Numerical Simulation of Inflow Performance for Perforated Horizontal Wells

Numerical Simulation of Inflow Performance for Perforated
Horizontal Wells
WANG Haijing1, XUE Shifeng2
1. College of Aerospace Engineering, Nanjing University of Aeronautics & Astronautics, China, 210016
2. College of Pipeline & Civil Engineering, University of Petroleum, China, 257061
[email protected]
Abstract: Accurate prediction of inflow performance for perforated horizontal wells is of great
significance to reservoir engineering analysis and perforation design optimization. A reservoir/wellbore
coupling method using a numerical reservoir model and a semi-analytical wellbore model was proposed
in this paper. To solve to the modeling and calculating problem caused by the large scale difference
between large-scale reservoir and small-scale perforation width or wellbore diameter, high-permeability
line elements and nodes on the well trajectory were used to model perforations and wellbore respectively.
The wellbore flow model considered the pressure loss inside the horizontal well and the effect of radial
influx at the perforations. An explicit iteration scheme was developed to achieve pressure and mass
continuity at the interface. With the proposed model, the inflow performance of perforated horizontal
wells was investigated. The inflow profile and the streamlines around the perforations were presented.
Sensitivity studies were conducted to identify the impact of perforation depth and permeability
heterogeneity on the inflow profile and productivity.
Keywords: Perforated horizontal wells, Inflow performance, Reservoir/wellbore coupling method,
Numerical simulation
1 Introduction
Perforated completion is a widely used method in horizontal well completions. Perforations are the only
channel between the wellbore and the producing formation. Accurate prediction of inflow performance
for perforated horizontal wells is of great significance to reservoir engineering analysis and perforation
design optimization. A considerable amount of analytical and numerical work has been done on the
inflow performance of open-hole horizontal wells and perforated vertical wells. Chen Chongxi et al.
[ 1,3 ]
presented a conception of equivalent hydraulic conductivity (EHC) to couple Darcy flow in the
reservoir and non-Darcy flow in the wellbore with numerical method. Ouyang [2] proposed a
reservoir/wellbore coupling model using an analytical reservoir model and a semi-analytical wellbore
flow model. Akim Kabir et al.[4]proposed an explicit method to couple a 3D numerical reservoir
simulator with a wellbore simulator. Yildiz[5] decompose the3D flow into a perforated horizontal well
into a transient 3D model for flow into selectively completed horizontal well, and a perforation
totalpseudoskin model to construct a simple and computationally fast model. Joseph Ansah et al.
[6 ]
developed a finite-element well inflow model with asymmetric, spiral distribution of cone-shaped
perforations around a wellbore to predict the flow performance of perforated vertical well completions.
Turhan Yildiz[7]presented a 3D analytical model to investigate the transient pressure characteristics of
perforated vertical wells. All these studies provided insight into the inflow behaviors of open-hole
horizontal wells and perforated vertical wells. However, fewer studies have been made on the perforated
horizontal wells. Erdal Ozkan[8]presented 3D analytical models to investigate the transient pressure
behavior of perforated horizontal wells. Liu Xiangping[9] derived 3D steady state pressure distribution of
oil flow due to a perforated horizontal well base on Green’s function and superposition principle. Yula
Tang[10] developed a semi-analytical reservoir/wellbore coupling model for perforated completed
horizontal wells. These analytical or semi-analytical methods have the limitations that they are rather
complicated and cumbersome to calculate, and not applicable to heterogeneous computational domain.
Therefore, numerical methods are required to solve this problem. In this paper, we proposed a numerical
318
reservoir/wellbore coupling method to investigate the inflow performance of horizontal perforated wells.
2 Coupled Reservoir/Wellbore Simulation
The fluid flow from the reservoir to the well influences the pressure distribution in the wellbore. On the
other hand, the pressure distribution in the wellbore conversely influences the flow from the reservoir.
Therefore, a comprehensive model coupling reservoir and horizontal well is of great importance to the
accurate description of inflow performance.
2.1 Simplified treatment of wellbore and perforations
For a perforated horizontal well, the perforation width is very small, about 10 mm. However, the
dimension of the drainage area is very large, usually several hundred meters or more. So big scale
difference increases the difficulties in modeling and calculating of this problem. Considering that the
perforation channel width is very small, and the permeability is very high, a high-permeability line
element is used to model the perforation. The horizontal wellbore can be represented by nodes on the
well trajectory. A segment of discrete reservoir/wellbore model is presented in Fig.1. The
high-permeability line elements are shown in red. The contribution of perforation to the flow can be
implemented by the superposition of high-permeability line-element stiffness matrix to the reservoir
stiffness matrix. The pressure of the wellbore can be calculated by the pressure loss between every two
adjacent nodes. This new treatment can simplify the modeling and meshing significantly.
Reservoir grid High-permeability line element
Wellbore node
Figure 1 A segment of discrete reservoir/wellbore model
2.2 Model description
2.2.1 Reservoir flow model
Assume that a single-phase slightly compressible Newtonian fluid flows in a heterogeneous and
anisotropic reservoir under isothermal conditions. Furthermore, the flow obeys Darcy's law and the
porosity of the reservoir is independent of pressure. With these assumptions, the 2D partial differential
equation:
K ∂P
∂
K ∂P
∂
∂ (P − P0 )
(ρ x
) + (ρ y
) = ρφ c f
(1)
µ ∂x ∂y
µ ∂y
∂x
∂t
Where ρ is fluid density, kg/m3; K is reservoir permeability, m2; µ is fluid viscosity, Pa·s; P is
reservoir pressure, Pa; φ is reservoir porosity; c f is fluid compressibility, Pa-1; P0 is initial pressure,
Pa; x, y are coordinates.
2.2.2 Wellbore Flow model
Wellbore flow differs from regular pipe flow in that radial inflow affects both wall friction and kinetic
energy change. The pressure loss in the horizontal well can be calculated using the single-phase
wellbore-flow model and wall-friction-factor correlations for pipe flow with perforation influx
developed by Ouyang11. The overall pressure gradient consists of four different components: the
pressure gradient caused by kinetic-energy change, the frictional-pressure gradient, the
319
inflow-direction-pressure gradient, the gravitational pressure gradient.
dpw
ρ fv 2 2n ρ Al
n ρ Al
g
=−
−
sin 2γ vl2 − ρ sin θ
vl v +
A
gc
dx
2d
2 Bgc A
(2)
Where Pw is well pressure, Pa; d is pipe diameter, m; n is perforation density, shot/m; ρ is fluid
density, kg/m3; Al is cross-sectional area in each perforation,m2; A is pipe cross-sectional area, m2; vl is
inflow velocity in each perforation, m/s; v is velocity averaged over a cross section, m/s; B is
momentum correction factor; gc is conversion factor; γ is inflow direction angle; θ is wellbore
inclination angle from horizontal.
The wall friction factor with both axial flow in the pipe and inflow through perforations takes the form:
For laminar flow
0.6142
f = f0 (1 + 0.04304 N Re,
(3)
w )
For turbulent flow
0.3978
f = f0 (1 + 0.0153N Re,
(4)
w )
Where N Re,w is wall Reynolds number, based on pipe inner diameter and equivalent inflow velocity; f0
is no-wall-flow friction factor.
2.2.3 Initial and boundary conditions
Let Ω and Γ represent the flow region and boundary of reservoir. Assume the pressure at the heel of the
well is constant. The initial condition and boundary conditions are:
Initial condition:
P t =0, Ω = P0
(5)
：
Boundary conditions
Pw
P
−
k
x=0
t ,Γ P
= Pw 0
(6)
=P
(7)
r
∇P ⋅ n t ,Γ = q
(8)
µ
Where P0 is initial reservoir pressure, Pa; Pw 0 is the well pressure at the heel of the well, Pa; P is
v
the boundary pressure, Pa. q is flow rate, m/s.
2.3 The coupling approach
The finite element program generator (FEPG) was chosen as the finite element code to calculate the
reservoir/wellbore model. The reservoir model was run with pressure boundary condition at the interface.
In order to achieve continuity at the interface, an iteration scheme was used which will converge quickly.
Fig.2 shows the flowchart of the algorithm used.
320
Preprocessor
Set initial pressure for reservoir,
Assume uniform pressure profile for wellbore
Time step T=dT
Iterative step
i =1
i=i+1
T=T+dT
Calculate the pressure of reservoir
Calculate the flow rate for each wellbore node,
Get the total flow rate at this iterative step Qi
Calculate pressure drop between each two
neighboring wellbore nodes,
get the pressure of each wellbore node
No
i >1
Yes
Criteria for convergence
|Qi-Qi-1|/Qi-1<
ε
No
Yes
T<Tmax
Yes
No
Postprocessor
Figure 2 Flow chart describing reservoir/wellbore coupling algorithm
3 Example Runs
In this section, results of the coupling model are presented and discussed.
A horizontal well was placed at the center of a rectangular shaped reservoir. The wellbore was strictly
horizontal and parallel to the x axis. The pressure at the heel of the wellbore is specified. Reservoir and
fluid parameters are listed in Table 1.
Table 1 Basic reservoir and fluid parameters
Parameter
Value
Reservoir dimension in x direction, m
200
Reservoir dimension in y direction, m
100
Permeability in x direction, mD
3000
Permeability in y direction, mD
3000
Reservoir porosity
0.3
Initial reservoir pressure,MPa
10
3
Fluid density, g/cm
0.9
Fluid compressiblitity,Pa-1
1.0×10-9
Wellbore radius, mm
139.7
321
Perforation shot density, shot/m
Perforation channel depth, mm
Fluid viscosity, mPas
Perforation length,m
Drawdown at the hell, Mpa
10
400
50
100
0.5
Fig.3 shows the inflow profile along the well. One can see that, due to the larger drainage area, the
specific inflow (flowrate per unit meter into the wellbore) at the two ends of the well is much higher
than that in the middle. In addition, influenced by the pressure loss in the wellbore, the drawdown at the
heel of the well is higher than that at the toe. Therefore, the inflow rate at the heel of the well is a little
higher than that at the toe.
Specific inflow m3/d/m
20
15
10
5
0
0
20
40
60
Well position m
80
100
Figure 3 Inflow profile along the well
Figure 4 Streamlines around a perforation
Fig.4 shows streamlines around a perforation. The length of the vector is proportional to the flow rate.
One can see that, away from the perforation, the flow streamlines are approximately parallel to the axis
of the perforation. However, around the perforation, the flow streamlines become almost perpendicular
to the axis of the perforation. Besides, the flow rate decreased gradually from the toe to the heel of the
perforation.
Figures 5 shows how the inflow rate changes with well position for different perforation depths. Five
cases were considered. In these cases, all other parameters are fixed except for the perforation depth. In
the first case, the perforation depth was 400mm. In the second case, the perforation depth was 600mm.
In the third case, the perforation depth is 800mm. In the fourth case, the perforation depth was 1000mm.
In the fifth case, the perforation depth is 1200mm. It was found that, with the increase of perforation
depth, there is a slight increase in the specific inflow at the two ends of the well and the well
productivity. Neglecting the impact of formation damage, when the perforation depth is larger than
400mm, the increase of perforation depth has little influence on the inflow profile and productivity of
perforated horizontal wells.
Figures 6 shows how the flow rate changes with well position for different permeability distribution.
Three cases were considered. In these cases, all other parameters are fixed except for the permeability.
In the first case, the permeability increased linearly along the well from 3000mD at the heel to 4500mD
at the toe. In the second case, the permeability along the well kept constant. In the third case, the
permeability decreased linearly along the well from 3000mD at the heel to 1500mD at the toe. It was
found that, when all other parameters are fixed, the specific inflow and the well productivity increase
with the increase of permeability. The permeability heterogeneity has a great influence on the inflow
profile of perforated horizontal wells.
322
Figure 5 Variation of inflow distribution with
perforation depth
Figure 6 Variation of inflow distribution with
permeability
4 Conclusion
A reservoir/wellbore coupling method using a numerical reservoir model and a semi-analytical wellbore
model has been proposed. Simplified treatment of perforations and wellbore was presented. The
wellbore flow model considered the pressure loss inside the horizontal well and the effect of axial influx
at the perforations. An explicit iteration scheme was developed to achieve pressure and mass continuity
at the interface. This model provides an important method for productivity prediction and perforation
design optimization of perforated horizontal wells.
With the proposed model, the inflow performance of perforated horizontal wells was investigated. The
streamlines around the perforation were presented. Sensitivity studies using this model were conducted
to identify the impact of deviation angle and permeability heterogeneity on the inflow profile and
productivity. Based on the results presented in this study, the following conclusions are obtained:
(1) For a perforated horizontal well in a homogeneous reservoir, due to end effects, the specific inflow
(flowrate per unit meter into the wellbore) at the two ends of the well is much higher than that in the
middle. In addition, influenced by the pressure loss in the wellbore, the drawdown at the heel of the
wellbore is higher than that at the toe, the inflow rate at the toe of the wellbore is a little lower than
that at the heel.
(2) Away from the perforation, the flow streamlines are approximately parallel to the axis of the
perforation. However, around the perforation, the flow streamlines become almost perpendicular to
the axis of the perforation. Besides, the flow rate decreased gradually from the toe to the heel of the
perforation.
(3) Specifying the pressure at the heel of the well and neglecting the impact of formation damage, when
the perforation depth is larger than 400mm, both the specific inflow at the ends of the well and the
well productivity increase slightly with the increase of perforation depth, the increase of perforation
depth has little influence on the inflow profile and productivity of perforated horizontal wells.
(4) Specifying the pressure at the heel of the well, both the specific inflow and the well productivity
increase with the increase of permeability. The permeability heterogeneity has a great influence on
the inflow profile and productivity of perforated horizontal wells.
References
[1]. Chen Chongxi. Groundwater Flow Model and Simulation Method in Triple Media of Karstic
Tube-Fissure-Pore. Earth Science—Journal of China University of Geosciences, 1995, 20(4):
361~366(in Chinese)
[2]. Ouyang, L., Aziz, K.. A Simplified Approach to Couple Wellbore Flow and Reservoir Inflow for
323
Arbitrary Well Configurations. Paper SPE48936 presented at the SPE Annual Technical
Conference and Exhibition, New Orleans, United States (Sep. 27-30, 1998)
[3]. Wu Shuhong, Liu Xiange, Guo Shangping. A Simplified Model of Flow in Horizontal Wellbore.
Petroleum Exploration and Development, 1999, 26(4): 64~65(in Chinese)
[4]. Kabir. A., Sanchez, G.. Accurate Inflow Profile Prediction of Horizontal Wells through Coupling
of a Reservoir and a Wellbore Simulator. Paper SPE 119095 presented at the SPE Reservoir
Simulation Symposium, The Woodlands, United States (Feb. 2-4, 2009)
[5]. Yildiz, T. and Ozkan, E., Pressure-Transient Analysis for Perforated Wells. SPE Journal, 1999.
4(2): 167~176.
[6]. Ansah, J., Proett, M. A. and Soliman, M. Y.. Advances in Well Completion Design: A New 3D
Finite-Element Wellbore Inflow Model for Optimizing Performance of Perforated Completions.
Paper SPE73760 presented at the International Symposium and Exhibition on Formation Damage
Control, Lafayette, United States (Feb. 20-21, 2002)
[7]. Yildiz, T.. Inflow Performance Relationship for Perforated Horizontal Wells. SPE Journal, 2004,
9(3): 265~279
[8]. Ozkan, E., Yildiz, T., R. Raghavan. Pressure-Transient Analysis of Perforated Slant and Horizontal
Wells. Paper SPE56421 presented at the SPE Annual Technical Conference and Exhibition,
Houston, United States (Oct. 3-6, 1999)
[9]. Liu Xiangping, Jiang Zhixiang, Jiang Ruyi et al.. Inflow Performance Relationship for Perforated
Horizontal Wells. Petroleum Exploration and Development, 1999, 26(2): 71~78(in Chinese)
[10]. Tang, Y., Ozkan, E., Kelkar, M. et al. Performance of Horizontal Wells Completed with Slotted
Liners and Perforations. Paper SPE65516 presented at the International Conference on Horizontal
Well Technology, Calgary, Canada(Nov. 6-8, 2000)
[11]. Ouyang, L., Arbabi, S., Aziz, K.. General Wellbore Flow Model for Horizontal, Vertical, and
Slanted Well Completions. SPE Journal, 1998, 3(2): 124~133
324