Flux Flummoxed: a Proposal for Consistent Usage by Philip H

Technical Commentary/ Flux Flummoxed: A Proposal for Consistent Usage by Philip H. Stauffer

I would like to bring to the attention of the hydro- 1983). Continuum mechanics and vector analysis are the geology community an ongoing inconsistency in the pub- primary tools used to construct and solve the governing lished literature concerning the use of the term flux. The equations of ground water flow and solute transport (Bear definition of flux that is most pertinent to Ground Water 1972; de Marsily 1986; Anderson and Woessner 1992; readers comes from the field of transport phenomena, Fetter 1999). For example, because it is a vector, mass where the flux of some quantity (e.g., mass, energy, flux ( fm) can be used in transport equations such as the momentum, entropy) is defined as the flow rate of that statement of mass continuity as r( fm) 2 (mass accumu- quantity per unit area. For example, a mass flow rate, lation rate) ¼ 0, where r is the divergence operator and which has SI units of kg/s, when referenced to a unit area, bold type is used to denote a vector quantity (Boas 1983). results in mass flux having SI units of kg/(m2 s). Because Conversely, the EM definition leads to magnetic or elec- this definition includes a direction (i.e., the surface nor- tric flux being a scalar quantity that has no time compo- mal of the unit area), flux is a vector. I would like to pro- nent. These two distinct definitions have been used for pose that the community agrees to consistently use this more than 100 years. Recently, however, there has been definition in technical presentations, published literature, a trend to mix these two definitions without being clear and, most importantly, classrooms. To support this pro- as to which usage is intended. One possible explanation posal, I first present details on the two primary definitions for this trend is that some EM textbooks use the flow of flux currently used in physics, describe the fluxes most of water as an analogy for magnetic flux, apparently commonly found in hydrogeology, continue with a brief unaware that the transport definition of flux is quite dif- history of the usage of the term flux, and then give some ferent (Lorrain and Corson 1962). Additional confusion examples of uses that are either incorrect or confusing. 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 mathemat- Fluxes in Hydrogeology ical frameworks. First, from the field of unified transport The most common flux used in the field of hydro- phenomena (momentum, heat, and mass transport), the geology is the volumetric flux (q), expressed by the gen- flux of some quantity is defined as the flow rate of that eralized form of Darcy’s law as follows: q ¼ 2Kr(h), quantity per unit area (Bird et al. 2002). Second, the field with SI units of m3/(m2 s) where K is the hydraulic con- of electromagnetism (EM) defines flux as the surface ductivity tensor as a function of water content and h is the integral of a vector field (Lorrain and Corson 1962). hydraulic head (Evans et al. 2001). Although the correct Because transport flux is defined with respect to the out- English term is volumetric flux, I would like to suggest ward normal of the reference area, flux is a vector at each that for consistency with mass flux, energy flux, solute point in space (Carslaw and Jaeger 1959; Bird et al. flux, and heat flux, we use the term volume flux to 2002). Vectors have special properties and implied mean- describe q in Darcy’s law. An added benefit of this con- ing in both mathematics and continuum mechanics (Boas sistent 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 Mail Stop T003, Earth and Environmental Sciences Division, volume flux (q), thermal energy flux (qt), and diffusive Los Alamos National Laboratory, Los Alamos, NM 87545; (505) chemical flux (j) are related to the spatial gradients of 665 4638; [email protected] scalar fields multiplied by a function of material proper- Received October 2005, accepted November 2005. Journal compilation ª 2006 National Ground Water Association. ties (e.g., qt ¼ 2Ktr (temperature) and j ¼ 2Dr (con- No claim to original US government works. centration) where Kt and D are the thermal conductivity doi: 10.1111/j.1745-6584.2006.00197.x 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 flux is meant ‘rate of flow per unit area.’ Momentum flux based on the notion of the spatial gradient of a scalar field, then has units of momentum per unit area per unit time.’’ many fluxes, including mass flux with a high Reynolds Also discussed in Bird et al. (1960) is Fickian mass flux, number and neutron flux, are not related to a scalar field. 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 History of Flux in Transport Phenomena found in Potter (1967). In this engineering sciences hand- One of the first instances of the use of the term flux book, heat flux, work flux, and mass flux are defined as to describe transport was by Maxwell (1891):‘‘the flux of vectors. Furthermore, the author notes that the difficult heat at any point of a solid body may be defined as the concept of fluid velocity (nonporous) can be more easily quantity of heat which crosses a small area drawn perpen- understood as the mass flux divided by the fluid density dicular to that direction divided by that area and by the (qf). Interestingly, the nonporous fluid velocity, fm/qf, re- time. Here the flux is referenced to an area.’’ Maxwell duces to the volume flux of the fluid, acknowledging that (1891) also states that ‘‘In the case of fluxes, we have to the true velocity (dx/dt) of a fluid molecule is not so take the integral, over a surface, of the flux through every clearly defined (e.g., turbulent flow). Bear (1972), in his element of the surface. The result of this operation is development of the equations of ground water flow, called the Surface integral of the flux. It represents the makes a careful distinction between the specific discharge quantity which passes through the surface.’’ Thus, the and the specific discharge vector. However, after the integral of the flux of some quantity is defined as the flow introductory material, he refers to the specific discharge rate of that quantity through the surface of integration. vector as simply ‘‘specific discharge.’’ Turcott and Shubert Carslaw and Jaeger (1959) state that ‘‘The rate at which (1982) are very clear that heat flux is a vector with SI heat is transferred across any surface S at a point P, per units of W/m2. The authors also differentiate heat flux unit area per unit time, is called the flux of heat.’’ The au- from heat flow, which has units of W or energy per time. thors note that the units of heat flux are energy per unit Anderson and Woessner (1992) clearly state that q in area per unit time. Additionally, after presenting the one- Darcy’s law is a vector. Fetter (1999) distinguishes be- dimensional Fourier equation for heat flux, they develop tween the vector q and a one-dimensional q, and note that the general form of the equation in three dimensions and in the general form of the transport equations, specific arrive at the heat flux vector (qt) that is, for the rest of the discharge is a vector. I would argue that q is a vector in book, referred to as simply ‘‘heat flux.’’ In one of the most either case, because if the one-dimensional Darcy equa- cited transport textbooks, Bird et al. (1960) state that ‘‘By 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 negative a vector, unless the normal to the area of integration sign that indicates the direction of flow. In all these exam- is given. Use of the term specific flux is redundant as it is ples, the flux of a quantity (e.g., energy, mass, moles) is implicit in the transport definition of flux that the flow is defined as a vector. The historical record shows that per unit area. Additionally, in the field of thermodynam- although usage of the term flux began in a fairly loose ics, use of the term specific as a modifier has a long asso- manner, referring to both the magnitude of a vector and ciation with meaning ‘‘per mass,’’ as in specific volume the vector itself, usage quickly evolved to define flux as (m3/kg), specific surface area (m2/kg), specific enthalpy a vector. (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 Flummoxed flux is vague to readers outside the hydrogeology field, In the subsurface flow literature, volume flux is suggesting that this term be carefully defined in pub- described using many very different terms such as water lications as a volume flux. The term Darcy velocity flux (Jury et al. 1992), volumetric flux vector (Evans et al. should not be used as this confuses the actual average 2001), specific flux vector (Bear 1972), volumetric flow water velocity (also called seepage velocity) and the vol- rate per unit area (Turcott and Shubert 1982), Darcy ume flux; although both have units of length per time, velocity (Domenico and Schwartz 1990), fictitious velocity each has a very distinct meaning. The terms fictitious (Davis and DeWiest 1966), superficial velocity (Lake velocity, filtration velocity, and superficial velocity are 1989), specific discharge (Fetter 1980), specific discharge also poor choices for describing volume flux as they vector (Bear 1972), discharge rate (Anderson and Woessner imply a velocity that is not real. The term flux density 1992), flux density (Hillel 1982), filtration velocity (de apparently is a combination of the EM and transport defi- Marsily 1986), or Darcy flux (Fetter 1999). These terms nitions. The EM literature discusses the flux density of are all used to refer to the same quantity and lead to much a magnetic field as the magnetic field strength per area. confusion not just among students but also among re- Flux density has no meaning with respect to the transport searchers in the field. Of all these terms, the most com- definition of flux. Another common misuse is the term monly used is specific discharge, which is generally flux rate, which is redundant because all transport fluxes defined as follows: Q/A, where Q is the volumetric flow are rates (i.e. per unit time). This is akin to saying veloc- rate (m3/s) and A is the area over which that volume is ity rate. Another confusing usage, although somewhat flowing. Although this term is often used synonymously infrequent, is to call the volumetric flow rate the flux of with volume flux, specific discharge is not necessarily 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 Bear, J. 1972. Dynamics of Fluids in Porous Media. New York: such as Turcott and Shubert (1982). Although the authors Dover Publications. make careful definitions of heat flux (J/(m2 s)) and heat Bejan, A. 1995. Convective Heat Transfer, 2nd ed. New York: J. Wiley and Sons. flow (J/s) in equations 4-1 and 4-5, throughout the text Bird, R.B., W.E. Stewart, and E.N. Lightfoot. 2002. Transport they mix these terms in a loose and imprecise manner. Phenomena, 2nd ed. New York: J. Wiley and Sons. Bird, R.B., W.E. Stewart, and E.N. Lightfoot. 1960. Transport Phenomena. New York: J. Wiley and Sons. Summary Boas, M.L. 1983. Mathematical Methods in the Physical Scien- ces, 2nd ed. New York: J. Wiley and Sons. This technical commentary shows that the most ap- Carslaw, H.S., and J.C. Jaeger. 1959. Conduction of Heat in propriate definition of flux, as related to ground water, is Solids, 2nd ed. Oxford, UK: Clarendon Press. that used in the field of transport phenomena. With re- Chen, W.F. 1995. The Civil Engineering Handbook. Boca Raton, spect to transport phenomena, the flux of some quantity is Florida: CRC Press. Davis, S.N., and R.J.M. DeWiest. 1966. Hydrogeology.New defined as the flow rate of that quantity per unit area. York: John Wiley and Sons. Because flux is defined with respect to a direction, flux is de Marsily, G. 1986. Quantitative Hydrogeology: Groundwater a vector at each point in space. Vectors have special prop- Hydrology for Engineers. San Diego, California: Academic erties and implied meaning in both mathematics and con- Press. tinuum mechanics. I propose that the transport definition Domenico, P.A., and F.W. Schwartz. 1990. Physical and Chem- ical Hydrogeology, 2nd ed. New York: John Wiley and Sons. of flux be used consistently in technical presentations, Evans, D.D., T.C. Rasmussen, and T.J. Nicholson. 2001. Flow published papers, books, and earth science classrooms. and Transport through Unsaturated Fractured Rock, Consistent usage will help in teaching the next generation 2nd ed., Geophysical Monograph 42. Washington, D.C.: of hydrogeologists that flux has a definite meaning in American Geophysical Union. science. Fetter, C.W. 1999. Contaminant Hydrogeology, 2nd ed. Upper Saddle River, New Jersey: Prentice-Hall. Fetter, C.W. 1980. Applied Hydrogeology. Columbus, Ohio: Charles E. Merrill Publishing Company. Acknowledgments Freeze, R.A., and J.A. Cherry. 1979. Groundwater. Englewood Reprinted (heavily revised) with permission from the Cliffs, New Jersey: Prentice-Hall. October 2005 issue of The Hydrogeologist, Newsletter of Hillel, D. 1982. Introduction to Soil Physics. Orlando, Florida: Academic Press. the Geological Society of America Hydrogeology Divi- Jury, W.A., D. Russo, G. Streile, and H.E. Abd. 1990. Evalua- sion. Special thanks to Eleanor Dixon, Don Neeper, Carl tion of volatilization by organic chemicals residing below Gable, Ioannis Tsimpanogiannis, Kay Birdsell, Bruce the soil surface. Water Resources Research 26, no. 1: 13–20. Robinson, George Zyvoloski, Dan Levitt, Josh Stein, Andy Lake, L.W. 1989. Enhanced Oil Recovery. Englewood Cliffs, Fisher, Chris Neuzil, David Deming, and Mary Anderson New Jersey: Prentice-Hall. Lorrain, P., and D.R. Corson. 1962. Electromagnetic Fields and for their very helpful and interactive reviews of this Waves. San Francisco, California: Freeman and Company. commentary. Maxwell, J.C. 1891. A Treatise on Electricity and Magnetism, 3rd ed. Dover Publications. Oxford, Clarendon Press. Potter, J.H. 1967. Handbook of the Engineering Sciences, Vol- ume II: The Applied Sciences. Princeton, New Jersey: References D. Van Nostrand Company. Anderson, M.P., and W.W. Woessner. 1992. Applied Groundwa- Turcott, D.L., and G. Shubert. 1982. Geodynamics Applications ter Modeling, Simulation of Flow and Advective Transport. of Continuum Physics to Geological Problems. New York: San Diego, California: Academic Press. John Wiley and Sons.

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