Higher-order effects on flow and transport in randomly heterogeneous porous media

Kuo Chin Hsu, Dongxiao Zhang, Shlomo P Neuman

Research output: Contribution to journalArticle

62 Citations (Scopus)

Abstract

A higher-order theory is presented for steady state, mean uniform saturated flow and nonreactive solute transport in a random, statistically homogeneous natural log hydraulic conductivity field Y. General integral expressions are derived for the spatial covariance of fluid velocity to second order in the variance σ2 of Y in two and three dimensions. Integrals involving first-order (in σ) fluctuations in hydraulic head are evaluated analytically for a statistically isotropic two-dimensional Y field with an exponential autocovariance. Integrals involving higher-order head fluctuations are evaluated numerically for this same field. Complete second- order results are presented graphically for σ2 = 1 and σ2 = 2. They show that terms involving higher-order head fluctuations are as important as those involving lower-order ones. The velocity variance is larger when approximated to second than to first order in σ2. Discrepancies between second- and first-order approximations of the velocity autocovariance diminish rapidly with separation distance and are very small beyond two integral scales. Transport requires approximation at two levels: the flow level at which velocity statistics are related to those of Y, and the advection level at which macrodispersivities are related to velocity fluctuations. Our results show that a second-order flow correction affects transport to a greater extent than does a second-order correction to advection. Asymptotically. the second-order transverse macrodispersivity tends to zero as does its first- order counterpart. An approximation of advection alone based on Corrsin's conjecture, coupled with either a first- or a second-order flow approximation, leads to a transverse macrodispersivity which is significantly larger than that obtained by standard perturbation and tends to a nonzero asymptote. Published Monte Carlo results yield macrodispersivities that lie significantly below those predicted by first- and second-order theories. Considering that Monte Carlo simulations often suffer from sampling and computational errors, that standard perturbation approximations are theoretically valid only for σ2 < 1, and that Corrsin's conjecture represents the leading term in a renormalization group perturbation which contains contributions from an infinite number of high-order terms, we find it difficult to tell which of these approximations is closest to representing transport in strongly heterogeneous media with σ2 ≤ 1.

Original languageEnglish (US)
Pages (from-to)571-582
Number of pages12
JournalWater Resources Research
Volume32
Issue number3
DOIs
StatePublished - Mar 1996

Fingerprint

porous media
Porous materials
porous medium
Advection
advection
perturbation
saturated flow
Solute transport
soil transport processes
heterogeneous medium
hydraulic head
Hydraulic conductivity
solute transport
hydraulic conductivity
fluid mechanics
statistics
Hydraulics
effect
Statistics
Sampling

ASJC Scopus subject areas

  • Aquatic Science
  • Environmental Science(all)
  • Environmental Chemistry
  • Water Science and Technology

Cite this

Higher-order effects on flow and transport in randomly heterogeneous porous media. / Hsu, Kuo Chin; Zhang, Dongxiao; Neuman, Shlomo P.

In: Water Resources Research, Vol. 32, No. 3, 03.1996, p. 571-582.

Research output: Contribution to journalArticle

@article{8bdf8355a7a54f28b80f362ce985d204,
title = "Higher-order effects on flow and transport in randomly heterogeneous porous media",
abstract = "A higher-order theory is presented for steady state, mean uniform saturated flow and nonreactive solute transport in a random, statistically homogeneous natural log hydraulic conductivity field Y. General integral expressions are derived for the spatial covariance of fluid velocity to second order in the variance σ2 of Y in two and three dimensions. Integrals involving first-order (in σ) fluctuations in hydraulic head are evaluated analytically for a statistically isotropic two-dimensional Y field with an exponential autocovariance. Integrals involving higher-order head fluctuations are evaluated numerically for this same field. Complete second- order results are presented graphically for σ2 = 1 and σ2 = 2. They show that terms involving higher-order head fluctuations are as important as those involving lower-order ones. The velocity variance is larger when approximated to second than to first order in σ2. Discrepancies between second- and first-order approximations of the velocity autocovariance diminish rapidly with separation distance and are very small beyond two integral scales. Transport requires approximation at two levels: the flow level at which velocity statistics are related to those of Y, and the advection level at which macrodispersivities are related to velocity fluctuations. Our results show that a second-order flow correction affects transport to a greater extent than does a second-order correction to advection. Asymptotically. the second-order transverse macrodispersivity tends to zero as does its first- order counterpart. An approximation of advection alone based on Corrsin's conjecture, coupled with either a first- or a second-order flow approximation, leads to a transverse macrodispersivity which is significantly larger than that obtained by standard perturbation and tends to a nonzero asymptote. Published Monte Carlo results yield macrodispersivities that lie significantly below those predicted by first- and second-order theories. Considering that Monte Carlo simulations often suffer from sampling and computational errors, that standard perturbation approximations are theoretically valid only for σ2 < 1, and that Corrsin's conjecture represents the leading term in a renormalization group perturbation which contains contributions from an infinite number of high-order terms, we find it difficult to tell which of these approximations is closest to representing transport in strongly heterogeneous media with σ2 ≤ 1.",
author = "Hsu, {Kuo Chin} and Dongxiao Zhang and Neuman, {Shlomo P}",
year = "1996",
month = "3",
doi = "10.1029/95WR03492",
language = "English (US)",
volume = "32",
pages = "571--582",
journal = "Water Resources Research",
issn = "0043-1397",
publisher = "American Geophysical Union",
number = "3",

}

TY - JOUR

T1 - Higher-order effects on flow and transport in randomly heterogeneous porous media

AU - Hsu, Kuo Chin

AU - Zhang, Dongxiao

AU - Neuman, Shlomo P

PY - 1996/3

Y1 - 1996/3

N2 - A higher-order theory is presented for steady state, mean uniform saturated flow and nonreactive solute transport in a random, statistically homogeneous natural log hydraulic conductivity field Y. General integral expressions are derived for the spatial covariance of fluid velocity to second order in the variance σ2 of Y in two and three dimensions. Integrals involving first-order (in σ) fluctuations in hydraulic head are evaluated analytically for a statistically isotropic two-dimensional Y field with an exponential autocovariance. Integrals involving higher-order head fluctuations are evaluated numerically for this same field. Complete second- order results are presented graphically for σ2 = 1 and σ2 = 2. They show that terms involving higher-order head fluctuations are as important as those involving lower-order ones. The velocity variance is larger when approximated to second than to first order in σ2. Discrepancies between second- and first-order approximations of the velocity autocovariance diminish rapidly with separation distance and are very small beyond two integral scales. Transport requires approximation at two levels: the flow level at which velocity statistics are related to those of Y, and the advection level at which macrodispersivities are related to velocity fluctuations. Our results show that a second-order flow correction affects transport to a greater extent than does a second-order correction to advection. Asymptotically. the second-order transverse macrodispersivity tends to zero as does its first- order counterpart. An approximation of advection alone based on Corrsin's conjecture, coupled with either a first- or a second-order flow approximation, leads to a transverse macrodispersivity which is significantly larger than that obtained by standard perturbation and tends to a nonzero asymptote. Published Monte Carlo results yield macrodispersivities that lie significantly below those predicted by first- and second-order theories. Considering that Monte Carlo simulations often suffer from sampling and computational errors, that standard perturbation approximations are theoretically valid only for σ2 < 1, and that Corrsin's conjecture represents the leading term in a renormalization group perturbation which contains contributions from an infinite number of high-order terms, we find it difficult to tell which of these approximations is closest to representing transport in strongly heterogeneous media with σ2 ≤ 1.

AB - A higher-order theory is presented for steady state, mean uniform saturated flow and nonreactive solute transport in a random, statistically homogeneous natural log hydraulic conductivity field Y. General integral expressions are derived for the spatial covariance of fluid velocity to second order in the variance σ2 of Y in two and three dimensions. Integrals involving first-order (in σ) fluctuations in hydraulic head are evaluated analytically for a statistically isotropic two-dimensional Y field with an exponential autocovariance. Integrals involving higher-order head fluctuations are evaluated numerically for this same field. Complete second- order results are presented graphically for σ2 = 1 and σ2 = 2. They show that terms involving higher-order head fluctuations are as important as those involving lower-order ones. The velocity variance is larger when approximated to second than to first order in σ2. Discrepancies between second- and first-order approximations of the velocity autocovariance diminish rapidly with separation distance and are very small beyond two integral scales. Transport requires approximation at two levels: the flow level at which velocity statistics are related to those of Y, and the advection level at which macrodispersivities are related to velocity fluctuations. Our results show that a second-order flow correction affects transport to a greater extent than does a second-order correction to advection. Asymptotically. the second-order transverse macrodispersivity tends to zero as does its first- order counterpart. An approximation of advection alone based on Corrsin's conjecture, coupled with either a first- or a second-order flow approximation, leads to a transverse macrodispersivity which is significantly larger than that obtained by standard perturbation and tends to a nonzero asymptote. Published Monte Carlo results yield macrodispersivities that lie significantly below those predicted by first- and second-order theories. Considering that Monte Carlo simulations often suffer from sampling and computational errors, that standard perturbation approximations are theoretically valid only for σ2 < 1, and that Corrsin's conjecture represents the leading term in a renormalization group perturbation which contains contributions from an infinite number of high-order terms, we find it difficult to tell which of these approximations is closest to representing transport in strongly heterogeneous media with σ2 ≤ 1.

UR - http://www.scopus.com/inward/record.url?scp=0029844130&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0029844130&partnerID=8YFLogxK

U2 - 10.1029/95WR03492

DO - 10.1029/95WR03492

M3 - Article

AN - SCOPUS:0029844130

VL - 32

SP - 571

EP - 582

JO - Water Resources Research

JF - Water Resources Research

SN - 0043-1397

IS - 3

ER -