TY - JOUR
T1 - Non-local and localized analyses of non-reactive solute transport in bounded randomly heterogeneous porous media
T2 - Theoretical framework
AU - Morales-Casique, Eric
AU - Neuman, Shlomo P.
AU - Guadagnini, Alberto
N1 - Funding Information:
This work was supported by NSF/ITR Grant EAR-0110289 and through a scholarship granted to the lead author by CONACYT of Mexico.
PY - 2006/8
Y1 - 2006/8
N2 - Solute transport in randomly heterogeneous media is described by stochastic transport equations that are typically solved via Monte Carlo simulation. A promising alternative is to solve a corresponding system of statistical moment equations directly. We present exact (though not closed) implicit conditional first and second moment equations for advective-dispersive transport in finite domains. The velocity and concentration are generally non-stationary due to possible trends in heterogeneity, conditioning on data, temporal variations in velocity, fluid and/or solute sources, initial and boundary conditions. Our equations are integro-differential and include non-local parameters depending on more than one point in space-time. To allow solving these equations, we close them by perturbation and develop recursive moment equations in Laplace space for the special case of steady state flow, to second order in σY where σY2 is a measure of (natural) log hydraulic conductivity variance. We also propose a higher-order iterative closure. Our recursive equations and iterative closure are formally valid for mildly heterogeneous media, or well-conditioned strongly heterogeneous media in which the random component of heterogeneity is relatively small. The non-local moment equations suggest (and a companion paper [Morales Casique E, Neuman SP, Guadagnini A. Nonlocal and localized analyses of nonreactive solute transport in bounded randomly heterogeneous porous media: computational analysis. Adv Water Resour, submitted for publication] demonstrates numerically) that, in general, transport cannot be validly described by means of Fick's law with a (constant or variable) macrodispersion coefficient. We show nevertheless that, under a limited set of conditions, the mean transport equation can be localized to yield a familiar-looking advection-dispersion equation with a conditional macrodispersion tensor that varies generally in space-time. In a companion paper [Morales Casique E, Neuman SP, Guadagnini A. Nonlocal and localized analyses of nonreactive solute transport in bounded randomly heterogeneous porous media: computational analysis. Adv Water Resour, submitted for publication] we present a high-accuracy computational algorithm for our iterative non-local and recursive localized moment equations, assessing their accuracy and computational efficiency in comparison to unconditional and conditional Monte Carlo simulations.
AB - Solute transport in randomly heterogeneous media is described by stochastic transport equations that are typically solved via Monte Carlo simulation. A promising alternative is to solve a corresponding system of statistical moment equations directly. We present exact (though not closed) implicit conditional first and second moment equations for advective-dispersive transport in finite domains. The velocity and concentration are generally non-stationary due to possible trends in heterogeneity, conditioning on data, temporal variations in velocity, fluid and/or solute sources, initial and boundary conditions. Our equations are integro-differential and include non-local parameters depending on more than one point in space-time. To allow solving these equations, we close them by perturbation and develop recursive moment equations in Laplace space for the special case of steady state flow, to second order in σY where σY2 is a measure of (natural) log hydraulic conductivity variance. We also propose a higher-order iterative closure. Our recursive equations and iterative closure are formally valid for mildly heterogeneous media, or well-conditioned strongly heterogeneous media in which the random component of heterogeneity is relatively small. The non-local moment equations suggest (and a companion paper [Morales Casique E, Neuman SP, Guadagnini A. Nonlocal and localized analyses of nonreactive solute transport in bounded randomly heterogeneous porous media: computational analysis. Adv Water Resour, submitted for publication] demonstrates numerically) that, in general, transport cannot be validly described by means of Fick's law with a (constant or variable) macrodispersion coefficient. We show nevertheless that, under a limited set of conditions, the mean transport equation can be localized to yield a familiar-looking advection-dispersion equation with a conditional macrodispersion tensor that varies generally in space-time. In a companion paper [Morales Casique E, Neuman SP, Guadagnini A. Nonlocal and localized analyses of nonreactive solute transport in bounded randomly heterogeneous porous media: computational analysis. Adv Water Resour, submitted for publication] we present a high-accuracy computational algorithm for our iterative non-local and recursive localized moment equations, assessing their accuracy and computational efficiency in comparison to unconditional and conditional Monte Carlo simulations.
KW - Advection
KW - Dispersion
KW - Moments
KW - Random media
KW - Stochastic
KW - Transport
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U2 - 10.1016/j.advwatres.2005.10.002
DO - 10.1016/j.advwatres.2005.10.002
M3 - Article
AN - SCOPUS:33746357914
SN - 0309-1708
VL - 29
SP - 1238
EP - 1255
JO - Advances in Water Resources
JF - Advances in Water Resources
IS - 8
ER -