Reply to the discussion by D. Hansen on "On the use of the Kozeny–Carman equation to predict the hydraulic conductivity of soils"

Abstract

Chapuis and Aubertin 996 The authors thank David Hansen for his unusually long discussion. The reply will be limited, however, to key points and relevant issues, treated in their order of appearance in his discussion. Different groups of scientists and engineers, not only chemical engineers, use the Kozeny–Carman equation with different expressions and for different purposes. The Kozeny– Carman equation is used for a low Reynolds number (e.g., groundwater), the Ergun equation for a high Reynolds number (e.g., pressure drop in packed bed reactors), and the Burke–Plummer equation for a very high Reynolds number. The chemical industry uses air permeability tests to estimate the specific surface of powders, an essential parameter in chemical processes. On the other hand, groundwater engineers and scientists use the specific surface to estimate the hydraulic conductivity of soils, either saturated or unsaturated (e.g., Mbonimpa et al. 2002; Aubertin et al. 2003). In continuum mechanics, expressing specific area as the ratio of surface area per unit bulk volume complicates differential equations, since both the numerator and denominator vary when a set of solid particles (any representative elementary volume) contracts or expands. The authors used the more practical (and scientifically relevant) ratio of surface area per unit mass of solids, where only the numerator varies with porosity. As expressed, eq. [D1] should not be viewed as fundamental for porous media. The hydraulic mean radius is physically well defined only for a constant-flow section. This is not the case with most porous media. An equivalent hydraulic mean radius may be defined as the ratio of two integrals along flow paths, but a few assumptions are needed to obtain eq. [D1] which may be either acceptable or too simplistic, depending on the pore space structure. Besides the hydraulic radius approach (eq. [D1]), other approximations (e.g., Aissen and Saint-Venant) can also be used (White 1991). The discusser also omits to mention that passing from eq. [D2b] to eq. [D2c] not only requires neglecting the contact areas, but also implies the assumption that all voids participate in the flow process, i.e., there are no disconnected voids, dead-end voids, or imprisoned immobile water. The physics behind eqs. [D1]–[D10] can be found in Dullien (1992). Equation [D12] is similar to the Seelheim (1880) equation. It served as the basis for several equations (Hazen 1892; Chapuis 2004a) that are well adapted for specific soils that have geometrically similar pore spaces. It should be noted that, historically, research has moved from the concept of eq. [D12] towards eq. [1], in a direction opposite to that suggested in the discussion. Thus the Kozeny–Carman equation (eq. [1]) is a particular theoretical (and partly empirical) solution to the conceptual eq. [D12], which can in principle be applied to any type of soil. The discusser nevertheless prefers eq. [D12] to eq. [1]. The authors can understand why the concept of m may be attractive at first glance. However, they do not share this preference, being aware that eq. [D12], which can work for artificial one-size media or specific soils, was historically pushed aside for general application with soils because it gave too much room for personal interpretation. In fact, some textbooks (e.g., Freeze and Cherry 1979; Domenico and Schwartz 1997) still refer to equations similar to eq. [D12] as the “Kozeny-Carmen” equation, but do not explain how to use it in practice. The improper spelling of Carman could have been avoided by reading any of the numerous publications by Carman (1937, 1938a, 1938b, 1939, 1956). It must be emphasized again that the Kozeny–Carman equation is a particular theoretical solution which shows that physical parameters other than particle size should be used as m in eq. [D12]. By using eq. [1], the authors provide a practical means that can be used for obtaining m with most natural soils. It could even be said that professionals might put themselves in a delicate position by using eq. [D12], due to its lack of practical clarity. The discusser also states that “the uninformed user will tend to answer with the median diameter D50” to the question of “what particle size to use as m in eq. [D12].” Shepherd (1989) answered with a power function of D50, with an exponent between 1.5 and 2 (in-