Transport equations and
MT3DMS use
Advection
U
v
U=q/A (m/s)
Darcy velocity
ou unit flux
v
x = v.t
A
A’=w.A
v = U/w
pore velocity (v>U)
or tracer velocity
Dispersion
v
DT: transverse dispersion
DL 
 L2
2t

 L2v
2x
DL   L v
C
DT   T v
x
DL: longitudinal dispersion
: dispersivity
Sediments heterogeneity
Instantaneous and continuous injection
t0
C
t1
t2
x
C
x
Basic equation with no reaction
 qC  

t
xi

 

C
qDij


vi C   qs Cs

 x

x
j 
i

xi : coordinates (i=1,2,3)
C : concentration
t : time
Dij : dispersion tensor
q : porosity
vi : pore velocity
qs : water flux at source points
Cs : source concentration
Link with flow model
K i h
vi 
q xi
The MT3DMS formulation
• Can include retardation (R)
• SSM : Sink and source Mixing (wells,
river...)
• RCT : "Reaction" sorption (linear,
freundlich...), radioactive decay
Why using distributed models?
A complex function is
linearized (using Taylor
approx.)
and solved locally using
only neighbouring cells
This requires :
- Initial conditions on the grid
- Boundary conditions
- A matrix solver
Initial conditions
• For concentrations, most of the time
Cinitial=0 over the whole domain
• Some time for restart, the
concentrations on a plume modelled in
a first simulation
Link with modflow
• A no-flow boundary in MODFLOW is automatically
treated as zero mass flux boundary
• All other boundaries (including specified-flow;
head-dependent; and constant-head) in
MODFLOW are automatically treated as specified
mass flux boundaries
• Mass flux= QC, where Q may be specified by user
or determined by MODFLOW. Dispersive flux
across boundary is always assumed to be
negligible.
Boundary conditions
• Play a very important role on results
• two major types :
– Specified concentration (Dirichlet)
– Specified total mass flux
A simple example (1)
MODFLOW
specified flux
Specified Head
Specified Head
No Flow (automatic)
MT3DMS
specified Conc
Specified Conc
All information for MT3DMS is
located on Modflow BC :
 no BC for MT3DMS
 concentrations specified at
these BC
Specified Conc.
No Flow
A simple example (2)
MODFLOW
Specified Head
Specified Head
No Flow (automatic)
MT3DMS
set BC +
specified
concentration
Specified Conc
For Modflow BC same as
previously. For the contaminant
source :
 define a BC zone for
MT3DMS
 concentration specified at this
BC
Specified Conc.
No Flow
Solving
• Fundamental difficulty: the need to treat
simultaneously the first-order (hyperbolic)
advection term and the second-order
(parabolic) dispersion term.
• “Embarrassingly difficult…” Mitchell (1984)
• “…a problem no numerical method has fully
overcome.” Baptista (1987)
Basic of transport solutions
• Classical Eulerian technique (FD)
– good for dispersion
– Mass conservation
– if cell size > dispersivity : too smooth
fronts
• Lagrangian technique (Particles)
– good for advection : sharp fronts
– numerical instability
– no mass conservation
Mixed strategies
• MOC (Method Of Characteristic) :
particle based but keeps information
on mass balance
• TVD (Total Variaiton Diminishing) :
Euclidian approach but higher order
=> able to keep sharp fronts
Finite difference
• 1st order approach (Taylor)
Cr  1
Adv./disadv FD
• advantages
– mass conservative,
– computationally efficient for dispersion dominated
problems,
– no numerical difficulty for distorted model grids or in the
presence of many sinks/sources
• disadvantages
– suffer from numerical dispersion errors for advectiondominated problems (upstream weighting) or from artificial
oscillation (central-in-space weighting)
– to minimize numerical dispersion errors, fine spatial
discretization may be necessary, i.e., to satisfy the Peclet
number constraint (Pe ~1)
FD example
Tritium
z
y
x
x
TVD (total variation diminish.)
• Takes 5 points (instead of 3)
Adv./disadv. TVD
• advantages
– mass conservative
– minimal numerical dispersion and artificial oscillation for
advection-dominated problems
– no numerical difficulty for distorted model grids or in the
presence of many sinks/sources
• disadvantages
– could be computationally demanding
– may not be as effective as MOC in eliminating numerical
dispersion for purely advective problems
MOC (method of characteristics)
• Each particle carries a volume
and a concentration
• Particle pattern: random or
fixed
• NPL: number of particle per
cell at low conc. gradient
• NPH: number of particle per
cell at high conc. gradient
• DCEPS: critical concentration
gradient for switching
• between NPL and NPH
MMOC (modified)
• Concentration at point P
is interpolated from those
at neighboring nodes
HMOC (Hybrid)
• MOC is most effective when advection
dominates over dispersion, while MMOC is
only suitable when advection is less
dominant
• Apply MOC where advection dominates,
dynamically adding particles as needed
• Automatically switch to MMOC where
advection is less dominant, removing
particles that are no longer needed
Adv./disadv. Particle Methods
• advantages
– virtually eliminate numerical dispersion
– computationally efficient for highly advection-dominated
problems
• disadvantages
– may have mass balance discrepancy problems,
particularly when the model grid is highly irregular
– calculated concentration breakthrough curves may be
“rough”
– computer memory intensive, particularly for multi-species
simulations, with a set of particles for each species
Comparison of solutions
Analytical 800j
TVD
F.Diff
Comparison of solutions
Analytical 800j
TVD
F.Diff
MT3DMS modules
• BTN : major package : geometry,
porosity, initial concentrations
• ADV : choice of solution method (FD,
TVD, MOC) and characteristic of
these methods
• DSP : dispersion and diffusion
parameters
• GCG : matrix solver (to look inside if
convergence problems)
Rôle de l’écoulement
12
13
14
6
85,80 m
18
7
19
2
8
85,90 m
20
3
21
1
15
9
4
16
10 m
5
17
10
11
Burnham N.Zélande, Lipping-Pang et al. Groundwater 1998