REVIEW

• What processes are represented in the governing equation that we use to represent solute transport through porous media?

Advection, dispersion, chemical reactions

• How is advection quantified?

By the average linear velocity (v) or specific discharge (q)calculated from Darcy’s law.

v = q/ = -KI/

• How is the dispersion process quantified?

Using Fick’s law of diffusion.

• How do we express Fick’s law of diffusion?

s

ccADF dDiff

12 where Dd is the effective

diffusion coefficient.

s

ccDfD

12For dispersion: Where D is the

dispersion coefficient

Which of these terms represents advection?Which represents dispersion?Which represents chemical reactions?

t

cRRc

cq

x

cv

x

cD

ss

2

2

• According to theory (i.e., assuming Fickian dispersion) andassuming uniform flow (v = a constant) and an instantaneoussource, the concentration profile isalways Gaussian. (True or False?)

• According to theory, in a uniform flow field, the breakthrough curve is always Gaussian. (True or False?)

• In the general case, how many components are in thedispersion coefficient tensor? nine

• Under what conditions does the dispersion coefficienttensor reduce to one component?

1D flow with a line source

Longitudinal dispersion

• Under conditions of 1D flow, how do we quantify thelongitudinal dispersion coefficient?

DL = L v + D*where L is dispersivityD* is the diffusion coefficient

• What are the units of dispersivity and physically what isit supposed to represent?

It is a “mixing length” that represents the deviations from the average linear velocity caused by mixing withinpore spaces (microdispersion) and mixing owing to thepresence of heterogeneities (macrodispersion).

Units are in length.

• What process is represented by the retardation factor, R?

sorption

• Give two equations used to quantify R underlinear sorption.

R = v/vc d

bKR

1

where Kd = c/c

• The classic example of a 1st order rate reactionis radioactive decay. Biodegradation of some organiccompounds can also be represented as 1st order reactions.

What is the relation between the 1st order rate constant ()and half-life?

• Describe two types of initial conditions.

c (x,y,z,0) = 0

c (x,y,z,0) = co(x,y,z)

specified concentration

no mass flux

Specified mass flux

• Name two general categories of boundary conditions.

specified concentrationspecified mass flux

• Which of these is a “free mass outflow” boundarycondition?

specified concentration

no mass flux

Specified mass flux

• Name the two types of numerical errors typicallyencountered when using conventional finite differencemethods to solve the advection-dispersion equation.

Numerical dispersion

Artificial oscillation/overshoot

• What is the Courant number and how is it used in numerical models that simulate transport?

x

tvCr

It is used to control the time step as the Courantnumber is usually less than or equal to one. Bycontrolling the time step, numerical errors are minimized.

• What is the Peclet number and how is it used in numerical models that simulate transport?

x

D

xvPe

Used to control numerical dispersion & oscillation

What is meant by an explicit finite difference approximation?

The space derivatives are evaluated at the known time level;the FD equation contains only one unknown–the concentration at the next (unknown) time level.

In an implicit approximation, the space derivativesare evaluated at the unknown time level.

• The central difference approximation for the advective term in the ADE causes instability in:

a) explicit approximationsb) implicit approximationsc) both explicit and implicit approximations

Multiple Choice. Select the correct answer.

• In upstream weighting, the space derivative in theadvective term in the ADE is formulated by using node cj and

a) node cj at the next time levelb) the immediately adjacent node in the upgradient directionc) the immediately adjacent nodes in both the upstream anddownstream directions

• Compared to central differences, the use of upstreamweighting for the advection term in the ADE causes more

a) numerical error in approximating the space derivativeb) artificial oscillationc) numerical dispersiond) a and be) a and cf) a, b, and c