Easy technique for calculating productivity index of horizontal wells

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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.
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 a€ected by the two parameters: the shape factor group and the pseudo-skin factor. Shape factor group is mainly a€ected by the drainage area con®guration while pseudoskin factor is mainly a€ected by vertical penetration. The study con®rms that the productivity index is a€ected 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 a€ect the pseudo-skin factor and vertical penetration ratio does not a€ect 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

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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 a€ected 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 di€usivity 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 e€ect 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 coecient 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 e€ects 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 e€ects on well performance. They developed numerical model for the productivity index for the cases of underbalance and overbalance drilling. The e€ect 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 e€ect 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 e€ect 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

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

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:



q

…1†

1P

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

q P…xm ; ym ; zm ; t; zw ; Lw ; h† ˆ c

Zt

Sxyz …xm ; ym ; zm ; t ÿ ; zw ; Lw ; h†d

…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 ˆ xeD LD



tRD

("

1

0

‡

11 X

4 

yeD nˆ1 n

exp…ÿ



n yeD D

2 2 2

4

† sin…n



yeD 2

† cos…n

 2 2 2 1 P  n x D xwD n eD † cos…n 2 † cos… 2 …xD xeD ‡ xwD †† 2 1 ‡ 2 exp…ÿ 4 nˆ1   1 P 1

‡ 2 exp…ÿn

LD D †

2 2 2

nˆ1





ywD 2

#

n

† cos… 2 …yD yeD ‡ ywD †† 2 …3†

cos…n zwD† cos…n …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 ˆ 2tDA ‡ xeD LD

‰XD ‡ YD ‡ ZD ‡ XDYD ‡ XDZD ‡ XDYD ZDŠ d D ˆ PDi ‡ PDa …4† 

0

tDA

ˆ

kx ct A

A ˆ 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

ˆ2



tDA

…7†

PDa is the dimensionless pressure drop between average reservoir pressure

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Easy Technique for Calculating Productivity Index of Horizontal Wells

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

ZD 

PDa ˆ xeD LD

‰XD ‡ YD ‡ ZD ‡ XDYD ‡ XDZD ‡ YD ZD ‡ XDYDZD Š 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

ˆ

‡

…†

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

ˆ

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 e€ect of the vertical penetration. While the three solutions …XD ; YD ; XD YD † represent the

…

solutions exist. Four of these solutions ZD ; XD ZD

;

e€ect 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

ˆ



xeD LD

‰XD ‡ YD ‡ XDYD Š 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 nˆ1 mˆ1 m…n2 x2 ‡ m2 y2 † sin…m 2 † cos…m 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 ˆ xeD LD 66 cos…n 2 † cos… 2 …xD xeD ‡ xwD †† ‡ 2 2 2 7 2 2  x eD nˆ1 n 6 7 1 6 7 16 X 1 yeD ywD m 4 5 ‡ 3 y3 m3 sin…m 2 † cos…m 2 † cos… 2 …yD yeD ‡ ywD ††array eD mˆ1 32



1

;

…12†

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

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

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

…

ˆ

†

It can be seen that the shape factor group has two linear relationships for di€erent 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

†

…

†

…

…

ˆ

†

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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Fig. 2. Shape factor group for di€erent

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Easy Technique for Calculating Productivity Index of Horizontal Wells

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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 e€ect 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

ˆ



xeD LD

‰YDZD ‡ ZD ‡ XD ZD ‡ XDYDZD Š d D 

0

In dimensionless form, the pseudo-skin factor is:

…13†

342

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

2

1 X

yeD

ywD

3

m

sin…m 2 † cos…m 2 † cos… 2 …yDyeD ‡ ywD†† 7 6 3 y 7 6 eD nˆ1 mˆ1 lˆ1 m…n2 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 mˆ1 lˆ1 m…m2 y2eD ‡ 4l2 L2D † 2 2 2 7 6 7 6 7 6 Sp ˆ xeD LD 6 cos…lzwD † cos…l…zD 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 nˆ1 lˆ1 …n2 xeD ‡ 4l2 LD † 7 6 7 6 7 6 cos…lzwD † cos…l…zD LD ‡ zwD †† 7 6 7 6 1 7 6 2 X1 5 4 ‡ 2 L2 l2 cos…nzwD † cos…n…zD LD ‡ zwD †† D lˆ1 64

;

1

;

;

…14†

;

The pseudo-skin factor in the above model represents the e€ect 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):

…

ˆ

†

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 e€ect of the instantaneous solution of the vertical direction ZD and the compound e€ect of the vertical direction instantaneous solution and the two horizontal instantaneous solutions XD ZD ; YD ZD ; XD YD ZD :

… †

…

†

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Easy Technique for Calculating Productivity Index of Horizontal Wells

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Fig. 8. Pseudo-skin factor for di€erent

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Fig. 10. Pseudo-skin factor for di€erent

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Fig. 12. Pseudo-skin factor for di€erent

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344

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

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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Fig. 14. Pseudo-skin factor for di€erent

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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.

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Fig. 15. Pseudo-skin factor for di€erent

Fig. 16. Shape factor group for di€erent

horizontal well vertical locations

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 e€ect 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 e€ect 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 a€ected by the penetration and location in the vertical direction rather than the horizontal direction.

…

ˆ

…

†

>сϴ͘Ϭ

ˆ †

>сϴ͘Ϭ

ϭϬϬ

ϭϬϬϬϬ

ϭϬϬϬ

,&

WϭϬ

^



ϭϬϬ

džǁсϬ͘ϱ͕ϭ͘ϱ

džǁсϬ͘ϱ͕ϭ͘ϱ

džǁсϭ͘Ϭ

džǁсϭ͘Ϭ

ϭϬ

ϭ Ϭ͘ϭ

ϭ

džĞ

ϭϬ

ϭϬϬ

Ϭ͘ϭ

ϭ

džĞ

ϭϬ

Fig. 17. Pseudo-skin factor for di€erent

Fig. 18. Shape factor group for di€erent

eccentricities

eccentricities

ϭϬϬ

346

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

EFFECT OF WELLBORE LENGTH The wellbore length has two di€erent 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 a€ected 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.

džĞсϭ͘Ϭ͕LJĞсϬ͘ϱ

džĞсϭ͘Ϭ͕LJĞсϬ͘ϱ

ϭϬϬϬ

ϭϬϬ

ϭϬϬ

&,

Ɖ ^



ϭϬ

ϭϬ

ϭ

ϭ ϭ

ϭϬ

ϭϬϬ

ϭ

ϭϬ

>

ϭϬϬ

>

Fig. 19. Shape factor group for di€erent

Fig. 20. Pseudo-skin factor for di€erent

wellbore lengths

wellbore lengths

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

Jp

ˆ 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.

… ˆ †

… ˆ †

Easy Technique for Calculating Productivity Index of Horizontal Wells >сϰ͘Ϭ͕ƐŵсϬ͘Ϭ

347

>сϲϰ͘Ϭ͕ƐŵсϬ͘Ϭ

ϭ ϭ

Ϭ͘ϭ

Ϭ͘ϭ

Ϭ͘Ϭϭ

W:

Ϭ͘Ϭϭ

W Ϭ͘ϬϬϭ : Ϭ͘ϬϬϭ LJĞсϬ͘ϭ

LJĞсϬ͘Ϯ

LJĞсϬ͘ϯ

LJĞсϬ͘ϰ

LJĞсϬ͘ϱ

LJĞсϬ͘ϲ

LJĞсϬ͘ϳ

LJĞсϬ͘ϴ

LJĞсϬ͘ϵ

LJĞсϭ͘Ϭ

Ϭ͘ϬϬϬϭ

Ϭ͘ϬϬϬϭ

^ĞƌŝĞƐϭ

^ĞƌŝĞƐϮ

^ĞƌŝĞƐϯ

^ĞƌŝĞƐϰ

^ĞƌŝĞƐϱ

^ĞƌŝĞƐϲ

^ĞƌŝĞƐϳ

^ĞƌŝĞƐϴ

^ĞƌŝĞƐϵ

^ĞƌŝĞƐϭϬ

Ϭ͘ϬϬϬϬϭ

Ϭ͘ϬϬϬϬϭ

Ϭ͘ϬϬϬϬϬϭ Ϭ͘ϭ

ϭ

džĞ

ϭϬ

ϭϬϬ

Ϭ͘ϭ

ϭ

džĞ

ϭϬ

ϭϬϬ

Fig. 21. Pseudo-productivity index for di€erent Fig. 22. Pseudo-productivity index for di€erent reservoir con®gurations

reservoir con®gurations

PRODUCTIVITY INDEX The productivity index can be written as:

J ˆ CJp

ˆ 2C ‡ CS ‡ S 3 HF p m

…16†

p kx kz …2ye † Cˆ

where

…17†

141:2B

The mechanical skin factor negatively a€ects 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 e€ect of the mechanical skin factor on the pseudo-productivity index for di€erent horizontal wellbore. >сϰ͘Ϭ͕Ɛŵсϰ͘Ϭ

>сϲϰ͘Ϭ͕Ɛŵсϰ͘Ϭ

ϭ

ϭ

Ϭ͘ϭ Ϭ͘ϭ

Ϭ͘Ϭϭ

:W

Ϭ͘Ϭϭ

:W Ϭ͘ϬϬϭ

Ϭ͘ϬϬϭ LJĞсϬ͘ϭ

LJĞсϬ͘Ϯ

LJĞсϬ͘ϭ

LJĞсϬ͘Ϯ

LJĞсϬ͘ϯ

LJĞсϬ͘ϰ

LJĞсϬ͘ϯ

LJĞсϬ͘ϰ

LJĞсϬ͘ϱ

LJĞсϬ͘ϲ

LJĞсϬ͘ϱ

LJĞсϬ͘ϲ

LJĞсϬ͘ϳ

LJĞсϬ͘ϴ

LJĞсϬ͘ϳ

LJĞсϬ͘ϴ

LJĞсϬ͘ϵ

LJĞсϭ͘Ϭ

LJĞсϬ͘ϵ

LJĞсϭ͘Ϭ

Ϭ͘ϬϬϬϭ

Ϭ͘ϬϬϬϭ

Ϭ͘ϬϬϬϬϭ

Ϭ͘ϬϬϬϬϭ

Ϭ͘ϬϬϬϬϬϭ Ϭ͘ϭ

ϭ

džĞ

ϭϬ

ϭϬϬ

Fig. 23. The e€ect of mechanical skin factor

Ϭ͘ϭ

ϭ

džĞ

ϭϬ

ϭϬϬ

Fig. 24. The e€ect of mechanical skin factor

348

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

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

ˆ 0 75 xeD ˆ 0 375 sm ˆ 0 0 LD ˆ 37 5

yeD

:

:

:

:

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

CHF

ˆ 128 5 :

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

SP

†

that has been generated for the above

ˆ 2 15 :

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

C ˆ 113:3 Using equation (16):

349

Easy Technique for Calculating Productivity Index of Horizontal Wells

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

ϭϬ

& , ϭϬϬ

^W



ϭϬ

ϭ Ϭ͘ϭ

ϭ

ϭϬ

ϭϬϬ

Ϭ͘ϭ

ϭ

džĞ

ϭϬ

ϭϬϬ

džĞ

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.

ϭϬϬϬĨƚ

ϮϱϬĨƚ

ϳϱϬĨƚ

ϭϬϬϬĨƚ

ϯϬϬϬĨƚ

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

350

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

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

ˆ05 ywD ˆ 0 75 xeD ˆ 0 25 xwD ˆ 1 5 zwD ˆ 0 5 sm ˆ 0 0 LD ˆ 2 5 yeD

:

:

:

:

:

:

:

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

CHF

ˆ 25 8 :

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

SP

†

that has been generated for the above

ˆ75 :

Using equation (17):

C ˆ 113:3 Using equation (16):

J ˆ 33:9STB/D/psi The calculated productivity index by Lee et al. 2003 is 33.5. The di€erence 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

ϭϬϬ

&, ϭϬϬ

W ^ ϭϬ



ϭϬ Ϭ͘ϭ

ϭ

ϭϬ

ϭ

ϭϬϬ

Ϭ͘ϭ

džĞ

ϭ

ϭϬ

ϭϬϬ

džĞ

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 a€ected by the drainage area con®guration. 3 - Pseudo-skin factor is mainly a€ected by the vertical penetration. 4 - The productivity index of horizontal wells is strongly a€ected 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 a€ect the pseudo-skin factor. Vertical penetration ratio does not a€ect 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

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

kx ky kz

permeability in the X-direction, md permeability in the Y-direction, md permeability in the Z-direction, md pressure di€erence, 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 di€usivity 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:

…

P…xm ; ym ; zm ; t; zw ; Lw ; h† ˆ

…

q c

Zt



†

Sxyz …xm ; ym ; zm ; t ÿ ; zw ; Lw ; h†d

…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

…

†ˆ … †2 … †2 … †

… ÿ †

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

q ˆ Q2Lw

…A ÿ 3†

Easy Technique for Calculating Productivity Index of Horizontal Wells

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: S…x; t† ˆ

"

1

‡2

1

2xe

1 1 X nˆ1

n

exp…ÿ

n x t † 4x2e

2 2 2



…n 2xxw † e

cos

…n 2xx † e

cos

#

…A ÿ 4†

Sy …y; t†can be written as follows:

"

1 Lw 4ye X 1 S…y; t† ˆ 1‡ n ye Lw nˆ 1

exp…ÿ

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: S…z; t† ˆ

1

"

1

h

‡2

1 X nˆ1

exp…ÿ



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 ˆ xeD LD



tRD 0

("

1

‡

4 

11 X

yeD nˆ1 n

exp…ÿ



n yeD D

2 2 2

4

yeD

† sin…n 2 † cos…n 

 2 2 2 1 P exp…ÿ  n x4eD D† cos…n xwD2 † cos…n2 …xDxeD ‡ xwD †† 2 nˆ1   1 P 2 2 2 1 ‡ 2 exp…ÿn  LD D † cos…nzwD † cos…n…zD LD ‡ zwD † dD nˆ1

x ÿ xw xD ˆ Lw

2

n

#

† cos… 2 …yD yeD ‡ ywD †† 2 … A ÿ 7†

‡

1 2

where:



ywD

s ky kx

…A ÿ 8†

354

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

yD

ˆ y ÿL yw

z ÿ zw zD ˆ Lw rwD

ˆ Lrw w

PD

ˆ2

ky t ct L2w

s ky kx

…A ÿ 11†

xwD

ˆ xxw

…A ÿ 13†

ywD

ˆ yyw

…A ÿ 14†

e

e

s kz ky

…A ÿ 15†

ˆ Lyw

…A ÿ 16†

e

ˆ Lxw e

s kx ky

…A ÿ 17†

ˆ Lyt where y ˆ 

p kx kz …2ye † P…xm



…A ÿ 10†

…A ÿ 12†

xeD



ky kz

ˆ zhw

yeD

ˆ

s

zwD

Lw LD ˆ 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 ˆ xeD LD 2

Z

D

‰1 ‡ XD ‡ YD ‡ ZD ‡ XDYD ‡ XD ZD ‡ YD ZD ‡ XD YDZDŠ 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

ˆ

… 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:

Easy Technique for Calculating Productivity Index of Horizontal Wells

ZD

355



ˆ

PDa

xeD LD



‰XD ‡ YD ‡ ZD ‡ XDYD ‡ XD ZD ‡ XD YDZDŠ d D …A ÿ 22† 

0

where:

YD ˆ

yeD

exp…ÿ m m

1

1 X 1





4

yeD D

2 2 2 

4



2

2 2 



†cso…n



† sin…m

L2D D †



yeD 2

cos…l

xwD 2

† cos…n2 …xDxeD ‡ xwD †† …A ÿ 23† 

ywD

†cso…n





zwD †

2

† cos…m2 …yD yeD ‡ ywD †† …A ÿ 24† 

cos…l …zDLD ‡ zwD†† 

…m yeD ‡ n xeD † †D † sin…m yeD † cos…m 4 2 2 2

yeD mˆ1 nˆ1

2 2

cos…n





xwD 2





ywD



4

2

2

cos…l zwD † cos…l …zD LD ‡ zwD ††

…A ÿ 27†





1 1 X

…A ÿ 26†



2 2 2 2 2 1 exp…ÿ  …n xeD ‡ 4l LD †D † cos…n xwD † cos…n …x x ‡ x †† X D eD wD 1

…A ÿ 25†

m …yD yeD ‡ ywD †† † cos … 2 2

† cos…n2 …xDxeD ‡ xwD ††

…m yeD ‡ 4l LD † D† sin…m yeD † cos…m exp …ÿ yeD m 4 2 cos…l zwD † cos…l …zD LD ‡ zwD †† 8

2



2 2

2 2







1

ywD

m …yD yeD ‡ ywD†† …A ÿ 28† † cos … 2 2 





XD YD ZD ˆ

xeD D

2 2 2

exp…ÿl

1 1 exp…ÿ 8 XX

XD ZD ˆ 4

YD ZD ˆ

1

ˆ2

ZD

XD YD ˆ

exp…ÿ n

1 1 X

4 

1 X

ˆ2

XD

1 1 X

exp…ÿ …m yeD ‡ n 4xeD ‡ 4l LD † D† sin…m y2eD† cos…m ywD2 † cos…m2 …yD yeD ‡ ywD †† yeD m …A ÿ 29† xwD n † cos… 2 …xDxeD ‡ xwD†† cos…l zwD† cos…l …zDLD ‡ zwD †† cso…n 2 16





1

2

2 2

2 2

2 2

















Integrating and approximating (for long time):

XD

ˆ

8

1 X

1

xeD nˆ1 n

2 2 

2

cos…n



xwD 2

† cos…n2 …xDxeD ‡ xwD †† 

…A ÿ 30†

356

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

YD

1 X

16

ˆ

yeD mˆ1 m

3

3

1 1 X



2

L2D lˆ1 l2

yeD

† cos…m 2



cos…l



ywD

ˆ 16

XD ZD



2



yeD mˆ1 lˆ1 m…m yeD ;

‡ 4l L † sin…m



2 2

D

† cos…n



yeD

† cos…m 2

1 X

1

2

2

† cos…m2 …yDyeD ‡ ywD††

xwD





…A ÿ 33†

† cos…n2 …xDxeD ‡ xwD††

ywD



…A ÿ 34†

m …yDyeD ‡ ywD†† † cos … 2 2

sin…m

yeD nˆ1 mˆ1 lˆ1 m…n2 x2eD ‡ m2 y2eD ‡ 4l2 L2D † …m …yDyeD ‡ ywD†† …n xwD† …n …xDxeD ‡ xwD††

cos

ywD



…A ÿ 35†



3







cos…l zwD† cos…l …zDLD ‡ zwD †† 64

cos…m

2 …n xeD ‡ 4l LD cos…l zwD† cos…l …zDLD ‡ zwD†† 2

2

1



XD YD ZD ˆ

1

2 2

nˆ1;lˆ1

2 2

…A ÿ 32†

2

1 X

1 X

32 3 

cos

2





sin

;

2

† cos…m2 …yDyeD ‡ ywD †† …A ÿ 31†

cos…l …zDLD ‡ zwD††

zwD †

1 …m y2eD† yeD nˆ1 mˆ1 m…n2 x2eD ‡ m2 y2eD † …n xwD† …n …xD xeD ‡ xwD††

cos

YD ZD ˆ



1 X

32 

sin…m

2

ˆ

ZD

XD YD ˆ

1

3 3 

;

;

cos

2

cos

2



yeD 2

† cos…m



ywD 2

†

cos…l zwD† cos…l …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 E€ects 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.

Easy Technique for Calculating Productivity Index of Horizontal Wells

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. E€ects 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

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

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