Table 1. Summary of tortuosity–porosity... | Emerald Publishing

Table 1.

Summary of tortuosity–porosity equations in published literature

Researcher	Equation	Applicable conditions
Koponen et al. (1997)	$τ = 1 + 0 \cdot 65 \frac{1 - ϕ}{{(ϕ - ϕ_{t})}^{0 \cdot 19}}$	τ becomes infinite at a threshold porosity ϕ_t = 0·33
Mota et al. (2001)	$τ = ϕ^{- β}$	β is an exponent. The suggested value is 0·4; for saturated sand, the value is 1·33 (Millington & Quirk, 1961)
Yu & Li (2004)	$τ = \frac{1}{2} [\begin{matrix} 1 + \frac{1}{2} \sqrt{1 - ϕ} + \\ \frac{\sqrt{{(1 - \sqrt{1 - ϕ})}^{2} + (1 - ϕ) / 4}}{1 - \sqrt{1 - ϕ}} \end{matrix}]$	It only enables flow between square particles in an equilateral-triangle configuration with streamlined direction changes limited to multiples of 90° (Ghanbarian et al., 2013).
Matyka et al. (2008)	$τ = 1 - Pln (ϕ)$	P is a constant with specific values: 0·49 for beds with high porosity (Mauret & Renaud, 1997), 0·41 for spheres that are both monosized and polydisperse (Comiti & Renaud, 1989), and 0·77 for laminar fluid flow in a 2-D porous media comprising freely overlapping solid squares (Matyka et al., 2008).
Ahmadi et al. (2011)	$τ = \sqrt{\frac{2 ϕ}{3 [1 - B {(1 - ϕ)}^{2 / 3}]} + \frac{1}{3}}$	B is a fixed value of 1·209 for cubic packings and 1·108 for tetrahedral packings. τ becomes infinite at porosity values of 0·248 and 0·143, respectively.
Conzelmann et al. (2022)	$τ = 3 \cdot 27 - 2 \cdot 4 ϕ$	A linear model was developed based on artificial aggregates of different shapes. However, it gives τ = 0·95 at ϕ = 1, which is physically unreasonable as τ must be 1 in this limit.

Researcher	Equation	Applicable conditions
Koponen et al. (1997)	$τ = 1 + 0 \cdot 65 \frac{1 - ϕ}{{(ϕ - ϕ_{t})}^{0 \cdot 19}}$	τ becomes infinite at a threshold porosity ϕ_t = 0·33
Mota et al. (2001)	$τ = ϕ^{- β}$	β is an exponent. The suggested value is 0·4; for saturated sand, the value is 1·33 (Millington & Quirk, 1961)
Yu & Li (2004)	$τ = \frac{1}{2} [\begin{matrix} 1 + \frac{1}{2} \sqrt{1 - ϕ} + \\ \frac{\sqrt{{(1 - \sqrt{1 - ϕ})}^{2} + (1 - ϕ) / 4}}{1 - \sqrt{1 - ϕ}} \end{matrix}]$	It only enables flow between square particles in an equilateral-triangle configuration with streamlined direction changes limited to multiples of 90° (Ghanbarian et al., 2013).
Matyka et al. (2008)	$τ = 1 - Pln (ϕ)$	P is a constant with specific values: 0·49 for beds with high porosity (Mauret & Renaud, 1997), 0·41 for spheres that are both monosized and polydisperse (Comiti & Renaud, 1989), and 0·77 for laminar fluid flow in a 2-D porous media comprising freely overlapping solid squares (Matyka et al., 2008).
Ahmadi et al. (2011)	$τ = \sqrt{\frac{2 ϕ}{3 [1 - B {(1 - ϕ)}^{2 / 3}]} + \frac{1}{3}}$	B is a fixed value of 1·209 for cubic packings and 1·108 for tetrahedral packings. τ becomes infinite at porosity values of 0·248 and 0·143, respectively.
Conzelmann et al. (2022)	$τ = 3 \cdot 27 - 2 \cdot 4 ϕ$	A linear model was developed based on artificial aggregates of different shapes. However, it gives τ = 0·95 at ϕ = 1, which is physically unreasonable as τ must be 1 in this limit.