Page 91 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

78 4 Algorithm for Simulating Fluid-Rock Interaction Problems

the diffusion/dispersion term are identically equal to zero, for solid minerals in the
rock mass. In order to solve these two types of transport equations simultaneously,
we need to develop the new concept of the generalized concentration for the solid
mineral. The generalized concentration of a solid mineral is defined as the moles of
the solid mineral per unit total volume (i.e. the volume of void plus the volume of
solid particles) of the porous rock mass. Using this new concept, the general form
of the second type of transport equation to be solved in this study can be expressed
as follows:

∂C Gi
= φR i (i = 1, 2,... , m), (4.8)
∂t
where C Gi is the generalized concentration of solid mineral/species i; φ is the poros-
ity of the porous rock mass; R i is the reaction rate of solid mineral/species i and m
is the number of the solid minerals to be considered in the system.
For the reactive aqueous species, the general form of the first type of trans-
port equation in a two dimensional pore-fluid saturated, isotropic and homogeneous
porous medium reads

∂C Ci ∂C Ci ∂C Ci e 2
φ + u x + u y = D ∇ C Ci + φR i (i = m + 1, m + 2,..., n),
i
∂t ∂x ∂y
(4.9)
where C Ci is the conventional concentration of aqueous species i; u x and u y are the
e
Darcy velocities of pore-fluid in the x and y directions respectively; D is of the
i
following form:
e
D = φD i , (4.10)
i
where D i is the dispersivity of aqueous species i.
It needs to be pointed out that under some circumstances, where the reactant
chemical species constitute only a small fraction of the whole matrix in a porous
medium (Phillips 1991), the total flux of pore-fluid flow in a horizontal aquifer may
be approximately considered as a constant. For example, in a groundwater supply
system, groundwater can be pumped out from a horizontal aquifer at a constant flow
rate. In a geological system, topographically induced pore-fluid flow can also flow
through a horizontal aquifer at a constant flow rate. This indicates that in order to
satisfy the mass conservation requirement of the pore-fluid in the above-mentioned
aquifers, the total flux of the pore-fluid flow should be constant through all vertical
cross-sections, which are perpendicular to the direction of pore-fluid flow. There-
fore, the Darcy velocity in the flow direction is constant if the horizontal aquifer has
a constant thickness. The reason for this is that the Darcy velocity is the velocity
averaged over the total area of a representative elementary area (Zhao et al. 1998e,
Phillips 1991, Nield and Bejan 1992, Zhao et al. 1994c), rather than over the pore
area of the representative elementary area. In other words, the Darcy velocity in
the above-mentioned aquifers can be maintained at a constant value, even though