1 Solute (and Suspension) Transport in Porous Media Patricia J Culligan Civil Engineering & Engineering Mechanics, Columbia University.

1

Solute (and Suspension) Transport in Porous Media

Patricia J CulliganCivil Engineering & Engineering Mechanics,

Columbia University

2

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

3

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

4

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

5

SECTION I

Building the Balance Equation

6

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

7

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

8

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

9

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

10

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

11

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

12

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

13

Balance for E in a Volume U

uE

Control volume, U

Element of control surface ∂S

Flux of E across ∂S = euE.

14

Div(flux) = excess of outflow over inflow

E

15

Term (a)

€

∂∂t

edUU

∫ =∂e

∂tdU

U

∫

Rate of accumulation of E within U

Amount of E in each dU

16

Term (b)

€

− eu E.ν dSS

∫

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

This can be re-written as

€

− ∇.euE dUU

∫

17

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

18

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

19

Balance for E per unit volume of continuous phase

€

∂e

∂t= −∇.(eu + JEu )+ ρΓE

Advective Flux

“Diffusive” Flux

E

20

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.

21

Macroscopic Balance Equation

Volume Averaging

E

22

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

23

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

24

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

25

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

26

€

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

27

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

28

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.

29

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

30

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

31

Section II

Development of a Working Mathematical Model for Solute

Transport at the Macroscopic Scale

32

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

33

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)

34

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

35

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

36

Continuous Source

c =1 c = 0

uf

c =1 c = 0

uf

t = t1

c

t = 0

Transition zone

Sharp front

c = 0.5

37

Point Source

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

38

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

39

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

40

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

41

Tortuosity

Dd* < Dd because the

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

Dd* = T Dd , where T < 1

42

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

43

One-Dimensional Flow

Dh/Dd versus Pe

€

Pe =u f d

Dd

0.4 10

Dh = D + Dd*

44

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

45

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

46

For saturated medium, s = (1-n)

€

fαβ =∂S

∂t=

∂(1− n)ρ sF

∂t(no other sources)

47

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

48

b) non-linear equilibrium isotherm

€

∂F

∂t= Kc α

Langmuir isotherm

€

F =K3c α

1+ K4c α

49

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.

50

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

51

Saturated medium, conservative tracer

€

∂(nc)

∂t= −∇.n(cu − D h∇c)

Rigid, uniform medium

€

∂c

∂t= −∇.(cu − Dh∇c)

Advection - Dispersion Equation

52

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

53

Influence of Various Processes

Initial conditions

Advection only

Advection + Dispersion

Advection , Dispersion, Sorption

Advection , Dispersion, Sorption, Decay

54

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

55

Some Challenges

€

∂(θα c)

∂t= −∇.θα (cu − D∇c − Dd

*∇c) −∂S

∂t+ other sources

Working assumption

Little understood

?Deforming medium