User's guide of Proceessing Modflow 5.0

176 Processing Modflow
where
n [-] is the effective porosity, and
-1
v x1, v x2, v y1, v y2, v z1 , and v z2 [LT ] are the average pore velocity components across each cell
face.
Pollock’s semi-analytical particle tracking scheme is based on the assumption that each
directional velocity component varies linearly within a model cell in its own coordinate direction.
The semi-analytical particle tracking algorithm uses simple linear interpolation to compute the
principle velocity components at any points within a cell. Given the starting location (x, y, z) of
the particle and the starting time t , the velocity components are expressed in the form
1
v x (t 1 ) ' A x (x & x 1 ) % v x1
v y (t 1 ) ' A y (y & y 1 ) % v y1
v z (t 1 ) ' A z (z & z 1 ) % v z1
A x ' (v x2 & v x1 ) / )x
A y ' (v y2 & v y1 ) / )y
A z ' (v z2 & v z1 ) / )z
x(t 2 ) ' x 1 % (v x (t 1 )@e A x @)T & v x1 ) / A x
y(t 2 ) ' y 1 % (v y (t 1 )@e A y @)T & v y1 ) / A y
z(t 2 ) ' z 1 % (v z (t 1 )@e A z @)T & v z1 ) / A z
4.1 The Semi-analytical Particle Tracking Method
(4.4a)
(4.4b)
(4.4c)
-1
where x 1, y 1 and z 1 are defined in Fig. 4.3. A x , A y and A z [T ] are the components of the velocity
gradient within the cell,
(4.5a)
(4.5b)
(4.5c)
Using a direct integration method described in Pollock (1988) and considering the movement of
the particle within a cell, the particle location at time t is
2
where )T = t - t
2 1
(4.6a)
(4.6b)
(4.6c)
For steady-state flow fields, the location of the particle at time t must be still within the same
2
cell as at time t . Given any particle’s starting location within a cell at time t , Pollock’s
1 1
algorithm allows to determine the particle’s exit time t and exit point from the cell directly,
2
without having to calculate the actual path of the particle within the cell.
The particle tracking sequence is repeated until the particle reaches a discharge point or until
a user-specified time limit is reached. Backward particle tracking is implemented by multiplying