Supplementary Discussion
1. Derivation of airflow solution in Laplace domain
For convenience, the governing equations, as well as the boundary and initial conditions,
are transformed into dimensionless forms using the definitions as follows:
PD1 
3 
2
2
2
2
2
2
 t
z
P1  P0


P2  P0
P3  P0
;
;
; t D  02 ; z D  ; 1  1 ;  2  2 ;
P

P

D2
D3
2
2
2
B
0
0
B
P0
P0
P0
B
k
B
 3 B1
k
 b1 ; 2  b2 ; 3  b3 ; b1  b2  b3  1 ;   1 ;   3 .
;
B
B
k2
0 B
k2
(S-1)
Therefore, the governing equations are transferred into
1 PD1  2 PD1

,
2
1 t D
z D
(S-2)
1 PD 2  2 PD 2

,
2
 2 t D
z D
(S-3)
1 PD 3  2 PD 3

,
2
 3 t D
z D
(S-4)
and the boundary and initial conditions are
PD1
z D
PD1 z
PD 2
PD 3
0,
(S-5)
z D 0
D b1
 PD 2
z D b1  b2
z D 1
z D b1
 PD 3
and 
z D b1  b2
PD1
z D
and

z D b1
PD 2
z D
PD 2
z D
,

z D  b1  b2
PD 3
z D
 f D (t D ) ,
PD1  PD 2  PD 3 t
D 0
(S-6)
z D b1
,
(S-7)
z D  b1  b2
(S-8)
 0,
(S-9)
where the subscript “D” denotes the dimensionless terms; PD is the dimensionless squared air
pressure; b1, b2 and b3 are the dimensionless thicknesses of the deep, middle and shallow
1
layers, respectively; 1 ,  2 ,  3 ,  and are defined in Eq. S-1; f D (t D ) is the dimensionless
squared atmospheric pressure fluctuations.  0 in Eq. S-1 is an arbitrary reference air diffusivity.
Without losing generality, one can choose the middle layer as a reference to calculate  0 . For
instance, for P0=105 Pa, k2 =10-12 m2,  2  0.5 , 2  0.5 , and   1.8  105 Pa·s, one has
 0  2.22  10 2 m2/s.
With the Laplace transform upon the dimensionless equation group Eqs. (S-2) to (S-4),
the airflow governing equations in Laplace domain can be expressed as
~
d 2 PD1
s ~

P ,
2
 1 D1
dz D
(S-10)
~
d 2 PD 2
s ~

P ,
2
2 D2
dz D
(S-11)
~
d 2 PD 3
s ~

P ,
2
3 D3
dz D
(S-12)
~
~
~
where the PD1 , PD 2 and PD 3 are the dimensionless subsurface pressure in Laplace domain
and s is the Laplace parameters.
The boundary conditions are transformed into Laplace domain accordingly
~
dPD1
dz D
~
PD1
~
PD 2
~
PD 3
 0,
(S-13)
z D 0
z D b1
~
 PD 2
z D b1  b2
z D 1
z D b1
~
 PD 3
~
dPD1
and 
dz D
z D b1  b2
and
z D b1
dPD 2
dz D
~
dPD 2

dz D
,

zD b1  b2
(S-14)
z D b1
dPD 3
dz D
~
 f D ( s) ,
,
(S-15)
z D b1  b2
(S-16)
~
where f D ( s ) is time-dependent function of the atmospheric pressure fluctuations in Laplace
domain.
2
Combining above boundary conditions, the solution to airflow in Laplace domain can be
obtained
 s 
~
PD1  C1  cosh 
z D  ,

 1 
(S-17)
 s

 s

~
PD 2  C2  cosh 
z D   C3  sinh 
z D  ,
 2 
 2 
(S-18)
 s 
 s 
~
PD 3  C4  cosh 
zD   C5  sinh 
z D  ,
 3 
 3 
(S-19)
where C1, C2, C3, C4 and C5 are unknown parameters in the solution, which can be determined
based on the boundary conditions in Laplace domain.
Substituting Eqs. (S-17) and (S-18) into the continuity condition in Eq. (S-14), one
obtains
æ s ö
æ s ö
æ s ö
C1 × cosh ç
b1 ÷ - C2 × cosh ç
b1 ÷ - C3 ×sinh ç
b1 ÷ = 0 ,
è b1 ø
è b2 ø
è b2 ø
C1 
(S-20)
 s 
 s 
 s 
s
s
sinh 
b1   C2 
sinh 
b1   C3 
cosh 
b1   0 . (S-21)
1





2
2
 1 
 2 
 2 
s
Substituting Eqs. (S-18) and (S-19) into the continuity condition in Eq. (S-15), one
obtains
3




 s



b1  b2   C3  sinh  s b1  b2   C4  cosh  s b1  b2   C5  sinh  s b1  b2   0 , (S-22)
C2  cosh 
 2

 2

 3

 3





 s



b1  b2   C3  s cosh  s b1  b2   C4  s sinh  s b1  b2   C5  s cosh  s b1  b2   0 .
sinh 
2
2
3
3
 2

 2

 3

 3

s
C2 
(S-23)
Substituting Eq. (S-19) into the upper boundary condition in Eq. (S-16), one obtains
 s 

 ~
  C5  sinh  s   f D ( s ) .
C 4  cosh 



 3 
 3 
(S-24)
The constants C1 to C5 are obtained correspondingly as
C1 

M
 b 
 b 
2 ~
 22
 ~  2
f D ( s)
cosh  A1  cosh  A2 1   
sinh  A1  sinh  A2 1  ;
f D ( s) ; C2 
M
1 3
3
  3
 b2 
 b2 
C3 
 
 b 
 b 
 22
f D ( s)  2 cosh  A1 sinh  A2 1   
sinh  A1 cosh  A2 1  ;
M
1  3
  3
 b2 
 b2 
C4 
 

1 ~
 22

f D ( s)  2 cosh  A1 cosh  A2 cosh  A4   
sinh  A1 sinh  A2 cosh  A4   cosh  A1 sinh  A2 sinh  A4    2 sinh  A1 cosh  A2 sinh  A4  ;
M
13
1
 3

C5 



 22
2
1 ~
f D ( s)   2 cosh  A1  cosh  A2  sinh  A4   
sinh  A1  sinh  A2  sinh  A4   cosh  A1  sinh  A2  cosh  A4   
sinh  A1  cosh  A2  cosh  A4  ;
M
3
1  3
1


 ~
where
 

 22
2
M   2 cosh  A1 cosh  A2 cosh  A3   
sinh  A1 sinh  A2 cosh  A3   cosh  A1 sinh  A2 sinh  A3   
sinh  A1 cosh  A2 sinh  A3  ;
1  3
1
  3

4
A1  b1
s
1
;
A2  b2
s
2
;
A3  b3
s
3
;
A4  b1  b2 
s
3
. Therefore, the
solution to airflow in layered unsaturated zone is obtained by the substitution of C1 to
C5 into Eqs. (S-17) to (S-19).
2. Verification of the solution
In this section, a comparison between the solution developed in section 2 and a
numerical simulation in a three-layer unsaturated zone is demonstrated in order to
justify the solution derivation as well as the accuracy of the numerical inverse Laplace
transform. The numerical simulation is conducted in COMSOL Multiphysics using
the built-in partial differential equation (PDE) mode, which runs the finite element
analysis together with adaptive meshing and error control. The airflow in the
comparison is one-dimensional vertical airflow in a three-layer unsaturated zone
depicted in Fig. S-1. The parameters used in this comparison are listed in Table S-1.
Meanwhile, two kinds of atmospheric pressure fluctuations are employed as the
boundary condition at the ground surface. One is a single-component sinusoidal
function with diurnal cycle and the other one is a summation of diurnal sinusoidal
function and semidiurnal sinusoidal function. The two kinds of atmospheric pressure
fluctuations are as following
f (t )  P0  A1 sin( 1t ) ,
(S-25)
f (t )  P0  A1 sin( 1t )  A2 sin( 2t ) .
(S-26)
The amplitudes and frequencies of the diurnal and semidiurnal sinusoidal functions
are listed in Table S-2.
In the numerical simulation, the three layers are vertically divided into 166
5
elements. The element density in the layer near the ground surface is designed denser
than those in the other two layers. The reason for such element configurations is to
capture the details of pressure changes near the ground surface due to atmospheric
pressure fluctuations. Besides, we have doubled and tripled the elements in the
vertical coordinate, but no obvious improvement in accuracy is observed in the
numerical simulation.
The pressure evaluations at three different depths deviating from P0 under two
kinds of atmospheric pressure fluctuations are plotted in Figs. S- 2 (a) and (b),
respectively. The three depths are z =5, 15 and 25 m which are located in the shallow,
middle and deep layers respectively. In Figs. S-2 (a) and (b), it is apparent that the
pressure evaluations using the solution developed in this study agree well with those
in the numerical simulation using COMSOL Multiphysics. Thus, the derivation of
solution based on Laplace transform shows adequate accuracy to calculate the
subsurface pressure in layered unsaturated zones. Meanwhile, the solution can be used
under flexible boundary conditions at the ground surface, which can be a
representation of atmospheric pressure using superposition of sinusoidal functions.
3.
Convergence of the Fourier series analysis
In this section, convergence of the Fourier series analysis on the discrete air
pressure data is checked. For convenience, the discrete air pressure data are
transformed into a summation of a series of harmonic functions through the Fourier
series analysis. For instance, the harmonic functions have the frequency of 2n T ,
where n=1, 2, 3…… and T is a reference period of the basic sinusoidal function.
6
These harmonic functions have decreasing amplitudes with increasing frequencies.
Fig. S-3 is a plot of the leading five harmonic functions representing the atmospheric
pressure fluctuations corresponding to one barometric pumping test reported in Shan
(1995) (with the deep observation point of 24.5 m). With the increase of frequency,
the amplitude of each individual mode decreases. In Fig. S-4, summations containing
different numbers of harmonic functions are plotted, indicating that the accuracy of
data representation increases with the number of terms included. For the atmospheric
pressure fluctuations discussed in this study, approximation using the leading 100
terms of the Fourier series is sufficiently accurate. Thus, the discrete pressure data can
be well represented by the Fourier series summation with proper numbers of the
harmonic functions.
Reference
Shan, C.: Analytical solutions for determining vertical air permeability in unsaturated
soils. Water Resour Res 31(9), 2193-2200 (1995)
7
Tables
Table S-1 Default parameters for the three-layer unsaturated zone.
Parameters
Value
Unit
B1
10
m
Thickness of the deep layer
B2
10
m
Thickness of the middle layer
B3
10
m
Thickness of the shallow layer
B
30
m
Total thickness: B=B 1+B 2+B 3
k1
5×10-13
m2
Air permeability in the deep layer
k2
-12
2
Air permeability in the middle layer
2
Air permeability in the shallow layer
Porosity in three layers
Air filled saturation in three layers
Air dynamic viscosity
k3
1×10
-13
φ
8×10
0.4
0.5
μ
1.8×10
P0
1×10
ε
-5
5
m
m
Pa·s
Pa
Definition
Mean atmospheric pressure
8
Table S-2 The amplitudes and frequencies of the diurnal and semidiurnal sinusoidal
functions for atmospheric pressure fluctuations.
Parameters
Value
Unit
Definition
A1
500
Pa
Amplitude of diurnal component in
atmospheric pressure fluctuations
A2
250
Pa
Amplitude of semi-diurnal component in
atmospheric pressure fluctuations
ω1
2π/86400
s-1
Frequency of diurnal component in
atmospheric pressure fluctuations
ω2
4π/86400
s-1
Frequency of semi-diurnal component in
atmospheric pressure fluctuations
9
Figures
Fig. S-1
Schematic diagram of vertical air flow in a three-layer unsaturated zone.
10
Atmospheric pressure fluctuations
Numerical simulation in COMSOL
Solution in this study
0.6
0
Pressure deviation from P (KPa)
0.8
0.4
z=5 m
0.2
z=15 m
0
z=25 m
-0.2
-0.4
-0.6
0
12
24
36
Time (hour)
Fig. S-2 (a)
11
48
60
72
0
Pressure deviation from P (KPa)
1
Atmospheric pressure fluctuations
Numerical simulation in COMSOL
Solution in this study
0.8
0.6
z=5 m
0.4
z=15 m
0.2
z=25 m
0
-0.2
-0.4
-0.6
-0.8
0
12
24
36
48
60
72
Time (hour)
Fig. S-2 (b)
Figure S-2 Comparison of subsurface pressure at different depths evaluated using the
solution in this study and numerical solution in COMSOL Multiphysics (a) the
atmospheric pressure is a single sinusoidal function with diurnal cycle; (b) the
atmospheric pressure is a summation of a diurnal sinusoidal function and a
semidiurnal sinusoidal function.
12
Harmonic functions
n=1
n=2
n=3
n=4
n=5
0
0.5
1
1.5
t
2
D
Fig. S-3
The leading five harmonic functions in the Fourier series analysis of the atmospheric
pressure fluctuations corresponding to the pressure data at the deep observation point
of 24.5 m in Shan (1995).
13
0.015
0.01
PD
0.005
0
-0.005
Field data
n=10
n=50
n=100
-0.01
-0.015
0
0.5
1
1.5
t
2
D
Fig. S-4
Atmospheric pressure fluctuations approximated through the Fourier series analysis
with different numbers of harmonic function terms. These atmospheric pressure
fluctuations are corresponded to the pressure data at the deep observation point of
24.5 m in Shan (1995)
14