International Journal of Geosciences, 2010, 1, 149-154
doi:10.4236/ijg.2010.13019 Published Online November 2010 (http://www.SciRP.org/journal/ijg)
Ground Rupturing Due to Entrapped Air/Gas in the
Unconfined Zone
Manas Banerjee1*, Vimla Prasad Singh1, Hridaya Narain Singh1,
Daya Shankar2, Sunjay1, Uma Shanker Singh1
1Department of Geophysics, Banaras Hindu University, Varanasi, India
2Department of Earthquake Engineering, University of Roorkee, Roorkee, India
Received June 6, 2010; revised August 10, 2010; accepted August 30, 2010
Abstract
The sudden and large oscillation of pressure of compressed air/gas entrapped in porous medium due to the
changes in the actual pore-fluid pressure, during recharge of water following intense rainfall after a prolonged
period of dryness such that the rainfall intensity exceeding infiltration capacity, leads to the generation of
hydo-tremors. These hydro-tremors cause ground rupturing, subsidence, developments of cracks in the building,
etc. A theoretical model has been presented to estimate the successive values of compressed air/gas pressures
due to the successive development of actual pore-fluid pressures and effective stresses during recharge of water
of the unconfined zone during the onset of the summer monsoon of 2008 in the northern parts of India.
Keywords: Unconfined Zone, Compressed Air/Gas, Pore-Fluid Pressure, Hydro-Tremor, Ground Rupturing,
Effective Stress
1. Introduction
The phenomenon of hydro-seismicity is caused by the
changes in pore-fluid pressure during subsurface re-
charge of water following intense rainfall [1-6]. The re-
charge of water to a depth 2-6 km initiates hydro- seis-
micity caused by the hydrologic triggering of earthquake
activity on critically stressed faults [3,7,8]; such tectonic
earthquakes may be of magnitude Mw > 3.
The tremors which has magnitude Mw < 3 and con-
fined within the unsaturated zone is referred to here as
hydro tremors. The hydro-tremors have been explained
by a mechanism that takes into account the entrapped
air/gas in pore spaces of soil above water table which
gets compressed maximum due to the actual pore-fluid
pressure following heavy rainfall, and upon relaxation of
pore-fluid pressure (due to the horizontal diffusion of
near surface water), the pressure of the compressed
air/gas oscillates, and this causes hydro-tremors to gen-
erate. The instant at which the air/gas expands suddenly
due to the relaxation of compressive pore-fluid pressure,
it develops effective stress [9] along the horizontal direc-
tion during its escape from the capillary channels through
the soil surface, which ruptures the ground surface.
It has been observed that the increased difference of
energy of the compressed air/gas inside the soil produces
noisy sound energy during its escape. The associated
hydro-tremor generates elastic waves which shakes the
ground. These are the observed facts. The incidences of
ground rupturing have been found in areas, experiencing
depleted amount of rainfall over the years associated
with mining of subsurface water, and have been reported
from many parts of the northern states of India in the
initial phases of heavy rainfall during the onset of 2008
summer monsoon.
2. Materials and Method
2.1. Observation
The observed features of ground rupturing have been
shown in Figures 1(a-b). Figures 1(a-b) show the rup-
turing that occurred in the Kakori area of Lucknow dis-
trict (on 17.06.2008) and in the Cantonment area of Va-
ranasi district (on 25.06.2008) of Uttar Pradesh, India,
respectively. The rupture dimensions were 0.1 to 0.5 m
wide and 1 to 100 m long. During rupturing there was
ejection of warm air/gas and water which was associated
*Corresponding Author

150 M. BANERJEE ET AL.
with shaking of ground. Also, a loud sound was heard
during acoustic emission.
2.2. Analytical Model
It has been observed that after a prolonged period of
dryness during summer season, the unsaturated soil is
generally air/gas filled with the lowering of water table
to a larger depth (Figure 2). The flow of water during
recharge in such unconfined zone following a sudden
heavy rainfall can be represented by the one-dimensional
diffusion equation of the form:
2' '
2t
z





 (1)
(a)
(b)
Figure 1. (a) The rupture on the surface occurred on 17.06.
2008 in Kakori area of Lucknow district of Uttar Pradesh,
India; (b) The rupture of the surface occurred or 25.06.2008
in Cantonment area Varanasi district of Uttar Pradesh,
India.
With the boundary conditions, such as:
1) at z = 0;
'
0cos(2/ );tT
  
and
2
) '0


at z = ∞
where '

is the pore-fluid pressure variation, 0

is
the amplitude,

is the hydraulic diffusivity which is
expressed as:
()
s
at
s
K
S

  (2)
And is assumed constant (

= 0.30256 m2/s). ()
s
at
K

is
the saturated hydraulic conductivity for saturated water
content (
s
at

), and
s
S is the specific storage of the
saturated layer.
The solution of equation is written as:
'/
0
2
.cos(/ )
zT t
ez
T

T




 (3)
When there occurs intense rainfall for sometimes after a
prolonged period of dryness, a constant height of water
(a) stands at the ground surface (Figure 3). The unsatu-
rated soil profile starts getting recharged, and the moving
wet-front goes on compressing the filled air/gas of the
soil above the water table (Figure 3). The hydrostatic
pressure thus developed due to recharge of water can be
expressed as:
Figure 2. The unsaturated soil column is filled with air/gas
before recharge above the water table. The depth of water
table (L) is 40 m.
Figure 3. The compression of filled air/gas due to the re-
charge of soil under the constant height of standing water
(a) at the ground surface following intense rainfall.

M. BANERJEE ET AL.
151
).(
hw ag
g
zg Lz



  (4)
where: h

is the hydrostatic pressure; L is the depth to
the water table; z is the depth to which the wet-front has
moved;
 
2
LLz

  , a factor by which the density
(ag

) increases due to compression; w

and ag

are
the densities of water and air/gas, respectively; and g is
the acceleration due to the gravity.
The development of actual pore-fluid pressure (
f

)
after a substantial amount of time can be expressed as:
'
fh

 (5)
After dividing
f

by 0

, we can write the normal
value of the actual pore-fluid pressure as:
0
f
n



 (6)
where: 0

= 1.01325 × 105 pa. When the recharge
water moves downward towards water table, it goes on
compressing the air/gas till the pressure of the air/gas
(ag

) is balanced against the actual pore-fluid pressure
(
f

), such that agf

 . Now, the normal value of
the actual pressure of the air/gas can also be written as:
00
ag f
n




(7)
When rainfall ceases, the percolation and the horizontal
diffusion of water occurs this causes relaxation in the
hydrostatic pressure (Figure 4). The relaxation process
leads to the oscillation of the level of water, and ulti-
mately level falls. And, the relaxed pore-fluid pressure
rf

can be written as
'
rf hrh

  (8)
where:
(
rhw ag
gl )

  (9)
In Equation (9), is the depth to which water level
falls during relaxation in pore-fluid pressure. Now, if we
divide rf
l

by 0

and call it the normal relaxed
pore-fluid pressure '
n

which can be expressed as:
'
0
rf
n



 (10)
When the magnitude of rh

is negligible, '
nn



= 0; and when the magnitude of relaxation (rh

) in ac-
tual pore-fluid pressure is large, nn
'

  > 0, the
compressed air/gas expands and causes the rupture on
the ground surface according to a Mohr-Coulomb crite-
rion.
In terms of effective stress, we state that the actual
pore-fluid pressure
f

is equivalent to the effective
stress '
ij

in the Bishop’s stress equation (Equation (10)
of [9]). We can write their equation (for 1D problem by
taking 1


for saturation) and dropping kronecker
delta as
'
ii
u  
w
(11)
where: i

is the total stress (due to the hydrostatic
pressure) and w is the pore water pressure. Now, the
Equation (11) can be written for the total vertical (
u
'
v

)
and the total horizontal ('
h

) stresses as:
'
v
zu 
w

 (12)
And
'.
ho w
K
zu 

 (13)
where: 0
K
is the lateral earth coefficient ('
hv
'


),
which is constant at rest.
2.2.1. Effective Stress during Sudden Heavy Rainfall
As we have considered in the present treatment that
when an unsaturated soil becomes saturated due to sud-
den heavy rainfall, the effective vertical stress is due to
()
fag


and can be written as:
'
vh'



 (14)
And the effective horizontal stress can be expressed
as:
'
.
hoh
K'



 (15)
where: '

and h

are given by Equation (3) and
Equation (4), respectively.
At balanced stress state, when '
nn

 , the normal
stress is given as:
'' ''
cos 2
22
vh vh
  


(16)
And the shear stress is given as:
''
sin 2
2
vh




 (17)
Figure 4. The percolation and horizontal diffusion of water
occurs after the cessation of rainfall with the depletion of
water level.

152 M. BANERJEE ET AL.
2.2.2. Effective Stress upon Cessation of Rainfall
As stated earlier that when rainfall ceases, there occurs
relaxation in pore-fluid pressure and thus affecting the
effective stress. In this situation, the effective vertical
stress may be written by combining Equation (8) and
Equation (14) as:
''
vr v rh

  (18)
Similarly the effective horizontal stress is written by
combining Equation (8) with Equation (15) as:
'' .
hrhorh
K

   (19)
where: '
vr

and '
hr

are the effective vertical and ef-
fective horizontal stresses during relaxation in pore-fluid
pressure.
At relaxed stress state, when '
nn

  > 0, the nor-
mal stress is expressed as:
'' ''
cos 2
22
vr hrvr hr
r
 



 (20)
And the shear stress is given as:
''
sin 2
2
vr hr
r




 (21)
3. Discussion
The annual cycle of pore-fluid pressure, '

can be es-
timated by Equation (3) for z = 0 (the top boundary con-
dition). The normal pore-fluid pressure ('
0

 ) varia-
tion is shown in Figure 5 (for the period Jan - Dec). In
this condi tion the soil is extremely dry with an increased
depth of water table in the unsaturated zone. The strength
of the soil can be represented by the stress field (
s

) as:
0.
s
f
c

  (22)
where: 0

is the cohesive stress binding the soil parti-
cles together;

is the normal stress; and
f
c is the
frictional coefficient [10].
A soil being a poro-elastic medium, it holds water in
its pore spaces under saturation. And, intermittent filling
and draining of saturated water of soils are governed by
the phenomena called sorption and de-sorption under con-
dition of flux and no-flux of water, respectively, through
the soil surface. The extreme conditions of sorption and
de-sorption in nature occur during the rainy season and
summer season, respectively. It has been observed that
there occur ruptures on ground surface following re-
charge of soil under the condition of constant depth of
water standing at the surface due to heavy rainfall during
the initial phase of summer monsoon (in the northern
parts of India). The dimension (length x width) of such
surface ruptures are found to be in the range from (1 m ×
0.01 m) to (100 m × 0.50 m).
To explain such phenomenon of ground rupturing in
the initial phases of intense summer monsoon rainfall,
the model estimates the successive values of the com-
pressed air/gas pressures due to the successive develop-
ment of actual pore fluid pressures during the recharge of
water in the group of capillaries forming channels with
the constant height of water (a) at the surface of the un-
saturated soil. As recharge of water continues (Figure 3),
the compression of air/gas increases with the increase of
the hydrostatic pressure. The successive values of the
build up pressure are shown in Figure 6 for recharge of
water up to a depth of z when the downward movement
of recharge water ceases. The curve X corresponds to the
pressure of the compressed air/gas which is balanced
against the actual pore-fluid pressure (n
'
n

 , for
0
rh


) that developed during the wet-front movement
to a depth of z, not allowing the air/gas to escape from
the pores (Equation (7)). Here, according to Equation (7),
n

is the normalized pressure of air/gas now and '
n

is the normalized pressure of the pore-fluid during re-
laxation (0
rh


). The movement of water ceases when
Figure 5. The annual cycle of the normal porefluid pressure
versus time at the ground surface.
Figure 6. The variation of the normal actual porefluid
pressure (ψn) of the entrapped compressed air/gas with time.

M. BANERJEE ET AL.
153
the entrapped air/gas cannot be compressed further, a
critical stage is reached (i.e., when the condition
n
'
n

 
0
rh
> 0 is about to occur and any decrease in the
normal stress and increase in shear stress together with
decrease in shear strength will lead to rupture of the soil).
This occurs when the differential horizontal diffusion of
water of the surface and near surface pore spaces takes
place (Figure 4). The resulting hydrostatic pressure
starts fluctuating which leads the compressed air/gas to
oscillate till the percolation and the differential horizon-
tal diffusion continues. When the magnitude of relaxa-
tion in actual pore-fluid pressure is sufficiently large
(

), '
n

reduces. A plot of the normalized relaxed
pore-fluid pressure variation with time has been shown
in Figure 7. For a situation when the balance between
the relaxed hydrostatic pressure and the compressed
air/gas pressure breaks down, the condition n
'
n

  >
0 occurs, and the entrapped air/gas expands suddenly
which generates compressional waves at the surface. The
rupturing can be explained in terms of the stress law us-
ing the effective vertical (vr
'
&
v'

 ) and horizontal
(hr
'
&
h'

 ) stresses, the normal stress (

), and the ac-
tual shear stresses (&r

 ); where the shear stress (

)
and the shear stress (r

) are associated with balanced
stress state and relaxed stress state, respectively. When
r

 , the rupture occurs as the effective strength of the
soil reduces that can be expressed as:
0.( )
s
f
cag

  (23)
By combining the Equations (16), (17) and (22), we
obtain:
Figure 7. The normal pore-fluid pressure variation with
time after the cessation of rainfall.
''
.2
v
so f
ch




 (24)
Now combining the Equation (22) with Equations (18),
(19), (20) and (21), we get
''
.(1
2
vh
rsofrh o
cK

  )



 



 (25)
Or,
.(1 )
rrho
K



 (26)
The actual shear stress (r

) is now greater than the
shear stress (

). Therefore, during the process of escape
of expanding air/gas from the pore-channels of group of
capillaries through the ground surface (Figure 8), there
would be an increase in shear stress (Equation 26) and
decrease in normal stress r

(Equation 20) which leads
to the failure because of reduced strength of the soil
(Equation (23)). The failure according to the Mohr-Cou-
lomb criterion is depicted in Figure 9. In Figure 9, the
normal stress (Equation (16)) for '
v

and '
h

has been
shown in Mohr circle 1 and the shear stress is

. Since
the relaxation in actual pore-fluid pressure is by an amount
rh

which causes the effective vertical stress and the ef-
fective horizontal stress to decrease to '
vr

and'
hr

, re-
spectively. Now the normal stress r

(Equation (20))
for '
vr

and '
hr

has been shown in Mohr circle 2
which moves the circle closer to the Mohr-Coulomb
failure criterion line and now the shear stress is r

; and
as r

 , the rupture occurs. The rupture offset at the
ground surface has been shown in Figure 10. The two
Figure 8. The escape of expanding air/gas from the pore
channels of group of capillaries through the ground surface.
Figure 9. The relationship between the normal and shear
stresses and the Mohr-Coulomb failure envelope.

M. BANERJEE ET AL.
154
Figure 10. The rupture offset due to the ruptu
gures (Figure 8 and Figure 10) also depict the resu
ring of
ground surface.
fi ltant
force vectors due to the escaping compressed air/gas
which are perpendicular to the rupture offset boundaries.
The rupture offset is proportional to the magnitude of the
energy (E) released which is generally of the order of 107
J to 107.2 J. (The energy is calculated using the formula:
2.
w
EghA  ; where w

= 1000 kg/m3, g = 9.8 m/s2, h
= 100 . This is equivalent of energy
magnitude (Mw = 1.2 to 1.3). The energy magnitude is
calculated as(log5.24) /1.44
w
ME . It has been
observed that ith ejection of
warm air/gas and water, and sometimes, with subsidence
at places. And, during rupturing there generates the hy-
dro-tremors.
= 40 m and A
t
m2)
cehis pross is associated w
4. Conclusion
The mechanism of the proposed model could explai
he authors are thankful to the Computer Centre, B
n the [8] L
. W. Wolf, C. A. Rowe and R. B. Horner, “Periodic
Seismicity near Mt. Ogden on the Alska-British Colum-
bia Border: A Case for Hydrologically Triggered Earth-
quakes,” Bulletin of Seismological Society of America,
Vol. 87, 1997, pp. 1473-1483.
incidence of ground rupturing in the areas experiencing
depleted amount of rainfall over the years associated
with mining of subsurface water, and have been reported
from many parts of the northern states of India in the
initial phases of heavy rainfall during the onset of 2008
summer monsoon.
5. Acknowledgements
T ana-
ras Hindu University for providing needful facilities for
computational works.
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