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3. Analytic Solutions of Hydrodynamic Dispersion Equations
The limitation of analytic solutions is obvious. The conditions of existing
an anlytic solution are very strict: the flow region must be regular, the structure of governing equation must be simple, all model parameters must be
constant. Although these requirements can never be satisfied in practice,
the analytic solution is still important in the study of advection-dispersion
problems. It has at least the following two major uses:
1. Since the analytic solutions are exact solutions, we can use it to verify the
accuracy of a numerical method and test the correctness of a program.
2. Analytic solutions can be used to simulate some simple tests conducted in
the field and in the laboratory for better understanding the mechanics of
dispersion and identifying dispersion parameters.
Exercises
3.1. Assume that there are two instantaneous point sources located at
(X1'Yl,zd and (X 2'Y2,Z2)' respectively. Find the concentration distribution by superposition.
3.2. Assume that there is an instantaneous point source located at the center
of two parallel walls in the x-Y plane. The lengths of the walls are infinite.
Find the concentration distribution by the image method.
3.3. Derive Eq. (3.1.21) from Eq. (3.1.19).
3.4. Write a subroutine for calculating the concentration distribution of Eq.
(3.1.31) for the continuous injection problem with input parameters:
D l l , D 22 , D 33 , U,!VI and ().
3.5. Write a subroutine to calculate C(x, t) in Eq. (3.2.15) with input parameters Co, D L , and V. Let V = 1, change D L from 0.01 to 10, draw the
breakthrough curves for t = 30.
3.6. Draw a figure to display the model given in Eqs. (3.2.33) and (3.2.34).
3.7. Derive the radial model (3.2.43) from the general advection-dispersion
Eq. (2.6.10). What assumptions are used in deriving the radial model
(3.2.43)?
3.8. Extend the mass transport model presented in Section 3.2.4, from the
case of a single fracture to the case of a system of parallel fractures.
3. Analytic Solutions of Hydrodynamic Dispersion Equations
The limitation of analytic solutions is obvious. The conditions of existing
an anlytic solution are very strict: the flow region must be regular, the structure of governing equation must be simple, all model parameters must be
constant. Although these requirements can never be satisfied in practice,
the analytic solution is still important in the study of advection-dispersion
problems. It has at least the following two major uses:
1. Since the analytic solutions are exact solutions, we can use it to verify the
accuracy of a numerical method and test the correctness of a program.
2. Analytic solutions can be used to simulate some simple tests conducted in
the field and in the laboratory for better understanding the mechanics of
dispersion and identifying dispersion parameters.
Exercises
3.1. Assume that there are two instantaneous point sources located at
(X1'Yl,zd and (X 2'Y2,Z2)' respectively. Find the concentration distribution by superposition.
3.2. Assume that there is an instantaneous point source located at the center
of two parallel walls in the x-Y plane. The lengths of the walls are infinite.
Find the concentration distribution by the image method.
3.3. Derive Eq. (3.1.21) from Eq. (3.1.19).
3.4. Write a subroutine for calculating the concentration distribution of Eq.
(3.1.31) for the continuous injection problem with input parameters:
D l l , D 22 , D 33 , U,!VI and ().
3.5. Write a subroutine to calculate C(x, t) in Eq. (3.2.15) with input parameters Co, D L , and V. Let V = 1, change D L from 0.01 to 10, draw the
breakthrough curves for t = 30.
3.6. Draw a figure to display the model given in Eqs. (3.2.33) and (3.2.34).
3.7. Derive the radial model (3.2.43) from the general advection-dispersion
Eq. (2.6.10). What assumptions are used in deriving the radial model
(3.2.43)?
3.8. Extend the mass transport model presented in Section 3.2.4, from the
case of a single fracture to the case of a system of parallel fractures.