Author Correction: The origin of hysteresis and memory of two-phase flow in disordered media

Correction to: Communications Physics https://doi.org/10.1038/s42005-020-00492-1, published online 4 December 2020.

The Supplementary Information (Supp. Note 3) was amended to replace a minus sign with a plus in Eq. (21). This affects the expression of the effective width $w_e$ in Eq. (22), also appearing as Eq. (7) in the article’s Methods. The amended Supplementary Information (SI) includes the corrected equations and a short analysis demonstrating that the snap-off in drainage occurs at the critical external head $H_c$, in agreement with our simulations. The Methods in the main article was corrected in a similar manner; also, the linear approximation in Eq. (6) was replaced by the exact solution. Below we provide a detailed list of changes to Supplementary Information and main article Methods (by order of appearance):

• 1. Supplementary Information —text below Eq. (14): changed from “for $H_c \ge d - a_0\left[ {1 - {{{{{{{\mathrm{exp}}}}}}}}\left( { - w/\ell _g} \right)} \right]/\rho g_e$”.

  to “for $H_e\, > \, H_c$, where the critical height is $H_c = d - a_0\left[ {1 - {{{{{{{\mathrm{exp}}}}}}}}\left( { - w/\ell _g} \right)} \right]/2\rho g_e$”.

  And the following text has been added afterwards:

  “Beyond that point, i.e. once $H_e = H_c$, the interface snaps. This is because as long as the defect is fully wet, interfacial tension and gravity are in equilibrium with the capillary forces (Eq. [2]). When the external head is lowered below $H_c$, the defect becomes partially wet, gravity and surface tension overwhelm the weakened capillary force, and the interface snaps.

  The snap-off at $H_c$ can also be demonstrated by the following analysis”.

• 2. Supplementary Information —text below Eq. (16): changed from

  “Note that the defect becomes weaker as it becomes narrower. Thus we do not expect that the effective width goes to zero until the release of the interface, or in other words, the interface is going to be released at finite $w_e\, > \, 0$”.

  To “The fact that the defect becomes weaker as it becomes narrower, provides the intuitive reasoning for the snap-off at $H_e = H_c$”.

• 3. Supplementary Information —in Eq. (21) the minus sign was replaced with a plus sign to correct an error. Hence, the old equation appeared as

  $$\frac{{dw_e\left( {H_e} \right)}}{{dH_e}}\left[ {1 - \frac{{\partial f\left( {H_e,w_e} \right)}}{{\partial w_e}}} \right] = \frac{{\partial f\left( {H_e,w_e} \right)}}{{\partial H_e}}$$

  This has now been replaced by

  $$\frac{{dw_e\left( {H_e} \right)}}{{dH_e}}\left[ {1 + \frac{{\partial f\left( {H_e,w_e} \right)}}{{\partial w_e}}} \right] = \frac{{\partial f\left( {H_e,w_e} \right)}}{{\partial H_e}}$$

  Supplementary Information —as a consequence of the change of sign in Eq. (21), in Eq. (22) the numerator and denominator flipped, namely the last term was changed from

  $$\frac{{d - H_c}}{{d - H_e}}$$

  to

  $$\frac{{d - H_e}}{{d - H_c}}$$

• 4. Supplementary Information —the text below Eq. (22) was changed accordingly from “Thus, for $H_0 \le H_e < \, H_c$ the interface height is given by Eq. [13] where $w$ is replaced by Eq. [22]. The interface leaves the defect at $H_0$, which is obtained by setting $h\left( {x = 0} \right) = d$”.

  to “Equation [22] indicates that $w_e$ would increase with decreasing $H_e$, which is not possible because it cannot exceed the physical width $w$. Another explanation for the snap-off at $H_c$ is that during drainage the interface is described by Eq. [13] with $a_e = a_0$. This solution is valid as long as the defect is fully wet. The defect remains fully wet while $h\left( {\pm w/2} \right) \ge\, d$, namely while the equilibrium pressure head $H_e = \left( {H + p_c^0/\rho g} \right)/{{{{{{{\mathrm{sin}}}}}}}}\alpha$ remains above the critical position, $H_e\, > \, H_c$. Once the pressure head falls below this critical value, $H_e \, < \, H_c$, the defect would become partially wet, such that the effective width $w_e$ would be smaller than the defect width $w$. The equilibrium position for this partially wet configuration would satisfy

  $$\begin{array}{*{20}{r}} \hfill {H_e = d - \frac{{a_0}}{{2\rho g_e}}\left[ {1 - {{{{{{{\mathrm{exp}}}}}}}}\left( { - w_e/\ell _g} \right)} \right].}\end{array}$$

  (23)

  Since $w_e\, < \, w$, this implies that $H_e\, > \, H_c$, in contradiction with the above. Therefore, the interface will snap at $H_c$”.

• 5. Methods—Eq. (6) which provided the linear approximation of $\eta _{{{{{\rm{c}}}}}}$ was replaced by the exact solution. Thus, Eq. (6) changed from

  $${\eta} _{{{{{\rm{c}}}}}} = \frac{{\ell _{{{{{\rm{g}}}}}}^2{{{{{{{\mathrm{cos}}}}}}}}\theta }}{{b_0^2}}2\left[ {1 - {{{{{{{\mathrm{exp}}}}}}}}\left( { - w/2\ell _{{{{{\rm{g}}}}}}} \right)} \right]\delta b,$$

  to

  $$\eta _{{{{{\rm{c}}}}}} = \frac{{a_0}}{{\rho g_{{{{{\rm{e}}}}}}}}\left[ {1 - {{{{{{{\mathrm{exp}}}}}}}}\left( { - w/2\ell _{{{{{\rm{g}}}}}}} \right)} \right].$$

  Consequently, the following text which appeared after Eq. (6), became redundant and was removed:

  “where we used that $a_0/\rho g_{{{{{\rm{e}}}}}} = 2\ell _{{{{{\rm{q}}}}}}^2{{{{{{{\mathrm{cos}}}}}}}}\theta \delta b/b_0\left( {b_0 - \delta b} \right)$ and set $b_0 - \delta b \approx b_0$”.

• 6. Methods—Eq. (7) is identical to Eq. (22) in the SI, and was thus corrected similarly, namely the last term was changed from

  $$\frac{{d - H_{{{{{\rm{c}}}}}}}}{{d - H_{{{{{\rm{e}}}}}}}}$$

  to

  $$\frac{{d - H_{{{{{\rm{e}}}}}}}}{{d - H_{{{{{\rm{c}}}}}}}}$$

• 7. Methods—accordingly, text above Eq. (7) changed from

  “$H_{{{{{\rm{e}}}}}} \ge H_{{{{{\rm{c}}}}}} = d - a_0\left[ {1 - {{{{{{{\mathrm{exp}}}}}}}}\left( { - w/\ell _{{{{{\rm{g}}}}}}} \right)} \right]/\rho g_{{{{{\rm{e}}}}}}$. For $H_0 \le H_{{{{{\rm{e}}}}}}\, < \, H_{{{{{\rm{c}}}}}}$,”.

  To “$H_{{{{{\rm{e}}}}}}\, > \, H_{{{{{\rm{c}}}}}} = d - a_0\left[ {1 - {{{{{{{\mathrm{exp}}}}}}}}\left( { - w/\ell _{{{{{\rm{g}}}}}}} \right)} \right]/2\rho g_{{{{{\rm{e}}}}}}$. For $H_e\, < \, H_c$,”

• 8. Methods—accordingly, text below Eq. (7) changed from

“The effective height The effective height $H_0$ at which the interface leaves the defect is obtained by setting $h\left( {x = 0} \right) = d$,

$$\begin{array}{*{20}{r}} \hfill {H_0 + \frac{a}{{\rho g_{{{{{\rm{e}}}}}}}}\left( {1 - {{{{{{{\mathrm{exp}}}}}}}}\left[ { - \frac{{w_{{{{{\rm{e}}}}}}\left( {H_0} \right)}}{{2\ell _{{{{{\rm{g}}}}}}}}} \right]} \right) = d.}\end{array}$$

to

“Equation (7) indicates that $w_{{{{{\rm{e}}}}}}$ would increase with decreasing $H_{{{{{\rm{e}}}}}}$, which is not possible because it cannot exceed the physical width $w$. Therefore, the interface will snap-off at $H_{{{{{\rm{c}}}}}}$”.

Note that Eq. (8) was eliminated, as it is no longer relevant.

The above corrections have been implemented in the HTML and PDF version of the article and the correct version of Supplementary Information document can be found as Supplementary Information associated with this Correction.