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Solute (and Suspension) Transport in Porous Media

Patricia J CulliganCivil Engineering & Engineering Mechanics,

Columbia University

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Broad Definitions

A solute is a substance that is dissolved in a liquide.g., Sodium Chloride (NaCl) dissolved in water

A suspension is a mixture in which fine particles are suspended in a fluid where they are supported by buoyancy

e.g., Sub-micron sized organic matter in water

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Approach to Modeling

Section I: a) Build a microscopic balance equation for an Extensive

Quantity in a single phase of a porous mediumb) Use volume averaging techniques to “up-scale” the

microscopic balance equation to a macroscopic level - described by a representative elementary volume of the porous medium

c) Examine balance equations for a two extensive quantities: a) fluid mass; b) solute mass

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Section II:• Examine examine each specific term in the macroscopic

balance equation for solute mass• Consider a few simplified versions of the solute mass

balance equation

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SECTION I

Building the Balance Equation

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Extensive Quantity, E

A quantity that is additive over volume, U

e.g., Fluid Mass, m

m = 1000 kgm = 2000 kg

U = 1 m3 U = 2 m3

water

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Porous MediumA material that contains a void space and a solid phase

The void space can contain several fluid phases:

Gas phase - airAqueous liquid - waterNon-aqueous liquid - oil

A porous medium is a multi-phase material

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Continuum ApproachAt the micro-scale, a porous medium is heterogeneous

At any single point, 100% of one phase (e.g., solid phase) and 0% of all other phases (e.g., fluid phases)

Continuum approach assumes that all phases are continuous within a REV of the porous media

100% Solid s solidf fluid

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Representative Elementary Volume (REV)

A sub-volume of a porous medium that has the “same” geometric configuration as the medium at a macroscopic scale

Porosity, nUvoids/U

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Microscopic Balance Equation

Consider the balance of E within a volume U of a continuous phase

[visualize the balance of mass in a volume U of water]

Velocity of E = uE

E uE

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Total Flux of E, JtE

Unit normal area

uE Total amount of E that passes through a unit area (A = 1) normal to uE per unit time (t = 1)

If e = density of E (e = E/U), then amount of E that passes A

€

=e uE t( ) A

€

JtE = euE

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Advective & Diffusive Flux of E

If the phase carrying E has a velocity u then

€

JtE = euE = eu + JEu

JEu

euE

eu

Flux of E relative to the advective flux -

Diffusive flux

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Balance for E in a Volume U

uE

Control volume, U

Element of control surface ∂S

Flux of E across ∂S = euE.

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Div(flux) = excess of outflow over inflow

E

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Term (a)

€

∂∂t

edUU

∫ =∂e

∂tdU

U

∫

Rate of accumulation of E within U

Amount of E in each dU

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Term (b)

€

− eu E.ν dSS

∫

Net Influx of E into U through S (influx - outflux)

This can be re-written as

€

− ∇.euE dUU

∫

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Term (c)

Net production of E within U

€

ρΓE

U

∫ dU

Where ρ is the mass density of the phase and ΓE is the rate

of production of E per unit mass of the continuous phase

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Balance Equation

€

(∂e

∂t+∇.eu E − ρΓE )dU

U

∫ = 0

Shrink U to zero - balance equation for E at a point in a phase

€

∂e

∂t= −∇.(eu + JEu )+ ρΓE

€

∂ρ∂t

= −∇.(ρu)Fluid mass: e = ρ

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Balance for E per unit volume of continuous phase

€

∂e

∂t= −∇.(eu + JEu )+ ρΓE

Advective Flux

“Diffusive” Flux

E

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Microscopic Processes

We have thus defined three basic mechanisms which allow the density, e, of an extensivequantity E to change at a point in a continuous phase within a porous medium, namely:

1 Advection with the average velocity u of the continuous phase,2 Diffusion, and3 Production (or decay) within the continuous phase.

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Macroscopic Balance Equation

Volume Averaging

E

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Continuous Phase = Phase

REV, volume Uo

phase

phase

u

Use volume averaging to covert balance equation for E in the phase to a balance equation for E in REV

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Consider uE

A

B

(uE)A ≠ (u

E)B

At the micro-scale, quantities within Uo are heterogeneous

Idea of volume averaging is to define an average value for u

E that represents this quantity for the REV

REV, Uo

-phase

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Intrinsic Phase Average

We will use intrinsic phase averages in our balance equation for E in the REV

The intrinsic phase average of e in the phase is

€

e α

This is the total amount of E in the phase averaged over the volume Uo of the phase

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If phase is a fluid phase and E = fluid mass m, e = density of the fluid mass in the phase, ρ

€

e α = average density of the fluid in the fluid phase of the REV

€

mass

Uoα

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€

e(x', t : x) = eα(x, t) + ˆ e α (x', t : x)

REV is centered at x at time t

€

e α is associated with x

Intrinsic phase average of e Deviation from average

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General Macroscopic Balance Equation

€

∂(θα eα

α)

∂t= −∇.θα (eα

αuα

α+ ˆ e α ˆ u α

α+ JEα uα

α

)+θα ρα ΓEαα

−1

Uo

(eαSαβ

∫ (uα − uαβ )+ JEα uα ).νdS

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Macroscopic Processes

We have now identified five terms that can contribute toward a change in themacroscopic density of a component within the phase of an REV:

1 Advection with the average (macroscopic) velocity of the α phase;2 Dispersion relative to the average advective flux;3 Diffusion at the macroscopic level;4 Production (or decay) within the phase itself, and5 Macroscopic sources (or sinks) at the phase boundaries.

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Mass Balance for phaseE = m, e = ρ and no internal or external sources or sinks for mass within the REV

€

∂(θα ρα

α)

∂t= −∇.θα (ρα

αuα

α+ ˆ ρ α ˆ u α

α+ Jmα uα

α

)

Normal to assume that the advective flux dominates

€

∂(θα ρα

α)

∂t= −∇.θα (ρα

αuα

α)

Solution of the mass balance equation provides

€

uα

α

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Mass Balance for a Component in the phase

E = m the mass of solute in the phase and e = ρ = c where c is the concentration of the solute (or suspension)

€

∂(θα cα

)

∂t= −∇.θα (c

αuα

α+ ˆ c ̂ u α

α+ Jα

dγα

)

+θα ρα Γmαγα

−1

Uo

(cSαβ

∫ (uα − uαβ )+ Jαdγ ).νdS

- Divergence of Fluxes

Sources in phase Sinks at phase boundary

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Section II

Development of a Working Mathematical Model for Solute

Transport at the Macroscopic Scale

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Approach

Examine each of the terms that can contribute to a change in the average concentration of a solute c, within the fluid phase of an REV

•Advective Transport•Dispersion•Diffusion•Sources and Sinks within the REV

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Advective Transport of a SoluteThe rate at which solute mass is advected into a unit volume of porous medium is given by

€

−∇.θα (c α u αα )

For a saturated medium = n, the porosity of the medium. If n does not change with time (rigid medium):

€

uα

α= u f = −

k

nμ f

(∇Pf − ρ f g)

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Steady-State uf

Advective transport describes the average distance traveled by the solute mass in the porous medium

uf

uf

LSolute mass transported an average distance L = uft by advection at constant uf

t = 0 t = L/uf

c = 1

c = 0

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Phenomenon of Dispersion

The dispersive flux of solute mass is represented by

€

( ˆ c ̂ u αα

)

Examine the behavior of a tracer (conservative solute) during transport at a steady-state velocity

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Continuous Source

c =1 c = 0

uf

c =1 c = 0

uf

t = t1

c

t = 0

Transition zone

Sharp front

c = 0.5

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Point Source

Observe spreading of solute mass in direction of flow and perpendicular to the direction of flow - hydrodynamic dispersion

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Reasons for Spreading

Microscopic heterogeneity in fluid velocity and chemical gradients

Some solute mass travels faster than average, while some solute mass travels slower than average

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Modeling Dispersion

€

ˆ c ̂ u α

= −D.∇c α

It is a working assumption that

Where D is a dispersion coefficient (dim L2/T).

For uniform porous media, D is usually assumed to be a product of a length (dispersivity) that characterizes the pore scale heterogeneity and fluid velocity

For one-dimensional flow D = aL ux

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Macroscopic Diffusion

The solute flux due to average macroscopic diffusion

€

Jαdγ

α

is described by Fick’s Law

€

Jαdγ

α

= −Dd* .∇c α

Diffusion transports solute mass from regions of high c to regions of lower c

Dd* = effective diffusion

coefficient

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Tortuosity

Dd* < Dd because the

phenomenon of tortuosity decreases the gradient in concentration that is driving the diffusion

Dd* = T Dd , where T < 1

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Hydrodynamic DispersionBoth macroscopic dispersive and diffusive fluxes are assumed to be proportional to

€

∇c α

€

D h = D + D d*

Hence, their effects are combined by joining the two dispersion/ diffusion coefficients is a single Hydrodynamic Dispersion Coefficient

The Behavior of Dh as a function of fluid velocity, u has been the subject of study for decades

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One-Dimensional Flow

Dh/Dd versus Pe

€

Pe =u f d

Dd

0.4 10

Dh = D + Dd*

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Sources and Sinks - at Solid Phase Boundary

u

Solute particle reaches solid surface and possibly adheres to it

€

fαβ =1

Uo

(cSαβ

∫ (uα − uαβ ) + Jαdγ ).νdS

Average rate of accumulation of solute mass on solid surface, S, per unit volume of porous medium as a result of flux from fluid phase

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Macroscopic Equation for ∂S/∂t

Define F: average mass of solute on solid phase per unit mass of solid phase

€

S = Fms

Us

Us

Uo

= Fρ sθ s

€

∂(θ sρ sF)

∂t= fαβ + other sources

Transfer across surface Other sources/ sinks

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For saturated medium, s = (1-n)

€

fαβ =∂S

∂t=

∂(1− n)ρ sF

∂t(no other sources)

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Defining F or ∂F/∂t

F or ∂F/∂t are usually linked to c, the solute concentration in the fluid phase, via sorption isotherms

a) Equilibrium isotherms

€

F = Kd c αLinear Equilibrium isotherm

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b) non-linear equilibrium isotherm

€

∂F

∂t= Kc α

Langmuir isotherm

€

F =K3c α

1+ K4c α

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Sources/ Sinks Within Fluid Phase

May be due to any of the processes listed below:

1 The actual injection or withdrawal of the phaseitself.2 Radioacti ve dec .ay3 Biodegradationorgrowt h duet o bactertial activities.Chemicalreactionoft hesolut e withanother(possible)componentof t hephase.

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Mass Balance Equation for a Single Component

€

∂(θα c)

∂t= −∇.θα (cu − D.∇c − Dd

* .∇c) −∂(θsρ sF)

∂t+ Qα c i −θα kα

γ c

Rate of increase of solute mass per unit volume of pm

-div (Fluxes)

Solute mass transfer to solid phase

Sources/ sinks for solute mass in fluid phase

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Saturated medium, conservative tracer

€

∂(nc)

∂t= −∇.n(cu − D h∇c)

Rigid, uniform medium

€

∂c

∂t= −∇.(cu − Dh∇c)

Advection - Dispersion Equation

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1-D Transport, Rigid Medium, Linear Equilibrium Sorption

€

∂c

∂t= D h

∂ 2c

∂x2− u

∂c

∂x−

(1− n)ρ sKd

n

∂c

∂t

Rd

€

Rd = 1+(1− n)ρ sKd

n

€

Rd

∂c

∂t= D h

∂ 2c

∂x2− u

∂c

∂x

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Influence of Various Processes

Initial conditions

Advection only

Advection + Dispersion

Advection , Dispersion, Sorption

Advection , Dispersion, Sorption, Decay

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SummaryMicroscale change in solute concentration at a point in a fluid is due to:

Advection at fluid velocityDiffusionProduction/ Decay within fluid phase

Macroscale change in average solute concentration within the fluid phase of the REV is due to:

Advection at average fluid velocityDispersionDiffusionProduction/ Decay within fluid phaseSorption on solid phase

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Some Challenges

€

∂(θα c)

∂t= −∇.θα (cu − D∇c − Dd

*∇c) −∂S

∂t+ other sources

Working assumption

Little understood

?Deforming medium