Numerical simulations of preasymptotic transport in heterogeneous porous media: Departures from the Gaussian limit

Trefry, M. G.; Feng, Ruoqiang; McLaughlin, Dennis K.

doi:10.1029/2001wr001101

Cited by 58 publications

(88 citation statements)

References 28 publications

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“…In this section we present the simulation results for the Eulerian and Lagrangian statistical characteristics, and compare our results with the data reported in [26]. We recall that the problem is considered in the dimensionless form where the correlation length scale equals to 1.…”

Section: Simulation Resultsmentioning

confidence: 90%

“…Two typical variants of correlation functions B Y (r) = Y (x + r)Y (x) are: the Gaussian correlation function (e.g., see [26], [4]):…”

Section: The Darcy Equationmentioning

confidence: 99%

“…However in this problem, the iterative methods are generally slowly convergent (e.g., see [5], [26]). Iterative methods of multigrid type are often efficient, well-suited to regular grids, but sensitive to condition number.…”

Section: Finite-difference Approximationmentioning

confidence: 99%

“…To validate our stochastic simulation algorithm we first check the conservation law, and then compare our calculations with the results due to Trefry et al [26]. In all calculations, except for the simulations presented in the last two figures, (Figures 11 and 12) we used the Gaussian correlation function (6).…”

Section: Validationmentioning

confidence: 99%

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Stochastic Flow Simulation and Particle Transport in a 2D Layer of Random Porous Medium

Kurbanmuradov

Sabelfeld

2010

Transp Porous Med

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A stochastic numerical method is developed for simulation of flows and particle transport in a 2D layer of porous medium. The hydraulic conductivity is assumed to be a random field of a given statistical structure, the flow is modeled in the layer with prescribed boundary conditions. Numerical experiments are carried out by solving the Darcy equation for each sample of the hydraulic conductivity by a direct solver for sparse matrices, and tracking Lagrangian trajectories in the simulated flow. We present and analyze different Eulerian and Lagrangian statistical characteristics of the flow such as transverse and longitudinal velocity correlation functions, longitudinal dispersion coefficient, and the mean displacement of Lagrangian trajectories. We discuss the effect of long-range correlations of the longitudinal velocities which we have found in our numerical simulations. The related anomalous diffusion is also analyzed.

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Section: Simulation Resultsmentioning

confidence: 90%

“…Two typical variants of correlation functions B Y (r) = Y (x + r)Y (x) are: the Gaussian correlation function (e.g., see [26], [4]):…”

Section: The Darcy Equationmentioning

confidence: 99%

Section: Finite-difference Approximationmentioning

confidence: 99%

Section: Validationmentioning

confidence: 99%

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Stochastic Flow Simulation and Particle Transport in a 2D Layer of Random Porous Medium

Kurbanmuradov

Sabelfeld

2010

Transp Porous Med

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“…To simulate superFickian dispersion in porous media, we anticipate that the largest particle motions are ahead of the mean velocity, so that b = 1 [see also Schumer et al, 2001;Zhang et al, 2006a] and thus the dispersive flux reduces to F = ÀD@ aÀ1 C/@x aÀ1 . This term simulates faster-than-Fickian plume evolution through a permeability field with longrange dependence and high sample variance [Herrick et al, 2002;Grabasnjak, 2003;Trefry et al, 2003;Kohlbecker et al, 2006]. The fractional order a decreases if the medium contains higher probabilities of high velocities.…”

Section: Model 1: the Ff-adementioning

confidence: 99%

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

Zhang

Benson

Meerschaert

et al. 2007

Water Resources Research

126

123

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[1] To model the observed local variation of transport speed, an extension of the homogeneous space-fractional advection-dispersion equation (fADE) to more general cases with space-dependent coefficients (drift velocity V and dispersion coefficient D) has been suggested. To provide a rigorous evaluation of this extension, we explore the underlying physical meanings of two proposed, and one other possible form, of the fADE by using the generalized mass balance law proposed by Meerschaert et al. (2006). When the classical Fick's law is replaced by its generalized form in the first-order mass conversation law, the original fADE with constant parameters extends to the advection-dispersion equation (ADE) with fractional flux (denoted as FF-ADE). When the net inflow of dispersive flux is from nonlocal concentration gradients following a fractional divergence, we get the ADE with fractional divergence (FD-ADE). When the total net inflow from both nonlocal advection and nonlocal concentration gradients follows a fractional form, the fADE contains fully fractional divergence (FFD-ADE). These three fADEs with constant parameters can also be obtained by proper choice of the two memory kernels in the nonlocal dispersive constitutive theory proposed by Cushman et al. (1994), while the space-variable fADEs correspond to the conditional nonlocal theory proposed by Neuman (1993) after specifying the general (Lévy-type) functional form of the random distribution. The corresponding Langevin Markov models can be found in many cases, where the Lagrangian stochastic processes can be conditioned directly on local aquifer properties at any practical, measurable level and resolution. The resulting Lagrangian random-walk particle tracking methods, along with previous numerical solutions using implicit Euler finite differences, distinguish and elucidate the plume behavior described by these fADEs. The fractional models are applied to fit the tritium plumes measured at the Macrodispersion Experiment test site. When the local parameters gleaned from the hydraulic conductivity (K) distribution are varied even slightly (i.e., a two-zone model), allowing the use of observed hydraulic conditions at the site, the model fits are well within the variability of the data. The extended fADEs describe the fast and space-dependent leading edges of measured plumes in the regional-scale alluvial system, which was underestimated and could not be fully captured by the original fADE with constant parameters. Applications also favor the FD-ADE model because of the ease of implementation and consistency with previous analysis of the K statistics.

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On the formation of multiple local peaks in breakthrough curves

Siirila‐Woodburn

Sánchez-Vila

Fernàndez-García

2015

Water Resources Research

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The analysis of breakthrough curves (BTCs) is of interest in hydrogeology as a way to parameterize and explain processes related to anomalous transport. Classical BTCs assume the presence of a single peak in the curve, where the location and size of the peak and the slope of the receding limb has been of particular interest. As more information is incorporated into BTCs (for example, with high-frequency data collection, supercomputing efforts), it is likely that classical definitions of BTC shapes will no longer be adequate descriptors for contaminant transport problems. We contend that individual BTCs may display multiple local peaks depending on the hydrogeologic conditions and the solute travel distance. In such cases, classical definitions should be reconsidered. In this work, the presence of local peaks in BTCs is quantified from high-resolution numerical simulations in synthetic fields with a particle tracking technique and a kernel density estimator to avoid either overly jagged or smoothed curves that could mask the results. Individual BTCs from three-dimensional heterogeneous hydraulic conductivity fields with varying combinations of statistical anisotropy, heterogeneity models, and local dispersivity are assessed as a function of travel distance. The number of local peaks, their corresponding slopes, and a transport connectivity index are shown to strongly depend on statistical anisotropy and travel distance. Results show that the choice of heterogeneity model also affects the frequency of local peaks, but the slope is less sensitive to model selection. We also discuss how solute shearing and rerouting can be determined from local peak quantification.

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Numerical simulations of preasymptotic transport in heterogeneous porous media: Departures from the Gaussian limit

Cited by 58 publications

References 28 publications

Stochastic Flow Simulation and Particle Transport in a 2D Layer of Random Porous Medium

Stochastic Flow Simulation and Particle Transport in a 2D Layer of Random Porous Medium

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

On the formation of multiple local peaks in breakthrough curves

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