Solute (and Suspension)
Transport in Porous Media
Patricia J Culligan
Civil Engineering & Engineering Mechanics,
Columbia University
1
Broad Definitions
A solute is a substance that is dissolved in a liquid
e.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
2
Approach to Modeling
Section I:
a) Build a microscopic balance equation for an Extensive
Quantity in a single phase of a porous medium
b) 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
3
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
4
SECTION I
Building the Balance Equation
5
Extensive Quantity, E
A quantity that is additive over volume, U
e.g., Fluid Mass, m
water
m = 2000 kg
m = 1000 kg
U = 1 m3
U = 2 m3
6
Porous Medium
A material that contains a void space and a solid phase
The void space can contain several
fluid phases:
Gas phase - air
Aqueous liquid - water
Non-aqueous liquid - oil
A porous medium is a multi-phase material
7
Continuum Approach
At 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
qs solid
qf fluid
8
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, n
Uvoids/U
9
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
10
Total Flux of E,
uE
Unit normal
area
tE
J
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
J  eu
tE

E
11
Advective & Diffusive Flux of E
If the phase carrying E has a velocity u then
J tE  euE  eu  J Eu
euE

JEu
eu
Flux of E
relative to the
advective flux -
Diffusive flux
12
Balance for E in a Volume U
Element
of control
surface ∂S

uE
Flux of E across ∂S
= euE.
Control
volume, U
13
E
Div(flux) = excess of outflow over inflow
14
Term (a)
Rate of accumulation of E within U

edU 

t U
e
 t dU
U
Amount of E in each dU
15

Term (b)
Net Influx of E into U through S (influx - outflux)
  eu . dS
E
S
This can be re-written as
  .euE dU
U
16
Term (c)
Net production of E within U
  dU
E
U
Where  is the mass density of the phase and E is the rate of
production of E per unit mass of the continuous phase

17
Balance Equation
e
E
E
 ( t  .eu   )dU  0
U
Shrink U to zero - balance equation for E at a point in a phase

e
Eu
E
 .(eu  J ) 
t
Fluid mass: e = 


 .(u)
t
18
Balance for E per unit volume of
continuous phase
E
e
Eu
E
 .(eu  J ) 
t

Advective
Flux
“Diffusive”
Flux
19
Microscopic Processes
We have thus defined three basic mechan is ms whic h a llow the den sit y, e, of an extens ive
quan tit y E to change at a point in a continuou s phase within a porous medium, na me ly:
1 Advection with the average v elocity u of the con tinuou s pha se,
2 Diffusion, and
3 Production (or decay) within the continuous phas e.
20
Macroscopic Balance Equation
E
Volume Averaging
21
Continuous Phase = a Phase

phase
REV, volume
Uo

a
a
phase
ua
Use volume averaging to covert balance equation for E
in the a phase to a balance equation for E in REV
22
Consider ua
REV, Uo
E
At the micro-scale, quantities
within Uo are heterogeneous
(uaE)A ≠ (uaE)B
A
B
a-phase
Idea of volume averaging is
to define an average value
for uaE that represents this
quantity for the REV
23
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 a phase is
e
a
This is the total amount of E in the a phase averaged
over the volume Uoa of the a phase

24
If a phase is a fluid phase and E = fluid mass m,
e = density of the fluid mass in the a phase, a
e
a = average density of the fluid in the fluid
phase of the REV
mass
U oa
25
REV is centered at
x at time t
e
a
is associated with x
a
a
ˆ
e(x',t : x)  e (x,t)  e (x',t : x)

Intrinsic phase average of e

Deviation from average
26
General Macroscopic Balance
Equation
a
a
a
a
a
a
 (qa ea )
Ea ua
Ea
 .qa (ea ua  eˆa uˆ a  J
)qa a 
t
1
Ea ua

(e
(u

u
)
J
).dS

a
a
a
Uo S a
27
Macroscopic Processes
We have now identifi ed five terms that can contribute toward a change in the
macroscopic densit y o f a componen t within the a phase of an R EV:
1
2
3
4
5
Advection with the average (macroscopic) velocit y o f the a phase ;
Dispersion relative to the ave rage advec tive flux ;
Diffusion at the macroscopic leve l;
Production (or decay) wit hin the pha se it self, and
Macroscopic sources (or sinks) at the pha se bounda ries.
28
Mass Balance for a phase
Ea = ma, ea = a and no internal or external sources or sinks
for mass within the REV
a
a
a
a
a
(qa a )
ma ua
ˆ a uˆ a  J
 .qa ( a ua  
)
t
Normal to assume that the advective flux dominates

a
a
a
 (qa a )
 .qa ( a ua )
t
Solution of the mass balance equation provides ua

a
29
Mass Balance for a g Component
in the a phase
Ea = mga the mass of solute in the a phase and ea = ag = c
where c is the concentration of the solute (or suspension)
a
a
a
a
a
 (qa c )
dg
 .qa (c ua  cˆuˆ a  Ja )
t
 qa a 
m ag
a
Sources in a phase
- Divergence of Fluxes
1
dg

(c
(u

u
)
J

a
a
a ).dS
Uo S a
Sinks at a phase boundary
30
Section II
Development of a Working
Mathematical Model for Solute
Transport at the Macroscopic Scale
31
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
32
Advective Transport of a Solute
The rate at which solute mass is advected into a unit
volume of porous medium is given by
a
a
.qa (c ua )
For a saturated medium qa = n, the porosity of the medium.
If n does not change with time (rigid medium):

ua
a
k
 uf  
(Pf  f g)
n f
33
Steady-State uf
Advective transport describes the average distance
traveled by the solute mass in the porous medium
c=1
L
uf
c=0
t=0
uf
Solute mass
transported an
average
distance L = uft
by advection at
constant uf
t = L/uf
34
Phenomenon of Dispersion
The dispersive flux of solute mass is represented by
a
( cˆuˆ a )
Examine the behavior of a tracer (conservative solute) during
transport at a steady-state velocity

35
Continuous Source
c =1
c=0
uf
c =1
c=0
uf
Sharp front
Transition
zone
c
c = 0.5
t=0
t = t1
36
Point Source
Observe spreading of solute mass in direction of flow and
perpendicular to the direction of flow - hydrodynamic dispersion
37
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
38
Modeling Dispersion
It is a working assumption that
a
cˆuˆ  D.c a
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
39
Macroscopic Diffusion
dg
The solute flux due to average macroscopic diffusion Ja
a
is described by Fick’s Law
dg
Ja

a
 D .c
*
d
a
Dd* = effective diffusion

coefficient
Diffusion transports solute
mass from regions of high c to
regions of lower c
40
Tortuosity
Dd* < Dd because the
phenomenon of
tortuosity decreases
the gradient in
concentration that is
driving the diffusion
Dd* = T Dd , where T < 1
41
Hydrodynamic Dispersion
Both macroscopic dispersive and diffusive fluxes are assumed
to be proportional to c a
Hence, their effects are combined by joining the two dispersion/
diffusion coefficients is a single Hydrodynamic Dispersion
Coefficient
D h  D  D *d

The Behavior of Dh as a function of fluid velocity, u has
been the subject of study for decades
42
One-Dimensional Flow
Dh/Dd versus Pe
Pe 

0.4
10
uf d
Dd
Dh = D + Dd*
43
Sources and Sinks - at Solid
Phase Boundary
Solute particle reaches
solid surface and possibly
adheres to it

ua
fa
1
dg

(c
(u

u
)

J

a
a
a ).dS
Uo Sa
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
44
Macroscopic Equation for ∂S/∂t
Define F: average mass of solute on solid phase per unit mass
of solid phase
ms U s
SF
 Fsqs
Us Uo


 (q ssF)
 fa  other sources
t
Transfer across a surface
Other sources/ sinks
45
For saturated medium, qs = (1-n)
fa
S (1 n)sF


t
t
(no other sources)

46
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  K d ca
Linear
Equilibrium
isotherm

47

Langmuir isotherm
a
K 3c
F
1 K 4 c a
b) non-linear equilibrium isotherm
F
a
 Kc
t
48
Sources/ Sinks Within Fluid
Phase
May be due to any of the processes list ed below:
1 The actual injection or withdra wal of the a phase itself.
2 Radio active decay.
3 Biodegradation or growth due to bactertial activities.
Chemical reaction of the solute with another (possible) component of the a phase .
49
Mass Balance Equation for a
Single Component
(qa c)
(qssF)
*
g
 .qa (cu  D.c  Dd .c) 
 Qa c i  qa ka c
t
t
-div (Fluxes)
Rate of increase of
solute mass per
unit volume of pm
Solute mass
transfer to
solid phase
Sources/ sinks
for solute mass
in fluid phase
50
Saturated medium, conservative
tracer
 (nc)
 .n(cu  D hc)
t
Rigid, uniform medium

c
 .(cu  Dh c)
t
Advection - Dispersion Equation
51

1-D Transport, Rigid Medium,
Linear Equilibrium Sorption
c
 c
c (1 n) sK d c
 Dh 2  u 
t
x
x
n
t
2
(1 n)sK d
Rd  1
n
Rd

c
 2c
c
Rd  D h 2  u
t
x
x
52
Influence of Various Processes
Initial conditions
Advection only
Advection + Dispersion
Advection , Dispersion, Sorption
Advection , Dispersion,
Sorption, Decay
53
Summary
Microscale change in solute concentration at a point in a
fluid is due to:
Advection at fluid velocity
Diffusion
Production/ Decay within fluid phase
Macroscale change in average solute concentration within
the fluid phase of the REV is due to:
Advection at average fluid velocity
Dispersion
Diffusion
Production/ Decay within fluid phase
Sorption on solid phase
54
Some Challenges
(qa c)
S
*
 .qa (cu  Dc  Dd c)   other sources
t
t
Working assumption
Little understood
Deforming medium
?
55