COMPARISON OF VARIOUS TECHINQUES FOR COMPUTING
WELL INDEX
A REPORT SUBMITTED TO THE DEPARTMENT OF PETROLEUM ENGINEERING
OF STANFORD UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE
By Jones Shu
August 2005
iii
I certify that I have read this report and that in my opinion it is fully adequate, in scope and in quality, as partial fulfillment of the degree of Master of Science in Petroleum Engineering.
__________________________________
Prof. Khalid Aziz (Principal Advisor)
v
Abstract
The Well Index (WI) plays a key role in reservoir simulation as it defines the relationship between well pressure and flow rate to reservoir properties and pressure. Currently, there are many models for computing well indices based on different assumptions and well configurations. A clear and complete understanding of these models and restrictions is essential as the application of miscalculated well indices lead to erroneous results rendering the simulation model ineffective as a prediction tool.
This work presents a comparison between three different methods for computing well indices: Peacemans model, the Projection method and a Semi-Analytical approach. This evaluation compares numerical simulation results using WI computed by each method with a semi-analytical reference solution. The well and reservoir models presented in this work include horizontal and vertical wells, 2D and 3D slanted wells, isolated wells and wells near boundaries or other wells. The models are all homogeneous (isotropic and anisotropic) reservoirs with uniform Cartesian grids. All simulations were done with single-phase flow and have closed (no flow) boundary conditions.
For each of these models, the values of the well indices computed by each method were compared. The influence of permeability, grid size, spherical flow, interference and boundary effects was studied.
Peacemans WI led to significant errors for all slanted wells and this was also true where spherical flow or interference of other wells and boundaries is dominant. The Projection method, a practical correction to Peacemans model for deviated wells, provides a reasonable approximation for slanted wells. However, it also leads to significant errors when spherical flow, interference and boundary effects are present. The Semi-Analytical approach computes exact well indices for any type of well in any scenario and their use in simulators exactly reproduce the semi-analytical reference solution.
vii
Acknowledgments
I would like to thank Professor Khalid Aziz, my advisor, who first of all opened his doors to me and brought me to be his student and part of his team. Throughout my two years in Stanford, I had the chance to learn and grow a lot, as an engineer, as a professional, and as a person. I am very thankful to Dr. Aziz for contributing to and being a part of my education.
I am very thankful to Huanquan Pan, who worked with me and supported me during the completion of this work. And special thanks to Dr. Jonathan Holmes who not only contributed to this work but whose enthusiasm has always brought motivation to our group.
Also, I would like to thank Prof. Lou Durlofsky, who has always been a great reference for me and has always given me important support and encouragement.
My love goes to Ftima and Isabella Dias, who while physically distant lived this entire experience with me. For sure my life in Stanford is strongly associated with our history together.
Many thanks to Anson, Gianluca and Lisa, as their friendship and companionship are the greatest gifts I received here in Stanford.
Also, I would like to thank all my friends in Brazil who have cheered for me from start to end, especially Joo Batista Csar Neto and his family, Prof. Denis Schiozer, Dionysio Moriconi, Fome, Marcos Borges and Marina Parahyba. And, of course, all my dear friends in Stanford who walked all the way along with me!
I am grateful to companies supporting Stanford University Petroleum Research Institute program in Reservoir Simulation (SUPRI-B), and Advanced Wells (SUPRI-HW), which made my graduate studies at Stanford possible.
Finally, I would send my love to my parents and sister Brenda, as they for 28 years have stood besides me, loved and cheered for me.
And mostly, I would like to thank God for his love, blessings, care and mercy to me.
ix
Contents
Abstract ............................................................................................................................... v
Acknowledgments............................................................................................................. vii
Contents ............................................................................................................................. ix
List of Figures .................................................................................................................... xi
1. Introduction................................................................................................................. 1
2. Current Methods for Computing Well Index.............................................................. 3
2.1. Peacemans Well Index....................................................................................... 3 2.2. Projection Well Index ......................................................................................... 4 2.3. Semi-Analytical Well Index................................................................................ 6
3. Simulation Results Using Different Well Indices....................................................... 9
3.1. Vertical and Horizontal Wells ............................................................................ 9 3.2. Slanted Wells .................................................................................................... 13 3.3. Partial Penetration............................................................................................. 19 3.4. Interference Among Wells and Boundary Effects ............................................ 20
4. Comparison of Different Well Indices...................................................................... 23
4.1. Spherical Flow .................................................................................................. 23 4.2. Normalization of Well Indices.......................................................................... 24 4.3. Different Permeabilities .................................................................................... 25 4.4. Different Grid Sizes .......................................................................................... 27 4.5. Partial Penetration............................................................................................. 28 4.6. Boundary Effects............................................................................................... 30
5. Conclusions and Future Work .................................................................................. 31
5.1. Conclusions....................................................................................................... 31 5.2. Future Work ...................................................................................................... 32
Nomenclature.................................................................................................................... 33
References......................................................................................................................... 34
xi
List of Figures
Figure 2-1: Peaceman WI Assumptions: Single Isolated Well, Fully Penetrating Grid
Block, Aligned with Grid, Single Phase Radial Flow......................................................... 3
Figure 2-2: (a) Well Trajectory Projected into the Axis, (b) Projection of Well Segments 5 Figure 2-3: Analytical Solution (Well Inflow and Pressure) Coupled with Numerical Grid (Block Pressures) ................................................................................................................ 6 Figure 3-1: Well Flow Rate: Isolated Vertical Well, Aligned with Grid, Fully Penetrating
Isotropic Reservoir............................................................................................................ 10
Figure 3-2: Well Pressure Distribution: Isolated Vertical Well, Aligned with Grid, Fully
Penetrating Isotropic Reservoir......................................................................................... 10
Figure 3-3: Well Flow Rate: Isolated Vertical Well, Aligned with Grid, Partially
Penetrating Isotropic Reservoir......................................................................................... 11
Figure 3-4: Well Pressure Distribution: Isolated Vertical Well, Aligned with Grid,
Partially Penetrating Isotropic Reservoir .......................................................................... 11
Figure 3-5: Flow Rate per Unit Length: Isolated Vertical Well, Aligned with Grid,
Partially Penetrating Anisotropic Reservoir...................................................................... 12
Figure 3-6: Well Pressure Distribution: Isolated Vertical Well, Aligned with Grid,
Partially Penetrating Anisotropic Reservoir...................................................................... 12
Figure 3-7: Flow Rate per Unit Length: Isolated Horizontal Well, Aligned with Grid,
Partially Penetrating Anisotropic Reservoir...................................................................... 13
Figure 3-8: Well Configuration: Horizontally Slanted Well (Deviated in 2D), Cutting 7 Grid Blocks ....................................................................................................................... 14
Figure 3-9: Flow Rate: Isolated Horizontally Slanted (2D) Well, Partially Penetrating Isotropic Reservoir............................................................................................................ 14
Figure 3-10: Flow Rate per Unit Length: Isolated Horizontally Slanted (2D) Well, Partially Penetrating Isotropic Reservoir .......................................................................... 15
Figure 3-11: Flow Rate: Isolated Horizontally Slanted (2D) Well, Partially Penetrating Anisotropic Reservoir ....................................................................................................... 15
xii
Figure 3-12: Flow Rate per Unit Length: Isolated Horizontally Slanted (2D) Well, Partially Penetrating Anisotropic Reservoir...................................................................... 16
Figure 3-13: Flow Rate: Isolated Horizontally and Vertically Slanted (3D) Well, Partially Penetrating Isotropic Reservoir......................................................................................... 17
Figure 3-14: Flow Rate per Unit Length: Isolated Horizontally and Vertically Slanted
(3D) Well, Partially Penetrating Isotropic Reservoir ........................................................ 17 Figure 3-15: Flow Rate: Isolated Horizontally and Vertically Slanted (3D) Well, Partially Penetrating Anisotropic Reservoir .................................................................................... 18
Figure 3-16: Flow Rate per Unit Length: Isolated Horizontally and Vertically (3D) Slanted Well, Partially Penetrating Anisotropic Reservoir............................................... 18
Figure 3-17: Well Pressure: Vertical Well Completed at Single Block, Fading Penetration
........................................................................................................................................... 19
Figure 3-18: Configuration of Horizontal Well (longer well) and Neighboring Vertical Well (shorter well) ............................................................................................................ 20 Figure 3-19: Flow Rate: Horizontal Well under Influence of another Vertical Well
(Producers) ........................................................................................................................ 21 Figure 3-20: Well Pressure: Horizontal Well under Influence of No-Flow Boundary..... 22
Figure 4-1: Spherical Flow Effect in the Semi-Analytical Well Index............................. 23
Figure 4-2: Well Indices Distributed by Respective Well Trajectory Length................... 24 Figure 4-3: Well Indices: Different Permeabilities Change Magnitude but Keeps Same
Pattern ............................................................................................................................... 25
Figure 4-4: Normalized WI of Wells in Different Isotropic Media .................................. 26
Figure 4-5: Normalized WI of Wells in Different Anisotropic Media ............................. 26
Figure 4-6: Normalized WI of Wells in Different Grids Sizes ......................................... 27
Figure 4-7: WI Sensitivity for Different Parameters for Partial Penetration .................... 28
Figure 4-8: Well Indices Relation Regarding Partial Penetration with Varying Parameters
........................................................................................................................................... 29
Figure 4-9: Well Indices Relation Regarding Partial Penetration with Varying Anisotropy
........................................................................................................................................... 29
Figure 4-10: Boundary Effect in the Well Index Value .................................................... 30
1
Chapter 1
1. Introduction
In reservoir simulation, flow models that define the relation between wells and reservoirs play a key role. A thorough understanding of the equations that govern these models is essential to compute correct results, such as inflow and pressure distributions along the wells.
In numerical models, the well pressure is different from the pressure of the grid block containing the well, due to large difference in scale of the well and the grid block. In order to define this relationship, a coefficient known as Well Index or Well Transmissibility is used. This coefficient accounts for the geometric characteristics of the well and the surrounding reservoir properties, as well as any interaction with other wells and boundaries. The Well Index is defined as the ratio of the well flow rate and the difference between the reservoir block and wellbore pressures (Eq. 1-1):
( )wiiw
ii pp
qWI
=
(1-1)
Because of its importance, many techniques for computing the well index have been developed. In this study, three methods have been considered; the classic approach known as Peacemans model [1,2,3], the Projection technique developed by J. Holmes (Schlumberger)[4] and the Semi-Analytical procedure [6,7,8,9,10] developed by the Department of Petroleum Engineering at Stanford University.
This study evaluates the well indices computed by the different methods regarding their performance and applicability in different well configurations and reservoir models. This analysis is done by comparing numerical simulation results using the three different well indices to an analytical reference solution.
This report proceeds as follow. In Chapter 2, different methods for calculating well indices are reviewed. In Chapter 3, the models are compared based on simulation results through sample cases that illustrate the WIs performances in diverse well and reservoir property scenarios. In Chapter 4, an evaluation of the values of the WI themselves is presented and their correlations and main discrepancies outlined. Finally, conclusions and recommendations for future work are listed in Chapter 5.
3
Chapter 2
2. Current Methods for Computing Well Index
2.1. Peacemans Well Index
The Peaceman WI [1,2,3] is the classical technique. All well index calculation techniques, including Peacemans method, are based on single phase flow.
The main assumption of this model is that it is derived for a vertical well in a uniform Cartesian grid, fully penetrating the grid block, with single-phase radial flow and no interaction with boundaries or other wells.
h
h
Figure 2-1: Peaceman WI Assumptions: Single Isolated Well, Fully Penetrating Grid Block, Aligned with Grid, Single Phase Radial Flow
For computing the well index, based on the single-phase steady-state radial flow equation (Eq. 2-1), Peaceman [1] introduced the equivalent well block radius ro, defined as the radial position at which the computed block pressure is equal to the pressure obtained from the analytical radial solution (Eq. 2-2). The equivalent radius is not a physical quantity [3], but rather an intermediate variable that makes the well model (Eq. 2-2) work.
( )
+=
w
ww
r
r
khqprp ln2pi
(2-1)
4
( )
=
w
o
w
ow
r
r
ppkhqln
2pi
(2-2)
To obtain the values of ro, both analytical and numerical solutions were used. For non-square grid blocks and anisotropic permeability, Peaceman [2] defined the WI as:
+
=
sr
r
zkkWI
w
o
yx
ln
2pi (2-3)
Where,
41
41
21
221
221
28.0
+
+
=
y
x
x
y
y
x
x
y
o
kk
kk
ykk
xkk
r (2-4)
This model is the most common and is the standard in commercial simulators.
2.2. Projection Well Index
The Projection WI is based on Peacemans model and therefore is limited by the same assumptions. However, the Projection WI, developed by Jonathan Holmes (Schlumberger) [4], corrects the model for slanted (not aligned with the grid) wells.
In this method, the well trajectory is projected onto three orthogonal axis, as shown in Figure 2-2a. Using the three projected lengths and using Peacemans equation for WI and ro, WI values are calculated for each direction (Eqs. 2-5 and 2-6). The WI for the well segment in the block is the square root of the sum of the squares of these partial well indices (Eq. 2-7)
5
Figure 2-2: (a) Well Trajectory Projected into the Axis, (b) Projection of Well Segments
iw
yo
yzxy
sr
r
LkkWI
+
=
,ln
2pi
iw
xo
xzyx
sr
r
LkkWI
+
=
,ln
2pi
iw
yo
yzxy
sr
r
LkkWI
+
=
,ln
2pi
iw
yo
yzxy
sr
r
LkkWI
+
=
,ln
2pi
iw
xo
xzyx
sr
r
LkkWI
+
=
,ln
2pi
iw
yo
yzxy
sr
r
LkkWI
+
=
,ln
2pi
(2-5)
41
41
21
221
221
,28.0
+
+
=
y
z
z
y
y
z
z
y
xo
kk
kk
zkk
zkk
r41
41
21
221
221
,28.0
+
+
=
z
x
x
z
z
x
x
z
yo
kk
kk
zkk
xkk
r
41
41
21
221
221
,28.0
+
+
=
y
x
x
y
y
x
x
y
zo
kk
kk
ykk
xkk
r
41
41
21
221
221
,28.0
+
+
=
y
z
z
y
y
z
z
y
xo
kk
kk
zkk
zkk
r41
41
21
221
221
,28.0
+
+
=
z
x
x
z
z
x
x
z
yo
kk
kk
zkk
xkk
r
41
41
21
221
221
,28.0
+
+
=
y
x
x
y
y
x
x
y
zo
kk
kk
ykk
xkk
r
(2-6)
222zyx
pj WIWIWIWI ++= (2-7)
In case of segmented wells where there are more than one segments within the same grid block, the projected length for the well index is the sum of the projections of all segments in that direction (Eq. 2-8, Fig. 2-2b).
=jsegment
kjkdirection LL_
,_ (2-8)
The Projection WI approach is part of Schlumberger Schedule tool, a pre-processor of GeoQuest ECLIPSE.
x
z
6
2.3. Semi-Analytical Well Index
The Semi-Analytical WI is obtained using well pressure and inflow distributions calculated semi-analytically and well block pressures obtained from a numerical simulator. The WI can be computed directly using:
( )wiiw
ii pp
qWI
=
(2-9)
The general approach used in this work was developed by many previous researchers at Stanford University: Valvatne [8], Serve [9], Wolfsteiner et al. [6,7] and others. A brief description is given here.
The first step of this framework is to compute accurately well inflows and well pressures for each segment representing the well by applying a semi-analytical procedure based on Greens functions to solve the single phase flow problem (Eq. 2-10). To obtain this reference solution, only the well, not the reservoir, is discretized into well segments. After this solution is obtained for each segment, the reference well flow rate for each block that is intercepted by the well is determined by an intersection algorithm. From these rates, using a single-phase numerical simulator, block pressures for each well block are obtained.
( )t
ck
= . (2-10)
w
iw
iw
jw
j pqpq ,,
blocki
w
iw
i ppq ,w
jw
j pq ,
!"# $
%
w
iw
iw
jw
j pqpq ,,
blocki
w
iw
i ppq ,w
jw
j pq ,
!"# $
%
Figure 2-3: Analytical Solution (Well Inflow and Pressure) Coupled with Numerical Grid (Block Pressures)
7
The Semi-Analytical WI is finally obtained by combining the well rates and pressures, from the semi-analytical solution, and the respective block pressures from the single phase numerical simulation (Eq.2-9). In order to obtain one single WI value for each block, the rates and pressures must be obtained under steady-state or pseudo-steady-state conditions.
In order to include near-well permeability heterogeneities, this framework also includes the s-k* model, creating a representation of heterogeneity through a constant background permeability k* obtained by power averaging or some other upscaling technique and a local skin s.
This allows accurate modeling of wells of any trajectory that intersect any grid arbitrarily. It also accounts for spherical flow at well ends and interaction among wells and boundaries.
This entire framework is implemented in AdWell 2.1, a research simulator developed by the SUPRI-HW team.
9
Chapter 3
3. Simulation Results Using Different Well Indices
In this chapter, the applicability of the previously introduced methods for computing well indices is discussed. Each of these comparisons shows different well configurations and reservoir models, with the idea of analyzing the applicability of each method in each scenario. This analysis is done by comparing the analytical reference solution and numerical simulation results using the three different well index calculation procedures.
The analytic reference solution is provided by AdWell 2.1, a computer program developed by SUPRI-HW for research purpose. It was used to solve the well model semi-analytically (well inflow and pressure distribution). Also, making use of the extended features implemented in AdWell, the Semi-Analytical WI was computed in pseudo-steady state (rate control) conditions. The Projection WI was also calculated by this application. Both of these indices were calculated using the techniques presented in the previous chapter. The numerical simulation results using the different WIs were obtained using GeoQuest ECLIPSE 100 - 2004a, a commercial reservoir simulator.
For each well and reservoir model, four simulation results are compared: ECLIPSE using its default WI (Peacemans WI), ECLIPSE using the Semi-Analytical WI, ECLIPSE using the Projection WI and AdWells analytical reference solution. The well and reservoir models included in this work are horizontal, vertical, 2D and 3D slanted wells, in homogeneous (isotropic and anisotropic) reservoirs with uniform Cartesian grids. All simulations were with single-phase flow and closed boundaries.
3.1. Vertical and Horizontal Wells
In this example, the condition in which Peacemans WI was derived is reproduced. A single isolated vertical well is inserted in an isotropic reservoir, which is discretized by a uniform Cartesian grid. The well is aligned with the grid, fully penetrates the entire reservoir (from top to bottom) and is located away from other boundaries.
The reservoir is 1300 ft x 1300 ft x 60 ft, with a grid of 13 x 13 x 6, permeability equals 200 md. The well radius is 0.25 ft and accounts only for pressure drop due to hydrostatic head. Well is located at the center of the grid in blocks i = 7, j = 7, k = 1-6. The well operates under constant rate control, producing 1500 bbl/day with an initial reservoir pressure of 3000 psi.
10
As Figures 3-1 and 3-2 show, as expected, the results of all WI match perfectly. Because the well fully penetrates the reservoir, only radial flow is present. In this case, the inflow is distributed homogeneously throughout the six blocks (Figure 3-1) while the well pressure varies due to the hydrostatic head only (Figure 3-2). These values were taken on day 100 of the simulation run.
0
50
100
150
200
250
300
0 1 2 3 4 5 6 7Grid Block
Flo
w R
ate
(STB
/da
y)
Eclipse with SA WIEclipse with PJ WIEclipse Default (PM)AdWell
Figure 3-1: Well Flow Rate: Isolated Vertical Well, Aligned with Grid, Fully Penetrating Isotropic Reservoir
600
615
630
645
660
675
690
0 1 2 3 4 5 6 7
Grid Block
Pre
ssu
re (p
sia
)
AdWellEclipse with SA WIEclipse with PJ WI Eclipse Default (PM)
Figure 3-2: Well Pressure Distribution: Isolated Vertical Well, Aligned with Grid, Fully Penetrating Isotropic Reservoir
In this next example, the isolated vertical well is not fully perforating the reservoir. Conditions similar to those in the previous case apply: rectangular reservoir grid (3100 ft x 3100 ft x 165 ft discretized into 31 x 31 x 11 blocks) and isotropic permeability (300 md). The well is aligned with the grid and fully perforates 4 blocks (blocks i = 16, j = 16, k = 4-8).
11
From Figures 3-3 and 3-4, it is clear that Peacemans WI (ECLIPSE Default) and the Projection WI, which has the same value as Peacemans for aligned wells, do not give the correct result since they do not account for spherical flow. The analytical reference solution, in this case, shows a result 30 bbl/day higher in blocks k = 1 and k = 4 (well ends) where spherical flow takes place, and 30 bbl/day lower in the two center well blocks. The Semi-Analytical WI, as expected, matches the analytical solution for both inflow and well pressure distributions. This is true for all scenarios.
450
460
470
480
490
500
510
520
530
540
0 1 2 3 4 5Grid Block
Flo
w Ra
te (S
TB/d
ay)
Eclipse with SA WIEclipse Default (PM)Eclipse with PJ WIAdWell
Figure 3-3: Well Flow Rate: Isolated Vertical Well, Aligned with Grid, Partially Penetrating Isotropic Reservoir
32003210322032303240325032603270328032903300
0 1 2 3 4 5Grid Block
Pres
su
re (ps
ia)
Eclipse with SA WIEclipse Default (PM)Eclipse with PJ WIAdWell
Figure 3-4: Well Pressure Distribution: Isolated Vertical Well, Aligned with Grid, Partially Penetrating Isotropic Reservoir
Using similar well configuration in an anisotropic reservoir (kx = 200md, ky = 200 md, kz = 20 md), Figure 3-5 shows that the anisotropy ratio (kv/kh) of 0.1 decreases the influence of spherical flow. But it still creates a mismatch between Peacemans and Projection WIs with the reference solution (AdWell) and the Semi-Analytical WI.
12
25
26
27
28
29
0 10 20 30 40 50 60 70 80Location Along Well (ft)
Flo
w Ra
te pe
r Le
ngt
h (S
TB/d
ay/le
ngt
h)Eclipse with SA WIEclipse with PJ WIEclipse Default (PM)AdWell
Figure 3-5: Flow Rate per Unit Length: Isolated Vertical Well, Aligned with Grid, Partially Penetrating Anisotropic Reservoir
3050
3060
3070
3080
3090
3100
0 1 2 3 4 5 6
Grid Block
Pres
sure
(ps
ia)
AdWellEclipse with SA WIEclipse with PJ WIEclipse Default (PM)
Figure 3-6: Well Pressure Distribution: Isolated Vertical Well, Aligned with Grid, Partially Penetrating Anisotropic Reservoir
The same behavior as seen in the vertical well examples was also observed for horizontal wells. Figure 3-7 shows an example. This figure shows the results of an isolated horizontal well, aligned with the grid, with no friction or acceleration pressure drop taken into account in the well model, in an anisotropic media (kx = 300 md, ky = 300 md, kz = 20 md) with reservoir dimensions 3100 ft x 3100 ft x 165 ft (i = 31, j = 31, k = 11 blocks). But in this example, the well cuts 10 blocks (i = 16, j = 11-20, k = 6), which is much more than in the previous vertical examples.
13
2.6
2.8
3.0
3.2
3.4
3.6
0 200 400 600 800 1000Location Along Well (ft)
Flo
w R
ate
pe
r Le
ngt
h (S
TB/d
ay/
len
gth)
Eclipse with SA WIEclipse with PJ WIEclipse Default (PM)AdWell
Figure 3-7: Flow Rate per Unit Length: Isolated Horizontal Well, Aligned with Grid, Partially Penetrating Anisotropic Reservoir
Once again deviation from the reference model at the ends of the well due to spherical flow is observed, although the inflow differences are more spread among well blocks.
The well pressures were constant throughout the well since no friction or acceleration was taken into account, and all methods matched each other.
3.2. Slanted Wells
In this section, simulation results of slanted wells wells not aligned with the grid are discussed.
The first example is a horizontally slanted (deviated in 2D) well, cutting through 7 grid blocks (Figure 3-8, i = 10-12, j = 9-13, k = 6). The reservoir is isotropic (k = 200 md) and its dimensions are 2100 ft x 2100 ft x 550 ft (21 x 21 x 11 blocks).
Evaluating the flow rate per block (Figure 3-9), it is evident that Peacemans WI is not suitable for modeling this well configuration. The Projection WI also gives a very similar result and shows that it is a good approximation for slanted wells. The flow rate of each grid block is also related to the well length within the block: where the flow rate is high, the well length in that block is also high.
14
x
y
x
y
Figure 3-8: Well Configuration: Horizontally Slanted Well (Deviated in 2D), Cutting 7 Grid Blocks
0
100
200
300
400
500
600
700
0 1 2 3 4 5 6 7 8
Grid Block
Flo
w R
ate
(S
TB/d
ay)
Eclipse with SA WIEclipse with PJ WIEclipse default (PM)Adwell
Figure 3-9: Flow Rate: Isolated Horizontally Slanted (2D) Well, Partially Penetrating Isotropic Reservoir
Figure 3-10 shows the flow rate per unit length. This plot is appropriate for visualizing the correct and expected results for the well inflow distribution. In this figure, the reference solution and the Semi-Analytical approach give a smooth distribution, while the Projection method is a very good numerical approximation, but Peacemans method gives an unphysical (oscillatory) solution.
Again, the well pressures were constant throughout the well, as no friction or acceleration was taken into account.
15
0
2
4
6
8
10
12
0 100 200 300 400 500 600 700Location along Well
Flo
w R
ate
per
Len
gth
(STB
/day
/ f
t)
Eclipse with SA WIEclipse with PJ WIEclipse default (PM)AdWell
Figure 3-10: Flow Rate per Unit Length: Isolated Horizontally Slanted (2D) Well, Partially Penetrating Isotropic Reservoir
In an anisotropic case (kx = 600md, ky = 300md, kz = 60 md) with same well and grid configuration as the previous example, simulation results (Figures 3-11, 3-12) were similar to the isotropic case but with some important differences.
0
100
200
300
400
500
600
700
0 1 2 3 4 5 6 7 8Grid Block
Flo
w R
ate
(STB
/day
)
Eclipse with SA WIEclipse with PJ WIEclipse Default (PM)AdWell
Figure 3-11: Flow Rate: Isolated Horizontally Slanted (2D) Well, Partially Penetrating Anisotropic Reservoir
16
Peacemans method, although already giving unrealistic results due to the deviated trajectory, was further impacted by reservoir anisotropy and the fact that the well is not aligned with the permeability tensor. (Eq. 2-3 and 2-4). On the other hand, the projection method is not influenced by the anisotropy, as it weights and calculates the different partial WI for each direction, making it independent of the anisotropy.
It is also apparent that the spherical flow causes a discrepancy in inflow distribution between the Projection WI and the reference solution. The well blocks at the extremes have lower flow rates and are compensated for by higher rates from the intermediate well blocks.
0
2
4
6
8
10
12
14
0 100 200 300 400 500 600 700Location Along Well (ft)
Flo
w R
ate
per
Len
gth
(STB
/day
/ f
t) Eclipse with SA WIEclipse with PJ WIEclipse default (PM)AdWell
Figure 3-12: Flow Rate per Unit Length: Isolated Horizontally Slanted (2D) Well, Partially Penetrating Anisotropic Reservoir
Slanted wells in 3D (deviated both vertically and horizontally) showed the same behavior as the ones in 2D (deviated vertically or horizontally). Figure 3-13 shows the flow rate per well block for a slanted well that cuts 10 blocks (i = 10-12, j = 9-12, k = 9-13) in an isotropic reservoir (k = 100 md) with a uniform Cartesian grid (2100 ft x 2100 ft x 1050 ft, 21 x 21 x 21 blocks). Figure 3-14 shows the flow rate per unit length for this well. As observed in the 2D cases, the Projection method provides a good approximation of the well index for modeling deviated wells, although the end effect is not reproduced.
17
0
50
100
150
200
250
300
350
400
450
0 2 4 6 8 10 12Grid Block
Flo
w R
ate
(STB
/day
)
Eclipse with SA WIEclipse with PJ WIEclipse default (PM)AdWell
Figure 3-13: Flow Rate: Isolated Horizontally and Vertically Slanted (3D) Well, Partially Penetrating Isotropic Reservoir
-
1.0
2.0
3.0
4.0
5.0
6.0
7.0
0 2 4 6 8 10Location at Well (ft)
Flo
w Ra
te pe
r Le
ngt
h(STB
/da
y / f
t)
Eclipse with SA WIEclipse with PJ WIEclipse default (PM)AdWell
Figure 3-14: Flow Rate per Unit Length: Isolated Horizontally and Vertically Slanted (3D) Well, Partially Penetrating Isotropic Reservoir
Furthermore, 3D deviated wells in anisotropic reservoirs also gave similar results to 2D cases, with Peacemans WI being dependent on the well and permeability tensor orientation while the other two techniques are not. Figure 3-15 and 3-16 show flow rate
18
distributions of an isolated well with same configuration as the previous example and permeability field of kx = 100 md, ky = 100 md, kz = 50 md.
0
50
100
150
200
250
300
350
400
450
0 2 4 6 8 10Grid Block
Flo
w R
ate
(STB
/day
)
Eclipse with SA WI
Eclipse with PJ WI
Eclipse default (PM)AdWell
Figure 3-15: Flow Rate: Isolated Horizontally and Vertically Slanted (3D) Well, Partially Penetrating Anisotropic Reservoir
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
0 200 400 600 800 1000Location at Well (ft)
Flo
w Ra
te pe
r Le
ngt
h (S
TB/d
ay/ft
)
Eclipse with SA WIEclipse default (PM)Eclipse with Proj WIAdwell
Figure 3-16: Flow Rate per Unit Length: Isolated Horizontally and Vertically (3D) Slanted Well, Partially Penetrating Anisotropic Reservoir
19
3.3. Partial Penetration
Several simulation runs were performed in order to investigate the performance of different well indices with respect to partial penetration of wells in grid blocks. One example is presented in this section to illustrate the results.
In this example, an anisotropic, single-layered reservoir was used. Its dimensions are 2100ft x 2100ft x 20ft (21 x 21 x 1 blocks) and permeability field kx = 400 md, ky = 400 md and kz = 100 md. An isolated vertical well completed in a single block (at center of the grid, i = 11, j = 11, k = 1) was simulated for a range of penetration of 0% to 100%. The well operated under flow rate control (1500 bbl/day) and the different output well pressures are compared (day 100, Figure 3-17).
It is observed that for the Semi-Analytical WI, as the well penetration is extrapolated to zero, the well pressure also approaches zero, just as expected. On the other hand, the Projection WI shows the same behavior as the Semi-Analytical WI, but results in a faster pressure drop as the penetration decreases. The differences between the methods increase as the penetration goes lower than 50%. For example, at 10% penetration the well pressure difference is 650 psi, which is significant. It is also noticed that the well pressure, given by the Projection WI decreases faster than expected as it extrapolates to a well pressure of 0 psi at a penetration of 5% and not 0%.
These differences between the results of the Projection method and the Semi-Analytical method are due to the fact that spherical flow is taken into account by the Semi-Analytical WI but not by the Projection method. This effect becomes larger as the penetration becomes smaller.
0
400
800
1200
1600
2000
0% 20% 40% 60% 80% 100%Penetration
Wel
l Pre
ssu
re (ps
ia)
Eclipse with PM WIEclipse with PJ WIEclipse with SA WIAdWell
Figure 3-17: Well Pressure: Vertical Well Completed at Single Block, Fading Penetration
20
Peacemans WI was kept constant for all partial penetrations. No multipliers were used to weight it. By weighting it with the penetration length, it would give the same value as the Projection WI
These simulation results show that, while the Semi-Analytical WI gives the expected solution, the Projection WI gives a faster pressure because it neglects spherical flow.
3.4. Interference Among Wells and Boundary Effects
Another relevant effect for well modeling is interference of other wells and boundaries. In this section, examples of each are presented.
Figure 3-18 illustrates a horizontal well, which fully penetrates the reservoir (no end effect at extremes) and is influenced by a vertical producer which is located 3 blocks away from its center.
Figure 3-18: Configuration of Horizontal Well (longer well) and Neighboring Vertical Well (shorter well)
The reservoir grid is 700 ft x 2100 ft x 100 ft (7 x 21 x 5 blocks) and its permeability field is kx = 200 md, ky = 500 md, kz = 30 md. The horizontal well is aligned with the grid and is completed in 7 blocks (i = 1-7, j = 11, k = 3) while the vertical well is completed in 5 blocks (i = 4, j = 14, k = 1-5).
The simulation results for the horizontal well (Figure 3-19) show that, although Peacemans method does not take account for the interference between these wells, the block pressures most affected by the depletion of both producers do influence inflow distribution. There is a mismatch between the Projection WI results and the analytical reference solution, mostly where the wells are closer (central block, 300 ft, 3 blocks apart).
21
390
400
410
420
430
440
450
0 100 200 300 400 500 600 700Location Along Well (ft)
Inflo
w pe
r Le
ngt
h (S
TB/d
ay/ft
) Eclipse with PM WIEclipse with PJ WIEclipse with SA WIAdWell
+
Figure 3-19: Flow Rate: Horizontal Well under Influence of another Vertical Well (Producers)
Another example of interference is regarding boundaries. In the next example, a horizontal well (600ft long, cuts 6 blocks) is placed close (150 ft, i = 2, j = 1-6, k = 8) to a no-flow boundary. The reservoir is 3100 ft x 600 ft x 300 ft (31 x 6 x 15 blocks) and its permeability field is kx = 600 md, ky = 300 md and kz = 30 md. Figure 3-20 shows the well pressure changes as the well is moved closer and closer to the boundary (moves from i = 2 to i = 1, from 150ft progressively to 5 ft from the boundary) .
There are significant differences between Peacemans WI and the analytical solution when the well gets very close to the boundary (in this case, less than 50 ft). Peacemans and the Projection methods overestimate the well pressures for these scenarios.
22
2482
2486
2490
2494
2498
0 10 20 30 40 50 60 70Distance from Boundary (ft)
Wel
l Pre
ssu
re (p
sia
)
Eclipse with PM WIEclipse with PJ WIEclipse with SA WIAdWell
Figure 3-20: Well Pressure: Horizontal Well under Influence of No-Flow Boundary
23
Chapter 4
4. Comparison of Different Well Indices
In this chapter, the values of well indices computed by the different methods are compared. The purpose is to investigate how these values are related to each other, rather than looking at their simulation results as in the previous chapter. The objective is to see whether there is any correlation between the Semi-Analytical and Projection WI.
To do this, first a reference to normalize the well indices was selected. After normalizing, well indices were computed and compared for various scenarios. The influence of parameters and other effects, such as permeability, grid size and spherical flow were studied and are presented here.
4.1. Spherical Flow
Spherical flow plays a key role when modeling wells as seen in Chapter 3. In this example, well indices of an isolated horizontal well aligned with the grid (550 ft x 1550 ft x 110 ft, 11 x 31 x 11 blocks) are presented (Figure 4-1).
10.0
10.5
11.0
11.5
0 1 2 3 4 5 6 7 8 9 10 11 12Grid Block
SA W
I
1 block3 blocks5 blocks7 blocks9 blocks11 blocks
Figure 4-1: Spherical Flow Effect in the Semi-Analytical Well Index
The well length faded from penetrating the entire reservoir (11 blocks, i = 1-11, j = 16, k = 6) to only penetrating the center (1 block, i = 6, j = 16, k = 6). In the fully penetrating
24
case (11 blocks), it is seen that only radial flow is present. The Semi-Analytical WI is the same as the Projection WI along the entire well. For wells partially penetrating the reservoir, Figure 4-1 shows how the spherical flow impacts the well index at the well extremes, giving it higher values. In this example, it increased the WI values by 4% to 12%.
From this comparison, it is clear that spherical flow should be considered for the correct modeling of wells, mostly when short wells are completed in just few grid blocks. Moreover, while the Projection and Peacemans WI ignore spherical flow, the Semi-Analytical WI correctly captures this effect.
4.2. Normalization of Well Indices
A normalization procedure is used to better compare the well indices for diverse scenarios and parameters, such as permeability and grid sizes. The idea is to create a scale that gives results of the same magnitude, independent of the well and reservoir parameter values.
A pattern was observed in plots of the value of well indices vs. the well trajectory length in the respective block. Figure 4-2 shows this pattern for a long horizontal deviated well (4500ft long, cuts 61 grid blocks, i = 4-41, j = 13-34, k = 17)) in an anisotropic reservoir (4500 ft x 4500 ft x 175 ft, 45 x 45 x 33 blocks, kx = 200 md, ky = 200 md and kz = 50 md).
0
5
10
15
20
0 20 40 60 80 100 120Completion length
We
ll In
dex
PM WIPJ WISA WI
Figure 4-2: Well Indices Distributed by Respective Well Trajectory Length
Peacemans WI is constant and independent of the well trajectory length as it is always calculated considering a fully penetrated block, even when it is not. The Projection WI, on the other hand, shows a linear relationship, that would start from the origin, cross
25
Peacemans value at exactly 100% and continue until the biggest trajectory length case (the possible maximum being the 3D diagonal). The Semi-Analytical WI, however, shows a distinct slightly concave curve, resulting in values lower than those calculated for the Projection WI. In this case, the exception comes from the WI of the blocks at the well ends, where spherical flow is dominant and the WI value is higher than the pattern for radial flow (outlier point in SA WI curve). It is important to emphasize that the points in Figures 4-2 and 4-3 are not ordered by location in the well, but rather by the well trajectory length in each grid block.
0
10
20
30
40
50
60
70
80
0 20 40 60 80 100 120Completion length in cell
We
ll In
dex
kx200 ky200 kz50 kx200 ky200 kz200 kx400 ky400 kz400 kx1000 ky1000 kz50
Figure 4-3: Well Indices: Different Permeabilities Change Magnitude but Keeps Same Pattern
Based on these observations, it was decided to normalize the WIs by taking ratios of the Semi-Analytical WIs to Peacemans WI (SAWI / PMWI) and the Projection WI to Peacemans WI (PJWI / PMWI).
4.3. Different Permeabilities
In this comparison, several wells were modeled with different reservoir permeability values.
For isotropic reservoirs, it was observed that the normalized WIs for various permeabilities collapse to a single curve. Figure 4-4 shows results of 33 horizontally slanted wells in 5 different grids and different isotropic media (permeabilities varying from 30 md to 1000 md). From these results, it is apparent that the Semi-Analytical WI and the Projection WI keep the same relation, independent of the permeability values for isotropic cases.
26
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2( PJ WI / PM WI)
(SA
WI /
PM
W
I )
Figure 4-4: Normalized WI of Wells in Different Isotropic Media
Moreover, if the reservoir is anisotropic, the ratio of the well indices changes. Figure 4-5 shows a horizontally deviated well, cutting 36 grid blocks, in different anisotropic media. The results show that although the same pattern is maintained, the values of the normalized WI change.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8( PJ WI / PM WI )
(SA
W
I / PM
W
I )
k 100 x 300 x 30k 100 x 200 x 50k 200 x 400 x 50k 200 x 600 x 50k 200 x 900 x 30k 300 x 100 x 30k 400 x 200 x 50k 500 x 300 x 50k 1000 x 500 x 50
Figure 4-5: Normalized WI of Wells in Different Anisotropic Media
27
It is also observed that for scenarios where the anisotropic ratio for the wells direction (horizontal well, x-y direction in this example) is the same (i.e. k = 100 x 200 x 50 md and k = 200 x 400 x 50 md), SAWI/PMWI and PJWI/PMWI keep the same relationship. This indicates that the anisotropic ratio impacts the relation between the Semi-Analytical WI and the Projection WI. This influence increases for PJWI/PMWI larger than 0.7.
4.4. Different Grid Sizes
Another important geometric property is the grid size. Here, well indices were computed for several wells and reservoirs under diverse course and fine grids. The block sizes varied from 20 to 200 ft, in uniform Cartesian grids. Figure 4-6 shows results from 16 different horizontally deviated wells in different reservoir grids.
From Figure 4-6 it is observed that the grid does not influence the relation between the Semi-Analytical WI and the Projection WI. The SAWI/PMWI and the PJWI/PMWI keep similar patterns for the cases studied, except that there is more scatter among the SAWI/PMWI results for PJWI/PMWI greater than 0.7.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.2 0.4 0.6 0.8 1 1.2 1.4( PJ WI / PM WI)
(SA
WI /
PM
W
I )
Figure 4-6: Normalized WI of Wells in Different Grids Sizes
28
4.5. Partial Penetration
The influence of partial penetration in the relationship between different WI values was also studied. For this, well indices were calculated by using different methods for vertical wells aligned with the grid, where its grid block penetration varied from 0 to 100%.
It is observed that the semi-analytical technique gives a higher WI value than the projection method for all cases (Figure 4-7). This is expected since the Semi-Analytical WI reproduces the well end effect of spherical flow, while the Projection WI does not.
02468
101214161820
0% 20% 40% 60% 80% 100%Penetration
Wel
l In
dex
SA WIPJ WI
Nx 15 Ny 35 rw = 0.2 k= 600
rw = 0.35
base case
k= 100
&'
"'(((((
'(( '(( '((
)#'(*+
Figure 4-7: WI Sensitivity for Different Parameters for Partial Penetration
Also, as shown in Figure 4-7, even when geometric parameters are changed such as well radius, grid size or permeability, the absolute WI value varies but not the pattern. This suggests that a normalization of the WI values would be appropriate for this comparison.
Comparing normalized WI values as before (SAWI/PMWI and PJWI/PMWI), it becomes clear that there is a strong relation between these two well indices with respect to partial penetration. Figure 4-8 shows a comparison of the SAWI/PMWI and PJWI/PMWI for varying parameters, with all cases aligning along the same curve. The largest difference is around 30%-70% penetration, where the SAWI/PMWI and PJWI/PMWI are about 20% different.
One property, however, made a significant difference: the anisotropy ratio. Figure 4-9 shows some normalized well indices and how they differ for various scenarios. The variation comes from the influence of spherical flow on the model.
Anisotropic ratios that promote spherical rather than radial flow give a larger difference between the Semi-Analytical WI to Projection WI.
29
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0(PJ WI / PM WI)
(SA
WI /
PM
W
I)k 100k 500k 1000N 21 x 15N 15 x 15N 15 x 35rw 0.2rw 0.35N 15 x 35, rw 0.2, k 600k 200, N 21 x 21, rw 0.25
Figure 4-8: Well Indices Relation Regarding Partial Penetration with Varying Parameters
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0(PJ WI / PM WI)
(SA
WI /
PM
W
I)
k 50 x 200 x 200k 200 x 100 x 200k 500 x 500 x 50k 1000 x 50 x 300k 1000 x 300 x 300
Figure 4-9: Well Indices Relation Regarding Partial Penetration with Varying Anisotropy
30
4.6. Boundary Effects
The Semi-Analytical WI, as shown in Chapter 3, captures the influence of boundary effects. Here, different well indices are compared for cases where the distance of the well to the boundary varies.
Figure 4-10 shows the well indices of a vertical well in a reservoir 1050ft x 1050ft x 50ft (21 x 21 x 3 blocks) with a permeability field of 400 md x 400 md x 20 md. As the well was moved from the center blocks (i = 11, j = 11, k = 1-3) towards a corner (i = 1, j = 21, k = 1-3), the Semi-Analytical WI only started to change as it got half-block (25ft) from the boundary.
The projection and Peacemans methods are not influenced by the boundary effects and therefore their values do not vary with distance to the boundary. However, the Semi-Analytical WI decreases significantly as it gets closer to the boundary.
0
1
2
3
4
5
6
7
8
0 5 10 15 20 25 30 35 40Distance from Boundary (ft)
Wel
l In
dex
SA WI
PJ WI
Figure 4-10: Boundary Effect in the Well Index Value
31
Chapter 5
5. Conclusions and Future Work
5.1. Conclusions
In this work, three different methods for computing well index - Peacemans model, the Projection method and the Semi-Analytical approach are compared for different well configurations and reservoir models. The well and reservoir models presented in this work included horizontal, vertical, 2D and 3D slanted wells, in homogeneous (isotropic and anisotropic) reservoirs with uniform Cartesian grids.
The values of the well indices computed by each method were also compared with the goal of finding whether there is any correlation between the Semi-Analytical and Projection WI. The influence of various parameters, such as permeability, grid size and proximity to the boundaries were studied.
The following main conclusions can be drawn from this work:
- Use of Peacemans method can lead to significant errors for slanted wells and when spherical flow or interference of other wells and boundaries are dominant.
- The Projection method is a practical correction to Peacemans model for slanted wells. However, this technique breaks down when spherical flow is present (i.e. block at well ends). It underestimates the true WI where spherical flow is present and overestimates it elsewhere. Also, it does not consider interference and boundary effects, which becomes critical when the well is very close (half block away) to a boundary.
- The Semi-Analytical approach is appropriate for computing exact well indices for any type of well.
- By comparing normalized values of the Semi-Analytical and Projection WI, it was observed that there is a correlation between them, unless spherical flow at well ends is present or the well interferes with other wells or boundaries.
32
5.2. Future Work
This study suggests that there is a correlation between the Semi-Analytical WI and the Projection well index. It may be possible to derive a correction factor that can be applied to the Projection WI to further improve this numerical approximation.
Also, further tests should be performed with more complicated and real-field scenarios, including non-uniform grids, multi-lateral wells and heterogeneous reservoirs. Other methods for computing the well index for slanted wells or partial penetration could also be included in the comparison, such as methods proposed by Cinco-Ley and Ramey [17] and Babu and Odeh [18].
33
Nomenclature
c
Compressibility, 1/psi h
Height, ft k
Permeability, md L
Length, ft p Pressure, psi pi Block Pressure, psi pw Well Pressure, psi PJWI Projected Well Index PMWI Peaceman Well Index qw
Well Flow Rate, bbl/d r Radius, ft ro effective well-block radius, ft rw Wellbore Radius, ft s
Skin SAWI Semi-Analytical Well Index t
Time, days x,y,z
Coordinate, ft WI
Well Index unit
Viscosity, cp Potential, psi
Porosity
Subscripts/Superscripts i Block j Segment x,y,z
Coordinate w Well
34
References
1. Peaceman, D.W.: Interpretation of Well-Block Pressures in Numerical Reservoir Simulation, paper SPE 6893, presented at the SPE-AIME 52nd Annual Fall Technical Conference and Exhibition, Denver, Oct 9-12, 1977
2. Peaceman, D.W.: Interpretation of Well-Block Pressures in Numerical Reservoir Simulation With Nonsquare Grid Blocks and Anisotropic Permeability, paper SPE 10528, presented at the 1982 SPE Symposium on Reservoir Simulation, New Orleans, Jan 31-Feb 3
3. Peaceman, D.W.: Interpretation of Wellblock Pressures in Numerical Reservoir Simulation: Part3 Off-Center and Multiple Wells Within a Wellblock, paper SPE 16976, presented at the 1987 SPE Annual Technical Conference and Exhibition, Dallas, Sept 27-30
4. Schlumberger ECLIPSE: Schedule User Guide 2004A, Chapter 6 Technical Description (2004)
5. Schlumberger ECLIPSE: ECLIPSE Technical Description 2004A, Chapter 64 Well Inflow Performance (2004)
6. Wolfsteiner, C., Aziz, K. and Durlofsky, L. J.: Modeling Conventional and Non-Conventional Wells, presented at Sixth International Forum on Reservoir Simulation, Hof/Salzburg, Austria, Sept 3-7 2001
7. Wolfsteiner, C., Durlofsky, L. J. and Aziz, K.: Calculation of Well Index for Nonconventional Wells on Arbitrary Grids, Computational Geosciences, 7, 61-82, 2003
8. Valvatne, P.H.: A Framework for Modeling Complex Well Configurations, Masters Report, Stanford University, 2000
9. Serve, J.: An Enhanced Framework for Modeling Complex Well Configurations, Masters Report, Stanford University, 2002
10. Maizeret, P.D.: Well Indices for Non-Conventional Wells, Masters Report, Stanford University, 1996
11. Williamson, A.S. and Chappelear, J.E.: Representing Wells in Numerical Reservoir Simulation: Part 1 Theory, paper SPE 7697, presented at the SPE Fifth Symposium on Reservoir Simulation, Denver, Jan31-Feb 2, 1979
12. Peaceman, D.W.: Representation of a Horizontal Well in Numerical Reservoir Simulation, paper SPE 21217 (1991), Advanced Technology Series, Vol1, No.1
13. Aziz, K., Durlofsky, L.J. and Gerritsen, M.: Notes for Petroleum Reservoir Simulation, Stanford University, 2004
35
14. Aziz, K. and Settari, A.: Petroleum Reservoir Simulation, Applied Science Publisher, 1979
15. Pan, H.Q.: AdWell 2.1 User Manual, Stanford University, 2005 16. Schlumberger ECLIPSE: ECLIPSE Reference Manual 2004A, 2004 17. Cinco-Ley, H. and Ramey Jr., H. J.: Pseudo-Skin Factors for Partially-
Penetrating Directionally-Drilled Wells, paper SPE 5589, presented at the SPE_AIME 50th Annual Fall Meeting, Dallas, Sept 28 Oct 1, 1975
18. Babu, D.K. and Odeh, A.S.: Productivity of a Horizontal Well, paper SPE 18298, presented at 1988 SPE Annual Technical Conference and Exhibition, Houston, Oct 2-5
