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**Solute (and Suspension) Transport in Porous Media**

Patricia J Culligan Civil Engineering & Engineering Mechanics, Columbia University

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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

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**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

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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

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**Building the Balance Equation**

SECTION I Building the Balance Equation

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**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

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**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

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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

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**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

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**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

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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

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**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

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**Balance for E in a Volume U**

uE Control volume, U Element of control surface ∂S Flux of E across ∂S = euE.n

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E Div(flux) = excess of outflow over inflow

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

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

This can be re-written as

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**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

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

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**Balance for E per unit volume of continuous phase**

Advective Flux “Diffusive” Flux

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**Microscopic Processes**

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**Macroscopic Balance Equation**

Volume Averaging

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**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

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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

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**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

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**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

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**REV is centered at x at time t**

is associated with x Intrinsic phase average of e Deviation from average

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**General Macroscopic Balance Equation**

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**Macroscopic Processes**

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**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

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**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

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

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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

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**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):

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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

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**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

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**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

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

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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

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**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

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**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

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Tortuosity Dd* < Dd because the phenomenon of tortuosity decreases the gradient in concentration that is driving the diffusion Dd* = T Dd , where T < 1

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**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

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One-Dimensional Flow Dh/Dd versus Pe Dh = D + Dd* 0.4 10

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**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

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**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

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**For saturated medium, qs = (1-n)**

(no other sources)

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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

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Langmuir isotherm b) non-linear equilibrium isotherm

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**Sources/ Sinks Within Fluid Phase**

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**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

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**Saturated medium, conservative tracer**

Rigid, uniform medium Advection - Dispersion Equation

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

Rd

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**Influence of Various Processes**

Initial conditions Advection only Advection + Dispersion Advection , Dispersion, Sorption Advection , Dispersion, Sorption, Decay

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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

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**Some Challenges Working assumption Little understood Deforming medium**

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