# Solute (and Suspension) Transport in Porous Media

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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 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 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

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

Building the Balance Equation
SECTION I Building the Balance Equation

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

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 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)
A sub-volume of a porous medium that has the “same” geometric configuration as the medium at a macroscopic scale Porosity, n Uvoids/U

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

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

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

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

E Div(flux) = excess of outflow over inflow

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

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

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

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

Microscopic Processes

Macroscopic Balance Equation
Volume Averaging

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

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

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

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

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

General Macroscopic Balance Equation

Macroscopic Processes

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

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

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 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

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 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

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

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 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
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

Tortuosity 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
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 Dh/Dd versus Pe Dh = D + Dd* 0.4 10

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

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

For saturated medium, qs = (1-n)
(no other sources)

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

Langmuir isotherm b) non-linear equilibrium isotherm

Sources/ Sinks Within Fluid Phase

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
Rigid, uniform medium Advection - Dispersion Equation

1-D Transport, Rigid Medium, Linear Equilibrium Sorption
Rd

Influence of Various Processes