- Email: [email protected]
A semi-analytical model for predicting inflow profile of horizontal wells in bottom-water gas reservoir
Accepted Manuscript A semi-analytical model for predicting inflow profile of horizontal wells in bottom-water gas reservoir Haitao Li, Yongsheng Tan, Beibei Jiang, Yongqing Wang, Nan Zhang PII:
S0920-4105(17)30845-8
DOI:
10.1016/j.petrol.2017.10.067
Reference:
PETROL 4391
To appear in:
Journal of Petroleum Science and Engineering
Received Date: 23 May 2017 Revised Date:
12 September 2017
Accepted Date: 23 October 2017
Please cite this article as: Li, H., Tan, Y., Jiang, B., Wang, Y., Zhang, N., A semi-analytical model for predicting inflow profile of horizontal wells in bottom-water gas reservoir, Journal of Petroleum Science and Engineering (2017), doi: 10.1016/j.petrol.2017.10.067. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT 1
A semi-analytical model for predicting inflow profile of
2
horizontal wells in bottom-water gas reservoir Haitao Li1
Yongsheng Tan1 Beibei Jiang1 Yongqing Wang1 Nan Zhang1
RI PT
3
1 State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation,
5
Southwest Petroleum University, Chengdu, China
6
Abstract
7
In this study, a semi-analytical model which couples the reservoir and wellbore, is
8
developed to predict the inflow profile of horizontal wells in a box-shaped gas
9
reservoir with a bottom water zone. The developed gas reservoir flow model is based
10
on a 3D volumetric source model which considers the reservoir properties in the
11
region near the wellbore, in terms of anisotropy, heterogeneity and formation damage.
12
Meanwhile, pressure drops in the wellbore flow model was also considered, due to
13
fluid flow in the horizontal wellbore. The model was calculated by a field case, and
14
the results of the calculation were compared with a commercial simulator: ECLIPSE
15
(Schlumberger 2011). The comparison between the developed model and ECLIPSE
16
indicates that, a good match was obtained, so that, the model derived in this study can
17
be applied to predict the inflow profile of horizontal well in a gas reservoir with
18
bottom water zone. Moreover, sensitivity analysis is conducted by using various
19
wellbore pressure drops, wall roughness, skin factors, wellbore diameters, anisotropy
20
and producing pressure drops.
21
Keywords：Semi-analytical model, inflow profile, horizontal well, gas reservoir,
AC C
EP
TE D
M AN U
SC
4
1
ACCEPTED MANUSCRIPT 1
bottom-water zone
1. Introduction
3
Gas recovery factor from a gas reservoir with a bottom water zone is often lower than
4
that of depleted gas reservoirs (Geffen et al., 1952; Keelan and Pugh, 1975). The
5
water influx occurs when the pressure drop at the gas-water interface in the process of
6
gas reservoir production (Ahmed, 2000). Nowadays, horizontal well is essential to
7
explore the oil/gas resources, because the horizontal well can generate a larger contact
8
area between the horizontal well and the reservoir, so that the oil production can be
9
enhanced. However, disadvantage of the horizontal well which applied in the
10
heterogeneous reservoir was observed in the oil/gas field, that is, the productivities of
11
the wells would decrease when an uneven distribution of inflow profile along the
12
wellbore occurred, due to heterogeneity appears near the wellbore and pressure drop
13
along the wellbore (Tatar et al. 2014, Song et al. 2015, Naderi et al. 2015). In order to
14
improve the recovery of bottom-water gas reservoirs, inflow profile of horizontal
15
wells in bottom-water gas reservoir has become an important task.
SC
M AN U
TE D
EP
The conventional production logging is widely used to predict the inflow profile
AC C
16
RI PT
2
17
wells，but there are some shortages in this approach. For instance, the inflow profile
18
along the wellbore cannot be tested, if the horizontal well is too deep; and the test
19
results are not accurate because of the irregular horizontal wellbore, etc. Moreover,
20
the service fee of dynamic gas producing profile is too high，so the theoretical model
21
for predicting the inflow profile along the wellbore is required.
22
It has been widely recognized that, the pressure drop along the horizontal well 2
ACCEPTED MANUSCRIPT significantly affect the production performance of the horizontal wells (Dikken, 1990).
2
A one dimensional reservoir/wellbore coupling model was developed to predict the
3
performance of horizontal wells using a productivity index (Dikken, 1990; Landman,
4
1994; Novy, 1995; Penmatcha and Aziz, 1998), but this one dimension model could
5
not reflect the state of the three-dimensional flow in the reservoir. To study the state of
6
the
7
reservoir/wellbore coupling model using a three-dimensional reservoir model and a
8
wellbore model to calculate productivity in an isotropic or anisotropic reservoir, but in
9
this model, the reservoir and near well wellbore rock property variations were not take
10
into account. Ozkan et al. (1999) applied a semi-analytical approach to study the
11
reservoir/wellbore coupling, an important assumption was made, that is, compared to
12
the reservoir pressure loss, the pressure drops along the wellbore is negligible. Vicente
13
et al.(2000) presented a reservoir/wellbore coupling model, considering an
14
oversimplified wellbore pressure drop model. Based on the experimental results,
15
Ouyang et al. (2005) proposed a coupled reservoir/wellbore formulation, however,
16
there is a certain limitation in the practical application. Souza et al.(2014) developed a
17
new numberical method to solve the well-reservoir model, taking into account
18
isotropic and anisotropic, wellbore length, completion scheme and a damaged region
19
around the wellbore. But it spends more computational time than semi-analytical
20
models(Luo et al., 2015). In recent years, some scholars have also studied on
21
reservoir/wellbore coupling models. Johansen and Khoriakov (2007) presented a new
22
method in simulation of multi-phase flow and in the near well region in the reservoir.
flow,
Penmatcha
and
Aziz
(1998)
developed
a
AC C
EP
TE D
M AN U
SC
three-dimensional
RI PT
1
3
ACCEPTED MANUSCRIPT Hasan and Kabir (2009) proposed an effective model for a two-phase flow in
2
geothermal wells by means of the drift-flux method. Adesina et al.(2016) presented an
3
improved reservoir/well model which considers pressure drop due to acceleration. But
4
the effect of the formation damage near wellbore zone has not been effectively
5
addressed in these papers.
RI PT
1
Point source is the most used techniques to build a reservoir/well model
7
(Khoriakov et al., 2012), however, the shortcoming of the point-source is the inherent
8
singularity of the solution, because no volume assumption was made in this method
9
(Valkó and Amini, 2007). The Distributed Volumetric Source (DVS) method was
10
proposed to predict production performance of the horizontal wells located in a
11
fractured reservoir, and high accuracy and computational efficiency were obtained
12
(Valkó and Amini, 2007; Amini and Valkó, 2010; Jiang et al., 2016). Meanwhile, the
13
volumetric-source model had been treated successfully for the inherent singularity of
14
the solution in the point-source. More importantly, rare studies on reservoir/well
15
model in a gas reservoir with bottom water zone, were proposed.
EP
TE D
M AN U
SC
6
This study of inflow profile prediction of horizontal wells is focused on the
17
reservoir/wellbore coupling model based on volumetric source solutions in a
18
box-shaped gas reservoir with a bottom water zone. And sensitivity study has
19
presented using various wellbore pressure drop, wall roughness, skin factor, wellbore
20
diameter, anisotropy and producing pressure drop. This model can be used to predict
21
inflow profile of horizontal wells in gas reservoir with a bottom water zone, and to
22
horizontal well completion design and optimization or production prediction.
AC C
16
4
ACCEPTED MANUSCRIPT 1
2. Semi-analytical model and solution
2
2.1 Assumptions The semi-analytical model, which focuses on the gas production performance in
4
the horizontal well located in the reservoir with a bottom water zone, is based on the
5
following assumptions:
6
2.2 Reservoir Flow Model
9
EP
8
Fig. 1. Schematic of a horizontal well in a box-shaped gas reservoir
(1) It is assumed that the gas reservoir is homogeneous.
AC C
7
TE D
M AN U
SC
RI PT
3
10
(2) The horizontal well is located in the middle of a box-shaped reservoir, as showed
11
in Fig. 1. A constant pressure boundary is applied at the bottom of the model,
12 13 14 15
closed boundary conditions are conducted in other five sides. (3) Single-phase and steady-state flow is considered, the production process is isothermal, and governed by Darcy's law. As we can see a gas reservoir（big box）in Fig. 1, 5
the sizes of the box-shaped
ACCEPTED MANUSCRIPT gas reservoir in three directions are defined as xe, ye, ze, respectively. A volumetric
2
source (small box in big box) with a strength flow rate q is applied in the gas reservoir.
3
The center coordinate of the volumetric source is (cx, cy, cz), so that the sizes of the
4
volumetric source in three directions are 2wx, 2wy, 2wz, respectively. The volumetric source model can be expressed as:
7
The boundary conditions: ∂ ( ∆ψ )
8
∂x ∂ ( ∆ψ )
9
∂y ∂ ( ∆ψ )
10
15 16 17 18 19
Where
AC C
14
(2)
x = 0, x = xe
=0
(3)
y = 0, y = ye
=0
(4)
z = ze
TE D ∆ψ
12 13
(1)
The initial condition:
EP
11
∂z
=0
SC
qg ∂ 2 ( ∆ψ ) ∂ 2 ( ∆ψ ) ∂ 2 ( ∆ψ ) + + + h ( x, y , z ) = 0 2 2 2 ∂x ∂y ∂z 86.4kVsource
6
M AN U
5
RI PT
1
z=0
ψ = 2∫
pini
p
=0
(5)
p dp µZ
(6)
∆ψ = ψ i −ψ g
(7)
pscZT qsc pgTsc
(8)
qg =
Vsource = 8wx wy wz h( x, y, z) = H( x−cx −wx ) −H( x−cx +wx ) ×H( y −cy −wy ) −H( y −cy +wy )× H( z −cz −wz ) −H( z −cz +wz ) H ( x − x0 ) is Heaviside function: 6
(9) (10)
ACCEPTED MANUSCRIPT 1 H ( x − x0 ) = 0
1 2
x > x0
(11)
x ≤ x0
With:
ψ i = Initial gas reservoir pseudo-pressure, MPa
4
ψ g = Pseudo-pressure of a point in the gas reservoir, MPa
5
q g = strength of the volumetric source in gas reservoir, m3/d
6
Vsource = Geometric volume of volumetric source in gas reservoir, m3
7
k= Reservoir permeability, µm2
8
p= Formation pressure in the gas reservoir, MPa
9
pini = Initial gas reservoir pressure, MPa
M AN U
SC
RI PT
3
µ = Gas fluid viscosity, mPa s
11
Z = Deviation factor of gas reservoir
12
psc = Initial gas reservoir pressure, MPa
13
Tsc = Temperature of ground standard condition, K
14
T = Temperature of gas reservoir, K
17 18 19
EP
16
The detailed derivation of the volumetric source model solution is provided in the Appendix. The solution of the volumetric source model can be presented as:
AC C
15
TE D
10
∆ψ =
qsc pT 86.4kVsourceTsc
σl ( x) σ m ( y ) σ n ( z ) γ l =0 m=0 n=0 ∞
∞
∞
∑∑∑
(12)
where
2wx x σl ( x) = e lπ cx lπ wx lπ x 4cos cos sin xe xe xe lπ
7
l =0
(13) l ≠0
ACCEPTED MANUSCRIPT m=0
(14) m≠ 0
π + nπ ) z π + nπ ) c π + nπ ) w ( ( ( 2 2 4sin cos sin 2 z
σn ( z) =
ze
ze
z
ze
π + nπ 2
n = 0,1, 2 ⋅⋅⋅⋅
(15)
SC
2
RI PT
1
2 wy ye σm ( y) = mπ c y mπ wy 4cos mπ y cos sin ye ye ye mπ
M AN U
3
Fig. 2. Schematic of the horizontal well and gas reservoir division
5
The cylindrical wellbore is approximate by rectangular shape (Fig.1), and this
6
approximation does not have big impact for steady state flow. And the horizontal well
7
is divided into N segments (Fig.2), each segment is taken as a volumetric sink, the
8
length of i segment is Li, the radius of wellbore is rw, the center coordinate of i
9
segment is (xi, yi, zi). Hence, the dimensions in three directions and coordinate of i
11 12 13 14 15
EP
AC C
10
TE D
4
segment can be expressed as:
2wxi = Lhi
(16)
2wyi = 2wzi = π rw2
(17)
(c ,c
(18)
xi
yi
, czi ) = ( xi , yi , zi )
As we can see in Fig. 2, the gas reservoir is divided into N blocks in response to segmentation of the horizontal well. The pressure drawdown at arbitrary point M(x, y, z) 8
ACCEPTED MANUSCRIPT 1
in the gas reservoir can be regarded as a linear superposition of the pressure in all
2
blocks (N) of the horizontal well, by using the superposition method:
4
σli ( x) σ mi ( y ) σ ni ( z ) γ l =0 m=0 n=0 ∞
∞
∞
∑∑∑
As shown in Fig. 2, Lr1 = Lh1 +
(19)
xe − L x −L and LrN = LhN + e , the other blocks are 2 2
RI PT
N qgsci pT ∆ψ Mi = ∑ i =1 86.4kVsourceTsc
3
the same as the horizontal well section. And there is no cross flow between reservoir
6
segments. According to Equation (19), the point M(x, y, z) is located in block i of the
7
gas reservoir, the pressure drop of point M can be achieved:
8
10 11 12
∞
∞
obtained: M
j =1
86.4kVsourceTsc
ψ g =ψ i − ∑
qgsci pT
∞
∞
∞
∑∑∑
σlj ( cxi ) σ mi ( cyi ) σ ni ( czi + wzi )
l =0 m=0 n=0
γ
（21）
The parameters of anisotropic in the gas reservoir can be obtained based on the
xe' = xe
EP
14
17
(20)
Substituting equation (20) into equation (7), the pressure in block i can be
Besson correction method (Besson, 1990):
16
γ
l =0 m=0 n=0
13
k k k , ye' = ye , ze' = ze kx ky kz
(22)
c xi' = c xi
k k k , c yi' = c yi , c zi' = c zi kx ky kz
(23)
wxi' = wxi
k k k , w'yi = wyi , wzi' = wzi kx ky kz
(24)
AC C
15
σlj ( cxi ) σmi ( cyi ) σ ni ( czi + wzi )
∑∑∑
TE D
9
∞
M AN U
qgsci pgT ∆ψi = ∑ j =1 86.4kVsourceTsc N
SC
5
k = kx k y kz
(25)
18
Reservoir heterogeneity is the main reason for the imbalance of inflow profile in
19
horizontal wells (Wang et al. 2000), so we can not ignore this factor. The s-k* 9
ACCEPTED MANUSCRIPT approach was proposed by Wolfsteiner et al.(2000), to model the heterogeneity in the
2
near well region. This method indicates the relationship of the average permeability
3
k* of the whole gas reservoir and an effective skin caused by varies along the
4
wellbore trajectory. Based on Hawkins' method (Hawkins, 1956), the effective
5
near-well skin of block i can be defined as:
RI PT
1
k r Shi = −1 ln a ki rw
(26)
SC
6
During drilling and completion process, all reservoirs are susceptible to
8
formation damage (Kersey, 1986), the degree of damage can be expressed by the loss
9
of permeability. Frick and Economides proposed the elliptical-cone-shape model to
10
describe damage distribution along the horizontal wellbore (Frick and Economides,
11
1993; Furui et al., 2002). In this study, the formation damage skin in block i is
12
expressed as:
13 Where
17 18
I ani =
AC C
15 16
k 1 rdh i + = − 1 ln k di I an i + 1 rw
EP
14
sdi
TE D
M AN U
7
2 rdh i 2 + I a ni − 1 rw
kx k y
(27)
(28)
kz
Therefore, the total skin of block i can be expressed as:
S ti = S h i + S d i
(29)
The total skin of block i is substituted in equation (29) in the form of additional
19
pressure drop. Therefore, the pressure of block i in the gas reservoir can be proposed
20
as:
21
M
j =1
86.4kVsourceTsc
ψg =ψi − ∑
qgsci pT
∞
∞
∞
σlj ( cxi ) σmi ( cyi ) σni ( czi + wzi )
∑∑∑
γ
l =0 m=0 n=0
10
qgsci pT Sti − 86.4kLhiTsc
(30)
ACCEPTED MANUSCRIPT
2 3
4
Substituting equation (6) into the equation (30), the pressure of block i of the gas reservoir can also be written as: M Z q pT pri2 = pini2 − 2µ∑ i gsci j =1 86.4kVsourceTsc
σlj ( cxi ) σ mi ( cyi ) σ ni ( czi + wzi ) µZi qgsci pT Sti (31) − γ l =0 m=0 n=0 43.2kLhi Tsc ∞
∞
∞
∑∑∑
RI PT
1
2.3 Wellbore Flow Model
As shown in Fig. 3, the horizontal well is divided into n segments. And segment
6
1 closes to the heel of the horizontal well, while segment n closes to the toe of the
7
horizontal well. In the wellbore flow, the friction and acceleration pressure drop
8
should be considered, because of the well trajectory and the gas reservoir inflow.
TE D
M AN U
SC
5
9 10
EP
12
As shown in Fig. 4, it is assumed that the single fluid flows between the two nodes.
AC C
11
Fig. 3 Schematic of wellbore flow in horizontal well
13 14
Fig. 4 Flow model in horizontal wellbore
11
ACCEPTED MANUSCRIPT The conservation of mass can be expressed as:
ρv
2 3
π D2 4
∂v π D 2 + ρ v Rπ D d x − ρ v + dx =0 ∂ x 4
According to Equation (32), we can get the following equation:
∂v 4vR = ∂x D
4
9
10 11
12
13 14 15
16
SC
= 1.0 × 10
4τ ∂ v ∂ v 2 w + ρ 2v + dx ∂ ∂ D x x
(34)
f ρv2 , and substituting τ w and equation (33) into equation (34), the 8
M AN U
8
Set τ w =
dx
−6
pressure drops can be expressed the following equation (Li et al., 2009):
−
d pw d xi
f ρ v 2 8 ρ vv R 16 ρ v R2 = 1.0 × 10 − 6 + + dx D D2 2D
(35)
In which, the friction factor f can be expressed by using the Reynolds number
TE D
7
−
d pw
under laminar flow and turbulent flow:
64 Re f = −2 ε − 0.9 1.14−2lg +21.25 Re d
EP
6
(33)
Then the momentum principles can be expressed as:
laminar flow ( Re ≤ 2300)
AC C
5
(32)
RI PT
1
（36）
turbulent flow ( Re ≥ 4000)
The ideal gas law is applied to calculate the properties of the gas, in terms of,
pressure, density and gas flow rate:
ρ= v=
M air γ g p R TZ
ρ sc q sc T Z p π rw2 Tsc 12
(37)
(38)
ACCEPTED MANUSCRIPT 1
velocity of segment i from gas reservoir to wellbore can be written as:
v Ri =
3 4
6
(39)
Therefore, the final pressure drops can be obtained from equation 31-39:
RI PT
5
p sc q sci TZ 2π rw p Tsc Li
2 2 16Mairγ g psc2 qscwi TZi 64Mairγ g psc2 qscwiqsciTZi 32Mairγ g psc2 qsci TZi p − p =1.0×10 fi ∆lj + + (40) 2 4 2 4 2 4 π π π RD RD RD T T T sc sc sc 2 w1
2 w2
−6
With:
SC
2
Around the wellbore, the uniform fluid flow was assumed, then the gas flow
Z i = Deviation factor of segment i.
8
Where the first item is the frictional pressure drop, and the second and three
M AN U
7
9
items are acceleration pressure drops.
10
2.4 Coupled model and solution
In this work, gas reservoir flow model and wellbore flow model are coupled to
12
study the restrains and effects of the fluid flow in both the formation and the wellbore.
13
For flow rate control:
15 16 17 18
EP
n
Qmax = ∑ qi
(41)
p1 = pwf ,min
(42)
AC C
14
TE D
11
i =1
For bottom-hole pressure control:
Where Qmax is the desired maximum flow rate, m3/d; pwf ,min is the specified
minimum bottom-hole pressure, Mpa.
19
Equations 31-42 constitute the coupled model, and the Newton Raphson
20
method is used to solve this model, due to the equations in the model are nonlinear. 13
ACCEPTED MANUSCRIPT 1
The procedure for solving the model is summarized as follows: (1) Assume the sandface pressure distribution is pri. So that, the sandface inflow
3
rate along the horizontal well can be calculated by reservoir flow model
4
(Equation 31).
5
(2) Substitute qri into equations 36-42 to calculate the horizontal well pressure
6
distribution (pwi), and the new horizontal well pressure distribution is equivalent
7
to the sandface pressure distribution (pri).
8
(3) Compare the new sandface pressure distribution with the calculated results of
9
the last iteration. If the error is less than the allowable range, stop the process and
10
output the results; otherwise, assume the sandface pressure distribution again and
11
repeat the iteration, until achieve the requirements in accuracy.
SC
M AN U
3. Applications
13
3.1 Model validation
TE D
12
EP
Table 1 Reservoir Properties for Homogeneous Gas Reservoir Reservoir length (m) Reservoir width (m) Reservoir thickness (m) Horizontal well length (m) Distance of wellbore to WGC (m) Wellbore radius(m) Initial formation pressure (Mpa) Porosity (%) Formation volume factor(m3/m3)
1000 600 50 680 35 0.12 63 14 2.72
Gas density (g/cm3)
0.68
AC C
14
RI PT
2
3
Water density (g/cm ) Gas viscosity (mPa s) Permeability kx (mD) Permeability ky (mD) Permeability kz (mD)
1.05 0.029 50 50 5 14
ACCEPTED MANUSCRIPT 560
Reservoir Simulation ECL Semi-analytical model
540
500
480
RI PT
3
Inflow rate (m /d)
520
460
440
420 0
50
100 150 200 250 300 350 400 450 500 550 600 650 700
1
Fig. 5 Comparison of inflow rate between semi-analytical model and ECL Semi-analytical model Reservoir Simulation ECL
M AN U
62.8
62.6
62.4
62.2
TE D
Pressure along the well (Mpa)
63.0
0
4
100
200
300
400
500
600
700
Distance from heel (m)
Fig. 6 Comparison of pressure along the well between our model and ECL
EP
3
SC
2
Distance from heel (m)
A gas field case is introduced in this study to validate the calculation results of
6
the developed model. A strong bottom-water drive gas reservoir, with an initial
7
pressure of 63 MPa was selected, and a horizontal well with horizontal length of
8
680m, is located in the middle of the gas reservoir. The properties of the real gas
9
reservoir data are listed in Table 1.
AC C
5
10
The calculation results of the developed model are compared with a commercial
11
software, ECLIPSE (Schlumberger 2011) to check the accuracy. The grid dimension
12
is 100×60×30, the grid sizes for x, y, and z directions are 10 m, 10 m and 2 m, 15
ACCEPTED MANUSCRIPT respectively，and the time step is 20 days. The inflow and pressure distributions along
2
the horizontal well are computed by using both of the developed model and the
3
commercial simulator, as shown in Fig. 5 and Fig. 6, good match results are obtained.
4
These results suggest that, the developed model has potential practical application in
5
predicting inflow profile of horizontal wells in bottom-water gas reservoir.
6
3.2 Influential factors analysis
SC
RI PT
1
Steady state flow is the predominant flow in many gas reservoirs. In this study,
8
some parameters were analyzed in terms of wellbore pressure drop, wall roughness,
9
skin factor, wellbore diameter, anisotropy and producing pressure drop at 500 days,
10
when the flow in the reservoir being considered has reached steady state, as indicated
11
in Table 1.
12
3.2.1 Effect of wellbore pressure drop
TE D
M AN U
7
Fig. 7 indicates the inflow profile of the horizontal well with and without
14
pressure drop. From Fig. 7, it is clear that: (1) for the case without pressure drop along
15
the wellbore, the inflow profile is symmetric with inflow distribution of around
16
512.06m3/d; (2) when the pressure drop was considered along the wellbore, the inflow
17
profile is asymmetric, with a flow rate under 541.3m3/d in the heel and 514.6m3/d in
18
the toe. Therefore, the pressure drop of the wellbore affects the inflow profile of the
19
horizontal well.
AC C
EP
13
16
ACCEPTED MANUSCRIPT 550 540
No wellbore pressure drop t=500d (Semi-analytical model) Wellbore pressure drop t=500d (Semi-analytical model) No wellbore pressure drop t=500d (Reservoir Simulation ECL) Wellbore pressure drop t=500d (Reservoir Simulation ECL)
530
510 500 490 480 470 460 450 440 0
100
200
1
500
600
700
Fig. 7 Inflow profile of the horizontal well for with and without pressure drop
SC
3
400
3.2.2 Effect of wall roughness
M AN U
2
300
Distance from heel (m)
RI PT
3
Inflow rate (m /d)
520
Fig. 8 shows the inflow profiles of the horizontal well for various wall roughness.
5
For a horizontal well with a flow rate of 20000m3/d, grater wall roughness causes a
6
higher non-uniform heel biased inflow profile. This is because a much greater
7
frictional resistance prevents the fluid flow along the wellbore. As expected, effects of
8
wall roughness can be reduced by using smaller wall roughness in horizontal well.
TE D
4
700
e=0.02mm (Semi-analytical model) e=0.002mm (Semi-analytical model) e=0.0002mm(Semi-analytical model) e=0.02mm (Reservoir Simulation ECL) e=0.002mm (Reservoir Simulation ECL) e=0.0002mm(Reservoir Simulation ECL)
EP
650
AC C
3
Inflow rate (m /d)
600
550
500
450
400 0
9 10
100
200
300
400
500
600
700
Distance from heel (m)
Fig. 8 Inflow profile of the horizontal well for various wall roughness
17
ACCEPTED MANUSCRIPT 1
3.2.3 Effect of skin factor
2
The formation skin factor ( S ) because of formation damage is considered in the
3
homogeneous gas reservoir, inflow profile of the horizontal well for various skin
4
factors are shown in Fig. 9. With the same producing pressure drop, the lower skin
5
factor gains higher inflow rate along the horizontal well, hence, the skin factor should
6
be taken into account in the case, due to the effect is remarkable.
RI PT
di
800
SC
Without considering skin factor s=0 (Semi-analytical model) Considering drilling pollution skin factor s=2 (Semi-analytical model) Considering drilling pollution skin factor s=5 (Semi-analytical model) Without considering skin factor s=0 (Reservoir Simulation ECL) Considering drilling pollution skin factor s=2 (Reservoir Simulation ECL) Considering drilling pollution skin factor s=5 (Reservoir Simulation ECL)
750 700
M AN U
3
Inflow rate (m /d)
650 600 550 500 450 400
0
7 8
100
200
300
400
500
600
700
Distance from heel (m)
Fig. 9 Inflow profile of the horizontal well for various skin factor
3.2.4 Effect of wellbore diameter
EP
9
TE D
350
Fig. 10 shows the effect of well diameter on inflow distribution along the well
11
under steady state (at a production time of 500 days). The flow profile becomes more
12
skewed as the well diameter is getting smaller, the wellbore drop will reduce and
13
results in inflow distribution nearly uniform with the increasing of the wellbore
14
diameter.
AC C
10
18
600 590 580 570 560 550 540 530 520 510 500 490 480 470 460 450 440 430
d=300mm (Semi-analytical model) d=250mm (Semi-analytical model) d=200mm (Semi-analytical model) d=300mm (Reservoir Simulation ECL) d=250mm (Reservoir Simulation ECL) d=200mm (Reservoir Simulation ECL)
0
300
400
500
Distance from heel (m)
600
700
SC
3
200
Fig. 10 Inflow profile of the horizontal well for various wellbore diameter
3.2.5 Effect of anisotropy
M AN U
1 2
100
RI PT
3
Inflow rate (m /d)
ACCEPTED MANUSCRIPT
As shown Fig. 11, the effect of anisotropy also can be represented by a skin term.
5
Fig. 11 indicates the inflow profile of the horizontal well with different anisotropy.
6
For the well with the same producing pressure drop of 5MPa, it can be observed that
7
higher anisotropy causes higher inflow rate along the horizontal well. As shown in Fig.
8
11, the inflow distribution is more inclined to the heel section with the increase of
9
anisotropy.
EP
TE D
4
Iani=0.2 (Semi-analytical model)
700
Iani=0.4 (Semi-analytical model) Iani=0.6 (Semi-analytical model)
650
Iani=0.8 (Semi-analytical model) Iani=0.2 (Reservoir Simulation ECL) Iani=0.4 (Reservoir Simulation ECL) Iani=0.6 (Reservoir Simulation ECL)
550
Iani=0.8 (Reservoir Simulation ECL)
3
Inflow rate (m /d)
AC C
600
500 450 400 350 300 250
0
10 11
100
200
300
400
500
600
700
Distance from heel (m)
Fig. 11 Inflow profile of the horizontal well for various anisotropy
19
ACCEPTED MANUSCRIPT 1
3.2.6 Effect of producing pressure drop Fig. 12 shows the inflow profile of the horizontal well for various producing
3
pressure drops. As the producing pressure drop increases, the inflow rate along the
4
horizontal well becomes higher. Similar to the effect of anisotropy, the inflow
5
distribution is more inclined to the heel section with the increase of producing
6
pressure drop.
RI PT
2
SC
800
∆p=1Mpa (Semi-analytical model) ∆p=3Mpa (Semi-analytical model) ∆p=5Mpa (Semi-analytical model) ∆p=1Mpa (Reservoir Simulation ECL) ∆p=3Mpa (Reservoir Simulation ECL) ∆p=5Mpa (Reservoir Simulation ECL)
750
M AN U
650
3
Inflow rate (m /d)
700
600 550 500
0
10 11 12 13
200
300
400
500
600
700
Distance from heel (m)
Fig. 12 Inflow profile of the horizontal well for various producing pressure drop
4. Conclusions
EP
9
100
(1) A semi-analytical model which coupled reservoir/wellbore, is developed for
AC C
7 8
TE D
450
predicting the inflow profile of the horizontal wells in a box-shaped gas reservoir with a bottom-water zone. The gas reservoir flow model is based on a 3D volumetric source model that considers the reservoir anisotropy, heterogeneity
14
and formation damage in the near well region. And the pressure drops along the
15
wellbore is considered.
16
(2) Comparison between the results of the developed model and a commercial
20
ACCEPTED MANUSCRIPT simulator (ECLIPSE), a good match result is obtained, so the new model has
2
potential practical application in predicting inflow profile of horizontal wells in
3
gas reservoir with a bottom-water zone. And sensitivity analysis is conducted by
4
using various wellbore pressure drops, wall roughness, skin factors, wellbore
5
diameters, anisotropy and producing pressure drops.
RI PT
1
(3) The semi-analytical model can be used as a basis, not only for predicting inflow
7
profile of horizontal wells in bottom-water gas reservoir, but also for horizontal
8
well completion design and optimization, to and production prediction.
Acknowledgements
M AN U
9
SC
6
The authors would like to acknowledge the National Science and Technology
11
Major Project (Grant Nos. 2016ZX05017005-006), for its technical and financial
12
support.
13
Appendix. Solution of volumetric source model
16 17
∂2 E ∂2 E ∂2 E ∂x 2 + ∂y 2 + ∂z 2 = − λ E ∂E ∂E x=0 = x = xe = 0 ∂x ∂x ∂E ∂E = =0 ∂y y = 0 ∂y y = y e ∂E E z =0 = 0 z = ze = 0 ∂z
EP
15
From Equation (1)-(5), the characteristic equation can be proposed as:
AC C
14
TE D
10
(A.1)
Set E = X ( x ) Y ( y ) Z ( z ) ， three one-dimensional eigenvalue problems are obtained by variables separation:
21
ACCEPTED MANUSCRIPT
1
X '' + µ X = 0 ' ' X x = 0 = X x = xe = 0
2
where λ =µ + ν + θ
6
mπ νl = , ye
7
π + nπ θn = 2 ze
X l = cos
lπ , xe
X l = cos
mπ , ye
13
14 15
16
n = 0,1, 2 ⋅⋅⋅⋅
ze
TE D
2 2 lπ mπ π 2 + nπ λ= + + ze x e y e
(A.5)
(A.6)
2
(A.7)
EP
The corresponding characteristic function system is: Elmn ( x, y , z ) =cos
π + nπ lπ mπ cos sin 2 xe ye ze
AC C
12
Zn
π + nπ ) ( = sin 2 ,
(A.4)
The corresponding characteristic value is:
9
11
2
,
m = 0,1, 2 ⋅⋅⋅⋅
M AN U
2
l = 0,1, 2 ⋅⋅⋅⋅
SC
2
lπ µl = , xe
10
(A.3)
three one-dimensional eigenvalue are obtained through Equation (A.2):
5
8
(A.2)
RI PT
3 4
Z '' + θ Z = 0 ' Z z = ze = 0 Z z = 0 = 0
Y '' + ν Y = 0 ' ' Y x = 0 = Y x = xe = 0
l , m, n = 0,1, 2, ⋅⋅⋅
(A.8)
The corresponding 2-norm of eigen function is:
El , m,n ( x, y, z ) = 2
wx wy wz 8
Al Bm
l , m = 0,1, 2 ⋅⋅⋅⋅
(A.9)
m=0 m≠0
(A.10)
Where Al =
{
2 1
l=0 l≠0
Bm =
{
2 1
Using the characteristic function system Equation (A.8) as the transform kernel, 22
ACCEPTED MANUSCRIPT 1
the orthogonal transformation is introduced to solve the volumetric source model.
2
Here, the orthogonal transformation is described as:
ψ = G ( ∆ψ ) = ∫
xe
0
0
El ,m ,n ( x, y , z ) ∆pd x d y d z
ze
0
∞
∞
∆ψ = ∑∑∑
5
l = 0 m =0 n =0
8
ψ =-
(A.12)
(A.13)
Equation (A.13) into the Equation (A.12):
-El,m,n ( x, y, z) G[（ h x, y, z） ] = qsc pT 2 l =0 m=0 n=0 γ 86.4kVsourceTsc El,m,n ( x, y, z) ∞
∞
∞
∞
2wx x σl ( x) = e lπ cx lπ wx lπ x 4cos cos sin xe xe xe lπ
AC C
Where
σl ( x) σm ( y) σn ( z) ∑∑∑ γ l =0 m=0 n=0 ∞
EP
qsc pT 86.4kVsourceTsc
1
TE D
11
14
ψ
The solution of the volumetric source model can be obtained by Substituting
∞
13
2
qsc pT G [（ h x, y , z）] γ 86.4kVsource Tsc 1
∆ψ = G( ∆ψ ) = ∑∑∑
12
El ,m,n ( x, y, z )
SC
equation can be obtained:
10
El ,m,n ( x, y, z )
Taking the orthogonal transformation of equations (1) – (5), the following
7
9
(A.11)
The inverse transformation of equation (A.11) can be written as: ∞
6
l , m, n = 0,1, 2 ⋅⋅⋅⋅
RI PT
4
ye
∫ ∫
M AN U
3
2 wy y σm ( y) = e lπ c y lπ wy 4 cos lπ y cos sin ye ye ye mπ
23
(A.14)
l =0
(A.15) l ≠0
m=0
(A.16) m≠ 0
ACCEPTED MANUSCRIPT 1
σn ( z) =
4sin
(π 2 + nπ ) z cos (π 2 + nπ ) c sin (π 2 + nπ ) w z
ze
ze
z
ze
π + nπ 2
n = 0,1, 2 ⋅⋅⋅⋅
Nomenclature
3
xe, ye, ze gas reservoir dimensions, m
4
cx, cy, cz central coordinates of the volumetric source, m
5
wx, wy, wz dimensions of the volumetric source, m
6
qg strength of the volumetric source in gas reservoir, m3/d
7
Vsource geometric volume of volumetric source in gas reservoir, m3
8
T temperature of gas reservoir, K
9
Tsc temperature of ground standard condition, K
M AN U
SC
RI PT
2
µg gas fluid viscosity, mPa s
11
k reservoir permeability, mD
12
kx, ky, kz reservoir permeability at three directions of x, y, z, mD
13
p pressure of a point in the gas reservoir, MPa
14
ψg pseudo-pressure of a point in the gas reservoir, MPa
15
pini initial gas reservoir pressure, MPa
16
ψi initial gas reservoir pseudo-pressure, MPa
17
γg gas relative density, dimensionless
18
Li length of segment i, m
19
rw wellbore radius, m
20
xi, yi, zi central coordinates of segment i, m
21
qri fluid inflow rate of segment i , m3/d
AC C
EP
TE D
10
24
(A17)
ACCEPTED MANUSCRIPT k* effective reservoir permeability, µm2
2
kai reservoir permeability of segment i in the altered zone, µm2
3
ra radius of the altered permeability zone, m
4
shi skin of segment i caused by permeability heterogeneity, dimensionless
5
kdi reservoir permeability of segment i in the damaged zone, µm2
6
amax half-length of the horizontal axes of damaged zone at the heel, m
7
Iani permeability anisotropic coefficient, dimensionless
8
Si distance between segment i and the toe of horizontal well, m
9
L horizontal well length, m
M AN U
SC
RI PT
1
Sdi skin of segment i caused by formation damage, dimensionless
11
Sti total skin of segment i, dimensionless
12
pri sandface pressure at the center of segment i, MPa
13
ρ gas density in wellbore, g/cm3
14
g acceleration due to gravity, m/s2
15
f friction factor in the tubing, dimensionless
16
v gas flow velocity, m/s
17
vR gas flow velocity from reservoir to wellbore, m/s
18
pwi pressure of wellbore segment i, Mpa
19
M air relative molecular mass of air
20
D diameter of horizontal wellbore, mm
21
R universal gas constant, J·mol-1·K-1
22
NRe Reynolds number
AC C
EP
TE D
10
25
ACCEPTED MANUSCRIPT pwi pressure of segment i in the horizontal wellbore, Mpa
2
qgsci fluid inflow rate of segment i in standard condition, m3/d
3
qscwi wellbore flow rate of segment i in standard condition, m3/d
4
ε roughness factor
5
Reference
EP
TE D
M AN U
SC
Adesina, F., Paul, A., Oyinkepreye, O., & Adebowale, O., 2016. An Improved Model for Estimating Productivity of Horizontal Drain Hole[C]//SPE-169386-MS. SPE Latin America and Caribbean Petroleum Engineering Conference, Maracaibo, Venezuela Ahmed, T., 2000. Reservoir Engineering Hand Book. Gulf Pub. Co, Houston, TX. Amini, S. and P.P. Valkó, 2010. Using distributed volumetric sources to predict production from multiple-fractured horizontal wells under non-darcy-flow conditions. J. SPEJ 15(1): 105-115. Besson J., 1990. Performance of slanted and horizontal wells on an anisotropic medium[C]//SPE-20965-MS. European Petroleum Conference, The Hague, Netherlands. Chen, Y., 2004. A Decomposition Method of Skin Factor for Gas Well. Xinjiang Pet. Geol. 25(2), 013. (in Chinese) Dikken, B. J., 1990. Pressure drop in horizontal wells and its effect on production performance. J. Pet. Technol. 42(11): 1426-1433. Dumkwu, F.A., A.W. Islam and E.S. Carlson, 2012. Review of well models and assessment of their impacts on numerical reservoir simulation performance. J. Pet. Sci. Eng. s 82– 83(1): 174–186. Frick, T.P., Economides, M.J., 1993. Horizontal well damage characterization and removal. SPE Prod. Fac. 8 (1), 15-22. Furui, K., D. Zhu and A.D. Hill, 2002. A rigorous formation damage skin factor and reservoir inflow model for a horizontal well. SPE Prod. Fac. 18(3): 151-157. Geffen, T.M., Parrish, D.R., Haynes, G.W., Morse, R.A., 1952. Efficiency of gas displacement from porous media by liquid flooding. J. Pet. Technol. 4 (2): 29-38. Keelan, D.K., Pugh, V.J., 1975. Trapped-gas saturations in carbonate formations. Soc. Pet. Eng. J. 15: 149-160. Hasan A R, Kabir C S.,2009. Modeling Two-Phase Fluid and Heat Flows in Geothermal Wells. J. Pet. Technol. 71(1-2):77-86. Hawkins, M.F., 1956. A note on the skin effect. J. Trans. AIME 207 (12), 356-357. Jiang, B., H. Li, Y. Zhang, Y. Wang, J. Wang and S. Patil, 2016. Multiple fracturing parameters optimization for horizontal gas well using a novel hybrid method. J. Nat. Gas Sci. Eng. 34: 604-615. Johansen, T.E. and V. Khoriakov, 2007. Iterative techniques in modeling of multi-phase flow in advanced wells and the near well region. J. Pet. Sci. Eng. 58(1–2): 49-67. Jr, H. J. Remay., 1965. Non-Darcy Flow and Wellbore Storage Effects in Pressure Build-Up
AC C
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39
RI PT
1
26
ACCEPTED MANUSCRIPT
EP
TE D
M AN U
SC
RI PT
and Drawdown of Gas Wells. J. Pet. Technol. 17(2):223-233. Kersey, D. G., 1986. The role of petrographic analyses in the design of nondamaging drilling, completion, and stimulation programs[C]//International Meeting on Petroleum Engineering, Beijing, China Khoriakov, V., A.C. Johansen and T.E. Johansen, 2012. Transient flow modeling of advanced wells. J. Pet. Sci. Eng. s 86–87: 99–110. Landman, M.J., 1994. Analytic modelling of selectively perforated horizontal wells. J. Pet. Sci. Eng. 10(3): 179-188. Li, S., Lian, P. Q., Li, X., 2009. Nonsteady state model of gas reservior and horizontal well wellbore coupling. J. Southwest. Pet. Univ. 31(1):53-57. (in Chinese) Luo, W., Li, H. T., Wang, Y. Q., Wang, J. C., 2015. A new semi-analytical model for predicting the performance of horizontal wells completed by inflow control devices in bottom-water reservoirs. J. Nat. Gas Sci. Eng. 27: 1328-1339. Naderi, M., B. Rostami, M. Khosravi., 2015. Effect of heterogeneity on the productivity of vertical, deviated and horizontal wells in water drive gas reservoirs. J. Nat. Gas Sci. Eng. 23: 481-491. Novy, R.A., 1995. Pressure drops in horizontal wells: When can they be ignored?. SPE Reservoir Eng. 10(1): 29-35. Ouyang L B, Huang W S B., 2005. A Comprehensive Evaluation of Well-Completion Impacts on the Performance of Horizontal and Multilateral Wells[C]//SPE Annual Technical Conference and Exhibition, Dallas, Texas. Ozkan, E., C. Sarica and M. Haci, 1999. Influence of pressure drop along the wellbore on horizontal-well productivity. J. SPEJ 4(3): 288-301. Penmatcha, V.R. and K. Aziz, 1998. A comprehensive reservoir/wellbore model for horizontal wells. J. SPEJ 4(3): 224-234. Wang H. J., Xue S. F., Gao C. F., Tong X. H., 2012. Profile control method for perforated horizonta wells in heterogeneous reservoirs. J. China Univ. Pet. 36(3): 135-139. (in Chinese) Song, H., Y. Cao, M. Yu, Y. Wang, J. E. Killough, J. Leung., 2015. Impact of permeability heterogeneity on production characteristics in water-bearing tight gas reservoirs with threshold pressure gradient. J. Nat. Gas Sci. Eng. 22: 172-181. Souza, G.D., A.P. Pires and E. Abreu, 2014. Well-reservoir coupling on the numerical simulation of horizontal wells in gas reservoirs[C]//SPE Latin America and Caribbean Petroleum Engineering Conference, Maracaibo, Venezuela. Tatar, A., M. R. Yassin, M. Rezaee, A. H. Aghajafari, A. Shokrollahi, 2014. Applying a robust solution based on expert systems and GA evolutionary algorithm for prognosticating residual gas saturation in water drive gas reservoirs. J. Nat. Gas Sci. Eng. 21: 79-94. Valkó, P.P. and S. Amini, 2007. The method of distributed volumetric sources for calculating the transient and pseudosteady-state productivity of complex well-fracture configurations[C]//SPE Hydraulic Fracturing Technology Conference, College Station, Texas, U.S.A. Vicente, R., C. Sarica and T. Ertekin, 2000. A numerical model coupling reservoir and horizontal well flow dynamics: Transient behavior of single-phase liquid and gas flow. J. SPEJ 7(1): 70-77.
AC C
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
27
ACCEPTED MANUSCRIPT
EP
TE D
M AN U
SC
RI PT
Wolfsteiner, C., L.J. Durlofsky and A. Khalid, 2000. Approximate model for productivity of nonconventional wells in heterogeneous reservoirs. J. SPEJ 5(2): 218-226.
AC C
1 2
28
ACCEPTED MANUSCRIPT
Highlights Based on volumetric source method, this article presents reservoir/wellbore
model
for
predicting
inflow
RI PT
a
performance of horizontal wells in bottom-water gas reservoir.
SC
This proposed model considers anisotropy, heterogeneity and formation damage in near well region.
M AN U
This proposed model has been treated successfully for the
AC C
EP
TE D
inherent singularity of the solution in the point source.
S0920-4105(17)30845-8
DOI:
10.1016/j.petrol.2017.10.067
Reference:
PETROL 4391
To appear in:
Journal of Petroleum Science and Engineering
Received Date: 23 May 2017 Revised Date:
12 September 2017
Accepted Date: 23 October 2017
Please cite this article as: Li, H., Tan, Y., Jiang, B., Wang, Y., Zhang, N., A semi-analytical model for predicting inflow profile of horizontal wells in bottom-water gas reservoir, Journal of Petroleum Science and Engineering (2017), doi: 10.1016/j.petrol.2017.10.067. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT 1
A semi-analytical model for predicting inflow profile of
2
horizontal wells in bottom-water gas reservoir Haitao Li1
Yongsheng Tan1 Beibei Jiang1 Yongqing Wang1 Nan Zhang1
RI PT
3
1 State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation,
5
Southwest Petroleum University, Chengdu, China
6
Abstract
7
In this study, a semi-analytical model which couples the reservoir and wellbore, is
8
developed to predict the inflow profile of horizontal wells in a box-shaped gas
9
reservoir with a bottom water zone. The developed gas reservoir flow model is based
10
on a 3D volumetric source model which considers the reservoir properties in the
11
region near the wellbore, in terms of anisotropy, heterogeneity and formation damage.
12
Meanwhile, pressure drops in the wellbore flow model was also considered, due to
13
fluid flow in the horizontal wellbore. The model was calculated by a field case, and
14
the results of the calculation were compared with a commercial simulator: ECLIPSE
15
(Schlumberger 2011). The comparison between the developed model and ECLIPSE
16
indicates that, a good match was obtained, so that, the model derived in this study can
17
be applied to predict the inflow profile of horizontal well in a gas reservoir with
18
bottom water zone. Moreover, sensitivity analysis is conducted by using various
19
wellbore pressure drops, wall roughness, skin factors, wellbore diameters, anisotropy
20
and producing pressure drops.
21
Keywords：Semi-analytical model, inflow profile, horizontal well, gas reservoir,
AC C
EP
TE D
M AN U
SC
4
1
ACCEPTED MANUSCRIPT 1
bottom-water zone
1. Introduction
3
Gas recovery factor from a gas reservoir with a bottom water zone is often lower than
4
that of depleted gas reservoirs (Geffen et al., 1952; Keelan and Pugh, 1975). The
5
water influx occurs when the pressure drop at the gas-water interface in the process of
6
gas reservoir production (Ahmed, 2000). Nowadays, horizontal well is essential to
7
explore the oil/gas resources, because the horizontal well can generate a larger contact
8
area between the horizontal well and the reservoir, so that the oil production can be
9
enhanced. However, disadvantage of the horizontal well which applied in the
10
heterogeneous reservoir was observed in the oil/gas field, that is, the productivities of
11
the wells would decrease when an uneven distribution of inflow profile along the
12
wellbore occurred, due to heterogeneity appears near the wellbore and pressure drop
13
along the wellbore (Tatar et al. 2014, Song et al. 2015, Naderi et al. 2015). In order to
14
improve the recovery of bottom-water gas reservoirs, inflow profile of horizontal
15
wells in bottom-water gas reservoir has become an important task.
SC
M AN U
TE D
EP
The conventional production logging is widely used to predict the inflow profile
AC C
16
RI PT
2
17
wells，but there are some shortages in this approach. For instance, the inflow profile
18
along the wellbore cannot be tested, if the horizontal well is too deep; and the test
19
results are not accurate because of the irregular horizontal wellbore, etc. Moreover,
20
the service fee of dynamic gas producing profile is too high，so the theoretical model
21
for predicting the inflow profile along the wellbore is required.
22
It has been widely recognized that, the pressure drop along the horizontal well 2
ACCEPTED MANUSCRIPT significantly affect the production performance of the horizontal wells (Dikken, 1990).
2
A one dimensional reservoir/wellbore coupling model was developed to predict the
3
performance of horizontal wells using a productivity index (Dikken, 1990; Landman,
4
1994; Novy, 1995; Penmatcha and Aziz, 1998), but this one dimension model could
5
not reflect the state of the three-dimensional flow in the reservoir. To study the state of
6
the
7
reservoir/wellbore coupling model using a three-dimensional reservoir model and a
8
wellbore model to calculate productivity in an isotropic or anisotropic reservoir, but in
9
this model, the reservoir and near well wellbore rock property variations were not take
10
into account. Ozkan et al. (1999) applied a semi-analytical approach to study the
11
reservoir/wellbore coupling, an important assumption was made, that is, compared to
12
the reservoir pressure loss, the pressure drops along the wellbore is negligible. Vicente
13
et al.(2000) presented a reservoir/wellbore coupling model, considering an
14
oversimplified wellbore pressure drop model. Based on the experimental results,
15
Ouyang et al. (2005) proposed a coupled reservoir/wellbore formulation, however,
16
there is a certain limitation in the practical application. Souza et al.(2014) developed a
17
new numberical method to solve the well-reservoir model, taking into account
18
isotropic and anisotropic, wellbore length, completion scheme and a damaged region
19
around the wellbore. But it spends more computational time than semi-analytical
20
models(Luo et al., 2015). In recent years, some scholars have also studied on
21
reservoir/wellbore coupling models. Johansen and Khoriakov (2007) presented a new
22
method in simulation of multi-phase flow and in the near well region in the reservoir.
flow,
Penmatcha
and
Aziz
(1998)
developed
a
AC C
EP
TE D
M AN U
SC
three-dimensional
RI PT
1
3
ACCEPTED MANUSCRIPT Hasan and Kabir (2009) proposed an effective model for a two-phase flow in
2
geothermal wells by means of the drift-flux method. Adesina et al.(2016) presented an
3
improved reservoir/well model which considers pressure drop due to acceleration. But
4
the effect of the formation damage near wellbore zone has not been effectively
5
addressed in these papers.
RI PT
1
Point source is the most used techniques to build a reservoir/well model
7
(Khoriakov et al., 2012), however, the shortcoming of the point-source is the inherent
8
singularity of the solution, because no volume assumption was made in this method
9
(Valkó and Amini, 2007). The Distributed Volumetric Source (DVS) method was
10
proposed to predict production performance of the horizontal wells located in a
11
fractured reservoir, and high accuracy and computational efficiency were obtained
12
(Valkó and Amini, 2007; Amini and Valkó, 2010; Jiang et al., 2016). Meanwhile, the
13
volumetric-source model had been treated successfully for the inherent singularity of
14
the solution in the point-source. More importantly, rare studies on reservoir/well
15
model in a gas reservoir with bottom water zone, were proposed.
EP
TE D
M AN U
SC
6
This study of inflow profile prediction of horizontal wells is focused on the
17
reservoir/wellbore coupling model based on volumetric source solutions in a
18
box-shaped gas reservoir with a bottom water zone. And sensitivity study has
19
presented using various wellbore pressure drop, wall roughness, skin factor, wellbore
20
diameter, anisotropy and producing pressure drop. This model can be used to predict
21
inflow profile of horizontal wells in gas reservoir with a bottom water zone, and to
22
horizontal well completion design and optimization or production prediction.
AC C
16
4
ACCEPTED MANUSCRIPT 1
2. Semi-analytical model and solution
2
2.1 Assumptions The semi-analytical model, which focuses on the gas production performance in
4
the horizontal well located in the reservoir with a bottom water zone, is based on the
5
following assumptions:
6
2.2 Reservoir Flow Model
9
EP
8
Fig. 1. Schematic of a horizontal well in a box-shaped gas reservoir
(1) It is assumed that the gas reservoir is homogeneous.
AC C
7
TE D
M AN U
SC
RI PT
3
10
(2) The horizontal well is located in the middle of a box-shaped reservoir, as showed
11
in Fig. 1. A constant pressure boundary is applied at the bottom of the model,
12 13 14 15
closed boundary conditions are conducted in other five sides. (3) Single-phase and steady-state flow is considered, the production process is isothermal, and governed by Darcy's law. As we can see a gas reservoir（big box）in Fig. 1, 5
the sizes of the box-shaped
ACCEPTED MANUSCRIPT gas reservoir in three directions are defined as xe, ye, ze, respectively. A volumetric
2
source (small box in big box) with a strength flow rate q is applied in the gas reservoir.
3
The center coordinate of the volumetric source is (cx, cy, cz), so that the sizes of the
4
volumetric source in three directions are 2wx, 2wy, 2wz, respectively. The volumetric source model can be expressed as:
7
The boundary conditions: ∂ ( ∆ψ )
8
∂x ∂ ( ∆ψ )
9
∂y ∂ ( ∆ψ )
10
15 16 17 18 19
Where
AC C
14
(2)
x = 0, x = xe
=0
(3)
y = 0, y = ye
=0
(4)
z = ze
TE D ∆ψ
12 13
(1)
The initial condition:
EP
11
∂z
=0
SC
qg ∂ 2 ( ∆ψ ) ∂ 2 ( ∆ψ ) ∂ 2 ( ∆ψ ) + + + h ( x, y , z ) = 0 2 2 2 ∂x ∂y ∂z 86.4kVsource
6
M AN U
5
RI PT
1
z=0
ψ = 2∫
pini
p
=0
(5)
p dp µZ
(6)
∆ψ = ψ i −ψ g
(7)
pscZT qsc pgTsc
(8)
qg =
Vsource = 8wx wy wz h( x, y, z) = H( x−cx −wx ) −H( x−cx +wx ) ×H( y −cy −wy ) −H( y −cy +wy )× H( z −cz −wz ) −H( z −cz +wz ) H ( x − x0 ) is Heaviside function: 6
(9) (10)
ACCEPTED MANUSCRIPT 1 H ( x − x0 ) = 0
1 2
x > x0
(11)
x ≤ x0
With:
ψ i = Initial gas reservoir pseudo-pressure, MPa
4
ψ g = Pseudo-pressure of a point in the gas reservoir, MPa
5
q g = strength of the volumetric source in gas reservoir, m3/d
6
Vsource = Geometric volume of volumetric source in gas reservoir, m3
7
k= Reservoir permeability, µm2
8
p= Formation pressure in the gas reservoir, MPa
9
pini = Initial gas reservoir pressure, MPa
M AN U
SC
RI PT
3
µ = Gas fluid viscosity, mPa s
11
Z = Deviation factor of gas reservoir
12
psc = Initial gas reservoir pressure, MPa
13
Tsc = Temperature of ground standard condition, K
14
T = Temperature of gas reservoir, K
17 18 19
EP
16
The detailed derivation of the volumetric source model solution is provided in the Appendix. The solution of the volumetric source model can be presented as:
AC C
15
TE D
10
∆ψ =
qsc pT 86.4kVsourceTsc
σl ( x) σ m ( y ) σ n ( z ) γ l =0 m=0 n=0 ∞
∞
∞
∑∑∑
(12)
where
2wx x σl ( x) = e lπ cx lπ wx lπ x 4cos cos sin xe xe xe lπ
7
l =0
(13) l ≠0
ACCEPTED MANUSCRIPT m=0
(14) m≠ 0
π + nπ ) z π + nπ ) c π + nπ ) w ( ( ( 2 2 4sin cos sin 2 z
σn ( z) =
ze
ze
z
ze
π + nπ 2
n = 0,1, 2 ⋅⋅⋅⋅
(15)
SC
2
RI PT
1
2 wy ye σm ( y) = mπ c y mπ wy 4cos mπ y cos sin ye ye ye mπ
M AN U
3
Fig. 2. Schematic of the horizontal well and gas reservoir division
5
The cylindrical wellbore is approximate by rectangular shape (Fig.1), and this
6
approximation does not have big impact for steady state flow. And the horizontal well
7
is divided into N segments (Fig.2), each segment is taken as a volumetric sink, the
8
length of i segment is Li, the radius of wellbore is rw, the center coordinate of i
9
segment is (xi, yi, zi). Hence, the dimensions in three directions and coordinate of i
11 12 13 14 15
EP
AC C
10
TE D
4
segment can be expressed as:
2wxi = Lhi
(16)
2wyi = 2wzi = π rw2
(17)
(c ,c
(18)
xi
yi
, czi ) = ( xi , yi , zi )
As we can see in Fig. 2, the gas reservoir is divided into N blocks in response to segmentation of the horizontal well. The pressure drawdown at arbitrary point M(x, y, z) 8
ACCEPTED MANUSCRIPT 1
in the gas reservoir can be regarded as a linear superposition of the pressure in all
2
blocks (N) of the horizontal well, by using the superposition method:
4
σli ( x) σ mi ( y ) σ ni ( z ) γ l =0 m=0 n=0 ∞
∞
∞
∑∑∑
As shown in Fig. 2, Lr1 = Lh1 +
(19)
xe − L x −L and LrN = LhN + e , the other blocks are 2 2
RI PT
N qgsci pT ∆ψ Mi = ∑ i =1 86.4kVsourceTsc
3
the same as the horizontal well section. And there is no cross flow between reservoir
6
segments. According to Equation (19), the point M(x, y, z) is located in block i of the
7
gas reservoir, the pressure drop of point M can be achieved:
8
10 11 12
∞
∞
obtained: M
j =1
86.4kVsourceTsc
ψ g =ψ i − ∑
qgsci pT
∞
∞
∞
∑∑∑
σlj ( cxi ) σ mi ( cyi ) σ ni ( czi + wzi )
l =0 m=0 n=0
γ
（21）
The parameters of anisotropic in the gas reservoir can be obtained based on the
xe' = xe
EP
14
17
(20)
Substituting equation (20) into equation (7), the pressure in block i can be
Besson correction method (Besson, 1990):
16
γ
l =0 m=0 n=0
13
k k k , ye' = ye , ze' = ze kx ky kz
(22)
c xi' = c xi
k k k , c yi' = c yi , c zi' = c zi kx ky kz
(23)
wxi' = wxi
k k k , w'yi = wyi , wzi' = wzi kx ky kz
(24)
AC C
15
σlj ( cxi ) σmi ( cyi ) σ ni ( czi + wzi )
∑∑∑
TE D
9
∞
M AN U
qgsci pgT ∆ψi = ∑ j =1 86.4kVsourceTsc N
SC
5
k = kx k y kz
(25)
18
Reservoir heterogeneity is the main reason for the imbalance of inflow profile in
19
horizontal wells (Wang et al. 2000), so we can not ignore this factor. The s-k* 9
ACCEPTED MANUSCRIPT approach was proposed by Wolfsteiner et al.(2000), to model the heterogeneity in the
2
near well region. This method indicates the relationship of the average permeability
3
k* of the whole gas reservoir and an effective skin caused by varies along the
4
wellbore trajectory. Based on Hawkins' method (Hawkins, 1956), the effective
5
near-well skin of block i can be defined as:
RI PT
1
k r Shi = −1 ln a ki rw
(26)
SC
6
During drilling and completion process, all reservoirs are susceptible to
8
formation damage (Kersey, 1986), the degree of damage can be expressed by the loss
9
of permeability. Frick and Economides proposed the elliptical-cone-shape model to
10
describe damage distribution along the horizontal wellbore (Frick and Economides,
11
1993; Furui et al., 2002). In this study, the formation damage skin in block i is
12
expressed as:
13 Where
17 18
I ani =
AC C
15 16
k 1 rdh i + = − 1 ln k di I an i + 1 rw
EP
14
sdi
TE D
M AN U
7
2 rdh i 2 + I a ni − 1 rw
kx k y
(27)
(28)
kz
Therefore, the total skin of block i can be expressed as:
S ti = S h i + S d i
(29)
The total skin of block i is substituted in equation (29) in the form of additional
19
pressure drop. Therefore, the pressure of block i in the gas reservoir can be proposed
20
as:
21
M
j =1
86.4kVsourceTsc
ψg =ψi − ∑
qgsci pT
∞
∞
∞
σlj ( cxi ) σmi ( cyi ) σni ( czi + wzi )
∑∑∑
γ
l =0 m=0 n=0
10
qgsci pT Sti − 86.4kLhiTsc
(30)
ACCEPTED MANUSCRIPT
2 3
4
Substituting equation (6) into the equation (30), the pressure of block i of the gas reservoir can also be written as: M Z q pT pri2 = pini2 − 2µ∑ i gsci j =1 86.4kVsourceTsc
σlj ( cxi ) σ mi ( cyi ) σ ni ( czi + wzi ) µZi qgsci pT Sti (31) − γ l =0 m=0 n=0 43.2kLhi Tsc ∞
∞
∞
∑∑∑
RI PT
1
2.3 Wellbore Flow Model
As shown in Fig. 3, the horizontal well is divided into n segments. And segment
6
1 closes to the heel of the horizontal well, while segment n closes to the toe of the
7
horizontal well. In the wellbore flow, the friction and acceleration pressure drop
8
should be considered, because of the well trajectory and the gas reservoir inflow.
TE D
M AN U
SC
5
9 10
EP
12
As shown in Fig. 4, it is assumed that the single fluid flows between the two nodes.
AC C
11
Fig. 3 Schematic of wellbore flow in horizontal well
13 14
Fig. 4 Flow model in horizontal wellbore
11
ACCEPTED MANUSCRIPT The conservation of mass can be expressed as:
ρv
2 3
π D2 4
∂v π D 2 + ρ v Rπ D d x − ρ v + dx =0 ∂ x 4
According to Equation (32), we can get the following equation:
∂v 4vR = ∂x D
4
9
10 11
12
13 14 15
16
SC
= 1.0 × 10
4τ ∂ v ∂ v 2 w + ρ 2v + dx ∂ ∂ D x x
(34)
f ρv2 , and substituting τ w and equation (33) into equation (34), the 8
M AN U
8
Set τ w =
dx
−6
pressure drops can be expressed the following equation (Li et al., 2009):
−
d pw d xi
f ρ v 2 8 ρ vv R 16 ρ v R2 = 1.0 × 10 − 6 + + dx D D2 2D
(35)
In which, the friction factor f can be expressed by using the Reynolds number
TE D
7
−
d pw
under laminar flow and turbulent flow:
64 Re f = −2 ε − 0.9 1.14−2lg +21.25 Re d
EP
6
(33)
Then the momentum principles can be expressed as:
laminar flow ( Re ≤ 2300)
AC C
5
(32)
RI PT
1
（36）
turbulent flow ( Re ≥ 4000)
The ideal gas law is applied to calculate the properties of the gas, in terms of,
pressure, density and gas flow rate:
ρ= v=
M air γ g p R TZ
ρ sc q sc T Z p π rw2 Tsc 12
(37)
(38)
ACCEPTED MANUSCRIPT 1
velocity of segment i from gas reservoir to wellbore can be written as:
v Ri =
3 4
6
(39)
Therefore, the final pressure drops can be obtained from equation 31-39:
RI PT
5
p sc q sci TZ 2π rw p Tsc Li
2 2 16Mairγ g psc2 qscwi TZi 64Mairγ g psc2 qscwiqsciTZi 32Mairγ g psc2 qsci TZi p − p =1.0×10 fi ∆lj + + (40) 2 4 2 4 2 4 π π π RD RD RD T T T sc sc sc 2 w1
2 w2
−6
With:
SC
2
Around the wellbore, the uniform fluid flow was assumed, then the gas flow
Z i = Deviation factor of segment i.
8
Where the first item is the frictional pressure drop, and the second and three
M AN U
7
9
items are acceleration pressure drops.
10
2.4 Coupled model and solution
In this work, gas reservoir flow model and wellbore flow model are coupled to
12
study the restrains and effects of the fluid flow in both the formation and the wellbore.
13
For flow rate control:
15 16 17 18
EP
n
Qmax = ∑ qi
(41)
p1 = pwf ,min
(42)
AC C
14
TE D
11
i =1
For bottom-hole pressure control:
Where Qmax is the desired maximum flow rate, m3/d; pwf ,min is the specified
minimum bottom-hole pressure, Mpa.
19
Equations 31-42 constitute the coupled model, and the Newton Raphson
20
method is used to solve this model, due to the equations in the model are nonlinear. 13
ACCEPTED MANUSCRIPT 1
The procedure for solving the model is summarized as follows: (1) Assume the sandface pressure distribution is pri. So that, the sandface inflow
3
rate along the horizontal well can be calculated by reservoir flow model
4
(Equation 31).
5
(2) Substitute qri into equations 36-42 to calculate the horizontal well pressure
6
distribution (pwi), and the new horizontal well pressure distribution is equivalent
7
to the sandface pressure distribution (pri).
8
(3) Compare the new sandface pressure distribution with the calculated results of
9
the last iteration. If the error is less than the allowable range, stop the process and
10
output the results; otherwise, assume the sandface pressure distribution again and
11
repeat the iteration, until achieve the requirements in accuracy.
SC
M AN U
3. Applications
13
3.1 Model validation
TE D
12
EP
Table 1 Reservoir Properties for Homogeneous Gas Reservoir Reservoir length (m) Reservoir width (m) Reservoir thickness (m) Horizontal well length (m) Distance of wellbore to WGC (m) Wellbore radius(m) Initial formation pressure (Mpa) Porosity (%) Formation volume factor(m3/m3)
1000 600 50 680 35 0.12 63 14 2.72
Gas density (g/cm3)
0.68
AC C
14
RI PT
2
3
Water density (g/cm ) Gas viscosity (mPa s) Permeability kx (mD) Permeability ky (mD) Permeability kz (mD)
1.05 0.029 50 50 5 14
ACCEPTED MANUSCRIPT 560
Reservoir Simulation ECL Semi-analytical model
540
500
480
RI PT
3
Inflow rate (m /d)
520
460
440
420 0
50
100 150 200 250 300 350 400 450 500 550 600 650 700
1
Fig. 5 Comparison of inflow rate between semi-analytical model and ECL Semi-analytical model Reservoir Simulation ECL
M AN U
62.8
62.6
62.4
62.2
TE D
Pressure along the well (Mpa)
63.0
0
4
100
200
300
400
500
600
700
Distance from heel (m)
Fig. 6 Comparison of pressure along the well between our model and ECL
EP
3
SC
2
Distance from heel (m)
A gas field case is introduced in this study to validate the calculation results of
6
the developed model. A strong bottom-water drive gas reservoir, with an initial
7
pressure of 63 MPa was selected, and a horizontal well with horizontal length of
8
680m, is located in the middle of the gas reservoir. The properties of the real gas
9
reservoir data are listed in Table 1.
AC C
5
10
The calculation results of the developed model are compared with a commercial
11
software, ECLIPSE (Schlumberger 2011) to check the accuracy. The grid dimension
12
is 100×60×30, the grid sizes for x, y, and z directions are 10 m, 10 m and 2 m, 15
ACCEPTED MANUSCRIPT respectively，and the time step is 20 days. The inflow and pressure distributions along
2
the horizontal well are computed by using both of the developed model and the
3
commercial simulator, as shown in Fig. 5 and Fig. 6, good match results are obtained.
4
These results suggest that, the developed model has potential practical application in
5
predicting inflow profile of horizontal wells in bottom-water gas reservoir.
6
3.2 Influential factors analysis
SC
RI PT
1
Steady state flow is the predominant flow in many gas reservoirs. In this study,
8
some parameters were analyzed in terms of wellbore pressure drop, wall roughness,
9
skin factor, wellbore diameter, anisotropy and producing pressure drop at 500 days,
10
when the flow in the reservoir being considered has reached steady state, as indicated
11
in Table 1.
12
3.2.1 Effect of wellbore pressure drop
TE D
M AN U
7
Fig. 7 indicates the inflow profile of the horizontal well with and without
14
pressure drop. From Fig. 7, it is clear that: (1) for the case without pressure drop along
15
the wellbore, the inflow profile is symmetric with inflow distribution of around
16
512.06m3/d; (2) when the pressure drop was considered along the wellbore, the inflow
17
profile is asymmetric, with a flow rate under 541.3m3/d in the heel and 514.6m3/d in
18
the toe. Therefore, the pressure drop of the wellbore affects the inflow profile of the
19
horizontal well.
AC C
EP
13
16
ACCEPTED MANUSCRIPT 550 540
No wellbore pressure drop t=500d (Semi-analytical model) Wellbore pressure drop t=500d (Semi-analytical model) No wellbore pressure drop t=500d (Reservoir Simulation ECL) Wellbore pressure drop t=500d (Reservoir Simulation ECL)
530
510 500 490 480 470 460 450 440 0
100
200
1
500
600
700
Fig. 7 Inflow profile of the horizontal well for with and without pressure drop
SC
3
400
3.2.2 Effect of wall roughness
M AN U
2
300
Distance from heel (m)
RI PT
3
Inflow rate (m /d)
520
Fig. 8 shows the inflow profiles of the horizontal well for various wall roughness.
5
For a horizontal well with a flow rate of 20000m3/d, grater wall roughness causes a
6
higher non-uniform heel biased inflow profile. This is because a much greater
7
frictional resistance prevents the fluid flow along the wellbore. As expected, effects of
8
wall roughness can be reduced by using smaller wall roughness in horizontal well.
TE D
4
700
e=0.02mm (Semi-analytical model) e=0.002mm (Semi-analytical model) e=0.0002mm(Semi-analytical model) e=0.02mm (Reservoir Simulation ECL) e=0.002mm (Reservoir Simulation ECL) e=0.0002mm(Reservoir Simulation ECL)
EP
650
AC C
3
Inflow rate (m /d)
600
550
500
450
400 0
9 10
100
200
300
400
500
600
700
Distance from heel (m)
Fig. 8 Inflow profile of the horizontal well for various wall roughness
17
ACCEPTED MANUSCRIPT 1
3.2.3 Effect of skin factor
2
The formation skin factor ( S ) because of formation damage is considered in the
3
homogeneous gas reservoir, inflow profile of the horizontal well for various skin
4
factors are shown in Fig. 9. With the same producing pressure drop, the lower skin
5
factor gains higher inflow rate along the horizontal well, hence, the skin factor should
6
be taken into account in the case, due to the effect is remarkable.
RI PT
di
800
SC
Without considering skin factor s=0 (Semi-analytical model) Considering drilling pollution skin factor s=2 (Semi-analytical model) Considering drilling pollution skin factor s=5 (Semi-analytical model) Without considering skin factor s=0 (Reservoir Simulation ECL) Considering drilling pollution skin factor s=2 (Reservoir Simulation ECL) Considering drilling pollution skin factor s=5 (Reservoir Simulation ECL)
750 700
M AN U
3
Inflow rate (m /d)
650 600 550 500 450 400
0
7 8
100
200
300
400
500
600
700
Distance from heel (m)
Fig. 9 Inflow profile of the horizontal well for various skin factor
3.2.4 Effect of wellbore diameter
EP
9
TE D
350
Fig. 10 shows the effect of well diameter on inflow distribution along the well
11
under steady state (at a production time of 500 days). The flow profile becomes more
12
skewed as the well diameter is getting smaller, the wellbore drop will reduce and
13
results in inflow distribution nearly uniform with the increasing of the wellbore
14
diameter.
AC C
10
18
600 590 580 570 560 550 540 530 520 510 500 490 480 470 460 450 440 430
d=300mm (Semi-analytical model) d=250mm (Semi-analytical model) d=200mm (Semi-analytical model) d=300mm (Reservoir Simulation ECL) d=250mm (Reservoir Simulation ECL) d=200mm (Reservoir Simulation ECL)
0
300
400
500
Distance from heel (m)
600
700
SC
3
200
Fig. 10 Inflow profile of the horizontal well for various wellbore diameter
3.2.5 Effect of anisotropy
M AN U
1 2
100
RI PT
3
Inflow rate (m /d)
ACCEPTED MANUSCRIPT
As shown Fig. 11, the effect of anisotropy also can be represented by a skin term.
5
Fig. 11 indicates the inflow profile of the horizontal well with different anisotropy.
6
For the well with the same producing pressure drop of 5MPa, it can be observed that
7
higher anisotropy causes higher inflow rate along the horizontal well. As shown in Fig.
8
11, the inflow distribution is more inclined to the heel section with the increase of
9
anisotropy.
EP
TE D
4
Iani=0.2 (Semi-analytical model)
700
Iani=0.4 (Semi-analytical model) Iani=0.6 (Semi-analytical model)
650
Iani=0.8 (Semi-analytical model) Iani=0.2 (Reservoir Simulation ECL) Iani=0.4 (Reservoir Simulation ECL) Iani=0.6 (Reservoir Simulation ECL)
550
Iani=0.8 (Reservoir Simulation ECL)
3
Inflow rate (m /d)
AC C
600
500 450 400 350 300 250
0
10 11
100
200
300
400
500
600
700
Distance from heel (m)
Fig. 11 Inflow profile of the horizontal well for various anisotropy
19
ACCEPTED MANUSCRIPT 1
3.2.6 Effect of producing pressure drop Fig. 12 shows the inflow profile of the horizontal well for various producing
3
pressure drops. As the producing pressure drop increases, the inflow rate along the
4
horizontal well becomes higher. Similar to the effect of anisotropy, the inflow
5
distribution is more inclined to the heel section with the increase of producing
6
pressure drop.
RI PT
2
SC
800
∆p=1Mpa (Semi-analytical model) ∆p=3Mpa (Semi-analytical model) ∆p=5Mpa (Semi-analytical model) ∆p=1Mpa (Reservoir Simulation ECL) ∆p=3Mpa (Reservoir Simulation ECL) ∆p=5Mpa (Reservoir Simulation ECL)
750
M AN U
650
3
Inflow rate (m /d)
700
600 550 500
0
10 11 12 13
200
300
400
500
600
700
Distance from heel (m)
Fig. 12 Inflow profile of the horizontal well for various producing pressure drop
4. Conclusions
EP
9
100
(1) A semi-analytical model which coupled reservoir/wellbore, is developed for
AC C
7 8
TE D
450
predicting the inflow profile of the horizontal wells in a box-shaped gas reservoir with a bottom-water zone. The gas reservoir flow model is based on a 3D volumetric source model that considers the reservoir anisotropy, heterogeneity
14
and formation damage in the near well region. And the pressure drops along the
15
wellbore is considered.
16
(2) Comparison between the results of the developed model and a commercial
20
ACCEPTED MANUSCRIPT simulator (ECLIPSE), a good match result is obtained, so the new model has
2
potential practical application in predicting inflow profile of horizontal wells in
3
gas reservoir with a bottom-water zone. And sensitivity analysis is conducted by
4
using various wellbore pressure drops, wall roughness, skin factors, wellbore
5
diameters, anisotropy and producing pressure drops.
RI PT
1
(3) The semi-analytical model can be used as a basis, not only for predicting inflow
7
profile of horizontal wells in bottom-water gas reservoir, but also for horizontal
8
well completion design and optimization, to and production prediction.
Acknowledgements
M AN U
9
SC
6
The authors would like to acknowledge the National Science and Technology
11
Major Project (Grant Nos. 2016ZX05017005-006), for its technical and financial
12
support.
13
Appendix. Solution of volumetric source model
16 17
∂2 E ∂2 E ∂2 E ∂x 2 + ∂y 2 + ∂z 2 = − λ E ∂E ∂E x=0 = x = xe = 0 ∂x ∂x ∂E ∂E = =0 ∂y y = 0 ∂y y = y e ∂E E z =0 = 0 z = ze = 0 ∂z
EP
15
From Equation (1)-(5), the characteristic equation can be proposed as:
AC C
14
TE D
10
(A.1)
Set E = X ( x ) Y ( y ) Z ( z ) ， three one-dimensional eigenvalue problems are obtained by variables separation:
21
ACCEPTED MANUSCRIPT
1
X '' + µ X = 0 ' ' X x = 0 = X x = xe = 0
2
where λ =µ + ν + θ
6
mπ νl = , ye
7
π + nπ θn = 2 ze
X l = cos
lπ , xe
X l = cos
mπ , ye
13
14 15
16
n = 0,1, 2 ⋅⋅⋅⋅
ze
TE D
2 2 lπ mπ π 2 + nπ λ= + + ze x e y e
(A.5)
(A.6)
2
(A.7)
EP
The corresponding characteristic function system is: Elmn ( x, y , z ) =cos
π + nπ lπ mπ cos sin 2 xe ye ze
AC C
12
Zn
π + nπ ) ( = sin 2 ,
(A.4)
The corresponding characteristic value is:
9
11
2
,
m = 0,1, 2 ⋅⋅⋅⋅
M AN U
2
l = 0,1, 2 ⋅⋅⋅⋅
SC
2
lπ µl = , xe
10
(A.3)
three one-dimensional eigenvalue are obtained through Equation (A.2):
5
8
(A.2)
RI PT
3 4
Z '' + θ Z = 0 ' Z z = ze = 0 Z z = 0 = 0
Y '' + ν Y = 0 ' ' Y x = 0 = Y x = xe = 0
l , m, n = 0,1, 2, ⋅⋅⋅
(A.8)
The corresponding 2-norm of eigen function is:
El , m,n ( x, y, z ) = 2
wx wy wz 8
Al Bm
l , m = 0,1, 2 ⋅⋅⋅⋅
(A.9)
m=0 m≠0
(A.10)
Where Al =
{
2 1
l=0 l≠0
Bm =
{
2 1
Using the characteristic function system Equation (A.8) as the transform kernel, 22
ACCEPTED MANUSCRIPT 1
the orthogonal transformation is introduced to solve the volumetric source model.
2
Here, the orthogonal transformation is described as:
ψ = G ( ∆ψ ) = ∫
xe
0
0
El ,m ,n ( x, y , z ) ∆pd x d y d z
ze
0
∞
∞
∆ψ = ∑∑∑
5
l = 0 m =0 n =0
8
ψ =-
(A.12)
(A.13)
Equation (A.13) into the Equation (A.12):
-El,m,n ( x, y, z) G[（ h x, y, z） ] = qsc pT 2 l =0 m=0 n=0 γ 86.4kVsourceTsc El,m,n ( x, y, z) ∞
∞
∞
∞
2wx x σl ( x) = e lπ cx lπ wx lπ x 4cos cos sin xe xe xe lπ
AC C
Where
σl ( x) σm ( y) σn ( z) ∑∑∑ γ l =0 m=0 n=0 ∞
EP
qsc pT 86.4kVsourceTsc
1
TE D
11
14
ψ
The solution of the volumetric source model can be obtained by Substituting
∞
13
2
qsc pT G [（ h x, y , z）] γ 86.4kVsource Tsc 1
∆ψ = G( ∆ψ ) = ∑∑∑
12
El ,m,n ( x, y, z )
SC
equation can be obtained:
10
El ,m,n ( x, y, z )
Taking the orthogonal transformation of equations (1) – (5), the following
7
9
(A.11)
The inverse transformation of equation (A.11) can be written as: ∞
6
l , m, n = 0,1, 2 ⋅⋅⋅⋅
RI PT
4
ye
∫ ∫
M AN U
3
2 wy y σm ( y) = e lπ c y lπ wy 4 cos lπ y cos sin ye ye ye mπ
23
(A.14)
l =0
(A.15) l ≠0
m=0
(A.16) m≠ 0
ACCEPTED MANUSCRIPT 1
σn ( z) =
4sin
(π 2 + nπ ) z cos (π 2 + nπ ) c sin (π 2 + nπ ) w z
ze
ze
z
ze
π + nπ 2
n = 0,1, 2 ⋅⋅⋅⋅
Nomenclature
3
xe, ye, ze gas reservoir dimensions, m
4
cx, cy, cz central coordinates of the volumetric source, m
5
wx, wy, wz dimensions of the volumetric source, m
6
qg strength of the volumetric source in gas reservoir, m3/d
7
Vsource geometric volume of volumetric source in gas reservoir, m3
8
T temperature of gas reservoir, K
9
Tsc temperature of ground standard condition, K
M AN U
SC
RI PT
2
µg gas fluid viscosity, mPa s
11
k reservoir permeability, mD
12
kx, ky, kz reservoir permeability at three directions of x, y, z, mD
13
p pressure of a point in the gas reservoir, MPa
14
ψg pseudo-pressure of a point in the gas reservoir, MPa
15
pini initial gas reservoir pressure, MPa
16
ψi initial gas reservoir pseudo-pressure, MPa
17
γg gas relative density, dimensionless
18
Li length of segment i, m
19
rw wellbore radius, m
20
xi, yi, zi central coordinates of segment i, m
21
qri fluid inflow rate of segment i , m3/d
AC C
EP
TE D
10
24
(A17)
ACCEPTED MANUSCRIPT k* effective reservoir permeability, µm2
2
kai reservoir permeability of segment i in the altered zone, µm2
3
ra radius of the altered permeability zone, m
4
shi skin of segment i caused by permeability heterogeneity, dimensionless
5
kdi reservoir permeability of segment i in the damaged zone, µm2
6
amax half-length of the horizontal axes of damaged zone at the heel, m
7
Iani permeability anisotropic coefficient, dimensionless
8
Si distance between segment i and the toe of horizontal well, m
9
L horizontal well length, m
M AN U
SC
RI PT
1
Sdi skin of segment i caused by formation damage, dimensionless
11
Sti total skin of segment i, dimensionless
12
pri sandface pressure at the center of segment i, MPa
13
ρ gas density in wellbore, g/cm3
14
g acceleration due to gravity, m/s2
15
f friction factor in the tubing, dimensionless
16
v gas flow velocity, m/s
17
vR gas flow velocity from reservoir to wellbore, m/s
18
pwi pressure of wellbore segment i, Mpa
19
M air relative molecular mass of air
20
D diameter of horizontal wellbore, mm
21
R universal gas constant, J·mol-1·K-1
22
NRe Reynolds number
AC C
EP
TE D
10
25
ACCEPTED MANUSCRIPT pwi pressure of segment i in the horizontal wellbore, Mpa
2
qgsci fluid inflow rate of segment i in standard condition, m3/d
3
qscwi wellbore flow rate of segment i in standard condition, m3/d
4
ε roughness factor
5
Reference
EP
TE D
M AN U
SC
Adesina, F., Paul, A., Oyinkepreye, O., & Adebowale, O., 2016. An Improved Model for Estimating Productivity of Horizontal Drain Hole[C]//SPE-169386-MS. SPE Latin America and Caribbean Petroleum Engineering Conference, Maracaibo, Venezuela Ahmed, T., 2000. Reservoir Engineering Hand Book. Gulf Pub. Co, Houston, TX. Amini, S. and P.P. Valkó, 2010. Using distributed volumetric sources to predict production from multiple-fractured horizontal wells under non-darcy-flow conditions. J. SPEJ 15(1): 105-115. Besson J., 1990. Performance of slanted and horizontal wells on an anisotropic medium[C]//SPE-20965-MS. European Petroleum Conference, The Hague, Netherlands. Chen, Y., 2004. A Decomposition Method of Skin Factor for Gas Well. Xinjiang Pet. Geol. 25(2), 013. (in Chinese) Dikken, B. J., 1990. Pressure drop in horizontal wells and its effect on production performance. J. Pet. Technol. 42(11): 1426-1433. Dumkwu, F.A., A.W. Islam and E.S. Carlson, 2012. Review of well models and assessment of their impacts on numerical reservoir simulation performance. J. Pet. Sci. Eng. s 82– 83(1): 174–186. Frick, T.P., Economides, M.J., 1993. Horizontal well damage characterization and removal. SPE Prod. Fac. 8 (1), 15-22. Furui, K., D. Zhu and A.D. Hill, 2002. A rigorous formation damage skin factor and reservoir inflow model for a horizontal well. SPE Prod. Fac. 18(3): 151-157. Geffen, T.M., Parrish, D.R., Haynes, G.W., Morse, R.A., 1952. Efficiency of gas displacement from porous media by liquid flooding. J. Pet. Technol. 4 (2): 29-38. Keelan, D.K., Pugh, V.J., 1975. Trapped-gas saturations in carbonate formations. Soc. Pet. Eng. J. 15: 149-160. Hasan A R, Kabir C S.,2009. Modeling Two-Phase Fluid and Heat Flows in Geothermal Wells. J. Pet. Technol. 71(1-2):77-86. Hawkins, M.F., 1956. A note on the skin effect. J. Trans. AIME 207 (12), 356-357. Jiang, B., H. Li, Y. Zhang, Y. Wang, J. Wang and S. Patil, 2016. Multiple fracturing parameters optimization for horizontal gas well using a novel hybrid method. J. Nat. Gas Sci. Eng. 34: 604-615. Johansen, T.E. and V. Khoriakov, 2007. Iterative techniques in modeling of multi-phase flow in advanced wells and the near well region. J. Pet. Sci. Eng. 58(1–2): 49-67. Jr, H. J. Remay., 1965. Non-Darcy Flow and Wellbore Storage Effects in Pressure Build-Up
AC C
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39
RI PT
1
26
ACCEPTED MANUSCRIPT
EP
TE D
M AN U
SC
RI PT
and Drawdown of Gas Wells. J. Pet. Technol. 17(2):223-233. Kersey, D. G., 1986. The role of petrographic analyses in the design of nondamaging drilling, completion, and stimulation programs[C]//International Meeting on Petroleum Engineering, Beijing, China Khoriakov, V., A.C. Johansen and T.E. Johansen, 2012. Transient flow modeling of advanced wells. J. Pet. Sci. Eng. s 86–87: 99–110. Landman, M.J., 1994. Analytic modelling of selectively perforated horizontal wells. J. Pet. Sci. Eng. 10(3): 179-188. Li, S., Lian, P. Q., Li, X., 2009. Nonsteady state model of gas reservior and horizontal well wellbore coupling. J. Southwest. Pet. Univ. 31(1):53-57. (in Chinese) Luo, W., Li, H. T., Wang, Y. Q., Wang, J. C., 2015. A new semi-analytical model for predicting the performance of horizontal wells completed by inflow control devices in bottom-water reservoirs. J. Nat. Gas Sci. Eng. 27: 1328-1339. Naderi, M., B. Rostami, M. Khosravi., 2015. Effect of heterogeneity on the productivity of vertical, deviated and horizontal wells in water drive gas reservoirs. J. Nat. Gas Sci. Eng. 23: 481-491. Novy, R.A., 1995. Pressure drops in horizontal wells: When can they be ignored?. SPE Reservoir Eng. 10(1): 29-35. Ouyang L B, Huang W S B., 2005. A Comprehensive Evaluation of Well-Completion Impacts on the Performance of Horizontal and Multilateral Wells[C]//SPE Annual Technical Conference and Exhibition, Dallas, Texas. Ozkan, E., C. Sarica and M. Haci, 1999. Influence of pressure drop along the wellbore on horizontal-well productivity. J. SPEJ 4(3): 288-301. Penmatcha, V.R. and K. Aziz, 1998. A comprehensive reservoir/wellbore model for horizontal wells. J. SPEJ 4(3): 224-234. Wang H. J., Xue S. F., Gao C. F., Tong X. H., 2012. Profile control method for perforated horizonta wells in heterogeneous reservoirs. J. China Univ. Pet. 36(3): 135-139. (in Chinese) Song, H., Y. Cao, M. Yu, Y. Wang, J. E. Killough, J. Leung., 2015. Impact of permeability heterogeneity on production characteristics in water-bearing tight gas reservoirs with threshold pressure gradient. J. Nat. Gas Sci. Eng. 22: 172-181. Souza, G.D., A.P. Pires and E. Abreu, 2014. Well-reservoir coupling on the numerical simulation of horizontal wells in gas reservoirs[C]//SPE Latin America and Caribbean Petroleum Engineering Conference, Maracaibo, Venezuela. Tatar, A., M. R. Yassin, M. Rezaee, A. H. Aghajafari, A. Shokrollahi, 2014. Applying a robust solution based on expert systems and GA evolutionary algorithm for prognosticating residual gas saturation in water drive gas reservoirs. J. Nat. Gas Sci. Eng. 21: 79-94. Valkó, P.P. and S. Amini, 2007. The method of distributed volumetric sources for calculating the transient and pseudosteady-state productivity of complex well-fracture configurations[C]//SPE Hydraulic Fracturing Technology Conference, College Station, Texas, U.S.A. Vicente, R., C. Sarica and T. Ertekin, 2000. A numerical model coupling reservoir and horizontal well flow dynamics: Transient behavior of single-phase liquid and gas flow. J. SPEJ 7(1): 70-77.
AC C
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
27
ACCEPTED MANUSCRIPT
EP
TE D
M AN U
SC
RI PT
Wolfsteiner, C., L.J. Durlofsky and A. Khalid, 2000. Approximate model for productivity of nonconventional wells in heterogeneous reservoirs. J. SPEJ 5(2): 218-226.
AC C
1 2
28
ACCEPTED MANUSCRIPT
Highlights Based on volumetric source method, this article presents reservoir/wellbore
model
for
predicting
inflow
RI PT
a
performance of horizontal wells in bottom-water gas reservoir.
SC
This proposed model considers anisotropy, heterogeneity and formation damage in near well region.
M AN U
This proposed model has been treated successfully for the
AC C
EP
TE D
inherent singularity of the solution in the point source.