Reply to the comment on"Avalanches and Non-Gaussian Fluctuations of the Global Velocity of Imbibition Fronts"

In [R. Planet, S. Santucci and J. Ortin, Phys. Rev. Lett. 102, 094502 (2009)], we reported that both the size and duration of the global avalanches observed during a forced imbibition process follow power law distributions with cut-offs. Following a comment by G. Pruessner, we discuss here the right procedure to perfom, in order to extract reliable exponents characterising those pdf's.

[1] we reported that both the size and duration of the global avalanches observed during a forced imbibition process follow power law distributions with cut-offs, . While the exponent of the power law appears robust within the quoted error bars, the cut-off of the pdf's depends on experimental control parameters such as the injection rates v.
When adimensionalising the variables, u = x/ x , we observed a collapse of the various pdf's P u (u). Thus, the exponent m x should be found equal to one, as explained in [2]. Indeed, we observed clearly an average power law exponent α = 1.00 ± 0.06 for the avalanche size pdf's. However, for the avalanche duration we reported a slightly larger value within the large dispersion. This can be attributed to the poorer statistics for the avalanche duration, as we explained in [1], affecting the quality of the collapse and/or the fitting procedure. This illustrates the difficulty of extracting accurate values of the power law exponents with experimental data with a limited statistics.
In order to extract reliable exponents, the right procedure -as described in [2]-is to find the power law exponent m * that provides the best collapse of Y = P x (x) x m/(2−m) as a function of X = x/ x 1/(2−m) . In order to quantify the quality of this collapse, first, we as a function of m. For various velocities (clip level C = 0), for the size distribution, the minimum value of ǫ (thus, the best collapse) is obtained for α = 1.00 ± 0.15, while for the avalanche durations we obtain τ = 1.25 ± 0.25, as shown in Fig. 1. The insets display the collapse P x (x)x m as a function of x/ x 1/(2−m) , showing that the scaling function is very well approximated by a decaying exponential. Then it is not difficult to show that the joint distribution of sizes and durations can be properly analyzed using u ′ = S/ S 1/(2−α) and w ′ = T / T 1/(2−τ ) , with the values of the exponents previously found. We show in Fig. 2 that u ′ ∝ w ′γ , where γ = 1.33 ± 0.12.
It is important to notice that the values obtained here are in agreement with the original ones obtained by a direct fitting of the pdf's of the raw and dimensionless data. Since the actual exponents m are close to one P (x/ x ) = P (x) x ≃ P (x) x m 2−m and x/ x ≃ x/ x 1 2−m , and due to both the experimental noise and limited statistics, the collapses previously observed were reasonably good and nearly indistinguishable from the present ones.
[2] suggests that the problem of forcedflow imbibition might belong to the quenched Edwards-Wilkinson (QEW) or equivalently the C-DP universality class [3], a conjecture based on a possible similarity of the values of the exponents. This should be taken with some caution, however, because forced-flow imbibition is a non-local process [4], while the QEW interfacial equation describes a local interfacial dynamics. Moreover, the various values of the exponents reported here are in very good agreement with the ones obtained from phase-field simulations of a non-local interfacial process [5]. [2] G. Pruessner, Comment LPK1047 submitted to Phys.