Page 232 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

Summary Statements 223

efficiently. Third, the use of consistent interpolation enables the interpolated
solution to be compatible with an original solution and therefore guarantees
the interpolated solution of extremely high accuracy. The related results from
the test problem have demonstrated the generality, accuracy, effectiveness, effi-
ciency and robustness of the proposed consistent point-searching integration
algorithm.
(4) To effectively and efficiently use the finite element method for solving fluid-
rock interaction problems of subcritical Zhao numbers in pore-fluid saturated
hydrothermal/sedimentary basins, A term splitting algorithm on the basis of a
new concept of the generalized concentration of a solid mineral has been pre-
sented to deal with the following three fundamental issues associated with the
fluid-rock interaction problems. First, since the fluid-rock interaction problem
involves heterogeneous chemical reactions between reactive aqueous chemical
species in the pore-fluid and solid minerals in the rock masses, it is necessary
to develop a new concept of the generalized concentration of a solid mineral,
so that two types of reactive mass transport equations, namely the conven-
tional mass transport equation for the aqueous chemical species in the pore-
fluid and the degenerated mass transport equation for the solid minerals in the
rock mass, can be solved simultaneously in computation. Second, because the
reaction area between the pore-fluid and mineral surfaces is basically a function
of the generalized concentration of the solid mineral, there is a definite need
to appropriately consider the dependence of the dissolution rate of a dissolv-
ing mineral on its generalized concentration in the numerical analysis. Third,
to consider porosity evolution with time in the transient analysis of fluid-rock
interaction problems, the concept of the equivalent source/sink terms in mass
transport equations needs to be developed to convert the problem of variable
mesh Peclet number and Courant number into the problem of constant mesh
Peclet and Courant numbers. The related numerical results have demonstrated
the usefulness and robustness of the proposed term splitting algorithm for solv-
ing fluid-rock interaction problems of subcritical Zhao numbers in pore-fluid
saturated hydrothermal and sedimentary basins.
(5) The chemical-dissolution-front propagation problem exists ubiquitously not
only in ore forming systems within the upper crust of the Earth, but also in
many other scientific and engineering fields. To solve this problem, it is nec-
essary to deal with a coupled system between porosity, pore-fluid pressure and
reactive chemical-species transport in fluid-saturated porous media. Due to the
morphological instability of a chemical dissolution front, this problem needs
to be solved numerically. A segregated algorithm on the basis of a combi-
nation of the finite element and finite difference methods has been proposed
for simulating the morphological evolution of chemical dissolution fronts in
reactive transport systems of critical and supercritical Zhao numbers. A set of
analytical solutions have been derived for a benchmark problem to verify the
proposed numerical procedure. Not only can the derived analytical solutions
be used to verify any numerical method before it is used to solve this kind
of chemical-dissolution-front propagation problem, but also they can be used