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Modeling species transport by concentrated brine in aggregated porous media

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Abstract

Basic equations governing the transport of species by concentrated brine flowing through an aggregated porous medium are developed. Some simple examples are solved numerically. The medium is considered to be composed of porous rock aggregates separated by ‘macropores’ through which the brine flows and transport of salt and low-concentration species takes place. The aggregates contain dead-end pores, cracks, and stationary pockets collectively called ‘micropores’. The micropore space does not contribute to the flow, but it serves as a storage for salt and species. Adsorption of fluid species takes place at internal surface of aggregates where it is assumed that a linear equilibrium isotherm describes the process. The effects of high salt concentrations are accounted for in the brine density relation, the viscosity relation, Darcy's and Fick's laws, and the rate of mass transfer between macropores and micropores. Mass balance equations, supplemented by extended forms of Darcy's and Fick's laws, are employed to arrive at two sets of equations. One set consists of seven coupled equations for the salt mass fraction and fluid density in macropores, salt mass fraction in micropores, fluid velocity vector, and the fluid pressure. The other set consists of two coupled equations to be solved for the mass fractions of low-concentration species in micropores and macropores. Based on these equations, a mathematical model called TORISM is developed. Using this model, the potential significance of modifications to Darcy's Law are demonstrated.

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Abbreviations

a :

shape factor in Equation (30)

b :

shape factor in Equation (30)

da :

infinitesimal element of area

D f :

coefficient of density flow in the modified Darcy's law, [L2/T]

D i :

the diffusion-dispersion tensor for low-concentration species, [L2/T]

D s :

the diffusion-dispersion tensor for the salt component, [L2/T]

D ia :

macropore-micropore mass transfer rate coefficient for species, [T-1]

D is :

in the modified Fick's law, is the coefficient of low-concentration species transport due to the salt movement, [L2/T]

D sa :

macropore-microport mass transfer rate coefficient for salt, [T-1]

D supiinfm :

the coefficient of molecular diffusion of low-concentration species, [L2/T]

D supsinfm :

the coefficient of molecular diffusion of salt, [L2/T]

e g :

direction vector of the gravity

E f :

rate of net exchange of mass between micropores and macropores, [M/L3/T]

E i :

rate of exchange of mass of low-concentration species between micropores and macropores, [M/L3/T]

E s :

rate of exchange of mass of salt between micropores and macropores, [M/L3/T]

g :

magnitude of gravity vector, [L/T2]

g :

gravity vector

G :

dimensionless constant defined in Equations (39)

j s :

microscopic diffusive mass flux of salt, [M/L2/T]

ĵ s :

effective microscopic diffusive mass flux of salt, [M/L2/T]

J if :

macroscopic diffusive-dispersive mass flux of low-concentration species, [M/L2/T]

k :

permeability coefficient of the porous medium, [L2]

K id :

‘distribution coefficient’ for adsorption of species on soil grains, [L2/M]

l :

a resistance coefficient employed in Equation (26), [L]

L :

macroscopic characteristic length of the porous medium, [L]

n :

effective porosity of the porous medium (i.e., volume fraction of macropores)

n fp :

normal unit vector at a micropore-macropore interface pointing into the micropore

p :

(micropore or macropore) fluid pressure [M/L/T2]

p 0 :

a reference pressure

q :

Darcy velocity of macropore fluid, [L/T]

q r :

reference flow velocity defined in Equations (39), [L/T]

r :

characteristic size of micropores, [L]

R :

characteristic size of macropores, [L]

R f :

retardation factor defined in Equation (16)

S 0 :

specific surface of the porous medium, [L-1]

S i :

rate of adsorption of low-concentration species on soil grains, [M/L3/T]

t :

time

t 0 :

time at which species are introduced at the lower boundary of the domain, [T]

t 1 :

time at which species mass fraction at the lower boundary of the domain is set to zero again, [T]

t 1/2 :

half-life of radioactive species, [T]

t r :

reference time defined in Equation (39)

v :

microscopic velocity of the fluid, [L/T]

v i :

microscopic velocity of the low-concentration component of the fluid, [L/T]

v s :

microscopic velocity of the salt component of the fluid, [L/T]

v f :

macroscopic mean velocity of the fluid, [L/T]

v if :

macroscopic velocity of the low-concentration component of macropore fluid, [L/T]

v sf :

macroscopic velocity of the salt component of macropore fluid, [L/T]

w :

(microscopic) velocity of the fluid-aggregate interface, [L/T]

X 1 :

defined in Equation (42a)

X 2 :

defined in Equation (42b)

y 1 :

defined in Equation (42c)

Y 2 :

defined in Equation (42d)

αi :

dispersivity of low-concentration species, [L/T]

αs :

dispersivity of salt, [L/T]

β:

coefficient of compressibility of brine, [LT2/M]

γ:

coefficient of dependence of brine density on salt mass fraction

δA af :

total area of macropore-aggregate interfaces within an averaging volume

δA fg :

total area of macropore-grain boundaries within an averaging volume

δA fp :

total area of micropore-macropore interfaces within an averaging volume

δA pg :

total area of micropore-grain interfaces within an averaging volume

δV :

volume of the averaging volume

ε:

total porosity of the porous medium

λi :

rate of decay of species, [T-1]

μ:

dynamic viscosity of brine, [M/L/T]

μ0 :

dynamic viscosity of fresh water, [M/L/T]

ϱ:

microscopic mass density of the fluid, [M/L3]

ϱ0 :

mass density of fresh water, [M/L3]

ϱi :

microscopic concentration of the low-concentration species, [M/L3]

ϱf :

microscopic mass density of the macropore fluid, [M/L3]

ϱp :

macroscopic mass density of the micropore fluid, [M/L3]

ϱs :

microscopic concentration of salt, [M/L3]

ϱ supginf0 :

macroscopic mass density of soil grains, [M/L2]

ϱif :

macroscopic concentration of species in macropores, [M/L3]

ϱip :

macroscopic concentration of species in micropores, [M/L3]

ϱsf :

macroscopic concentration of salt in macropores, [M/L3]

ϱip :

macroscopic concentration of salt in micropores, [M/L3]

ωs :

microscopic mass fraction of salt, [M/M]

∼ωs :

deviation of microscopic mass fraction of salt from its average value in the macropores, [M/M]

\(\hat \omega ^s\) :

difference between average mass fraction of salt in macropores and micropores, [M/M]

ω supiinf0 :

a reference mass fraction for low-concentration species, [M/M]

ωif :

mass fraction of low-concentration species in macropores, [M/M]

ωig :

mass fraction of low-concentration species in soil grains, [M/M]

ωip :

mass fraction of low-concentration species in micropores, [M/M]

ωsf :

mass fraction of salt in macropores, [M/M]

ωsp :

mass fraction of salt in micropores, [M/M]

a :

aggregates

f :

macropore fluid

g :

soil grains

i :

low-concentration species

p :

(micro)pore fluid

s :

salt component

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Authors and Affiliations

  1. National Institute of Public Health and Environmental Protection (RIVM), PO Box 1, 3720, Bilthoven, BA, The Netherlands

    Hassanizadeh S. Majid

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Hassanizadeh, S.M. Modeling species transport by concentrated brine in aggregated porous media. Transp Porous Med 3, 299–318 (1988). https://doi.org/10.1007/BF00235333

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  • DOI: https://doi.org/10.1007/BF00235333

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