Vol. 48, 01, 05ok.qxd


195

ANNALS OF GEOPHYSICS, VOL. 48, N. 1, February 2005

Key  words  method – radon – soil – convective
velocity – radon flux density

1. Introduction

A knowledge of certain radon transport
characteristics (depth distribution function
A(z) of the soil gas radon concentration, radon
flux density q(z)|z = 0 from the Earth’s surface,
convective radon flux velocity υ in soil, etc.) is
essential in solving a number of radioecologi-
cal and geophysical problems. Radon transport
in soil is described by the well-known diffu-
sive-convective equation (Nazaroff and Nero,
1988). The solution to this equation is an ex-
ponential soil radon concentration distribution
with depth. The exponential coefficient varies
with the physical-geological soil parameters
and weather conditions. The latter affect the
convective radon flux velocity in soil.

It has been verified experimentally (Fleischer,
1997; Abumurad and Al-Tamimi, 2001; Jönsson,
2001) that the depth distribution of the soil gas
radon concentration obeys the exponential law in
the case of a relatively homogeneous geological
structure and a great depth of occurrence of water-
bearing horizons. Given the radon concentration
distribution function we can readily determine the
following parameters: equilibrium soil gas radon
concentration A∞, characterizing the radon poten-
tial of a given area, depth at which the equilibrium
concentration is found, soil gas radon concentra-
tion gradient specifying the radon flux density ac-
cording to Fick’s law, and convective velocity.

The central problem is to find experimental-
ly the function A(z). Reconstruction of the verti-
cal profile of the soil gas radon concentration re-
quires that measurements be performed at differ-
ent depths. The number of measurements varies
with prescribed accuracy. The measurements can
be very expensive and difficult to perform. 

However, the number of radon concentration
measurements can be reduced to two measure-
ments, using the properties of the exponential law. 

The measurements should be performed at
shallow depths (≤ 1 m deep), because here the
gradient is very step.

Mailing address: Dr. Valentina S. Yakovleva, Tomsk
Polytechnic University, pr. Lenin 30, Tomsk, 634050 Russia;
e-mail: jak@interact.phtd.tpu.edu.ru

A theoretical method for estimating 
the characteristics of radon transport 

in homogeneous soil

Valentina S. Yakovleva
Tomsk Polytechnic University, Tomsk, Russia

Abstract
A theoretical method for estimating the characteristics of radon transport in homogeneous soil is developed. The
method allows the following characteristics to be estimated: depth distribution function of the soil gas radon con-
centration, equilibrium radon concentration in the soil air, depth at which the radon concentration reaches its
equilibrium value, radon flux density from the Earth’s surface, and convective radon transport velocity. The
method is based on soil gas radon concentration measurements and is appropriate in the case of relatively uni-
form geology.



196

Valentina S. Yakovleva

In this work, a method for estimating the
radon transport characteristics in soil is devel-
oped. The approach under review is based on the
above-mentioned diffusive-convective radon
transport model and in situ radon concentration
measurements at two depths.

2. Methodology

Solving the stationary diffusive-convective
radon transport equation in the quasi-homoge-
neous approximation, we will get a depth distri-
bution of the radon concentration in the soil air
(Jönsson, 1997) for the z-axis directed down-
ward from the Earth’s surface. Thus,

D D D2 2e e e
3

+ +
y m y

2

z-

( )A z A= exp1 -

J

L

K
KK

_b
N

P

O
OO

i l
(2.1)

where A(z) is the radon concentration per unit
volume of the soil air (Bqm–3), υ is the convec-
tive radon flux velocity (ms–1), De is the effec-
tive radon diffusion coefficient (m2s–1), and λ is
the radon decay constant (s–1).

The equilibrium soil gas radon concentra-
tion depends solely on the physical-geological
soil parameters, and we have

A
K A 1em sRa

=
-

h
t h

3

^ h
(2.2)

where Kem is the radon emanation coefficient (rel.
units), ARa is the specific activity of 226Ra (Bqkg–1),
ρs is the solid soil particle density (kgm–3), and η
is the soil porosity (rel. units).

Let us denote the soil gas radon concentra-
tion measured at a depth h1 by A1 and that meas-
ured at a depth h2 = 2h1 by A2. Substituting A1
and A2 into eq. (2.1), we will arrive at the fol-
lowing equation

A
1-

A
A

A

2

ln
h A

1

2

1

1 1

1

1

2

-
( ) expA z 1= -

z-

J

L

K
K

J

L

K
KK

J

L

K
K
K
K
K

N

P

O
O

N

P

O
OO

N

P

O
O
O
O
O

. (2.3)

It is evident from eq. (2.3) that the equilibrium

soil gas radon concentration generally found at
a great depth and characterizing the soil radon
potential (Yakovleva, 2002) can be estimated
from as few as two measurements near the
Earth’s surface. Thus we obtain (Yakovleva and
Ryzhakova, 2002)

A

A
A

A

2
1

2

1=
-

3 . (2.4)

The depth (zeq) at which A∞ is found is deter-
mined by introducing the parameter (rel. units)
X A z Aeq= 3_ i . The parameter specifies the de-
gree to which the soil gas radon concentration ap-
proaches its equilibrium value. For example, with
X = 0.95, the soil gas radon concentration at the
depth sought will be only 5% lower than its equi-
librium value. Then we can find zeq from the fol-
lowing equation

( )
.

ln

ln
z h

A
A

X

1

1
eq 1

1

2

=
-

-

c m

(2.5)

The radon flux density from the Earth’s sur-
face is defined by the following relation
(Ryzhakova and Yakovleva, 2002)

( )
( )

ln

q z D
z

A z

D

A
A

A
h

A
A

2

1

1

1

z e

e

0

1

2

1

1

1

2

$ $ $

2

2
= - =

=
- -

h

h

=

J

L

K
K
KK

^

N

P

O
O
OO

h

(2.6)

and the convective radon flux velocity is ex-
pressible as

ln
ln

h
D

A
A

A
A

h

1

1

1

e

1

1

2

1

2

1=
-

+
-

y
m

J

L

K
K
KK c

N

P

O
O
OO m

. (2.7)

The radon flux density and convective velocity
can be determined by eqs. (2.6) and (2.7). To
this end, we need to know the radon diffusion
coefficient in addition to two measured values
of the soil gas radon concentration. The choice
of the diffusion coefficient presents no special



197

A theoretical method for estimating the characteristics of radon transport in homogeneous soil

problems. For the majority of sedimentary rocks
constituting the surface layer, the diffusion coef-
ficient varies within a small range and is, on av-
erage, 0.03 cm2s–1 (Durrani and Ilić, 1997).

The measurements of A1 and A2 should be
performed concurrently (by means of any con-
ventional devices and techniques) at two points
spaced 0.5 – 1 m apart. There is a limitation on
the maximum separation between the two
measuring points (∼1 m). This is due to the fact
that the soil properties at the measuring points
should be the same. A minimum point separa-
tion of 0.5 is needed to avoid a possible influ-
ence on the results of the two measurements.
Moreover, measurements for a smaller point
separation present some technical problems. 

It is recommended that both of the measure-
ments should be performed at depths between
0.3 and 1 m for the following reasons: i) the soil
gas radon concentration varies comparatively
rapidly at these depths, which enables us to re-
duce the error in determination of the function
A(z), ii) the depth h1 should not be smaller than
0.3 m because of a great influence of atmos-
pheric conditions, which reduces the reliability
of the results obtained, and iii) an increase in
the measurement depth above 1 m would not be
economically attractive.

The method under review is applicable for
areas with a relatively homogeneous geological
structure. In the case of radon anomalies (rocks
with a high content of uranium, large fractures
in the Earth’s crust, etc.), the method will be not
suitable.

3. Preliminary results of practical evaluation
of the method under review

The method was tested in a small survey
area with a homogeneous geological structure
(surface soil layer is loam). The area is locat-
ed in Lagernii sad (camp garden) in Tomsk
(West Siberia, Russia). Two holes spaced 0.5
m apart were drilled by a customized soil
auger. One hole was 35 cm deep (h1), and the
other was 70 cm deep (2h1). The hole diameter
was 5.5 cm.

Radon radiometers with track etch detectors
of LR-115 type III-b (Nikolaev and Ilić, 1999)

were placed in the holes. Then the holes were
covered to provide air-tightness and allowed to
stay for 72 h. The soil gas radon concentration
(A1 and A2) was determined as directed by op-
erating instructions for the AIST–TRAL com-
plex. The etching and track counting methodol-
ogy are described in Nikolaev et al. (1993).

The measured soil gas radon concentrations
A1 and A2 were 6.8 and 11.4 kBqm–3. The equi-
librium radon concentration A∞ calculated by
eq. (2.4) was 21.0 kBqm–3. This value is twice
as high as the measured value A2 at a depth of
70 cm, which is usually recommended for
radon concentration measurements. 

We have also estimated A∞ by eq. (2.2) to get
20 kBqm–3. To this end, soil samples were taken,
and their density, porosity and 226Ra specific ac-
tivity were determined (Karataev et al., 2000;
Yakovleva, 2002). The radon emanation coeffi-
cient was taken to be 0.2. The values of A∞ calcu-
lated by eqs. (2.2) and (2.4) agree very closely. 

The depth at which the soil gas radon con-
centration accounts for 95% of its equilibrium
value is 2.7 m. The radon flux density from the
Earth’s surface is 33.8 kBqm–2s–1, and the con-
vective flux velocity is 1.7·10–4 cms–1.

4. Concluding remarks

We have developed a method for estimating
the radon transport characteristics in soil. The
approach under review has the following prac-
tical benefits: i) versatility since only two meas-
urements of the soil gas radon concentration are
needed to determine a number of radon trans-
port characteristics; ii) validity for any conven-
tional devices and techniques used for measur-
ing the soil gas radon concentration; and iii)
low cost since it requires neither a large number
of measurements to determine the function A(z)
nor detailed information on the physical-geo-
logical soil parameters. 

This method is useful in different fields of
applied research such as: radioecology, for im-
proving the reliability of potential soil radon
risk estimates and for reducing the weather
conditions effect on results of soil radon con-
centration monitoring; geophysics, for studying
the convective gas flow velocity; etc.



198

Valentina S. Yakovleva

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