A review and numerical assessment of the random walk particle tracking method.

We review the basic mathematical concepts of random walk particle tracking (RWPT) and its advantages and limitations. Three different numerical approaches to overcome the local mass conservation problem of the random walk methodology are examined: (i) the interpolation method, (ii) the reflection principle, and (iii) the generalized stochastic differential equations (GSDE). Analytical solutions of the spatial moments for a two-layer system are compared to model predictions using the different techniques and results demonstrate that the interpolation method reproduces correctly average velocity, but fails to reproduce macrodispersion at higher hydraulic conductivity contrasts between the two layers. On the contrary, the reflection principle and the GSDE approach underestimate average velocity, but reproduce macrodispersion better for high contrasts. The different behavior is based on an artificial shift of mass for increasing heterogeneities for the GSDE approach and the reflection principle, whereas the interpolation method suffers from the smoothing of the dispersion tensor. The behavior of these approaches was furthermore analyzed in two-dimensional heterogeneous hydraulic conductivity fields, which are characterized by different random function models. Solute transport was simulated correctly by all three approaches for the reference fields having Gaussian structures or non-Gaussian structures with an isotropic spatial correlation, even for a variance of the natural log of hydraulic conductivity of sigma(lnK)(2)=4. However, for the non-Gaussian model with a strong anisotropic spatial correlation and a variance of sigma(lnK)(2)=2 and higher, the interpolation method was the only technique modelling solute transport correctly. Furthermore, we discuss the general applicability of random walk particle tracking in comparison to the standard transport models and conclude that in advection-dominated problems using a high spatial discretization or requiring the performance of many model runs, RWPT represents a good alternative for modelling contaminant transport.

[1]  Harihar Rajaram,et al.  Assessment of the predictive capabilities of stochastic theories in a three‐dimensional laboratory test aquifer: Effective hydraulic conductivity and temporal moments of breakthrough curves , 2005 .

[2]  D. Misra,et al.  Modeling plume behavior for nonlinearly sorbing solutes in saturated homogeneous porous media , 1998 .

[3]  H. Hoteit,et al.  Three-Dimensional Modeling of Mass Transfer in Porous Media Using the Mixed Hybrid Finite Elements and the Random-Walk Methods , 2002 .

[4]  Charles F. Harvey,et al.  When good statistical models of aquifer heterogeneity go bad: A comparison of flow, dispersion, and mass transfer in connected and multivariate Gaussian hydraulic conductivity fields , 2003 .

[5]  J. Gómez-Hernández,et al.  To be or not to be multi-Gaussian? A reflection on stochastic hydrogeology , 1998 .

[6]  Graham E. Fogg,et al.  Random-Walk Simulation of Transport in Heterogeneous Porous Media: Local Mass-Conservation Problem and Implementation Methods , 1996 .

[7]  Adrian E. Scheidegger,et al.  Statistical Hydrodynamics in Porous Media , 1954 .

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

[9]  J. Quastel,et al.  Diffusion theory for transport in porous media: Transition‐probability densities of diffusion processes corresponding to advection‐dispersion equations , 1998 .

[10]  Harihar Rajaram,et al.  Differences in the scale dependence of dispersivity and retardation factors estimated from forced‐gradient and uniform flow tracer tests in three‐dimensional physically and chemically heterogeneous porous media , 2005 .

[11]  R. Aris.  On the dispersion of a solute in a fluid flowing through a tube , 1956, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[12]  F. Molz,et al.  An Analysis of Dispersion in a Stratified Aquifer , 1984 .

[13]  Ahmed E. Hassan,et al.  Monte Carlo study of conservative transport in heterogeneous dual-porosity media , 2003 .

[14]  G.J.M. Uffink,et al.  Analysis of dispersion by the random walk method , 1990 .

[15]  J. Nieber,et al.  Modeling the effects of nonlinear equilibrium sorption on the transport of solute plumes in saturated heterogeneous porous media , 2000 .

[16]  G. Fogg,et al.  Role of Molecular Diffusion in Contaminant Migration and Recovery in an Alluvial Aquifer System , 2001 .

[17]  E. Frind,et al.  Delineation of Three‐Dimensional Well Capture Zones for Complex Multi‐Aquifer Systems , 2002, Ground water.

[18]  G. J. M. Uffink,et al.  A random walk method for the simulation of macrodispersion in a stratified aquifer , 1985 .

[19]  P. Ackerer,et al.  Random-Walk Method to Simulate Pollutant Transport in Alluvial Aquifers or Fractured Rocks , 1988 .

[20]  Andrew F. B. Tompson,et al.  Numerical simulation of solute transport in three-dimensional, randomly heterogeneous porous media , 1990 .

[21]  G. Pinder,et al.  Computational Methods in Subsurface Flow , 1983 .

[22]  E. Foufoula‐Georgiou,et al.  Modeling kinetic non-equilibrium using the first two moments of the residence time distribution , 1991 .

[23]  P. Ackerer,et al.  Comment on “Diffusion theory for transport in porous media: Transition‐probability densities of diffusion processes corresponding to advection‐dispersion equations” by Eric M. LaBolle et al. , 2000 .

[24]  W. Kinzelbach,et al.  Computation of stochastic wellhead protection zones by combining the first-order second-moment method and Kolmogorov backward equation analysis , 2000 .

[25]  Bruce A. Robinson,et al.  New form of dispersion tensor for axisymmetric porous media with implementation in particle tracking , 2002 .

[26]  Arlen W. Harbaugh,et al.  A modular three-dimensional finite-difference ground-water flow model , 1984 .

[27]  A. Scheidegger.  General Theory of Dispersion in Porous Media , 1961 .

[28]  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 .

[29]  Amvrossios C. Bagtzoglou,et al.  Projection functions for particle‐grid methods , 1992 .

[30]  K. Semra,et al.  Three dimensional groundwater quality modelling in heterogeneous media , 1970 .

[31]  Harihar Rajaram,et al.  Differences in the scale-dependence of dispersivity estimated from temporal and spatial moments in chemically and physically heterogeneous porous media , 2005 .

[32]  E. Frind,et al.  Simulation of contaminant transport in three dimensions: 2. Dimensionality effects , 1987 .

[33]  J. C. Jaeger,et al.  Conduction of Heat in Solids , 1952 .

[34]  J. Jaime Gómez-Hernández,et al.  ISIM3D: and ANSI-C three-dimensional multiple indicator conditional simulation program , 1990 .

[35]  Andrew F. B. Tompson,et al.  Impacts of Physical and Chemical Heterogeneity on Cocontaminant Transport in a Sandy Porous Medium , 1996 .

[36]  Peter K. Kitanidis,et al.  Macroscopic behavior and random‐walk particle tracking of kinetically sorbing solutes , 2000 .

[37]  C. R. Cole,et al.  Multicomponent mass transport model: theory and numerical implementation (discrete-parcel-random-walk version) , 1977 .

[38]  Georg Teutsch,et al.  Heterogeneity patterns of Quaternary glaciofluvial gravel bodies (SW-Germany): application to hydrogeology , 2003 .

[39]  T. A. Prickett,et al.  A "random-walk" solute transport model for selected groundwater quality evaluations , 1981 .

[40]  Graham E. Fogg,et al.  Diffusion processes in composite porous media and their numerical integration by random walks: Generalized stochastic differential equations with discontinuous coefficients , 2000 .

[41]  M. Celia,et al.  Large-scale natural gradient tracer test in sand and gravel, Cape Cod, Massachusetts, 2, Analysis of spatial moments for a nonreactive tracer , 1991 .

[42]  C. Zheng,et al.  A dual‐domain mass transfer approach for modeling solute transport in heterogeneous aquifers: Application to the Macrodispersion Experiment (MADE) site , 2000 .

[43]  C. Zheng.  Analysis of Particle Tracking Errors Associated with Spatial Discretization , 1994 .

[44]  Robert D. Falgout,et al.  Analysis of subsurface contaminant migration and remediation using high performance computing , 1998 .

[45]  Gaisheng Liu,et al.  Limits of applicability of the advection‐dispersion model in aquifers containing connected high‐conductivity channels , 2004 .

[46]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[47]  A. Valocchi,et al.  Stochastic analysis of the transport of kinetically sorbing solutes in aquifers with randomly heterogeneous hydraulic conductivity , 1993 .

[48]  Xian-Huan Wen,et al.  The Constant Displacement Scheme for Tracking Particles in Heterogeneous Aquifers , 1996 .

[49]  D. W. Pollock.  Semianalytical Computation of Path Lines for Finite‐Difference Models , 1988 .

[50]  Ahmed E. Hassan,et al.  On using particle tracking methods to simulate transport in single-continuum and dual continua porous media , 2003 .

[51]  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 .

[52]  Brian Berkowitz,et al.  Anomalous transport in laboratory‐scale, heterogeneous porous media , 2000 .

[53]  G. De Josselin De Jong,et al.  Longitudinal and transverse diffusion in granular deposits , 1958 .

[54]  Kiyosi Itô.  Stochastic Differential Equations , 2018, The Control Systems Handbook.

[55]  J. W. Mercer,et al.  Contaminant transport in groundwater. , 1992 .

[56]  Andrew F. B. Tompson,et al.  Numerical simulation of chemical migration in physically and chemically heterogeneous porous media , 1993 .

[57]  R. L. Stratonovich.  A New Representation for Stochastic Integrals and Equations , 1966 .

[58]  Paul P. Wang,et al.  MT3DMS: A Modular Three-Dimensional Multispecies Transport Model for Simulation of Advection, Dispersion, and Chemical Reactions of Contaminants in Groundwater Systems; Documentation and User's Guide , 1999 .