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