368

HYDRAULIC GRADIENT

Bibliography
Introduction to Environmental Soil Physics (First Edition) 2003
Elsevier Inc. Daniel Hillel (ed.) http://www.sciencedirect.com/
science/book/9780123486554

HYDRAULIC GRADIENT
The slope of the hydraulic grade line which indicates thechange in pressure head per unit of distance.

HYDRAULIC HEAD
The sum of the pressure head (hydrostatic pressure relative to atmospheric pressure) and the gravitational head(elevation relative to a reference level). The gradient of
the hydraulic head is the driving force for water flow inporous media.

Bibliography
Introduction to Environmental Soil Physics (First Edition) 2003
Elsevier Inc. Daniel Hillel (ed.) http://www.sciencedirect.com/
science/book/9780123486554

HYDRAULIC PROPERTIES OF UNSATURATED SOILS
Martinus Th. van Genuchten1, Yakov A. Pachepsky2
1
Department of Mechanical Engineering, COPPE/LTTC,
Federal University of Rio de Janeiro, UFRJ, Rio deJaneiro, Brazil
2
Environmental Microbial and Food Safety Laboratory,
Animal and Natural Resource Institute, Beltsville
Agricultural Research Center, USDA-ARS, Beltsville,
MD, USA

Definition
Hydraulic Properties of Unsaturated Soils. Properties
reflecting the ability of a soil to retain or transmit waterand its dissolved constituents.
Introduction
Many agrophysical applications require knowledge of thehydraulic properties of unsaturated soils. These properties
reflect the ability of a soil to retain or transmit water andits dissolved constituents. For example, they affect thepartitioning of rainfall and irrigation water into infiltration
and runoff at the soil surface, the rate and amount of redistribution of water in a soil profile, available water in thesoil root zone, and recharge to or capillary rise fromthe groundwater table, among many other processes in
the unsaturated or vadose zone between the soil surface
and the groundwater table. The hydraulic properties arealso critical components of mathematical models forstudying or predicting site-specific water flow and solutetransport processes in the subsurface. This includes using
models as tools for designing, testing, or implementing
soil, water, and crop management practices that optimize
water use efficiency and minimize soil and water pollution
by agricultural and other contaminants. Models areequally needed for designing or remediating industrial
waste disposal sites and landfills, or assessing the forlong-term stewardship of nuclear waste repositories.
Predictive models for flow in variably saturated soils are
generally based on the Richards equation, which combines
the Darcy–Buckingham equation for the fluid flux witha mass conservation equation to give (Richards, 1931):


@yðhÞ
@
@h
¼
KðhÞ  KðhÞ
(1)
@t
@z
@z
in which y is the volumetric water content (L3 L3), h is
the pressure head (L), t is time (T), z is soil depth (positive
down), and K is the hydraulic conductivity (L T1).
Equation 1 holds for one-dimensional vertical flow; similar
equations can be formulated for multidimensional flow
problems. The Richards equation contains two constitutive
relationships, the soil water retention curve, y(h), and theunsaturated soil hydraulic conductivity function, K(h).
These hydraulic functions are both strongly nonlinear functions of h. They are discussed in detail below.

Water retention function
The soil water retention curve, y(h), describes the relationship between the water content, y, and the energy status ofwater at a given location in the soil. Many other names
may be found in the literature, including soil moisturecharacteristic curve, the capillary pressure–saturation
relationship, and the pF curve. The retention curve historically was often given in terms of pF, which is defined as
the negative logarithm (base 10) of the absolute value ofthe pressure head measured in centimeters. In the unsaturated zone, water is subject to both capillary forces in soil
pores and adsorption onto solid phase surfaces. This leads
to negative values of the pressure head (or matric head)relative to free water, or a positive suction or tension.
As opposed to unsaturated soils, the pressure head h is
positive in a saturated system. More formally, the pressure
head is defined as the difference between the pressures of
the air phase and the liquid phase. Capillary forces are theresult of a complex set of interactions between the solidand liquid phases involving the surface tension of the liquid phase, the contact angle between the solid and liquid
phases, and the diameter of pores.
Knowledge of y(h) is essential for the hydraulic characterization of a soil, since it relates an energy density(potential) to a capacity (water content). Rather than using
the pressure head (energy per unit weight of water), manyagrophysical applications use the pressure or matricpotential (energy per unit volume of water, usually measured in Pascal, Pa), cm = rwgh, where rw is the densityof water (ML3) and g the acceleration of gravity (L T2).

HYDRAULIC PROPERTIES OF UNSATURATED SOILS

Figure 1 shows typical soil water retention curves for
relatively coarse-textured (e.g., sand and loamy sand),medium-textured (e.g., loam and sandy loam), and finetextured (e.g., clay loam, silty loam, and clay) soils. The
curves in Figure 1 may be interpreted as showing the equilibrium water content distribution above a relatively deep
water table where the pressure head is zero and the soil
fully saturated. The plots in Figure 1 show that coarsetextured soils lose their water relatively quickly (at small
negative pressure heads) and abruptly above the watertable, while fine-textured soils lose their water much more
gradually. This reflects the particle or pore-size distribution of the medium involved. While the majority of poresin coarse-textured soils have larger diameters and thus
drain at relatively small negative pressures, the majorityof pores in fine-textured soils do not drain until very large
tensions (negative pressures) are applied.
As indicated by the plots in Figure 1, the water content
varies between some maximum value, the saturated watercontent, ys, and some small value, often referred to as
the residual (or irreducible) water content, yr . As a first
approximation and on intuitive ground, the saturated
water content is equal to the porosity, and yr equal to zero.
In reality, however, the saturated water content, ys, of soils
is generally smaller than the porosity because of entrappedand dissolved air. The residual water content yr is likely to
be larger than zero, especially for fine-textured soils with
their large surface areas, because of the presence ofadsorbed water. Most often ys and especially yr are treatedas fitting parameters without much physical significance.
Soil water retention curves such as shown in Figure 1
are not unique but depend on the history of wetting anddrying. Most often, the soil water retention curve is determined by gradually desaturating an initially saturated soil
by applying increasingly higher suctions, thus producing
a main drying curve. One could similarly slowly wet an

Pressure head, |h| (cm)

100,000
10,000
1,000
100
10
1
0.0

0.1

0.2

0.3

0.4

0.5

Volumetric water content [–]Hydraulic Properties of Unsaturated Soils, Figure 1 Typicalsoil water retention curves for relatively coarse- (solid line),medium- (dashed line), and fine-textured (dotted line) soils. The
curves were obtained using Equation 6a assuming hydraulicparameter values as listed in Table 1.

369

initially very dry sample to produce the main wetting
curve, which is generally displaced by a factor of
1.5–2.0 toward higher pressure heads closer to saturation.
This phenomenon of having different wetting and drying
curves, including primary and secondary scanning curves
is referred to hysteresis. Hysteresis is caused by the factthat drainage is determined mostly by the smaller pore ina certain pore sequence, and wetting by the larger pores(this effect is often referred to as the ink bottle effect).
Other factors contributing to hysteresis are the presenceof different liquid–solid contact angles for advancing
and receding water menisci, air entrapment during wetting, and possible shrink–swell phenomena of some soils.

Hydraulic conductivity function
The hydraulic conductivity characterizes the ability ofa soil to transmit water. Its value depends on many factors
such as the pore-size distribution of the medium, and thetortuosity, shape, roughness, and degree of interconnectedness of the pores. The hydraulic conductivity decreases
considerably as soil becomes unsaturated since less porespace is filled with water, the flow paths become increasingly tortuous, and drag forces between the fluid and thesolid phases increase.
The unsaturated hydraulic conductivity function gives
the dependency of the hydraulic conductivity on the watercontent, K(y), or pressure head, K(h). Figure 2 presents
examples of typical K(y) and K(h) functions for relatively
coarse-, medium-, and fine-textured soils. Notice that thehydraulic conductivity at saturation is significantly larger
for coarse-textured soils than fine-textured soils. This difference is often several orders of magnitude. Also noticethat the hydraulic conductivity decreases very significantly
as the soil becomes unsaturated. This decrease, when
expressed as a function of the pressure head (Figure 2;
right), is much more dramatic for the coarse-textured soils.
The decrease for coarse-textured soils is so large that at
a certain pressure head the hydraulic conductivity becomes
smaller than the conductivity of the fine-textured soil. The
water content where the conductivity asymptotically
becomes zero (Figure 2; left) is often used as an alternative
working definition for the residual water content, yr.
Soil water diffusivity
Another hydraulic function often used in theoretical andmanagement application of unsaturated flow theories is
the soil water diffusivity, D(y), (L2 T1), which is defined as
 
dh
(2)
DðyÞ ¼ KðyÞ :dyThis function appears when Equation 1 is transformed
into a water-content-based equation in which y is now
the dependent variable:


@y
@
@h
¼
DðyÞ  KðyÞ :
(3)
@t @z
@z

HYDRAULIC PROPERTIES OF UNSATURATED SOILS

4

4

2

2

log (K), [cm/day]log (K), [cm/day]

370

0
−2
−4
−6
0.0

0
−2
−4
−6

0.1
0.2
0.3
0.4
Volumetric water content, [–]

0.5

0

1

2

3
4
log(|h|, [cm])56

Hydraulic Properties of Unsaturated Soils, Figure 2 Typical curves of the hydraulic conductivity K, as a function of the pressure
head (left) and water content (right) for coarse- (solid line), medium- (dashed line), and fine-textured (dotted line) soils. The curves wereobtained using Equation 6b assuming hydraulic parameter values as listed in Table 1.

Equation 3 is very attractive for approximate analyticalmodeling of unsaturated flow processes, especially formodeling horizontal (without the K(y) gravity term) andvertical infiltration (e.g., Philip, 1969; Parlange, 1980).
However, the water-content-based equation is less attractive
for more comprehensive numerical modeling of flow inlayered media, flow in media that are partially saturated
and partially unsaturated, and for highly transient flow
problems.

Analytical representations
To enable their use in analytical or numerical models forunsaturated flow, the soil hydraulic properties are often
expressed in terms of simplified analytical expressions.
A large number of functions have been proposed over
the years to describe the soil water retention curve, y(h),and the hydraulic conductivity function, K(h) or K(y).
A comprehensive review of the performance of some ofmany these models is given by Leij et al. (1997). The functions range from completely empirical equations tomodels based on the simplified conceptual picture thatsoils are made up of a bundle of equivalent capillary tubes
that contain and transmit water.
While extremely simplistic as indicated by Tuller andOr (2001) among others, conceptual models that view
a soil as a bundle of capillaries of different radii are still
useful for explaining the shape of the water retention curvefor different textures, as well as to provide a means forpredicting the hydraulic conductivity function from soil
water retention information. These models typicallyassume that pores at a given pressure head are either
completely filled with water, or empty, depending upon
the applied suction. Flow in each water-filled capillary
tube is subsequently calculated using Poiseuille’s law forflow in cylindrical pores. By adding the contribution of
all capillaries that are still filled with water at a particularpressure head, making some assumption about how small
and large capillaries connect to each other in sequence(using a cut-and-paste concept of a cross-section of themedium containing different-sized pores), and then integrating over all water-filled capillaries leads to the hydraulic conductivity of the complete set of capillaries, andconsequently of the soil itself. The approach allows information of the soil water retention curve to be translated in
predictive equations for the unsaturated hydraulic conductivity. Many theories of this type, often referred to also as
statistical pore-size distribution models, have been proposed in the past, including Childs and Collis-George(1950), Burdine (1953), Millington and Quirk (1961),
and Mualem (1976). A review of the different approaches
is given by Mualem (1992). Examples of analytical y(h)and K(h) equations resulting from this approach are thehydraulic functions of Brooks and Corey (1964), basedon the approach by Burdine (1953), and equations by
van Genuchten (1980) and Kosugi (1996), based on thetheory of Mualem (1976).
The classical equations of Brooks and Corey (1964) fory(h), K(h), and D(y) are given by

y¼

 l
yr þ ðys  yr Þhhe ysKðhÞ ¼ Ks Se2=lþlþ1DðyÞ ¼

Ks
S 1=lþl
aðys  yr Þ e

h < heh  he

(4a)
(4b)
(4c)

where, as before, yr is the residual water content (L3 L3),
ys is the saturated water content (L3 L3), he is often
referred to as the air-entry value (L), l is a pore-size distribution index characterizing the width of the soil pore-size
distribution, Ks is the saturated hydraulic conductivity(LT1), l a pore-connectivity parameter assumed to be
2.0 in the original study of Brooks and Corey (1964),
and Se = Se(h) is effective saturation given by

HYDRAULIC PROPERTIES OF UNSATURATED SOILS

Se ðhÞ ¼

yðhÞ  yr
ys  yr(5)

For completeness we have given here also the expression for the soil water diffusivity, D(y). Note that Equations 4b and 4c contain parameters that are also present
in Equation 4a, in particular yr and ys through Equation 5,
as well as he and l. The value of l in Equation 4a reflects
the steepness of the retention function and is relatively large
for soils with a relatively uniform pore-size distribution (generally coarse-textured soils such as those shown in Figures 1
and 2), but small for soils having a wide range of pore sizes.
One property of Equation 4a is the presence of a sharp
break in the retention curve at the air-entry value, he. This
break (or discontinuity in the slope of the function) is
often visible in retention data for coarse-textured soils,
but may not be realistic for fine-textured soils and soils
having a relatively broad pore- or particle-size distribution. A sharp break is similarly present in the hydraulicconductivity function when plotted as a function of thepressure head, but not versus the water content. As analternative, van Genuchten (1980) proposed a set of equations that exhibit a more smooth sigmoidal shape. The van
Genuchten equations for y(h), K(h), and D(y) are given by:yðhÞ ¼ yr þ

ys  yrm ðm ¼ 1  1=n; n > 1Þ
ð1 þ jahjn Þ
(6a)

h

m i2
KðhÞ ¼ Ks Sel 1  1  Se1=mDðyÞ ¼(6b)

m
ð1  mÞKs l1=m hSe
1  Se1=mamðys  yr Þmi
þ 1  Se1=m  2(6c)respectively, where a (L1), n (), and m (= 1–1/n) ()are shape parameters, and l is the pore-connectivity
parameter (). The parameter n in Equation 6 tends tobe large for soils with a relatively uniform pore-size distribution and small for soils having a wide range of poresizes. The pore-connectivity parameter l in Equation 6b
was estimated by Mualem (1976) to be about 0.5 as anaverage for many soils. However, many other values

371

for l have been suggested in various studies. Based on an
analysis of a large data set from the UNSODA database,Schaap and Leij (2000) recommended using l equalto 1 as a more appropriate value for most soil textures.
Equations 6a, 6b, and 6c assume the restrictive relationship m = 1  1/n, which simplifies the predictive K(h)expression compared to leaving m and n as independent
parameters in Equation 6b. In particular, the convex andconcave curvatures at the high and low pressure heads in
Figure 1 have then a particular relationship with eachother. Other restrictions on Equation 6a have been used
also. For example, Haverkamp et al. (2005) used therestriction m = 1  2/n in connection with Equation 6aand Burdine’s (1953) model to produce a different expression for K(h). The restrictions are not formally needed,since they limit the flexibility of Equation 6a in describing
experimental data. However, the predicted K(h) function
obtained with the theories of Burdine or Mualem becomesthen extremely complicated by containing incomplete
beta or hypergeometric functions, thus limiting the practicality of the analytical functions.
Rawls et al. (1982) provided average values of theparameters in the Brooks and Corey (1964) soil hydraulicparameters for 11 soil textural classes of the U.S. Department of Agriculture (USDA) textural triangle. Carsel andParrish (1988) gave similar values for the van Genuchten(1980) parameters for 12 USDA soil textural classes. In
Table 1, we list typical van Genuchten hydraulic parametervalues for relative coarse-, medium-, and fine-textured soils.
The data in this table were actually used to calculate thewater retention and hydraulic conductivity functions,shown in Figures 1 and 2, respectively, with Equations 6a,b. Average values such as those given in Table 1, or provided in more detail by Rawls et al. (1982) and Carsel andParrish (1988), are often referred to as textural class averaged pedotransfer functions. Pedotransfer functions are
relationships that use more easily measured of readily available soil data to estimate the unsaturated soil hydraulicparameters or properties (Bouma and van Lanen, 1987;
Leij et al., 2002; Pachepsky and Rawls, 2004).
We note that Equations 4 and 6 provide only two examples in which the hydraulic properties are describedanalytically. Many other combinations (Leij et al., 1997;
Kosugi et al., 2002) are possible and have been used.
For example, the combination of Equation 6a for y(h) witha simple expression like

Hydraulic Properties of Unsaturated Soils, Table 1 Typical values of the soil hydraulic parameters in the analytical functionsof van Genuchten (1980) for relatively coarse-, medium-, and fine-textured soils. The parameters were used to calculate thehydraulic properties plotted in Figures 1 and 2 using Equations 6a and 6b, respectively
Soil textureyr(cm day1)

ys(cm3 cm3)a(cm3 cm3)

n(cm1)Ks()

CoarseMedium
Fine

0.045
0.057
0.020

0.430
0.410
0.540

0.145
0.124
0.0010

2.68
2.28
1.2

712.8
350.2
45.0

372

HYDRAULIC PROPERTIES OF UNSATURATED SOILSKðhÞ ¼ Ks Seb(7)which is essentially identical to Equation 4b, for K(h) is
also very realistic. Another attractive alternative equation
for K(h) is of the form (e.g., Vereecken et al., 1989)KðhÞ ¼Ks
1 þ jahjb(8)

Many alternative expressions have been used also forthe soil water diffusivity function, D(y), mostly to facilitate simplified analytical analyses of unsaturated flow
problems (e.g., Parlange, 1980).

Experimental procedures
A large number of experimental techniques can be used toestimate the hydraulic properties of unsaturated soils.
A direct approach for the water retention function would
be to measure a number of water content (y) and pressure
head (h) pairs, and then to fit a particular retention function to the data. Direct measurement techniques include
methods using a hanging water column, pressure cells,pressure plate extractors, suction tables, soil freezing,and many other approaches. Comprehensive reviews ofvarious methods are given by Gee and Ward (1999) andDane and Hopmans (2002). Once the pairs of y and hdata are obtained, the data may be analyzed in terms ofspecific analytical water retention and conductivity functions such as those discussed earlier. Several convenient
software packages are available for this purpose (vanGenuchten et al., 1991; Wraith and Or, 1998). Alternatively, the data can be analyzed without assuming specificanalytical functions for y(h) and K(h) or K(y). This could
be done using linear, cubic spline, or other interpolation
techniques (Kastanek and Nielsen, 2001; Bitterlich et al.,2004).
Similar direct measurement approaches involving pairsof conductivity (or diffusivity) and pressure head (or watercontent) data are also possible for the K(h) and D(y)functions, at least in principle (Dane and Topp, 2002),
including for the saturated hydraulic conductivity, Ks.
The saturated hydraulic conductivity can be measured in
the laboratory using a variety of constant or falling head
methods, and in the field using single or double ring
infiltrometers, constant head permeameters, and various
auger-hole and piezometer methods (Dane and Topp,
2002). Unfortunately, because of the strongly nonlinear
nature of the soil hydraulic properties, pairs for the K(h)and D(y) data are not easily measured directly, especiallyat relatively low (negative) pressure heads, unless more
specialized techniques are used such as centrifuge
methods (Nimmo et al., 2002). Even then, the data are
generally not distributed evenly over the entire water content range of interest. Consequently, unsaturated hydraulicconductivity properties are most often estimated using
inverse or parameter estimation procedures.
Parameter estimation methods generally involve themeasurement during some experiment of one or several

capacity or flow attributes (e.g., water contents, pressure
heads, boundary fluxes), which are then used in combination with a mathematical solution (generally numerical) toobtain estimates of the hydraulic parameters such as
those that appear in Equations 4 and 6, or other functions.
Popular methods include one-step and multi-step outflow
methods (Kool et al., 1987; van Dam et al., 1994), tension
infiltrometers methods (Šimůnek et al., 1998a), and evaporation methods (Šimůnek et al., 1998b), although manyother laboratory and field methods also exist or can besimilarly employed (Hopmans et al., 2002). This also pertains to different approaches for minimizing the objectivefunction, including quantification of parameter uncertainty (Abbaspour et al., 2001; Vrugt and Robinson,2007). Very attractive now also is the use of combined
hard (e.g., directly measured) and soft (e.g., indirectly estimated) data, including hydrogeophysical measurements
and information derived from pedotransfer functions, toextract the most out of available information (e.g.,
Kowalski et al., 2004; Segal et al., 2008).

Hydraulic properties of structured soils
The Richards equation 1 typically predicts a uniform flow
process in the vadose zone. Unfortunately, the vadosezone can be extremely heterogeneous at a range of scales,from the microscopic (e.g., pore scale) to the macroscopic(e.g., field or larger scale). Some of these heterogeneities
can lead to a preferential (or bypass) flow process thatmacroscopically is very difficult to capture with the standard Richards equation. One obvious example of preferential flow is the rapid movement of water and dissolved
solutes through soil macropores (e.g., between soil aggregates, or created by earthworms or decayed root channels
or rock fractures), with much of the water bypassing(short-circuiting) the soil or rock matrix. However, manyother causes of preferential flow exist, such as flow
instabilities caused by soil textural changes or water repellency (Hendrickx and Flury, 2001; Šimůnek et al., 2003;
Ritsema and Dekker, 2005), and lateral funneling of wateralong sloping soil layers (e.g., Kung, 1990).
While uniform flow in granular soils is traditionallydescribed with a single-porosity model such as theRichards equation given by Equation 1, flow in structured
media can be described using a variety of dualporosity, dual-permeability, multi-porosity, and/or multipermeability models (Šimůnek and van Genuchten,
2008; Köhne et al., 2009). While single-porosity models
assume that a single pore system exists that is fully accessible to both water and solute, dual-porosity and dualpermeability models both assume that the porous medium
consists of two interacting pore regions, one associatedwith the inter-aggregate, macropore, or fracture system,and one comprising the micropores (or intra-aggregate
pores) inside soil aggregates or the rock matrix. Whereas
dual-porosity models assume that water in the matrix isstagnant, dual-permeability models allow also for waterflow within the soil or rock matrix.

HYDRAULIC PROPERTIES OF UNSATURATED SOILS

provide more realistic simulations of field data than thestandard approach using unimodal hydraulic propertiesof the type shown in Figures 1 and 2. In soils, the two partsof the conductivity curves may be associated with soil
structure (near saturation) and soil texture (at lower negative pressure heads).
The use of composite hydraulic functions such as thoseshown in Figure 2 is consistent with field measurementssuggesting that the macropore conductivity of soils at
saturation is generally about one to two orders of magnitude larger than the matrix conductivity at saturation,
depending upon texture. These findings were confirmed
by Schaap and van Genuchten (2006) using a detailed
neural network analysis of the UNSODA unsaturated soil
hydraulic database (Leij et al., 1996). The analysis
revealed a relatively sharp decrease in the conductivity
away from saturation and a slower decrease afterward.
Schaap and van Genuchten (2006) suggested an improved
composite function for K(h) to account for the effects ofmacropores near saturation as follows:

To avoid over-parameterization of the governing equations, one useful simplifying approach is to assume instantaneous hydraulic equilibration between the fracture andmatrix regions. In that case, the Richards equation can still
be used, but now with composite hydraulic properties ofthe form (e.g., Peters and Klavetter, 1988)yðhÞ ¼ wf yf ðhÞ þ wm ym ðhÞ(9a)KðhÞ ¼ wf Kf ðhÞ þ wm Km ðhÞ(9b)where the subscripts f and m refer to the fracture(macropore) and matrix (micropore) regions, respectively,
and where wi are volumetric weighting factors for the twooverlapping regions such that wf + wm = 1. Rather thanusing Equations 6a,b directly in Equations 9a and 9b,
Durner (1994) proposed a slightly different set of equations for the composite functions as follows
Se ðhÞ ¼

wfwm
yðhÞ  yr¼
m þm½1 þ jaf hjnf  f ½1 þ jam hjnm  mys  yr(10a)KðhÞ ¼KðSe Þ ¼
o

l n
mm 2
1=m mfm
wf Sef þ wm Sem wf af ½1  ð1  Sef f Þ  þwm am ½1  ð1  Se1=m
Þ 
m
Ks

2
wf af þ wm am

where

(10b)

RðhÞ ¼

where ai, ni, and mi (=1  1/ni) are empirical parameters of
the separate hydraulic functions (i = f,m). An example ofcomposite retention and hydraulic conductivity functions
based on Equations 10a and 10b is shown in Figure 2 forthe following set of parameters: yr = 0.00, ys = 0.50,
l = 0.5, Ks = 1 cm d1, am = 0.01 cm1, nm = 1.50, wm =0.975, wf = 0.025, af = 1.00 cm1, and nf = 5.00. The
fracture domain in this case represents only 2.5% of theentire pore space, but accounts for almost 90% of thehydraulic conductivity close to saturation (Figure 3).
While still leading to uniform flow, models using suchcomposite media properties do allow for faster flow andtransport during conditions near saturation, and as suchKs
Km ðhÞ

RðhÞKm ðhÞ

(11a)

8
<

0
h < 40 cm:
0:2778 þ 0:00694h 40  h < 4 cm:
1 þ 0:1875h4  h  0 cm
(11b)

and where Km(h) is the traditional hydraulic conductivity function for the matrix as given by Equation 6b.
Equations 11a and 11b were found to produce very small
systematic errors between the observed (UNSODA) andcalculated hydraulic conductivities across a wide rangeof pressure heads between saturation and 150 m.
While the macropore contribution was most significant
between pressure heads 0 and 4 cm, its influence on
the conductivity function extended to pressure heads as
low as 40 cm (Equation 11b).

0.5
0
0.4
log(K), [cm/day])Volumetric water content, [–]

373

0.3
0.2
0.1
0.0
−1

0

1
2
log(|h|, [cm])

3

4

−2
−4
−6
−8

−10
−1

0

1
2
log(|h|, [cm])34

Hydraulic Properties of Unsaturated Soils, Figure 3 Bimodal water retention (left) and hydraulic conductivity (right) functions as
described with the composite soil hydraulic model of Durner (1994).

374

HYDRAULIC PROPERTIES OF UNSATURATED SOILS

Multiphase constitutive relationships
The use of Equation 1 implies that the air phase has noeffect of water flow. This is a realistic assumption for mostflow simulations, except near saturation in relatively
closed systems where air may not move freely. The
resulting situation may need to be described using two
flow equations, one for the air phase and one for the liquid
phase. The same is true for multiphase air, oil, and water
systems in which the fluids are not fully miscible. Flowin such multiphase systems generally require flow equations for each fluid phase involved. Two-phase air-water
systems hence could be modeled also using separate equations for air and water. This shows that the standard
Richards equation is a simplification of a more complete
multiphase (air-water) approach in that the air phase isassumed to have a negligible effect on variably saturated
flow, and that the air pressure varies only little in space
and time. This assumption appears adequate for most variably saturated flow problems. Similar assumptions, however, are generally not possible when nonaqueous phaseliquids (NAPLs) are present. Mathematical descriptionsof multiphase flow and transport hence in general require
flow equations for each of the fluid phases involved.
Assuming applicability of the van Genuchten hydraulicfunctions and ignoring the presence of residual air andwater, the hydraulic conductivity functions for the liquid(wetting) and air phase (non-wetting) phases are givenby (e.g., Luckner et al., 1989; Lenhard et al., 2002):
h

m i2
(12a)
Kw ðSe Þ ¼ Kw Sel 1  1  Se1=mhi2mKa ðSe Þ ¼ Ka ð1  Se Þl 1  Se1=m(12b)where the subscripts w and a refer to the water and air
phases, respectively, and Kw and Ka are the hydraulic conductivities of the medium to water and air when filled
completely with those fluids. A detailed overview of various approaches for measuring and describing the hydraulic properties of multi-fluid systems is given by Lenhard
et al. (2002).

A look ahead
The unsaturated soil hydraulic properties are key factors
determining the dynamics and movement of water andits dissolved constituents in the subsurface. Reliable estimates are needed for a broad range of agrophysical applications, including for subsurface contaminant transport
studies. A large number of approaches are now available
for describing and measuring the hydraulic properties,
especially for relatively homogeneous single-porosity
soils. This includes direct measurement of discrete y(h),and K(h) or K(y) data points and fitting appropriate analytical models to the data, and the use of increasingly sophisticated inverse methods.
Considerable challenges remain in the description andmeasurement of the hydraulic properties of structured

media (macroporous soils and fractured rock). The
hydraulic properties of such media may require special
provisions to account for the effects of soil texture and soil
structure on the shape of the hydraulic functions near saturation, thus leading to dual- or multi-porosity formulations as indicated by Schaap and van Genuchten (2006)
and Jarvis (2008), among others. Estimation of the effective
properties of heterogeneous (including layered) field soil
profiles also remains an important challenge. Very promising here is the increased integration of hard (directly
measured) data and soft (indirectly estimated) information
for improved estimation of field- or larger-scale hydraulicproperties, including the use of noninvasive geophysicalinformation. New noninvasive technologies with enormous
potential range from neutron and X-ray radiography andmagnetic resonance imaging at relatively small (laboratory)
scales, to electrical resistivity tomography and ground
penetrating radar at intermediate (field) scales, to passive
microwave remote sensing at regional or larger scales.
Challenges remain on how to optimally integrate,
assimilate, or otherwise fuse such information with direct
laboratory and field hydraulic measurements (Yeh andŠimůnek, 2002; Kowalski et al., 2004, Looms et al.,2008; Ines and Mohanty, 2008), including the optimaland cost-effective use of pedotransfer function and soil
texture information, and resultant quantification of uncertainty (Minasny and McBratney, 2002; Wang et al., 2003).
These various integrated technologies undoubtedly willfurther advance in the near future, as well as the use ofincreasingly refined pore-scale modeling approaches(e.g., Tuller and Or, 2001) at the smaller scales for more
precise simulation of the basic physical processes
governing the retention and movement of water in unsaturated media.

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Cross-references
Bypass Flow in Soil
Databases of Soil Physical and Hydraulic Properties
Ecohydrology
Evapotranspiration
Field Water Capacity
Hydropedological Processes in Soils
Hysteresis in Soil
Infiltration in Soils
Layered Soils, Water and Solute Transport
Pedotransfer Functions
Physics of Near Ground Atmosphere
Pore Size Distribution
Soil Hydrophobicity and Hydraulic Fluxes
Soil Water Flow
Soil Water Management
Sorptivity of Soils
Surface and Subsurface Waters
Tensiometry
Water Balance in Terrestrial Ecosystems
Water Budget in Soil
Wetting and Drying, Effect on Soil Physical Properties

HYDRODYNAMIC DISPERSION
The tendency of a flowing solution in a porous mediumthat is permeated with a solution of different composition
to disperse, due to the non-uniformity of the flow velocity
in the conducting pores. The process is somewhat analogous to diffusion, though it is a consequence of
convection.

Bibliography
Introduction to Environmental Soil Physics (First Edition) 2003
Elsevier Inc. Daniel Hillel (ed.) http://www.sciencedirect.com/
science/book/9780123486554

HYDROPEDOLOGICAL PROCESSES IN SOILS
Svatopluk Matula
Department of Water Resources, Food and Natural
Resources, Czech University of Life Sciences, Prague,
Czech Republic

Synonyms
Soil hydrophysical processes; Soil physical processes;
Soil water processes; Soil water–soil morphology
interactions
Definition
The hydropedological processes in the broader sense are all
soil processes in which flowing or stagnant water acts as theenvironment or agent or the vehicle of transport. These processes affect the visible or otherwise discernible morphological features of the soil profile and analogous features on thepedon, polypedon, catena, and soil landscape or soil series
scales. These features can be distinguished and categorizedaccording to various pedological classification systems(Lal, 2005) and, vice versa, used to identify and semiquantify the soil water processes (e.g., Stewart and Howell,2003) that have produced or affected them. In the narrower
sense, only those processes in which water itself (its content,energy status, movement, and balance) is in the focus are
regarded as hydropedological processes.
Introduction
Pedology is the branch of soil science dealing with soil
genesis, morphology, and classification. In some parts of
the world, however, the world pedology has been or stillis used to denote the whole of soil science. Under these
conditions, it was quite natural to name that branch of soil
science that deals with soil water (e.g., Stewart andHowell, 2003) and is otherwise referred to, for example,
as soil physics, soil water physics, physics of soil water,or soil hydrology as hydropedology, notwithstanding its
relations (or rather the absence of such relations) to soil