Water Resources Research

A new capillary and thin film flow model for predicting the hydraulic conductivity of unsaturated porous media

Authors


Abstract

[1] Most classical predictive models of unsaturated hydraulic conductivity conceptualize the pore space as either bundles of cylindrical tubes of uniform size or assemblies of cylindrical capillary tubes of various sizes. As such, these models have assumed that liquid configuration is the same in both the wet and dry ranges and that a single concept can be used to describe water transport over the entire range of matric head. Yet theoretical and experimental findings suggest that water transport in wet media, which mostly occurs in water saturated capillaries, is quite different from that in dry media, which occurs in thin liquid films. Following these observations, this paper proposes a new model for predicting the hydraulic conductivity of porous media that accounts for both capillary and thin film flow processes. As with other predictive models, a mathematical relationship is established between hydraulic conductivity and the water retention function. The model is mathematically simple and can easily be integrated into existing numerical models of water transport in unsaturated soils. In sample calculations, the model provided very good agreement with hydraulic conductivity data over the entire range of matric head. Two other well-supported models, on the other hand, were unable to conform to the experimental data.

1. Introduction

[2] The description of soil-water transport processes in unsaturated porous media often hinges on the adequacy of the highly nonlinear functions that represent water retention and hydraulic conductivity. The literature abounds with parametric models that describe both water retention and hydraulic conductivity data. These models range from simple mathematical expressions for unstructured media, with unimodal pore size distributions [Leij et al., 1997], to more complex formulations for structured media with multimodal pore size distributions [Othmer et al., 1991; Ross and Smettem, 1993; Durner, 1994]. Models have also been proposed to describe water retention [Khlosi et al., 2008] and hydraulic conductivity [Peters and Durner, 2008] over the entire range of matric head. In many cases, however, measurements of unsaturated hydraulic conductivity are unavailable. As an alternative to direct measurements, models have been proposed to estimate hydraulic conductivity from water retention data, which is more easily measured. In general, these models may be classified as those based on (1) similarity between Darcy-Buckingham's law and other viscous flow equations such as those of Navier-Stokes and (2) statistical representations of pore space [Brutsaert, 1967; Mualem, 1986]. It follows that the models either conceptualize the pore space as cylindrical tubes of uniform size, with coexisting wetting and nonwetting fluids, or as assemblies of cylindrical capillary tubes of various sizes, which are either completely filled with wetting or nonwetting fluid. As such, these models have assumed that liquid configuration is the same in both the wet and dry ranges. Although it has generally been found that these models are successful in the wet to moderately wet range (high values of matric head), where water is mostly held by capillary forces, they have been found to underestimate hydraulic conductivity under dry conditions [Goss and Madliger, 2007; Jansik, 2009]. As pointed out by Tuller and Or [2001], this shortcoming may be attributed to the lack of consideration of soil-water flow in thin films, likely to become important at low values of matric head.

[3] Motivated by this and other limitations, such as the lack of consideration of adsorptive forces and unrealistic cylindrical representation of pore space, Tuller and Or [2001] proposed an alternative approach for the derivation of hydraulic conductivity functions for homogeneous porous media. Their model is based on liquid configurations in both angular and slit-shaped spaces, and accounts for capillary, corner and thin film flows. Although of great scientific interest, the model is mathematically complex and must be used in conjunction with the water retention model of Or and Tuller [1999], which often fails to describe experimental data in the intermediate saturation range because of the limited flexibility of the probability distribution function used to characterize the pore size distribution. Tokunaga [2009] also used the principles of interfacial physical chemistry and the concept of flow in thin films to predict water transport in dry monodisperse porous media.

[4] In this study, theoretical considerations and experimental observations are used to partition water retention into capillary and adsorptive components. These components are associated with two different liquid configurations. The first is an assembly of water or air filled capillary tubes of various diameters whereas the second is a thin film of water stretched over the surface of the solid particles. A conventional statistically based model is used to predict hydraulic conductivity within the water filled capillaries while a theoretical model of soil-water flow in thin films is used to predict hydraulic conductivity within the thin liquid films. In so doing, the model accounts for both capillary and adsorptive forces as well as for dual occupancy of wetting and nonwetting phases.

[5] The paper shows the effectiveness of the new model in using water retention data to predict hydraulic conductivity over the entire range of matric head. Model performance is subsequently evaluated by comparing results with those of other well-supported models.

2. Proposed Model

2.1. Water Retention

[6] Most popular among the parametric models that have been proposed to describe water retention data are the equations of Brooks and Corey [1964] and van Genuchten [1980]. An often problematic element in these equations is the use of a parameter called residual water content. While this parameter does not consistently correspond to a recognized physical entity, it is speculated to represent the largest water content for which pore water is held primarily by adsorptive rather than capillary forces [Corey and Brooks, 1999]. Most studies consider residual water content as a fitting parameter and assume that liquid flow processes are negligible at water contents smaller than a residual value. This approach has generally been successful at large values of matric head, where flow mostly occurs in water saturated capillaries. Yet, it remains physically unrealistic to consider water retention data as never becoming less than the residual water content. In fact, there is ample experimental evidence showing that water content ultimately approaches zero as the soil equilibrates with moisture-free air [Schofield, 1935; Campbell and Shiozawa, 1992].

[7] In essence, theoretical studies of water retention have led to believe that multilayer adsorption progressively dominates capillary retention, and that adsorption is the dominating force that holds water in its condensed state when the soil is dry. Generally speaking, adsorption is a process in which molecules of water vapor move onto a solid or liquid surface. This process occurs until a thermodynamic equilibrium is reached between the gaseous phase and the adsorbed layer. In very dry soils, the experimental studies of Orchiston [1953], Amali et al. [1994], and de Seze et al. [2000] have shown that water content is best described with the Brunauer-Emmett-Teller (BET) theory, which extends the Langmuir theory for monolayer adsorption of gas molecules on a solid surface to multilayer adsorption [Brunauer et al., 1938]. The BET theory is generally adequate for values of relative humidity less than approximately 35%, which translates into a matric head of −104 m when using Kelvin's equation at 298.15 K. For larger values of matric head, adsorption gives rise to multilayer films with thicknesses of the order of nanometers. These films can be regarded as part of a thinned liquid phase [Derjaguin et al., 1987; Churaev, 2000]. The state of this thinned liquid phase can be described by means of the disjoining pressure isotherm, as discussed later in the paper.

[8] In response to these and other observations, new models of water retention and physically based modifications of existing models have been suggested by a number of researchers [Ross et al., 1991; Campbell and Shiozawa, 1992; Rossi and Nimmo, 1994; Fredlund and Xing, 1994; Fayer and Simmons, 1995; Morel-Seytoux and Nimmo, 1999; Webb, 2000; Groenevelt and Grant, 2004; Khlosi et al., 2006; Silva and Grifoll, 2007]. The seminal work of Campbell and Shiozawa [1992] is of particular importance as it was the first to note that water retention at low water content shows a linear behavior on a semilog plot. In a later study, Fayer and Simmons [1995] used this relationship to develop a method of extending most parametric models to zero water content conditions. In essence, the method consists in replacing residual water content by the log linear form observed by Campbell and Shiozawa [1992]. The resulting equation can be expressed in a manner that highlights the mechanisms (capillarity and adsorption) of water retention:

equation image

where θ is the volumetric water content, θc is the volumetric water content ascribed to capillary forces, θs is the saturated volumetric water content, and θa is the volumetric water content ascribed to adsorptive forces. It follows that the terms on the right-hand side of this equation are the capillary and adsorptive components of volumetric water content, respectively.

[9] Although a number of parametric models may be used to describe the volumetric water content ascribed to capillary forces in soils with unimodal and multimodal pore structures, it is herein represented with a simplified form of Kosugi's [1996] two-parameter equation with zero residual water content:

equation image

where erfc( ) is the complementary error function, hm is the matric head, hm,median is the matric head that corresponds to the median capillary pore radius, and σ is the standard deviation of the log-transformed capillary pore radius distribution.

[10] The volumetric water content ascribed to adsorptive forces is described by a form of the equation proposed by Campbell and Shiozawa [1992]:

equation image

where θo is the volumetric water content due to adsorption at a matric head of −1 m, and hm,dry is the matric head at oven dryness. The value of matric head at oven dryness depends on prevailing laboratory conditions, i.e., temperature, pressure and relative humidity. However, experimental results have shown that oven dryness generally corresponds to a finite matric head of −105 m [Schofield, 1935; Russam, 1958; Croney and Coleman, 1961; Campbell and Shiozawa, 1992].

[11] Figure 1 expresses water content in terms of degree of saturation, and highlights the different components that are accounted for in the water retention model: (1) a capillary component with a sigmoidal behavior in the wet range and (2) an adsorption component with a logarithmic behavior in the dry range. As the mechanisms responsible for these components of water retention are not related, the presence of one component does not exclude or preclude the other. Hence, in the moderately wet range, smaller pores may be completely filled with water while a thin film of water may cover the surface of the larger pores.

Figure 1.

Water retention model with underlying capillary and adsorptive components.

2.2. Hydraulic Conductivity

[12] According to the preceding description of water retention, water movement in moderately wet porous media may occur through the entire section of smaller water-filled pores as well as through thin films of water surrounding larger pores. In dry media, on the other hand, liquid water movement may only occur through the continuous water film that covers the solid particles as the central portion of the pores is occupied by the nonwetting phase. As a result, the pores cannot be conceptualized solely as an assembly of parallel tubes, which are either completely filled with water or air. To account for the presence of a thin liquid film, it is herein suggested that the liquid phase be conceived as both an assembly of water filled capillary tubes of various diameters, and a thin film of liquid stretched over the solid particles. As illustrated in Figure 2, tubes of various radii are used to represent capillarity. The radii of the fully saturated tubes are inferred from the capillary component of the water retention function using the Young-Laplace equation. In contrast, adsorption is represented by a thin liquid film of varying thickness. Based on the dependence of film thickness on matric head, the thickness of the film is shown to be greatest when the porous medium is wet and smallest when it is dry. The extent of the film, on the other hand, is shown to vary as a function of the capillary component of water retention.

Figure 2.

Liquid configurations for the different components of water retention.

[13] It follows that two different models must be used simultaneously to describe the hydraulic conductivity of the porous media: (1) a capillary flow model and (2) a thin film flow model. Peters and Durner [2008] recently suggested that relative hydraulic conductivity can be described as the weighted average of the contributions of capillary and thin film flows. However, in the presence of a partitioned water retention function, with both capillary and adsorptive components, it is suggested that water transport models be combined as follows:

equation image

where kr is the relative hydraulic conductivity, kr,c is the relative hydraulic conductivity ascribed to capillary flow, and kr,a is the relative hydraulic conductivity ascribed to thin film flow. As formulated, the terms on the right-hand side of this equation represent the capillary and adsorptive (or thin film) components of relative hydraulic conductivity, respectively. In contrast to using a weighted average, equation (4) combines the capillary and thin film flow models without requiring an additional fitting parameter.

2.2.1. Capillary Conductivity Model

[14] Capillary flow models conceptualize the pores as an assembly of parallel capillary tubes, which may or may not be sectioned and randomly rejoined. Under this assumption, a number of fully saturated capillary tubes transmit water while other tubes remain empty. For the sake of simplicity, the portion of fully saturated tubes is obtained from the water retention function by means of the Young-Laplace equation. Hydraulic conductivity at the continuum scale is then determined by integrating the contributions of the fully saturated tubes, as calculated with the Hagen-Poiseuille equation. In the literature, models have evolved from very simple representations of pore space to more complex descriptions that account for tortuosity and variations in tube geometry [Childs and Collis-George, 1950; Gates and Lietz, 1950; Fatt and Dykstra, 1951; Burdine, 1953; Wyllie and Gardner, 1958; Mualem, 1976a; Alexander and Skaggs, 1986]. Although many different models may be used, Mualem's [1976a] model has been shown to provide a reasonable description of the relative hydraulic conductivity due to capillary flow. Inserting equation (2) into Mualem's model yields the following expression of relative hydraulic conductivity [Kosugi, 1996]:

equation image

where kc is the hydraulic conductivity due to capillary flow, ks is the saturated hydraulic conductivity, and l is a lumped parameter that is generally assumed to account for pore tortuosity and pore connectivity, which Mualem [1976a] empirically set equal to 0.5 for predictive purposes. The physical meaning of parameter l is often questioned given that its fitted value is frequently found to be negative [Schaap and Leij, 2000]. It is to be noted that by using equation (5), capillary conductivity at the continuum scale is estimated with the portion of pore water that is retained solely by capillary forces. This is significant given that these models are based on the assumption of capillary flow and that there is no reason to expect them to be valid when adsorptive forces dominate.

2.2.2. Thin Film Conductivity Model

[15] Given the similarity between Darcy-Buckingham's law and other viscous flow equations (solutions of the Navier-Stokes equations for various geometries), fluid flow has often been imagined to occur in the form of annuli or thin planar films. In these cases, the wetting fluid is assumed to be a continuous component stretched over the solid particles whereas the nonwetting fluid is assumed to occupy the central portion of the pores. Hence, the wetting and nonwetting fluids are assumed to coexist within the pores. In recent years, a number of researchers have applied the concept of flow in thin films to the study of hydraulic conductivity in unsaturated porous media [Tuller and Or, 2001; Tokunaga, 2009].

[16] Inspired by these studies, a theoretical model of thin film flow in porous media is obtained by solving the Navier-Stokes equations for steady state laminar flow through planar liquid films. Most critical to this approach is understanding the dependence of thin film thickness on potential. By nature, thin liquid films are composed of many transition layers within which intensive properties and composition may differ from those in the bulk liquid phase. The combined effect of these interfacial interactions results in larger pressures in the thin liquid films than in the bulk liquid phase. For planar films, the difference in potential may be described by means of the disjoining pressure isotherm [Derjaguin et al., 1987]:

equation image

where Π is the disjoining pressure, δ is the film thickness, Pg is the pressure in the gas phase, and Po is the pressure in the bulk liquid phase. Based on this definition, disjoining pressure in thin planar films (without capillary interactions) is equal to the negative value of matric potential. It follows that matric head is equal to −Π(δ)/(ρog) where ρo is the density of the liquid phase and g is the gravitational acceleration constant. The form of the disjoining pressure isotherm is contingent on the type of surface forces at play. In porous media, the major types of surface forces are ionic-electrostatic, molecular, structural, and adsorptive. Yet, in the study of thin films, it is common practice to consider only the effects of ionic-electrostatic and molecular forces. Hence, disjoining pressure may be defined as [Derjaguin and Churaev, 1974]:

equation image

where Πe and Πm are the ionic-electrostatic and molecular components of disjoining pressure, respectively.

[17] In order to define the ionic-electrostatic component of disjoining pressure, let us consider a planar film of ionic solution bounded by a substrate of given potential. Given its charge, the substrate produces a much higher concentration of ions near the surface. This concentration forces the ions to diffuse away from the surface to equalize the concentration throughout the solution. The charged surface and the distributed charge in the adjacent solution are termed the diffuse double layer, which may be described with the Poisson-Boltzmann equation. Solutions to this equation may be found for various cases that include truncated and extended diffuse double layers with symmetric or asymmetric ionic solutions. Langmuir [1938] solved the Poisson-Boltzmann equation for a low-concentration symmetric ionic solution with a high potential substrate, and found the following expression for the ionic-electrostatic component of disjoining pressure:

equation image

where ɛr is the static relative permittivity of the liquid phase, ɛo is the permittivity of free space, kB is the Boltzmann constant, T is the absolute temperature, e is the electron charge, and Z is the valence change.

[18] The molecular component of disjoining pressure arises from long-range fluctuating dipole bonds, or van der Waals bonds, between condensed bodies. Once again, let us consider the simple case of a planar film, and assume that the film is held in place only by long-range molecular forces. In this case, the Gibbs free energy is calculated by adding liquid/liquid, liquid/solid, and solid/solid interactions between two macroscopic bodies [Israelachvili, 1992]. The disjoining pressure is subsequently calculated as the negative value of the derivative of the Gibbs free energy per unit area, which yields

equation image

where Asvl is the Hamaker constant for solid-vapor interactions through the intervening liquid. Theoretical and experimental values of the Hamaker constant for solid-liquid-air systems range from −10−19 to −10−20 J [Israelachvili, 1992]. In real soils, the Hamaker constant is expected to differ slightly from these values due to the effects of surface geometry and heterogeneity [Tuller and Or, 2005].

[19] Equations (7) to (9) provide the basis for computing film thickness as a function of disjoining pressure (or matric head), and vice versa. Figure 3 illustrates the contributions of the ionic-electrostatic and molecular components of disjoining pressure in the case of a thin film of water on a substrate composed primarily of crystalline and amorphous silicates, such as the soil solid phase. Under these conditions, films in excess of 10 nm are mostly held by ionic-electrostatic forces whereas thinner films may be ascribed to molecular forces.

Figure 3.

Disjoining pressure isotherm for water films on a planar substrate mainly composed of crystalline and amorphous silicates at room temperature.

[20] Let us now focus on the hydrodynamic behavior of these thin liquid films. As shown in Figure 4, it is assumed that flow is parallel to the plate in the x direction, and that other components of flow are nonexistent. Recalling that velocity is finite at z = 0 and null at z = δ, the solution to the Navier-Stokes equations yields a parabolic distribution of velocity. Integrating the velocity distribution over the film thickness results in the following expression of volumetric flow rate per unit length [Bird et al., 1960]:

equation image

where Qa is the volumetric flow rate per unit length of film, and μo is the dynamic viscosity of the bulk liquid. In contrast to thicker films, flow in very thin films (thinner than 10 nm) is likely to be influenced by modified liquid viscosity close to the plate. For flow in these films, Or and Tuller [2000] introduced an exponential viscosity function into the Navier-Stokes equations and found the following expression for volumetric flow rate per unit length of film:

equation image

and B, a function of film thickness and temperature, is given by

equation image

where a = 1.621 · 10−7/T (m), and Ei(−x) = −E1(x) = −equation image [exp(−t)/t]dt is the exponential integral, which can be evaluated using Taylor series [Gautschi and Cahill, 1964]. Note that as an alternative to equations (11) and (12), flow in very thin films can be described by a form of equation (10) in which the dynamic viscosity of the bulk liquid is replaced by the average dynamic viscosity of the liquid film.

Figure 4.

Adsorption on a planar surface with schematic representation of velocity distribution.

[21] To obtain the superficial, or Darcy-Buckingham flux, the volumetric flow rate per unit length must be multiplied by the specific perimeter (perimeter of the solid particles per unit cross-sectional area). For monodisperse samples of spherical particles, Tokunaga [2009] found the specific perimeter to be equal to [12(1 − n)]/(πD) where n is the porosity, and D is the diameter of the spherical particles. It follows that the Darcy-Buckingham flux in a monodisperse sample of spherical particles can be expressed as

equation image

where ka is the hydraulic conductivity due to film flow. Using the experimental value of saturated hydraulic conductivity, the relative hydraulic conductivity due to film flow can be defined as

equation image

[22] This equation extends the work of Tokunaga [2009] to films thinner than 10 nm. To further generalize this work to polydisperse samples, the diameter of the spherical particles can be replaced by an equivalent diameter, De, which is the diameter of a spherical particle whose monodisperse sample has the same specific surface area per unit volume as the polydisperse sample. Based on simple geometrical considerations, the specific surface area per unit volume of a monodisperse sample of equivalent particles is found to be equal to 6/De. The equivalent diameter can therefore be expressed in terms of the specific surface area per unit volume of the polydisperse sample, i.e., De = 6/Ss,polydisperse. Although the specific surface area of the polydisperse sample can be determined experimentally, it can also be estimated from water retention data at low values of matric head [Tuller and Or, 2005]. Based on this estimation, the equivalent diameter can be expressed as

equation image

where θm is the volumetric water content at matric head hm,m for which capillary condensation due to surface roughness is negligible [Tuller and Or, 2005]. As an approximation, this data point can be taken as the water retention function value closest to hm = −103 m.

[23] The preceding model builds on analyses of interfacial physical chemistry to predict relative hydraulic conductivity in the dry range without the use of any adjustable parameters. Figure 5 highlights the contribution of this model to relative hydraulic conductivity. As shown, relative hydraulic conductivity is dominated by thin film flow in the dry range while capillary flow is the primary means of water transport in the wet range. The point of crossover separates the relative hydraulic conductivity curve into capillary and film flow dominated regions. A smooth transition is observed as thin film flow starts to govern water transport at a matric head of −2.1 m. At this point, flow is mostly ascribed to the portion of the film which is held by ionic-electrostatic forces. A further decrease in matric head leads to the predominance of the portion of film which is held by long-range molecular forces. In general, this reveals that thin film flow, which is often ignored in conventional models, plays an important role in water transport at low values of matric head.

Figure 5.

Hydraulic conductivity model with underlying capillary and thin film flow components.

3. Model Evaluation (or Testing)

3.1. Evaluation Data Sets

[24] A comprehensive search of relevant literature yielded only a small number of data sets suitable to evaluate the capabilities of the proposed model. Of these data sets, one is reported by Pachepsky et al. [1984], one is described by Mehta et al. [1994], another is reported by Fujimaki and Inoue [2003a, 2003b], five are listed in Mualem's [1976b] catalog of unsaturated soils, and four are selected from the UNSODA database [Nemes et al., 2001]. These data sets of mostly undisturbed soils, with a wide range of textures and origins, were selected because the experimental measurements covered both wet and dry regions of the water retention and hydraulic conductivity functions. A list of the different data sets can be found in Table 1.

Table 1. Measured Soil Properties for the Testing Data Sets
Data SetReferenceaMeasured Soil Properties
nequation imagesks (m/s)
Sandy loamA0.430.439.26 × 10−7
Shonai sandB0.430.431.09 × 10−4
Masa loamy sandC0.390.345.90 × 10−6
Gilat loamD0.440.442.00 × 10−6
Pachapa loamD0.460.462.00 × 10−6
Adelanto loamD0.430.434.50 × 10−7
Pachapa fine sandy clayD0.330.331.40 × 10−6
Rehovot sandD0.400.401.33 × 10−4
Berlin medium sand 4E0.400.398.01 × 10−5
Poederlee loamy sandE0.420.422.79 × 10−5
Poederlee sandE0.420.421.90 × 10−5
Hupsel sandE0.360.368.16 × 10−5

3.2. Illustrative Examples

[25] The essential features of the new model are demonstrated with the various evaluation data sets, which range from loam to sand. A summary of the model parameters obtained by fitting the water retention model (equations (1) through (3)) to the testing data sets is given in Table 2. Curve fitting was conducted with a hybrid genetic-simplex algorithm, which combined the exploratory capacity of a genetic algorithm with the convergence behavior of the downhill simplex, or Nelder-Mead, algorithm to meet requirements of accuracy, reliability and computing time. The set of best fit parameters obtained from the water retention data, was then used to predict relative hydraulic conductivity using equations (4), (5), and (14) with equations (7), (8), and (9) defining film thickness and equation (15) describing the equivalent diameter of the porous media. Table 3 summarizes the disjoining pressure parameters that were used to compute film thickness. Also note that parameter values for the bulk liquid were taken as those of pure water at 298.15 K (ρo = 997.04 kg/m3 and μo = 8.90 × 10−4 N s/m2).

Table 2. Fitted Parameters for the Testing Data Sets
Data SetReferenceaFitted Parameters
equation imageohm,median (m)σ
Sandy loamA0.07−2.061.54
Shonai sandB0.06−0.240.38
Masa loamy sandC0.08−1.181.23
Gilat loamD0.15−0.670.55
Pachapa loamD0.13−2.270.86
Adelanto loamD0.26−3.760.72
Pachapa fine sandy clayD0.11−1.140.82
Rehovot sandD0.02−0.270.59
Berlin medium sand 4E0.05−0.370.36
Poederlee loamy sandE0.12−0.590.89
Poederlee sandE0.06−0.630.97
Hupsel sandE0.04−0.330.47
Table 3. Disjoining Pressure Parameters
ParameterDefinitionUnitValue
AsvlHamaker constant(J)−6.00 × 10−20
eElectron charge(C)1.60 × 10−19
kBBoltzmann constant(J/K)1.38 × 10−23
TAbsolute temperature(K)298.15
ZValence change 1
ɛoPermittivity of free space(C2/(J m))8.85 × 10−12
ɛrStatic relative permittivity of water 78.41

[26] Figure 6 illustrates the capabilities of the new model for the Gilat loam data set. As shown, the water retention model is in very good agreement with the liquid saturation data, which ranges from saturation to a matric head of approximately −103 m. As formulated, the water retention model separates liquid saturation into individual contributions due to adsorptive and capillary forces. The point of crossover of these curves separates liquid saturation into capillary and adsorptive dominated regions. For the loam under consideration, adsorptive forces start to dominate water retention at a matric head of −1.0 m. At lower values of matric head, adsorption is the predominant force that holds water in the porous medium. The hydraulic conductivity model also shows very good agreement with the experimental data. By considering thin film flow, the model is able to reproduce the change in slope of the experimental data, which occurs at a matric head of −1.5 m. It is to be emphasized that such film contributions have generally been overlooked in conventional predictive models of hydraulic conductivity. Figure 7 shows somewhat similar results for the Pachapa fine sandy clay data set. In this case, however, the hydraulic conductivity model slightly underestimates the experimental data in both wet and dry ranges.

Figure 6.

Model results for Gilat loam. (a) Fitted water retention. (b) Predicted relative hydraulic conductivity.

Figure 7.

Model results for Pachapa fine sandy clay. (a) Fitted water retention. (b) Predicted relative hydraulic conductivity.

[27] In contrast to the finer-grained soils, the sandy loam data set presents a much smaller quantity of adsorbed water. Yet, as shown in Figure 8, the water retention model shows very good agreement with the experimental data over the entire range of matric head. The hydraulic conductivity model also predicts the experimental data quite well. For this soil, the point of transition from capillary-dominated flow to film-dominated flow occurs at a matric head of −10.1 m. The model also shows close agreement with the experimental data for other coarse-grained soils. As shown in Figures 9, 10, and 11, the model captures the distinctive features of both the liquid saturation and relative hydraulic conductivity data.

Figure 8.

Model results for Sandy loam. (a) Fitted water retention. (b) Predicted relative hydraulic conductivity.

Figure 9.

Model results for Rehovot sand. (a) Fitted water retention. (b) Predicted relative hydraulic conductivity.

Figure 10.

Model results for Berlin medium sand 4. (a) Fitted water retention. (b) Predicted relative hydraulic conductivity.

Figure 11.

Model results for Hupsel sand. (a) Fitted water retention. (b) Predicted relative hydraulic conductivity.

[28] Figures 12 and 13 show experimental data for Poederlee loamy sand and Poederlee sand, respectively. In both cases, the water retention model is in close agreement with the data while the hydraulic conductivity model shows significant differences with the experimental data in the dry range. For these data sets, the hydraulic conductivity data in the dry range was determined with the hot air method, which has been shown to overestimate hydraulic conductivity [Stolte et al., 1994]. The experimental error can be ascribed to thermal water (liquid and vapor) transport as well as redistribution and evaporation loss during sampling. According to van Grinsven et al. [1985], the effects of water redistribution can be expected to be less pronounced in finer-grained soils, such as the Poederlee loamy sand.

Figure 12.

Model results for Poederlee loamy sand. (a) Fitted water retention. (b) Predicted relative hydraulic conductivity.

Figure 13.

Model results for Poederlee sand. (a) Fitted water retention. (b) Predicted relative hydraulic conductivity.

[29] Figure 14 illustrates the features of the new model for four (4) data sets in which hydraulic conductivity is expressed as a function of liquid saturation. As shown, the water retention model is in good agreement with the measured data, capturing the behavior in both wet and dry regions. For most of the data sets, the hydraulic conductivity model is also in close agreement with the measurements. The discrepancy observed in some of the data sets can be attributed to the fact that film thickness is computed as a function of matric head, and not liquid saturation. Hence, hydraulic conductivity in the dry range is computed in terms of matric head, and subsequently expressed in terms of liquid saturation using the water retention function. Model errors are therefore compounded by small inaccuracies in the water retention function.

Figure 14.

Model results for various soils.

[30] On the whole, these examples show that the new theoretically based model, which uses simplified conceptualizations of pore space, has a great deal of explanatory power, and that it is generally consistent with that found by Tuller and Or [2001, 2002].

3.3. Comparison With Other Models

[31] In the following paragraphs, inferences on model performance are drawn from model predictions for the evaluation data sets. In adhering to the method of multiple working hypotheses, three (3) well-supported models are considered: (1) Kosugi-Mualem, K1-M, (2) extended Kosugi-Mualem, K2-M, and (3) extended Kosugi-Lebeau and Konrad, K2-LK. These models are formed with one of two parametric models of water retention. The first is the well-documented parametric model of Kosugi [1996], which is herein referred to as the K1 model, and the second is the extended form of the Kosugi model, which is referred to as the K2 model. In terms of hydraulic conductivity, two different models are used. The first is the conventional capillary model of Mualem [1976a], and the second is the capillary and thin film flow model presented earlier in this paper. These models are referred to as the M and LK models, respectively. For the sake of completeness, a detailed description of the K1-M and K2-M models is given in Appendix A.

[32] Figure 15 presents a visual comparison of the various models for the Gilat loam data set. As shown, the K1-M model is essentially unable to describe the experimental data in the dry range. Moreover, the model adjusts residual water content to an unrealistically large finite value, which is in no way related to the quantity of adsorbed water. A significant improvement is observed when using the K2-M model, which describes water retention from saturation to oven dryness. Although this model adequately describes the liquid saturation data and imparts a change in the slope of the relative hydraulic conductivity curve, it is unable to conform to the experimental hydraulic conductivity data in the dry range. This shortcoming is ascribed to the fact that conventional models of hydraulic conductivity omit thin film flow. This is highlighted by the fact that the K2-LK model, which accounts for thin film flow, shows much closer agreement with the experimental data.

Figure 15.

Comparison of model results with those of other well-supported models for Gilat loam. (a) Fitted water retention. (b) Predicted relative hydraulic conductivity.

[33] Such visual observations can be translated into quantifiable terms by means of the mean squared error, MSE, which is an indicator of the overall magnitude of the residuals (differences between the observed and predicted data). In this study, the data is log-transformed to ensure that the complete range of relative hydraulic conductivity is well represented, and the mean squared error is computed as follows:

equation image

where N is the number of observations in the data set, kr,i is the ith observed relative hydraulic conductivity, and equation imager,i is the ith predicted relative hydraulic conductivity. Table 4 summarizes the MSE of the various models for each of the data sets. For the sake of interpretation, the lowest values of MSE are highlighted in bold. As shown, the K2-LK model is the best of the candidate models for nine (9) out of the twelve (12) evaluation data sets, including the Gilat loam data set. This overall performance is reflected in the mean value of the mean squared error, MMSE, which is equal to 1.79. As such, the MMSE is 25 times smaller than that for the K1-M model and 2 times smaller than that for the K2-M model. These significant differences suggest that omitting thin film flow entails a large loss in predictive capability. This is clearly illustrated in the scatterplots of observed versus predicted values shown in Figure 16. The predicted values of the K2-M model show a general tendency to underestimate hydraulic conductivity in the dry range. In contrast, the predicted values of the K2-LK model are close to the observed values, and the data generally falls along a straight line with a slope of one.

Figure 16.

Scatterplot of observed versus predicted values for the various models.

Table 4. Model Prediction Statistics
Data SetReferenceaMean Squared Errorb
K1-McK2-MdK2-LKe
Sandy loamA60.3312.362.03
Shonai sandB19.956.172.02
Masa loamy sandC0.990.160.79
Gilat loamD165.723.530.29
Pachapa loamD3.673.420.67
Adelanto loamD18.601.951.92
Pachapa fine sandy clayD44.841.061.01
Rehovot sandD25.863.312.55
Berlin medium sand 4E121.551.560.89
Poederlee loamy sandE21.200.962.84
Poederlee sandE34.013.972.85
Hupsel sandE30.472.323.59
  45.603.401.79

4. Conclusion

[34] Most classical predictive models of unsaturated hydraulic conductivity have been derived by conceptualizing the pore space as either bundles of cylindrical tubes of uniform size or assemblies of cylindrical capillary tubes of various sizes. As such, these models have assumed that a single concept can be used to describe water transport in both the wet and dry ranges. In contrast, theoretical and experimental findings have shown that thin film flow most likely prevails over capillary flow in dry porous media. Motivated by this shortcoming, a new approach for modeling unsaturated hydraulic conductivity in both wet and dry ranges was proposed. As a starting point, water retention was partitioned into capillary and adsorptive components. These components were then associated with two different liquid configurations: (1) an assembly of water or air filled capillary tubes of various diameters and (2) a thin film of adsorbed water stretched over the solid particles. A conventional statistically based model was subsequently used to determine the conductivity of the water filled capillaries whereas a theoretical model of soil-water flow in thin films was used to describe the conductivity of the adsorbed water. This is significant given that there is no reason to expect conventional statistical models to be valid when adsorptive forces dominate. Finally, the components of conductivity were coupled to form a hydraulic conductivity function that accounts for both capillary and thin film flows. In this manner, conductivity at the continuum scale was determined with both the portions of pore water that are retained by capillary and adsorptive forces.

[35] Sample calculations for a number of testing data sets showed the prevalence of thin film flow in the dry range. In general, the proposed predictive model provided very good agreement with hydraulic conductivity data over the entire range of matric head. Moreover, comparison with two other well-supported models revealed significant advantages of the proposed model. Overall, the new model shows promise, and warrants further comparison to experimental data as it becomes available. At present, model testing is limited by the paucity of combined measurements of water retention and hydraulic conductivity in the dry range.

Appendix A

[36] Numerous capillary flow models have been proposed to predict unsaturated hydraulic conductivity from the water retention function. Among the most popular, the model put forward by Mualem [1976a] may be written as follows:

equation image

where kr is the relative hydraulic conductivity, k is the unsaturated hydraulic conductivity, ks is the saturated hydraulic conductivity, Θ = (θ − θr)/(θs − θr) is the normalized volumetric water content, θ is the volumetric water content, θr is the residual volumetric water content, θs is the saturated volumetric water content, X is a dummy variable of integration representing volumetric water content, and hm is the matric head. As with other capillary flow models, the model expressed by equation (A1) can be used with any parametric model of water retention. It is nonetheless more convenient to perform the integration along the logarithm of the absolute value of the matric head axis when water retention is described from saturation to oven dryness. In such a case, the following variant of equation (A1) is preferred:

equation image

where Θ = θ/θs is the normalized volumetric water content, Y is a dummy variable of integration representing the logarithm of the absolute value of matric head, and hm,dry is the matric head at oven dryness.

[37] The following paragraphs present the equations obtained when Mualem's [1976a] capillary flow model is used in conjunction with the extended and nonextended forms of the Kosugi [1996] water retention model.

A1. Kosugi Retention Model (Nonextended Form)

[38] The equation of Kosugi [1996] is among the most widely used parametric models of water retention, and is generally expressed as

equation image

where erfc( ) is the complementary error function, hm,median is the matric head that corresponds to the median pore radius, and σ is the standard deviation of the log-transformed pore radius distribution. By integrating this model into equation (A1), the following closed form expression for relative hydraulic conductivity may be derived:

equation image

The combination of the water retention and hydraulic conductivity models expressed by equations (A3) and (A4) is herein referred to as the K1-M model.

A2. Kosugi Retention Model (Extended Form)

[39] Fayer and Simmons [1995] presented a method of extending most parametric models to zero water content conditions by replacing residual water content with an equation proposed by Campbell and Shiozawa [1992]. By applying this method to the Kosugi model, the following expression is found [Khlosi et al., 2006]:

equation image

where θa = θo [1 − ln∣hm∣/ln∣hm,dry∣] is the volumetric water content ascribed to adsorptive forces, and θo is the volumetric water content due to adsorptive forces at a matric head of −1 m. Using this water retention model in conjunction with equation (A2) does not result in a closed form expression for relative hydraulic conductivity. For this reason, relative hydraulic conductivity must be computed by numerical integration. It is to be noted that the combined use of equations (A2) and (A5) is herein referred to as the K2-M model.

Notation
a

Viscosity constant (L).

Asvl

Hamaker constant for solid-vapor interactions through the intervening liquid (ML2 T−2).

B

Thin film variable viscosity function (L3).

D

Diameter of the spherical particles (L).

De

Equivalent diameter (L).

e

Electron charge (TI).

g

Gravitational acceleration constant (LT−2).

hm

Matric head (L).

hm,dry

Matric head at oven dryness (L).

hm,m

Matric head for which capillary condensation due to surface roughness is negligible (L).

hm,median

Matric head that corresponds to the median capillary pore radius (L).

k

Hydraulic conductivity (LT−1).

ka

Hydraulic conductivity ascribed to film flow (LT−1).

kB

Boltzmann constant (ML2 T−2 Θ−1).

kc

Hydraulic conductivity ascribed to capillary flow (LT−1).

kr

Relative hydraulic conductivity.

kr,a

Relative hydraulic conductivity ascribed to thin film flow.

kr,c

Relative hydraulic conductivity ascribed to capillary flow.

kr,i

ith observed relative hydraulic conductivity.

equation imager,i

ith predicted relative hydraulic conductivity.

ks

Saturated hydraulic conductivity (LT−1).

l

Capillary model parameter.

MSE

Mean squared error.

MMSE

Mean value of the mean squared error.

n

Porosity.

N

Number of observations in the data set.

Pg

Pressure in the gas phase (ML−1 T−2).

Po

Pressure in the bulk liquid phase (ML−1 T−2).

qa

Superficial volumetric flux due to film flow (LT−1).

Qa

Volumetric flow rate per unit length of film (L2 T−1).

Sr

Degree of saturation.

Ss,polydisperse

Specific surface area per unit volume of the polydisperse sample (L−1).

T

Absolute temperature (Θ).

vf

Film velocity (LT−1).

x,z

Spatial coordinates (L).

Z

Valence change.

δ

Film thickness (L).

ɛo

Permittivity of free space (M−1 L−3 T4 I2).

ɛr

Static relative permittivity of the liquid phase.

θ

Volumetric water content.

θa

Volumetric water content ascribed to adsorptive forces.

θc

Volumetric water content ascribed to capillary forces.

θm

Volumetric water content at matric head hm,m.

θo

Volumetric water content due to adsorption at a matric head of −1 m.

θr

Residual volumetric water content.

θs

Saturated volumetric water content.

Θ

Normalized volumetric water content.

μo

Dynamic viscosity of the bulk liquid (ML−1 T−1).

Π

Disjoining pressure (ML−1 T−2).

Πe

Ionic-electrostatic component of disjoining pressure (ML−1 T−2).

Πm

Molecular component of disjoining pressure (ML−1 T−2).

ρo

Density of the liquid phase (ML−3).

σ

Standard deviation of the log-transformed capillary pore radius distribution.

Acknowledgments

[40] The authors wish to acknowledge the financial participation of Natural Sciences and Engineering Research Council of Canada (NSERC) Industrial Research Chair in the Operation of Infrastructures Exposed to Freezing. The authors also extend their appreciation to Nicolas Venkovic for his help in programming the hybrid genetic-simplex algorithm. Finally, the authors thank John R. Nimmo, Andre Peters, and the anonymous reviewer for their very insightful comments.

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