A new theoretical model accounting for film flow in unsaturated porous media

Authors


Abstract

[1] Commonly used soil hydraulic models that account for capillary forces often do not perform well when water flow occurs in thin liquid films in unsaturated porous media. A new theoretical soil water retention curve model was formulated based on the film thickness function that accounts for matric potential. Coupling this new model with a modified Tokunaga model that considers hydraulic conductivity yields a theoretical and mathematically simple soil hydraulic model that accounts for film flow in unsaturated porous media. Inverse modeling of an evaporation experiment showed that the new model provides a good description of water flow at low water contents. The combination of van Genuchten model with the modified Tokunaga model performed best in inverse modeling; however, the fitted parameters did not describe water flow under drier conditions because of the higher hydraulic conductivity and sharper water potential gradient when the water content approached the residual water content, which caused severe overestimation of water loss. Simulation of the water dynamic in a 2 m profile using the modified Tokunaga model combined with the new theoretical soil water retention curve model provided reliable results for water changes under field conditions, including very dry conditions.

1. Introduction

[2] The unsaturated zone in soil acts as a link between the surface system and the groundwater system and plays a crucial role in water and solute cycling. In semiarid and arid regions where precipitation is low and long-term evaporation is high, soil water is important for its key position in controlling plant growth and other biological processes.

[3] The Richard's equation is the most wildly used model for simulating variably saturated water and solute flow in soil [Vanclooster et al., 2004], the solution of which depends greatly on the soil water retention curve (SWRC) h(θ) and the hydraulic conductivity curve K(h). However, the commonly used soil hydraulic models that account for capillary forces do not deal well with the flow process that occurs under very dry conditions [e.g., Rossi and Nimmo, 1994; Silva and Grifoll, 2007]. One important reason for the poor performance in dry conditions is that the common conceptual models for unsaturated hydraulic conductivity on which these flow models rely, e.g., the frequently used model of Mualem [1976a], are unsuitable under dry conditions. The Mualem and similar models often rely on oversimplified representation of pore space in a given medium as a bundle of cylindrical capillaries and assume that the matric potential is attributed to capillary forces only [Tuller and Or, 2001]. Experimental evidence has shown that the models for K(h) often underestimate the hydraulic conductivity in the medium to dry range [Mualem, 1976b; Pachepsky et al., 1984]. Goss and Madliger [2007] also noted this phenomenon in field scale.

[4] Pore bundle models may underestimate hydraulic conductivity under dry conditions, because they neglect the flow in thin liquid films, which can be the dominant process at low matric potentials [Li and Wardlaw, 1986; Lenormand, 1990; Toledo et al., 1990]. On the basis of this assumption and previous studies, Tuller and Or [2001] proposed a K(h) model that accounts for both capillary and film flow using hydrodynamic considerations and specific pore geometry assumptions. However, this model, which is meant to be coupled with the retention model developed by Or and Tuller [1999], is little used due to its complexity. Peters and Durner [2008] presented a model that combines Mualem's capillary flow model with a simple film flow function using film flow conductivity as a function of the effective saturation. The disadvantages of this simple model are that it lacks a physical basis and that it must be coupled with a common SWRC models (e.g., the van Genuchten model used in that paper), which may be invalid at lower water contents [Nimmo, 1991; Rossi and Nimmo, 1994; Silva and Grifoll, 2007]. By combining the film model developed by Langmuir [1938] with scaling analysis, Tokunaga [2009] developed a formula for estimating hydraulic conductivity that results from film flow in media consisting of smooth uniform spheres. This type of flow is highly dependent on film thickness f and accounts for matric potential h on a grain scale [Wan and Tokunaga, 1997]. Zhang [2011] coupled Tokunaga's conductivity model with a modified van Genuchten [1980] model and obtained a well-fitted K(h) curve in very dry ranges by introducing a soil-dependent factor. Lebeau and Konrad [2010] proposed a new model, developed from a modified version of Tokunaga's [2009] model, for predicting the hydraulic conductivity of porous media. This model accounts for both capillary and thin film flow processes, and establishes a mathematical relationship between hydraulic conductivity and the water retention function; however, it targets media with smooth spherical grains.

[5] The models developed to date are either mathematically complex or need to be coupled with common SWRC models, which do not perform well in drier ranges. Measured hydraulic conductivity data are often needed to obtain proper parameters and these data are difficult to obtain in dry conditions. In this paper, we discuss how we developed a theoretical model based on Tokunaga's formula and to obtain the parameters needed through traditional evaporation experiments using an inverse method. Finally, we discuss the implications of the different models on water dynamic.

2. Materials and Methods

2.1. Theory

2.1.1. Flow Equation

[6] Water flow in homogeneous, rigid porous media with variable saturations is usually described by Richards' equation, which, for one spatial dimension, is written as

display math(1)

where θ (L3 L−3) is the water content (in this paper, all the water content used means the volumetric water content), t (T) is time, z(L) is the vertical coordinate, positive downward, h(L) is the pressure head, and K(h)(L T−1) is the hydraulic conductivity as a function of pressure head.

2.1.2. h(θ) and K(h) Accounting for Film Flow

[7] By combining Langmuir's [1938] film model with scaling analysis, Tokunaga [2009] developed the following formula for estimating hydraulic conductivity resulting from film flow in smooth uniform spheres:

display math(2)

where K is hydraulic conductivity due to film flow, h is matric potential (in Pa), hc is the critical matric potential below which thin film flows control flow and transport through the aqueous phase, ρ is the fluid density, η is fluid viscosity (1.005 × 10−3 Pa s at 293 K), f is film thickness, NA is the number of grains intercepting a unit cross-sectional area, written as

display math(3)

where n is porosity and λ is grain diameter.

[8]  inline image in equation (2) is the average intercepted perimeter per grain, written as

display math(4)

where φ is the polar angle associated with the sphere-plane intersection. However, equations (3) and (4) are for smooth uniform spheres (smooth spheres with the same λ). Thus, the hydraulic conductivity of natural porous media due to film flow can be expressed by introducing a dimensionless correction factor ω0 accounts for the surface roughness and particle nonuniformity, as

display math(5)

[9] It should be noted that the correction factor ω0 must be greater or equal to 1, and typically it is far larger than 1 [Tokunaga et al., 2003]. According to Wan and Tokunaga [1997], the film thickness f can be expressed as

display math(6)

where ε is the relative permittivity of water (78.54), ε0 is the permittivity of free space (8.85 × 10−12 C2 J−1 m−1), kB is the Boltzmann constant (1.381 × 10−23 J K−1), T is the Kelvin temperature, z is the ion change, e is the electron charge (1.602 × 10−12 C), and σ is surface tension (7.27 × 10−2 N m−1 at 293 K).

[10] Equation (6) is expressed at grain scale and assumes smooth uniform spheres, whereas equation (1) is valid only at spatial scales larger than the representative elementary volume (REV), which is far larger than grain diameter. Because an assumption of smooth uniform spheres is unrealistic in the field, we modified equation (6) to be suitable for the Richards' model. First, we proposed that all the grains in a REV with a volume of V have the same diameter λ and are in an equilibrium state (with the same h, the mean matric potential of the REV can be written as hREV = N0S0h/(N0S0) = h, S0 is the surface area of an individual grain). As such, according to equation (6), they should have the same film thickness f and the mean film thickness of the REV can be written as fREV = N0S0f/(N0S0) = f. The number of grains in the REV N0 can be written as

display math(7)

[11] The total surface area, S, of grains in the REV is calculated as

display math(8)

[12] The volumetric water content θ can be expressed as

display math(9)

[13] To determine hydraulic conductivity, we also introduced a dimensionless correction factor ω1 that accounts for surface roughness and particle nonuniformity. It should be noted that ω1 may be different than ω0 (e.g., the surface roughness factors for equations (8) and (4) are surface area weighted mean value and perimeter-weighted mean value, respectively). The volumetric water content of natural porous media can be written as

display math(10)

[14] Substituting equation (10) into equation (5) enables us to calculate of K(θ) for film flow as

display math(11)

[15] Equations (10) and (11) are REV-scale formulas for SWRC and hydraulic conductivity that accounts for film flow. Here, we expressed the hydraulic conductivity as a function of θ by eliminating the film thickness that accounts for matric potential. Tokunaga [2009, 2011] found that the water film thickness was impacted by surface and solution chemistry in unsaturated soils, and the Langmuir model was derived for low ionic strength solutions (≤0.1 mol m−3), while higher salinity systems will have thinner water films. Thus, the K(θ) form hydraulic conductivity is a meaningful and more convenient function to use. Clearly, equation (11) can be easily coupled with other SWRC models by introducing only one more parameter ω (ω = ω0λ213).

[16] However, in this paper, our new SWRC model still relies on the mean film thickness fREV (=f of individual uniform grain with same diameter) that accounts for the mean matric potential hREV (=h of individual grain at equilibrium state). For smooth uniform grains, the mean film thickness fREV and mean matric potential hREV should take the form of equation (6), and can be written as

display math(12)

[17] Although the Langmuir model was derived for low ionic strength solutions, the exponential form between film thickness fREV (or water content θ) and matric potential hREV (4σ/λ << hREV at low water contents) of different soils were proved to be reliable not only in other SWRC models that account for film flow (e.g., the θ(h) model introduced by Tuller and Or [2005]), but also in laboratory experiments (e.g., the measured h(f) of different soils at very dry ranges in Resurreccion et al. [2011]).

[18] For the sand soil used in this paper, we measured the θ(h) relationship at dry ranges using WP4-T (Decagon Devices, Inc., Pullman, Washington), details of the experiment will be introduced in the following experimental section.

[19] The measured θ(h) of very dry samples is shown in Figure 1, in the form of

display math(13)
Figure 1.

Measured matric potential (in Pa) and volumetric water content (multiply measured gravimetric water content by bulk density, 1.6 g cm−3) of the sand samples using WP4-T.

[20] Equation (13) shows that very good correlation between water content θ and (4σ/λ − h) using the same form of the modified Langmuir's film model (equations (9) and (12)), although a different exponent may apply (rather than −0.5).

[21] Thus, the SWRC of natural porous media accounting for film flow should have the form of

display math(14)

where ω1 and τ account for the total impact of surface roughness, particle nonuniformity, and ionic strength on θ and f.

2.1.3. Parameter Optimization

[22] The objective function ψ to be minimized during the parameter estimation process is defined as

display math(15)

where Nh is the number of data pairs for the retention function, wi is the weight of the matric potential data (all with the same value of 1 in this paper), and hi and inline image are the measured and model predicted values, respectively.

[23] To evaluate the goodness of fit, an important value r2 was introduced, expressed as

display math(16)

where yi and inline image are the observed and fitted value, respectively. The r2 value is a measure of the relative magnitude of the total sum of squares associated with the fitted equation. A value of 1 indicates a perfect correlation between the fitted and observed values.

[24] The measured porosity n and grain diameter λ (surface area weighted mean value) of the soil samples are 0.38 and 2.0 × 10−4 m, respectively. The critical matric potential is hc ≈ −26σ/λ ≈ −95 cm during drainage processes [Haines, 1930]. As such, only three parameters, ω0, ω1, and τ, need to be defined in θ(h) and K(θ) functions (equations (11) and (14)); if the grains have the same surface roughness, we can propose ω0 = ω1.

[25] For comparison, five pairs of hydraulic models were used to evaluate the measured data (Table 1). The VM case, as a combination of Van Genuchten model (θ(h)) with Mualem model (K(θ)), was used to test the performance of common hydraulic model that accounts for capillary forces at low water contents. In other cases, the hydraulic conductivity models (K(θ)) we used were all the modified Tokunaga model that accounts for film flow (equation (11)). The VT case was to test the performance of Van Genuchten model (θ(h)) when thin liquid films dominating the water flow, while the cases of FT0, FT1, and FTM were all to test our new SWRC model (expressed as SWRCFFM, equation (14)), but with different parameters to fit.

Table 1. Summary of Hydraulic Models Used to Evaluate Measured Data
Caseθ(h) ModelK(θ) ModelEstimated ParametersParameters Not Estimated
VMVan GenuchtenMualemθr, θs, α, l, n, Ks 
VTVan GenuchtenModified T.Kθr, θs, α, l, n, ω0 = ω1λ
FT0SWRCFFMModified T.Kω0, ω1, τ, λ 
FT1SWRCFFMModified T.Kω0 = ω1, τ, λ 
FTMSWRCFFMModified T.Kω0 = ω1, τλ

[26] The inverse solution is provided by UCODE_2005 [Poeter et al., 2005] in combination with a modified Hydrus-1D model [Simunek et al., 2008].

2.2. Experimental Methods

2.2.1. Measured θ(h) of Very Dry Samples

[27] The sand soils used in this paper were sampled in Tengeer desert. The physical properties are given in Table 2. The surface area averaged grain diameter λ was measured by laser diffraction (Mastersizer 2000, Malvern).

Table 2. Physics Properties of the Soil Considered for This Study
Bulk Density (g cm−3)PorosityClay (<0.002 mm)Silt (0.002–0.06 mm)Fine Sand (0.06–0.2 mm)Medium Sand (0.2–0.6 mm)Grain Diameter (mm)
1.60.380033.6866.320.2

[28] We used a chilled-mirror WP4-T Dewpoint Potentiameter (Decagon Devices, Inc., Pullman, Washington) to measure the matric potential. The details of our experiment were the same as those of Resurreccion et al. [2011]. First, air-dried samples of 10 g were prepared and a small amount of water was added. Then, the soil samples were mixed, sealed in small airtight plastic bags, and left to equilibrate for 4 weeks. Following the equilibration/desiccation period, the matric potential was measured with WP4-T. The water content was determined gravimetrically at each measured matric potential condition. We provided the volumetric water content by multiplying the measured gravimetric water content by bulk density (1.6 g cm−3).

2.2.2. Evaporation Experiment

[29] The evaporation experiments were conducted using a ku-pF Apparatus DT 04-01 system (Umwelt-Geräte-Technik Ltd., Germany). The details of our evaporation experiment were the same as those of Bohne and Salzmann [2002], with a column height of 6.2 cm and a volume of 245 cm3. Before starting the analysis, the sand samples were saturated for about 12 h. Two Tensio 130 tensiometers, after calibration, were installed at depths of 1.8 and 4.8 cm beneath the soil surface and the mass of the soil cores were recorded automatically every half hour. The soil cores were sealed at the bottom and allowed to evaporate from the surface without using heating or using a fan. The tensiometers were accurate to ±1 cm. To obtain more information about soil behavior in the very dry range, we continued the experiment until the lower tensiometer failed. Five soil cores of sand were measured in experiment.

2.3. Impact on Water Dynamic

[30] To evaluate the effects of the different model fits on water dynamics at the field scale, we simulated an evaporation scenario using different soil hydraulic functions.

[31] The simulation used a 2 m vertical soil column, with a 0.2 cm d−1 evaporate flux at the surface. The lower boundary condition had free drainage, as the water table is often very deep in arid and semiarid regions [McCord, 1991]. The initial water content of the soil profile was 0.06 and the simulation duration was 10 years.

3. Results and Discussion

3.1. Parameters Fitted

[32] The observed matric potential data from the two tensiometers and the mean volumetric water content of the soil core are shown in Figure 2. It is clear that there is an inflexion for both matric potential and water content at the potential of about −100 cm, after which the matric potential of the upper tensiometer had a sharp decrease and the loss of water content slowed.

Figure 2.

Observed matric potential values and mean volumetric water content of soil core.

[33] The critical matric potential hc of the samples is about −95cm. For the sand used here, vapor flow may be significant when matric potential is less than −500 cm [Bohne and Salzmann, 2002], so the fitted range of matric potential is −95 to about −500 cm. Because all five soil samples exhibited the same result, we included the results for one sample only. Figure 3a shows the fitted result for VG model combination. This model combination does not accurately describe the observed data, especially when the matric potential is less than −200 cm. That is, the commonly used hydraulic models that account for capillary forces could not describe the water dynamic at low matric potentials (more negative) when film flow may become the dominant process. However, it should be noted that, for the film flow to be the dominant process, desaturation of pores and moderately low magnitude matric potentials must be met. And these two required conditions can only be satisfied in media with moderately large pore sizes or fracture apertures [Tokunaga, 2009, 2011]. The best fits between predicted and observed data are VT and FT0 model pair variations, with r2 values of 0.999 and 0.998 (Figures 3b and 3c), respectively. Model variations FT1 and FTM (Figure 3d), which pair the same models as FT0 but propose ω0 = ω1 also showed good fitted results, but with lower accuracies. The r2 values are 0.987 (fitted λ) and 0.969 (measured λ), respectively. These low accuracies may indicate that the surface roughness could be an important factor in describing the hydraulic models. Although the VT model provides a best fitted result, the existence of residual water content in SWRC may be physically unrealistic and would not be suitable under very dry conditions.

Figure 3.

Observed and fitted matric potential of upper tensiometer.

[34] For the VT model, Figure 4 shows that when water content approaching the residual water content of 0.005, it becomes nearly constant; thus, a small change in moisture content could cause a dramatic change in matric potential.

Figure 4.

Fitted soil water retention curves.

[35] The fitted results of all cases illustrate that the modified TK model can describe film flow under dry conditions very well. Table 3 shows all fitted parameters and relative r2 values.

Table 3. Fit Parameters
CaseθrθsαlnKs (cm/h)ω0ω1τλ (mm)r2
VM0.0480.1570.1210.0051.84568.22    0.845
VT0.0050.2880.1620.5002.026 17.42   0.999
FT0      7.80534.84−0.3760.1980.998
FT1      41.31 −0.4290.1500.987
FTM      50.31 −0.436 0.969

3.2. Testing Fitted Parameters in Drier Ranges

[36] The parameters for all model variations fitted actual data well when the matric potential is greater than about −10 m. Under field conditions, however, the matric potential could be less than −1000 m. To check the performance of the fitted parameters under more drier conditions (less than −10 m), the observed potential data from lower tensiometer were used as test values after the upper tensiometer failed (when the lower tensiometer failed, the surface matric potential may have been less than −1.0 × 105 m as predicted by the FT0 model variation). Figure 5 shows that the VT model variation provides a very poor prediction of results when water content approaches the residual water content. The FT0 model variations shows predict values well, but at a lower accuracy than the fitted result of the FT0 model variation. This poor performance may be due to the presence of vapor flow, which would be significant under very dry conditions [Bohne and Salzmann, 2002], and hydraulic nonequilibrium [Bohne and Salzmann, 2002; Tokunaga, 2009], caused by very low hydraulic conductivities.

Figure 5.

Observed and predicted matric potential of lower tensiometer (FT0 predicted values mean the simulation results of matric potential using the fitted parameters from upper tensiometer).

3.3. Impact on Water Dynamic

[37] Figure 6 shows the simulation results of the water dynamic in a 2 m vertical soil column in 10 years, using different pairs of hydraulic models. Compared to other model variations, the VG combination greatly underestimates water loss, because it underestimates conductivity accounting for capillary flow and because it has a high fitted value of residual water content (0.045). The water loss predicted by VT model is quite rapid, with the dry front reaching to about 130 cm in 30 days, which is not realistic under field conditions [e.g., Scanlon et al., 2003; Ma et al., 2009]. This rapid water loss contributes to the higher hydraulic conductivity (≥K(θr)) and sharper water potential gradients when the surface water content approaches the residual water content, as shown in Figure 4. Model variations FT0 and FTM show similar changes, with surface water contents decreasing to near zero quickly, followed by the hydraulic conductivity of the surface dry layer dropping to extremely low values, slowing the water loss from the underlying profile.

Figure 6.

Simulation of water flow for a 2 m long profile using different pairs of hydraulic models (a: VG; b: VT; c: FT0; d: FTM). (Water content profile of t = 365 d and t = 3650 d are the same in VT.)

[38] Thus, neglecting film flow can greatly underestimate the water loss during evaporation. The commonly used SWRC model with residual water content is not suitable for describing the water dynamic in very dry ranges, even when it is coupled with the hydraulic conductivity function to account for film flow. Our new theoretical SWRC model can provide accurate and reliable results for water dynamics when coupled with the modified Tokunaga K(θ) model.

4. Conclusions

[39] Traditional soil hydraulic models accounting for capillary forces may perform well in the wet to medium moisture range, but provide poor results for water flow at lower water contents where film flow may become the dominant process in porous media. Based on the film thickness function accounting for matric potential, we formulated a new theoretical SWRC model. We coupled this new SWRC with a modified Tokunaga model that accounts for hydraulic conductivity and yields a theoretical and mathematically simple soil hydraulic model that can be used to describe film flow in unsaturated porous media. Additionally, we obtained a K(θ) form hydraulic conductivity function by eliminating the film thickness. This K(θ) form hydraulic conductivity is a meaningful and more convenient function to use as the water film thickness is impacted by surface and solution chemistry in unsaturated soils.

[40] We used evaporation experiments to test the performance of the new model in obtaining proper parameters using an inverse method. A comparison of different pairs of soil hydraulic models showed that the new model may describe the water flow at low water contents well. Although the combination of van Genuchten model with modified Tokunaga model performed best in inverse modeling, the fitted parameters could not describe the water flow under drier conditions due to the higher hydraulic conductivity and sharper water potential gradient when water content approached the residual water content.

[41] A 10 years water dynamic simulation in a 2 m vertical soil column showed that the commonly used hydraulic model (the VG-Mualem model in this paper) that accounts for capillary force may cause serious underestimation of water loss during the evaporation at low water contents (dependent on the magnitude of the residual water content), while the commonly used SWRC model with residual water content overestimates water loss in very dry ranges when coupled with the hydraulic conductivity function accounting for film flow. Additionally, the combination of the new theoretical SWRC model with the modified Tokunaga model can provide proper and reliable results of the water change.

[42] However, additional measured data from different soils are needed to test the performance of this new model, both in laboratory and field experiments.

Acknowledgments

[43] The research was supported by the Keygrant Project of Chinese Ministry of Education (310005) and the National Science Foundation of China (41272039). This work also forms part of International Collaboration projects (B06026) and the wider UK-China collaboration. We wish to thank Jingfang Wang, Jin Zhang, and Bing Jia for the field works and laboratory analysis.

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