Introduction to MT3DMS All equations & illustrations taken from the MT3DMS manual.
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Transcript of Introduction to MT3DMS All equations & illustrations taken from the MT3DMS manual.
![Page 1: Introduction to MT3DMS All equations & illustrations taken from the MT3DMS manual.](https://reader035.fdocuments.us/reader035/viewer/2022062805/5697c0281a28abf838cd70c6/html5/thumbnails/1.jpg)
Introduction to MT3DMS
All equations & illustrations takenfrom the MT3DMS manual
![Page 2: Introduction to MT3DMS All equations & illustrations taken from the MT3DMS manual.](https://reader035.fdocuments.us/reader035/viewer/2022062805/5697c0281a28abf838cd70c6/html5/thumbnails/2.jpg)
Refer to the document onthe course homepage entitled
“MT3DMS Solution Methods and Parameter Options”
(Look under the MT3DMS tab on the homepage)
![Page 3: Introduction to MT3DMS All equations & illustrations taken from the MT3DMS manual.](https://reader035.fdocuments.us/reader035/viewer/2022062805/5697c0281a28abf838cd70c6/html5/thumbnails/3.jpg)
General form of the ADE:
Expands to 9 terms
Expands to 3 terms
(See eqn. 3.48 in Z&B)
![Page 4: Introduction to MT3DMS All equations & illustrations taken from the MT3DMS manual.](https://reader035.fdocuments.us/reader035/viewer/2022062805/5697c0281a28abf838cd70c6/html5/thumbnails/4.jpg)
9 DispersionCoefficients
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This schematic assumes that
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MODFLOW
MT3DMS
MT3DMS time steps are selected by the code consideringstability constraints, if any, and Courant numbers.
![Page 7: Introduction to MT3DMS All equations & illustrations taken from the MT3DMS manual.](https://reader035.fdocuments.us/reader035/viewer/2022062805/5697c0281a28abf838cd70c6/html5/thumbnails/7.jpg)
![Page 8: Introduction to MT3DMS All equations & illustrations taken from the MT3DMS manual.](https://reader035.fdocuments.us/reader035/viewer/2022062805/5697c0281a28abf838cd70c6/html5/thumbnails/8.jpg)
![Page 9: Introduction to MT3DMS All equations & illustrations taken from the MT3DMS manual.](https://reader035.fdocuments.us/reader035/viewer/2022062805/5697c0281a28abf838cd70c6/html5/thumbnails/9.jpg)
Dispersion, sink/source, chemical reactions
Advection
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![Page 11: Introduction to MT3DMS All equations & illustrations taken from the MT3DMS manual.](https://reader035.fdocuments.us/reader035/viewer/2022062805/5697c0281a28abf838cd70c6/html5/thumbnails/11.jpg)
MT3DMS Solution Options
1
2
3
4
![Page 12: Introduction to MT3DMS All equations & illustrations taken from the MT3DMS manual.](https://reader035.fdocuments.us/reader035/viewer/2022062805/5697c0281a28abf838cd70c6/html5/thumbnails/12.jpg)
j-1 j j+1
x
j-1/2 j+1/2
![Page 13: Introduction to MT3DMS All equations & illustrations taken from the MT3DMS manual.](https://reader035.fdocuments.us/reader035/viewer/2022062805/5697c0281a28abf838cd70c6/html5/thumbnails/13.jpg)
Upstream weighting
Central differences
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MT3DMS Solution Options
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Explicit Approximation
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Stability constraintsfor explicit solutions
Courant Number
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Courant Numberx
tvCr
Cr < 1
6 Courant Numbers
One for each face ofthe cell block
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MT3DMS Solution Options
Use GCG Solver
Use GCG Solver
Use GCG Solver
![Page 19: Introduction to MT3DMS All equations & illustrations taken from the MT3DMS manual.](https://reader035.fdocuments.us/reader035/viewer/2022062805/5697c0281a28abf838cd70c6/html5/thumbnails/19.jpg)
![Page 20: Introduction to MT3DMS All equations & illustrations taken from the MT3DMS manual.](https://reader035.fdocuments.us/reader035/viewer/2022062805/5697c0281a28abf838cd70c6/html5/thumbnails/20.jpg)
Implicit Approximationfor advection term
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MT3DMS Solution Options
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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.
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Steps in the TVD Method
Correctionfor oscillationerrors
Check foroscillationerrors
oscillation
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TVD ULTIMATE METHOD
In one dimension
Compare with an equation for alower order explicit approximation
nj
nj
nj
nj ccc
x
tvc
)( 1
1
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MT3DMS Solution Options
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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.
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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
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• 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.
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Dynamic Particle Allocation
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Breakthrough curve for example problemin the MT3DMS manual
Compare with Fig. 7.26 in Z&B
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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
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MT3DMS Solution Options
1
2
3
4
PS#2