A Fast Fourier transform stochastic analysis of the contaminant transport problem

Deng, Fei-Wen; Cushman, John H.; Delleur, J. W.

doi:10.1029/93wr01236

Cited by 68 publications

(58 citation statements)

References 30 publications

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“…The covariance macrodispersive flux integral solution in equation (13), on setting R = x or R = :~, is unfortunately not a convolution like the mean concentration macrodispersive flux integral solution in (5) is, on setting ±=x. Therefore the Fourier-Laplace methodology of Deng et al [1993], [1995], and Cushman and Hu [1995], which relates the macrodispersive flux and the mean concentration field by simple algebraic expressions in Fourier-Laplace space, is not directly applicable to the covariance macrodispersive flux calculation.…”

Section: Relationship With Other Formulationsmentioning

confidence: 99%

Advection-diffusion in spatially random flows: Formulation of concentration covariance

Kapoor

Kitanidis

1997

Stochastic Hydrol Hydraul

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Abstract:The concentration c(x,t) of a nonreactive solute undergoing advection and diffusion in a spatially random divergence-free flow field is analyzed. A leading order formulation for the spatial covariance of the concentration field, c'(x, t)c'(:~, t), is made. That formulation includes the velocity variability induced macrodispersive flux of the covariance field, and the smoothing effects of diffusion. Previous formulations of the concentration covariance had dropped at least one of these effects. It is shown that both these effects need to be included to obtain a qualitatively correct description of the concentration fluctuations.

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Section: Relationship With Other Formulationsmentioning

confidence: 99%

Advection-diffusion in spatially random flows: Formulation of concentration covariance

Kapoor

Kitanidis

1997

Stochastic Hydrol Hydraul

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“…Like the above-mentioned finite element method McLaughlin, 1989a, b, 1991], the nonstationary spectral method is able to evaluate all first and second concentration moments, but it takes about same order of operations as the finite element method, though it may require less storage. Deng et al [1993] derived analytical expressions for the ensemble mean concentration and macrodispersive flux terms resulting from an unbounded stationary random velocity field in the Fourier transform domain, where the coupled equations of mean concentration and macrodispersive flux become decoupled. The decoupled mean concentration and macrodispersive equations can be evaluated by numerical techniques such as fast Fourier transform or numerical inverse Laplace transform.…”

Section: Introductionmentioning

confidence: 99%

Stochastic analysis of solute transport in heterogeneous aquifers subject to spatiotemporal random recharge

Li¹,

Graham²

1999

Water Resources Research

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Abstract. This paper extends the work of Li and Graham [1998] to deal with the unconditional moments of head, velocity, and concentration under transient flow conditions, which are assumed to be caused by a spatiotemporally random recharge. Semianalytical solutions are derived for the unconditional covariances for transient velocity with a constant mean recharge using a Fourier transform approach. Results demonstrate that the velocity covariance derived for the steady state random recharge field is a limiting case of the spatiotemporally variable velocity covariance with an infinite temporal correlation scale. Another limiting case indicates that introduction of temporally random but spatially uniform recharge has no effect on the velocity covariances (over that induced by the mean recharge on the mean head gradient). Thus for this limiting case there is no increased effect on the ensemble mean concentration plume spreading or the concentration prediction uncertainty. Following Deng et al. [1993], the equations for mean concentration and macrodispersive flux under zero mean transient recharge are decoupled in the Laplace-Fourier domain and solved using a fast Fourier transform algorithm, which significantly reduces the computational demand. The first-order concentration variance is solved using three different approximate techniques: an approximate fast Fourier transform technique, a finite element method, and a direct numerical integration. The simulation results show that introduction of a spatiotemporally random recharge enhances both longitudinal and lateral mean concentration plume spreading compared to the no recharge case. However, transient recharge produces less spreading and less concentration prediction uncertainty than the steady state spatially random recharge case. Hydraulic conductivity heterogeneity generates variability in hydraulic head and pore water velocity and thus results in solute transport prediction uncertainty. Temporal and spatial fluctuations in recharge or discharge impact the magnitude and direction of the head gradient and thus increase the variability in pore water velocity resulting in increased solute transport prediction uncertainty. Li and Graham [1998] examined the impact of steady state spatially random recharge on the predictions of flow and transport in a two-dimensional aquifer. They derived closed-form expressions for the unconditional, nonstationary autocovariances for head and velocity and the cross covariances between velocity, head, recharge, and log transmissivity based on a system of coupled first-order partial differential moment equations. These equations were then used to solve approximate unconditional moment equations for a concentration plume in a two-dimensional spatially random transmissivity field subject to spatially random recharge and nonuniform (linear trending) velocity using a finite element algorithm. The results showed that a constant positive mean recharge yields a mean velocity gradient which enhances the ensemble mean plume spreading in the lo...

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“…In order to rearrange system (15)- (18) in a workable form we make use of a perturbative expansion of the stochastic unknowns fu; pg in terms of a dimensionless parameter which is selected as s Y to represent a measure of the standard deviation of the fluctuation of the logconductivity Y [20,21]. The perturbation expansion aims at providing a recursive solution of the stochastic poroelastic problem with order of accuracy parametrized by a power of the dimensionless number s Y : The form of the closure issue discussed herein consists in establishing an N th-order accuracy in the s Y and truncate the series, neglecting the higher terms in Oðs…”

Section: Asymptotic Developmentsmentioning

confidence: 99%

“…In evolving media, the covariance grows with the size of the region being sampled resulting in the appearance of phenomena such as anomalous diffusion (an increase of the macrodispersivity with travel distance), asymptotic scaling laws in fluid mixing regimes (References [16,17]) and non-local effects (see e.g. References [13,[18][19][20][21][22][23]). …”

Section: Introductionmentioning

confidence: 99%

Stochastic computational modelling of highly heterogeneous poroelastic media with long‐range correlations

Frias

Murad

Pereira

2003

Num Anal Meth Geomechanics

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SUMMARYThe compaction of highly heterogeneous poroelastic reservoirs with the geology characterized by long-range correlations displaying fractal character is investigated within the framework of the stochastic computational modelling. The influence of reservoir heterogeneity upon the magnitude of the stresses induced in the porous matrix during fluid withdrawal and rock consolidation is analysed by performing ensemble averages over realizations of a log-normally distributed stationary random hydraulic conductivity field. Considering the statistical distribution of this parameter characterized by a coefficient of variation governing the magnitude of heterogeneity and a correlation function which decays with a power-law scaling behaviour we show that the combination of these two effects result in an increase in the magnitude of effective stresses of the rock during reservoir depletion. Further, within the framework of a perturbation analysis we show that the randomness in the hydraulic conductivity gives rise to non-linear corrections in the upscaled poroelastic equations. These corrections are illustrated by a self-consistent recursive hierarchy of solutions of the stochastic poroelastic equations parametrized by a scale parameter representing the fluctuating log-conductivity standard deviation. A classical example of land subsidence caused by fluid extraction of a weak reservoir is numerically simulated by performing Monte Carlo simulations in conjunction with finite elements discretizations of the poroelastic equations associated with an ensemble of geologies. Numerical results illustrate the effects of the spatial variability and fractal character of the permeability distribution upon the evolution of the Mohr-Coulomb function of the rock.

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A Fast Fourier transform stochastic analysis of the contaminant transport problem

Cited by 68 publications

References 30 publications

Advection-diffusion in spatially random flows: Formulation of concentration covariance

Advection-diffusion in spatially random flows: Formulation of concentration covariance

Stochastic analysis of solute transport in heterogeneous aquifers subject to spatiotemporal random recharge

Stochastic computational modelling of highly heterogeneous poroelastic media with long‐range correlations

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