Easy technique for calculating productivity index of horizontal wells

Journal of Engg. Research Vol. 1 - (1) June 2013 pp. 335-358, 2013

Easy technique for calculating productivity index of horizontal wells JALAL F. OWAYED*, SALAM AL-RBEAWI**, AND DJEBBAR TIAB**

* Kuwait University, P.O. Box 5969, Safat 13060, Kuwait ** University of Oklahoma, 100 E. Boyd St., Norman, OK, 73019

ABSTRACT In recent years, horizontal well technology have evolved as the more favorable option in the state of Kuwait over the conventional vertical and deviated wells. Several models have been published in the literature to estimate the productivity index of horizontal wells. Generally, all of these models require two factors which are the shape and pseudo-skin factors. Also, most of these models require parameters that are not always easy to determine. This study presents easy and quick technique for calculating the productivity index of a horizontal well. The new technique has been established based on the instantaneous source solutions for the pressure response of a horizontal well. The pseudo-steady state ¯ow is expected to develop because the horizontal well is assumed to be acting in ®nite reservoirs. Two parameters were derived and their in¯uences on the productivity index were investigated. The ®rst one is the pseudo-skin factor due to asymmetry of a horizontal well. The second one is the shape factor group. The study emphasizes that the productivity index for horizontal wells are strongly aected by the two parameters: the shape factor group and the pseudo-skin factor. Shape factor group is mainly aected by the drainage area con®guration while pseudoskin factor is mainly aected by vertical penetration. The study con®rms that the productivity index is aected by the penetration ratio in the horizontal plane and reservoir geometry. In addition, square-shaped reservoir produces at maximum productivity index while channel-shaped reservoir produces at minimum productivity index. The study ®nds that wellbore eccentricity (wellbore location in the horizontal plane) does not aect the pseudo-skin factor and vertical penetration ratio does not aect the shape factor group. The results obtained from the new technique have been compared with the results from Babu & Odeh model and Economides model. Numerical examples will be included in the paper.

Keywords: Productivity index; horizontal well; pseudo-skin factor; shape factor group; wellbore eccentricity.

INTRODUCTION Productivity index of oil and gas wells represents one of the important parameters in reservoirs management and development. It is de®ned simply as the production rate corresponding to the pressure drawdown. Typically, it is

336

Jalal F. Owayed, Salam Al-Rbeawi, and Djebbar Tiab

estimated after long-time of production when the pseudo-steady state or steady state is developed. For a horizontal well, the productivity index is aected by several parameters such as the length of the horizontal wellbore, formation properties, ¯uid properties, and the geometry of reservoir drainage area. Several models have been established in the literatures for the productivity index of horizontal wells. Generally, all these models were included two factors: the shape and skin factor. Babu & Odeh (1988) introduced a detail study for the productivity index and proposed several analytical models for these two factors. These models have been derived based on the instantaneous solutions for the diusivity equation when the well reaches pseudo-steady state after long production time. Goode & Kuchuk (1991) presented the in¯ow performance models of horizontal wells for the cases of no-¯ow and constant-pressure boundary. They derived a general solution for the pseudo-steady state pressure drop of a horizontal well producing from a rectangular reservoir of uniform thickness. Mukherjee & Economides (1991) compared the productivity index of horizontal wells with those resulted from the hydraulic fractures. They stated that the horizontal wells based on the productivity index are not the best choices for all formations. Economides et al. (1996) developed new analytical model for the productivity index of horizontal wells taking into consideration the well con®gurations and reservoir anisotropy. They studied the eect of the wellbore orientation on its deliverability and established new horizontal plane shape factor. Helmy & Wattenbarger (1998) introduced new shape factors for wells produced at constant pressure. The new factors were developed for bounded reservoirs produced by wells operating at constant bottom hole pressure. They stated that the shape factor is a function of the inner boundary condition in addition to the reservoir shape and well location. The productivity index of horizontal wells de®antly increases as the wellbore length increase. However, at a certain length, the productivity index is no longer increases with length because of the friction losses. Valko & Blasingame (2000) generalized the concept of the pseudo-steady productivity index for the case of multiple wells producing from or injecting into a closed rectangular reservoir of constant thickness. They explained that the multi-well productivity index provides a useful analytical tool which expands the capabilities of the reservoir/production engineers. Chao & Shah (2001) presented an approach for speci®c productivity index to predict the production rate considering the friction losses in long horizontal wells. Levitan et al. (2004) presented the results of comprehensive analysis of BP horizontal wells productivity and completion performance data. They introduced the well-productivity coecient as an attempt to remove the in¯uences of reservoir and ¯uid properties and to present the well productivity index in dimensionless form. The most challenge issue in the mathematical formulation for the productivity index and the in¯ow performance of horizontal

Easy Technique for Calculating Productivity Index of Horizontal Wells

337

wells is the existing of two-phase ¯ow. Wiggins & Wang (2005) investigated the in¯ow performance data generated for a horizontal wells producing in a solution gas drive reservoirs. Based on their simulated data, two in¯ow performance relationships have been generated one was the generalized IPR and the other one was the IPR that is a function of reservoir recovery. Tang et al. (2005) studied the eects of formation damage and high-velocity ¯ow on the productivity of perforated horizontal wells. They developed a comprehensive semi-analytical model in their study. The model incorporates the additional pressure drop caused by formation damage and high-velocity ¯ow into a semianalytical coupled wellbore/reservoir model. Diyashev & Economides (2006) presented for the ®rst time the idea of the dimensionless productivity index as a general approach to wells evaluation. They analyzed more than 100 wells drilled in Siberia and compared the actual productivity index with the calculated one. Ding et al. (2006) studied the nearwellbore formation damage eects on well performance. They developed numerical model for the productivity index for the cases of underbalance and overbalance drilling. The eect of selective perforated horizontal wells was investigated by Yildiz (2006). He stated that the changes in ¯ow rate, pseudosteady state productivity, and cumulative production for a given perforation design can be computed using the solution of his analytical model. He explained that the performance of wells treated with oriented perforation would be in¯uenced by the orientation of the perforation and the reservoir anisotropy. Aulisa et al. (2009) addressed the eect of nonlinearity of ¯ow on the value of productivity index. They presented a rigorous framework in their study based on several sets of experiments to measurer the index of a well for nonlinear Forchheimer ¯ow. The developed technique combines the generalized Darcy equation and easy-to-apply numerical and analytical methods. Tabatabaei & Ghalambor (2011) introduced a new method to predict performance of horizontal and multilateral wells. They derived semi-analytical model that couples the ¯ow from box-shaped drainage volume to the ¯ow in the wellbore. They stated that the ignoring of wellbore pressure drop may cause overestimation for the productivity index while ignoring the ¯uid-in¯ow eect can result in the underestimation of well productivity. Hagoort (2011) presented two exact analytical formulas for the semi-steady state productivity index of an arbitrarily positioned well in a closed rectangular reservoir. The ®rst model is for the constant rate and the second one is for the constant pressure.

MATHEMATICAL MODELING Horizontal well technology has added signi®cant values to the petroleum industry in terms of increased deliverability, injectivity, and increased ultimate recovery. The great contact area between horizontal wellbore and rock matrix

338


allows reservoir ¯uids to ¯ow freely to the wellbore. The production rate from horizontal wells is a function of total surface area of the perforated sections and the pressure drop. For a constant production rate, the pressure drop at any point in the formations depends on several parameters: permeability, homogeneity, isotropy, formation drainage area con®guration, reservoir ¯uid properties, and wellbore length. The production rate and the pressure drop at the wellbore are the two items required for estimating the productivity index. Generally, the simple model for the productivity index can be written as:

J

q

1

1P

For constant sandface production rate, the pressure drop is the main parameter that aects the productivity index. Pressure drop can be modeled using the instantaneous source solutions as:

q Pxm ; ym ; zm ; t; zw ; Lw ; h c

Zt

Sxyz xm ; ym ; zm ; t ÿ ; zw ; Lw ; hd

2

0

In dimensionless form, the pressure drop a horizontal well acting in a ®nite reservoir using uniform ¯ux solution can be written as:

PD xeD LD

tRD

("

1

0

11 X

4

yeD n1 n

expÿ

n yeD D

2 2 2

4

sinn

yeD 2

cosn

2 2 2 1 P n x D xwD n eD cosn 2 cos 2 xD xeD xwD 2 1 2 expÿ 4 n1 1 P 1

2 expÿn

LD D

2 2 2

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ywD 2

#

n

cos 2 yD yeD ywD 2 3

cosn zwD cosn zDLD zwD d D

The above model represents the sum of the pressure drop due to transient period and the pressure drop due to pseudo-steady state. It can be written as:

ZD

PD 2tDA xeD LD

XD YD ZD XDYD XDZD XDYD ZD d D PDi PDa 4

0

tDA

kx ct A

A 4xe ye

5 6

In equation (4), PDi is the dimensionless pressure drop from initial reservoir pressure to average reservoir pressure that represents the transient period. It can be written as:

PDi

2

tDA

7

PDa is the dimensionless pressure drop between average reservoir pressure

339


and reservoir pressure at any point and any time. This pressure drop represents the pseudo-steady state. It can be written as:

ZD

PDa xeD LD

XD YD ZD XDYD XDZD YD ZD XDYDZD d D

8

0

The model given in equation (8) can be solved for long time pseudo-steady state. It can be written as follows: 9 PDa CHF Sp

CHF is de®ned as the shape factor group, and Sp is de®ned as the pseudo-skin factor. Appendix A shows the derivation of the above pressure drop models and the dimensionless groups (XD ; YD ; ZD ; XD YD ; XD ZD ; YD ZD andXD YD ZD .

SHAPE FACTOR GROUP FOR HORIZONTAL WELLS The dimensionless pressure of horizontal wells can be approximated for long time to the following model: 1 4A PD 2tDA 10 2 2

CA r w

ln

The term

1 2

ln

4A

CA r2w

in equation (10) is the shape factor group because it contains

the shape factor CA . From equation (8), it can be recognized that seven instantaneous

YD ZD ; XD YD ZD represent the eect of the vertical penetration. While the three solutions XD ; YD ; XD YD represent the

solutions exist. Four of these solutions ZD ; XD ZD

;

eect of the reservoir drainage area in the horizontal plane where the shape factor is determined by this area. Therefore, the shape factor group from equation (8) can be written as:

ZD

CHF

xeD LD

XD YD XDYD d D

11

0

For long time approximation (pseudo-steady state), the group can be de®ned as:

2

1 P

yeD

ywD

m

3

6 3 yeD n1 m1 mn2 x2 m2 y2 sinm 2 cosm 2 cos 2 yD yeD ywD 7 6 7 eD eD 6 7 1 X xwD n 8 1 xwD n 6 7 7 x cos cos n x x D eD wD CHF xeD LD 66 cosn 2 cos 2 xD xeD xwD 2 2 2 7 2 2 x eD n1 n 6 7 1 6 7 16 X 1 yeD ywD m 4 5 3 y3 m3 sinm 2 cosm 2 cos 2 yD yeD ywD array eD m1 32

1

;

12

The results of the shape factor group given in equation (12) are plotted in

340


Figures 1 to 6, for dierent wellbore length and dierent reservoir 0:5 con®gurations. They are designed for symmetrical horizontal wells zwD where the well extends at the midpoint of the formation height.

It can be seen that the shape factor group has two linear relationships for dierent reservoir geometries. The ®rst linear relationship has negative unit slope lines when the reservoir boundary normal to the wellbore is less than one xeD < 1:0 :This fact indicates that the horizontal wells that extend in squareshaped or in wide rectangular reservoirs have the highest productivity index. The second linear relationship has positive unit slope line when the reservoir boundary normal to the wellbore is more than one xeD > 1:0 :This fact indicates that the productivity index of narrow or channel reservoirs is low. In addition, the maximum productivity index is obtained when the wellbore fully 1:0 : penetrates the formation in the horizontal plane yfD

In general, the following conclusions about shape factor group are correct: 1 - Shape factor group increases as the wells partially penetrate the formation in the horizontal plane. Therefore the productivity index increases as the wells fully penetrate the formation horizontally. 2 - Shape factor group increases as the wellbore length increases. However, the productivity index in this case depends on both values of shape factor group and the pseudo-skin factor. >сϭ͘Ϭ

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PSEUDO-SKIN FACTOR The pseudo-skin factor for horizontal wells can be found from the instantaneous solutions that include the vertical direction. It represents the eect of vertical penetration (i.e. the point in the formation height where the horizontal wells are extending) on pressure behavior. Therefore, the pseudo-skin factor can be written as:

ZD

Sp

xeD LD

YDZD ZD XD ZD XDYDZD d D

0

In dimensionless form, the pseudo-skin factor is:

13

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2

1 X

yeD

ywD

3

m

sinm 2 cosm 2 cos 2 yDyeD ywD 7 6 3 y 7 6 eD n1 m1 l1 mn2 x2eD m2 y2eD 4l2 L2D 7 6 7 6 xwD n 7 6 x cos n cos x x cos l z cos l z L z D eD wD wD D D wD 7 6 2 2 7 6 7 6 32 X 1 1 yeD ywD m 7 6 7 6 3 y cos m cos y y sin m D eD wD 7 6 yeD m1 l1 mm2 y2eD 4l2 L2D 2 2 2 7 6 7 6 7 6 Sp xeD LD 6 coslzwD coslzD LD zwD 7 7 6 7 6 1 7 6 16 X 1 xwD n 7 6 x cos cos n x x D eD wD 7 6 2 2 2 2 2 7 6 n1 l1 n2 xeD 4l2 LD 7 6 7 6 7 6 coslzwD coslzD LD zwD 7 6 7 6 1 7 6 2 X1 5 4 2 L2 l2 cosnzwD cosnzD LD zwD D l1 64

;

1

;

;

14

;

The pseudo-skin factor in the above model represents the eect of the partial penetration in the vertical direction and the location of the horizontal wellbore. For a well extending in the midpoint of the formation thickness zwD 0:5 ; the following comments can be inferred from Fig. (7) to Fig. (12):

1 - Pseudo-skin factor increases as the well penetrates partially the formation in the horizontal plane. Physically, the partial penetration in the horizontal plane can be explained as the loss of production that can be obtained if the well fully penetrates the formation. 2 - Pseudo-skin factor increases slightly as the width of formation or the distance to the boundary normal to the wellbore increases. 3 - Pseudo-skin factor decreases as the length of the wellbore increases. Even though the pseudo-skin factor is de®ned as a function of the partial penetration in the vertical direction; the formation geometry and the wellbore length have some impact on it. Mathematically this is true because of the eect of the instantaneous solution of the vertical direction ZD and the compound eect of the vertical direction instantaneous solution and the two horizontal instantaneous solutions XD ZD ; YD ZD ; XD YD ZD :

343


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EFFECT OF PARTIAL PENETRATION IN HORIZONTAL PLANE The penetration ratio of horizontal wells in the horizontal plane represents the length of the formation that can be reached by the wellbore. This ratio, represented by yeD , has great similar impact on both shape factor group and pseudo-skin factor as shown in Fig. (13) and Fig. (14). As the wellbore reaches the tips of the formation in the horizontal plane, the shape factor group and the pseudo-skin factor will be in their minimum values. This leads to high productivity index. Conversely, when the horizontal well partially penetrates the formation, both shape factor group and pseudo-skin factor will increases and the productivity index will decrease.

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penetration ratios

penetration ratios

EFFECT OF PARTIAL PENETRATION IN VERTICAL DIRECTION The location of the horizontal wells with respect to the formation height or the point where the well is extending in the formation has some impact on pseudoskin factor as shown in Fig. (15). However it has no impact on the shape factor group as shown in Fig. (16). It can be seen from Fig. (15) that pseudo-skin factor increases slightly when the well extends at a point above or below the midpoint of formation height. Therefore symmetrical wells are the best choice for the productivity index.

Easy Technique for Calculating Productivity Index of Horizontal Wells >сϭ͘Ϭ

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horizontal well vertical locations

EFFECT OF ECCENTRICITY The eccentricity refers to the well location in the reservoir horizontal plane. The optimum location of the well is the middle of the formation width where it has equal distance to the boundary parallel to the wellbore direction. Fig. (17) shows the eect of eccentricity for the well in the middle of the formation xwD 1:0 or the one close to the boundary xwD 0:5; 1:5 on shape factor group. It can be seen that the shape factor group is changed signi®cantly with the location of the well in the horizontal plane due to the impact of the drainage area on shape factor group. Fig. (18) shows the eect of eccentricity on the pseudo-skin factor. It can be seen that the eccentricity does not have great impact on the pseudo-skin factor due to the fact that the pseudo-skin factor is aected by the penetration and location in the vertical direction rather than the horizontal direction.

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eccentricities

eccentricities

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EFFECT OF WELLBORE LENGTH The wellbore length has two dierent impacts on shape factor group and pseudo-skin factor. The Shape factor group is linearly proportional with the wellbore length. The slope of the line is 1 as shown in Fig. (19). The pseudo-skin factor is conversely aected by the wellbore length as shown in Fig. (20). It decreases slightly as the wellbore length increases. The productivity index in this case depends on the compound values of shape factor group and pseudo-skin factor.

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PSEUDO-PRODUCTIVITY INDEX The pseudo-skin factor can be de®ned as:

Jp

2C 1S S 3 HF p m

15

Sm is the mechanical skin factor. Fig. (21) and Fig. (22) show the pseudo-productivity index for two horizontal wells LD 4 and LD 64 respectively. It is clear that the productivity index can be increased if the penetration ratio in the horizontal plane increases.

Easy Technique for Calculating Productivity Index of Horizontal Wells >сϰ͘Ϭ͕ƐŵсϬ͘Ϭ

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Fig. 21. Pseudo-productivity index for dierent Fig. 22. Pseudo-productivity index for dierent reservoir con®gurations


PRODUCTIVITY INDEX The productivity index can be written as:

J CJp

2C CS S 3 HF p m

16

p kx kz 2ye C

where

17

141:2B

The mechanical skin factor negatively aects the productivity index. It increases the resistance of the formation to the ¯ow of ¯uid from the drainage area close to the wellbore. Fig. (23) and Fig. (24) show the eect of the mechanical skin factor on the pseudo-productivity index for dierent horizontal wellbore. >сϰ͘Ϭ͕Ɛŵсϰ͘Ϭ

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Fig. 23. The eect of mechanical skin factor

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Fig. 24. The eect of mechanical skin factor

348


APPLICATIONS

Exampl-1

Table 1 includes the required data of the formation and reservoir ¯uid properties (Economides et al. 1996). The well is extending at the midpoint of the formation either in the vertical plane or in the horizontal plane.

Table 1. Formation and well data for Example-1 Pay zone thickness

20 ft

Oil viscosity

1.0 cp

Horizontal wellbore length (2Lw Wellbore radius (rw

Mechanical skin factor (sm

1500 ft 0.4 ft

0

Reservoir Permeability: kx

10 md

ky

10 md

kz

10 md

Formation volume factor Bo

1.25 res-bbl./STB

Formation width (2ye

2000 ft

Formation length (2xe

4000 ft

0 75 xeD 0 375 sm 0 0 LD 37 5

yeD

:

:

:

:

Fig. (25) shows the plot of (CHF ) hat has been generated for the above conditions. It can be found that:

CHF

128 5 :

Fig. (26) shows the plot of (Sp conditions. It can be found that:

SP

that has been generated for the above

2 15 :

Both CHF ; SP are calculated using MATLAB simulator due to the diculties of using the mathematical model for the two parameters given in Eqs. (12) and (14) respectively. Using equation (17):

C 113:3 Using equation (16):

349


J 0:87STB/D/psi The calculated productivity index by Economides et al. 1996 is 0.88. ϭϬϬϬ

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Fig. 25. Shape factor group for Example-1

Fig. 26. Pseudo-skin factor for example-1

Exampl-2 Table 2 includes the required data of the formation and reservoir ¯uid properties (Lee et al. 2003). The well is extending at the midpoint of the formation in the vertical direction. Fig. (27) shows the well and reservoir geometry.

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Fig. 27. Well and reservoir geometry for Example-2

350


Table 2. Formation and well data for Example-2 Pay zone thickness

100 ft

Oil viscosity Horizontal wellbore length (2Lw Wellbore radius (rw

Mechanical skin factor (sm

1.0 cp

1000 ft 0.25 ft

0

Reservoir Permeability: kx

200 md

ky

200 md

kz

50 md

Formation volume factor Bo

1.25 res-bbl./STB

Formation width (2ye

2000 ft

4000 ft

Formation length (2xe

05 ywD 0 75 xeD 0 25 xwD 1 5 zwD 0 5 sm 0 0 LD 2 5 yeD

:

:

:

:

:

:

:

Fig. (28) shows the plot of (CHF ) hat has been generated for the above conditions. It can be found that:

CHF

25 8 :

Fig. (29) shows the plot of (Sp conditions. It can be found that:

SP

that has been generated for the above

75 :

Using equation (17):

C 113:3 Using equation (16):

J 33:9STB/D/psi The calculated productivity index by Lee et al. 2003 is 33.5. The dierence between the two values results because the model that used by Lee et al. (2003) in their calculation is the model presented by Babu & Odeh (1988). In this model, both (ZD ) and (YD ZD ) were assumed equal to zero.

Easy Technique for Calculating Productivity Index of Horizontal Wells ϭϬϬϬ

351

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Fig. 28. Shape factor group for Example-2.

Fig. 29. Pseudo-skin factor for example-2.

CONCLUSIONS 1 - The productivity index for horizontal wells can be calculated from the pseudo-steady state pressure model using long time approximation. Shape factor group and pseudo-skin factor are the main parameters in the productivity index model. 2 - Shape factor group is mainly aected by the drainage area con®guration. 3 - Pseudo-skin factor is mainly aected by the vertical penetration. 4 - The productivity index of horizontal wells is strongly aected by the penetration ratio in the horizontal plane. The high penetration ratio, the high productivity index. 5 - Square-shaped reservoir produces at maximum productivity index. Channel-shaped reservoir produces at minimum productivity index. 6 - Wellbore eccentricity does not aect the pseudo-skin factor. Vertical penetration ratio does not aect the shape factor group.

NOMENCLATURES B

formation volume factor, res-bbl/STB

c

compressibility factor, psiÿ1

CA

shape factor, dimensionless

CHF

shape factor group, dimensionless

h

formation height, ft

J

productivity index, dimensionless

Jp

Pseudo-productivity index, dimensionless

Lw

Wellbore half length, ft

352


kx ky kz

permeability in the X-direction, md permeability in the Y-direction, md permeability in the Z-direction, md pressure dierence, psi total ¯ow rate, STB/D ¯ow rate, STB/D wellbore radius, ft pseudo-skin factor, dimensionless mechanical skin factor, dimensionless time, hrs reservoir half width, ft reservoir half length, ft X-Coordinate of the monitoring point Y-Coordinate of the monitoring point Z-Coordinate of the monitoring point X-Coordinate of the production point (wellbore) Y-Coordinate of the production point (wellbore) Z-Coordinate of the production point (wellbore)

1P Q q rw Sp Sm t xe ye xm ym zm xw yw zw

Greek Symbols

porosity viscosity, cp diusivity dummy variable of time

APPENDIX A - MODELS DERIVATION Consider a horizontal well having wellbore length (2Lw) extends in a ®nite reservoir as shown in Fig. (A-1). The formation thickness is (h). The formation has a length of (2ye ) and a width of (2xe ). The unsteady state pressure drop created by production from the wellbore at any point in the reservoir xm ; ym ; zm is:

Pxm ; ym ; zm ; t; zw ; Lw ; h

q c

Zt

Sxyz xm ; ym ; zm ; t ÿ ; zw ; Lw ; hd

A ÿ 1

0

where Sxyz xm ; ym ; zm ; t; zw ; Lw ; h is the instantaneous source function that can be determined as follows: Sxyz xm ; ym ; zm ; t; zf ; hf ; xf ; h Sx x; t Sy y; t S z z; t A 2

2 2

ÿ

q is the ¯uid withdrawal per unit surface area per unit time can be determined as follows:

q Q2Lw

A ÿ 3


353

h

z Y

2Lw 2xe

x

2ye

Fig. A-1. Horizontal well extends in a ®nite reservoir Sx x; t can be written as follows: Sx; t

"

1

2

1

2xe

1 1 X n1

n

expÿ

n x t 4x2e

2 2 2

n 2xxw e

cos

n 2xx e

cos

#

A ÿ 4

Sy y; tcan be written as follows:

"

1 Lw 4ye X 1 Sy; t 1 n ye Lw n 1

expÿ

n y t 4y2e

2 2 2

n 2Lyw e

n 2yyw e

sin

cos

n x t h2

n zhw

Sz z; t can be written as follows: Sz; t

1

"

1

h

2

1 X n1

expÿ

2 2 2

cos

cos

#

n 2yy e #

n hz

cos

A ÿ 5

A ÿ 6

Substituting Eqs. (A-4), (A-5), and (A-6) in Eq. (A-2) ®rst and then substitute Eqs. (A-2) and (A-3) in Eq. (A-1) gives the pressure drop in dimensionless form as:

PD xeD LD

tRD 0

("

1

4

11 X

yeD n1 n

expÿ

n yeD D

2 2 2

4

yeD

sinn 2 cosn

2 2 2 1 P expÿ n x4eD D cosn xwD2 cosn2 xDxeD xwD 2 n1 1 P 2 2 2 1 2 expÿn LD D cosnzwD cosnzD LD zwD dD n1

x ÿ xw xD Lw

2

n

#

cos 2 yD yeD ywD 2 A ÿ 7

1 2

where:

ywD

s ky kx

A ÿ 8

354


yD

y ÿL yw

z ÿ zw zD Lw rwD

Lrw w

PD

2

ky t ct L2w

s ky kx

A ÿ 11

xwD

xxw

A ÿ 13

ywD

yyw

A ÿ 14

e

e

s kz ky

A ÿ 15

Lyw

A ÿ 16

e

Lxw e

s kx ky

A ÿ 17

Lyt where y

p kx kz 2ye Pxm

A ÿ 10

A ÿ 12

xeD

ky kz

zhw

yeD

s

zwD

Lw LD h

tD

A ÿ 9

w

2

;

w

ky ct

A ÿ 18

; ym ; zm ; t; zw ; Lw ; h q

A ÿ 19

To solve the above model given in Eq. (A-7):

PD xeD LD 2

Z

D

1 XD YD ZD XDYD XD ZD YD ZD XD YDZD d D A ÿ 20

0

In this equation, PDi is the dimensionless pressure drop from initial reservoir pressure to average reservoir pressure. It can be given as: A 21 PDi xeD LD tD

ÿ

while PDa is the dimensionless pressure drop between average reservoir pressure and reservoir pressure at any point and any time. It is given by:


ZD

355

PDa

xeD LD

XD YD ZD XDYD XD ZD XD YDZD d D A ÿ 22

0

where:

YD

yeD

expÿ m m

1

1 X 1

4

yeD D

2 2 2

4

2

2 2

cson

sinm

L2D D

yeD 2

cosl

xwD 2

cosn2 xDxeD xwD A ÿ 23

ywD

cson

zwD

2

cosm2 yD yeD ywD A ÿ 24

cosl zDLD zwD

m yeD n xeD D sinm yeD cosm 4 2 2 2

yeD m1 n1

2 2

cosn

xwD 2

ywD

4

2

2

cosl zwD cosl zD LD zwD

A ÿ 27

1 1 X

A ÿ 26

2 2 2 2 2 1 expÿ n xeD 4l LD D cosn xwD cosn x x x X D eD wD 1

A ÿ 25

m yD yeD ywD cos 2 2

cosn2 xDxeD xwD

m yeD 4l LD D sinm yeD cosm exp ÿ yeD m 4 2 cosl zwD cosl zD LD zwD 8

2

2 2

2 2

1

ywD

m yD yeD ywD A ÿ 28 cos 2 2

XD YD ZD

xeD D

2 2 2

expÿl

1 1 expÿ 8 XX

XD ZD 4

YD ZD

1

2

ZD

XD YD

expÿ n

1 1 X

4

1 X

2

XD

1 1 X

expÿ m yeD n 4xeD 4l LD D sinm y2eD cosm ywD2 cosm2 yD yeD ywD yeD m A ÿ 29 xwD n cos 2 xDxeD xwD cosl zwD cosl zDLD zwD cson 2 16

1

2

2 2

2 2

2 2

Integrating and approximating (for long time):

XD

8

1 X

1

xeD n1 n

2 2

2

cosn

xwD 2

cosn2 xDxeD xwD

A ÿ 30

356


YD

1 X

16

yeD m1 m

3

3

1 1 X

2

L2D l1 l2

yeD

cosm 2

cosl

ywD

16

XD ZD

2

yeD m1 l1 mm yeD ;

4l L sinm

2 2

D

cosn

yeD

cosm 2

1 X

1

2

2

cosm2 yDyeD ywD

xwD

A ÿ 33

cosn2 xDxeD xwD

ywD

A ÿ 34

m yDyeD ywD cos 2 2

sinm

yeD n1 m1 l1 mn2 x2eD m2 y2eD 4l2 L2D m yDyeD ywD n xwD n xDxeD xwD

cos

ywD

A ÿ 35

3

cosl zwD cosl zDLD zwD 64

cosm

2 n xeD 4l LD cosl zwD cosl zDLD zwD 2

2

1

XD YD ZD

1

2 2

n1;l1

2 2

A ÿ 32

2

1 X

1 X

32 3

cos

2

sin

;

2

cosm2 yDyeD ywD A ÿ 31

cosl zDLD zwD

zwD

1 m y2eD yeD n1 m1 mn2 x2eD m2 y2eD n xwD n xD xeD xwD

cos

YD ZD

1 X

32

sinm

2

ZD

XD YD

1

3 3

;

;

cos

2

cos

2

yeD 2

cosm

ywD 2

cosl zwD cosl zD LD zwD

A ÿ 36

REFERENCES Aulisa, E., Ibragimov, A. & Walton, J.R. 2009.

A New Method for Evaluating the Productivity Index of Nonlinear Flows. SPE Journal, 693-706.

Babu, D.K. & Odeh, A.S. 1988. Productivity of a horizontal Well. SPE 18298 presented at the 63rd Annual Technical Conference and Exhibition held in Houston, TX, USA, 2-5 October.

Cho, H. & Subhash, S.N. 2001. Prediction of Speci®c Index for Long Horizontal Wells. SPE 67237

presented at the SPE Production and Operation Symposium held in Oklahoma City, OK, USA, 24-27 March.

Ding, Y., Herzhaft, B. & Renard, G. 2006.

Near-Wellbore Formation Damage Eects on Well Performance: A comparison Between Underbalance and Overbalance Drilling. SPE Production and Operation, 51-57, SPE 86588.

Diyashev, I. & Economides, M.J. 2006.

The Dimensionless Productivity Index as a General approach to Well Evaluation. SPE Production and Operation, 394-401.


Economides, M.J., Brand, C.W. & Frick, T.P. 1996.

357

A Well Con®guration in Anisotropic

Reservoirs. SPE Formation Evaluation, 257-262.

Goode, P.A. & Kuchuk, F.J. 1991.

In¯ow Performance of Horizontal Wells. SPE reservoir

Engineering, 319-323.

Hagoort, J. 2011. Semisteady-State Productivity of a Well in a Rectangular Reservoir Producing at constant Rate or Constant Pressure. SPE Reservoir Evaluation and Engineering, 677-686.

Helmy, M.W. & Wattenbarger, R.A. 1998.

New Shape Factor for Well Produced at Constant Pressure. SPE 39970 presented at the SPE Gas Technology Symposium held in Calgary, Canada, 15-18 March.

Lee, J., Rollins, J.B. & Spivey, J.P. 2003.

Pressure Transient Testing.The Society of Petroleum

Engineer, Richardson, TX, USA.

Levitan, M.M., Clay, P.L. & Gilchrist, J.M. 2004.

Do Your Horizontal Wells Deliver Their Expected Rates? SPE Drilling and Completion, 40-45.

Mukherjee, H. & Economides, M.J. 1991. A Parametric Comparison of Horizontal and Vertical Well Performance. SPE Formation Evaluation, 209-216.

Tabatabaei, M. & Ghalambor, A. 2011. A New Method to Predict Performance of Horizontal and Multilateral Wells. SPE Production and Operation, 75-87.

Tang, Y., Yildiz, T. & Ozkan, E. 2005. Eects of Formation Damage and High-Velocity Flow on

the Productivity of Perforated Horizontal Wells. SPE Reservoir Evaluation and Engineering, 315-324.

Valko, P.P., Doublet, L.E. & Blasingame, T.A. 2000.

Development and Application of the Multiwell Productivity Index (MPI) . SPE Journal, V.5(1), 21-31.

Wiggins, M.L. & Wang, H.S. 2005.

A Two Phase IPR for Horizontal Oil Wells. SPE 94302 presented at the SPE Production and Operation Symposium held in Oklahoma City, OK, USA, 17-19 April.

Yildiz, T. 2006.

Productivity of Selectively Perforated Horizontal Wells. SPE Production and Operation, 75-80.

Submitted : 19/12/2012 Revised : 11/3/2013 Accepted : 20/3/2013

358


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