1

Solutions to the Solutions to the Advection-Dispersion Advection-Dispersion

EquationEquation

2

Road Map to SolutionsRoad Map to Solutions

We will discuss the following solutionsWe will discuss the following solutionsInstantaneous injection in infinite and semi-infinite Instantaneous injection in infinite and semi-infinite

1-dimensional columns1-dimensional columnsContinuous injection into semi-infinite 1-D columnContinuous injection into semi-infinite 1-D columnInstantaneous point source solution in two-Instantaneous point source solution in two-

dimensions (line source in 3-D)dimensions (line source in 3-D)Instantaneous point source in 3-dimensionsInstantaneous point source in 3-dimensions

Keep an eye on:Keep an eye on:the initial assumptionsthe initial assumptionssymmetry in space, asymmetry in timesymmetry in space, asymmetry in time

We will discuss the following solutionsWe will discuss the following solutionsInstantaneous injection in infinite and semi-infinite Instantaneous injection in infinite and semi-infinite

1-dimensional columns1-dimensional columnsContinuous injection into semi-infinite 1-D columnContinuous injection into semi-infinite 1-D columnInstantaneous point source solution in two-Instantaneous point source solution in two-

dimensions (line source in 3-D)dimensions (line source in 3-D)Instantaneous point source in 3-dimensionsInstantaneous point source in 3-dimensions

Keep an eye on:Keep an eye on:the initial assumptionsthe initial assumptionssymmetry in space, asymmetry in timesymmetry in space, asymmetry in time

3

Recall the Governing EquationRecall the Governing Equation

What have we assumed thus far?What have we assumed thus far?Dispersion can be expressed as a Fickian processDispersion can be expressed as a Fickian processDiffusion and dispersion can be folded into a single Diffusion and dispersion can be folded into a single

hydrodynamic dispersionhydrodynamic dispersionFirst order decayFirst order decay

What do we need next?What do we need next?More Assumptions!More Assumptions!

What have we assumed thus far?What have we assumed thus far?Dispersion can be expressed as a Fickian processDispersion can be expressed as a Fickian processDiffusion and dispersion can be folded into a single Diffusion and dispersion can be folded into a single

hydrodynamic dispersionhydrodynamic dispersionFirst order decayFirst order decay

What do we need next?What do we need next?More Assumptions!More Assumptions!

4

Adding SorptionAdding Sorption

Thus far we have addressed only the solute behavior Thus far we have addressed only the solute behavior in the liquid state.in the liquid state.

We now add sorption using a linear isothermWe now add sorption using a linear isotherm

Recall the linear isotherm relationshipRecall the linear isotherm relationship

where cwhere cll and c and css are in mass per volume of water and are in mass per volume of water and

mass per mass of solid respectivelymass per mass of solid respectively

The total concentration is thenThe total concentration is then

Thus far we have addressed only the solute behavior Thus far we have addressed only the solute behavior in the liquid state.in the liquid state.

We now add sorption using a linear isothermWe now add sorption using a linear isotherm

Recall the linear isotherm relationshipRecall the linear isotherm relationship

where cwhere cll and c and css are in mass per volume of water and are in mass per volume of water and

mass per mass of solid respectivelymass per mass of solid respectively

The total concentration is thenThe total concentration is then

5

Retardation factorRetardation factor

We haveWe have

Which may be written asWhich may be written as

where we have defined the retardation where we have defined the retardation factor R to be factor R to be

We haveWe have

Which may be written asWhich may be written as

where we have defined the retardation where we have defined the retardation factor R to be factor R to be

C b kd c l

R =kb d1

6

Putting this all togetherPutting this all togetherThe ADE with 1st order decay & linear isothermThe ADE with 1st order decay & linear isotherm

What do we need now? More assumptions!What do we need now? More assumptions! constant in space (pull from derivatives + cancel)constant in space (pull from derivatives + cancel)D’ constant in space (slide it out of derivative)D’ constant in space (slide it out of derivative)R constant in time (slide it out of derivative)R constant in time (slide it out of derivative)

Use the chain rule:Use the chain rule:the divergence of the divergence of

a scalar =gradient;a scalar =gradient;divergence of a divergence of a

constant is zeroconstant is zero

The ADE with 1st order decay & linear isothermThe ADE with 1st order decay & linear isotherm

What do we need now? More assumptions!What do we need now? More assumptions! constant in space (pull from derivatives + cancel)constant in space (pull from derivatives + cancel)D’ constant in space (slide it out of derivative)D’ constant in space (slide it out of derivative)R constant in time (slide it out of derivative)R constant in time (slide it out of derivative)

Use the chain rule:Use the chain rule:the divergence of the divergence of

a scalar =gradient;a scalar =gradient;divergence of a divergence of a

constant is zeroconstant is zero

0R)'()()R(

llll ccct

c

33 Du

l

l

lll

cu

cu

uccucu

3

3

333

)(

)()(

7

Applying the previous assumptionsApplying the previous assumptions

The divergence operators turn into The divergence operators turn into gradient operators since they are applied gradient operators since they are applied to scalar quantities.to scalar quantities.

What does this give us? The new ADEWhat does this give us? The new ADE

The divergence operators turn into The divergence operators turn into gradient operators since they are applied gradient operators since they are applied to scalar quantities.to scalar quantities.

What does this give us? The new ADEWhat does this give us? The new ADE

clt

u3

Rcl

D3'

R2c l cl 0

8

Looking at 1-D case for a momentLooking at 1-D case for a moment

To see how this retardation factor works, take t* = To see how this retardation factor works, take t* = t/R, and t/R, and = 0. With a little algebra, = 0. With a little algebra,

The punch line: The punch line: the spatial distribution of solutes is the same in the the spatial distribution of solutes is the same in the

case of non-adsorbed vs. adsorbed compounds! case of non-adsorbed vs. adsorbed compounds! For a given boundary condition and time t*, the solution For a given boundary condition and time t*, the solution

is unique and independent of Ris unique and independent of R

To see how this retardation factor works, take t* = To see how this retardation factor works, take t* = t/R, and t/R, and = 0. With a little algebra, = 0. With a little algebra,

The punch line: The punch line: the spatial distribution of solutes is the same in the the spatial distribution of solutes is the same in the

case of non-adsorbed vs. adsorbed compounds! case of non-adsorbed vs. adsorbed compounds! For a given boundary condition and time t*, the solution For a given boundary condition and time t*, the solution

is unique and independent of Ris unique and independent of R

d cld t

u

R

d cld x

DL

R

d2 cld x2 cl 0

9

1-D infinite column Instantaneous Point Injection1-D infinite column Instantaneous Point Injection

Column goes to +Column goes to +and -and -Area AArea Asteady velocity usteady velocity umass M injected at x = o and t=0mass M injected at x = o and t=0

(boundary condition)(boundary condition)initially uncontaminated column. initially uncontaminated column.

i.e. c(x,0) = 0 (initial condition)i.e. c(x,0) = 0 (initial condition)linear sorption (retardation R)linear sorption (retardation R)first order decay (first order decay ())

Column goes to +Column goes to +and -and -Area AArea Asteady velocity usteady velocity umass M injected at x = o and t=0mass M injected at x = o and t=0

(boundary condition)(boundary condition)initially uncontaminated column. initially uncontaminated column.

i.e. c(x,0) = 0 (initial condition)i.e. c(x,0) = 0 (initial condition)linear sorption (retardation R)linear sorption (retardation R)first order decay (first order decay ())

x = 0

velocity u

cL(x,t) M

2AR Dt / Rexp

x ut / R 2

4Dt / R

exp t

10

Features of solution:Features of solution:Gaussian, symmetric in space, Gaussian, symmetric in space, 22 = Dt/R = Dt/RExponential decay of pulseExponential decay of pulseExcept for decay, R only shows up as t/RExcept for decay, R only shows up as t/R

Features of solution:Features of solution:Gaussian, symmetric in space, Gaussian, symmetric in space, 22 = Dt/R = Dt/RExponential decay of pulseExponential decay of pulseExcept for decay, R only shows up as t/RExcept for decay, R only shows up as t/R

cL(x,t) M

2AR Dt / Rexp

x ut / R 2

4Dt / R

exp t

0.00

0.05

0.10

0.15

0.20

0.25

0.30

-5 0 5 10 15

Position (m)

Con

cent

ratio

n

t = 0.1

t = 0.5

t = 2

t = 4

t = 8

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 2 4 6 8 10

Time

Con

cent

ratio

n

Center of Mass2 hr

Peak at 1.23 hr

Spatial Distribution Temporal Distribution

Upstream solutes

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Putting this in terms of Pore VolumesPutting this in terms of Pore Volumes

•If solute transport is dominIf solute transport is dominaated by advection ted by advection and dispersion, then the process is really and dispersion, then the process is really only dependent on total water displacementonly dependent on total water displacement ((the two transport processes scale linerarly the two transport processes scale linerarly with timewith time)). . •This would not be true if there were mass This would not be true if there were mass transfer between the mobile solution and transfer between the mobile solution and liquid within the particles, or if diffusion were liquid within the particles, or if diffusion were significant.significant.

•If solute transport is dominIf solute transport is dominaated by advection ted by advection and dispersion, then the process is really and dispersion, then the process is really only dependent on total water displacementonly dependent on total water displacement ((the two transport processes scale linerarly the two transport processes scale linerarly with timewith time)). . •This would not be true if there were mass This would not be true if there were mass transfer between the mobile solution and transfer between the mobile solution and liquid within the particles, or if diffusion were liquid within the particles, or if diffusion were significant.significant.

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Pore Volume form of SolutionPore Volume form of Solution

    Want Want concentration at concentration at x=x=L, in L, in vs.vs. pore- pore-volumes passed through the system, P.volumes passed through the system, P.

  Key substitutions:Key substitutions:

Coefficient n relates time and volume: nt=PCoefficient n relates time and volume: nt=P

At P=1, ut=L. This implies that u=Ln.At P=1, ut=L. This implies that u=Ln.

This gives us:This gives us:

Now how do we get the n out?Now how do we get the n out?

nDP

LPx

nDPA

MtxcL /4

exp/2

),(2

13

Pore Volume Form of SolutionPore Volume Form of Solution

Note that D/n = DL/u = Note that D/n = DL/u = lluL/u = uL/u = llLL

With this the result is entirely written in terms With this the result is entirely written in terms of the pore volumes and the media of the pore volumes and the media dispersivity, as we desired! dispersivity, as we desired!

Note that D/n = DL/u = Note that D/n = DL/u = lluL/u = uL/u = llLL

With this the result is entirely written in terms With this the result is entirely written in terms of the pore volumes and the media of the pore volumes and the media dispersivity, as we desired! dispersivity, as we desired!

nDP

LPx

nDPA

MtxcL /4

exp/2

),(2

LP

LPx

LPA

Mtxc

ll

L 4 exp

2),(

2

14

What about Retardation?What about Retardation?

No Problem, just put it in as before:No Problem, just put it in as before:No Problem, just put it in as before:No Problem, just put it in as before:

RLP

RLPx

RLPAR

Mtxc

ll

L /4

/ exp

/2),(

2

15

1-D semi-infinite Instantaneous Point Injection1-D semi-infinite Instantaneous Point Injection

How do we handle a surface application?How do we handle a surface application?Use the linearity of the simplified ADEUse the linearity of the simplified ADECan add any two solutions, and still a solutionCan add any two solutions, and still a solutionBy uniqueness, any solution which satisfies initial By uniqueness, any solution which satisfies initial

and boundary conditions is THE solutionand boundary conditions is THE solution

Boundary and initial conditionsBoundary and initial conditions

How do we handle a surface application?How do we handle a surface application?Use the linearity of the simplified ADEUse the linearity of the simplified ADECan add any two solutions, and still a solutionCan add any two solutions, and still a solutionBy uniqueness, any solution which satisfies initial By uniqueness, any solution which satisfies initial

and boundary conditions is THE solutionand boundary conditions is THE solution

Boundary and initial conditionsBoundary and initial conditions

16

Semi-infinite solutionSemi-infinite solution

Upward pulse and downward.Upward pulse and downward.Only include region of x > 0 in domainOnly include region of x > 0 in domainSolution symmetric about x = 0, therefore slope of Solution symmetric about x = 0, therefore slope of

dc(x=0,t)/dx = 0 for all t, as requireddc(x=0,t)/dx = 0 for all t, as requiredCompared to infinite column, c starts twice as high, Compared to infinite column, c starts twice as high,

but in time goes to same solution but in time goes to same solution

Upward pulse and downward.Upward pulse and downward.Only include region of x > 0 in domainOnly include region of x > 0 in domainSolution symmetric about x = 0, therefore slope of Solution symmetric about x = 0, therefore slope of

dc(x=0,t)/dx = 0 for all t, as requireddc(x=0,t)/dx = 0 for all t, as requiredCompared to infinite column, c starts twice as high, Compared to infinite column, c starts twice as high,

but in time goes to same solution but in time goes to same solution

17

Continuous injection, 1-DContinuous injection, 1-D

Since the simplified ADE is linear, we use Since the simplified ADE is linear, we use superposition. Basically get a continuous superposition. Basically get a continuous injection solution by adding infinitely many injection solution by adding infinitely many infinitely small Gaussian plumes. infinitely small Gaussian plumes.

Use the complementary error function: erfc Use the complementary error function: erfc

Since the simplified ADE is linear, we use Since the simplified ADE is linear, we use superposition. Basically get a continuous superposition. Basically get a continuous injection solution by adding infinitely many injection solution by adding infinitely many infinitely small Gaussian plumes. infinitely small Gaussian plumes.

Use the complementary error function: erfc Use the complementary error function: erfc

18

Plot of erfc(x)Plot of erfc(x)

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

2.00

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

x

erf

c(x)

19

Solution for continuous injection, 1-DSolution for continuous injection, 1-D

1 4R /u

20

Plot of solutionPlot of solution

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 10 20 30 40 50

Distance

Con

cen

trati

on

(C

/Co)

t = 1

t = 10

t=20

t=30

t=40

= 0.1, R = 1, = 0.02, u = 1.0, and m = 1•

21

1-D, Cont., simplified1-D, Cont., simplified

With no sorption or degradation this reduces to With no sorption or degradation this reduces to With no sorption or degradation this reduces to With no sorption or degradation this reduces to

cl (x, t) m.

2Auerfc

x ut2 ut

exp

x

erfc

x ut2 ut

22

2-D and 3-D instantaneous solutions2-D and 3-D instantaneous solutions

tRtD

y

RtD

Rutx

DDt

Mtyxc

tltl

L

exp

/4/4

/exp

4),,(

22 tRtD

y

RtD

Rutx

DDt

Mtyxc

tltl

L

exp

/4/4

/exp

4),,(

22

Note:Note:

- Same Gaussian form as 1-D - Same Gaussian form as 1-D

- Note separation of longitudinal and - Note separation of longitudinal and transverse dispersion transverse dispersion

Note:Note:

- Same Gaussian form as 1-D - Same Gaussian form as 1-D

- Note separation of longitudinal and - Note separation of longitudinal and transverse dispersion transverse dispersion

23

Review of AssumptionsReview of Assumptions

Assumption Effects if Violated constant in space -R higher where lower

-Velocity varies inversely with D constant in space -Increased overall dispersion due

to heterogeneity

D independent of scale - Plume will grow more slowly at

first, then faster.

Reversible Sorption - Increase plume spreading and

overall region of contamination

Equilibrium Sorption - Increased “tailing” and spreading

Linear Sorption - Higher peak C and faster travel

Anisotropic media - Stretching & smearing along beds

Heterogeneous Media - Greater scale effects of D and ALL

EFFECTS DISCUSSED ABOVE

Assumption Effects if Violated constant in space -R higher where lower

-Velocity varies inversely with D constant in space -Increased overall dispersion due

to heterogeneity

D independent of scale - Plume will grow more slowly at

first, then faster.

Reversible Sorption - Increase plume spreading and

overall region of contamination

Equilibrium Sorption - Increased “tailing” and spreading

Linear Sorption - Higher peak C and faster travel

Anisotropic media - Stretching & smearing along beds

Heterogeneous Media - Greater scale effects of D and ALL

EFFECTS DISCUSSED ABOVE