Correction to: Arabian Journal of Geosciences (2020) 13:801

https://doi.org/10.1007/s12517-020-05811-y

The paper includes certain errors in the original version. The main errors are described subsequently.

First, Eq. 2 is a well-known equation for unsaturated soil, but the minus sign should be added to the equation for the precise using in this paper. The equation should be corrected as follows. Kr is a dimensionless and notation has an error.

$${v}_{n}=-\frac{{k}^{s}{K}_{r}\left({\theta }_{n}\right)}{{\theta }_{n}}\frac{dh}{dz}$$
(2)

In relation to that, minus sign should be deleted of Eq. 13 as follows. The sentence below the Eq. 13, at the left portion of page 7 should be changed as follows; Another situation for the left of Fig. 5a and b is reversed to add the minus sign of Eq. (13), as flow direction is from a drier portion to a wetter portion.

$$Q=\left|{v}_{1}{\theta }_{1}-{v}_{2}{\theta }_{2}\right|={k}^{s}\left[{K}_{r}\left({\theta }_{1}\right){\left.\frac{dh}{dz}\right|}_{n=1}-{K}_{r}\left({\theta }_{2}\right){\left.\frac{dh}{dz}\right|}_{n=2}\right]$$
(13)

Equation (24a) and (24b) are similar.

$${v}_{n,x}=-\frac{{{k}_{x}}^{s}{K}_{r}({\theta }_{n})}{{\theta }_{n}}\frac{dh}{dx}$$
(24a)
$${v}_{n,y}=-\frac{{{k}_{y}}^{s}{K}_{r}\left({\theta }_{n}\right)}{{\theta }_{n}}\frac{dh}{dy}$$
(24b)

Second, the paper uses the mathematical symbol mistakenly; \(\left|h-\Delta z\right|\) should be changed to \(h-\Delta z\). The related equations (Eq. 6, Eq. 11, and Eq. 12), sentences, and figures (Fig. 4 and Fig. 6) should be corrected. Titles and sentences of the figures are grammatically proofed. Figure 6 is slightly adjusted regarding the arrangement of the arrow. Figure 7a and 7b

Fig. 4
figure 1

Boundaries of consistency under known theories among h, Δz, and Δψ

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Fig. 6
figure 2

Rearranged Fig. 4 and estimated actual range of continuous flow

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Fig. 7a
figure 3

Concept of the tentative seepage length, referring to the right of Fig. 5a

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Fig. 7b Concept of the tentative hydraulic head loss, referring to the right of Fig. 5a

Equation (6) should be corrected as below:

$$h-\Delta z\geq\Delta\psi\;\mathrm w\mathrm h\mathrm e\mathrm n\;\mathrm\Delta\mathrm\psi\geq0,$$
(6.1)
$$h-\Delta z\le \Delta \psi \mathrm{ when} \Delta \psi \le 0,$$
(6.2)
$$\Delta \psi = \Delta P/\rho g.$$
(6.3)

Equation (11) and (12) should be corrected as below:

$$1>\frac{{{\theta }_{1}K}_{r}\left({\theta }_{2}\right)}{{{\theta }_{2}K}_{r}\left({\theta }_{1}\right)}>{\left.\frac{dh}{dz}\right|}_{n=1}/{\left.\frac{dh}{dz}\right|}_{n=2},\mathrm{ if }{\theta }_{1}> {\theta }_{2}\mathrm{ and }h-\Delta z<\Delta \psi .$$
(11)
$$1>\frac{{{\theta }_{2}K}_{r}\left({\theta }_{1}\right)}{{{\theta }_{1}K}_{r}\left({\theta }_{2}\right)}>{\left.\frac{dh}{dz}\right|}_{n=2}/{\left.\frac{dh}{dz}\right|}_{n=1},\mathrm{ if }{\theta }_{2}> {\theta }_{1}\mathrm{ and }h-\Delta z>\Delta \psi .$$
(12)

The first sentence in the Conclusions section in Chapter 6 of the original paper should be corrected as follows; A simple criterion under a one-dimensional condition is proposed in this study; when fr > 0 and \((h-\Delta z-\Delta \psi )\Delta \psi \ge 0\) are fulfilled, the condition is regarded as consistent to the existing theories of unsaturated soil. \(\left|h-\Delta z\right|\ge 0\) in the original sentence is changed to \((h-\Delta z-\Delta \psi )\Delta \psi \ge 0\).

The appropriate Fig. 4 and 6 are illustrated as follows.

Third, in Fig. 7a and 7b, the places of \(n=1 and n=2\) accompanying the hydraulic gradient are swapped. The corrected figure is as shown below. Titles of the figures are grammatically proofed.

Moreover, Eq. (10) contains some errors. The equation should be corrected as below.

$$\left(\left|{v}_{1}\right|-\left|{v}_{2}\right|\right)\left({\theta }_{1}- {\theta }_{2}\right)\equiv \left(\left|\frac{{K}_{r}\left({\theta }_{1}\right)}{{\theta }_{1}}{\left.\frac{dh}{dz}\right|}_{n=1}\right|-\left|\frac{{K}_{r}\left({\theta }_{2}\right)}{{\theta }_{2}}{\left.\frac{dh}{dz}\right|}_{n=2}\right|\right)\left({\theta }_{1}- {\theta }_{2}\right)>0$$
(10)
FormalPara Acknowledgements

The author would like to thank Editage [http://www.editage.com] for editing and reviewing this manuscript for English language.