Page 42 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

28 2 Simulating Steady-State Natural Convective Problems
2.6.2 Three-Dimensional Convective Pore-Fluid Flow Problems

The proposed progressive asymptotic approach procedure can be straightforwardly
extended to the simulation of three-dimensional convective pore-fluid flow problems
in fluid-saturated porous media (Zhao et al. 2001a, 2003a). Since the streamline
function is not available for three dimensional fluid flow problems, it is necessary
to use the particle tracking technique to show particle movements in three dimen-
sional fluid flow systems. In the particle tracking technique, a fundamental problem,
which needs to be solved effectively, is as follows. If the initial (known) location of
a particle is point A (x A , y A , z A ), where x A , y A and z A are the coordinate compo-
nents of point A in the x, y and z directions of a global coordinate system, then we

need to determine where the new location (i.e. point A ) of this particle is after a
given time interval, Δt. Clearly, if the velocity of the particle at point A is known,

then the coordinate components of point A in the global coordinate system can be
approximately determined for a small Δt as follows:

x A = x A + u A Δt, (2.67)

y A = y A + v A Δt, (2.68)

z A = z A + w A Δt, (2.69)

where x A , y A and z A are the coordinate components of point A in the x, y and

z directions of the global coordinate system; u A , v A and w A are the velocity com-
ponents of point A in the x, y and z directions of the global coordinate system,
respectively.
In general cases, the location of point A is not coincident with the nodal points
in a finite element analysis so that the consistent interpolation of the finite element
solution is needed to determine the velocity components at this point. For this pur-
pose, it is essential to find the coordinate components of point A in the local coordi-
nate system from the following equations for an isoparametric finite element.
n

x A = φ i (ξ A ,η A ,ζ A )x i , (2.70)
i=1
n

y A = φ i (ξ A ,η A ,ζ A )y i , (2.71)
i=1
n

z A = φ i (ξ A ,η A ,ζ A )z i , (2.72)
i=1

where x A , y A and z A are the coordinate components of point A in the x, y and z direc-
tions of the global coordinate system; ξ A , η A and ζ A are three coordinate components