Statistical Formulation of Generalized Tracer Retention in Fractured Rock

Cvetković, Vladimir

doi:10.1002/2017wr021187

Cited by 8 publications

(23 citation statements)

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“…The novel form of a truncated power law partition function

g

, combined with the statistical representation of retention (Cvetkovic, ) enabled derivation of analytical expressions for estimating

T_{1}

. However, inferring

T_{1}

from the group

\overset{τ}{true} false/ \sqrt{T_{1}}

is only the first step toward reliable upscaling.…”

Section: Discussionmentioning

confidence: 99%

“…Hence, we consider the special case of a truncated power law form as presented in Cvetkovic () with exponent 1/2:

\overset{g}{true} false(s false) = \frac{1}{s \sqrt{T_{1}}} (][, \sqrt{\frac{1}{T_{2}} + s} - \frac{1}{\sqrt{T_{2}}})

g false(t false) = \frac{1}{\sqrt{T_{1} T_{2}}} ()(, Erf (][, \sqrt{t false/ T_{2}}) - 1) + \frac{{normale}^{- t false/ T_{2}}}{\sqrt{π T_{1} t}}

where

T_{1}

[T] is a characteristic retention (or trapping) time, and

T_{2}

[T] is a characteristic return time that controls asymptotic behavior and the extent of the tailing. A

g

that is derived by explicitly solving for diffusion into a matrix can be shown to be exactly and for an infinite matrix (

T_{2} \to \infty

), and approximately and for a finite matrix, where the associated parametric relationships have been identified by method of moments (Cvetkovic, ) (see Appendix for a summary). Note that in Cvetkovic ()

k_{1} \equiv 1 false/ T_{1}

and

k_{2} \equiv 1 false/ T_{2}

where used and referred to as “rates” in view of their dimension [1/T].…”

Section: Theorymentioning

confidence: 99%

“…To test the applicability of with

\overset{τ}{true}

estimated using , we extract from Table in Cvetkovic ()

h^{⋆} = 0.0006315

^{- 1}

and

t_{p}^{⋆} = 442

hr. Inserting these numbers into yields

\overset{τ}{true} = 198

hr, while yields

T_{1} = 26.7

hr which are close to previously obtained values (Figure ).…”

Section: Illustration Examplesmentioning

confidence: 99%

“…where ΔT i are independent and identically distributed random variables with a PDF p(ΔT i ), and N( ) is distributed following a Poisson process with the mean conditioned on the water travel time asN = ∕T 1 Water Resources Research 10.1029/2019WR025266 (Cvetkovic, 2017). The PDF of the underlying process ΔT i is defined in the Laplace domain asp = 1 − sĝ T 1 , for any suitable g; inversion ofp withĝ (6) yields…”

Section: Statistical Representation Of Retentionmentioning

confidence: 99%

“…We compare here different retention models for granitic rocks. Specifically, we compare the RTD in form of the cumulative distribution function (CDF), for two geometrical models with alternative boundary conditions (zero flux or zero concentration), the TOSS model as proposed in Cvetkovic (2017) and used in this work, and finally the first-order model. The interest in this comparison is to show consistent effects of the characteristic retention times T 1 and T 2 across models.…”

Section: Appendix B: Rtdmentioning

confidence: 99%

See 4 more Smart Citations

Inference of Retention Time From Tracer Tests in Crystalline Rock

Cvetković

Poteri

Selroos

et al. 2020

Water Resources Research

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A statistical parametrization of transport combined with a new, general partition function for diffusive mass transfer (Cvetkovic, 2017, https://doi.org/10.1002/2017WR021187) is here developed into a practical tool for evaluating tracer tests in crystalline rock. The research question of this study is how to separate the characteristic times of retention and advection, using tracer test information alone; this decoupling is critical for upscaling of transport. Three regimes are identified based on the unconditional mean number of trapping events. Analytical expressions are derived for inferring transport‐retention parameters; these are first tested on a series of generic examples and then using two sets of tracer test data. Our results indicate that the key transport‐retention parameters can be inferred separately with reasonable accuracy by a few simple steps, provided that the macrodispersion is not too large and retention not too strong. Of particular interest is inference of the retention time from the breakthrough curve peak that avoids costly asymptotic monitoring. Finally, we summarize the retention times as inferred from a series of nonsorbing tracer tests in the Swedish granite, demonstrating the uncertainties when estimating retention based on material and structural properties from samples. Possible strategies for reducing these uncertainties that combine improved understanding of crystalline rock evolution with numerical simulations are noted as topics for future research.

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“…The novel form of a truncated power law partition function