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Web End = Deliverability equation in pseudo-steady state for fractured vertical well in tight gas

Shaoyuan Mo1 Shunli He1 Gang Lei1 Shaohua Gai1

Received: 11 October 2014 / Accepted: 22 March 2015 / Published online: 9 April 2015 The Author(s) 2015. This article is published with open access at Springerlink.com

Abstract Based on non-Darcy ow theory in tight gas reservoirs, a new deliverability equation of fractured vertical gas well in pseudo-steady state is presented with the consideration of the stress-sensitive effect, and the open ow capacity calculation formula of gas well has been also derived. With the new deliverability equation, the effects of stress-sensitive coefcient, fracture parameters and matrix permeability on the productivity of gas well have been analyzed. The computation across an instance shows that due to the stress-sensitive effect, the IPR curves bend over to the pressure axis and the productivity of gas well is lower than that derived from the equation without consideration of stress-sensitive effect under the same pressure drop. As the stress-sensitive coefcient increasing, the well productivity becomes lower, the decline rate of production is higher and the IPR curve bends over in earlier stage with a greater bending. Besides, the productivity is affected by and has a positive correlation with the length and conductivity of fracture, namely that it becomes lower as the length and conductivity of fracture decreasing. Matrix permeability has an apparent impact on the productivity. If matrix permeability is extremely low, gas well cannot achieve the industrial production even after fracturing. As the matrix permeability increasing, stimulation results are signicant.

Keywords Tight gas Pseudo-steady state Non-Darcy

ow Stress-sensitive effect Fractured vertical well

List of symbolsQ The gas production, m3/d

qAOF The absolute open ow, m3/d re The drainage radius, mrw The well bore radius, mp The reservoir pressure, MPapw The bottom-hole pressure, MPa

p The average reservoir pressure, MPa u The porosity, %

r The radial distance, mb The velocity coefcient describing the effect of turbulent ow in porous media, m-1

v The percolation velocity, m/dt The producing time, dh The effective thickness, mk The permeability, lm2cg The gas compressibility, MPa-1 l The gas viscosity, mPa s

q The gas density, kg/m3T The temperature, Kz The gas compressibility factora The stress-sensitive coefcient, MPa-1

Bg The bulk coefcient of gas in reservoir condition l The arc length of ellipse, mxf The half-length of fracture, mwf The width of fracture, mn, g The elliptic coordinates

Subscript0 Initial momentsf Fracturesc Standard condition g Gase Edge/boundary

& Shaoyuan [email protected]

1 MOE Key Laboratory of Petroleum Engineering, ChinaUniversity of Petroleum, Beijing, China

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34 J Petrol Explor Prod Technol (2016) 6:3338

Introduction

Tight gas reservoirs have features of tiny pores, narrow throats and extremely low permeability. Wells produce at a low natural productivity so that most of wells need the stimulation treatment of fracturing. Recently, the performance of fractured vertical well has been widely studied by many researchers. With conformal transformation, Wang et al. (2003) and Jiang (2001) derived the deliverability equation of fractured vertical well and studied the corresponding pressure performance. Li et al. (2008) established the trinomial deliverability equation of fractured vertical gas well in ultra-low-permeability reservoirs by linear regression with bottom-hole pressure and gas production. Yang et al. (2003) proposed a model by considering the gas ow in reservoirs after fracturing as a combination of radial ow and linear ow and obtained the deliverability equation by integral calculation. In their study, the effects of fracture length, effective permeability and effective thickness on the production after fracturing were investigated. Yin et al. (2005) built up a mathematic model of bilinear ow for fractured vertical well and presented the solution. They also plotted and analyzed the theoretical IPR curves of fractured vertical well. Using elliptic ow model, Li et al. (2005) derived the deliverability equation of fractured vertical well with the consideration of threshold pressure gradient. Based on the theory of steady-state ow, Xiong et al. (2012) used the concept of pseudo-pressure to obtain the deliverability equation for predicting the performance of fractured vertical well in low-permeability gas reservoirs. The equation in their study featured the assumptions of nite conductivity fracture and the consideration of threshold pressure, slippage effect and stress sensitivity. Based on the characteristics and ow law of ellipse system, Luo et al. (2005) proceeded from the generalized Darcy law and synthesized the ow inside and outside the fracture to derive the deliverability equation in steady state for gas reservoirs and obtain the corresponding open ow capacity of fractured vertical gas well. According to the concept of disturbance ellipse and the idea of equivalent rectangle system, Zhou et al. (2009) gained the productivity predictions for the gas well after fracturing in low-permeability reservoirs and plotted the corresponding IPR curves for analysis. Study on productivity of fractured vertical well in above documents all contain the assumptions that boundary conditions are constant pressure and the ow is in steady state, which cannot be appropriate for the pseudo-steady state in reservoirs.

During the depletion in oil or gas reservoirs, ow in pseudo-steady state begins when the pressure wave reaches the closed boundary with the feature that the decrements of pressure at points from borehole walls to boundary gradually tend to be the same. Thus, it is signicant for the

development of tight gas to investigate the characteristics of IPR curve in this pseudo-steady state. Lian et al. (2013) derived a deliverability equation in pseudo-steady state for oil reservoirs with closed outer boundary. They considered the effects of stress sensitivity and degasication in bottom, and also investigated the stress-sensitive effect on production. Li et al. (2009) took the characteristics of non-linear ow into account to derive the deliverability equation in pseudo-steady state for gas well. Liao (1998) investigated the characteristics of pressure response in pseudo-steady state of deviated wells by considering the oil reservoirs as dual porous media. In this paper, a percolation model of pseudo-steady ow with the consideration of stress-sensitive effect was proposed to derive a deliverability equation of fractured vertical gas well featuring the non-linear ow of gas. The analysis we made on the effects of stress-sensitive coefcient, matrix permeability, fracture length and fracture conductivity on the productivity of fractured vertical gas well is signicant for the tight gas development.

Model establishment

Assumptions

Fracturing for vertical gas well induces the decrease of percolation resistance and variation of ow pattern in which the gas ow is pseudo-radial inside the fracture and linear outside the fracture (Abbaszadeh and Hegeman 1990). Assumptions specically include the following: (1) formation thickness is equal, outer boundary is closed, production rate is constant and percolation reached the pseudo-steady state. (2) Vertical fracture is symmetrical about the borehole and vertically penetrating the formation.(3) Flow inside the fracture is one-way turbulent and gas owing to borehole is only along the fracture. (4) Flow outside the fracture is pseudo-radial; isobars are elliptical and perpendicularly crossed by streamlines. (5) The damage to fracture surface is not considered.

Flow outside the fracture

During the production of fractured vertical well, elliptical percolation in 2D plane is induced, namely that elliptical isobars and corresponding conjugate hyperbolic streamlines form by focusing on end points of fracture. The relation between Cartesian coordinates and elliptic coordinates can be written as

x xfchn cos g; y xfshn sin g: 1 By Eq. (1), equations of elliptical isobars and hyperbolic streamlines can be obtained and written as below

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J Petrol Explor Prod Technol (2016) 6:3338 35

qg l n

: 2

Based on the concept of disturbance ellipse, elliptical isobars can be described by equivalent rectangle patterns as follows.

x xfchn;

2 p

x2 xfchn

< 2

y2 xfshn

x2xf cos g

: 2

y2 xf sin g

where

l n

4xfchn

1 1 th2n

q sin2 tdt:

With the chain rule, we can obtain dpdy

dp d

Z p 0

dpdn : 11

Introduce the variable of pseudo-pressure

U p

p p0

Z p=2

0 xfshn sin gdg

dp dn

dn d

p 2xfchn

2xfp shn 3

In consideration of the stress-sensitive effect and the nonlinear ow in tight gas, the equation of the ow to fracture is written asdpdy

l k0ea p p

pea p p0

lz dp, and substitute Eqs. (9), (10) and (11) into Eq. (4), we can obtain

0 k0h 1

chn l n

sh 2n

sh 2ne

v bqv2: 4

According to the denition of gas isothermal compressibility, the total elastic volume in control area of a certain gas well is as follows

V t cgVt p0

p t

4xfpscT

pTsc

qsc

dn:

With the boundary conditions, in the boundary of well controlling area, there is

Ue Ujrre

Z pe

bqscq2scea p p0

lh2l2 n

2chn

sh 2n

sh 2ne

5 where

V t p x2fshnechne xfwf

pea p p0

ne ln re=xf

re=xf

q 2 1

And the gas production can be expressed as

qsc t Bg

dV t

lz dp: 13

Solving Eq. (13), the pressure in an isopiestic ellipse with semi-major axis of xfchn and semi-minor axis of xfshn can be obtained

U n

4xfpscT

pTsc

dt cgVt

p t

h d

p t

cgp

x2fsh 2ne

2 xfwf

qsc

k0h

Z ne

chn l n

sh 2n

sh 2ne

dt : 6

Rate of ow through any isopiestic pressure section in the well controlling area is expressed as

qsc n; t Bg cgp

x2fsh 2ne

x2fsh 2n

2dn

bqscq2scea p p0

lh2

Z ne

chn l2 n

sh 2n

sh 2ne

h d

p t

dt : 7

The average pseudo-pressure in gas supply area is

R ne

0 sh2n ch2n

U n

Due to xfsh(2ne) 2wf, it can obtain Eq. (8) as

qsc n; t

x2fsh 2ne

x2fsh 2n

x2fsh 2ne

2xfwf

: 15

Substitute Eq. (15) into Eq. (14) to obtain Eq. (16)

U Ue

8xfpscT

pTscsh2ne

sh ne

ch ne

qsc t

qsc t : 8

Transform qsc(n, t) to qg in subsurface conditions

psczT

sh 2n

sh 2ne

Z ne

0 qsc sh2n ch2n

chn0 l n0

dn0

sh 2n0

sh 2ne

Z ne

n 1

qsc t : 9

For an isopiestic ellipse with a semi-major axis of xfchn and a semi-minor axis of xfshn, the ow velocity can be written as

Tscp 1

sh 2n

sh 2ne

sh 2n0

sh 2ne

bqscq2scea p p

lh2

sh2nch2n

2 chn0 l2 n0

dn0

Z ne

n 1

dn:

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36 J Petrol Explor Prod Technol (2016) 6:3338

With Eq. (16), we can obtain the pseudo-pressure Uw1 in

n = 0. And combine Uw1 and Eq. (16), we can obtain the follow equation

U Uw1 Aqsc Bq2sc 17

where

4xfpscT

pk0hTsc

chn l n

Z ne

0 1

sh 2n

sh 2ne

Z ne

0 sh2n ch2n

8xfpscT

pk0hTscsh 2ne

Z ne

n 1

chn0 l n0

dn0

sh 2n0

sh 2ne

4xfpscT

pTsc

bqscea p p0

lh2

Z ne

chn l2 n

2dn

sh 2n

sh 2ne

Z ne

0 sh2n ch2n

8xfpscT

pTsc

bqscea p p0

lh2sh 2ne

! dn:

Flow inside the fracture

After owing into the fracture, gas ows to borehole along the fracture. According to the ow formula in fracture, the below equation can be derived through integral calculation.

Uw1 Uw A1qsc B1q2sc 18

where

pscTxf

2 chn0 l2 n0

dn0

sh 2n0

sh 2ne

Z ne

n 1

bqscpscTxfea p p0

lTscw2fh2 :

Combining the ow equations outside the fracture, Eq. (17) and inside the fracture, Eq. (18); we can obtain the Eq. (19)

U Uw A A1

qsc B B1

q2sc: 19

With Eq. (19), the gas production and the open ow capacity, both taking stress-sensitive permeability into account, are derived respectively as

qsc

A A1

4kfTscwfh ; B1

A A1

24 B B1

U Uw

=0 =0.001MPa-1

=0.005MPa-1

=0.01MPa-1

=0.02MPa-1

2 B B1

BHP, MPa

qAOF

A A1

24 B B1

Applications and discussions

In this section, we programmed by MATLAB to calculate the productivity of fractured vertical gas well via the equations that have been derived above across an instance

where the parameters are from a tight gas in china and we also investigated the effects on gas production. The parameters of a certain tight gas in china mainly include initial pressure of 31.889 MPa, bottom-hole pressure of16.55 MPa, average formation pressure of 31.05 MPa, effective thickness of 18.75 m, initial absolute permeability of 0.089 9 10-3 lm2, formation temperature of 395.6 K, average uids viscosity of 0.027 mPa s, average gas compressibility factor of 0.89, fracture half-length of 400 m, fracture width of 0.005 m, fracture permeability of 40 lm2, wellbore radius of 0.1015 m, drainage radius of 500 m and stress-sensitive coefcient of 0.01 MPa-1.

Effect of stress-sensitive coefcient on IPR curves

Productivity curves of fractured vertical gas well in different stress-sensitive coefcients are plotted in Fig. 1. From which we can see that as the stress-sensitive coefcients increase, well productivity decreases and the decline rate of gas production is higher, meanwhile the productivity curves bend over to the pressure axis in the earlier stage with a greater bending. This is because the larger stress-sensitive coefcient indicates more intense of stress-sensitive effect to induce more pronounced decrease of permeability with the decline of reservoir pressure, which impacts signicantly on well productivity (McKee et al. 1988).

Effect of matrix permeability on IPR curves

Figure 2 contains productivity curves of fractured vertical gas well in different matrix permeabilities, where the gas production under the bottom-hole pressure of 0 MPa is the absolute open ow. Figure 2 shows that well productivity after fracturing increases apparently with the matrix

0.0 0.5 1.0 1.5 2.0 2.5

Gas Production, 104m3/d

Fig. 1 The IPR curves of different stress-sensitive coefcients

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J Petrol Explor Prod Technol (2016) 6:3338 37

0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8

Gas Production, 104m3/d

Matrix Permeability,10-3 m

Fracture conductivity, D.cm

10D.cm 15D.cm 20D.cm

25D.cm 30D.cm

0.059

0.069

0.079

0.089

0.1

BHP, MPa

0 0.0 0.5 1.0 1.5 2.0 2.5

Gas Production, 104m3/d

Fig. 2 The IPR curves of different matrix permeabilities

Fig. 4 The IPR curves of different fracture conductivities

xf=200m xf=250m xf=300m xf=350m xf=400m

BHP,MPa

0 0.0 0.4 0.8 1.2 1.6 2.0 2.4

Fig. 3 The IPR curves of different fracture half-lengths

Gas Production, 104m3/d

permeability. The open ow rate is 1.53 9 104 m3/d with

the matrix permeability of 0.059 9 10-3 lm2 and in

creases to 2.52 9 104 m3/d as the matrix permeability up to 0.10 9 10-3 lm2. It indicates that extremely low matrix permeability may lead to a poor gas production that cannot reach the industrial value for exploitation even if fracturing. On the contrary, the stimulation of fracturing has a positive effect in higher matrix permeability. Thus, those gas wells in relatively high matrix permeability should be chosen for fracturing.

Effects of fracture half-length and conductivity on IPR curves

Productivity curves of fractured vertical gas well with different fracture half-lengths are displayed in Fig. 3 and those with different fracture conductivities are plotted in

Fig. 4. In general, Figs. 3 and 4 give the impressions that gas production has a positive relation with the fracture half-length and conductivity. The increase of gas production is non-linear with the linear variations of fracture half-length and conductivity. It is signicant that the increment of gas production gradually ascends with the fracture half-length but descends with the fracture conductivity. Namely, increasing fracture half-length could more benet the gas production than increasing the fracture conductivity.

Effects of the factors on open ow

The effects of stress-sensitive coefcient, matrix permeability, fracture half-length and conductivity on the open ow capacity of gas well are investigated. The analysis is shown in Table 1. It demonstrates that stress-sensitive coefcient, matrix permeability, fracture half-length and conductivity impact the open ow in varying degrees. Open ow has a positive correlation with matrix permeability, fracture half-length and conductivity, but a negative relation with stress-sensitive coefcient. In these factors we investigated, matrix permeability and fracture half-length lead the rst and second place for most affecting the open ow, respectively. Meanwhile, fracture conductivity contributes the slightest impact on gas production. Thereby, an instruction is drawn that the longer fracture half-length and the choosing of gas well in favorable geological condition are primary for fracturing stimulation.

Conclusions

Productivity equations in pseudo-steady state for fractured vertical gas well embedded with the stress-sensitive effect have been derived and applied across an instance. And the effects of stress-sensitive coefcient, matrix permeability,

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38 J Petrol Explor Prod Technol (2016) 6:3338

Table 1 The open ow capacity of gas well with different sensitive parameters

Stress-sensitive coefcient (M Pa-1)

Open ow (104 m3/d)

Matrix permeability (10-3 lm2)

Open ow (104 m3/d)

Fracture half-length (m)

Open ow (104 m3/d)

0 2.51 0.059 1.53 200 1.45 10 2.15

0.001 2.49 0.069 1.77 250 1.50 15 2.22

0.005 2.38 0.079 2.02 300 1.62 20 2.260.01 2.26 0.089 2.26 350 1.85 25 2.29

0.02 2.05 0.10 2.52 400 2.26 30 2.30

Open ow (104 m3/d)

Fracture conductivity (lm2 cm)

fracture half-length and conductivity on gas production have been investigated, which draws a signicant instruction for fracturing stimulation. Gas production and open ow are affected in different extents by stress-sensitive coefcient, matrix permeability, fracture half-length and conductivity. They are positively correlative with matrix permeability, fracture half-length and conductivity, but negatively related with stress-sensitive coefcient. In the factors we investigated, matrix permeability leads the list of the effects on gas production and open ow. The gas well in extremely low-permeability area cannot achieve the production of industrial value for exploitation even after fracturing unless multistage fracturing was conducted or complicated fractures were generated.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http:// creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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Li S, Li X, Zeng ZL et al (2005) W ell production evaluation of vertical fracture in low permeable reservoir. PGODD 24(1):5456

Li W, Yu SQ, Zheng LK (2008) New method to determine the deliverability equation factor of tri-item equation for low permeability fracturing gas wells. PGODD 27(4):6163

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King Abdulaziz City for Science and Technology 2016

Abstract

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Based on non-Darcy flow theory in tight gas reservoirs, a new deliverability equation of fractured vertical gas well in pseudo-steady state is presented with the consideration of the stress-sensitive effect, and the open flow capacity calculation formula of gas well has been also derived. With the new deliverability equation, the effects of stress-sensitive coefficient, fracture parameters and matrix permeability on the productivity of gas well have been analyzed. The computation across an instance shows that due to the stress-sensitive effect, the IPR curves bend over to the pressure axis and the productivity of gas well is lower than that derived from the equation without consideration of stress-sensitive effect under the same pressure drop. As the stress-sensitive coefficient increasing, the well productivity becomes lower, the decline rate of production is higher and the IPR curve bends over in earlier stage with a greater bending. Besides, the productivity is affected by and has a positive correlation with the length and conductivity of fracture, namely that it becomes lower as the length and conductivity of fracture decreasing. Matrix permeability has an apparent impact on the productivity. If matrix permeability is extremely low, gas well cannot achieve the industrial production even after fracturing. As the matrix permeability increasing, stimulation results are significant.

Details

Title

Deliverability equation in pseudo-steady state for fractured vertical well in tight gas

Author

Mo, Shaoyuan; He, Shunli; Lei, Gang; Gai, Shaohua

Pages

33-38

Publication year

2016

Publication date

Mar 2016

Publisher

Springer Nature B.V.

ISSN

21900558

e-ISSN

21900566

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1007/s13202-015-0164-z

ProQuest document ID

1765333802

King Abdulaziz City for Science and Technology 2016

Deliverability equation in pseudo-steady state for fractured vertical well in tight gas

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