J. Cent. South Univ. (2013) 20: 1928−1937
DOI: 10.1007/s11771-013-1692-7
Mechanism model for shale gas transport considering diffusion,
adsorption/desorption and Darcy flow
WEI Ming-qiang(魏明强)1, DUAN Yong-gang(段永刚)1, FANG Quan-tang(方全堂)1,
WANG Rong(王容)1, YU Bo-ming(郁伯铭)2, YU Chun-sheng(于春生)1
1. State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University),
Chengdu 610500, China;
2. School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China
© Central South University Press and Springer-Verlag Berlin Heidelberg 2013
Abstract: To improve the understanding of the transport mechanism in shale gas reservoirs and build a theoretical basic for further
researches on productivity evaluation and efficient exploitation, various gas transport mechanisms within a shale gas reservoir
exploited by a horizontal well were thoroughly investigated, which took diffusion, adsorption/desorption and Darcy flow into account.
The characteristics of diffusion in nano-scale pores in matrix and desorption on the matrix surface were both considered in the
improved differential equations for seepage flow. By integrating the Langmuir isotherm desorption items into the new total
dimensionless compression coefficient in matrix, the transport function and seepage flow could be formalized, simplified and
consistent with the conventional form of diffusion equation. Furthermore, by utilizing the Laplace change and Sethfest inversion
changes, the calculated results were obtained and further discussions indicated that transfer mechanisms were influenced by diffusion,
adsorption/desorption. The research shows that when the matrix permeability is closed to magnitude of 10−9D, the matrix flow only
occurs near the surfacial matrix; as to the actual production, the central matrix blocks are barely involved in the production; the closer
to the surface of matrix, the lower the pressure is and the more obvious the diffusion effect is; the behavior of adsorption/desorption
can increase the matrix flow rate significantly and slow down the pressure of horizontal well obviously.
Key words: shale gas; diffusion; adsorption/desorption; transport mechanism; horizontal well
1 Introduction
Recently, interest in developing shale gas has grown
tremendously. Along with the economic success comes
the complexity of understanding shale gas transport
mechanism. Many scholars have been working on
investigation and exploration of the shale gas transport
mechanism. However, most of the studies only
considered single impact factor that affects the gas
tranport, so the understanding of the shale gas transport
mechanisms remains limited.
Based on the results of shale core experiment,
BUMB and MCKEE [1] found that the adsorption and
desorption follow the Langmuir isothermal adsorption
theory, and by integrating the Langmuir equation into
total matrix compression coefficient, they derived the
differential equation for single-phase gas seepage flow in
the shale gas well and obtained the solution of pressure
drop in the condition of production quotas. FATHI et al
[2] pointed out that the output process of gas from
low-permeability shale formations could be divided into
three stages, including desorption stage, diffusion stage
and Darcy flow stage. LUFFEL et al [3] developed three
laboratory methods to measure matrix gas permeability
(km) of Devonian shale cores and drill cuttings, and the
experiment results show that km ranges from about 0.1 to
10−8 mD. GAO and JOHN [4] were the first to consider
the adsorption and desorption of shale gas in differential
equations with seepage flow and obtained the analytical
solution based on the multi-layer shale gas well test.
Later on, the researches demonstrated that the diffusion
may be another feature which affects the gas flow,
especially in low-permeability reservoirs [5−7]. Thus, in
the study of low and ultra-low permeability gas reservoir,
the diffusion should not be ignored. Researches of shale
gas transport suggested that the primary mechanisms of
shale gas flow include Knudsen diffusion, the slip flow
in nano-scale pores, Darcy flow in large pores and
micro-fractures and the desorption on the matrix surface
[8−9]. BEYGI and RASHIDI [10] proposed an analytical
solution for the unsteady flow in homogeneous tight gas
Foundation item: Project(PLN1129)supported by Opening Fund of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest
Petroleum University), China
Received date: 2012−04−08; Accepted date: 2012−08−25
Corresponding author: DUAN Yong-gang, Professor, PhD; Tel: +86−28−83037532; E-mail: [email protected]
J. Cent. South Univ. (2013) 20: 1928−1937
reservoirs by taking the effect of gas diffusion into
account. Later, FREEMAN et al [11−12] used the
dusty-gas model to invest microscale flow behavior in
tight gas and shale gas reservoir systems and identified
the potential for using changes in measured gas
composition over time to measure effective flow
boundaries. Furthermore, OZKAN et al [13] developed a
dual-mechanism model for fluid transfer from shale
matrix to fracture network and presented a new transfer
function. Their model takes multiple mechanisms into
account, including diffusion and Darcy flow in the
matrix and the stress sensitivity in the fracture. Based on
physical simulation of slippage effect, GAO et al [14]
established a gas well productivity formula considering
both artificial fracturing of gas wells and gas slippage
effect in a pay zone, and analyzed the influence degree of
slippage coefficient on gas well productivity and
pressure drop at different formation pressures. Later,
based on the scanning electron microscopy image and
drainage experiment of shale, SAKHAEE-POUR and
BRYANT [15] analyzed the effects of adsorbed layers of
CH4 and gas slippage at pore walls on the flow behavior
in individual conduits of simple geometry and in
networks of such conduits, and pointed out that at high
pressure such as typical initial shale gas reservoir
pressures, the effect of the adsorbed layer dominates the
effect of slip on gas phase permeability, while slip
dominates the flow at lower pressures typical of those
after longer periods of production. Numerous studies
showed that the main difference between production
from shale gas reservoir and conventional natural gas
reservoir is the existence of adsorption/desorption and
diffusion flow. Most of the current studies are basically
restricted to a single factor which influences the shale
gas transport, therefore, the shale gas transport and
output mechanisms are still not well understood.
Thus, in this work, the dual-mechanism model
established by OZKAN et al [13] is improved and a
novel triple-mechanism model for shale gas transfer in
the shale matrix is proposed. The new model is coupled
with the shale gas Darcy flow, diffusion flow and
desorption from the matrix surface. By integrating the
Langmuir isotherm desorption items into the new total
dimensionless compression coefficient in matrix, the
transport function and seepage flow can be formalized,
simplified and consistent with the conventional form of
diffusion equation. Furthermore, the impacts of gas
diffusion and adsorption/desorption on the flow rate in
matrix and horizontal well-bore pressure drop are
analyzed and discussed by using the analytical solution
of pressure response of the horizontal well. It is helpful
to improving the understanding of the transport
mechanism in shale gas reservoirs, providing a
1929
theoretical basic for further researches on productivity
evaluation and efficient exploitation.
2 Apparent permeability of diffusion
Gas diffusion is an important transport mechanism
in low permeability gas reservoirs. The lower the
permeability is, the stronger the diffusion influence is
[5−6]. It has been shown that the shale gas reservoirs fall
into unconventional gas reservoirs with dual-porosity
and ultra-low permeability. Therefore, a dual porosity
medium with spherical matrix blocks of uniform radius,
rw, and the uniform surface of matrix block are assumed
in this work. By adopting the theory of diffusion effect as
well as the approach taken by ERTEKIN et al [16], who
defined the total radial component of the flow velocity
(νrm) in matrix as the sum of Darcy radial flow velocity
(νprm) and the gas slip velocity (νsrm), we have
vrm  vprm  vsrm
(1)
The Darcy radial flow velocity caused by the
macro-pressure gradient is given by
vprm 
k m pm
(
)
 g r
(2)
where km is the matrix permeability, μm2; pm is the
pressure in matrix, 0.1 MPa; r is radial distance in
spherical coordinates, cm; μg is the gas viscosity, mPa·s.
The gas slip velocity caused by the concentration
gradient in molecular diffusion flow is given by
vsrm 
M g Dg  Cm 


 g  r 
(3)
where Mg is the relative molecular mass of gas, g/mol;
Dg is the diffusion coefficient of gas in matrix, cm2/s; Cm
is the molar concentration of gas.
Inserting Eqs. (2) and (3) into Eq. (1), the equation
can be obtained as follows:
vrm 
k m  p m  M g Dg  Cm 




 g  r 
 g  r 
(4)
According to the gas equation of state, the molar
concentration of single-phase gas can be written as
Cm 
g
Mg

pm
zRgT
(5)
where z is the gas deviation factor, dimensionless; Rg is
gas constant, 8.314 472 cm3·MPa/(K·mol); T is reservoir
temperature, K.
Then, Eq. (4) can be rewritten as
vrm 
k m  pm  M g Dg  ( pm /z ) 




 g  r   g RgT  r 
(6)
J. Cent. South Univ. (2013) 20: 1928−1937
1930
Coupled with the gas equation of state, Eq. (6) can
be rewritten as
vrm 
k m  pm 
 p 

c D  m  
 g  r  g g  r 
k m   g cg Dg
1 
 g 
km
 pm 

 r 

(7)
According to ERTEKIN et al [16], the apparent gas
slippage factor is defined as
bam 
Dg  g cg p m
(8)
km
Fig. 2 Schematic of dual-porosity medium model with
spherical shape [16]
And the absolute permeability of the matrix is given by
matrix, and the gas slippage factor at the interface is
given by
 b
k am  k m 1  am
pm

0, r  rm
bam (r )  
bm , r  rm



(9)
The shale matrix permeability is generally in the
orders of magnitude of nD [9]. So, the error
((kam−km)/km×100%) can be obtained by using Eqs. (8),
(9) and (11). Figure 1 shows the matrix permeability
error, which is caused by ignoring the diffusion flow for
the relative density of 0.7, at temperature of 373.15 K
and the molar mass of 16 g/mol of methane gas. As can
be seen from Fig. 1, the lower the pressure and the
permeability are, the more the contribution of diffusion
flow is. Therefore, for extraordinarily low permeability
reservoirs, diffusion is an important transport mechanism
that has a great effect on the production of shale gas.
In this work, the formula proposed by ERTEKIN
et al [16] is used to calculate the diffusion coefficient Dg
(m2/D):
Dg 
The existence of slippage velocity is due to the
concentration gradient in the matrix. While the
concentration gradient at the interface between matrix
and fracture does not exist, the slip velocity is thus
approximated to be zero at the interface. Figure 2 shows
the schematic of a dual-porosity medium model with
spherical shape [16] which denotes the fracture and
0.894
Mg
k m0.67
(11)
3 Adsorption/desorption of shale gas
The greatest difference between shale gas and
conventional natural gas reservoir is that the shale gas is
mainly adsorbed and stored in the shale matrix. Many
literatures pointed out that the Langmuir isotherm curve
can well describe the shale reservoirs’ adsorption/
desorption relationship with constant temperature.
Adsorption and desorption process within shale gas
matrix is reversible desorption, thus the shale gas
desorption can also be characterized by the isothermal
adsorption curve. Accordingly, the Langmuir monolayer
adsorption equation [17] is used here to describe the
adsorption/desorption of shale gas, and the equation is
given by
VE 
Fig. 1 Matrix permeability error caused by ignoring diffusion
flow
(10)
VL P
PL  P
(12)
where VL is Langmuir volume (the maximum volume of
adsorbed gas), cm3/cm3; PL is Langmuir pressure (the
pressure at the maximum adsorption capacity of 50%),
MPa; P is the reservoir pressure, MPa; VE is the
Langmuir volume (the volume at the maximum
adsorption capacity of 50%), cm3/cm3.
4 Transport function of dual-porosity model
with spherical shape
Diffusion is also an important transport mechanism
J. Cent. South Univ. (2013) 20: 1928−1937
1931
of shale gas. This work assumes that gas flows from
shale matrix to fracture network, as shown in Fig. 2.
The assumptions and relevant definitions in this
work are as follows:
Assume that the pressure is uniform at the interface
between matrix and fracture, i.e.
p m (rm , Rm , t )  pf ( Rm , t )
(13)
The pseudo-pressures are respectively defined as
follows.
In the matrix system:
pi
mm  2 
pm 
b (r )  p
pm
dpm  2  1  am 
dp (14a)
0
g z
pm   g z

m
pm
In the fracture system:
mf  2
p
pf
0
g z
dp
(14b)
where mf is pseudo-pressure in the fracture system,
MPa2/(mPa·s); mm is pseudo-pressure in the matrix
system, MPa2/(mPa·s).
Using Eqs. (10), (13) and (14), the expression can
be obtained:
mm (rm , Rm , t )  mf ( Rm , t )
(15)
Equation (15) indicates that the pseudo-pressures
(mm and mf) are equal at the interface between matrix and
fracture.
We assume that the flow rate in matrix has
instantaneous and uniform distribution in half the
fracture volume, Vf/2, which surrounds the matrix. Let
Q(R, t) represent the mass influx of fracture per unit
volume and per unit time, and qm(r=rm, Rm, t) denotes the
outflow rate per unit time in matrix. According to
Swaan’s model [15], we can obtain the following
expression:
Q( R, t )   ( R, t )q~ ( R, t ) 
g
m
  g ( R, t ) q m ( R , t ) 



Vf / 2

 ( r rm , Rm , t )

k p
4π
(r 2  g m m )
(Vf / 2)
 g r
(16)
( r rm , Rm , t )
h
h
1
(17)
Vf  Am f  4 πrm2 f
2
2
2
Using the continuous condition of Eq. (16) for flow
rate at the interface between matrix and fracture, the
equation can be obtained:
2
hf
 k m p m 
 g

  g r 

 ( r rm , Rm , t )
Since diffusion in the matrix system and desorption
on the matrix surface are the key transport mechanisms
of shale gas, the transport mechanism model has been
established, in which the diffusion, Darcy flow and
adsorption/desorption of shale gas are included (Eqs. (9)
and (12)). In order to derive a triple-mechanism seepage
model for shale gas transport, some assumptions are
made as follows:
1) The gas reservoir is homogeneous with dual
porosity.
2) The temperature keeps constant in the gas
reservoirs, i.e. the flow is isothermal.
3) The flow in the fracture system is Darcy flow,
and the flow in the matrix is non-Darcy flow.
4) The gas is single-phase methane gas, and the
force of gravity and capillary effect are neglected.
5) Desorption near the surface of matrix is
instantaneous.
The improved differential equation for seepage flow
in the matrix system is given by

1  2

(r  gVrm )  (  g m   gVE )
t
r 2 r
(18)
(19)
The differential equation for seepage flow in the
fracture system is

1 
( R g vRf )  Q( R, t )  (  gf )
R R
t
(20)
where R is the distance from the well, cm; t is time, s; νRf
is the velocity in the fracture, mm/s; f is the fracture
porosity; m is the matrix porosity.
For convenience and further simplification of the
equations, the relevant definitions are as follows [16]:
m , D 
In addition, the approximate fracture volume that
encircles the matrix block has a uniform average
thickness hf, and the equation is given by
Q ( R, t )  
5 Triple-mechanism seepage model for shale
gas transport
πk fi hfiTsc
m
qsc pscT
(  f or m)
m ( p )  m , i ( pi )  m ( p )  2
pi
p

(21)
p
g z
dp 
(  m or f )
m  1 
tD 
bam (r )
and  f  1
pm
k fi
(22)
t
(23)
q mD ( rD , RD , t D )  nf ( q msc /qsc ) ( r , R , t )
(24)
(f ctf  g ) L2
 mD 
m ctm  g k fi rm2
(f c tf  g ) i k am L2
(25)
J. Cent. South Univ. (2013) 20: 1928−1937
1932
 fD 
f ctf  g k fi
(f c tf  g ) i k f
1
rD  r/rm
(27)
RD  R/L
(28)
where ctm is the matrix compressibility, MPa−1; ctf is the
fracture compressibility, MPa−1; ctfi is the initial fracture
compressibility, MPa−1; hfi is the uniform initial thickness
of the fracture, cm; kf is the fracture permeability, μm2; kfi
is the initial fracture permeability, μm2; L is
characteristic length of the system (well radius, halflength of fracture, etc), cm; mmD is the dimensionless
pseudo-pressure in the matrix system; mfD is the
dimensionless pseudo-pressure in the fracture system;
qmD is the dimensionless volume flow rate of the matrix
block, qmsc is the matrix flow rate under the standard
conditions, cm3/s; qsc is the well production flow rate
under the standard conditions, cm3/s; rD is the
dimensionless radial distance in spherical coordinates;
RD is the dimensionless radial coordinate of the radial
distance; tD is the dimensionless time; Tsc is the standard
conditions temperature, K; ηmD is the dimensionless
diffusivity coefficient in matrix system; ηfD is the
dimensionless diffusivity coefficient in fracture system;
gi is the initial gas viscosity, mPa·s; fi is the initial
fracture porosity.
By applying the spherical dual-porosity medium
model and according to OZKAN et al’s methodology
[13] for definition of the transmissivity coefficient and
elastic storage ratio, the transmissivity coefficient and
elastic storage ratio are defined, respectively, i.e.
The transmissivity:
3 

 k amiVm 
2 k ami ( 4 / 3) πrm
L





2
 k fi (Vf / 2) 
 k fi 4 πrm (hf / 2) 
  L2 
 2k ami rm
 3k fi hf
L2 



In the matrix system:
(26)
rD2

rD
 mD   mDi 

15
rm2
2
(f c tf  g ) i k ami L
mmD (rD , RmD , s ) 
 15


(33)
sinh( s mDi rD )
rD sinh( s mDi )
mfD ( RmD , s )
2
q mD (rD , RmD , s )  srmD
  2 mmD
 rD
rD
5s 



 ( rD , RmD , tD )
2
srmD
[ f (rD , s )  1]mfD ( RmD , s )
(35)
where rmD is the dimensionless radius at spherical
coordinates in matrix; RmD is the dimensionless radial
distance at matrix surface in spherical coordinates.
Equation (35) presents the flow rate in matrix in the
Laplace space at somewhere in the matrix.
The expression f(rD, s) in Eq. (35) is given by
 
sinh(
1 
5s 
15

15

srD coth(

s rD ) 


15
s rD )
15

(36)
s)
Since shale gas reservoirs reflect the characteristics
of dual-porosity media, the flow equation in fracture
system can be written as
~ m
m
1 
( RD  fD )  2 ( D )
RD RD
rD ( r
RD
D 1, RmD , tD )
  fD
mfD
t D
(37)
−1
where ctmi is the initial matrix compressibility, MPa ;
kami is the initial absolute permeability of the matrix, μm2;
Vm is the matrix volume in spherical coordinates, cm3;
mi is the initial fracture porosity.
The derivation of the seepage equations in matrix
and fracture system is given in detail in Appendix A and
B, respectively. Here they are directly written as follows.
(34)
where mmD is the dimensionless matrix pseudo-pressure
in Laplace domain; mfD is the dimensionless fracture
pseudo-pressure in Laplace domain; s is the Laplace
transformation of the time variable.
According to Eq. (34), the dimensionless volume
flow rate in the matrix block in the Laplace space can be
obtained:
sinh(
(31)
( m c tm  g ) i k fi rm2
Using Eqs. (A-7)−(A-15) in Appendix, the solution
of Eq. (32) is
(29)
The expression of the shape factor in Eq. (29) is
(32)
is
f (rD , s )  1 
(30)

mmD
   mD
t D

In order to linearize Eq. (32), the assumed condition
The elastic storage ratio:
(c t ) mi Vm
2(c t ) mi rm


(c t ) fi (Vf / 2) 3(c t ) fi hf
 2 mmD
 rD
rD

Assuming that
 fD   fDi  1
~

k m rm L2
k fi hf rm2

(38)

10
(39)
Then, Eq. (37) can be simplified into the following
J. Cent. South Univ. (2013) 20: 1928−1937
1933
linear diffusion equation in Laplace domain:
mfD
1 
( RD
)  sf ( s )mfD  0
RD
RD RD
(40)
The definition of transport function f(s) between
matrix and fracture in Eq. (40) is given by
f ( s )  f (rD  1, s )  1 
 
1 
5s 
15

s coth(

s )


(41)
15
It is well known that the horizontal well is an
effective technology for exploitation of shale gas
reservoirs. Thus, we analyze the shale gas transport
mechanism under the horizontal well. In this work, we
assume that the horizontal well is at the center of the
cylindrical reservoir with top and bottom impermeable,
and the model [18] is shown in Fig. 3.
the higher the matrix flux will be.
2) Case 2
Figure 5 illustrates that the non-dimensional matrix
flow rate changes with the dimensionless time at the
matrix surface rD=1, when the permeability is constant
(km=10−8 mD). It is shown that the plateau of considering
Table 1 Cases considering effects of adsorption/desorption and
diffusion on matrix flow rate, pseudo-pressure in horizontal
wellbore
Case No.
Study content
1
With constant tD, relationship between nondimension flux rate in matrix system and spherical
matrix dimensionless radius, rD
2
Adsorption/desorption
impacts
dimensionless flux, at rD=1
3
Flow rate error neglecting diffusion flow at
different spherical matrix dimensionless radius, rD
4
Flux at different spherical matrix dimensionless
radius, rD influenced by adsorption/ desorption
5
Pseudo-pressure
in
horizontal
influenced by adsorption/desorption
on
matrix
wellbore
Fig. 3 Model on cylindrical reservoir with top and bottom
impermeable
Furthermore, as shown in Appendix C, the transport
function Eq. (41) is introduced and extended to the
horizontal well pressure response [19] of Eq. (C-4) of
Appendix in the shale gas reservoir, and then, the effects
of adsorption/desorption and diffusion on the
non-dimensional flow rate and non-dimensional wellbore
pressure are discussed in the next section.
Fig. 4 Non-dimension flux rate in matrix versus non-dimension
radius at tD=100 000
6 Results and discussion
By utilizing the above theory, five cases (Table 1)
are discussed on the cylindrical reservoir with
impermeable top and bottom exploited by the horizontal
well.
1) Case 1
Figure 4 shows the non-dimension flux rate qmD in
matrix versus the non-dimension radius rD when the
non-dimension time tD is 100 000. In Fig. 4, only the
Darcy flow in the matrix is considered, and the flow rate
in Fig. 4 suggests that even if the dimensionless time tD
is 100 000, most matrix flux comes from the surface of
matrix block. The closer the surface of matrix blocks is,
Fig. 5 Comparison of matrix flux rates with and without
adsorption/desorption at rD=1
1934
adsorption/desorption is longer than that not considering
the adsorption/desorption. Therefore, the results confirm
that the shale gas adsorption/desorption can extend the
time of the stable matrix production. Also, this can well
explain the phenomenon that shale gas production life is
longer and the deliverability of post-production declines
slowly, compared with conventional reservoirs.
3) Case 3
Figure 6 shows the influence of permeability on the
percentage of the production with the diffusion flow
being ignored when the non-dimension time tD=100 000.
It is seen from Fig. 6 that the flow rate error increases
with rD and reaches its possible maximum value near the
matrix surface (rD=1.0) at lower matrix permeability, and
the error of matrix flow rate is the greatest at the
minimum permeability (such as 10−8 mD). The reason is
that the production increases with rD and reaches its
maximum value at rD=1.0 (also see Fig. 4), and the lower
the pressure near the matrix surface is, the stronger the
diffusion will be (Fig. 1). Thus, we can confirm that
diffusion is an important factor which significantly
influences the shale gas productivity.
J. Cent. South Univ. (2013) 20: 1928−1937
Fig. 7 Comparison of flux rates in matrix with and without
adsorption/desorption
Fig. 8 Comparison of pseudo-pressure versus time in shale
matrix with and without adsorption-desorption
7 Conclusions
Fig. 6 Flow rate error by neglecting diffusion flow in matrix
4) Case 4
Figure 7 compares flux rates based on Eq. (35) in
matrix with and without adsorption/desorption. It can be
seen from Fig. 7 that the adsorption/desorption can
significantly increase the flow rate in matrix. Thus, the
characteristic of adsorption/desorption is an important
factor for commercial exploitation of shale gas.
5) Case 5
Figure 8 denotes a comparison of the pseudopressures in the horizontal well based on Eq. (C-4) in
Appendix with and without adsorption/desorption. It is
clearly seen from Fig. 8 that when the permeability is
constant (km=10−8 mD) (based on Eq. (C-4) in Appendix),
the adsorption/desorption can significantly decrease the
pressure drop. This also shows the feature of stable
production of shale gas.
1) When the matrix permeability tends to the
magnitude of nD, the flow only occurs near the surface
of matrix, and the centre of matrix blocks is almost not
involved in actual production.
2) The flow in matrix only occurs near the surface
of matrix. The closer to the surface of matrix, the lower
the pressure is and the stronger the diffusion effect is.
3) As the production rises and the formation
pressure drops, desorption will occur on the surface of
shale matrix, and the desorption behavior can
significantly increase the flow rate in matrix.
4) The desorption can significantly decrease
pressure drop in the horizontal well, and this is also a key
factor that determines the long and stable production for
shale gas wells.
5) To achieve the commercial exploitation of shale
gas reservoirs, on the basis of the shale gas transport
mechanism proposed, further studies on the single well
productivity with shale gas reservoir should be made in
the future.
J. Cent. South Univ. (2013) 20: 1928−1937
1935
the pressure is uniform at the interface between matrix
and fracture, i.e. ∆pm(r=rm, Rm, t)=∆pf(Rm, t). The
dimensionless conversion based on the above conditions
by using Eqs. (14), (21), (22), (23) and (28) results in
Appendix A
In matrix system:
The improved seepage differential equation is
1  2

(r  gVrm )  (  gm   gVE )
 2
t
r r
(A-1)
Inserting the equation of gas state and Langmuir
isothermal adsorption equation into Eq. (A-1), we can
obtain
1   2 k m  bam
1 
r
pm
r 2 r   g 
mmD (rD , RmD , t D )  0
(A-8)
mmD (rD  0, RmD , t D )  mmD0 (t D )  finite
(A-9)
mmD (rD  1, RmD , t D )  mfD ( RmD , t D )
In order to linearize Eq. (A-7) and make convenient
for the solution, apply the following equation:
 mD (rD , RmD , t D )  rD mmD (rD , RmD , t D )
 p m M pm


 zRgT t

Mpm
  Mpm
m 
VE 

t  RgTz
RgTz 
(A-2)
(A-10)
(A-11)
Substituting Eq. (A-11) into Eq. (A-7) through
Eq. (A-10) and taking the Laplace transform of the
resulting equation, the following equations can be
obtained:
For deriving the total compressibility coefficient of
matrix, Eq. (A-2) can be rewritten as
 2u mD
 s mDi u mD  0
(A-12)
1   2 k m  bam  p m p m 
1 

r

p m  z r 
r 2 r   g 

 p m p m
VL p m c g
VL p L

m  (cg  cm ) 

2 
( p L  pm ) ( p L  pm )   z t

(A-3)
u mD (rD  0, RmD , s)  0
(A-13)
u mD (rD  1, RmD , s )  mfD ( RmD , s )
(A-14)
The definition of new total compression coefficient
of matrix is
ctm  (cg  cm ) 
VL pm cg
( pL  pm )m

VL pL
( pL  pm ) 2 m
(A-4)
Inserting Eqs. (9), (22) and (A-4) into Eq. (A-3), we
can obtain
1   2 mm  ctm  g mm
r

r 
k am
t
r 2 r 
(A-5)
Assuming the change of physical property of
formation is little, the approximate value of the
diffusivity coefficient is
 mD 
 g m ctm k fi rm2
k am (f c tf  g ) L2
  mDi
(A-6)
Due to Eqs. (21), (23), (27) and (A-6), Eq. (A-5)
can be simplified as

2

rD rD
1
 2 mmD
 rD
rD


mmD
   mDi
t D

(A-7)
The initial condition for matrix is ∆Pm(r, Rm, t=0);
the constraint conditions: the pressure drop at r=0 is
bounded, that is, ∆Pm(r=0, Rm, t)=finite.
The boundary condition on the matrix block surface:
rD2
The solution of (A-12) by using the condition of
(A-13) and (A-14) yields
mmD (rD , RmD , s ) 
sinh( s mDi rD )
rD sinh( s mDi )
mfD ( RmD , s ) (A-15)
The volume rate through the radius r in the matrix
block is

 g k am pm 


qmsc (r , Rm , t )  4π r 2
  gsc  g r 

 ( r , Rm , t )
 2πk m
Tsc  2 mm 

r
pscT 
r  ( r , R , t )
m
(A-16)
By using Eqs. (21), (24), (27) and (31) as well as by
applying the Laplace transform, Eq. (A-16) can be
changed into Eq. (35).
Appendix B
In fracture system:
The seepage differential equation for fracture
system [13] is
1 

( R g vRf )  Q( R, t )  (  gf )
R R
t
(B-1)
Inserting Eqs. (14), (18) and the equation of gas
state into (B-1), the expression can be obtained:
mf
2k m mm
1   mf 
R

( r rm , Rm , t )  f c tf
R R  r  k f hf  m r
t
(B-2)
J. Cent. South Univ. (2013) 20: 1928−1937
1936
Inserting dimensionless Eqs. (21), (22), (23), (27)
and (28) into Eq. (B-2), and due to the fact that the
concentration gradient at the interface between matrix
and fracture is 0, there is no diffusion at the interface.
Then, the diffusion equation is obtained:

mfD  2k m L2 mmD
 RD

RD  k f hf rm rD

 gf c tf
k fi
mfD
kf
(f c tf  g ) i t D
1 
RD RD
( r  rm , Rm , t )

(B-3)
In order to simplify Eq. (B-3), the relevant
definitions are given as follows [13]:
~

k m rm L2
k fi hf rm2

k ami rm L2
k fi hf rm2


Inserting Eqs. (21) and C-2) into Eq. (C-1) results in
the following equation:
1 
RD RD
mfD ( p ) 
 fD
1 1
 K 0 (rD sf ( s ) ) 
2s  1
I 0 (rD sf ( s ) ) K1 (reD sf ( s ) ) 
 d 
I1 (reD sf ( s ) )


1
Z
1 
Z
n2π2
cos nπ cos nπ w   [ K 0  rD sf ( s )  2


1
s n1
h
h
hD

Due to the little change of physical property of
formation, the approximate value of the diffusivity
coefficient in fracture is
f ctf  g k fi

  fi  1
(f ctf  g ) i k f
I 0 (rD sf ( s ) 
(B-5)

~ mmD
mfD 
 RD
  2

R
rD
D 

( r  rm , Rm , t )
mfD
(B-6)

t D
Applying the Laplace transform of (B-6) yields
1 
RD RD

~ mmD
mfD 
 RD
  2

rD
R
D 

( r  rm , Rm , t )
 smfD (B-7)
Due to Eq. (A-15), Eq. (B-7) becomes
1 
RD RD

mfD 
 RD
  sf ( s )mfD  0
RD 

(B-8)
hD2
) K1 (reD sf ( s ) 
n2π2
hD2
n2π2
hD2




)
] d
)
(C-4)
By introducing and extending the transport function
Eq. (41) into the horizontal well pressure response of
Eq. (C-4), Eq. (C-4) can represent the horizontal well
pressure response which considers the diffusion,
adsorption/desorption and Darcy flow in shale gas
reservoir.
References
[1]
BUMB A C, MCKEE C R. Gas-well testing in the pressure of
desorption for coalbed methane and Devonian shale [J]. SPE
Appendix C
Formation Evaluation, 1988(3): 179−185.
To get higher shale gas deliverability, the horizontal
well technique is usually used to exploit the shale gas
reservoir.
The seepage differential equation [18] for horizontal
well is
2k m mm
1   mf   2 mf

( r rm , Rm , t ) 

R
2
R R  r  Z
k f hf  m r
mf
 gctf
(C-1)
t
[2]
FATHI E, YUCEL AKKUTLU I. Nonlinear sorption kinetics and
surface diffusion effects on gas transport in low-permeability
formations
[C]//
SPE124478.
2009
SPE
Annual
Technical
Conference and Exhibition. New Orleans, Louisiana, USA, 2009:
4−7.
[3]
LUFFEL D L, HOPKINS C W, SCHETTLER P D. Matrix
permeability measurement of gas productive shales [C]// SPE 26633.
68th Annual Technical Conference and Exhibition of the Society of
Petroleum Engineers. Houston, Texas, USA, 1993: 3−6.
[4]
GAO Chao, JOHN L W. Modeling multilayer gas reservoirs
including sorption effects [C]// Paper SPE29173-MS. SPE Eastern
Regional Conference & Exhibition. Charleston, West Virginia, 1994:
8−10.
where Z is the coordinate in the vertical, cm.
The dimensionless quantities are defined as
 RD  R/L

rD  r/rm
 Z  Z /L
 D
n2π2
I1 (reD sf ( s ) 
Inserting Eqs. (B-4) and (B-5) into Eq. (B-3), the
equation becomes
1 
RD RD
(C-3)
This work uses the horizontal well pressure
response solution in the Laplace space [19] with closed
circular boundary as well as top and bottom closed
boundary, and the equation is expressed as
(B-4)
10

mfD   2 mfD

 RD
 sf ( s )mfD  0
RD  Z D2

[5]
AYALA H L F, ERTEKIN T, DEWUMIN M. Compositional
modeling
of
retrograde
gas-condensate
reservoirs
in
multi-mechanistic flow domains [J]. SPEJ, 2006, 11(4): 480−487.
(C-2)
[6]
AYALA H L F, ERTEKIN T, DEWUMI M. Numerical analysis of
multi-mechanistic flow effects in naturally fractured gas-condensate
systems [J]. J Pet Sci Eng, 2007(58): 13−29.
J. Cent. South Univ. (2013) 20: 1928−1937
[7]
1937
RICHARD F S, BIN Qin. Examination of the importance of self
SPE Annual Technical Conference and Exhibition. Florence, Italy,
diffusion in the transportation of gas in shale gas reservoirs [J].
Petrophysics, 2008, 49(3): 301−305.
[8]
2010: 19−22.
[14]
JAVADPOUR F, FISHER D, UNSWORTH M. Nanoscale gas flow
effect on shale gas well productivity [J]. Natural Gas Industry, 2011,
in shale gas sediments [J]. JCPT, 2007, 46(10): 16−21.
[9]
JAVADPOUR F. Nanopores and apparent permeability of gas flow in
31(4): 55−58. (in Chinese)
[15]
mudrocks (shales and siltstone) [J]. Canadian Petroleum Technology,
2009, 16(8): 16−21.
[10]
formations [J]. SPE Formation Evaluation, 1986(2): 43−52.
[17]
FREEMAN C M, MORIDIS G J, BLASINGAME T A. A Numerical
[12]
Chinese)
[18]
FREEMAN C M. A numerical study of microscale flow behavior in
Conference and Exhibition. Florence, Italy, 2010: 19−22.
[13]
OZKAN E, RAGHAVAN R, APAYDIN O G. Modeling of fluid
transfer from shale matrix to fracture network [C]// SPE 134830.
LI Xiao-ping, ZHANG Lie-hui, WU Feng, LI Yun, LIU Qi-guo. A
new method for well test analysis of horizontal gas well [J]. Acta
tight gas and shale gas reservoir systems [C]// SPE141125. SPE
International Student Paper Contest at the SPE Annual Technical
SCHEIDEGGC A E. Physics of flow through porous media [M].
WANG Hong-xun Tr. Beijing: Petroleum Industry Press, 1982. (in
study of microscale flow behavior in tight gas and shale gas reservoir
systems [J]. Transport in Porous Media, 2011, 90(1): 253−268.
ERTEKIN T, KING G R, SCHWERER F C. Dynamic gas slippage:
A unique dual-mechanism approach to the flow of gas in tight
in low permeability porous media [J]. J Springer Science+Business
[11]
SAKHAEE-POUR A, BRYANT S. Gas permeability of shale [J].
SPE Reservoir Evaluation & Engineering, 2012, 15(4): 401−409.
[16]
BEYGI M E, RASHIDI F. Analytical solutions to gas flow problems
Media, 2010 B.V: 421−436.
GAO Shu-sheng, YU Xing-he, LIU HUA-xun. Impact of slippage
Petrolia Sinica, 2008, 29(6): 903−906. (in Chinese)
[19]
OZKAN E. Performance of horizontal wells [D]. Oklahoma, USA:
Tulsa University, 1988.
(Edited by YANG Bing)