Vadose Zone Journal | Advancing Critical Zone Science

Considering Surface Roughness Effects in a Triangular Pore Space Model for Unsaturated Hydraulic ConductivityWenjuan Zheng, Xuan Yu, and Yan Jin*Quantifying the unsaturated hydraulic conductivity of a porous medium has been a great interest in the fields of hydrology, environmental engineering, and petroleum engineering. Previous research has shown that rough sur-faces enhance liquid retention and conductance of flow in the form of liquid film. We present a pore-scale-based water retention and hydraulic conduc-tivity model considering surface roughness effects. In the proposed model, a porous medium is simplified as a bundle of statistically distributed capillaries with triangular cross-sections. Surface roughness effects are characterized by a roughness factor, which accounts for increased film thickness under relatively wet conditions due to capillary effects and increased film area under relatively dry conditions. The model significantly improved the predic-tion of hydraulic conductivity across the entire range of matric potentials for the illustrative soils compared with the van Genuchten–Mualem model (VGM), while maintaining the same number of adjustable parameters. The improved performance of the proposed model demonstrates the advan-tage of incorporating surface roughness in the pore-scale-based models. Furthermore, sandy soils and loams showed distinct roughness factors and pore-size distribution functions. Sandy soils tended to have smaller rough-ness factors and greater mean pore sizes than loams.

Abbreviations: R-TPSM, roughness–triangular pore space model; VGM, van Genuchten–Mualem model; WRC, water retention curve.

A common approach to modeling flow in porous media is to use the Richards equation, where reliable prediction requires an accurate and effective description of the water retention curve (WRC) and hydraulic conductivity function. This approach is based on the continuum theory in which the WRC and hydraulic conductivity function are established based on the concept of a representative elementary volume (REV). Many geological formations and soils contain microscale heterogeneities on solid-phase surfaces at a length scale below the REV scale. However, the prevailing approaches for modeling the WRC and hydraulic conductivity function have limited representation of microscale heterogeneities (e.g., surface roughness).

Several past studies have conceptualized soil as a bundle of cylindrical capillaries (BCC) (Burdine, 1953; Childs and Collis-George, 1950; Mualem, 1976; van Genuchten, 1980). The BCC-based water retention and hydraulic conductivity models, e.g., the van Genuchten–Mualem model (VGM), are widely used in the practice of hydrology or environmental engineering due to their simple mathematical formulation. However, the BCC-based models could not be rigorously verified against experimental data at low satu-rations (Nimmo, 1991). The models’ unsatisfactory performance has been attributed, in part, to the oversimplified pore geometry, which does not allow adequate accounting of liquid retention and transport mechanisms (Tuller and Or, 2001).

The BCC-based models have been improved by an alternative triangular pore space model (TPSM), in which the pore space is represented by a triangular central pore (Tuller and

Surfaces of soil particles are rough rather than smooth, and surface roughness can enhance water retention and conductance of flow in the form of water films. We modeled water storage and movement in soils under vari-able water saturation conditions, considering surface roughness effects. Our model significantly improves the predictive capa-bil ity compared with the van Genuchten–Mualem model.

W. Zhang and Y. Jin, Dep. of Plant and Soil Sciences, Univ. of Delaware, Newark, DE 19716; and X. Yu, Dep. of Civil and Environmental Engineering, Pennsylvania State Univ., University Park, PA 16802. *Corresponding author (yjin@udel.edu).

Vadose Zone J. doi:10.2136/vzj2014.09.0121Received 17 Sept. 2014.Accepted 20 Feb. 2015.Supplemental material online.

Original Research

© Soil Science Society of America 5585 Guilford Rd., Madison, WI 53711 USA.

All rights reserved. No part of this periodical may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information sto-rage and retrieval system, without permission in writing from the publisher.

VZJ | Advancing Critical Zone Science p. 2 of 13

Or, 2001). The TPSM enables dual occupancy of the pore space by liquid and gas, as liquid remains in the corners after the center pore is drained or accumulates in corners before the central pore is completely filled during imbibition. Another advantage of the TPSM is that film flow on planar surfaces is controlled by an adsorptive force so that both adsorption and capillary processes are accentuated. However, the TPSM often had poor predictions in the intermediate range of saturation in the WRC, which was attributed to the limited flexibility of the applied pore size distribu-tion function (Lebeau and Konrad, 2010). Recently, BCC-based models have been developed accounting for adsorptive film and film conductivity (Peters and Durner, 2008; Lebeau and Konrad, 2010; Zhang, 2011; Peters, 2013). In general, the BCC-based models and the TPSM assume ideally smooth solid surfaces in the derivations of the WRC and unsaturated hydraulic conductivity. Ignoring surface roughness effects may be associated with errors in WRC and hydraulic conductivity modeling.

Flow along surfaces in unsaturated fractures has been found to be sensitive to surface roughness (Sweeney et al., 1993; Kim et al., 2012; Kibbey, 2013). Direct measurements of water film on smooth and rough surfaces showed that film thickness, strongly controlled by surface roughness, was orders of magnitude greater than that based on the calculation of adsorptive forces. A linear correlation was reported between the film thickness and the root mean square roughness at a constant potential (Kibbey, 2013). The observed thick films on rough surfaces were attributed to the fact that nearly all surface-associated film was held by capillarity rather than controlled by adsorption. Therefore, Kibbey (2013) suggested using “inter-granular capillary area” and “surface-associated area,” in place of capillary and film area, respectively. Or and Tuller (2000) developed a rough fracture surface model with a tractable geometry for calculations of surface liquid retention and hydraulic conductivity. In their model, surface roughness was represented by a statistical distribution of pits. Water storage on the surface and hydraulic conductivity were calculated, which were found to be in excellent agreement with experimental data. However, the surface fracture model of Or and Tuller (2000) did not consider flow between two surfaces, which is an essential element for flow in porous media. Recently, pore network models have been used as a platform to investigate pore space heterogeneity effects (Ataie-Ashtiani et al., 2001; Mirzaei and Das, 2007; Aggelopoulos and Tsakiroglou, 2008). In the pore network models, soils are represented by pore bodies connected to each other through pore throats. The pore net-work models provide an elegant way to simulate multiphase flow (Blunt, 2001; Joekar-Niasar and Hassanizadeh, 2012). Nevertheless, the small rep-resentative sample volume (usually 1–a few mm3) limits the applicability of these models to large-scale problems (Blunt et al., 2002).

We developed a new model for WRC and unsaturated hydraulic conductivity considering surface roughness effects. In this study, sensitivity analysis was conducted to test the model robustness. To further demonstrate its capability, the model was applied to fit WRCs and to predict the hydraulic conductivity of different textured soils.

6Model DescriptionThe proposed roughness-TPSM (R-TPSM) assumes that soil pores are composed of a bundle of capillaries with triangular cross-sec-tions and with a statistical pore size distribution. The description of the model includes three parts: (i) a single-pore geometric rep-resentation and retention curve and hydrodynamic consideration for a single pore; (ii) a pore size distribution function; and (iii) upscaling of retention curve and hydraulic conductance from pore scale to sample scale.

Single-Pore Representation and CalculationSingle-Pore GeometryIn the R-TPSM, a single pore space is represented by a capillary tube with a triangular cross-section. We assume that the surface of the tube wall is rough, so that a single pore is characterized by an equivalent side length L and an average height of roughness features Ra (Fig. 1a), which characterizes the magnitude of the surface roughness.

Liquid Configuration at Equilibrium during Drainage and ImbibitionLiquid saturation of a single pore involves different filling stages during drainage or imbibition. We consider a triangular tube that is initially filled with a perfectly wetting liquid, where the liquid drains from one end (Fig. 1b). The removal of a very small amount of liquid at the tube entrance causes a rise in the capillary pressure, thus

Fig. 1. Schematics of the (a) cross-section, showing the increase in arc menisci curva-ture at corners during drainage and a conceptual sketch for the shifted Young–Laplace approach, and (b) longitudinal section of A–A in a single pore space representation, where L is the equivalent side length of the triangle. Roughness features are shown on the walls of the pore, and Ra is the mean roughness height defining the magnitude of surface roughness. The radius of interfacial curvature r is shifted by the film thickness h of surface-associated liquid.

VZJ | Advancing Critical Zone Science p. 3 of 13

an invading meniscus is formed at the other end. When the matric potential m decreases (i.e., becomes more negative) and reaches the threshold of md, the liquid is displaced by the gas phase from the cen-tral region, leaving liquid wedges in corners and liquid films on solid walls. Away from the invading meniscus, the liquid–gas interface in the plane of the tube cross-section can be approximated as a circular arc. The curvature of the arc increases and the thickness of the film decreases as the matric potential continues to decrease. Gravitational effects are neglected because the tube size is small, thus the curvature remains constant at a given matric potential.

During an imbibition process, the tube is relatively dry at the beginning, with liquid films adsorbed on the walls and liquid wedges in the corners. As the matric potential increases to a criti-cal value mi, the accumulation of liquid causes the circular arcs to begin to touch each other, followed by a spontaneous filling of the pore (i.e., the pore snap-off). The threshold matric poten-tial mi during imbibition is different from the threshold md during drainage, which is the main mechanism causing hysteresis between imbibition and drainage.

Or and Tuller (2000) proposed a shifted Young–Laplace approach to describe the liquid configuration in angular pores. In the shifted Young–Laplace equation, adsorptive films and liquid wedges in cor-ners were treated separately. The film adsorbed on all exposed solid surfaces was controlled by van der Waals forces, while the radius of the interface curvature for liquid wedges, calculated from the classic Young–Laplace equation, was shifted by the film thickness h (Fig. 1a).

The R-TPSM modifies the shifted Young–Laplace equation by con-sidering capillarity-held film so that the film thickness is controlled

by both adsorptive forces and capillary forces under relatively wet conditions. Moreover, surface roughness not only affects the film thickness but also increases the film area under relatively dry con-ditions. Figure 2 schematically illustrates liquid configurations on a substrate surface structured with dual-scale roughness at differ-ent degrees of saturation. Under relatively wet conditions, water films tend to be of uniform thickness regardless of the shape of the underlying substrates (Fig. 2a). To differentiate from adsorp-tive film, we use the term surface-associated film (Kibbey, 2013) to indicate the inclusion of capillary effects on retained water films. We propose the following empirical equation to account for rough-ness effects on a thick film (>10 nm; Or and Tuller, 2000):

( ) ( ) svl3t ad t 6A

h X h Xm = m =prm

[1]

where had(m) is the adsorptive film thickness (m) estimated by adsorp-tive forces (Tuller and Or, 2001); m is the matric potential (J kg−1) (note that the matric potential is usually defined in the unit of joules per cubic meter; however, to apply the shifted Young–Laplace equa-tion, we converted joules per cubic meter into joules per kilogram by dividing the matric potential by the density of water r, 998.21 kg m−3, but keeping the same name, matric potential); Asvl is the Hamaker constant for solid–gas interaction through the intervening liquid (the predicted Hamaker constants for common soil minerals range from −0.6 ´ 10−20 to −4.9 ´ 10−20 J [Tokunaga, 2011]; we used the value of −6 ́ 10−20 J to approximately incorporate the addi-tional contribution from electric double-layer influences and surface hydration influences [Tokunaga, 2011]); Xt is the thickness factor, which quantifies the magnitude of increased film thickness due to roughness effects. The thickness factor Xt is proposed based on the experimental study of Kibbey (2013). According to Kibbey (2013),

Fig. 2. Schematics of thick, intermediate, and thin water films on a dual-scale-structured rough surface: (a) under relatively wet conditions, a thick layer of water film is adsorbed due to capillarity and adsorptive forces, where h is the film thickness on a rough surface, had is the adsorptive film thickness estimated by adsorptive forces, Lf is the film length (in a two-dimensional geometry), and X is a constant roughness factor; (b) under relatively dry conditions, film thickness and film length are affected by roughness effects, with a thickness factor Xt and an area factor Xa; (c) under very dry condi-tions, the film length of a thin film equals the medium surface length (area); and (d), the thickness factor Xt and area factor Xa are functions of X and h.

VZJ | Advancing Critical Zone Science p. 4 of 13

the microscopically observed film thickness on a rough surface is proportional to the mean roughness height of the surface at a fixed matric potential value. In other words, assuming that the thickness of an adsorptive film on a relatively smooth surface is determined only by the van der Waals forces, the thickness of a film on a rough surface can be predicted by the film thickness on the smooth surface and the mean roughness heights of the smooth and rough surfaces. Kibbey (2013) also showed that the ratio between film thickness and mean roughness height varied at different matric potentials, which was not considered in this study; therefore, the thickness factor for thick films for a given type of soil is a constant, i.e., Xt = X.

Under very dry conditions, the film area increases dramatically compared with that in Fig. 2a due to surface roughness effects (Fig. 2c). An extreme case of thin films is where a monolayer water film in the equilibrium state distributes evenly across soil surfaces and within pores and interlayers that are greater than the size of water molecules. The film area is assumed to equal the surface area of the substrate where capillarity has minimal effects on increas-ing its thickness. We consider that a thin film area adsorbed on a rough surface is increased by an area factor Xa relative to a smooth surface. To simplify the calculations, the magnitude of the film area increase for very thin films is assumed to be the same as the magnitude of the film thickness increase for thick films, i.e., Xa = Xt = X. In fact, as shown below, this assumption is reasonable because the calculated surface area values using the R-TPSM and those based on a clay fraction method (Leão and Tuller; 2014) are comparable. The thickness of a very thin film is determined only by adsorptive forces (i.e., h = had).

In addition, Fig. 2b shows that both film area and film thickness increase due to the first-level and second-level roughness, respec-tively, for a film with a thickness at the scale comparable to the height of the surface roughness. We consider an intermediate film thickness between 0.35 and 10 nm, where both film thickness and film area are affected by roughness effects with roughness factors of Xt and Xa, respectively. These roughness factors are related to the film thickness h, and fulfill the conditions of Xa = 1 and Xt = X at h = 10 nm, and Xa = X and Xt = 1 at h = 0.35 nm, to ensure conti-nuity across the entire range of matric potentials. We assume that the roughness factor Xa is an exponential function of h and allow Xt ´ Xa = X, so that Xa = (10−8/h)lnX/(ln200−ln7) and Xt = X/Xa, where h has the unit of meters.

In summary, the relationship between film thickness and matric potential is determined by a constant roughness factor X and the adsorptive film thickness:

( )( )

( )

( )

ad 1ln / ln200 ln 7

ad 2 18

ad 2

( ) for

for10

for

X

Xh

hh X h

h

-

-

ì m m³mïïïïï æ öï ÷çm = m m <m<m÷í ç ÷çï è øïïï m m£mïïî

[2]

where m1 = −3.19 J kg−1 and m2 = −7.44 ´ 104 J kg−1, which are the matric potentials corresponding to adsorptive films with thick-nesses of 10 and 0.35 nm, respectively.

The radius of the curved liquid–gas interface r for liquid wedges is expressed as a function of m based on the Young–Laplace relationship:

( )r sm =-

rm [3]

where s is the surface tension of the liquid (s = 0.0728 N m−1).

Retention Curve for a Single PoreA pore is completely filled before (or after) the pore snap-off during drainage (or imbibition), so the saturation Sw = 1. Under unsatu-rated conditions, Sw is reduced and is equal to the ratio of the area occupied by liquid (the sum of liquid films and wedges) to the total area on the cross-sectional face of the capillary tube. The expression for Sw as a function of m , which includes the liquid in surface-associated films and corners, is written as (Or and Tuller, 2000):

( )( ) ( ) ( )2

3w d i2 2

3 3

3or

h L h r FS

A L A L

é ùm - m më ûm = + m<m m<m [4]

where A3 and F3 are pore-shape-dependent parameters. For an equivalent triangle, A3 = Ö(3)/4 and F3 = 3Ö3 − p. We assume Xt ´ Xa = X for films with intermediate thickness and Xa = X for thin films, which allows saturation for both conditions to be calculated by Eq. [4].

Hydrodynamic Consideration for a Single PoreBecause local flow remains at a relatively slow rate in soil pores, a stable and fully developed laminar flow in modeled single pores is expected (Tuller and Or, 2001). In the situation of complete pore saturation, liquid flows through filled capillary channels (ducts). Under partially saturated conditions, flow occurs in films on solid surfaces and in corners. For film flow and corner flow, we assume fixed liquid–gas interfaces because breakup or significant deforma-tion of the interface requires relatively large viscous forces, which is unlikely under very negative matric potential conditions (Tuller and Or, 2001). The average flow velocity is obtained by solving the Navier–Stokes equation with corresponding boundary condi-tions. For flow through filled ducts with an equilateral triangular cross-section, the average flow velocity can be written as (Tuller and Or, 2001)

2

d0

dDuct flow:

80 dpLvz

æ ö÷ç= - ÷ç ÷çè øh [5]

Film flow is solved with a no-slip boundary condition on the planar surface and a no-shear liquid–gas interface (Tuller and Or, 2001):

2

f0

dFilm flow:

3 dphvz

æ ö÷ç= - ÷ç ÷çè øh [6]

VZJ | Advancing Critical Zone Science p. 5 of 13

Flow in corners for different corner angles and degrees of shear at air–water interfaces was solved by Ransohoff and Radke (1988) using a general form for the average velocity:

2

c0

dCorner flow:

dprvz

æ ö÷ç= - ÷ç ÷çè øeh [7]

where h0 is the viscosity of the bulk phase (h0 = 1.002 ´ 10−3 kg m−1 s−1), −dp/dz is the pressure head gradient in the flow direc-tion z (N m−3), and e = 31.62 is the dimensionless flow resistance parameter for the corner flow with a no-shear air–water interface and with a corner angle of 60° (Ransohoff and Radke, 1988; Tuller and Or, 2001). Flow in intermediate and thin films is likely to be affected by the modified liquid viscosity close to the solid surfaces. Or and Tuller (2000) derived velocity of flow in intermediate and thin films (<10 nm) by introducing an exponential viscosity func-tion into the Navier–Stokes equation. Film flow was also described by Eq. [6] in which the viscosity of the bulk liquid is replaced by an average dynamic viscosity of the liquid film (Lebeau and Konrad, 2010). We applied an elevated viscosity for intermediate and thin film flow in the R-TPSM, which is 100 times greater than the vis-cosity for the bulk phase.

Flow conductance by ducts, films, and corners are derived by incor-porating the average flow velocity in a single pore into Darcy’s law:

dd

Q pKvA g z

æ ö÷ç= = - ÷ç ÷çè ør [8]

where Q is the volumetric flow rate (m3 s−1), A is the cross-sectional area of the water phase (m2), K is the hydraulic conductivity (m s−1), and g is the acceleration due to gravity (m s−2). By inserting Eq. [5–7] into Eq. [8] dp/dz is eliminated, and thus flow conductance for ducts, films, and corners can be written, respectively, as

2D

0

180

gK L

r=

h [9]

( )2

F0 3

hgK

mr=

h [10]

( )2

C0

rgK

mr=

h e [11]

Pore Size DistributionTo facilitate the derivation of functions of sample-scale saturation and unsaturated hydraulic conductivity, we apply a gamma distri-bution function to describe the statistical property of a population of pores. The density gamma function is expressed as

( ) 1 exp!

L Lf Lx

x+

æ ö÷ç= - ÷ç ÷çè øwx w [12]

The two parameters used in the gamma distribution function are the shape factor x, which is limited to integer values, and the scale factor w; L is the length of the equivalent triangle’s sides (m).

Upscaling from Pore Scale to Sample ScaleA conceptual schematic of the applied upscaling scheme is depicted in Fig. 3. For illustrative purpose, we used a population of pores with sizes from L1 to L5 that satisfy a gamma distribution func-tion. During drainage, pores are gradually drained from large (L5) to small pores (L1). At a certain matric potential (e.g., m3), larger pores are partially filled (e.g., pores with the size of L4 or L5). The shape of a liquid–gas interface in unsaturated pores at each matric potential (m1–m5) is determined by the shifted Young–Laplace equation using Eq. [2] and [3]. The total sample-scale satura-tion and hydraulic conductivity are calculated as the respective weighted sums at different pore-filling stages as described below.

Sample-Scale SaturationWe express the sample-scale saturation (Sw) as the sum of the three terms that are related to filled ducts (Swd), films (Swf), and corners (Swc), respectively. The expressions for Sw are derived according to the upscaling scheme (Fig. 3):

( ) ( ) ( ) ( )w wd wf wcS S S Sm = m + m + m [13]

( ) ( )1

min

( )wd d

L

LS f L L

mm =ò [14]

( )( ) ( )

( )max

1wf 2( )

3

3d

L

L

h L hS f L L

A Lm

é ùm - më ûm =ò [15]

( )( )

( )max

1

23

wc 2( )3

dL

L

r FS f L L

A Lm

mm =ò [16]

Equation [14] describes the portion of saturation contributed by the completely filled pores, which is expressed as the integration between the smallest pore size Lmin (=10−9 m) and a certain pore size denoted by L1(m).The value of L1(m) is calculated from the radius of the capillary interface curvature at the onset of pore snap-off (Tuller and Or, 2001):

Fig. 3. Schematic of the applied upscaling scheme in the model: (a) gamma distribution of pore length with five hypothetical bins; (b) pore drainage process for a single triangular pore (represented by L1–L5) under matric potential from m1 (wet) to m5 (dry).

VZJ | Advancing Critical Zone Science p. 6 of 13

( )1 3L Csm =-

rm [17]

where C3 is the drainage radius of curvature (Tuller and Or, 2001), and C3 = 2[Ö3 + Ö(p/Ö3)].

The contribution of film flow and corner flow from the partially filled pores under a certain matric potential m are described by Eq. [15] and [16]. The two limits of the integration are defined between L1(m) and Lmax. During the drainage process, the largest pore achieves the onset of pore snap-off at the air-entry value md. Therefore, the largest pore size Lmax present in the porous medium can be related to the air-entry value md as (Tuller and Or, 2001)

( ) ( )max 1 d d2L L h= m + m [18]

Sample-Scale Hydraulic ConductivitySimilar to the sample-scale saturation, the sample-scale hydraulic conductivity (Kw) is determined based on the upscaling scheme (Fig. 3) by adding each contribution from duct flow, film flow, and corner flow:

( )( ) ( ) ( )

( )wd wf wc

rs

K K KK

Km + m + m

m =m

[19]

( ) ( )max

mins DK K f L Lm = fò [20]

( ) ( )1

min

( )wd D d

L

LK K f L L

mm = fò [21]

( )

( ) ( )( )

( ) ( )( )

( ) ( )( )

max

1

max

1

max

1

F 2( )3

1

ta F 2( )wf 3

2 1

2( )3

2

3 2 3d

for

3 2 3d

for

3 2 3d

for

L

L

L

L

LFL

h L rK f L L

A L

h X L rX K f L LK A L

h XL rX K f L L

A L

m

m

m

ì é ùï m - mï ê úë ûï fïïïïïï m³mïïï é ùï m - mï ê úë ûï fïïm =íïïï m <m<mïïïï é ùm - mï ê úï ë ûï fïïïïï m£mïïî

ò

ò

ò

[22]

( )( )

( )max

1

23

wc 2( )3

dL

CL

r FK K f L L

A Lm

mm = fò [23]

where f is the porosity of the porous medium. The term Ks is the saturated hydraulic conductivity, where all pores are completely filled and thus only duct flow is considered. The terms Kwd, Kwf, and Kwc are defined as the portions from duct flow, film flow, and corner flow, respectively. It should be noted that the film flow equation (Eq. [22]) applies only to the portion of the film that lies between the curved corner menisci (Tuller and Or, 2001). The rela-tive hydraulic conductivity Kr is simply the quotient of Kw and Ks.

Surface AreaThe specific surface area is defined as the mass normalized surface area. Based on the calculated pore geometric parameters, the spe-cific surface area (m2 kg−1) is written as

( )max

min2

3

SA dL

L

XL f L LA L

=rfò [24]

6Sensitivity Analysis and Model ApplicationSensitivity AnalysisTo better understand the relationship between the input and output variables in the developed R-TPSM as well as to test the robustness of model predictions in the presence of uncertainty, we conducted a sensitivity analysis before applying the model to mea-sured soil data. The sensitivity of the model to the key parameters (w, X, or md) was explored. For these simulations, values of x, w, X, and md were set equal to 1, 10−4, 500, and −5 J kg−1, respectively. These values were selected to reflect the typical model fitting results for sandy and loamy soils. We refer to the modeling results based on these parameter settings as the measurements. Each time, we scaled one of the three key parameters to 0.5, 0.7, 0.9, 1.1, 1.3, and 1.5 times the initial setting while keeping all other parameters at the preset values. We investigated the behaviors of the WRC and relative hydraulic conductivity according to each parameter change. It should be noted that the parameter x was not treated as a free parameter in the model, so sensitivity analysis was not conducted for x. However, the model was calculated with x = 1, x = 2, and x = 6 for each set of the testing data to test the flexibility of the proposed model.

Model ApplicationEstimation of Model Parameters from Water Retention CurveThe optimization of the estimated model parameters (i.e., w , X, or md) was achieved by using the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) (Hansen et al., 2003), which is an evolutionary algorithm for nonlinear multiparameter optimiza-tion problems in a continuous domain. The objective function used is

( )2w,m w,c1

w,RMSE

Ni S S

SN

=-

=å

[25]

where Sw,m and Sw,c are measured and calculated saturation, respectively, and N is the number of data points. A MATLAB version of the source code was used to conduct the inverse fitting.

Calculation of Unsaturated Hydraulic ConductivityThe estimated model parameters from WRCs were used to cal-culate the relative hydraulic conductivity using Eq. [19–23]. The

VZJ | Advancing Critical Zone Science p. 7 of 13

model performance was evaluated by the root mean square error (RMSE):

( )2r,m r,c1

r,RMSEln lnN

i K KK

N=

-=

å [26]

where Kr,m and Kr,c are measured and calculated relative hydraulic conductivities, respectively, and N is the number of data points.

Soil PropertiesThe proposed model was tested against 20 sets of experimental data obtained from the Unsaturated Soil Hydraulic Database (UNSODA) (Leij et al., 1996) and other literature (Lebeau and Konrad, 2010). We present the results from nine soils, which ranged from coarse sand to sandy clay, as illustrative examples to demonstrate the R-TPSM. The physical properties of the selected soils are summarized in Table 1 (and the additional 11 soils are listed in Supplemental Table S1). All 20 sets of data were included in hydraulic conductivity calculations and model parameterization. These soils were chosen because they have fairly complete measurements of both WRC (drainage process) and hydraulic conductivity.

6ResultsSensitivity AnalysisFigure 4 presents results of the sensitivity analysis of the three free model parameters (w, X, and md) for the WRC and unsaturated hydraulic conductivity. The scale factor had a pronounced effect on the WRC across the entire range of the drainage process (Fig. 4a) and had a moderate effect on unsaturated hydraulic conductivity at low saturations (Fig. 4b). The roughness factor X significantly influenced the WRC and unsaturated hydraulic conductivity at low saturations (Fig. 4c and 4d). The air-entry value denoted by

md had a great impact on the intermediate level of saturation in the WRC (Fig. 4e), while unsaturated hydraulic conductivity was considerably affected by md across the entire drainage process at all saturation levels (Fig. 4f).

Modeling Results with the Roughness–Triangular Pore Space ModelPrimary Features of the Roughness– Triangular Pore Space ModelThe primary features of the model are demonstrated and described using the Poederlee sand (Soil 1) as an example. A set of model parameters (x = 1, w = 4.68 ´ 10−5, X = 304.4, and md =

−3.28 J kg−1) was estimated by fitting the model to the measured WRC. Individual contributions of the inter-granular capillary liquid (liquid retained in filled ducts and corners) and surface-associated film to liquid retention in soil pores are shown in Fig. 5 (Soil 1-a). Inter-granular capillary liquid dominated at high satura-tions, while surface-associated film was the main form of retained liquid in soil pores at low saturations.

Relative hydraulic conductivity was readily predicted using the same set of model parameters estimated from the WRC. For the example, individual contributions of flow related to filled ducts, films, and corners, which are called duct flow, film flow, and corner flow, respectively, are depicted separately in Fig. 5 (Soil 1-b). Flow in filled ducts absolutely dominated near saturation, but its con-tribution to hydraulic conductivity decreased with decreasing matric potential. At the point where film flow started to domi-nate (m » −102 J kg−1), a change in the slope of the logKr–logm curve was observed.

Test of the Roughness–Triangular Pore Space Model on Soil DataThe developed R-TPSM was used to predict relative hydraulic con-ductivity from the measured WRCs of the nine test soils (four

sandy soils in Fig. 5, four loamy soils in Fig. 6, and one clay soil in Fig. 7). The R-TPSM fitted well the WRCs of all the test soils. In addition, the R-TPSM was able to impart a change in the slope of the relative hydraulic conductivity at low saturations, which provided reasonable predictions for most of the measurements. The estimated model parameters (w , X, and md) are listed in Table 2. The values of w and X were on the orders of 10−5 and 103 or 102, respectively. The values of the specific surface area for two loams were consistently greater than those of the four sandy soils (Table 3).

Table 1. Properties of the soils investigated in this study, including saturated water content (qs), saturated hydraulic conductivity (Ks), clay, silt, and sand contents, and porosity.

Data set† Soil no. qs Ks Clay Silt Sand Porosity

cm3 cm−3 cm d−1 ————————— % —————————

Poederlee sand 1 0.420 164.0 2.5 10.1 87.4 0.40‡

Poppel loamy sand 2 0.398 60.1 5.4 12.8 81.8 0.45‡

Retie sand 3 0.352 175.0 1.8 2.4 95.8 0.40‡

Bordenan sand 4 0.428 1140.5 4.0 4.0 92.0 0.43

Rosdorf loess silt loam 5 0.390 2.2 12.3 82.2 5.5 0.471

Northen silt loam 6 0.394 2.4 18.0 76.0 6.0 0.394

Gilat loam 7 0.440 17.3 – – – –

Sandy loam 8 0.430 8.0 – – – –

Pachapa fine sandy clay 9 0.330 12.1 – – – –

† Sources: Soils 1–6 from UNSODA (Leij, 1996); Soils 7–9 from Lebeau and Konrad (2010). The soil codes defined in UNSODA for Soils 1–6 are 4021, 4130, 4142, 4661, 3380, and 4672, respectively.

‡ Porosity values estimated based on the typical bulk density for sandy soils (1.5–1.7 g cm−3), which calculates to a porosity between 0.43 to 0.36.

VZJ | Advancing Critical Zone Science p. 8 of 13

Comparison with the van Genuchten–Mualem ModelWe compared the R-TPSM with the VGM for the test soils (Fig. 5–7). Model parameters for the VGM were estimated from the same measured WRCs (Table 2) and were used to calculate relative hydraulic conductivity. The RMSE values for the nine test soils are presented in Table 3. Both the VGM and R-TPSM achieved excellent fits to the measured retention data. However, predic-tions of the relative hydraulic conductivity of the illustrative soils showed that the VGM considerably underestimated the hydraulic conductivity, especially at low saturations, and failed to capture the change in slope of the logm–logKr curve, whereas the R-TPSM provided improved fitting to the measurements in most cases (e.g., Soils 2, 4, and 6–9).

6DiscussionSignificance of Film FlowIn the R-TPSM, liquid is retained in filled ducts and duct flow prevails at high saturations, whereas liquid film contributes to the majority of the retained liquid and film conductivity dominates at

low saturations. The model adequately describes retention data and imparts a change in the slope of the relative hydraulic conductivity, which is a typical feature of measured hydraulic conductivity data. The improved performance of the R-TPSM is attributed to the consideration of flow in different regimes, especially the surface-associated film flow.

The importance of film flow has also been recognized in other hydraulic conductivity models. Tuller and Or (2001) elucidated the contribution of film flow to overall conductivity with the proposed angular pore space model. Lebeau and Konrad (2010) proposed a hydraulic conductivity model considering both cap-illary flow and thin film flow based on cylindrical pore space. Peters (2013) coupled the hydraulic conductivity model with a water retention model that incorporated an adsorptive saturation function by Fredlund and Xing (1994). Results from these models emphasize that film flow prevails at low saturations, which agrees well with the conclusion from this study.

Comparison with the VGM highlights that neglecting liquid film retention and film conductivity can lead to considerable errors for

Fig. 4. Sensitivity analysis of the roughness–triangular pore space model (R-TPSM) for the water retention curve and relative hydraulic conductivity: the effects of (a,b) the scale factor w, (c,d) the roughness factor X, and (e,f ) the air-entry value md. Other parameters were fixed at shape factor x = 1, w = 10−4, X = 500, and md = −5 J kg−1.

VZJ | Advancing Critical Zone Science p. 9 of 13

hydraulic conductivity modeling. In addition, the VGM fitted the retention data of sandy soils only when the residual water content was set to a nonzero value (Table 2), which treats flow as negligible at water contents lower than the resid-ual value (van Genuchten, 1980). However, the residual water content is not a consistently physi-cally recognized concept because water flow has been observed under conditions near the residual water content (Nimmo, 1991).

Film Thickness and Surface AreaThe developed R-TPSM explicitly takes into account roughness effects on film thickness under relatively wet conditions and on film area under relatively dry conditions. An empirical expression of film thickness (Eq. [1]) is for-mulated based on the results of a microscopic observation (Kibbey, 2013), which showed that increased film thickness on rough surfaces was due to capillary forces. Therefore, the surface-associated film is determined by capillary forces as well as the van der Waals force, electric double-layer influences, and surface hydration influences, where the latter three are incorporated in the Hamaker constant in the R-TPSM. Previous hydraulic conductivity models considered only thin film controlled by adsorptive forces (Lebeau and Konrad, 2010; Peters, 2013).

As demonstrated by the fitted roughness factors from WRCs (Table 2), surface roughness can dramatically increase film thickness by two to three orders of magnitude when capillary effects are considered, compared with the thickness of the adsorptive film governed by the van der Waals force alone. This result is in good agree-ment with measured film thicknesses by Kibbey (2013), who reported that measured film thick-nesses were nearly two orders of magnitude larger than the estimation by adsorptive forces for the matric potential range of 0 to 10 J kg−1. Figure 8 depicts the estimated film thicknesses for four sandy soils and two loams at different matric potentials using the R-TPSM. The thick-ness decreases with increasing matric potentials as the air–water interface approaches the solid surface. Film thickness tends to be greater for loams than for sandy soils at the same matric potential, with the exception of Soil 4 (a sandy soil), which has thicker films than Soil 6. The inserted figure also displays the observed thick-ness of films on a rough sand surface across the

Fig. 5. Measured and calculated water retention curves (left) and relative hydraulic conductiv-ity (right) for four sandy soils (Soils 1–4) using the roughness–triangular pore space model (R-TPSM) and the van Genuchten–Mualem model (VGM).

VZJ | Advancing Critical Zone Science p. 10 of 13

matric potential range of 0 to 10 J kg−1 (Kibbey, 2013) and compared with the thicknesses predicted by the R-TPSM and when only adsorptive forces are assumed. The measured film thicknesses are slightly smaller than those of three sandy soils (Soils 1, 2, and 3) estimated by the R-TPSM and far greater than those of the adsorptive film at different matric potentials. It should be noted that the sand used in the experiment was commercial pure sand while the calcu-lation was conducted for natural soils. It is possible that natural soils contain more het-erogeneity due to chemical, biological, and geological processes.

Leão and Tuller (2014) reported a sig-nificant correlation between the specific surface area and clay fraction based on data sets of 21 test soils with clay contents ranging from 0.6 to 52.2%. We derived a regression function between the two (SA = 2.8Clay% + 5.5) to calculate the specific surface area of each soil used in this study based on its clay fraction and found that they are comparable to the specific surface area values determined by Eq. [24]. This result confirms that it is reasonable to assume that the magnitude of the increased film thickness under wet conditions equals the magnitude of the increased film area under relatively dry conditions due to roughness effects (Xa = X and Xa ´ Xt = X).

Film ConductivityPrediction of the relative hydraulic conduc-tivity using the R-TPSM revealed that film flow prevails over flow in filled capillaries at very low saturations. In addition, we found that using an elevated film viscosity that equals 100 times the viscosity of the bulk phase provided much closer predictions of the relative hydraulic conductivity to the test data sets. Liquid viscosity close to the solid surfaces tends to be increased relative to the bulk liquid due to short- and long-range interfacial forces (Low, 1976, 1979; McBride and Baveye, 1995). The irregularities of the surface can cause viscous dissipation or even bring the fluid to rest as fluid passes over rough features (Richardson, 1973). Wraith and Or (1999) approximated that

Fig. 6. Measured and calculated water retention curves (left) and relative hydraulic conductivity (right) for four loamy soils (Soil 5–8) using the roughness–triangular pore space model (R-TPSM) and the van Genuchten–Mualem model (VGM).

VZJ | Advancing Critical Zone Science p. 11 of 13

the viscosity for films with a thickness of <10 nm increased as an exponen-tial function of the distance from the solid surface. Zou (2012) applied the exponential function of Wraith and Or (1999) in a macroscopic model for relative hydraulic conductivity and concluded that heavier textured soils had a greater viscosity for pore water than coarse-textured soils. However, recent studies have indicated that the magnitude of the viscosity increase near solid surfaces remains highly uncertain (Feibelman, 2013; Goertz et al., 2007). Therefore, the simplistic approach of using an elevated viscosity value, espe-cially for water films with thickness of >1 to 2 nm, in the R-TPSM may not accurately capture the physical pro-cesses associated with film flow near surfaces. Rather, this approach should be seen as an effective means that lumps roughness effects and other factors or effects that have not been considered in the model. For example, the irregu-larities of the surface can cause viscous dissipation or even bring the fluid to rest as fluid passes over rough features (Richardson, 1973).

Surface Roughness EffectsPore-scale flow behavior can be greatly affected by surface roughness. As shown by the sensitivity analysis, the roughness factor X defined in the pro-posed R-TPSM significantly influences the WRC and hydraulic conductivity at low saturations (Fig. 4a and 4b).

Surface roughness on soil particles can vary widely depending on various fac-tors, e.g., mineralogy, organic matter coatings on solid particles, and bio-logical factors. In this study, smaller roughness factors were observed for sandy soils than for loamy soils. The box plot of the roughness factor for the 20 soils demonstrates that the average roughness factor was 546 for sandy soils, while it was 1119 for loamy soils (Fig. 9a), indicating that the roughness factor is closely related to soil texture.

Fig. 7. Measured and calculated water retention curves (left) and relative hydraulic conductivity (right) for Pachapa fine sandy clay using the roughness–triangular pore space model (R-TPSM) and the van Genuchten–Mualem model (VGM).

Table 2. Fitted model parameters with the roughness–triangular pore space model (R-TPSM), including scale factor w, roughness factor X and the threshold matric potential during drainage (md), and the van Genuchten–Mualem model (VGM), including the residual water content (qr) and shape factors n and a .

Soil texture Soil no.

R-TPSM VGM

w X md qr n a

J kg−1 cm3 cm−3

Sandy 1 4.68 × 10−5 304.4 −3.28 0.033 2.05 0.26

2 1.66 × 10−5 308.9 −0.46 0.052 1.64 0.11

3 4.36 × 10−5 157.5 −0.99 0.021 2.53 0.20

4 2.13 × 10−4 1257.9 −0.52 0.034 1.72 1.31

Loamy 5 4.18 × 10−5 1562.1 0.00 0.000 1.27 0.09

6 7.58 × 10−6 490.3 −19.24 0.000 1.15 0.08

7 3.36 × 10−5 423.9 −1.22 0.085 2.54 0.15

8 1.62 × 10−5 247.23 −10.12 0.006 1.51 0.14

Clay soil 9 2.32 × 10−5 349.9 −3.28 0.062 2.069 0.12

Table 3. Root mean square errors (RMSE) for the water retention curve (WRC) and relative hydraulic conductivity (Kr), accumulated gamma distribution ò max

min

LL f(L)dL, and surface area (SA) for the rough-

ness–triangular pore space model (R-TPSM) and the van Genuchten–Mualem model (VGM).

Soil texture Soil no.

RMSE

ò max

min

LL f(L)dL SAR-TPSM SAclay

WRC Kr

R-TPSM VGM R-TPSM VGM

——— 103 m2 kg−1 ———

Sandy 1 0.014 0.010 2.544 3.241 0.809 11.9 12.5

2 0.014 0.013 2.624 1.237 1.000 34.9 20.6

3 0.008 0.007 2.226 2.937 1.000 6.9 10.5

4 0.020 0.010 0.759 6.109 0.928 11.6 16.7

Loamy 5 0.006 0.011 1.285 1.601 1.000 86.9 39.9

6 0.012 0.004 1.088 3.239 0.892 107.3 55.9

7 0.017 0.019 1.948 9.981 1.000 NA† NA

8 0.020 0.009 1.196 4.696 0.793 NA NA

Clay 9 0.009 0.009 1.140 3.332 0.986 NA NA

† NA, not available.

VZJ | Advancing Critical Zone Science p. 12 of 13

Soil Pore Size DistributionThe proposed R-TPSM applied a gamma distribution func-tion to describe the statistical distribution of pore sizes. The gamma function contains two parameters, the shape factor w and the scale factor x. We showed that the best fit for all the soils was mostly obtained with x = 1 (Supplemental Table S3), which indicates that the gamma function with x = 1 [i.e., f(L) = (L/w2)exp(−L/w)] is the most representative of the pore size dis-tribution of the test soils. It should be noted that in the objective function for fitting WRCs, we did not minimize the tails of the gamma distribution during the optimization process (Tuller and Or, 2001) because a posterior test of the accumulated pore size distribution [ò max

min

LL f(L)dL] showed that most values were >0.9

(Table 3). The accumulated pore size distribution demonstrates

that the optimization in the R-TPSM is effective without minimizing the tails of the gamma function.

The scale factor w was estimated from the measured WRCs (Table 2). The average scale factor for the test sandy soils was 6.43 ´ 10−5, while a smaller average scale factor (2.46 ´ 10−5) was obtained for the loamy soils (Fig. 9b). The relatively large pore size of sandy soils can be attributed to larger particle sizes because the pore size distribution is related to the particle size distribution (Hwang and Choi, 2006).

6ConclusionsIn this study, a R-TPSM for the WRC and hydraulic conductivity was developed with detailed consid-eration of surface roughness effects. The model is physically based and all the parameters have clear physi-cal meanings. The model parameters, which are sensitive to model output,

are estimated by fitting the model to the measured WRC and then used to predict the relative hydraulic conductivity with-out additional parameters. The developed R-TPSM is able to effectively predict the relative hydraulic conductivity of soils of different textures with a known WRC and saturated hydraulic conductivity. Consideration of film conductivity in the R-TPSM allows a smaller slope of the logKr–logm curve at lower satu-rations, which is in good agreement with the typical pattern of most measurements. Therefore, the prediction of hydraulic conductivity is significantly improved compared with the VGM.

One of the primary advantages of the proposed model is the con-sideration of capillary-held surface-associated film due to surface

roughness. Capillary forces cause film thickness on natural (rough) surfaces to be orders of magnitude greater than what consideration of adsorp-tive forces alone would suggest. In the presence of surface roughness, capil-larity bridges roughness features so that capillary-held film commonly dominates in liquid retention and flow in the vadose zone except under extremely dry conditions. In addition, the R-TPSM allows the film area to increase to the soil surface area at the monolayer water thickness, thus

Fig. 8. Film thicknesses at different matric potentials calculated by the roughness–triangular pore space model (R-TPSM). The inserted figure compares calculated values with the results of Kibbey (2013) and the values determined only by the adsorptive forces.

Fig. 9. Estimated roughness factors X and scale factors w for sandy and loamy soils. On each box, the cen-tral mark is the median, the edges of the box are the 25th and 75th percentiles, and outliers more distant from the median than 1.5 times the interquartile range (the length of the rectangle) are represented as a + symbol.

VZJ | Advancing Critical Zone Science p. 13 of 13

roughness effects on an increasing surface area under extremely dry conditions are also considered.

One limitation of the model is that the effects of surface roughness are reflected only in the film thickness or area, which is probably incomplete. Factors such as contact angle and the hydrodynamic boundary condition of film flow could also be influenced by sur-face roughness. In addition, the R-TPSM considered only the triangular pore space and neglected the slit-shaped space area, which could lead to underestimation of the surface area, especially for finer textured soils. Further improvement may be achieved by considering additional mechanisms and effects due to surface roughness and/or by using more realistic pore space representa-tions, e.g., the pore network model.

AcknowledgmentsWe gratefully acknowledge funding by the USDA National Research Initiative (NRI# 2008- 00653) and USDA Hatch funds for the W2188 Multi-State Project. We are very grateful for the insightful and constructive comments, from reviewer Dr. Marc Lebeau of Laval University and associated editor Dr. Tetsu Tokunaga of Lawrence Berkeley National Laboratory, on the earlier versions of the manuscript, which are critical to its significant improvement.

ReferencesAggelopoulos, C.A., and C.D. Tsakiroglou. 2008. The effect of mi-

cro-heterogeneity and capillary number on capillary pressure and relative permeability curves of soils. Geoderma 148:25–34. doi:10.1016/j.geoderma.2008.08.011

Ataie-Ashtiani, B., S.M. Hassanizadeh, M. Oostrom, M.A. Celia, and M.D. White. 2001. Effective parameters for two-phase flow in a porous medium with periodic heterogeneities. J. Contam. Hydrol. 49:87–109. doi:10.1016/S0169-7722(00)00190-X

Blunt, M.J. 2001. Flow in porous media: Pore-network models and multiphase flow. Curr. Opin. Colloid Interface Sci. 6:197–207. doi:10.1016/S1359-0294(01)00084-X

Blunt, M.J., M.D. Jackson, M. Piri, and P.H. Valvatne. 2002. Detailed phys-ics, predictive capabilities and macroscopic consequences for pore-network models of multiphase flow. Adv. Water Resour. 25:1069–1089. doi:10.1016/S0309-1708(02)00049-0

Burdine, N.T. 1953. Relative permeability calculations from pore size distribution data. Trans. Am. Inst. Min. Metall. Eng. 198:71–78. doi:10.2118/225-G

Childs, E.C., and N. Collis-George. 1950. The permeability of porous mate-rials. Proc. R. Soc. London Ser. A 201:392–405.

Feibelman, P.J. 2013. Viscosity of ultrathin water films confined between aluminol surfaces of kaolinite: Ab initio simulations. J. Phys. Chem. C 117:6088–6095. doi:10.1021/jp312152h

Fredlund, D.G., and A. Xing. 1994. Equations for the soil-water characteris-tic curve. Can. Geotech. J. 31:521–532. doi:10.1139/t94-061

Goertz, M.P., J.E. Houston, and X.-Y. Zhu. 2007. Hydrophilicity and the viscos-ity of interfacial water. Langmuir 23:5491–5497. doi:10.1021/la062299q

Hansen, N., S.D. Muller, and P. Koumoutsakos. 2003. Reducing the time complexity of the derandomized evolution strategy with co-variance matrix adaptation (CMA-ES). Evol. Comput. 11:1–18. doi:10.1162/106365603321828970

Hwang, S., and S. Choi. 2006. Use of a lognormal distribution model for es-timating soil water retention curves from particle-size distribution data. J. Hydrol. 323:325–334. doi:10.1016/j.jhydrol.2005.09.005

Joekar-Niasar, V., and S.M. Hassanizadeh. 2012. Analysis of fundamen-tals of two-phase flow in porous media using dynamic pore-network models: A review. Crit. Rev. Environ. Sci. Technol. 42:1895–1976. doi:10.1080/10643389.2011.574101

Kibbey, T.C.G. 2013. The configuration of water on rough natural sur-faces: Implications for understanding air–water interfacial area, film

thickness, and imaging resolution. Water Resour. Res. 49:4765–4774. doi:10.1002/wrcr.20383

Kim, T.W., T.K. Tokunaga, D.B. Shuman, S.R. Sutton, M. Newville, and A. Lanzirotti. 2012. Thickness measurements of nanoscale brine films on silica surfaces under geologic CO2 sequestration conditions us-ing synchrotron X-ray fluorescence. Water Resour. Res. 48:W09558. doi:10.1029/2012WR012200

Leão, T.P., and M. Tuller. 2014. Relating soil specific surface area, wa-ter film thickness, and water vapor adsorption. Water Resour. Res. 50:7873–7885. doi:10.1002/2013WR014941

Lebeau, M., and J. Konrad. 2010. A new capillary and thin film flow model for predicting the hydraulic conductivity of unsaturated porous me-dia. Water Resour. Res. 46:W12554. doi:10.1029/2010WR009092

Leij, F.J., W.J. Alves, M.Th. van Genuchten, and J.R. Williams. 1996. The UNSODA unsaturated soil hydraulic database: User’s manual. Version 1.0. USEPA National Risk Manage. Res. Lab., Cincinnati, OH.

Low, P.F. 1976. Viscosity of interlayer water in montmorillonite. Soil Sci. Soc. Am. J. 40:500–505. doi:10.2136/sssaj1976.03615995004000040017x

Low, P.F. 1979. Nature and properties of water in montmo-rillonite–water systems. Soil Sci. Soc. Am. J. 43:651–658. doi:10.2136/sssaj1979.03615995004300040005x

McBride, M.B., and P. Baveye. 1995. Mobility of anion spin probes in hec-torite gels: Viscosity of surficial water. Soil Sci. Soc. Am. J. 59:388–394. doi:10.2136/sssaj1995.03615995005900020017x

Mirzaei, M., and D.B. Das. 2007. Dynamic effects in capillary pressure–saturations relationships for two-phase flow in 3D porous media: Im-plications of micro-heterogeneities. Chem. Eng. Sci. 62:1927–1947. doi:10.1016/j.ces.2006.12.039

Mualem, Y. 1976. New model for predicting hydraulic conductiv-ity of unsaturated porous media. Water Resour. Res. 12:513–522. doi:10.1029/WR012i003p00513

Nimmo, J.R. 1991. Comment on the treatment of residual water content in “A consistent set of parameteric models for the two-phase flow of immiscible fluids in the subsurface” by L. Luckner et al. Water Resour. Res. 27:661–662. doi:10.1029/91WR00165

Or, D., and M. Tuller. 2000. Flow in unsaturated fractured porous media: Hydraulic conductivity of rough surfaces. Water Resour. Res. 36:1165–1177. doi:10.1029/2000WR900020

Peters, A. 2013. Simple consistent models for water retention and hydrau-lic conductivity in the complete moisture range. Water Resour. Res. 49:6765–6780. doi:10.1002/wrcr.20548

Peters, A., and W. Durner. 2008. A simple model for describing hydraulic conductivity in unsaturated porous media accounting for film and cap-illary flow. Water Resour. Res. 44:W11417. doi:10.1029/2008WR007136

Ransohoff, T.C., and C.J. Radke. 1988. Laminar flow of a wetting liquid along the corners of a predominantly gas-occupied noncircular pore. J. Colloid Interface Sci. 121:392–401. doi:10.1016/0021-9797(88)90442-0

Richardson, S. 1973. On the no-slip boundary condition. J. Fluid Mech. 59:707–719. doi:10.1017/S0022112073001801

Sweeney, J.B., T. Davis, L.E. Scriven, and J.A. Zasadzinski. 1993. Equilibrium thin films on rough surfaces: 1. Capillary and disjoining effects. Lang-muir 9:1551–1555. doi:10.1021/la00030a021

Tokunaga, T.K. 2011. Physicochemical controls on adsorbed water film thickness in unsaturated geological media. Water Resour. Res. 47:W08514. doi:10.1029/2011WR010676

Tuller, M., and D. Or. 2001. Hydraulic conductivity of variably saturated porous media: Film and corner flow in angular pore space. Water Re-sour. Res. 37:1257–1276. doi:10.1029/2000WR900328

van Genuchten, M.Th. 1980. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44:892–898. doi:10.2136/sssaj1980.03615995004400050002x

Wraith, J.M., and D. Or. 1999. Temperature effects on soil bulk dielectric permittivity measured by time domain reflectometry: Experimental evidence and hypothesis development. Water Resour. Res. 35:361–369. doi:10.1029/1998WR900006

Zhang, Z.F. 2011. Soil water retention and relative permeability for condi-tions from oven-dry to full saturation. Vadose Zone J. 10:1299–1308. doi:10.2136/vzj2011.0019

Zou, Y. 2012. A macroscopic model for predicting the relative hydrau-lic permeability of unsaturated soils. Acta Geotech. 7:129–137. doi:10.1007/s11440-011-0153-4