Page 231 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

222 Summary Statements

of observed geoscience phenomena, rather than to describe the observed phe-
nomena themselves.
(2) Convective pore-fluid flow, known as the steady-state Horton-Rogers-Lapwood
problem, is an important mechanism to control ore body formation and miner-
alization in hydrothermal systems within the upper crust of the Earth. This kind
of problem belongs to a kind of bifurcation problem, from a nonlinear math-
ematics point of view. A progressive asymptotic approach procedure has been
presented for solving the steady-state Horton-Rogers-Lapwood problem in a
fluid-saturated porous medium. This problem possesses a bifurcation and there-
fore makes the direct use of conventional finite element methods difficult. Even
if the Rayleigh number is high enough to drive the occurrence of natural con-
vection in a fluid-saturated porous medium, the conventional methods often pro-
duce a trivial non-convective solution. This difficulty can be overcome using the
progressive asymptotic approach procedure associated with the finite element
method. The method considers a series of modified Horton-Rogers-Lapwood
problems in which gravity is assumed to tilt a small angle away from ver-
tical. The main idea behind the progressive asymptotic approach procedure
is that through solving a sequence of such modified problems with decreas-
ing tilt, an accurate non-zero velocity solution to the Horton-Rogers-Lapwood
problem can be obtained. This solution provides a very good initial prediction
for the solution to the original Horton-Rogers-Lapwood problem so that the
non-zero velocity solution can be successfully obtained when the tilted angle
is set to zero. Comparison of numerical solutions with analytical ones to a
benchmark problem of any rectangular geometry has demonstrated the use-
fulness of the proposed progressive asymptotic approach procedure for deal-
ing with convective pore-fluid flow problems within the upper crust of the
Earth.
(3) To deal with coupled problem between material deformation, pore-fluid flow,
heat transfer, mass transport and chemical reactions in hydrothermal sys-
tems within the upper crust of the Earth, the combined use of two or more
commercially available computer codes is a favorable choice. A consistent
point-searching algorithm for solution interpolation in unstructured meshes
consisting of 4-node bilinear quadrilateral elements has been presented to trans-
late and transfer solution data between two totally different meshes that are used
in two different computer codes, both commercially available. The proposed
algorithm has the following significant advantages: first, the use of a point-
searching strategy allows a point in one mesh to be accurately related to an
element (containing this point) in another mesh. Thus, to translate/transfer the
solution of any particular point from the mesh used in one computer code to
that in another computer code, only one element needs to be inversely mapped.
This certainly minimizes the number of elements, to which the inverse map-
ping is applied, so that the present algorithm is very effective and efficient.
Second, analytical solutions to the local coordinates of any point in a four-node
quadrilateral element, which are derived in a rigorous mathematical manner,
make it possible to carry out an inverse mapping process very effectively and