In this presentation, an exact analytical solution to steady state evaporation from porous media is introduced. The solution is presented in terms of a set of infinite series. An advantage of this solution compared to previous derivations is that the infinite series can be very closely approximated using a closed-form solution (i.e., excluding integrals or series).

1. An Analytical Solution to
Soil Steady-State
Evaporation
Morteza Sadeghi
Utah State University
Ferdowsi University of Mashhad
Nima Shokri
Boston University
Scott B. Jones
Utah State University

2. Motivation

3. Unsaturated soil
Surface water
Water table
Ground water

4. Steady State Evaporation
1 – Near surface water table (Phase one):
2- Deeper water table (Phase two):

5. Darcy’s law:
e
Drying Front
Film
Region
z
Saturated
Region
Suction
Air-entry
Liquid flow region (Dmax)
Water table depth (D)
Gas Region
dh
K ( h)
1
dz

6.  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.

7. Analytical solutions have been developed using:
Gardner function
Ks
K
1
h / hb
P
Brooks-Corey function
K
K: Unsaturated conductivity
Ks: Saturated Conductivity
h: suction head
hb : Air-entry suction head
P: Shape parameter
Ks
K s h / hb
(h
P
hb )
(h > hb )

8. Literature Review

9. In this research:

10. Mathematical Derivations
Darcy:
K
dh
K e
z
z: depth to water table
K: Unsaturated Conductivity
h: suction head
e: evaporation rate
hb : Air-entry suction head
he : h (K=e)
hDF : h at the Drying front
Defining variables:
T
e/ K
(hb <h he )
U
K /e
(he <h hDF )
Applying Brooks-Corey model for K(h):
K
Ks
K s h / hb
(h
P
hb )
(h > hb )

11. Mathematical Derivations
1
1 e / Ks
z
1
1 e / Ks
1
1 e / Ks
h
(h hb )
dh
hb 1 T
he dh
hb 1 T
h
h
h
( hb< h he)
h
he
Udh
1 U
(he < h hDF )
Maclaurin series expansion for |x| < 1 as
(1 – x)-1 = 1 + x + x2 + x3 +…
1 e / Ks
z
1 e / Ks
1 e / Ks
1
1
1
h
h
h
(h hb )
h
hb
he
hb
1 T T 2 ... dh
1 T T
2
... dh
(hb <h he )
h
he
U U 2 U 3 ... dh (he <h hDF )

12. Mathematical Derivations
Exact Solution
1
e
1
Ks
z
h
1
e
1
Ks
z he
(h hb )
hb
1
he
n
h / he
i 0
1
he
i 1
1 iP
h / he
1 iP
1
he
1 iP
i 1
1 iP
i 0
1
he
i 1
n
hb / he
1 iP
(hb <h he )
1 iP
i 1
1 iP
(he <h hDF )
Suction head distribution above the water
table as a function of hydraulic properties and

13. Mathematical Derivations
Closed-form Solution
1
e
1
Ks
h
hb
e
e
)
Ks
Ks
e
1 P
1
Ks
hb
e
e
ln(1
)
Ks
Ks
e
1 P
1
Ks
(h
ln(1
z
1 P
1
h
1 P
e
Ks
Ks
h ln 1
(h / hb )
e
P
1/ P
1
e
h ln 1
(h / hb ) P
Ks
ln 2
1 P
hb )
(hb <h
he )
2
/12 ln 2
ln 2
1
P 1 P
1 P
(he <h
hDF )

14. Dmax
hb
e
e
ln(1
)
Ks
Ks
e
1 P
1
Ks
2
e
Ks
1/ P
Gas Region
Film Region
z
Saturated
Region
Suction
Dmax
Drying Front
ln 2
1 P
ln 2
12
P 1 P
ln 2
1
1 P

15. Results & Discussions
Suction head distribution
3.0
2.5
z/hb
2.0
1.5
Warrick [1988], Brooks-Corey K(h)
Warrick [1988], Gardner K(h)
New solution, Exact
New solution, Approximate
Salvucci [1993]
h=h
h = he
b
1.0
0.5
0.0
0.001
0.01
0.1
1
h/hb
10
100

16. Results & Discussions
Suction head distribution
3.0
2.5
z/hb
2.0
1.5
Warrick [1988], Brooks-Corey K(h)
Warrick [1988], Gardner K(h)
New solution, Exact
New solution, Approximate
Salvucci [1993]
1.0
0.5
0.0
0.001
0.01
0.1
1
h/hb
10
100

17. Results & Discussions
Suction head distribution
2.0
z/hb
1.5
1.0
Warrick [1988], Brooks-Corey K(h)
Warrick [1988], Gardner K(h)
New solution, Exact
New solution, Approximate
Salvucci [1993]
0.5
0.0
0.01
0.1
1
h/hb
10
100

18. Results & Discussions
Dmax (cm), Approximate solution
Liquid flow region
200
Chino
Pachappa
1.02 mm
0.48 mm
0.16 mm
coarse sand
fine sand
silt
150
100
50
0
0
50
100
150
Dmax (cm), Exact solution
200

19. Results & Discussions
When D > Dmax, evaporation rate decreases
significantly due to hydraulic discontinuity.
1.0
D = Dmax
Chino
Pachappa
1.02 mm
0.48 mm
0.16 mm
coarse sand
fine sand
silt
0.8
e/e0
0.6
0.4
0.2
0.0
0
1
2
D/Dmax
3
4
5

20. Conclusions

21. For more Details read: