Refer to the document onthe course homepage entitled

“MT3DMS Solution Methods and Parameter Options”

(Look under the MT3DMS tab on the homepage)

Dispersion, sink/source, chemical reactions

Advection

MT3DMS Solution Options

1

2

3

4

MT3DMS Solution Options

Stability constraintsfor explicit solutions

Courant Number

MT3DMS Solution Options

Use GCG Solver

Use GCG Solver

Use GCG Solver

MT3DMS Solution Options

TVD ULTIMATE METHODa higher order FD method

Conventional FD methodsuse 3 nodes in the FDapproximation. The TVDmethod uses 4 nodes withupstream weighting. Thisessentially eliminatesnumerical dispersion.

Steps in the TVD Method

Correctionfor oscillationerrors

Check foroscillationerrors

oscillation

TVD ULTIMATE METHOD

In one dimension

Compare with an equation for alower order explicit approximation

nj

nj

nj

nj ccc

x

tvc

)( 1

1

MT3DMS Solution Options

Eulerian vs Lagrangian Methods

• Eulerian: fixed coordinate system with mass flux through an REV

• Lagrangian: moving particles; each particle carries mass. The Random Walk method is a Lagrangian method.

• Mixed Eulerian-Lagrangian methods use particles to solve the advection portion of the ADE and an Eulerian method to solve the rest of the equation.

Method of Characteristics(MOC)

1

where is a weighting factor to weight concentration between time level n and an intermediate time level n*, normally = 0.5

2

3

1*1 nm

nm

nm CCC4Step 1 is a Lagrangian method;

Step 3 is a Eulerian method.

Also update concentration of each particle. For example,

for particles in cell m:11 n

mnp

np CCC

• MOC uses multiple particles per cell.

• MMOC uses one particle per cell.• HMOC uses multiple particles in high concentration regions and one particle per cell elsewhere.

Dynamic Particle Allocation

Breakthrough curve for example problemin the MT3DMS manual

Compare with Fig. 7.26 in Z&B

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20

Time (years)

Co

nce

ntr

atio

n

TVD

HMOC

Upstream weighting

Central FD

Upstream FD

Central FD

TVD

MT3DMS Solution Options

1

2

3

4

PS#2