2.6. Extensions
39
For two-dimensional flow, we have
Dll = (()(J7/ + ()(TVl)/V + Dd T
D12 = D21 = (()(L - ()(T)V1 V2 /V
D22 = (()(TV/ + ()(LVl)/V + DdT.
2.6 Extensions and Subsidiary Conditions of the
Hydrodynamic Dispersion Equation
(2.5.18)
2.6.1 Hydrodynamic Dispersion Equations in Orthogonal
Curvilinear Coordinate Systems
In this section we will derive expressions for the hydrodynamic dispersion
equation in different co ordinate systems, the source and sink terms, and the
relevant initial and boundary conditions, in order to obtain a complete
mathematical statement of a hydrodynamic dispersion problem. Since we are
concentrating only on the problems relevant to porous media, all of the
parameters under consideration are their macroscopic averages. Hence, the
bars above the symbols will be omitted. For example, unless otherwise specified, parameters V, c, p and J-l, etc., all denote their macroscopic means.
Let (x l ' x 2 , x 3 ) be the coordinates of point P in the Cartesian co ordinate
system and (u t , U 2 , u 3 ) be the coordinates of the same point in a curvilinear
coordinate system. The relationship between the two systems is
(2.6.1)
We require that the Jacobian
OXt oX 2 oX3
oU 1 OUt
OUt
D(x t ,X 2 ,X 3 )
oX 1 OX2 oX 3
J=
=
D(u t ,U 2 ,U 3 )
oU 2 oU 2 oU 2
(2.6.2)
OXt oX 2 oX 3
oU 3 oU 3 oU 3
is non-zero, so that the two coordinate systems have a one-to-one
correspondence.
If the three co ordinate directions of the curvilinear co ordinate system are
perpendicular to each other, the system is called an orthogonal curvilinear
coordinate system. Let
hi = I :~J i = 1, 2, 3,
(2.6.3)
39
For two-dimensional flow, we have
Dll = (()(J7/ + ()(TVl)/V + Dd T
D12 = D21 = (()(L - ()(T)V1 V2 /V
D22 = (()(TV/ + ()(LVl)/V + DdT.
2.6 Extensions and Subsidiary Conditions of the
Hydrodynamic Dispersion Equation
(2.5.18)
2.6.1 Hydrodynamic Dispersion Equations in Orthogonal
Curvilinear Coordinate Systems
In this section we will derive expressions for the hydrodynamic dispersion
equation in different co ordinate systems, the source and sink terms, and the
relevant initial and boundary conditions, in order to obtain a complete
mathematical statement of a hydrodynamic dispersion problem. Since we are
concentrating only on the problems relevant to porous media, all of the
parameters under consideration are their macroscopic averages. Hence, the
bars above the symbols will be omitted. For example, unless otherwise specified, parameters V, c, p and J-l, etc., all denote their macroscopic means.
Let (x l ' x 2 , x 3 ) be the coordinates of point P in the Cartesian co ordinate
system and (u t , U 2 , u 3 ) be the coordinates of the same point in a curvilinear
coordinate system. The relationship between the two systems is
(2.6.1)
We require that the Jacobian
OXt oX 2 oX3
oU 1 OUt
OUt
D(x t ,X 2 ,X 3 )
oX 1 OX2 oX 3
J=
=
D(u t ,U 2 ,U 3 )
oU 2 oU 2 oU 2
(2.6.2)
OXt oX 2 oX 3
oU 3 oU 3 oU 3
is non-zero, so that the two coordinate systems have a one-to-one
correspondence.
If the three co ordinate directions of the curvilinear co ordinate system are
perpendicular to each other, the system is called an orthogonal curvilinear
coordinate system. Let
hi = I :~J i = 1, 2, 3,
(2.6.3)