4.1. Finite Difference Methods
79
where Di+1/2 and Di- 1/2 are the dispersion coefficients between node i and its
adjacent nodes i + 1 and i - 1, respectively. The corresponding difference
equations can be derived without difficulty. For example, we have the following implicit difference scheme:
C i,n+1 - Ci,n _ D i+1/2 C i+1,n+1 - (Di+1/2 + D i- 1/2 )Ci,n+1 + D i- 1/2 C i-l,n+1
!l.t
-
(!l.X)2
_ VCi+1,n+1 - Ci - 1 ,n+1
I
2!l.x
.
(4.1.29)
The node values of D may be used to express Di+1/2 and Di - 1/2 , e.g., we may
use the harmonic mean, i.e., let
2D.D·+ 1
D
-
I
I
i+1/2 - D + D '
i
i+1
2D;Di-l
Di - 1/2 = D
D .
i + i-1
The method mentioned above can also be used to solve the radial dispersion
problem:
oC
IXLA 02C
A oC
ot - -r- or2 - r Fr'
If the implicit scheme is adopted, we have the following finite difference
approximation:
C i,n+1 - Ci,n = ~[IX Ci+1,n+1 - 2Ci,n+1 + C i- 1,n+1 _ Ci+I,n+1 - C i - I ,n+1].
At
ri
L
(!l.r)2
2!l.r
After rearranging, we have
= ri(!l.r)2 C.
A!l.t I,n'
(4.1.30)
where r i is the radial distance of node i. Eq. (4.1.30) can be written for
all internal nodes. After combining with boundary conditions, a set of tridiagonal equations are formed. The concentrations of all nodes at the next
time step can be obtained using the Thomas algorithm. If the sizes of !l.t
and !l.r are appropriately controlled, the result obtained by this algorithm is
very dose to the exact solution. The program based on this algorithm is
simple and efficient, and can be used for simulating the field tests of tracer
injection in order to identify the longitudinal dispersion coefficient.
Now let us study the difficulties and problems which may be encountered
when FDM is used for solving the advection-dispersion equations.
79
where Di+1/2 and Di- 1/2 are the dispersion coefficients between node i and its
adjacent nodes i + 1 and i - 1, respectively. The corresponding difference
equations can be derived without difficulty. For example, we have the following implicit difference scheme:
C i,n+1 - Ci,n _ D i+1/2 C i+1,n+1 - (Di+1/2 + D i- 1/2 )Ci,n+1 + D i- 1/2 C i-l,n+1
!l.t
-
(!l.X)2
_ VCi+1,n+1 - Ci - 1 ,n+1
I
2!l.x
.
(4.1.29)
The node values of D may be used to express Di+1/2 and Di - 1/2 , e.g., we may
use the harmonic mean, i.e., let
2D.D·+ 1
D
-
I
I
i+1/2 - D + D '
i
i+1
2D;Di-l
Di - 1/2 = D
D .
i + i-1
The method mentioned above can also be used to solve the radial dispersion
problem:
oC
IXLA 02C
A oC
ot - -r- or2 - r Fr'
If the implicit scheme is adopted, we have the following finite difference
approximation:
C i,n+1 - Ci,n = ~[IX Ci+1,n+1 - 2Ci,n+1 + C i- 1,n+1 _ Ci+I,n+1 - C i - I ,n+1].
At
ri
L
(!l.r)2
2!l.r
After rearranging, we have
= ri(!l.r)2 C.
A!l.t I,n'
(4.1.30)
where r i is the radial distance of node i. Eq. (4.1.30) can be written for
all internal nodes. After combining with boundary conditions, a set of tridiagonal equations are formed. The concentrations of all nodes at the next
time step can be obtained using the Thomas algorithm. If the sizes of !l.t
and !l.r are appropriately controlled, the result obtained by this algorithm is
very dose to the exact solution. The program based on this algorithm is
simple and efficient, and can be used for simulating the field tests of tracer
injection in order to identify the longitudinal dispersion coefficient.
Now let us study the difficulties and problems which may be encountered
when FDM is used for solving the advection-dispersion equations.