Random‐Walk Simulation of Transport in Heterogeneous Porous Media: Local Mass‐Conservation Problem and Implementation Methods

Abstract

The random-walk method for simulating solute transport in porous media is typically based on the assumption that the velocity and velocity-dependent dispersion tensor vary smoothly in space. However, in cases where sharp interfaces separate materials with contrasting hydraulic properties, these quantities may be discontinuous. Normally, velocities are interpolated to arbitrary particle locations when finite difference or finite element methods are used to solve the flow equation. The use of interpolation schemes that preserve discontinuities in velocity at material contacts can result in a random-walk model that does not locally conserve mass unless a correction is applied at these contacts. Test simulations of random-walk particle tracking with and without special treatment of material contacts demonstrate the problem. Techniques for resolving the problem, including interpolation schemes and a reflection principle, are reviewed and tested. Results from simulations of transport in porous media with discontinuities in the dispersion tensor show which methods satisfy continuity. Simulations of transport in twodimensional heterogeneous porous media demonstrate the potentially significant effect of using a nonconservative model to compute spatial moments and breakthrough of a solute plume. Introduction The random-walk particle method (RWPM) has been used successfully for years to simulate conservative and reactive transport in porous media [Ahlstrom et al., 1977; Prickerr et al., 1981; Uffink, 1985; Tompson e! al., 1987; Tompson, 1993]. This method is computationally appealing because it is grid independent and therefore, given the proper conditions, will require little computer storage relative to finite element, finite difference, and method of characteristic models. In addition, this method does not Suffer from numerical dispersion in problems dominated by advection. Traditional finite element and finite difference models generally perform poorly under such conditions unless the computational grid is highly resolved. As a result, a random walk is often the method of choice for simulating transport in large, heterogeneous flow systems [Tompson and Gelhat, 1990; Tompson, 1993; Tompson et al., 1994]. Global mass conservation is compulsory with the RWPM because particles cannot disappear. This distinct advantage of the RWPM, however, is often overstated; accurate solutions require local as well as global mass conservation.In practice, the flow Problem is often solved numerically, and velocities are interpolated to arbitrary particle locations.Advective particle tracking models can be made mass conservative by using a divergence-free velocity interpolation scheme [Schafer-Perini and Wilson, 199i]. However, additional criteria are necessary to formulate a mass conservative random-walk model. For example, discontinuities in the velocity or effective porosity may yield a dispersion tensor that is discontinuous in space. Local mass conservation conditions for the RWPM require that the dispersion tensor be continuous in spa...

Space‐fractional advection‐dispersion equations with variable parameters: Diverse formulas, numerical solutions, and application to the Macrodispersion Experiment site data

“…The (a À 1)-order Lévy random noise scaled by the gradient of the dispersion coefficient, as indicated by the third term on the right-hand side (RHS) of equation (10), captures the influence of the spatial variation of dispersion coefficient on the drift of solutes. This compares directly to the additional drift that arises from dD(x)/dx in the traditional ADE [LaBolle et al, 1996]. If the effective porosity also varies in space, then the spatial variability in porosity will also affect the drift, but not the dispersion [LaBolle …”

“…[23] Note that when a = 2, all of the above fADEs [equations (4), (11), and (15)] reduce to the second-order ADE (if the upstream boundary remains clean from the contaminants in the last two fADEs), and all above Markov processes [equations (10), (13), and (17)] reduce to the traditional Markov process used to simulate the secondorder ADE [LaBolle et al, 1996[LaBolle et al, , 1998]. …”

Toward efficiency in heterogeneous multispecies reactive transport modeling: A particle‐tracking solution for first‐order network reactions

“…In this context, Particle Tracking Methods (PTMs) offer a convenient numerical solution particularly efficient in dealing with heterogeneities [e.g., Wen and GomezHernandez, 1996;LaBolle et al, 1996;Salamon et al, 2007;Riva et al, 2008] and a large variety of complex transport processes such as non-Fickian transport [Delay and Bodin, 2001;Cvetkovic and Haggerty, 2002;Berkowitz et al, 2006;Zhang and Benson, 2008;Dentz and Castro, 2009] and multiple porosity systems [Salamon et al, 2006b;Benson and Meerschaert, 2009;Tsang and Tsang, 2001;Huang et al, 2003;Willmann et al, 2013]. Moreover, this methodology, which is always mass conservative, avoids some of the inherent numerical difficulties associated with Eulerian approaches, i.e., numerical dispersion and oscillations due to truncation errors [Salamon et al, 2007;Boso et al, 2013].…”

Numerical Study of Spatial Behavior of Solute Particle Transport in Single Fracture with Variable Apertures

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“…A number of studies on fluid flow and solute transport in a single fracture has been conducted so far and include laboratory or field experiments [1][2][3][4][5][6][7][8] and numerical experiments [9][10][11][12][13].…”