Large‐time behavior of concentration variance and dilution in heterogeneous formations

Pannone, Marilena; Kitanidis, Peter K.

doi:10.1029/1998wr900063

Cited by 56 publications

(57 citation statements)

References 30 publications

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“…In order to handle it, the originally laterally confined domain will be transformed into an unbounded, laterally periodic flow field where the requirement of nonabsorbing boundaries is fulfilled by the method of the images [e.g., Pannone and Kitanidis, 1999]. In this case, a mirror image of the velocity profile is envisioned about each confining bank (y = 0 and y = B), yielding…”

Section: Formulationmentioning

confidence: 99%

On the exact analytical solution for the spatial moments of the cross‐sectional average concentration in open channel flows

Pannone¹

2012

Water Resources Research

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This paper shows how an exact analytical solution for the transient‐state spatial moments of the cross‐sectional average tracer concentration in large open channel flows can be derived from the depth‐averaged advection‐diffusion equation resorting to the method of Green's functions, without any simplifying assumption about the regularity of the actual concentration field, the smallness of the fluctuations, or the large space‐time scale of variation of the average concentration gradient (justifying the a priori localization of the problem), which were the basis of the classic Taylor dispersion theory. The results reveal that in agreement with the findings by Aris (1956) and later by others for flows within a conduit, there are an initial centroid displacement and a variance deficit dependent on the specific position and dimension of the initial injection. The second central moment asymptotically tends to the linearly increasing function predictable on the basis of Taylor's classic theory, and the skewness, which is constantly zero for the cross‐sectionally uniform injection, in the case of nonuniform initial distributions tends to slowly vanish after having reached a maximum. Thus, the persistent asymmetry exhibited by the field concentration data, as well as the retardations and the accelerations in the peak trajectory, can be justified without making any a priori assumption about the physical mechanism underlying their appearance, like transient storage phenomena, just by rigorously solving the governing equation for the cross‐sectional average concentration in the presence of nonuniform, asymmetrically located solute injections.

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Section: Formulationmentioning

confidence: 99%

On the exact analytical solution for the spatial moments of the cross‐sectional average concentration in open channel flows

Pannone¹

2012

Water Resources Research

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“…The coefficient or variation (or its square) has been noted from time to time as a useful quantifier of tracer dispersion characteristics [2,27,33]. The coefficient of variation is particularly relevant for parallel systems because it is a distance-invariant constant when x is sufficently large for advective effects to dominate, as noted in Section 3.…”

Section: Estimating the Large-distance Gðt; Xþ Coefficient Of Variationmentioning

confidence: 99%

Temporal moments of a tracer pulse in a perfectly parallel flow system

Bardsley

2003

Advances in Water Resources

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Perfectly parallel groundwater transport models partition water flow into isolated one-dimensional stream tubes which maintain total spatial correlation of all properties in the direction of flow. The case is considered of the temporal moments of a conservative tracer pulse released simultaneously into N stream tubes with arbitrarily different advective-dispersive transport and steady flow speeds in each of the stream tubes. No assumptions are made about the form of the individual stream tube arrival-time distributions or about the nature of the between-stream tube variation of hydraulic conductivity and flow speeds. The tracer arrival-time distribution gðt; xÞ is an N -component finite-mixture distribution, with the mean and variance of each component distribution increasing in proportion to tracer travel distance x. By utilising moment relations of finite mixture distributions, it is shown (to r ¼ 4) that the rth central moment of gðt; xÞ is an rth order polynomial function of x or /, where / is mean arrival time. In particular, the variance of gðt; xÞ is a positive quadratic function of x or /. This generalises the well-known quadratic variance increase for purely advective flow in parallel flow systems and allows a simple means of regression estimation of the large-distance coefficient of variation of gðt; xÞ. The polynomial central moment relation extends to the purely advective transport case which arises as a largedistance limit of advective-dispersive transport in parallel flow models. The associated limit gðt; xÞ distributions are of N-modal form and maintain constant shapes independent of travel distance. The finite-mixture framework for moment evaluation is also a potentially useful device for forecasting gðt; xÞ distributions, which may include multimodal forms. A synthetic example illustrates gðt; xÞ forecasting using a mixture of normal distributions.

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“…The crucial role of local dispersion (sometimes simply named 'diffusion') in solute macro-dispersion and dilution was already explored in the context of subsurface flow and transport by Pannone and Kitanidis [25] and in the context of river-flow and transport by Pannone [26,27]. Overall, these studies showed that, in the case of heterogeneous structures characterized by short-range correlations, macro-dispersion and dilution are singularly driven by the interplay of advective heterogeneities and diffusive-like mechanisms.…”

Section: Introductionmentioning

confidence: 99%

An Analytical Model of Fickian and Non-Fickian Dispersion in Evolving-Scale Log-Conductivity Distributions

Pannone

2017

Water

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Abstract:The characteristics of solute transport within log-conductivity fields represented by power-law semi-variograms are investigated by an analytical Lagrangian approach that accounts for the automatic frequency cut-off induced by the initial contaminant plume size. The transport process anomaly is critically controlled by the magnitude of the Péclet number. Interestingly enough, unlike the case of fast-decaying correlation functions (i.e., exponential or Gaussian), the presence of intensive transverse diffusion acts as an antagonist mechanism in the process of Fickian regime achievement. On the other hand, for markedly advective conditions and finite initial plume size, even the ergodic longitudinal dispersion coefficient turns out to be asymptotically constant, and the corresponding expected concentration distribution can therefore be obtained by conventional mathematical methods.

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Large‐time behavior of concentration variance and dilution in heterogeneous formations

Abstract: FormulationConsider the advective-dispersive transport of a passive solute through porous geological formations characterized by translational invariance. Translational invariance signifies the 623

Cited by 56 publications

References 30 publications

On the exact analytical solution for the spatial moments of the cross‐sectional average concentration in open channel flows

On the exact analytical solution for the spatial moments of the cross‐sectional average concentration in open channel flows

Temporal moments of a tracer pulse in a perfectly parallel flow system

An Analytical Model of Fickian and Non-Fickian Dispersion in Evolving-Scale Log-Conductivity Distributions

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