116
5. Finite Element Methods for Solving Hydrodynamic Dispersion Equations
vertical direction, and the concentration C in Eq. (5.2.1) is defined as the
average concentration over the whole thickness of the aquifer.
Assume that (D) is a domain encircled by an arbitrary curve (L) in the flow
region. Integrating Eq. (5.2.1) over (D) and using Green's formula, we then
have:
- r mDgradC'ndl + r VmC'ndl
J(L)
J(L)
= f r [o(mC) + C' WJ dx dy,
J(Dl ot
n
(5.2.2)
where n is the unit internal normal vector of (L). The first term on the
left-hand side of this equation represents the solute flux passing through
(L) into (D) resulting from hydrodynamic dispersion, while the second term
represents the solute flux resulting from advection. The right-hand side
shows the increment of solute mass in domain (D). Therefore, Eq. (5.2.2) is
just the mass conservation equation of solute in domain (D).
In Cartesian coordinates, the scalar form of Eq. (5.2.2) is
IL) m[ (Dxx ~~ + D xy ~~)dY - (D xy ~~ + D yy ~~)dX ]
+ r mC(Yydx - Vxdy)
JL)
= f r [a(mC) + C'W]dXd Y .
J(D)
ot
n
(5.2.3)
Sun and Yeh (1983) derived a system of discrete equations from the
conservation equation (5.2.3) directly with the aid of basis functions used in
the FEM. The mathematics associated with this method is very simple and
the errors in local and global mass conservations are quite small. This method can be used for unconfined aquifers and the saturated-unsaturated zone,
as weIl as for confined aquifers. This flexibility makes the method very useful
in practice.
Bear (1979) named a numerical method, which is based on element mass
conservation, as "Multiple Cell Balance Method." We use the name because
it vividly describes the essence of this kind of method.
5.2.2 An Algorithm Based on Multiple Cell Balance
First, partition the domain into a tri angular net and consider an arbitrary
element (e) (see Figure 5.1 in Section 5.1). Let the no des be numbered as i,j, k,
and their coordinates (Xi' y;), (Xj' Yj), (xk, Yd. In this element, the concentra-
5. Finite Element Methods for Solving Hydrodynamic Dispersion Equations
vertical direction, and the concentration C in Eq. (5.2.1) is defined as the
average concentration over the whole thickness of the aquifer.
Assume that (D) is a domain encircled by an arbitrary curve (L) in the flow
region. Integrating Eq. (5.2.1) over (D) and using Green's formula, we then
have:
- r mDgradC'ndl + r VmC'ndl
J(L)
J(L)
= f r [o(mC) + C' WJ dx dy,
J(Dl ot
n
(5.2.2)
where n is the unit internal normal vector of (L). The first term on the
left-hand side of this equation represents the solute flux passing through
(L) into (D) resulting from hydrodynamic dispersion, while the second term
represents the solute flux resulting from advection. The right-hand side
shows the increment of solute mass in domain (D). Therefore, Eq. (5.2.2) is
just the mass conservation equation of solute in domain (D).
In Cartesian coordinates, the scalar form of Eq. (5.2.2) is
IL) m[ (Dxx ~~ + D xy ~~)dY - (D xy ~~ + D yy ~~)dX ]
+ r mC(Yydx - Vxdy)
JL)
= f r [a(mC) + C'W]dXd Y .
J(D)
ot
n
(5.2.3)
Sun and Yeh (1983) derived a system of discrete equations from the
conservation equation (5.2.3) directly with the aid of basis functions used in
the FEM. The mathematics associated with this method is very simple and
the errors in local and global mass conservations are quite small. This method can be used for unconfined aquifers and the saturated-unsaturated zone,
as weIl as for confined aquifers. This flexibility makes the method very useful
in practice.
Bear (1979) named a numerical method, which is based on element mass
conservation, as "Multiple Cell Balance Method." We use the name because
it vividly describes the essence of this kind of method.
5.2.2 An Algorithm Based on Multiple Cell Balance
First, partition the domain into a tri angular net and consider an arbitrary
element (e) (see Figure 5.1 in Section 5.1). Let the no des be numbered as i,j, k,
and their coordinates (Xi' y;), (Xj' Yj), (xk, Yd. In this element, the concentra-