A heat pulse technique for measuring water flow in soil

Tyson Ochsner

USDA-ARSSoil and Water Management Research Unit

St. Paul, MN

Application of the heat pulse technique

Upstream sensor

Heater

Downstream sensor

1 cm

The heat transfer occurs by conduction and convection.

Heat transfer equations

2

2

2

2

x

TV

y

T

x

T

t

T

C

CJV ww

where Jw is the water flux.

• The governing heat transfer equation is

• The solution of this governing equation leads to a simple relationship between water flux and the temperature increase ratio.

u

d

ww T

T

CxJ ln

0

We tested the technique in the lab using packed columns of sand, sandy loam, and silt loam soil.

Laboratory experiments

Time (s)

0 20 40 60 80 100

Td

/ Tu

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.2 0.40.91.72.65.39.617.727.037.2

Heat pulse signals converted to Td/Tu

The temperature increase ratio increases with flow rate.

Sand

Flux (cm h-1)

u

d

ww T

T

CxJ ln

0

Flux measured at outlet (cm h-1)

0 10 20 30 40

Est

ima

ted

flu

x b

ase

d on

ln T

d / T

u (c

m h

-1)

0

10

20

30

40SandSandy loamSilt loam

Resulting water flux estimates

Strong linear relationship between flux measured by the sensor and that measured at the column outlet.

Soil typeSlope

(S)Intercept

Std. Err.

r2

cm hr-1 cm hr-1

Sand 0.765 0.605 0.390 0.998

Sandy loam 0.232 0.452 0.125 0.998

Silt loam 0.400 2.88 0.071 0.999

Linear regression results

Regression results indicate good precision and linearity, but the slopes are less than one.

u

d

ww T

T

Cxb

aJ ln

0

Sb

a 1

abVx

T

T

u

d 0expx

TbV

y

T

x

Ta

t

T

2

2

2

2

Linking theory and measurements

Choosing a = 1/S and b = 1 leads to an “enhanced conduction” model.

Choosing a = 1 and b = S leads to a “reduced convection” model.

Original theory Enhanced conduction Reduced convection

Silt loam

16.1 cm h-1

Time (s)

0 20 40 60

Tem

pera

ture

Inc

reas

e (K

)

0.0

0.4

0.8

1.2

Time (s)

0 20 40 60

Sandy loam

36.5 cm h-1

Tem

pera

ture

Inc

reas

e (K

)

0.0

0.4

0.8

1.2

Time (s)

0 20 40 60 80

Sand

37.2 cm h-1

Tem

pera

ture

Inc

reas

e (K

)

0.0

0.4

0.8

1.2

Td

Tu

Modeled Td

Modeled Tu

Results

• The smallest flux detected with the heat pulse technique was 0.12 cm hr-1.

• The relationship between heat pulse flux estimates and the outflow flux was linear up to 40 cm hr-1.

• The standard heat transfer model over-predicted the sensitivity of the heat pulse sensor to water flux.

• A reduced convection model accounted for the discrepancies between the measured data and the standard model.

Closing statements

• “Uncorrected” heat pulse measurements are fairly accurate in sand.

• Theoretical over-predictions of instrument response to convective heat transfer are common.

• A reliable procedure for correcting heat pulse measurements is needed.

• The heat pulse approach for measuring soil water flux warrants further development.

Acknowledgements

• Collaborators are Robert Horton, Gerard J. Kluitenberg, and Quanjiu Wang.

• This work was conducted at Iowa State University.