Reply to comment by N. Shokri and D. Or on “A simple model for describing hydraulic conductivity in unsaturated porous media accounting for film and capillary flow”
2010; Wiley; Volume: 46; Issue: 6 Linguagem: Inglês
10.1029/2010wr009181
ISSN1944-7973
AutoresAndré Peters, Wolfgang Durner,
Tópico(s)Geophysical and Geoelectrical Methods
ResumoWater Resources ResearchVolume 46, Issue 6 CommentariesFree Access Reply to comment by N. Shokri and D. Or on “A simple model for describing hydraulic conductivity in unsaturated porous media accounting for film and capillary flow” A. Peters, A. Peters [email protected] Institut für Ökologie, Technische Universität Berlin, Berlin, GermanySearch for more papers by this authorW. Durner, W. Durner Institut für Geoökologie, Technische Universität Braunschweig, Braunschweig, GermanySearch for more papers by this author A. Peters, A. Peters [email protected] Institut für Ökologie, Technische Universität Berlin, Berlin, GermanySearch for more papers by this authorW. Durner, W. Durner Institut für Geoökologie, Technische Universität Braunschweig, Braunschweig, GermanySearch for more papers by this author First published: 22 June 2010 https://doi.org/10.1029/2010WR009181Citations: 4 This is a reply to DOI:10.1029/2009WR008917. AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL 1. Introduction [1] We thank Shokri and Or [2010] (hereafter referred to as SO) for their comment on our paper [Peters and Durner, 2008b], which introduced a new simple parametric model to account for both, capillary and film flow in porous media. We are pleased that SO commend our basic approach to describe the essential shape of the mechanically based function of Tuller and Or [2001] by a versatile and yet simple conductivity function, which can be easily incorporated into existing simulation codes. We also appreciate their comments and thoughtful ideas concerning the selected application example and inferences from the model made in our paper. However, as will be outlined in the following, we do not agree with all of their objections. Their main points may be summarized as follows: (1) unwarranted extrapolation of experimentally obtained hydraulic functions in a relatively small range of capillary pressures (in our paper and in the following called matric heads) to a broader range, (2) unwarranted assumption of hydraulic continuity during evaporation from a water table 5 m below the surface and (3) neglect of vapor transport in simulation of an evaporation scenario. [2] As a general remark on the chosen scenario (5 m long soil column with homogeneous hydraulic properties, 3000 days of identical atmospheric evaporative demand) it should be clear that it was not our purpose to simulate or predict a realistic field scenario. As stated in the original paper, we rather wanted to give an indication how the choice of the hydraulic parameterization affects a water flux simulation based on the Richards equation in the dry range. To answer this question we chose a simple setup and compared the modeling results obtained for the widely used capillary-based conductivity model of Mualem [1976] with our model. We now will discuss the three points of concern. Extrapolation of Hydraulic Functions [3] We agree that extrapolation of experimentally obtained functions is always critical and should be avoided if possible. However, on one hand the hydraulic functions must be known or assumed for the dry moisture range if evaporation scenarios shall be simulated. This is also the case for the analytical modeling scenarios of Gardner [1958], which were used by SO to show that our scenario overestimates the fluxes. These scenarios even assume the K(h) function to be known up to h → −. We will come back to that point later. On the other hand, the typical measurement range for the K(h) relationship is limited by the measurement range of tensiometers (h > −1000 cm). Often, only the water retention characteristics, θ(h) and the saturated conductivity, Ks, are determined in the laboratory and the relative hydraulic conductivity relationship, Kr(h), is deduced from the θ(h) function with a capillary bundle model [e.g., Mualem, 1976]. Thus, our aim was to compare different extrapolation approaches for calculating evaporation-induced soil water fluxes (with film flow and without film flow), on the same experimental basis. [4] The quotation by Shokri and Or [2010] from our paper on the evaporation method [Peters and Durner, 2008a, p. 151] “The range was restricted to avoid misleading errors that might occur in the extrapolated range of the fitted functions” is not applicable in this context. It was stated in an investigation of the systematic effects of linearization assumptions in the simplified evaporation method. There we compared fitted functions with “true” functions. It was not stated in the context of extrapolation in modeling scenarios. [5] SO claim that we used measurements in the pressure head range of 0.3 m to deduce the hydraulic behavior in the pressure head range of many meters. In fact the measurements were in the range of 3 m [Peters and Durner, 2008b, Figure 3 (top)], as were the measurements of Schindler and Müller [2006, Figure 8]. (Note that unfortunately the abscissa of Figure 8 of Schindler and Müller is erroneously labeled in hPa instead of kPa due to a typing error.) As mentioned above, the figures of Peters and Durner [2008a], to which SO also refer, display synthetic data and have no relation to the data used by Peters and Durner [2008b]. Assumption of Hydraulic Continuity [6] Shokri and Or [2010] argue that hydraulic continuity over a 5 m long sand column is not warranted. We think that this is a crucial part of the comment and admit that we do not know whether liquid continuity is given under such circumstances or not. They confirm this statement by experimental results shown by Malik et al. [1989], who determined “maximum” heights of capillary rise for several sands and fitted an analytical solution of steady state evaporation equations, similar to the one of Gardner [1958]. In our simulations we predict very low water flux densities, which would be hardly detectable in the laboratory. Malik et al. [1989] state that they continued the measurements until capillary rise ceased practically, that is until a further water flux was not detected. The measured fluxes for the 4 sands with the maximum capillary rises given in Table 1 of Shokri and Or [2010] were ∼3 × 10−4 cm d−1 to ∼1.5 × 10−2 cm d−1 [see Malik et al., 1989, Figure 1]. Our simulated fluxes were in the range of ∼1 × 10−4 cm d−1 for the film flow model to ∼1 × 10−5 cm d−1 for the capillary model, neglecting film flow. Furthermore, the measurement setup of Malik et al. [1989] did not allow evaporation. Hence, we conclude, that their measurements are not fully comparable to our modeling scenarios. [7] We note that the characteristic length, describing the length of the “thick film region” [Lehmann et al., 2008, Figure 1] is deduced from very simple pore geometries; that is, SO compare our model assumptions with the model assumptions of Lehmann et al. [2008]. The onset of a “true” residual water content, θr would indeed mark the cessation of hydraulic continuity. However, with θr = 0.08 in a sand [Peters and Durner, 2008b, Figure 3 (top)], a complete cessation of liquid flow appears unlikely. Since from a physical point of view, the assumption of a real residual water content is questioned [Rossi and Nimmo, 1994] we treat it as a mere fitting parameter due to a lack of data in the very dry range. For a correct description of θ(h) in that moisture range (and thus for a sound derivation of a characteristic length) a more complex water retention model than the van Genuchten model should be used [e.g., Fayer and Simmons, 1995]. Without more experimental examination there is no reason to prefer the set of assumptions of SO to ours. [8] We believe that the analytical steady state solution of evaporation fluxes of Gardner [1958] (SO, Figure 2) is not suitable to show that our model solution is misleading. First, this equation oversimplifies hydraulic conductivity in porous media, especially if film and capillary flow is considered. Second, the parameter β was simply derived from the retention curve, without reference to direct conductivity measurements. Third, the analytical solution for steady state evaporation presumes the knowledge of K(h) up to h → −; that is, this is an extrapolation far beyond our approach, whereas less knowledge about the K(h) relationship is made use of. Moreover, the ordinate of SO's Figure 2 gives no resolution for distinguishing between no flux and the small flux in our simulated scenario, which is ∼1 × 10−4 cm d−1. [9] In order to give a comparison of the steady state evaporation scenarios with the different model approaches, we present the analytical solution of Gardner [1958] together with numerical solutions based on the conductivity functions of our paper. Assuming constant maximal flux, qmax [cm d−1], and the vertical coordinate, z [cm] positively defined upward, the Darcy-Buckingham equation can be rearranged to [10] Beginning with h = 0 [cm] at the groundwater table, and a certain maximum flux, qmax the height above the groundwater where h exceeds a certain threshold, hcrit can be iteratively calculated. We used the fourth-order Runge-Kutta method to solve equation (1). As a reasonable value we defined hcrit = −106 cm, corresponding to relative air humidity of 0.5 at 20°C. Since the computational burden is small for this problem we chose the value for Δz = 0.01 cm. Figure 1 shows the resulting maximum evaporative fluxes (qmax) as a function of water table depth for the different conductivity models. In Figure 1 (left), were we chose the same scale for the maximum fluxes as SO in their Figure 2, the differences between the different models are almost not visible. To illustrate that the differences between our film flow model (AII), the classic capillary bundle model (A0*), and the Gardner model (GA) become important just in the dry moisture range and thus at low fluxes, we depict a logarithmic transformation of the maximum fluxes in Figure 1 (right). Figure 1Open in figure viewerPowerPoint (left) Maximum evaporative steady state fluxes, qmax, as a function of height above groundwater level for the different conductivity models. (right) Logarithmic transformation of qmax. Solid lines, solutions considering liquid flow only (equation (1)); dashed lines, solutions considering both liquid and vapor flow (equation (8)). GA, analytical solution of Gardner's equation as used by Shokri and Or [2010, Figure 2]; A0*, numerical solution for capillary model, neglecting film flow; AII, numerical solution for our model with film flow. The extension +vap indicates that both liquid and vapor flow are considered. Figure 2Open in figure viewerPowerPoint Hydraulic conductivity and vapor conductivity functions of soil 4 from Peters and Durner [2008b, Figure 3]. Solid lines, liquid conductivity functions (K); dotted lines, vapor conductivity functions (Kv). Notation as in Figure 1. [11] We note that the maximum fluxes of the sand for the heights from ∼150 to ∼200 cm displayed by the squares in Figure 2 of Shokri and Or [2010] have all higher fluxes as predicted from any of the models in this study. Hence, the selected experimental data do not support SO's statement that we overestimated the fluxes with the film flow model. However, these data were taken from Willis [1960] who used rather a loamy sand with ∼22% of silt and ∼4% of clay content, leading to different hydraulic functions and thus to different maximum fluxes. Neglecting Vapor Transport [12] SO state that vapor flow can not be distinguished from liquid flow for our flow scenario. The vapor flux equation from Philip and de Vries [1957] for describing isothermal vapor flux in soils, as described by Saito et al. [2006], is given by where Kv [cm d−1] is the isothermal vapor conductivity: [13] In equation (3), ρsv [kg m−3] and ρw [kg m−3] (ρw = 1000 kg m−3) are the saturated vapor density and the liquid density of water, respectively, M [kg mol−1] (M = 0.018015 kg mol−1) is the molecular weight of water, g [m s−2] (g = 9.81 m s−2) is the gravitational acceleration, R [J mol−1 kg−1] (R = 8.314 J mol−1 kg−1) is the universal gas constant, T [K] is the absolute temperature, D [m2 s−1] is the vapor diffusivity, and Hr (dimensionless) is the relative humidity. D and Hr were calculated for a temperature of 20°C according to [Saito et al., 2006] and where θa (dimensionless) is the volumetric air content, Da [m2 s−1] is the diffusivity of water vapor in air and τ (dimensionless) is the tortuosity factor; τ was calculated after Millington and Quirk [1961]: where θs (dimensionless) is the saturated water content. Da = 2.44 × 10−5 m2 s−1 and ρsv = 0.017 kg m−3 at 20°C. With this parametrization we get a Kv(h) characteristic for the soil from Schindler and Müller [2006] as shown in Figure 2 (dotted lines). In the simulation range (i.e., −3000 ≤ h ≤ 0 cm) vapor transport would indeed play a dominant role for the capillary model, neglecting film flow (A0*). For the film flow model (AII), Kv(h) is more than one order of magnitude below K(h). Thus, we conclude that vapor transport did not play a role in our chosen simulation scenario. [14] In order to investigate the influence of vapor transport in the vicinity of the very dry soil surface we simulated the combined vapor and liquid flow for steady state conditions. According to Saito et al. [2006] the isothermal water and vapor flow equation for steady state conditions can be written as the sum of the two fluxes: where qlv is the sum of vapor and liquid flow. This equation can again be rearranged to and numerically solved; qmaxlv is now the maximum flux density of the sum of liquid and vapor flow. Note that setting Kv to 0 yields equation (1). The dashed lines in Figure 1 (right) indicate the numerical steady state solutions of the combined liquid and vapor flow for the two hydraulic functions displayed in Figure 2. The vapor flux does indeed play an important role for the case where film flow is neglected (A0*). However, for the model accounting for film flow (AII) the vapor flow contribution is only small. The combined fluxes are now in the same order of magnitude for both models. Note that the capillary model (A0*) does not represent the measured data in the range from about −1.3 × 103 to about −3 × 103 cm (Figure 2). Thus, in that scenario vapor flow is probably highly overestimated, which has an impact on the hydraulic state inside the column. In Figure 3 we display the steady state depth profiles of matric heads and conductivities for the coupled liquid and vapor fluxes following Figure 4 in our original paper [Peters and Durner, 2008b]. In these simulations, vapor flux dominates the water transport only in the upper 1 m for our model but in the upper 3 m for the model neglecting film flow. Figure 3Open in figure viewerPowerPoint Steady state solutions of evaporation from a 5 m column accounting for liquid and vapor flow (equation (8)). (left) Steady state distribution of pressure head and (right) steady state distribution of unsaturated liquid (K) and vapor (Kv) hydraulic conductivity. Notation as in Figure 1. As in Figure 1, the extension +vap indicates that both liquid and vapor flow are considered in the simulations. 2. Concluding Remarks [15] Our objective in the application part of Peters and Durner [2008b] was to illustrate the sensitivity of a numerical simulation with a prescribed evaporative flux boundary condition on the kind of the hydraulic conductivity model. As Figure 2 illustrates, the part of the hydraulic conductivity that is primarily affected by film flow lies in the moderate dry region, i.e., at the pressure heads between about −102.5 and −104 cm. We believe that a more adequate description in this region is important to better simulate water transport processes that are dominated by liquid flow, and can help to explain differences between observations and current models as found, for example, by Goss and Madliger [2007]. Acknowledgments [16] We thank Sascha Iden for critical reading and helpful discussions on the manuscript. References Fayer, M. J., and C. S. Simmons (1995), Modified soil water retention functions for all matric suctions, Water Resour. Res., 31, 1233– 1238. Gardner, W. R. (1958), Some steady state solutions of the unsaturated moisture flow equation with application to evaporation from a water table, Soil Sci., 85, 228– 232. Goss, K.-U., and M. Madliger (2007), Estimation of water transport based on in situ measurements of relative humidity and temperature in a dry Tanzanian soil, Water Resour. Res., 43, W05433, doi:10.1029/2006WR005197. Lehmann, P., S. Assouline, and D. Or (2008), Characteristic lengths affecting evaporative drying of porous media, Phys. Rev. E, 77(5), 056309, doi:10.1103/PhysRevE.77.056309. Malik, R. S., S. Kumar, and R. K. Malik (1989), Maximal capillary rise flux as a function of height from the water table, Soil Sci., 148, 322– 326. Millington, R., and J. P. Quirk (1961), Permeability of porous solids, Trans. Faraday Soc., 57(8), 1200– 1207. Mualem, Y. (1976), A new model for predicting the hydraulic conductivity of unsaturated porous media, Water Resour. Res., 12, 513– 521. Peters, A., and W. Durner (2008a), Simplified evaporation method for determining soil hydraulic properties, J. Hydrol., 356(1–2), 147– 162, doi:10.1016/j.jhydrol.2008.04.016. Peters, A., and W. Durner (2008b), A simple model for describing hydraulic conductivity in unsaturated porous media accounting for film and capillary flow, Water Resour. Res., 44, W11417, doi:10.1029/2008WR007136. Philip, J. R., and D. A. de Vries (1957), Moisture movement in porous media under temperature gradients, Eos Trans. AGU, 38(2), 222. Rossi, C., and J. R. Nimmo (1994), Modeling of soil water retention from saturation to oven dryness, Water Resour. Res., 30, 701– 708. Saito, H., J. Simunek, and B. P. Mohanty (2006), Numerical analysis of coupled water, vapor, and heat transport in the vadose zone, Vadose Zone J., 5, 784– 800, doi:10.2136/vzj2006.0007. Schindler, U., and L. Müller (2006), Simplifying the evaporation method for quantifying soil hydraulic properties, J. Plant Nutr. Soil Sci., 169, 623– 629, doi:10.1002/jpln.200521895. Shokri, N., and D. Or (2010), Comment on “A simple model for describing hydraulic conductivity in unsaturated porous media accounting for film and capillary flow” by A. Peters and W. Durner, Water Resour. Res., 46, W06801, doi:10.1029/2009WR008917. Tuller, M., and D. Or (2001), Hydraulic conductivity of variably saturated porous media: Film and corner flow in angular pore space, Water Resour. Res., 31, 1257– 1276. Willis, W. O. (1960), Evaporation from layered soils in the presence of a water table, Soil Sci. Soc. Am. Proc., 24(4), 239– 242. Citing Literature Volume46, Issue6June 2010 FiguresReferencesRelatedInformation
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