Perhaps the most significant change in the field of metamorphic fluid flow
over the last quadrennium was the trend toward abandonment of fluid-rock
ratios in favor of time-integrated fluid fluxes as the principal measures
of ``amounts'' of fluid that flowed during metamorphism. Although both
parameters can serve as relative measures of the degree of fluid-rock
interaction, fluid fluxes are preferred over fluid-rock ratios because
numerical values for the latter are difficult to interpret in terms of
hydrodynamics. For example, a reactive rock column that has experienced
steady one-dimensional advection of fluid for an arbitrary time interval
can record fluid-rock ratios ranging from 0 to 
along the flow
path while simultaneously recording a single time-integrated flux of
fluid. The latter quantity has more physical significance than the former
in that it was stipulated that fluid did indeed pass everywhere through
the entire length of the column.
The shift away from fluid-rock ratios and toward fluid fluxes was due largely to a paper by Baumgartner and Ferry [1991]. They showed that the time-integrated form of the equation of continuity, or mass balance equation, for a species i in the fluid phase,
can be rearranged to yield a straightforward equation that
relates time-integrated molar fluid flux 
(mol/m
) to
reaction progress 
:
where 
is the total moles of fluid species i produced or
consumed per unit rock volume, 
is the mole fraction of i in the
fluid phase, and 
are stoichiometric coefficients
for fluid species j in the reaction. Implicit is a low porosity and
predominance of advection of fluid over dispersive transport. In cases of
more complicated reactions, 
can be written as a sum over each
of the component linearly independent reactions that describe the complete
reaction history of the rock [ Young and Morrison, 1992].
Assuming that equilibrium between fluid and rock is maintained throughout
the flow path, Baumgartner and Ferry expanded 
to yield:
The specified changes in 
with T and pressure P are
constrained by phase equilibria owing to the assumption of local
equilibrium. The expanded equation thus describes gradient reactions where
and 
are the driving forces. The flux J
is
a measure of fluid available for reaction, but regardless of its magnitude
no reaction is permitted for 
because of the
assumption of local equilibrium. Usually, 
and so
the distribution and extent of fluid-rock reaction is controlled by
and J
. Phillips [1991] noted that the product 
is sometimes referred to as the ``rock alteration
index'' for this reason.
One of the principal uses of the equation presented by
Baumgartner and Ferry [1991] has been assessment of the direction of
fluid flow relative to thermal gradients in contact and regional
metamorphic terranes. For this purpose only the signs of the various terms
are decisive. In many common metamorphic terranes phase equilibria
dictates positive 
. The sign of the rock
alteration index, and hence the polarity of reaction progress, is then
determined by the sign of 
(flux is positive by definition).
Baumgartner and Ferry argued that for
forward progress of many metamorphic reactions to have occurred, as
indicated by field observations, 
must have been positive,
i.e., flow must have been in the direction of increasing temperature.
Dipple and Ferry [1992] presented equations and interpretations similar to those of Baumgartner and Ferry, but formulated explicitly for stable isotope exchange reactions between fluid and rock. Although not expressed in the same fashion, for purposes of illustration one can show that the salient features of the model of Dipple and Ferry are embodied in a time-integrated flux-reaction progress equation analogous to that of Baumgartner and Ferry:
where 
is the equilibrium rock/fluid isotope ratio
fractionation factor (essentially 1), 
is a
time-integrated flux of fluid oxygen (as traced by 
,
parameter Bt of Dipple and Ferry), and 
(
is the per mil deviation of 
O
O
from a standard)
signifies the initial 
for the
rock at a particular position along the flow path. Dipple and Ferry
presented detailed 
distance profiles for a
variety of circumstances. The principal implications of these
profiles are gleaned from the equation above. The equation is seen
immediately to apply to gradient-controlled exchange of 
and 
O between fluid and rock; in the absence of a
temperature gradient 
, no shift in rock 
occurs and the extent of isotopic alteration is a function of
the rock alteration index J
.
For the typical case of flow of H
O through carbonate or
quartzo-feldspathic lithologies, 
is positive so that flow up temperature, or
positive 
, results in a decrease in rock 
at any location on the flow path, while flow down temperature, or
negative 
, results in an increase in rock 
.
In their analysis of isotope exchange Dipple and Ferry gave a cogent
illustration of the effects of exhaustion of a reactant during progress of
a gradient reaction in a continuum. In this case it is the 
exchange capacity of the rock that is depleted. An
isotopic front then develops. Unlike fluid-driven fronts, the shape of the
gradient reaction fronts are strongly influenced by the rock alteration
index. Because the infiltrated fluid enters the system in equilibrium with
the rock there is no step in 
to be propagated
downstream, as there is in a fluid-driven front. The gradient front is
therefore defined by a discontinuity in 
rather than a
step. The difference between fluid-driven and gradient exchange
fronts is topologically reminiscent of the
dissemblance between first-order and second-order reaction enthalpies,
respectively (for the thermodynamically inclined reader).
Upstream of the gradient isotope exchange front 
is constant and equal to the initial value at the inlet of the
flow system. Rock 
values are controlled by this
constant initial fluid 
and temperature. Downstream
the system exhibits the gradual changes typical of gradient reactions.