Invariant Solutions of Richards’ Equation for Water Movement in Dissimilar Soils

SSSAJ: Volume 76: Number 1 • January–February 2012

1

Soil Sci. Soc. Am. J. 76:1–9

Posted online 16 Nov. 2011

doi:10.2136/sssaj2011.0275

Received 29 July 2011.

*Corresponding author ([email protected]).

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

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reprinting the material contained herein has been obtained by the publisher.

Invariant Solutions of Richards’ Equation for Water Movement in Dissimilar Soils

Soil Physics

At er Miller and Miller (1956), who introduced the “similar media” con-

cept, scaling methods were invented and have been frequently used in soil

physics studies, for example, for describing soils variability in terms of

soil hydraulic properties (Warrick et al., 1977; Ahuja and Williams, 1991; Kosugi

and Hopmans, 1998; Shouse, and Mohanty, 1998; Tuli et al., 2001; Das et al.,

2005; Nasta et al., 2009), or obtaining generalized solutions to a variety of soil–

water phenomena (Simmons et al., 1979; Sharma et al., 1980; Shukla et al., 2002,

Rasoulzadeh and Sepaskhah, 2003; Kozak and Ahuja, 2005; Roth, 2008).

One important aspect of scaling methods is to scale RE so that a single

solution will sui ce for numerous specii c cases of water l ow in a wide range of

unsaturated soils. Hence, these methods considerably reduce the calculations

required for heterogeneous soils (Warrick and Hussen, 1993). Some methods for

scaling RE were described by Reichardt et al. (1972), Warrick and Amoozegar-

Fard (1979), Warrick et al. (1985), Sposito and Jury (1985), Vogel et al. (1991),

Kutilek et al. (1991), Warrick and Hussen (1993), Nachabe (1996), Wu and Pan

(1997), and Sadeghi et al. (2011). Using specii c scaling factors, these methods

suggest linear transformations of RE variables to achieve invariant solutions for a

set of soils and/or conditions. However, satisfying the “similarity condition” for

the soils and conditions is a necessity in all these methods. h e similarity may be

M. Sadeghi*Dep. of Water EngineeringCollege of AgricultureFerdowsi Univ. of MashhadMashhad, IranandDep. of Plants, Soils, and ClimateUtah State Univ.Logan, UT 84322-4820

B. GhahramanDep. of Water EngineeringCollege of AgricultureFerdowsi Univ. of MashhadMashhad, Iran

A.N. ZiaeiDep. of Water Engineering,College of AgricultureFerdowsi Univ. of MashhadMashhad, Iran

K. DavaryDep. of Water EngineeringCollege of AgricultureFerdowsi Univ. of MashhadMashhad, Iran

K. ReichardtLab. of Soil PhysicsCenter for Nuclear Energy in AgricultureUniv. of São PauloPiracicaba, Brazil

Scaling methods allow a single solution to Richards’ equation (RE) to sufi ce for numerous specii c cases of water l ow in unsaturated soils. During the past half-century, many such methods were developed for similar soils. In this paper, a new method is proposed for scaling RE for a wide range of dissimilar soils. Exponential-power (EP) functions are used to reduce the dependence of the scaled RE on the soil hydraulic properties. To evaluate the proposed method, the scaled RE was solved numerically considering two test cases: ini ltration into relatively dry soils having initially uniform water content distributions, and gravity-dominant drainage occurring from initially wet soil proi les. Although the results for four texturally different soils ranging from sand to heavy clay (adopted from the UNSODA database) showed that the scaled solution were invariant for a wide range of l ow conditions, slight deviations were observed when the soil proi le was initially wet in the ini ltration case or deeply wet in the drainage case. The invariance of the scaled RE makes it possible to generalize a single solution of RE to many dissimilar soils and conditions. Such a procedure reduces the numerical calculations and provides additional opportunities for solving the highly nonlinear RE for unsaturated water l ow in soils.

Abbreviations: BC, Brooks–Corey; EP, exponential-power; PM, proposed method; RE, Richards’ equation; WHM, Warrick–Hussen method.

2 SSSAJ: Volume 76: Number 1 • January–February 2012

dei ned based on microscopic-scale geometry (Miller and Miller,

1956), shape of soil hydraulic functions (Simmons et al., 1979),

or a linear variability concept (Vogel et al., 1991).

h e similarity condition is dii cult to truly hold in reality

and will be a limitation for the application of previous scaling

methods to real soils. Focusing on this limitation, the objective

of this study was mainly developing a method to scale RE

applicable to dissimilar soils and specii cally extending the

method of Warrick and Hussen (1993). Adopting Brooks-Corey

soil hydraulic models, Warrick and Hussen (1993) developed

scaled solutions of RE invariant to the saturated and residual

volumetric water contents, saturated hydraulic conductivity,

and air-entry pressure head, as well as boundary or initial

conditions. However, the scaled solutions were still dependent

on the shape parameters of the Brooks–Corey functions, λ and

v. In other words, equality in λ and v is a requirement (i.e., the

similarity condition) in this method and the scaled solutions will

be invariant only for soils having equal values of λ and v. Here,

we handle this limitation and present a new method to scale RE

considering dissimilar soils.

THEORY

Richards’ (1931) equation is obtained by combining Darcy’s

law, q = -K(;h/;z –1), and the mass conservation law, ;θ/;t =

-;q/;z, which, in one-dimensional form, is written as:

hK K

t z z

θ¶ ¶ ¶æ ö= -ç ÷¶ ¶ ¶è ø [1]

where q [LT−1] is the water l ux density, θ [L3L–3] the soil wa-

ter content, h [L] the pressure head, K [LT–1] the unsaturated

hydraulic conductivity, t [T] the time, and z [L] the soil depth

positive downward.

Warrick–Hussen Method for Scaling Richards’ Equation

Warrick and Hussen (1993) adopted Brooks and Corey,

hereinat er BC, (1964) soil hydraulic functions as follows:

( ) , ( 0)r s r b

b

hh h

h

λ

θ θ θ θ-

æ ö= + - < <ç ÷

è ø [2]

, ( 0)

v

s b

b

hK K h h

h

-æ ö

= < <ç ÷è ø

[3]

where θs and θr are the saturated and residual volumetric water

contents, respectively, Ks the saturated hydraulic conductivity, hb

the air-entry pressure head, and λ and v the shape parameters.

Considering a boundary or initial water content, θ0, and

dei ning the absolute value of h0 = h(θ0) (i.e., the pressure head

corresponding to θ0) as a length scaling factor, z0 [L], Warrick and

Hussen proposed the following scaled functions and variables:

*

0

r

r

θ θθθ θ-

=-

[4]

*

0

hh

z= [5]

*

0

KK

K= [6]

*

0

zz

z= [7]

( )* 0

0 0r

K tt

zθ θ=

- [8]

where K0 represent K(θ0) (i.e., the hydraulic conductivity cor-

responding to θ0). Considering Warrick and Hussen dei nitions

of θ0 (boundary water content in ini ltration and initial water

content in drainage— discussed in detail later), in this method,

θ* and K* range between 0 and 1 and h* ranges from −1 to -Y.

Substituting the dimensionless functions and variables, Eq.

[4] to [8], into Eq. [1] yields a scaled form of RE as follows:

* ** *

* * *

hK K

t z z

θ æ ö¶ ¶ ¶= -ç ÷¶ ¶ ¶è ø

[9]

with the following scaled forms of BC hydraulic functions:

( )* *hλ

θ-

= - [10]

( )* * vK h

-= - [11]

Equation [9] is expressed in a form independent of θ0, θs,

θr, Ks, and hb. However, λ and v are the hydraulic parameters

remaining in Eq. [10] and [11] which make Eq. [9] dependent

on the soil properties. h erefore, the scaled solutions will be

invariant only for the similar soils (i.e., with the same values of

λ and v).

Combining Eq. [9], [10], and [11] yields the following form

of the scaled RE rearranged based on θ*:

2

2* * * 2 * * **

* * * * **

dD dKD

t d z d zz

θ θ θ θθ θæ ö¶ ¶ ¶ ¶

= + -ç ÷¶ ¶ ¶¶è ø [12]

where D* is the scaled form of dif usivity, D = Kdh/dθ[L2T–1],

as follows:

** *

*

dhD K

dθ= [13]

h e i rst two terms in the right hand side of Eq. [12] are

called hereinat er dif usivity terms and the third one is called

gravity term. Combining Eq. [10–13], the scaled RE of the

Warrick–Hussen method (WHM) is given in terms of the shape

parameters, λ and v:

( )( )

( )( ) ( )( )2

2* *2 1 /*

* 2 *

2 * *1 / /* *

**

1

1

v

v v

v

t z

v

zz

λ λ

λ λ λ λ

θ λ θθ

λ

θ θθ θ

λ λ

- -

- - -

æ ö¶ - - ¶= +ç ÷¶ ¶è ø

¶ ¶-

¶¶

[14]

SSSAJ: Volume 76: Number 1 • January–February 2012 3

As can be seen, in the WHM, all the three terms are

dependent on the shape parameters. h erefore, this method does

not give opportunity to scale two dissimilar soils (e.g., a sand and

a clay) which have signii cantly dif erent shape parameters. In

the following, we propose a scaling method by which scaling of

dissimilar soils is expectable under some special scenarios.

Proposed Method for Scaling Richards’ EquationExperimental (Nielsen et al., 1973) and theoretical

(Hunt, 2004) studies indicate that the following exponential

conductivity model can be applied adequately for many soils:

( ) ( )exp , s s tK K s θ θ θ θ= - ³é ùë û [15]

where θt is a threshold soil water content below which Eq. [15]

does not apply, and s is a i tting parameter. Adopting Eq. [15] in

combination with the power relationship between K and h, Eq.

[3], yields the following water retention model:

( ) ( )exp , b s t

sh h

vθ θ θ θé ù= - ³ê úë û

[16]

and dif usivity model:

( ) ( ) ( )1

exp s bs t

s vK h sD

v vθ θ θ θ

-é ù=- - ³ê ú

ë û [17]

Equations [15 −17] which also keep the power relationships

between h, K, and D are called hereinat er EP models.

In the PM, we consider soil water phenomena in which the

soil proi le never dries beyond θt. We dei ne the following scaled

soil water content and time:

* 1

0 1

θ θθθ θ-

=- [18]

( )* 0

0 1 0

K tt

zθ θ=

- [19]

where θ1 is a soil water content between θs and θt (discussed in

detail later), and the length scaling factor, z0, is dif erently de-

i ned in this method as follows:

( )0 0 1

0

0

Dz

K

θ θ-= [20]

where D0 = D(θ0). Using Eq. [16], [17], and [18], it can be indi-

cated that Eq. [20] results in:

0 0 1lnz h H=- [21]

where H1 = h(θ1)/h0. Scaled h, K, and z are kept the same as

dei ned in Eq. [5] to [7], although with the newly dei ned length

scaling factor, z0, as Eq. [20] or [21]. Substituting the proposed

scaled functions and variables into Eq. [1], the resulting scaled

RE remains in the form of Eq. [9], however; the following scaled

hydraulic functions are applied instead of Eq. [10] and [11]

(proven in Appendix A):

( )* *1

1

1exp ln 1

lnh H

Hθ

- é ù= -ë û [22]

( )* * *1exp ln 1K K θé ù= -ë û [23]

where ( ) ( )* *1 1 1 0/K K K Kθ θ= = and is given as follows based on

Eq. [15] (see Appendix A):

( )*1 0 1exp K s θ θ= - -é ùë û [24]

In this method, scaled dif usivity can also be dei ned as follows

which is in agreement with the previous dei nition in Eq. [13]:

*

0

DD

D= [25]

It should be noticed that Eq. [25] is not the case in WHM

(i.e., Eq. [13] and [25] result in dif erent scaled functions in

WHM). Based on Eq. [17] and [25], the scaled dif usivity model

of the PM is yielded as follows (see Appendix A):

( )* * *1exp ln 1D D θé ù= -ë û [26]

where ( ) ( )* *1 1 1 0/D D D Dθ θ= = and is given as follows based on

Eq. [17]:

( ) ( )*1 0 1

1exp

s vD

v

-é ù= - -ê ú

ë û [27]

Substituting Eq. [24] and [26] into Eq. [12] yields the

scaled RE of the PM as:

( ) ( )

( )

2

2* * 2 ** * * * *

1 1 1* * *

** * *

1 1 *

ln exp ln 1 exp ln 1

ln exp ln 1

D D Dt z z

K Kz

θ θ θθ θ

θθ

æ ö¶ ¶ ¶é ù é ù=- - + - +ç ÷ë û ë û¶ ¶ ¶è ø¶é ù-ë û ¶

[28]

h erefore, in the PM, soil-dependency of the scaled

RE is dei ned in terms of *1K and *

1D rather than the shape

parameters of the hydraulic functions as is the case in WHM,

Eq. [14]. h is property allows us to i nd conditions under which

scaling of dissimilar soils (even with signii cantly dif erent shape

parameters) is expected.

Two special scenarios are considered here under which the

scaled RE, Eq. [12] or Eq. [28], is invariant (soil-independent)

and the scaling method is applicable to dissimilar soils: (i) when *

1D is the same for dif erent soils and when gravity is negligible

(i.e., the l ow regime is capillarity-dominated such as ini ltration

into initially relatively dry soils), and (ii) when *1K is the same

for dif erent soils and when dif usion (capillarity) is negligible

(i.e., the l ow regime is gravity-dominated such as free drainage

from an initially wet soil).

Equality of *1D in scenario A or *

1K in scenario B for

dif erent soils is obtained through a proper choice of θ1 according

to Eq. [24] or [27], respectively. h erefore, the θ1 value plays an

important role in the PM to determine the conditions under

which scaling of dissimilar soils is possible.


MATERIALS AND METHODSFour texturally dif erent soils that cover a wide texture

range regarding the shape parameters (λ, v, and s) were selected

from the UNSODA database (Leij et al., 1996). h e soils in

the UNSODA have been named with codes 3142 (sand), 2680

(loam), 1360 (silty clay), and 1400 (clay). Although the PM

applies for θ ≥ θt, for determining the EP models parameters, we

adopted the following piece-wise model which modii es Eq. [17]

for entire range of θ:

( ) ( )

( ) ( )

exp

exp exp < <

b s t

tb s t r t

r

sh h

v

sh h

v

θ θ θ θ

θ θθ θ δ θ θ θ

θ θ

ì é ù= - ³ï ê úë ûïí é ùæ ö-é ùï = - ê úç ÷ê úï -ë û è øë ûî

[29]

where δ is calculated from the equality of the derivatives of the

two pieces at θ = θt:

( )t r

s

vδ θ θ= - [30]

h e BC and EP parameters of the soils, presented in Table

1, were determined by i tting Eq. [2], [3], and [29] to the

experimental data. Since the best i tting hb was dif erent for each

equation, we considered a dif erent hb for each of them and then

assumed the geometrical mean of the three values for each soil.

However, identical θs and θr were considered for Eq. [2] and

[29]. Goodness of i t of BC and EP models was approximately

identical as shown, for example, for the sand (3142) and the clay

(1400) soils, in Fig. 1 in which the models are in agreement.

To show the improvements of the PM to scale RE, Warrick

and Hussen (1993) evaluations were repeated for both methods.

h e scaled RE, Eq. [9], was solved numerically

with the scaled hydraulic functions of [10] and

[11] for the WHM, and [22] and [23] for the

PM. h e numerical calculations were performed

using the i nite dif erence method with the fully

implicit scheme identical to that of HYDRUS-1D

(Simunek et al., 2005). To do so, a computer code

was written in MATLAB.

Two test cases were considered: (i) ini ltration

with constant water content at the soil surface into a uniformly dry

soil, and (ii) drainage of a uniformly wet soil with no l ow at the

soil surface. θ0 was dei ned in a similar way as Warrick and Hussen

(1993) did (i.e., upper boundary water content for ini ltration

and initial water content for drainage). We set θ1 as initial water

content in the ini ltration case (Scenario i). Hence, ini ltration

solutions can be scaled for dif erent soils, each having a specii c

value of initial water contents (θ1) so that gives an identical *1D

based on Eq. [26]. For the drainage case (Scenario ii), θ1 can be

considered as an arbitrary value between θs and θt so that gives

an identical *1K for dif erent soils based on Eq. [24]. Table 2

summarizes the boundary and initial conditions in the two cases.

In this table, L is the length of the soil column in drainage.

Considering Darcy’s law, q = –K(;h/;z –1), the three Eq.

of [5], [6], and [7] suggest the following scaled l ux density, q*,

in both methods:

*

0

qq

K= [31]

Applying Eq. [4], [6], and [31], the scaled boundary

and initial conditions as reported in Table 3 are attained. In

the WHM, the scaled lower boundary and initial conditions

in ini ltration, *1 , is dependent on θ1 and inl uences the

scaled solutions. h erefore, in the WHM, an identical *1 was

considered for all soils. h e scaled length of the soil column

in drainage, L*, also inl uences the scaled solutions in both

methods. All other scaled boundary and initial conditions are

invariant in the PM. However, as discussed earlier, the value of θ1

in any case inl uences the scaled solutions of the PM.

Another requirement regarding *

1D and *1K is that they should

be corresponding to θ1 values not

less than θt. Hence, *1D and *

1K

should not be less than a minimum

value according to Eq. [24] and

[27], respectively. Based on these

equations, when θ0 and θ1 are equal

to their maximum and minimum

allowable values (i.e., θs and θt,

respectively), *1D and *

1K take

their minimum value, for example,

8.94E-3 and 2.26E-3, respectively,

for 3142 and, 1.63E-5 and 4.5E-10,

respectively, for 1360.

Table 1. Brooks–Corey and exponential-power hydraulic parameters of the selected soils.

Code Texture θs

θt

θr

hb K

sλ v s

cm cm d–1

3142 sand 0.355 0.137 0.0657 21.5 35.00 1.1153 4.43 27.95

2680 loam 0.593 0.190 0.0252 5.4 76.00 0.1853 2.31 41.23

1360 sily clay 0.439 0.103 0 55.6 0.35 0.0934 2.05 64.05

1400 clay 0.439 0.065 0 66.23 0.20 0.0814 1.69 60.8

Fig. 1. Soil water retention curves of soils (a) 3142 (sand), and (b) 1400 (clay).


RESULTS AND DISCUSSIONIni ltration

Scaled solutions of the WHM for the constant-θ ini ltration,

considering two scaled initial water content, *1 , of 0.02 and 0.2,

are presented in Fig. 2. h e i gure shows

plot of the scaled water content vs. the

scaled depth at t* = 0.1 which, assuming

θ0 = θs, is corresponding to t = 0.42 and

349 h in soils 3142 and 1400, respectively.

Due to the wide range of λ and v for the

four selected soils (see Table 1), the scaled

solutions of the WHM are signii cantly

dif erent. h is is quantitatively shown by

the mean absolute error (MAE) values

appearing in the i gure which represent

an average of the absolute deviations of θ*

from a mean curve along the wetting zone

(the mean curve is obtained by averaging

θ* of dif erent soils at each z*). Since the

WHM is expected to give invariant scaled

solutions for similar soils (i.e., with equal

values of λ and v), these selected soils are

indicating a high degree of dissimilarity

which provides an opportunity to

seriously evaluate the PM.

Figure 3 shows the scaled solutions of

the PM for ini ltration with two specii c *

1D of 1E-4 and 1E-2. h e scaled water

content proi les are shown at t* = 0.1 and

0.3 (For θ0 = θs and *1D = 1E-2, t* = 0.1

is corresponding to t = 0.43 and 865 h in

soils 3142 and 1400, respectively). Since

1E-4 is less than the minimum allowable

value of *

1D in soil 3142, this soil is not

included in Fig. 3a.

Although the soils are highly

dissimilar, Fig. 3 indicates that the scaled

solutions are nearly invariant for all soils

and show a unii ed curve. However, a slight

deviation exists in Fig. 3b in which the

initial soil represents a wetter condition

especially when time increases and the

soil proi le becomes deeply saturated. As

discussed earlier, this is due to the ef ect

of the gravity term which, when *1D is

identical for all soils, remains as the only

soil-dependent term of Eq. [28] and is

more pronounced when the wetness

increases within the soil proi le.

Figure 4 presents scaled cumulative

ini ltrated water, I*, curves corresponding

to Fig. 3 which were calculated using Eq.

[7] and [18] as follows:

Table 2. Boundary and initial conditions in the two test cases.

Case Upper boundary Lower boundary Initial

Ini ltration θ = θ0 θ = θ1 θ = θ1

Drainage q = 0 Free drainage (q = K) θ = θ0 (z < L)

Table 3. Scaled boundary and initial conditions in the two test cases.

Case

Warrick–Hussen method Proposed method

Upperboundary

Lowerboundary

InitialUpper

boundaryLower

boundaryInitial

Ini ltration θ* = 1 θ* = θ*1† θ* = θ*1 θ* = 1 θ* = 0 θ* = 0

Drainage q* = 0 q* = K* θ* = 1 (z* < L*‡) q* = 0 q* = K* θ* = 1 (z* < L*)

† θ*1 = (θ1 – θr)/(θ0 – θr).

‡ L* = L/z0.

Fig. 2. Scaled water content vs. scaled depth obtained using the Warrick–Hussen method for constant-θ ini ltration with (a) *

1 = 0.02, and (b) *1 = 0.2 at t* = 0.1.

Fig. 3. Scaled water content vs. scaled depth obtained using the proposed method for constant-θ ini ltration with (a) *

1D = 1E-4, and (b) *1D = 1E-2.


( )* * *

00 1 0

II dz

zθ

θ θ

¥= =

-ò [32]

where ( )10

I dz¥

= -ò [L] is the cumulative ini ltrated water.

Similarly, the scaled curves of Fig. 4a are invariant, while those

of Fig. 4b deviate at large times. It is noticeable that the scaled

curves of Fig. 4b are invariant until about t* = 0.2 at which

ini ltration rate approaches a constant value (i.e., the i nal

ini ltration rate) and gravitational force becomes signii cant

(Assuming θ0 = θs, t* = 0.2 equals 0.86 and 1730 h in soils 3142

and 1400, respectively).

As can be realized from Eq. [9], the signii cance of the

dif usivity and gravity terms of Eq. [12] is af ected by the gradient

;h*/;z*. With increasing ini ltration time, the magnitude of

;h*/;z* along the soil proi le decreases and consequently, the

impact of gravity increases. Hence, to quantify the range of

applicability of the PM during ini ltration, we consider gradient

of h* along the entire l ow zone as G = |Δh*|/Δz* ( = |Δh|/Δz).

In this case, Δz* equals the scaled wetting front depth, *fz (i.e.,

zf/z0, where zf is the wetting front depth), and |Δh*| represents

the dif erence between |h*| at the soil surface and z = zf.

h e values of G in Fig. 5 corresponding to the scaled results

presented in Fig. 3 and 4 quantify the scaling performance. It

is obvious that G, and as a result

scaling performance of the PM,

diminishes when: (i) ini ltration

time increases, (ii) initial soil water

content (θ1) increases, and (iii) soil

texture becomes coarser. Based on

the information derived from Fig. 3,

4, and 5, we conclude that whenever

values of G roughly exceed 10, the

impact of gravity will be negligible

and the PM applicable.

Combining Eq. [7], [21], and

[22], the magnitude of G during

ini ltration becomes:

1 01

*1

1

lnf f

h hHG

z H z

--= = [33]

Hence, whenever *

1 10.1 1 /lnf

z H H< - or 1 00.1fz h h< - , we

know that G is larger than 10 and the PM is valid.

Drainageh e WHM scaled water content proi les for drainage at t*

= 0.1 are presented in Fig. 6 considering two cases of L* = L/

z0 = 0.1 and 0.5 (For θ0 = θs, L* = 0.5 is corresponding to L

= 10.75 and 33.1 cm in soils 3142 and 1400, respectively). h e

scaled results in the three i ner soils (2680, 1360, and 1400) are

closed to each other, however, a large deviation is observed for

the sand (3142).

h e scaled proi les of the PM for drainage are shown in

Fig. 7 for L* = 0.1 and 0.5 and for *1K = 5E-3 considering the

minimum allowable values of *1K for the four soils (For θ0 = θs,

L* = 0.5 is corresponding to L = 12.9 and 101.25 cm in 3142

and 1400, respectively). h e scaled proi les are invariant in the

shorter column, while deviate near the soil surface in the longer

column. As far as the θ* proi les are vertical (as can be seen in Fig.

7a), the scaled results are invariant and when the proi les depart

Fig. 4. Scaled cumulative ini ltrated water vs. scaled time obtained using the proposed method for constant-θ ini ltration with (a)

*1D = 1E-4, and (b)

*1D = 1E-2.

Fig. 5. Values of G = |∆h*|/∆z* corresponding to the proposed method (PM) scaled results shown in (a) Fig. 3a and (b) Fig. 3b.


from the vertical line, as occurs in

long columns (Fig. 7b), the scaled

solutions will become apart.

Since the no l ow upper

boundary condition and free

drainage lower boundary condition

are maintained during drainage,

|;h*/;z*| always equals unity at z

= 0 and zero at z = L. h erefore,

G (equal to |Δh*|/L* in this case)

should always manifest values

between 0 and 1. Evaluating the

results presented in Fig. 7, we

ascertain that values of G remaining

between 0.4 and 0.5 do not

signii cantly change with time, soil,

or conditions.

We learn that although the

drainage l ow regime is always

dominated by the force of gravity,

the small, ever-present capillary force

(with a nearly constant ratio to that

of gravity) af ects the reliability of the PM, particularly for the

drainage of longer columns discussed in the following paragraph.

Since G is roughly constant, |Δh*| increases when L*

increases and for each specii c L*, |Δh*| is almost identical for

dif erent soils (consider |Δh*| = GL*). Due to the pressure head

dif erence between the top and bottom of the column, θ* proi les

deviate from the vertical line. Having the same Δh*, this deviation

is more in coarser soils and less in i ner soils due to their dif erent

water retention capability. h at is why with increasing L* and

as a result increasing |Δh*|, ef ect of the existent capillary drive

becomes more pronounced and the scaled solutions become

more apart from each other.

Based on the above discussion,

the applicability range of the PM in

this case can be judged only by L*

value. As Fig. 7 shows, the scaled results

start to deviate in the case of L* = 0.5.

Hence, as far as L* < 0.5 or, based on

Eq. [21], ( )0 1 00.5 ln /L h h h<- , the

PM can be applicable in this case.

Regenerating the Solutions of Richards’ Equation

h e invariance of the scaled

RE of the proposed method makes

it possible to generalize a single

solution of RE to many dissimilar

soils and conditions. In this

section, by an example, we show

the applicability of the PM for

regenerating RE solution for a specii c soil when the solution is

known for another soil.

Assume that RE solution is known for the constant-θ

ini ltration, for example, in the loamy soil of 2680 with θ0 = 0.9θs

and θ1 = θt. We call this case as the i rst case. Using the solution

for this case, we intend to estimate the RE solution in a second

case, for example, in the clayey soil of 1400 with the upper

boundary water content of θ0 = θs. For equality of *1D in the

two cases (as a requirement in the PM), the following equation

obtained from Eq. [27] should be held:

( ) ( ) ( ) ( )0 1 0 1

first case second case

1 1s v s v

v vθ θ θ θ

- -é ù é ù- = -ê ú ê ú

ë û ë û [34]

Fig. 6. Scaled water content vs. scaled depth obtained using the Warrick–Hussen method for drainage in a proi le with scaled depth of (a) 0.1 and (b) 0.5 at t* = 0.1.

Fig. 7. Scaled water content vs. scaled depth obtained using the proposed method for drainage in a proi le with scaled depth of (a) 0.1 and (b) 0.5 assuming

*1K = 5E-3.


by which θ1 in the second case is required to be equal to 0.108.

As can be seen in Table 1, this value of θ1 is within the valid range

(between θs and θt) for soil 1400. Note that if the calculated θ1

was less than θt, the i rst case solution could not be applied to the

second case. Hence, such an application of the PM is limited to a

specii c range of initial water content (θ1) dependent on the soil

properties and θ0 of the i rst and second case.

First, RE was solved for the i rst case. h e solutions, in

terms of cumulative ini ltration, were scaled using Eq. [8] and

[32] by application of the properties (θ0, θ1, z0, and K0) of 2680.

Assuming that the scaled solutions of the two cases are the same,

these scaled solution were de-scaled (converted to the real scale)

using Eq. [8] and [32], this time, by application of the properties

of 1400. h e descaled solution in comparison with the direct

solution of RE for 1400 are presented in Fig. 8 which shows that

both solutions are in a good agreement.

CONCLUSIONSIn this paper, a new method has been proposed for scaling RE

for water movement in dissimilar soils. Considering a wide range

of soils from sand to heavy clay, solutions of the proposed scaled

RE are approximately invariant for ini ltration and drainage.

However, the scaled solutions deviate when gravitational force

increases (G roughly becomes less than 10) in the ini ltration

case or column depth increases (L* roughly becomes more than

0.5) in the drainage case.

A primary limitation of this method is that it applies only for

the EP hydraulic functions, Eq. [15−17]. Perhaps, future studies

can involve other forms of the hydraulic functions. Also, this

scaling method is applicable only to the soil water phenomena in

which the soil proi le never dries beyond a threshold, θt.

Applicability of the proposed method to regenerate

RE solutions has been indicated. h is kind of application

introduces this method as a promising tool to reduce tedious and

complicated numerical calculations and opens a new window to

easily obtain approximate solutions to the highly nonlinear RE for

water l ow in unsaturated soils, within prescribed levels of error.

APPENDIX A: DERIVATION OF THE SCALED HYDRAULIC FUNCTIONS OF THE PROPOSED METHOD (EQUATIONS [22], [23], AND [26])

Rewriting Eq. [15] for θ0 yields:

( )0 0exp s sK K s θ θ= -é ùë û [A1]

Dividing [15] by [A1] yields:

( )*0

0

exp K

K sK

θ θ= = -é ùë û [A2]

which equals:

( ) ( )

( )( )

* 0 11 0 1 0

1 0 0 1

*1 0

exp exp 1

=exp 1

K s s

s

θ θ θ θθ θ θ θ

θ θ θ θ

θ θ θ

é ùé ù æ ö- -= - = - -ê úç ÷ê ú- -ë û è øë û

é ù- -ë û

[A3]

Rewriting [A2] for θ1 yields Eq. [24] which also can be written

in the following form:

( ) *1 0 1 =ln s Kθ θ- [A4]

Combining Eq. [A3] and [A4], Eq. [23] is obtained.

Starting from Eq. [17] and going through a similar way,

Eq. [26] can be easily proven. Using Eq. [16], the following

relationship can be derived similarly:

( )*1

0 0

exp ln 1hh

h h

é ù= -ê ú

ë û [A5]

Combining Eq. [21] and Eq. [A5], Eq. [22] can be derived:

( )* *1

0 0 1 1

1exp ln 1

ln ln

h hh H

z h H Hθ

- é ù= = = -ë û- [22]

ACKNOWLEDGMENTSWe gratefully acknowledge Professors A.W. Warrick, D.R. Nielsen, and A.R. Sepaskhah for their thorough review and helpful comments. Mr. Behzad Ghanbarian-Alavijeh is also thanked for his contribution regarding Eq. [29]—a modii cation of EP models.

REFERENCESAhuja, L.R., and R.D. Williams. 1991. Scaling water characteristic and hydraulic

conductivity based on Gregson-Hector-McGowan approach. Soil Sci. Soc. Am. J. 55:308–319. doi:10.2136/sssaj1991.03615995005500020002x

Brooks, R.H., and A.T. Corey. 1964. Hydraulic properties of porous media. Hydrolo. Paper 3. Colorado State Univ., Fort Collins.

Das, B.S., W.N. Haws, and P.S.C. Rao. 2005. Dei ning geometric similarity in soils. Vadose Zone J. 4:264–270. doi:10.2136/vzj2004.0113

Hunt, A.G. 2004. An explicit derivation of an exponential dependence of the hydraulic conductivity on relative saturation. Adv. Water Resour. 27:197–201. doi:10.1016/j.advwatres.2003.11.005

Kosugi, K., and J.W. Hopmans. 1998. Scaling water retention curves for soils with lognormal pore-size distribution. Soil Sci. Soc. Am. J. 62:1496–1504. doi:10.2136/sssaj1998.03615995006200060004x

Kozak, J.A., and L.R. Ahuja. 2005. Scaling of ini ltration and redistribution of water across soil textural classes. Soil Sci. Soc. Am. J. 69:816–827. doi:10.2136/sssaj2004.0085

Kutilek, M., K. Zayani, R. Haverkamp, J.Y. Parlange, and G. Vachaud. 1991. Scaling of Richards’ equation under invariant l ux boundary conditions. Water Resour. Res. 27:2181–2185. doi:10.1029/91WR01550

Leij, F.J., W.J. Alves, M. h van Genuchten, and J.R. Williams. 1996. h e

Fig. 8. Cumulative ini ltrated water vs. time in soil 1400 as obtained by direct solution of Richards’ equation and descaling the scaled solutions of 2680.


UNSODA unsaturated soil hydraulic database, version 1.0. EPA Rep. EPA/600/R-96/095. U.S. Environ. Protection Agency, Cincinnati, OH.

Miller, E.E., and R.D. Miller. 1956. Physical theory for capillary l ow phenomena. J. Appl. Phys. 27:324–332. doi:10.1063/1.1722370

Nachabe, M.H. 1996. Macroscopic capillary length, sorptivity, and shape factor in modeling the ini ltration rate. Soil Sci. Soc. Am. J. 60:957–962.

Nasta, P., T. Kamai, G.B. Chirico, J.W. Hopmans, and N. Romano. 2009. Scaling soil water retention functions using particle-size distribution. J. Hydrol. 374:223–234. doi:10.1016/j.jhydrol.2009.06.007

Nielsen, D.R., J.W. Biggar, and K.T. Erh. 1973. Spatial variability of i eld-measured soil-water properties. Hilgardia 42:215–259.

Rasoulzadeh, A., and A.R. Sepaskhah. 2003. Scaled ini ltration equations for furrow irrigation. Biosyst. Eng. 86:375–383. doi:10.1016/j.biosystemseng.2003.07.004

Reichardt, K., D.R. Nielsen, and J.W. Biggar. 1972. Scaling of horizontal ini ltration into homogeneous soils. Soil Sci. Soc. Am. J. Proc. 36:241–245. doi:10.2136/sssaj1972.03615995003600020014x

Richards, L.A. 1931. Capillary conduction of liquids through porous mediums. Physics 1:318–333. doi:10.1063/1.1745010

Roth, K. 2008. Scaling of water l ow through porous media and soils. Eur. J. Soil Sci. 59:125–130. doi:10.1111/j.1365-2389.2007.00986.x

Sadeghi, M., B. Ghahraman, K. Davary, S.M. Hasheminia, and K. Reichardt. 2011. Scaling to generalize a single solution of Richards’ equation for soil water redistribution. Sci. Agric. (Piracicaba, Braz.) 68:582–591.

Sharma, M.L., G.A. Gander, and C.G. Hunt. 1980. Spatial variability of ini ltration in a watershed. J. Hydrol. 45:101–122. doi:10.1016/0022-1694(80)90008-6

Shouse, P.J., and B.P. Mohanty. 1998. Scaling of near-saturated hydraulic conductivity measured using disc ini ltrometers. Water Resour. Res. 34:1195–1205. doi:10.1029/98WR00318

Shukla, M.K., F.G. Kastanek, and D.R. Nielsen. 2002. Inspectional analysis of convective-dispersion equation and application on measured breakthrough

curves. Soil Sci. Soc. Am. J. 66:1087–1094. doi:10.2136/sssaj2002.1087Simmons, C.S., D.R. Nielsen, and J.W. Biggar. 1979. Scaling of i eld-measured

soil-water properties. Hilgardia 47:77–173.Simunek, J., M.h . van Genuchten, and M. Sejna. 2005. h e HYDRUS-1D

sot ware package for simulating the movement of water, heat, and multiple solutes in variably saturated media. Version 3.0. HYDRUS Sot w. Ser. 1. Dep. of Environ. Sci., Univ. of California, Riverside.

Sposito, G., and W.A. Jury. 1985. Inspectional analysis in the theory of water l ow through unsaturated soils. Soil Sci. Soc. Am. J. 49:791–798. doi:10.2136/sssaj1985.03615995004900040001x

Tuli, A., K. Kosugi, and J.W. Hopmans. 2001. Simultaneous scaling of soil water retention and unsaturated hydraulic conductivity functions assuming lognormal pore-size distribution. Adv. Water Resour. 24:677–688. doi:10.1016/S0309-1708(00)00070-1

Vogel, T., M. Cislerova, and J.W. Hopmans. 1991. Porous media with linearly hydraulic properties. Water Resour. Res. 27:2735–2741. doi:10.1029/91WR01676

Warrick, A.W., and A. Amoozegar-Fard. 1979. Ini ltration and drainage calculations using spatially scaled hydraulic properties. Water Resour. Res. 15:1116–1120. doi:10.1029/WR015i005p01116

Warrick, A.W., and A.A. Hussen. 1993. Scaling of Richards’ equation for ini ltration and drainage. Soil Sci. Soc. Am. J. 57:15–18. doi:10.2136/sssaj1993.03615995005700010004x

Warrick, A.W., D.O. Lomen, and S.R. Yates. 1985. A generalized solution to ini ltration. Soil Sci. Soc. Am. J. 49:34–38. doi:10.2136/sssaj1985.03615995004900010006x

Warrick, A.W., G.J. Mullen, and D.R. Nielsen. 1977. Scaling of i eld measured hydraulic properties using a similar media concept. Water Resour. Res. 13:355–362. doi:10.1029/WR013i002p00355

Wu, L., and L. Pan. 1997. A generalized solution to ini ltration from single-ring ini ltrometers by scaling. Soil Sci. Soc. Am. J. 61:1318–1322. doi:10.2136/sssaj1997.03615995006100050005x

Invariant Solutions of Richards’ Equation for Water Movement in Dissimilar Soils

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