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Generalized inflow performance relationships for horizontal gas wells
Journal of Natural Gas Science and Engineering 2 (2010) 132e142
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Generalized inflow performance relationships for horizontal gas wells M. Tabatabaei, D. Zhu* Texas A&M University, TX, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 19 April 2010 Accepted 6 May 2010 Available online 8 June 2010
The analytical inflow performance relationships (IPRs) of horizontal gas wells are presented in this paper for different reservoir boundary conditions. Even though reservoir simulation models (numerical models) may give more flexibility and detailed results of oil and gas production, analytical models are commonly used in the field for quick, practical and reasonable estimation of well performance. The analytical models are especially attractive when working on single well design and performance optimization. Similar to vertical well models, the analytical models for horizontal wells are developed for specific conditions. The IPR equations for horizontal gas wells are categorized into three boundary conditions; constant boundary pressure (steady-state flow condition), no-flow boundary (pseudosteady-state flow condition), infinite acting reservoir (transient flow condition). For each condition, the IPR equations of horizontal gas wells are presented and the limitation and appropriate application are discussed carefully. Moreover, this paper discusses the effect of critical parameters such as permeability anisotropy, wellbore length, non-Darcy flow and near wellbore formation damage on the inflow performance of horizontal gas wells. These equations provide the reservoir and production engineers with an invaluable tool in well structure design, development plan, and their daily practice dealing with horizontal wells. Published by Elsevier B.V.
Keywords: Horizontal Gas well IPR Non-Darcy flow Anisotropy
1. Introduction Predicting and evaluating the well performance is one of the critical steps in developing new fields, designing new wells, or optimizing the performance of existing wells. Well performance can be predicted by either numerical models or analytical models. Although numerical models in general give more accurate and detailed results, they require extensive input preparation and need more time and effort to be applied, compared to analytical models. Therefore, in practice, analytical models e referred as Inflow Performance Relationship equations (IPR equations) e are used more often, especially in single well studies. Analytical models are developed based on the assumptions about the reservoir boundary conditions, boundary conditions at the well, the flow pattern from the reservoir to the well, and the properties of the reservoir fluid. In general, these models assume a constant pressure throughout the well, so the pressure drop along the wellbore should be small compared to the drawdown. In order to develop IPR equations for horizontal wells, the same reservoir boundary conditions applied in vertical well models can be used; steady-state condition for constant boundary pressure,
* Corresponding author. E-mail address: [email protected] (D. Zhu). 1875-5100/$ e see front matter Published by Elsevier B.V. doi:10.1016/j.jngse.2010.05.002
pseudo-steady-state condition for no-flow boundary condition, and transient flow for infinite acting reservoir or short flow time. The main differences between horizontal well productivity models and vertical well productivity models are the drainage and flow pattern and the effect of permeability anisotropy. In this paper, we will summarize the gas IPR equations for horizontal wells. And since IPR equations for gas wells are developed analogous to the IPR equations for oil wells with the same reservoir boundary conditions, we describe how oil well IPR equations can be modified to be used for gas wells. This paper also discusses the effect of some critical parameters on productivity of horizontal gas wells, such as permeability anisotropy, wellbore length, near wellbore formation damage and non-Darcy flow. 2. Steady-state flow equation The steady-state condition is defined as the reservoir pressure at the drainage boundary being a constant. With this assumption, certain geometries of the reservoir drainage area are assumed to generate the analytical IPR equations. In 1994, applying the superposition principle, Butler presented a steady-state model to predict the productivity of a fully penetrating horizontal well in a box-shaped undersaturated oil reservoir. Fig. 1 shows the geometry used to develop this model. Butler’s model can be expressed as
M. Tabatabaei, D. Zhu / Journal of Natural Gas Science and Engineering 2 (2010) 132e142
133
Fig. 2. Flow pattern described by Furui et al. (2003).
By substituting the definition of the gas formation volume factor, Darcy’s equation for gas can be written in the following from.
qg psc TZ k dp ¼ C ; m dx pTsc Af Fig. 1. Geometry model used in Butler’s (1994) model and Furui et al.’s model.
7:08 103 kH L pe pwf ! qo ¼ : i hIani pyb mo Bo ½Iani ln þ h 1:14Iani þ s rw ðIani þ 1Þ
(1)
Later in 2003, Furui et al. presented another analytical model for fully penetrating horizontal well in a box-shaped reservoir with noflow boundaries at the top and bottom of the reservoir and constant pressure boundaries at the sides. In this model, it is assumed that the flow to a horizontal well can be divided into two regimes, a radial flow regime near the wellbore and a linear flow regime away from the wellbore. Similar to Butler’s model, skin factor caused by formation damage or well completion effects is incorporated in the model. Furui et al.’s model can be written as
qo ¼
7:08
103 kL
pe pwf
mo Bo ½lnðrw ðIhIanianiþ1Þ þ hIpyanib
i; 1:224 þ s
(2)
(3)
Eq. (2) can be rearranged by substituting the effective permeability by [kHkV]0.5.
7:08 103 kH L pe pwf i: qo ¼ mo Bo ½Iani lnðrw ðIhIanianiþ1Þ þ phyb Iani ð1:224 þ sÞ
(4)
qg Bg k dp ¼ C : m dx Af
qg 2 psc TZ
pTsc ðLhÞ
¼ C
kH dp ; m dy
(7)
and
qg psc TZ ¼ C pTsc ð2prLÞ
pffiffiffiffiffiffiffiffiffiffiffi kH kV dp : m dr
(8)
Integrating over the length for linear flow gives qg 2 psc T
Tsc ðLhÞkH
Z ðyb yt Þ ¼ C
pe
p
dp;
(9)
p dp: Z pwf m
(10)
pt mZ
and for radial flow we have, pt
Using the definition of real gas pseudo-pressure function, m(p), presented by Al-Hussainy et al. (1966),
Z mðpÞ ¼ 2
p
p
p0 Z mg
dp:
(11)
Eqs. (9) and (10) can be written as follows.
As we can observe, with completely different approaches, Butler’s model and Furui et al.’s model yield very similar expression for inflow performance of horizontal wells, except the constants in the denominators (1.14 for Butler and 1.224 for Furui et al.). Following describes how Furui et al.’s approach can be used to develop an inflow equation for a horizontal gas well in steady-state condition. Considering the flow pattern illustrated in Fig. 2, flow from the reservoir boundary to the wellbore can be described using Darcy’s law over each part of the flow pattern, linear and radial, separately. General form of Darcy’s equation is
v ¼
where qg, is gas flow rate, C is the unit conversion factor, Tsc and psc are the temperature and pressure at the standard condition and Af is the area open to flow, where for linear flow Af ¼ Lh and for radial flow Af ¼ 2prL, substituting the corresponding area, and considering the direction of flow and the defined coordinate, Darcy’s equation for linear and radial flow part of the flow pattern shown in Fig. 2 can be written as
Z qg psc T rt ¼ C ln Tsc ð2pLÞk rw
where, k is the effective permeability and can be defined as
pffiffiffiffiffiffiffiffiffiffiffi k ¼ kH kV :
(6)
(5)
qg psc T ðy yt Þ ¼ C½mðpe Þ mðpt Þ; Tsc ðLhÞkH b
(12)
and
i h 2qg psc T rt ¼ C mðpt Þ m pwf ; ln Tsc ð2pLÞk rw
(13)
In definition of the real gas pseudo-pressure function, m(p), p0 is the reference pressure and can be any convenience base pressure. Determination of the pseudo-pressure at a given pressure requires knowledge of gas viscosity and gas compressibility factor as functions of pressure and temperature. As these functions are complicated and not explicit, a numerical integration technique is frequently used.
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Combining Eqs. (12) and (13), and considering the values of yt and rt as described by Furui et al., the gas flow from the reservoir boundary to the wellbore in oil field unit can be described as
L mðpe Þ m pwf pffiffiffi3: qg ¼ 2 y p hb 0:5 ln h2rw2 5 1424T 4 þ kH k
(14)
0 rw ¼ rw
(22)
In this equation the first term on the right hand side is the Darcy or viscous component while the second is the non-Darcy component. Eq. (22) can be rewritten as
m dp ¼ ð1 þ Fo Þ v; dx k
To account for the effect of permeability anisotropy, Muskat’s (1937) general transformation and Brigham’s (1990) transformation, can be used as follows.
"
m dp ¼ v þ brv2 : dx k
#
Iani þ 1 pffiffiffiffiffiffiffi ; 2 Iani
(23)
where Fo is the Forchheimer number which is defined as
Fo ¼
brkv : m
(24)
pffiffiffiffiffiffiffi h0 ¼ h Iani ;
(16)
In the above equations, b is the coefficient of inertial resistance or known as turbulence factor and has the dimension [L1]. A common correlation for this coefficient is presented in the following form.
y y0 ¼ pffiffiffiffiffiffiffi: Iani
(17)
b¼
(15)
Including the effect of permeability anisotropy and also the effect of near wellbore formation damage as a skin factor, the IPR equation for a horizontal gas well in steady-state condition can be written as
kL mðpe Þ m pwf i: h qg ¼ pyb 1:224 þ s 1424T lnðrw ðIhIanianiþ1Þ þ hI
(18)
ani
As the real gas pseudo-pressure is difficult to evaluate without a computer program, approximations to Eq. (11) are usually used in the natural gas industry (Guo and Ghalambor, 2005). At pressures lower than 2000 psia, gas pseudo-pressure can be approximated as
Z mðpÞ ¼ 2
p
p
p0 Z mg
dp ¼
p2 p2b Z mg
:
(19)
b ; ka
(25)
a and b in Eq. (25) depend on properties of the formation. There are some other correlations (Geertsma, 1974) which also include the effect of porosity such as
b¼
b : ka fc
(26)
The non-Darcy component in Eq. (22) is negligible at low flow velocities; therefore, it is generally ignored in liquid flow equations. For a given pressure drawdown, however, the velocity of gas is at least an order of magnitude greater than the one for oil, due to the lower viscosity of the former, therefore, the non-Darcy component is often included in equations describing the flow of a real gas through a porous medium. The non-Darcy flow effect can be modeled as an additional skin term, which in literature (Economides et al., 1994; Golan and Whitson, 1991) it is mostly presented in terms of flow rate as,
Using this approximation, Eq. (18) can then be simplified using a pressure-squared approach and it becomes
st ¼ s þ Dq;
kL p2e p2wf h i: qg ¼ pyb 1:224 þ s 1424Z mg T lnðrw ðIhIanianiþ1Þ þ hI
where D is the non-Darcy coefficient. Including the effect of non-Darcy flow as an additional skin term, the general form of the productivity model for horizontal gas wells can be expressed as
(20)
ani
According to this approximation, the IPR equations for gas wells do not have a linear relationship between the flow rate and drawdown, as appeared in the equations for oil wells. At pressures higher than 3000 psia, highly compressed gases behave like liquids. Therefore, Eq. (18) can be approximated using pressure approach as
kL pe pwf h i: qg ¼ pyb 1:224 þ s 141:2 103 mg Bg lnðrw ðIhIanianiþ1Þ þ hI
(21)
ani
For gas wells, the flow velocity is usually much higher than the one for oil wells, especially near the wellbore. At higher flow velocities, in addition to the viscous force component represented by Darcy’s equation, there is also an inertial force acting to convective accelerations of the fluid particles in passing through the pore spaces which is known as non-Darcy flow effect. Deviations from Darcy’s law, observed at high velocities, can be added in mathematical terms in several ways. The most widely accepted model is Forchheimer’s equation (Forchheimer, 1901):
(27)
kL mðpe Þ m pwf h i: qg ¼ pyb 1424T lnðrw ðIhIaniþ1Þ þ hI 1:224 þ s þ Dqg ani
(28)
ani
There are many discussions about the non-Darcy coefficient, D. It can be obtained from lab experimental data, or from empirical correlations. Using Forchheimer’s equation, Furui (2004) developed an equation for the non-Darcy coefficient in horizontal wells considering the permeability anisotropy and formation damage near the wellbore. For horizontal gas well and in field unit the nonDarcy coefficient presented by Furui is expressed as follow.
2 D ¼ 1:4 1014
bgk 6bd b 4 þ 1 d b 2prw Lm b
13
0 B @ rdH rw
þ
Iani þ 1 C7 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A5; rdH rw
2
2 1 þIani
(29)
M. Tabatabaei, D. Zhu / Journal of Natural Gas Science and Engineering 2 (2010) 132e142
where the turbulence factor for undamaged and damaged zone b and bd, can be determined by the correlation presented by Firoozabadi and Katz (1979).
b¼
2:6 1010 : k1:2
(30)
Therefore, for horizontal well
2:6 1010
b ¼ pffiffiffiffiffiffiffiffiffiffiffi1:2 ;
(31)
kH kV
and
2:6 1010
bd ¼ pffiffiffiffiffiffiffiffiffiffiffi1:2 : kH kV
(32)
d
Note that the calculation of formation damage skin factor in horizontal wells is different from vertical wells due to permeability anisotropy. For a horizontal well, the formation damage skin factor can be calculated as (Furui, 2004)
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 rdH 2 1 dH þ þIani rw rw k A: s ¼ 1 ln@ Iani þ 1 kd 0r
(33)
3. Pseudo-steady-state flow equation Pseudo-steady-state models of inflow performance assume that reservoir is bounded by no-flow boundaries and pressure declines in a uniform fashion in the reservoir. In this case an average reservoir pressure is introduced in the IPR equations. Pressure decline curves, if available, can be used to calculate the average pressure as a function of time, and therefore to obtain a production history. Babu and Odeh (1989) presented a horizontal well IPR model under the pseudo-steady-state condition. The model treated the reservoir as a box-shaped drainage area with horizontal well-drilled parallel to one side of the reservoir. Fig. 3 illustrates the geometry used in Babu and Odeh’s model. Location of wellbore in drainage area in this model is arbitrary. Their model is generally based on the radial flow in the yez plane, with the deviation of the drainage area from a circular shape in this plane accounted for with a geometry factor, and flow from beyond the wellbore in the xdirection accounted for with a partial penetration skin factor. Note that the geometry factor used in Babu and Odeh’s model is related
Fig. 3. Geometry model used in Babu and Odeh’s (1989) model.
135
inversely to the commonly used Dietz (1965) shape factor. Thus, Babu and Odeh’s inflow equation is
pffiffiffiffiffiffiffiffiffi kx kz b p pwf h pffiffiffi i: qo ¼ 141:2mo Bo ln rwA þ lnCH 0:75 þ s þ sR
(34)
where A is the drainage area (ah), CH is the shape factor, sR is the partial penetration skin factor and s is any other kind of skin factors, such as completion or damage skin effects. The shape factor, CH, accounts for the deviation of the shape of the drainage area from cylindrical and the departure of the wellbore location from the middle of the system. The partial penetration skin, sR, accounts for the flow from the reservoir located beyond the end points of the well in the x-direction, and it is equal to zero for a fully penetrating horizontal well. Follow the approach used in previous section, the IPR equation for single-phase gas well for pseudo-steady-state condition can be derived from the oil well pseudo-steady-state IPR equation. Considering the non-Darcy flow effect, the resulting equation is
i pffiffiffiffiffiffiffiffiffi h kx kz b mðpÞ m pwf h pffiffiffi i qg ¼ 1424T ln rwA þ lnCH 0:75 þ s þ sR þ Dqg
(35)
And the appropriate equations for calculating the shape factor, CH, and the partial penetration skin, sR, are presented in the Appendix. As pointed out in the original paper, the model assumed that the thickness of the formation, h, is generally much smaller than the other two dimensions of the drainage box, a and b. If this condition does not apply, it should be examined first if a horizontal well is the right application for the field development. The equation is very helpful when used to examine the effects of the reservoir and well parameters on well performance, and therefore to optimize well design and operation. It also should be noted that, the results from previously discussed single-phase IPR models, Butler and Furui et al., shouldn’t be compared with Babu and Odeh’s model since the assumption of the boundary conditions used to develop the models are absolutely different. 4. Transient flow equation This type of flow, also known as infinite-acting flow, occurs when the boundary effect is not observed. This flow ends when all the outer boundaries of the reservoir are reached by the propagating pressure disturbance. During this period, the well behaves as if it was placed in a reservoir with infinite size.
Fig. 4. Geometry model used in Kuchuk et al.’s (1991) model.
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M. Tabatabaei, D. Zhu / Journal of Natural Gas Science and Engineering 2 (2010) 132e142
The transient flow period provides engineers with an invaluable tool to determine reservoir parameters and well productivity. Therefore, it was the subject of many studies for many years (Clonts and Ramey, 1986; Ozkan et al., 1989; Odeh and Babu, 1990; Kuchuk et al., 1991). Kuchuk et al. (1991) presented an analytical solution for transient pressure behavior of a horizontal well completed in an
4.2. Second radial flow period This is a hemicylindrical flow period that follows the first radial flow. This flow period may occur when the well is not centered with respect to the top and bottom boundaries. The equation for the second flow period is
pffiffiffiffiffiffiffiffiffiffiffi kH kV L1=2 pi pwf qo ¼ qffiffiffiffi ii: h pffiffiffiffiffiffiffiffi h k k t 162:6Bo mo log fm Hc rV2 3:2275 þ 0:4343s log 1 þ kkH zrww
(39)
V
o t w
infinite anisotropic medium bounded above and below by horizontal planes (Fig. 4), using Laplace transform. Their solution is based on the uniform-flux, line-source solution and also averaging the pressure along the length of the well. They identified four different flow periods during the transient flow of a horizontal well and presented specific equations for each of them. Applying the same approach used earlier, the transient flow equation of the horizontal gas wells for each flow period can be derived from the corresponding oil well transient flow equation. 4.1. First radial flow period The very first flow pattern for horizontal wells is elliptic-cylindrical. After some time, the elliptic-cylindrical flow period becomes
This flow period starts to appear at
fmo ct
t ¼
0:0002637pkV
o n min z2w ; ðh zw Þ2 ;
(40)
n o max z2w ; ðh zw Þ2 :
(41)
and ends at
fmo ct
t ¼
0:0002637pkV
And for a horizontal gas well, the equation for this flow period can be expressed as
pffiffiffiffiffiffiffiffiffiffiffi kH kV L1=2 mðpi Þ m pwf qg ¼ qffiffiffiffi ii: h pffiffiffiffiffiffiffiffi h k k t 1639:8T log fm Hc rV2 3:2275 þ 0:4343 s þ Dqg log 1 þ kkH zrww V
g t w
approximately radial. This radial flow around the wellbore may continue until the effect of the nearest boundary is felt at the wellbore. The behavior of this period is equivalent to the behavior of fully penetrating vertical well in an infinite reservoir. The equation for the first radial flow period in oilfield units may be written as
4.3. Intermediate-time linear flow period If the horizontal well is much longer than the reservoir thickness, for the no-flow boundary case, this flow period may develop after the effects of the upper and lower boundaries are felt at the wellbore. The equation for the linear flow is
pffiffiffiffiffiffiffiffiffiffiffi 2 kH kV L1=2 pi pwf qo ¼ qffiffiffiffi: h pffiffiffiffiffiffiffiffi q k k t 162:6Bo mo log fm Hc rV2 3:2275 þ 0:8686s 2log12 4kkH þ 4 kkV o t w
V
The start of the effect of the nearest boundary (no-flow or constant pressure) or the end of this flow period can be determined as
t ¼
fmo ct
0:0002637pkV
o n min z2w ; ðh zw Þ2 :
The corresponding equation for a horizontal gas well is
(37)
(36)
H
pi pwf sffiffiffiffiffiffiffiffiffiffiffiffiffi qo ¼ " #; mo t 8:128 141:2mo 70:6mo pffiffiffiffiffiffiffiffiffiffiffis Bo þ sz þ kH h 2hL1=2 fct kH L1=2 kH kV
(43)
where
pffiffiffiffiffiffiffiffiffiffiffi 2 kH kV L1=2 mðpi Þ m pwf qg ¼ qffiffiffiffi: h pffiffiffiffiffiffiffiffi q k k t 1639:8T log fm Hc rV2 3:2275 þ 0:8686 s þ Dqg 2log12 4kkH þ 4 kkV g t w
(42)
V
H
(38)
M. Tabatabaei, D. Zhu / Journal of Natural Gas Science and Engineering 2 (2010) 132e142
sz ¼ 1:1513
sffiffiffiffiffiffi sffiffiffiffiffiffi! # " pz prw kH h kV w 1þ : log sin kV L1=2 h kH h
5. Discussion
(44)
This flow period starts to develop at
t ¼
fmo ct h2 ; 0:0002637kV
(45)
and ends at
t ¼
fmo ct L21=2 0:0002637kH
:
(46)
This flow period will not appear for wells with a gas cap or a bottom aquifer. The corresponding equation for a horizontal gas well can be expressed as
One of the main advantages of the analytical productivity models is that they provide us with an invaluable tool to investigate the effect of different parameters on productivity of a well and then to optimize well design and operation. In this section we first show a brief comparison between the analytical models and numerical solution, and then discuss the effect of some critical parameters on performance of horizontal gas wells. 5.1. Comparison of analytical models and numerical solution Kamkom and Zhu (2006) showed that when the flowing bottomhole pressure, pwf, is not too low, the analytical models for productivity of horizontal gas wells give the similar results compared to the simulation results (numerical solution). Fig. 5 shows such a comparison for a steady-state case. The comparison
mðpi Þ m pwf " #: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qg ¼ 4:064 t 1 1 pffiffiffiffiffiffiffiffiffiffiffi s þ Dqg 1424T þ sz þ 141:2hL1=2 fmg ct kH kH h 2L1=2 kH kV 4.4. Late-time radial flow period After a sufficiently long time, the pressure front will become approximately radial in the xey plane and a third radial flow pattern will develop. Similar to intermediate-time linear flow period, this period does not exist for wells with a gas cap or bottom aquifer. The equation for late-time radial flow is
"
qo ¼ Bo mo
pi pwf
137
(47)
for the studied condition showed that at low to moderate flow rate (less than 50 MMscf/day), the analytical solution matches the simulation results very well. When flow rate increases (higher drawdown) the results of analytical solution starts deviating from the simulation results, and the deviation is more pronounced at higher flow rate.
" ! # #; 162:6 kH t 141:2 70:6 p ffiffiffiffiffiffiffiffiffiffi ffi log 2:5267 þ þ s s z kH h kH h fmo ct L21=2 L1=2 kH kV
(48)
5.2. Effect of permeability anisotropy
where
sffiffiffiffiffiffi sffiffiffiffiffiffi! # " pz prw kH h kV w 1þ log sin kV L1=2 h kH h 2 2 k h 1 zw zw 0:5 H 2 þ 2 : kV L1=2 3 h h
sz ¼ 1:1513
(49)
The onset of this flow period is
20fmo ct L21=2 tz : 0:0002637kH
(50)
And for a horizontal gas well we have
Consider a fully penetrating horizontal well in a gas reservoir with no-flow boundary and reservoir properties described in Table 2. The effect of permeability anisotropy on well performance can be studied applying Babu and Odeh’s model since there is no external pressure support in this case. Fig. 6 illustrates the variation of the productivity index with permeability anisotropy, Iani, for two different scenarios. In the first scenario, horizontal permeability was kept constant (kH ¼ 1 md) and vertical permeability was decreased to increase Iani, and in other one, vertical permeability was kept constant (kV ¼ 0.1 md) and horizontal permeability was increased to increase Iani. As we can see, the variation in J with Iani is different for these two
mðpi Þ m pwf " #: ! # " qg ¼ 1:151 kH t 1 1 pffiffiffiffiffiffiffiffiffiffiffi s þ Dqg log 2:5267 þ 1424T sz þ kH h kH h fmg ct L21=2 2L1=2 kH kV
The equations for inflow performance relationship of horizontal wells under different boundary conditions are summarized in Table 1.
(51)
scenarios. In first one, productivity index decreases with increase in Iani because the effective permeability ([kHkV]0.5) is decreasing as a result of decrease in the vertical permeability. In second one,
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M. Tabatabaei, D. Zhu / Journal of Natural Gas Science and Engineering 2 (2010) 132e142
Table 1 Summary of IPR equations for horizontal wells.
Oil
Steady-state flow Augmented Joshi qo ¼
141:2mo Bo
kH hðpe pwf Þ ffii n h pffiffiffiffiffiffiffiffiffiffiffiffi h 2 2 aþ
ln
a ðL=2Þ
I
h
þ aniL
L=2
ln
Iani h rw ðIani þ1Þ
Pseudo-steady-state flow pffiffiffiffiffiffiffiffiffi k k bðp pwf Þ h pffiffixffi z i qo ¼ 141:2mo Bo ln rwA þ ln CH 0:75 þ s þ sR
i o þs
Butler "
qo ¼
mo Bo
ln CH ¼ 6:28I
7:08 103 kH Lðpe pwf Þ # Iani ln rw ðIhIanianiþ1Þ þ phyb 1:14Iani þ s
h
a
ani h
1 3
xa0 þ
2 i x0 a
ln sinphz0 12ln I a h 1:088 ani
Furui et al. "
qo ¼
mo Bo Gas
qg ¼
7:08 103 kLðpe pwf Þ # py b ln rw ðIhIanianiþ1Þ þ hI 1:224 þ s ani
kLðmðpe Þ mðpwf ÞÞ # py b 1424T ln rw ðIhIanianiþ1Þ þ hI 1:224 þ s þ Dqg "
qg ¼
pffiffiffiffiffiffiffiffiffi kx kz b½mðpÞ mðpwf Þ
pffiffiffi 1424T½lnð rwAÞ
þ lnCH 0:75 þ s þ sR þ Dqg
ani
Transient flow of slightly compressible fluid (oil well) First radial
Second radial
pffiffiffiffiffiffiffiffiffiffiffi 2 kH kV L1=2 ðpi pwf Þ # qffiffiffiffi qffiffiffiffi pffiffiffiffiffiffiffiffi k k t 162:6Bo mo log fm Hc rV2 3:2275 þ 0:8686s 2log 12 4 kkH þ 4 kkV V H o t w pffiffiffiffiffiffiffiffiffiffiffi kH kV L1=2 ðpi pwf Þ " " # qo ¼ qffiffiffiffi pffiffiffiffiffiffiffiffi k k t 162:6Bo mo log fm Hc rV2 3:2275 þ 0:4343s log 1 þ kkH zrww "
qo ¼
V
o t w
"
Intermediate-time linear qo ¼ Bo
ðpi pwf Þ qffiffiffiffiffiffiffiffiffi mo t 141:2mo 8:128 fc k þ k h s z þ 2hL 1=2
sz ¼ 1:1513 Late-time radial
t H
"
qffiffiffiffi
"
Bo mo 162:6 kH h sz ¼ 1:1513
log
kH t fmo ct L21=2
fmo ct minfz2w ; ðh zw Þ2 g t 0:0002637pkV o n fmo ct max z2w ; ðh zw Þ2 0:0002637pkV fmo ct L21=2 fmo ct h2 t 0:0002637kV 0:0002637kH
# mo 70:6 pffiffiffiffiffiffiffiffi s L1=2 kH kV
þ
qffiffiffiffi kV Þsinðphzw Þ k H
ðpi pwf Þ # s þ 2:5267 þ141:2 k h z H
#
qffiffiffiffi kH h log phrw ð1 þ kkV Þsinðphzw Þ 0:5kkH k L1=2 V
H
t
70:6 pffiffiffiffiffiffiffiffis L1=2 kH kV
"
qffiffiffiffi
fmo ct minfz2w ; ðh zw Þ2 g 0:0002637pkV
#
kH h log phrw ð1 kV L1=2
"
qo ¼
H
t
V
h2 1 L21=2 3
z2
zhw þ hw2
20fmo ct L21=2 0:0002637kH
Transient flow of compressible fluid (gas well) First radial
Second radial
pffiffiffiffiffiffiffiffiffiffiffi fmg ct 2 kH kV L1=2 ðmðpi Þ mðpwf ÞÞ # t minfz2w ; ðh zw Þ2 g qffiffiffiffi qffiffiffiffi pffiffiffiffiffiffiffiffi 0:0002637pkV kH kV t 4 4 k k H V 1639:8T log fm c r2 3:2275 þ 0:8686ðs þ Dqg Þ 2log12 þ kV kH g t w pffiffiffiffiffiffiffiffiffiffiffi fmg ct kH kV L1=2 ðmðpi Þ mðpwf ÞÞ " ## " qg ¼ minfz2w ; ðh zw Þ2 g t qffiffiffiffi pffiffiffiffiffiffiffiffi 0:0002637pkV kH kV t kH zw 1639:8T log fm c r2 3:2275 þ 0:4343ðs þ Dqg Þ log ð1 þ k Þrw o n V fmg ct g t w max z2w ; ðh zw Þ2 0:0002637pkV "
qg ¼
ðmðpi Þmðpwf ÞÞ
"
Intermediate-Time Linear qg ¼ 1424T
4:064 141:2hL1=2
sz ¼ 1:1513 Late-time radial
1424T
t
fmg ct kH
1:151 kH h
log
1 ffiffiffiffiffiffiffi p ðsþDqg Þ kH kV
#
þ
qffiffiffiffi kV Þsinðphzw Þ k H
ðmðpi Þ mðpwf ÞÞ # 2:5267 þk1hsz þ
fmg ct L21=2 fmg ct h2 t 0:0002637kV 0:0002637kH
#
2L1=2
H
kH h log phrw ð1 kV L1=2
"
sz ¼ 1:1513
þk 1 hsz þ
"
qffiffiffiffi
"
qg ¼
qffiffiffiffiffiffiffiffiffiffiffi
kH t fmg ct L21=2
H
# 2L1=2
# qffiffiffiffi prw pz w kH h kV log Þ 0:5kkH 1 þ sinð k L1=2 h k h
qffiffiffiffi V
"
H
V
productivity index increases because effective permeability is increasing as a result of increase in the horizontal permeability. This example reveals that, in absence of the non-Darcy flow effect, kH has a more pronounced and constant effect on productivity (linear relationship) compared with the kV effect where it declines very
1 ffiffiffiffiffiffiffiffiðs þ Dq Þ p g kH kV
h2 1 L21=2 3
z2
zhw þ hw2
t
20fmg ct L21=2 0:0002637kH
fast and the effect diminishes as kV approaches a small value. This is because the denominator in Eq. (35) is related to permeability anisotropy through the shape factor and as we can observe from Fig. 7, ln(CH) declines rapidly with the increase in Iani and it approaches almost a constant value for higher permeability
M. Tabatabaei, D. Zhu / Journal of Natural Gas Science and Engineering 2 (2010) 132e142
139
Fig. 6. Effect of permeability anisotropy on productivity of a horizontal gas well. Fig. 5. Comparison of the analytical model for horizontal gas wells and the numerical simulation results.
anisotropy values. Consequently, for higher permeability anisotropy values (greater than 4) we can show the relationship between productivity index and Iani is
k Jf H or JfkV Iani : Iani
therefore, productivity of well increases. It should be mention that, to truly investigate the effect of wellbore length, it is necessary to consider the effect of pressure drop along the wellbore on productivity of well. The benefit of longer horizontal well length will be jeopardized for long horizontal wells with high inflow rate. When frictional pressure drop in the wellbore is significant, production becomes tubing-limited (Hill and Zhu, 2007)
(52)
The above relationships clearly explain the behavior we observed in Fig. 6. However, if we include the effect of non-Darcy flow (see Fig. 6) we would see that these relationships would no longer be valid since the denominator in Eq. (35) cannot be considered constant in higher permeability anisotropies and is changing with flow rate. The deviation from the relationship described in Eq. (52) would be even more dramatic if productivity index was higher (due to higher drawdown or higher permeability values). 5.3. Effect of wellbore length Similar to previous case, consider a horizontal well penetrated a gas reservoir with no-flow boundary. Using the same reservoir properties described in Table 2 and assuming horizontal permeability and vertical permeability of the reservoir is 1 md and 0.1 md, respectively. The effect of wellbore length can be studied again by using Babu and Odeh’s model. Fig. 8 shows the effect of wellbore length, L, and penetration degree, L/b on productivity of this horizontal gas well. In this case b is constant and L/b is changing by change of L. As we expected, productivity of the well increases by increasing the wellbore length or penetration degree. This is because by increasing the penetration degree, partial penetration skin factor, sR, decreases (see Fig. 8),
5.4. Effect of near wellbore formation damage and non-darcy flow Consider a fully penetrating horizontal well in a gas reservoir described in Table 3. The pressure at the drainage boundary of this reservoir is 5600 psi and it is assumed to be constant due to external pressure support. Fig. 9 illustrates the IPR curve of this well for the following cases. (a) Assuming there is no near wellbore formation damage and ignoring the non-Darcy flow effect. (b) Assuming there is no near wellbore formation damage, but this time considering the non-Darcy flow effect. (c) Assuming a uniform damage along the entire wellbore length, with the damage extending 6 inches beyond the well in the horizontal direction and damage permeability is 10% of undamaged permeability (kd ¼ 0.1k), and ignoring the nonDarcy flow effect.
Table 2 Reservoir and well parameters used in the study. Reservoir length Reservoir width Reservoir thickness Wellbore diameter Average reservoir pressure Bottomhole pressure Reservoir temperature Gas gravity Skin factor
2000 ft 1200 ft 50 ft 4 in 4500 psi 500 psi 210 F 0.7 (air ¼ 1) 0
Fig. 7. Effect of permeability anisotropy on shape factor.
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Fig. 8. Effect of wellbore length and penetration ratio (L/b) on productivity.
Table 3 Reservoir and well parameters in study the effect formation damage and non-Darcy flow. Reservoir length Reservoir width Reservoir thickness Wellbore diameter Reservoir pressure Horizontal permeability Vertical permeability Reservoir temperature Gas gravity
2000 ft 1500 ft 50 ft 4 in 5600 psi 1 md 0.1 md 210 F 0.7 (air ¼ 1)
(d) Assuming a uniform damage along the entire wellbore length as described in case “c”, and also considering the non-Darcy flow effect. As shown in Fig. 9, near wellbore formation damage has a significant effect on the well performance. For this case it reduces the productivity of well by about 25%. This figure also reveals that non-Darcy flow effect is very insignificant in absence of near wellbore formation damage but presence of formation damage can intensify its effect. For this case, in high rates, we would lose about 2% of the production rate when there is no formation damage, but about 10% of the production rate in presence of formation damage. However, for both cases the effect of non-Darcy flow is insignificant in low rates. But this is not the case for vertical wells. Fig. 10 shows the inflow performance of a vertical well penetrates a reservoir with exactly
Fig. 10. Effect of near wellbore formation damage and non-Darcy flow on productivity of a vertical gas well (pseudo-steady-state case).
the same geometry, properties, pressure and the same fluid properties. As we can see, again there is a significant reduction is production rate due to the near wellbore formation damage, even more than the case of horizontal well (about 60% for this case). Although production rate is much lower than the case of horizontal well, we can see a significant reduction in production rate due to non-Darcy flow either in absence or presence of formation damage (20%). This is because the area open to flow in case of vertical well is so much smaller than horizontal wells. Therefore, non-Darcy flow plays a less significant rule in productivity of horizontal wells compared to vertical wells. To investigate more about the effect of non-Darcy flow on productivity of horizontal gas wells, we considered two other examples. One pseudo-steady state flow case and one transient flow case. Tables 4 and 5 present the reservoir and wellbore parameters used for these two examples. Fig. 11 shows the results for pseudo-steady state flow condition. As we can see, similar to the steady state example discussed Table 4 Reservoir and well parameters used to study the effect formation damage and nonDarcy flow (pseudo-steady-state case). Reservoir length Reservoir width Reservoir thickness Wellbore length Wellbore diameter Reservoir pressure Horizontal permeability Vertical permeability Reservoir temperature Gas gravity
2000 ft 1200 ft 50 ft 1500 ft 4 in 4500 psi 1 md 0.1 md 210 F 0.7 (air ¼ 1)
Table 5 Reservoir and well parameters used to study the effect formation damage and nonDarcy flow (transient flow case).
Fig. 9. Effect of near wellbore formation damage and non-Darcy flow on productivity of a horizontal gas well (steady-state case).
Reservoir length Reservoir width Reservoir thickness Wellbore length Wellbore diameter Initial reservoir pressure Horizontal permeability Vertical permeability Reservoir temperature Gas gravity Reservoir porosity Total compressibility of reservoir
5000 ft 2500 ft 100 ft 2500 ft 4 in 5200 psi 1 md 0.1 md 210 F 0.7 (air ¼ 1) 20% 5 105 psi1
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141
This paper discussed the effect of some critical parameters on productivity of horizontal gas wells such as, permeability anisotropy, wellbore length, near wellbore formation damage and nonDarcy flow. It is concluded that to study the effect of the permeability anisotropy on productivity of a horizontal well it is important to consider the effect of horizontal and vertical permeability separately. It was also observed that the effect of non-Darcy flow in horizontal wells is not as significant as vertical wells. However, in transient flow case with high flow rate, non-Darcy flow can significantly affect the productivity of well and reduce the production rate. Moreover, we found out that the near wellbore formation damage would intensify the non-Darcy flow effect. In other words, reduction in production rate due to the non-Darcy flow effect is much more in presence of formation damage. This would suggest that stimulation of a horizontal gas well will help to minimize the effect of non-Darcy flow. Fig. 11. Effect of near wellbore formation damage and non-Darcy flow on productivity of a horizontal gas well (pseudo-steady-state case).
Fig. 12. Effect of near wellbore formation damage and non-Darcy flow on productivity of a horizontal gas well (transient flow case).
previously, in absence of the near wellbore formation damage, nonDarcy flow has an insignificant effect on productivity of a horizontal gas well in pseudo-steady state flow condition. Also we can see, the same as previous example, non-Darcy flow plays a bigger rule when there is formation damage around the wellbore. Fig. 12 illustrates the effect of non-Darcy flow in transient flow. This figure reveals that, even in absence of the near wellbore formation damage, non-Darcy flow effect significantly reduces the production rate in this case. This is because non-Darcy flow is a function of the flow rate. The higher rate, the more significant the effect of non-Darcy flow would be. Therefore, it encounters more reduction in production rate in this case since flow rate is much higher.
6. Summary Evaluating horizontal well performance can be done by either numerical models or analytical models. When used correctly, the analytical models give reasonable predictions of horizontal well performances and they can be very helpful in designing, operating and optimizing horizontal wells.
Nomenclature A, drainage area of horizontal well, ft2 Af, area open to flow, ft2 a, extension of drainage volume of horizontal well in x-direction, ft aH,max, horizontal axis of damage ellipse, ft Bo, oil formation volume factor, rb/STB Bg, gas formation volume factor, res ft3/scf b, extension of drainage volume of horizontal well in y-direction, ft C, unit conversion factor CH, geometric factor in Babu and Odeh’s model ct, total compressibility, psi1 D, non-Darcy flow coefficient, day/Mscf Fo, Forchheimer number, dimensionless h, reservoir thickness, ft Iani, permeability anisotropy, dimensionless J, productivity index, STB/day/psi k, effective permeability, md kd, effective permeability in damaged zone, md kx, permeability in x-direction, md ky, permeability in y-direction, md kz, permeability in z-direction, md kH, horizontal permeability, md kV, vertical permeability, md L, horizontal wellbore length, ft L1/2, half length of horizontal wellbore, ft m(p), pseudo-pressure, psi2/cp p, reservoir pressure, psi p0, reference pressure, psi pavg, average reservoir pressure, psi psc, pressure at standard condition, psi pe, reservoir pressure at boundary, psi pi, initial pressure, psi pwf, bottomhole flowing pressure, psi qo, oil production rate, STB/day qg, gas production rate, Mscf/day rdH, radius of damaged zone in horizontal direction, ft re, radius of outer boundary of the reservoir, ft rw, wellbore radius, ft s, skin factor, dimensionless sR, partial penetration skin factor, dimensionless T, reservoir temperature, R Tsc, temperature at standard condition, R t, ime, hr v, velocity, ft/s
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M. Tabatabaei, D. Zhu / Journal of Natural Gas Science and Engineering 2 (2010) 132e142
x0, x coordinate of center of well, ft ymid, midpoint location along well length, ft yb, half length of the reservoir width perpendicular to the wellbore, ft y1, y coordinate of beginning of well, ft y2, y coordinate of end of well, ft Z, gas compressibility factor Zavg, average gas compressibility factor z0 z coordinate of center of well, ft zw, distance of wellbore from the lower boundary, ft b, turbulence factor, ft1 bd, turbulence factor for damaged zone, ft1 r, density, lbm/ft3 m, viscosity, cp mo, oil viscosity, cp mg, gas viscosity, cp mg,avg, average gas viscosity, cp gg, gas gravity, dimensionless, (air ¼ 1) 4, porosity, dimensionless
FðxÞ ¼ ðxÞ 0:145 þ lnðxÞ 0:137ðxÞ2 :
(A-6)
If arguments are >1, then:
FðxÞ ¼ ð2 xÞ 0:145 þ lnð2 xÞ 0:137ð2 xÞ2 :
(A-7)
where x can be p 4yffiffiffiffiffi orffiffiffiffiffi 4ymid pL=2b. mid þ L=2b p ffiffiffiffiffi Case 2: If b= ky > 1:33a= kx [h= kz , then
sR ¼ Pxyz þ Py þ Pxy ;
(A-8)
where
Pxyz ¼
Py ¼
pz b h kx 1 ln þ0:25ln ln sin 0 1:84 ; kz h L rw
6:28b2 ah
! pffiffiffiffiffiffiffiffiffi" # L L kx kz 1 ymid y2mid 3 ; þ 2 þ 3 24b b b ky b
(A-9)
(A-10)
and Appendix
The heart of Babu and Odeh’s model are procedures for calculating the shape factor and the partial penetration skin factor. These parameters were obtained by simplifying the solution of the diffusivity equation for parallelepiped reservoir geometry and comparing it with the assumed inflow equation (Eq. (34)). Babu and Odeh solved the 3D diffusivity equation with a wellbore boundary condition of constant flow rate (uniform flux) at the well and no flow across the reservoir boundaries using the Green’s function approach. In this manner, the following correlations for the shape factor and the partial penetration skin factor were obtained.
pz a 1 x 0 x 0 2 ln sin 0 ln CH ¼ 6:28 þ Iani h 3 a a h 1 a 1:088: ln 2 Iani h
(A-1)
Calculation of the partial penetration skin depends on the geometry and the permeability anisotropy of reservoir. Babu and Odeh defined two thisp parameter. pffiffiffiffifficases for calculating ffiffiffiffiffi pffiffiffiffiffi Case 1: If a= kx 0:75b= ky [0:75h= kz , then 0 sR ¼ Pxyz þ Pxy ;
(A-2)
where
Pxyz ¼
b 1 L
pz h kx ln þ0:25ln ln sin 0 1:84 ; kz h rw
(A-3)
and 0 Pxy
2b2 ¼ LhIani
! L 4ymid þL 4ymid L þ0:5 F F ; F 2b 2b 2b (A-4)
where, ymid ¼ 0:5ðy1 þy2 Þ is the midpoint along the well length and
F
" 2 # L L L L ¼ 0:145 þ ln 0:137 : 2b 2b 2b 2b
(A-5)
The evaluation of Fð4ymid þ L=2bÞ and Fð4ymid L=2bÞ depends on their arguments. If arguments are 1, then:
Pxy ¼
0 sffiffiffiffiffi1 ! b 6:28a kz A 1 x0 x20 @ : 1 þ kx L h 3 a a2
(A-11)
References Al-Hussainy, R., Ramey, H.J., Crawford, P.B., 1966. The flow of real gases through porous media. Journal of Petroleum Technology 18 (5), 624e636. SPE-1243-PA. Babu, D.K., Odeh, A.S., 1989. Productivity of a horizontal well. SPE Resorvoir Engineering 4 (4), 417e421. SPE-18298-PA. Brigham, W.E., 1990. Discussion of productivity of a horizontal well. SPE Resorvoir Engineering 5 (2), 254e255. SPE-20394-PA. Butler, R.M., 1994. Horizontal Wells for the Recovery of Oil, Gas and Bitumen, Monograph No. 2. Petroleum Society of Canadian Institute of Mining, Metallurgy, and Petroleum. Clonts, M.D., Ramey, H.J., Jr. 1986. Pressure-transient analysis for wells with horizontal drainholes. In: Paper SPE 15116 Presented at the SPE California regional Meeting, Oakland, California, 2e4 April. Dietz, D.N., 1965. Determination of average reservoir pressure from build-up survey. Journal of Petroleum Technology 17 (8), 955e959. SPE-1156-PA. Economides, M.J., Hill, A.D., Ehlig-Economides, C., 1994. Petroleum Production Systems. Prentice Hall Inc., Englewood Cliffs, New Jersey. Firoozabadi, A., Katz, D.L., 1979. An analysis of high velocity gas flow through Porous Media. JPT 31 (2), 211e216. SPE-6827-PA. Forchheimer, P., 1901. Wasserbewegung druch Boden. Z. Vereines Deutcher Ingenieure 45 (50), 1782–1788 Furui, K., 2004. A Comprehensive Skin Factor model For Well Completions Based on Finite Element Simulations. PhD dissertation, University of Texas, Austin, Texas. Furui, K., Zhu, D., Hill, A.D., 2003. A rigorous formation damage skin factor and reservoir inflow model for a horizontal well. SPE Production and Facilities 18 (3), 151e157. SPE-84964-PA. Geertsma, J., 1974. Estimating the coefficient of inertial resistance in fluid flow through porous media. SPE Journal 14 (5), 445e450. SPE-4706-PA. Golan, M., Whitson, C.H., 1991. Well Performance. Prentice Hall Inc., Englewood Cliffs, New Jersey. Guo, B., Ghalambor, A., 2005. Natural Gas Engineering Handbook. Gulf Publishing Co., Houston, Texas. Hill, A.D., Zhu, D., June 2007. The relative importance of wellbore pressure drop and formation damage in horizontals well. Nafta e Exploration, Production, Processing, Petrochemisty. SPE Production and Operations, 334e338. Kamkom, R., Zhu, D., 2006. Generalized horizontal well inflow relationships for liquid, gas, or two-phase flow. In: Paper SPE 99712 Presented at the SPE Symposium on Improved Oil Recovery, Tulsa, Oklahoma, 22e26 April. Kuchuk, F.J., Goode, P.A., Wilkinson, D.J., Thambynayagam, R.K.M., 1991. Pressuretransient behavior of horizontal wells with and without gas cap or aquifer. SPE Formation Evaluation 6 (1), 86e94. SPE-17413-PA. Muskat, M., 1937. The Flow of Homogeneous Fluids Through Porous Media. McGraw-Hill Book Co. Inc, New York City. Odeh, A.S., Babu, D.K., 1990. Transient flow behavior of horizontal wells: pressure drawdown and buildup analysis. SPE Formation Evaluation 5 (1), 7e15. SPE18802-PA. Ozkan, E., Raghvan, R., Joshi, S.D., 1989. Horizontal well pressure analysis. SPE Formation Evaluation 4 (4), 567e575. SPE 16378-PA.
Contents lists available at ScienceDirect
Journal of Natural Gas Science and Engineering journal homepage: www.elsevier.com/locate/jngse
Generalized inflow performance relationships for horizontal gas wells M. Tabatabaei, D. Zhu* Texas A&M University, TX, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 19 April 2010 Accepted 6 May 2010 Available online 8 June 2010
The analytical inflow performance relationships (IPRs) of horizontal gas wells are presented in this paper for different reservoir boundary conditions. Even though reservoir simulation models (numerical models) may give more flexibility and detailed results of oil and gas production, analytical models are commonly used in the field for quick, practical and reasonable estimation of well performance. The analytical models are especially attractive when working on single well design and performance optimization. Similar to vertical well models, the analytical models for horizontal wells are developed for specific conditions. The IPR equations for horizontal gas wells are categorized into three boundary conditions; constant boundary pressure (steady-state flow condition), no-flow boundary (pseudosteady-state flow condition), infinite acting reservoir (transient flow condition). For each condition, the IPR equations of horizontal gas wells are presented and the limitation and appropriate application are discussed carefully. Moreover, this paper discusses the effect of critical parameters such as permeability anisotropy, wellbore length, non-Darcy flow and near wellbore formation damage on the inflow performance of horizontal gas wells. These equations provide the reservoir and production engineers with an invaluable tool in well structure design, development plan, and their daily practice dealing with horizontal wells. Published by Elsevier B.V.
Keywords: Horizontal Gas well IPR Non-Darcy flow Anisotropy
1. Introduction Predicting and evaluating the well performance is one of the critical steps in developing new fields, designing new wells, or optimizing the performance of existing wells. Well performance can be predicted by either numerical models or analytical models. Although numerical models in general give more accurate and detailed results, they require extensive input preparation and need more time and effort to be applied, compared to analytical models. Therefore, in practice, analytical models e referred as Inflow Performance Relationship equations (IPR equations) e are used more often, especially in single well studies. Analytical models are developed based on the assumptions about the reservoir boundary conditions, boundary conditions at the well, the flow pattern from the reservoir to the well, and the properties of the reservoir fluid. In general, these models assume a constant pressure throughout the well, so the pressure drop along the wellbore should be small compared to the drawdown. In order to develop IPR equations for horizontal wells, the same reservoir boundary conditions applied in vertical well models can be used; steady-state condition for constant boundary pressure,
* Corresponding author. E-mail address: [email protected] (D. Zhu). 1875-5100/$ e see front matter Published by Elsevier B.V. doi:10.1016/j.jngse.2010.05.002
pseudo-steady-state condition for no-flow boundary condition, and transient flow for infinite acting reservoir or short flow time. The main differences between horizontal well productivity models and vertical well productivity models are the drainage and flow pattern and the effect of permeability anisotropy. In this paper, we will summarize the gas IPR equations for horizontal wells. And since IPR equations for gas wells are developed analogous to the IPR equations for oil wells with the same reservoir boundary conditions, we describe how oil well IPR equations can be modified to be used for gas wells. This paper also discusses the effect of some critical parameters on productivity of horizontal gas wells, such as permeability anisotropy, wellbore length, near wellbore formation damage and non-Darcy flow. 2. Steady-state flow equation The steady-state condition is defined as the reservoir pressure at the drainage boundary being a constant. With this assumption, certain geometries of the reservoir drainage area are assumed to generate the analytical IPR equations. In 1994, applying the superposition principle, Butler presented a steady-state model to predict the productivity of a fully penetrating horizontal well in a box-shaped undersaturated oil reservoir. Fig. 1 shows the geometry used to develop this model. Butler’s model can be expressed as
M. Tabatabaei, D. Zhu / Journal of Natural Gas Science and Engineering 2 (2010) 132e142
133
Fig. 2. Flow pattern described by Furui et al. (2003).
By substituting the definition of the gas formation volume factor, Darcy’s equation for gas can be written in the following from.
qg psc TZ k dp ¼ C ; m dx pTsc Af Fig. 1. Geometry model used in Butler’s (1994) model and Furui et al.’s model.
7:08 103 kH L pe pwf ! qo ¼ : i hIani pyb mo Bo ½Iani ln þ h 1:14Iani þ s rw ðIani þ 1Þ
(1)
Later in 2003, Furui et al. presented another analytical model for fully penetrating horizontal well in a box-shaped reservoir with noflow boundaries at the top and bottom of the reservoir and constant pressure boundaries at the sides. In this model, it is assumed that the flow to a horizontal well can be divided into two regimes, a radial flow regime near the wellbore and a linear flow regime away from the wellbore. Similar to Butler’s model, skin factor caused by formation damage or well completion effects is incorporated in the model. Furui et al.’s model can be written as
qo ¼
7:08
103 kL
pe pwf
mo Bo ½lnðrw ðIhIanianiþ1Þ þ hIpyanib
i; 1:224 þ s
(2)
(3)
Eq. (2) can be rearranged by substituting the effective permeability by [kHkV]0.5.
7:08 103 kH L pe pwf i: qo ¼ mo Bo ½Iani lnðrw ðIhIanianiþ1Þ þ phyb Iani ð1:224 þ sÞ
(4)
qg Bg k dp ¼ C : m dx Af
qg 2 psc TZ
pTsc ðLhÞ
¼ C
kH dp ; m dy
(7)
and
qg psc TZ ¼ C pTsc ð2prLÞ
pffiffiffiffiffiffiffiffiffiffiffi kH kV dp : m dr
(8)
Integrating over the length for linear flow gives qg 2 psc T
Tsc ðLhÞkH
Z ðyb yt Þ ¼ C
pe
p
dp;
(9)
p dp: Z pwf m
(10)
pt mZ
and for radial flow we have, pt
Using the definition of real gas pseudo-pressure function, m(p), presented by Al-Hussainy et al. (1966),
Z mðpÞ ¼ 2
p
p
p0 Z mg
dp:
(11)
Eqs. (9) and (10) can be written as follows.
As we can observe, with completely different approaches, Butler’s model and Furui et al.’s model yield very similar expression for inflow performance of horizontal wells, except the constants in the denominators (1.14 for Butler and 1.224 for Furui et al.). Following describes how Furui et al.’s approach can be used to develop an inflow equation for a horizontal gas well in steady-state condition. Considering the flow pattern illustrated in Fig. 2, flow from the reservoir boundary to the wellbore can be described using Darcy’s law over each part of the flow pattern, linear and radial, separately. General form of Darcy’s equation is
v ¼
where qg, is gas flow rate, C is the unit conversion factor, Tsc and psc are the temperature and pressure at the standard condition and Af is the area open to flow, where for linear flow Af ¼ Lh and for radial flow Af ¼ 2prL, substituting the corresponding area, and considering the direction of flow and the defined coordinate, Darcy’s equation for linear and radial flow part of the flow pattern shown in Fig. 2 can be written as
Z qg psc T rt ¼ C ln Tsc ð2pLÞk rw
where, k is the effective permeability and can be defined as
pffiffiffiffiffiffiffiffiffiffiffi k ¼ kH kV :
(6)
(5)
qg psc T ðy yt Þ ¼ C½mðpe Þ mðpt Þ; Tsc ðLhÞkH b
(12)
and
i h 2qg psc T rt ¼ C mðpt Þ m pwf ; ln Tsc ð2pLÞk rw
(13)
In definition of the real gas pseudo-pressure function, m(p), p0 is the reference pressure and can be any convenience base pressure. Determination of the pseudo-pressure at a given pressure requires knowledge of gas viscosity and gas compressibility factor as functions of pressure and temperature. As these functions are complicated and not explicit, a numerical integration technique is frequently used.
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M. Tabatabaei, D. Zhu / Journal of Natural Gas Science and Engineering 2 (2010) 132e142
Combining Eqs. (12) and (13), and considering the values of yt and rt as described by Furui et al., the gas flow from the reservoir boundary to the wellbore in oil field unit can be described as
L mðpe Þ m pwf pffiffiffi3: qg ¼ 2 y p hb 0:5 ln h2rw2 5 1424T 4 þ kH k
(14)
0 rw ¼ rw
(22)
In this equation the first term on the right hand side is the Darcy or viscous component while the second is the non-Darcy component. Eq. (22) can be rewritten as
m dp ¼ ð1 þ Fo Þ v; dx k
To account for the effect of permeability anisotropy, Muskat’s (1937) general transformation and Brigham’s (1990) transformation, can be used as follows.
"
m dp ¼ v þ brv2 : dx k
#
Iani þ 1 pffiffiffiffiffiffiffi ; 2 Iani
(23)
where Fo is the Forchheimer number which is defined as
Fo ¼
brkv : m
(24)
pffiffiffiffiffiffiffi h0 ¼ h Iani ;
(16)
In the above equations, b is the coefficient of inertial resistance or known as turbulence factor and has the dimension [L1]. A common correlation for this coefficient is presented in the following form.
y y0 ¼ pffiffiffiffiffiffiffi: Iani
(17)
b¼
(15)
Including the effect of permeability anisotropy and also the effect of near wellbore formation damage as a skin factor, the IPR equation for a horizontal gas well in steady-state condition can be written as
kL mðpe Þ m pwf i: h qg ¼ pyb 1:224 þ s 1424T lnðrw ðIhIanianiþ1Þ þ hI
(18)
ani
As the real gas pseudo-pressure is difficult to evaluate without a computer program, approximations to Eq. (11) are usually used in the natural gas industry (Guo and Ghalambor, 2005). At pressures lower than 2000 psia, gas pseudo-pressure can be approximated as
Z mðpÞ ¼ 2
p
p
p0 Z mg
dp ¼
p2 p2b Z mg
:
(19)
b ; ka
(25)
a and b in Eq. (25) depend on properties of the formation. There are some other correlations (Geertsma, 1974) which also include the effect of porosity such as
b¼
b : ka fc
(26)
The non-Darcy component in Eq. (22) is negligible at low flow velocities; therefore, it is generally ignored in liquid flow equations. For a given pressure drawdown, however, the velocity of gas is at least an order of magnitude greater than the one for oil, due to the lower viscosity of the former, therefore, the non-Darcy component is often included in equations describing the flow of a real gas through a porous medium. The non-Darcy flow effect can be modeled as an additional skin term, which in literature (Economides et al., 1994; Golan and Whitson, 1991) it is mostly presented in terms of flow rate as,
Using this approximation, Eq. (18) can then be simplified using a pressure-squared approach and it becomes
st ¼ s þ Dq;
kL p2e p2wf h i: qg ¼ pyb 1:224 þ s 1424Z mg T lnðrw ðIhIanianiþ1Þ þ hI
where D is the non-Darcy coefficient. Including the effect of non-Darcy flow as an additional skin term, the general form of the productivity model for horizontal gas wells can be expressed as
(20)
ani
According to this approximation, the IPR equations for gas wells do not have a linear relationship between the flow rate and drawdown, as appeared in the equations for oil wells. At pressures higher than 3000 psia, highly compressed gases behave like liquids. Therefore, Eq. (18) can be approximated using pressure approach as
kL pe pwf h i: qg ¼ pyb 1:224 þ s 141:2 103 mg Bg lnðrw ðIhIanianiþ1Þ þ hI
(21)
ani
For gas wells, the flow velocity is usually much higher than the one for oil wells, especially near the wellbore. At higher flow velocities, in addition to the viscous force component represented by Darcy’s equation, there is also an inertial force acting to convective accelerations of the fluid particles in passing through the pore spaces which is known as non-Darcy flow effect. Deviations from Darcy’s law, observed at high velocities, can be added in mathematical terms in several ways. The most widely accepted model is Forchheimer’s equation (Forchheimer, 1901):
(27)
kL mðpe Þ m pwf h i: qg ¼ pyb 1424T lnðrw ðIhIaniþ1Þ þ hI 1:224 þ s þ Dqg ani
(28)
ani
There are many discussions about the non-Darcy coefficient, D. It can be obtained from lab experimental data, or from empirical correlations. Using Forchheimer’s equation, Furui (2004) developed an equation for the non-Darcy coefficient in horizontal wells considering the permeability anisotropy and formation damage near the wellbore. For horizontal gas well and in field unit the nonDarcy coefficient presented by Furui is expressed as follow.
2 D ¼ 1:4 1014
bgk 6bd b 4 þ 1 d b 2prw Lm b
13
0 B @ rdH rw
þ
Iani þ 1 C7 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A5; rdH rw
2
2 1 þIani
(29)
M. Tabatabaei, D. Zhu / Journal of Natural Gas Science and Engineering 2 (2010) 132e142
where the turbulence factor for undamaged and damaged zone b and bd, can be determined by the correlation presented by Firoozabadi and Katz (1979).
b¼
2:6 1010 : k1:2
(30)
Therefore, for horizontal well
2:6 1010
b ¼ pffiffiffiffiffiffiffiffiffiffiffi1:2 ;
(31)
kH kV
and
2:6 1010
bd ¼ pffiffiffiffiffiffiffiffiffiffiffi1:2 : kH kV
(32)
d
Note that the calculation of formation damage skin factor in horizontal wells is different from vertical wells due to permeability anisotropy. For a horizontal well, the formation damage skin factor can be calculated as (Furui, 2004)
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 rdH 2 1 dH þ þIani rw rw k A: s ¼ 1 ln@ Iani þ 1 kd 0r
(33)
3. Pseudo-steady-state flow equation Pseudo-steady-state models of inflow performance assume that reservoir is bounded by no-flow boundaries and pressure declines in a uniform fashion in the reservoir. In this case an average reservoir pressure is introduced in the IPR equations. Pressure decline curves, if available, can be used to calculate the average pressure as a function of time, and therefore to obtain a production history. Babu and Odeh (1989) presented a horizontal well IPR model under the pseudo-steady-state condition. The model treated the reservoir as a box-shaped drainage area with horizontal well-drilled parallel to one side of the reservoir. Fig. 3 illustrates the geometry used in Babu and Odeh’s model. Location of wellbore in drainage area in this model is arbitrary. Their model is generally based on the radial flow in the yez plane, with the deviation of the drainage area from a circular shape in this plane accounted for with a geometry factor, and flow from beyond the wellbore in the xdirection accounted for with a partial penetration skin factor. Note that the geometry factor used in Babu and Odeh’s model is related
Fig. 3. Geometry model used in Babu and Odeh’s (1989) model.
135
inversely to the commonly used Dietz (1965) shape factor. Thus, Babu and Odeh’s inflow equation is
pffiffiffiffiffiffiffiffiffi kx kz b p pwf h pffiffiffi i: qo ¼ 141:2mo Bo ln rwA þ lnCH 0:75 þ s þ sR
(34)
where A is the drainage area (ah), CH is the shape factor, sR is the partial penetration skin factor and s is any other kind of skin factors, such as completion or damage skin effects. The shape factor, CH, accounts for the deviation of the shape of the drainage area from cylindrical and the departure of the wellbore location from the middle of the system. The partial penetration skin, sR, accounts for the flow from the reservoir located beyond the end points of the well in the x-direction, and it is equal to zero for a fully penetrating horizontal well. Follow the approach used in previous section, the IPR equation for single-phase gas well for pseudo-steady-state condition can be derived from the oil well pseudo-steady-state IPR equation. Considering the non-Darcy flow effect, the resulting equation is
i pffiffiffiffiffiffiffiffiffi h kx kz b mðpÞ m pwf h pffiffiffi i qg ¼ 1424T ln rwA þ lnCH 0:75 þ s þ sR þ Dqg
(35)
And the appropriate equations for calculating the shape factor, CH, and the partial penetration skin, sR, are presented in the Appendix. As pointed out in the original paper, the model assumed that the thickness of the formation, h, is generally much smaller than the other two dimensions of the drainage box, a and b. If this condition does not apply, it should be examined first if a horizontal well is the right application for the field development. The equation is very helpful when used to examine the effects of the reservoir and well parameters on well performance, and therefore to optimize well design and operation. It also should be noted that, the results from previously discussed single-phase IPR models, Butler and Furui et al., shouldn’t be compared with Babu and Odeh’s model since the assumption of the boundary conditions used to develop the models are absolutely different. 4. Transient flow equation This type of flow, also known as infinite-acting flow, occurs when the boundary effect is not observed. This flow ends when all the outer boundaries of the reservoir are reached by the propagating pressure disturbance. During this period, the well behaves as if it was placed in a reservoir with infinite size.
Fig. 4. Geometry model used in Kuchuk et al.’s (1991) model.
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The transient flow period provides engineers with an invaluable tool to determine reservoir parameters and well productivity. Therefore, it was the subject of many studies for many years (Clonts and Ramey, 1986; Ozkan et al., 1989; Odeh and Babu, 1990; Kuchuk et al., 1991). Kuchuk et al. (1991) presented an analytical solution for transient pressure behavior of a horizontal well completed in an
4.2. Second radial flow period This is a hemicylindrical flow period that follows the first radial flow. This flow period may occur when the well is not centered with respect to the top and bottom boundaries. The equation for the second flow period is
pffiffiffiffiffiffiffiffiffiffiffi kH kV L1=2 pi pwf qo ¼ qffiffiffiffi ii: h pffiffiffiffiffiffiffiffi h k k t 162:6Bo mo log fm Hc rV2 3:2275 þ 0:4343s log 1 þ kkH zrww
(39)
V
o t w
infinite anisotropic medium bounded above and below by horizontal planes (Fig. 4), using Laplace transform. Their solution is based on the uniform-flux, line-source solution and also averaging the pressure along the length of the well. They identified four different flow periods during the transient flow of a horizontal well and presented specific equations for each of them. Applying the same approach used earlier, the transient flow equation of the horizontal gas wells for each flow period can be derived from the corresponding oil well transient flow equation. 4.1. First radial flow period The very first flow pattern for horizontal wells is elliptic-cylindrical. After some time, the elliptic-cylindrical flow period becomes
This flow period starts to appear at
fmo ct
t ¼
0:0002637pkV
o n min z2w ; ðh zw Þ2 ;
(40)
n o max z2w ; ðh zw Þ2 :
(41)
and ends at
fmo ct
t ¼
0:0002637pkV
And for a horizontal gas well, the equation for this flow period can be expressed as
pffiffiffiffiffiffiffiffiffiffiffi kH kV L1=2 mðpi Þ m pwf qg ¼ qffiffiffiffi ii: h pffiffiffiffiffiffiffiffi h k k t 1639:8T log fm Hc rV2 3:2275 þ 0:4343 s þ Dqg log 1 þ kkH zrww V
g t w
approximately radial. This radial flow around the wellbore may continue until the effect of the nearest boundary is felt at the wellbore. The behavior of this period is equivalent to the behavior of fully penetrating vertical well in an infinite reservoir. The equation for the first radial flow period in oilfield units may be written as
4.3. Intermediate-time linear flow period If the horizontal well is much longer than the reservoir thickness, for the no-flow boundary case, this flow period may develop after the effects of the upper and lower boundaries are felt at the wellbore. The equation for the linear flow is
pffiffiffiffiffiffiffiffiffiffiffi 2 kH kV L1=2 pi pwf qo ¼ qffiffiffiffi: h pffiffiffiffiffiffiffiffi q k k t 162:6Bo mo log fm Hc rV2 3:2275 þ 0:8686s 2log12 4kkH þ 4 kkV o t w
V
The start of the effect of the nearest boundary (no-flow or constant pressure) or the end of this flow period can be determined as
t ¼
fmo ct
0:0002637pkV
o n min z2w ; ðh zw Þ2 :
The corresponding equation for a horizontal gas well is
(37)
(36)
H
pi pwf sffiffiffiffiffiffiffiffiffiffiffiffiffi qo ¼ " #; mo t 8:128 141:2mo 70:6mo pffiffiffiffiffiffiffiffiffiffiffis Bo þ sz þ kH h 2hL1=2 fct kH L1=2 kH kV
(43)
where
pffiffiffiffiffiffiffiffiffiffiffi 2 kH kV L1=2 mðpi Þ m pwf qg ¼ qffiffiffiffi: h pffiffiffiffiffiffiffiffi q k k t 1639:8T log fm Hc rV2 3:2275 þ 0:8686 s þ Dqg 2log12 4kkH þ 4 kkV g t w
(42)
V
H
(38)
M. Tabatabaei, D. Zhu / Journal of Natural Gas Science and Engineering 2 (2010) 132e142
sz ¼ 1:1513
sffiffiffiffiffiffi sffiffiffiffiffiffi! # " pz prw kH h kV w 1þ : log sin kV L1=2 h kH h
5. Discussion
(44)
This flow period starts to develop at
t ¼
fmo ct h2 ; 0:0002637kV
(45)
and ends at
t ¼
fmo ct L21=2 0:0002637kH
:
(46)
This flow period will not appear for wells with a gas cap or a bottom aquifer. The corresponding equation for a horizontal gas well can be expressed as
One of the main advantages of the analytical productivity models is that they provide us with an invaluable tool to investigate the effect of different parameters on productivity of a well and then to optimize well design and operation. In this section we first show a brief comparison between the analytical models and numerical solution, and then discuss the effect of some critical parameters on performance of horizontal gas wells. 5.1. Comparison of analytical models and numerical solution Kamkom and Zhu (2006) showed that when the flowing bottomhole pressure, pwf, is not too low, the analytical models for productivity of horizontal gas wells give the similar results compared to the simulation results (numerical solution). Fig. 5 shows such a comparison for a steady-state case. The comparison
mðpi Þ m pwf " #: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qg ¼ 4:064 t 1 1 pffiffiffiffiffiffiffiffiffiffiffi s þ Dqg 1424T þ sz þ 141:2hL1=2 fmg ct kH kH h 2L1=2 kH kV 4.4. Late-time radial flow period After a sufficiently long time, the pressure front will become approximately radial in the xey plane and a third radial flow pattern will develop. Similar to intermediate-time linear flow period, this period does not exist for wells with a gas cap or bottom aquifer. The equation for late-time radial flow is
"
qo ¼ Bo mo
pi pwf
137
(47)
for the studied condition showed that at low to moderate flow rate (less than 50 MMscf/day), the analytical solution matches the simulation results very well. When flow rate increases (higher drawdown) the results of analytical solution starts deviating from the simulation results, and the deviation is more pronounced at higher flow rate.
" ! # #; 162:6 kH t 141:2 70:6 p ffiffiffiffiffiffiffiffiffiffi ffi log 2:5267 þ þ s s z kH h kH h fmo ct L21=2 L1=2 kH kV
(48)
5.2. Effect of permeability anisotropy
where
sffiffiffiffiffiffi sffiffiffiffiffiffi! # " pz prw kH h kV w 1þ log sin kV L1=2 h kH h 2 2 k h 1 zw zw 0:5 H 2 þ 2 : kV L1=2 3 h h
sz ¼ 1:1513
(49)
The onset of this flow period is
20fmo ct L21=2 tz : 0:0002637kH
(50)
And for a horizontal gas well we have
Consider a fully penetrating horizontal well in a gas reservoir with no-flow boundary and reservoir properties described in Table 2. The effect of permeability anisotropy on well performance can be studied applying Babu and Odeh’s model since there is no external pressure support in this case. Fig. 6 illustrates the variation of the productivity index with permeability anisotropy, Iani, for two different scenarios. In the first scenario, horizontal permeability was kept constant (kH ¼ 1 md) and vertical permeability was decreased to increase Iani, and in other one, vertical permeability was kept constant (kV ¼ 0.1 md) and horizontal permeability was increased to increase Iani. As we can see, the variation in J with Iani is different for these two
mðpi Þ m pwf " #: ! # " qg ¼ 1:151 kH t 1 1 pffiffiffiffiffiffiffiffiffiffiffi s þ Dqg log 2:5267 þ 1424T sz þ kH h kH h fmg ct L21=2 2L1=2 kH kV
The equations for inflow performance relationship of horizontal wells under different boundary conditions are summarized in Table 1.
(51)
scenarios. In first one, productivity index decreases with increase in Iani because the effective permeability ([kHkV]0.5) is decreasing as a result of decrease in the vertical permeability. In second one,
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Table 1 Summary of IPR equations for horizontal wells.
Oil
Steady-state flow Augmented Joshi qo ¼
141:2mo Bo
kH hðpe pwf Þ ffii n h pffiffiffiffiffiffiffiffiffiffiffiffi h 2 2 aþ
ln
a ðL=2Þ
I
h
þ aniL
L=2
ln
Iani h rw ðIani þ1Þ
Pseudo-steady-state flow pffiffiffiffiffiffiffiffiffi k k bðp pwf Þ h pffiffixffi z i qo ¼ 141:2mo Bo ln rwA þ ln CH 0:75 þ s þ sR
i o þs
Butler "
qo ¼
mo Bo
ln CH ¼ 6:28I
7:08 103 kH Lðpe pwf Þ # Iani ln rw ðIhIanianiþ1Þ þ phyb 1:14Iani þ s
h
a
ani h
1 3
xa0 þ
2 i x0 a
ln sinphz0 12ln I a h 1:088 ani
Furui et al. "
qo ¼
mo Bo Gas
qg ¼
7:08 103 kLðpe pwf Þ # py b ln rw ðIhIanianiþ1Þ þ hI 1:224 þ s ani
kLðmðpe Þ mðpwf ÞÞ # py b 1424T ln rw ðIhIanianiþ1Þ þ hI 1:224 þ s þ Dqg "
qg ¼
pffiffiffiffiffiffiffiffiffi kx kz b½mðpÞ mðpwf Þ
pffiffiffi 1424T½lnð rwAÞ
þ lnCH 0:75 þ s þ sR þ Dqg
ani
Transient flow of slightly compressible fluid (oil well) First radial
Second radial
pffiffiffiffiffiffiffiffiffiffiffi 2 kH kV L1=2 ðpi pwf Þ # qffiffiffiffi qffiffiffiffi pffiffiffiffiffiffiffiffi k k t 162:6Bo mo log fm Hc rV2 3:2275 þ 0:8686s 2log 12 4 kkH þ 4 kkV V H o t w pffiffiffiffiffiffiffiffiffiffiffi kH kV L1=2 ðpi pwf Þ " " # qo ¼ qffiffiffiffi pffiffiffiffiffiffiffiffi k k t 162:6Bo mo log fm Hc rV2 3:2275 þ 0:4343s log 1 þ kkH zrww "
qo ¼
V
o t w
"
Intermediate-time linear qo ¼ Bo
ðpi pwf Þ qffiffiffiffiffiffiffiffiffi mo t 141:2mo 8:128 fc k þ k h s z þ 2hL 1=2
sz ¼ 1:1513 Late-time radial
t H
"
qffiffiffiffi
"
Bo mo 162:6 kH h sz ¼ 1:1513
log
kH t fmo ct L21=2
fmo ct minfz2w ; ðh zw Þ2 g t 0:0002637pkV o n fmo ct max z2w ; ðh zw Þ2 0:0002637pkV fmo ct L21=2 fmo ct h2 t 0:0002637kV 0:0002637kH
# mo 70:6 pffiffiffiffiffiffiffiffi s L1=2 kH kV
þ
qffiffiffiffi kV Þsinðphzw Þ k H
ðpi pwf Þ # s þ 2:5267 þ141:2 k h z H
#
qffiffiffiffi kH h log phrw ð1 þ kkV Þsinðphzw Þ 0:5kkH k L1=2 V
H
t
70:6 pffiffiffiffiffiffiffiffis L1=2 kH kV
"
qffiffiffiffi
fmo ct minfz2w ; ðh zw Þ2 g 0:0002637pkV
#
kH h log phrw ð1 kV L1=2
"
qo ¼
H
t
V
h2 1 L21=2 3
z2
zhw þ hw2
20fmo ct L21=2 0:0002637kH
Transient flow of compressible fluid (gas well) First radial
Second radial
pffiffiffiffiffiffiffiffiffiffiffi fmg ct 2 kH kV L1=2 ðmðpi Þ mðpwf ÞÞ # t minfz2w ; ðh zw Þ2 g qffiffiffiffi qffiffiffiffi pffiffiffiffiffiffiffiffi 0:0002637pkV kH kV t 4 4 k k H V 1639:8T log fm c r2 3:2275 þ 0:8686ðs þ Dqg Þ 2log12 þ kV kH g t w pffiffiffiffiffiffiffiffiffiffiffi fmg ct kH kV L1=2 ðmðpi Þ mðpwf ÞÞ " ## " qg ¼ minfz2w ; ðh zw Þ2 g t qffiffiffiffi pffiffiffiffiffiffiffiffi 0:0002637pkV kH kV t kH zw 1639:8T log fm c r2 3:2275 þ 0:4343ðs þ Dqg Þ log ð1 þ k Þrw o n V fmg ct g t w max z2w ; ðh zw Þ2 0:0002637pkV "
qg ¼
ðmðpi Þmðpwf ÞÞ
"
Intermediate-Time Linear qg ¼ 1424T
4:064 141:2hL1=2
sz ¼ 1:1513 Late-time radial
1424T
t
fmg ct kH
1:151 kH h
log
1 ffiffiffiffiffiffiffi p ðsþDqg Þ kH kV
#
þ
qffiffiffiffi kV Þsinðphzw Þ k H
ðmðpi Þ mðpwf ÞÞ # 2:5267 þk1hsz þ
fmg ct L21=2 fmg ct h2 t 0:0002637kV 0:0002637kH
#
2L1=2
H
kH h log phrw ð1 kV L1=2
"
sz ¼ 1:1513
þk 1 hsz þ
"
qffiffiffiffi
"
qg ¼
qffiffiffiffiffiffiffiffiffiffiffi
kH t fmg ct L21=2
H
# 2L1=2
# qffiffiffiffi prw pz w kH h kV log Þ 0:5kkH 1 þ sinð k L1=2 h k h
qffiffiffiffi V
"
H
V
productivity index increases because effective permeability is increasing as a result of increase in the horizontal permeability. This example reveals that, in absence of the non-Darcy flow effect, kH has a more pronounced and constant effect on productivity (linear relationship) compared with the kV effect where it declines very
1 ffiffiffiffiffiffiffiffiðs þ Dq Þ p g kH kV
h2 1 L21=2 3
z2
zhw þ hw2
t
20fmg ct L21=2 0:0002637kH
fast and the effect diminishes as kV approaches a small value. This is because the denominator in Eq. (35) is related to permeability anisotropy through the shape factor and as we can observe from Fig. 7, ln(CH) declines rapidly with the increase in Iani and it approaches almost a constant value for higher permeability
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139
Fig. 6. Effect of permeability anisotropy on productivity of a horizontal gas well. Fig. 5. Comparison of the analytical model for horizontal gas wells and the numerical simulation results.
anisotropy values. Consequently, for higher permeability anisotropy values (greater than 4) we can show the relationship between productivity index and Iani is
k Jf H or JfkV Iani : Iani
therefore, productivity of well increases. It should be mention that, to truly investigate the effect of wellbore length, it is necessary to consider the effect of pressure drop along the wellbore on productivity of well. The benefit of longer horizontal well length will be jeopardized for long horizontal wells with high inflow rate. When frictional pressure drop in the wellbore is significant, production becomes tubing-limited (Hill and Zhu, 2007)
(52)
The above relationships clearly explain the behavior we observed in Fig. 6. However, if we include the effect of non-Darcy flow (see Fig. 6) we would see that these relationships would no longer be valid since the denominator in Eq. (35) cannot be considered constant in higher permeability anisotropies and is changing with flow rate. The deviation from the relationship described in Eq. (52) would be even more dramatic if productivity index was higher (due to higher drawdown or higher permeability values). 5.3. Effect of wellbore length Similar to previous case, consider a horizontal well penetrated a gas reservoir with no-flow boundary. Using the same reservoir properties described in Table 2 and assuming horizontal permeability and vertical permeability of the reservoir is 1 md and 0.1 md, respectively. The effect of wellbore length can be studied again by using Babu and Odeh’s model. Fig. 8 shows the effect of wellbore length, L, and penetration degree, L/b on productivity of this horizontal gas well. In this case b is constant and L/b is changing by change of L. As we expected, productivity of the well increases by increasing the wellbore length or penetration degree. This is because by increasing the penetration degree, partial penetration skin factor, sR, decreases (see Fig. 8),
5.4. Effect of near wellbore formation damage and non-darcy flow Consider a fully penetrating horizontal well in a gas reservoir described in Table 3. The pressure at the drainage boundary of this reservoir is 5600 psi and it is assumed to be constant due to external pressure support. Fig. 9 illustrates the IPR curve of this well for the following cases. (a) Assuming there is no near wellbore formation damage and ignoring the non-Darcy flow effect. (b) Assuming there is no near wellbore formation damage, but this time considering the non-Darcy flow effect. (c) Assuming a uniform damage along the entire wellbore length, with the damage extending 6 inches beyond the well in the horizontal direction and damage permeability is 10% of undamaged permeability (kd ¼ 0.1k), and ignoring the nonDarcy flow effect.
Table 2 Reservoir and well parameters used in the study. Reservoir length Reservoir width Reservoir thickness Wellbore diameter Average reservoir pressure Bottomhole pressure Reservoir temperature Gas gravity Skin factor
2000 ft 1200 ft 50 ft 4 in 4500 psi 500 psi 210 F 0.7 (air ¼ 1) 0
Fig. 7. Effect of permeability anisotropy on shape factor.
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Fig. 8. Effect of wellbore length and penetration ratio (L/b) on productivity.
Table 3 Reservoir and well parameters in study the effect formation damage and non-Darcy flow. Reservoir length Reservoir width Reservoir thickness Wellbore diameter Reservoir pressure Horizontal permeability Vertical permeability Reservoir temperature Gas gravity
2000 ft 1500 ft 50 ft 4 in 5600 psi 1 md 0.1 md 210 F 0.7 (air ¼ 1)
(d) Assuming a uniform damage along the entire wellbore length as described in case “c”, and also considering the non-Darcy flow effect. As shown in Fig. 9, near wellbore formation damage has a significant effect on the well performance. For this case it reduces the productivity of well by about 25%. This figure also reveals that non-Darcy flow effect is very insignificant in absence of near wellbore formation damage but presence of formation damage can intensify its effect. For this case, in high rates, we would lose about 2% of the production rate when there is no formation damage, but about 10% of the production rate in presence of formation damage. However, for both cases the effect of non-Darcy flow is insignificant in low rates. But this is not the case for vertical wells. Fig. 10 shows the inflow performance of a vertical well penetrates a reservoir with exactly
Fig. 10. Effect of near wellbore formation damage and non-Darcy flow on productivity of a vertical gas well (pseudo-steady-state case).
the same geometry, properties, pressure and the same fluid properties. As we can see, again there is a significant reduction is production rate due to the near wellbore formation damage, even more than the case of horizontal well (about 60% for this case). Although production rate is much lower than the case of horizontal well, we can see a significant reduction in production rate due to non-Darcy flow either in absence or presence of formation damage (20%). This is because the area open to flow in case of vertical well is so much smaller than horizontal wells. Therefore, non-Darcy flow plays a less significant rule in productivity of horizontal wells compared to vertical wells. To investigate more about the effect of non-Darcy flow on productivity of horizontal gas wells, we considered two other examples. One pseudo-steady state flow case and one transient flow case. Tables 4 and 5 present the reservoir and wellbore parameters used for these two examples. Fig. 11 shows the results for pseudo-steady state flow condition. As we can see, similar to the steady state example discussed Table 4 Reservoir and well parameters used to study the effect formation damage and nonDarcy flow (pseudo-steady-state case). Reservoir length Reservoir width Reservoir thickness Wellbore length Wellbore diameter Reservoir pressure Horizontal permeability Vertical permeability Reservoir temperature Gas gravity
2000 ft 1200 ft 50 ft 1500 ft 4 in 4500 psi 1 md 0.1 md 210 F 0.7 (air ¼ 1)
Table 5 Reservoir and well parameters used to study the effect formation damage and nonDarcy flow (transient flow case).
Fig. 9. Effect of near wellbore formation damage and non-Darcy flow on productivity of a horizontal gas well (steady-state case).
Reservoir length Reservoir width Reservoir thickness Wellbore length Wellbore diameter Initial reservoir pressure Horizontal permeability Vertical permeability Reservoir temperature Gas gravity Reservoir porosity Total compressibility of reservoir
5000 ft 2500 ft 100 ft 2500 ft 4 in 5200 psi 1 md 0.1 md 210 F 0.7 (air ¼ 1) 20% 5 105 psi1
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This paper discussed the effect of some critical parameters on productivity of horizontal gas wells such as, permeability anisotropy, wellbore length, near wellbore formation damage and nonDarcy flow. It is concluded that to study the effect of the permeability anisotropy on productivity of a horizontal well it is important to consider the effect of horizontal and vertical permeability separately. It was also observed that the effect of non-Darcy flow in horizontal wells is not as significant as vertical wells. However, in transient flow case with high flow rate, non-Darcy flow can significantly affect the productivity of well and reduce the production rate. Moreover, we found out that the near wellbore formation damage would intensify the non-Darcy flow effect. In other words, reduction in production rate due to the non-Darcy flow effect is much more in presence of formation damage. This would suggest that stimulation of a horizontal gas well will help to minimize the effect of non-Darcy flow. Fig. 11. Effect of near wellbore formation damage and non-Darcy flow on productivity of a horizontal gas well (pseudo-steady-state case).
Fig. 12. Effect of near wellbore formation damage and non-Darcy flow on productivity of a horizontal gas well (transient flow case).
previously, in absence of the near wellbore formation damage, nonDarcy flow has an insignificant effect on productivity of a horizontal gas well in pseudo-steady state flow condition. Also we can see, the same as previous example, non-Darcy flow plays a bigger rule when there is formation damage around the wellbore. Fig. 12 illustrates the effect of non-Darcy flow in transient flow. This figure reveals that, even in absence of the near wellbore formation damage, non-Darcy flow effect significantly reduces the production rate in this case. This is because non-Darcy flow is a function of the flow rate. The higher rate, the more significant the effect of non-Darcy flow would be. Therefore, it encounters more reduction in production rate in this case since flow rate is much higher.
6. Summary Evaluating horizontal well performance can be done by either numerical models or analytical models. When used correctly, the analytical models give reasonable predictions of horizontal well performances and they can be very helpful in designing, operating and optimizing horizontal wells.
Nomenclature A, drainage area of horizontal well, ft2 Af, area open to flow, ft2 a, extension of drainage volume of horizontal well in x-direction, ft aH,max, horizontal axis of damage ellipse, ft Bo, oil formation volume factor, rb/STB Bg, gas formation volume factor, res ft3/scf b, extension of drainage volume of horizontal well in y-direction, ft C, unit conversion factor CH, geometric factor in Babu and Odeh’s model ct, total compressibility, psi1 D, non-Darcy flow coefficient, day/Mscf Fo, Forchheimer number, dimensionless h, reservoir thickness, ft Iani, permeability anisotropy, dimensionless J, productivity index, STB/day/psi k, effective permeability, md kd, effective permeability in damaged zone, md kx, permeability in x-direction, md ky, permeability in y-direction, md kz, permeability in z-direction, md kH, horizontal permeability, md kV, vertical permeability, md L, horizontal wellbore length, ft L1/2, half length of horizontal wellbore, ft m(p), pseudo-pressure, psi2/cp p, reservoir pressure, psi p0, reference pressure, psi pavg, average reservoir pressure, psi psc, pressure at standard condition, psi pe, reservoir pressure at boundary, psi pi, initial pressure, psi pwf, bottomhole flowing pressure, psi qo, oil production rate, STB/day qg, gas production rate, Mscf/day rdH, radius of damaged zone in horizontal direction, ft re, radius of outer boundary of the reservoir, ft rw, wellbore radius, ft s, skin factor, dimensionless sR, partial penetration skin factor, dimensionless T, reservoir temperature, R Tsc, temperature at standard condition, R t, ime, hr v, velocity, ft/s
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x0, x coordinate of center of well, ft ymid, midpoint location along well length, ft yb, half length of the reservoir width perpendicular to the wellbore, ft y1, y coordinate of beginning of well, ft y2, y coordinate of end of well, ft Z, gas compressibility factor Zavg, average gas compressibility factor z0 z coordinate of center of well, ft zw, distance of wellbore from the lower boundary, ft b, turbulence factor, ft1 bd, turbulence factor for damaged zone, ft1 r, density, lbm/ft3 m, viscosity, cp mo, oil viscosity, cp mg, gas viscosity, cp mg,avg, average gas viscosity, cp gg, gas gravity, dimensionless, (air ¼ 1) 4, porosity, dimensionless
FðxÞ ¼ ðxÞ 0:145 þ lnðxÞ 0:137ðxÞ2 :
(A-6)
If arguments are >1, then:
FðxÞ ¼ ð2 xÞ 0:145 þ lnð2 xÞ 0:137ð2 xÞ2 :
(A-7)
where x can be p 4yffiffiffiffiffi orffiffiffiffiffi 4ymid pL=2b. mid þ L=2b p ffiffiffiffiffi Case 2: If b= ky > 1:33a= kx [h= kz , then
sR ¼ Pxyz þ Py þ Pxy ;
(A-8)
where
Pxyz ¼
Py ¼
pz b h kx 1 ln þ0:25ln ln sin 0 1:84 ; kz h L rw
6:28b2 ah
! pffiffiffiffiffiffiffiffiffi" # L L kx kz 1 ymid y2mid 3 ; þ 2 þ 3 24b b b ky b
(A-9)
(A-10)
and Appendix
The heart of Babu and Odeh’s model are procedures for calculating the shape factor and the partial penetration skin factor. These parameters were obtained by simplifying the solution of the diffusivity equation for parallelepiped reservoir geometry and comparing it with the assumed inflow equation (Eq. (34)). Babu and Odeh solved the 3D diffusivity equation with a wellbore boundary condition of constant flow rate (uniform flux) at the well and no flow across the reservoir boundaries using the Green’s function approach. In this manner, the following correlations for the shape factor and the partial penetration skin factor were obtained.
pz a 1 x 0 x 0 2 ln sin 0 ln CH ¼ 6:28 þ Iani h 3 a a h 1 a 1:088: ln 2 Iani h
(A-1)
Calculation of the partial penetration skin depends on the geometry and the permeability anisotropy of reservoir. Babu and Odeh defined two thisp parameter. pffiffiffiffifficases for calculating ffiffiffiffiffi pffiffiffiffiffi Case 1: If a= kx 0:75b= ky [0:75h= kz , then 0 sR ¼ Pxyz þ Pxy ;
(A-2)
where
Pxyz ¼
b 1 L
pz h kx ln þ0:25ln ln sin 0 1:84 ; kz h rw
(A-3)
and 0 Pxy
2b2 ¼ LhIani
! L 4ymid þL 4ymid L þ0:5 F F ; F 2b 2b 2b (A-4)
where, ymid ¼ 0:5ðy1 þy2 Þ is the midpoint along the well length and
F
" 2 # L L L L ¼ 0:145 þ ln 0:137 : 2b 2b 2b 2b
(A-5)
The evaluation of Fð4ymid þ L=2bÞ and Fð4ymid L=2bÞ depends on their arguments. If arguments are 1, then:
Pxy ¼
0 sffiffiffiffiffi1 ! b 6:28a kz A 1 x0 x20 @ : 1 þ kx L h 3 a a2
(A-11)
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