Technical Commentary/
Flux Flummoxed: A Proposal for Consistent Usage
by Philip H. Stauffer
I would like to bring to the attention of the hydrogeology community an ongoing inconsistency in the published literature concerning the use of the term flux. The
definition of flux that is most pertinent to Ground Water
readers comes from the field of transport phenomena,
where the flux of some quantity (e.g., mass, energy,
momentum, entropy) is defined as the flow rate of that
quantity per unit area. For example, a mass flow rate,
which has SI units of kg/s, when referenced to a unit area,
results in mass flux having SI units of kg/(m2 s). Because
this definition includes a direction (i.e., the surface normal of the unit area), flux is a vector. I would like to propose that the community agrees to consistently use this
definition in technical presentations, published literature,
and, most importantly, classrooms. To support this proposal, I first present details on the two primary definitions
of flux currently used in physics, describe the fluxes most
commonly found in hydrogeology, continue with a brief
history of the usage of the term flux, and then give some
examples of uses that are either incorrect or confusing.
1983). Continuum mechanics and vector analysis are the
primary tools used to construct and solve the governing
equations of ground water flow and solute transport (Bear
1972; de Marsily 1986; Anderson and Woessner 1992;
Fetter 1999). For example, because it is a vector, mass
flux ( fm) can be used in transport equations such as the
statement of mass continuity as r( fm) 2 (mass accumulation rate) ¼ 0, where r is the divergence operator and
bold type is used to denote a vector quantity (Boas 1983).
Conversely, the EM definition leads to magnetic or electric flux being a scalar quantity that has no time component. These two distinct definitions have been used for
more than 100 years. Recently, however, there has been
a trend to mix these two definitions without being clear
as to which usage is intended. One possible explanation
for this trend is that some EM textbooks use the flow
of water as an analogy for magnetic flux, apparently
unaware that the transport definition of flux is quite different (Lorrain and Corson 1962). Additional confusion
comes from sources such as Chen (1995), who mistakenly
applies an EM style integral definition of flux to the
transport of mass through a finite surface.
Flux Defined
In the various subfields of physics, there exist two
common usages of the term flux with rigorous mathematical frameworks. First, from the field of unified transport
phenomena (momentum, heat, and mass transport), the
flux of some quantity is defined as the flow rate of that
quantity per unit area (Bird et al. 2002). Second, the field
of electromagnetism (EM) defines flux as the surface
integral of a vector field (Lorrain and Corson 1962).
Because transport flux is defined with respect to the outward normal of the reference area, flux is a vector at each
point in space (Carslaw and Jaeger 1959; Bird et al.
2002). Vectors have special properties and implied meaning in both mathematics and continuum mechanics (Boas
Mail Stop T003, Earth and Environmental Sciences Division,
Los Alamos National Laboratory, Los Alamos, NM 87545; (505)
665 4638; stauffer@lanl.gov
Received October 2005, accepted November 2005.
Journal compilation ª 2006 National Ground Water Association.
No claim to original US government works.
doi: 10.1111/j.1745-6584.2006.00197.x
Fluxes in Hydrogeology
The most common flux used in the field of hydrogeology is the volumetric flux (q), expressed by the generalized form of Darcy’s law as follows: q ¼ 2Kr(h),
with SI units of m3/(m2 s) where K is the hydraulic conductivity tensor as a function of water content and h is the
hydraulic head (Evans et al. 2001). Although the correct
English term is volumetric flux, I would like to suggest
that for consistency with mass flux, energy flux, solute
flux, and heat flux, we use the term volume flux to
describe q in Darcy’s law. An added benefit of this consistent usage is that students will have a much easier time
making transitions between discussions of Darcy’s law,
Fourier’s law, and Fick’s law. In these three cases, the
volume flux (q), thermal energy flux (qt), and diffusive
chemical flux (j) are related to the spatial gradients of
scalar fields multiplied by a function of material properties (e.g., qt ¼ 2Ktr (temperature) and j ¼ 2Dr (concentration) where Kt and D are the thermal conductivity
tensor and chemical diffusion tensor, respectively) (Bejan
Vol. 44, No. 2—GROUND WATER—March–April 2006 (pages 125–128)
125
1995). Although one might be tempted to define flux
based on the notion of the spatial gradient of a scalar field,
many fluxes, including mass flux with a high Reynolds
number and neutron flux, are not related to a scalar field.
History of Flux in Transport Phenomena
One of the first instances of the use of the term flux
to describe transport was by Maxwell (1891):‘‘the flux of
heat at any point of a solid body may be defined as the
quantity of heat which crosses a small area drawn perpendicular to that direction divided by that area and by the
time. Here the flux is referenced to an area.’’ Maxwell
(1891) also states that ‘‘In the case of fluxes, we have to
take the integral, over a surface, of the flux through every
element of the surface. The result of this operation is
called the Surface integral of the flux. It represents the
quantity which passes through the surface.’’ Thus, the
integral of the flux of some quantity is defined as the flow
rate of that quantity through the surface of integration.
Carslaw and Jaeger (1959) state that ‘‘The rate at which
heat is transferred across any surface S at a point P, per
unit area per unit time, is called the flux of heat.’’ The authors note that the units of heat flux are energy per unit
area per unit time. Additionally, after presenting the onedimensional Fourier equation for heat flux, they develop
the general form of the equation in three dimensions and
arrive at the heat flux vector (qt) that is, for the rest of the
book, referred to as simply ‘‘heat flux.’’ In one of the most
cited transport textbooks, Bird et al. (1960) state that ‘‘By
126
P.H. Stauffer GROUND WATER 44, no. 2: 125–128
flux is meant ‘rate of flow per unit area.’ Momentum flux
then has units of momentum per unit area per unit time.’’
Also discussed in Bird et al. (1960) is Fickian mass flux,
defined as mass flow rate per unit area, with SI units of
kg/(m2 s). A well-written summary of flux as related to
the laws of conservation in heat and mass transfer can be
found in Potter (1967). In this engineering sciences handbook, heat flux, work flux, and mass flux are defined as
vectors. Furthermore, the author notes that the difficult
concept of fluid velocity (nonporous) can be more easily
understood as the mass flux divided by the fluid density
(qf). Interestingly, the nonporous fluid velocity, fm/qf, reduces to the volume flux of the fluid, acknowledging that
the true velocity (dx/dt) of a fluid molecule is not so
clearly defined (e.g., turbulent flow). Bear (1972), in his
development of the equations of ground water flow,
makes a careful distinction between the specific discharge
and the specific discharge vector. However, after the
introductory material, he refers to the specific discharge
vector as simply ‘‘specific discharge.’’ Turcott and Shubert
(1982) are very clear that heat flux is a vector with SI
units of W/m2. The authors also differentiate heat flux
from heat flow, which has units of W or energy per time.
Anderson and Woessner (1992) clearly state that q in
Darcy’s law is a vector. Fetter (1999) distinguishes between the vector q and a one-dimensional q, and note that
in the general form of the transport equations, specific
discharge is a vector. I would argue that q is a vector in
either case, because if the one-dimensional Darcy equation were really just giving the magnitude of the flux
vector (a scalar), there would be no need for the negative
sign that indicates the direction of flow. In all these examples, the flux of a quantity (e.g., energy, mass, moles) is
defined as a vector. The historical record shows that
although usage of the term flux began in a fairly loose
manner, referring to both the magnitude of a vector and
the vector itself, usage quickly evolved to define flux as
a vector.
Flux Flummoxed
In the subsurface flow literature, volume flux is
described using many very different terms such as water
flux (Jury et al. 1992), volumetric flux vector (Evans et al.
2001), specific flux vector (Bear 1972), volumetric flow
rate per unit area (Turcott and Shubert 1982), Darcy
velocity (Domenico and Schwartz 1990), fictitious velocity
(Davis and DeWiest 1966), superficial velocity (Lake
1989), specific discharge (Fetter 1980), specific discharge
vector (Bear 1972), discharge rate (Anderson and Woessner
1992), flux density (Hillel 1982), filtration velocity (de
Marsily 1986), or Darcy flux (Fetter 1999). These terms
are all used to refer to the same quantity and lead to much
confusion not just among students but also among researchers in the field. Of all these terms, the most commonly used is specific discharge, which is generally
defined as follows: Q/A, where Q is the volumetric flow
rate (m3/s) and A is the area over which that volume is
flowing. Although this term is often used synonymously
with volume flux, specific discharge is not necessarily
a vector, unless the normal to the area of integration
is given. Use of the term specific flux is redundant as it is
implicit in the transport definition of flux that the flow is
per unit area. Additionally, in the field of thermodynamics, use of the term specific as a modifier has a long association with meaning ‘‘per mass,’’ as in specific volume
(m3/kg), specific surface area (m2/kg), specific enthalpy
(J/kg), and specific heat (J/(kg C)). The expression volumetric flux vector is also redundant because when using
the transport definition, flux is a vector. The use of Darcy
flux is vague to readers outside the hydrogeology field,
suggesting that this term be carefully defined in publications as a volume flux. The term Darcy velocity
should not be used as this confuses the actual average
water velocity (also called seepage velocity) and the volume flux; although both have units of length per time,
each has a very distinct meaning. The terms fictitious
velocity, filtration velocity, and superficial velocity are
also poor choices for describing volume flux as they
imply a velocity that is not real. The term flux density
apparently is a combination of the EM and transport definitions. The EM literature discusses the flux density of
a magnetic field as the magnetic field strength per area.
Flux density has no meaning with respect to the transport
definition of flux. Another common misuse is the term
flux rate, which is redundant because all transport fluxes
are rates (i.e. per unit time). This is akin to saying velocity rate. Another confusing usage, although somewhat
infrequent, is to call the volumetric flow rate the flux of
water (de Marsily 1986) or volumetric flux (Freeze and
P.H. Stauffer GROUND WATER 44, no. 2: 125–128
127
Cherry 1979). Finally, confusion is introduced in books
such as Turcott and Shubert (1982). Although the authors
make careful definitions of heat flux (J/(m2 s)) and heat
flow (J/s) in equations 4-1 and 4-5, throughout the text
they mix these terms in a loose and imprecise manner.
Summary
This technical commentary shows that the most appropriate definition of flux, as related to ground water, is
that used in the field of transport phenomena. With respect to transport phenomena, the flux of some quantity is
defined as the flow rate of that quantity per unit area.
Because flux is defined with respect to a direction, flux is
a vector at each point in space. Vectors have special properties and implied meaning in both mathematics and continuum mechanics. I propose that the transport definition
of flux be used consistently in technical presentations,
published papers, books, and earth science classrooms.
Consistent usage will help in teaching the next generation
of hydrogeologists that flux has a definite meaning in
science.
Acknowledgments
Reprinted (heavily revised) with permission from the
October 2005 issue of The Hydrogeologist, Newsletter of
the Geological Society of America Hydrogeology Division. Special thanks to Eleanor Dixon, Don Neeper, Carl
Gable, Ioannis Tsimpanogiannis, Kay Birdsell, Bruce
Robinson, George Zyvoloski, Dan Levitt, Josh Stein, Andy
Fisher, Chris Neuzil, David Deming, and Mary Anderson
for their very helpful and interactive reviews of this
commentary.
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