4.2. The Method of Characteristics
91
a one-dimensional dispersion problem (stated in section 3.2.1, Problem 1).
Konikow (1977) used the method of characteristics to simulate the migration
of chloride ions in an alluvial aquifer located in the Rocky Mountains of the
United States of America.
To extend the MOC for solving three-dimensional advection-dispersion
problems is straight. Some modified techniques of the MOC will be given in
Chapter 6.
4.2.5 The Random-Walk Model
Prickett et al. (1981) proposed a Random-Walk model for solving the
problems of one-dimensional advection-dispersion. Its basic idea is that the
mass transport in porous media may be looked upon as an average result of
the movements of a large number of tracer particles. Each particle is engaged
in two kinds ofmovements: one is advection, represented by the movement of
the tracer particle with an average velocity in the flow field; the other is
dispersion, which may be seen as a random fluctuation around the average
movement. The development of modern computational techniques enables
us to trace the two movements of a huge number of tracer particles by using
a computer. As long as the number of tracer particles is large enough, mass
transport in porous media can be described.
The Random-Walk method is the same as the method of characteristics in
dealing with the advection of tracer particles, but is different with regard to
the treatment of dispersion. In this method, the movement of each particle is
constituted by the advection and an additional random movement, so that
the new location of a particle can be simply determined without solving any
equation in association with the dispersion. It is unnecessary to assign concentrations to a difference grid system within each time step and to modify
the concentrations of the particles themselves. Thus, the computation is
greatly simplified.
Similar to the method of characteristics, the Random-Walk considers the
movements of a group of particles. If the location of a particle P at time t k
is (Xp,k,Yp,k)' its location based on the advection movement after time ll.t will
be
X:,k+l = Xp,k + Vx,p,k ·ll.t,
Y:,k+l = Yp,k + v",p,k 'M.
(4.2.15)
(4.2.16)
These two equations have the same meaning as Eq. (4.2.7) and Eq. (4.2.8).
The superscript * is used to denote that (X:,k+l' Y:,k+l) is just the result of
advection, and not the finallocation of the particle at time tk+l = t k + M.
Its final location (X p,k+1' Yp,k+d may be looked upon as the result of the
advection of particle P plus a dispersion, i.e.,
X p,k+1 = X:,k+l + «5Xp,k+l'
Y p,k+l = Y;,k+l + «5YP,k+l'
(4.2.17)
(4.2.18)
91
a one-dimensional dispersion problem (stated in section 3.2.1, Problem 1).
Konikow (1977) used the method of characteristics to simulate the migration
of chloride ions in an alluvial aquifer located in the Rocky Mountains of the
United States of America.
To extend the MOC for solving three-dimensional advection-dispersion
problems is straight. Some modified techniques of the MOC will be given in
Chapter 6.
4.2.5 The Random-Walk Model
Prickett et al. (1981) proposed a Random-Walk model for solving the
problems of one-dimensional advection-dispersion. Its basic idea is that the
mass transport in porous media may be looked upon as an average result of
the movements of a large number of tracer particles. Each particle is engaged
in two kinds ofmovements: one is advection, represented by the movement of
the tracer particle with an average velocity in the flow field; the other is
dispersion, which may be seen as a random fluctuation around the average
movement. The development of modern computational techniques enables
us to trace the two movements of a huge number of tracer particles by using
a computer. As long as the number of tracer particles is large enough, mass
transport in porous media can be described.
The Random-Walk method is the same as the method of characteristics in
dealing with the advection of tracer particles, but is different with regard to
the treatment of dispersion. In this method, the movement of each particle is
constituted by the advection and an additional random movement, so that
the new location of a particle can be simply determined without solving any
equation in association with the dispersion. It is unnecessary to assign concentrations to a difference grid system within each time step and to modify
the concentrations of the particles themselves. Thus, the computation is
greatly simplified.
Similar to the method of characteristics, the Random-Walk considers the
movements of a group of particles. If the location of a particle P at time t k
is (Xp,k,Yp,k)' its location based on the advection movement after time ll.t will
be
X:,k+l = Xp,k + Vx,p,k ·ll.t,
Y:,k+l = Yp,k + v",p,k 'M.
(4.2.15)
(4.2.16)
These two equations have the same meaning as Eq. (4.2.7) and Eq. (4.2.8).
The superscript * is used to denote that (X:,k+l' Y:,k+l) is just the result of
advection, and not the finallocation of the particle at time tk+l = t k + M.
Its final location (X p,k+1' Yp,k+d may be looked upon as the result of the
advection of particle P plus a dispersion, i.e.,
X p,k+1 = X:,k+l + «5Xp,k+l'
Y p,k+l = Y;,k+l + «5YP,k+l'
(4.2.17)
(4.2.18)