DiscussionReply to comments on “Column-scale unsaturated hydraulic conductivity estimates in coarse-textured homogeneous and layered soils derived under steady-state evaporation from a water table” [J. Hydrol. 519 (2014), 1238–1248]
Introduction
We thank Lehmann et al. (2015) for their comments on our recent paper about a new simple analytical solution to soil steady-state evaporation (Sadeghi et al., 2014) and appreciate the opportunity to clarify and discuss the concerns raised. The following discussion focuses on: (i) the definition of pressure head at the drying front where liquid continuity is disrupted; (ii) the approximation of the spatial extent of hydraulic continuity; (iii) the assumption of hydraulic continuity across 10’s of meters; and (iv) the applicability of the proposed approach for layered coarse-textured soils.
First, it should be noted that Sadeghi et al. (2014) should be primarily viewed as a mathematical solution for steady-state evaporation rate (e) from a water table given as Eq. (1) in Sadeghi et al. (2014):where h is the absolute value of the pressure head, z is the vertical distance between the water table (WT) and soil surface (positive upward), and Kl, Kv, and K are the liquid, vapor, and total hydraulic conductivities, respectively.
Note that by considering hydraulic conductivity as the sum of liquid and vapor hydraulic conductivities, Eq. (R1) is equivalent to the Buckingham–Darcy law for the liquid flow domain where Kv is assumed to be negligible, and it takes the same form as the vapor diffusion law for the vapor flow domain where Kl is negligible. The latter was previously demonstrated, for example, in Bittelli et al. (2008).
Although effects due to temperature gradients are not considered, we postulate that Eq. (R1) is adequate to describe steady-state evaporation from a water table, which was demonstrated in Sadeghi et al. (2014). We used independent experimental data from 9 differently-textured soils to demonstrate the efficacy of Eq. (R1), which predicted measured data reasonably well. Furthermore, we show in the following that the solution for the hydraulic continuity domain, Dmax, developed by Lehmann et al. (2008) and denoted as Lc in their paper, is essentially the same as existing analytical solutions to Eq. (R1), though it was derived with a different approach (i.e. from force balance considerations).
Section snippets
The concept of hmax (reply to Section 2 in Lehmann et al. (2015))
Lehmann et al. (2015) argue that hmax in Sadeghi et al. (2014) is different from and not an “alternative” to their definition of hcrit = hb + Δhcap, where hb is the air-entry pressure head and Δhcap is the maximum capillary driving force (see Fig. 4 of Lehmann et al. (2008) for definition of hb and Δhcap). In this section, we (i) provide proof that hmax is equivalent to hcrit, (ii) demonstrate that hmax in Sadeghi et al. (2014) was correctly formulated based on Eq. (R1), (iii) discuss how the
Approximation of Dmax based on D (reply to Section 3 in Lehmann et al. (2015))
It has been shown that D ≈ Dmax when the evaporation rate is higher than 0.01 cm/day. This was concluded based on experimental data for 9 soils with varying textures (i.e. not limited to coarse-textured soils) depicted in Fig. 9 in Sadeghi et al. (2014). Note that this conclusion was entirely based on the observation that the numerical solution of Eq. (R1) for the DF depth was adequate to predict the experimental e(D) data. In addition, the coincidence of e(D) and K(h) experimental data in Fig. 9
Applicability of the proposed approach for layered coarse soils (reply to Section 4 in Lehmann et al. (2015))
Lehmann et al. (2015) questioned the applicability of our proposed approach for inferring effective unsaturated hydraulic conductivity for layered coarse-textured soils. They challenged the approach for two reasons: (i) the e–D and e–δ relationships (δ: DF depth) in Fig. 9 are not consistent with those in Figs. 10 and 11; and (ii) “The process is not only affected by the soil hydraulic properties but also by the order and thickness of the layers”.
The difference between Fig. 9 and Figs. 10 and
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