A Note on the Forchheimer Approach for Estimating Soil Air Permeability

Abstract

Important soil pore characteristics may be revealed from air permeability data. Recent research has quantified significant bias in estimates of the true Darcian permeability when frequently reported pneumatic pressure differences to drive the convective flow are used. An alternative to measurement at infinitesimal pressure differences is the Forchheimer approach, including a polynomial regression of corresponding values of the superficial air velocity (v) and the pressure gradient (G) applied. However, in situations with Darcian flow at low pressure gradients, this procedure may theoretically give an overestimation of the Darcian permeability. We constructed sample plastic cores (∼3.5 cm high) with different numbers (n = 1–19) of drilled tubes (holes of 1, 2, 4.5, and 5.8 mm diameter). Gas diffusivity was measured with a transient‐state method. Air permeability was measured at four pneumatic pressure differences (0.5, 1, 2, and 5 hPa), and the Darcian permeability was estimated with the Forchheimer approach. The ratio of apparent permeability at 5 hPa pneumatic pressure difference to the Forchheimer‐estimated Darcian permeability was significantly lower than unity (0.06–0.52) for all test samples. For all 1‐ and 2‐mm hole samples and for all four levels of G, the Reynolds number indicated nonturbulent flow, whereas turbulence was predicted for samples with 4.5‐ and 5.8‐mm diameters. A model combining relative gas diffusivity, air permeability, and the space available for gas transport indicated that the Forchheimer estimates of Darcian permeability were correct in situations with nonturbulent flow, whereas erroneous estimates were generated for the larger tubes with turbulence. Core Ideas We tested the Forchheimer approach for simple samples with straight tube pores. For nonturbulent flow conditions, the Forchheimer approach provides accurate results. Air flow at 5 hPa pressure difference implies significant underestimation of permeability.

Soil pore system evaluated from gas measurements and CT images: A conceptual study using artificial, natural and 3D‐printed soil cores

“…Air permeability calculated from the dimensions of the drilled holes of Tubes using a combination of Darcy and Poiseuille's law as suggested by Ball (1981) was k a‐Poiseuille = 5,355 μm 2 (Equation ). Revisiting k a‐Darcy measurements for tubular pores of various diameters from Schjønning (2019), we observed a good agreement between k a‐Darcy and k a‐Poiseuille for tubular pores up to 2 mm in diameter, and a lower k a‐Darcy than k a‐Poiseuille for tubular pores with larger diameters (Figure 4). This supports the above theory: (a) that there was an underestimation of k a‐Darcy for Tubes, probably because of turbulences in the 3‐mm holes; and (b) that the contribution of each tubular hole of Tubes could be added to approximate k a‐Poiseuille (Equation ).…”

“…As shown by Schjønning (2019), an increase in air velocity is related to an increase in turbulence during the flow. For all soil cores but S1 and S8, the printing led to an increase in average air pore velocity and a decrease of R S (Figure 7).…”

“…Schjønning (2019) reported that Ball's d B calculation using RD and k a‐Darcy was accurate for tubular pores of 1 and 2 mm diameter, but was not able to estimate the diameter for tubular pores of 4.5 and 5.8 mm. For such large pores, the average pore air velocity becomes too large to retain a Darcian flow, thus disqualifying even the Forchheimer methodology for estimation of k a‐Darcy (Schjønning, 2019). This means that, for Tubes, the too low d B as compared to the physical core relates to an underestimation of k a‐Darcy due to non‐Darcian flow.…”

“…Air permeability calculated from the dimensions of the drilled holes of tubes using Poiseuille's law, k a‐Poiseuille , as suggested by Ball et al (1981) (Equation )) against the Darcian air permeability measurements for tubular pores of various diameters, k a‐Darcy , from Schjønning (2019)…”

Ratio of Non‐Darcian to Darcian Air Permeability as a Marker of Soil Pore Organization

“…Please consult Schjønning and Koppelgaard (2017) for a detailed explanation of the measurement and calculation procedures. The Forchheimer approach for estimating Darcian air permeability has recently been validated (Schjønning, 2019).…”