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Solute (and Suspension) Transport in Porous Media. Patricia J Culligan Civil Engineering & Engineering Mechanics, Columbia University. Broad Definitions. A solute is a substance that is dissolved in a liquid e.g., Sodium Chloride (NaCl) dissolved in water

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Solute and suspension transport in porous media

Solute (and Suspension) Transport in Porous Media

Patricia J Culligan

Civil Engineering & Engineering Mechanics, Columbia University

Broad definitions
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

Approach to modeling
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

Solute and suspension transport in porous media

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

Section i


Building the Balance Equation

Extensive quantity e
Extensive Quantity, E

A quantity that is additive over volume, U

e.g., Fluid Mass, m

m = 2000 kg


m = 1000 kg

U = 1 m3

U = 2 m3

Porous medium
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

Continuum approach
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

Representative elementary volume rev
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


Microscopic balance equation
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



Total flux of e j te


Unit normal area

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)

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

Advective diffusive flux of e




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

Balance for e in a volume u


Element of control surface ∂S

Control volume, U

Balance for E in a Volume U

Flux of E across ∂S

= euE.n

Solute and suspension transport in porous media


Div(flux) = excess of outflow over inflow

Term a
Term (a)

Rate of accumulation of E within U

Amount of E in each dU

Term b
Term (b)

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

This can be re-written as

Term c
Term (c)

Net production of E within U

Where r is the mass density of the phase and GEis the rate of production of E per unit mass of the continuous phase

Balance equation
Balance Equation

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

Fluid mass: e = r

Balance for e per unit volume of continuous phase


Balance for E per unit volume of continuous phase

Advective Flux

“Diffusive” Flux

Macroscopic balance equation


Macroscopic Balance Equation

Volume Averaging

Continuous phase a phase
Continuous Phase = a Phase

REV, volume Uo

 phase



 phase

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

Consider u a e
Consider uaE


At the micro-scale, quantities within Uo are heterogeneous


(uaE)A ≠ (uaE)B

Idea of volume averaging is to define an average value for uaE that represents this quantity for the REV



Intrinsic phase average
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

Solute and suspension transport in porous media

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

Solute and suspension transport in porous media

REV is centered at x at time t

is associated with x

Intrinsic phase average of e

Deviation from average

Mass balance for a phase
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

Mass balance for a g component in the a phase
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

Section ii

Section II

Development of a Working Mathematical Model for Solute Transport at the Macroscopic Scale


  • 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

Advective transport of a solute
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):

Steady state u f




Steady-State uf

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

c = 1

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

c = 0

t = 0

t = L/uf

Phenomenon of dispersion
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

Continuous source
Continuous Source

c =1

c =1

c = 0

c = 0



Sharp front

Transition zone


c = 0.5

t = 0

t = t1

Point source
Point Source

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

Reasons for spreading
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

Modeling dispersion
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

Macroscopic diffusion
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


Dd* < Dd because the phenomenon of tortuosity decreases the gradient in concentration that is driving the diffusion

Dd* = T Dd , where T < 1

Hydrodynamic dispersion
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

One dimensional flow
One-Dimensional Flow

Dh/Dd versus Pe

Dh = D + Dd*



Sources and sinks at solid phase boundary


Sources and Sinks - at Solid Phase Boundary

Solute particle reaches solid surface and possibly adheres to it

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

Macroscopic equation for s t
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

Solute and suspension transport in porous media

For saturated medium, qs = (1-n)

(no other sources)

Defining f or f t
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

Solute and suspension transport in porous media

b) non-linear equilibrium isotherm

Langmuir isotherm

Mass balance equation for a single component
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

Saturated medium conservative tracer
Saturated medium, conservative tracer

Rigid, uniform medium

Advection - DispersionEquation

Influence of various processes

Initial conditions

Advection only

Advection + Dispersion

Advection , Dispersion, Sorption

Advection , Dispersion,

Sorption, Decay

Influence of Various Processes


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

Advection at fluid velocity


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



Production/ Decay within fluid phase

Sorption on solid phase

Some challenges
Some Challenges

Working assumption

Little understood

Deforming medium