Displacement Mechanism of the Two-Phase Flow Model for Water

Displacement Mechanism of the Two-Phase Flow Model for Water
and Gas Based on Adsorption and Desorption in Coal Seams
CHEN Shikuo, YANG Tianhong, WEI Chenhui
Center for rock instability and seismicity research Northeastern University, Shenyang, 110819
[email protected]
Abstract: On the basis of seepage theory of gas flow in coal seams, the mechanics of fluids in porous
media and the continuum mechanics, a continuum theory of multiphase porous media is developed to
analyze the displacement mechanism of the two-phase flow in a porous medium saturated by a mixture
of two immiscible, viscous, compressible fluids. The water injection induced the gas content changes,
gas adsorption/desorption, the associated characteristics of relative permeability, water injection
efficiency and the gas content changes in coal seam during this process were numerically simulated. The
numerical simulations are helpful for understanding the displacement mechanism of the two-phase flow
and taking effective measures to prevent the occurrences of gas outbursts in collieries.
Keywords: Two-phase flow, Relative permeability, Numerical simulation, Adsorption and desorption
1 Introduction
Having studied on two-phase flow in porous media follows and the flowing law of the gas in the coal
seams, the gas outburst can be prevented by water injection and gas outburst hazard can be effectively
relieved. It is very complicated of the displacement mechanism of the two-phase flow for water and gas
through this process. Gas migration in coal seams includes adsorption and desorption, diffusion and
seepage, together with the dynamic coupling relationship between stress and seepage field [1-8], a
mathematical model for two-phase flow model for water and gas in coal seams was established.
Furthermore, the displacement mechanism of the two-phase flow model for water and gas based on
adsorption and desorption in coal seams will be getting in the research to give some scientific advice to
coal mining and prevention of coal and gas outburst.
2 Governing Equations and Boundary Conditions
Two-phase flow in porous media follows separate equations for the wetting (w) and non-wetting (nw)
fluids. Assuming that the porous medium is non-deformable, the wetting fluid is incompressible and that
cross-product permeability terms associated with the viscous drag tensor can be neglected, the general
form of the two-fluid flow equations is described by the two-fluid, volume-averaged momentum and
continuity equations.
K k

∂(φSew )
(1)
− ∇ ⋅ int r , w ⋅ (∇p + ρ g∇D) = 0
∂t
Vi


w
w
 ηw

K int k r ,i
=−
⋅ (∇pi + ρ i g∇D )
ηi
(2)
where ϕ is the total porosity or saturated volume fraction; Sew is effective saturation function; t is time
(s); Vi (i = w, nw)is Darcy’s velocity of gas phase (m·s-1); Kint is the intrinsic permeability of the porous
medium (m2); kr,i is the relative permeability function for a given fluid; ηi is the fluid’s dynamic
viscosity (kg/(m·s)); pi is pressure (kg/(m·s2)); ρi is the fluid density (kg/m3); g is acceleration of gravity;
and D is the coordinate (for example, x, y, or z) of vertical elevation (m).
∂m
+ ∇ ⋅ (ρ g q g ) = Q p
∂t
(3)
Under isothermal conditions, the gas flow in porous media is governed by a mass balance equation,
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where m is the coal seam gas content (kg·m-3), ρg is gas density (kg·m-3), qg is Darcy’s velocity of gas
phase (m·s-1), Qp is source term (kg·m-3·s-1), and t is time (s).
For gas as the non-wetting fluid, it is furthermore assumed that its density is a linear function of the
pressure head, or
ρ
∂p
∂ρ nw
λ
(4)
ρ nw = ρ 0,nw + ( 0, nw )hnw
=
⋅ nw
h0
ρ w g w ∂t
∂t
where ρ0, nw and h0 are the reference density and pressure head (at atmospheric pressure), respectively.
The ratio ρ0, nw/h0 is defined as the compressibility λ (λ=1.24e-6 g/cm4) [9]. Subsequently, using eqn (4),
and writing water potential in terms of pressure head yields the flow equation for the non-wetting fluid.
The material is composed of a solid matrix which contains interstitial pore space filled with a freely
diffusing pore gas. The absorption or desorption of gas may occurs when the gas pressure and porosity
of coal seam are changed. The methane content reserved in the rock matrix can be described by the
Langmuir’s equation:
 φSe nw
aa ρ 
(5)
m = ρ nw 
+ 1 2 s  p nw
1 + a 2 p nw 
 p0
where m is the methane content (kg·m-3), Φ is porosity, pnw is gas pressure (Pa), p0 is unit atmospheric
pressure (Pa), which is 101325Pa, ρs is the density of coal (kg·m-3) , a1 and a2 are Langmuir’s constants
with units of kg·m-3 and Pa-1, respectively. Actually, this equation defines the sorption and desorption of
gas from coal seam under changing gas pressure, where the sorption isotherm defines the relationship
between gas content and pressure only.
The gas flow follows Darcy’s flaw, and so the Darcy’s velocity of the gas phase is defined as eqn (2).
Substituting Equations (2), (4) and (5) into Equation (3), we have

 φSe
φSenw
aa ρ
p λ  ∂p
aa ρ
a1 a 22 ρ s
+ 1 2 s −
⋅ p nw  + ( nw + 1 2 s ) ⋅ nw  nw
 ρ nw 
2
p0
1 + a2 p nw ρ w g w  ∂t
1 + a2 pnw (1 + a 2 p nw )


 p0
+
φρ nw p nw ∂ ( Senw )
p0
∂t

− ∇ ⋅  ρ nw

K int k r ,nw
µ nw
(6)

⋅ (∇p nw + ρ nw g nw ∇D ) = 0

If the fluid distribution is continuous, neither fluid ever completely fills the coal seam, giving a volume
fraction for the wetting phase, θw, and non-wetting phase, θnw, at all times. For the wetting phase, θ
varies from zero or a small residual value θr to the total porosity, θs. The effective saturation, Se, comes
from scaling θ with respect to θs and θr and so varies from 0 to 1. Both θ and Se are functions of the
pressures of all fluids in the system [5-7, 10-12]. We define capillary pressure: Pc=Pnw-Pw, The pore space
can be completely filled with one fluid at a given time: Sew+Senw=1, θw+θnw=1. How effective saturation
changes with capillary pressure, therefore, is
∂Sew
(7)
=φ
C = −C
p,w
p , nw
∂Pc
where C is the specific capacity, and the subscript “p” denotes units of pressure.
C p,w
K k

∂
(Pnw − Pw ) − ∇ ⋅  int r ,w ⋅ (∇pw + ρ w g∇D ) = 0
∂t
η
w


(8)

 ∂p
 φSe nw
 φSenw
aa ρ
a1 a 22 ρ s
aa ρ
λ
+ 1 2 s −
⋅ p nw  + (
+ 1 2 s )⋅
p nw  nw
 ρ nw 
2
p0
1 + a 2 p nw ρ w g w
1 + a 2 p nw (1 + a 2 p nw )

 ∂t (9)
 p 0

K k


ρ p
∂
− nw nw C p , w ( Pnw − Pw ) − ∇ ⋅  ρ nw int r , nw ⋅ (∇p nw + ρ nw g nw ∇D ) = 0
p0
∂t
µ


nw
We can solve this system of equations simultaneously for pw and pnw. In this example, the water is
incompressible, but the gas is compressible. Initially, the water and gas in the coal seam follow
hydrostatic distributions. The boundary conditions allow the gas to exit only from the boundary of the
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coal seam. Because water enters at the coal seam boundary but never exits, the boundary conditions for
the wetting phase and the non-wetting phase are
 K int k r , nw

⋅ (∇p nw + ρ nw g∇D ) = 0
n ⋅ −
η
nw


pnw = pnwr
∂Ω
Outlet
pw = pw (t )
∂Ω
∂Ω
Sides
(10)
Inlet
where n is the unit vector normal to the boundary.
The existing model uses retention and permeability relationships from Ref. 3, 4 and 6 that express
changes in θ, C, Se, and kr as a function of pc. Because pc is large and because changes in θ, C, Se, and
kr are small, these expressions transform capillary pressure to the equivalent height of water or capillary
pressure head as in Hc = pc / (ρw·g-1). The hydraulic properties relative to the wetting fluid are
Hc > 0
θ r , w + Se w (θ s , w − θ r , w )
θ w= 
θ s,w
Hc ≤ 0

1


Se w =  1+ | αH c | n

1

[
Hc > 0
]
m
Hc ≤ 0
1
1 m
 αm



m  1 − Se w m 
θ
θ
(
)
−
Se
w
r ,w
C p, w = 1 − m s , w



0
2

1 m

Se w L 1 − 1 − Se w m  
k r ,w = 
 
 

1

(11)
Hc > 0
Hc ≤ 0
Hc > 0
Hc ≤ 0
where α, n, m, and L are the van Genuchten parameters that denote soil characteristics. Note that with
two-phase flow, the van Genuchten-Mualem formulas hinge on the value of Hc.
θ nw= θ s ,w − θ w
Se nw = 1 − Se w
(12)
C p ,nw = −C p ,w
2m
1


k r ,nw = (1 − Se w ) L 1 − Se w m 


For the non-wetting fluid, the properties arise naturally from the definitions for the wetting phase.
3 Numerical Model and Calculation Scheme
3.1 Model Descriptions
Coal seam water injection in large-scale full-mechanized caving is an effective means to prevent gas
outburst. The flowing process of two immiscible fluids in coal seams is very complicated. This paper
describes the mechanism of the coal seam infusion: it is the synthetical result of the numerator pervasion,
the capillary dint and the pressure of infusion. The incoming water forces the gas toward the outlet at the
right boundary of the coal seam. At the inlet, water pressure increases by steps in time, and only saturate
gas exits through the coal seam. Neither the gas nor the water can pass through the vertical coal seam
walls. The gas pressure around the water injection hole, which changes in time, corresponds to the
injecting rate. The model has a total length of 20m, and 3m thick. The test covers about 1 hour.
According to the process of actual coal seam water injection in Yangquan coal mine, a 2D numerical
model is established (Fig. 1(a)). On the assumption that the left boundary of model is working face and
middle line is the water injection hole. It is 13m and the front 3m is sealed borehole length which
prevents the water to flow to the working face. Because of boundary constraint of seal section is difficult
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working face
to control and not the focus of our attention in the analysis, so we have a simplified model (Fig.1 (b))
instead of the first one.
hole sealing
water injection hole
hole sealing
(a)
full calculating model
water injection hole
(b)
simplified model
Fig.1 Numerical model
(1) Boundary conditions: set the working face, upper surface and under surface as zero flux/symmetry
boundary. Geostatic stress affected the permeability is taken into accounted.
(2) Initial conditions: There is initial gas pressure of 2Mpa within the model and the water pressure
increases by steps in time and maximize to 15Mpa.
(3) Time steps: It is increased by non-equal series steps: the initial value and terminating value is 0.1h
and 2h separately.
(4) Calculating Parameters: As shown in table 1.
water density
ρw
(kg/m3)
1000
viscosity of the
water
(Pa·s)
1e-3
Table 1 The relevant parameters of model
gas density
viscosity of the Seam density
ρnw
gas
ρs
(kg/m3)
(Pa·s)
(kg/m3)
0.716
1.81e-4
1400
absolute
permeability
(mD)
3.75
3.2 Analysis of Numerical Simulation Results
As stated above, Darcy’s velocity, momentum equation and mass equation need to be coupled together.
Two phases, wetting and non-wetting were coupled by capillary pressure which is the pressure head
difference of these two phases. This discussion lays out the two-phase flow simulation in the following
sections. The relative permeability curve was shown in Fig.2.
1
1
kr,nw
tyi 0.8
li
ba
em 0.6
re
p
ev
it 0.4
al
er
eh 0.2
t
0
0
0.2
0.4
0.6
water saturation
kr,w
y 0.8
t
i
l
i
b
a
e
m
0.6
r
e
p
e
v
i
0.4
t
a
l
e
r
e
0.2
h
t
0.8
1
0
0
0.2
Fig. 2 The relative permeability curve
600
0.4
0.6
water saturation
0.8
1
Fig. 3 show gas saturation distributions curves at different time. It is obvious that the range of gas
pressure/saturation was changed evidently with water injecting. It is found that the gas saturation
appeared a peak at the gas-water interface and the gas migrates to the boundary with the passage of
water injection time, gas content decreased and gas pressure increased accordingly.
Fig. 4 shows that gas saturation/pressure distribution with time. It is obvious that water injection is a
greatly efficient prevention of coal and gas outburst. At the time of 1h, the gas content is effectively
reduced, initial reservoir pore space were occupied by water and velocity of gas around gas-water
interface is fast due to the local increasing gas pressure. With respect to the condition of water pressure
stepwise loading, range of action is different slightly and the water/gas mass velocity is changed
correspondingly.
(a) Gas pressure vs. vertical direction
(b) Gas saturation vs. vertical direction
Fig. 3 Gas pressure/saturation distributions curves with time
t=0.2h
t=0.5h
t=0.8h
t=1.0 h
t=1.2h
Fig. 4 Gas saturation/content distribution with time
3.3 Discussion
In this paper, gas adsorption, desorption, compression and fully coupling mechanism between stress,
permeability and seepage are investigated. The relations between relative permeability, saturation and
gas content can be established through the eqns of (8) and (9), we have a complex coupled equations
about two phase flow for water and gas based on adsorption and desorption in coal seams, furthermore,
the two coupled equations show the change of water injection leads to the change of water/gas relative
permeability and gas content consists of free and desorption, respectively. But complete desorption is
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improbable, residual gas is taken into account; however, the residual gas content is not constant and
changed at different gas pressure. The effective method is to use a modified system of Langmuir’s
equation or double-porosity model.
4 Conclusion
Making use of this numerical model, the two-phase immiscible flow for water and gas based on
adsorption and desorption in coal seams is conducted and the evolvement of gas content/pressure at the
different water injection time as well as the changing law of gas-water interface around the water
injecting hole is analyzed. It is found that gas-water interface migrates to the boundary with the passage
of water injection time, gas content decreased and local gas pressure increased accordingly and initial
reservoir pore space were occupied by water and velocity of gas around gas-water interface is fast due to
the local increasing gas pressure. A better understanding of the displacement mechanism of the
two-phase flow model for water and gas based on adsorption and desorption in coal seams has an
important theoretical and practical significance to the coal mining and prevention of coal and gas
outburst and controlling measures.
Acknowledgements:
The work presented in this paper was financially jointly supported from the General Project of the
National Natural Science Foundation of PR China (Grant No. 50504005, 50490274, 50674025, 107033
and 10872046), National Basic Research Program of China (Grant No. 2005CB221503,
2006CB202201-6, 2006CB202204-2), the Creative Team Construction Projects of NEU of China (No.
90101001) and the Doctor Programs Foundation of NEU of China (No. 90601003)..
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