A mass balance based numerical method for the fractional advection‐dispersion equation: Theory and application

Zhang, Xiaoxian; Crawford, John W.; Deeks, Lynda K.; Stutter, Marc; Bengough, A. Glyn; Young, Iain M.

doi:10.1029/2004wr003818

Cited by 76 publications

(63 citation statements)

References 41 publications

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“…We discretized the three-dimensional pore architecture images to provide a numerical mesh for the computational modelling of transport coefficients. Using the lattice Boltzmann approach, which has been successfully applied to the prediction of the saturated hydraulic conductivity of soil directly from images of structure [20], we calculated the conductivity of the images corresponding to both the sterile and the fungal-inoculated soil. We assume in the modelling that the presence of microbial activity had no effect on the properties of the water, and so any calculated differences are solely due to structure changes.…”

Section: Self-organization Is Important For Soil Watermentioning

confidence: 99%

Microbial diversity affects self-organization of the soil–microbe system with consequences for function

Crawford

Deacon

Grinev

et al. 2011

J. R. Soc. Interface.

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139

110

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Soils are complex ecosystems and the pore-scale physical structure regulates key processes that support terrestrial life. These include maintaining an appropriate mixture of air and water in soil, nutrient cycling and carbon sequestration. There is evidence that this structure is not random, although the organizing mechanism is not known. Using X-ray microtomography and controlled microcosms, we provide evidence that organization of pore-scale structure arises spontaneously out of the interaction between microbial activity, particle aggregation and resource flows in soil. A simple computational model shows that these interactions give rise to self-organization involving both physical particles and microbes that gives soil unique material properties. The consequence of self-organization for the functioning of soil is determined using lattice Boltzmann simulation of fluid flow through the observed structures, and predicts that the resultant micro-structural changes can significantly increase hydraulic conductivity. Manipulation of the diversity of the microbial community reveals a link between the measured change in micro-porosity and the ratio of fungal to bacterial biomass. We suggest that this behaviour may play an important role in the way that soil responds to management and climatic change, but that this capacity for self-organization has limits.

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Section: Self-organization Is Important For Soil Watermentioning

confidence: 99%

Microbial diversity affects self-organization of the soil–microbe system with consequences for function

Crawford

Deacon

Grinev

et al. 2011

J. R. Soc. Interface.

Self Cite

139

110

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“…When a = 2, equation (1) reduces to the second-order ADE with constant parameters. The nonlocal fADE (1) has been used to simulate solute transport through unsaturated soils [Pachepsky et al, 2001;Zhang et al, 2005], saturated porous media [Zhou and Selim, 2003;Chang et al, 2005;Huang et al, 2006], streams and rivers [Deng et al, 2004;Zhang et al, 2005;Kim and Kavvas, 2006], and overland flow [Deng et al, 2006].…”

Section: Introductionmentioning

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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“…Many alternative frameworks for understanding transport have been proposed, including the Fractional Advection-Dispersion Equation (FADE) [15][16][17][18][19][20][21][22], and the continuous time random walk (CTRW) [10,11,13,14,[23][24][25]. Since the FADE still underestimates solute arrivals at short times [11] and the dispersion coefficient of the FADE can still be scale-dependent [17], its relevance may be chiefly in the field of mathematics rather than to actual solute transport problems.…”

Section: Existing Models For Solute Transport In Porous Mediamentioning

confidence: 99%

Saturation dependence of dispersion in porous media

2012

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In this study, we develop a saturation-dependent treatment of dispersion in porous media using concepts from critical path analysis, cluster statistics of percolation, and fractal scaling of percolation clusters. We calculate spatial solute distributions as a function of time and calculate arrival time distributions as a function of system size. Our previous results correctly predict the range of observed dispersivity values over ten orders of magnitude in experimental length scale, but that theory contains no explicit dependence on porosity or relative saturation. This omission complicates comparisons with experimental results for dispersion, which are often conducted at saturation less than 1. We now make specific comparisons of our predictions for the arrival time distribution with experiments on a single column over a range of saturations. This comparison suggests that the most important predictor of such distributions as a function of saturation is not the value of the saturation per se, but the applicability of either random or invasion percolation models, depending on experimental conditions.

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A mass balance based numerical method for the fractional advection‐dispersion equation: Theory and application

Cited by 76 publications

References 41 publications

Microbial diversity affects self-organization of the soil–microbe system with consequences for function

Microbial diversity affects self-organization of the soil–microbe system with consequences for function

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

Saturation dependence of dispersion in porous media

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