Seepage Mechanism and Transient Pressure Analysis of Shale Gas

doi:10.4236/am.2013.41A030

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Applied Mathematics, 2013, 4, 197-203

http://dx.doi.org/10.4236/am.2013.41A030 Published Online January 2013 (http://www.scirp.org/journal/am)

Seepage Mechanism and Transient Pressure Analysis of

Shale Gas*

Xiao Guo, Weifeng Wang

State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Nanchong, China

Email: [email protected]

Received October 9, 2012; revised November 9, 2012; accepted November 17, 2012

ABSTRACT

The current research of nonlinear seepage theory of shale-gas reservoir is still in its infancy. According to the charac-

teristics of shale gas in adsorption-desorption, diffusion, slippage and seepage during accumulation, migration and pro-

duction, a mathematical model of unstable seepage in dual-porosity sealed shale-gas reservoir was developed while

considering Knudsen diffusion, slip-flow effect and Langmuir desorption effect. By solving the model utilizing the

Stehfest numerical inversion and computer programming in Laplace space, several typical curves of bottomhole pres-

sure were obtained. In this paper, we discussed the effects of several parameters on the pressure dynamics, i.e. storativ-

ity ratio, Langmuir volume, Langmuir pressure, adsorption-desorption, tangential momentum accommodation coeffi-

cient, flow coefficient, boundary. The results show that the desorbed gas extends the time for fluid to flow from matrix

system to fracture system; the changes of Langmuir volume and Langmuir pressure associated with desorption and ad-

sorption effect are the internal causes of the storativity ratio change; when the tangential momentum accommodation

coefficient decreases, the time for pressure wave to spread to the border reduces; interporosity flow coefficient deter-

mines the occurrence time of the transition stage; boundary range restricts the time for pressure wave to spread to the

border.

Keywords: Shale Gas; Seepage Mechanism; Mathematical Model; Knudsen Diffusion; Slippage; Langmuir Desorption

1. Introduction

With the successful development of shale gas in North

America and the sharp increase in demand of natural gas

in China, Chinese government and petroleum companies

are attaching more and more attention on shale gas de-

velopment. Studying shale gas percolation mechanism is

the foundation and premise of effective development.

However, there are only a few articles on exploitation of

shale gas. Besides, most scholars still use conventional

theories and methods to study shale-gas reservoirs.

So far, some studies have been made in shale gas ex-

ploration. The idea of applying desorption theories of

coalbed methane in shale-gas reservoir has been sug-

gested several years ago [1]. And a capacity descending

chart had been plotted by Duan Yonggang et al. (2011).

In their study shale gas adsorption and desorption were

considered [2].

In the past years, some scholars established a new gas

well productivity formula. In their study, they considered

the effect of artifici al fracturing effect and gas slippage

effect. Besides, some of their researches were conducted

by applying physical simulation [3].

To fully understand the mechanism of seepage in

shale-gas reservoir, observation of nano-pores is dispen-

sable. And for the first time, F. Javadpour (2009) used

nanoscope to observe nano-pores in the shale-gas reser-

voir. He calculated the apparent Darcy permeability,

while considering Knudsen diffusion. In addition, V.

Shabr et al. (2011) introduced a new surface mass bal-

ance law to model transient desorption. In his research,

he studied nanoscale seepage mechanism in pores, while

considering Knudsen diffusion and Langmuir desorption.

In some studies, many prominent results were achieved,

but on the other hand, Darcy’s law was used [4,5]. In fact,

in nanoscale pores, Darcy’s law is no longer valid [6].

According to the characteristics of shale gas in adsorp-

tion-desorption, diffusion, slippage and seepage during

accumulation, migration and production, a mathematical

model of unstable seepage in dual-porosity confined

shale-gas reservoir was built in this paper, while consid-

ering Knudsen diffusion, slip-flow effect and Langmuir

desorption effect [7]. By solving the model utilizing the

Stehfest numerical inversion and computer programming

in Laplace space, several typical curves of bottomhole

pressure were obtained [8-11]. And we discussed the

effects of several parameters on the pressure dynamics,

*Special Issue—Numerical Analysis

X. GUO, W. F. WANG

198

i.e. storativity ratio, Langmuir volume, Langmuir pres-

sure, adsorption-desorption, tangential momentum accom-

modation coefficient, flow coefficient, boundary.

2. Mathematical Model and Solutions

2.1. Assumptions

To simplify the mathematical model, and make it con-

venient to solve, the following assumptions were made:

1) The shale-gas reservoir is dual-porosity;

2) The entire seepage process in shale-gas reservoir is

isothermal;

3) Unsteady seepage in porous media doesn’t follow

Darcy law;

4) Flow in artificial fracture follows Darcy law;

5) The gas diffusi on flux in artificial fracture can be

neglected when calculated with Darcy flow flux.

2.2. Mathematical Model

In nanoscale shale matrix pores, the mean free path of

gas molecule is comparable or slightly less than pore

diameter. The major forms of migration of shale gas in

pores are Knudsen diffusion and slippage, while Knud-

sen diffusion inside the pore and slippage on its inner

face [4,12-15].

Take a pore as example. Inside the pore, the mass flux

of diffusing gas is described as following:

poro gk

rM RTMD

ZRMZR T





(1)

where 2

3π

poro g

 is Knudsen diffusion coeffi-

cient [13].

On the inner face of the pore, the mass flux is de-

scribed by

π8

8π8

porog poro

rRTFr











 p





. (2)

Where 8

2π

1π

poro

Frp M











 is slippage coeffi-

cient [14]. Here, Darcy flow item is not considered.

Quantitative description of the relationship between

shale gas adsorption and desorption under constant tem-

perature is obtained by using Langmuir isothermal ad-

sorption equation [15].

. (3)

According to the mass conservation law and Langmuir

isothermal adsorptio n equation and considering the Knud-

sen diffusion effect and slippage effect, the unsteady

flow mathematical model of the dual-porosity sealed

shale-gas reservoir is established. We assumed the matrix

unit as spherical, and its diameter is r1. In the center of

the spherical matrix unit, we assumed the pressure is zero,

and at the interface of matrix system and fracture system,

the pressure equals to that of fracture system.

The seepage in fractures is described as

1ff ffffg

pp pCpRT

rr ZrKZtMK













 

 (4)

where qm is the generation term (the volume of gas that

outflow from matrix unit).

mrr

VMD

qrr rZRTr





 



. (5)

The unsteady seepage in spherical matrix is described



poro

gsc LLm

mm Lm

rp Zr

Vp p



















































. (6)

By substituting Equations (1)-(3) into the mass con-

servation law, we can get



poro

gsc LLm

mm Lm

rp Zr

Vp p





















































. (7)

Boundary conditions:



00; ,

rmrr

pprtp

r



f



(8)

Definitions

For convenience, we define following variables

pseudopressure d



;

dimensionless pseudopressure



2πfi

sc i i

hm m

mqB





;

dimensionless time



tVC r







;

dimensionless diameter of fracture fD

;

dimensionless diameter of matrix

;

interporosity flow coefficient

X. GUO, W. F. WANG

199

15 15

app poro

wwk

rrD

rrK







 





By substituting Equation (13) into Equation (14), cou-

pled equation of fluid seepage in fractures and diffusion

and slippage in matrix is deduced:

matrix apparent permeability

2π21

838 π

poroporo g

app

kpp









 











fD fD







 (15)

matrix total compressibility coefficient where the interporosity flow function is



gsc LL

tm m

mm Lm





 



 

15π2

15 1151

coth 1

fs s











 



















. (16)

elastic storativity ratio



tfftf

tfm gsc LL

ftffm

mm Lm

VC C

VC Vp





 





















2.3. Solutions

Introduce pseudopressure in Laplace space, and then

the unsteady seepage in spherical matrix and the bound-

ary conditions can be described as Provided that there is a vertical well in the sealed dual-

porosity shale-gas reservoir, the boundary conditions

corresponding with pseudopressure in Laplace space are

as follows(consider wellbore storage effect CD and skin

factor S):

mD mD







 (9)

r





 (10)

DwD D

fD r

Csmr rs



















(17)



,mD

mrs m

fD

(11)

fD D

r



 (18)

where



2151







. (12)

wD fD

fD r

mmS

r

















. (19)

s is Laplace variable.

By substituting Equations (10) and (11) into Equation

(9), pressure between spherical matrix and fracture can

be described as

Equation (15) is the generalized Bessel equation, its

general solution is



fD fD

Kfsr

BIfsr

m

 (20)



1coth 1

mWWm

r



D

. (13)

Transform Equation (4) into dimensionless equation

and make Laplace transformation, and then we can de-

scribe seepage in fracture as: Substitute Equations (17) and (18) into Equation (20),

and then factor A can be calculated: (see Equation (21))

15π2

fD fD

fD r

sm r



























(14)

BMA



. (22)

As to circle sealed formation









fsR

IfsR

 (23)



 



00 11

AsCsKf sMIf sCSsKfsMIf sf s

 

 











(21)

X. GUO, W. F. WANG

200

Substitute A and B into Equation (20), then the distri-

bution of pseudopressure in Laplace space can be de-

scribed as (see Equation (24)).

Substitute Equation (24) and its derivation function

into Equation (19), the unsteady pseudopressure (dimen-

sionless) can be calculated, while considering C D and S.

(see Equation (25)).

3. Typical Curves and Analysis

By applying Stehfest numerical inversion on the pseudo-

pressure in Equation (13) in Laplace space, the relation-

ship between and



wD D

tC is obtained. Ac-

cording to Equation (13), main parameters are ω, VL, PL,

α, λ, RD, set CD = 0.8, S = 1.2.

The effects of each parameter on pressure dynamics

will be discussed next.

Figure 1 shows that the elastic storativity ratio ω de-

termines the width and depth of the concavity in the

shale gas pressure derivative curve. In another words, the

smaller the storativity ratio, the longer the transition time

and the lower the pressure.

Figure 2 indicates that the influence of Langmuir

volume on the transition time for fluid to flow from ma-

trix system to fracture system. Under the same Langmuir

pressure, with the Langmuir volume increasing, the con-

cavity in the curve is deepened. When the Langmuir

volume increases to a certain degree, the concavity am-

plitude will reduce.

Figure 3 shows the influence of Langmuir pressure on

the transition stage (presents the stage when interporosity

fluid flow from matrix system to fracture system). Under

the same Langmuir volume, with the Langmuir pressure

increasing, the concavity is deepened. But when the

Langmuir pressure increases to a certain degree, the

concavity amplitude will reduce.

Figure 4 shows that the shale gas desorbed from ma-

trix system enlarges the storage capacity of the matrix

system, and prolongs the occurrence of the transition

stage. It is also the reason why the storativity ratio de-

creases and the concavity deepens.

-2

-1.5

-1

-0.5

0.5

1.5

-20246

lg(t

)

lg(m

),lg (m

)

ω=0.1

ω=0.05

ω=0.01

ω=0.001

ω=0.007

=0.0002

α=0.8

=10000

=0.8

S=1.2

Figure 1. Effect of storativity ratio on pressure dynamics.

-2

-1.5

-1

-0.5

0.5

1.5

-20 24 6 8

lg(t

)

lg(m

),lg(m'

)

=30m

=10m

=1m

=0.0002

α=0.8

=10000

=0.8

S=1.2

Figure 2. Effect of Langmuir volume on pressure dynamics.



 



1 1

00 1

1 1

fD fD

D D

mrs

KfsR

Kfsr Ifsr

IfsR

sKfsR KfsR

CsKfsIfsCSsfs KfsIfs

IfsR IfsR



 















(24)



 



 



1 1

0011

1 1

00 1

1 1

D D

KfsRKf sR

KfsIfsS fsKfsIfs

IfsRIfsR

sKfsR KfsR

CsKfsIfsCSsfs KfsIfs

IfsRIfsR











 















(25)

X. GUO, W. F. WANG 201

-2

-1.5

-1

-0.5

0.5

1.5

-20246

lg(t

)

lg(m

),lg(m'

)

ω=0.007

=0.0002

α=0.8

=10000

=0.8

S=1.2

=3×10

=1×10

Figure 3. Effect of Langmuir pressure on pressure dyna-

mics.

-2

-1.5

-1

-0.5

0.5

1.5

-20 24 6 8

lg(t

)

lg(m

),lg(m'

)

=0.0002

α=0.8

=10000

=0.8

=1.2

Without desorption

With desoption

Figure 4. Effect of gas desorption on pressure dynamics.

Figure 5 describes the influence of the tangential

momentum accommodation coefficient upon the pressure

dynamics. With the tangential momentum accommoda-

tion coefficient decreasing, the concavity in the typical

curve deviates to right and becomes shallower, and the

time for pressure wave to spread to the border reduces.

Figure 6 shows that the interporosity flow coefficient

determines the occurrence time of the transition stage too.

With the interporosity flow coefficient increasing, the

transition stage appears earlier, the concavity deviates to

left and the interporosity flow gets more intense.

Figure 7 reflects how boundary range influences the

pressure dynamics. With the range getting narrower, the

time for pressure wave to spread to the border reduces.

When the pressure wave reaches the boundary, the pres-

sure curve as well as its derivative curve upturns, finally

the two curves are tangent into a straight line, whose

slope turn to be 1.

From all the figures, we can conclude that the pressure

change of unsteady seepage in vertical wells in sealed

shale-gas reservoir can be divided into three stages:

 Early wellbore storage;

 Transition stage: interporosity flow (from matrix sys-

tem to fracture system) appears. Concavity of the

-2

-1

-202468

lg(t

)

lg(m

),lg(m

)

ω=0.007

=0.0002

=10000 C

=0.8

S=1.2

α=0.8

α=0.5

α=0.2

Figure 5. Effect of tangential momentum accommodation

coefficient on pressure dynamics.

-2

-1.5

-1

-0.5

0.5

1.5

-2 02468

lg(t

)

lg(mwD),lg(m'wD*tD/

ω=0.007

α=0.8

=10000

=0.8

S=1.2

=0.01

=0.001

=0.0001

Figure 6. Effect of interporosity flow coefficient on pres-

sure dynamics.

-2

-1

-202468

lg(t

)

lg(m

),lg(m

)

ω=0.007

=0.0002

α=0.8

=0.8

S=1.2

=10000

=7000

=4000

Figure 7. Effect of boundary on pressure dynamic s.

curve appears in this stage too;

 The radial flow of dual porosity system: this stage

presents the homogeneous characteristics of forma-

tion. Pressure derivative curve shows a horizontal line,

whose value is 0.5.

4. Conclusions

1) According to the characteristics of shale gas in ad-

sorption-desorption, diffusion, slippage and seepage dur-

ing accumulation, migration and production, a mathe-

X. GUO, W. F. WANG

202

matical model of unsteady seepage in dual-porosity sealed

shale-gas reservoir was built while considering Knudsen

diffusion, slip-flow effect and Langmuir desorption ef-

fect. By solving the model utilizing the Stehfest numeri-

cal inversion and computer programming in Laplace

space, several typical curves of bottomhole pressure were

obtained.

2) In the sealed shale-gas reservoir, the stage when

flow in fracture system exists only is extremely transient,

the transition stage appears immediately after the well-

bore storage stage. So in fracture system, radial flow

doesn’t appear. The pressure dynamics of unsteady seep-

age appear only in storage stage, the radial flow stage,

and the stage when the pressure is unsteady.

3) The typical curves of bottomhole pressure are pre-

sented to discuss the influences of several sensitive pa-

rameters upon pressure behavior. These sensitive pa-

rameters include elastic storativity ratio, Langmuir vol-

ume, Langmuir pressure, adsorption-desorption, tangen-

tial momentum accommodation coefficient, interporosity

flow coefficient, and boundary range. The smaller the

storativity ratio, the longer the transition stage. The

changes of Langmuir volume and Langmuir pressure, as

well as desorption and adsorption mechanisms are the

internal causes of the storativity ratio change. The tan-

gential momentum accommodation coefficient describes

smoothness of the pores’ inner face, With the tangential

momentum accommodation coefficient decreasing, the

concavity of typical curves deviates right and becomes

shallower, and the time for pressure wave to spread to the

border reduces; The interporosity flow coefficient deter-

mines the occurrence time of the transition stage. With

the interporosity flow coefficient increasing, the transi-

tion stage appears earlier, the concavity deviates to left

and the interporosity flow gets more intense; with the

boundary range getting narrower, the time for pressure

wave to spread the border reduces.

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X. GUO, W. F. WANG 203

Nomenclature

VE: total volume of gas adsorbed of per unit volume of

the reservoir in equilibrium at pressure P, m3/t;

VL: Langmuir volume, the maximum sorption capacity

of the shale, m3/t;

PL: Langmuir pressure, at which the total gas volume

adsorbed, VE, is equal to 50% of the Langmuir volume,

Pa;

P: pressure, Pa;

Pi: initial pressure, Pa;

p: average pressure, Pa;

ω: dimensionless storativity ratio;

λ: dimensionless interporosity flow coefficient;

K: absolute permeability, m2;

M: molar mass, kg/mol;

Z: gas compressibility, fraction;

Rg: universal gas constant, 8.314Pa·m3/ (mol·K);

T: temperature, K;

ρ: density, kg/m3

μ: viscosity, Pa·s;



: average viscosity Pa·s;

φ: porosity, fraction;

C: gas compressibility, Pa−1;

α: tangential momentum accommodation coefficient,

depending on the smoothness of the pores’ inner face,

gas type, temperature and pressure, fraction (0 ~ 1);

r: radial distance in spherical coordinates, m;

rporo: matrix system pore radius, m;

r1: spherical matrix block radius, m;

rw: gas well radius, m;

R: gas reservoir boundary, m;

S: skin coefficient, fraction;

CD: dimensionless wellbore storage factor;

m: pseudopressure;

mD: dimensionless pseudopressure;

m: dimensionless pseudopressure in Laplace space;

m: dimensionless bottom hole pseudopressure in

Laplace space;

s: Laplace transform variable;

f(s): interporosity flow function;

Iυ: υ order of first species modified Bessel function,

Kυ: υ order of second species modified Bessel function,

υ = 0.1.