﬀorts in recent years (e.g. Nordbotten and Aavatsmark,2005;Le Potier, 2005b; Mlacnikand Durlofsky,2006;Edwards and Zheng,2008;Yuan and Sheng,2008;Lipnikov et al.,2009). To this day, no single solution provides a scheme that can be used in generalscenarios.Although, small oscillations can be usually tolerated in conservative transport simula-tions, they are unacceptable in reactive transport simulations because they can resultin unstable solutions in presence of non-linear chemistry. For example, Steefel and Mac-Quarrie (1996) discusses the eﬀect of small oscillations in a problem involving organic car-bon degradation via sulfate reduction coupled to two equilibrium dissolution-precipitationreactions. They simulate solute transport using a high-order FCT method that intro-duces spurious oscillations that do not produce problems in a tracer simulation, but thatproduce unstable results when chemical reactions are included.151.3.3.3 PerformanceNumerical simulations of reactive transport in porous media are computationally de-manding. Although, the increasingly availability of high-performance computers hasmade feasible detailed simulations of reactive transport in two- and three-dimensionaldomains (e.g. PFLOTRAN, retrieved on August 12, 2009), it is still not possible to re-solve practical problems with enough detail to capture all the scales of heterogeneity thatare relevant for reactive transport (Steefel et al.,2005).Performance is also important when a large number of scenarios must be simulated.For example, because of our inability to observe all the scales of physical and chemicalheterogeneity present in natural porous media, reactive transport simulations include ahigh degree of uncertainty. A standard way to deal with uncertainty is based on a MonteCarlo approach that involves simulating many equally probable scenarios to determineprobability distributions for possible outcomes (Steefel et al., 2005). Simulating a largenumber of realizations is also a requirement of some methods to estimate the eﬀects ofvariations in the input parameters on the model results (sensitivity analysis) or of someautomatic parameter calibration frameworks (Hill and Tiedeman,2007).Nowadays, high performance requires to use numerical methods that can be implementedin algorithms that are amenable to parallelization. While RWPT methods can be imple-mented using an embarrassingly parallel algorithm, eﬃcient implementations of mesh-based algorithms, although possible, are more diﬃcult to obtain (Mills et al.,2007).Onthe other hand, RWPT have very low convergence and very large numbers of particles (ofthe order of billions (Suciu et al., 2006)) are required to obtain accurate results, whichcounterbalances its parallel advantages.Finally, high-order multidimensional mesh-based solvers for advection are usually im-plemented using explicit schemes (Steefel and MacQuarrie, 1996). Explicit solvers havestability limits on the time step size given by the Courant–Friedrichs–Lewy (CFL) condi-tion,∆t ≤∆/|v|, where∆is the cell size. Therefore, the stability limit is more restrictivefor finer grids and finer discretizations require smaller time steps, hence, computationaleﬀort. Since mesh-based multidimensional solvers introduce a global coupling betweenconcentration values, the stability limit is global and given by the maximum velocity inthe grid. Thus, it is not possible to take advantage of the irregular velocity distributionto use greater time steps in slower areas of the domain.161.4 Lagrangian Numerical Methods1.4.1 Meshless MethodsKernel interpolation methods simulate mass transport using a collection of particles thatmove according to the velocity field and carry and exchange solute mass with surroundingparticles. Particle locations are used as quadrature points to evaluate integral interpola-tions of variables and their derivatives. Importantly, these schemes are able to incorporatediﬀusive eﬀects and mixing without using a grid or mesh, so they are also called mesh-less methods. Some examples of this type of method are Vortex methods (Cottet andKoumoutsakos, 2000) and particle strength exchange (PSE) method (Degond and Mas-Gallic,1989a;Zimmermann et al., 2001). Methods based on this approach give accurateand stable results if a remeshing technique is used to control errors that result from ir-regular spatial particle distributions produced by non-uniform flow fields. The remeshingstep introduces numerical dispersion that can be controlled but not completely avoidedby using suitable interpolation schemes (Cottet and Koumoutsakos,2000;Chaniotis et al.,2002).Smoothed Particle Hydrodynamics (SPH) methods are another type of kernel-based in-terpolation scheme (Gingold and Monaghan,1977;Lucy,1977).Cleary and Monaghan(1999) presented a SPH scheme that allows one to solve the multidimensional advection-dispersion equation, assuming isotropic dispersion, using an integral interpolation of thedispersion operator that it is supposed to be less sensitive to particle disorder than tra-ditional kernel interpolation schemes. Since the method can handle dispersion withoutremapping the concentration field onto a grid, it is free of numerical dispersion and gridorientation eﬀects.As discussed in Chapter 2, a key property of kernel interpolation methods is that theytrack concentration values, in contrast to RWPT approaches which fundamentally trackparticles with fixed masses. This feature allows one to evaluate reactions at individualparticles (e.g. Chaniotis et al., 2003). Heterogeneous reactions that include the solid(immobile) phase can be handled by introducing additional fixed particles (Tartakovskyet al.,2007).171.4.2 Streamline-Based SimulationsStreamline-based methods have been successfully used to simulate oil migration (Thieleet al., 1996, 1997) and multidimensional solute transport (Crane and Blunt,1999;Di Do-nato and Blunt,2004;Obi and Blunt, 2004, 2006). These methods use a numerical gridthat adapts to the flow field, which reduces numerical dispersion and grid orientationeﬀects. Because of its adaptation to the flow and its ability to minimize numerical dis-persion, the method is well suited for simulations of advection-dominated transport asfound in heterogeneous porous media (Di Donato et al., 2003). Moreover, the use ofstreamlines allows the transformation of a multidimensional transport equation to a setof individual one-dimensional transport problems. The numerical solution of the result-ing set of one-dimensional transport problems allows the use of more eﬃcient numericalsolvers, more relaxed stability constraints and it is amenable to parallelization (Craneand Blunt,1999;Bandilla et al., 2009). Because of the eﬃciency of the method, it is pos-sible to simulate large-scale domains with fine spatial and temporal resolution (Di Donatoet al.,2003;Obi and Blunt, 2004, 2006). In addition, chemical reactions, including ho-mogeneous and heterogeneous reactions can be easily handled (Crane and Blunt,1999;Di Donato and Blunt,2004).1.5 DiscussionThe previous analysis demonstrated that no single traditional numerical method presentsthe three main features sought in reactive transport simulations. Moreover, one of theconditions, monotonicity, is not satisfied by any of the current methods if anisotropicdispersion is considered.Because of the limitations of current numerical schemes, reactive transport simulationsmust make some trade-oﬀs to obtain practical results. For example, low-order approx-imations for advection are preferred over more accurate high-order approximations, be-cause they do not suﬀer numerical oscillations. Similarly, simulations that include local-dispersion assume isotropic dispersion or remove the cross-terms to avoid introducingnon-physical artifacts, e.g. negative concentrations. In both cases, monotonicity comesat the cost of tolerating additional numerical errors.In our opinion, most of the problems that aﬀect current numerical methods are caused bythe use of a grid or mesh, either to compute approximations of concentrations derivatives18as in finite diﬀerence or finite element methods or, to compute cell averaged concentrationvalues as in RWPT, MOC and other hybrid approaches. As discussed, the use of a singlemultidimensional grid introduces grid orientations eﬀects due to the non-uniform flowdirection, artificial mixing because of the computation of cell averaged concentrations,and global stability restrictions. Therefore, it seems that to overcome many of the prob-lems that plague current numerical schemes, one should develop numerical methods thatdo not require a rigid multidimensional grid. Numerical methods based on a Lagrangiandescription of solute transport satisfy that condition and are attractive alternatives tosimulate reactive solute transport.Because of their ability to control numerical dispersion and grid orientation eﬀects andtheir eﬃciency, meshless methods and streamline-based simulations are attractive alter-natives for the simulation of reactive transport in natural porous media. However, theyalso present some deficiencies that can be problematic for their use in reactive transportcodes.The standard SPH approximation for diﬀusion (Cleary and Monaghan, 1999) can onlybe used to simulate isotropic dispersion. Therefore, the use of this type of methods inreactive transport modeling in porous media requires deriving new expressions to simulateanisotropic dispersion.Although, approximations for longitudinal dispersion along individual streamlines arestraightforward, transverse mixing between streamlines is more diﬃcult to simulate. Twoapproaches have been used to incorporate transverse dispersion in streamline-based simu-lations. In the first one, solute transport is solved using a flow-oriented grid and transversedispersion is included as a flux component perpendicular to the streamlines (Frind andGermain,1986;Frind et al.,1987;Cirpka et al., 1999). This approach has been suc-cessfully used in two-dimensional simulations (Frind et al.,1987;Cirpka et al., 1999b),but it has not been extended to three-dimensions. A second alternative consists in usinga hybrid approach (Obi and Blunt, 2004). First, advection is solved along streamlines.Then, concentration values are mapped onto a grid where a mesh-based solver is used tosolve for dispersion. Finally, concentration values are interpolated back from the grid tothe streamlines. The interpolation from and to streamlines introduces some numericalerror that is diﬃcult to quantify (Obi and Blunt, 2004). Because the interpolation mustbe done at each time step, the cumulative eﬀect can be important even if an accurateinterpolation scheme is used.The limitations of the meshless SPH and streamline-based methods for the simulation19of multidimensional reactive transport in porous media provide the motivation for theresearch in this thesis.1.6 ObjectivesThe main objective of this thesis is to develop, implement, and evaluate new numericalschemes based on meshless methods and streamline-based simulations to simulate reactivetransport. The main objective includes the following specific objectives:1. To derive and implement a meshless approximation for conservative transport inheterogeneous porous media. This includes deriving expressions to approximateisotropic and anisotropic local-scale dispersion.2. To devise schemes to incorporate local-scale dispersion (longitudinal and transverse)in multidimensional streamline-based simulations.3. To evaluate the newly derived numerical schemes in terms of accuracy, monotonicityand performance.4. To compare the new schemes with others current numerical methods such as: high-order finite volume, method of characteristics and random-walk particle trackingmethods.5. To evaluate the suitability of using the new meshless and streamline-based schemesin reactive transport simulations.1.7 OrganizationThis thesis is organized in four additional chapters and four appendices. Chapters 2,3, and 4 correspond to manuscripts that have been published or will be submitted forpublication.Chapter 2 presents the application of a meshless numerical method based on smoothedparticle hydrodynamics (SPH) for the simulation of conservative transport in heteroge-neous geological formations assuming isotropic dispersion. The chapter includes ana-lytical and numerical results that demonstrate that the new proposed scheme is stable,20accurate, and conserves global mass. Appendix A presents details of the derivation ofthe SPH-based numerical approximation.In Chapter 3, we extend the SPH-based approximation implemented in Chapter 2 tosimulate anisotropic dispersion. In addition, we compare the new approximation withanother meshless method (particle strength exchange) and a mesh-based finite volumescheme to simulate the dispersion of a two-dimensional contaminant plume under diﬀerentscenarios. We conclude that, although attractive to simulate conservative transport, thenew SPH-based approximation is unsuitable for reactive transport simulations because ofspurious oscillations that arise if the dispersion tensor is anisotropic. The new numericalapproximation is based on a SPH approximation for mixed second order derivatives,which is derived in detail in Appendix B.Chapter 4 presents the derivation of a new numerical scheme to incorporate dispersion– including transverse dispersion – in streamline simulations. A key element of themethod is that dispersion is approximated in a flow oriented grid using a combinationof a one-dimensional finite diﬀerence scheme and a meshless approximation for isotropicdispersion. We demontrate through analytical and numerical results that the resultingapproximation is always monotonic and, hence, suitable for reactive transport simula-tions. Some key issues that arise in streamline-based simulations such as: streamlinetracing, streamline spatial distribution and streamline discretization, are discussed inAppendix D.Finally, Chapter 5 summarizes the main conclusions of the three preceding chapters andincludes recommendations for future research directions.211.8 ReferencesCleaning Up the Nation’s Wastes Sites: Markets and Technology Trends, Tech. Rep.542R96005A, U.S. Environmental Protection Agency, 1996.Federal Contaminated Sites Inventory, http://www.tbs-sct.gc.ca/fcsi-rscf/home-accueil.aspx, retrieved on August 10, 2009.Ahlstrom, S., H. Foote, R. Arnett, C. 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Wang, MT3DMS: A Modular Three-Dimensional Multispecies Trans-port Model for Simulation of Advection, Dispersion, and Chemical Reactions of Con-taminants in Groundwater Systems; Documentation and User’s Guide, Contract ReportSERDP-99-1, US Army Engineer Research and Development Center, Vicksburg, MS,1999.Zimmermann, S., P. Koumoutsakos, and W. Kinzelbach, Simulation of pollutant trans-port using a particle method, J. Comput. Phys., 173,322–347,2001.28Chapter 2A Meshless Method to SimulateSolute Transport in HeterogeneousPorous Media12.1 Introduction2.1.1 BackgroundContaminant transport in natural aquifers is a complex, multiscale process that is fre-quently studied using numerical methods. Conservative solute transport is typicallymodeled using the advection-dispersion equation (ADE). Despite the large number ofavailable numerical methods that have been developed to solve it, the accurate numericalsolution of the ADE still presents formidable challenges. In particular, current numeri-cal solutions of multidimensional advection-dominated transport in non-uniform velocityfields are aﬀected by one or all of the following problems: numerical dispersion that in-troduces artificial mixing and dilution, grid orientation eﬀects, and unphysical numericaloscillations (Herrera and Valocchi,2006).To correctly capture the basic mechanisms that control conservative solute transport innatural aquifers an ideal numerical method should be able to: (i) accurately capture1A version of this chapter has been published. P. Herrera, M. Massabo, and R. Beckie (2009) Ameshless method to simulate solute transport in heterogeneous porous media. Adv. Water Resour.,32:413–429.29the eﬀect of small-scale velocity fluctuations upon solute distribution, (ii) simulate theeﬀect of small values of local-scale dispersion in advection-dominated transport, (iii)reproduce small-scale concentration fluctuations and (iv) allow the eﬃcient simulation ofproblems at moderate to large scales. In what follows we only discuss numerical methodsto simulate conservative solute transport but the same set of requirements should besatisfied by any successful numerical method used to simulate reactive transport.The objective of this paper is to develop and test a meshless method to simulate contam-inant transport in porous media. This work was primarily motivated by our experiencein theoretical investigations of solute mixing and plume dilution in heterogeneous porousmedia. Those investigations require an eﬃcient numerical method that is able to accu-rately simulate solute transport in multidimensional non-uniform velocity fields. We arealso interested in developing a meshless method that can be used to simulate conservativeand non-conservative solute transport with minimum modification.The main contributions of this paper are: (i) to derive a meshless approximation for thedispersion operator in heterogeneous porous media flow, (ii) to compare the meshlessapproximation with other numerical methods traditionally used to simulate conserva-tive solute transport in heterogeneous multidimensional velocity fields, (iii) to studythe convergence properties of the meshless approximation of the dispersion operator fordiﬀerent spatial node distributions, and (iv) to demonstrate that the proposed mesh-less approximation can be used to solve the ADE in diﬀerent scenarios ranging fromadvection-dominated to dispersion-dominated solute transport with accuracy better thanor comparable to standard numerical methods. Although we address only conservativetransport in this paper we highlight the advantages of the proposed scheme for simulatingreactive transport.2.1.2 Numerical MethodsA detailed discussion of every numerical method used to solved the ADE is beyondthe scope of this manuscript, however we briefly discuss some of them to motivate thedevelopment of our new meshless method.Grid or mesh-based methods such as finite diﬀerence, finite volume or finite element meth-ods are relatively easy to implement, their convergence, stability and accuracy propertiesare well understood, and it is possible to develop formulations that are mass conserva-tive. On the other hand, mesh-based methods have diﬃculty in accurately simulating30multidimensional advection-dominated problems because of numerical dispersion that in-troduces excessive artificial mixing and dilution of the contaminant plume. Therefore,grid-based methods are only advised for problems with low Péclet number (Zheng andWang,1999).Hybrid schemes were developed to address the limitations of grid-based methods. Thefollowing hybrid schemes are based on the same general concept – the use of streamlinesor particles to handle advection and a grid-based method to handle dispersion: hybridEulerian-Lagrangian methods (Neuman, 1981, 1984), method of characteristics (MOC)and hybrid method of characteristics (HMOC) in the popular MT3DMS (Zheng andWang, 1999) and MOC3D (Konikow et al., 1996), Eulerian-Lagrangian localized adjointmethods (ELLAM) (Celia et al.,1990;Russell and Celia, 2002), and three-dimensionalhybrid streamline-grid approaches (Obi and Blunt, 2004). At each time step, these meth-ods solve the advection–dispersion equation in two steps. First, changes in concentrationsdue to advection are computed using some suitable scheme such as particle-tracking orby solving the transport equation along streamlines. Then, concentration values are in-terpolated onto a grid. Next, grid concentration values are used to solve for dispersion,and eventually reactions, using some traditional mesh-based solver. Multiple variationsof this approach exist depending on the interpolation methods and tracking algorithm.However, all of them require mapping concentrations between cell centers and particlelocations or streamline nodes. This remapping step introduces numerical dispersion thatis diﬃcult to quantify and control in multidimensional simulations. Moreover, the nu-merical dispersion due to the remapping step is more important in simulations with largegrid Péclet number where the eﬀect of dispersion is not enough to smooth the sub-gridscale concentration distribution.The limitations of grid-based and hybrid methods to simulate advection dominated prob-lems have motivated the development of Lagrangian and meshless methods includingthose based on a random-walk and those based on integral interpolations.Random-walk particle-tracking methods (RWPT) have long been used to simulate con-servative solute transport in porous media (Ahlstrom et al.,1977;Smith and Schwartz,1980; Pickens and Grisak, 1981). In this type of model, solute mass is distributed amonga set of particles that move carried by the flow velocity and by a random drift that mod-els dispersive transport (Delay et al.,2005;Salamon et al., 2006). Solute concentrationsare estimated by averaging the mass contained in the particles found in some specifiedvolume. Therefore, concentration values depend upon the total number of particles, sizeof the averaging volume, and spatial particle distribution. The numerical precision of31the computed concentration is limited by the finite number of particles used in a simula-tion and the calculated concentration usually exhibits numerical oscillations that can beamplified in presence of non-linear reactions (Tompson and Dougherty,1992;Tompson,1993). Therefore, the use of the method is limited to conservative transport or to reac-tive transport simulations with simple reactions that can be modeled by changing thestate or phase of individual particles, e.g. sorption (Valocchi and Quinodoz, 1989). Thepopularity of RWPT is due to its natural capacity to accurately simulate advection, easeof implementation, and relatively moderate computational requirements. Because of itsadvantages RWPT has become the de facto standard method used in numerical studiesof plume spreading and dilution.In contrast to RWPT methods that simulate dispersion in a fluid by a random movementof particles that carry solute mass, kernel interpolation methods simulate mass transportusing a collection of particles that move according to the velocity field and carry and ex-change solute mass with surrounding particles. Particle locations are used as quadraturepoints to evaluate integral interpolations of variables and their derivatives. Importantly,these schemes are able to incorporate diﬀusive eﬀects and mixing without using a grid ormesh, so they are also called meshless methods. Some examples of this type of methodare vortex methods (Cottet and Koumoutsakos, 2000) and particle strength exchange(PSE) method (Degond and Mas-Gallic,1989a).Zimmermann et al. (2001) present, tothe best of our knowledge, the only application of this type of method to simulate solutetransport in porous media at the continuum scale. Their results indicate that the methodgives accurate and stable results for the flow configuration considered if a remeshing tech-nique is used to control errors that result from irregular spatial particle distributions. Asin other methods, the remeshing step introduces numerical dispersion that can be con-trolled but not completely avoided by using a suitable interpolation scheme (Cottet andKoumoutsakos,2000;Chaniotis et al.,2002).Smoothed particle hydrodynamics (SPH) methods are another type of kernel-based in-terpolation scheme (Gingold and Monaghan,1977;Lucy,1977).Cleary and Monaghan(1999) presented a SPH scheme that allows one to solve the multidimensional ADE us-ing an integral interpolation of the dispersion operator that is less sensitive to particledisorder than traditional kernel interpolation schemes. Since the method can handledispersion without remapping the concentration field onto a grid, it is free of numericaldispersion and grid orientation eﬀects. As we show later, a key property of SPH is thatthe method tracks concentration values, in contrast to RWPT approaches which funda-mentally track particles with fixed masses. This feature of SPH allows one to simulate32reactive transport directly with individual particles (e.g. Chaniotis et al.,2003).We next propose a method to simulate solute transport in porous media based on the SPHformalism. We show that this SPH-based method is advantageous because of its inherentability to solve advection, its capacity to approximate dispersion in a meshless fashion,and its ability to reproduce smooth fine-scale concentration distributions appropriate forreactive transport simulations. After deriving the method in Section 2.2, we compare andcontrast it to a conventional RWPT method and other traditional mesh-based methodsin Section 2.3.2.2 Monte Carlo SPH methodIn application of meshless methods such as SPH it is appropriate to reformulate the ADEin terms of a Lagrangian coordinate system asdrdt= v(r) (2.1)dCdtr= ∇·(D∇C)|r(2.2)where C is the solute concentration, D is the local-scale dispersion tensor, v is the porewater velocity, and r is the position vector of a small fluid volume or material point.As we describe next, in SPH methods the concentration field is represented using a setof particles that carry concentration information and are distributed through the domain– even in areas where solute concentration is zero. Practically all meshless methodsincluding SPH use a particle-tracking approach to integrate (2.1) in the same way asdone in RWPT (Delay et al.,2005;Salamon et al., 2006). The key distinction betweenSPH and RWPT is how dispersion is approximated. In SPH methods, dispersion – thesolution of equation (2.2) – is evaluated using a kernel interpolation approximation.The “smoothed” in SPH comes from the representation of a scalar or vector field by asmoothed integral interpolation. The smoothed interpolation AS(r) of a field A(r) isdefined as the integral (Gingold and Monaghan,1977)AS(r)=ˆA(r)W(r−r,h)dr(2.3)33where W (r−r,h) is a kernel function with compact support around r and smoothinglength h that satisfies some normalization properties (Gingold and Monaghan,1977).Spline polynomials with finite support are usually used as kernel functions because oftheir practical advantages (Price, 2004). For those functions the compact support volumeof the kernel depends upon h. Figure 2.1 shows values of the Gaussian and cubic-splinekernels and their derivatives as function of the smoothing length.−3 −2 −1 0 1 2 30.00.20.40.60.8r/hW(r/h)−3 −2 −1 0 1 2 3−0.8−0.6−0.4−0.20.0r/hW’(r/h)−3 −2 −1 0 1 2 3−2.0−1.5−1.0−0.50.00.51.0r/hW’’(r/h)−3 −2 −1 0 1 2 3−2.0−1.5−1.0−0.50.0r/hF(r/h)Figure 2.1: Kernel function W, first derivative W’, second derivative W’’ and sym-metric F function defined in Appendix A, for Gaussian (solid line) andcubic spline (dashed line) kernels.The numerical approximation of AScan be evaluated using a Monte Carlo integrationscheme by sampling the integrand A(r)W(r) at a limited set of disordered points orparticle locations. To evaluate the integral one must consider that the set of scatteredpoints is not uniformly distributed, therefore their spatial distribution must be explicitlytaken into account in the integral evaluation to get the following modified form of (2.3),As(r)=ˆA(r)p(r)W(r−r)p(r)dr(2.4)34where p(r) is the probability density of finding a particle in a given unit volume withunits of one over volume [1/L3] (Gingold and Monaghan,1982;Tartakovsky and Meakin,2005). Thus, p(r)dr can be interpreted as a non-uniform density to sample the modifiedintegrand A(r)W(r)/p(r) (Press et al., 1992, p. 316). The exact evaluation of p(r)for a set of scattered points in multiple dimensions is a very diﬃcult task, but it canbe estimated using a density estimation by kernels approach (Schaback and Wendland,2006), which yields,p(r)=1NpNpj=1W (r−rj) (2.5)Finally, the Monte Carlo approximation of (2.4) isAN(r)=1NpNpjA(rj)W (r−rj)p(rj)±O1Np(2.6)where Npis the total number of points that eﬀectively contribute to the integral. Thislast expression is a valid approximation of the integral for any set of scattered points.We note that in practice the error estimate in (2.6) is an upper bound and that theactual error also depends upon p(r), i.e. the spatial distribution of the points, and thelocal smoothness properties of A. For example, numerical studies have shown that theactual error is much smaller for points that are reasonably well distributed (Cleary andMonaghan,1999;Monaghan,2005).Traditional SPH simulations used to solve hydrodynamics equations consider that the setof locations rjrepresent the positions of a set of fluid particles with constant mass mj.In that case, the fluid mass density ρ(r) is proportional to the particle density p(r) andit can be estimated by using an expression similar to (2.5) where the mass of individualparticles appears explicitly (Monaghan, 1992, 2005). We note that the formulation givenby (2.5) and (2.6) is equivalent to the standard SPH formulation if the total mass in thesystem is one and it is uniformly distributed among particles. This is consistent withSPH simulations where, in general, the fluid mass assigned to each particle is consideredas a constant scaling parameter that is set at the beginning of individual simulations totune the spatial fluid density distribution (Dilts,1999).35There are two SPH approaches to approximate transport equations involving second orderderivatives. First, second order derivatives can be easily evaluated by diﬀerentiating (2.6).The resulting expression involves the second derivative of the kernel, so it is very sensitiveto particle disorder (Cleary and Monaghan, 1999). In that case particle positions mustbe periodically reinitialized on a regular lattice to achieve acceptable accuracy (Chaniotiset al., 2002). A second approach is based on an integral approximation of the dispersivefluxes that depends only upon the first derivative of the kernel (Brookshaw,1985;Clearyand Monaghan, 1999). Considering isotropic dispersion, i.e. D(r)=D(r)I, where I isthe identity matrix, the approximation of (2.2) isdCidt=1NpNpj=11pj(Dj+ Di)(Cj−Ci)F(rj−ri) (2.7)where Ciis the solute concentration at position riand F(rj−ri) is a function of theseparation vector and first derivative of the kernel that has spherical symmetry. AppendixA presents the derivation of (2.7).Equation (2.7) indicates that the magnitude of the contribution of solute from particle jto particle i is equal to the contribution of particle j to particle i, i.e. the mass flux is anti-symmetric fij= −fji, only if pi= pj. In general, for a set of irregularly spaced particlespi= pj, so (2.7) does not satisfy a basic property of the dispersion operator (Kuzminand Turek, 2002). Thus, it is necessary to replace the denominator by a symmetricapproximation of the form ˆpij= g(pi,pj), e.g. arithmetic or harmonic average. Ingeneral, this correction is relatively small because particles that eﬀectively contribute tothe summation are within few smoothing lengths due to the compact support property ofthe kernel. Similar corrections to recover symmetry are used in standard SPH simulationsthat consider variable smoothing lengths (Monaghan, 2005). Our MC-SPH method isbased upon this formulation to approximate dispersive fluxes that results indCidt=1NpNpj=11ˆpij(Dj+ Di)(Cj−Ci)F(rj−ri) (2.8)We refer to this approximation as the Monte Carlo SPH (MC-SPH) formulation for dis-persion. Equation (2.8) satisfies two important physical constraints. First, dispersivefluxes are anti-symmetric. Second, mass transfer occurs from higher to lower concentra-tions because for typical kernels, F(rj−ri) < 0 as shown in Figure 2.1.36In standard SPH simulations there is another approach to recover the symmetry of thefluxes that consists in including the fluid density on the right hand side of (2.2). Inthis case, density is placed inside operators following the “second golden rule of SPH”(Monaghan, 1992). The resulting numerical approximation isdCidt=Npj=1mjρiρj(ρjDj+ ρiDi)(Cj−Ci)F(rj−ri) (2.9)where ρiis the numerical approximation of the fluid mass density at positionri. Equation(2.9) is the standard SPH approximation for dissipative or dispersive transport and ithas been used to simulate heat conduction in compressible gases (Cleary and Monaghan,1999; Español and Revenga,2003;Jubelgas et al., 2004); to simulate viscous eﬀects inlow Reynolds number flows (Morris et al.,1997;Zhu et al., 1999); and to simulate solutedispersion at the pore scale (Zhu and Fox,2002;Tartakovsky and Meakin,2005). Inwhat follows we refer to this approximation as weakly compressible SPH (WC-SPH)formulation for dispersion. Although this approach is reasonable in simulations thatconsider variable fluid density so that density must be explicitly incorporated inside thedispersion operator, its use in incompressible flow simulations with constant density is, atleast, questionable. Moreover, as shown by equation (2.8), it is not necessary to use sucha pragmatic approach to develop a numerical formulation that satisfies basic physicalprinciples and conserves solute mass.We close this section by summarizing the main distinctive properties of the MC-SPHformulation. The approach considers that fluid particles represent a constant fluid volumelarger than a representative elementary volume (REV) such that Darcy’s velocities canbe defined and that particle trajectories can be computed by integrating (2.1). Solutemass is distributed among a set of particles that carry concentration values. Local-scaledispersion that occurs at scales much smaller than the REV, is modeled as a Fickian solutemass transfer process between particles. Numerically, local-dispersion is approximatedby a local integral interpolation of the dispersion operator in (2.2). We note that theparticle fluid volume does not explicitly appear in the numerical formulation and that,from a practical point of view, fluid particles can be regarded as numerical nodes.2.2.1 Time IntegrationThe time integration of the system of equations (2.1) and (2.2) requires the use of asequential procedure. First, at the beginning of each time step node locations and con-37centrations are recorded. Then, new locations are calculated by integrating (2.1) usingan explicit time-marching scheme (e.g. Runge–Kutta methods) or a particle-tracking al-gorithm (e.g. Pollock, 1988). After the new locations are computed, new concentrationsare calculated by integrating (2.2). This term can be integrated using explicit or implicitschemes. For example, using a first-order approximation it can be approximated byCt+∆ti−Cti∆t=jαijC∗j−C∗i(2.10)whereαij=1ˆpj(Di+ Dj)F (rj−ri) ≥ 0 (2.11)where C∗i= Ctior C∗i= Ct+∆tifor explicit and implicit time integration, respectively.Then, the first-order implicit approximation is given by1+∆tjαijCt+∆ti−∆tjαijCt+∆tj= Cti(2.12)It is easy to demonstrate that this integration scheme is unconditionally stable and posi-tivity preserving. Although possible, implicit schemes are seldom used because of compu-tational overhead. Since nodes move with the flow, the connectivity list, i.e. the numberof nodes within the kernel support volume of a given node changes at each time step.The memory requirements to store the associated matrix and the computational cost togenerate it and computing its inverse can be quite large, depending upon the averagenumber of nodes per smoothing length. That is why we used conditionally stable ex-plicit time integration schemes in the simulations presented in Section 2.3. We motivatethe discussion about the stability of explicit schemes by writing the first-order explicitapproximation of (2.10)Cti=1−∆tjαijCti+∆tjαijCtj(2.13)which is stable and positivity preserving if 1/αij≥ ∆t. Empirical tests have shownthat other explicit solutions are stable for time steps∆t such that (Cleary and Monaghan,1999)38∆t ≤h2D(2.14)where is a coeﬃcient that depends upon the kernel function. In our simulations weused a cubic-spline kernel and =0.1 to get stable results using an explicit second-orderRunge–Kutta scheme. From a physical point of view, equation (2.12) indicates that thetime step must be smaller than a dispersion time scale given by the dispersion coeﬃcientand the kernel support volume.2.2.2 Accuracy and Spatial ResolutionErrors in meshless approximations based on kernel interpolants come from two sources(Brackbill,2005;Quinlan et al., 2006). The integral interpolation in (2.3) introduces asmoothing error that depends on the shape and smoothing length of the kernel. Addition-ally, there is a numerical integration error that depends upon the number and locationof the nodes and the smoothness of the real function (Schaback and Wendland,2006).There is a tradeoﬀ between both sources of error because the accuracy of the smoothedquantity increases as the smoothing length decreases, while the numerical integration er-ror increases as the number of nodes per support volume decreases. Therefore, the onlyway to simultaneously reduce both sources of error is to decrease the smoothing lengthwhile increasing the total number of nodes to keep the same average number of nodes persupport volume. In general, for a given function and node distribution there is a criticalsmoothing length hcsuch that for h>hcthe smoothing error dominates and that forh<hcthe integration error is more important. The determination of hcfor irregularlyspaced points is very diﬃcult, if not impossible.The numerical integration error also depends upon the regularity of the node distribution.Meshless approximations such as SPH methods produce very accurate results in situa-tions where particles are regular or uniformly distributed (Monaghan, 2005). In thosesituations the error aﬀecting the simulation is much smaller than the theoretical error es-timate of (2.6) which considers a random particle distribution (Monaghan, 2005). Whenparticles are uniformly distributed the leading error term is due to the interpolation er-ror and is controlled by the kernel smoothing length that sets the spatial resolution. Forexample, numerical studies have shown that for a set of equispaced particles the error for(2.9) converges with h2(Cleary and Monaghan,1999).39The previous discussion indicates that the accuracy of the MC-SPH method to simulatelocal-dispersion will evolve during the simulation. At early times the node distributionis similar to the initial regular distribution, so the leading error term is due to theinterpolation approximation. As nodes are redistributed in space by the non-uniformflow velocity, they become clustered in diﬀerent zones, so the numerical integration errorbecomes much more important. Then, the accuracy of the method during the simulationmust be controlled by an appropriate choice of the kernel smoothing length and thenumber of nodes per kernel support volume or initial average spacing. The choice ofthose parameters is not trivial, particularly in multidimensional simulations, and requiressome trial and error. In the simulations presented below, we selected those parametersusing the following steps: (i) setup an initial node configuration, (ii) simulate advectiononly, (iii) check node distribution and, particularly, number of nodes per kernel supportvolume, (iv) if node distribution was considered too sparse, increase the initial number ofnodes, (v) repeat until an acceptable final node distribution is produced. Because particletracking is very eﬃcient, the determination of the optimum initial node configurationdemanded little time and eﬀort relative to the overall simulation time.2.2.3 Mass ConservationIn the proposed MC-SPH scheme solute mass is distributed in space as a finite set ofconcentration values at node positions, thus the total mass in the system equal to theintegral of the concentration over the domain can be approximated asM =ˆΩC(r)dr≈CV + O1Np(2.15)where we have used a Monte Carlo integration approach and V is the volume of thedomain Ω. As discussed above, the accuracy of the integral depends upon the totalnumber of points Npand their spatial distribution. In most practical SPH simulationsthe number of particles is quite large and (2.15) is a good approximation. Therefore, wecan study the evolution of the total solute mass in the system by writingdMdt≈ VdCdt=VNpNpi=1dCidt(2.16)40Finally, substituting expression (2.10) used to compute the temporal derivative of theconcentration, we getdMdt=VNpijαij(Cj−Ci) =0 (2.17)since αij= −αji, and thus solute mass is globally conserved. In the analyses that followwe use this expression to characterize mass balance.2.3 Numerical evaluation of the MC-SPH methodTo test the capacity of the MC-SPH method to provide reasonably accurate solutionsfor dispersive transport we compare it with the analytical solutions of simple one- andtwo-dimensional dispersion problems and with other numerical solutions for simulatingadvective-dispersive transport in non-uniform velocity fields.2.3.1 One-Dimensional DispersionWe consider a simple one-dimensional problem to illustrate the behavior of the erroraﬀecting our new MC-SPH approximation in (2.8) as function of the particle distributionand smoothing length. We simulate the dispersion of a one-dimensional Gaussian plumewhere the concentration as function of position and time is given byC(x,t)=C0s0se−(x−x0)22s2(2.18)where s =s20+2Dt, C0is the maximum initial concentration, s0is a constant thatcontrols the size of the initial plume, and x0is the position of the plume center.Particles were initially distributed over a regular equispaced grid with spacing ∆x.Tostudy the eﬀect of the particle distribution, we generated a non-uniform particle distri-bution by adding a normally distributed perturbation with standard deviation σ. Then,we computed the numerical solution using a Gaussian kernel with cutoﬀ at 4h, i.e. onlyparticles within 4h contribute to the kernel summation. The error introduced by thisapproximation is small because of the rapid falloﬀ of the Gaussian function.41Figure (2.2) shows the maximum normalized error defined aserror = CNumerical−CAnalytical2/N (2.19)where N = Npis the number of particles, versus h for two diﬀerent ratios ∆x/h forsolutions computed using our MC-SPH and the traditional WC-SPH, formulations. Foruniform particle distribution (i.e., σ/∆x =0) and ∆x/h =0.66 the error increases as h2.As particles become disordered the error increases for any value of h. Particle disorder isless important for large smoothing lengths and errors for diﬀerent σ are similar. Largervalues of the ratio ∆x/h, which is equivalent to fewer particles per support volume,produces larger error even for the uniform particle distribution. For large particle disorder(σ/∆x =1) the interpolation error is dominant and the error of the numerical solutionsis almost independent of h. Figure (2.2) also shows that the use of the new MC-SPHapproximation instead of the standard WC-SPH does not make a diﬀerence as bothsolutions produce results with similar accuracy.2.3.2 Two-Dimensional DispersionIn this section we consider the simulation of the dispersion of a two-dimensional Gaus-sian plume using the RWPT and MC-SPH numerical methods. Despite the fact thatboth methods are based on a Lagrangian formulation of the solute transport problemand use a particle-tracking algorithm to integrate the advection equation, there are im-portant diﬀerences in their conceptual approaches, accuracy, numerical implementation,and computational performance that must be considered to evaluate their merits. Table2.1 summarizes the main diﬀerences between both methods and Appendix C gives detailsabout our implementation of the RWPT method.To make things simple we assume that the Gaussian plume is within a square two-dimensional domain and that the maximum concentration, C0, occurs at the center ofthe domain. In this case, the concentration as function of position and time is given byC(x,y,t)=C0s20s2e−(x−x0)22s2−(y−y0)22s2(2.20)To calculate a reasonable value for the local-dispersion coeﬃcient we assume a porewater velocity v =10−7m/s and dispersivity α =1cmwhich results in a local-dispersion4210−310−210−110−710−610−510−410−310−210−1hError MC σ/Δx = 1WC σ/Δx = 1MC σ/Δx = 0.3WC σ/Δx = 0.3MC σ/Δx = 0.1WC σ/Δx = 0.1 σ/dx = 0Δx/h=1.010−310−210−110−710−610−510−410−310−210−1hError MC σ/Δx = 1WC σ/Δx = 1MC σ/Δx = 0.3WC σ/Δx = 0.3MC σ/Δx = 0.1WC σ/Δx = 0.1 σ/Δx = 0Δx/h=0.66Figure 2.2: Error for one-dimensional simulations as a function of the smoothinglength h for Monte Carlo (MC) and Weakly Compressible (WC) formula-tions defined by equations (2.8) and (2.9), respectively. Error magnitudeis shown for diﬀerent ratios of particle spacing over smoothing length∆x/h and for diﬀerent perturbations over particle spacing, σ/∆x. Uni-form particle spacing corresponds to σ/∆x =0.43MC-SPH RWPTParticles carry solute concentration Particles carry solute massIt is possible to compute chemicalreactions at individual particlesChemical reactions must be evaluatedat some cell scaleIt is more complex to implement fulldispersion tensorIt is easy to implement full dispersiontensorNumerical precision to representconcentration values up to hardwarerepresentationNumerical precision to representconcentration values given by numberof particlesSimulates solute mass transfer betweenparticlesSimulates solute mass transfer at cellscaleIt demands more computational eﬀortbut it is possible to get very goodaccuracy with moderate use of memoryIt demands less computational eﬀortbut it requires more memory to gethigher accuracyTable 2.1: Comparison of MC-SPH and RWPT methods.coeﬃcient equal to D = v · α =10−9m2/s. We consider that the plume is centered ina square domain of side L =100m and that s0=5m, so that boundary eﬀects arenegligible. To integrate the solution in time we use a time step ∆t =11.6 days and weconsider a total simulation time T =500∆t =15.9 years.We performed a series of simulations to evaluate the relative performance and conver-gence properties of the RWPT and our MC-SPH methods. First, we solved the problemusing a RWPT method with diﬀerent combinations of averaging volumes and number ofparticles to represent the mass in the cell with the maximum concentration as summa-rized in Table 2.2. Then, we simulated the same situation using MC-SPH consideringdiﬀerent combinations of number of particles and kernel smoothing length which definesthe average number of particles per kernel support volume as summarized in Table 2.3.We note that the direct comparison of both methods is diﬃcult because of the diﬀerencesin the way they calculate concentrations that result in diﬀerent spatial resolutions andaccuracies for the same number of particles. For example, particles in the RWPT simu-lations are only located within the plume edges while in MC-SPH simulations particles,as explained below, must cover a larger area. In the MC-SPH simulations presentedhere particles are quasi-randomly distributed in all the domain. However, it would bepossible to improve the spatial resolution by distributing the same number of particlesin a smaller area. Nevertheless, we believe that the results presented next constitute afair comparison of both methods.44Simulation # cells NrNpRW1 50 x 50 10 329RW2 50 x 50 100 3,957RW3 50 x 50 1000 40,713RW4 100 x 100 10 1,257RW5 100 x 100 100 15,345RW6 100 x 100 1000 157,972RW7 200 x 200 10 4,993RW8 200 x 200 100 60,965RW9 200 x 200 1000 627,153Table 2.2: Parameters used in RWPT simulations: number of cells used to calculateconcentrations, number of particles used to represent the mass within thecell with maximum concentration Nr, and total number of particles Np.Simulation hNpNkSPH1 0.5 10,000 3SPH2 0.5 20,000 6SPH3 0.5 40,000 13SPH4 0.5 60,000 19SPH5 0.5 80,000 25SPH6 1.0 10,000 13SPH7 1.0 20,000 25SPH8 1.0 40,000 50SPH9 1.0 60,000 75SPH10 1.0 80,000 101SPH11 2.0 10,000 50SPH12 2.0 20,000 101SPH13 2.0 40,000 201SPH14 2.0 60,000 302SPH15 2.0 80,000 402Table 2.3: Parameters used in SPH simulations: smoothing length h, total number ofparticles Np, and average number of particles per kernel support volumeNk.452.3.2.1 Initial particle and concentration distributionTo apply the RWPT method to this problem we must first map the spatial concentrationdistribution to a regular Cartesian grid. Next, we must set the number of particles thatrepresent a given mass to compute the equivalent particle distribution. In this examplewe use diﬀerent numbers of particles to represent the mass contained in the cells withhighest concentration value. Particles within each cell are initially distributed using aquasi-random distribution to generate a uniform spatial coverage. Figure 2.3 shows thecorresponding particle distribution and the equivalent cell concentrations for some exam-ple configuration. We note that particles are only present in cells where concentrationvalues are greater than some numerical threshold equal to the ratio between the mass ofindividual particles and the cell volume (see Appendix C for details). Because of the av-eraging procedure used to compute cell concentrations, the maximum cell concentrationvalue is less than C0. Similar diﬀerences between cell values and actual concentrationsoccur in the rest of the domain and they are relatively more important near the plumeedges where cells contain fewer particles.Figure 2.3: In RWPT simulations concentration values are approximated accordingto the spatial distribution of particles. The left figure show the initialparticle distribution corresponding to a Gaussian plume with maximumconcentration C0at the domain center. The right figure shows concen-tration values computed according to the number of particles in each cell.The initialization of particle positions and concentration values in SPH simulations areindependent. Particle positions are assigned such that the resulting particle distributioncovers the region of interest. For example, given an initial Gaussian plume and the samenumber of particles as used in Figure 2.3, particles can be quasi-randomly distributed in a46rectangular volume or distributed in a uniform rectangular lattice within a circular regionas shown in Figure 2.4. We observe that the maximum concentration value is within 0.1%of the actual maximum concentration value C0in both cases. This shows that the MC-SPH method provides a better numerical resolutionto represent concentration values thanthe RWPT using the same number of particles. In this simple example, particles are onlycreated within a region of the numerical domain where concentrations are greater thana given threshold value plus a surrounding buﬀer zone. The buﬀer zone is necessary toprovide additional space to allow dispersion to distribute the initial solute mass in a largervolume. In real simulations it is important to prevent the existence of isolated particleswith non-negligible concentration at the edge of the particle cloud to avoid numericalerrors. There are three possible alternatives to control this source of error: (i) generate aparticle distribution that covers all the domain, (ii) use a dynamical scheme that insertsparticles as needed, (iii) generate an initial particle distribution with a buﬀer zone largeenough to guarantee an appropriate particle distribution during the simulation. Thefirst alternative is very simple to implement but it can become prohibitively expensivein large-scale simulations or in simulations that require fine-spacing between particles.The second solution works well but it is more diﬃcult to implement and introduces somecomputational overhead because it requires more sophisticated data structures to storeand manage the particle set. The third alternative combines the advantages of the othertwo because it is very easy to implement and requires fewer particles than the first option.Figure 2.4: In MC-SPH simulations concentration values are directly assigned to eachparticle. Figures show two possible initial particle distributions and cor-responding concentration values equivalent to a Gaussian plume withmaximum concentration C0at the domain center.472.3.2.2 PerformanceThe overall number of floating operations and memory requirements of the RWPTmethod using a background grid to compute concentration scales linearly with the to-tal number of particles, i.e. it is O(N). However, the application of (C.3) to computesmoother concentration distributions requires O(N2) operations. On the other hand,the evaluation of the summation in (2.6) corresponds to an n-body problem which naiveimplementation scales as O(N2) (Greengard, 1994). However, because the compact sup-port of the kernel the actual work required to compute the summation can be reducedto O(NNk), where Nkcorresponds to the average number of particles per kernel sup-port volume. The implementation of such algorithms requires an eﬃcient method forsearching near-neighbor particles. Such algorithms are based on data structures used toclassify particles according to their spatial coordinates. For constant smoothing lengthimplementations as presented in this paper, the background grid algorithm is the mosteﬃcient method (Viccione et al., 2008). For spatially varying smoothing lengths, moresophisticated data structures based on octrees or binary trees must be used (e.g. Barnesand Hut,1986;Waltz et al., 2002). An explicit implementation of the proposed meshlessmethod as discussed in Section 2.2.1 requires an amount of memory that scales with thenumber of particles (O(N)).Figure 2.5 shows the CPU time required to complete a single time step as function of thetotal number of particles in RWPT simulations. As expected the computational cost ofthe method grows linearly with the total number of particles. On the other hand, Figure2.6 shows that in SPH simulations the CPU time depends in a non-linear fashion on thetotal number of particles and the kernel smoothing length. Larger smoothing lengths,equivalent to more particles per kernel support volume, result in longer simulation timesfor the same total number of particles. Figure 2.6 also shows that, as expected, theCPU time required to complete a single time step scales linearly with the product ofthe total number of particles and the average number of particles per kernel support,NPK = NpNk. The diﬀerences observed between the curves corresponding to diﬀerentsmoothing lengths for the same product NPK are due to diﬀerences in performance ofthe routine that evaluates the changes in concentration at each time step as result ofdiﬀerent combinations of Npand Nk.480 1 2 3 4 5 6 7x 105020040060080010001200NpNormalized CPU TimeFigure 2.5: Normalized CPU time required to solve one time step using RWPT asfunction of the total number of particles Np. Computational cost ofRWPT simulations is proportional to the total number of particles Np.491 2 3 4 5 6 7 8x 104050100150200250300NpNormalized CPU Time h = 0.5h = 1.0h = 2.00 0.5 1 1.5 2 2.5 3 3.5x 107050100150200250300Np Nk Normalized CPU Time h = 0.5h = 1.0h = 2.0Figure 2.6: Normalized CPU time required to solve a single time step as function ofthe total number of particles Np, kernel smoothing lenght h, and averagenumber of particles per kernel support volume Nk. Computational costof MC-SPH simulations is proportional to the product of Npand Nk.502.3.2.3 AccuracyWe used two criteria to compare the accuracy and convergence properties of RWPTand MC-SPH. Figure 2.7 compares the temporal evolution of the simulated maximumconcentration with the theoretical result according to (2.20). The simulated RWPTresults exhibit unphysical oscillations and large errors. Such unphysical oscillations wouldcreate serious problems if an operator splitting approach was used to simulate reactivetransport simulations where these errors would be amplified by non-linear reactions.Local errors in RWPT simulations do not only depend upon the total number of particlesbut also on the number of particles at each cell and the cell volume. For example,simulation RW3 with Np=40713produces maximum concentration values that are closerto the true value than the results of RW6 and RW9 with Np=157972 and Np=627153,respectively. Simulations with lower Nras defined in Appendix C, not shown in thefigure, produced results with even larger errors. Figure 2.7 also shows a comparisonof the maximum concentration simulated with MC-SPH considering Np=40,000 andthe analytical solution. All of the simulations give solutions that are free of unphysicaloscillations, however, simulations with low average number of particles per kernel supportvolume such as SPH3 (Nk=13) can result in considerable local errors. Local errors canbe made negligible by choosing an appropriate combination of total number of particlesand kernel smoothing length to obtain larger average number of particles that eﬀectivelycontribute to the numerical integration, e.g. simulations SPH8 (Nk=50) and SPH13(Nk=201). Results of other simulations with larger number of particles, not shown inthe figure, produced smaller errors.Figure 2.8 shows the global error as function of the total number of particles in thesimulation and CPU time per time step. It is clear that using these metrics numericalsolutions computed using MC-SPH converge faster to the true solution than the RWPTsolutions. Figure 2.8 also shows that MC-SPH solutions have diﬀerent convergence ratesdepending upon the kernel smoothing length used. Moreover, curves corresponding todiﬀerent smoothing lengths intersect indicating the transition between regions where theerror is controlled by the average number of particles per kernel support volume Nk(small Np) and the region where the error depends upon the spatial resolution given bythe kernel smoothing length (large Np). For simulations that require low CPU time, theconvergence rate for MC-SPH simulations is faster than the one for RWPT simulations.However, the convergence rate of both methods becomes similar for simulations withlarger number of particles that require longer CPU times to complete a single time step.510 50 1001502002503003504004505000.60.70.80.91Time StepCmax/C0 AnalyticalRW3RW6RW90 50 1001502002503003504004505000.60.650.70.750.80.850.90.951Time StepCmax/C0 AnalyticalSPH3SPH8SPH13Figure 2.7: Maximum simulated concentration versus time step. RWPT simulationswith resolution number Nr= 1000 and MC-SPH simulations with Np=40000. Estimated concentrations in RWPT simulations present numericaloscillations due to representing the solute mass distribution as a finite setof particles. MC-SPH simulations compute concentrations that are freeof numerical oscillations.5210210310410510610−610−510−410−3NpError RWPTSPH h = 0.5SPH h = 1.0SPH h = 2.010010110210310410510−610−510−410−3Normalized CPU TimeError RWPTSPH h= 0.5SPH h = 1.0SPH h = 2.0Figure 2.8: Normalized global error versus total number of particles and normalizedCPU time. Error computed as defined in (2.19) substituting N by thenumber of cells for RWPT and the total number of particles for MC-SPHsimulations. The convergence rate, measured as function of the totalnumber of particles Npor the CPU time required to complete a singletime step, is faster for MC-SPH simulations than for RWPT simulations.However, the convergence rate of both methods become similar for simu-lations that use more particles and demands longer CPU times.532.3.3 Advection–Dispersion in Heterogeneous Porous MediaThe objective of this section is to evaluate the performance of the MC-SPH approach tosimulate solute transport in two-dimensional heterogeneous porous media. To verify ourMC-SPH code we compared it with the ULTIMATE total variation diminishing (TVD)finite diﬀerence solver and the hybrid method of characteristics (HMOC) particle-meshsolver included in the popular MT3DMS package (Zheng and Wang, 1999). We focusour analysis on verifying if the numerical results satisfy some basic physical requirementssuch as: avoiding numerical dispersion, providing positive solution free of oscillations,and mass conservation.2.3.3.1 SetupBefore solving the solute transport problem we generated a velocity field as follows: (i)generate a moderately heterogeneous random hydraulic conductivity field, (ii) calculate avelocity field by solving the saturated flow problem using MODFLOW (Harbaugh,2000)considering a constant hydraulic head gradient from left to right, and no-flow boundariesattopandbottom. Table 2.4shows the parameters usedtogenerate the randomhydraulicconductivity field and to solve the flow problem. We used the resulting velocity field tosimulate conservative transport of a square initial plume with constant concentration,C0. Table 2.5 shows the parameters used to solve the transport problem. Figure 2.9shows a schematic of the simulation setup. In all the simulations discussed below weonly considered constant isotropic dispersion.Description Symbol ValueVariance of ln(K) σY1.4Correlation length of ln(k) IY2.5Domain dimension (Lx,Ly)(20IY,50IY)Grid size ∆ IY/5Mean velocity in x U 0.81Max. velocity in x umax6.22Table 2.4: Parameter and results of flow model.For the TVD simulations we used the same grid discretization that was used to solvethe flow problem. For the HMOC simulations the allowed total maximum number of54Description Symbol ValueInitial plume size (Lpx,Lpy)(20IY,20IY)Initial plume center (Xp,Yp)(36IY,25IY)Péclet number Pe= UIY/D [20,200,∞]Dimensionless time step τ = U∆t/∆6.5 · 10−3Mean CFL number CFLmean=U∆t/∆0.03Max. CFL number CFLmax=umax∆t/∆0.25Table 2.5: Parameter values used in transport model.Figure 2.9: Domain dimensions, square initial plume, and breakthrough observationpoints P1 and P2 along centerline.55particles was 6 · 105and the number of particles per cell in cells where the relativeconcentration gradient (Zheng and Wang, 1999, p. 65) was higher than 1 · 10−5was15. The threshold value, DCHMOC, which controls if the forward or backward MOCmethod is used for an individual cell according to its relative concentration gradient wasset equal to 1 · 10−4(Zheng and Wang, 1999, p. 73). In the MC-SPH simulations thetotal number of particles was constant during the simulation and equal to 3.8·105whichwas equivalent to an initial number of particles per grid cell equal to 8. Particles whereinitially distributed in a rectangular lattice within a rectangular region of size 68IYin thedirection of the flow and 28IYin the transverse direction centered at the initial plumecenter. The kernel smoothing length was constant and equal to half the grid size used inthe MT3DMS models, so the spatial resolution of the three methods was comparable.The results of both, TVD and HMOC, methods correspond to concentration values atthe center of grid cells. We interpolated the MC-SPH results which correspond to con-centration values at scattered points onto a similar grid to compare them. We stressthat this interpolation was needed only for comparison purposes and it is usually notnecessary in SPH simulations. We used the following expression to compute interpolatedvalues AI,AI(ri) =jAjˆW (|ri−rj|)jˆW (|ri−rj|)=jAjψij(2.21)where ψijare Shepard functions (Shepard, 1968) and summations are over all particles.Although the interpolation in (2.21) is valid for any kernel functionˆW, we used the samekernel used in MC-SPH simulations to get interpolated quantities with similar spatialresolution.2.3.3.2 ResultsFigure 2.10 shows the spatial concentration distribution at the end of the simulation forthe advection-only case. Solutions given by the three methods are very diﬀerent. TheTVD solver is not able to avoid numerical dispersion so the plume exhibits large dilu-tion and the initial plume mass is distributed within a much larger volume. There areonly few areas where the solute concentration is similar to the initial concentration. The56HMOC produces less numerical dispersion and the plume edges are clearly distinguish-able. However, the eﬀects of numerical dispersion are clear in zones located between fastor slower fingers and in front and behind the main plume. The existence of those arti-ficial low concentration zones has some important practical implications. For example,in presence of reactions controlled by mixing such as biodegradation those low concen-tration zones could potentially extend the reactive zone near the plume edge (Cirpkaet al., 1999b). The concentration distribution generated by the MC-SPH code is freeof numerical dispersion and the plume edges are very sharp as expected in absence oflocal-dispersion. Zones without contaminant located between fast and slow fingers areobservable surrounding all the plume volume. It is interesting to notice some isolatedhigh concentration spots in the front and back of the plume as result of connected highand low permeability regions.Figure 2.11 shows concentration values along the domain centerline at the end of thesimulation, i.e., after the plume center has traveled 62 integral scales. The three methodsproduce very similar results for low Péclet numbers. As expected, the results producedby TVD and HMOC methods are identical considering that both methods share the samedispersion solver routine. It is more interesting to notice the good agreement betweenMC-SPH and the other two methods for low Péclet values. It is diﬃcult to say if thesmall diﬀerences observed at the plume edges are due to diﬀerences between the meth-ods or the interpolation method used to map the MC-SPH results onto a grid. Moreimportant diﬀerences are observed for the more strongly advection-dominated scenarios.For Péclet number equal to 200, the two Lagrangian based methods, HMOC and MC-SPH, perform similarly while the TVD solution is smoothed by numerical dispersion.This comparison clearly shows that even high-order Eulerian mesh-based methods suchas TVD cannot compete with particle-based methods for advection-dominated problems.For the advection-only case the three methods give solutions that are clearly distinguish-able. TVD results show little diﬀerence with respect to the situation for Pe=200.Onthe other hand, HMOC and MC-SPH results are closer to the expected sharp profilewith concentration values equal to the initial concentration or zero. The MC-SPH solu-tion seems to perform better close to the plume edges where the HMOC solution givesa smoother profile as consequence of the accumulated numerical dispersion due to theinterpolation of concentration values from scattered points to the cell centers at eachtime step.Figures 2.12 and 2.13 show breakthrough curves for points P1 and P2 located along thedomain centerline at 26IYand 42IYdownstream from the initial plume center, respec-57Figure 2.10: Spatial concentration distribution at dimensionless time τ = Ut/IY=62 for Pe = ∞. TVD solver (top), HMOC solver (middle), and SPHsolution mapped onto rectangular grid (bottom). Only dimensionlessvalues C/C0> 0.001 are shown.580 10 20 30 40 50 60 70 8000.20.40.60.81C / C0 Pe = 200 10 20 30 40 50 60 70 8000.20.40.60.81C / C0Pe = 2000 10 20 30 40 50 60 70 8000.20.40.60.81d / IYC / C0Pe = ∞TVDHMOCSPHFigure 2.11: Concentration versus accumulated distance along centerline at dimen-sionless time τ = Ut/IY= 62.59tively. Breakthrough curves for low Péclet values equal to 20 given by the three methodsare almost identical. This confirms that the MC-SPH method is able to simulate sit-uations where dispersion is important with accuracy that is comparable to well-testedmesh-based methods as used in MT3DMS. As observed in the longitudinal profile com-parison, the diﬀerence between HMOC and MC-SPH solutions and the TVD solutionincreases as the transport process becomes controlled by advection. For Péclet numberequal to 200, HMOC and MC-SPH produces similar results while the corresponding TVDbreakthrough curves have a completely diﬀerent shape typical of much higher dispersioncoeﬃcients. Finally, for the purely-advective case the breakthrough curves correspondingto each method become very diﬀerent. For the point located closer to the plume center,P1, HMOC and TVD predict a much earlier arrival time and longer tail. The earlierarrival time is due to the lateral mixing produced by the numerical dispersion, whichtransfers solute concentration from faster plume fingers that pass close to point P1. Thelonger tail is also due to lateral and longitudinal numerical mixing. The breakthroughcurve corresponding to the MC-SPH solution is not aﬀected by numerical dispersion,thus it exhibits a rectangular shape as expected. For point P2, the three breakthroughcurves have similar arrival time indicating that a set of fast streamlines forming a fastadvancing front passes through P2, so the lateral mixing does not have a significant ef-fect. However, lateral and longitudinal numerical dispersion produce much longer tailsin the corresponding HMOC and TVD curves. The analysis of the breakthrough curvesat both points indicates that neither the HMOC nor TVD schemes are able to correctlypredict mixing and dilution in situations where advection is much more important thandispersion.The value of the maximum plume concentration is an important parameter because itis usually used as criteria for regulatory purposes. It has also been used in theoreticalstudies to characterize and measure mixing and dilution processes. Figure 2.14 showsthe maximum plume concentration value as a function of time. It contains two curvesthat correspond to the MC-SPH solution. SPH-particles correspond to the maximumconcentration value observed at any particle position while SPH-mesh corresponds to themaximum concentration value after interpolation onto a grid. For Péclet value equal to20 the three methods predict the same maximum concentration confirming the previouslyobserved behavior. For higher Péclet values the TVD solution predicts lower maximumvalues than the other two methods, thus it over-predicts the dilution of the plume.As discussed above it is diﬃcult to directly compare the mass balance properties of mesh-based methods like TVD, Eulerian–Lagrangian methods such as HMOC, and meshless600 10 20 30 40 50 6000.20.40.60.81C / C0 Pe = 20TVDHMOCSPH0 10 20 30 40 50 6000.20.40.60.81C / C0Pe = 2000 10 20 30 40 50 6000.20.40.60.81τ = U t / IYC / C0Pe = ∞Figure 2.12: Breakthrough curve at point P1 located 26IYdownstream from initialplume center.610 10 20 30 40 50 6000.20.40.60.81C / C0 Pe = 20TVDHMOCSPH0 10 20 30 40 50 600.20.40.60.81C / C0Pe = 2000 10 20 30 40 50 6000.20.40.60.81τ = U t / IYC / C0Pe = ∞Figure 2.13: Breakthrough curve at point P2 located 42IYdownstream from initialplume center.620 10 20 30 40 50 600.80.850.90.951C / C0 Pe = 20TVDHMOCSPH−MeshSPH−Particles0 10 20 30 40 50 600.80.850.90.951C / C0Pe = 2000 10 20 30 40 50 600.80.850.90.951τ = U t / IYC / C0Pe = ∞Figure 2.14: Dimensionless maximum concentration versus dimensionless time. Be-cause of numerical dispersion, the maximum concentration for simula-tions with large Pe is smaller in TVD than in HMOC and MC-SPHsolutions.63methods such as MC-SPH. In general, the only comparison that can be made consists inlooking at the total solute mass distributed in the domain volume. For grid-based meth-ods with equispaced grid and uniform cell concentration, Ci, as used in the simulationsdiscussed here, the total mass in the domain is equal to M =´CdV =∆ViCi, where∆V is the cell volume. In that case the mean concentration defined as¯C = M/V is equalto the arithmetic average of the cell concentrations, i.e.¯C =∆VViCi=1ncellsiCi.Since the total domain volume is constant, the total solute mass is conserved only if themean concentration¯C is constant. Figure 2.15 shows the mean solute concentration ver-sus dimensionless time for the three Péclet number considered. In the three scenarios themean concentration computed using the TVD and MC-SPH solution at particle positionsis constant, indicating that the total mass in the domain is conserved. On the other hand,the mean concentration given by the HMOC method presents small fluctuations (<1%)which are more important for the advection-dominated case. Small mass diﬀerences dueto the interpolation scheme used in HMOC are expected and this error is often citedas the main disadvantage of the method (Zheng and Wang, 1999). There is a tempo-ral change in the mean concentration computed using interpolated values given by theMC-SPH solution. The change increases with time and it is higher for higher dispersioncoeﬃcients. The temporal variation is probably due to the increasing interpolation errordue to the more irregular particle distribution. On the other hand, higher dispersion co-eﬃcients increase the dilution of the plume, so the total solute mass is distributed amonga larger number of particles with lower concentration values which are more sensible tointerpolation errors.2.4 ConclusionsWe derived a new SPH formulation based on the Monte Carlo nature of the original SPHmethod to simulate solute transport in heterogeneous porous media. We demonstratedthat the new MC-SPH method is able to accurately simulate scenarios of practical andtheoretical interest where the combined action of flow heterogeneity and local-scale dis-persion aﬀects the plume movement, mixing and dilution. The study of those situationsusing traditional numerical methods is very diﬃcult if not impossible due to numericaldispersion and other numerical artifacts such as unphysical oscillations, that not onlydegrade the accuracy of the numerical solution but also modify the basic physical mech-anisms that control solute transport in porous media.640 10 20 30 40 50 600.9911.011.021.03C / C0 Pe = 200 10 20 30 40 50 600.9911.011.021.03C / C0Pe = 2000 10 20 30 40 50 600.9911.011.021.03τ = U t / IYC / C0Pe = ∞TVDHMOCSPH−MeshSPH−ParticlesFigure 2.15: Mean concentration versus dimensionless time. While the mean concen-tration value of SPH-Mesh (concentrations interpolated onto grid) is notconstant, the mean concentration at particle locations (SPH-Particles)is constant throughout the simulation which demonstrates that the MC-SPH formulation conserves total mass.65We demonstrated through numerical benchmarks that the numerical error of the MC-SPH method behaves in a complex way and it depends upon several factors such assmoothing length, particle spacing and solution smoothness. However, numerical resultsindicate that the overall accuracy of the method can be controlled with an adequate choiceof those parameters. The overall performance of the proposed method compares favor-ably with existent numerical methods such as RWPT, higher-order Eulerian and hybridEulerian–Lagrangian schemes for the set of simulations discussed in this paper. Due toits Lagrangian nature the MC-SPH method performs very well for advection-dominatedproblems. 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Phys., 173,322–347,2001.71Chapter 3Evaluation of ParticleApproximations to SimulateAnisotropic Dispersion13.1 IntroductionSolute transport in natural porous media is commonly modeled using an advection-dispersion equation (ADE). In most real situations, the transport process is advection-controlled and the resulting parabolic partial diﬀerential equation exhibits more of ahyperbolic character. On the other hand, the natural heterogeneity of geological forma-tions results in rapid changes of the magnitude and direction of the flow velocity. Thosefeatures make the numerical solution of the resulting transport equation with traditionalmesh based methods very challenging. The numerical solution of ADE that representssolute transport in porous media is further complicated by the fact that the dispersioncoeﬃcient is a second-order tensor with principal axes that are oriented parallel andperpendicular to the flow velocity (Bear, 1988), so that the spreading of a contaminantplume is anisotropic: faster in the flow direction than in the transverse direction.Particles methods oﬀer advantages for the simulation of solute transport in natural porousmedia because of their natural ability to adapt to the flow velocity and to simulate soluteadvection without introducing numerical dispersion and artificial mixing. Thus, there1A version of this chapter will be submitted for publication. P. Herrera, M. Massabó, and R. Beckie.Evaluation of Particle Approximations to Simulate Anisotropic Dispersion.72has been a long dated interest in the use of particle methods to simulate solute transportin the subsurface, e.g. (Kinzelbach, 1988) and references therein. The main challenge forthe use of particle methods is to derive an accurate approximation for dispersion thatcan simulate solute mixing and dilution, while avoiding numerical oscillations that plaguemost traditional numerical approximations of parabolic or elliptic equations that includemixed derivatives or “cross-terms” (Crumpton et al.,1995;Le Potier, 2005b; Nordbottenand Aavatsmark,2005;Mlacnik and Durlofsky,2006;Yuan and Sheng,2008;Edwardsand Zheng,2008;Lipnikov et al.,2009).Recent approaches to incorporate diﬀusion or viscous eﬀects in particle simulations arebased on a integral approximation of second order derivatives (Degond and Mas-Gallic,1989a; Cleary and Monaghan,1999;Eldredge et al., 2002). Particle locations are usedas quadrature points to discretize the integral approximation. When used to simulatesolute transport, these types of methods approximate the local dispersion operator usingconcentration values at a set of scattered particles or nodes (Zimmermann et al.,2001;Herrera et al., 2009b). The eﬀects of dispersion are incorporated by modifying concen-tration values of individual particles as the result of mass exchange between neighboringparticles. Therefore, important physical mechanisms such as dilution and solute mixingare easily incorporated.Zimmermann et al. (2001) investigated the use of the particle strength exchange (PSE)method (Degond and Mas-Gallic, 1989a,?) to simulate solute transport in homogeneousporous media considering anisotropic dispersion and uniform and non-uniform flow con-ditions. Their results showed that the PSE approximation provides accurate results fora set of benchmark problems if a remeshing procedure was used to control the irregularparticle distribution due to the flow velocity.Herrera et al. (2009b) compared a smoothed particle hydrodynamics (SPH) approxi-mation to simulate conservative transport in heterogeneous porous media with a high-order finite volume and a hybrid method of characteristics (HMOC) solvers consideringisotropic dispersion. Herrera et al. (2009b) used a SPH approximation for isotropic dis-persion, first introduced by Cleary and Monaghan (1999) to simulate thermal conduction,that only involves the first derivative of the kernel, so it is less sensitive to particle disorderthan other SPH approximations for second derivatives that require remeshing (Chaniotiset al., 2002). The results presented in (Herrera et al., 2009b) clearly show the advan-tages of the SPH approximation for simulating advection-dominated solute transport inheterogeneous porous media.73The first objective of this paper is to derive a SPH expression to approximate anisotropicdispersion to extend our previous work presented in (Herrera et al., 2009b). The secondobjective is to evaluate the accuracy with which the two particle methods, SPH and PSE,and a standard finite volume formulation can simulate isotropic and anisotropic disper-sion under diﬀerent conditions. In particular, we are interested in understanding theconvergence properties of both particle methods, the factors that control their accuracy,and their relative performance in comparison with a well established mesh-based solver.Additionally, we discuss the monotonicity properties of both particle approximations fordiﬀerent degrees of anisotropy of the dispersion tensor.3.2 Mathematical FormulationThe Lagrangian formulation of conservative solute transport in porous media involvesthe following system of diﬀerential equations,drdt= v(r,t) (3.1)dC(r,t)dt= ∇·(D(r)∇C(r,t)) (3.2)where r is the position of a fluid particle, C(r,t) is the solute concentration [M/L3] andD(r) is the hydrodynamic dispersion coeﬃcient [L2/T]. The first equation describes themovement of a fluid particle due to the flow velocity, while the second equation describesthe change in concentration due to dispersion. Generally, the flow field is computedexternally and it is an input parameter for the transport simulation.In isotropic porous media the components of the tensor D are given by (Bear,1988)Dij= (αT|v| + Dm)δij+ (αL−αT)vivj|v|(3.3)where αLand αTare the longitudinal and transverse dispersivity [L], respectively; Dmis the molecular diﬀusivity [L2/T], and v is the pore water velocity [L/T]. In general,the longitudinal dispersivity is at least one order of magnitude larger than the transversedispersivity, i.e. αT/αL 1.74The solution of (3.1) can be easily evaluated using a semi-analytical particle-trackingscheme (Pollock, 1988) or an explicit time integration scheme. In the rest of thismanuscript we focus our discussion on the numerical solution of (3.2), which representsa much more challenging problem in the context of particle methods.3.3 Smoothed Particle Hydrodynamics (SPH) Ap-proximation3.3.1 BackgroundIn the standard SPH formulation the smoothed interpolation AS(r) of a variable A(r) isdefined as the integral (Gingold and Monaghan,1977;Lucy,1977)AS(r)=ˆA(r)W(r−r,h)dr(3.4)where W (r−r,h) is a kernel function with smoothing length h that satisfies (Monaghan,1992)ˆW(r−r,h)dr=1 (3.5)limh→0W(r−r,h)=δ(r−r) (3.6)Spline polynomials with compact support are usually used as kernel functions because oftheir practical advantages (Monaghan,1992).In the standard SPH formulation the numerical approximation of the integral in (3.4) isevaluated asA(ra)=b1pbA(rb)W(|ra−rb|) (3.7)75where the numerical density pbis a measurement of the spatial particle distribution. Inmost cases, it is approximated aspa=bW(|ra−rb|) (3.8)When computing approximations for first and second order derivatives it is also usefulto introduce the scalar function F(r) such that the gradient of a spherically symmetrickernel can be evaluated as (Cleary and Monaghan,1999;Jubelgas et al.,2004)∇W(r)=rF(r) (3.9)3.3.2 SPH Approximation for Tensorial DispersionTo derive a SPH expression to approximate the dispersion term (3.2), we use the followingidentityij∂∂xiDij∂C∂xj=12ij∂2∂xi∂xj(DijC)−C∂2Dij∂xi∂xj+ Dij∂2C∂xi∂xj(3.10)that is valid for any symmetric tensor D. This expression is the generalization of theidentity used by Jubelgas et al. (2004) to derive a SPH approximation for thermal con-duction.Second derivatives of a scalar field A can be evaluated using (Español and Revenga,2003;Monaghan,2005)∂2A∂xi∂xja=b1pb(Aa−Ab)F (ra−rb)Γ(r−r)i(r−r)j|r−r|2−δij(3.11)where Γ=4in two dimensions and Γ=5in three dimensions.Finally, substituting (3.11) into (3.10), we arrive at our SPH approximation for equation(3.2),76dCadt=12b1pab(Ca−Cb)F(|ra−rb|)D(ra,rb) (3.12)whereD(ra,rb) =ijDaij+ Dbij4(rb−ra)i(rb−ra)j|rb−ra|2−δij=ijDabijΘij(rb−ra)where Dais the dispersion tensor at position raand we have replaced the density pbby a symmetric expression pab= f(pa,pb), e.g. the arithmetic average of paand pb,toensure a symmetric approximation (Herrera et al., 2009b). This expression reduces tothe standard SPH approximation for diﬀusion (Tartakovsky and Meakin,2005;Herreraet al., 2009b) or thermal conduction (Cleary and Monaghan,1999;Jubelgas et al., 2004), ifD = DI, where I is the identity matrix. In simulations that consider variable coeﬃcients,the term Dabij=(Dbij+ Daij) can be substituted by an eﬀective coeﬃcient of the formDabij=2DaijDbij/(Daij+ Dbij) , which has given more robust results in thermal conductionsimulations(Cleary and Monaghan,1999;Jubelgas et al.,2004).The approximation (3.12) has two sources of error. First, the SPH integral interpolant(3.4) introduces an error that grows with the smoothing length (O(h2)). Second, thenumerical discretization of the integral introduces an error that depends on the numberand position of the particles that contribute to the summation in (3.7). This source oferror is related to the ratio between the average number of particles per kernel smoothinglength γ, which is equivalent to the ratio between the smoothing length and the averageparticle spacing ∆x, i.e. γ = h/∆x. In general, a larger number of particles per kernelsupport volume (larger γ), results in a better approximation of the integral. However,the use of large γ values while controlling h to minimize the interpolant error requiresan increasingly small particle spacing and, hence, a large number of particles. Therefore,one must make a trade-oﬀ between γ and h to obtain reasonable error while controllingthe number of particles and computational eﬀort (Cleary and Monaghan,1999).3.3.3 MonotonicityIt is well known that traditional numerical approximations of parabolic or elliptic equa-tions of the form (3.2) that consider the oﬀ-diagonal terms of the dispersion tensor do not77satisfy the monotonicity properties of the solution, e.g. see (Herrera and Valocchi,2006)and references therein for details. The development of numerical approximations thatovercome those numerical issues is still the object of intense research (Le Potier, 2005b;Mlacnik and Durlofsky,2006;Nakshatrala and Valocchi,2008;Yuan and Sheng,2008;Lipnikov et al., 2009). Therefore, it is important to study the monotonicity properties ofthe SPH approximation derived above.First, we notice that (3.12) has the formdCidt=j=iβij(Cj−Ci)=jˆβijCj(3.13)withˆβii= −j=iβij, thusjˆβij=0.Then, we can use the local extremum diminishing (LED) criteria (Jameson,1995)tostudy the monotonicity of this type of numerical discretization. A numerical approxima-tion such (3.13) satisfies the LED criteria if βij≥ 0,i= j (Kuzmin and Turek,2002),which is a suﬃcient condition to obtain monotonic solutions as can be easily demon-strated by the following rationale. If the concentration at node i, Ci, is a minimum thetemporal derivative of the concentration at that node is positive or zero. Therefore, aminimum concentration can only increase or stay constant. Similar arguments can beused to prove that a maximum value cannot increase.In the case of (3.12), we have thatβab= −12b1pabF(|ra−rb|)D(ra,rb) a = b (3.14)with F(r) ≤ 0 because of the kernel properties.Then, the LED criteria requires that D(ra,rb) ≥ 0. This condition cannot be demon-strated for the general case of an irregular node distribution or non-uniform flow, butit can be studied for the simple case of equispaced nodes in a square lattice in uniformflow field. To make the analysis simpler we use a polar coordinate system such that θis the angle formed by the vector connecting two nodes located at raand rband the xaxis. Then, we obtain that Θxx=4cos2θ−1, Θyy= 4sin2θ−1, and Θxy= 4sinθcosθ.In a square lattice, θ =[0,π/4,π/2] or a multiple of those numbers. For θ =0orθ = π/2 there is only one term that is not zero and it is positive. If θ = nπ/4 with n78integer, we have that D(ra,rb)=Dabxx+ Dabyy+4Dabxy, which can be positive or negativebecause of the change in sign of Dxx, Dyy, and Dxywith the flow orientation accordingto (3.3). Figure 3.1 shows the value of D(ra,rb) as a function of the velocity directionfor θ = nπ/4. The figure shows that D(ra,rb) 0 for all possible flow orientations. Thisimplies that the SPH discretization with nodes distributed in a square lattice does notsatisfy the LED criteria and that the numerical solution of (3.12) might exhibit negativeconcentrations depending upon the flow orientation. This is confirmed by the results ofnumerical simulations presented below.!!!"#$""#$!!#$%%#$&'()**" !+% ! &+%! %!",+"-*.*!#"",+"-*.*"#!",+"-*.*"#"!xy!Figure 3.1: Coeﬃcient D(ra,rb)=Dxx+ Dyy+4Dxyfor |v| =1and such thatr =rb−raforms an angle of 45◦with the x axis as function of the angleβ formed by the flow velocity and the x axis.793.4 Particle Strength Exchange (PSE) Approxima-tionThe PSE approximation of (3.2) is also based on an integral expression to compute thedispersion operator (Degond and Mas-Gallic, 1989a). An approximation for anisotropicdispersion derived in (Degond and Mas-Gallic, 1989) and used in (Zimmermann et al.,2001) isdCadt=(∆x)2ε6b(Cb−Ca)K(rab,ε)ijMij(ra,rb)(ra−rb)i(ra−rb)j(3.15)where ∆x is the representative inter-particle spacing, K(rab,ε) is a cutoﬀ function thatsatisfies so-called moment conditions, ε is known as the core size which defines the sizeof the area of influence of each particle, and the components of the matrix M(ra,rb) aregiven byMij(ra,rb)=12(mij(ra)+mij(rb)) (3.16)wherem(r)=D(r)−14tr(D(r))I (3.17)with tr(D)=iDii. Zimmermann et al. (2001) provide expressions for second, fourthand sixth order cutoﬀ kernels and Eldredge et al. (2002) discuss the details of the kernelproperties and provide expressions to compute kernels that are up to eighth order in oneand two dimensions.The same analysis used in the previous section to study the monotonicity properties ofthe SPH approximation can be used to demonstrate that (3.15) does not guarantee themonotonicity of the solution when the full dispersion tensor is considered as discussedby Degond and Mas-Gallic (1989) and confirmed through numerical simulations by Zim-mermann et al. (2001).80Because of the similarities between the SPH and PSE methods it is possible to establish adirect parallel between the kernel and cutoﬀ functions and between the smoothing lengthand core size in SPH and PSE, respectively. In the rest of this manuscript, we will usethe terms kernel or cutoﬀ function to refer to the function K and the terms core size orsmoothing length to refer to ε. We will also use W to refer to the cutoﬀ K and h insteadof ε to refer to the core size whenever such change helps to simplify notation.3.5 Numerical TestsWe next evaluate the accuracy of our SPH anisotropic dispersion approximation and thePSE method from (Degond and Mas-Gallic, 1989) and (Zimmermann et al.,2001).Weuse the simulation of the instantaneous release of a solute mass ∆M in an unboundeddomain with a temporally and spatially constant velocity as benchmark problem to studythe accuracy and controls on error of the dispersion approximations for SPH and PSE.We also use a standard 9-points finite volume scheme (FV) in a Cartesian grid (Zheng andBennet,1995;Herrera and Valocchi, 2006) to define a base case to compare the relativeperformance of both particle methods. A similar problem has been previously used tostudy the convergence properties of the PSE (Zimmermann et al., 2001) and diﬀusionvelocity methods (Beaudoin et al.,2003).Since we are interested in numerical approximations of dispersion, we simplify the prob-lem and neglect the contribution of advection. In this case, the transport process dependson the flow only through the relation of the dispersion tensor and the flow velocity givenby (3.3). Because advection can be easily incorporated within a particle framework with-out introducing additional errors, the results of our analysis can be directly extrapolatedto more realistic situations.The analytical solution for the solute concentration as function of position and time isgiven by,c(x,t)=C1C4exp−X2(2tDyy+ w2)−Y2(2tDxx+ w2)+4XYtDxy8t2C2+4w2tC3+2w4(3.18)where X = x−x0and Y = y −y0, (x0,y0) is the position of the initial solute release,w is a measure of the size of the initial input, the constant C1 is related to the initialmass ∆M, and the other constants are C2=DxxDyy− D2xy, C3=Dxx+ Dyyand81C4=√4t2C2+2twC3+w4. To simplify the presentation of the results, we chooseC1=C0w2such that the maximum initial concentration is equal to C0. Table 3.1 showsa summary of the parameters used to setup the test problem.Parameter Symbol Value UnitReleased mass ∆M 107gInitial plume width w 44 mMaximum initial concentration C0320 mg/LLength numerical domain L 2000 mLong. Dispersivity αL10 mTime step ∆t 1 dayTotal time T 300 daysTable 3.1: Parameters used in all simulations.The three solutions are computed using an explicit fourth-order Runge-Kutta solver tointegrate in time. The use of an explicit solver imposes restrictions on the size of thetime step to obtain stable solutions. The three methods have stability limits of the form∆t ≤ CT∆2Dxx+ Dyy(3.19)where ∆ is the grid size for the finite volume, core size for the PSE (Zimmermann et al.,2001), and smoothing length for the SPH approximations(Cleary and Monaghan,1999),respectively. The constant CTis equal to 0.5 for the finite volume approximation, andit depends upon the kernel or cutoﬀ functions for the SPH and PSE. Higher order cutoﬀfunctions result in slightly more restrictive stability conditions, for example Zimmermannet al. (2001) found that CT≈ 2.5 and CT≈ 1.2 for second and fourth order cutoﬀfunctions, respectively. Additionally, the stability limits of both particle methods dependsupon the particle distribution. We found through numerical experiments that the SPHsolution is stable if CT=0.1 and use this value to compute a time step that satisfies thestability restrictions of three methods for the case of equispaced particles.The PSE and SPH approximations require that the area of influence or support of par-ticles overlap. Thus, one must use a core size for PSE or smoothing length for SPHthat is larger than the average particle spacing. Additionally, the error of the solution82given by both methods depends upon the ratio of the smoothing length to the averageparticle spacing. In our simulations, we used diﬀerent ratios to test the influence of thatparameter on the error of the solution. On other hand, higher order kernels and cutoﬀfunctions have larger support volume as shown in Figure 3.2, which results in larger areasof influence and number of neighboring particles for a given smoothing length or core size.Eﬃcient implementations of the PSE and SPH solvers require a fast algorithm to identifynear neighbor particles. The SPH implementation is based on kernels that have compactsupport, so an individual particle interacts only with particles that are within the kernelsupport volume. In that case it is easy to use a background grid to classify particlesin space. The cell size of that grid is related to the kernel smoothing length such thatneighbor particles are always at most one cell apart (Welton, 1998). Kernels used inthe PSE approximation are modified Gaussian functions which have infinite support.Therefore, in theory, all particles interact with each other. However, PSE kernels fallrapidly with distance and one can assume that they have an eﬀective compact supportthat is few times the core size as shown in Figure 3.2. In our implementation, we haveassumed that the eﬀective compact support of the PSE kernels is equal to five times thekernel core size and we have applied the same strategy as in SPH to search for neighborparticles.3.5.1 Simulation CasesTo test the performance of the three numerical methods we define diﬀerent scenariosbased on the values of the parameters summarized in Table 3.2. The ranges of values ofthose parameters are similar to the ones used in previous studies or were selected basedon reasonable physical assumptions. For example, we use αT/αLin the range [0.001,1.0]with αL=10mand β equal to [0◦,45◦,53◦], which are similar to the values reported in(Zimmermann et al.,2001;Beaudoin et al., 2003). We use value for h/∆x and ε/∆x inthe range [1.0,1.6], which is similar to values used in other numerical studies to studythe convergence properties of the SPH approximation for thermal conduction (Cleary andMonaghan, 1999) and PSE for solute dispersion (Zimmermann et al.,2001).83Figure 3.2: Cubic, quartic, and quintic SPH kernels, W, with finite compact support(Price, 2004) and second-, fourth-, and sixth-order cutoﬀ functions, K,used in PSE simulations (Zimmermann et al., 2001) as function of theratio between distance and kernel core size or smoothing length, h. Allkernels fall rapidly with distance and have an eﬀective support equal tofew smoothing lengths.84Parameter ExplanationαT/αLDispersivity ratioh/∆x Smoothing length or core size over average particle spacingβ Angle formed by velocity and x axisSPH Kernel Three diﬀerent SPH kernels: cubic, quartic and quinticPSE Cutoﬀ Three diﬀerent cutoﬀ functions: 2nd, 4th and 6th orderTable 3.2: Parameters used to define diﬀerent simulation scenarios to evaluate ap-proximations for anisotropic dispersion.To study the convergence of the three methods with respect to the particle or grid spacing,we define a set of runs with diﬀerent number of cells or particles as summarized in Table3.3. To assign the position of particles and cells we assume a large square domain withside L. We assign the same number of particles and cells in each direction, Nc,forsimulations that consider equispaced particles. For simulations that consider random orquasi-random particle distributions, the total number of particles, N, is calculated suchthat the average number of particles in each direction is equal to Nc. We compute twoerrors, E2=je2j/N and E∞= max(|ei|), where ejis the diﬀerence between analyticaland numerical solutions at node j, to measure the accuracy of the numerical solutions. Wealso look at the temporal evolution of the diﬀerence between the maximum concentrationvalues of the numerical and analytical solutions. In the discussion that follows we reporterrors after 200 time steps unless explicitly indicated.Run Nc∆xR1 40 50.0R2 60 33.3R3 80 25.0R4 100 20.0R5 120 16.7R6 140 14.3R7 160 12.5Table 3.3: Definition of diﬀerent runs used to study convergence properties. Each runis defined by the number of cells or average number of particles in eachdirection, Nc, which results in a grid or average particle spacing, ∆x.853.5.2 Equispaced ParticlesWe first consider the case of equispaced particles in a square lattice. This scenario isuseful because it allows the direct comparison of the particle methods and the finitevolume approximation. Besides, the accuracy of both particle methods is expected to beoptimal for this configuration, thus the results of this section provide a best case estimateof the error of the SPH and PSE methods.Unless explicitly specified, all the results reported for equispaced particles were computedusing a cubic spline SPH kernel and second order PSE cutoﬀ functions.3.5.2.1 Eﬀect of particle spacingFigure 3.3 shows the error E2versus the particle or grid spacing. For isotropic dispersion(αT/αL=1.0), the convergence rate of the three methods is similar, but the meshbased FV approximation has in average an error that is one order of magnitude smallerthan the SPH approximation and almost two orders of magnitude smaller than the PSEapproximation for the range of particle or grid spacing considered.For αT/αL=0.01 (anisotropic case) the analysis is more complicated. For all the casesthe mesh-based FV solver is more accurate than both particle methods but the diﬀerenceis smaller than for the isotropic case. The PSE and FV methods exhibit good convergencein all cases, while the SPH approximation is very sensitive to the value of the ratio h/∆x.The SPH solution converges much faster for larger number of particles per kernel supportvolume (larger h/∆x). Nevertheless, the convergence rate of the SPH solution for small∆x is lower than for the other two methods.Figure 3.4 shows the error E∞divided by the maximum initial concentration versus theparticle or grid spacing. The situation is similar to the previously discussed for the errorE2. The three methods have smaller errors for the isotropic case than for the anisotropicone. The approximation FV has consistently lower error than the two particle methodsfor all the situations analyzed, however the diﬀerence is smaller in the anisotropic case.For the isotropic situation the maximum absolute error is around 1% of C0for the PSEapproximation and less than 1% for the SPH and FV methods. For the anisotropic casethe error is around 1% of the initial maximum concentration for the two particle methodsand less than that for FV solution. For small ∆x the convergence rates of the PSE andFV approximations are comparable, while the SPH solution has a lower rate.8610 15 20 25 30 35 40 45 5010−410−310−210−1100101102∆xE2 FVSPH h/∆x=1.2SPH h/∆x=1.6PSE h/∆x=1.2PSE h/∆x=1.6αT/αL=1.010 15 20 25 30 35 40 45 5010−210−1100101102∆xE2 FVSPH h/∆x=1.2SPH h/∆x=1.6PSE h/∆x=1.2PSE h/∆x=1.6αT/αL=0.01Figure 3.3: Error E2as function of particle or grid spacing for equispaced particles.The three methods exhibit good convergence for the isotropic case in-dependently of the ratio h/∆x. For the anisotropic case, the PSE andFV solutions exhibit good convergence in all cases. However, the SPHapproximation is very sensitive to the value of h/∆x. The convergencerate of the FV and PSE methods for small ∆x is higher than the SPHone for the anisotropic case.8710 15 20 25 30 35 40 45 5010−310−210−1100∆xE∞/C0 FVSPH h/∆x=1.2SPH h/∆x=1.6PSE h/∆x=1.2PSE h/∆x=1.6αT/αL=1.010 15 20 25 30 35 40 45 5010−310−210−1100∆xE∞/C0 FVSPH h/∆x=1.2SPH h/∆x=1.6PSE h/∆x=1.2PSE h/∆x=1.6αT/αL=0.01Figure 3.4: Normalized error E∞as function of particle or grid spacing for equispacedparticles. The three methods exhibit good convergence for the isotropiccase independently of the ratio h/∆x. For the anisotropic case, the PSEand FV solutions exhibit good convergence in all cases. However, the SPHapproximation is versy sensitive to the value of h/∆x. The convergencerate of the FV and PSE methods for small ∆x is higher than the SPHone for the anisotropic case.883.5.2.2 Maximum concentrationFigure 3.5 shows the diﬀerence between the maximum concentration values of the ana-lytical and numerical solutions as function of the number of time steps. For the isotropicand anisotropic cases the diﬀerence increases at early time until reaching a maximumvalue. For later times, as the initial plume smooths out, the error decreases to an asymp-totic value. The SPH solution with h/∆x =1.2 is the exception to this pattern since theerror grows unboundedly with time. The diﬀerence between the numerical and analyti-cal solutions after 300 time steps is less than 1% for the FV and the best SPH run andaround 1% for the PSE solution.3.5.2.3 Negative concentrationsFigure 3.6 shows a comparison of the analytical and numerical solutions after 300 daysfor run R7 and αT/αL=0.01. The three numerical solutions are similar to the analyt-ical solution. However, the three numerical solutions exhibit negative concentrations inbands that tend to be aligned with the main direction of the flow. Figure 3.7 shows thespatial distribution of the diﬀerence between the analytical and numerical solutions. Ingeneral, the FV and SPH approximations overestimate the concentration values in thecenter of the plume in a region parallel to the flow direction and they underestimate theconcentration in areas outside the plume center along a line that is perpendicular to theflow. The spatial distribution of the error of the PSE solution follows a diﬀerent pattern.Concentration values are underestimated in the central region of the plume and they areoverestimated in two separate regions that are located near the plume edge along theplume center line. Therefore, the spatial distribution of the error of the three methodsdepends upon the flow orientation.890 50 100 150 200 250 300−0.0100.010.020.030.040.05Time Step(CmaxN − CmaxA)/C0 αT/αL=1.0FVSPH h/∆x=1.2SPH h/∆x=1.6PSE h/∆x=1.2PSE h/∆x=1.60 50 100 150 200 250 300−0.04−0.03−0.02−0.0100.010.020.030.04Time Step(CmaxN − CmaxA)/C0 αT/αL=0.01FVSPH h/∆x=1.2SPH h/∆x=1.6PSE h/∆x=1.2PSE h/∆x=1.6Figure 3.5: Diﬀerence between maximum concentration values of numerical and an-alytical solutions as function of time for equispaced particles and runR7. For the isotropic case the error of the three methods increases atearly time. As the concentration field smooths out the error decreasesat later times. For the anisotropic case the error of the SPH approxima-tion for h/∆x =1.2 grows unboundedly. However, the SPH solution forh/∆x =1.6 performs similar to the FV approximation.90Figure 3.6: Concentration distribution after 300 time steps for run R7, αT/αL=0.01,and β = 45◦. All three methods exhibit negative concentrations (darkbands). Minimum concentration values are −1.8·10−2for FV, −6.9·10−4for SPH, and −3.2 · 10−1for PSE.91Figure 3.7: Diﬀerence between analytical and numerical solutions (Error = CN−CA)after 300 time steps for run R7, αT/αL=0.01, and β = 45◦. The spatialpattern of the error of the three methods depends upon the flow velocitydirection.3.5.2.4 Eﬀect of ratio between smoothing length and particle spacingAs discussed above, the error of the SPH approximation for the integral in 3.4 dependsupon the number of particles per support volume, which is related to the ratio betweenthe smoothing length and the average particle spacing, γ = h/∆x. Previous numeri-cal studies have shown that the SPH approximation for scalar diﬀusion (isotropic case)provides accurate results even for small values of γ in the case of reasonably distributedparticles(Cleary and Monaghan, 1999). On the other hand, the stability of the PSEapproximation requires that particles overlap, i.e. the core size must be always largerthan the representative particle spacing. However, the accuracy of the approximationdecreases as the core size increases, thus it provides optimal solutions for small h suchthat h/∆x>1. Figure 3.8 shows errors E2and E∞of the SPH and PSE solutionsfor run R6 as function of γ. We observe that as expected the error of the PSE solu-tion increases monotonically with γ for the isotropic and anisotropic cases. In contrast,the SPH solution exhibits a more interesting behavior. The error of the SPH solutionfor the anisotropic case decreases with γ, which indicates that the error of the integral92approximation controls the overall error in that case.3.5.2.5 Eﬀect of anisotropy ratioFigure 3.9 shows E2and normalized E∞versus the anisotropy ratio for run R7. The errorof the SPH and FV approximations is larger for smaller αT/αLratio, while the error ofthe PSE method is almost constant for the range of dispersivity ratios considered. TheFV approximation has the smallest error in all the cases, while the PSE solution is moreaccurate than the SPH solution for all the situations that consider anisotropic dispersion(i.e. αT/αL=1). We note that these results consider γ =1.2 and that accordingour previous discussion, one would expect that the SPH solution would behave betterif a larger γ is used. However, the results of our simulations indicate that the trend ofincreasing error for larger anisotropy ratios of the SPH and FV methods is independentof the other parameters considered in this study.3.5.2.6 Eﬀect of kernel functionTable 3.4 presents a summary of the E2and E∞errors for run R5 for scenarios thatconsider diﬀerent SPH kernels and PSE cutoﬀ functions. For the isotropic case, the usehigher-order SPH kernels does not have a clear impact on the accuracy of the solution,while the use of higher-order PSE cutoﬀ functions results in smaller errors. In particular,the diﬀerence between the second and fourth order cutoﬀ functions is quite importantand it confirms that the error of the PSE approximation can be eﬀectively improved usinghigher-order cutoﬀ functions as discussed by Eldredge et al. (2002). For the anisotropiccase, the use of higher-order SPH kernels improve the solution but the eﬀect is lessimportant than one observed using diﬀerent cutoﬀ functions in the PSE case. Moreover,the use of higher-order cutoﬀ functions improve the PSE approximation and makes itmore accurate than the FV method for the anisotropic case.931 1.1 1.2 1.3 1.4 1.5 1.610−410−310−210−1100101h/∆xE2 SPH αT/αL=1.0SPH αT/αL=0.01PSE αT/αL=1.0PSE αT/αL=0.011 1.1 1.2 1.3 1.4 1.5 1.600.010.020.030.040.050.06h / ∆ xE∞ / C0 SPH αT/αL=1.0SPH αT/αL=0.01PSE αT/αL=1.0PSE αT/αL=0.01Figure 3.8: Error E2for run R6 versus the ratio between smoothing length or coresize and particle spacing, γ = h/∆x. While the error of the PSE solutiongrows monotonically with γ for the isotropic and anisotropic cases, theerror of the SPH solution for the anisotropic case decreases with it.9410−310−210−110010−410−310−210−1100αT/αLE210−310−210−110010−310−210−1αT / αLE∞ / C0 FVSPHPSEFigure 3.9: Error as function of the anisotropy ratio αT/αLfor run R7 and h/∆x =1.2. FV and SPH methods are less accurate for higher levels of anisotropy(lower αT/αL). The error of the PSE solution is almost constant for therange of anisotropy ratios tested.95SPH PSEαT/αLKernel E2E∞Kernel E2E∞1 Cubic 7.0 2.7 2nd order 103.4 11.11 Quartic 4.1 2.1 4th order 4.1 2.11 Quintic 7.6 2.9 6th order 2.6 1.50.01 Cubic 17.4 3.9 2nd order 5.2 2.50.01 Quartic 7.1 2.6 4th order 0.2 0.50.01 Quintic 4.7 2.2 6th order 0.1 0.3Table 3.4: Normalized error for diﬀerent SPH kernels and PSE cutoﬀ functions. Er-ror of the SPH and PSE numerical solutions divided by the error of theFV approximation for run R5 and h/∆x =1.2 considering diﬀerent SPHkernels and PSE cutoﬀ functions.One would expect that higher order polynomials used as SPH kernels have the advantageof smoother derivatives which, in combination with the increased size of support volume,could decrease the sensitivity of the kernel to the degree of particle disorder (Price,2004).However, the results of our simulations show that the use of higher-order kernels doesnot provide a significant improvement of the numerical solution in the simple case ofequispaced particles.3.5.2.7 Eﬀect of velocity orientationIt is well known that the error of numerical methods based on grids or meshes used tosolve (3.2) that include the oﬀ-diagonal terms of the dispersion tensor exhibit numericalartifacts that depend upon the flow orientation with respect to the grid axes (Herreraand Valocchi, 2006). Therefore, it is interesting to test if the error of the two particlemethods changes for diﬀerent flow orientations. Table 3.5 summarizes the results for runR6 assuming diﬀerent flow orientations. As expected, all three methods are not sensitiveto the flow direction for the isotropic case. However, for the anisotropic case the meshbased FV method exhibits diﬀerences of up to two orders of magnitude in the E2errorand one order of magnitude in the E∞depending on the flow direction. The error of theSPH solution also depends upon the flow direction but it only shows small diﬀerences fordiﬀerent velocity directions. On the other hand, the error of the PSE solution is almostindependent of the flow direction.96SPH PSE FVαT/αLβ◦E2E∞E2E∞E2E∞1 45 0.0061 1.8051 0.0728 6.6221 0.0007 0.59641 0 0.0061 1.8051 0.0728 6.6221 0.0007 0.59641 53 0.0061 1.8051 0.0728 6.6221 0.0007 0.59640.01 45 0.5062 10.4855 0.0880 5.1519 0.0172 2.13330.01 0 0.6919 14.9257 0.0880 5.0957 0.0006 0.55520.01 53 0.4495 9.7445 0.0880 5.2464 0.0154 1.9997Table 3.5: Error versus flow velocity direction for R5 and h/∆x =1.2. The errorof the numerical solutions is independent of the flow direction if isotropicdispersion is considered. However, the error of the FV and SPH solutiondepends on the flow velocity direction if anisotropic dispersion is simulated.3.5.3 Irregularly Spaced ParticlesIt is well known that the accuracy and stability of the PSE and SPH methods dependsupon the spatial distribution of particles (Cleary and Monaghan,1999;Zimmermannet al.,2001;Chaniotis et al., 2002). In general, at the beginning of a simulation particlesare distributed in a uniform fashion, e.g. rectangular lattice. As particles move carried bythe flow, high velocity gradients result in the distortion of the initial regular distributionas shown in Figure 3.10. In general, the continuity property of the flow prevents particlesfrom moving randomly and the particle set maintains some regularity (Monaghan,2005).However, some non-uniform flows can result in very irregular particle distributions.97Figure 3.10: Particle distortion due to flow velocity. High gradients in fluid velocity(arrows) result in distortion of the initial regular particle distribution(black circles). However, the continuity property of the flow preventsthat particles become randomly distributed.To evaluate eﬀect of the particle disorder on the accuracy and stability of the SPHand PSE solutions, we set up a set of simulations that evaluate the numerical solutionusing randomly and quasi-randomly distributed particles. An example of the diﬀerencebetween the distributions is shown in Figure 3.11. A random distribution results in largecontrasts in the spatial density of particles in diﬀerent areas of the domain. On the otherhand, a quasi-random distribution results in an irregular but uniform spatial particle98Figure 3.11: Particle locations for run R1 considering random and quasi-random dis-tributions. The random distribution has large contrasts in particle den-sity, while the quasi-random distribution has an irregular but uniformspatial particle density.density. Based on our experience, particle distributions as result of real flow fields fallbetween these two extreme cases. Therefore, simulations that consider these two spatialdistributions allow us to estimate upper and lower bounds for the performance of thePSE and SPH approximations for more realistic simulations.For the simulations that consider irregularly distributed particles, we used a cubic SPHkernel and a sixth order PSE cutoﬀ function.3.5.3.1 Isotropic caseFigure 3.12 shows the error E2versus the average particle spacing for the same scenariobut diﬀerent particle spatial distribution. As expected the error increases with the degreeof disorder. As seen by the slope of the curves in Figure 3.12, the convergence rate ofthe two methods decreases as particles become more disordered. Both methods convergevery slowly for the case of randomly distributed particles and the convergence rate isnot monotonic. It is interesting to notice that while a larger ratio h/∆x results in largererror for the case of equispaced particles, it actually helps to control the error in thecase of random and quasi-random particle distributions. Overall, the PSE method is lesssensitive to the disorder of the nodes than the SPH approximation.9910 15 20 25 30 35 40 45 5010−310−210−1100101102∆xE2 Equispaced h/∆x=1.2Equispaced h/∆x=1.6Random h/∆x=1.2Random h/∆x=1.6Quasi−Random h/∆x=1.2Quasi−Random h/∆x=1.6a10 15 20 25 30 35 40 45 5010−310−210−1100101∆xE2 Equispaced h/∆x=1.2Equispaced h/∆x=1.6Random h/∆x=1.2Random h/∆x=1.6Quasi−Random h/∆x=1.2Quasi−Random h/∆x=1.6bFigure 3.12: E2error of (a) SPH and (b) PSE numerical solutions versus averageparticle spacing for αT/αL=1.0 using equispaced, random, and quasi-random particle distributions. The convergence rate of boths methodsis lower for irregular particle distributions. In both cases, the errorincreases with the degree of particle disorder.100The previous observations are confirmed by Figure 3.13, which shows the normalized E∞error as function of the average particle spacing. As a first approximation, we can saythat for small particle spacing the error increases by one order of magnitude between theequispaced and quasi-random distribution and by another order of magnitude betweenthe quasi-random and random distributions.Figure 3.14 shows the temporal evolution of the diﬀerence between the maximum concen-tration of the analytical and numerical solutions. The error of the PSE solution is muchsmaller than the error of the SPH approximation for the case of randomly distributedparticles. The error of both methods is smaller for quasi-randomly distributed particles.This figure confirms that the use of larger ratios between smoothing length or core sizeand particle spacing result in smaller errors when particles are irregularly distributed. Inparticular, the error of the PSE method is almost constant and less than 1% of the initialconcentration if a ratio h/∆x =1.6 is used. This error is comparable to the maximumerror of the PSE approximation for the case of equispaced particles with a second-ordercutoﬀ function (see Figure 3.5).10110 15 20 25 30 35 40 45 5010−310−210−1100∆xE∞/C0 Equispaced h/∆x=1.2Equispaced h/∆x=1.6Random h/∆x=1.2Random h/∆x=1.6Quasi−Random h/∆x=1.2Quasi−Random h/∆x=1.6a10 15 20 25 30 35 40 45 5010−310−210−1100∆xE∞/C0 Equispaced h/∆x=1.2Equispaced h/∆x=1.6Random h/∆x=1.2Random h/∆x=1.6Quasi−Random h/∆x=1.2Quasi−Random h/∆x=1.6bFigure 3.13: Normalized E∞error of (a) SPH and (b) PSE numerical approxima-tions versus average particle spacing for αT/αL=1.0 using equispaced,random, and quasi-random particle distribution. The convergence rateof both methods is lower for irregular particle distributions.1020 50 100 150 200 250 300−0.04−0.0200.020.040.060.080.10.120.140.16Time Step(CmaxN − CmaxA)/C0 aSPH h/∆x=1.2SPH h/∆x=1.6PSE h/∆x=1.2PSE h/∆x=1.60 50 100 150 200 250 300−0.0100.010.020.030.040.050.060.070.08Time Step(CmaxN − CmaxA)/C0 bSPH h/∆x=1.2SPH h/∆x=1.6PSE h/∆x=1.2PSE h/∆x=1.6Figure 3.14: Diﬀerence between maximum concentration values of analytical and nu-merical solutions as function of the number of time steps for run R7,αT/αL=1.0, and (a) random and (b) quasi-random particle distribu-tions.1033.5.3.2 Anisotropic caseBoth particle approximations proved to be much more sensitive to particle disorder whensimulating anisotropic dispersion than for the isotropic case. SPH simulations with ran-dom and quasi-random distributions and h/∆x =1.2 became unstable after few timesteps. For the other scenarios, the errors of both methods stay almost constant as theaverage particle spacing decreases, as shown in Figure 3.15. The use of larger smoothinglengths or core sizes results in lower errors, but it does not significantly improve theconvergence rate of the numerical approximations. The minimum E∞corresponds to thePSE solution for quasi-randomly distributed particles and h/∆x =1.6 is approximately7% of the initial maximum concentration C0.Figure 3.16 shows the concentration field for run R7 and quasi-randomly distributedparticles at the end of the simulation. Both solutions exhibit negative concentrations(dark bands) that, as for equispaced particles, are located in regions almost parallel tothe flow direction. While the maximum magnitude of the negative values in the PSEsolution (−0.28) is very similar to the one observed for equispaced particles (−0.32), itis five orders of magnitude larger for the SPH solution, −16.86 for quasi-random and−6.9 · 10−4for equispaced particles.3.6 ConclusionsWe present the derivation of SPH approximation to simulate anisotropic dispersion. Wealso present an analytical analysis of the monotonicity properties of the new approxima-tion. In addition, we compare the new approximation to the particle strength exchangemethod and a standard 9-point finite volume scheme to simulate the dispersion of acontaminant plume in two-dimensions under diﬀerent dispersivity ratios and flow orien-tations. Furthermore, we test the numerical properties of the three methods by evaluatingthe sensitivity of the solution to a variety of numerical parameters such as particle andgrid spacing, kernel and cutoﬀ functions, and ratio of smoothing length or core size toparticle spacing.Based on the results of the numerical simulations presented above, we conclude thefollowing:1. Simulations that consider anisotropic dispersion are troublesome for all three meth-ods. The error of the numerical solution is larger and the convergence rate lower10410 15 20 25 30 35 40 45 5010−210−1100101102∆xE2 SPH Random h/∆x=1.6SPH Quasi−Random h/∆x=1.6PSE Random h/∆x=1.2PSE Quasi−Random h/∆x=1.2PSE Random h/∆x=1.6PSE Quasi−Random h/∆x=1.610 15 20 25 30 35 40 45 5000.10.20.30.40.50.60.7∆xE∞/C0 SPH Random h/∆x=1.6SPH Quasi−Random h/∆x=1.6PSE Random h/∆x=1.2PSE Quasi−Random h/∆x=1.2PSE Random h/∆x=1.6PSE Quasi−Random h/∆x=1.6Figure 3.15: Error as function of average particle spacing for αT/αL=0.01 usingrandomly and quasi-randomly distributed particles. The use of largerratio h/∆x decreases the error, but convergence rates of both methodsare much lower than for equispaced particles or for the isotropic case.105Figure 3.16: Concentration distribution after 300 time steps for run R7, αT/αL=0.01 and quasi-randomly distributed particles. Dark bands indicate ar-eas of negative concentrations. Minimum concentration values are -16.86for SPH and -0.28 for PSE.than for the corresponding isotropic case. Furthermore, the numerical solutionscomputed with any of the three methods independent of the particle distributionexhibit artificial oscillations and negative concentrations.2. For equispaced particles, the convergence rate of both particle methods is similarto that of the standard 9-point finite volume scheme. However, in contrast to thefinite volume scheme, the convergence rate and the overall accuracy of the SPH andPSE methods does not only depend on the number of particles or average particlespacing used, but also on other additional parameters such as kernel function andsmoothing length.3. The spatial distribution of particles is the most important factor that controls theaccuracy of the numerical solutions computed with the PSE or SPH approximations.The accuracy of the solution decreases as the degree of disorder of the particlesincreases. This eﬀect is more important for simulations that include anisotropicdispersion than for simulations of isotropic dispersion. To a certain extent, theloss of accuracy of the numerical solution can be controlled by using larger ratiosbetween smoothing length or core size to average particle spacing. Overall, thePSE method is less sensitive to particle disorder than the SPH method.106Previous studies (Zimmermann et al.,2001;Chaniotis et al., 2002) have demonstratedthat the periodic remeshing of particles can help to control the loss of accuracy of particleformulations due to the particle disorder caused by the flow velocity. Our numericalresults indicate that using a remeshing step is likely beneficial in simulations that consideranisotropic dispersion. However, the loss of accuracy of the particle methods for isotropicdispersion is less important, thus the benefits of remeshing could be counter balanced bythe additional computational cost and artificial diﬀusion that it introduces. Moreover,the remeshing procedure would not prevent the occurrence of negative concentrations.Those numerical oscillations can be particularly troublesome if particle methods are usedto simulate reactive transport. In that case, negative values can be amplified by non-linearchemical reactions. Therefore, although the PSE and SPH schemes may be compellingalternatives to simulate conservative solute transport in porous media, they may not beappropriate for reactive solute transport simulations.1073.7 ReferencesBear, J., Dynamics of fluids in porous media,Dover,1988.Beaudoin, A., S. Huberson, and E. Rivoalen, Simulation of anisotropic diﬀusion by meansof a diﬀusion velocity method, J. Comput. Phys., 186,122–135,2003.Chaniotis, A., D. Poulikakos, and P. Koumoutsakos, Remeshed smoothed particle hydro-dynamics for the simulation of viscous and heat conducting flows, J. Comput. Phys.,182,67–90,2002.Cleary, P. W., and J. J. Monaghan, Conduction modelling using smoothed particle hy-drodynamics, J. Comput. Phys., 148,227–264,1999.Crumpton, P., G. Shaw, and A. Ware, Discretisation and Multigrid Solution of EllipticEquations with Mixed Derivative Terms and Strongly Discontinuous Coeﬃcients, J.Comput. Phys., 116,343–358,1995.Degond, P., and S. Mas-Gallic, The weighted particle method for convection-diﬀusionequations. Part 1: The case of an isotropic viscosity, Math. Comput., 53,485–507,1989a.Degond, P., and S. Mas-Gallic, The weighted particle method for convection-diﬀusionequations. II: The anisotropic case, Math. Comp, 53, 485,508, 1989b.Edwards, M., and H. Zheng, A quasi-positive family of continuous Darcy-flux finite-volume schemes with full pressure support, J. Comput. Phys.,2008.Eldredge, J. D., A. Leonard, and T. Colonius, A General Deterministic Treatment ofDerivatives in Particle Methods, J. Comput. Phys., 180,686–709,2002.Español, P., and M. Revenga, Smoothed dissipative particle dynamics, Phys. Rev. E, 67,026,705–12, 2003.Gingold, R. A., and J. J. Monaghan, Smoothed particle hydrodynamics: Theory andapplication to non-spherical stars, Mon. Not. R. Astron. Soc., 181,375–389,1977.Herrera, P., and A. Valocchi, Positive solution of two-dimensional solute transport inheterogeneous aquifers, Ground Water, 44,803–813,2006.Herrera, P., M. Massabo, and R. Beckie, A meshless method to simulate solute transportin heterogeneous porous media, Adv. Water Resour., 32,413–429,2009.108Jameson, A., Positive schemes and shock modelling for compressible flows, Int. J. Numer.Methods Fluids, 20,1995.Jubelgas, M., V. Springel, and K. Dolag, Thermal conduction in cosmological SPH sim-ulations, Mon. Not. R. Astron. Soc., 351,423–435,2004.Kinzelbach, W., The random walk method in pollutant transport simulation, in Ground-water flow and quality modelling, edited by E. Custodio, A. Gurgui, and J. P. Lobo,pp. 227–246, 1988.Kuzmin, D., and S. Turek, Flux correction tools for Finite Elements, J. Comput. Phys.,175,525–558,2002.Le Potier, C., Finite volume monotone scheme for highly anisotropic diﬀusion operatorson unstructured triangular meshes, Comptes Rendus Mathématique, 341,787–792,2005b.Lipnikov, K., M. Shashkov, D. Svyatskiy, and Y. Vassilevski, Monotone finite volumeschemes for diﬀusion equations on unstructured triangular and shape-regular polygonalmeshes, J. Comput. Phys., 227,492–512,2007.Lipnikov, K., D. Svyatskiy, and Y. Vassilevski, Interpolation-free monotone finite volumemethod for diﬀusion equations on polygonal meshes, J. Comput. Phys., 228,703–716,2009.Lucy, L., A numerical approach to the testing of the fission hypothesis, Astron. J., 82,1013–1024, 1977.Mlacnik, M., andL.Durlofsky, Unstructuredgridoptimizationforimprovedmonotonicityof discrete solutions of elliptic equations with highly anisotropic coeﬃcients, J. Comput.Phys., 216,337–361,2006.Monaghan, J., Smoothed particle hydrodynamics, Rep. Prog. Phys., 68,1703–1759,2005.Monaghan, J. J., Smoothed particle hydrodynamics, Annu. Rev. Astron. Astrophys., 30,543–574, 1992.Nakshatrala, K., and A. Valocchi, Non-negative mixed finite element formulations for atensorial diﬀusion equation, Arxiv preprint arXiv:0810.0322,2008.Nordbotten, J., and I. Aavatsmark, Monotonicity conditions for control volume methodson uniform parallelogram grids in homogeneous media, Computat. Geosci., 9,61–72,2005.109Pollock, D., Semianalytical computation of path lines for Finite-Diﬀerence models,Ground Water, 26,743–750,1988.Price, D. J., Magnetic Fields in Astrophysics, PhD thesis, Institute of Astronomy, Uni-versity of Cambridge, 2004.Tartakovsky, A., and P. Meakin, A smoothed particle hydrodynamics model for miscibleflow in three-dimensional fractures and two-dimensional Rayleigh-Taylor instability, J.Comput. Phys., 207,610–624,2005.Tompson, A., Numerical simulation of chemical migration in physically and chemicallyheterogeneous porous media, Water Resour. Res., 29,3709–3726,1993.Welton, W., Two-dimensional PDF/SPH simulations of compressible turbulent flows, J.Comput. Phys., 139,410–443,1998.Yuan, G., and Z. Sheng, Monotone finite volume schemes for diﬀusion equations onpolygonal meshes, J. Comput. Phys.,2008.Zheng, C., and G. Bennet, Applied Contaminant Transport Modelling: Theory and Prac-tice, Van Nostrand Reinhold, New York, 1995.Zimmermann, S., P. Koumoutsakos, and W. Kinzelbach, Simulation of pollutant trans-port using a particle method, J. Comput. Phys., 173,322–347,2001.110Chapter 4A MultidimensionalStreamline-Based Method toSimulate Reactive Solute Transportin Heterogeneous Porous Media14.1 Introduction4.1.1 MotivationDespite considerable eﬀorts made during the last decades to advance the state of theart in numerical modeling of conservative and reactive solute transport in porous media,current numerical methods still have serious limitations to provide accurate and eﬃcientsimulations of situations of practical interest. The use of grid-based methods such as finitediﬀerence or finite elements to simulate conservative and reactive transport in porousmedia is problematic for several reasons. Solute transport in porous media is typicallyadvection-dominated, thus grid-based methods are aﬄicted by numerical dispersion thatcan be mitigated but not avoided by using high-order numerical schemes. In addition,spurious oscillations arise because of the application of non-linear high-order methods1A version of this chapter will be submitted for publication. P. Herrera, A. Valocchi, and R. Beckie.A Multidimensional Streamline-Based Method to Simulate Reactive Solute Transport in HeterogeneousPorous Media.111to solve advection (Steefel and MacQuarrie,1996;Cirpka et al., 1999) and numericalapproximations of oﬀ-diagonal entries in the dispersion tensor (Herrera and Valocchi,2006; Lipnikov et al., 2007). Finally, explicit time integration schemes can result invery restrictive global stability criteria, especially in highly heterogeneous velocity fields(Thiele et al.,1997;Crane and Blunt,1999).Particle methods based on random-walk schemes are an attractive alternative to simulatesolute transport in porous media because of their natural ability to simulate advection-dominated transport, their simplicity, and their inherent advantages for scalable andeﬃcient parallel implementations. However, those methods also have important disad-vantages. First, they have problems in tracking low concentrations and producing smoothconcentration distributions (Tompson,1993;Obi and Blunt,2004;Herrera et al., 2009b).Second, they require a background grid to simulate the eﬀects of local-scale dispersionon the mixing of diﬀerent chemical compounds and to compute concentrations to es-timate reaction rates. The computation of averaged concentration values over the cellvolumes can introduce artificial mixing that can compromise the original advantages ofthe method. Third, their performance can be degraded in situations that include multi-ple chemical species because of the large number of particles required to track multipleconcentrations.Reactive solute transport simulations impose even more severe restrictions on the per-formance of numerical methods than do conservative solute transport simulations. First,numerical artifacts such as spurious oscillations and numerical dispersion can be ampli-fied in presence of non-linear reactions (Cirpka et al., 1999b). Second, in many situationsof practical interest — e.g. biodegradation of contaminant plumes — chemical reactionsmainly occur in areas of low solute concentrations that can be diﬃcult to model accu-rately with methods that have been successfully applied to simulate conservative solutetransport, e.g. random-walk methods (Tompson, 1993). Last, numerical methods ap-plied to simulate reactive transport must be computationally eﬃcient to allow for thesimulation of multiple species at fine spatial and temporal scales.Recently, Herrera et al. (2009b) presented a meshless approach based on smoothed parti-cle hydrodynamics (SPH) (Gingold and Monaghan,1977;Lucy, 1977), hereafter referredto as the Monte-Carlo SPH (MC-SPH) method, to simulate conservative solute transportinheterogeneous porousmedia. MC-SPHisaLagrangianmethodthatuses akernel-basedinterpolation scheme to represent dispersion (Cleary and Monaghan,1999;Jubelgas et al.,2004). In MC-SPH simulations, individual particles move along instantaneous stream-lines carrying solute concentration and exchanging solute mass with nearby particles, as112shown in Figure 4.1. Since the method can handle dispersion without remapping theconcentration field onto a grid, it is free of numerical dispersion and grid orientationeﬀects. In addition, because particles carry concentration information and not mass, themethod accurately resolves low concentration values and produces smooth concentrationdistributions.Figure 4.1: Meshless MC-SPH method. In MC-SPH simulations a given particle(black circle) moves along an instantaneous streamline (dashed line) whileexchanging solute mass with particles that are within its “area of in-fluence” defined by the support volume of the kernel function (shadedcircles).Although promising, MC-SPH also presents problems. Because it is based on an integralinterpolation scheme, the accuracy of the solution depends on the spatial distributionof the particles, which can become a problem in presence of heterogeneous flow fieldswhere particles accumulate in stagnant zones (Herrera et al., 2009a). The evaluation ofthe interpolation scheme requires identifying near neighbor particles at each time step,which introduces a computational overhead in comparison with other numerical schemes.Moreover, the method had been only used to simulate isotropic dispersion until the recentwork of Herrera et al. (2009a) who introduced a method to handle the full dispersiontensor in three dimensions. They demonstrated that their approximation worked wellfor anisotropic dispersion, but can produce negative concentrations for the full tensordispersion with oﬀ-diagonal terms. Therefore, this approximation is not suitable forreactive transport simulations and this motivates us to turn to a new approach based113upon streamline methods.Streamline methods have been successfully used to simulate oil migration (Thiele et al.,1996, 1997) and multidimensional solute transport (Crane and Blunt,1999;Di Donatoand Blunt,2004;Obi and Blunt, 2004, 2006). These methods use a numerical gridthat adapts to the flow field, which reduces numerical dispersion and grid orientationeﬀects. The use of streamlines allows the transformation of a multidimensional trans-port equation to a set of individual one-dimensional transport problems. Because of itsadaptation to the flow and its ability to minimize numerical dispersion, the method iswell suited for simulations of advection-dominated transport as found in heterogeneousporous media (Di Donato et al., 2003). In addition, the numerical solution of the re-sulting one-dimensional transport problem also allows the use of more eﬃcient numericalsolvers and more relaxed stability constraints (Crane and Blunt, 1999). Because of theeﬃciency of the method, it is possible to simulate large-scale domains with fine spatialand temporal resolution (Di Donato et al.,2003;Obi and Blunt,2004,2006).Although longitudinal dispersion along individual streamlines can be easily incorporated,transverse mixing between streamlines is more diﬃcult to simulate. Since many impor-tant reactions in situations of practical interest occur along the fringes of contaminantplumes and are controlled by transverse dispersion (Ham et al., 2004), it is crucial toincorporate transverse mixing in a streamline-based formulation to obtain a general sim-ulation framework that can be applied to a wide range of reactive transport problems.To best of our knowledge, two approaches have been used to incorporate transversedispersion in streamline-based simulations. In the first, solute transport is solved using aflow-oriented grid and transverse dispersion is included as a flux component perpendicularto the streamlines (Cirpka et al., 1999). This approach has been successfully used in two-dimensional simulations (Cirpka et al., 1999b), but it has not been extended to three-dimensions. A second alternative employs a hybrid approach Obi and Blunt (2004). First,advection is solved along streamlines. Then, concentration values are mapped onto a gridwhere a mesh-based solver is used to solve for dispersion. Finally, concentration valuesare interpolated back from the grid to the streamlines. The interpolation from and tostreamlines introduces some numerical error that is diﬃcult to quantify (Obi and Blunt,2004). Because the interpolation must be done at each time step, the cumulative eﬀectcan be important even if an accurate interpolation scheme is used.114Figure 4.2: Hybrid streamline-SPH method. The method combines ideas taken fromstreamlines simulations and MC-SPH. Solute advection and longitudinaldispersion are handled as in traditional streamline simulations. Nodesalong streamlines are used as interpolations points to apply the samemeshless approximation for local-scale dispersion as the one used in MC-SPH, so an individual node (black circle) exchanges solute mass withother nodes (grey circles) that are within the support volume of the kernelfunction (shaded region).4.1.2 ObjectivesThe main objective of this paper is to present a new hybrid numerical method thatcombines some of the most important advantages of streamline methods and MC-SPH.Streamlines are used to discretize the domain and to define the location of a set ofnodes that is used to evaluate the numerical solution. Advection is solved in tradi-tional streamline-based simulations, i.e. as a composite of one-dimensional solutions.Anisotropic dispersion is accommodated as the sum of a three-dimensional isotropic dis-persion contribution handled using a meshless integral approximation, and longitudinaldispersion solved along each streamlines using a standard finite diﬀerence formulation.Figure 4.2 shows a schematic of the proposed hybrid method.The main advantages of the proposed method are: (i) like traditional streamlines meth-ods, it is well suited to simulate advection-dominated transport in heterogeneous porousmedia because of the elimination of artificial mixing due to numerical dispersion, (ii) itprovides a robust mechanism to incorporate transverse dispersion between streamlineswithout requiring additional interpolation steps, and (iii) the use of a flow oriented gridto approximate dispersion results in a numerical scheme that is monotone and positivitypreserving even for full dispersion tensors, thereby avoiding negative concentrations and115spurious oscillations that plague many other numerical methods.4.2 Mathematical Formulation4.2.1 Governing EquationAt the local-scale reactive solute transport in porous media is modeled by a system ofpartial diﬀerential equations, which is given as follows for the case of constant porosityand an incompressible fluid (Bear,1988):∂Ck∂t= ∇·(D∇Ck)−v·∇Ck+ Rk(c) k =1,...,m∂Ck∂t= Rk(c) k = m +1,...M (4.1)where Ck[M/L3] is the solute concentration of species or component k, D [L2/T] is thehydrodynamic dispersion tensor, v [L/T] is the pore water velocity, Rk[M/L3/T] is thetotal reaction rate for species or component k, c = (C1,...,CM) is the concentrationvector, m is the number of species or components in the aqueous (mobile) phase, and Mis the total number of species or components.In what follows we will focus our attention on the numerical solution of the first group ofequations that describe the migration of chemical species in the aqueous phase, which cor-respond to a set of advection-dispersion-reaction (ADR) equations. However, as demon-strated by others authors (Di Donato and Blunt, 2004) and in Section (4.4), the proposedstreamline formulation can also handle situations that include immobile species in thesolid phase.The most common expression to compute the components of the dispersion tensor D foran isotropic porous medium considering a Cartesian coordinate system is (Bear,1988;Lichtner et al.,2002)Dij= (αTq + Dm)δij+ (αL−αT)vivjq(4.2)116where Dm[L2/T] is the molecular diﬀusivity, δijis Kronecker’s delta, q = |v| [L/T] isthe magnitude of the pore water velocity, and αLand αT[L] are the longitudinal andtransverse dispersivity of the medium, respectively.4.2.2 Streamline FormulationTraditional streamline models neglect dispersion and rewrite the multidimensional trans-port equation (4.1) as a one-dimensional transport equation along streamlines using thefollowing identity (Thiele et al.,1997;Crane and Blunt,1999)v·∇≡|v|∂∂s=∂∂τ(4.3)where s is the arc length coordinate and τ is the time of flight (TOF), defined as thetime required to reach a point located at a distance s along a streamline (Thiele et al.,1996; Crane and Blunt, 1999). Mathematically,τ =sˆ01|v|dξ (4.4)If we consider a local coordinate system with components ˆxithat are parallel and perpen-dicular to the flow direction as shown in Figure 4.3, then the oﬀ-diagonal terms in (4.2)are equal to zero and the diagonal terms simplify toD11= αL|v| andD22=D33= αT|v|.Therefore, the multidimensional ADR equations can be written using a flow-orientedcoordinate system and the TOF to get∂Ck∂t= ∇·D∇Ck−∂Ck∂τ+ Rk(c) k =1,...m (4.5)or in terms of the arc length coordinate, s,toget∂Ck∂t= ∇·D∇Ck−|v|∂Ck∂s+ Rk(c) k =1,...m (4.6)The formulations given by (4.5) and (4.6) are equivalent, but they each present distinctchallenges for numerical methods.117ˆx1ˆx2s0,!s0+!s,"0+"Figure 4.3: Flow oriented coordinate system. The transport equation can be writtenconsidering using a coordinate system with coordinates ˆxithat are paralleland perpendicular to the direction of the flow. Streamline simulationsalso define other two coordinates along individual streamlines, the timeof flight τ and the arc length s.4.2.3 Numerical ApproximationThe numerical solution of equations (4.5) or (4.6) requires numerical approximationsfor the advection, dispersion and reactions terms. The advection term can be approx-imated within a traditional streamline framework whereas the dispersion term can beevaluated using a combination of a one-dimensional finite diﬀerence approximation alongstreamlines for the longitudinal component and a kernel-based interpolation for the othercomponents. Finally, the reaction component can be approximated using diﬀerent solversdesigned to integrate initial value problems (e.g. see Oran and Boris,2000).In the following description of the proposed streamline-based formulation we assumethat: (i) the flow velocity field is externally computed, thus it is an input parameter;and (ii) the flow field is steady-state. The second assumption allows us to simplify thediscussion about the implementation of the method but it does not represent a reallimitation, because transient flow fields can be easily handled by tracing new streamlineswhen flow conditions change and, then, interpolating concentrations from the old to thenew streamline locations(Thiele et al.,1996;Thiele,2005).In the discussion that follows we drop the sub-index k and assume a single species orcomponent to simplify notation.1184.2.3.1 Advection along streamlinesFor simple problems that involve only advection and longitudinal dispersion, concentra-tion values along individual streamlines can be exactly computed using one-dimensionalanalytical solutions (Thiele et al., 1996). However, more general scenarios require theuse of numerical approximation schemes.Standard streamline simulations generate a numerical grid in TOF space by trackingparticles and recording the TOF when particles enter and exit from individual cells (e.g.Crane and Blunt,1999;Obi and Blunt, 2004). Because of the heterogeneity of the porousmedium and, hence, the flow velocity, the time required to cross individual cells may bevery diﬀerent and node separation in TOF space is very irregular. In general, the inte-gration of the advective term in (4.5) over the irregularly spaced nodes in TOF spacerequires some regularization of the one-dimensional grid to avoid excessive numerical dis-persion, to relax the stability restrictions of explicit solvers, and to simplify its numericalsolution by keeping a uniform grid spacing (Crane and Blunt,1999;Thiele,2003).Forexample, Crane and Blunt (1999) refined the original TOF grid to get uniformly spacednodes and used a regularization algorithm to conserve mass balance in the interpolatedconcentration values. Another approach is to only remove cells that are considered toosmall to significantly improve the spatial resolution of the method and may introduce un-necessary numerical constraints (Thiele,2003).Thiele et al. (1997) used a second-orderin space and an explicit first-order in time total variation diminishing (TVD) solver tocompute the solution, while Crane and Blunt (1999) used a first-order finite diﬀerencemethod with upstream diﬀerences for space and an implicit (backward Euler) diﬀerencefor time.A second option, which is the one we have implemented in our code, is to formulatethe one-dimensional advection-dispersion equation along a streamline in terms of the arclength coordinate using (4.6). In that case the numerical solution can be computed usinga grid defined by equispaced nodes along individual streamlines. The generation of sucha grid involves several steps. First, one must numerically integrate the fluid particletrajectories recording particle positions instead of TOF. Next, one can approximate thetotal length of a streamline as the sum of the distances of the arcs that connect adjacentnodes. The approximation of the arc length segments by the arc connecting adjacentnodes is second-order accurate for nodes that are relatively close (Aris, 1989). Then,the total arc length is divided into a number of segments such that the length of eachsegment is close to a given target spacing ∆s, while the sum of the individual segments119is equal to the original streamline length and the starting and end points are the sameas in the original streamline.The solution of the advective term in (4.6) is equivalent to solving the following PDE∂C∂t+ q(s)∂C∂s=0 (4.7)which corresponds to an advection equation with variable coeﬃcients also known as thecolor equation (LeVeque, 2002). The last equation can be recast as a conservation lawwith flux F = qC plus a source term due to the change in velocity, to obtain∂C∂t+∂∂s(qC) = C∂q∂s(4.8)The last solution can be solved with any one-dimensional solver for hyperbolic equations,e.g. low- or high-resolution schemes and explicit or implicit time discretizations.The use of a uniformly spaced grid in s instead of one spaced in τ allows more control ofthe physical distribution of nodes and the possibility of refining the streamline grid bydecreasing the node spacing independently of the resolution of the grid used to computethe flow velocity.4.2.3.2 DispersionThe ratio between longitudinal and transverse dispersivities in porous media is equal toor greater than one. Then, one can make the change of variable αL= αT+ αLwith˜αL≥ 0. Therefore, the dispersion tensor with principal directions that are parallel andperpendicular to the flow velocity,D, can be rewritten asD =DL+D 000D 000D(4.9)whereDL= qαLandD = qαT+Dm. Then, the dispersion term in (4.6) can be rewrittenas,120∇·D∇C= ∇·D∇C+∂∂ˆx1DL∂C∂ˆx1(4.10)where ˆx1is the coordinate aligned with the flow direction (see Figure 4.3).Using a streamline discretization the derivative along ˆx1can be easily evaluated using asecond-order finite diﬀerence approximation to obtain,∂∂ˆx1DL∂C∂ˆx1=1si+1/2−si−1/2Di+1/2Lsi+1−si(Ci+1−Ci)−Di−1/2Lsi−si−1(Ci−Ci−1)(4.11)To derive this expression we have assumed that the diﬀerence between the indices ofconsecutive nodes along a streamline is equal to one. Thus, (4.11) may be written as∂∂ˆx1DL∂C∂ˆx1= γi+1Ci+1+ γi−1Ci−1−γiCi(4.12)with γi≥ 0.The first term on the right-hand-side of (4.10), which is equivalent to isotropic dispersionor a diﬀusion process, can be evaluated using a MC-SPH approximation to obtain (Clearyand Monaghan,1999;Jubelgas et al.,2004;Herrera et al., 2009b),dCidt= −j1ˆpijDi+Djrij|rij|2∇W(rij)(Cj−Ci) (4.13)where rij= ri−rjis the separation vector between nodes i and j,Diis the modifieddispersion coeﬃcient at node i, W is a kernel function that satisfies some normalizationconditions and that, in general, has compact support, and ˆpijis a symmetric approxima-tion of the node density at nodes i and j (for details see Cleary and Monaghan,1999;Monaghan,2005;Herrera et al., 2009b, and references therein). The node density atnode i is evaluated as,pi=jW(rij) (4.14)121Then, we can rewrite (4.13) asdCidt=jβij(Cj−Ci) (4.15)with βij≥ 0 because for typical kernels ∇W ≤ 0.Therefore, the numerical approximation of (4.10) can be writtendCidt= γi+1Ci+1+ γi−1Ci−1−γiCi+jβij(Cj−Ci) (4.16)To study the monotonicity properties of this approximation (4.16), we notice that it hasthe formdCidt=j=iˆβij(Cj−Ci)=jˆβijCj(4.17)withjˆβij=0. The local extremum diminishing (LED) criteria (Jameson,1995;Kuzminand Turek, 2002) establishes that numerical approximations such as (4.17) that satisfyˆβij≥ 0, ∀i = j preserves the monotonicity of the solution, because the temporal deriva-tive of the concentration at a maximum can only be negative and similarly, the con-centration at a minimum can only increase(Kuzmin and Turek, 2002). Therefore, thediscretization (4.16) preserves the positivity of the concentration distribution and pro-vides solutions that are free of spurious oscillations and negative concentrations.4.3 Implementation Details4.3.1 Streamline TracingAn important part of streamline-based simulations consists in tracing streamlines givena velocity field. Given the location of an initial seed, the geometry of the streamline thatpasses through that point is generated using a particle tracking method to integrate thefluid trajectory (in the forward or backward directions) until reaching an inlet or outlet122face or a sink or source cell. The arc length and TOF are recorded at discrete intervalsalong the fluid particle trajectory.Pollock’s semi-analytical method Pollock (1988) is a popular choice to track streamlineswhen the flow velocity is known in a Cartesian staggered grid, because of its high per-formance and accuracy due to its semi-analytical character. The method has also beenextended to handle situations when the flow velocity is computed in an unstructured grid(Prevost et al., 2002). It is also possible to use explicit schemes to integrate streamlinetrajectories, e.g. Runge-Kutta schemes (Zheng and Wang, 1999) or adaptive algorithmse.g. (Bensabat et al.,2000).In our experience it is possible to obtain a similar accuracy with Pollock’s method or withan adaptive explicit scheme. Although, Pollock’s method is generally faster than explicitschemes, the performance diﬀerences are negligible when compared to the total timerequired for a simulation. On the contrary, explicit schemes are simpler to program andmore numerically robust because they avoid floating point errors that can be problematicfor semi-analytical methods. In our streamline solver we have implemented both Pollock’salgorithm and an explicit adaptive scheme, however we use the second one as our defaultparticle tracking method.The spatial distribution of streamlines is another key issue in streamline-based simula-tions. Without an adequate choice of the initial seed particle locations and in the presenceof heterogeneous flow fields or sources or sinks, the spatial streamline distribution can bevery irregular with large areas of the domain that do not contain any streamlines. Thiscan be particularly problematic in situations that require mapping concentrations onto abackground grid (Obi and Blunt, 2004). It can also be problematic for the implementa-tion of our hybrid scheme, because the MC-SPH approximation for dispersion assumesthat the area of influence of nodes in neighboring streamlines always overlap. Thus, itis important to assure a minimum density of streamlines in every region of the domain.A common solution of this problem is to use a background grid. After a set of initialstreamlines has been traced, new streamlines are traced starting at each grid cell thatdoes not contain one (Batycky et al.,1997;Crane and Blunt, 1999). This is the approachwe have implemented in our streamline simulator.1234.3.2 Time IntegrationIn principle, it is possible to use an explicit, implicit or a hybrid scheme to integrate(4.16). Moreover, depending on the numerical scheme used to approximate advection,the solution of the advection-dispersion-reaction equation can be solved fully-implicitly.A fully-implicit solution would result in more relaxed stability restrictions for the timestep size. On the other hand, as discussed below, a fully-implicit implementation alsohas disadvantages such as larger memory requirements that make it less attractive. Asa consequence, in our streamline simulator we use an operator-splitting (OS) approachto solve the advection and dispersion terms. The OS approach is also used in currentstreamline simulators to incorporate dispersion (Obi and Blunt, 2004), to decouple thesaturation and pressure equations in streamline simulations of oil migration (Thiele et al.,1997), and in numerical packages to solve solute transport in porous media (e.g. Zhengand Wang,1999).The OS approach provides ample flexibility in the selection of numerical algorithms toobtain high accuracy and reasonable performance. An OS approach also allows use ofdiﬀerent time steps to solve the advective and dispersive terms. This can be an advantageto overcome some of the limitations of explicit solvers. Given a global time step ∆t, onecan integrate the dispersion term using ∆tD≤ ∆t such that many sub-steps may benecessary to complete a global step. A similar idea can be applied to integrate theadvection term. This is the approach we apply in our streamline simulator: a singleglobal step can involve many sub-steps to integrate the dispersive and advective terms.4.3.3 Advection SolutionWe use a first-order explicit TVD finite diﬀerence approximation to solve (4.8) (LeVeque,2002). Although formally first-order accurate, the high-resolution approximation per-forms better than low-order approximations such as upstream finite diﬀerence (LeVeque,2002). The implementation of the one-dimensional TVD solver can be done assumingthat the flow velocity is evaluated at the node positions or at the midpoint between them.In our implementation we evaluate the flow velocity at the node positions using a linearinterpolation scheme for the velocity components (Pollock, 1988), however any velocityreconstruction scheme that provides a continuous velocity field that satisfies the originalmass balance equation may be used.124We use diﬀerent time steps along each streamline to satisfy the stability constraints of theexplicit solver while minimizing numerical dispersion and computational requirements.We use nitime steps of size ∆tito integrate (4.8) along streamline i during a global timestep ∆t such that ∆t = ni∆ti, and the internal time step ∆tiis such that the maximumCFL number along streamline i is less than or equal to one, i.e. max(q∆ti/∆si) ≤ 1.Therefore, small time steps are necessary only along streamlines that cross fast flow re-gions, while large time steps can be applied to integrate along slow streamlines. Thisconstitutes an important performance advantage of streamline simulations versus multi-dimensional mesh-based solvers.4.3.4 MC-SPH Solution4.3.4.1 SPH kernelIn our streamline simulator we use a cubic-spline kernel (Monaghan,1992),W(r,h)=σhnd1−32rh2+34rh30 ≤rh< 1142−rh31 ≤rh< 20rh≥ 2(4.18)where r is the magnitude of the separation vector, h is the so called smoothing length,σ is a normalization constant, and nd is the number of dimensions. Hence, the size ofthe kernel support volume is given by the selection of h. In our implementation we use aconstant smoothing length for all the nodes. We select h based on the spatial distributionof streamlines and nodes such that the number of nodes per kernel support volume stayswithin a range that guarantees reasonable accuracy (Herrera et al., 2009b,a).4.3.4.2 Neighbor searchThe number of nodes that eﬀectively contribute to the summation in (4.13) depends uponthe support volume of the kernel function. Thus, evaluation of the temporal derivativein (4.13) can be accelerated by discarding the nodes that are beyond the kernel supportvolume. In that case the evaluation of the dispersion term for all the nodes becomes a125problem of order O(NkN), where Nkis the average number of nodes per kernel volumeand N is the total number of nodes (Herrera et al., 2009b).There are diﬀerent methods to identify near neighbor nodes that use data structuresto classify nodes according to their spatial location (Waltz et al.,2002;Viccione et al.,2008), e.g. linked lists (Welton, 1998) or hierarchical trees (Hernquist and Katz,1989).In our streamline simulator we use a background grid together with linked lists to storea list of the nodes located at each cell. Then, the evaluation of (4.13) for a specific nodeonly requires looping over nodes that are within adjacent cells (Welton, 1998). While inSPH particle simulations the node lists must be updated at each time step, in our hybridmethods the lists must be only updated each time that the streamlines are traced whichhappens much less often.4.3.4.3 Time integrationNumerical experiments show implicit schemes to solve (4.16) are unconditionally stableindependent of the time step utilized. Although attractive because of its stability prop-erties, the implicit solution of (4.16) can become impractical when some nodes have alarge number of neighbor nodes, e.g. three-dimensional problems or highly heterogeneoussystem. In those cases, the number of nodes that contribute to the summation in (4.17)and, hence, the number of non-zero coeﬃcientsˆβijcan be quite large. As consequence,the memory required to store the matrix of the implicit solution can become prohibitiveeven for a moderate number of nodes. A possible solution to this problem would be touse a variable smoothing length such that the number of neighbor nodes stays relativelyconstant (Monaghan,2005).Based on the above considerations, we use a first-order explicit formulation to integratethe dispersion term. The stability restriction of the explicit solver requires a time stepsuch that (Cleary and Monaghan,1999;Herrera et al., 2009b)∆t ≤ h2maxD(4.19)where is constant factor. In the simulations presented in the next section we find that =0.1 provides stable solutions.It is important to notice that the maximum time step given by (4.19) is inversely pro-portional to the magnitude ofD, which, for typical porous media applications is small126compared to the scale of the numerical discretization given by h. Thus, in general, thecondition imposed by (4.19) on the time step is less restrictive that the stability require-ments of explicit solvers for advection.4.3.5 Longitudinal Dispersion4.3.5.1 Interface coeﬃcientsThe numerical approximation of the longitudinal dispersion terms requires the evaluationof the coeﬃcientsDi±1/2Lin (4.11). Those coeﬃcients can be directly evaluated if the flowvelocity is known at the interface position i±1/2. Alternatively,Di±1/2Lcan be evaluatedas the harmonic average of the coeﬃcients at the nodes i and i±1 as in a standard cell-centered finite diﬀerence approximation (Zheng and Bennet, 1995). Both approximationsresult in symmetric expressions to compute dispersive numerical fluxes and guarantee fluxcontinuity across the interface. Because we evaluate the flow velocity at node positions,we have implemented the second alternative in our streamline simulator.4.3.5.2 Time integrationThe use of an OS strategy allows decoupling of the temporal integration of the longitu-dinal and transverse dispersion components that appear in (4.16). If each component isintegrated separately then the longitudinal term can be approximated using an explicit orimplicit approximations. An explicit approximation for the one-dimensional dispersionequation is conditionally stable and must satisfy the following stability restriction∆t ≤ 0.5(∆si)2maxDL(4.20)On the other hand, an implicit approximation has the advantage of being unconditionallystable. The resulting linear system is tri-diagonal and can be eﬃciently solved usinga direct solver, e.g. one based on the Thomas algorithm (Wang and Anderson,1982).However, such a splitting strategy would introduce an additional operator-splitting error.Alternatively, one can apply a single step to integrate (4.17), which combines the lon-gitudinal and transverse dispersion terms. This is the approach implemented in ourstreamline simulator which uses an explicit time marching scheme.127Figure 4.4 shows a diagram of the overall solution strategy implemented in our streamline-based simulator.4.4 Numerical ExamplesIn this section, we present four examples that we use to compare our implementation ofthe proposed streamline-based method and a finite diﬀerence package, MT3DMS (Zhengand Wang, 1999), which is a well-established solute transport simulator. The examplescorrespond to diﬀerent hydraulic conditions and solute release mechanisms. We considerdiﬀerent dispersivity values to test the quality of our new streamline-based approximationfor dispersion. In all the examples, the velocity field was computed with an external finitevolume package, MODFLOW (Harbaugh and McDonald,1996).MT3DMS is a well-tested and robust numerical package that provides several solvers tosimulate solute advection (Zheng and Wang, 1999). We use the multidimensional TVDand method of characteristics (MOC) solvers to simulate the examples presented below.The multidimensional TVD solver is a natural candidate for a comparison with our one-dimensional TVD solver used to advect solute along streamlines. On the other hand, theMOC solver minimizes numerical dispersion when transport is advection-dominated atthe cost of introducing additional mass balance errors and numerical oscillations (Zhengand Wang, 1999). Thus, the two solvers provide a range of solutions that are a goodrepresentation of the performance of the state of the art numerical solvers used to simulatesolute transport in porous media. In the rest of this section, we focus our analysison the ability of our streamline simulator to incorporate dispersion more than on therelative advantages of the MT3DMS or streamline solvers to simulate advection, sinceour principal objective is to introduce and validate our new formulation to incorporatedispersion in streamline-based simulations.ThecomparisonoftheperformanceofthesolversavailableinMT3DMSandourstreamline-based simulator is diﬃcult because of diﬀerences in their implementations and capabili-ties. For example, the solvers included in MT3DMS are implemented in the FORTRANprogramming language using single precision, while our streamline simulator is imple-mented in Java using double precision. There are other implementation details that canalso result in additional performance diﬀerences. Furthermore, the simulations presentedin this section are for relatively small spatial domains and we expect that the observeddiﬀerences in performance would be diﬀerent and, probably, more important for larger128Initialize SimulationNewTimeStepSolve AdvectionSolve DispersionSolve ReactionsMoretimesteps?Endof SimulationFor each speciesk in aqueousphase, solve∂Ck∂t+q(s)∂Ck∂s=0 k =1,...,malongeach streamlinei using∆ti≤ min(∆si/q) andanexplicitone-dimensionalTVDsolver.Advancesolutionalongeachstreamlineuntilcomplet-inga globaltimestep.For each speciesk in aqueousphase, solve∂Ck∂t−∇·D∇Ck−∂∂ˆx1DL∂Ck∂ˆx1=0 k =1,...,mateach nodeusinga hybridMC-SPHfinitediﬀerenceschemewithanexplicitin timeapproximation.Computemaximumtimestep,∆tD,tosatisfystabilityrequirements. If∆tD<∆t, loopoveruntilcompletinga globaltimestep.For each speciesk, solve∂Ck∂t=Rk(c) k =1,...,Mat each node.yesTimeStep,∆tnoFigure 4.4: Overall solution approach implemented in streamline-based simulator.An operator splitting approach is used to decouple the solution of theadvective, dispersive and reactive terms.129domains or finer grids. Finally, our streamline simulator and the solvers implementedin MT3DMS are for serial computer platforms, thus we do not take into account theobvious advantages of the streamline formulation to transform the solution of advectioninto multiple independent one-dimensional problems that are amenable to parallelization.Therefore, one should keep in mind that the performance comparisons reported beloware limited to specific implementations of the methods and only a few problem sizes.4.4.1 Example 1: Continuous Solute Release in Uniform FlowThe first example corresponds to the continuous release of a contaminant in a uniformtwo-dimensional flow. The domain is square with each side equal to 1000 m. The finitediﬀerence grid has 100 x 100 cells, thus the cell spacing∆is equal to 10 m. The streamlinegrid consists of 100 streamlines with nodes spaced every 10 m. Thus, the number of nodesin the streamline grid is equal to the number of cells in the finite diﬀerence grid.The flow velocity is constant and equal to 1 m/d and parallel to the x axis. The so-lute is continuously released with constant concentration equal to 1 mg/L from a smallregion in the center of the inlet boundary of the domain. We use a time step equalto 3.65 d and we simulate the solute migration for a total of 150 time steps. In thestreamline simulations, we use a smoothing length equal to 12 m. For this simple quasione-dimensional problem we use the MT3DMS multidimensional TVD solver; which, forthis flow configuration, is equivalent to the one-dimensional TVD solver implemented inour the streamline simulator. Then, diﬀerences between both numerical solutions areonly due to the approximation for dispersion.We define four diﬀerent scenarios depending upon the longitudinal and transverse disper-sivity values as summarized in Table 4.1. The dispersivity values considered correspond tolongitudinal (PeL=∆/αL) and transverse (PeT=∆/αT) grid Péclet numbers between1 to 10 and 2 to 100, respectively.Figure 4.5 shows contours of concentration values equal to 0.2 and 0.8 mg/L. The TVDand streamline solutions are identical independently of the longitudinal and transversedispersivity values used in each scenario. This example demonstrates that the newstreamline-based approximation for anisotropic dispersion performs well independentlyof the dispersivity values or anisotropy ratio.130Scenario αLαTPeLPeT1A 10 1 1 101B 10 5 1 21C 1 0.1 10 1001D 1 0.5 10 20Table 4.1: Dispersivity and equivalent longitudinal (PeL) and transverse (PeT) gridPéclet values used in Example 1.Figure 4.5: Comparison of simulated concentrations for Example 1. Contours forconcentration values equal to 0.2 and 0.8 mg/L at the end of runs 1A (a),1B (b), 1C (c) and 1D (d). Solid lines correspond to finite diﬀerence andsquares to streamline solutions, respectively.1314.4.2 Example 2: Quarter Five-Spot in Heterogeneous Medium4.4.2.1 SetupThe second example corresponds to the well-known quarter five-spot well configuration.We consider a square domain with an injection well located on the lower left corner andan extraction well on the upper right corner. The four faces of the domain have no flowboundary conditions. The domain is a square of 64 x 64 m and is discretized using auniform grid with 64 cells in each direction. We compute the flow velocity field assumingsteady-state conditions and a heterogeneous hydraulic conductivity field. The spatialdistribution of the natural logarithm of the hydraulic conductivity, Y = ln(K) with Kin units of m/d, is generated assuming an exponential covariance model with mean value¯Y =1.0, variance σY=1.0 and correlation length in both directions equal to five timesthe grid spacing. The injection and extraction rates at the wells is set equal to 10 m3/d.Streamlines are first generated from 100 equispaced points located over the diagonal linethat connects the upper left and lower right corners. Additional streamlines are tracedsuch that each cell of the grid used to compute the velocity field is crossed by at least onestreamline resulting in a total of 146 streamlines. The generated hydraulic conductivityand streamlines are shown in Figure 4.6.Figure 4.6: Spatial distribution of the natural logarithm of the hydraulic conductivityand streamlines in Example 2.132To test the sensitivity of the TVD and MOC solutions with respect to the grid size, werefine the original grid to obtain a fine grid with 128 cells in each direction. We assignthe hydraulic conductivity values computed in the coarse grid to the corresponding cellsin the the fine grid, i.e. one value computed in the coarse grid is assigned to four cellsin the fine grid. We then solve the flow and transport problems in the new grid. Thereare diﬀerences between the flow solutions computed in the coarse and fine grid as resultof the diﬀerent spatial discretizations, but they are rather small. For example, the meanvelocity computed in the fine grid is less than 1% higher than the one computed inthe coarse grid. However, the simulation results presented below demonstrate that suchdiﬀerences have a negligible impact on the solute transport solution. In the streamlinesimulations nodes are uniformly distributed along streamlines with an average spacingof 1 m. Additional parameters required to set up the MOC solver are listed in Table4.2. Those parameters were chosen to minimize numerical oscillations observed in somepreliminary simulations.Parameter ValueMax. # of cells a particle can move in one time step 1Relative cell concentration gradient (DCCELL) 0.00001Number of particles in cells with relative gradient >DCCELL32Number of particles in cells with relative gradient <DCCELL2Number of particles in sources or sink cells 64Tracking algorithm 4th-orderRunge-KuttaTable 4.2: Parameters used in MOC simulations. For a detailed explanation see(Zheng and Wang, 1999).We assume that the contaminant is injected through the lower left well with concentra-tion equal to 1 mg/L and that the initial concentration is zero. The period simulatedcorresponds to 300 days and it is discretized using diﬀerent time steps according to thestability restrictions of the explicit multidimensional TVD, MOC and streamline solversas summarized in Table 4.3. The time step used in the streamline simulations was chosensuch that the product of the mean flow velocity times the time step is equal to the averagearc length spacing.133Method # Nodes or Cells Time Step # Time stepsStreamlines 15,106 0.500 600TVD Coarse 4,096 0.109 2820TVD Fine 16,384 0.034 8520MOC Coarse 49,116 0.193 1560MOC Fine 203,439 0.054 5520Table 4.3: Number of nodes or cells, time step size and number of time steps used insimulations of Example 2. Number of nodes reported for MOC correspondsto the maximum number of particles used during the simulation since thisis the factor that controls the computational requirements, i.e. CPU timeand memory.We use the three solvers to simulate four scenarios that represent diﬀerent dispersivityvalues as summarized in Table 4.4. For the streamline simulations that include dispersionwe use a smoothing length equal to 1.2 m.Scenario αL(m) αT(m) PeLPeT2A 0.1 0 10 ∞2B 0.1 0.01 10 1002C 0.1 0.1 10 102D 0 0 ∞∞Table 4.4: Dispersivity and equivalent longitudinal (PeL) and transverse (PeT) gridPéclet values used in Example 2.4.4.2.2 Simulated concentrationsFigure 4.7 shows simulated concentrations for scenarios 2A and 2C after the injectionof 0.4 pore volumes of solute. MOC and TVD solutions were computed using the finegrid. Simulated concentration values using the streamline simulator were interpolatedfrom nodes along streamlines to the cell centers of the fine grid for comparison purposes.In general, the numerical solutions computed with all three methods are similar.For scenario 2A the simulated concentrations indicate the presence of a slow flow regionalong the diagonal line that connects the two wells, which creates a sharp interface134between two fast moving fingers with concentration equal to the source concentrationand a central region with concentration equal to zero. When transverse dispersion isincluded (run 2C) the slow central flow region is filled with concentrations that are about50% of the source value. Transverse dispersion also produces a wider mixing zone atthe edges of the advancing contaminant front where concentration values lie between thesource and the background values.Figure 4.7: Simulated concentration values after injection of 0.4 pore volume of con-taminated fluid for Example 2. Interpolated streamline (first row), MOC(second row) and TVD (third row) solutions for scenarios 2A (left col-umn) and 2C (right column). The solutions computed with the threemethods are similar for both scenarios.135We also recorded the simulated concentration versus time at two observation points.To avoid introducing additional numerical dispersion in the streamline results, we chosethe locations of the observation points such that they coincide with two nodes in thestreamline grid. The first observation point, P1, is located in the region of rapid flowchanges near the plume center-line (see Figure 4.7) and the second one, P2, in a regionwhere the advance of the plume is relatively homogeneous.Figures 4.8 and 4.9 show simulated breakthrough curves for the four scenarios consideredat P1 and P2, respectively. While all three numerical methods simulate concentrationvalues that are similar for point P2, there are important diﬀerences in the simulatedcurves for P1. At P1, the solution is very sensitive to numerical dispersion because of thepresence of a slow flow region between two fast advancing plume fingers, which createshigh concentration gradients.Simulated concentrations with the MOC and streamline-based solvers at point P1 forthe purely-advective case (2D) agree well, with the exception of numerical oscillationin the MOC solution. On the other hand, numerical dispersion is clearly observable inboth, coarse and fine, TVD curves. In general, both mesh-based solvers predict earlierbreakthrough for all the simulated scenarios because of additional transverse dispersiondue to computing cell average concentrations. For example, in absence of transverse dis-persion (2A) both mesh based solvers predict that the arrival of solute to the observationpoint P1 would occur around 80 days earlier than predicted by the streamline simulator.However, the diﬀerence between the streamline and mesh-based solvers becomes smalleras transverse dispersion increases and concentration values are smooth out at the gridscale, e.g. scenarios 2B and 2C. The solutions computed with the MOC and TVD solversalso become more similar as transverse dispersion increases and the advantages of theMOC solver to minimize numerical dispersion become less important.1360.2 0.3 0.4 0.5 0.6 0.7 0.800.20.40.60.81Pore VolumeC/C0 AMOC CoarseMOC FineTVD CoarseTVD FineStreamlines0.2 0.3 0.4 0.5 0.6 0.7 0.800.20.40.60.81Pore VolumeC/C0 BMOC CoarseMOC FineTVD CoarseTVD FineStreamlines0.2 0.3 0.4 0.5 0.6 0.7 0.800.20.40.60.81Pore VolumeC/C0 CMOC CoarseMOC FineTVD CoarseTVD FineStreamlines0.2 0.3 0.4 0.5 0.6 0.7 0.800.20.40.60.81Pore VolumeC/C0 DMOC CoarseMOC FineTVD CoarseTVD FineStreamlinesFigure 4.8: Breakthrough curves at observation point P1 in Example 2. Point P1 islocated in a region of rapid flow changes near the plume center-line. Eachfigure corresponds to one of the four scenarios simulated in Example 2:2A, 2B, 2C and 2D. At P1 simulated concentrations are very sensitiveto transverse dispersion. The numerical solutions computed with thestreamline solver in absence of transverse dispersion (A and D) diﬀersignificantly from the ones computed with the two mesh-based solvers.Those diﬀerences become smaller as transverse dispersion increases (Band C).Figure 4.10 shows a comparison of the simulated breakthrough curves at P1 using thestreamline and MOC (fine grid) solvers. The streamline solver predicts a breakthroughcurve for run 2A, which considers only longitudinal dispersion, that is similar to thecurve for the purely-advective case (2D), but has earlier breakthrough and reaches thepeak concentration at later time. This is the expected behavior for that situation, whichcorresponds to a quasi one-dimensional transport problem. When transverse dispersionis included (run 2B), the streamline simulator predicts a change in the first part of thecurve as consequence of the transfer of solute mass from fast streamlines to slower ones,1370.2 0.3 0.4 0.5 0.6 0.7 0.800.20.40.60.81Pore VolumeC/C0 AMOC CoarseMOC FineTVD CoarseTVD FineStreamlines0.2 0.3 0.4 0.5 0.6 0.7 0.800.20.40.60.81Pore VolumeC/C0 BMOC CoarseMOC FineTVD CoarseTVD FineStreamlines0.2 0.3 0.4 0.5 0.6 0.7 0.800.20.40.60.81Pore VolumeC/C0 CMOC CoarseMOC FineTVD CoarseTVD FineStreamlines0.2 0.3 0.4 0.5 0.6 0.7 0.800.20.40.60.81Pore VolumeC/C0 DMOC CoarseMOC FineTVD CoarseTVD FineStreamlinesFigure 4.9: Breakthrough curves at observation point P2 in Example 2. Point P2 islocated in a relatively homogeneous flow region where the advance of thesolute front is relatively uniform. Each figure corresponds to one of thefour scenarios simulated in Example 2: 2A, 2B, 2C and 2D. Simulatedconcentrations with the streamline simulator and the two mesh-basedsolvers are similar independent of the longitudinal and transverse disper-sivity values.138which results in an earlier breakthrough and change in slope with respect to the curves forruns 2A and 2D. In contrast, the breakthrough curves computed with the MOC solver,which performs well for the purely-advective case, do not show a clear distinction betweenthe addition of longitudinal or transverse dispersion. When longitudinal dispersion is in-cluded the breakthrough curve shifts to the left of the curve for the purely-advectivecase, which is not consistent with the situation analyzed. Adding transverse dispersionresults in an additional shift of the curve to the left, but there are no clearly distin-guishable changes in slope as observed in the curve obtained with the streamline-basedsolver. This example demonstrates some of the advantages of the streamline simulator tostudy situations of theoretical interest such as the eﬀect of transverse dispersion in thetransport of solutes in heterogeneous porous media.0.2 0.3 0.4 0.5 0.6 0.7 0.800.20.40.60.81Pore VolumeC/C0 aRun ARun BRun CRun D0.2 0.3 0.4 0.5 0.6 0.7 0.800.20.40.60.81Pore VolumeC/C0 bRun ARun BRun CRun DFigure 4.10: Comparison of simulated breakthrough curves at observation point P1for Example 2, (a) streamline simulator and (b) MOC solver using finegrid.Figure 4.11 shows a comparison of the simulated breakthrough curves at P2 using thestreamline and MOC (fine grid) solvers for the four transport scenarios analyzed. Withthe exception of the purely-advective case simulated with the MOC solver, the simu-lated breakthrough curves are similar independently of the solver and dispersivity valuesconsidered.1390.2 0.3 0.4 0.5 0.6 0.7 0.800.20.40.60.81Pore VolumeC/C0 aRun ARun BRun CRun D0.2 0.3 0.4 0.5 0.6 0.7 0.800.20.40.60.81Pore VolumeC/C0 bRun ARun BRun CRun DFigure 4.11: Comparison of simulated breakthrough curves at observation point P2for Example 2, (a) streamline simulator and (b) MOC solver using finegrid.4.4.2.3 Numerical oscillationsFigure 4.12 shows simulated concentrations after injection of 0.4 pore volumes of solutefor runs 2A and 2B. Negative and greater than source concentration values are coloredblack, thus dark areas within the domain boundaries indicate zones where numericaloscillations occur. Numerical oscillations present in the solutions computed with theMOC and TVD solvers cover most of the simulation domain including areas that havenot been reached by the solute front.Table 4.5 and 4.6 lists the normalized maximum and minimum simulated concentrations,respectively. Minimum concentration values simulated with both mesh-based solvers,TVD and MOC, are negative for scenarios that include anisotropic dispersion (A, B) dueto the presence of the oﬀ-diagonal terms of the dispersion tensor. The TVD solver alsoproduces small negative values for the advection-only case (D), but their magnitude ismuch smaller than for the runs that included anisotropic dispersion. In contrast, solutionscomputed with the streamline solver are always positive. Moreover, maximum concen-tration values computed with the TVD and MOC solvers are greater than the sourceconcentration for some of the scenarios simulated because of spurious numerical oscilla-tions. The magnitude of the oscillations is greater for scenarios that consider anisotropicdispersion (A and B), probably because of the presence of negative concentrations val-ues that arise due to the presence of the cross-dispersion terms. The magnitude of theoscillations decreases when only advection (D) or isotropic dispersion are simulated (C).Solutions computed with the streamline solver are free of spurious oscillations in all cases.140Figure 4.12: Numerical oscillations in simulated concentrations for Example 2. Con-centration values after injection of 0.4 pore volume of contaminatedfluid. Interpolated streamline (first row), MOC (second row) and TVDsolutions (third row) for scenarios 2A (left column) and 2B (right col-umn). Negative and greater than source concentration values are shownas dark areas within the domain boundaries. Solutions computed withthe MOC and TVD solver exhibit numerical oscillations that cover mostof the domain.141Method\Run 2A 2B 2C 2DStreamlines 0000TVD Coarse -0.0285 -0.0171 0 <-0.0001TVD Fine -0.0236 -0.0111 0 <-0.0001MOC Coarse -0.0042 -0.0011 0 0MOC Fine -0.0043 -0.0010 0 0Table 4.5: Normalized minimum simulated concentration values for Example 2.Method\Run 2A 2B 2C 2DStreamlines 1.0000 1.0000 1.0000 1.0000TVD Coarse 1.0272 1.0147 1.0037 1.0063TVD Fine 1.0588 1.0276 1.0000 1.0034MOC Coarse 1.0435 1.0112 1.0000 1.0000MOC Fine 1.0561 1.0076 1.0000 1.0000Table 4.6: Normalized maximum simulated concentration values for Example 2.4.4.2.4 Performance comparisonTable 4.7 summarizes the normalized CPU time required to simulate the four diﬀerentscenarios analyzed in Example 2. The streamline solver is faster to solve advection. It isup three times faster than the TVD or MOC solvers for the coarse grid (64x64 cells) andmore than thirty times faster for the fine grid (128x128 cells). However, the mesh-basedsolvers are faster if dispersion is included and the solution is computed using the coarsegrid. Yet, the streamline solver is faster by a factor of about two, if the fine grid is used.This demonstrates that the streamline solver performs much better than mesh-basedsolvers to simulate two-dimensional problems. The diﬀerence in performance is moreimportant when advection and longitudinal dispersion are simulated, which can be veryvaluable in many situations of practical interest. For example, simulating scenarios thatconsider only advection and longitudinal dispersion can be useful in the first stages ofa model calibration, sensitivity analysis, evaluation of parameters uncertainty (Hill andTiedeman, 2007) or in any situation where a large number of solute transport simulationsis required.142Method\Run 2A 2B 2C 2DStreamlines 1.1 28.6 28.6 1.0TVD Coarse 4.4 4.4 4.4 2.5TVD Fine 55.1 55.1 48.1 29.0MOC Coarse 5.5 5.5 5.5 3.4MOC Fine 81.0 81.0 81.0 50.2Table 4.7: Normalized CPU time required to simulate Example 2 for diﬀerent scenar-ios. Streamline solver is faster than mesh-based solvers to solve advectionindependently of the grid size. It is also faster to solve advection-dispersionthan mesh-based solvers using a fine grid.4.4.3 Example 3: Quarter Five-Spot in Heterogeneous Mediumwith Rate-Limited SorptionAs a third example we consider the transport of a dissolved solute in groundwater that re-acts with the porous medium and sorbs onto the solid grains. This example demonstratesthe capacity of streamline-based simulations to easily incorporate heterogeneous chemicalreactions that involve the aqueous and solid phases. Di Donato and Blunt (2004) studieda similar problem in the context of the migration of a reactive solute through fracturedrocks.Sorption is usually modeled assuming local equilibrium, i.e. that solute sorption occursalmost instantaneously relative to the solute transport time scale. However, the localequilibrium assumption is not always valid (Valocchi, 1985) and it is necessary to modifythe equilibrium model to incorporate rate-limited mass transfer eﬀects. In those cases,sorption can be modeled as a first-order reversible kinetic reaction of the form (Haggertyand Gorelick,1995)∂S∂t=βρbC −SKd(4.21)and∂C∂t= −βηC −SKd(4.22)143where β is the first-order mass transfer rate between the dissolved and solid phases [1/T],Kdis the distribution coeﬃcient for the solid phase [M/L3], ρbis the bulk density of thesolid, η is the soil porosity, and S is the amount of mass in the solid phase [M/M]. As βdecreases, mass transfer rates become smaller and sorption becomes negligible. On theother hand, as β increases the solution becomes similar to the one obtained assuminglocal equilibrium. A similar mathematical model can be used to model mass transferbetween mobile and immobile flow regions (Haggerty and Gorelick,1995).In our streamline-based simulator, we compute the solution of the system of diﬀerentialequations given by (4.21) and (4.22) at each node of the streamline grid using a first-order implicit (backward Euler) discretization in time. To verify our implementation,we simulate the advective transport of a solute that undergoes rate limited sorptionunder four diﬀerent scenarios summarized in Table 4.8. We set the soil density valueequal to 1,500 kg/m3and select Kdvalues such that the equivalent retardation factors,R =1+(ρb/η)Kd, are equal to 1.15 and 1.30.Run Kd(L/mg) β (1/d)3A 1 · 10−71 · 10−13B 1 · 10−71 · 10−43C 2 · 10−71 · 10−13D 2 · 10−71 · 10−4Table 4.8: Parameters of the rate-limited sorption model used in Example 3.We compare concentrations simulated with the our streamline simulator and MOC andTVD solvers using the coarse grid. Figure 4.13 shows the simulated breakthrough curvesat point P2. Solutions computed with the streamline solver and the MOC solver behavesimilarly. While the curves for scenarios 3A and 3C are retarded, the curves for scenarios3B and 3D do not show retardation because in those cases sorption is limited by the smallβ values used. As in the previous example simulations performed with the streamlinesolver were almost three times faster than the ones computed with the MOC solver.1440.2 0.3 0.4 0.5 0.6 0.7 0.800.20.40.60.81Pore VolumeC/C0 aKd=1x10−7 β=10−1Kd=1x10−7 β=10−4Kd=2x10−7 β=10−1Kd=2x10−7 β=10−40.2 0.3 0.4 0.5 0.6 0.7 0.800.20.40.60.81Pore VolumeC/C0 bKd=1x10−7 β=10−1Kd=1x10−7 β=10−4Kd=2x10−7 β=10−1Kd=2x10−7 β=10−4Figure 4.13: Breakthrough at observation point P2 for diﬀerent mass transfer (β)and partition (Kd) coeﬃcients considered in Example 3. Concentrationssimulated with (a) streamline-based and (b) MOC solvers.1454.4.4 Example 4: Natural Biodegradation in Three-dimensionalHeterogeneous Porous MediaIn this example, we simulate the natural biodegradation of a mixture of volatile organiccompounds such as benzene, toluene, ethylbenzene, and xylenes (BTEX) in an heteroge-neous aquifer with oxygen as electron acceptor.We simulate the aerobic reaction using an instantaneous explicit reaction model (Bordenand Bedient,1986;Rifai et al.,1987;Rifai and Bedient, 1990). The concentrations ofoxygen ([O]) and hydrocarbon ([H]) at time t+1given their values at time t are computedusing (Clement et al.,1998)[H](t +1)=[H](t)−[O](t)/F if [H](t) > [O](t)/F0 if [H](t) ≤ [O](t)/F(4.23)and[O](t +1)=0 if [H](t) > [O](t)/F[O](t)−F · [H](t) if [H](t) ≤ [O](t)/F(4.24)where F is the stoichiometric reaction ratio between oxygen and the hydrocarbon, in thiscase BTEX, and it is assumed equal to 3.0 (Rifai et al.,1987;Clement et al.,1998).4.4.4.1 SetupThe domain is three-dimensional and rectangular with dimensions of 50 m in the xdirection, 21 m in the y direction and 5 m in the vertical direction (z) and it is discretizedusing a regular Cartesian grid such that the cell spacing in the three directions is equalto 1 m.The hydrocarbon source is a well located at coordinates x=8.5 m, y=10.5 m and z=2.5 mthat releases 0.2 m3/d of water with a BTEX concentration equal to 2 mg/L.Oxygenis initially present in the natural groundwater and enters the domain with concentrationequal to 5 mg/L.146The spatial distribution of the natural logarithm of the hydraulic conductivity, Y =ln(K), was generated assuming an exponential covariance model with mean value¯Y =1.0with K in units of meter per day (m/d), variance σY=1.5 and correlation length equalto four times the grid spacing in the x and y directions and two times in the verticaldirection.We solve the flow equation assuming steady-state conditions and a constant head diﬀer-ence of 0.4 m between the planes define by x=0and x=50and no flow conditions onthe other faces. The spatial distributions of the generated hydraulic conductivity andcomputed flow velocity are shown in Figure 4.14.To evaluate the sensitivity of the solutions computed with the mesh-based solvers to thegrid size, we refined the original coarse grid by a factor of two in each direction to obtaina fine grid with 100 cells in the longitudinal, 42 cells in the transverse and 10 cells in thevertical directions, respectively. We applied the same procedure as in the Example 2 tocompute a flow solution that is similar to the one computed in the coarse grid.We consider three transport scenarios summarized in Table 4.9: one that includes onlyadvection, a second one that considers advection and biodegradation, and a third onethat also includes dispersion.Run αLαTReaction4A 0 0 Yes4B 0.1 0.01 Yes4C 0 0 NoTable 4.9: Definition of three scenarios simulated in Example 4: advective transportwith biodegradation (4A), advective-dispersive transport with biodegrada-tion (4B) and conservative advective transport (4C).The streamline grid includes 281 streamlines with average node spacing along streamlinesequal to 1 m. We use a constant smoothing length equal to 1m to ensure that everynode in the domain has at least a minimum of 10 neighboring nodes. To compute theMOC solution we use the same parameters listed in Table 4.2.We simulate this problem using our streamline simulator and the RT3D numerical modelClement (1997), which is a modified version of MT3DMS that adds the capability to147Figure 4.14: Spatial distribution of natural logarithm of hydraulic conductivity andflow velocity magnitude used in Example 4. Magnitude of hydraulicconductivity and flow velocity vary more than six and two orders ofmagnitude, respectively.148simulate chemical reactions. RT3D and our streamline solver utilize the same operator-splitting approach to incorporate chemical reactions.We simulate a total of 10,000 days which is longer than the time required for the plumessimulated with the two mesh-based solvers to reach steady-state conditions. We chosetime step values to satisfy the stability or accuracy criteria required by each method.Table 4.10 summarizes the numerical discretizations and their associated time steps.Method # Nodes or Cells Time Step (d) # Time StepsStreamlines 13,539 2.0 5,000TVD Coarse 5,250 3.2 3,300TVD Fine 42,000 0.4 27,816MOC Coarse 75,562 5.0 2,000MOC Fine 565,464 0.6 16,000Table 4.10: Spatial and temporal discretizations used to simulate Example 4.Because of the flow configuration there are two diﬀerent regions separated by a surfacethat divides the flow between flow that enters the domain through the well and naturalgroundwater (Cirpka et al., 1999b). In absence of longitudinal and transverse mixing, thesubstrate (BTEX) and oxygen are physically segregated and occupy two diﬀerent regionsand biodegradation cannot take place. If longitudinal dispersion is included, biodegra-dation only occurs until all the oxygen initially present in the area located downgradientfrom the source is depleted or the plume exits the domain. Thus, in the long termbiodegradation is only possible if transverse dispersion mixes the natural groundwaterand the contaminated water (Cirpka et al., 1999b; Ham et al., 2004). It is expected that asteady-state plume will be established when the mass of substrate entering the domain isequal to the amount that is consumed through biodegradation. Since the reaction rate iscontrolled by transverse mixing between the two flow regions, the length of steady-stateplumes depends only on the transverse dispersivity value (Ham et al.,2004).4.4.4.2 Simulated concentrationsFigure 4.15 shows simulated BTEX and oxygen concentrations at the nodes that definethe streamline grid after 10,000 days since the initial release of BTEX for the scenario149that includes advection and biodegradation (4A). As expected, oxygen is depleted in theregion located downgradient of the BTEX spill. Because of the absence of transverse dis-persion there is a sharp transition in BTEX and oxygen concentrations between two flowregions that correspond to BTEX-contaminated water flowing from the well and naturalgroundwater. BTEX concentration values along most of the streamlines are equal to thesource concentration, which indicates that flow coming from the spill has completely re-placed the original groundwater. Yet, few slower streamlines present concentrations thatare lower than the source concentration as a result of biodegradation due to numericaldispersion along streamlines. The presence of those lower concentration values indicatesthat flow originating at the contaminant source and moving along slow streamlines hasnot reached the outlet face before the end of the simulation.Figure 4.16 shows simulated concentration for the scenario that includes advection, dis-persion and biodegradation (4B). Oxygen is depleted in the region located downgradientof the BTEX spill. However, in contrast to the situation without dispersion shown inFigure 4.15, BTEX and oxygen concentrations change gradually between the two flowregions because of mixing due to transverse dispersion. In this case, natural degradationtakes place in most of the plume volume and BTEX concentration values gradually de-crease along streamlines. At the outlet face they are between 25 to 50% lower than thesource concentration.Figure 4.17 shows simulated concentrations in the vertical plane defined by the coordinatey=11.5 at the end of the simulation using the streamline-based solver and the MOCand TVD solvers using the fine grid. For comparison purposes the streamline solutionwas interpolated onto a grid that is similar to the one used in the MOC and TVDsimulations. For the conservative advective case the streamline and MOC solutions aresimilar, however, the TVD solver produces lower concentrations after some distance fromthe source as result of numerical dispersion. When aerobic degradation is included,hydrocarbon concentrations simulated with the TVD and MOC solvers are much lowerthan for the conservative case after some distance from the source. The diﬀerence betweenthe conservative and reactive case is due to numerical dispersion that mixes BTEX andoxygen even in absence of physical dispersion and results in the degradation of most ofthe released BTEX before it exits the domain.The streamline solution for the case that includes biodegradation is similar to the onefor the conservative scenario, with the exception of minor diﬀerences near the top centerof the domain. Those diﬀerences are due to slow streamlines that cross that region. Theleading BTEX concentration along those streamlines is lower than in the conservative150Figure 4.15: Simulated concentrations at nodes along streamlines after 10,000 dayssince the initial release of BTEX for the scenario that includes advectivetransport with biodegradation in Example 4. Only 10,000 nodes areshown. Oxygen is depleted in the region located downgradient of theBTEX spill. Because of the absence of transverse dispersion there isa sharp transition in BTEX and oxygen concentrations between twoflow regions that correspond to flow originating at the well and naturalgroundwater.151Figure 4.16: Simulated concentrations at nodes along streamlines after 10,000 dayssince the initial release for the scenario that includes advection, disper-sion and biodegradation in Example 4. Only 10,000 nodes are shown.Oxygen is depleted in the region located downgradient of the BTEXspill. However, in this case BTEX and oxygen concentrations changegradually between the two flow regions because of mixing due to trans-verse dispersion.152case because of longitudinal numerical dispersion that mixes oxygen and BTEX (seeFigure 4.15). However, the artificial longitudinal mixing results in small errors whencompared with the large errors observed in the simulated concentrations obtained withthe two mesh-based solvers.When transverse dispersion is included, the streamline solver, as expected, predicts lowerconcentrations after some distance from the source as consequence of transverse mixingthat provides oxygen to degrade BTEX that flows through the center of the plume. TheTVD and MOC solvers also predict lower concentrations than for the advective case withbiodegradation, but the eﬀect on the plume shape is relatively minor.Figure 4.17: Simulated concentration values for Example 4 at vertical plane definedby y=11.5 m. Streamlines (top), MOC (middle), and TVD (bottom) foradvective transport (left column), advective transport with biodegrada-tion (middle column) and advective-dispersive transport with biodegra-dation (right column). Streamline solution shows concentration valuesinterpolated onto a grid of 100x42x10 cells. TVD and MOC solutionscomputed using the fine grid. Vertical scale is exaggerated by a factorof two.Figure 4.18 shows simulated concentrations after 10,000 days of BTEX injection in thehorizontal plane that crosses the domain center defined by coordinate z=2.5 m. Thisfigure confirms the results observed in Figure 4.17. The simulated plume with the MOCand TVD solvers for the case that included advection and biodegradation are shorterand thinner than for the conservative case. Including transverse dispersion has onlyan small eﬀect in the overall shape and extension of the simulated plume. In contrastthe streamline solver predicts similar plumes for the purely advective and the advective-reactive cases and a smaller plume when transverse dispersion is included.153Figure 4.18: Simulated concentration values for Example 4 at horizontal plane de-fined by z=2.5 m. Streamlines (top), MOC (middle), and TVD (bot-tom) for advective transport (left column), advective transport withbiodegradation (middle column) and advective-dispersive transport withbiodegradation (right column). The streamline solution shows concen-tration values interpolated onto a grid of 100x42x10 cells. TVD andMOC solutions were computed using fine grid.Figure 4.19 shows contours of concentration values equal to 1 mg/L equivalent to 50% of the source concentration for the horizontal plane shown in Figure 4.18. For theconservative case the simulated concentrations with the streamline solver and the TVDand MOC solvers using the coarse and fine grids are similar. However, the simulatedplumes with the mesh-based solvers using the coarse and fine grids are diﬀerent whenbiodegradation is included. The coarse solution predicts a shorter plume, while thefine grid solutions predict a longer plume that looks more similar to the one simulatedwith the streamline-based solver. The observed convergence of the mesh-based solutionstowards the streamline-based solution as the grid spacing decreases demonstrates that,for this problem, a streamline-based formulation may provide a high level of accuracy ata fraction of the memory requirements and, as discussed below, computational cost thatwould be required by a mesh-based solver.154Figure 4.19: Comparison of simulated concentration values for Example 4 at horizon-tal plane defined by z=2.5 m. Contours of BTEX concentration equalto 50% of the source concentration for advective transport (first row),advective transport with biodegradation (second row), and advective-dispersive transport with biodegradation (third row). Streamline solu-tion (black), TVD solution (red) and MOC solution (blue) computed incoarse (left column) and fine (right column) grids. Mesh-based solutionsoverestimate biodegradation as consequence of numerical transverse dis-persion. The error is larger if the solution is computed in the coarse grid.In absence of physical transverse dispersion, the mesh-based solvers pre-dict plumes that are similar to the ones computed including transversedispersion.The eﬀects of numerical dispersion on the simulated concentration values using the mesh-based solvers can be more easily understood by analyzing a transverse profile as shownin Figure 4.20. That figure shows normalized BTEX and oxygen concentration valuesalong the profile parallel to the y direction at coordinates x=35 m and z=2.5 m simulatedusing the MOC and TVD solvers using the fine grid. The figure shows that in absenceof transverse dispersion and reaction, there is an artificial mixing region where BTEXand oxygen are present. That mixing region is due to two factors. First, mesh-basedsolvers compute average concentration values over a volume (cell or element) that is notnecessarily aligned with the direction of the flow, thus it is impossible to capture the flowdivide that separates the two flow regions Cirpka et al. (1999b). Second, because of the155heterogeneity of the medium the flow velocity is non-uniform and solvers based on finitevolume formulations, such as the multidimensional TVD solver included in MT3DMS,suﬀer grid orientation eﬀects that introduce additional numerical dispersion. Both causesare related but are independent as demonstrated by a comparison of the profiles simulatedwith the MOC and TVD solvers. Because of its hybrid Eulerian-Lagrangian nature, theMOC solver is able to control the numerical dispersion due to variations in the flowvelocity, however it cannot remove the eﬀect of the concentration averaging over a cellvolume. Therefore, the width of the simulated mixing region is equal to two or threetimes the dimension of a grid cell. On the other hand, the solution computed with theTVD solver is greatly aﬀected by numerical dispersion caused by the non-uniform flowvelocity, which results in a much larger area (approximately ten cells) where BTEX andoxygen overlap.If biodegradation is included, degradation of BTEX takes place within the artificial mix-ing region. Thus, the simulated plume is thinner than in the conservative case. Although,the profiles shown in Figure 4.20 are illustrative of the eﬀect of numerical dispersion at agiven control plane, the overall shape of the plume is the result of the cumulative eﬀect ofnumerical dispersion and biodegradation that occur between the source and the controlplane. For example, the MOC solution for the conservative case shows much less overlapof BTEX and oxygen than the TVD solution. However, the widths of the simulatedplumes for the reactive scenarios are similar. This confirms that in presence of chemicalreactions even small numerical errors observed in conservative transport simulations canbe amplified to produce an overall solution that is very similar to the one computed withless accurate methods, e.g. TVD.Finally, we must mention that simulated concentrations with the MOC and TVD solversinclude negative values and values greater than the source or natural groundwater con-centrations. The magnitude of those numerical oscillations is larger for the scenario thatincludes dispersion because of the presence of the cross-terms in the dispersion approxi-mation.4.4.4.3 Breakthrough curvesAs an additional comparison between the streamline, TVD and MOC solvers, we alsorecorded the simulated concentrations versus time at two observation locations P1 andP2. We chose the location of those observations points such that they coincide with theposition of nodes along streamlines and that are located close to the flow divide that1560 5 10 15 2000.20.40.60.81y(m)C/C0 aBTEX−AOxygen−ABTEX−BOxygen−BBTEX−COxygen−C0 5 10 15 2000.20.40.60.81y(m)C/C0 bBTEX−AOxygen−ABTEX−BOxygen−BBTEX−COxygen−CFigure 4.20: Simulated concentration values for Example 4 along the profile parallelto y direction at coordinates x=35 m and z=2.5 m. Normalized BTEXand oxygen concentration values computed using (a) MOC and (b) TVDsolvers in fine grid for advective transport with biodegradation (A),advective-dispersive transport with biodegradation (B), and advectivetransport (C). In absence of transverse mixing and reaction, there is anoverlap (mixing region) of BTEX and oxygen (black lines) because ofnumerical dispersion. If biodegradation is included, BTEX and oxygenreact within that numerical mixing region.157separates the flow originating at the spill and the natural groundwater flow. Figure 4.21shows the location of the two observation points with respect to the contaminant sourceand simulated plume defined by the 1 mg/L BTEX contour. Point P1 is located atapproximately 7 m downgradient of the BTEX source along the main direction of theflow while point P2 is located 10 m farther downgradient from P1.Figure 4.21: Location of contaminant source (red dot) and the two observation wells(black dots) in Example 4. Solid line corresponds to simulated 1 mg/LBTEX concentration in plane defined by coordinate z=2.5 m.Figure 4.22 shows simulated BTEX breakthrough curves at the two observation points forthe scenario that includes only advection. Concentrations simulated with the streamline,MOC and TVD solvers in the coarse and fine grids are similar close to the locationof the source, i.e. at P1. However, simulated concentrations farther from the source,at point P2, are diﬀerent. Curves that correspond to the TVD and MOC solvers showearlier breakthrough. Furthermore, solutions computed with the TVD solver in the coarseand fine grid do not reach the source concentration at point P2, which indicates artificialdilution due to numerical mixing that is more important for the solutions computed usingthe coarse grid. Both solutions computed with the MOC solver show large numericaloscillations.On the other hand, the breakthrough curve corresponding to the streamline-based solveralso shows some longitudinal numerical dispersion. To demonstrate that that error can beeasily avoided, we also simulated this scenario using a streamline grid that has the samenumber of streamlines but has a smaller average node spacing equal to one fifth of theoriginal one, i.e. 0.2 m. The breakthrough curve simulated using the refined streamlinegrid is very sharp and concentration values go from zero to the source concentration ina very short time, which is the expected behavior for this scenario that does not include158transverse or longitudinal dispersion. Nevertheless, as discussed above, errors due tolongitudinal dispersion along streamlines are minor in comparison to the errors causedby the variable flow velocity orientation with respect to the main axes of the numericalgrid associated to the MOC and TVD solvers.Figure 4.23 shows simulated BTEX breakthrough curves at the two observation pointsfor the scenario that includes advection and biodegradation. As in the conservativecase, simulated concentrations with the streamline, MOC and TVD solvers in the finegrid are similar close to the location of the BTEX release at P1. However, simulatedconcentrations with the TVD and MOC solvers at point P2 are much lower than thesource concentration because of the combined action of numerical transverse mixing andbiodegradation. Simulated concentrations are even smaller when computed in the coarsegrid because of larger numerical errors. A comparison with Figure 4.22 shows that in thiscase diﬀerences between the simulated concentrations using the MOC and TVD solversand the streamline-based solver are much more important that for the conservative case.This demonstrates how the addition of chemical reactions can amplify errors observed inconservative solute transport simulations.4.4.4.4 PerformanceTable 4.11 summarizes the normalized CPU time required to simulate the two scenariosthan include biodegradation. The streamline solver is faster than the mesh-based solversto simulate advection and biodegradation independent of the grid size and number oftime steps. However, it is much slower than the mesh-based solvers using the coarsegrid and it is slightly slower than the TVD solver using the fine grid when dispersion isincluded.1590 0.1 0.2 0.3 0.400.20.40.60.81Pore VolumeC/C0 P1Coarse TVDFine TVDCoarse MOCFine MOCStreamlines ∆ s = 1.0Streamlines ∆ s = 0.20 0.1 0.2 0.3 0.400.20.40.60.81Pore VolumeC/C0 P2Coarse TVDFine TVDCoarse MOCFine MOCStreamlines ∆ s = 1.0Streamlines ∆ s = 0.2Figure 4.22: Simulated BTEX concentration versus time for Example 4 assumingadvective transport only. Concentrations simulated with the streamline,MOC and TVD solvers in the coarse and fine grids are similar close to thelocation of the BTEX release (P1). However, simulated concentrationsfarther from the source at point P2 are diﬀerent because of the numericaldispersion that aﬀects the solution computed with the TVD and MOCsolvers.1600 0.1 0.2 0.3 0.400.20.40.60.81Pore VolumeC/C0 P1Coarse TVDFine TVDCoarse MOCFine MOCStreamlines0 0.1 0.2 0.3 0.400.20.40.60.81Pore VolumeC/C0 P2Coarse TVDFine TVDCoarse MOCFine MOCStreamlinesFigure 4.23: Simulated BTEX concentration versus time for Example 4 assumingadvective transport and biodegradation. Concentrations simulated withthe streamline, MOC and TVD solvers in the fine grid are similar closeto the location of the BTEX release (P1). However, simulated con-centrations with the TVD and MOC solvers farther from the source,at point P2, are much lower than the source concentration because ofthe combined action of numerical transverse mixing and biodegradation.Simulated concentrations are smaller when computed in the coarse gridbecause of the larger numerical errors.161Method\Run 4A 4BStreamlines 1.0 132.7TVD Coarse 1.4 1.7TVD Fine 90.4 117.1MOC Coarse 5.0 5.7MOC Fine 341.5 385.8Table 4.11: Normalized CPU time required to simulate Example 4 for the two scenar-ios than include biodegradation. The streamline solver is faster than themesh-based solvers to solve advection and biodegradation independent ofthe grid size. However, it is much slower than the mesh-based solversusing the coarse grid and it is slightly slower than the TVD solver usingthe fine grid when dispersion is included.It is somehow surprising that the streamline solver using more than 13,000 nodes and5,000 time steps is faster than the TVD solver using a grid that has only 5,250 cells and3,300 time steps (see Table 4.10) to simulate scenario 4A. This diﬀerence is due to thestreamline-formulation to solve advection that allows the utilization of more eﬃcient andsimple data structures to solve multiple one-dimensional problems versus more complexand costly ones required by multidimensional solvers. That advantage of the streamline-based formulation makes it even more attractive for parallel implementations.The advantages of the streamline-based solver are even more important when comparedto mesh-based solutions of similar level of accuracy, which require using a grid with finediscretization. At the same time that the computational cost and memory requirementsincrease because of the larger number of cells due to the finer discretization, the stabilityrestrictions of the explicit mesh-based solvers require using smaller time steps. The neteﬀect is a rapid increase in the simulation time with each level of grid refinement. Forexample, the explicit multidimensional TVD solver in MT3DMS requires that (Zhengand Wang,1999),∆t ≤1|vx|∆x+|vy|∆y+|vz|∆z(4.25)The restriction on the time step is global and it is computed using the maximum velocityin the grid. The flow velocity distribution in heterogeneous porous media is skewed andmost cells have velocities that are much lower than the maximum value as shown in Figure1624.24. Thus, the use of a single time step imposes an unnecessarily restrictive condition inmost of the domain. However, because of the global coupling of the concentration valuesintroduced by use of a multidimensional solver, it is impossible to use diﬀerent time stepsaccording to the local flow velocities. The distribution of flow velocities is even moreasymmetric for more heterogeneous porous media, thus the global stability restriction iseven costlier in those cases.0 0.010.020.030.040.050.060.070100200300400500600700800900Velocity MagnitudeNumber of Cellsa0 0.010.020.030.040.050.060.07020004000600080001000012000Velocity MagnitudeNumber of CellsbFigure 4.24: Distribution of cells according to the flow velocity magnitude for the(a) coarse grid and (b) fine grid. Explicit mesh-based solvers such asthe multidimensional TVD solver in MT3DMS must satisfy a globalstability restriction, thus concentration values must be updated usinga single time step for all the cells in the grid. The global time step iscomputed based on the maximum velocity at any of the cells in the grid.However, the flow velocity distribution in heterogeneous porous mediais skewed and most cells have velocities that are much lower than themaximum value.The distribution of the flow velocity at the streamline nodes follows a distribution thatis similar to the one of the magnitude of the flow velocity at the grid cells as shown inFigure 4.25. Yet, the explicit solution of advection must satisfy a local stability restrictionthat is given by the maximum velocity along individual streamlines, which has a muchmore uniform distribution. While short time steps must be used to advect concentrationvalues along few fast streamlines, larger time steps can be used along many other slowerstreamlines. The possibility of using variable time steps according to the flow regionssampled by individual streamlines can result in important performance advantages andconstitutes one of the main advantages of streamline-based simulations.1630 0.010.020.030.040.050.060.070.080200400600800100012001400160018002000Velocity MagnitudeNumber of Nodesa0 0.010.020.030.040.050.060.070.080510152025303540Max. Velocity MagnitudeNumber of StreamlinesbFigure 4.25: Distribution of (a) nodes along streamlines according to the flow velocitymagnitude and (b) number of streamlines based on the maximum veloc-ity magnitude along individual streamlines. The distribution of the flowvelocity at the streamline nodes follows a distribution that is similar tothe one of the magnitude of the flow velocity at the grid cells shown inFigure 4.24. However, the explicit solution of advection along individualstreamlines must satisfy a local stability restriction that is given by themaximum velocity along individual streamlines. While short time stepsmust be used to advect concentration values along few fast streamlines,larger time steps can be used along many other slower streamlines.The increase in the time required to complete the streamline-based simulations for sce-nario 4A and 4B is due to the approximation for the dispersion terms. Because of theirregularity of the flow, streamlines and nodes along them are irregularly distributed inspace. Thus, the number of neighbor nodes that contribute to the SPH approximationfor dispersion, if a constant smoothing length is used, varies in diﬀerent regions of thedomain and ranges from 10 in divergent flow regions or close to the boundaries to 138 inconvergent flow regions or near the contaminant source as shown in Figure 4.26. Whilefew nodes have less than 20 neighbors which is considered a reasonable number (Clearyand Monaghan, 1999), most nodes have many more neighbors resulting in a large compu-tational overhead. That computational disadvantage can be overcome by using a variablesmoothing length that is automatically adjusted to get a relatively constant number ofneighboring nodes (Monaghan, 2005). However, the use of an adaptive smoothing lengthwould result in a more complicated implementation.1640 20 40 60 80 100 120 140020040060080010001200Number of NeighborsNumber of NodesMin. 10Max. 138Figure 4.26: Spatial distribution of nodes and number of nodes according to the num-ber of neighboring nodes that contribute to the SPH summation to ap-proximate dispersion. The selection of a constant smoothing length toguarantee a reasonable number of neighboring nodes for any node inthe domain results in an irregular distribution of the number of nodesthat contribute to the summation in (4.17). While few nodes have lessthan 20 neighbors which is considered a reasonable number Cleary andMonaghan (1999), most nodes have many more neighbors resulting in alarge computational overhead.165Although, the SPH-based approximation for dispersion introduces a large computationaloverhead, the time required by the streamline-base solver to compute the solution of theoverall transport equation that includes advection and dispersion is comparable to theone required by the fastest of the two mesh-based solvers (TVD) using a fine grid and itis much lower than the one required by the slowest one (MOC) .4.5 ConclusionsWe derived and implemented a numerical scheme to incorporate dispersion, includingtransverse dispersion, in multidimensional streamline simulations. Because the newscheme does not require the mapping of concentration values onto a grid it does notintroduce additional numerical dispersion.Because the new scheme is able to simulate transverse dispersion and completely avoidnumerical dispersion, it is an attractive alternative to obtain accurate simulations ofproblems that involve chemical reactions that are controlled by transverse mixing suchas bio-remediation or natural attenuation of contaminant plumes.Furthermore, we demonstrated theoretically and by numerical simulations that the pro-posed scheme guarantees solutions that are free of numerical oscillations even when thedispersion tensor is anisotropic. This is an important advantage of the new method overtraditional numerical approximations that suﬀer numerical artifacts that lead to negativeconcentration values if the cross-terms of the dispersion tensor are included. 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Wang, MT3DMS: A Modular Three-Dimensional Multispecies Trans-port Model for Simulation of Advection, Dispersion, and Chemical Reactions of Con-taminants in Groundwater Systems; Documentation and User’s Guide, Contract ReportSERDP-99-1, US Army Engineer Research and Development Center, Vicksburg, MS,1999.171Chapter 5ConclusionsPrevious chapters presented the derivations, implementation and evaluation of two newnumerical methods for the simulation of conservative and reactive solute transport innatural porous media. Both approximations are based on a continuum description ofreactive transport, thus parameters and variables are defined at the REV or local-scale.Chapter 2 presents the derivation and implementation of a meshless approximation forthe advection-dispersion equation derived from smoothed particle hydrodynamics (SPH).Although, SPH formulations had been previously used to simulate solute transport at thepore (Zhu et al.,1999;Zhu and Fox,2001,2002;Tartakovsky et al., 2007) and laboratory(Tartakovsky et al., 2008) scales, Chapter 2 presents one of the first attempts reported inthe literature to use meshless methods to simulate solute transport in porous media atthe field-scale together with the work of Zimmermann et al. (2001), Li et al. (2003) andPraveen Kumar and Dodagoudar (2008).Chapter 3 presents the derivation of a new SPH approximation to simulate anisotropicdispersion, which extends the traditional SPH scheme for diﬀusion and thermal conduc-tion (Cleary and Monaghan,1999;Zhu and Fox,2001;Jubelgas et al., 2004) used inChapter 2. Results of numerical simulations demonstrated that the accuracy of the newapproximation depends upon multiple parameters such as: average particle spacing, ker-nel function, smoothing length, and, most importantly, degree of particle disorder. Thisfeature is common to other kernel interpolation methods such as the particle strengthexchange (PSE) method (Degond and Mas-Gallic,1989;Zimmermann et al., 2001). An-other important conclusion of this chapter is that SPH and PSE methods provide moreaccurate approximations for isotropic dispersion than for anisotropic dispersion, which isalso similar to the behavior of other standard numerical schemes.172Chapter 4 presents a hybrid scheme to simulate solute transport in a flow oriented grid:advection and longitudinal dispersion are solved along streamlines, while transverse dis-persion is handled using the meshless approximation presented in Chapters 2 and 3. Thehybrid scheme overcomes two main limitations of previous streamline-based simulations.First, it allows simulating transverse dispersion without using a background grid andintroducing numerical dispersion. Second, it can be used to simulate three-dimensionalproblems, which was not possible with previous methods that used orthogonal flow-oriented grids and were restricted to two-dimensional problems (e.g. Frind,1982;Cirpkaet al., 1999). Furthermore, the hybrid approach also represents one of the few and, inour opinion the simplest, monotonic approximation for anisotropic dispersion (see Mlac-nik and Durlofsky,2006;Lipnikov et al., 2009, for examples of alternative monotonicschemes). Moreover, benchmarking results demonstrated that the streamline-based for-mulation minimizes numerical dispersion in the longitudinal direction and it completelyavoids it in the transverse direction. Because of these features, the new hybrid schemeis suitable to accurately simulate conservative and reactive solute transport includingreactions controlled by transverse mixing. Furthermore, the new hybrid formulation, asothers streamline-based schemes (Di Donato and Blunt, 2004), also allows the simulationof general heterogeneous chemical reactions that involve species or components in thesolid phase, which constitutes one of the main advantages of this type of method overparticle based methods.5.1 Limitations of Proposed Numerical SchemesAlthough, the two proposed numerical schemes represent attractive alternatives versusmost traditional numerical methods, they also have their own limitations.First, as demonstrated in Chapter 3, the accuracy of meshless approximations for dis-persion, including SPH and PSE, depends upon the spatial distribution of particles ornumerical nodes. Although, the distribution of particles can become very irregular inpresence of non-uniform velocity fields, the numerical simulations presented in Chapter2 demonstrated that it is still possible to obtain reasonable accurate results in simula-tions that consider large Péclet numbers. Under those conditions, even large errors inthe approximation of dispersion are relatively minor in comparison to other numericalerrors, e.g. due to solution of advection. Nevertheless, we believe that the eﬀect of par-ticle disorder should always be considered and, if possible, estimated whenever meshlessmethods like the ones presented in Chapters 2 and 3 are used.173Second, meshless methods require searching for nearby nodes and evaluating pair inter-actions between neighboring nodes, which can result in computational overhead. Thisproblem can be controlled by an adequate choice of the radius of interaction of the nu-merical nodes (e.g. smoothing length in SPH simulations). As demonstrated by thenumerical simulations presented in Chapter 2, which include hundreds of thousands ofnodes, this is an eﬀective alternative for two-dimensional simulations. However, this canbe a much more serious limitation in three-dimensional scenarios as demonstrated bythe simulation of aerobic biodegradation presented in Chapter 4, because the number ofneighboring nodes can increase dramatically going from two- to three-dimensional sce-narios. Moreover, this issue becomes more important when simulations consider veryheterogeneous velocity fields and/or sources and sinks. In those cases, particles concen-trate in slow flow regions, while streamlines converge towards high flow velocity zonesand sources or sinks. An obvious solution to this problem is to adapt the radius of in-fluence according to the local node density such that the number of neighboring nodesstays relatively constant, which is the standard approach in SPH simulations of fluidswith large density variations (Monaghan, 2005). That modification would introduce ad-ditional complexity to the overall algorithm, but we believe that the performance gainswould be enough to justify that cost. Moreover, an implementation of such algorithmwould provide a method with a grid that automatically adapt to the flow field and thatprovides spatially varying resolution according to the flow characteristics.Last, all the simulations presented in the previous chapters considered steady-state flowconditions, however, many real problems involve transient flows. While, the extension ofthe SPH-based method to transient flows is direct and it does not require additional mod-ifications, the streamline-based approach would require additional changes. Streamline-based simulations of transient flows are standard in reservoir simulations (Thiele,2005).Streamlines are updated each time than the flow field changes and concentrations aremapped from the old to the new streamlines. As other schemes that require interpola-tions, the mapping concentrations from the old to the new streamline locations introducesnumerical dispersion. However, changes of the flow field occur over time periods that arerelatively long in comparison to typical transport time steps, thus the mapping of con-centrations is only necessary few times during a simulation and the cumulative eﬀect isrelatively minor.1745.2 General ConclusionsIn addition to the specific conclusions discussed above, the results discussed in the pre-vious chapters also confirmed the following general results that are relevant for reactivetransport modeling:1. The simulation of aerobic biodegradation presented in Chapter 4 is a good exampleof situations were relatively small errors that can be tolerated in conservative so-lute transport simulations, can result in much larger errors if chemical reactions areincluded. Thus, as pointed out by Steefel and MacQuarrie (1996), traditional meth-ods that seem to work well for simulations of conservative transport are unsuitablefor simulating reactive transport.2. Thediﬀerencebetweennumericalsolutionscomputedwithdiﬀerentnumericalmeth-ods can be very large. This is particularly true for realistic problems that involveheterogeneous media and chemical reactions. Since, in general, it is not possible toderive analytical solutions for such problems it is diﬃcult to decide a priori whichnumerical algorithm provides the best solution. This indicates the need for defin-ing a set of benchmark problems that can be used to verify the performance ofnumerical methods and their implementations.3. Current numerical methods are inadequate to simulate reactive transport in highlyheterogeneous porous media under common field conditions. Mesh-based methodssuﬀer excessive numerical dispersion and spurious oscillations. Similarly, hybridsschemes like the method of characteristics (MOC) that use a background grid tocompute concentrations also suﬀer numerical mixing and require using a very largenumber of particles to obtain smooth solutions. Finally, random walk particletracking methods have problems to simulate low concentrations and also suﬀernumerical oscillations (overshooting).4. The main advantages of the proposed Lagrangian schemes are due to the fact thatthey track concentrations defined at the REV or local-scale in contrast to traditionalEulerian or Eulerian-Lagrangian formulations that compute grid-scale concentra-tions. Although, usually overlooked, the equations that describe the evolution ofconcentrations at the grid and local-scale are diﬀerent (Beckie, 1998). Most of thenumerical errors that plague traditional numerical approximations are a manifes-tation of those diﬀerences.1755.3 PerspectivesAlthough, the implementations of both new numerical schemes use an SPH-based ap-proximation for isotropic dispersion, any other meshless approximation for diﬀusion mayalso be used. SPH and PSE methods are only two of multiple meshless methods de-veloped during recent years. For example, Fries and Matthies (2004) and Schaback andWendland (2006) present exhaustive reviews of many others meshless numerical meth-ods. We believe that others meshless methods may have advantages over SPH, e.g. higherconvergence, which could make them better alternatives for new implementations of theproposed Lagrangian numerical schemes.The two proposed numerical methods are also attractive alternatives to study fundamen-tal issues related to mixing and reactions in heterogeneous porous media. For example,Tartakovsky et al. (2008) present a novel multiscale approach to model mixing in hetero-geneous porous media. They use a SPH scheme similar to the one presented in Chapter2, to simulate solute transport. The novel characteristics of their approach is that theevolution of particle positions is given by an stochastic Langevin equation that modelvelocity fluctuations at the local-scale. The resulting model is simple and theoreticallyappealing because it provides a clear separation between the two main components ofmixing in porous media: local-dispersion and spreading due to local-scale velocity vari-ations. On the other hand, the streamline-based method represents a unique tool toevaluate the potential eﬀect of transverse dispersion and flow heterogeneity, e.g. conver-gent flow regions, on the enhancement of mixing and dilution rates in natural aquifers ashypothesized by Werth et al. (2006).The two numerical methods derived in this research allow using diﬀerent spatial reso-lutions to simulate flow and reactive transport. This flexibility can be an importantadvantage in the implementation of multiscale or upscaling methods that require solv-ing transport and flow with diﬀerent levels of spatial resolution as demonstrated by thenested gridding approach of Gautier et al. (1999) or the multiscale method of Tartakovskyet al. (2008). This constitutes a significant advantage over mesh-based methods that, ingeneral, solve flow and transport using the same spatial discretization.1765.4 Final RemarksAlthough, a large portion of this thesis has been focused on discussing specific methodsand implementation details, we think that its most important contribution would beto promote and demonstrate the advantages of Lagrangian approaches – be they SPH,streamline-based simulations or other – to simulate reactive transport in natural porousmedia. We hope that the results of this research would help to highlight those advantagesand contribute to the adoption of Lagrangian schemes in current and future reactivetransport codes.1775.5 ReferencesBeckie, R., Scale dependence and scale invariance in hydrology, chap. Analysis of scaleeﬀects in large-scale solute-transport models, pp. 314–334, Cambridge University Press,1998.Benson, D., and M. Meerschaert, Simulation of chemical reaction via particle tracking:Diﬀusion-limited versus thermodynamic rate-limited regimes, Water Resour. Res., 44,12, 2008.Cirpka, O., E. Frind, and R. Helmig, Numerical methods for reactive transport on rect-angular and streamline-oriented grids., Adv. Water Res., 22,711–728,1999.Cleary, P. W., and J. J. Monaghan, Conduction modelling using smoothed particle hy-drodynamics, J. Comput. Phys., 148,227–264,1999.Cottet, G., and P. Koumoutsakos, Vortex methods: Theory and practice., CambridgeUniversity Press, 2000.Degond, P., and S. Mas-Gallic, The weighted particle method for convection-diﬀusionequations. II: The anisotropic case, Math. Comp, 53, 485,508, 1989.Di Donato, G., and M. Blunt, Streamline-based dual-porosity simulation of reactive trans-port and flow in fractured reservoirs, Water Resour. Res., 40,2004.Fries, T., and H. Matthies, Classification and overview of meshfree methods, Tech. rep.,Institute of Scientific Computing, Technical University Braunschweig, 2004.Frind, E., The principal direction technique: a new approach to groundwater contam-inant transport modeling, in Proceedings, Fourth International Conference on FiniteElements in Water Resources, Tech. Univ. Hannover, Germany. Springer Verlag, NewYork, vol. 13, pp. 25–42, 1982.Frind, E., E. Sudicky, and S. Schellenberg, Micro-scale modelling in the study of plumeevolution in heterogeneous media, Stoch. Hydrol. Hydraul., 1,263–279,1987.Gautier, Y., M. Blunt, and M. Christie, Nested gridding and streamline-based simulationfast reservoir performance predicition, Computat. Geosci., pp. 295–320, 1999.Jubelgas, M., V. Springel, and K. Dolag, Thermal conduction in cosmological SPH sim-ulations, Mon. Not. R. Astron. Soc., 351,423–435,2004.178Li, J., Y. Chen, and D. Pepper, Radial basis function method for 1-d and 2-d groundwatercontaminant transport modeling, Comput. Mech., 32,10–15,2003.Lipnikov, K., D. Svyatskiy, and Y. Vassilevski, Interpolation-free monotone finite volumemethod for diﬀusion equations on polygonal meshes, J. Comput. Phys., 228,703–716,2009.Mlacnik, M., andL.Durlofsky, Unstructuredgridoptimizationforimprovedmonotonicityof discrete solutions of elliptic equations with highly anisotropic coeﬃcients, J. Comput.Phys., 216,337–361,2006.Monaghan, J., Smoothed particle hydrodynamics, Rep. Prog. Phys., 68,1703–1759,2005.Praveen Kumar, R., and G. Dodagoudar, Two-dimensional modelling of contaminanttransport through saturated porous media using the radial point interpolation method(RPIM), Hydrogeol. J., 16,1497–1505,2008.Schaback, R., and H. Wendland, Kernel techniques: From machine learning to meshlessmethods, Acta Numerica, pp. 1–97, 2006.Steefel, C., and K. MacQuarrie, Approaches to modeling of reactive transport in porousmedia, Reviews in Mineralogy and Geochemistry, 34,85–129,1996.Suciu, N., C. Vamos, J. Vanderborght, H. Hardelauf, and H. Vereecken, Numerical in-vestigations on ergodicity of solute transport in heterogeneous aquifers, Water Resour.Res, 42,1–17,2006.Tartakovsky, A., P. Meakin, T. Scheibe, and B. Wood, A smoothed particle hydrody-namics model for reactive transport and mineral precipitation in porous and fracturedporous media, Water Resour. Res., 43, W05,437, 2007.Tartakovsky, A., D. Tartakovsky, and P. Meakin, Stochastic Langevin model for flow andtransport in porous media, Phys. Rev. Lett., 101, 44,502, 2008.Thiele, M., Streamline simulation, in 8th International Forum on Reservoir Simulation,2005.Tompson, A., and D. Dougherty, Particle-grid methods for reacting flows in porous mediawith applications to Fisher’s equation, Appl. Math. Modelling, 16,374–383,1992.Werth, C. J., O. A. Cirpka, and P. Grathwohl, Enhanced mixing and reaction throughflow focusing in heterogeneous porous media, Water Resour. Res., 42, W12,414, 2006.179Zhu, Y., and P. Fox, Smoothed Particle Hydrodynamics Model for Diﬀusion throughPorous Media, Transport Porous Med., 43,441–471,2001.Zhu, Y., and P. Fox, Simulation of pore-scale dispersion in periodic porous media usingsmoothed particle hydrodynamics, J. Comput. Phys., 182,622–645,2002.Zhu, Y., P. Fox, and J. Morris, A pore-scale numerical model for flow through porousmedia, Int. J. Numer. Anal. Meth. Geomech., 23,881–904,1999.Zimmermann, S., P. Koumoutsakos, and W. Kinzelbach, Simulation of pollutant trans-port using a particle method, J. Comput. Phys., 173,322–347,2001.180Appendix ADerivation of SPH Approximationfor Isotropic DispersionTo solve the PDE equation given bydAdt=1α∇·(βΦ∇A) (A.1)where A is an scalar variable, and α, β and Φ are scalar parameters; we can use theidentity1α∇·(βΦ∇A) =12α∇2(βΦA)−A∇2(βΦ) + βΦ∇2A(A.2)Thus, the solution of (A.1) requires only an expression to evaluate the Laplacian. Ap-plying a Taylor series expansion and some algebraic manipulation it is possible to showthat the Laplacian of a scalar field S can be approximated by (Jubelgas et al.,2004)∇2Sr≈−2ˆ[S(r)−S(r)](r−r)|r−r|2·∇W (r−r) dr+ O(h2) (A.3)If this last expression is integrated using a traditional SPH approach, then∇2Sri= −2NpNpj=11pj[S(rj)−S(ri)]F (rj−ri) (A.4)181whereF(rj−ri)=(rj−ri)|rj−ri|2·∇W (rj−ri) (A.5)Finally, substituting (A.4) into (A.2), we getdAdtri=1NpNpj=11αipj(βiφi+ βjφj)(Aj−Ai)F (rj−ri) (A.6)ReferencesJubelgas, M., V. Springel, and K. Dolag, Thermal conduction in cosmological SPH sim-ulations, Mon. Not. R. Astron. Soc., 351,423–435,2004.182Appendix BDerivation of SPH Approximationfor Second Order DerivativesEspañol and Revenga (2003)and Monaghan (2005)provide expressions tocompute secondderivatives using a smoothed particle hydrodynamics (SPH) formulation. Here, we givea detailed derivation of those expressions.We start by recalling some of the properties of SPH kernels. The gradient of a sphericallysymmetric SPH kernel, W(r), can be expressed as ∇W(r)=−rF(r), where F(r) is aspherically symmetric scalar function and r and r are the separation vector and itsmagnitude, respectively. Additionally, the kernel satisfies the normalization condition´W(r)dr =1and has compact support, i.e. W(r)=0 ∀r>he, where heis a finiteconstant.The Taylor series approximation of a function A around x is,A(x)=A(x)+(x−x)T·∇A|x+12(x−x)T·∇∇A|x· (x−x)+O(|x−x|3) (B.1)To simplify notation, we use r = x−x =(r1,r2,r3), such thatA(x)−A(x) ≈iri∂A∂xix+12ijrirj∂2A∂xi∂xjx(B.2)183Now, we multiply by Γαβ(r)=F(r)rαrβ/r2and integrate to obtain,ˆ(A(x)−A(x))Γαβ(r)dx=i∂A∂xiˆF(r)rirαrβr2dx+12ij∂2A∂xi∂xjˆF(r)rirjrαrβr2dx(B.3)We note that dr =dxand use spherical coordinates to evaluate the integrals, to obtain,ˆF(r)r1r1r2r2dr =∞ˆ02πˆ0πˆ0F(r)r3cos2(θ)sin(θ)sin3(φ)dφdθdr =0 (B.4)because´2π0cos2(θ)sin(θ)dθ =0. Similarly, the other terms multiplying first derivativesvanish. Thus,ˆ(A(x)−A(x))Γαβ(r)dx=12αβ∂2A∂xα∂xβΛαβ(B.5)where only coeﬃcients of the form Λαβ=´F(r)rαrαrβrβ/r2dr are not zero. Therefore,2ˆ(A(x)−A(x))Γαα(r)dr =Λαα∂2A∂x2α+β=αΛαβ∂2A∂x2β=(Λαα−Λαβ)∂2A∂x2α+Λαβ∇2A(B.6)and2ˆ(A(x)−A(x))Γαβ(r)dr =Λαβ∂2A∂xα∂xβ+Λβα∂2A∂xβ∂xα=2Λαβ∂2A∂xα∂xβ(B.7)The coeﬃcients Λαβcan be easily evaluated, for example,184Λxx=∞ˆ02πˆ0πˆ0F(r)r4cos4(θ)sin5(φ)dφdθdr=45π∞ˆ0F(r)r4dr(B.8)andΛxy=∞ˆ02πˆ0πˆ0F(r)r4cos2(θ)sin2(θ)sin5(φ)dφdθdr=415π∞ˆ0F(r)r4dr(B.9)The coeﬃcients related to the other directions have one of these two forms. To computethe integral in r, we recall that rF(r)=−∂W(r)/∂r, thus∞ˆ0F(r)r4dr = −∞ˆ0r3∂W(r)∂rdr =3∞ˆ0r2W(r)dr (B.10)after integration by parts. Finally, we use the normalization condition of the kernelˆW(r)dr =πˆ02πˆ0∞ˆ0r2W(r)drdθdφ =4π∞ˆ0r2W(r)dr =1 (B.11)to obtain∞ˆ0F(r)r4dr =34π (B.12)Then,Λαα=35Λαβ=15(B.13)185We observe that Λαα−Λαβ=2Λαβ, thus equations B.6 and B.7 can be combined toobtain,2ˆ(A(x)−A(x))Γαβ(x−x)dx=Λαβ∇2Aδαβ+2Λαβ∂2A∂xα∂xβ(B.14)where δαβis Kronecker delta. This last expression gives the following approximation forthe Laplacian,∇2A =2ˆ(A(x)−A(x))F(x−x)dx(B.15)which is identical to an expression previously derived to simulate thermal conduction(Cleary and Monaghan,1999;Jubelgas et al.,2004).Finally, substituting B.15 into B.14 and rearranging terms, we obtain∂2A∂xα∂xβx=ˆ(A(x)−A(x))F(x−x)1Λαβrαrβr2−δαβdx(B.16)Using an SPH approximation to evaluate the integral,∂2A∂xα∂xβxa=b1pb(A(xb)−A(xa))F(|xb−xa|)Θαβ(xb−xa) (B.17)where Θαβ(x−x)=1Λαβ(x−x)α(x−x)β|x−x|2−δαβand pa=bW(xb−xa). Substituting1/Λαβ=5in three dimensions and 1/Λαβ=4in two dimensions, we obtain the expres-sions given by Español and Revenga (2003) and Monaghan (2005), respectively.ReferencesCleary, P. W., and J. J. Monaghan, Conduction modelling using smoothed particle hy-drodynamics, J. Comput. Phys., 148,227–264,1999.Español, P., and M. Revenga, Smoothed dissipative particle dynamics, Phys. Rev. E, 67,026,705–12, 2003.186Jubelgas, M., V. Springel, and K. Dolag, Thermal conduction in cosmological SPH sim-ulations, Mon. Not. R. Astron. Soc., 351,423–435,2004.Monaghan, J., Smoothed particle hydrodynamics, Rep. Prog. Phys., 68,1703–1759,2005.187Appendix CRandom Walk Particle MethodRandom-walk particle-tracking (RWPT) methods are based on the equivalence in thelimit of a large number of particles between the ADE which describes continuum-scalemass conservation and the Fokker–Planck equation which describes the time evolutionof the probability density function of the position of a solute particle. Most numericalimplementations of the RWPT are based on the Îto integration of the equivalent Langevinequation (Gardiner, 1990) that written in matrix form corresponds toX(t +∆t)=X(t)+∆tA(t)+B(t)Z√∆t (C.1)where X(t) corresponds to the vector of particle position at time t and Z is a normallydistributed random vector with zero mean and unit variance that represents the Brownianmotion of the particles due to dispersion. The other two terms in (C.1) corresponds to adrift term given byA =v+∇·D (C.2)assuming constant porosity; and a displacement matrix, B, that depends on the disper-sion tensor. The relation of these terms with the ADE are discussed in detail in severalreferences (e.g. Lichtner et al.,2002;Delay et al.,2005;Salamon et al.,2006).Several approaches have been proposed to interpolate the flow velocity required to eval-uate the coeﬃcients appearing in (C.1) and (C.2). Particular attention has been paid to188methods that avoid local mass conservation problems due to discontinuities of the dis-persion tensor across sharp interfaces (LaBolle et al.,1996;Labolle et al.,2000;Salamonet al., 2006). In our implementation of the RWPT we use a hybrid approach to interpo-late velocity: a linear interpolation is used to evaluate the velocity vector that appearsin the drift term, and a trilinear interpolation is used to interpolate the velocity vectorused to evaluate the dispersion tensor. This interpolation scheme guarantees the spatialcontinuity of the advective velocity and the dispersion tensor (Salamon et al.,2006).ToapplytheRWPTmethodtopracticalsituations itisnecessarytomapsolute mass fromand to concentration values because initial and boundary conditions and geochemicalcomputations are usually expressed in concentration units. Given the total solute massin the system M, the mass of each particle is computed as mp= M/Np. In general, thetotal number of particles is chosen to satisfy some numerical resolution, for example suchthat the mass of a given number of particles Nris equal to some unit mass value M0, i.e.mpNr= M0. Given a set of particles, a continuum spatial concentration distribution canbe approximated as (Bagtzoglou et al.,1992;Tompson,1993)C(x)=ˆΩmpξ(x−x)dx≈Npj=1mpξ(x−xj) (C.3)where ξ(x) is a projection function. In theory, this expression allows the computationof the concentration values at any location. In practice, most implementations use abox function with value 1/Vcfor points within a cube of volume Vcaround x and 0otherwise(Tompson, 1993). In simple terms, the domain Ω is divided in a set of cubiccells with volume Vc, then concentration values are assigned to cell j by counting thenumber of particles within it, nj, so that the concentration value of the cell is computedas Cj= njmp/Vc.The value of Npaﬀects the accuracy of the method in two ways. First, the equivalencebetween the Langevin equation and the continuum ADE is valid for Np→∞. Second,concentration values can only be represented as an integer multiple of mp/Vc.Errorscanbe particularly important near the plume edges where the drop in the number of particlesproduces unphysical numerical oscillations in the computed concentrations that can beamplified in presence of non-linear chemical reactions if a splitting approach is used tosolve reactive transport (Tompson and Dougherty,1992;Tompson,1993).189ReferencesBagtzoglou, A., A. Tompson, and D. Dougherty, Projection functions for particle-gridmethods, Num. Meth. Part. Diﬀ. Eq., 8,325–340,1992.Delay, F., P. Ackerer, and C. Danquigny, Simulating Solute Transport in Porous orFractured Formations Using Random Walk Particle Tracking: A Review, Vadose ZoneJ., 4,360–379,2005.Gardiner, C., Handbook of Stochastic Methods for Physics, Chemistry and the NaturalSciences,1990.LaBolle, E. M., G. E. Fogg, and A. F. B. Tompson, Random-walk simulation of transportin heterogeneous porous media: Local mass-conservation problem and implementationmethods, Water Resour. Res., 32,583–594,1996.Labolle, E. M., J. Quastel, G. E. Fogg, and J. Gravner, Diﬀusion processes in compositeporous media and their numerical integration by random walks: Generalized stochasticdiﬀerential equations with discontinuous coeﬃcients, Water Resour. Res., 36,651,2000.Lichtner, P., S. Kelkar, and B. Robinson, New form of dispersion tensor for axisymmetricporous media with implementation in particle tracking., Water Resour. Res., 38,1146,2002.Salamon, P., D. Fernàndez-Garcia, and J. Gómez-Hernández, A review and numericalassessment of the random walk particle tracking method., J. Contam. Hydrol., 87,277–305, 2006.Tompson, A., Numerical simulation of chemical migration in physically and chemicallyheterogeneous porous media, Water Resour. Res., 29,3709–3726,1993.Tompson, A., and D. Dougherty, Particle-grid methods for reacting flows in porous mediawith applications to Fisher’s equation, Appl. Math. Modelling, 16,374–383,1992.190Appendix DStreamline TracingIn streamline simulations the value of variables such as time of flight and arc lengthtogether with the node positions along streamlines define the computational grid. There-fore, the accuracy of the results depends upon the accuracy of the computed streamlinestrajectory, arc length and time of flight. On the other hand, it may be necessary to tracea large number of streamlines to provide an adequate coverage of the domain in com-plex three-dimensional problems. Additionally, streamlines must be periodically updatedwhenever temporal changes of the flow field occur. Therefore, it is important to selecttracing algorithms that allow an accurate and eﬃcient computation of the streamlines.When tracing the streamlines used in a simulation it is sometimes useful to consider someof the characteristics of the flow field in order to optimize the location and minimize thenumber of streamlines necessary to obtain a given spatial resolution and accuracy.We discuss some of the issues associated with tracing and spatial distribution of stream-lines in the next sections.Streamline Tracing AlgorithmsMost current streamline simulators use a semi-analytical method to trace streamlinesthat was first introduced by Pollock (1988) to track fluid particles in groundwater simu-lations. This method is attractive because given a velocity field in a staggered Cartesiangrid, it provides analytical expressions to compute particle trajectories without introduc-ing additional numerical errors. Moreover, the method is simple to understand and, in191theory, easy to implement. In practice, implementations of the algorithm are complexand sensitive to errors due to floating point arithmetic. Additionally, the extension of themethod to unstructured grids, although possible using isoparametric coordinate trans-formations (Cordes and Kinzelbach,1992;Prevost et al., 2002), is much more involvedand diﬃcult to implement than the original method.Fluid particle trajectories that define streamlines can also be integrated using explicittime integration schemes, e.g. Runge-Kutta. This type of algorithm is straightforward,very easy to implement, numerically robust, and can be used with velocity fields givenin structured or unstructured grids without modifications. However, explicit integra-tion schemes introduce additional numerical errors that can be diﬃcult to quantify andcontrol.In this section, we compare Pollock’s method with an explicit adaptive first-order timeintegration scheme. We give a brief description of both methods and compare them ina set of benchmark problems. Finally, we comment on their relative advantages anddisadvantages.Pollock’s MethodGiven the components of the flow velocity in a three-dimensional staggered Cartesian grid,Pollock’s algorithm assumes that the components of the velocity vector v =(u,v,w),canbe approximated as a linear function of the velocity components at the cell faces, i.e.u(x,y,z)=Ax(x−x1) + u1v(x,y,z)=Ay(y−y1)+v1w(x,y,z)=Az(z−z1)+w1(D.1)where u1is the velocity component in the x direction at face 1 located at x1as shownin Figure D.1, and the slope Aiis computed as ratio of the diﬀerence of the cell facevelocities over the grid spacing, e.g. Ax= (u2−u1)/∆x. Similar definitions apply forthe other two directions.192B A C D !"#"u1u2vv1(x0,yt)1,1x2yyFigure D.1: Pollock’s method assumes that the components of the velocity within acell can be approximated as a linear function of the velocity componentsat the cell faces. The method provides analytical expressions to computethe new position D after an interval of time ∆t = t1−t0of a particleinitially at point C. However, the method is usually used to compute theexit point B from a cell given an entry point A.The fluid velocity and particle position in each direction are related by,dudt=dudxdxdt= Axu (D.2)Integrating this last expression one obtains,∆t =1Axlnu(x0+∆x,t0+∆t)u(x0,t0)(D.3)Taking the exponential of each side and substituting u(x0+∆x,t0+∆t) from (D.1),∆x =1Axu(x0)eAx∆t−u1(D.4)193Similar expressions are valid in the y and z directions.While equation (D.3) provides an expression to compute the travel time required by afluid particle at position x0to move a distance ∆x, (D.4) can be used to compute thenew position after an interval of time ∆t.A common application of Pollock’s method is to compute the time a fluid particle atposition xpwill need to exit from the current cell. Assuming all velocities are positive,∆tx=1Axlnu2u(xp)∆ty=1Aylnv2v(yp)∆tz=1Azlnw2w(zp)(D.5)Then, the particle will exit after∆texit= min(∆tx,∆ty,∆tz) (D.6)Thus, given an entry point to a cell it is possible to calculate the exact exit position insingle step and without introducing additional numerical errors.A simple inspection of (D.4) and (D.3) reveals that implementations of the algorithmmust take into account several possible issues:1. If the flow velocity is constant within a cell, then A =0, and expressions (D.3)and (D.4) become undefined. The situation is more complicated if floating pointarithmetic is used because the ratio 1/A can also overflow for small diﬀerences ofvelocity.2. The analytical expressions are not valid in cells with sources or sinks where theslope of the velocity changes sign within a single cell.3. Equations (D.3) and (D.4) are valid within a single cell. Thus, to compute the newposition of a particle after a time step ∆t, one must first check if the particle wouldexit the current cell before that time. If it exits, then the time step must be dividedin smaller sub steps.1944. Additionally, the algorithm assumes that it is always possible to exactly determinethe cell where an individual particle is located. Thus, particles located over or closeto one of the cell edges or corners can be problematic.Algorithm D.1 summarizes the steps necessary to find the position of a fluid particleinitially at position xpafter an interval of time ∆t. Algorithm D.2 presents the stepsused to obtain the remaining time to exit from the current cell. Finally, Algorithm D.3presents the steps to update the position of a fluid particle.Algorithm D.1 Pollock’s particle tracking algorithm.1: xp← initial position2: time ← 03: while time < ∆t and xpinside domain do4: cell ← get cell that contains xp5: if cell = sink or cell = source then6: break7: else8: vp← get velocity at xp9: // for each direction10: fi← get exit face in direction i11: ∆ti← get time to exit through face fi12:13: ∆texit← min(∆ti)14: if ∆texit> ∆t then15: xp← update position using ∆t16: break17: else18: xp← update position using texit19: // now xpis over one of the cell edges20: move xpto next cell21: time ← time + texit22: end if23: end if24: end while25: return xpExplicit Time IntegrationThe temporal evolution of the trajectory of a fluid particle initially at x0at time t0isgiven by the solution of the following diﬀerential equation,195Algorithm D.2 Function to compute exit time from current cell in Pollock’s method.1: x1,x2,xp← coordinates of face 1, 2 and fluid particle2: u1,u2,up← velocities at face 1, 2 and fluid particle position3:4: A ← (x2−x1)/(u2−u1)5: if exit through face 1 then6: if constant velocity then7: time ← (x1−xp)/up8: else9: time ← A· ln(u1/up)10: end if11: else if exit through face 2 then12: if constant velocity then13: time ← (x2−xp)/up14: else15: time ← A· ln(u2/up)16: end if17: end if18: return timeAlgorithm D.3 Function to update position in Pollock’s method.1: x1,x2,xp← coordinates of face 1, 2 and fluid particle2: u1,u2,up← velocities at face 1, 2 and fluid particle position3:4: if exit throug face 1 or 2 then5: // we have to move xpto new cell6: // thus we add to or substract from x1 or x2 a small number7: if exit through face 1 then8: xp← x1−9: else if exit through face 2 then10: xp← x2+ 11: end if12: else13: if constant velocity then14: xp← xp+ up∆t;15: else16: A =(x2−x1)/(u2−u1)17: xp← x1+ A[up· exp(∆t/A)−u1]18: end if19: end if20: return xp196dxdt= v(x,t) (D.7)with,x(t0)=x0(D.8)The numerical solution of (D.7) can be easily computed using an explicit time integrationscheme. For example, a first-order explicit approximation reads,x(t +∆t)=x0+v(x0,t0)∆t (D.9)Although simple to implement, (D.9) is seldom used in practice because of its relativelylarge error, which is proportional to the time step. In practice, (D.7) is solved usinga higher-order integration scheme, e.g. explicit second- and fourth-order Runge-Kutta,that have smaller errors for a given ∆t. Such schemes apply expressions similar to (D.9)to compute the particle position at intermediate steps. Then, the final position after ∆tis computed as a weighted combination of the the intermediate locations.The main drawback of explicit integrators is that it is diﬃcult to determine a priori avalue∆t such that the error remains below a given threshold. This problem is particularlyimportant in velocity fields that exhibit large diﬀerences in velocity magnitude and/ordirection within short distances as shown in Figure (D.2).197!"#"$"%"&"Figure D.2: Example of explicit time integration. A fluid particle starting at point Atravels along the instantaneous streamline (solid line). After a time step∆t, it is located at point B. If an explicit integration scheme is used tointegrate the particle trajectory numerical errors can become important.For example, if a large time step ∆t is used, the particle moves in thedirection of the instantaneous velocity at position A ending at point D.However, the error can be made smaller if two steps are used for theintegration. First, the particle moves with the initial velocity to E, andthen it moves to C along the direction of the velocity at point E.There are multiple possible solutions to control the error of explicit integration schemes.The first one is to use a very small constant time step such that the error in the worstcase stays below a specific threshold. In general, this solution is not acceptable becauseit introduces unnecessary computational overhead in slow regions.A second alternative consists in using an adaptive step size integration algorithm, e.g.Dormand-Prince (Dormand and Prince, 1980); which uses two methods with diﬀerentorder at each time step. The diﬀerence between both solutions is used to estimate thenumerical error and, if necessary, to adapt the size of the time step.A third alternative consists in recognizing that the accuracy of the explicit integrationalso depends on the spatial resolution of the reconstruction of the velocity field. Theerror of the velocity field reconstruction is related to the size of the numerical grid usedto compute the velocity components. Thus, an eﬃcient algorithm can limit the error ofthe integration by controlling how far a fluid particle can move in one time step relativeto the grid size. Therefore, many sub-steps may be necessary in areas of high velocity,198while one or few sub-steps may be enough in slow zones. Therefore, it is possible tosatisfy restrictions on accuracy and performance. We refer to this scheme as the first-order explicit adaptive particle tracking (FEAPT) method. Algorithm D.4 summarizesthe steps of the explicit method.Algorithm D.4 First-order explicit adaptive particle tracking.1: xp← initial position2: ∆ ← maximum distance a particle is allowed to move during one time step3: time ← 04: while time < ∆t and xpinside domain do5: cell ← get cell that contains xp6: if cell = sink or cell = source then7: break8: else9: vp← get velocity at xp10: t ← ∆/|vp|11: xp← xp+vp·t12: time ← time + t13: end if14: end while15: return xpNumerical ExamplesIn this section, we compare Pollock’s and the FEAPT methods in a set of benchmarkproblems.Homogeneous quarter five-spotThe first test problem corresponds to the well known quarter five-spot configurationin a homogeneous medium. An injection and extraction well are located in the lowerleft corner and upper right corner, respectively. The resulting streamline pattern iswell known and it has become a common test problem for streamline simulations (e.g.Matringe,2004).We consider a square domain of 25 m side and we use a 100 x 100 regular Cartesian gridto solve the flow problem. We trace streamlines from ten seeds placed along a diagonalstraight line that passes through the center of the domain and that connects the upperleft and lower right corners.199Figure D.3 shows streamlines traced with Pollock’s and the explicit adaptive scheme withtolerance equal to 5% and 50% of the cell size of the velocity field. The explicit schemeperforms similarly that Pollock’s method to trace the interior streamlines. However, theFEAPT solution with larger tolerance has problems tracing the exterior streamlines atthe lower right and upper left corners where the curvature of the trajectory is maximum.When using the larger tolerance particles travel too far and exit the domain. While theexplicit solution with 50% tolerance required the same time that Pollock’s method, theFEAPT solution with 5% tolerance needed three times longer.Figure D.3: Streamlines traced with Pollock’s method (solid lines) and with explicitadaptive scheme (crosses). Spacing between crosses is equal to four timesthe cell size of velocity field. Numbers indicate streamline labels usedin the text. Solutions for explicit time integration correspond to 5%(left) and 50% (right) tolerance. In both cases the position of the inte-rior streamlines is almost identical. However, the FEAPT solution withlarger tolerance has problems tracing the exterior streamlines at cornerswhere the streamline curvature is maximum.Heterogeneous quarter five-spotAs a second test problem, we consider the same well configuration but a heterogeneoushydraulic conductivity field. We generate two hydraulic conductivity fields assuminga spatial distribution of Y = ln(K) given by an exponential covariance with varianceσY=5and correlation length equal to four times the grid spacing (Figure D.5).2000 0.2 0.4 0.6 0.8 100.20.40.60.81Normalized Arc LengthNormalized Time of Flight streamline #2PollockExplicit 5%Explicit 50%0 0.2 0.4 0.6 0.8 100.20.40.60.81Normalized Arc LengthNormalized Time of Flight streamline #5PollockExplicit 5%Explicit 50%Figure D.4: Normalized time of flight versus normalized arc length computed withPollock’s and adaptive explicit time integration with tolerances equal to5% and 50% of grid spacing. Streamlines traced with any of the threemethods are similar.Figure D.5: Spatial distribution of hydraulic conductivity in fields K1 (left) and K2(right). Note that hydraulic conductivity values vary by more than 14and 18 orders of magnitude in field K1 and K2, respectively. In bothcases, the magnitude of the resulting flow velocity varies up to five ordersof magnitude.201Figure D.6: Streamlines traced with Pollock’s method (solid lines) and with explicitadaptive scheme (crosses) for fields K1 (left) and K2 (right). Spacingbetween crosses is equal to four times the cell size of the grid used tocompute the velocity field. Streamlines were traced with explicit timeintegration scheme using a tolerance equal to 5% of the grid spacing.As in the homogeneous case, we trace streamlines from ten seeds placed along a diagonalstraight line that passes through the center of the domain and that connects the upperleft and lower right corners. The resulting streamlines are shown in Figure D.6. With theexception of few location along streamline 8 in field K1, streamlines traced with Pollock’sand FEAPT method are very similar. As shown in Figure D.7, the small diﬀerences inthe streamline locations are also observed in curves that relate the time of flight and arclength along individual streamlines. For the strongly heterogeneous fields used in thisexample, it is necessary to use a very small tolerance (< 1% of grid spacing) in the explicitintegration scheme to obtain a perfect match between the FEAPT and Pollock’s methods.However, the FEAPT solution with tolerance equal to 1% of the grid produces resultsthat are also very close to the curves generated with Pollock’s method. Moreover, it is isclear that the FEAPT and Pollock’s solutions converge to the same curve as we decreasethe maximum distance than a fluid particle is allowed to advance in an individual timestep.2020 0.2 0.4 0.6 0.8 100.20.40.60.81Normalized Arc LengthNormalized Time of Flight streamline #6PollockExplicit 0.1%Explicit 1%Explicit 5%0 0.2 0.4 0.6 0.8 100.20.40.60.81Normalized Arc LengthNormalized Time of Flight streamline #8PollockExplicit 0.1%Explicit 1%Explicit 5%0 0.2 0.4 0.6 0.8 100.20.40.60.81Normalized Arc LengthNormalized Time of Flight streamline #4PollockExplicit 0.1%Explicit 1%Explicit 5%0 0.2 0.4 0.6 0.8 100.20.40.60.81Normalized Arc LengthNormalized Time of Flight streamline #7PollockExplicit 0.1%Explicit 1%Explicit 5%Figure D.7: Normalized time of flight versus normalized arc length for streamlinesin heterogeneous K1 (top) and K2 (down) hydraulic conductivity fields.The diﬀerence between the results of the adaptive explicit integrationscheme and Pollock’s method become negligible as the tolerance for theexplicit integration decreases to around 1% of the grid spacing.203PerformanceThe main advantage of Pollock’s method consists in its ability to provide accurate resultswith minimum computational overhead. To assess the diﬀerence between the performanceof the explicit integration scheme versus Pollock’s algorithm, we traced 600 streamlinesin the heterogeneous quarter five-spot configuration discussed in the previous section.The resulting streamlines are shown in Figure D.8. When using the explicit integrationalgorithm, nodes are recorded only if the separation distance between the new and thepreviously recorded node is equal to or larger than an specified arc length spacing ∆s.In Pollock’s method only entry and exit points to/from cells are recorded. All the testswere run in laptop computer with an Intel Core 2 Duo 2GHz processor with 3MB of L2cache and 2GB of RAM memory. The results of the comparison are summarized in TableD.1.The explicit streamline integration with tolerance equal to 1% of the grid spacing takesone order of magnitude longer than Pollock’s method, while reducing the tolerance to0.1% resulted in an extra order of magnitude increase in the computational time. Al-though the performance diﬀerence between both methods is important, it is very likelythat it would not be relevant in practical streamline simulations. In real simulations, thetime required to trace streamlines would be much smaller than the time required to solvethe flow and transport problems. In that case, even the slowest of the three methods,explicit integration with 0.1% tolerance, would require only a small fraction of the totalsimulation time.Method Time (sec) # Nodes ∆sPollock 0.3 99,361 −Explicit 1% 5.9 221,437 ∆x/2Explicit 1% 5.8 111,739 ∆xExplicit 0.1% 56.6 223,140 ∆x/2Explicit 0.1% 57.9 112,206 ∆xTable D.1: Comparison of the performance of Pollock’s and explicit adaptive algo-rithm to trace streamlines shown in Figure D.8. When using the explicitintegration algorithm, nodes are recorded only if the separation distancebetween the new and the previously recorded node is equal to or largerthan the arc length spacing ∆s. In Pollock’s method only entry to andexit points from cells are recorded.204Comments on the Selection of a Streamline Tracing AlgorithmBased on the examples presented above and our accumulated experience using Pollock’sand the explicit adaptive integration schemes, we have the following comments:1. Both methods are able to provide the same level of accuracy if a small tolerance isused for FEAPT method.2. However, as demonstrated with the heterogeneous quarter five-spot problem, Pol-lock’s method can be orders of magnitude faster than the explicit integration algo-rithm for the same level of accuracy.3. On the other hand, tracing streamlines is a relatively inexpensive part of a simula-tion when compared to the solution of the flow and transport problems. Moreover,the cost of the tracing step can be easily reduced by parallelizing both algorithms,making the relative cost even smaller. Therefore, performance should not be con-sidered as the sole factor in choosing a streamline tracing algorithm.4. Because of its semi-analytical formulation, the original Pollock’s method is re-stricted to flow fields computed in regular cell-centered Cartesian grids. Its ex-tension to unstructured grids, although possible, it involves much more complexexpressions and additional steps. Additionally, the method assumes a linear re-construction of the velocity components, which is a low order approximation. Onthe other hand, the FEAPT method only requires a routine to evaluate the flowvelocity at each location, thus it can be applied to any type of grid (structured orunstructured) and to any velocity approximation without modifications.In summary, Pollock’s method continue to be an attractive option because of its inherentaccuracy and relative low computational overhead. However, the explicit integrationschemes should be considered as a serious alternative to replace it in new streamlinesimulation packages because of its simplicity, flexibility, and numerical robustness.Streamline DistributionIn streamline simulations the resolution of the numerical grid is given by the distributionand number of streamlines. Good spatial resolution requires that streamlines cover all205the domain with some minimum density of lines crossing every region of the domain. Ingeneral, streamlines are traced starting from an initial location (seed) and then trackingbackward and forward until exiting the domain or hitting a sink or source cell. There-fore, it is possible, for simple flow fields, to optimize the streamline distribution with anadequate choice of the initial seeds. For example, Figure D.9 shows two possible dis-tributions of ten streamlines in the homogeneous quarter five-spot problem. It is clearthat the streamlines traced from equispaced seeds over the diagonal line provide a moreregular coverage of the domain than the ones traced from points located over a verticalline.The optimal selection of initial seeds becomes much more complicated in the case ofheterogeneous velocity fields as shown in Figure D.10. The figure shows four possibledistributions for hundred streamlines in the heterogeneous quarter five-spot problemconsidering field K1. We observe that independently of the distribution of the initialseeds, the distribution of streamlines is very irregular. Large areas of the domain containfew streamlines, while few small regions concentrate many of them. Streamlines occur lessoften in slow flow areas, while they concentrate in fast regions. Therefore, an attractivealternative to obtain a more uniform coverage is to launch some streamlines from theslowest cells in the domain. However, many of the slowest cells are located nearby, andseeding streamlines using such strategy produces a more irregular streamline distributionthan the one obtained using randomly distributed seeds.In many streamline simulations, it is impossible to tolerate regions of the domain with-out a crossing streamline. For example, in solute transport simulations that use anoperator-splitting approach and solve advection along streamlines and dispersion using abackground grid (Crane and Blunt,1999;Obi and Blunt, 2004), at least one streamlinemust cross each cell of the background grid in order to minimize interpolation errors.A common approach in those cases consists in using a background grid to control thestreamline distribution. First, streamlines are traced from specified seeds marking cellsof the grid that are crossed by at least one streamline. Second, new streamlines aretraced from the rest of cells. In general, the number of streamlines is dependant on theinitial seeds configuration and the order used to identify empty cells in the grid. As illus-tration, we use the quarter five-spot problem considering the field K1 and trace enoughstreamlines such as at least one streamline crosses every cell a 100 x 100 square grid. Weuse Pollock’s method and only record the entry and exit points in each cell. Table D.2summarizes the total number of streamlines and nodes required to cover the grid usingfive diﬀerent strategies for the distribution of initial seeds. We observe that independent206Figure D.8: Distribution of 600 streamlines traced to compare the performance ofPollock’s and the explicit adaptive integration scheme. Streamlines forfields K1 (left) and K2 (right).Figure D.9: Two possible distributions of ten streamlines in the homogeneous quarterfive-spot problem. Red squares indicate the position of initial seeds.Using a sensible choice of the initial seeds, it is possible to obtain a moreuniform coverage of the domain with the same number of streamlines.207Figure D.10: Four possible distributions for hundred streamlines in the heteroge-neous quarter five-spot problem considering field K1. Red squares indi-cate the position of initial seeds. Seed distributions correspond to: (i)equispaced points along diagonal line (top left), (ii) quasi-randomly dis-tributed points (top right), (iii) fifty equispaced points along diagonaland fifty quasi-randomly distributed points (down left), and (iv) fiftyequispaced points along diagonal and the location of the fifty (0.5%) ofthe slowest cells. Independent of the distribution of the initial seeds,the distribution of streamlines is very irregular in highly heterogeneousvelocity fields.208Initial seeds # Streamlines # Nodes100 points over diagonal line 1036 169,831100 quasi-random points 1034 169,25350 points over diagonal and50 quasi-random points 1016 166,274400 quasi-random points 1064 174,313without initial seeds(trace streamlines from each empty cell) 1035 169,128Table D.2: Comparison of diﬀerent strategies to trace streamlines such that at leastone streamline crosses each cell of a 100 x 100 Cartesian grid. The totalnumber of streamlines and nodes are very similar independently of thediﬀerent seed locations.of the seed distribution the required number of streamlines and nodes are similar withsmall variations around average values of 1035 streamlines and 169,000 nodes. It is im-portant to highlight that the average number of nodes is much larger than the numberof cells in the background grid (10,000), which is, in our experience, a common situationin streamline simulations. The large number of nodes required to provide an adequatecoverage of the domain is the price one must pay to obtain good spatial resolution andbetter accuracy. On the other hand, that cost is smaller in simulations that consider lessheterogeneous media.Streamline DiscretizationThere are three possible choices to distribute nodes along streamlines: (i) uniform spacingin the time of flight coordinate, (ii) uniform spacing in the arc length coordinate, and(iii) only record positions of entry and exit points in each cell. The selection of thestreamline discretization has important practical implications for the performance of thesimulation, selection of numerical solvers for the transport step, and stability restrictionsof the overall numerical method. For example, it is much easier to solve the advectionstep formulated in terms of the time of flight coordinate τ, if nodes along streamlines areseparated by a uniform step ∆τ(Crane and Blunt, 1999). However, there are also othertrade-oﬀs that must be considered such as limitations in the total number of nodes andspatial node distribution.209We use a simple flow scenario to motivate our discussion about the merits of the diﬀer-ent streamline discretization schemes. We consider flow in a homogeneous medium withhydraulic conductivity K0that contains a high permeability inclusion with K1=10K0.We set boundary conditions such that the mean flow goes from left to right. We solvethe flow problem using a regular Cartesian grid with 10 x 10 cells. Figure D.11 shows anschematic of the flow problem. We use the resulting velocity field to trace ten streamlinesusing the three diﬀerent discretization approaches. First, we trace streamlines recordingonly the entry and exit points to/from individual cells. This is the most common dis-cretization used in streamlines simulation because, as explained above, it arises naturallyfrom the use of Pollock’s method as the tracing algorithm. Second, we trace streamlinesrecording node positions that are equispaced in the time of flight coordinate with step∆τ, i.e. a fluid particle at node i needs ∆τ time to move to the next node i +1. Third,we trace streamlines recording only the entry and exit points to/from individual cellsas in the first discretization method, but we additionally apply a post-processing stepto uniformly distribute nodes in the arc length coordinate such as neighbor nodes areseparated by an arc length spacing ∆s.K010H1FLOW!Figure D.11: Schematic of flow problem used to discuss alternatives streamline dis-cretization. We consider a homogeneous medium with hydraulic con-ductivity K0that contains a high permeability inclusion with K1=10K0. Boundary conditions are set such that the mean flow goes fromleft to right.Figure D.12 shows the locations of nodes along streamlines traced using the first twodiscretization approaches. If only entry and exit points are recorded, then some of the210nodes located near the high permeability inclusion are separated by a very short distancebecause they enter and exit near one of the cell corners. Those small separations canintroduce unacceptable stability constrains for explicit solvers used to simulate advection(Thiele, 2003). Therefore, some post-processing step must be applied to remove nodeswith short separation (Crane and Blunt,1999;Thiele, 2003). On the other hand, adiscretization based on the time of flight with spacing ∆τ =∆x/U, where ∆x is thegrid spacing and U is the mean velocity; can result in a very irregular spatial nodedistribution. Nodes along streamlines cluster in slow flow zones, while they are separatedby long distances in regions with fast flow velocity. Therefore, a prohibitively largenumber of nodes may be required in order to keep a minimum spatial node density in allthe domain.Figure D.12: Comparison of streamline discretization approaches. Red dots indi-cate the position of nodes along streamlines. In traditional streamlinemethods, Pollock’s semi-analytical method is used to find nodes wherestreamlines enter or exit to/from individual cells (left). Alternatively,nodes can be distributed using a constant time of flight spacing (right).Figure D.13 presents a comparison of node locations if only entry and exit points arerecorded and for a uniform arc length discretization with spacing ∆s =∆x/2. The arclength based discretization provides a uniform coverage of the domain with a relativelysmall number of nodes in comparison with the time of flight based discretization. Ad-ditionally, the constant node spacing along individual streamlines relaxes the stabilityrestrictions of explicit solvers in comparison to the situation where entry and exit pointsare recorded. Because of its advantages, we selected the arc length based discretization211as the default option in our streamline simulator.Figure D.13: In traditional streamline methods, Pollock’s semi-analytical method isused to find nodes where streamlines enter or exit to/from individualcells (left). That information can be used to distribute nodes using aconstant arc length spacing along streamlines (right). Red dots indi-cate the position of nodes along streamlines. Note the short separationbetween some nodes in the left figure and the regular spatial node dis-tribution in the right one.ReferencesAris, R., Vectors, Tensors and the Basic Equations of Fluid Mechanics, Dover Publica-tions, 1989.Cordes, C., and W. Kinzelbach, Continuous groundwater velocity fields and path linesin linear, bilinear, and trilinear finite elements., Water Resour. Res., 28,2903–2911,1992.Crane, M., and M. Blunt, Streamline-based simulation of solute transport, Water Resour.Res., 35,3061–3078,1999.Dormand, J., and P. Prince, A family of embedded Runge-Kutta formulae, J. Comput.Appl. Math., 6,19–26,1980.Matringe, S., Accurate streamline tracing and coverage, Master’s thesis, Stanford Uni-versity, 2004.212Obi, E., and M. Blunt, Streamline-based simulation of advective-dispersive solute trans-port, Adv. Water Resour., 27,913–924,2004.Pollock, D., Semianalytical computation of path lines for Finite-Diﬀerence models,Ground Water, 26,743–750,1988.Prevost, M., M. Edwards, and M. Blunt, Streamline tracing on curvilinear structuredand unstructured grids, Soc. Petrol. Eng. J., 7,139–148,2002.Thiele, M., Streamline simulation, in 7th International Forum on Reservoir Simulation,2003.Thiele, M., R. Batycky, and M. Blunt, Simulating flow in heteroneous systems usingstreamtube and streamlines, SPE Reservoir Engineering, pp. 5–12, 1996.213
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Particle and streamline numerical methods for conservative and reactive transport simulations in porous.. Herrera, Paulo Andres Ricci 2009-11-30
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Title | Particle and streamline numerical methods for conservative and reactive transport simulations in porous media |
Creator |
Herrera, Paulo Andres Ricci |
Publisher | University of British Columbia |
Date Issued | 2009 |
Description | Reactive transport modeling has become an important tool to study and understand the transport and fate of solutes in the subsurface. However, the accurate simulation of reactive transport represents a formidable challenge because of the characteristics of flow, transport and chemical reactions that govern the migration of solutes in geological formations. In particular, solute transport in natural porous media is advection-controlled and dispersion is higher in the direction of flow than in the transverse direction. Both characteristics create difficulties for traditional numerical schemes that result in numerical dispersion and/or spurious oscillations. While these errors can often be tolerated in conservative transport simulations, they can be amplified in presence of chemical reactions resulting in much larger errors or unstable solutions. In this thesis, new Lagrangian based methods to simulate conservative and reactive transport in porous media are investigated. First, the derivation of a new meshless approximation based on smoothed particle hydrodynamics (SPH) to simulate conservative multidimensional solute transport, including advection and anisotropic dispersion, is presented. Second, a hybrid scheme that combines some of the advantages of streamline-based simulations and meshless methods and that allows simulating longitudinal and transverse dispersion without requiring a background grid is also derived. The numerical properties of both methods are analyzed analytical and numerically. Furthermore, both formulations are compared with existing numerical techniques in a set of two- and three-dimensional benchmark problems. It is demonstrated that the proposed schemes provide accurate and efficient solutions of physical transport processes in heterogeneous porous media and overcome most of the issues in existing numerical formulations. The new methods have the potential to remove or minimize numerical dispersion and grid orientation effects and, in the case of the hybrid streamline method, also eliminate spurious oscillations even in presence of large longitudinal to transverse dispersivity ratios. Therefore, the results presented in this thesis confirm that the Lagrangian formulations of solute transport investigated here are viable and compelling alternatives to simulate reactive transport versus more standard numerical techniques. |
Extent | 10878280 bytes |
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Thesis/Dissertation |
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Text |
File Format | application/pdf |
Language | eng |
Date Available | 2009-11-30 |
Provider | Vancouver : University of British Columbia Library |
DOI | 10.14288/1.0053008 |
URI | http://hdl.handle.net/2429/15967 |
Degree |
Doctor of Philosophy - PhD |
Program |
Geological Engineering |
Affiliation |
Science, Faculty of Earth, Ocean and Atmospheric Sciences, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 2010-05 |
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UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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