92
4. Finite DitTerence Methods
where I5xp .k+l and I5Yp .k+l are additional components caused by dispersion.
According to the statistical theory of dispersion phenomena, the dispersion
results in a normal distribution of tracer particle locations around their
average locations. This can be verified by the solution of one-dimensional
advection-dispersion problems.
For the equation,
the solution with a transient source was given in paragraph 3.l.3, i.e.,
J1./()
{(X - Vt)2}
C(x, t) =
;::-n-:exp
4
.
2y 1T.D L t
DLt
(4.2.19)
On the other hand, the density function of normal distribution is
1
{(X - m)2}
n(x) = fou exp -
2u2
'
(4.2.20)
where m is the mathematical expectation, and u is the variance. After comparing the above two equations we may find that for each given t, the
concentration C(x, t) is commensurate to a density function of normal distribution within a constant factor. Its corresponding mathematical expectation
and variance are respectively:
m = Vt; u = j2D L t.
(4.2.21)
For each time step, we assume that particle P first moves a distance VL\t
according to the advection, i.e., it arrives at the location of the mathematical
expectation. The particle then makes a random walk along the + x or - x
directions. The distance of the walk is
j2D L t· ANORM(O),
(4.2.22)
where ANORM(O) is a value of the normal distribution with mathematical
expectation 0 and variance l. ANORM(O) may be limited to the range of
( - 6, + 6) during calculation, because the probability of being more than 6
standard deviations from the mean is very smalI. Thus, the total displacement
of particle P is
V' L\t + j2D L t· ANORM(O).
(4.2.23)
This procedure is shown in Figure 4.1l.
The procedure is repeated for a large number of tracer particles. Their
displacements are different because the second term of Eq. (4.2.23) is stochastic. The more particles there are, the smaller the difference between their
distribution and normal distribution will be. It is assumed that a total number of particles No are instantaneously injected into the origin (x = 0) at the
initial time. After moving many time steps (the time is from 0 to t), the
4. Finite DitTerence Methods
where I5xp .k+l and I5Yp .k+l are additional components caused by dispersion.
According to the statistical theory of dispersion phenomena, the dispersion
results in a normal distribution of tracer particle locations around their
average locations. This can be verified by the solution of one-dimensional
advection-dispersion problems.
For the equation,
the solution with a transient source was given in paragraph 3.l.3, i.e.,
J1./()
{(X - Vt)2}
C(x, t) =
;::-n-:exp
4
.
2y 1T.D L t
DLt
(4.2.19)
On the other hand, the density function of normal distribution is
1
{(X - m)2}
n(x) = fou exp -
2u2
'
(4.2.20)
where m is the mathematical expectation, and u is the variance. After comparing the above two equations we may find that for each given t, the
concentration C(x, t) is commensurate to a density function of normal distribution within a constant factor. Its corresponding mathematical expectation
and variance are respectively:
m = Vt; u = j2D L t.
(4.2.21)
For each time step, we assume that particle P first moves a distance VL\t
according to the advection, i.e., it arrives at the location of the mathematical
expectation. The particle then makes a random walk along the + x or - x
directions. The distance of the walk is
j2D L t· ANORM(O),
(4.2.22)
where ANORM(O) is a value of the normal distribution with mathematical
expectation 0 and variance l. ANORM(O) may be limited to the range of
( - 6, + 6) during calculation, because the probability of being more than 6
standard deviations from the mean is very smalI. Thus, the total displacement
of particle P is
V' L\t + j2D L t· ANORM(O).
(4.2.23)
This procedure is shown in Figure 4.1l.
The procedure is repeated for a large number of tracer particles. Their
displacements are different because the second term of Eq. (4.2.23) is stochastic. The more particles there are, the smaller the difference between their
distribution and normal distribution will be. It is assumed that a total number of particles No are instantaneously injected into the origin (x = 0) at the
initial time. After moving many time steps (the time is from 0 to t), the