Simple consistent models for water retention and hydraulic conductivity in the complete moisture range

Authors


Abstract

[1] The commonly used hydraulic models only account for capillary water retention and conductivity. Adsorptive water retention and film conductivity is neglected. This leads to erroneous description of hydraulic properties in the dry range. The few existing models, which account for film conductivity and adsorptive retention are either difficult to use or physically inconsistent. A new set of empirical hydraulic models for an effective description of water dynamics from full saturation to complete dryness is introduced. The models allow a clear partitioning between capillary and adsorptive water retention as well as between capillary and film conductivity. The number of adjustable parameters for the new retention model is not increased compared to the commonly used models, whereas only one extra parameter for quantifying the contribution of film conductivity is required for the new conductivity model. Both models are mathematically simple and thus easy to use in simulation studies. The new liquid conductivity model is coupled with an existing vapor conductivity model to describe conductivity in the complete moisture range. The new models were successfully applied to literature data, which all reach the dry to very dry range and cannot be well described with the classic capillary models. The investigated soils range from pure sands to clay loams. A simulation study with steady-state water transport scenarios shows that neglecting either film or vapor conductivity or both can lead to significant underestimation of water transport at low water contents.

1. Introduction

[2] Modeling fluxes of water and solutes in soils are an essential means to address many problems in applied soil science, such as water, nutrient, and salinity management research. Typically, the Richards equation is used in order to model the behavior of water in unsaturated soils. An accurate knowledge of the soil hydraulic functions is required to solve this equation, i.e., the soil water retention function θ(h) and the hydraulic conductivity function K(h), where θ is the volumetric water content, h (L) is the suction, and K (L T−1) is the hydraulic conductivity. This knowledge implies both the appropriate soil hydraulic models and the correct parameterization of these models.

[3] The selection of the correct model combination is of crucial importance. In many cases, the well-established retention functions of Brooks and Corey [1964], van Genuchten [1980], or more recently Kosugi [1996] in combination with the capillary bundle models of Mualem [1976a] or Burdine [1953] for conductivity prediction are used and well suited for specific problems. However, a lot of studies show that other model combinations are of great benefit for specific soils or boundary conditions. Soils with structural elements are often better described by bimodal, multimodal [Ross and Smettem, 1993; Durner, 1994] or, for the most flexible description, free from spline retention functions [Iden and Durner, 2007].

[4] Usually, the water retention functions assume a distinct residual water content, θr, which is asymptotically reached at very high suctions. The residual water content is either interpreted as the water held by adsorptive forces [Corey and Brooks, 1999] or as a mere fitting parameter. In the very dry range, even the adsorptive water content finally reaches a value of 0 and thus the concept of residual water content is inappropriate. Several retention models have been proposed to account for this fact [Campbell and Shiozawa, 1992; Fredlund and Xing, 1994; Rossi and Nimmo, 1994; Fayer and Simmons, 1995; Khlosi et al., 2006]. Since the measured water contents are usually based on oven drying, it is conventional to assign a finite suction, at which the water content becomes zero (herein denoted as h0), to a value corresponding to oven dry conditions at 105°C. This yields a suction of ≈ 6.3 × 106 to 107 cm depending on laboratory conditions.

[5] The frequently used retention models of Fayer and Simmons [1995] or Khlosi et al. [2006], which account for water adsorption in the medium to dry range, fail for soils with wide pore-size distributions, because in these models θ does not reach 0 at h0. The same applies to the Zhang [2011] retention model. Fredlund and Xing [1994] developed a retention model, where θ equals 0 at h0, regardless of pore-size distribution. Unfortunately, their model does not allow a partition of capillary and adsorptive water. Moreover, Peters et al. [2011] showed that the models of the type introduced by Fayer and Simmons [1995] (including the model of Khlosi et al. [2006]) can lead to the physically unrealistic case of increasing water content with increasing suction. They solved this problem by introducing a slight modification together with appropriate parameter constraints.

[6] The capillary bundle models often fail to express hydraulic conductivity in the medium to dry range, because they neglect film and corner flow [Tuller and Or, 2001]. Extensions of the capillary models accounting for film flow can improve hydraulic conductivity prediction [Peters and Durner, 2008a; Lebeau and Konrad, 2010; Zhang, 2011]. In the very dry range, liquid water flow might completely cease, but water can still be conducted by vapor flow [Philip and de Vries, 1957; Saito et al., 2006]. An accurate description of θ(h) and K(h) in the medium to dry range is of great importance for simulating evaporation or root water uptake processes.

[7] Peters and Durner [2008a] described liquid conductivity in the complete moisture range with a weighted sum of capillary and film conductivity. In their model, capillary conductivity is described by a capillary bundle model [e.g., Mualem, 1976a] and film conductivity is given by a simple empirical power function of saturation. They showed that their film conductivity model is well suited to describe the measured conductivity data. However, both capillary and film conductivity depend on the capillary retention characteristics with residual water content. Thus, their film conductivity part is coupled with an inappropriate retention model. This conceptual inconsistency was first overcome by Lebeau and Konrad [2010]. They developed a more consistent model, where the retention model of Khlosi et al. [2006] is partitioned into a capillary and an adsorptive part. The capillary and film conductivities are linked to the capillary and adsorptive retention characteristics, respectively. In their model, capillary conductivity is again given by a capillary bundle model, whereas the film conductivity is described by a more complex hydrodynamic model. The same applies to the Zhang [2011] conductivity model. He also partitioned his slightly modified version of the Fayer and Simmons [1995] retention function into capillary and adsorptive water and linked capillary and film conductivity to the according retention parts.

[8] The physically based film conductivity in the model of Lebeau and Konrad [2010] is further partitioned into thick and thin film conductivity, accounting for different viscosities in the vicinity of solid surfaces. However, their model is difficult to implement due to its mathematical complexity. Furthermore, the used physical constants might differ from the literature values because of heterogeneities of mineral and organic matter surfaces as well as the usually unknown composition of the fluid.

[9] The film conductivity model of Zhang [2011] also uses a number of physical constants, which might differ from the literature values. Moreover, it is coupled to his retention function, which is difficult to use with regard to the determination procedure of the critical point distinguishing between capillary and adsorptive retention.

[10] Summarizing, the models accounting for water adsorption and film conductivity are either difficult to use or physically inconsistent. Up to now no physically consistent and easy-to-use hydraulic models for the complete moisture range exist.

[11] In this paper, a new retention model is presented allowing a direct partition of the capillary and adsorptive water and guaranteeing that θ = 0 at h0. A new empirical conductivity model is introduced, which links the capillary and film conductivity to capillary and adsorptive water retention. Both models are easy to use. Thus, the gap between simple straightforward-to-use models with physical inconsistencies on the one hand and physically consistent but more cumbersome-to-use models on the other hand is closed.

[12] The new liquid conductivity model is coupled with an existing vapor conductivity model to describe conductivity in the complete moisture range. The new models are tested with literature data reaching dry to very dry conditions. Finally, the contributions of capillary, film, and vapor conductivity to total flow are exemplarily shown in a steady-state simulation study.

2. Materials and Methods

2.1. Theory

2.1.1. New Retention Function

2.1.1.1. Complete Retention Function

[13] Total water retention is described by two different terms accounting for water held by capillary and adsorptive forces:

display math(1)

where the superscripts tot, cap, and ad mean total, capillary, and adsorptive, θ is the volumetric water content and h (L) is the suction, defined positive for unsaturated soils. This notation will be used throughout the rest of the paper. Saturation (S) can be defined for both fractions. Thus, the new total saturation model is given by the weighted sum of a capillary and an adsorptive saturation term:

display math(2)

where w is the weighting factor subjected to w ≤ 1. The soil water retention model is given by math formula, where θs is the saturated water content of the soil. The capillary and adsorptive bound water of equation (1) are given by math formula and math formula (see Figure 1). The capillary bound water is 0 at a total water content of math formula, which might be interpreted as the residual water content of the capillary part. This is the same water content at which the adsorptive fraction is saturated (see Figure 1 for illustration).

Figure 1.

Exemplary illustration of the contributions of capillary and adsorptive components to the proposed retention model (equation (7)) using the Kosugi function as basic function. Used parameter values are: hm = 20 cm, σ = 0.7, w = 0.8, θs = 0.5, and ha = hm. The capillary held water water θcap was replaced by math formula to emphasize the contributions of capillary and adsorptive retention to total water retention. Note that math formula is identical to the original Kosugi retention function with residual water content, which is given by θs(1−w).

[14] Note that saturation is defined as math formula here, where θs can be but does not have to be equivalent to porosity. Capillary saturation can be interpreted as math formula or math formula (see Figure 1), where math formula and math formula resemble the total and residual water contents in the commonly applied retention models with residual water content. Thus, Scap is equivalent to the effective saturation in the publications of e.g., van Genuchten [1980] or Kosugi [1996]. The adsorptive saturation is given by math formula.

2.1.1.2. Basic Functions

[15] In the following, the two parts of equation (2) will be explained in detail. The adsorptive water usually decreases linearly toward 0 on a semilog scale [Campbell and Shiozawa, 1992]. This is approximately given by a slight modification of the correction function of Fredlund and Xing [1994]:

display math(3)

with

display math

[16] where h0 (L) is the suction at water content of 0 and ha (L) is the suction below which X = 1. Xm is a fictitious parameter slightly greater than 1 adopted from the retention model of Vogel et al. [2001] (see Figure 2 for illustration). The adsorptive saturation is simply given by

display math(4)
Figure 2.

X as a function of suction (equation (3)) with fictitious parameter Xm. [2001]. Parameters were set to ha = 20 cm and h0 = 6.3 × 106 cm. See text for explanation.

[17] Since X is also used as correction for the general capillary saturation function (see below), it is distinguished between Sad and X. The introduction of the fictitious parameter Xm and a distinct value for ha, above which Sad is 1, is necessary to guarantee that with increasing suction water of the adsorptive fraction does not decrease before water of the capillary fraction.

[18] The basic saturation functions for the capillary part used here are the constrained function of van Genuchten [1980] and the function of Kosugi [1996]:

display math(5)

and

display math(6)

where α (L−1) and n in equation (5) are curve shape parameters. In equation (6), hm (L) is the suction corresponding to the median pore radius, σ is the standard deviation of the log-transformed pore-size distribution density function and erfc[] is the complementary error function. Since Γ is used as sole capillary saturation in the simple form of the new retention model and as part of the corrected capillary saturation function (see next sections), this study distinguishes between Scap and Γ.

2.1.1.3. Simple Form of New Retention Model

[19] Using the basic functions Γ as functions for capillary water retention, i.e., Scap = Γ and Sad = X, the new simplified retention model can be written as:

display math(7)

[20] If Γ is expressed by appropriate functions, Mualem's or Burdine's capillary conductivity models have analytical solutions. Setting w = 1 reduces equation (7) to the original saturation function.

[21] The parameter ha marks the suction below which saturation of the adsorptive part is 1. Since adsorptive water will not leave the soil before capillary water drains, ha should have a value above the air entry value. One possibility is to treat it as a free fitting parameter. Since the shape of the complete retention model is not very sensitive with regard to ha when fitting the new models to the data used in this study (not shown), it is set at a certain value. Here, ha is expressed in dependence on the basic capillary retention function. The choice is kept simple by defining ha = hm for the Kosugi function. This means that ha is given by h at Γ = 0.5. For the van Genuchten function, ha at Γ = 0.5 is given by math formula. For simplicity, ha is set here at ha = α−1. Parameter α−1 corresponds to a suction where 1 > Γ > 0.5, hence this choice is justified. Note that ha should not be interpreted in a strict physical sense but rather as a shape parameter of the soil water retention function.

[22] Setting h0 = 6.3 × 106 cm [see Schneider and Goss, 2012], only four parameters (either θs, w, hm, and σ or θs, w, α and n) are needed to describe the complete retention function. The contributions of capillary and adsorptive water retention to the complete retention characteristics are exemplarily shown in Figure 1. The course of θtot is dominated by θcap in the wet moisture range and by θad in the dry range. The model has a transition zone between these moisture ranges, where both capillary and adsorptive retention are important. This is in accordance to the concept of Lebeau and Konrad [2010] (see their Figure 2).

[23] The water content at h = h0 is given by θsΓ(h0). Thus, the same bias as in the retention models of Fayer and Simmons [1995], Khlosi et al. [2006], or Zhang [2011] is allowed. In case of narrow to medium pore-size distributions, Γ reaches approximately 0 at h0, hence math formula. If θ(h0) is significantly greater than 0, which is usually found for soils with wide pore-size distributions combined with a lack of data in the suction range close to h0, a correction is introduced as shown in the next section.

2.1.1.4. New Retention Function for Pore-Size Distributions of Any Width

[24] In case of wide pore-size distributions, the saturations of the basic retention functions (equations (5) and (6)), and thus the saturation of equation (7) do not reach a value of 0 at h = h0. In fact, this is true for any pore-size distribution, since θ only asymptotically reaches a value of 0 in equations (5) and (6). This problem is solved without extra adjustable parameters by the correction math formula [Fredlund and Xing, 1994], leading to:

display math(8)

[25] The benefit of this correction is shown in Figure 3 (top). For small values of n or high values of σ, Γ does not reach a value close to 0 if hh0. Thus, Scap(h) is not well represented by Γ. If hh0 then X(h) → 0. Thus, X(h)Γ(h) approaches also 0, regardless of the shape of Γ(h). For narrow pore-size distributions, X(h)Γ(h) ≈ Γ(h). The corresponding complete new saturation models are shown in Figure 3 (bottom). The disadvantage of this capillary saturation model is that no analytical solution for the Mualem conductivity model exists (see below).

Figure 3.

Sensitivity of capillary model with respect to pore-size distribution index. (top left) ΓVG and ΓVGX are the van Genuchten capillary saturation function and the corrected function; (top right) ΓKos and ΓKosX are the Kosugi capillary saturation function and the corrected function. (bottom) Complete new saturation functions (including adsorptive saturation) with correction. Numbers indicate values for parameters n and σ. Other parameters were set to w = 0.8, α = 0.005 cm−1, hm = 103 cm, h0 = 6.3 × 106 cm, and ha = α−1 (left) or ha = hm (right). See text for further explanations.

[26] A value of 10−3 for θ(h0), corresponding to one-tenth of the assumed measurement error of the water content data (see below), is accepted before switching from the new simple model (equation (7)) to the corrected model (equation (8)).

[27] Note that the parameters α and n or hm and σ have different values in equations (8) and (7). Therefore, they should be interpreted as mere shape parameters without further meaning when equation (8) is used.

2.1.2. New Conductivity Model

2.1.2.1. Complete Model for Liquid Conductivity

[28] The hydraulic conductivity function for liquid flow is expressed by the sum of capillary and film conductivity:

display math(9)

where the superscripts liq, cap, and film stand for liquid, capillary, and film. The relative hydraulic conductivity function for liquid flow can be expressed by the weighted sum of capillary and film conductivity [Peters and Durner, 2008a], where the capillary and film parts depend on the capillary and adsorptive water saturation functions:

display math(10)

[29] The entire hydraulic conductivity function for the liquid phase is:

display math(11)

where Ks (L T−1) is the saturated overall conductivity. Equations (9)-(11) can also be interpreted as follows:

display math(12)

where math formula and math formula are capillary and film conductivity at saturation given by math formula and math formula. Note that both terms in equation (10) were coupled with the complete capillary saturation function in the original Peters and Durner [2008a] model, whereas here the concepts of Lebeau and Konrad [2010] and Zhang [2011] are followed by defining them explicitly for the capillary and adsorptive water fractions, respectively. Figure 4 exemplarily shows the contributions of capillary and film conductivity together with isothermal vapor conductivity (see below). These three parts of the complete conductivity model will be explained in the next three sections.

Figure 4.

Exemplary illustration of the contribution of capillary, film, and vapor components to the proposed hydraulic conductivity model. Used parameter values are: τ = 0.5, Ks = 100 cm d−1, ω = 10−4, and a = −1.5. The parameters for water retention are the same as in Figure 1.

2.1.2.2. Model for Capillary Conductivity

[30] An expression for relative unsaturated hydraulic conductivity for capillary flow is derived from the capillary water retention characteristics by pore-bundle models. The general form of these models can be expressed as follows [Hoffmann-Riem et al., 1999]:

display math(13)

where math formula is the relative hydraulic conductivity and x is a dummy variable of integration. The parameters τ, κ, and β can be varied to get more specific functional expressions. For the Burdine model [Burdine, 1953], τ = 2, κ = 2, and β = 1. In Mualem's model [Mualem, 1976a], τ = 0.5, κ = 1, and β = 2, whereas in the model of Alexander and Skaggs [1986] τ = 1, κ = 1, and β = 1. The parameter τ accounts for tortuosity and connectivity in Mualem's original interpretation, hence in a physical sense τ must be positive. However, its physical meaning must be questioned [Hoffmann-Riem et al., 1999], and τ is often treated as a free fitting parameter that is frequently negative [Schaap and Leij, 2000]. Peters et al. [2011] derived boundaries for the lower allowed value of τ, which guarantee physical consistency of the complete function, whereas τ is interpreted as a mere shape parameter.

[31] If the new simple retention model (equation (7)) is used and Scap is expressed by equations (5) or (6), the analytical solutions of Mualem's capillary-bundle model are given by

display math(14)

and

display math(15)

where m = 1−1/n and erfc−1[] is the inverse of the complementary error function. If equation (8) is used instead, Mualem's model has to be solved by numerical integration.

2.1.2.3. New Model for Film Conductivity

[32] Most of the data in literature show that conductivity as a function of suction decreases more or less linearly on the log-log scale at low water contents (see data below). This is in accordance with the Langmuir-based film flow model for monodisperse particles derived by Tokunaga [2009]. The film conductivity is proportional to the third power of film thickness in his model, leading to a linear relationship between film conductivity and film thickness on the log-log scale. The logarithm of film thickness in turn depends not exactly but approximately linearly on log h (see Figure 3 in Lebeau and Konrad [2010]). Therefore, the hydrodynamically derived models [e.g., Tuller and Or, 2001; Tokunaga, 2009; Lebeau and Konrad, 2010; Zhang, 2011] usually show the exact or approximate linear relationship between film conductivity and log h. Tokunaga [2009] showed that film conductivity is proportional to h−1.5 for constant viscosity, low ionic strength, high surface electrostatic potentials and at high suctions. In other words, the slope on the log-log scale is −1.5.

[33] The exact value for film conductivity in real soils depends on many, usually unknown, soil and fluid properties like particle-size distribution, surface roughness, surface charge, ionic strength of the fluid, and so on. Furthermore, soils usually consist of a heterogeneous mixture of different minerals and organic matter of variable sizes and surface properties. Thus, the exact prediction of film conductivity in natural soils with a physically based model like the models of Tokunaga [2009] or Lebeau and Konrad [2010] might need a correction to account for the unknown properties of a certain soil. Zhang [2011] used the derivation of Tokunaga [2009] and introduced a correction factor for saturated film conductivity. Introducing a correction, which accounts for all the different soil and fluid properties, makes the use of different literature constants of surface and fluid properties questionable. Therefore, the linear relationship shall be accounted for but without further physical interpretation. This linearity at suctions greater than ha can be empirically described with a simple power function:

display math(16)

where a is the slope on the log-log scale. Following Tokunaga [2009], a is here set at −1.5. Since the dynamic viscosity of the fluid will increase at very high suctions [Or and Tuller, 2000] and the dominant forces acting on the water molecules switch from ionic electrostatic to molecular forces, the slope will slightly change as suction increases and thus thickness decreases [Lebeau and Konrad, 2010]. In this paper, it is assumed that a sufficient approximation is given by an effective slope, which encompasses the complete course of film conductivity. Note that in cases of increasing ionic strength of the fluid or decreasing surface electrostatic potential, the slope will deviate from −1.5.

[34] The capillary conductivity is usually given as a function of capillary saturation. Thus, expressing the film conductivity as a function of Sad is more convenient. Solving the unsaturated part of equation (3) with respect to h yields:

display math(17)

[35] Since h0/ha >> 1, Xm ≈ 1 and film flow is important when Sad < 1, equation (17) is approximately given by

display math(18)

which can be combined with equation (16) yielding:

display math(19)

where the slope on the semilog plot with respect to Sad is now given by math formula. The new film conductivity model as a function of either h or Sad is shown in Figure 5 with different values for ω.

Figure 5.

Contribution of relative film conductivity in equation (10) (left) as a function of suction and (right) as function of adsorptive water saturation, where math formula is given by equation (19). Numbers indicate values for parameter ω. Other parameters were set to h0 = 6.3 × 106 cm, ha = 20 cm, and a = −1.5.

2.1.2.4. Model for Vapor Conductivity

[36] In the very dry range, liquid conductivity will completely cease and water transport is solely governed by vapor flow [Saito et al., 2006]. Thus, when evaporation takes place long enough to dry out the soil surface toward suctions in the surrounding air, vapor transport might be the main process for water flow. The isothermal vapor conductivity Kvap (L T−1) is calculated as described in Appendix Prediction of Conductivity Due to Vapor Flow. Since the gradient in gravitational potential is negligible when vapor transport is the dominant flow process, the complete isothermal conductivity is approximately given by

display math(20)

where the superscript lv stands for liquid + vapor. Figure 4 shows the contributions of capillary, film, and vapor conductivity to total conductivity as a function of suction, combining equations (20), (10), (11), (15), (19), and (A1). There are three different suction ranges in which either capillary, film, or vapor conductivity predominates. The transition zones are relatively narrow. Note that most available data are in the zone where either capillary or film conductivity is dominant. Therefore, it is not the object of this paper to test the vapor conductivity model. Since the laboratory temperatures for most of the literature data are unknown, the temperature for vapor conductivity is assumed to be 20°C, which seems reasonable for typical laboratory conditions.

2.1.3. Parameter Estimation

[37] The models for θ(h) and K(h) were fitted to the data with a nonlinear regression algorithm by minimizing the sum of weighted squared residuals between model prediction and data pairs:

display math(21)

where r and k are the number of data pairs for the retention and the conductivity function, wθ and wK are the weights of water content and conductivity data, math formula, math formula, Ki, and math formula are the measured and model predicted values, respectively, and b is the parameter vector. In case of unknown Ks and θs, b consisted of seven adjustable parameters: math formula for the Kosugi and math formula for the van Genuchten function. With known Ks and θs, the size of b was reduced to 5 (see below).

[38] The predicted water contents in equation (21) were either calculated in a standard manner as the point water contents at mean suction (“classic method”) or, if the column height was known, as the mean water content of the whole column (“integral method”) to avoid systematic errors [Peters and Durner, 2006].

[39] For normally distributed and uncorrelated measurement errors with zero mean, the single weights can be set to the reciprocal of the variance of the measurement error. This is in accordance with the maximum likelihood principle for the method of least squares [Omlin and Reichert, 1999]. The errors for the retention and the log10(K) data were assumed to be σθ = 0.01 and math formula, leading to wθ = 10000 and wK = 16.

[40] A descriptive measure giving the mean deviation between model and data is the root-mean-square error:

display math(22)

where yi and math formula are measured and model predicted quantities. For a sound representation of the data by the model, the values of RMSE should be close to the assumed measurement error, i.e., 0.01 for the retention data and 0.25 for the conductivity data.

2.2. Test on Data

[41] Ten data sets were chosen in order to analyze and test the new models. Three samples stem from Pachepsky et al. [1984] (Sandy Loam (soil 1), Clay Loam (soil 2), and Silt Loam (soil3)) and three stem from Mualem [1976b] (Gilat Loam (soil 4), Rehovot Sand (soil 5), and Pachapa Fine Sandy Clay (soil 6)). For soils 1–6, the saturated conductivities and water contents were either available or the measured data reach values very close to saturation. Therefore, the parameters θs and Ks were treated as known and set at the known values. The soils, their references and the known properties are summarized in Table 1.

Table 1. Soils Used in This Study and Their Measured Properties
Data SetSoil NumberReferenceθsKs (cm d−1)
Sandy LoamSoil 1Pachepsky et al. [ 1984]0.438.0
Clay LoamSoil 2Pachepsky et al. [ 1984]0.500.65
Silt LoamSoil 3Pachepsky et al. [ 1984]0.533.07
Gilat LoamSoil 4Mualem [ 1976b]0.4417.3
Rehovot SandSoil 5Mualem [ 1976b]0.401.1 × 103
Pachapa Fine Sandy ClaySoil 6Mualem [ 1976b]0.3312.1
Minasny SandSoil 7Minasny and Field [ 2005]  
Minasny LoamSoil 8Minasny and Field [ 2005]  
Schindler SandSoil 9Schindler and Müller [ 2006]  
Berlin SandSoil 10Own data  

[42] The retention data of soils 1–3 are in the suction range from close to saturation to ≈ 106 cm (Figure 7, left). The conductivity data are in the range from close to saturation to ≈ 105 cm or ≈ 1.5 × 104 cm (Figure 7, right). The retention data of soils 4–6 were measured in the suction range from close to saturation to values of ≈ 104 cm to ≈ 105 cm (Figure 8, left), whereas the conductivity data reach suction values from ≈ 2 × 102 cm (soil 5) to ≈ 105 cm (soil 4) (Figure 8, right). These six data sets can also be found in Tuller and Or [2001], Peters and Durner [2008a], Lebeau and Konrad [2010], or Zhang [2011].

[43] Additionally, four evaporation experiments [Schindler, 1980] were evaluated according to the method of Peters and Durner [2008b]. The raw data stem from Minasny and Field [2005] (a packed sand (soil 7) and an undisturbed clayey topsoil (soil 8)), Schindler and Müller [2006] (an undisturbed sand (soil 9)), and from an experiment that was performed in our laboratory (a packed sandy soil (soil 10)). Due to a relatively high-temporal resolution of the evaporation experiments, these data sets contain more information in the measured suction range. However, since the experiments are conducted with tensiometers, the suction range is usually limited to values <103 cm. Moreover, this experiment type does not yield conductivity data close to saturation (see Figure 9). In this suction range, small uncertainties in tension measurements lead to high uncertainties in the determined conductivities [Peters and Durner, 2008b]. The reader is referred to the original publications for details of the soil properties and of the experimental procedures for soils 1–9.

[44] Soil 10 (in the following referred to as Berlin Sand) was a packed medium sand (texture: <63 μm: 1 wt%; 63–200 µm: 10 wt%; 200–630 µm: 77 wt%; 630–2000 µm: 12 wt%) without organic matter and with a bulk densitiy of 1.55 g cm−3. The evaporation measurement was conducted with the HYPROP® system (UMS, Munich, Germany), where the soil column has a volume of 250 cm3 and a height of 5 cm. Two tensiometers are vertically aligned and record the water potential at 1.25 and 3.75 cm from the bottom of the column. The column is placed on a scale and measured weights and tensions are automatically recorded with high-temporal resolution.

2.3. Modeling Steady-State Water Flux Scenarios

[45] A simple steady-state modeling study was conducted to investigate how neglecting film and/or vapor conductivity influences model outputs. The Darcy-Buckingham equation was rearranged to an ordinary differential equation [Peters and Durner, 2010]:

display math(23)

where qmax (cm d−1) is the maximum steady-state flux and z (cm) is the vertical coordinate defined positively upward. For solving equation (23), an initial condition hin is required. The value for hin was set to 0, simulating a groundwater table. Equation (23) was numerically solved using the fourth-order Runge-Kutta method for different values of qmax. When h exceeded the maximum allowed suction hcrit, the distance between groundwater and hcrit was determined, yielding a certain groundwater depth for each steady-state flux rate. This modeling is equivalent to solving the Richards equation for steady-state conditions with the given Dirichlet boundary conditions, i.e., h = 0 at the bottom and h = hcrit at the surface.

[46] Two values for the suction at the surface (hcrit) were chosen. The first value was 106 cm, which is the water potential in air at 20°C and 50% humidity (equation (A6)). This scenario simulates steady-state evaporation for different ground water depths. The second value was 104 cm, which is a typical suction close to wilting point. This scenario simulates the maximum capillary rise from the groundwater to the root zone in drought stress situations.

[47] K(h) was given in four different ways: (i) solely by capillary conductivity (Kcap), (ii) by capillary and vapor conductivity (Kcap+vap), (iii) by capillary and film conductivity (Kcap+film), and (iv) by capillary, film, and vapor conductivity (Kcap+film+vap). The modeling was conducted for soil 10 with the fitted new models using the Kosugi retention model and omitting the correction (i.e., equation (7) was used as retention model). Kcap was described by using the fitted parameters and setting ω = 0. Thus, the model is given by the original formulation of Kosugi-Mualem.

3. Results and Discussion

3.1. Test on Data

3.1.1. New Corrected Versus Uncorrected θ(h) Function

[48] The simple uncorrected form of the retention model (equation (7)) could be used for all 10 soils if the Kosugi function was the basic function. The maximum value of 1.3 × 10−5 for θ(h0) was found by fitting the new uncorrected retention model to the data of the Silt Loam (soil 3). This is far below the threshold value of 10−3 defined above.

[49] When the constrained van Genuchten function was used as basic function instead, the Silt Loam (soil 3) had to be described with the corrected form of the new retention model (equation (8)). Neither the uncorrected new model nor the Fayer and Simmons [1995] model, which is given in the modified version of Peters et al. [2011] here, met the requirement that θ(h0) ≤ 10−3 (Figure 6). Note that the uncorrected new model and the Fayer and Simmons model almost have the same shape and cannot be distinguished visually.

Figure 6.

Water retention data of Silt Loam (soil 3) and fitted retention functions. Constrained van Genuchten saturation function (equation (5)) was used as basic function. “New model no corr” means new retention model without correction, i.e., equation (7), “new model with corr” is given by equation (8), and “Fayer and Simmons” means the model of Fayer and Simmons [1995] modified by Peters et al. [2011].

Figure 7.

Water retention and conductivity data of soil 1 to soil 3 and fitted new model combination. The Kosugi saturation function without correction (equation (7)) was used as basic retention function.

Figure 8.

Water retention and conductivity data of soil 4 to soil 6 and fitted new model combination. The Kosugi saturation function without correction (equation (7)) was used as basic retention function.

Figure 9.

Water retention and conductivity data of soil 7 to soil 10 and fitted new model combination. Kosugi saturation function without correction was used as basic function.

[50] The RMSEθ is 0.0133, 0.0132, and 0.0149 for the Fayer and Simmons, the new simple, and the new corrected model. Thus, the requirement that θ(h0) = 0, leads to a slight loss of fitting accuracy. The new corrected retention model is probably even more important if no data in the dry moisture range are available, because then extrapolation of the capillary functions might lead to high values of θ(h0).

3.1.2. Fit of New θ(h) and K(h) Models to Data

3.1.2.1. Data With Known θs and Ks

[51] Figure 7 shows the retention and conductivity data of soils 1–3 with the fitted new model combination using the Kosugi function and omitting the correction. All data are well described by the new model combination. The transition zones between capillary and adsorptive water in the retention model are located at ≈ 103 cm for the Sandy Loam, ≈ 3.2 × 103 cm for the Clay Loam, and ≈ 104 cm for the Silt Loam. The transition zones between capillary-dominated and film-dominated conductivity are in the same order of magnitude.

[52] Parameter math formula, given by math formula and denoting the residual capillary water content or the saturated adsorptive water content math formula (see Figure 1), is approximately 0.08, 0.26, and 0.12 for soils 1–3. The conductivity data of soils 1 and 3 reach the transition zone from film-dominated to vapor-dominated conductivity. Thus, omitting the Kvap term in the fitting procedure would lead to biased parameter values for the liquid conductivity model. The conductivity data indeed show the linear decrease on the log-log scale in the film-dominated range. Setting the slope of the Kfilm(h) model at a = −1.5 provides acceptable agreement with experimental data. Treating it as a free fitting parameter does not essentially improve the fit (not shown).

[53] The data of soils 4–6 and the fitted new model combination using the Kosugi function as basic function are shown in Figure 8. The water contents decrease linearly on the semilog scale in the suction range beyond the transition zone between capillary and adsorption dominated water retention. Obviously, these data cannot be described with a retention model within the concept of residual water content. The new retention model is well suited to fit these data. The transition zone is located at ≈1.6 × 102 cm for the Gilat Loam (soil 4) and the Pachapa Fine Sandy Clay (soil 6), whereas it is located at a suction of ≈ 6.3 × 101 cm for the coarser Rehovot Sand (soil 5). The saturated adsorptive water content math formula is approximately 0.15, 0.03, and 0.1 for soils 4–6.

[54] Again, the new film flow model with fixed slope of −1.5 is well suited to describe the data in the film flow dominated suction range. The conductivity data of soil 4 reach values close to the vapor flow dominated region, whereas the conductivities of soils 5 and 6 are two to three orders of magnitude above maximum vapor conductivity. The estimated parameters for soils 1–6 are listed in Table 2. The goodness of fit, indicated by the minimum value of the objective function Φmin, and the RMSE for both hydraulic functions are listed in Table 4. The RMSE values are all close to the assumed measurement errors of 0.01 for the water retention and 0.25 for the logarithmic conductivity data.

Table 2. Estimated Parameters of the New Model Combination Using the Kosugi Model as Basic Function (Soils 1–6)
Soilhm (cm)σwτω
Soil 1201.11.240.82−0.481.3 × 10−4
Soil 2532.01.430.48−0.721.1 × 10−3
Soil 3660.42.280.788.753.1 × 10−6
Soil 468.00.550.661.125.3 × 10−4
Soil 526.90.460.930.485.7 × 10−7
Soil 6116.10.680.69−0.342.4 × 10−4
3.1.2.2. Data With Unknown θs and Ks

[55] Figure 9 shows the retention and conductivity data obtained from the simplified evaporation method (soils 7–10) and the fitted new model combination using the Kosugi function as basic function. As stated above, there is neither information for the conductivity in the suction range close to saturation nor for both retention and conductivity at suctions greater than ≈ 6.3 × 102 cm. However, the data density is high in the measured range. The retention data of the sands (soils 7, 9, and 10) are dominated by the adsorptive part above suctions of ≈ 102 cm, where the decrease of water content is linear on the semilog scale.

[56] All retention data are well described with the new model. The Minasny Sand and the Berlin Sand have math formula values of ≈ 0.04 cm, whereas math formula of the finer textured Schindler Sand is ≈ 0.1 cm. The retention data of the Minasny Loam do not show a clear transition between capillary and adsorptive water contents. However, the new retention model is well suited to effectively describe these data, although the linear extrapolation from the last data point at θ ≈ 0.3 to θ = 0 is questionable. The conductivity data show a clear bend in the transition zone and are well described by the new model with a =−1.5 for soils 7, 8, and 10. Only the slope of the conductivity data in the film-dominated range of soil 9 is different from −1.5 (see zoom in figure). The estimated parameters and the goodness of fit are listed in Tables 3 and 4, respectively.

Table 3. Estimated Parameters of the New Model Combination Using the Kosugi Model as Basic Function (Soils 7–10)
Soilhm (cm)σwθsτKs (cm d−1)ω
  1. a

    Estimated parameter reached boundary of parameter space.

Soil 738.00.230.900.43−1.00a11.41.3 × 10−4
Soil 814.41.060.270.540.061.8 × 1032.6 × 10−5
Soil 958.80.350.730.35−0.765.363.6 × 10−4
Soil 1020.00.400.890.31−0.8815.72.1 × 10−4
Table 4. Objective Function (equation (21)) at Minimum (Φmin) and Root-Mean-Square Error (RMSE) for the Fit of the New Model Combination Using the Kosugi or the Constrained van Genuchten Model as Basic Functiona
 Kosugivan Genuchten
SoilRMSEθRMSElogKΦminRMSEθRMSElogKΦmin
  1. a

    The lowest values of Φmin are highlighted in bold.

  2. b

    In this case, the new corrected retention model (equation (8)) had to be used.

Soil 10.01130.25339.400.01190.29149.49
Soil 20.01600.23146.020.01710.24652.44
Soil 30.01220.15328.780.01350.20542.96b
Soil 40.00730.17021.420.00570.16516.16
Soil 50.01130.33565.650.01070.481106.78
Soil 60.00960.14020.100.00920.11316.56
Soil 70.00590.15942.340.00480.07124.12
Soil 80.00370.04614.470.00370.03914.53
Soil 90.00440.09523.370.00350.07514.60
Soil 100.00620.07736.920.00540.06827.56

[57] The conductivity in the very dry range above suctions of ≈ 1.5 × 104 to 105 is dominated by vapor conductivity for all soils. This is important when simulating evaporation processes under atmospheric conditions, where the suction in the surrounding air is ≈ 106 cm at 20°C and 50% humidity (see equation (A6)). For soils with relatively wide pore-size distributions (soils 2, 4, 6, and 8), the additional contribution of vapor conductivity is small and a differentiation between vapor and film conductivity is not very distinct.

[58] The saturated adsorptive water contents math formula are increasing as the texture is getting finer. This is expected since finer textures lead to higher specific surfaces on which adsorption of water molecules takes place [Schneider and Goss, 2011].

[59] The performance of the new model using the van Genuchten function as basic function is not shown in detail. Table 4 shows that there are only small differences between the new model using the Kosugi or the van Genuchten function. This work focused on the Kosugi function, since it never required the more complex corrected retention model.

3.2. Steady-State Modeling of Water Transport

[60] The preceding sections showed that the new models accounting for capillary, adsorptive, and film components are well suited to describe soil hydraulic properties in the complete moisture range. The dominance of either capillary, film, or vapor conductivity in the different suction ranges could be distinguished. The importance of vapor transport was shown to be particularly important for sandy soils. In this section, the effect of neglecting film and/or vapor conductivity in modeling scenarios shall be investigated by simulating steady-state water flux scenarios.

[61] The results of this modeling analysis are exemplarily shown for the Berlin Sand (soil 10) in Figure 10. The hydraulic conductivity is described with the new model in which the Kosugi function is used to describe the capillary water retention. The used parameters are given in Table 3.

Figure 10.

(top) Steady-state flow rates at different distances (d) between groundwater table and surface calculated with the estimated parameters for soil 10. (bottom) Vertical distribution of suction for steady-state conditions at 100 cm groundwater depth. (left) Suction at surface (hcrit) is 106 cm, simulating typical suction in air at 20°C and 50% humidity. (right) Suction at surface (hcrit) is 104 cm, simulating typical suction close to wilting point at root surface.

[62] In the first case (left), the suction at the surface (hcrit) is 106 cm, simulating typical steady-state evaporation scenarios. If only the capillary conductivity is accounted for (Kcap), the predicted steady-state evaporation becomes very small as soon as the distance between groundwater and soil surface becomes larger than 100 cm (Figure 10, left top). If film conductivity is additionally accounted for (Kcap+film), the resulting steady-state evaporation is similar to the one for Kcap as long as groundwater depth is smaller than 50 cm. For larger groundwater depths, the difference between pure capillary driven flow and additionally allowed film flow increases drastically. The predicted fluxes only slightly increase, when additionally accounting for vapor conductivity (Kcap+film+vap). If liquid conductivity is solely given by capillary conductivity, the influence of vapor conductivity (Kcap+vap) becomes drastic again. Similar results have also been found by Peters and Durner [2010] when analyzing their film conductivity model [Peters and Durner, 2008a].

[63] The difference between neglecting film conductivity on the one hand (Kcap+vap) and vapor conductivity on the other hand (Kcap+film) is only approximately half an order of magnitude. This is surprising, since the courses of the conductivity functions are very different for the two cases (Figure 9, bottom right). Kcap+vap is given by the combination of the blue dashed and green dotted lines, and Kcap+film is given by the combination of the blue dashed and red dash-dotted lines. However, the smaller conductivity for capillary and vapor flow in the medium suction range is compensated by the higher conductivity in the dry range at math formula.

[64] In the evaporation simulations with vapor conductivity (dotted lines), a distinct drying front is formed (Figure 10, left bottom). The steady-state front depth is much deeper when film conductivity is neglected. If no vapor conductivity is accounted for (solid lines), there is no distinct drying front. Although often done in water transport simulation studies [e.g., Peters and Durner, 2008a], omitting vapor flow (Kcap and Kcap+film) is unrealistic when considering evaporation from soils with deeper groundwater levels (discussed by Shokri and Or [2010]).

[65] In the second case (Figure 10, right top), the suction at the surface (hcrit) is 104 cm, simulating the suction close to wilting point in the root zone. This scenario was chosen to simulate the maximum capillary rise from the groundwater to the root zone in drought stress situations. The maximum fluxes for the liquid conductivities (solid lines) are only slightly smaller than the fluxes in the evaporation simulations. Addition of vapor conductivity to film conductivity obviously does not influence the results, since in this case film conductivity is still more than one order of magnitude higher than vapor conductivity (Figure 9, bottom right). However, addition of vapor conductivity to sole capillary conductivity has a large impact again, since in this case vapor conductivity is several orders of magnitude larger than capillary conductivity.

[66] Now the difference between neglecting film conductivity on the one hand (Kcap+vap) and vapor conductivity on the other hand (Kcap+film) is three orders of magnitude. Thus, neglecting film conductivity leads to a drastic underestimation of water fluxes in this scenario. Water transport to the roots might be largely influenced by film flow in sandy soils, hence neglecting film conductivity might lead to errors in modeling water flow in soil-plant atmosphere systems.

[67] The vertical distributions of the single conductivity components for Kcap+vap and Kcap+film+vap with groundwater depth of 100 cm and hcrit = 106 cm are shown in Figure 11. In the simulation neglecting film conductivity (Kcap+vap), liquid flow is dominant from groundwater table to 65 cm above groundwater, where vapor flow starts to be dominant. The transition between liquid-dominated and vapor-dominated flow is very sharp. If film conductivity is additionally accounted for (Kcap+film+vap), water flow is dominated by capillary conductivity from the groundwater table to ≈ 55 cm above groundwater. From 55 to 90 cm, flow is dominated by film, and above 90 cm by vapor flow. Thus, liquid phase flow is dominant from 0 to 90 cm. In this case, the transition between liquid-dominated and vapor-dominated zones is rather smooth. It should be noted that the simulated vertical distribution of h and K, and thus of the height above groundwater where vapor flow starts to be dominant, is not only dependent on the selected model and hcrit but also on the distance between groundwater and soil surface (not shown here).

Figure 11.

Vertical distribution of conductivity with contributions of single components for simulated steady-state evaporation. (top) K was expressed as Kcap+vap and (bottom) K was expressed as Kcap+film+vap. Groundwater depth and hcrit were 100 cm and 106 cm (Figure 10, left, bottom).

[68] Summarizing, neglecting film conductivity caused significant underestimations of water fluxes when either evaporation from the soil surface or capillary rise to the root zone was simulated. Neglecting vapor conductivity caused a significant underestimation of fluxes in the evaporation simulations, whereas it had no influence on capillary rise to the root zone. Moreover, the simulated depths of the steady-state drying fronts were highly dependent on accounting for or neglecting film flow.

[69] These simulations have been conducted in the frame of continuum theory. Current research of evaporation from porous media suggests that liquid phase continuity completely ceases as the suction reaches the characteristic length of the medium and thus, water movement is exclusively governed by vapor flow [Lehmann et al., 2008; Shokri and Or, 2010; Or et al., 2013]. Moreover, two drying fronts are distinguished, the so-called primary (depths where the soil is close to saturation) and secondary drying fronts (depths where liquid phase continuity ceases and vapor flow is dominant) [see Or et al., 2013]. The simple simulations presented in this study show qualitatively similar results. The path between the two drying fronts can be interpreted as a capillary flow network [Lehmann et al., 2008], as the film flow dominated region (this study) or as a region where both processes occur concurrently.

[70] The aim of the steady-state simulation study was to investigate the impact of film and vapor flow on total flow in a strongly simplified system with the assumption of phase continuity in the complete moisture range. Homogeneous soil properties and isothermal conditions were assumed. Transient simulation studies with nonisothermal conditions were not aimed here, but might be important for further studies to see how water moves in the soil-plant-atmosphere system with realistic hydraulic properties and realistic boundary conditions. The reader is referred to the review of Or et al. [2013] and the references therein for an overview of recent development in soil evaporation physics.

4. Summary and Conclusions

[71] A new set of empirical hydraulic models for an effective description of water dynamics from full saturation to complete dryness was introduced. The new models are simple to use and to implement into simulation tools. The retention function allows partitioning of capillary and adsorptive water in a straightforward way. The corrected form of the function guarantees that the water content must be 0 at a suction corresponding to oven dryness for soils of any pore-size distribution and regardless of available data. The used unimodal capillary retention functions require only four adjustable parameters to describe the complete course of soil water retention. Thus, in comparison to the frequently used models of Fayer and Simmons [1995] or Khlosi et al. [2006] no additional fitting parameter is required. If the Kosugi model was used as basic function, the simple form of the new model, allowing usage of the analytical solution of Mualem's integral, was sufficient for all soils, which range from pure sands to a clay loam.

[72] The conductivity model links the capillary conductivity and film conductivity to capillary and adsorptive water retention. Film conductivity is described by a simple power function, expressing the linear course of film conductivity on the log-log scale with respect to suction. Only one adjustable parameter, which corresponds to the saturated film conductivity, is required. The new empirical model is well suited to describe data in the complete moisture range without the conceptual drawback of the original Peters and Durner [2008a] model, where no partition between adsorptive and capillary water was made, and also without the practical drawback of the hydrodynamic model of Lebeau and Konrad [2010], which is more difficult to implement. The new models can be used in a straightforward way for modeling purposes.

[73] Neglecting film and vapor flow can lead to significant underestimations of evaporation. Furthermore, film flow can already significantly influence water transport in the suction range where root water uptake takes place. Thus, neglecting film conductivity might lead to significant errors when modeling water transport in the soil-plant-atmosphere system, especially in sandy soils. It should be noted that the modeling part presumes liquid phase continuity in the complete moisture range.

[74] A so-called “enhancement factor” is often used to adequately describe the observed water fluxes, if a vapor transport model is coupled with a capillary water transport model [e.g., Saito et al., 2006]. Shokri et al. [2009] argued that the enhanced vapor transport might be a misinterpretation and that this transport might be better attributed to liquid flow. Further effort could be made in investigating whether the new film flow model is suited to make such an “enhancement factor” obsolete.

[75] In this stage, the new models are fitting functions, which are helpful if measured data of both retention and conductivity characteristics are available. Parameter w of the retention model, which partitions between capillary and adsorptive water, should not be interpreted in a strong mechanistic sense. In some cases, especially for sandy soils with data available for a broad moisture range, w might give a good measure of the portions of adsorptive and capillary fractions. In other cases, i.e., for fine-textured soils and available data in a limited moisture range (see for example soil 8), it is clear that w must be interpreted as a mere fitting parameter without physical meaning. Without further tests on large data sets, it is concluded that w should be interpreted as a fitting parameter in the same manner as math formula in the commonly used capillary retention models. In fact w is completely determined by math formula. However, since the new model is essentially an empirical fitting function, it is not recommended to calculate w from given capillary retention parameters but fitting the complete function to the complete data.

[76] The conductivity model cannot be used for sole predictive purposes without further investigations. Further measurements are required to test whether the remaining free fitting parameter of the film flow part ω can be related to either basic easy to measure soil properties, like texture, or to properties of the adsorptive water retention function, which might give a measure for cross-sectional area for film flow.

Appendix A: Prediction of Conductivity Due to Vapor Flow

[77] The isothermal vapor hydraulic conductivity is given by Saito et al. [2006]:

display math(A1)

where Kvap (m s−1) is isothermal vapor hydraulic conductivity, 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, 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 is the relative humidity. D is dependent on water content and is calculated according to Saito et al. [2006]:

display math(A2)

where θa is the volumetric air content, Da (m2 s−1) is the diffusivity of water vapor in air, and ζ is the tortuosity factor for gas transport, calculated according to Millington and Quirk [1961]:

display math(A3)

where ϕ is the porosity, which is here, for simplicity, assumed to be equal to θs. Da and ρsv are dependent on temperature:

display math(A4)

and

display math(A5)

Hr is calculated with the Kelvin equation:

display math(A6)

where h (m) is the suction. The isothermal water and vapor flow equation can be written as the sum of the two fluxes [Saito et al., 2006]:

display math(A7)

where Kliq (m s−1) is the liquid hydraulic conductivity and qlv (m s−1) is the sum of vapor and liquid flow. Vapor flow only plays a role in the very dry range, where the hydraulic gradient is practically solely determined by the suction gradient, hence math formula. Therefore, equation (A7) might be simplified to:

display math(A8)

where Klv (m s−1) is the effective isothermal hydraulic conductivity accounting for liquid and vapor hydraulic flow.

Acknowledgments

[78] This study was financially supported by the Deutsche Forschungsgemeinschaft (DFG grant WE 1125/29-1). I thank Marc Lebeau (Laval University, Quebec City, Canada), who kindly sent me the data and the literature for soils 1 to 6, Budiman Minasny (University of Sydney, Australia), and Uwe Schindler (Leibniz Center for Agricultural Landscape and Land Use Research, Müncheberg, Germany) for providing data of their evaporation experiments, Michael Facklam for measuring the soil properties of soil 10 and Doreen Zirkler for language correction. I also thank Tetsu Tokunaga as Associate Editor and three anonymous reviewers for their insightful comments and suggestions. Finally, I thank Gerd Wessolek for the fruitful discussions and financial support.

Ancillary