2.6. Extensions
45
and time t. To obtain a particular solution, we not only need the dispersion
equation, but also some other specifications, as described below:
1. The space domain (R) and time interval (0, T) of the considered problem
must be provided.
2. The flow field in the considered flow domain (R) and the distributions of
relevant parameters in the equation, such as mean flow velocity V(x, t),
o < t < T, and (XL' (XT' Dd T, Rd , Ä, W, and so forth, must be provided.
3. The initial condition for (R) and boundary conditions for its boundary
(B) must be provided. The initial condition is the concentration distribution at the instant t = 0:
C(X,O) = Co (x); X E (R),
(2.6.30)
where t = 0 is an arbitrary initial time. Co(x) is a known function of
position x. For example, if at t = 0, a solution with a tracer is injected
into (R), but no tracer had existed in (R) before, then Co(x) = O.
Similar to groundwater flow simulation, there are three types of boundary
condition.
The first type of boundary condition specifies concentration distribution
along the boundary, Le., it is given by
C(xB, t) = gl (x B , t),
o < t < T, X B E (BI)'
(2.6.31)
where (Bd is part of the boundary of (R), X B a point on the boundary, and
gl (XB, t) a known function. This type of boundary condition is often called
the Dirichlet boundary condition.
The second type of boundary condition gives a known dispersion flux
along the boundary. It is often called the Neumann boundary condition and
given by
ac I
-DiFa.ni
= g2(XB,t)
xJ XB
o < t < T, xB E (B2 ),
(2.6.32)
where (B 2 ) is also part of the boundary of (R), n i (i = 1,2,3) are components
of the unit normal vector of (B2 ), and g2(X B , t) is a known function.
The third type of boundary condition defines the solute transport flux
at the boundary surface. It is also called Cauchy boundary condition and
given by
(2.6.33)
where (B 3 ) is part of the boundary of (R), and g3(X B , t) a known function.