Page 52 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

38 3 Algorithm for Simulating Coupled Problems in Hydrothermal Systems

computational codes in combination, we attempt to use a combination of FIDAP
(Fluid Dynamics International 1997) and FLAC (Itasca Consulting Group 1995)
for solving a fully coupled problem between medium deformation, pore-fluid flow,
heat transfer and reactive species transport in a porous medium under high Rayleigh
number situations. FIDAP is a well developed, finite element method based, com-
putational fluid dynamics code, whereas FLAC, designed for civil engineering,
is based on a finite difference method, but can accommodate unstructured grids.
FIDAP can be used to model pore-fluid flow, heat transfer and reactive species trans-
port in a porous medium, but does not treat the medium deformation effects. On
the other hand, FLAC is very powerful in its modeling of geomechanical and geo-
logical deformation processes, especially for the simulation of large deformation
problems. However, the weakness of FLAC is that it cannot be used rigorously to
model steady-state pore-fluid convection and the related reactive species transport
in a fluid-saturated porous medium. Thus, it is very reasonable to envisage interac-
tively using FIDAP and FLAC for solving a fully coupled problem between medium
deformation, pore-fluid flow, heat transfer and reactive mass transport in a porous
medium under high Rayleigh number situations. In order to do this, it must be pos-
sible to relate accurately any point in the mesh for one code to the equivalent point
in the mesh for the other code.
To do this, we present a consistent point-searching interpolation algorithm, also
known as the consistent point-searching algorithm for solution interpolation in
unstructured meshes consisting of 4-node bilinear quadrilateral elements. The pro-
posed algorithm has the following significant advantages: (1) 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 mesh 2 to mesh 1, only one element in mesh 2 needs to
be inversely mapped. This certainly minimizes the number of elements to which the
inverse mapping is applied. In this regard, the proposed consistent algorithm is very
effective and efficient. (2) Analytical solutions to the local coordinates of any point
in a four-node quadrilateral element, which are derived in a rigorous mathemati-
cal manner in the context of this chapter, make it possible to carry out an inverse
mapping process very effectively and efficiently. (3) The use of consistent interpo-
lation enables the interpolated solution to be compatible with an original solution
and therefore guarantees the interpolated solution of extremely high accuracy. Since
the algorithm is very general and robust, it makes it possible to translate and transfer
data between FIDAP and FLAC, and vice versa.

3.1 Statement of the Coupled Problem and Solution Method

In terms of simulating the physical and chemical processes associated with ore
body formation and mineralization in hydrothermal systems within the upper crust
of the Earth, the fully coupled problem between material deformation, pore-fluid
flow, heat transfer and mass transport/chemical reactions can be divided into two
sub-problems (Zhao et al. 1999c, 2000b). For the first sub-problem, which is the