Darcy Velocity Is Not a Velocity.

Abstract

We write Darcy's law as q = −K dh/ds. Preferences of notation aside, this is an unambiguous equation, but the terminology for “q” is ambiguous and often outright confusing, if not misleading! In groundwater hydrology “q” as defined above is variously termed: Darcy velocity, Darcy flux, groundwater flux, seepage velocity, filtration velocity, fictitious velocity, discharge velocity, and specific discharge. Does it matter? Yes, it does matter for two important reasons. To begin with, “q” is not a velocity, but a volumetric flux or specific discharge (volume of water per unit time per unit area). Moreover, an average (linear) groundwater velocity is defined as v = q/n where “n” is the effective porosity (ratio of the volume of interconnected pore space available for flow to the total volume of porous material), and values of n are typically in the range 0.1–0.4. Hence, although v and q have the same units, “v” is about an order of magnitude larger than “q.” Secondly, some of the terms for “q” are also used for “v.” For instance, “seepage velocity” most often means “v,” but in the Russian literature is found to mean “q.” Some authors use Darcy flux and Darcy velocity interchangeably in the same article! Authors of modern hydrogeology textbooks recognize the difference between “q” and “v” and explain concepts clearly (e.g., see Section 3.2.1 entitled “Specific Discharge and Average Linear Velocity” in the textbook by Fitts (2013)). However, many textbook authors (including Fitts) continue to include the term “Darcy velocity” in their discussion. Why are we so stubbornly calling a flux a velocity, fictitious or otherwise? The most likely reason is that flux has units [L/T] associated with a velocity. Furthermore, there is some confusion in history starting with Darcy himself. Darcy (1856) used the letter “v” and somewhat confusingly (although correctly) pointed out that “v” was “the velocity of decrease of the height of water above the filter.” This velocity “v” then is the specific discharge q. Shortly thereafter, Dupuit (1863) reduced the equation for flow in open channels to Darcy's law, which he too wrote in terms of a velocity. However, his equation includes the porosity, thus in Dupuit's version of Darcy's law “v” is indeed a velocity and not a flux. The groundwater community is ill served by this confusing, indeed misleading, use of symbols and terms. Each time we find the term Darcy velocity (or seepage velocity, etc.) in an article or report, we have to dig down to find out what is truly meant, if it is to be found at all. A serious consequence of this confusion is when q is mistakenly used to compute travel times. We have seen examples where this resulted in erroneous time-of-travel capture zones (too small) and arrival times of contaminants (too late). Stauffer (2006) addressed this very same issue and presented a comprehensive overview of the use of fluxes in the groundwater flow and transport literature. We agree with Stauffer that “q” as defined in Darcy's law is best termed a (volume) flux or a specific discharge. Division by an effective porosity yields an average (linear) groundwater velocity. These terms are physically correct and unambiguous. The groundwater community would be well served if it adopted those terms.