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4.5 IMPLEMENTATION OF THE EULERIAN-LAGRANGIAN METHODS
4.5.3 The MOC Procedure
The first step in the method of characteristics is to generate representative particles in the finite-difference grid. Instead of placing a uniform number of particles in every cell of the grid, a dynamic approach is used in the MT3DMS transport model to control the
distribution of moving particles. The number of particles placed at each cell is normally set either at a high level or at a low level, according to the so-called “relative cell concentration gradient”, or, DCCELL, defined as:
CMIN CMAX
CMIN DCCELLi jk CMAXijk i jk
−
= , , − , ,
,
, (4.62)
where
( )
= +
−
= +
−
= +
−
= ii jjkk
i
i ii j
j jj k
k k kk j
i C
CMAX , ,
1
1 1
1 1
, 1
, max max max , is the maximum concentration in the immediate vicinity of the cell (i, j, k);
Fig. 4.9.
The fourth-order Runge-Kutta method. In each step, the velocity is evaluated four times: once at the initial point, twice at trial midpoints, and once at a trial endpoint. From these velocities a weighted velocity is calculated which is used to compute the final position of the particle (shown as a filled dot).
( )
, min min min , is the minimum concentration in the immediate vicinity of the cell (i, j, k);
CMAX is the maximum concentration in the entire grid; and CMIN is the minimum concentration in the entire grid.
With the dynamic approach, the user defines the criterion, DCEPS, which is a small number near zero; the higher number of particles, NPH, is placed in cells where the relative concentration gradient is greater than DCEPS, and the lower number of particles, NPL, in cells where the relative concentration gradient is less than DCEPS, i.e.,
Initially, if the concentration gradient at a cell is zero or small, (i.e., the concentration field is relatively constant near that cell), the number of particles placed in that cell is NPL, which may be zero or some small integer number; this is done because the concentration change due to advection between that cell and the neighboring cells will be insignificant. If the concentration gradient at a cell is large, which indicates that the concentration field near that cell is changing rapidly, then the number of particles placed in that cell is NPH.
As particles leave source cells or accumulate at sink cells, it becomes necessary to insert new particles at sources, or remove particles at sinks. At non-source or non-sink cells, it also becomes necessary to insert or remove particles as the cell concentration gradient changes with time. This is done in the dynamic insertion-deletion procedure by specifying the minimum and maximum numbers of particles allowed per cell, called NPMIN and NPMAX, respectively. When the number of particles in any cell, (source or non-source), becomes smaller than the specified minimum, NPMIN, new particles equal to NPL or NPH are inserted into that cell without affecting the existing particles. On the other hand, when the number of particle in any cell, (sink or non-sink), exceeds the specified maximum, or NPMAX, all particles are removed from that cell and replaced by a new set of NPH particles to maintain mass balance. To save computer storage, memory space occupied by the deleted particles is reused by newly inserted particles.
Figure 4.10 illustrates the dynamic particle distribution approach in contrast with the uniform approach in simulating two-dimensional solute transport from a continuous point source in a uniform flow field. Whereas the uniform approach inserts and maintains an approximately uniform particle distribution throughout the simulated domain, the dynamic approach adjusts the distribution of moving particles dynamically, adapting to the changing nature of the concentration field. In many practical problems involving contaminant transport modeling, the contaminant plumes may occupy only a small fraction of the finite-difference grid and the concentrations may be changing rapidly only at sharp fronts. In these cases, the number of total particles used is much smaller than that required in the uniform particle distribution approach, thereby dramatically increasing the efficiency of the method-of-characteristics model with little loss in accuracy.
Fig. 4.10.
Comparison of the uniform and dynamic approaches in controlling the distribution of moving particles.
Particles can be distributed either with a fixed pattern or randomly, as controlled by the user-specified option (see Figure 4.11). If the fixed pattern is chosen, the user determines not only the number of particles to be placed per cell, but also the pattern of the particle placement in plan view and the number of vertical planes on which particles are placed within each cell block. If the random pattern is chosen, the user only needs to specify the number of particles to be placed per cell. The program then calls a random number generator and distributes the required number of particles randomly within each cell block. (The selection of these options is discussed in Chapter 6: Input Instructions). The fixed pattern may work better if the flow field is relatively uniform. On the other hand, if the flow field is highly nonuniform with many sinks or sources in largely heterogeneous media, the random pattern may capture the essence of the flow field better than the fixed pattern does.
Fig. 4.11.
Initial placement of moving particles.
(a) Example of initial particle placement with fixed pattern (8 particles are placed on 2 vertical planes).
(b) Example of initial particle placement with random pattern (8 particles are placed randomly in the cell).
Each particle is associated with a set of attributes, that is, the x-, y-, and z-coordinates and the concentration. The initial concentration of the particle is assigned as the
concentration of the cell where the particle is initialized. At the beginning of each transport step, all particles are moved over the time increment, ∆t, using the particle tracking
techniques described previously. The x-, y-, and z-coordinates of the moving particles are then updated to reflect their new positions at the end of the transport step. The average concentration of a finite-difference cell at the end of the transport step due to advection alone, Ci j kn, ,* , can be obtained from the concentrations of all particles that are located at that cell. If a simple arithmetic averaging algorithm is used, the average concentration is expressed by the following equation:
∑
= >If the grid spacing is irregular, the volume-based averaging algorithm of Zheng (1993) is used as follows:
0 particles at the cell is zero, then the average concentration after particle tracking is set equal to the cell concentration at the previous time level because the concentration change at that cell over the time increment is either negligible or dominated by an external source:
0
It is necessary to locate the cell indices of any particle in the tracking and averaging calculations as described above. If the finite-difference grid is regular, it is straightforward to convert particle coordinates
(
xP,yP,zP)
to cell indices(
JP IP KP, ,)
according to the following formulas:( )
( )
( )
+
∆
=
+
∆
=
+
∆
=
1 /
1 /
1 /
z z INT KP
y y INT IP
x x INT JP
P P P
(4.67)
where INT(x) is a FORTRAN function, equal to the truncated value of x; and ∆ ∆ ∆x, y, z are the uniform grid spacings along the x-, y-, and z-axes. If the finite-difference grid is
irregular, then, the coordinate of the particle in any direction, say x, must be compared with the those of cell interfaces in the same direction to determine which cell the particle is located.
After the Cin,*j,k term is evaluated at every cell, it is used to calculate the concentration change due to dispersion, sink/source mixing and/or chemical reactions
(
∆Ci j kn, ,+1)
using the finite-difference method as discussed previously. The concentration of all active particles is then updated by adding the concentration change(
∆Ci j kn, ,+1)
calculated at the cell where each particle is located. Therefore, for moving particles located at cell (i, j, k):1 , ,
1 +
+ = +∆ injk n
l n
l C C
C (4.68)
where Cln+1 is the concentration of the l particle which is located at cell (i, j, k) at the new th time level. If ∆Cin,+j,1k is positive, equation (4.68) is applied directly. However, if ∆Cin,+j1,k is negative, the concentration of the moving particle may become negative if its concentration at the old time level, C , is zero or small. When this happens, all particles in the cell are ln removed and replaced by a new set of particles which are assigned the concentration of the cell.