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

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

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

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

Numerical Oscillation

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

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

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.

Numerical Dispersion

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.

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

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.

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.

Eulerian Problems

Eulerian methods have great difficulty in simulating sharp interfaces and stabilization methods that suppress oscillation often lead to smearing.

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.

Method of Characteristics

Modified Method of Characteristics

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.

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.

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

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!

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