Soil Steady-State Evaporation
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Transcript of Soil Steady-State Evaporation
An Analytical Solution to Soil Steady-State
EvaporationMorteza Sadeghi
Utah State UniversityFerdowsi University of Mashhad
Nima ShokriBoston University
Scott B. JonesUtah State University
Motivation
Soil Evaporation
a significant component of water
cycleaffects energy exchange between land and
atmosphere
Water tableSurface water
Unsaturated soil
Ground water
Having a shallow water table, a sustained water loss
will occur from soil evaporation.
So, quantifying evaporation in the presence of a
water table is considered as an important issue.
1 – Near surface water table (Phase one):
Steady State Evaporation
Liquid water flows along the soil profile.
Vaporization occurs at the soil surface.
2- Deeper water table (Phase two):
Liquid water flows up to a “drying front”,
and vaporizes at the drying front.
Vapor moves toward the surface by diffusion.
Suction
z
Gas Region
Film Region
Drying Front
Saturated Region
Wat
er ta
ble
dept
h (D
)
Liqui
d flo
w re
gion
(Dm
ax)
Air-entry
( ) 1dhe K hdz
Darcy’s law:
Conductivity
Suction head gradient
When D < Dmax (phase one), evaporation rate is high.
When D > Dmax (phase two), evaporation rate significantly decreases due to the hydraulic discontinuity between water table and soil surface.
So, a knowledge of Dmax seems to be so
important in water resources management.
Analytical solutions have been developed using:
Gardner function
1 /s
Pb
KK
h h
( )
/ ( > )s b
Ps b b
K h hK
K h h h h
Brooks-Corey function
K: Unsaturated conductivity Ks: Saturated Conductivityh: suction headhb : Air-entry suction headP: Shape parameter
Literature Review
Gardner [1958] developed a solution for Gardner
function only for integer values of P.
Warrick [1988] developed exact solutions for all non-
integer P, for both Gardner and Brooks-Corey functions.
The solutions were obtained in terms of an incomplete Beta
function and a hypergeometric function. They were not
closed-form.
Salvucci [1993] introduced closed-form approximate solutions for Gardner function. The solutions are not accurate for fine-textured soils.
we develop an exact solution to steady-state evaporation.
In this research:
We approximate the exact solution into a closed-
form (i.e., excluding integrals or series).
Mathematical Derivations
Kz dhK e
z: depth to water tableK: Unsaturated Conductivityh: suction heade: evaporation ratehb : Air-entry suction headhe : h (K=e) hDF : h at the Drying front
/ ( < )/ ( < )
b e
e DF
T e K h h hU K e h h h
Applying Brooks-Corey model for K(h):
Darcy:
Defining variables:
( )
/ ( > )s b
Ps b b
K h hK
K h h h h
Maclaurin series expansion for |x| < 1 as (1 – x)-1 = 1 + x + x2 + x3 +…
1
1
1
1 / ( )
1 / ( < )1
1 / ( < )1 1
b
e
b e
s b
h
s b eh
h h
s e DFh h
e K h h h
dhz e K h h h hT
dh Udhe K h h h hT U
1
1 2
1 2
1 / ( )
1 / 1 ... ( < )
1 / 1 ...
b
s b
h
s b eh
s
e K h h h
z e K h T T dh h h h
e K h T T dh
2 3 ... ( < )e
b e
h h
e DFh hU U U dh h h h
Mathematical Derivations
1
1 1 1
0 0
1 1
1 ( )
1 / 1 /1 ( < )
1 1
1 /1
bs
n iP n iPe b e
b e e b ei is
i iPe
e e
e h h hK
h h h hez h h h h h hK iP iP
h hz h h
i
1
1 1
1 ( < )
1
i
e e DFi i
h h h hP iP
Suction head distribution above the water table as a function of hydraulic properties and evaporation rate
Mathematical Derivations
Exact Solution
1
1
1 ( )
ln(1 )1 ln 1 ( / ) ( < )
1 1
ln(1 )
1 1
bs
Ps sb b b e
s
s
s sb
e h h hK
e eK K eh h P h h h h h heP K
Kze e
K Kh ePK
1/ 2
1
ln 2 /12 ln 2 ln 211 1 1
1 ln 1 ( / ) ( < )
P
s
s
Psb e DF
eK P P P P
KP h h h h h he
Closed-form Solution
Mathematical Derivations
21/
max
ln(1 ) ln 2ln 2 ln 212 11 1 1 11
P
s sb
s
s
e eK K eD h eP K P P P P
K
Suction
z
Gas Region
Film Region
Drying Front
Saturated Region
Dm
ax
Dmax = F(e, Ks, P)
Evaporation rate
Saturated conductivity
Power of BC function
Suction head distribution
Results & Discussions
h/hb 0.001 0.01 0.1 1 10 100
z/h b
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Warrick [1988], Brooks-Corey K(h)Warrick [1988], Gardner K(h)New solution, ExactNew solution, ApproximateSalvucci [1993] h = hb h = he
A clayey
soil
Results & Discussions
Suction head distribution
h/hb 0.001 0.01 0.1 1 10 100
z/h b
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Warrick [1988], Brooks-Corey K(h)Warrick [1988], Gardner K(h)New solution, ExactNew solution, ApproximateSalvucci [1993]
A loamy
soil
Results & DiscussionsSuction head distribution
h/hb 0.01 0.1 1 10 100
z/h b
0.0
0.5
1.0
1.5
2.0
Warrick [1988], Brooks-Corey K(h)Warrick [1988], Gardner K(h)New solution, ExactNew solution, ApproximateSalvucci [1993]
A sandy
soil
Dmax (cm), Exact solution0 50 100 150 200
Dm
ax (
cm),
App
roxi
mat
e so
luti
on
0
50
100
150
200ChinoPachappa1.02 mm0.48 mm 0.16 mmcoarse sand fine sandsilt
Results & Discussions
Liquid flow region
D/Dmax 0 1 2 3 4 5
e/e 0
0.0
0.2
0.4
0.6
0.8
1.0
ChinoPachappa1.02 mm0.48 mm 0.16 mmcoarse sand fine sandsilt
D = Dmax
When D > Dmax, evaporation rate decreases significantly due to hydraulic discontinuity.
Results & Discussions
A closed-form analytical solution to Darcy’s law has been developed during steady-state evaporation.
The solution closely matches the exact
solution for a wide range of soil texture.
This solution can be used for directly modeling the steady-state evaporation or for inversely determining the Brooks-Corey parameters.
Conclusions
For more Details read:
Sadeghi, M., N. Shokri, and S.B. Jones. 2012. A
novel analytical solution to steady-state
evaporation from porous media. Water
Resources Research. W09516.
Thanks
for your attention