Scaling in Soil Physics

Morteza Sadeghi

Department of Plants, Soils, and Climate, Utah State University

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Scaling in soil physics is based on Miller and Miller (1956) “Similar media” concept

“Similar media” Theory

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- are similar in their microscopic geometry and differ only in scale

- have identical porosities

Two similar media

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Two non-similar media

Identical particle size distributionDifferent pore size distribution

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Two dissimilar media

Different particle size distributionDifferent pore size distribution

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identical water content (%)

similar media in similar state

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Similar media are scalable into each other by a “scaling factor”, a ratio of two corresponding physical lengths.

λ2/λ1 can scale the first media into the second.

λ1

λ2

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Scaling soil-water suction, h

*1 1 2 2 ... n nh h h h

Capillary equation:1

pore radiush

Similar media in similar state:

Scaled suction head, h*, is the same for all similar media in similar state

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Scaling hydraulic conductivity, K

Poiseuille equation:

2pore radiusK

Similar media in similar state:

Scaled hydraulic conductivity, K*, is the same for all similar media in similar state

*1 22 2 2

1 2

...

n

n

KK KK

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Klute and Wilkinson (1958) tested Similar-Media Concept

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- Five similar media were made by sand particles

- Similarity was defined based on “shape of the particle size distributions”

- Mean particle size was used as “the physicals length scale” (scaling factor) of each soil

- Millers scaled h and K were calculated.

Identical porosity

104 125

2

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Scaled particle size distribution

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Unscaled retention curve Scaled retention curve

*h h

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Unscaled conductivity curve

Scaled conductivity curve

*2

KK

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Some conclusions found

1.Klute and Wilkinson (1958): Disagreement was apparent, particularly when the volumetric water content was greater than 0.3.

2. Elrick et al. (1959): Scaling theory worked well when the medium was clean sand, but much less well when the amount of colloid increased in the media.

3. Tillotson and Nielsen (1984): Application of scaling theory is restricted to use in sand or sandy soils.

Warrick et al. (1977) modifications

Art Warrick Don Nielsen

Owing to the fact that soils do not have

identical porosity, Warrick et al. (1977)

used “degree of saturation” (S = θ/θs)

rather than volumetric water content.

First

By this modification:

- Media do not need having identical

porosities for Scaling .

- Having identical degree of saturation

is enough for having “media in similar

state”.

There is no need to search for

“geometric similarity”.

Scaling factor can be obtained by a

least-square fitting to an average

curve.

Second

Assume r soils (locations) each having i

data points of retention curve, hr,i.

At a given degree of saturation, minimizing

following SS gives scaling factors (αr) of

each soil (location).

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, ,,

ˆr i r r i

r i

SS h h

Average curve Scaling factors

Individual curves

*rh h

201 1

ˆ 1 1 ... 1 nn

ah S S a S a S

S

Functional form of average curve:

This form was assumed for ease of mathematical derivations

Unscaled

Scaled

A similar procedure was followed for scaling hydraulic conductivity curves.

2

, ,,

ˆln 2 lnr i r r ir i

SS K K

Average curve Scaling factors

Individual curves

20 1 2

ˆln ... nnK S b b S b S b S

*2r

KK

Average curve:

*ln ln 2ln rK K

Unscaled

Scaled

Distribution of scaling factors was found to be

Log-normal.

- Scaling provides a tool for describing soil heterogeneity.

-The soil heterogeneity is approximated by a single stochastic parameter of scaling factor having a log-normal

distribution .

-Average soil hydraulic properties are described by the invariant scaled curves (the average

curves) .

- Scaling factors from K(s)

were not the same as those

calculated from h(s). But

they were highly correlated.

- Scaling factors from h(s)

showed less dispersion.

- Technology to measure K

is not developed to the

same degree as that for h.

Sadeghi and Ghahram

(2010) found a similar

result.

Sadeghi and Ghahraman (2010) introduced a Beta

parameter as:

2sK

Scaling factor from retention curve

Saturated hydraulic conductivity

They theoretically indicated that β must be the

same for all similar soils when simultaneous scaling

(equality of scaling factors from K and h data) is

expected.

Therefore,

- Similarity is “necessity” for validity of Millers

theory, but is not sufficient.

- Equality of β values gives the “sufficiency” for

this validity.

- This equality is related to the validity of capillary

and Poiseuille equations in real soils.

Simmons, C.S., D.R. Nielsen and J.W. Biggar. 1979. Scaling of field-measured soil-water properties. I. Methodology. II. Hydraulic conductivity and flux.

Hilgardia 47, 74-173.

Simmons et al. (1977) further developed a

scaling method.

They defined the “similarity” based on “shape

similarity of hydraulic functions”.

This definition helped Millers scaling theory to

be applied in reality.

The shape similarity can be easily investigated by

“shape parameters” in hydraulic models.

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s rr mn

h

For example, in van Genuchen model, n and m

are shape parameters in this model. Soils having

identical n and m would be called as “similar”

according to Simmons et al (1979).

For the purpose of scaling unsaturated flow, Simmons et

al. (1977) considered different scaling factors for h and K:

*

h

h

h

2

*K

K

K

For scaling unsaturated flow (e.g., scaling

Richards equation), equality of αh and αK is not

necessary. But, for describing soil variability,

the difference it is not desirable.

A similar idea of “linear variability concept” was

described by Vogel et al (1991).

1

s rr mn

h

Linear variability deals with variability only in

“scale parameters” (e.g., α, θs, and θr in van

Genuchten model).

Soils are scalable when their variability is linear .

Soils with nonlinear variability (e.g. different n and

m) are considered as “dissimilar soils .”

To scale unsaturated flow (Richards’ equation),

Vogel et al. (1977) considered different scaling

factors for h, K, and θ as:

* * * * * *

, , rh K

r

K h hh

h K h h

Scaling Richards’ Equation

Different methods have been proposed for scaling Richards’ equation: - Miller and Miller (1956)- Reichardt et al (1972)- Youngs and Price (1981)- Warrick and Amoozergar-Fard (1979)- Warrick et al (1985) - Kutilek et al (1991) - Vogel et al (1991)- Warrick and Hussen (1993) - Sadeghi et al (2011) - Sadeghi et al (2012a)- Sadeghi et al (2012b)- …

Four of these methods are introduced here, as representatives of different generations - Warrick et al (1985) - Kutilek et al (1991) - Warrick and Hussen (1993) - Sadeghi et al (2011) - Sadeghi et al (2012a)

Warrick et al. (1985)

Richards’ equation:

Scaled θ

Scaled timeScaled depth

Scaled conductivityScaled pressure head

Scaled diffusivity

Scaled Richards’ equation:

Scaled Hydraulic functions:

- Only n remains in the scaled RE and all other soil-dependent parameters (θr, θs, α and Ks) go out.

- Solution does not change by changing θr, θs, α and Ks.

- Soils having identical n may correspond “similar” soils of Millers.

Consider and infiltration process (the following IC and BC):

Philips’ Solution to the scaled form of RE:

- A, B, and C are functions of n and Wi (scaled initial water content).

A, B, and C were numerically calculated using the procedure of Philip (1968).

A, B, and C for van Genuchten functions.

A, B, and C for Brooks-Corey functions.

Comparing the solutions (points) with numerical solutions of Richards’ equation (line)

- Scaling provided a simple method for solving Richards’ equation.

- The solutions of Warrick et al. (1985) needs identical scaled initial and boundary conditions.

- To capture this limitations other methods were proposed.

Kutilek et al (1991)

Richards’ equation:

Initial and boundary conditions for a constant flux infiltration:

Proposed scaled variables:

q0: constant flux of infiltrationα, β, and ϒ: scaling constants

Soil hydraulic functions:

Scaled soil hydraulic functions:

Resulting scaled Richards’ equation:

For the following conditions, q0 goes out of the scaled RE (solutions get invariant with respect to infiltration flux):

Scaled Richards’ equation:

Invariant IC and BC:

Scaled solutions for three different q0 are the same.

Warrick and Hussen (1993) developed a more general method for constant-head and constant-flux infiltration and drainage from a wet soil column.

Warrick and Hussen (1993)

Richards’ equation:

Brooks-Corey soil hydraulic functions

θ0 was defined:

- to be soil water content (upper BC) in constant-head infiltration

- to be initial water content in drainage

- to give K(θ0) = q0, in the constant-flux infiltration (q0 is the constant flux).

Scaled variables:

where:

Scaled soil hydraulic functions:

Scaled Richards’ equation:

- Scaled BC and IC are invariant.

- Scaled RE depends only on v and m.

Scaled RE was solved for two different soils and different IC and BC.

m and v are identical (soils are similar)

Scaled results for constant-head infiltration

Scaled results for drainage

Scaled results for constant-flux infiltration

- Methods of Kutilek et al (1991) and Warrick and Hussen (1993) are limited to special form of Hydraulic functions.

- Sadeghi et al. (2011) developed a method in which all forms of hydraluc functions can be used.

Sadeghi et al. (2011)

Boundary conditions:

Initial conditions:

Redistribution process was assumed.

Scaled variables were defined based on initial conditions:

vfi is the initial velocity of the scaled wetting front movement:

An invariant scaled initial condition was obtained:

Richards’ equation was numerically solved considering van Genuchten functions.

Van Genuchten functions:

Twelve soils were considered.

Different initial conditions were assumed.

Scaled solutions were the same for medium- and fine-textured soils and different initial conditions.

- All the previous methods were proposed for similar soils. This limits application of these methods to real (dissimilar) soils.

- Sadeghi et al. (2012) developed a method for scaling Richards’ equation for “dissimilar soils”.

Sadeghi et al. (2012)

Richards’ equation:

Exponential-power hydraulic functions:

Constant-head infiltration and drainage processes were considered (following IC and BC):

For the drainage process, θ1 can be any arbitrary water content.

where K0 = K(θ0), D0 = D(θ0), and z0 is:

Scaled variables were defined as:

Scaled Richards’ equation:

Scaled hydraulic functions:

H1 = h(θ1)/h(θ0)

K*1 = K(θ1)/K(θ0)

D*1 = D(θ1)/D(θ0)

Scaled solutions are invariant when:

1 – D*1 is kept constant and flow regime is capillary-

dominant such as infiltration process.

2- K*1 is kept constant and flow regime is gravity-

dominant such as drainage process.

Four dissimilar soils (from sand to clay) were used for testing this method.

Scaled Richards’ equation was solved numerically.

Scaled solutions for infiltration

Effect of gravity

Scaled solutions for drainage