Download - Scaling in Soil Physics
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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.
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Warrick et al. (1977) modifications
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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
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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”.
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There is no need to search for
“geometric similarity”.
Scaling factor can be obtained by a
least-square fitting to an average
curve.
Second
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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).
2
, ,,
ˆr i r r i
r i
SS h h
Average curve Scaling factors
Individual curves
*rh h
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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
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Unscaled
Scaled
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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
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Unscaled
Scaled
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Distribution of scaling factors was found to be
Log-normal.
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- 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) .
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- 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.
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Sadeghi and Ghahram
(2010) found a similar
result.
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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.
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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.
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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.
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The shape similarity can be easily investigated by
“shape parameters” in hydraulic models.
1
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).
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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.
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A similar idea of “linear variability concept” was
described by Vogel et al (1991).
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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 .”
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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
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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)- …
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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)
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Warrick et al. (1985)
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Richards’ equation:
Scaled θ
Scaled timeScaled depth
Scaled conductivityScaled pressure head
Scaled diffusivity
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Scaled Richards’ equation:
Scaled Hydraulic functions:
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- 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.
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Consider and infiltration process (the following IC and BC):
Philips’ Solution to the scaled form of RE:
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- 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).
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A, B, and C for van Genuchten functions.
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A, B, and C for Brooks-Corey functions.
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Comparing the solutions (points) with numerical solutions of Richards’ equation (line)
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- 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.
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Kutilek et al (1991)
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Richards’ equation:
Initial and boundary conditions for a constant flux infiltration:
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Proposed scaled variables:
q0: constant flux of infiltrationα, β, and ϒ: scaling constants
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Soil hydraulic functions:
Scaled soil hydraulic functions:
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Resulting scaled Richards’ equation:
For the following conditions, q0 goes out of the scaled RE (solutions get invariant with respect to infiltration flux):
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Scaled Richards’ equation:
Invariant IC and BC:
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Scaled solutions for three different q0 are the same.
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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)
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Richards’ equation:
Brooks-Corey soil hydraulic functions
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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).
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Scaled variables:
where:
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Scaled soil hydraulic functions:
Scaled Richards’ equation:
- Scaled BC and IC are invariant.
- Scaled RE depends only on v and m.
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Scaled RE was solved for two different soils and different IC and BC.
m and v are identical (soils are similar)
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Scaled results for constant-head infiltration
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Scaled results for drainage
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Scaled results for constant-flux infiltration
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- 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)
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Boundary conditions:
Initial conditions:
Redistribution process was assumed.
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Scaled variables were defined based on initial conditions:
vfi is the initial velocity of the scaled wetting front movement:
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An invariant scaled initial condition was obtained:
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Richards’ equation was numerically solved considering van Genuchten functions.
Van Genuchten functions:
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Twelve soils were considered.
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Different initial conditions were assumed.
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Scaled solutions were the same for medium- and fine-textured soils and different initial conditions.
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- 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”.
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Sadeghi et al. (2012)
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Richards’ equation:
Exponential-power hydraulic functions:
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Constant-head infiltration and drainage processes were considered (following IC and BC):
For the drainage process, θ1 can be any arbitrary water content.
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where K0 = K(θ0), D0 = D(θ0), and z0 is:
Scaled variables were defined as:
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Scaled Richards’ equation:
Scaled hydraulic functions:
H1 = h(θ1)/h(θ0)
K*1 = K(θ1)/K(θ0)
D*1 = D(θ1)/D(θ0)
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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.
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Four dissimilar soils (from sand to clay) were used for testing this method.
Scaled Richards’ equation was solved numerically.
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Scaled solutions for infiltration
Effect of gravity
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Scaled solutions for drainage