Technical Commentary/

Flux Flummoxed: A Proposal for Consistent Usageby Philip H. Stauffer

I would like to bring to the attention of the hydro-geology community an ongoing inconsistency in the pub-lished literature concerning the use of the term flux. Thedefinition of flux that is most pertinent to Ground Waterreaders 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 thatquantity 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). Becausethis definition includes a direction (i.e., the surface nor-mal of the unit area), flux is a vector. I would like to pro-pose that the community agrees to consistently use thisdefinition in technical presentations, published literature,and, most importantly, classrooms. To support this pro-posal, I first present details on the two primary definitionsof flux currently used in physics, describe the fluxes mostcommonly found in hydrogeology, continue with a briefhistory of the usage of the term flux, and then give someexamples of uses that are either incorrect or confusing.

Flux DefinedIn the various subfields of physics, there exist two

common usages of the term flux with rigorous mathemat-ical frameworks. First, from the field of unified transportphenomena (momentum, heat, and mass transport), theflux of some quantity is defined as the flow rate of thatquantity per unit area (Bird et al. 2002). Second, the fieldof electromagnetism (EM) defines flux as the surfaceintegral of a vector field (Lorrain and Corson 1962).Because transport flux is defined with respect to the out-ward normal of the reference area, flux is a vector at eachpoint in space (Carslaw and Jaeger 1959; Bird et al.2002). Vectors have special properties and implied mean-ing in both mathematics and continuum mechanics (Boas

1983). Continuum mechanics and vector analysis are theprimary tools used to construct and solve the governingequations of ground water flow and solute transport (Bear1972; de Marsily 1986; Anderson and Woessner 1992;Fetter 1999). For example, because it is a vector, massflux ( fm) can be used in transport equations such as thestatement of mass continuity as r�( fm) 2 (mass accumu-lation rate) ¼ 0, where r� is the divergence operator andbold type is used to denote a vector quantity (Boas 1983).Conversely, the EM definition leads to magnetic or elec-tric flux being a scalar quantity that has no time compo-nent. These two distinct definitions have been used formore than 100 years. Recently, however, there has beena trend to mix these two definitions without being clearas to which usage is intended. One possible explanationfor this trend is that some EM textbooks use the flowof water as an analogy for magnetic flux, apparentlyunaware that the transport definition of flux is quite dif-ferent (Lorrain and Corson 1962). Additional confusioncomes from sources such as Chen (1995), who mistakenlyapplies an EM style integral definition of flux to thetransport of mass through a finite surface.

Fluxes in HydrogeologyThe most common flux used in the field of hydro-

geology is the volumetric flux (q), expressed by the gen-eralized form of Darcy’s law as follows: q ¼ 2Kr(h),with SI units of m3/(m2 s) where K is the hydraulic con-ductivity tensor as a function of water content and h is thehydraulic head (Evans et al. 2001). Although the correctEnglish term is volumetric flux, I would like to suggestthat for consistency with mass flux, energy flux, soluteflux, and heat flux, we use the term volume flux todescribe q in Darcy’s law. An added benefit of this con-sistent usage is that students will have a much easier timemaking transitions between discussions of Darcy’s law,Fourier’s law, and Fick’s law. In these three cases, thevolume flux (q), thermal energy flux (qt), and diffusivechemical flux (j) are related to the spatial gradients ofscalar fields multiplied by a function of material proper-ties (e.g., qt ¼ 2Ktr (temperature) and j ¼ 2Dr (con-centration) where Kt and D are the thermal conductivitytensor and chemical diffusion tensor, respectively) (Bejan

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 compilationª 2006 National Ground Water Association.No claim to original US government works.doi: 10.1111/j.1745-6584.2006.00197.x

Vol. 44, No. 2—GROUND WATER—March–April 2006 (pages 125–128) 125

1995). Although one might be tempted to define fluxbased on the notion of the spatial gradient of a scalar field,many fluxes, including mass flux with a high Reynoldsnumber and neutron flux, are not related to a scalar field.

History of Flux in Transport PhenomenaOne of the first instances of the use of the term flux

to describe transport was by Maxwell (1891):‘‘the flux ofheat at any point of a solid body may be defined as thequantity of heat which crosses a small area drawn perpen-dicular to that direction divided by that area and by thetime. Here the flux is referenced to an area.’’ Maxwell(1891) also states that ‘‘In the case of fluxes, we have totake the integral, over a surface, of the flux through everyelement of the surface. The result of this operation iscalled the Surface integral of the flux. It represents thequantity which passes through the surface.’’ Thus, theintegral of the flux of some quantity is defined as the flowrate of that quantity through the surface of integration.Carslaw and Jaeger (1959) state that ‘‘The rate at whichheat is transferred across any surface S at a point P, perunit area per unit time, is called the flux of heat.’’ The au-thors note that the units of heat flux are energy per unitarea per unit time. Additionally, after presenting the one-dimensional Fourier equation for heat flux, they developthe general form of the equation in three dimensions andarrive at the heat flux vector (qt) that is, for the rest of thebook, referred to as simply ‘‘heat flux.’’ In one of the mostcited transport textbooks, Bird et al. (1960) state that ‘‘By

flux is meant ‘rate of flow per unit area.’ Momentum fluxthen 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 ofkg/(m2 s). A well-written summary of flux as related tothe laws of conservation in heat and mass transfer can befound in Potter (1967). In this engineering sciences hand-book, heat flux, work flux, and mass flux are defined asvectors. Furthermore, the author notes that the difficultconcept of fluid velocity (nonporous) can be more easilyunderstood as the mass flux divided by the fluid density(qf). Interestingly, the nonporous fluid velocity, fm/qf, re-duces to the volume flux of the fluid, acknowledging thatthe true velocity (dx/dt) of a fluid molecule is not soclearly defined (e.g., turbulent flow). Bear (1972), in hisdevelopment of the equations of ground water flow,makes a careful distinction between the specific dischargeand the specific discharge vector. However, after theintroductory material, he refers to the specific dischargevector as simply ‘‘specific discharge.’’ Turcott and Shubert(1982) are very clear that heat flux is a vector with SIunits of W/m2. The authors also differentiate heat fluxfrom heat flow, which has units of W or energy per time.Anderson and Woessner (1992) clearly state that q inDarcy’s law is a vector. Fetter (1999) distinguishes be-tween the vector q and a one-dimensional q, and note thatin the general form of the transport equations, specificdischarge is a vector. I would argue that q is a vector ineither case, because if the one-dimensional Darcy equa-tion were really just giving the magnitude of the flux

126 P.H. Stauffer GROUND WATER 44, no. 2: 125–128

vector (a scalar), there would be no need for the negativesign that indicates the direction of flow. In all these exam-ples, the flux of a quantity (e.g., energy, mass, moles) isdefined as a vector. The historical record shows thatalthough usage of the term flux began in a fairly loosemanner, referring to both the magnitude of a vector andthe vector itself, usage quickly evolved to define flux asa vector.

Flux FlummoxedIn the subsurface flow literature, volume flux is

described using many very different terms such as waterflux (Jury et al. 1992), volumetric flux vector (Evans et al.2001), specific flux vector (Bear 1972), volumetric flowrate per unit area (Turcott and Shubert 1982), Darcyvelocity (Domenico and Schwartz 1990), fictitious velocity(Davis and DeWiest 1966), superficial velocity (Lake1989), specific discharge (Fetter 1980), specific dischargevector (Bear 1972), discharge rate (Anderson and Woessner1992), flux density (Hillel 1982), filtration velocity (deMarsily 1986), or Darcy flux (Fetter 1999). These termsare all used to refer to the same quantity and lead to muchconfusion not just among students but also among re-searchers in the field. Of all these terms, the most com-monly used is specific discharge, which is generallydefined as follows: Q/A, where Q is the volumetric flowrate (m3/s) and A is the area over which that volume isflowing. Although this term is often used synonymouslywith volume flux, specific discharge is not necessarily

a vector, unless the normal to the area of integrationis given. Use of the term specific flux is redundant as it isimplicit in the transport definition of flux that the flow isper unit area. Additionally, in the field of thermodynam-ics, use of the term specific as a modifier has a long asso-ciation 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 volu-metric flux vector is also redundant because when usingthe transport definition, flux is a vector. The use of Darcyflux is vague to readers outside the hydrogeology field,suggesting that this term be carefully defined in pub-lications as a volume flux. The term Darcy velocityshould not be used as this confuses the actual averagewater velocity (also called seepage velocity) and the vol-ume flux; although both have units of length per time,each has a very distinct meaning. The terms fictitiousvelocity, filtration velocity, and superficial velocity arealso poor choices for describing volume flux as theyimply a velocity that is not real. The term flux densityapparently is a combination of the EM and transport defi-nitions. The EM literature discusses the flux density ofa magnetic field as the magnetic field strength per area.Flux density has no meaning with respect to the transportdefinition of flux. Another common misuse is the termflux rate, which is redundant because all transport fluxesare rates (i.e. per unit time). This is akin to saying veloc-ity rate. Another confusing usage, although somewhatinfrequent, is to call the volumetric flow rate the flux ofwater (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 bookssuch as Turcott and Shubert (1982). Although the authorsmake careful definitions of heat flux (J/(m2 s)) and heatflow (J/s) in equations 4-1 and 4-5, throughout the textthey mix these terms in a loose and imprecise manner.

SummaryThis technical commentary shows that the most ap-

propriate definition of flux, as related to ground water, isthat used in the field of transport phenomena. With re-spect to transport phenomena, the flux of some quantity isdefined as the flow rate of that quantity per unit area.Because flux is defined with respect to a direction, flux isa vector at each point in space. Vectors have special prop-erties and implied meaning in both mathematics and con-tinuum mechanics. I propose that the transport definitionof flux be used consistently in technical presentations,published papers, books, and earth science classrooms.Consistent usage will help in teaching the next generationof hydrogeologists that flux has a definite meaning inscience.

AcknowledgmentsReprinted (heavily revised) with permission from the

October 2005 issue of The Hydrogeologist, Newsletter ofthe Geological Society of America Hydrogeology Divi-sion. Special thanks to Eleanor Dixon, Don Neeper, CarlGable, Ioannis Tsimpanogiannis, Kay Birdsell, BruceRobinson, George Zyvoloski, Dan Levitt, Josh Stein, AndyFisher, Chris Neuzil, David Deming, and Mary Andersonfor their very helpful and interactive reviews of thiscommentary.

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