Supercomputers and their use in modeling subsurface solute transport modeling

Supercomputers offer a wide range of possibilities for advancing knowledge and solving difficult, real-world problems. Solute transport-related theory and applications have long been an area of specialization within hydrology. In this review, recent applications of supercomputers to solute transport problems are presented. These applications are based mostly on the assumption of a dilute solute, allowing the governing equations to be decoupled. An application of supercomputers is the simulation of solute transport in large heterogeneous spatial domains by both direct solution of the governing equations and by particle- tracking methods. An impetus for this research has been the rapid increase in predictive dispersion models based on analytical stochastic theory. Up until now, relatively few applications have incorporated algorithms designed with a targeted supercomputer in mind, with the result that the machine’s potential cannot be exploited fully. Since many numerical problems are reduced to solutions of algebraic systems when they are implemented on computers, algorithms for linear algebraic systems of equations suitable for vector and parallel machines are discussed in some detail. Data analysis, necessary both for evaluating emerging theories and for more “routine” applications like determining model parameters, is identified as another general area that is ably handled by supercomputers. In particular, supercomputers offer a way to become less dependent on parametric assumptions and constraints, substituting, instead, raw computational power. Visualization techniques and associated facilities have flourished alongside the development of supercomputer centers. These offer the possibility for real-time viewing of solute transport processes, both real and simulated, although little has been produced thus far. It is expected that many more researchers involved with subsurface solute transport will make use of supercomputers in the future.

[1]  D. Sherrington.  Stochastic Processes in Physics and Chemistry , 1983 .

[2]  Stephen Fahrney McCormick.  Multigrid methods, theory applications, and supercomputing , 1990 .

[3]  G. Sposito,et al.  A derivation of the macroscopic solute transport equation for homogeneous, saturated, porous media: 2. Reactive solutes at low concentration , 1981 .

[4]  Harold S. Stone,et al.  An Efficient Parallel Algorithm for the Solution of a Tridiagonal Linear System of Equations , 1973, JACM.

[5]  E. Sudicky,et al.  Three-dimensional simulation of organic transport with aerobic biodegradation , 1989 .

[6]  R. Bhattacharya,et al.  Foundation theories of solute transport in porous media: a critical review , 1979 .

[7]  T. Illangasekare,et al.  A discrete kernel method of characteristics model of solute transport in water table aquifers , 1989 .

[8]  R. L. Naff,et al.  A note on the recent natural gradient tracer test at the Borden Site , 1988 .

[9]  Allan L. Gutjahr,et al.  Stochastic analysis of macrodispersion in a stratified aquifer , 1979 .

[10]  David J. Kuck,et al.  On Stable Parallel Linear System Solvers , 1978, JACM.

[11]  The Principal Direction Technique: A New Approach to Groundwater Contaminant Transport Modeling , 1982 .

[12]  William E. Lorensen,et al.  Marching cubes: A high resolution 3D surface construction algorithm , 1987, SIGGRAPH.

[13]  F. Gustavson,et al.  Implementing Linear Algebra Algorithms for Dense Matrices on a Vector Pipeline Machine , 1984 .

[14]  J. Parker,et al.  Analysis of the inverse problem for transient unsaturated flow , 1988 .

[15]  Apostolos Hadjidimos,et al.  The parallel solution of banded linear equations by the new quadrant interlocking factorisation (q.i.f.) method , 1981 .

[16]  Mary F. Wheeler,et al.  A modified method of characteristics technique and mixed finite elements method for simulation of groundwater solute transport , 1989 .

[17]  A. Peters,et al.  Finite element ground-water models implemented on vector computers , 1986 .

[18]  Peter A. Forsyth,et al.  Incomplete Factorization Methods for Fully Implicit Simulation of Enhanced Oil Recovery , 1984 .

[19]  David L. Freyberg,et al.  A natural gradient experiment on solute transport in a sand aquifer: 2. Spatial moments and the advection and dispersion of nonreactive tracers , 1986 .

[20]  C. Axness,et al.  Three‐dimensional stochastic analysis of macrodispersion in aquifers , 1983 .

[21]  S. P. Neuman,et al.  estimation of spatial covariance structures by adjoint state maximum likelihood cross validation: 1. Theory , 1989 .

[22]  H. L. Gray,et al.  The Generalised Jackknife Statistic , 1974 .

[23]  G. Dagan.  Statistical Theory of Groundwater Flow and Transport: Pore to Laboratory, Laboratory to Formation, and Formation to Regional Scale , 1986 .

[24]  W. Yeh.  Review of Parameter Identification Procedures in Groundwater Hydrology: The Inverse Problem , 1986 .

[25]  C. L. Winter,et al.  A Perturbation Expansion for Diffusion in a Random Velocity Field , 1984 .

[26]  T. A. Zang,et al.  Spectral Methods for Partial Differential Equations , 1984 .

[27]  Craig M. Bethke,et al.  A Numerical Model of Compaction-Driven Groundwater Flow and Heat Transfer and Its Application to the Paleohydrology of Intracratonic Sedimentary Basins , 1985 .

[28]  E. Frind.  Seawater intrusion in continuous coastal aquifer-aquitard systems , 1982 .

[29]  George E. Forsythe,et al.  Finite-Difference Methods for Partial Differential Equations , 1961 .

[30]  E. A. Sudicky,et al.  The Laplace Transform Galerkin Technique: A time‐continuous finite element theory and application to mass transport in groundwater , 1989 .

[31]  David J. Evans,et al.  Parallel Algorithms for the Iterative Solution to Linear Systems , 1982, Comput. J..

[32]  R. T. Cheng,et al.  On seawater encroachment in coastal aquifers , 1974 .

[33]  A. Herbert,et al.  Coupled groundwater flow and solute transport with fluid density strongly dependent upon concentration , 1988 .

[34]  W. Strawderman.  The Generalized Jackknife Statistic , 1973 .

[35]  G. Dagan.  Stochastic modeling of groundwater flow by unconditional and conditional probabilities: 2. The solute transport , 1982 .

[36]  G. Dagan.  Solute transport in heterogeneous porous formations , 1984, Journal of Fluid Mechanics.

[37]  John Noye,et al.  Finite Difference Techniques for Partial Differential Equations , 1984 .

[38]  John A. Cherry,et al.  Migration of contaminants in groundwater at a landfill: A case study: 1. Groundwater flow and plume delineation , 1983 .

[39]  Larry W. Lake,et al.  Precipitation and dissolution of solids attending flow through porous media , 1984 .

[40]  Jack C. Parker,et al.  Parameter estimation for coupled unsaturated flow and transport , 1989 .

[41]  E. C. Childs.  Dynamics of fluids in Porous Media , 1973 .

[42]  P. F. Andersen,et al.  Saltwater intrusion in aquifers: Development and testing of a three‐dimensional finite element model , 1987 .

[43]  Gérard M. Baudet,et al.  Asynchronous Iterative Methods for Multiprocessors , 1978, JACM.

[44]  C. Upson,et al.  Supercomputer Analysis of Sedimentary Basins , 1988, Science.

[45]  George F. Pinder,et al.  Transient Simulation of Saltwater Intrusion in Southeastern Florida , 1976 .

[46]  Kenny S. Crump,et al.  Numerical Inversion of Laplace Transforms Using a Fourier Series Approximation , 1976, J. ACM.

[47]  C. Carnahan.  Simulation of uranium transport with variable temperature and oxidation potential: The computer program THCC (Thermo-Hydro-Chemical Coupling) , 1986 .

[48]  C. W. Gardiner,et al.  Handbook of stochastic methods - for physics, chemistry and the natural sciences, Second Edition , 1986, Springer series in synergetics.

[49]  D. A. Barry,et al.  On the Dagan model of solute transport in groundwater: Application to the Borden site , 1988 .

[50]  H. T. Kung,et al.  Synchronized and asynchronous parallel algorithms for multiprocessors , 1976 .

[51]  E. Frind.  Simulation of long-term transient density-dependent transport in groundwater , 1982 .

[52]  John A. Cherry,et al.  Migration of contaminants in groundwater at a landfill: A case study: 4. A natural-gradient dispersion test , 1983 .

[53]  T. N. Narasimhan,et al.  Groundwater contamination from an inactive uranium mill tailings pile: 2. Application of a dynamic mixing model , 1984 .

[54]  Peter A. Forsyth,et al.  Comparison of Fast Iterative Methods for Symmetric Systems , 1983 .

[55]  N. S. Sridharan.  Computer generation of vertex-graphs , 1973, ACM Annual Conference.

[56]  B. Efron.  The jackknife, the bootstrap, and other resampling plans , 1987 .

[57]  Lynn W. Gelhar,et al.  Stochastic subsurface hydrology from theory to applications , 1986 .

[58]  G. Amdhal,et al.  Validity of the single processor approach to achieving large scale computing capabilities , 1967, AFIPS '67 (Spring).

[59]  E. A. Sudicky,et al.  Simulation of biodegradable organic contaminants in groundwater: 2. Plume behavior in uniform and random flow fields , 1990 .

[60]  D J Kuck,et al.  Parallel Supercomputing Today and the Cedar Approach , 1986, Science.

[61]  R. Ababou,et al.  Three-dimensional flow in random porous media , 1988 .

[62]  W. G. Gray,et al.  A Galerkin‐finite element technique for calculating the transient position of the saltwater front , 1975 .

[63]  David L. Freyberg,et al.  Simulation of one-dimensional correlated fields using a matrix-factorization moving average approach , 1990 .

[64]  Mark N. Goltz,et al.  Interpreting organic solute transport data from a field experiment using physical nonequilibrium models , 1986 .

[65]  D. W. Peaceman,et al.  Numerical Calculation of Multidimensional Miscible Displacement by the Method of Characteristics , 1964 .

[66]  T. N. Narasimhan,et al.  Redox‐controlled multiple‐species reactive chemical transport: 1. Model development , 1989 .

[67]  G. Sposito,et al.  A derivation of the macroscopic solute transport equation for homogeneous, saturated, porous media , 1980 .

[68]  D. Cox,et al.  A General Definition of Residuals , 1968 .

[69]  M. H. Quenouille.  NOTES ON BIAS IN ESTIMATION , 1956 .

[70]  Michael Hatzopoulos,et al.  A parallel linear system solver , 1979 .

[71]  James R. Wallis.  Hydrology—The computer revolution continues , 1987 .

[72]  S. P. Garabedian,et al.  Large-scale dispersive transport in aquifers : field experiments and reactive transport theory , 1987 .

[73]  Chris R. Jesshope,et al.  Parallel Computers 2: Architecture, Programming and Algorithms , 1981 .

[74]  Y. Saad,et al.  Practical Use of Polynomial Preconditionings for the Conjugate Gradient Method , 1985 .

[75]  D. A. Barry,et al.  On the Dagan Model of solute transport in groundwater: Foundational aspects , 1987 .

[76]  E. A. Sudicky,et al.  The Laplace Transform Galerkin Technique for large-scale simulation of mass transport in discretely fractured porous formations , 1992 .

[77]  David Andrew Barry,et al.  Application of the convection-dispersion model to solute transport in finite soil columns. , 1988 .

[78]  G. Dagan.  Time‐dependent macrodispersion for solute transport in anisotropic heterogeneous aquifers , 1988 .

[79]  Paul V. Roberts,et al.  A natural gradient experiment on solute transport in a sand aquifer: 4. Sorption of organic solutes and its influence on mobility , 1986 .

[80]  E. A. Sudicky,et al.  Micro-scale modelling in the study of plume evolution in heterogeneous media , 1987 .

[81]  P.K.W. Vinsome,et al.  Block iterative methods for fully implicit reservoir simulation , 1982 .

[82]  M. Hestenes,et al.  Methods of conjugate gradients for solving linear systems , 1952 .

[83]  S. P. Neuman,et al.  Stochastic theory of field‐scale fickian dispersion in anisotropic porous media , 1987 .

[84]  D. Marquardt.  An Algorithm for Least-Squares Estimation of Nonlinear Parameters , 1963 .

[85]  Thomas W. Crockett,et al.  Modeling Algorithm Execution Time on Processor Arrays , 1984, Computer.

[86]  Charles L. Seitz,et al.  Engineering limits on computer performance , 1984 .

[87]  T. N. Narasimhan,et al.  Redox-Controlled Multiple-Species Reactive Chemical Transport 2. Verification and Application , 1989 .

[88]  David L. Freyberg,et al.  Stochastic modeling of vertically averaged concentration uncertainty in a perfectly stratified aquifer , 1987 .

[89]  H. A. van der Vorst,et al.  Algorithms and applications on vector and parallel computers , 1987 .

[90]  J. Ortega,et al.  Solution of Partial Differential Equations on Vector and Parallel Computers , 1987 .

[91]  David S. Kershaw,et al.  Solution of Single Tridiagonal Linear Systems and Vectorization of the ICCG Algorithm on the Cray-1 , 1982 .

[92]  Bradley Efron,et al.  Computer-Intensive Methods in Statistical Regression , 1988 .

[93]  M. Shinozuka,et al.  Digital simulation of random processes and its applications , 1972 .

[94]  P. Bedient,et al.  Transport of dissolved hydrocarbons influenced by oxygen‐limited biodegradation: 1. Theoretical development , 1986 .

[95]  Peter J. Denning.  Speeding up parallel processing , 1988 .

[96]  D. Young,et al.  A Survey of Numerical Mathematics , 1988 .

[97]  Duncan H. Lawrie,et al.  High Speed Computer and Algorithm Organization , 1977 .

[98]  Joseph A. Fisher.  The VLIW Machine: A Multiprocessor for Compiling Scientific Code , 1984, Computer.

[99]  S. Majid Hassanizadeh,et al.  On the modeling of brine transport in porous media , 1988 .

[100]  C. Gardiner.  Handbook of Stochastic Methods , 1983 .

[101]  E. Sudicky.  A natural gradient experiment on solute transport in a sand aquifer: Spatial variability of hydraulic conductivity and its role in the dispersion process , 1986 .

[102]  S. Ashby,et al.  A numerical investigation of the conjugate gradient method as applied to three‐dimensional groundwater flow problems in randomly heterogeneous porous media , 1989 .

[103]  A. Mantoglou,et al.  The Turning Bands Method for simulation of random fields using line generation by a spectral method , 1982 .

[104]  Identification of Aquifer Dispersivities: Methods of Analysis and Parameter Uncertainty , 1987 .

[105]  Gedeon Dagan,et al.  Theory of Solute Transport by Groundwater , 1987 .

[106]  E. Sudicky,et al.  Simulation of biodegradable organic contaminants in groundwater. 1. Numerical formulation in principal directions. , 1990 .

[107]  John A. Sharp,et al.  An Introduction to Distributed and Parallel Processing , 1987 .

[108]  S. Swain.  Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences , 1984 .

[109]  William A. Jury,et al.  Fundamental Problems in the Stochastic Convection‐Dispersion Model of Solute Transport in Aquifers and Field Soils , 1986 .

[110]  D. A. Barry,et al.  Three‐dimensional statistical moment analysis of the Stanford/Waterloo Borden tracer data , 1990 .

[111]  Larry J. Ruff,et al.  How good are our best models? Jackknifing, bootstrapping, and earthquake depth , 1989 .

[112]  A.F.B. Tompson,et al.  Numerical simulation of solute transport in randomly heterogeneous porous media: Motivation, model development, and application , 1987 .

[113]  R. Ababou,et al.  Implementation of the three‐dimensional turning bands random field generator , 1989 .