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

2. 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

3. Approach to Modeling Section I:
Build a microscopic balance equation for an Extensive Quantity in a single phase of a porous medium
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
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. Building the Balance Equation
SECTION I
Building the Balance Equation

6. Extensive Quantity, E A quantity that is additive over volume, U
e.g., Fluid Mass, m
m = 2000 kg
water
m = 1000 kg
U = 1 m3
U = 2 m3

7. 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

8. 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
qs solid
qf fluid
100% Solid

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, n
Uvoids/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 uE E

11. Total Flux of E, JtE
Total amount of E that passes through a unit area (A = 1) normal to uE per unit time (t = 1)
Unit normal area uE
If e = density of E (e = E/U), then amount of E that passes A

12. Advective & Diffusive Flux of E
If the phase carrying E has a velocity u then
Flux of E relative to the advective flux -
Diffusive flux
JEu
euE eu

13. Balance for E in a Volume UuE
Control volume, U
Element of control surface ∂S 
Flux of E across ∂S
= euE.n

14. E
Div(flux) = excess of outflow over inflow

15. Term (a)
Rate of accumulation of E within U
Amount of E in each dU

16. Term (b) Net Influx of E into U through S (influx - outflux)
This can be re-written as

17. Term (c) Net production of E within U
Where r is the mass density of the phase and GE is the rate of production of E per unit mass of the continuous phase

18. Balance Equation
Shrink U to zero - balance equation for E at a point in a phase
Fluid mass: e = r

19. Balance for E per unit volume of continuous phase
Advective Flux
“Diffusive” Flux

20. Microscopic Processes

21. Macroscopic Balance Equation
Volume Averaging

22. Continuous Phase = a Phase
REV, volume Uo
 phase
 
u
 phase
Use volume averaging to covert balance equation for E in the a phase to a balance equation for E in REV

23. Consider uaE
REV, Uo
At the micro-scale, quantities within Uo are heterogeneous A
(uaE)A ≠ (uaE)B
Idea of volume averaging is to define an average value for uaE that represents this quantity for the REV B
a-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 a phase is
This is the total amount of E in the a phase averaged over the volume Uoa of the a phase

25. If a phase is a fluid phase and E = fluid mass m, e = density of the fluid mass in the a phase, ra
= average density of the fluid in the fluid phase of the REV

26. REV is centered at x at time t
is associated with x
Intrinsic phase average of e
Deviation from average

27. General Macroscopic Balance Equation

28. Macroscopic Processes

29. Mass Balance for a phase
Ea = ma, ea = ra and no internal or external sources or sinks for mass within the REV
Normal to assume that the advective flux dominates
Solution of the mass balance equation provides

30. Mass Balance for a g Component in the a phase
Ea = mga the mass of solute in the a phase and ea = rag = c where c is the concentration of the solute (or suspension)
- Divergence of Fluxes
Sinks at ab phase boundary
Sources in a phase

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 Solute
The rate at which solute mass is advected into a unit volume of porous medium is given by
For a saturated medium qa = n, the porosity of the medium. If n does not change with time (rigid medium):

34. Steady-State uf
Advective transport describes the average distance traveled by the solute mass in the porous medium uf L
c = 1
Solute mass transported an average distance L = uft by advection at constant uf
c = 0
t = 0
t = L/uf

35. Phenomenon of Dispersion
The dispersive flux of solute mass is represented by
Examine the behavior of a tracer (conservative solute) during transport at a steady-state velocity

36. Continuous Source c =1 c = 0 c =1 c = 0 uf uf c t = 0 t = t1 c = 0.5
Sharp front
Transition zone c
c = 0.5
t = 0
t = t1

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 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
is described by Fick’s Law
Dd* = effective diffusion coefficient
Diffusion transports solute mass from regions of high c to regions of lower c

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 Dispersion
Both macroscopic dispersive and diffusive fluxes are assumed to be proportional to
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
Dh = D + Dd*
0.4 10

44. Sources and Sinks - at Solid Phase Boundary
Solute particle reaches solid surface and possibly adheres to it 
u
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
Other sources/ sinks
Transfer across ab surface

46. For saturated medium, qs = (1-n)
(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
Linear Equilibrium isotherm

48. Langmuir isotherm
b) non-linear equilibrium isotherm

49. Sources/ Sinks Within Fluid Phase

50. Mass Balance Equation for a Single Component
-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

51. Saturated medium, conservative tracer
Rigid, uniform medium
Advection - Dispersion Equation

52. 1-D Transport, Rigid Medium, Linear Equilibrium SorptionRd

53. Influence of Various Processes
Initial conditions
Advection only
Advection + Dispersion
Advection , Dispersion, Sorption
Advection , Dispersion,
Sorption, Decay

54. 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
Sorption on solid phase

55. Some Challenges Working assumption Little understood Deforming medium?