# 4.5 Molecular, Knudsen, and Transition Regimes

Resistance to diffusion resulting from both molecule-molecule and molecule-particle collisions is included in Equations 14 and 15 but no hints have yet been offered as to the conditions that dictate their relative importance. A qualitative guide to the relative importance of the resistance terms is provided by the Knudsen number, defined as the ratio of mean free path length to the characteristic pore dimension, *K*_{n}* = λ/**λ*_{p}. The mean free path length, *λ*, is the average distance traveled by molecules between collisions with other molecules and can be calculated from elementary kinetic theory. The mean free path length is inversely proportional to gas pressure, but the gas pressure in most applications of interest to readers of this book will likely differ little from one atmosphere.

Various measures could be used to characterize pore dimension, but we elect to use the square root of intrinsic permeability, *λ*_{p}* = **k*^{0.5}*.* This is a natural choice because intrinsic permeability is proportional to the square of a characteristic pore dimension (Freeze and Cherry, 1979; Corey, 1994; Hubbert, 1956) and is determined or estimated in almost any study of fluid movement in the subsurface. Furthermore, it provides an immediate link between the Knudsen number and the range of porous media of interest to the earth scientist.

The terms involving the Knudsen diffusion coefficient in Equations 14 and 15 are negligible for the condition *K*_{n} << 1. From the point of view of the gas molecules, the diffusion space appears to be densely populated with molecules but only sparsely populated with solid particles for *K*_{n} << 1. For this condition, molecules are likely to experience a great many intermolecular collisions before encountering a solid particle. Thus, the resistance offered to diffusion of individual components via molecule particle-collisions is negligible relative to that offered by inter-molecular collision (i.e., ). The diffusion regime is said to be molecular in this case. Note that small resistance to diffusion is characterized by large diffusion coefficients, just as small resistance to viscous flow corresponds to large permeability.

At the other extreme, terms involving the molecular diffusion coefficient are negligible and the Knudsen regime prevails when *K*_{n} >> 1. Now it is the particles that densely populate the diffusion space and solid surfaces appear in close proximity to all of the gas molecules as measured relative to the mean free path length. The probability of molecule-particle collisions overwhelms the probability of molecule-molecule collisions and the diffusion is said to be occurring in the Knudsen regime. When neither molecular nor Knudsen diffusion dominates, diffusion occurs in the transition regime.

Suppose nitrogen gas at 20°C and one atmosphere pressure is present in clean coarse sand with an intrinsic permeability of 1 × 10^{–}^{10} m^{2}. For this case, *λ* is about 6.5 × 10^{–}^{8} m as calculated from *λ* = 1/(2^{0.5}*πσ*^{2}*ñ)*, where *σ* is the diameter of the molecule (3.75 × 10^{–}^{10} m for N_{2}; Daniels and Alberty, 1962) and *ñ* is the number of molecules per unit volume (2.5 × 10^{25} m^{–}^{3} for the given temperature and pressure). The square root of intrinsic permeability is *λ*_{p}* = *1 × 10^{–}^{5} m, and the Knudsen number is about 0.0065. Resistance to diffusion offered by molecule-particle collisions is likely negligible in this case and diffusion is said to be molecular diffusion. On the other hand, the Knudsen number would be 6.5 for the same gas present in a glacial till with an intrinsic permeability of 1 × 10^{–}^{16} m^{2}. This situation is likely in the transition regime in which both molecular and Knudsen diffusion is important.

The values of intrinsic permeability vary widely within the large range of materials that may be of interest to the earth scientist. Based on the calculations in the above paragraph, the molecular regime can be expected to prevail in materials with an intrinsic permeability greater than about 1 × 10^{−}^{12} m^{2}. The transition regime is likely to prevail in the remainder of the permeability range of interest to the practicing earth scientist. These guidelines are intended to provide the reader only with a rough idea of where the diffusion regimes might occur in the permeability range of interest.

*The above discussion might lead one to erroneously conclude that diffusion in the molecular regime is independent of molecule**–**particle collisions. Even when such collisions are negligible in the context of resistance to diffusion of individual components, molecule**–**particle collisions remain very important in the context of the momentum balance for the **gas as a whole, regardless** of the prevailing diffusion regime. Therefore, the coupling between the diffusive fluxes expressed in* *Equation**s** **20** **and** **21 must be satisfied in all diffusion regimes because they follow from the momentum balance for the **gas as a whole**.*