Copyright © the author(s). This work is licensed under a Creative Commons Attribution 4.0 International License. 

DOI: 10.14800/IOGR.1148  

Received December 30, 2019; revised February 23, 2020; accepted April 12, 2020. 

*Corresponding author: dongzhenzhen1120@hotmail.com                                                                              1 

Horizontal Well Productivity Evaluation 
For Stress Sensitive Elliptical Reservoirs 

 

Weidong Tian, Zhenzhen Dong* and Jiaen Lin, Xi’an Shiyou University, Xi’an, China 

 

 

Abstract 

Well productivity model is one of the vital tools required to evaluate well performance. Most horizontal 

well productivity models are idealistic in nature, mainly developed for homogeneous reservoirs and 

conventional reservoirs, and ignore the influences of the pore pressure and stress changes. However, as the 

capillary in low permeability porous media is tiny, the medium permeability is quite sensitive to pressure 

change. Thus, there is an urgent need for new realistic productivity models that describe the actual reservoir 

inflow performance behavior more efficiently than the available models. 

  This paper presents a new horizontal well productivity model which accounts for the stress sensitive 

permeability in an elliptical reservoir. Then, the proposed model was extended to investigate the effects of 

reservoir heterogeneity, eccentricity, and formation damage on horizontal well productivity. The results 

show that the thinner the formation is, the greater the impact of the horizontal well lengths on production. 

As the horizontal well length is longer, the impact of stress sensitivity on the production becomes more 

significant. Horizontal well would be a better well type option for elliptical reservoirs. The longer the 

horizontal well is, the more impact of heterogeneity, eccentricity distance, as well as skin factor on 

productivity. 

  The new model provides a simpler and more reliable means to optimize horizontal well length and 

efficiently forecast well behavior in stress sensitive reservoirs, such as tight gas reservoir and shale oil 

reservoirs, with respect to horizontal well productivity to vertical well productivity.  

Introduction 

To determine the economic feasibility of drilling a horizontal well, the engineers need reliable methods to 

estimate its expected productivity. There have been attempts to describe and estimate horizontal well 

productivity. 

  Joshi (1988a) further illustrates the principle of horizontal well production through electrical simulation, 

and the calculation of steady-state production of horizontal well was derived in detail. Up to now, most of 

the steady-state horizontal well productivity formulas proposed by many authors are similar to the formulas. 

Larsen (1996) proposed a method for calculating the productivity equation of multilateral wells, branch 

wells and other generalized wells. 

  Babu and Odeh (1989) calculated the productivity equation of the horizontal well in a pseudo-steady 

state. Billiter et al. (2001) proposed the dimensionless inflow dynamic curve of the non-fracture horizontal 

gas well. The flow equation of the Babu and Odeh’s horizontal well is transformed into the pseudo-pressure 

form of the gas well, and the non-darcy flow effect as well as the mechanical skin effect are considered. 

Furthermore, the pseudo-steady horizontal well productivity of anisotropic reservoir (Lu and Tiab 2007), 

mailto:dongzhenzhen1120@hotmail.com


2 

constant rate and constant pressure (Hagoort 2011) are given. Salam (2019) presents a new practical method 

for determining the start time of pseudo-steady flow and constant-behavior productivity index (PI). 

  Other scholars have considered the wellbore flow into the productivity formula. Penmatcha and Khalid 

(1999) proposed a semi-analytical model for homogeneous reservoirs that can quantify the impact of 

wellbore single-phase and two-phase oil and gas in wellbores on productivity. Anklam and Wiggins (2005) 

presented a model for estimating horizontal well productivity, which combined wellbore fluid dynamics to 

calculate the well pressure for the entire wellbore. Zhu et al. (2002) (for multilateral wells) and Yildiz (2003) 

(for perforated horizontal wells) have done similar research. Guo, et al. (2006) developed a general 

mechanistic model combining the fluid flow of a single branch. The model strictly considers the pressure 

drop in the vertical and inclined wellbore sections. 

  In recent years, scholars have been mainly devoted to the study of the productivity of horizontal wells 

with tight reservoirs, taking into account the factors such as multi-layer reservoir, capillary force, hydraulic 

fracturing and so on. Tabatabaei et al. (2009) in studying the yield of horizontal wells for hydraulic 

fracturing in Bakken shale reservoir, established an analytical model to predict the horizontal well yield for 

longitudinal fractures in multi-layer reservoirs. Kewen and Chen (2012) derived formulas for calculating 

water cut and dimensionless total and oil productivity indices (PIs) by considering capillary pressure, to 

study the effect of capillary pressure on production performance in low-permeability oil wells or reservoirs. 

Bin et al. (2015) present a new analytical solution to study the interplay between flowing pressure and 

production rate for horizontal well completed within stimulated reservoir volumes (SRV) in tight gas 

reservoirs. Chen et al. (2019) presented the calculation method of fracturing production of horizontal well 

through layer, considering the inter-slit interference and wellbore interference. Sun et al. (2019) set up a 

seepage model for the tiny reservoir by coupling the elliptical flow in the matrix and the near radial flow in 

the fracture. 

All of the above pseudo-steady state equations are either too complicated to use or very time consuming. 

Moreover, they all ignored the influences of the pore pressure and stress changes on horizontal wells. 

However, as the capillary in low permeability porous media is tiny, the medium permeability is quite 

sensitive to pressure change. The effect of pressure on permeability cannot be ignored, especially for the 

abnormally high pressure and low permeability reservoirs. This paper provides analytical equations to 

calculate productivity of horizontal wells in low-permeability reservoirs with considering the effect of stress 

on permeability. Then, the effect of shape of drainage area, heterogeneity, eccentricity and formation 

damage on the proposed horizontal well productivity model were studied. The work discussed here was 

carried out at Xi’an Shiyou University, from June to December 2019. 

Materials and Methods 

Physical model 

Figure 1 is a schematic of a horizontal well. The following assumptions are made: 

1. The horizontal well is in the middle of an elliptical reservoir.  

2. The horizontal well is in the middle of the formation, with an impermeable top and bottom boundary.  

3. The reservoir is homogeneous and isotropic.  

4. The length of horizontal well is L and the width of reservoir is h.  

5. The flow of fluid is slight compressible single-phase flow of oil which corresponds to the low-speed 

non-Darcy flow law.  

6. Ignore the effect of gravity and capillary forces.  

 



3 

 

Figure 1—The horizontal well scheme of low permeability reservoir 

Productivity Formula Derivation 

The horizontal flow calculations. The horizontal flow of well keeps its shape in ellipses, introducing the 

Ru koves Ki function  

𝑧

𝐿/2
=

1

2
(𝜔 +

1

𝜔
)...……………………..……………………………………………………………….(1) 

  Utilize conformal mapping and transfer the area of elliptical shape with semi-major axis of a as well as 

semi-minor axis of b into the circular area with radius of 
𝑎+𝑏

𝐿/2
. The segment from (-L/2, 0) to (+L/ 2, 0) is 

imaged into the unit circle, as shown in Figure 2. The flow on ω surface can be considered as the supply 

which is provided from the circular with radius of 
𝑎+𝑏

𝐿/2
 to a vertical well with radius of 1. 

 

 

Figure 2—Scheme of horizontal conformal mapping 

 

Most horizontal well productivity models ignored the influences of the pore pressure and stress changes. 

However, as the capillary in low permeability porous media is tiny, the medium permeability is quite 

sensitive to pressure change. The effect of pressure on permeability cannot be ignored, especially for the 

abnormally high pressure and low permeability reservoirs. Many research efforts have shown that the 

permeability changes exponentially with the pressure. Thus, 

 𝐾𝐷 =
𝑘

𝑘ℎ
= 𝑒−𝛼𝑘(𝑝𝑖−𝑝).……………………….…………………………………………………………(2) 

Consider the existence of starting pressure gradient, so the fluid velocity can be defined as  

  ν =
𝑘

𝜇
[
𝑑𝑝

𝑑𝑟
− 𝐺𝑝].………...………………………………………………………………………………(3) 

Substitute Eq. 2 into Eq. 3 and replace the fluid velocity with production to yield

 

 

 h/2

h

x

y

z

(-L/2,0) o (L/2,0) x

y













1

2

1

2/L

z

o 



z 

b 

a 



4 

 
QμB

85.2618×2πrhkh
= e−αk(pi−p) [

dp

dr
− Gp],……………………………………………….………………...(4) 

Based on the research of Chen et al. (2006, 2007), the oil production formula is as follows. 

Q =
𝑘ℎℎ

1.8665×10−3𝛼𝑘𝜇𝐵

1−𝑒{−𝛼𝑘[𝑝𝑖−𝑝𝑤−𝐺𝑝(𝑟𝑒−𝑟𝑤)]}

𝑙𝑛
𝑟𝑒
𝑟𝑤

....………………………………..……….………………(5) 

Consider the property of elliptical b = √𝑎2 − (𝐿/2)2, the horizontal flow of the production wells can 

be expressed as 

Q𝐻 =
𝑘ℎℎ

1.8665×10−3𝛼𝑘𝜇𝐵

1−𝑒{
 

 
−𝛼𝑘

[
 
 
 

𝑝𝑖−𝑝𝑤−𝐺𝑝

(

 
𝑎+√𝑎2−

𝐿2

4
𝐿/2

−1

)

 

]
 
 
 

}
 

 

𝑙𝑛(
𝑎+√𝑎2−

𝐿2

4
𝐿/2

)

,……....…………….……………………………(6) 

where 

𝑎 =
𝐿

2
√1
2
+√

1

4
+ (

2𝑟𝑒𝐻

𝐿
)
4
, 𝑟𝑒𝐻 = √𝐴/𝜋...……………..….…………………………...…………...…(7) 

  

The vertical flow calculations. The vertical flow of horizontal well can be regarded to be a junction of 

supply from the top and bottom boundaries. The diagram of vertical conformal transformation is shown in 

Figure 3.  

 

Figure 3—Vertical conformal transformation 

 

Convert the band-shaped region (-h/2<y<h/2) on z plane to the unit circle domain on ω plane. Then, the 

junction of (0, 0) on z plane will be converted to the B point (1, 0) on ω plane and the well radius of rw on 

z plane will be converted to ρw on ω plane.  

On ω plane, the complex potential of any points can be obtained through mirror image theory. Thus, 

U =
𝑞

2𝜋
𝑙𝑛(𝜔 − 1)(𝜔 + 1) + 𝐶 =

𝑞

4𝜋
𝑙𝑛[(𝜍2 + 𝜂2 + 1)2 − 4𝜍2] + 𝑖

𝑞

2𝜋
(𝑎𝑟𝑐𝑡𝑔

𝜂

𝜍−1
+ 𝑎𝑟𝑐𝑡𝑔

𝜂

𝜍+1
) + 𝑐2,……….(8) 

Where 𝜍2 = 𝑒
2𝜋𝑥

ℎ 𝑐𝑜𝑠2
𝜋𝑦

ℎ
; 𝜂2 = 𝑒

2𝜋𝑥

ℎ 𝑠𝑖𝑛2
𝜋𝑦

ℎ
. 

Therefore, the potential function (pressure function) is given as, 

𝛷 =
𝑞

4𝜋
𝑙𝑛 [(𝑒

2𝜋𝑥

ℎ + 1)
2

− 4𝑒
2𝜋𝑥

ℎ 𝑐𝑜𝑠2
𝜋𝑦

ℎ
] + 𝑐2,………………………………………………………..(9) 

o x

y

o 



h

z

e


 

A(-1,0) B(1,0)

h/2

h/2

z 



5 

When x=0，y=h/2, then 𝜙ℎ

2

=
𝑞

4𝜋
𝑙𝑛4 + 𝑐2. 

The potential of the junction B on the wall of well can be defined as Φw and it is always divided into two 

parts, which is brought by the junction A and junction B, respectively. The distance from junction A to 

junction B can be approximately equal to 2 and the radius of junction B is ρw. Thus,  

𝜌𝑤 =
𝑑(𝜔−1)(𝜔+1)

𝑑𝑧
|(0,0)𝑟𝑤 =

2𝜋

𝑑𝑧
𝑒
𝜋𝑧

ℎ |(0,0)𝑟𝑤 =
2𝜋𝑟𝑤

ℎ
,…………....…..………………………………….(10) 

𝛷𝑤 =
𝑞

2𝜋
[𝑙𝑛2 + 𝑙𝑛

2𝜋𝑟𝑤

ℎ
] + 𝑐2.…………………………….…………………………………………..(11) 

The Non-Darcy flow equation can be expressed as follows 

 
𝑑𝛷

𝑑𝑟
=

𝑘

𝜇
(
𝑑𝑝

𝑑𝑟
− 𝐺𝑝),…………………………………………………….………….…………………....(12) 

The flow on ω plane can be regarded as the unitary circular supply for a vertical well in one area with 

radius of ρw (which will be corrected to the effective caliper using Peaceman method (1990)) whose 

discharge area is a rectangle with h/2 as long side and h/2π as short side. The area of this rectangle is 

equivalent to the area of the circle with a radius of h/2π. Do integration to the Eq.11 within (rw, h/2) and 

according to the Dupuit formula, the shunt volume on the vertical direction of horizontal wells is  

Q𝑣 =
𝑘ℎ𝐿

1.8665×10−3𝛼𝑘𝜇𝐵

1−𝑒
{−𝛼𝑘[𝑝𝑒−𝑝𝑤−𝐺(1−𝑙𝑛

2𝜋𝑟𝑤
ℎ

)]}

𝑙𝑛(
ℎ

2𝜋(𝑟𝑤+1)
)

......…………………………………………………..(13) 

Horizontal well productivity. The oil flow resistance on horizontal direction (ΩH) and on vertical direction 

(ΩV) which are based on Eq. 6 and Eq.13 can be expressed as follows 

Ω𝐻 =
1.8665×10−3𝛼𝑘𝜇𝐵

𝑘ℎℎ
𝑙𝑛 (

𝑎+√𝑎2−
𝐿2

4

𝐿/2
)....……………………….……………………………………..(14) 

Ω𝑣 =
1.8665×10−3𝛼𝑘𝜇𝐵

𝑘ℎ𝐿
𝑙𝑛 (

ℎ

2𝜋(𝑟𝑤+1)
)....………………………….…………………………………….(15) 

Total production of the horizontal well resulting from these two flow components is obtained by adding 

the respective flow resistances, 

∆p

Q
= (

∆p

Q𝐻
) + (

∆p

Q𝑣
),…………………………………………...………………………………………..(16) 

  or pressure drawdown per unit volumetric flow rate, 

 Q = (𝑝𝑒 − 𝑝𝑤)/(Ω𝐻 − Ω𝑣),…………………………………………………………………………..(17) 

Thus, the horizontal well flow rate can be given as, 

Q =
𝑘ℎℎ

1.8665×10−3𝛼𝑘𝜇𝐵

1−𝑒{
 

 
−𝛼𝑘

[
 
 
 

𝑝𝑒−𝑝𝑤−𝐺

(

 
𝑎+√𝑎2−

𝐿2

4
𝐿/2

−𝑙𝑛
2𝜋𝑟𝑤
ℎ

)

 

]
 
 
 

}
 

 

𝑙𝑛(
𝑎+√𝑎2−

𝐿2

4
𝐿/2

)+
ℎ

𝐿
𝑙𝑛(

ℎ

2𝜋(𝑟𝑤+1)
)

.......…………………..……………………..(18)  

Expansion of the Proposed Productivity Model 

The proposed productivity model (Eq. 18) was extended to consider more practical conditions, such as 

heterogeneous reservoir, off-center wells, and formation damage.  

Heterogeneity. With considering reservoir heterogeneity, the steady flow percolation equation is given as, 

𝑘𝑥
𝜕2𝑝

𝜕𝑥2
+ 𝑘𝑦

𝜕2𝑝

𝜕𝑦2
+ 𝑘𝑧

𝜕2𝑝

𝜕𝑧2
= 0....………………………………………….…………………………...(19) 



6 

According to the assumption of Joshi (1988b), the horizontal wells flow in the three-dimensional system 

is always divided into the horizontal flow and the vertical flow. Therefore, the stable seepage can be 

simplified as,  

(
𝜕2𝑝

𝜕𝑥2
) +

𝑘𝑣

𝑘ℎ
(
𝜕2𝑝

𝜕𝑧2
) = 0..…………………………………………………………………………………(20)  

Introduce z = z√
𝑘ℎ

𝑘𝑣
, then the heterogeneous problem of Eq. 20 can be transformed into a homogeneous 

problem, expressed in Eq. 19. 

(
𝜕2𝑝

𝜕𝑥2
) + (

𝜕2𝑝

𝜕𝑧2
) = 0..……………………………………………………………………………………(21) 

Substitute ℎ√
𝑘ℎ

𝑘𝑣
 for h in Eq.20 and Eq.21, and substitute the average permeability√𝑘ℎ𝑘𝑣 

or ki in Eq.19, 

then the horizontal well production formula with considering heterogeneity can be expressed as, 

Q =
𝑘ℎℎ

1.8665×10−3𝛼𝑘𝜇𝐵

1−𝑒{
 

 
−𝛼𝑘

[
 
 
 

𝑝𝑒−𝑝𝑤−𝐺

(

 
𝑎+√𝑎2−

𝐿2

4
𝐿/2

−𝑙𝑛
2𝜋𝑟𝑤
ℎ

)

 

]
 
 
 

}
 

 

𝑙𝑛(
𝑎+√𝑎2−

𝐿2

4
𝐿/2

)+𝐼𝑎𝑛𝑖
ℎ

𝐿
𝑙𝑛(

ℎ𝐼𝑎𝑛𝑖
2𝜋(𝑟𝑤+1)

)

,……………………………………………..(22) 

Where 𝐼𝑎𝑛𝑖 = √
𝑘ℎ

𝑘𝑣
. 

Eccentricity. The eccentricity only affects the vertical plane of horizontal well productivity. Introducing 

transformation ω = z + iδ will transform the problem solving with eccentricity into the problem solving 

without eccentricity, seen in Figure 4. 

 

 

Figure 4—Vertical Conformal Transformation Affected by Eccentricity 

 

Introduce conformal transformation, 

ω =
1−𝑒

𝜋
ℎ
(z+iδ)

1+𝑒
𝜋
ℎ
(z+iδ)

,…………………………………………………………………………………………(23) 

and convert the band-shaped region with eccentricity on z plane to the unit circle domain on ω plane. Then, 

the junction of (0, 0) on z plane will be converted to the original point (0, 0) on ω plane and the production 

well radius of rw on z plane will be converted to ρw on ω plane. Thus, 

𝜌𝑤 =
𝑑𝜔

𝑑𝑧
|(0,0)𝑟𝑤 =

2𝜋𝑟𝑤

ℎ(2+𝑒
𝑖
𝜋δ
ℎ +𝑒

−𝑖
𝜋δ
ℎ )

=
𝜋𝑟𝑤

2ℎ𝑐𝑜𝑠2
𝜋δ

2ℎ
 ,…………………………………..…………………(24) 

According to the study of Yuan et al. (2009 and 2015) and Chen et al.(2009, 2012), for the reservoirs 

with anisotropic permeability, there is 

0.5h 

0.5h 

(0,0) 𝛿 𝑥 

𝑦 

0.5h 

0.5h (0,0) 

𝜂1 

Z w 

𝜁1 



7 

𝐽ℎ =
0.543𝑘ℎℎ/(𝜇𝑜𝐵𝑜)

𝑙𝑛(
𝑎+√𝑎2−

𝐿2

4
𝐿/2

)+(
𝛽ℎ

𝐿
)𝑙𝑛(

(
𝛽ℎ
2
)
2
+𝛽2𝛿2

𝛽ℎ
2
𝑟𝑤

)

,…………………………………………………………………(25) 

Therefore, horizontal well production formula considering the existence of eccentricity can be expressed 

as, 

Q =
𝑘ℎℎ

1.8665×10−3𝛼𝑘𝜇𝐵

1−𝑒{
 

 
−𝛼𝑘

[
 
 
 

𝑝𝑒−𝑝𝑤−𝐺

(

 
𝑎+√𝑎2−

𝐿2

4
𝐿/2

−𝑙𝑛
2𝜋𝑟𝑤
ℎ

)

 

]
 
 
 

}
 

 

𝑙𝑛(
𝑎+√𝑎2−

𝐿2

4
𝐿/2

)+𝐼𝑎𝑛𝑖
ℎ

𝐿
𝑙𝑛
ℎ𝐼𝑎𝑛𝑖+(2𝐼𝑎𝑛𝑖𝛿)

2

2𝜋(𝑟𝑤+1)

………………………………………………(26) 

Formation damage. Compared with vertical wells, the drilling time of horizontal wells in the oil layer is 

much longer, so the contact area between producing formation and drilling fluid is larger. Therefore, under 

the same drilling conditions, the contamination of horizontal well reservoir is much severer than that of the 

vertical wells reservoir, resulting in the production difference between horizontal wells and vertical wells. 

The parameter which is used to represent the horizontal wellbore damage is usually called skin factor, 

abbreviated as S. It is be obtained through drill-stem testing (DST) or pressure recovery testing. The value 

of S can affect the flow efficiency of the fluid, but it is not the unique factor and it does not exist 

independently. The flow efficiency of the fluid also can be affected by the vertical penetration level (Giger 

et al. 1984). Therefore, when the effect of formation damage on horizontal well productivity was studied, 

the effect of skin and permeability anisotropy should be considered. According to the study of Renard and 

Dupuy (1991), the value of skin can be estimate roughly based on the following equation, 

𝑆ℎ =
𝐼𝑎𝑛𝑖ℎ𝑆

𝐿
……….……………………………………………………………………………………(27) 

Thus, horizontal well production formula considering the existence of formation damage can be 

expressed as follows. 

Q =
𝑘ℎℎ

1.8665×10−3𝛼𝑘𝜇𝐵

1−𝑒{
 

 
−𝛼𝑘

[
 
 
 

𝑝𝑒−𝑝𝑤−𝐺

(

 
𝑎+√𝑎2−

𝐿2

4
𝐿/2

−𝑙𝑛
2𝜋𝑟𝑤
ℎ

)

 

]
 
 
 

}
 

 

𝑙𝑛(
𝑎+√𝑎2−

𝐿2

4
𝐿/2

)+𝐼𝑎𝑛𝑖
ℎ

𝐿
 (ln

ℎ𝐼𝑎𝑛𝑖+(2𝐼𝑎𝑛𝑖𝛿)
2

2𝜋(𝑟𝑤+1)
+𝑠)

 ………………………………………….(28) 

Results and Discussion 

In this section, the proposed production model (Eq. 28) was applied to study the impact of horizontal well 

length (L), reservoir thickness (h), medium deformation coefficient (ak), shape of drainage area (a/b), 

anisotropy index (𝐼𝑎𝑛𝑖), eccentric distance(𝛿) and formation damage (S) on horizontal well production (Q). 

The basic data that used to calculate production rate are listed in Table 1.  

 

  



8 

Table 1—Basic data used in the calculation 

Parameters Value Parameters Value 

kh, μm2 0.1 pw, MPa 8 

αk, MPa-1 0.1 G, MPa/m 0.001 

μ, mPa.s 0.9 rw, m 0.1 

B 1.2 A, m2 300000 

pe, MPa 10 h, m 40 

𝐼𝑎𝑛𝑖  1 𝛿,m 0 

S 0   

 

Horizontal well length and reservoir thickness. Figure 5 shows that the production increases as the 

reservoir thickness and horizontal well length increase. The thinner the formation is, the greater the impact 

of the horizontal well lengths on production. For instance, as the reservoir thickness increases from 5 m to 

50 m, the horizontal well production increases from 1280.7 to 9941.5 105m3/d (7.8 times) when horizontal 

well length is set to be 1000 m. While the well length is 3000 m, the production only increases by 5.5 times. 

Thus, horizontal well is suggested in a thin formation. 

 

 

Figure 5—Effect of reservoir thickness on production at different horizontal well length 

 

Medium deformation coefficient. Medium deformation coefficient, 𝛼𝑘, is an indicator of the degree of 

stress sensitivity (Eq. 2). Its influence on productivity is shown in Figure 6, which indicates that medium 

deformation coefficient has a negative linear correlation with production on the semi-log plot. The larger 

the coefficient, the smaller the production. In other words, the more stress sensitive the formation is, the 

lower the production. As the horizontal well length is longer, the impact of medium deformation coefficient 

on the production becomes more significant.  

 

0

1

2

3

4

5

6

7

5 10 15 20 25 30 35 40 45 50

P
ro

d
u
ct

io
n
,1

0
5
m

3
/d

Reservoir Thickness,m

L=1000m

L=1500m

L=2000m

L=2500m

L=3000m



9 

 

Figure 6—Effect of the medium deformation coefficient on production at different horizontal well length 

 

Shape of drainage area. The shape of drainage area is determined by the ratio a/b in Figure 2. The influence 

of the medium deformation coefficient on production with various ratio of a/b is shown in Figure. 7. The 

horizontal well length L was set to be 1000 m, other parameters are listed in Table 1. Fig 6 shows that the 

smaller a/b (the closer reservoir shape is to the circle), the lower the production of horizontal well. Thus, 

horizontal well would be a better well type for elliptical reservoirs. 

 

 

Figure 7—Effect of different a/b on production 

 

Anisotropy index. Anisotropy index is an indicator is an indicator of the degree of reservoir heterogeneity 

(Eq. 22). The influence of the permeability anisotropy index on production is shown in Figure 8 in terms 

of different horizontal well length. It shows heterogeneity has great impact on production. The smaller the 

vertical permeability (the greater the anisotropy index), the lower the horizontal well production. As 

0.1

1

10

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

P
ro

d
u
ct

io
n
,1

0
5
m

3
/d

Medium Deformation Coefficient

L=1000m

L=1500m

L=2000m

L=2500m

L=3000m

0

0.2

0.4

0.6

0.8

1

1.2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

P
ro

d
u
ct

io
n
,1

0
5
m

3

Medium Deformation Coefficient

a/b=1.5

a/b=2

a/b=2.5

a/b=3

a/b=3.5



10 

anisotropy index increases from 1 to 10, the production decreases from 41,024 to 6,019 105m3/d (6.8 times) 

when the horizontal well length is 3000m. While the horizontal well length is 1000m, the production only 

decreases 2.6 times. Thus, it can be concluded that the stronger the heterogeneity, the more impact of 

horizontal well length on productivity. 

 

 

Figure 8—Effect of the anisotropy index on production at different horizontal well length 

 

Eccentric distance. The influence of the eccentric distance (𝛿) on production was investigated versus 

various horizontal well length. The anisotropy index (Iani), was set to be 1. Figure 9 shows that the larger 

the eccentricity distance, the smaller the production, and the degree of influence gradually decreases. The 

longer the horizontal well is, the more impact of eccentricity distance on productivity. 

 

 

Figure 9—Effect of the eccentricity distance on production at different horizontal well length 

 

Formation damage. Figure 10 depicts the influence of formation damage. It indicates that the longer the 

horizontal well is, the more impact of skin factor(S) on horizontal well production. 

0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10

P
ro

d
u
ct

io
n
,1

0
5
m

3
/d

Anisotropy Index

L=1000m

L=1500m

L=2000m

L=2500m

L=3000m

0

1

2

3

4

5

6

0 2 4 6 8 10 12 14 16 18 20

P
ro

d
u
ct

io
n
,1

0
5

m
3

/d

Eccentricity Distance,m

L=1000m

L=1500m

L=2000m

L=2500m

L=3000m



11 

 

Figure 10—Effect of the formation damage on production at different horizontal well length 

Conclusions 

A new horizontal well productivity evaluation model based on stress sensitive permeability in an elliptical 

reservoir is established. And the proposed productivity model was extended to consider the effects of 

reservoir heterogeneity, eccentricity, and formation damage on horizontal well productivity. The following 

conclusions can be drawn: 

1. The thinner the formation is, the greater the impact of the horizontal well lengths on production. 

Thus, horizontal well is suggested in a thin formation. 

2. The more stress sensitive the formation is, the lower the production. As the horizontal well length 

is longer, the impact of stress sensitivity on the production becomes more significant.  

3. The closer reservoir shape is to the circle, the lower the production of horizontal well. Thus, 

horizontal well would be a better well type option for elliptical reservoirs. 

4. The longer the horizontal well is, the more impact of heterogeneity, eccentricity distance, as well as 

skin factor on productivity. 

Acknowledge 

Acknowledgment to the following projects for supporting the research work of this paper: Scientific 

Research Project of Key Laboratory of Education Department in Shaanxi Province: "Research on well test 

evaluation technology of unconventional horizontal well exploitation" (13JS090) 

Conflicts of Interest 

The author(s) declare that they have no conflicting interests. 

Nomenclature 

a =   the semi-major axis of the elliptical drainage area formed in the horizontal wells, m 

b =   the short half axis of the ellipsoidal drain region formed in the horizontal wells, m 

B =   formation volume factor 

h =   reservoir thickness, m 

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 3 6 9 12 15

P
ro

d
u
ct

io
n
,1

0
5
m

3
/d

Skin Factor

L=1000m

L=1500m

L=2000m

L=2500m

L=3000m



12 

Iani =   permeability anisotropy index 

kh =   horizontal permeability, μm2 

L =   horizontal well length, m 

Pe =   supply pressure, MPa 

pwf =   wellbore flowing pressure, MPa 

Q =   production, m3/d 

rw =   wellbore radius, m 

S =   pseudo-skin factor  

δ =   eccentric distance, m 

αk =   medium deformation coefficient, 1/MPa 

Gp =   starting pressure gradient, MPa/m 

reh =   equivalent radius of horizontal well, m 

μ =   viscosity, mPa.s 

Reference 

Anklam, E.G. and Wiggins, M.L. 2005. Horizontal Well Productivity and Wellbore Pressure Behavior Incorporating 

Wellbore Hydraulics. Paper presented at SPE Production Operations Symposium, Oklahoma City, Oklahoma, 16-

19 April. SPE-94316-MS. 

Babu, D.K. and Odeh, A.S. 1988. Productivity of a Horizontal Well. Paper presented at SPE Annual Technical 

Conference and Exhibition, Houston, Texas, 2-5 October. SPE-18334-MS.  

Billiter, T., Lee, J., and Chase, R. 2001. Dimensionless Inflow-Performance-Relationship Curve for Unfractured 

Horizontal Gas Wells. Paper presented at SPE Eastern Regional Meeting, Canton, Ohio, 17-19 October. SPE-

72361-MS. 

Chen, M., Pu, C., and Zhao, J. 2006. Calculation and Analysis of the True Productivity of a Well of Low Permeable 

Reservoirs of Deformation Medium. Journal of Xi’an Shiyou University (Natural Science Edition) 21(2):18-23. 

Chen, M., Zhang, M., and Pu, C. 2007. Productivity of Horizontal Well in Deformation Medium of Low-Permeability 

Reservoirs. Acta Petrolei Sinica 28(1): 107-110. 

Chen, Y. and Zou, C. 2009. Derivation, Comparison and Application of Horizontal Well Production Formula 

Considering Anisotropy and Eccentricity Effects. Xinjiang Petroleum Geology 30(4):486-489. 

Chen, Y., Hao, M., Sun, B., et al. 2012. Comparative Study of the Production Rate Formulas for Horizontal Wells. 

Xinjiang Petroleum Geology 33(5):566-569. 

Chen, X., Mu, L., and Wu, Z. 2019. Calculating Method of the Productivity for the Trans-Layer Fractured Horizontal 

Wells. Petroleum Geology & Oilfield Development in Daqing 38(3):65-72. 

Guo, B., Ling, K., and Ghalambor, A. 2006. A Rigorous Composite-IPR Model for Multilateral Wells. Paper 

Presented at SPE Annual Technical Conference and Exhibition, San Antonio, Texas, USA, 24-27 September. SPE-

100923-MS. 

Giger, F., Reiss, L.H., and Jourdan, A. 1984. The Reservoir Engineering Aspects of Horizontal Drilling. Paper 

presented at SPE Annual Technical Conference and Exhibition, Houston, Texas, 16-19 September. SPE-13024-

MS. 

Hagoort, J. 2011. Semisteady-State Productivity of a Well in a Rectangular Reservoir Producing at Constant Rate or 

Constant Pressure. SPE Reservoir Evaluation & Engineering 14(6):677-686. SPE-149807-PA.  

Joshi, S. D. 1988. Augmentation of Well Productivity with Slant and Horizontal. Journal of Petroleum Technology 

40(6):729-739. SPE-15375-PA.  

Joshi, S. D. 1988. Production Forecasting Methods for Horizontal Wells. Paper presented at International Meeting 

on Petroleum Engineering, Tianjin, China, 1-4 November. SPE-17580-MS. 



13 

Kewen, L. and Chen, Z. 2012. Computation of Productivity Index with Capillary Pressure Included and Its 

Application in Interpreting Production Data from Low-Permeability Oil Reservoirs. SPE Journal 17(4):1041-

1046. SPE-146648-PA.  

Larsen, L. 1996. Productivity Computations for Multilateral, Branched and Other Generalized and Extended Well 

Concepts. Paper presented at SPE Annual Technical Conference and Exhibition, Denver, Colorado, 6-9 October. 

SPE-36754-MS.  

Lu, J. and Tiab, D. 2007. A Simple Productivity Equation for a Horizontal Well in Pseudo-Steady State in A Closed 

Anisotropic Box Shaped Reservoir. Paper presented at Production and Operations Symposium, Oklahoma City, 

Oklahoma, U.S.A., 31 March-3 April. SPE-106970-MS.  

Peaceman, D.W. 1990. Discussion of Productivity of a Horizontal Well. SPE Reservoir Engineering 5(2): 252-253. 

SPE-31025-PA.  

Penmatcha, V.R. and Khalid, A. 1999. Effects of Pressure Drop in Horizontal Wells and Optimum Well Length. Paper 

presented at SPE Production Operations Symposium, Oklahoma City, Oklahoma, 9-11 March. SPE-37494-MS. 

Renard, G. and Dupuy, J. M. 1991. Formation Damage Effects On Horizontal-Well Flow Efficiency. Journal of 

Petroleum Technology 43(7):786-869. SPE-19414-PA.  

Salam, A. 2019. Revisiting Current Techniques for Analyzing Reservoir Performance: A New Approach for 

Horizontal-Well Pseudosteady-State Productivity Index. SPE Journal 24(1):71-91. SPE-191139-PA.  

Sun, R., Zhang, Y., and Zhang, X. 2019. Productivity Prediction Model and Parameter Analysis for Fractured 

Horizontal Wells in Double Medium Tight Oil Reservoirs. Science Technology and Engineering 19(10):89-96. 

Tabatabaei, M., Mack, D.J., and Daniels, N.R. 2009. Evaluating the Performance of Hydraulically Fractured 

Horizontal Wells in the Bakken Shale Play. Paper presented at SPE Rocky Mountain Petroleum Technology 

Conference, Denver, Colorado, 14-16 April. SPE-122570-MS. 

Yildiz, T. 2003. Mutilateral Pressure-Transient Response. SPE Journal 8(1):234-245. SPE-83631-PA.  

Yuan, B., Su, Y., and Rouzbeh, G.M. 2015. A New Analytical Multi-Linear Solution for Gas Flow toward Fractured 

Horizontal Wells with Different Fracture Intensity. Journal of Natural Gas Science and Engineering 

23(2015):227-238.  

Yuan, Y., Zhang, L., Wang, J., et al. 2009. A Binomial Deliverability Equation for Horizontal Gas Wells in Formations 

with Nonlinear Seepage Flow Features. Oil＆Gas Geology 30(1):122-126.  

Zhu, D., Hill, A. D., and Landtum, W. R. 2002. Evaluation of Crossflow Effects in Multilateral Well. Paper presented 

at SPE/DOE Improved Oil Recovery Symposium, Tulsa, Oklahoma, 13-17 April. SPE-75250-MS.  

 

Weidong Tian, is a master candidate in Petroleum Engineering department at Xi’an Shiyou University. He 

has focused his research in areas involving reservoir simulation, well testing, and production analysis. Tian 

holds an BS degree in Petroleum Engineering from Xi’an Shiyou University. 

 

Zhenzhen Dong is a Professor in the Petroleum Engineering Department at Xi’an Shiyou University. She 

worked as a reservoir engineer with Schlumberger from 2012 to 2016. Her research interests include 

unconventional resources/reserves estimates, reservoir simulation, well testing, and production analysis. 

Dong holds a bachelor’s degree in mathematics from Northeast Petroleum University, China; a master’s 

degree in petroleum engineering from Research Institute of Petroleum Exploration and Development, 

China; and a PhD degree in petroleum engineering from Texas A&M University. 

 

Jiaen Lin is a Professor of Petroleum Engineering Department at Xi’an Shiyou University, where he has 

worked as faculty for the last 30 years. He focuses his research in well testing and production analysis.