An unsteady seepage flow model considering kickoff pressure gradient for low-permeability gas reservoirs

PETROLEUM EXPLORATION AND DEVELOPMENT Volume 35, Issue 4, August 2008 Online English edition of the Chinese language journal Cite this article as: PETROL. EXPLOR. DEVELOP., 2008, 35(4): 457–461.

RESEARCH PAPER

An unsteady seepage flow model considering kickoff pressure gradient for low-permeability gas reservoirs FENG Guo-qing1,*, LIU Qi-guo1, SHI Guang-zhi2, LIN Zuo-hua3 1. State Key Lab. of Oil & Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu 610500, China 2. Research Institute of Exploration and Development, Daqing Oilfield Limited Company, PetroChina, Daqing 136172, China 3. Huabei Project Department, CNLC, CNPC, Renqiu 062552, China

Abstract: Kickoff pressure gradient occurs when gas flows in a low permeability reservoir. In order to describe the issue of unsteady seepage flow in this kind of reservoir, an unsteady-state seepage flow mathematical model which takes kickoff pressure gradient into consideration is built up based on previous research and the characteristic of start-up pressure gradient existing in low velocity non-Darcy percolation flow (the fluid flow boundary expands outwards constantly). Combining the Laplace space analytic solution by using Green function method with the numerical approximation, the nonlinear mathematical model can be resolved. An equation for calculating single-well control radius is established, and a single-well control radius chart board is drawn up which can be used to calculate single well control radius. Case study indicates that the unsteady seepage flow model taking kickoff pressure gradient into account can correctly present the percolation mechanism and real production performance of a low-permeability gas reservoir. Key words: starting pressure gradient; low-permeability gas reservoir; unsteady seepage; well control radius

1

Introduction

The pattern of fluid seepage flow is obviously different between low and high permeable reservoirs. When the permeability in low permeable reservoir is lower to certain extent, its seepage characteristic will no longer accord with Darcy law and such kind of flow is called nonDarcy seepage, which has kickoff pressure gradient as a distinguishing feature. The kickoff pressure gradient[1-4] is the pressure gradient when the pressure drawdown, which acts on fluid medium, has attained a certain level to make the fluid to overcome the viscous force and start to flow. If the pressure gradient is small, the flow velocity increase slowly and obey the nonlinear law, while the flow velocity increase quickly and obey the linear law if the pressure gradient exceeds the kickoff pressure gradient. In past decade, Chinese scholars have done a large number of mathematical simulations and empirical researches on low velocity and nonDarcy flow, and it is found that if the water saturation is bigger than the bound water saturation in the water-bearing rock, the low velocity and nonDarcy flow also exists for gas, and the lower the permeability and the higher the water saturation, the higher the kickoff pressure gradient and the more obvious the low

velocity and nonDarcy flow[5]. Based on previous research, an unsteady seepage flow model considering kickoff pressure gradient for low permeable gas reservoir has been built up and resolved in this article. Resolution equation for the control radius of single is also established, and the relevant chart board has been made which can be used to conveniently calculate the control radius of single well. Combined with the case calculation, it shows that the unsteady seepage model considering kickoff pressure gradient can reflect the seepage mechanism and production performance for low permeable gas reservoir correctly.

2 An unsteady seepage flow model considering kickoff pressure gradient for low permeability gas reservoirs The feature of low velocity and nonDarcy flow considering kickoff pressure gradient is that the fluid flow boundary can extend outward continuously, that is, the fluid flow boundary is controlled by kickoff pressure gradient, and the fluid which is beyond the flow boundary can not flow. According to this, the unsteady seepage flow model considering kickoff pressure gradient for low permeability gas reservoir has been established and it is shown as:

Received date: 05 October 2006; Revised date: 17 February 2008. * Corresponding author. E-mail: [email protected] Foundation item: Supported by CNPC Innovation Fundation(07E1016). Copyright © 2008, Research Institute of Petroleum Exploration and Development, PetroChina. Published by Elsevier BV. All rights reserved.

FENG Guo-qing et al. / Petroleum Exploration and Development, 2008, 35(4): 457–461

­ ° 1 w «ª rD w\ D »º  1 O\ BD ° rD wrD ¬ wrD ¼ rD ° °\ D (rD , tD 0) 0 ° w\ ° D 1  O\ BD ® ° wrD rD 1 ° ° w\ D O\ BD ° wrD rD rFD ( tD ) ° °¯\ D (rD ! rFD (tD )) 0

(1)

(2) (3)

r rw Khrw O\ B 0.012 73 Tqsc rD

3

(4)

Define

rD

derivative on







(rD  W  f)

g\ D (6)



let

f

(9)

1

\ D (rD , g )

firstly

partial

rD to obtain:

 g 

b g K1 1

O\ BD g

Where, the general solution of basic equation[9,11-13] in above equation is given by:



(1  W  rD )

and

(5)

Considering the infinite outer boundary, applying Laplace transform to the unsteady seepage flow model considering kickoff pressure gradient for low permeable gas reservoirs, Eq.(6) is obtained in the following:







1

w\ D (rD , g ) wrD r

I1

 g ³

f

1

 b g K1

K0

 gW dW

 g  ʌ O2 g I  g \BD

1

(10) Substitute Eq.(10) into the inner boundary condition to obtain:

b

 g / 2 g K  g

1  O\BD  ʌ O\BD I1 g

(11)

1

Substitute Eq.(11) into Eq.(9), to obtain:

\ D (rD , g )

 g / 2 K r g K  g

1  O\BD  ʌ O\BD I1 g

³

f

1

In Eq.(12), take

0

D

G (rD ,W ) d W

rD



g 

1

1,

(12)

and substitute Eq.(8) into, the

Laplace space analytic solution of bottomhole dimensionless pesudopressure can be evaluated as:

3.1 Laplace space analytic solution of Green functions

a I 0 rD g 



(7)

1

b K 0 rD g  ³ G (rD ,W ) d W

\ D (rD , g )

D

There are three types of methods available to solve this model: (1) Green functions[6-8] method to get Laplace space analytic solution for the model: this method needs approximate consideration of pressure propagation which can instantaneously reach infinitely long. (3) Discrete mathematic[9] to get numerical solution: this is according to the characteristic that flow boundary will elapse outward as time passes. (3) Combing power series analytic solution[10] with numerical approximation: this is to analyze the problems about the propagation front of pressure and the time in nonDarcy percolation flow with low velocity. This article combines method (1) with (3); i.e., using the Laplace space analytic solution from Green functions and numerical approximation to solve the problem of nonDarcy unsteady seepage with low velocity.

\ D (rD , g )



f

­ O\ BD ° g K 0 rD g I 0 W g ° ® ° O\ BD K W g I r g 0 0 D °¯ g

Solution of the model

­ 1 w ª w\ º 1 D O\BD ° «rD » w w r r r grD ° D D¬ D ¼ ° ®\ D (rD o f, g ) 0 ° 1 O ° w\ D   \BD ° wrD r 1 g g D ¯



(8) By considering the infinite external boundary, a = 0, Eq.(7) can be expressed as:

Kh '\ 0.012 73 Tqsc 3.6 K tD t IPCg rw2

O\ BD

Where

G (rD ,W )

Where

\D



b K 0 rD g  ³ G (rD ,W ) d W

w\ D wtD

\ WD ( g )

 g / 2 K  g  g g K  g I  g (13) g

1  O\BD  ʌ O\BD I1

0

1

ʌ O\BD 2g

0

In a similar way, the Laplace space analytic solution of bottomhole dimensionless pseudopressure in low permeability gas reservoir with homogeneous circular closed boundary and nonDarcy unsteady percolation flow with low velocity can be calculated as:

\ WD differential

· E rD § 1  O\ BD ¨¨  e  d ¸¸  g g F rD © ¹

H  rD  f

Where

 rD

1

(14)

FENG Guo-qing et al. / Petroleum Exploration and Development, 2008, 35(4): 457–461

I0

E (rD )









I1

c I0

O\ BD

c

g

I1





g rD g RD



K1 RD g

O\ BD

d

I1

g

g RD







g RD

(15)



g rD I 1

g RD



w\ D wrD



­ ° 1 w ª«rD w\ D º»  1 O\BD ° rD wrD ¬ wrD ¼ grD ° O ° w\ D  \BD ® g ° wrD rD rFD (tD ) ° 1 O ° w\ D   \BD ° wrD g g rD 1 ¯

(17)

³

RD

1

 g ³



RD

1



I0 W g d W



(18)



K 0 W g dW

(19)

g Where

 I0

E (rD )

 g ³

rFD



g rD K 0



F (rD )





g rD K 0

H rD 

O\ BD g

d

g

e

 g ³

g RD









g rD g RD

(21)

 rD



g rD I 0



g rD I 0





K 0 W g dW

 g I  gR

(22)

g RD



(23)

g RD



(25)

0



e

(24)

RD

1

1

H (rD )

§ O\ BD ¨¨ c  g g © I1



(26)

(27)

(28)

D

Numerical approximation for flow boundary model

In infinitely large formation, at certain time tD, and fluid flowing front radius rFD satisfies the following equation:

§ O · ¨ c  \BD ¸ I 0 g rD ¨ g g ¸¹ © I1 g rFD



· ¸¸ g I1 ¹ grFD



(32)

 g (33)



Where RD can be replaced by rFD(tD), and E(rD), F(rD), c, d are separately defined in Eq. (15), Eq.(16), Eq.(18), and Eq.(19). In Eq.(31), define rD=1, and then, substitute Eq.(8) into the Laplace space analytical solution of bottomhole dimensionless pseudopressure which is defined as:

\ WD

E  rD

1 § 1  O\ BD · ed¸ ¨ g F  rD 1 © g ¹

H (rD

³ I W g dW

c g I1 0

3.2



RD

1







I0



I1

g RD

c I0

K 0 RD g

O\ BD



g RD  K 1 I0

c



g RD  K 0 I0

I1



K 0 W g dW

(31)

1

0



RD

1

(30)

H (rD )  ³ G (rD ,W ) d W

RD

I0

g\ D

§ 1  O\ BD · ed¸ ¨ 1 © g ¹

g F  rD

(20)

· E rD § 1  O\ BD ¨¨  e  d ¸¸  H rD  g g F rD © ¹

O\ BD

E  rD

\D

The Laplace space solution for bottomhole dimensionless pseudopressure in low velocity and nonDarcy unsteady seepage flow of homogeneous circle supply boundary in low permeable gas reservoir can be evaluated as:

\ WD

(29)

Resolving Eq.(30) we can get:

D

1

O\BD rD rFD ( t D )

Supposing the boundary is fixed, a new model can be derived as:

(16)

 g I  gR I  g ³ K W g d W 0

g



g RD

1

O\BD



g rD I 1

c g I1

e f



g RD  K 1

g rD K 1

H (rD )



g RD  K 0 I1

 I1

F ( rD )



g rD K 1

1) 

O\ BD g

I0

 g ³

rFD

1





K 0 W g dW

(34)

So, numerical approximation of flow boundary model can be described as: first define dimensionless time as tD, and suppose at that time the fluid flowing front is rFD and then calculate dimensionless pseudopressure ȥD (rFD, tD) according to the analytical solution obtained from Eq.(31), if the calculated value is smaller than ȥBD (the dimensionless kickoff pseudopressure to help the fluid start to flow), decrease rFD, otherwise, increase rFD. Using interactive method to determine rFD, finally calculate bottomhole dimensionless pseudopressure ȥWD(tD). If the calculated flowing front radius (rFD) has expanded to be equal to the radius (RCD) of practical closed circle outer boundary, then calculate the typical curve data which the sequent time corresponds with; the used radius is no longer expanding and using Eq.(14) at that time in the equation RD=RCD. If the supply boundary is a circle and the calculated rFD has

FENG Guo-qing et al. / Petroleum Exploration and Development, 2008, 35(4): 457–461

Fig. 1 Variation of dimensionless radius with dimensionless time at different dimensionless kickoff pressure gradient Table 1 Calculation of control radius for each well Production time/d

Permeability/

Kickoff pressure

(10୉3 Pm2)

gradient/(MPa·m୉1)

1.45

0.011

A

396

0.50 2.00

0.016 0.010

3.70

0.009

B

418

1.00 0.50

0.013 0.016

0.60

0.015

C

1 493

1.00 0.10

0.013 0.026

0.52

0.016

D

591

1.00 0.10

0.013 0.026

0.27

0.019

E

365

0.60 1.00

0.015 0.013

4.20

0.009

F

126

2.00 1.00

0.011 0.013

Well

Control radius/m 1 118 1 109 1 125 1 238 1 229 1 215 1 134 1 139 1 109 1 163 1 168 1 094 653 658 662 872 870 859

expanded to be equal to RSD, then calculate the typical curve data which the sequent time corresponds with; the used radius is no longer expanding and using Eq.(22), where RD=RSD.

4 Applications of the model Making the chart with the solutions of above equations, we can get the correlation between dimensionless radius and dimensionless time under different dimensionless kickoff pressure gradient, as depicted in Fig. 1. All the curves shown in Fig. 1 separately present different dimensionless kickoff pressure gradient ȜpD in pressure form, which is defined as:

O pD

Khrw pi OB 6.637 u 10 3 qsc P TZ

(35)

In actual calculation, tD, rD, and ȜpD are needed to be calculated separately, then inquiry Fig. 1 and the control radius corresponding to each well at some certain time can be calculated. This kind of method has been applied to a gas reservoir in

Sichuan to calculate five wells’ control radii (Table 1). The first row of Table 1 shows each well’s radii which are separately calculated by its actual permeability and in other two rows, the sensitivity research were done to understand how the permeability effect control radius. It is shown in the result that the control radius of each well is over 1 100 m, except Well E and Well F; the reason why control radii of Well E and Well F are short is that the production time is short and pressure wave can only propagate to a limited area. Meanwhile, each well’s control radius will correspondingly decrease or increase with permeability’s decreasing or increasing. Take well A as example, when the permeability is 1.45×10-3 Pm2 its control radius is 1 118 m; when the permeability decreases to 0.5×10-3 Pm2 its control radius is 1 109 m, which has decreased by 9 m; whereas if the permeability increases to 2×10-3 Pm2 its control radius is 1 125 m, which has increased by 7m. However, in general, permeability’s change has a small affection on control radius if the permeability is in a low level (less than 5×10-3 Pm2).

5 Conclusions A nonlinear flow mathematical model considering kickoff pressure gradient for low permeable gas reservoir has been established based on previous research. Combine numerical approximation with the Laplace space analytical solution which is obtained using Green function to solve this established model and establish the resolution equation for single well’s control radius. Meanwhile, the relevant chart board which is used for calculating single well’s control radius has been made up, and it is very convenient to calculate single well’s control radius by using this chart board. It is shown by case calculation that this unsteady flow model which has considerable kickoff pressure can correctly reflect the seepage mechanism and production performance in low permeable gas reservoirs.

Nomenclature rD——dimensionless radius; ȌD——dimensionless pseudopressure; ȜȌBD——dimensionless kickoff pseudopressure gradient; tD——dimensionless time; rFD——radius of dimensionless flowing front; K——permeability of gas layer, Pm2; h——thickness of gas layer, m; T——temperature of gas reservoir, K; qsc——production rate of gas well, 104 m3/d; ଠȌ——differential pseudopressure, MPa2/(mPa·s); ij——rock porosity, f; ȝ——viscosity of natural gas, mPa·s; Cg——gas compressibility coefficient, MPa-1; rw——well radius, m; t——time, h; r——radial distance, m; ȜȌB——kickoff pseudopressure gradient, MPa2/(mPa·s);

\D

——dimensionless

pseudopressure of Laplace space; g——Laplace variable; a, b——coefficient; I0——The Zeroth-order deformation of first kind Bessel Function; K0——The Zeroth-order deformation of second kind Bessel Function; G(rD, IJ)——Green function; IJ——variable of integration; K1——The first order deformation of second kind Bessel Function; I1——The first order deformation of first kind Bessel Function;

\ WD ——bottomhole dimensionless pesudopressure for

Laplace space; E(rD), F(rD), H(rD), c, e, d, f——intermediate variable;

FENG Guo-qing et al. / Petroleum Exploration and Development, 2008, 35(4): 457–461

RCD——radius of dimensionless circular closed external boundary; RSD——radius of dimensionless circular supply external boundary; RD——radius of dimensionless external boundary; ȜpD——dimensionless kickoff pressure gradient; pi——initial formation pressure of gas reservoir, MPa; ȜB——kickoff pressure gradient, MPa/m; P ——natural gas viscosity under average pressure and temperature, mPa·s; Z——gas compressibility factor, dimensionless.

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