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Solute Transport Equation Form Consider the finite difference
approximation to the
one-dimensional transport (or convection-diffusion)
equation:
h or in more succinct PDE notation: DTxx vTx = Tth where x,t are
the independent variables, D and C are
the diffusion-dispersion coefficient the and
advective-convective velocity divided by a storage term,
respectively, and T(x,t) is some scalar quantity (temperature or
concentration for example). h We restrict the convective velocity
to be non-negative
(V 0), and obviously the diffusion coefficient is also
non-negative (D 0). The PDE is parabolic.
VD
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1D Finite Difference Equation Using an implicit second order FD
scheme in space
and time; the diffusive term is approximated by a central
difference and the convective term by a linear a central
difference.
A simple constant time step is used for the temporal derivative.
The FD approximation is :
h For the convection-diffusion equation, this is not a very
successful approximation for general application.h When D>>V
(diffusion dominated systems) it is
acceptable but as V approaches and exceeds D difficulties arise
with numerical oscillation.
D V
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FD Form Asymmetry Collecting terms gives:
h The convective term (V-term) makes the coefficient matrix
asymmetric and the asymmetry increases as convection becomes more
important. h Notice that when V=0, the coefficients of Tj-1 and
Tj+1
are identical and the coefficient matrix is symmetric.
V
V
V
V
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Oscillation Examining the difference equation:
h It is easy to see that the coefficient of Tj+1 remains
positive for Vx/D 2 (later we will see this is the grid Peclet
number for advection-dispersion problems). h When this term becomes
negative (when convection
dominates the system) the solution tends to oscillate and when
the absolute value of the negative coefficient of Tj+1 exceeds the
diagonal coefficient a solution cannot be obtained.h It is possible
to prevent this occurrence by controlling
x (that is, by refining the mesh) but this is sometimes
difficult to achieve.
V
V
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Numerical Oscillation
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Upwinding to avoid Oscillation One way to improve the finite
difference scheme is to
use a backward difference approximation for the convective term
while retaining a central difference approximation for the
diffusion term. This gives:
h This FD scheme is called upwinding because node j-1 is
upstream of node j.h Notice that the coefficient matrix remains
asymmetric
but all coefficients are positive and the matrix remains
diagonally dominant as long as C is positive.
V
V V
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Upwinding to Avoid Oscillation
With a large x, the upwind solution shows no oscillation where
the convective component is high near the x=1 boundary but is
inaccurate.
Reducing the grid spacing x by a factor of three, increases the
accuracy of the solution but is not as accurate as the
central-difference scheme.
2=xVD
32=xV
D
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Upwinding Pros and Cons The bottom line is that central
difference
methods place rather severe resolution requirements for
convection dominated flows and this has lead to the development of
upwind type methods in an attempt to circumvent this problem.
Although the backward difference method is only first order
accurate, the introduction of the upwind difference provides
oscillation free solutions even for large values of C.
Unfortunately the accuracy of the solution is not very good and
the method is overdiffuse, that is, it inherently generates too
much diffusive mixing.
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Numerical Dispersion
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Grid Courant Number For the advection-dispersion case (where C
is the
advective velocity), in the absence of upwinding or stabilized
FE schemes, mesh design requires that:
Ct 1x
Ct is called the grid Courant number.x
The Courant Number constraint can be simply stated in terms of
mesh size and time-step length.
It requires that the distance travelled by advection during one
time-step, t, is not larger than one spatial increment, x.
That is, an advected particle cannot cross more than one element
or cell boundary in a time-step.
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Grid Peclet Number For the advection-dispersion case (where D is
the
dispersion coefficient), in the absence of upwinding or
stabilized FE schemes, mesh design requires that:
Cx 2D
Cx is called the grid Peclet number.D
The Peclet number constraint can be simply stated in terms of
mesh size.
The Peclet number constraint requires that the spatial
discretization, x, of the flow regime is not larger than twice the
diffusion-dispersion potential of the medium.
That is, the mixing length for a dispersing or diffusing
particle, at the centre of a grid cell or element, is less
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Courant-Peclet TIPS
As a rule of thumb when the Peclet number is too big, the mesh
size, x, should be reduced, or alternatively, the material
dispersivity-diffusivity should be increased.
REMEMBER: Increasing the material property parameter will
mitigate the numerical problem at the expense of accuracy.
When the Courant number is too big, the time-step increment, t,
should be reduced.
REMEMBER: Reducing the time-step also increases the compute
time.
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Eulerian Methods
Eulerian methods (FD or FE) for solving the advection-dispersion
equation have disadvantages for advection dominated problems with
high Peclet number:1. Numerical dispersion is a numerical
artifact
analagous to mechanical dispersion generated by truncation error
as a result of discretization. The affect is to smear sharp
concentration fronts.
2. Numerical oscillation is another numerical artifact,
characteristic of advection dominated systems that can be addressed
by various upwinding schemes. The affect is for the solution to
oscillate in regions where advective component is dominant.
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Eulerian Problems
Eulerian methods have great difficulty in simulating sharp
interfaces and stabilization methods that suppress oscillation
often lead to smearing.
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Eulerian-Lagrangian Methods
Mixed Eulerian-Lagrangian methods are virtually free from the
problems of numerical dispersion and oscillation.
The method of characteristics (MOC) and modified method of
characteristics (MMOC) are typical examples of such methods.
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Method of Characteristics
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Modified Method of Characteristics
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E-L Approach
In essence, these mixed Eulerian-Lagrangianmethods split the
advection-dispersion problem into two parts:1.Advection is modelled
using a Lagrangian finite-
difference (or FE) scheme involving moving coordinates where the
change of solute concentration is predicted along a streamline.
This method is known as particle tracking.
2.Dispersion is modelled using a Eulerian finite-difference (or
FE) scheme with fixed coordinates.
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Disadvantages of Lagrangian Schemes
Unfortunately, unlike pure Eulerian finite-difference (and FE)
schemes, MOC and MMOC are not based on strict mass conservation
principles and large mass-balance discrepancies can arise.
These problems can be minimized (with a time penalty) if
higher-order interpolation schemes are used in particle
tracking.
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Recommendation for solution method
Characteristics of the model application Recommended method
Pronounced spatial concentration gradients (e.g. contaminant
plume with point source, lab study with contaminant pulse)
TVD, MOC, HMOC
Regional scale contaminant transport with small concentration
gradients (e.g. nitrate transport with distributed sources)
FD
Exact mass balance important (e.g. coupling of transport and
non-linear reactions)
FD, TVD
Large time steps required (e.g. long range solute transport over
decades)
FD implicit, HMOC
Large grid Peclet numbers (small dispersivities or large grid
spacing)
MOC, HMOC
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Boundary conditions
These are fairly simple in MT3D Fixed concentration Fixed Mass
flux (connected to flow boundary
conditions
Issue that the Boundary conditions do not admit dispersive
fluxes across the boundary!
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Remember
Even though groundwater flow may be treated at 2D, solute
transport is often 3D.
An FD grid suitable for flow is not likely to be suitable for
solute transport. Pecletnumber and Courant number criteria must be
satisfied.
Choice of method of solution is important for a sensible
representation of contaminant migration in an aquifer.
Solute Transport Equation Form1D Finite Difference EquationFD
Form AsymmetryOscillationNumerical OscillationUpwinding to avoid
OscillationUpwinding to Avoid OscillationUpwinding Pros and
ConsNumerical DispersionGrid Courant NumberGrid Peclet
NumberCourant-Peclet TIPSEulerian MethodsEulerian
ProblemsEulerian-Lagrangian MethodsMethod of
CharacteristicsModified Method of CharacteristicsE-L
ApproachDisadvantages of Lagrangian SchemesRecommendation for
solution methodBoundary conditionsRemember