Manual Perform

PERFORM 4.50

Well PERFORMance Analysis™ Technical Reference Manual

© Copyright 2002, Petroleum Information/Dwights LLC d/b/a IHS Energy Group. All rights reserved.

Copyright (c) 2002, Petroleum Information/Dwights LLC d/b/a IHS Energy Group. All rights reserved. The accompanying software may incorporate certain programs or applications, which are the copyrighted property of third parties.

Except as specifically authorized in the accompanying documentation, the software and this documentation may not be copied, reproduced, displayed, translated, transferred to any electronic medium or machine-readable format, transcribed, or stored in a retrieval system, in any form or by any means, whether electronic, mechanical, magnetic, optical, manual or otherwise, without the prior written permission of IHS Energy Group. IHS ENERGY GROUP MAKES NO REPRESENTATIONS OR WARRANTIES WITH RESPECT TO THE CONTENTS OF THE SOFTWARE AND/OR DOCUMENTATION, WHICH ARE PROVIDED "AS IS." IHS ENERGY GROUP SPECIFICALLY DISCLAIMS AND NEGATES ANY AND ALL EXPRESSED OR IMPLIED WARRANTIES, INCLUDING THE WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE, MERCHANTABILITY AND NONINFRINGEMENT.

Well PERFORMance Analysis is a trademark of Petroleum Information/Dwights LLC. (Certain product names used herein are the trade names of trademarks of their respective companies. Mention of these products is for informational purposes and constitutes neither an endorsement nor recommendation by IHS Energy Group. It is not IHS Energy Group's intent to use any of these trade names or trademarks generically and the reader is cautioned to investigate all claimed trademark rights before using any of such names or marks.)

©Copyright 2002, Petroleum Information/Dwights LLC d/b/a IHS Energy Group. All rights reserved. iii

Table of Contents

Table of Contents ...............................................................................................................iii

Table of Figures .................................................................................................................vi

1 System Analysis Overview.........................................................................................1

Nodal Analysis ...................................................................................................................2

2 Using System Analysis................................................................................................5

General Analysis Procedure ................................................................................................5

Applying System Analysis ...................................................................................................7

Reservoir Skin....................................................................................................................9

Completion Effects............................................................................................................10 Differential Graph...........................................................................................................11 Perforation Shot Density.................................................................................................12 Perforation Interval.........................................................................................................13

Tubing Size.......................................................................................................................14

Wellhead or Separator Pressure........................................................................................16

3 Reservoir Component ...............................................................................................19

Vertical IPR Types ...........................................................................................................20 User Enters PI ...............................................................................................................20 Vogel/Harrison (1968) ...................................................................................................21 Darcy ............................................................................................................................24 Jones et al. (1976) .........................................................................................................30 Jones—4-Point Test and Jones—Enter a and b..............................................................33 Back Pressure Eq (1930) and Back Pressure—4-Point Test...........................................34 Backpressure Four-Point Test (gas wells only)................................................................35 Transient Flow Equation.................................................................................................36 Fractured Well...............................................................................................................38 Datafile—2 Col ASCII ..................................................................................................43

Horizontally Completed Wells ...........................................................................................43

Horizontal IPR Types........................................................................................................48 Giger et al. (1984)..........................................................................................................49 Economides et al. (1991) ...............................................................................................52 Joshi (1988)...................................................................................................................54 Renard and Dupuy (1991)..............................................................................................57 Kuchuk (1988) ..............................................................................................................59

Table of Contents PERFORM Technical Reference Manual

iv ©Copyright 2002, Petroleum Information/Dwights LLC d/b/a IHS Energy Group. All rights reserved.

Babu and Odeh (1989) ..................................................................................................62 Goode and Thambynaya (1987).....................................................................................66

4 Completion Component.............................................................................................69

Open Hole Completion.....................................................................................................69

Open Perforation Completion............................................................................................70

Stable Perforation Completion...........................................................................................73

Collapsed Perforation Completion.....................................................................................76

Gravel Pack Completion...................................................................................................79 Gravel Pack Beta Turbulence Factor ..............................................................................81

Gravel Pack Open Hole Completion.................................................................................82

Gravel Pack Open Perforation Completion........................................................................83

Gravel Pack Stable Perforation Completion.......................................................................83

Gravel Pack Collapsed Perforation Completion.................................................................83

5 Wellbore and Flowline...............................................................................................85

Oil Well Vertical Flow......................................................................................................86 Category A....................................................................................................................87 Category B....................................................................................................................87 Category C....................................................................................................................88

Gas Well Vertical Flow.....................................................................................................90

Oil Well Horizontal Flow...................................................................................................91

Gas Well Horizontal Flow.................................................................................................92

Flow Through Restrictions.................................................................................................93 Critical Flow..................................................................................................................94 Subcritical Flow.............................................................................................................94 Critical and/or Subcritical Flow.......................................................................................97

Maximum Erosional and Minimum Unloading Velocity.....................................................101 Maximum Erosional Rate..............................................................................................102 Minimum Unloading Rate .............................................................................................103

Heat Transfer..................................................................................................................104 Linear Temperature Gradient........................................................................................104 Temperature Survey.....................................................................................................105 Heat Transfer Correlation.............................................................................................105 User-Entered Heat Transfer Coefficients.......................................................................106

Flow Assurance..............................................................................................................106 Scale: Oddo-Tomson method.......................................................................................106

PERFORM Technical Reference Manual Table of Contents

©Copyright 2002, Petroleum Information/Dwights LLC d/b/a IHS Energy Group. All rights reserved. v

Wellbore Deviation.........................................................................................................107 Example 1....................................................................................................................108 Example 2....................................................................................................................109 Example 3....................................................................................................................109

6 Downhole Network..................................................................................................111

References ......................................................................................................................115

Index................................................................................................................................121

©Copyright 2002, Petroleum Information/Dwights LLC d/b/a IHS Energy Group. All rights reserved. vi

Table of Figures Figure 1.1: Producing System....................................................................................................................................................... 1 Figure 1.2: Nodal Plot..................................................................................................................................................................... 3 Figure 2.1: System Analysis Plot with Multiple Conditions.................................................................................................... 6 Figure 2.2: Gradient Curves........................................................................................................................................................... 8 Figure 2.3: Effect of Formation Skin............................................................................................................................................. 9 Figure 2.4: Inflow Sensitivity on Skin........................................................................................................................................ 10 Figure 2.5: Differential Graph...................................................................................................................................................... 11 Figure 2.6: Effect of Perforation Shot Density (SPF)............................................................................................................... 12 Figure 2.7: Inflow Sensitivity on Perforation Shot Density (SPF)......................................................................................... 12 Figure 2.8: Effect of Perforation Interval................................................................................................................................... 13 Figure 2.9: Inflow Sensitivity on Perforation Interval............................................................................................................. 14 Figure 2.10: Effect of Tubing Size .............................................................................................................................................. 15 Figure 2.11: Outflow Sensitivity on Tubing Size ..................................................................................................................... 15 Figure 2.12: Effect of Wellhead Pressure .................................................................................................................................. 16 Figure 2.13: Outflow Sensitivity on Wellhead Pressure ......................................................................................................... 17 Figure 3.1: Reservoir Component............................................................................................................................................... 19 Figure 3.2: User Enters PI............................................................................................................................................................ 20 Figure 3.3: Vogel Solution Gas Drive with Flow Efficiency.................................................................................................... 23 Figure 3.4: Square Reservoir....................................................................................................................................................... 38 Figure 3.5: Horizontally Completed Well .................................................................................................................................. 44 Figure 3.6: Schema for Giger, Joshi, Renard & Dupuy, and Economides correlations ...................................................... 49 Figure 3.7: Schema for Kuchuk and Babu & Odeh correlations............................................................................................ 59 Figure 3.8: Schema for Goode & Thambynaya correlation .................................................................................................... 66 Figure 4.1: Open Hole Completion............................................................................................................................................. 69 Figure 4.2: Open Perforation Completion.................................................................................................................................. 70 Figure 4.3: Open Perforation....................................................................................................................................................... 70 Figure 4.4: Collapsed Perforation (Spherical Flow Model)..................................................................................................... 76 Figure 4.5: Gravel Pack Schematic.............................................................................................................................................. 79 Figure 6.1: Multilayer................................................................................................................................................................. 111 Figure 6.2: Multilateral............................................................................................................................................................... 112

©Copyright 2002, Petroleum Information/Dwights LLC d/b/a IHS Energy Group. All rights reserved. 1

1 System Analysis Overview

The primary objective of the system analysis technique is to maximize well productivity by analyzing and optimizing the complete producing well system. The analysis can lead to increased profitability from oil and gas investments by improving completion design, increasing well productivity, and increasing producing efficiency.

System analysis is essentially a simulator of the producing well system. The system, illustrated in Figure 1.1, includes flow between the reservoir and the wellhead (separator if a flowline is included), and contains the following components:

• Flow through the reservoir to the sandface

• Flow through the completion

• Flow through the bottomhole restrictions

• Flow through the tubing

• Flow through the surface flowline restrictions

• Flow through the flowline into the separator

Figure 1.1: Producing System

System Analysis Overview PERFORM Technical Reference Manual

2 ©Copyright 2002, Petroleum Information/Dwights LLC d/b/a IHS Energy Group. All rights reserved.

As system analysis simulates the entire system, it models each component within the system using equations or correlations to determine the pressure loss through the component as a function of flow rate. The total pressure loss through the system for a given flow rate is the summation of the pressure losses through all components. Minimizing pressure loss in individual components within the system results in less overall pressure loss and increased flow rate from a well.

The total pressure loss is ultimately realized as the overall difference between average reservoir pressure, Pr, and the wellhead or separator pressure, Pwh or Psep. The average reservoir pressure and wellhead or separator pressure constitute the endpoints of the system (inlet and outlet), and are the only pressures in the system that do not vary with flow rate.

Nodal Analysis

System analysis analyzes the entire system by focusing on one point within the series of components. This point generally is referred to as a node, hence the term nodal analysis. The final solution is independent of the location of the node.

For manual calculations, the primary interest of the application generally dictates the location of the node. For example, if the main interest is an investigation of the effects of the components near the surface (such as flowline or surface choke), then the node is chosen at the wellhead or separator. If the effects of the downhole components are the primary interest (such as the bottomhole flowing pressure), then the node is chosen at downhole.

In PERFORM, you can use a sensitization technique that allows you to see the effects of changing parameters. In this way, you can usually choose the node at a point inside the wellbore directly adjacent to the perforations. This point is designated as wellbore flowing bottomhole pressure, Pwf.

The producing system is divided into two segments at the node. The upstream, or inflow, segment is comprised of all components between the node and the reservoir boundary. The downstream, or outflow, segment consists of the components between the node and the separator.

After isolating the node in the system, both of the following fundamental requirements at the node must be met:

• Only one pressure exists at the node at any given flow rate (Pinflow = Poutflow)

• Only one flow rate exists through the node (Qinflow = Qoutflow)

PERFORM Technical Reference Manual System Analysis Overview


Because the producing system consists of interacting components that each contributes pressure loss independently as a function of flow rate, the procedure necessary to find the unique flow rate that satisfies the two requirements at the node is iterative. To simplify the procedure, the system analysis approach uses a graphical solution in which the pressure at the node is shown as a function of the producing rate for both the inflow and outflow segments. The system analysis plot, or nodal plot, illustrated in Figure 1.2 contains both the inflow and outflow relationships.

Figure 1.2: Nodal Plot

The inflow curve bends downward. This illustrates that as flow rate increases through the inflow segment, pressure loss increases so that there is less pressure available at the node (or the downstream side of the inflow segment).

The outflow curve bends upward. This illustrates that for a fixed separator pressure, the pressure required at the node (inlet to the outflow segment) increases as flow rate increases.

Although each segment is exclusive of the other at varying flow rates, the two requirements listed previously (only one pressure and flow rate exist at the node) dictate that only one solution exists for the system at a particular set of conditions. On the nodal plot, this solution is the intersection of the inflow and outflow curves. This intersection indicates the producing capacity

System Analysis Overview PERFORM Technical Reference Manual


of the system and provides both the flow rate, Q, and the corresponding bottomhole pressure, Pwf.


2 Using System Analysis

General Analysis Procedure

A general procedure for solving most cases involves the following steps.

1 Make a specific objective for the case, such as determining the size of tubing to use in a well.

2 Determine the type of analysis needed to solve the problem, such as a Systems Analysis.

3 Determine the components needed (reservoir, wellbore, completion, and flowline) and the correlations desired.

4 Find all required data, make educated guesses for unknown values, and enter the data for each component.

5 Calculate the case and check the output graphically.

6 Interpret the output based on the type of case. Test the results for confidence by comparing the results with the data you have found.

7 Adjust the input and calculate again to improve the output results as needed.

8 Repeat from step 1 for the next objective of the case.

You can use a general analysis procedure to determine the producing capacity of a well system for a set of well conditions. More importantly, you can use the procedure to determine the quantitative effect and importance of each variable within the system on the overall system performance. The system components use the variables in either equations or correlations.

Although some values generally do not change during the well's life (for example, reservoir thickness, permeability, and total depth), many values are variable. The ability to change the values that directly affect system performance and well productivity allows you to achieve complete well optimization.

Using System Analysis PERFORM Technical Reference Manual


One of the underlying advantages of system analysis is its ability to predict the result caused by changes in the design variables. The alteration in well performance is seen directly on the systems plot through multiple inflow or outflow curves (each at a different set of conditions) and multiple intersection points. The Q and Pwf values at each intersection represent the producing status at that particular condition. The simplified systems plot in Figure 2.1 illustrates a typical scenario with multiple inflow curves at different reservoir pressures and multiple outflow curves at various tubing diameters.

Figure 2.1: System Analysis Plot with Multiple Conditions

As mentioned, the primary node used in most system analysis applications is the node at the bottom of the wellbore. Furthermore, although the system is comprised of many interacting components, it usually is simplified to four primary components:

• Flow through the reservoir

• Flow through the completion

• Flow up the tubing and any restrictions (vertical flow)

• Flow through the flowline and any restrictions (horizontal flow)

PERFORM Technical Reference Manual Using System Analysis


Applying System Analysis

System analysis can be applied in both new and existing wells. In new wells, the technique can be used to simulate anticipated conditions and plan the optimum completion and well design. In existing wells, the technique is used first to model existing conditions, then to evaluate areas of potential improvement.

To use system analysis on new wells, you must estimate data from offset wells, regional experience, and common sense. Because you do not have measured test data for comparison, the system analysis solution should cover the entire range of input variables. For example, you can select a preliminary tubing size for a new well by calculating a system analysis solution for a well using a broad range of inflow curves generated with "most pessimistic," "most likely," and "most optimistic" values of formation permeability. Although this type of solution is not meant to be entirely accurate, it provides a general idea of anticipated conditions.

Although using system analysis for existing wells can be slightly more complex than for proposed wells, the results obtained are more complete and accurate. The primary difference between the two cases is the ability in existing wells to model current conditions using actual data so that you can adjust input variables accordingly to better predict system performance. After reliably matching existing conditions, the effect of varied well conditions can be predicted with a higher degree of confidence.

In both cases, you must completely understand each component in the system to fully use the system analysis technique. In order to understand a particular component, you must have a quantitative description of each of the variables used to model the component. The pressure loss through the component is a direct function of the magnitude of these variables. In the design and implementation of an efficient producing well system, you can alter many of the variables that directly affect the producing capacity of the well. This flexibility is the basis of well optimization through system analysis.

Existing producing conditions in a well can be modeled by matching either a producing rate or pressure. If no producing bottomhole pressure is known, the system can be modeled simply by calculating both a rate and pressure and comparing the rate to known conditions. In the event that a producing bottomhole pressure is known, either through a single pressure or a flowing gradient survey, the tubing performance can be modeled directly. This procedure is especially beneficial in an oil well case, where there are many different correlations available but only one provides the best solution for the well. The use of an improper correlation in a system analysis solution can cause serious error.



In addition to correlation selection, the gradient match is also helpful in confirming input data that may not be exact. Figure 2.2 is an example illustrating the use of the gradient curve to match actual well data for an oil well by varying wellhead pressure.

Figure 2.2: Gradient Curves

As mentioned earlier, the system analysis approach can be understood as simulation of the producing system. Once the data is entered to create a base case of the well system (and confirmed through matching, if possible), the technique can be used to simulate varied conditions and solve a "what if" scenario. The effect of design and completion variables on total system performance can be predicted. Many variables can be simulated and optimized. The importance of each depends on specific well conditions. The items used most often in system analysis to optimize oil and gas wells include the following:

• Reservoir Skin

• Completion Effects

• Tubing Size

• Wellhead or Separator Pressure



Reservoir Skin

The reservoir skin is a deviation from Darcy flow generally caused by damage near the wellbore from drilling and completion fluids or from enhancement through stimulation. The effect of altering skin is really the effect of removing damage through stimulation. In system analysis, you can do this by reviewing several inflow cases, each at an improved skin value. Figures 2.3 and 2.4 illustrate this case, where a highly damaged formation with a skin of 32 is analyzed after stimulation with skins of 20, 5, 0, -3, and -6.

Figure 2.3: Effect of Formation Skin



Figure 2.4: Inflow Sensitivity on Skin

Completion Effects

The following items induce a similar response in the system performance and are variables in the completion design that are generally subject to change and optimize:

• Perforation shot density

• Perforation size

• Perforation diameter

• Perforation length

• Perforation interval

• Gravel pack size

• Gravel pack permeability

• Damaged zone radius and permeability

• Perforation crushed zone effects

• Perforated interval



Differential Graph

The differential graph, Figure 2.5, is especially helpful in emphasizing the completion effects of a well. The differential graph has two main curve types. The first type, shown bending downward to the left, represents the difference between the pressure remaining after flowing through the reservoir (Pws) and the pressure needed to flow through the outflow segment. The difference is the pressure available to produce through the completion. The curves shown bending upward to the left are the actual pressure losses through the completion as a function of rate.

Figure 2.5: Differential Graph

Similar to the standard system analysis graph, the intersection of these two curves dictates the producing capacity of a well for a given set of conditions. Although both example plots in this section illustrate the effect of varied perforation shot density, you can vary and display any of the completion variables listed in the same manner.



Perforation Shot Density

A typical analysis applied to completion design is shown in Figures 2.6 and 2.7, which illustrate the effect of perforation shot density.

Figure 2.6: Effect of Perforation Shot Density (SPF)

Figure 2.7: Inflow Sensitivity on Perforation Shot Density (SPF)



Perforation Interval

The perforation interval is the measured length of formation interval that is actually perforated. In many completions, the perforation interval is somewhat less than the formation thickness. This can be the result of:

• Well problems that result in the inability to completely penetrate the producing formation

• Reduced perforation interval aimed at lowering completion cost

• Altered perforation intervals to accommodate subsequent stimulation treatments

A reduced perforation interval affects the inflow segment in two ways. First, if reservoir turbulence is taken into account (i.e., Jones equation), the reduced interval increases the pressure loss encountered as the flow converges in the reservoir into the perforation interval. Second, the reduced perforation interval reduces the number of actual perforations available for flow into the wellbore, thereby increasing pressure loss through the completion. Both of these effects result in less productivity from a well, as illustrated in Figures 2.8 and 2.9.

Figure 2.8: Effect of Perforation Interval



Figure 2.9: Inflow Sensitivity on Perforation Interval

Tubing Size

Properly sized tubing is very important in an efficiently designed well system. In an oil well, pressure loss through the tubing can constitute the majority of the pressure loss through the entire system. If the tubing size is too small, friction loss will become excessive. If the tubing size is too large, additional pressure loss will be encountered due to liquid loading. In some cases, this loading can prevent the well from flowing at all. Incorrectly sized tubing can result in less available production from a well and possibly reduced flowing periods.



Figures 2.10 and 2.11 show the effect of tubing size in an oil well. The reversal effect in the largest diameter as it actually crosses the next smaller diameter indicates less available production due to liquid loading. The tubing sizes sensitized are 2 3/8", 2 7/8", 3 1/2", 4", and 4 1/2" respectively.

Figure 2.10: Effect of Tubing Size

Figure 2.11: Outflow Sensitivity on Tubing Size



Wellhead or Separator Pressure

The wellhead pressure (if no flowline) or separator pressure (if a flowline is included) is the outlet pressure of the total system. In most cases, as the total system yields production as a function of the overall pressure differential, the lowering of this outlet pressure results in increased well capacity. Figures 2.12 and 2.13 illustrate the reduction of wellhead pressure system performance for a typical oil well. This is accomplished by installing larger chokes in the wellhead or installing a compressor to reduce the wellhead pressure.

Figure 2.12: Effect of Wellhead Pressure



Figure 2.13: Outflow Sensitivity on Wellhead Pressure




3 Reservoir Component

The reservoir component, illustrated in Figure 3.1, of the system is composed of the flow between the reservoir boundary and the sandface. This component is always upstream of the node and, in this discussion, is combined with the completion component to form the entire inflow segment.

Figure 3.1: Reservoir Component

The flow through the reservoir is often referred to as the inflow performance relationship (IPR) of a well. It is a measure of the reservoir's ability to produce fluid as a result of a pressure differential. This ability depends on many factors, including reservoir type, producing drive mechanism, reservoir pressure, formation permeability, and fluid properties.

Reservoir Component PERFORM Technical Reference Manual


Vertical IPR Types

User Enters PI

The inflow performance relationship for an oil well is often simplified as a constant inflow or productivity index (PI), where inflow is directly proportional to drawdown, in the form of:

Constant Productivity Index

wfr P PQ

= PI−

where: PI = Productivity index (stb/d/psi)

Q = Total liquid flow rate (stb/d)

Pr = Average reservoir pressure (psi)

Pwf = Bottomhole flowing pressure (psi)

Figure 3.2: User Enters PI

The constant productivity index is expressed on the system analysis plot as a straight line between Pr and Qmax (at Pwf = 0) with a slope of 1/PI. The Vogel equation can be used to correct the flow below the bubblepoint pressure with the user-entered PI to calculate the IPR above the bubblepoint pressure.

PERFORM Technical Reference Manual Reservoir Component


Vogel/Harrison (1968)

The productivity index concept relies on the assumptions that reservoir and fluid properties remain constant and are not a function of pressure. Although these assumptions are true in some cases, especially in single-phase liquid flow, wells that produce both oil and gas will be overestimated below the bubblepoint if you use the user-entered PI relationship.

In 1968, Vogel presented an IPR solution for wells producing both oil and gas from saturated reservoirs.5 Using the reservoir model proposed by Weller,35 Vogel used a computer to calculate IPR curves for several fictitious solution gas drive reservoirs that covered a wide range of oil PVT properties and reservoir permeability characteristics. He plotted these IPR curves as dimensionless IPR curves with each pressure value divided by the maximum shut-in pressure, and each flow rate divided by the maximum rate (Qmax at Pwf = 0). He combined these dimensionless curves into a general reference curve in the following form.

Vogel equation

2

r

wf

r

wf

max PP

8.0P

P2.00.1

QQ

−

−=

where: Q = Total liquid flow rate (stb/d)

Qmax = Maximum flow rate at Pwf = 0 (stb/d)



The Vogel relationship can be regarded as a general equation for solution gas drive reservoirs producing below the bubblepoint. Above the bubblepoint, the standard Darcy equation or user-entered straight line PI is considered adequate. In cases of undersaturated reservoirs where wellbore pressure may be above or below the bubblepoint, the Vogel equation can be used as a correction below the bubblepoint pressure in combination with the user-entered PI, Darcy, transient, and fractured well correlations. In this case, the selected correlation is used between reservoir pressure (Pr) and bubblepoint pressure (Pb), followed by the Vogel relationship below the bubblepoint pressure.

JOSELUIS

Resaltado

JOSELUIS

Resaltado



The Vogel equation is differentiated with respect to Pwf to give a secondary equation for Qmax. Vogel equation

1.8

P PI + Q = Q

bbmax

×

where:


Qb = Flow rate at bubblepoint (stb/d)

PI = Productivity index (stb/d/psi)

Pb = Bubblepoint pressure (psi)

The final form of the Vogel equation for wells producing above the bubblepoint is: Combination Vogel equation Pwf > Pb

( )wfr P P PI = Q −×


PI = Productivity index (stb/d)



The final form of the Vogel equation for wells producing below the bubblepoint is: Combination Vogel equation Pwf < Pb

( )wfrmaxb P' P PI Q + Q= Q −×=′

where: Q' = Flow rate below bubblepoint (stb/d)

Qb = Flow rate at bubblepoint (stb/d)


Pb = Bubblepoint pressure (psi)

P'wf = Bottomhole flowing pressure below bubblepoint (psi)



The Vogel equation was developed with the assumption that there is no skin effect or that flow efficiency (FE) equals one. Standing6,7 proposed a method to correct the Vogel relationship to account for non-unity flow efficiencies. In this correction, test pressures used in the Vogel equation are first modified as follows:

( ) ( )wfrwfwf P P FE 1 + P = 'P −−

where:

P'wf = Equivalent undamaged flowing pressure

FE = Flow efficiency, 0.5 to 1.5

This correction alters the bottomhole flowing pressure due to additional pressure loss through the damaged area around the wellbore. Note that a higher flow efficiency value reduces the well productivity.

Figure 3.3: Vogel Solution Gas Drive with Flow Efficiency

The previous equation presents a problem with high flow efficiencies and low flowing bottomhole pressures. The value of P'wf can calculate as a negative value, which cannot be used in the Vogel equation. A correction to the Vogel solution is to account for either positive or negative values of P'wf in the following equation.

)/P(1.792P')0.1FEmax(

rwfe2.02.1Q/Q −==

This equation is only used if the P'wf is negative, otherwise the normal Vogel equation is used.

JOSELUIS

Resaltado

JOSELUIS

Resaltado



Darcy

The basic equation used to describe the flow of fluid through a reservoir is the radial form of the Darcy equation. Henry Darcy originally developed the equation in 1856 to describe the flow through sand filter beds used in water purification. The basic Darcy concept describes flow through porous media as a function of pressure differential, cross-sectional area, fluid viscosity, flow distance, and permeability (the measure of the media's ability to transmit fluid). He developed the equation under the assumptions that only single-phase, laminar flow existed, and the fluid was essentially incompressible.

Although the original Darcy equation was developed for linear flow in the vertical direction, the equation has been modified to predict radial flow. The general Darcy equation for an oil well is:

Darcy equation—oil well

−µ

−

DQ + S + 43

(x)ln B

) P (Pkh (0.00708) = Q wsr


k = Effective permeability (md)

h = Net formation thickness (ft)


Pws = Flowing sandface pressure (psi)

µ = Average liquid viscosity (cp)

B = Average formation factor (rb/stb)

x = Drainage area factor re/rw or from area and shape factor

S = Skin effect

D = Non-Darcy turbulence factor (1/stb/d)

re = Reservoir radius (ft)

rw = Wellbore radius (ft)



The Darcy equation for a gas well is slightly different because of the dynamic behavior of gas properties as a function of rate and pressure, where pseudopressure, Ψ, is used and is:

Darcy equation—gas well

( ) ( )

( )

−

ψ−ψ

DQ + S + 43

xln T

h k 0.000703 = Q

g

wsrgg

where:

Qg = Gas flow rate (Mscf/d)

kg = Effective gas permeability (md)


Ψr = Avg. reservoir pseudopressure (psi2/cp)

Ψws = Flowing sandface pseudopressure (psi2/cp)

T = Average reservoir temp (°R)


S = Skin effect

D = Non-Darcy turbulence factor (1/Mscf/d)



The Vogel equation can be used to correct the Darcy equation below the bubblepoint pressure. PI is calculated from the following equation and is used in the Vogel equation described earlier.

Productivity Index—Darcy

( )( )

−µ S +

43

xln B

hk 0.00708 = PI

where: PI = Productivity Index (stb/d/psi)






S = Skin effect



JOSELUIS

Resaltado



Non-Darcy Turbulent Term

The non-Darcy turbulent term, D, in the Darcy equation is used to account for inflow turbulence. This term is sometimes referred to as the Ramey Turbulence, or Ramey D, term. The non-Darcy term is applied as an effective rate-dependent skin, shown as the DQ or DQg term in the denominator of the Darcy equation. The term is usually obtained through multi-rate testing of wells where skin is calculated as a function of rate.

Skin Effect

Skin can be defined as a correction to account for non-Darcy or non-homogeneous flow behavior. In many discussions, skin is defined as a total skin that is comprised of several individual components.

DQ)t,q(SS'S ++=

where: S' = Total skin

S = Physical skin caused by damage near the wellbore or enhancement through stimulation

S(q,t) = Rate- and time- dependent skin, generally caused by permeability alteration due to changing gas saturation near the wellbore

DQ = Rate-dependent skin, described as the non-Darcy flow term

The physical skin, S, is understood to be caused by a physical alteration to the reservoir, generally near the wellbore. This can be in the form of damage from the penetration of drilling and/or completion fluids, causing a positive skin. Conversely, this skin can be represented as a negative value, caused by stimulation of the well through fracturing or acidizing.

The rate- and time-dependent skin, S(q,t), is induced by two-phase fluid behavior at or near the wellbore. This skin can give the appearance of non-Darcy flow behavior. In general, this skin is a permanent condition (unless fluid conditions change), and cannot be altered with stimulation.

The non-Darcy term, DQ, is simply a representation of the energy loss due to turbulent behavior in the reservoir. The value can be determined by isochronal testing.

JOSELUIS

Resaltado

JOSELUIS

Resaltado

JOSELUIS

Resaltado

JOSELUIS

Resaltado



System analysis accounts for the total skin effect in several ways. Most inflow equations allow for a skin entry, which generally is the physical skin, S. If the skin entry is positive, it indicates damage. If the skin entry is negative, it indicates stimulation. The rate-dependent non-Darcy term is available for use in the Darcy equation. Because system analysis is an isochronal procedure, the rate- and time-dependent skin, S(q,t), becomes a function of rate only and logically can be included with the non-Darcy skin.

Note: In a system analysis solution, be sure not to include turbulent or physical skin more than once. If the skin effect is measured including the completion by transient testing within the wellbore, this skin takes into account completion effects. If this skin is used subsequently in the reservoir segment as a physical skin, S, or as a rate-dependent skin, DQ, or as both, additional pressure loss through the completion segment will cause an underestimated inflow curve. This situation exists for the four-point test (Jones and back pressure) for oil wells and gas wells and Vogel for oil wells.

Drainage Area and Shape Factor

The previous Darcy equations are actually slight modifications of the original equation. The ln(x) term is a modification of the standard ln(re/rw), which is a representation of the area of flow in the radial form of the equation. The ln(re/rw) value is applicable only for a well producing in the center of a circular drainage area. In the cases where the well is located in an irregularly shaped drainage area, ln(re/rw) is replaced by ln(x), where x is a reservoir size and shape factor that describes the actual drainage shape and well position.22

w

0.5areafactor

rS S

=x ×

where: X = Reservoir drainage factor

Sfacto

r

= Reservoir shape factor from table

Sarea = Reservoir area (ft2)


The following table of shape factors are available for use in the Darcy and Jones equations.

JOSELUIS

Resaltado

JOSELUIS

Resaltado

JOSELUIS

Resaltado



Reservoir Shape Factors

The reservoir shape factor is available with the Darcy and Jones IPR correlations to describe a well that is not necessarily in the center of a circular reservoir as assumed by the Darcy and Jones equations. The shape factor is the constant given in the following table. For example, the shape factor of a well in the center of a square is 0.571. The well in the center of a circle can either use the shape factor listed or directly use the reservoir radius and wellbore radius.

SHAPE SHAPE FACTOR

SHAPE SHAPE FACTOR

0.564 2

1

0.966

1

1

0.571 2

1

1.444

0.565 2

1

2.206

JOSELUIS

Resaltado



SHAPE SHAPE FACTOR

SHAPE SHAPE FACTOR

0.605 4

1

1.925

0.610 4

1

6.590

1/3

0.678 4

1

9.360

2

1

0.668 1

1

1.724

4

1

1.368 2

1

1.794



SHAPE SHAPE FACTOR

SHAPE SHAPE FACTOR

5

1

2.066

2

1

4.072

1

1

.884 2

1

9.523

1

1

1.485

10.135

Jones et al. (1976)

Turbulent flow in the reservoir, generally occurring near the area of wellbore convergence, can cause significant additional pressure loss. This is especially prevalent in high rate gas wells.

The basic Darcy equation was generated with the assumptions that only laminar flow existed through the porous media. As wells produce at relatively high rates, this assumption becomes invalid as turbulent flow begins to develop. The overall effect of this turbulence is added energy loss, which results in a lower flow rate for a given pressure differential. This turbulent flow is often referred to as non-Darcy flow behavior.

JOSELUIS

Resaltado

JOSELUIS

Resaltado

JOSELUIS

Resaltado



An equation suggested by Jones, Blount, and Glaze8 in 1976 accounts for turbulence in a producing oil or gas well. The equation, referred to as the Jones equation, is written in the following forms:

Jones equation—oil well

bQ + aQ = PP 2wsr −

where:

( )r h

B 10 2.30 = a

w2p

2-14 ρβ×; turbulent term

( )[ ]( )kh0.00708

S + 0.472xlnB = bµ

; laminar term

and where:




β = Turbulence coefficient (1/ft)


ρ = Fluid density (lb/ft3)

hp = Perforated thickness (ft)




S = Skin effect




JOSELUIS

Resaltado



Jones equation—gas well

g2gwsr bQ + aQ = ψ−ψ

where:

( )gw

2p

g2-1

r h

T 10 3.16 = a

µ

γβ×; turbulent term

( ) ( )[ ]hk

S + 0.472xlnT1424 = b

g

; laminar term

and where:

Ψr = Average reservoir pressure (psi2/cp)

Ψws = Flowing sandface pressure (psi2/cp)


β = Turbulence coefficient (1/ft)

γg = Gas specific gravity

T = Average reservoir temperature (°R)



µg = Gas viscosity (cp)


S = Skin effect

kg = Effective permeability (md)



For oil wells, you can also obtain the turbulent term, a, and the laminar term, b, by plotting (Pr - Pwf)/Q versus Q. For gas wells, plot (Pr

2 - Pwf2) / Qg versus Qg. The resulting slope will be the

turbulent term and the intercept will be the laminar term.

The laminar term is simply the Darcy equation. The turbulent term is the turbulent portion of the Jones equation and is shown as a function of rate. The contribution of this turbulent term tends to reduce the available flow rate from a well as rate increases. The term accounts for additional wellbore convergence effects caused by partial penetration or a limited perforated interval. This is accomplished with the use of the perforated interval, hp, instead of the gross formation interval, h, in the denominator of the turbulent term.

The turbulence coefficient, β , is a function of reservoir permeability.

JOSELUIS

Resaltado

JOSELUIS

Resaltado

JOSELUIS

Resaltado

JOSELUIS

Resaltado

JOSELUIS

Resaltado

JOSELUIS

Resaltado



Jones equation—turbulence coefficient

k10 2.33

= 1.201

10×β

where: β = Turbulence coefficient (1/ft)


The Jones equation is recommended in wells in which turbulence is assumed to be a factor. In gas wells, turbulence is almost always prevalent and the Jones equation is suggested. In oil wells, turbulence generally does not become significant unless rates are in excess of several thousand barrels per day. For oil wells, the Darcy equation is usually adequate. The Vogel equation can be used to adjust the Jones equation below the bubblepoint pressure for solution gas drive oil wells.

Jones—4-Point Test and Jones—Enter a and b

The Jones Four-Point Test IPR method calculates the Jones a and b terms from a given set of rate and flowing bottomhole pressure data points according to the equations discussed previously for the Jones equation for oil and gas wells.

Along with the four-point test, one IPR option is to use your own Jones a and b term. The a and b terms are based on different equations for gas wells and oil wells. The oil well equation was given previously. The gas well equation given previously is based on the equation involving pseudopressure. For gas wells, the Jones a and b terms are based on a difference in pressure squared and not pseudopressure as follows:

Jones 'a' and 'b' user-entered (gas wells only)

bQ + aQ = P P g2g

2ws

2r −

Note: For gas wells, do not use the resulting a and b terms from the four-point test IPR method in the Jones user-entered a and b IPR method because the Jones user-entered a and b terms are based on pressure squared and the Jones four-point test is based on pseudopressure. For oil wells, you can use the calculated Jones coefficients in the Jones a and b user-entered IPR because both IPR methods use the same equation.

JOSELUIS

Resaltado

JOSELUIS

Resaltado

JOSELUIS

Resaltado

JOSELUIS

Resaltado

JOSELUIS

Resaltado



Back Pressure Eq (1930) and Back Pressure—4-Point Test

The effects of reservoir turbulence can also be modeled using the backpressure equation:

( )n2ws

2r P P C = Q −×

where: Q = Flow rate (stb/d or Mscf/d)

C = Backpressure coefficient

Pr = Reservoir pressure (psi)

Pws = Sandface pressure (psi)

n = Turbulence coefficient

The turbulence coefficient, n, can be obtained from stabilized test data where (Pr2 - Pws

2) is plotted versus Q on a log-log scale. This method requires at least three and usually four flowing bottomhole pressure and flow rate data pairs (thus called a four-point test). The turbulence coefficient is determined from the inverse slope of the line, and is a measurement of the turbulent condition of the well.

Turbulent flow yields values of n between 0.5 (completely turbulent flow) and 1.0 (completely laminar flow). In some solution gas drive reservoirs, the 'n' value can be larger than 1.0.45 The backpressure equation is considered a valid inflow representation if turbulence is a factor and test data are available and suitable for confident prediction of n. Solve for the backpressure coefficient, C, using a point on the backpressure line. The Backpressure Four-Point Test method calculates the best fit of the four-point test data points to arrive at the n and C values.

The Backpressure equation is used to calculate the IPR from a known n and C value based on the results of a plot of (Pr

2 - Pws2) versus Q for both oil and gas wells. The Backpressure 4-Pt

Test IPR method involves the computer calculation of the backpressure n and C values based on user-entered test data. The results for oil wells and gas wells will be very different because pseudopressure is used in the gas well cases so that the equation becomes:

JOSELUIS

Resaltado

JOSELUIS

Resaltado

JOSELUIS

Resaltado

JOSELUIS

Resaltado

JOSELUIS

Resaltado

JOSELUIS

Resaltado



Backpressure Four-Point Test (gas wells only)

( )nwsrCQ ψ−ψ×=

where: Q = Flow rate (stb/d or Mscf/d)

C = Backpressure coefficient

ψr = Reservoir pseudopressure (psi2/cp)

ψws = Sandface pseudopressure (psi2/cp)

N = Turbulence coefficient

Note: For gas wells, do not use the resulting n and C values from this equation in the user-entered Backpressure equation. This restriction does not apply to oil wells because both methods use the difference in the pressure squared and not pseudopressure.

JOSELUIS

Resaltado

JOSELUIS

Resaltado



Transient Flow Equation

In many cases, an inflow is desired for a new well that has not reached pseudosteady state and is still producing in a transient condition. Both the Darcy equation and its derivative, the Jones equation, were developed under the assumption that the producing well has reached pseudosteady state. During the transient period, use the transient equation to predict the inflow performance for a well. Use the Vogel equation below the bubblepoint pressure to correct the transient equation for oil wells with a solution gas drive.

Transient equation—oil well

( )

−

µφ

µ

−

0.87S + 3.2275 r c

k tlog B 162.6

P Pkh = Q

2wt

wsr


K = Effective permeability (md)

H = Net formation thickness (ft)





T = Producing time (hrs)

φ = Porosity

ct = Total system compressibility (1/psi)


S = Skin effect

JOSELUIS

Resaltado

JOSELUIS

Resaltado

JOSELUIS

Resaltado

JOSE

Resaltado



Transient equation—gas well

−

φµ

ψ−ψ

0.87S + 3.2275 rc

tklog T 1638

)h( k = Q

2wtg

g

wsrgg

where:


kg = Gas effective permeability (md)


Ψr = Average reservoir pseudopressure (psi2/cp)

Ψws = Flowing sandface pseudopressure (psi2/cp)


t = Producing time (hrs)

φ = Porosity

µg = Average gas viscosity (cp)

ct = Total system compressibility (1/psi)


S = Skin effect

The pressure behavior of a reservoir during the transient period is essentially the same as that of an infinite acting reservoir. Use the following equation to estimate the length of time required to surpass this transient period and reach pseudosteady state:

Time to pseudosteady state

( )k0.001005rc

= (hrs) Time2etµφ

where: φ = Porosity

µ = Viscosity (cp)

ct = Total compressibility (1/psi)

re = Drainage radius (ft)


JOSELUIS

Resaltado



Fractured Well

PERFORM uses a digitized, constant rate, finite-conductivity, closed square, fractured well type-curve to calculate the effect of a vertically drilled well that has been hydraulically fractured. The type curve requires a dimensionless time, dimensionless fracture conductivity, and fracture penetration ratio to calculate a dimensionless pressure drop for a known wellbore pressure and time. The well is assumed to be in the center of a square reservoir with an aspect ratio 1:1.

Figure 3.4: Square Reservoir

A reservoir conductivity is calculated as: Oil well

B hk 0.00708

= R ct µ

Gas well

460 Thk 0.000703

= R ct +

where:

Rct = Reservoir conductivity

K = Reservoir permeability (md)

H = Reservoir thickness (ft)

µ = Fluid viscosity (cp)

B = Formation volume factor (rb/stb)

T = Reservoir temperature (°F)

JOSELUIS

Resaltado

JOSELUIS

Resaltado



The fracture penetration ratio is determined by the following formula and must evaluate to between 0.0 and 1.0 or an error message appears. Note that re is normally evaluated as a reservoir radius but in this case, it is the length of one side of a square reservoir divided by 2.

Fracture penetration ratio

e

fpr r

x = F

where:

Fpr = Fracture penetration ratio

xf = Fracture half length (ft)

re = Length of one side of square reservoir divided by 2 (ft)

The dimensionless fracture conductivity is calculated as follows and must evaluate between 0.01 and 500.0 or PERFORM displays an error message:

Dimensionless fracture conductivity

f

fcd k x

wk = F

where:

Fcd = Dimensionless fracture conductivity

kf = Fracture permeability (md)

W = Fracture width (ft)


xf = Fracture half width (ft)

JOSELUIS

Resaltado

JOSELUIS

Resaltado



The dimensionless time is calculated as follows and must be between 0.00001 and 1000.0 or PERFORM displays an error message:

Dimensionless time

2ft

Dxf xc

k t 0.000264 = t

µφ

where:

tDxf = Dimensionless time


T = Production time (hr)

φ = Porosity (pore volume/ bulk volume)



xf = Fracture half length (ft)

The type curve function interpolates the type curve to arrive at the dimensionless pressure drop in the fracture and reservoir as:

( )prcdfDxD F ,F ,tf = p

The flow rate is calculated as follows: Flow rate—oil well

( )D

wfr ct

pP P R

= Q−

Flow rate—gas well

( )D

wfrct g p

R = Q

Ψ−Ψ

where: Q = Flow rate at Pwf (stb/d or Mscf/d)

Rct = Oil or gas reservoir conductivity

Pr = Reservoir pressure (psia)

Pwf = Wellbore pressure (psia)

pD = Dimensionless pressure

Ψr = Reservoir pseudopressure (psi2/cp)

Ψwf = Wellbore pseudopressure (psi2/cp)

JOSELUIS

Resaltado

JOSELUIS

Resaltado



The flow rate above assumes that the well is in non-turbulent flow. To account for turbulence in the fracture that may occur, a non-Darcy flow rate adjustment is made to the flow rate according to the size of the proppant in the fracture itself as follows:

NON-DARCY FLOW FACTORS

PROPPANT SIZE a-TERM b-TERM

8 - 12 mesh 1.24 17423.61

10 - 20 mesh 1.34 27539.48

20 - 40 mesh 1.54 110470.39

40 - 60 mesh 1.60 69405.31

A turbulence beta factor is calculated as:

af

7

kb 10 3.088386

= ×

β

where: β = Turbulence factor

b = b term from the previous Non-Darcy Flow Factors table


a = a term from the previous Non-Darcy Flow Factors table

A flow velocity and Reynold's number is determined to calculate a revised fracture conductivity as follows:

Oil well velocity

whB Q 10 x 249.3

V oo5−

=

Gas well velocity

wh

B Q 10 x .7875V gg

3−

=

Reynold's number

µσβ× f

-11

REk V 10 1.5808

= N

JOSELUIS

Resaltado

JOSELUIS

Resaltado



Non-Darcy dimensionless fracture conductivity

( )REf

fcd N1k x

wkF

+=

where: V = Fracture flow velocity

Q = Liquid flow rate (stb/d)

Bo = Liquid volume factor (rb/stb)

h = Formation thickness (ft)

w = Fracture width (ft)


Bg = Gas volume factor (scf/stb)

NRE = Reynold's number

β = Fracture turbulence factor

σ = Fluid density (lbm/ft3)



k = Formation permeability (md)

An iteration technique is used to converge on a dimensionless pressure and flow rate using the type curve to arrive at a final non-Darcy flow rate at a given wellbore pressure. The same equations used above to calculate Q and Rct are used in the iteration until a convergence with the flow rate, Q, used in the above velocity equations gives the same flow rate from the type curve calculation. PERFORM allows a maximum of 20 iterations and displays an error message if unable to converge.

Oil well cases can also be adjusted for the Vogel relationship below the bubblepoint pressure using the Vogel equations. An instantaneous productivity index is calculated for the Vogel equation as:

Productivity Index

wfr

o

P PQ

= PI−

where: PI = Productivity index (stb/d/psi)

Qo = Liquid flow rate (stb/d)

Pr = Reservoir pressure (psi)

Pwf = Wellbore pressure (psi)



Datafile—2 Col ASCII

The Datafile—2 Col ASCII IPR type requires a data file of pressure and flow rate stored in a text file. The file cannot have header or column labels but must contain two columns of numerical ASCII text separated by a comma and/or spaces between the two column entries. The first column must contain pressure in decreasing order for production wells and increasing order for injection wells. The second column must be the flow rate in increasing order of flow rate.

If the file does not contain a point at a zero flow rate, PERFORM calculates the zero flow rate pressure by extrapolating a line through the first two data points back to a zero flow rate. This is assumed to be the static reservoir pressure. If the file does not contain an Absolute Open Flow (AOF) rate at the standard pressure, then the AOF point is extrapolated from a straight line passed through the last two data points.

The data points for calculation of the final inflow curve can include the completion component if the file contains at least 15 data points. The file must contain at least four points to make a fitted inflow curve and at least two points for an unfitted inflow curve. The file can contain no more than 23 points. The points can also be fitted to make a smoother inflow curve if desired. If the points cannot be adequately smoothed with a second order polynomial fit, PERFORM displays a warning message and attempts a first order fit. If the first order fit is not possible, PERFORM uses the raw data and displays a warning message.

Horizontally Completed Wells

Most wells drilled are configured as a vertical or semi-vertical wellbore that intercepts the reservoir interval either perpendicular to the formation or at an angle less than 90 degrees from horizontal. Horizontal wellbores are considered a special type of well whereby the well strikes the reservoir at 90 degrees from vertical and extends a tunnel through the reservoir for production.

Not all reservoirs are good candidates for horizontal technology. Reservoirs that typically are good candidates for horizontal wells are thin reservoirs (less than 500 ft thick), have lower productivity than vertical wells, have tight formations with horizontal as well as vertical permeability, may have fractures, and may have water-coning or gas-coning problems. Horizontal wells drilled in these types of reservoirs have shown to produce from 2 to 20 times the rates exhibited by vertical wells.

Drilling costs are typically two to four times higher than conventional vertical wells. Reservoirs with multiple pay zones that are separated by impermeable barriers may require drilling a horizontal well for each zone. Additional techniques are needed for workovers, logging, and tools because normal wireline operations are inadequate.



Pressure and flow rate calculations in horizontally completed wells are done in much the same ways as vertically drilled wells, with the added reservoir and wellbore geometry of the horizontal completion. Several IPR options are available for modeling the horizontal reservoir. The effects of the horizontal tunnel are determined by pressure drop calculations by Dikken.47

The basic assumptions for the reservoir IPR are a horizontal well with single phase (or a single mixed phase) and the well is in turbulent flow in the horizontal tunnel. The models are used for open hole or cased hole conditions depending on the completion type. Vertical and horizontal reservoir permeability and horizontal tunnel length play an important role in the IPR estimate.

Figure 3.5 shows a typical horizontal well configuration including the vertical and horizontal wellbore and reservoir. The kickoff point (KOP) is designated at any depth above the end of the tubing where an angle will be calculated. It is assumed that KOP-TMD/TVD value calculates an angle of the well from the surface to the KOP depth, therefore, KOP-TMD is entered the same as KOP-TVD to calculate a vertical well (angle of 0 degrees) from the surface to the KOP depth.

All angles and TMD/TVD pairs are entered in the Directional Survey dialog box. TBG-TMD/TVD is the depth of the end of the tubing string. This is used to calculate the well angle and determine the total length of the tubing for pressure profile and gradient calculations.

Figure 3.5: Horizontally Completed Well

If the completion type is open hole, then CS-TMD/TVD designates the depth of the casing shoe and is the total length of the casing segment and the location of the bottomhole node. This depth must be between the well total depth including the tunnel length and tubing end. The horizontal tunnel length will be the distance from the casing shoe to the end of the horizontal tunnel.



If the completion is a cased hole type, then TOP-TMD/TVD is the depth of the topmost perforation and is the location of the bottomhole node. The horizontal tunnel length is the distance from the top perforation to the last perforation.

You must match the values you enter between the reservoir and completion. For example, if you have a cased hole completion in a horizontal well, then you should enter the same value for the horizontal tunnel length in the Reservoir dialog box as the perforated interval in the Completion dialog box. The perforated interval actually determines the horizontal tunnel length in cased holes but the two values are not linked together in the calculations. If you want to sensitize the horizontal tunnel length, you should also sensitize the perforated interval at the same values as the horizontal tunnel length.

The special reservoir IPR types for horizontal wells include the User Enters PI and Vogel for oil wells, and Backpressure Equation and Datafile for oil and gas wells. The three correlations and the datafile option operate the same as discussed earlier for the vertical well type. However, an extra pressure loss is included for losses in the horizontal tunnel according to the value specified by the alpha term. This term is casing alpha for cased hole completions and open hole alpha for open hole completions.

The alpha term is used to judge the amount of turbulence induced in the horizontal tunnel. Alpha ranges from a value of zero (completely turbulent flow) to 0.25 (completely laminar flow). For most open hole completions, the roughness of the wellbore wall suggests that the alpha term be close to zero.

Cased hole completions require the casing ID and casing alpha term. In addition to the roughness of the casing surface, the effect of perpendicular inflow through the perforations in the cased hole suggests that turbulence will be dominant even in this situation so an alpha value close to zero is also suggested.

Dikken discusses the effect of total well rate as a function of horizontal tunnel length and shows that the total well rate increases with increasing well length for various values of well diameter. Regardless of diameter, all wells must produce at the same critical rate per foot and converge on a single rate versus length profile at low horizontal tunnel lengths. With increasing well length, the total rate levels off earlier for smaller diameter wellbores. By sensitizing on various values of horizontal length at different wellbore diameters, you can judge the optimum tunnel length for well completions during their design. The production performance is also sensitive to the alpha value.

Dikken also suggests using 80 percent of the infinite well length as an engineering criterion for the optimal length of the horizontal tunnel. By sensitizing on horizontal tunnel length and looking at the inflow sensitivity graph, you can estimate the optimum tunnel length from the graph at 80 percent of the maximum rate obtainable. This should ensure a sufficient flow rate to minimize the horizontal completion loss and help control the expense of the well.



For open hole completions, the outflow results are calculated from the surface to the node as in a deviated vertical well using the tubing ID and roughness down to the tubing end depth, Tbg-TMD/TVD, and then through the casing segment using the casing ID, roughness, and casing shoe depth. The casing segment length is the distance from the tubing end depth to the casing shoe depth. The open hole ID designates the diameter of the horizontal tunnel below the casing shoe.

In cased hole completions, the casing ID is used for the ID of the casing segment above the node position and for the ID of the horizontal tunnel. The casing roughness is only used in the casing segment above the node. Casing roughness is accounted for with the casing alpha term in the horizontal tunnel.

The special horizontal IPR types require vertical and horizontal reservoir permeability. Many reservoirs have a vertical permeability that is 10 percent of the horizontal permeability. You can sensitize the horizontal permeability and/or vertical permeability. To keep the same permeability ratio, sensitize both permeabilities using multiple sensitivity lines.

If a completion is included, then the pressure loss in the completion is calculated at the flow rate of the IPR and subtracted from the sandface pressure to arrive at a nominal wellbore pressure. This result is then modified by the pressure loss calculation for the horizontal tunnel to arrive at a flow rate and pressure value at the node. In the output, the horizontal tunnel effects and completion pressure loss are combined into a total completion pressure loss. The horizontal tunnel effect is calculated from the equations given by Dikken as:

α−−

ρ

µ××

ρ×=

d10495.1d

10259.2R

5

5

8

w

where:

Rw = Flow resistance of wellbore (psi-day/ft4)

ρ = Fluid density (lbm/ft3)

d = Tunnel or casing diameter (in.)


α = Wellbore turbulence factor



An integration constant is calculated: Integration constant—oil well

α−−α

−

α−−××=

31

w

sowsr

75

RJ)3(

)PP(10491.510434.5K

Integration constant—gas well

α−−α

−

α−−××=

31

w

sgwsr

63

R

J)3()PP(10329.710051.3K

where:

Jso = Productivity index per ft of tunnel (bbl/d/psi/ft)

Jsg = Productivity index per ft of tunnel (Mcf/d/psi/ft)

The dimensionless wellbore tunnel length is calculated: Dimensionless wellbore tunnel length—oil well

α−α−

×=3

RJK

1L10105.9x wso5

D

Dimensionless wellbore tunnel length—gas well

α−α−

×=3

RJ

K1

L10823.6x wsg4D

The dimensionless tunnel length, xD, and the alpha term are used in a digitized type curve to calculate a dimensionless total flow rate, qD. The final flow rate, Q, is calculated as:

Flow rate—oil well

α−−××=

12

6

D5

)K10840.1(

q10434.5Q

Flow rate—gas well

α−−××=

12

4

D3g

)K10277.3(

q10051.3Q

Two additional well parameters are necessary to calculate the horizontal tunnel effects depending on the type of completion. For open hole completions, you must enter an open hole ID that is an average drilled ID from the bit records or a caliper survey. Also required is the open hole alpha term.



Horizontal IPR Types

Horizontal reservoirs are modeled using a rectangular reservoir with a horizontal wellbore. Some of the IPR types require that the wellbore be in the center of the reservoir. Other types specify the location of the wellbore. Both vertical and horizontal permeability is needed for calculations.

Most of the correlations require the same input data with the additional wellbore position data for the Kuchuk and Goode & Thambynaya methods.

The following lists categorize the horizontal IPR types available in PERFORM.

Steady-State Flow

• Giger et al. (1984)

• Economides et al. (1991)

• Joshi (1988)

• Renard and Dupuy (1991)

Pseudosteady-State Flow

• Kuchuk (1988)

• Babu and Odeh (1989)

Transient Flow

• Goode and Thambynaya (1987)

Other

• Back Pressure Eq (1930)

• Datafile—2 Col ASCII

• No Inflow Calculated

• User Enters PI

• Vogel/Harrison (1968)



Figure 3.6: Schema for Giger, Joshi, Renard & Dupuy, and Economides correlations

Giger et al. (1984)

Giger55 et al. developed a steady-state reservoir model for calculating the sandface pressure and flow rate pairs for isotropic and anisotropic reservoirs. For anisotropic reservoirs, the Muskat method is used to calculate equivalent reservoir permeability and adjusted the rest of the parameters.

Isotropic equivalent parameters

verthorzavg kkk ×=

horz

avghorz k

kN =

vert

avgvert k

kN =

horzeq NLL ×=

verteq Nhh ×=

horzeeq Nrr ×=

Note: PERFORM displays an error message if the reservoir radius entered is less than half the horizontal tunnel length. This prevents the horizontal tunnel from extending past the reservoir boundary.



Horizontal drainage component—horizontal plane

eq

eq

eq

eq

2

eq

eq

horz h

L

r2

L

r2

L11

lnD ×

−+

=

Horizontal drainage component—vertical plane

π

=w

eqvert r2

hlnD

Reservoir storage term

SDDW verthorzs ++=

Reservoir rate calculation without Ramey D turbulent flow included Oil well

slo

wfreqavg

WB

)PP(Lk00708.0Q

µ

−=

Gas well

s

wfreqavgg W)460T(

)(Lk000703.0Q

+

ψ−ψ=

Reservoir rate calculation with Ramey D turbulent flow included Oil well

)DQW(B

)PP(Lk00708.0Q

slo

wfreqavg

+µ

−=

Gas well

)DQW)(460T(

)(Lk000703.0Q

gs

wfreqavgg ++

ψ−ψ=



where:

kavg = Equivalent horizontal/vertical permeability (md)

khorz = Horizontal reservoir permeability (md)

kvert = Vertical reservoir permeability (md)

Nhorz = Horizontal permeability coefficient

Nvert = Vertical permeability coefficient

Leq = Equivalent horizontal tunnel length (ft)

L = Horizontal tunnel length (ft)

heq = Equivalent reservoir thickness (ft)

h = Reservoir thickness (ft)

req = Equivalent reservoir radius (ft)


Dhorz = Horizontal drainage term

Dvert = Vertical drainage term

rw = Wellbore radius (in)

Ws = Reservoir storage term

S = Reservoir skin factor

Bo = Formation volume factor (rb/stb)

µl = Liquid viscosity (cp)

Qg = Gas rate (Mscf/d)

ψr = Reservoir static pseudopressure (psi2/cp)

ψrwf = Wellbore pseudopressure (psi2/cp)


Pr = Reservoir static pressure (psi)


D = Ramey D turbulence term (1/bpd or 1/Mscf/d)

Q = Total liquid rate (bbls/d)



Economides et al. (1991)

Economides51,52 used an augmented version of Joshi's equation for modeling the horizontal well performance as steady state considering anisotropic wells and a drainage ellipsoid. This IPR type is not suggested if half the tunnel length is greater than 0.9 times the effective drainage radius. The tunnel length must be greater than (kh/kv)1/2 times formation thickness.

Half of major axis of ellipse

4

e

2Lr

25.05.02L

a

++×=

Anisotropic factor

v

hani k

kI =

Drainage factor—horizontal plane

−+

=

2L

2L

aalnD

22

horz

Drainage factor—vertical plane

×+

×

=

12r

)1I(

hIln

Lh

IDw

ani

anianivert


SDDW verthorzs ++=


slo

wfrh

WB)PP(hk00708.0

Qµ

−=

Gas well

s

wfrhg W)460T(

)(hk000703.0Q

+ψ−ψ

=




+µ

−=

DQLh

IWB

)PP(hk00708.0Q

anislo

wfrh

Gas well

++

ψ−ψ=

ganis

wfrhg

DQLh

IW)460T(

)(hk000703.0Q

where: a = Ellipse major axis factor



Iani = Anisotropic factor

kh = Horizontal reservoir permeability (md)

kv = Vertical reservoir permeability (md)










Bo = Formation volume factor, (rb/stb)




ψwf = Wellbore pseudopressure (psi2/cp)





Joshi (1988)

The Joshi53,54 method is very similar to Economides with a different form of the Dvert equation for

steady-state flow. The gas well reservoir storage term calculation considers an effective wellbore radius for skin effect. This IPR type is not suggested if half the tunnel length is greater than 0.9 times the effective drainage radius.


4

e

2Lr

25.05.02L

a

++×=

Anisotropic factor

v

h

kk

=β


−+

=

2L

2L

aalnD

22

horz


×

β×

β=

12r

2

hln

Lh

Dw

vert



Reservoir storage term Oil well

SLh

DDW verthorzs ×

β++=

Gas well

L/h

w

2

e

we

12/r2h

a2L

11a

2L

rr

β

β

−+

=

Srr

lnWwe

es +

=

Reservoir rate calculation without Ramey D turbulent flow included

Oil well

slo

wfrh

WB)PP(hk00708.0

Qµ

−=

Gas well

s

wfrhg W)460T(

)(hk000703.0Q

+ψ−ψ

=




β+µ

−=

DQLh

WB

)PP(hk00708.0Q

slo

wfrh

Gas well

β++

ψ−ψ=

gs

wfrhg

DQLh

W)460T(

)(hk000703.0Q



re = Reservoir drainage radius (ft)

β = Anisotropic factor









rwe = Effective wellbore radius (ft)













Renard and Dupuy (1991)

The Renard and Dupuy56 IPR type considers a drainage shape factor for steady-state flow. This

IPR type is not suggested if half the tunnel length is greater than 0.9 times the effective drainage radius.


4

e

2Lr

25.05.02L

a

++×=

Calculate a drainage shape factor

La2

=∂

Calculate anisotropic factor

v

h

kk

=β


)(coshD 1horz ∂= −


π×+β

β×

β=

12r2

)1(

h2ln

Lh

Dw

vert


SLh

DDW verthorzs ×β++=


slo

wfrh

WB)PP(hk00708.0

Qµ

−=

Gas well

s

wfrhg W)460T(

)(hk000703.0Q

+ψ−ψ

=




β+µ

−=

DQLh

WB

)PP(hk00708.0Q

slo

wfrh

Gas well

β++

ψ−ψ=

gs

wfrhg

DQLh

W)460T(

)(hk000703.0Q



re = Reservoir drainage radius (ft)



















D = Ramey D turbulence term (1/bpd or I /Mscf/d)



Figure 3.7: Schema for Kuchuk and Babu & Odeh correlations

Kuchuk (1988)

Kuchuk57 makes an evaluation of a horizontal well performance where the position of the wellbore in the reservoir can be designated for pseudosteady-state flow. The Kuckuk IPR type assumes that the distance from the well to any lateral boundary must be large relative to the distance from the well to the top and bottom of the reservoir.

Distance and permeability ratios

parL d2

Ld =

y

x

par

perr k

kd

dK ×=

The above ratios are used in a dimensionless drainage calculation routine of tables to calculate a dimensionless fd term.

)K,d,d,d(ff rLyxd =

Effective horizontal permeability

yxh kkk ×=

Anisotropic factor

v

h

kk

=β




π×

β

+×π

−=hd

sin1

1h12

rlnD botw

horz


+−×

β=

2botbot

vert hd

hd

31

Lh2

D


[ ]S)DD(Lh2

fW verthorzds +−β×+=


slo

wfrh

WB6.70)PP(hk

Qµ−

=

Gas well

s

wfrhg W)460T(711

)(hkQ

+ψ−ψ

=



where:

dL = Distance ratio


dpar = Drainage distance parallel to wellbore (ft )

Kr = Horizontal permeability ratio

dper = Drainage distance perpendicular to wellbore (ft)

kx = Horizontal permeability in X direction (md)

ky = Horizontal permeability in Y direction (md)

fd = Dimensionless drainage factor

dx = Distance ratio to X

dy = Distance ratio to Y

kh = Effective horizontal permeability (md)






dbot = Distance to reservoir bottom (ft)














Babu and Odeh (1989)

The Babu and Odeh59 horizontal reservoir correlation uses a pseudosteady-state model with the horizontal tunnel position considered. For Babu and Odeh, one of the following conditions must be satisfied:

a)vx

x

y

y

kh75.0

k

L75.0

k

L>>≥

b) zy

y

x

x

kh

k

L33.1

k

L>>>

Calculate permeability ratios

hy

perper

k

d'k =

hx

parpar

k

d'k =

vver

kh

'k =

Two cases are possible:

Case 1: If k'per > 0.75>>0.75 k'ver

Pxyz term

−

π−

+

−= 84.1

hd

sinlnkk

ln25.0r

h12ln1

L

dP bot

v

hx

w

parxyz

Calculate P'xy

hx

vpar2

o kk

hLd2

F ×=

par1 d

L5.0F =

parx2 d

L5.0v2F +=

parx3 d

L5.0v2F −=



Evaluate Fi. If Fi is less than or equal to 1.0

( )( )2iiii F137.0F2ln145.0F'F −−+×−=

Otherwise

( ) ( ) ( )( )2iiii F2137.0F2ln145.0F2'F −−−+×−=

This results in three values of F'1, F'2, F'3. Then

( )( )3210xy 'F'F5.0'FF'P −+×=

Case 2: If k'par> 0.75 k'par >> k'ver then

−

+

+−=

par

par2

par

x

par

x

xper

par2

vhyy d24

3dL

L

dv

dv

31

khdd28.6

kkP

+−

−= y

2y

hy

vper

parxy vv

31

h

kk

d28.6

1L

dP

xyyxy PP'P +=

S'PPS xyxyzr ++=



Shape factor, CH

h

dCH per

ml =

hy

vr k

kk =

y2

y1 vv31

CH +−=

π

=hd

sinlnCH bot2

( )rml3 kCHln5.0CH ×=

088.1CHCHCHkCH28.6CH 321rml −−−=

Effective horizontal permeability

vhyhe kkk ×=

Drainage area—vertical plane

parvert dhD ×=



Calculate rate Oil well

+−+

µ

−=

rw

ll

wfrparhr

S75.0CH

12rA

lnB

)PP(rk00708.0Q

Gas well

+−+

+

ψ−ψ=

rw

wfrparhrg

S75.0CH

12rA

ln)460T(1422

)(rkQ

where: h = Reservoir thickness (ft)


dper = Perpendicular drainage distance (ft)

khy = Horizontal permeability in Y direction (md)

dpar = Parallel drainage distance (ft)

khx = Horizontal permeability in X direction (md)




vx = Distance ratio in X direction

vy = Distance ratio in Y direction






Bl = Liquid formation volume factor (rb/stb)

A = Drainage area (ft2)







Goode and Thambynaya (1987)

Figure 3.8: Schema for Goode & Thambynaya correlation

The Goode and Thambynaya58 IPR type uses a transient model for use in horizontal wells and takes into account the position of the wellbore.

Calculate anisotropy and position effect

4/1

hy

vww k

k12r

'r

×=

hy

hx

par

wx k

kd'r

v ×=

hy

vwz k

kh'r

v ×=

( )h

'r47.1dz wbot

s

×+=

Dimensionless time calculated Oil well

2wtl

hyd

'rc

tk000264.0t

φµ=

Gas well

2wtg

hyd

'rc

24tk000264.0t

φµ

×=



Position factors

( )par

sdnl d

dLX

+π=

par

sd2n d

dX

π=

( )h

'r2dX wbot

ml

+π=

( )h

'r2dX wbot

2m

−π=

Error function terms

dx1 tvE π=

dz2 tvE π=

Summation terms

( ) in2

1i11 xEerf

i1

S ∑∞

=

=

( ) ( )xim1i

22 zicosxEerfi1

S π= ∑∞

=

where:

( ) ( )Li

XisinXisinx 2nnl

in ××−×

=

( ) ( )w

2mmlim 'r4i

XisinXisinx

×××−×

=

Summation multipliers

x2

par2

1m vd

Sπ

=

π=

z

par2m Lv

hdS

Skin term

SL'r2

k

khd

Sw

v

hypar

h ×=



Rate Oil well

( )( )h2m21m1dwlo

wfrhypar

SSSSSt'rB4.282

PPhkdQ

+++πµ

−=

Gas well ( )

( )h2m21m1dw

wfrhyparg

SSSSSt)460T('r2844

hkdQ

+++π+

ψ−ψ=

where:



khy = Horizontal permeability in Y direction (md)

vx = X position factor

dpar = Parallel drainage distance (ft)

khx = Horizontal permeability in X direction (md)

vz = Z position factor


zs = Vertical position factor


r'w = Adjusted wellbore radius (in)

t = Producing time (hr)

φ = Reservoir porosity





dsd = Distance to reservoir side (ft)

S = Formation skin









4 Completion Component

The completion component is a critical part of an efficient producing system, yet it often is taken lightly. In the past, it has been found that many wells have produced at less than optimum levels due to inadequacies in the completion design.

There are five primary completion types for an oil or gas well. The well depth, well type and formation characteristics generally govern the decision of which completion method is used.

Open Hole Completion

In an open hole completion, illustrated in Figure 4.1, casing is set ,and usually cemented, directly above the producing horizon prior to penetration of the zone. This type of completion is the least expensive and, although common several years ago, is not used often in the industry today. The disadvantage of this completion method is the inability to isolate any part of the producing formation for stimulation, water shut-off, etc.

Figure 4.1: Open Hole Completion

In system analysis, the open hole is generally regarded with no pressure loss between the sandface and the wellbore. If any damage is incurred at the formation face, it can be accounted for as an additional skin effect within the reservoir. In the open hole completion, the sandface pressure, Pws, is considered equal to the wellbore pressure, Pwf.

Completion Component PERFORM Technical Reference Manual


Open Perforation Completion

The most common form of well completion used today is the perforated completion. In this case, pipe is set and cemented through the producing formation and subsequently perforated to allow the flow of fluids from the formation into the wellbore.

Figure 4.2: Open Perforation Completion

The variables that determine the efficiency of this completion method include the size and number of perforations, the distribution of perforations, and the integrity of the reservoir rock directly adjacent to the perforated tunnel. In 1983, Harry McLeod98 published a paper that provided a practical solution to the effects of a perforation on the well productivity. The approach was to treat each perforation as a miniature, horizontal wellbore surrounded by a crushed or compacted zone of reduced permeability, as shown in Figure 4.3.

Figure 4.3: Open Perforation

PERFORM Technical Reference Manual Completion Component


The perforation crushed or compacted zone is generally assumed to be 0.5 inches thick, with a permeability of 20 percent of formation permeability if shot over-balanced, and 60 percent of formation permeability if shot under-balanced. In order to predict the pressure loss through the perforation, the miniature wellbore is assumed to be infinite acting, and the Jones equation (discussed previously in the reservoir section) is modified as follows:

Open Perforated Completion—oil well

bQ + aQ = P P p2pwfws −

where:

( ) [ ]2p

cp2

p14

L

r/1r/1B1030.2a

−ρβ×=

−

( )[ ]( ) cp

pc

kL00708.0

r/rlnBb

µ=

where:



Qp = Liquid flow rate per perforation (stb/d/perf)

βp = Perforation turbulence factor (ft -1)



rp = Radius of perf tunnel (ft)

rc = Radius of compacted zone (ft)

Lp = Perforated tunnel length (ft)


kc = Compacted zone permeability (md)

JOSELUIS

Resaltado



Open Perforated Completion—gas well

bQ + aQ = P P p2p

2wf

2ws −

where:

( ) [ ]2p

cppg12

L

r/1r/1TZ1016.3a

−βγ×=

−

( ) ( )[ ]cp

pcp

kL

r/rlnTZ1424b

µ=

where:



Qp = Gas flow rate per perforation (Mscf/d/perf)


βp = Perforation turbulence factor (1/ft)


Z = Average gas compressibility factor

rp = Radius of perforated tunnel (ft)





It is suggested that the majority of pressure loss through a perforation is incurred as a result of the turbulent flow through the crushed or compacted zone.



Stable Perforation Completion

An alternative method for considering the pressure drop through the completion is given by modifying the previous equations to account for an additional damaged zone near the wellbore due to drilling or completion fluid incompatibilities with the formation, or due to formation damage from other sources. The equations used to account for the pressure loss through the completion using this model are:

Stable Perforation Completion—oil well

[ ]DQ + QShk

B 141.4 = P P 2

Pptotpr

wfwsµ

−

where:

( )LrN

hkB10 1.63 = D

2pp

2

prp-16

µ

ρβ×

and:

S + S + S = S dpdptot

( )[ ] ( )[ ][ ]drcrpcppdp k/kk/kr/rlnNL/hS −×=

( )[ ]1 k/kr/rln = S drwdd −

JOSELUIS

Resaltado



and where:





kr = Formation permeability (md)


Qp = Liquid flow rate per perforation (stb/d/perf)



N = Total number of perforations

rp = Radius of perf tunnel (ft)


Sp = Skin factor due to perforation geometry



kd = Damaged zone permeability (md)

rd = Radius of damaged zone (ft)

rw = Radius of the wellbore (ft)



Stable Perforation Completion—gas well

[ ]DQ + QShk

ZT1424 = P P 2

pptotpr

g2wf

2ws

µ−

where:

( )LrN

hk10 2.22 = D

2pgp

2

prgp-15

µ

γβ×

and:

S + S + S = S dpdptot

( )[ ] ( )[ ][ ]k/k k/kr/rlnN L/h = S drcrpcppdp −×

( )[ ]1 k/kr/rln = S drwdd −

where:





T = Formation temperature (R)



Qp = Gas flow rate per perforation (Mscf/d/perf)




rp = Radius of perforated tunnel (ft)


Sp = Skin factor due to perforation geometry




rd = Radius of damaged zone (ft)

rw = Radius of the wellbore (ft)



Collapsed Perforation Completion

An alternate method for determining the pressure loss through a perforated well is used when the damage caused by perforating is severe. This usually occurs when sands move or slough during perforation washing or acidizing. The method used to predict the completion pressure loss under these conditions assumes that the fluid is flowing spherically into each perforation. The resulting pressure drop is significantly higher than for 'normally' perforated wells or 'normal' perforated wells as shown by McLeod.41

Figure 4.4: Collapsed Perforation (Spherical Flow Model)



Collapsed Perforation Completion—oil well

( )DQ + QShk

B2141. = P P 2

totpr

wfwsµ

−

where:

( )µ

ρβ×

rNhkB10 5.42

= D3p

2

prp-17

and:

sdstot SSS +=

[ ] [ ][ ][ ]r48 1r1//Nh + /Nh45S ppp1.1

ps −=

[ ][ ][ ][ ]k/k k/kr/r 1r1//Nh = S drcrcpppsd −−

1.2d

10

pk

10 2.60 =

×β

121

+ r= r pc

where:







Q = Liquid flow rate (stb/d)




rp = Radius of perforation tunnel (ft)






Collapsed Perforation Completion—gas well

( )DQ + QShk

ZT1424 = P P 2

ggtotpr

gwf

2ws

2µ

−

where:

( )µ

γβ

gp32

prgp-16

rN

hk10 x 7.37 = D

and:

sdstot SS = S +

[ ] [ ][ ][ ]r48 1r1//Nh + /Nh45 = S ppp1.1

ps −

[ ][ ][ ][ ]k/k k/kr/r 1r1//Nh = S drcrcpppsd −−

1.2d

10

pk

10 2.60 =

×β

121

+ r = r pc

and where:





T = Formation temperature (oR)







rp = Radius of perforation tunnel (ft)


kc = Damaged zone permeability (md)



Gravel Pack Completion

In many wells, matrix cementing within the formation sand is insufficient to prevent sand from being produced when the well is exposed to a pressure drawdown. Problems that are realized because of this include restricted production, erosion of equipment and sand disposal problems. In order to eliminate this sand control problem, the gravel pack technique was developed.

In a standard gravel pack arrangement, illustrated in Figure 4.5, the well is perforated and sometimes washed to remove debris. A slotted liner or gravel pack screen is run on tubing and set across the perforations. High-permeability gravel is then pumped downhole and placed between the screen and the perforations to provide a barrier between the formation sand and wellbore.

Figure 4.5: Gravel Pack Schematic

In system analysis, the technique used to predict completion effects through a gravel pack is simply the pressure loss due to linear flow through the gravel. The flow of fluid is assumed to be linear through the perforation tunnel, through the gravel pack, and into the perforated or slotted liner. The effect of flow through a perforation-damaged zone is considered negligible due to the high permeability and unconsolidated nature of wells that are typically gravel packed. However, the effect of linear flow through the gravel filled perforation tunnel can cause significant non-Darcy pressure drop as shown by McLeod.41 The distance of linear flow generally is assumed to be from the outside of the cement sheath to the outside edge of the liner or screen.



The effect of the perforation tunnel itself is not considered in the gravel pack equation, however, it can be considered by simply adding the pressure loss of the open perforation equation to the gravel pack pressure loss to arrive at a total completion loss. The same method is used with the gravel pack stable perforation and gravel pack collapsed perforation completion models. It must be realized that in all cases the perforation tunnel past the cement sheath is NOT considered filled with sand with the exception of the collapsed perforation model. If you want to consider sand in the perforation tunnel, it is suggested that the gravel packed collapsed perforation model be used.

The gravel pack equation uses the Jones equation modified to predict the pressure loss through the completion in a gravel packed well. The equation is used in the linear form.

Gravel Pack Completion—oil well

bQ + aQ = P P2

wfws −

where:

( )

( ) Ak0.001127LB

= b

A

LB10 9.08 = a

g

g

2

g2

g13-

µ

ρβ×

and where:




βg = Gravel pack turbulence factor (1/ft)



Lg = Gravel pack linear flow length (ft)

A = Total area open to flow (ft2)(area/perf)(SPF)(Hp)


kg = Permeability of gravel (md)



Gravel Pack Completion—gas well

bQ + aQ = P P22

wf2ws −

where:

( )

( )Ak

LTZ8930 = b

A

LTZ10 1.247 = a

g

gg

2

ggg10-

µ

βγ×

and where:



Q = Gas flow rate (Mscf/d)

γg = Gas Specific gravity(area/perf)(SPF)(Hp)


T = Avg. reservoir temp (°R)

Z = Avg. gas compressibility factor

Lg = Gravel pack linear flow length (ft)

A = Total area open to flow (ft2)


kg = Effective gas permeability (md)

Gravel Pack Beta Turbulence Factor

The most common equations used to determine the gravel pack turbulence factor term, βg, are the Crawford-McLeod41 equation, the Cooke equation,36 the Saucier equation,11 and the Firoozabadi and Katz equation.37 Additionally, Tenneco has determined that the turbulence term for a resin pack can be predicted by using a separate equation from those listed.

Crawford-McLeod equation

( )0.5gravel

gk

10,000,000 = β

where:


kgrave

l

= Gravel permeability (md)



Cooke equation

( )Fgravel

g/1000k

E 30,920,000 = ×β

where E and F are dependent on the gravel pack sand size.

GRAVEL SIZE E F

8-12 3.32 1.24

10-12 2.63 1.34

20-40 2.65 1.54

40-60 1.10 1.60

Saucier equation

( )( )( )1000/klog 0.95 6.5g

gravel10 = ×−β

Firoozabadi and Katz equation

( ) 55.0gravel

g k4,700,0001

= β

Tenneco equation for resi n packs

( )( )/1000klog - 5.7g

gravel10 = β

Unless company policy or field experience dictates otherwise, the Firoozabadi and Katz equation is recommended for use in determining the gravel pack turbulence factor.

Gravel Pack Open Hole Completion

The gravel pack open hole completion is a gravel packed well that does not have casing or perforations across the producing zone.

Gravel Pack Open Hole—oil well

12L

r = r

h k 0.00708rr

log B Q = P P

gws

g

s

w

wfws

−

µ

−



Gravel Pack Open Hole—gas well

12L

r = r

h k 10 7.03rr

log ZT Q = P P

gws

g4-

s

wg

2wf

2ws

−

×

µ

−

where:



Q = Flow rate (stb/d or Mscf/d)

µ = Oil viscosity (cp)

B = Formation volume factor (rb/stb)


rs = Gravel pack screen radius (ft)

kg = Gravel pack sand permeability (md)

H = Reservoir thickness (ft)

Lg = Gravel pack linear flow length (in)


T = Avg. reservoir temp (°R)


Gravel Pack Open Perforation Completion

The gravel pack open perforation completion is a numerical addition of the gravel pack and open perforation models.

Gravel Pack Stable Perforation Completion

The gravel pack stable perforation completion is a numerical addition of the gravel pack and stable perforation models.

Gravel Pack Collapsed Perforation Completion

The gravel pack collapsed perforation completion is a numerical addition of the gravel pack and collapsed perforation models. Wells with collapsed perforations typically have sand control problems and are typically gravel packed. This model may be best used in most gravel packed well situations.




5 Wellbore and Flowline

The vertical flow component of the well system is necessary for calculating the wellbore or tubing curve for the system analysis at several flow rates. It is also used to calculate the gradient curve for the gradient analysis at a defined flow rate. For a system analysis, the calculation of the vertical flow is dependent on the node position. If the node is at the bottom of the well, then the vertical flow component is the outflow curve of the well system. If the node is at the wellhead, then the vertical flow component is part of the inflow curve. In this case, the outflow curves are nothing more than constant pressure curves if there is no flowline considered in PERFORM.

In system analysis, with the node at the top perforation in the wellbore, the outflow segment is defined as the summation of the components between the node and the downstream endpoint of the system, usually the separator (with a flowline) or wellhead (no flowline). Because this discussion designates the node at a point within the wellbore directly adjacent to the top perforation of the completion or the top of the reservoir interval in an open hole completion, the outflow segment is comprised of the following components:

• Flow through wellbore downhole safety valves or restrictions

• Flow up the tubing

• Flow through surface valves, restrictions, or chokes

• Flow through the flowline

In most producing well systems, flow up the tubing constitutes the majority of pressure loss in the outflow segment, if not the entire system. In fact, in some oil wells more than 80 percent of the pressure loss in the entire system occurs in the tubing as fluids are moved vertically from downhole to the surface.

The flowline component is usually the second most predominant pressure loss component in the outflow segment followed by the valves, chokes and other restrictions. In general, pressure loss through restrictions is minimal unless an obvious undersizing or similar abnormality is present. For this reason, the majority of this discussion will concentrate on the effects of the vertical flow (tubing) component of the outflow segment.

In a typical oil or gas well, predicting the pressure loss through the tubing (and flowline) is complicated by the fact that more than one fluid phase generally exists in the producing stream. This multiphase behavior causes a problem in determining the fluid characteristics necessary for the pressure drop calculation. Because of the complexity, the remainder of the discussion on the outflow segment will avoid theory and will concentrate on the results of the work done to date on the subject.

Wellbore and Flowline PERFORM Technical Reference Manual


Oil Well Vertical Flow

The general pressure gradient equation for vertical flow can be summarized as:

dZdP

+ dZdP

+ dZdP

= dZdP

accelfricelevtotal

The elevation component is a function of average liquid density calculated using a liquid holdup value. Holdup is defined as the volumetric fraction of the liquid phase to the total flowing fluid. The friction component requires the determination of a two-phase friction factor. The acceleration component is significant only in cases of extremely high flow velocities, and is generally considered negligible.

Many correlations have been developed over the years to predict the relationship of the gradient components to vertical multiphase flow. Beggs and Brill21 have summarized these correlations in three main categories, each varying in complexity and technique.

• Category A: No slip effect or flow regime considered

• Category B: Slip considered, no flow regime considered

• Category C: Slip and flow regime considered

Slip is defined as the movement of the gas phase by the liquid phase when the two phases are flowing independently at different velocities. Flow regimes have been suggested to describe these different types of flow patterns that can exist in multiphase flow. These include bubble, slug, transition, and mist flow.13

There have been many multiphase flow correlations developed to date. Yet, all of the investigators maintain that no correlation has been found to be superior to all others for all flow conditions. Individual well test data and experience in an area can be used to obtain the correlation that will best fit each well's characteristics. In lieu of having data to validate a particular correlation type, the Hagedorn and Brown correlation is suggested as the initial correlation to use in oil wells and the Orkiszewski correlation for gas wells with GLR's above 50,000 scf/bbl. Use the Gray correlation for gas condensate wells. The following sections describe some of the more predominant correlations by category type.

PERFORM Technical Reference Manual Wellbore and Flowline


Category A

CORRELATION EXPLANATION

Poettmann & Carpenter25

Used field data to prepare a correlation that treated the multiphase flow as though it were a single, homogeneous phase. Assumed that the flow had a high degree of turbulence and that flow would be independent of viscosity effects. It can be used with confidence for the following conditions.

• Tubing sizes, 2, 2.5, and 3 inches.

• Viscosities less than 5 cp.

• GLR less than 1500 scf bbl.

• Flow rates greater than 400 bpd

Baxendell & Thomas26

Used La Paz and Mara field (Venezuela) data to develop a revision of the Poettmann method to perform better at higher flow rates.

Fancher & Brown27 Used data generated from an 8,000-ft experimental well equipped with 2 3/8-in. plastic coated tubing to develop a revision to the Poettmann method to better match low rate, high GLR cases. Used for:

• GLR less than 5000 scf/bbi

• Flow rates less than 400 bpd

• Extended to 2 7/8 in. tubing

Category B


Hagedorn & Brown12 Developed experimentally using a 1500-ft test well with 1-in., 1.25-in., and 1.5-in. tubing. The correlation is used extensively throughout the industry and is recommended for wells with minimal flow regime effects and generally with GLR < 10,000 scf/bbl. The Griffith and Wallis correlation can be used for improved performance in bubble flow regimes.



Category C


Orkiszewski14 Developed using work from both Duns & Ros and Hagedorn & Brown. Used Griffith and Wallis30 method for bubble flow, a new method for slug flow, and Duns and Ros for transition and mist flow. The Triggia liquid distribution coefficient can be used if desired when the mixture velocity is greater than 10 ft/sec. It was developed to eliminate pressure discontinuities.

Duns & Ros13 The result of laboratory work where liquid holdup and flow regime were observed. Utilized a flow pattern map to determine the slip velocity (and consequently liquid holdup) and friction factor. This correlation is recommended for wells where high gas-liquid ratios and flow velocities have induced flow regime behavior.

Aziz, et al. 29 Presented new correlations for bubble and slug flow. Used Duns & Ros for transition and mist flow. Also revised the flow regime map.

Beggs & Brill21 This correlation was developed experimentally using 1-in. and 1.5-in. pipe, inclined at several angles. Correlations were made to account for inclined flow. The correlation is recommended for deviated wells or horizontal flow. The correlation is recommended for deviated wells or horizontal flow. You can use the Palmer correlation to correct for liquid holdup effects. Note that the Palmer correlation is unsuitable for single phase flow and should be used with caution.

Mukherjee & Brill28 Developed experimentally using 1.5-in. steel pipe inclined at several angles. Included downhill flow as a flow regime. Recommended for inclined or horizontal flow.

MONA49 Correlation requiring three flow coefficients to model vertical flow

from actual data to account for phase slippage. Coeff 1 is the relative velocity of the liquid phase. Coeff 2 represents the additional velocity of the gas phase over the liquid phase such that the gas velocity is (Coeff 1 X liquid velocity) + Coeff 2. Coeff 3 is a two-phase friction factor. Use 1.0 by default.

For nominal results, set Coeff 1 to 1.2, Coeff. 2 to 1.43, and Coeff. 3 to 1.00 for nominal results and change the Coeff 1 as needed to adjust the liquid holdup. For homogeneous flow with no slip, set coeff 1 to 1.0, coeff. 2 to 0.0, and coeff 3 to 1. For vertical slug flow, set coeff 1 to 1.2, coeff 2 to 0.35, and coeff 3 to 1.0.

MONA Modified49 Correlation requiring two flow coefficients to model vertical flow from actual data. Set Coeff 1 to 1.0 and Coeff. 2 to 0.0 for nominal results and change the Coeff 1 as needed to adjust the liquid holdup. Coeff. 2 is normally not changed. The Modified MONA omits Coeff. 3 because of the friction factor being calculated using the Moody factor with either the laminar flow or the Colebrook equations. If the flow is laminar, it uses the Blasius friction factor for the first guess in the Colebrook equation and therefore does not need Coeff. 3.

JOSE

Resaltado

JOSE

Resaltado




Sylvester & Yao Mechanistic70

Mechanistic and empirical combination model for predicting pressure traverses for two-phase flow using flow pattern prediction and a set of independent mechanistic models. Can be used for vertical and inclined flow.

Ansari Mechanistic74 Consists of a comprehensive model to predict flow behavior for upward two-phase flow composed of a model for predicting the flow patterns and independent models for predicting holdup and pressure drop dependent on the flow pattern. The model was compared to a 1,712 well data bank and found to match better than any of the other empirical or mechanistic models. Uses bubble flow, slug flow, and annular flow models.

In vertical multiphase flow calculations, the pipe is divided into small increments based either on a set length or pressure amount. The pressure loss in each increment is determined in a trial-and-error process using average pressure and temperature values to calculate fluid properties. The iterative procedure is necessary as flow regime and subsequent fluid and flow properties change continually through the pipe. As a result, computer solution is almost mandatory; however, curves have been prepared and published to aid hand calculations.

The pressure loss calculated over the entire pipe interval is related in part to the size and number of increments chosen. Each of the correlations listed relates to certain wells and well conditions. The determination of the best-suited correlation for a particular well is accomplished by first using the preliminary guidelines listed earlier, followed by testing and comparison to actual field results.

Many of the correlations presented actually use the methods of other authors in certain instances or flow regimes where the other author describes the pressure traverse in that environment. These correlation switches are documented by the author in his original or subsequent work. This manual does not document when or where these correlation switches are done, but they are done according to the correlation author's method.

JOSE

Resaltado

JOSE

Resaltado

JOSE

Resaltado



Gas Well Vertical Flow

Gas well vertical flow is very similar to the flow of oil wells except that special correlations have been developed when gas is the primary phase. Some correlations can be used in either oil or gas wells.


Gray17 Gas well correlation for wet gas condensate consideration. Calculates the dew point of the gas to predict flow behavior with a two-phase flow when the condensate condenses into liquid.

Modified Gray Uses the Gray correlation with ability to modify two coefficients used in the Gray correlation. Set Coeff. 1 to 2.314 and Coeff. 2 to 0.0 for the original Gray result.

Duns & Ros13 The result of laboratory work where liquid holdup and flow regime were observed. Utilized a flow pattern map to determine the slip velocity (and consequently liquid holdup) and friction factor. This correlation is recommended for wells where high gas-liquid ratios and flow velocities have induced flow regime behavior.

Hagedorn & Brown12 Developed experimentally using a 1500-ft test well with 1-in., 1.25-in., and 1.5-in. tubing. The correlation is used extensively throughout the industry and is recommended for wells with minimal flow regime effects and generally with GLR < 10,000 scf/bbl. The Griffith and Wallis correlation can be used for improved performance in bubble flow regimes.

Ros & Gray Combines the Duns & Ros flow regime maps for niist flow with the Gray correlation for wet gas wells that produce condensate.

Cullender & Smith50 Used for dry gas well calculations only for predicting dry gas pressure losses in vertical flow. Suggested for wells where the GLR is 100,000 scf/bbl or higher.

Fundamental Flow4 Uses a basic pressure gradient equation derived from an energy balance integrated over the entire flow distance to give an equation which is similar- to the Cullender and Smith equation. If liquids are also in the flow stream, suggest using the Fundamental flow adjusted correlation. Suggested for wells where the GLR 50,000 scf/bbl or greater.

Fundamental Flow Adjusted

Same as the Fundamental flow correlation except an adjustment to the gas gravity is made to account for liquids in the flow stream. The liquid gravity and GLR is combined to calculate an adjusted gas gravity. Suggested for wells where the GLR 50,000 scf/bbl or greater.



Oil Well Horizontal Flow

The following correlations are available for determining pressure losses in pipelines for oil wells. Some are for a single phase while others are for two-phase flow.


Xiao Mechanistic76 Comprehensive mechanistic model developed for gas-liquid two-phase flow in horizontal and near horizontal pipelines. The model first detects the existing flow pattern, predicts the flow characteristics (liquid holdup and pressure drop) for stratified, intermittent, annular, or dispersed bubble flow patterns.

Beggs, Brill, & Minami

Modification of the original Beggs & Brill correlation for horizontal flow only.

Dukler15 Simple horizontal flow correlation that does not require determination of flow patterns. It includes effects for single and two-phase flow in horizontal flow only.

MONA49 Correlation requiring three flow coefficients to model vertical flow from actual data to account for phase slippage. Coeff 1 is the relative velocity of the liquid phase. Coeff 2 represents the additional velocity of the gas phase over the liquid phase such that the gas velocity is (Coeff 1 X liquid velocity) + Coeff 2. Coeff 3 is a two-phase friction factor. Use 1.0 by default.

For nominal results, set Coeff 1 to 1.2, Coeff. 2 to 1.43, and Coeff. 3 to 1.00 for nominal results and change the Coeff 1 as needed to adjust the liquid holdup. For homogeneous flow with no slip, set coeff 1 to 1.0, coeff. 2 to 0.0, and coeff 3 to 1. For vertical slug flow, set coeff 1 to 1.2, coeff 2 to 0.35, and coeff 3 to 1.0.

MONA Modified49 Correlation requiring two flow coefficients to model vertical flow from actual data. Set Coeff 1 to 1.0 and Coeff. 2 to 0.0 for nominal results and change the Coeff 1 as needed to adjust the liquid holdup. Coeff. 2 is normally not changed. The Modified MONA omits Coeff. 3 because of the friction factor being calculated using the Moody factor with either the laminar flow or the Colebrook equations. If the flow is laminar, it uses the Blasius friction factor for the first guess in the Colebrook equation and therefore does not need Coeff. 3.

Mukherjee & Brill28 Developed experimentally using 1.5-in. steel pipe inclined at several angles. Included downhill flow as a flow regime. Recommended for inclined or horizontal flow.

Beggs & Brill21 This correlation was developed experimentally using 1-in. and 1.5-in. pipe, inclined at several angles. Correlations were made to account for inclined flow. The correlation is recommended for deviated wells or horizontal flow. The correlation is recommended for deviated wells or horizontal flow. You can use the Palmer correlation to correct for liquid holdup effects. Note that the Palmer correlation is unsuitable for single phase flow and should be used with caution.



Gas Well Horizontal Flow

Gas well horizontal flow is designed to predict pressure losses in horizontal flow. Some sections of pipelines are many times vertical, therefore some vertical flow correlations are available for these sections of the pipeline. Some correlations can be used in either oil or gas wells. The IGT, Weymouth, Panhandle Eastern , and Panhandle A correlations differ only by the coefficients used in the following equation.

5280zL)460T(

d1

P460T

f1a

1000qPP

3a/1

5a

4a

g

2a

SC

SCe

gU

2D

+

γ

+

×−=

where:

PD = Downstream pressure (psi)

Pu = Upstream pressure (psi)

qg = Gas rate (Mscf/d)

a = Coefficients

fe = Flow efficiency

Tsc = Standard temperature (°F)

Psc = Standard pressure (psia)

γg = Gas gravity (air=1.00)

d = Pipe diameter (in)

T = Gas temperature (°F)

z = Gas compressibility factor

L = Pipeline length (ft)




IGT a1=337.90 a2=1.1110 a3=0.5560 a4=0.4000 a5=2.667

Modified coefficients of Weymouth.

Panhandle Eastern

a1=737.00 a2=1.0200 a3=0.5100 a4=0.4901 a5=2.530

Use for larger diameter pipelines 16-in. or greater.

Panhandle A a1=435.87 a2=1.0788 a3=0.5394 a4=0.4604 a5=2.618

Recommended for smaller diameter pipelines less than 16-in. ID. Use the Panhandle Eastern correlation for larger diameter pipelines.

Weymouth a1=433.50 a2=1.0000 a3=0.5000 a4=0.5000 a5=2.667

Determined basic gas flow coefficients.

Cullender & Smith50

Used for dry gas well calculations only for predicting dry gas pressure loses in vertical flow. Suggested for wells where the GLR is 100,000 scf/bbl or higher. Use in vertical sections of pipeline only.

Fundamental Flow

Uses a basic pressure gradient equation derived from an energy balance integrated over the entire flow distance to give an equation which is similar- to the Cullender and Smith equation. If liquids are also in the flow stream, suggest using the Fundamental flow adjusted correlation. Suggested for wells where the GLR 50,000 scf/bbl or greater. Use in vertical flow sections of the pipeline only.

Fundamental Flow Adjusted

Same as the Fundamental flow correlation except an adjustment to the gas gravity is made to account for liquids in the flow stream. The liquid gravity and GLR is combined to calculate an adjusted gas gravity. Suggested for wells where the GLR 50,000 scf/bbl or greater. Use in vertical flow sections of the pipeline only.

Flow Through Restrictions

Pressure loss occurs when fluids flow through restrictions or chokes and needs to be accounted for in the wellbore or flowline of the system. The pressure loss calculations involve two distinct types of choke performance. These are sub-critical and critical flow.

The Perkins69 correlation is used for all calculations in the beginning to check if the flow is critical or sub-critical. If the correlation selected is Gilbert20, Ros66, Baxendell, Achong39 (critical flow only), and Perkins determined that the flow is sub-critical, then the Perkins results are used for the pressure drop disregarding the selected correlation. If the flow is critical then the selected correlation is used. If the selected correlation is API-14B33 (sub-critical only) and Perkins correlation determines that the flow is critical, then the Perkins correlation results are used for the pressure drop disregarding the selected correlation. Otherwise, the API-14B as selected is used.



Critical Flow

Pressure drops are highly dependent on the flow velocity through the restriction. If the velocity is at the speed of sound, a compressional wave is generated. This compressional wave prevents fluids from traveling faster than the wave generated. The flow rate at which this occurs is called the critical flow rate. At the critical flow rate, any adjustment to the pressure on the downstream side of the restriction does not affect the pressure distribution of the upstream side of the restriction. Various investigators have used a similar technique for predicting the upstream pressure for flow rates above or at the critical flow rate.20, 37-39 All take the form of:

D

Q GLR A = P c

liqB

u

××

where A, B, and C depend on the investigator. Pu is the upstream pressure at the liquid flow rate, Qliq, in bbls/day. D is the choke size in 64ths-in. and GLR is in Mscf/bbl or scf/bbl depending on the 'A' constant used. These methods are only valid if the node is at the bottom of the well. The following table summarize the coefficients.

INVESTIGATOR A B C

Gilbert20 10.00 0.546 1.89

Ros66 17.40 0.500 2.00

Baxendell 9.56 0.546 1.93

Achong39 3.82 0.650 1.88

Subcritical Flow

Under subcritical flow, the mass flow rate of a stream will be a function of the pressure downstream of the choke when the upstream pressure is held constant. If the pressure drop across the choke becomes sufficiently large, the flow regime will become critical and the mass flow rate will be independent of the downstream pressure when the upstream pressure is held constant.



API 14B

For subcritical flow (flow that is below the critical flow rate), the method used is API 14B (used in the API Subsurface Controlled Subsurface Safety Valve Design Program). It is used when specific restriction information is unavailable. This method uses an iterative technique that estimates a Y term based on an assumed gas pressure drop. A new gas pressure drop is calculated and compared to the previous estimation. If the difference between the two values is within a tolerable range, a liquid pressure drop is calculated based on the Bernoulli equation for incompressible flow. A two-phase pressure drop is then calculated. The equations used in this method are:33

2

DL2c

4

t

cL Cd 80083

Q

dd

1 = P

−ρ∆

∆

−

U

G

P

V

4

t

c

PP

CC

dd

0.35 + 0.41 1 = Y

2

DG

DLLG YC

C P = P

∆∆

1 YCC

f + 1 P = P2

DG

DLL

−

∆∆

( )( ) wwoogosg

gosg

Bq + Bq + BqR q

BqR q = f

−

−



where:

∆PL = Differential pressure drop for liquids (psia)

ρ = Weight density of fluid (lbs/ft3)

dc = Orifice or bean diameter (in)

dt = Flow tube diameter (in)

Q = Flow rate (stb/d)

Cv/Cp = Ratio of specific heats for gas at constant pressure Cp and constant volume Cv

CDL = Orifice discharge coefficient for liquid

Y = Net expansion factor for compressible flow through a bean

∆PG = Differential pressure drop for gas (psia)

PU = Bean entry pressure upstream (psia)

CDG = Orifice discharge coefficient for gas

∆P = Differential pressure drop (psia)

F = Free gas volumetric fraction

qg = Gas flow rate (scf/d)

Rs = Gas solubility ratio (scf/stb)

qo = Oil flow rate (stb/d)

Bg = Gas formation volume factor (bbl/scf)

Bo = Oil formation volume factorl (bbl/stb)

qw = Water flow rate (bbl/d)

Bw = Water formation volume factor (bbl/bbl)



Critical and/or Subcritical Flow

Perkins

Perkins uses a general energy equation to describe isentropic (adiabatic with no friction) flow of multiphase mixtures through chokes for both critical and sub-critical flow. The procedure determines whether the flow is critical or sub-critical from equations and physical property information for oil-water-gas systems for determining the mass flow rate of the choke. The method solves for upstream pressure if the mass flow rate and downstream pressure are known. Conversely, it solves for downstream pressure if the mass flow rate and upstream pressure are known.

Determine choke throat temperature

( )[ ] 460p460TT n/)1n(r12 −+= −

Where n is the polytropic expansion factor

vwwvoovgg

vwwvoovgg

CfCfCf

CfCfFCfn

++

++=

Calculate average temperature and pressure upstream of the choke

2pp

p 21 +=

2pp

T 21 +=

Recalculate the polytropic expansion factor exponent, n, at the average pressure and temperature Iterate on pr until the following equation is satisfied

( )[ ] ( ){ } ( ) ( ) ( )

( )

α+

α+

+

α+

α+

−−α+−λ

−

+−+−

−−

2

1n/1

rg

n/n1r

21gg

2

1

2n/n1r

g

2

1n/1

rg

1g2

1

2r1

n/1nr

pf

pf

n

f

AA

pn

f

pf

f

AA

1p12p12

( )

α+

−

λα+

α+

α+

−= −−

− 1n/1

r1n/1

rg

2

1n/1

rg

1g

1

2 pn

1npf

pf

f

AA

1

Calculate downstream pressure (p3)

( )85.1

d

c

4113

dd

1

pppp

−

−−=



If pr>p3, the flow is critical then use pr=p2/p1 in the following equations.

If pr<p3, the flow is sub-critical and pr=p3/p1 in the following equations for calculating isentropic velocity and mass flow rate.

Calculate lambda (λ) term

( )zR

MCfCfCff vwwvoovgg

g

+++=λ

Calculate alpha (α1) term

ρ

+ρ

ρ=αw

w

o

ol1

ff

Calculate isentropic velocity in the choke throat

[ ] ( )

α+

α+

−

−

ρ

+

ρ

+−λ

=

−

−

1n/1

rg

1g2

1

2

r1w

w

o

on/)1n(r11c

2

pf

f

AA

1

p1pff

p1vpg288

V

Calculate isentropic mass flow rate

[ ] ( )

( )2

1n/1

rg

2

1n/1

rg

1g2

1

2

r1n/)1n(

r

1

1c2i

pfpf

f

AA

1

p1p1v

pg288Aw

α+

α+

α+

−

−α+−λ=

−−

−

If the actual mass flow rate is not known, calculate the actual mass flow rate with a discharge coefficient (default is 0.826).

wa = 0.826 wi



Calculate downstream pressure

p3=pr × pl

where:

T2 = Choke throat temperature (°F)

T1 = Upstream temperature (°F)

pr = Pressure ratio

n = Polytropic expansion exponent

fg = Weight gas fraction

F = Cp/Cv

Cvg = Gas heat capacity at constant volume

fo = Weight oil fraction

Cvo = Oil heat capacity at constant volume

fw = Weight water fraction

Cvw = Water heat capacity at constant volume

p1 = Upstream pressure (psia)

p2 = Choke throat pressure (psia)

λ = Calculation factor

α = Upstream calculation factor

A2 = Choke throat area (ft2)

A1 = Upstream pipe are (ft2)

p3 = Choke outlet pressure (psia)

p4 = Pipe downstream pressure (psia)

dc = Choke diameter (ft)

dd = Downstream pipe diameter (ft)

M = Molecular weight


R = Universal gas constant (ft-lbf/lbm/mol/°R)

ρ l = Liquid density (lbm/ft3)

ρo = Oil density (lbm/ft3)

ρw = Water density (lbm/ft3)

V2 = Choke throat velocity (ft/sec)

gc = Gravity acceleration

vl = Upstream specific volume (ft3/lbm)

wi = Isentropic mass flow rate (lbm/sec)

wa = Actual mass flow rate (lbm/sec)



Ashford and Pierce

The Ashford and Pierce68 choke calculation for critical and subcritical flow is very similar to the Perkins method. The method considers polytropic expansion of the gas phase of the fluid expanding through the choke. First, an expression is written relating the flowing fluid specific volume and velocity to the mass flow rate. Second, an independent equation is used to incorporate the behavior of the gaseous phase of the fluid with pressure and using the energy balance equation. The method solves for upstream pressure if the mass flow rate and downstream pressure are known. Conversely, it solves for downstream pressure if the mass flow rate and upstream pressure are known. The critical ratio is defined as the ratio of the upstream pressure to the downstream pressure.

The critical ratio is solved for indirectly by the relationship

( ) ( )( ) 1n

Rn1n2nn

R1n2

n/1cc −

=

−+

ε++ε −

The total flow rate is iterated until the critical ratio is minimized

wgotf qqqq ++=

Thus

+

−

+= wosc1

11scsootf F

TpzTp

615.5RR

Bqq

The oil flow rate

10102

eo Cd51.3q βα=

where

( ) 2/1woo10 FB −+=α

and

( ) ( ) [ ]

( ) [ ]wwogon/1

sl

ll

2/1

wwosgoln

1n

s11

10

FR000217.0RRpzT

6.198

FR000217.01p6.1981RRzT1n

n

λ+λ+λ

ε−+

λ+λ+λ

ε−+

ε−−

−=β

−

−



where:

εc = Critical downstream pressure/upstream pressure

n = Cp/Cv

R = Producing GOR (scf/stb)

qtf = Total fluid flow rate (bbls/d)

qo = Oil flow rate (stb/day)

qg = Gas flow rate (bbls/day)

qw = Water flow rate (bbls/day)

Bo = Oil formation volume factor (rb/stb)

Rs = Solution GOR (scf/stb)

psc = Standard pressure (psia)

Tl = Upstream temperature (°R)

zl = Gas compressibility factor at Tl and pl

pl = Upstream pressure (psia)

Tsc = Standard temperature (°R)

Fwo = Water-oil ratio

C = Orifice coefficient

de = Choke diameter (64th-in.)

ε = Downstream pressure/upstream pressure

λo = Oil specific gravity (water=1)

λg = Gas specific gravity (air=1)

λw = Water specific gravity

Maximum Erosional and Minimum Unloading Velocity

In multiphase production environments, specifically wet gas wells, excessive fluid velocity can become a significant factor in the erosion of pipe walls. The erosional process is primarily caused by small bubbles in the flow stream that form and break as vapor pressure of the liquid is reached. These bubbles ultimately strike the pipe with considerable force usually causing erosion of the corrosion inhibitor films, but may strike with a force great enough to erode the pipe wall. An accepted value for maximum allowable velocity is 50 feet per second.



Maximum Erosional Rate

A correlation method is available to calculate the associated gas flow rate that corresponds to the maximum allowable velocity to prevent pipe wall erosion. The correlation is a function of surface tubing diameter, pressure, temperature, and gas properties. The maximum erosional velocity generally occurs at the wellhead where pressure is at a minimum.

z460T460T

P144

VAP4.86Q

sc

whsc

eroswhma

++

=

where:

Q ma = Maximum erosional rate (Mscf/d)

A = Tubing are at surface (in2)

Pwh = Wellhead pressure (psi)

Veros = Erosional velocity (ft/sec)

Psc = Standard pressure (psia)

Twh = Wellhead temperature (°F)

Tsc = Standard temperature (°F)


The erosional velocity, Veros, can be designated or calculated from a common industry correlation.

meros

CV

ρ=

where C = Predetermined constant usually 100

ρm = Mixture density (lbm/ft3)

The C constant for erosional limit ranges from 100 to 110. This constant is arbitrarily set by engineers for keeping production velocities below the erosional limit. A higher value of C will cause a higher erosional limit, thus increasing the maximum erosional rate.



Minimum Unloading Rate

Minimum unloading velocities represent the pipe velocity needed to effectively unload liquids from gas wells for continuous flow. Efficient production is maintained by producing the well at a sufficiently high rate and tubing velocity to keep all of the produced liquids cleaned out of the tubing. If the rate and velocity fall below the minimum unloading rate for either water or condensate, then the liquids will begin to build up in the tubing and eventually choke off the well to extremely low or no flow conditions.

The unloading velocity and rate is calculated for two liquid phases being 100 percent produced condensate and 100 percent produced water. The actual unloading minimum for wells that produce both water and condensate will fall somewhere between these two unloading minimum limits. The unloading velocity can be either entered directly or calculated using Turner's34

correlation. As a default, it is suggested that the unloading minimum velocity for water is 7 feet per second and for condensate it is 4 feet per second when not using Turner's correlation.

The velocity and corresponding rates calculated at these unloading minimum velocities represent a theoretical minima on the tubing curve based on the theory of Turner. et al.

Turner unloading rate

( ) z460T144APV3060

qwh

whunlunl ×+×

×××=

Modified Turner unloading velocity

These equations assume a gas gravity of 0.6, surface tension of condensate at 40 dynes/cm, and a surface tension of water of 60 dynes/cm.

( )( ) 5.0

wh

25.0whw

unlP00279.0

P00279.03.5)water(V

−ρ=

( )( ) 5.0

wh

25.0whc

unlP00279.0

P00279.003.4)condensate(V

−ρ=

where:

qunl = Unloading rate (Mscf/d)

Vunl = Unloading minimum velocity (ft/sec)

Pwh = Wellhead pressure (psia)

A = Tubing flow are at surface (in2)



ρw = Water density )lbm/ft3)

ρc = Condensate density (lbm/ft3)



Heat Transfer

Flowing temperature distribution or heat transfer is used to model the changes in temperature of the produced flow stream along the path of flow. The fluid temperature is normally a constant in the reservoir, however, when the fluid starts up the wellbore toward the surface, the temperature decreases. The produced fluid heat is dissipated from the fluid to the surrounding environment as it flows to the surface. For gases, the reduction in pressure as the fluid reaches the surface will also cause a reduction in temperature.

Rigorous prediction of wellbore and pipeline temperature distribution is a complex issue. It requires solutions for momentum, continuity, and energy balance. The solution is further complicated by thermal environmental reactions, especially from the reservoir. For this reason, rigorous analytical solutions are impossible, therefore, numerical algorithms or approximate analytical solutions have been developed.

Linear Temperature Gradient

The linear temperature gradient is a simple method of calculating the temperature at any point along the flow stream by determining a temperature gradient between two known points a known distance apart. For the wellbore, a bottomhole and surface temperature is given along with the depth of the well. The temperature gradient is calculated as:

( )tvd

whbht M

TT100g

−=

The temperature at any depth is found by:

whtvdt

d T100Mg

T +=

where:

gt = Temperature gradient (°F/100ft)

Tbh = Bottomhole temperature (°F)


Mtvd = True vertical depth (ft)

Td = Temperature at depth Mtvd (°F)

The flowline linear temperature gradient uses the same equations by substituting the flowline length for the Mtvd value and temperature at the separator for Twh and temperature of the wellhead for Tbh.



Temperature Survey

The temperature survey is used to indicate the temperature or temperature gradient for the flowline or wellbore. Straight line interpolation is used along the temperature path to calculate a temperature at any wellbore depth or distance from the wellhead for the flowline. The first temperature point in the wellbore is at the wellhead and does not need to be entered in the survey because it was entered on the Wellbore dialog box. The last point in the wellbore is the bottomhole temperature also not entered into the table because it was entered in the Reservoir dialog box. The first temperature point in the flowline is the separator temperature and the last point is the wellhead temperature. Neither of these values needs to be entered into the survey table but are carried over from the Flowline and Wellbore dialog boxes.

Heat Transfer Correlation

The heat transfer correlations use environmental data to determine the amount of heat transfer using either empirical or numerical solutions.


Alves et al. unified model60 A unified temperature distribution model for either the flowline or wellbore in production or injection modes. Uses general and unified equations with conservation laws of mass, momentum, and energy balance solved for with simplified yet sound assumptions. This correlation is highly recommended.

Sagar et al. simplified model63

A simple temperature profile model for wellbores only for two-phase flow. The model, developed with measured temperature data from 392 wells, assumes that the heat transfer with the wellbore is steady state. The average absolute error is suggested to be 2.4°F when the mass flow rate is greater than 5 Ibm/sec and 3.9°F otherwise.

Shiu & Beggs63,72 Empirical correlation for wellbores only that determines the relaxation constant defined by Ramey's work. The method is an attempt to avoid the complex calculation of the overall heat-transfer coefficient in the wellbore and the transient heat behavior of the reservoir. Although this correlation simplifies the Ramey method, it should be used with caution as a rough approximation.

Ramey61 Proposed the classic method for temperature prediction in wellbores only. The method couples heat transfer mechanisms in the wellbore and transient thermal behavior of the reservoir. Equations for injection or production of single-phase fluids were derived.

Coulter & Bardon62 Equations developed to predict the thermodynamic behavior of the flowing fluid in a rigorous approach. However, the assumptions of steady-state heat transfer with a constant temperature environment and horizontal flow limit this method to pipeline or horizontal flow only.



User-Entered Heat Transfer Coefficients

Use this heat transfer method to enter known heat transfer coefficients for the wellbore and flowline segments separately. A table of oil and gas specific heats for a range of temperatures is needed. The heat transfer coefficients and algorithms are used to calculate the temperature at the specified segments needed in wellbore and flowline correlations.

Flow Assurance

Scale: Oddo-Tomson method

Scales are solids deposited in wellbores and pipelines due to precipitation of minerals from produced brines or injected water. These solids may cause formation damage and pipe blockage, with a subsequent loss of production.

The most common scales found in oil and gas fields are calcium carbonate (CaCO3), calcium sulfates (CaSO4), barium sulfate (BaSO4) and strontium sulfate (SrSO4). Carbonate scales are formed mainly after pressure and temperature changes in the system, which causes the escape of CO2 and H2S from the gas and an increase in the pH of the water. Sulfates are formed basically after breakthrough of injected incompatible waters or mixing of different brines from diverse zones in the formation.

Scale minerals tend to precipitate after pressure drops in the system. As for temperature, an increase will cause calcite deposition, whereas a decrease will cause barite deposition.

Scales are generally predicted with saturation indexes, which compare the amount of scaling constituents in solution to the solubility. Oddo and Tomsom developed a prediction model that is based on the produced water chemistry and production data such us pressure and temperature, flow rate and percentage of CO2 in the gas at surface. In this method, the degree of saturation of a scale is related to the saturation index, SI, and is defined as the log of the product of the concentrations of the scaling minerals divided by the conditional solubility product of the particular scale, Kc.

For instance, for calcium sulfate, the saturation index is:

),,(

]4][[log

22

IspTKcSOCa

SI−+

=

Kc was derived from literature data and is a function of temperature, pressure and ionic strength, Is. The use of this parameter avoids the calculation of the activity coefficients for the metal ions and the anions in the scales. The resulting equations are of the form:

-LogKc = a + bT + cT2 + dP + eIs0.5 + fIs + gIs1.5 + hTIs0.5



Oilfield waters may have an important concentration of carboxylic acids, normally represented as equivalent concentration of acetic acid. When this happens, the HCO3 concentration is corrected to adjust the total alkalinity of the water.

Even though a theoretical value of saturation index above zero indicates a tendency of scale deposition, Oddo and Tomson suggest a value above 0.4 for actual deposition to take place. For calcium sulfate this number may be lower, about 0.2.

Wellbore Deviation

You can use PERFORM to analyze deviated wells using the wellbore profile. The directional survey dialog from the wellbore dialog has entries for entering the directional data as either a TVD (true vertical depth) and MD (measured depth) data pair or an MD and angle. For vertical wells, the directional survey is not needed. For deviated wells however, the MD to the top-perf depth will be larger than the TVD of the well. The directional survey is used to tell PERFORM how much to deviate the well by either entering MD/TVD depth pairs or angles at MD.

There is a difference in how this information is used by PERFORM.

• PERFORM assumes that the well is vertical if no directional survey data is entered.

• If you want to deviate the wellbore, you can select type of data to enter in the directional survey as either "Measured vs. Vertical Depth" or "Measured Depth vs. Angles."

• If you choose "Measured Depth vs. Vertical Depth," then PERFORM will calculate the angle of the wellbore for the segments listed when calculations are done. The listing of data pairs should be completed all the way to the top perforation depth as the final data entry. If the first segment of the wellbore near the surface is vertical, you should enter the first data pair as the kickoff point with the MD and TVD values equal.

• If you choose "Measured Depth vs. Angles," the angles entered are used in the segment below the measured depth entered. If the first segment of the wellbore is vertical, you should enter the first data element as a measured depth of the kickoff point with an angle greater than zero degrees from vertical to start deviating the wellbore from that depth and below.

PERFORM uses angles in the calculation of the wellbore segments whether entered directly or calculated from the MD/TVD pairs. You must enter the deviation information correctly. The following examples show how PERFORM interprets the wellbore deviation to model a deviated well.



Example 1

Example 1 demonstrates using the directional survey and an angle to deviate the well. Note that the TVD and MD in the wellbore segment are set to the same value to allow PERFORM to calculate the TVD itself. The kickoff point is 3000 ft.

The Directional Survey dialog box contains the following information:

MD TVD ANGLE

3000 12 kickoff depth

6270 0 top perforation measured depth

An angle of 0 degrees is set from the surface to 3,000 ft MD as the kickoff depth. The wellbore deviates from 3,000 ft MD to 6,270 ft MD at 12 degrees. The new TVD of the top perforation is calculated as:

( ) ( ) '6199 = 3000 6270 12cos + 3000 = calc TVD −×o



Example 2

Example 2 demonstrates what happens if the kickoff depth segment is not entered using the MD and TVD data entry option.


MD TVD ANGLE

5 5

6270 6199 Top perforation measured depth

PERFORM assumes that the well is deviated starting at about 5 ft. The resulting angle from the surface is calculated as:

°

−−

φ 8.63 = 5627056199

cos = 1-

Example 3

Example 3 demonstrates how to correctly enter the directional data using a kickoff point.


MD TVD ANGLE 3000 3000 Kickoff depth

6270 6199 Top perforation measured depth

The angle calculated from the kickoff depth to the top perforation is:

( )( ) °

−−

φ 12 = 3000 62703000 6199

cos = 1-

This is good practice where a kickoff depth is entered in the Directional Survey dialog box.




6 Downhole Network

Use the Downhole Network dialog box to optimize the performance of multilayers and multilaterals for oil and gas wells. Multilayers, Figure 6.1, are the wells with a single wellbore penetrating several production zones. Multilaterals, Figure 6.2, are the wells with several wellbores penetrating a single or multiple production zones or reservoirs.

The Downhole Network dialog box makes reference to nodes and links. A node is a reference point where flow enters, leaves, or merges. A link is a connection between two nodes in which a single stream flows.

Figure 6.1: Multilayer

Downhole Network PERFORM Technical Reference Manual


Figure 6.2: Multilateral

PERFORM Technical Reference Manual Downhole Network


In general, the calculation procedure is the same for both multilayer and multilateral. It starts from the reservoir, continues for the links, and ends at the bottom of the main wellbore. From the wellbore to the surface, the calculation is similar to that for a single wellbore and has the same options. You can change the node position for calculation from separator, to wellhead, and to the bottomhole, which in this case is the bottom of the main wellbore.

The following procedure is used to perform the calculation for downhole network.

1 For each individual reservoir, the Inflow Performance Relationship (IPR) is calculated by considering the fluid, reservoir, and completion. The calculation procedure for each reservoir is similar to the calculation for single reservoir. These IPR curves present the inflow performance at the node closest to the reservoir, the reservoir node. These IPR curves for the reservoirs are presented in the Inflow Graph by Reservoir.

2 If there is a link between the reservoir node and the next node, the pressure drop for each rate is calculated. The IPR at the reservoir node is adjusted to present a new IPR at the next node, link IPR. The link IPR presents the pressure and flow rate immediately before the flow enters the next node. This procedure is similar to calculating the pressure drop in the single wellbore and setting the node at the wellhead or top of the link for downhole network. If there is no link, then the new IPR will be the same as the IPR for the reservoir node. These IPR curves are presented in Inflow Graph by Node.

3 Flow is merged at the node, and three simultaneous calculations are performed:

• The fluid property of the mixed fluid is determined as a function of individual fluid properties, temperature, and flow rate at in-situ condition.

• The temperature at the node is calculated based on oil, gas, and water flow rates and temperature gradient.

• The composite IPR curve is calculated from individual IPR of step 2. The composite IPR is different than IPR for single reservoir because the temperature and/or fluid properties may vary with each rate. Therefore, each rate may have different fluid properties and temperature.

Note: If you selected the option to handle crossflow in the Analysis Settings dialog box, PERFORM determines which link or layer has the highest pressure. Each of the other links joined at the node is assumed to be experiencing injection until pressure drops to the pressure of that link or layer. For the same case, if you have not selected crossflow instead of injection, PERFORM assumes zero rate.

Downhole Network PERFORM Technical Reference Manual


4 Steps 2 and 3 are repeated to establish the final composite IPR at the bottom of the main wellbore. The final composite IPR curve is presented in System Graph for Total System.

5 The rest of the calculation is for the wellbore and flowline and is the same for single wellbore and downhole network.

To illustrate these steps, consider the bilateral case in Figure 6.2. In step 1, the IPR curves for reservoirs R1 and R2 at the nodes n1 and n2 are calculated. In step 2, the pressure losses for the links from n1 and n2 to n3 are calculated and included in new IPR curves for each link at n3. In step 3, the fluid properties for the mixture, temperature, and composite IPR for n3 is calculated. The composite IPR is the combination of IPR curves for Link1 and Link2.


References

1. Brown, K. E.: Technology of Artificial Lift Methods, vol. 1, PennWell Publishing Co., Tulsa, OK (1980).

2. Brown, K. E.: Technology of Artificial Lift Methods, vol. 4, PennWell Publishing Co., Tulsa, OK (1980).

3. Brill, J. P. and Beggs, H. D.: Two-Phase Flow in Pipes, University of Tulsa, Tulsa, OK (1978).

4. Beggs, H. D.: Production Optimization Using Nodal Analysis, Society of Petroleum Engineers, Richardson, TX.

5. Vogel, J. V.: "Inflow Performance Relationships for Solution Gas Drive Wells," JPT, (January 1968) 83-93.

6. Standing, M. B.: "Inflow Performance Relationships for Damaged Wells Producing by Solution Gas Drive Reservoirs," JPT (November 1970) 1399-1400.

7. Standing, M. B.: "Concerning the Calculation of Inflow Performance of Wells Producing from Solution Gas Drive Reservoirs," JPT (September 1971) 1141-1142.

8. Jones, Loyd G., Blount, E. M., and Glaze, O. H.: "Use of Short Term Multiple Rate Flow Test to Predict Performance of Wells Having Turbulence," SPE 6133, SPE of AIME, (1976).

9. McLeod, H. O., Jr.: "The Effect of Perforating Conditions on Well Performance," JPT (January 1983).

10. Crouch, E. C. and Pack, K. J.: "System Analysis Use for the Design and Evaluation of High-Rate Gas Wells," SPE 9424, SPE of AIME (September 21-24, 1980).

11. Saucier, R. J.: "Gravel Pack Design Considerations," SPE 4030, SPE of AIME (October 8-11, 1972).

12. Hagedorn, A. R. and Brown, K. E.: "Experimental Study of Pressure Gradients Occurring During Continuous Two-Phase Flow in Small Diameter Vertical Conduits," JPT (April 1965) 475.

13. Duns, H., Jr. and Ros, N. C. J.: "Vertical Flow of Gas and Liquid Mixtures in Wells," 6th World Petroleum Congress, Frankfurt, Germany.

References PERFORM Technical Reference Manual


14. Orkiszewski, J.: "Predicting Two-Phase Pressure Drops in Vertical Pipe," JPT (June 1967).

15. Dukler, A. E. et al.: "Gas-Liquid Flow in Pipelines," Research Results, vol. 1. American Gas Association, American Petroleum Institute, (May 1969).

16. Eaton, B. A. et al.: "The Prediction of Flow Patterns, Liquid Holdup and Pressure Losses Occurring During Continuous Two-Phase Flow in Horizontal Pipelines," Trans., AIME (1966).

17. Gray, H. E.: "Vertical Flow Correlations for Gas Wells," User Manual 14B Subsurface Controlled Safety Valve Sizing Computer Program, (June 1974).

18. Flanigan, O.: "Effect of Uphill Flow on Pressure Drop in Design of Two-Phase Gathering Systems," Oil & Gas J. (10 March 1958) 132.

19. Cullender, M. H. and Smith, R. V.: "Practical Solution of Gas Flow Equations for Wells and Pipelines with Large Temperature Gradients," Trans., AIME (1956).

20. Gilbert, W. E.: "Flowing and Gas-Lift Well Performance," API Drilling and Production Practice, (1954) 126.

21. Beggs, H. D. and Brill, J. P.: "A Study of Two Phase Flow in Inclined Pipes," JPT (May 1973) 607.

22. Odeh, A. S.: "Pseudosteady-State Flow Equation and Productivity Index for a Well with Noncircular Drainage Area," SPE-AIME, Mobil Research Development Corp., Dallas, TX.

23. Lescarboura, J. A.: "Handheld Calculator Program Finds Minimum Gas Flow For Continuous Liquids Removal," Oil & Gas J. (16 April 1984) 68-70.

24. Bradburn, J. B.: "Velocity in Gas Lines Erosional Velocity," internal correspondence, Tenneco Oil Company, (7 November 1980).

25. Poettmann, F. H.: "The Multiphase Flow of Gas, Oil, and Water Through Vertical Flow Strings with Application to the Design of Gas Lift Installations," Drill. & Prod. Prac. (1952) 257.

26. Baxendell, P. B. and Thomas, R.: "The Calculation of Pressure Gradients in High-Rate Flowing Wells," JPT (October 1961) 1023.

27. Fancher, G. H., Jr. and Brown, K. E.: "Prediction of Pressure Gradients for Multiphase Flow in Tubing," SPEJ (March 1963) 59.

28. Mukherjee, H. and Brill, J. P.: "Liquid Holdup Correlations for Inclined Two-Phase Flow," JPT (May 1983) 1003.

PERFORM Technical Reference Manual References


29. Aziz, K., Govier, G. W. and Fogarasi, M.: "Pressure Drop in Wells Producing Oil and Gas," J. Cdn. Pet. Tech. (July-September 1972) 38.

30. Griffith, P. and Wallis, G. B.: "Two-Phase Slug Flow," J. of Heat Transfer (August 1961) 307.

31. Lawson, J. D. and Brill, J. P.: "A Statistical Evaluation of Methods Used to Predict Pressure Loses for Multiphase Flow in Vertical Oil Well Tubing," JPT (August 1974) 903.

32. Vohra, I. R., Robinson, J. R. and Brill, J. P.: "Evaluation of Three New Methods for Prediction Pressure Losses in Vertical Oil Well Tubing," JPT (August 1974) 829.

33. "API Users Manual for API 14B-Subsurface Controlled Safety Valve Sizing Program," API Manual 14BM Second Edition, American Petroleum Institute, Washington, D. C. (January 1978).

34. Turner, R. G., Hubbard, M. G., and Dukler, A. E.: "Analysis of Prediction of Minimum Flow Rate for the Continuous Removal of Liquids from Gas Wells," JPT (November 1969) 1475.

35. Weller, W. T.: "Reservoir Performance During Two-Phase Flow," JPT (September 1973) 210-246.

36. Cooke, C. E. Jr., "Conductivity of Fracture Proppants in Multiple Layers," JPT (September 1973) 1101-1107.

37. Firoozabadi, A. and Katz, D. L.: "An Analysis of High-Velocity Gas Flow Through Porous Media," JPT (February 1973) 211-216.

38. Ros, N. C. J.: "Simultaneous Flow of Gas and Liquid as Encountered in Well Tubing," JPT (October 1961) 1037.

39. Achong, I. B.: "Revised Bean and Performance Formula For Lake Maracaibo Wells," University of Zulia, Maracaibo, Venezuela.

40. Beggs, H. D.: "Gas Production Operations," O. G. C. I. Publications, Tulsa, OK (1984).

41. McLeod, H. O., Jr. and Crawford, H. R.: "Gravel Packing for High Rate Completions," SPE 11008, SPE of AIME (26-29 September 1982).

42. Brown, K.E.: Technology of Artificial Lift Methods, vol. 2A, PennWell Publishing Co.: Tulsa, OK (1980).

43. Winkler, H. W. and Smith, S. S.: CAMCO Gas Lift Manual, Houston, TX (1962).



44. Brown, K. E.: Gas Lift Theory and Practice, PennWell Publishing Co., Tulsa, OK (1973).

45. Camacho, R.G.V. and Raghavan, R.: "Inflow Performance Relationships for Solution Gas Drive Reservoirs," SPE 16204, JPT (May 1989) 54.

46. Mukherjee, H. and Economides, Michael J.: "A Parametric Comparison of Horizontal and Vertical Well Performance," SPE 18303, Proceedings for 63rd Annual Technical Conference of Society of Petroleum Engineers, Houston, Texas (2-5 October 1988).

47. Dikken, B. J.: "Pressure Drop in Horizontal Wells and Its Effect on Production Performance," JPT (November 1990) 1426-1433.

48. Cinco-Ley, Heber: "Evaluation of Hydraulic Fracturing by Transient Pressure Analysis Methods," SPE 10043, SPE of AIME (1982).

49. Ascheim: "MONA Correlation: An Accurate Two-Phase Well Flow Model Based on Slippage," Transactions for SPE European Conference.

50. Cullender, M. H. and Smith, R. V.: "Practical Solution of Gas Flow Equations for Wells and Pipelines with Large Temperature Gradients," Trans., AIME (1956).

51. Economides et al.: "Comprehensive Simulation of Horizontal Well Performance," SPEFE (December 1991), 418-421.

52. Economides et al.: Petroleum Production Systems, PTR Prentice Hall (1994) Chap. 2 and 4.

53. Joshi, S. D.: "Augmentation of Well Productivity with Slant and Horizontal Wells," JPT (June 1988) 729-739.

54. Joshi, S. D.: Horizontal Well Technology, PennWell Book Publishing Co. (1991) 75, 91, 224-226, 344.

55. Giger, F. M. et al.: "The Reservoir Engineering Aspects of Horizontal Drilling," paper SPE 13024, presented at the 59th Annual Technical Conference and Exhibition , Houston, TX, 16-19 September 1984.

56. Renard, G. and Dupuy, J. M.: "Formation Damage Effects on Horizontal Well Flow Efficiency," JPT (July 1991) 786-869.

57. Kuchuk, F. J. et al: "Pressure Transient Analysis and Inflow Performance for Horizontal Wells," paper SPE 18300, Houston, TX, 2-5 October 1988.

58. Goode, P. A. and Thambynayagam, R. K. M.: "Pressure Drawdown and Buildup Analysis of Horizontal Wells in Anisotropic Media," May 1987.

PERFORM Technical Reference Manual References


59. Babu, D. K. and Odeh, A. S.: "Productivity of a Horizontal Well," SPERE (November 1989) 417-421.

60. Alves, I. N., Alhanati, F. J. S., and Shoham, O.: "A Unified Model for Predicting Flowing Temperature Distribution in Wellbores and Pipelines," JPT (November 1992) 363-367.

61. Ramey, H. J., Jr.: "Wellbore Heat Transmission," paper SPE 96, April 1962.

62. Coulter, D. M. and Bardon, M. F.: "Revised Equation Improves Flowing Gas Temperature Prediction," Oil & Gas J. (26 February 1979).

63. Sagar, Rajiv: "Predicting Temperature Profiles in a Flowing Well," JPT (November 1991) 441-448.

64. Bradley, H. B.: Petroleum Engineering Handbook, 46-4 and 46-5.

65. Brill, J. P. and Beggs, H. D.: "Two-Phase Flow in Pipes," U. of Tulsa, Tulsa, OK, (December 1988), 1-31 to 1-36.

66. Ros, N. C. J.: "An Analysis of Critical Simultaneous Gas/Liquid Flow through a Restriction and its Application to Flowmetering," Appl. Sci. Rev. (1960) 9, Sec. A, 374.3.

67. Ashford, F. E.: "An Evaluation of Critical Multiphase Flow Performance through Wellhead Chokes," JPT (August 1974).

68. Ashford, F. E. and Pierce, P. E.: "Determining Multiphase Pressure Drops and Flow Capacities in Downhole Safety Valves," JPT (September 1975).

69. Perkins, T. K.: "Critical and Subcritical Flow of Multiphase Mixtures through Chokes," paper SPE 20633 (December 1993).

70. Fortunati, F.: "Two-Phase Flow Through Wellhead Chokes," paper SPE 3742 (May 1972).

71. Omana, R. et al.: "Multiphase Flow Through Chokes," paper SPE 2682 (September 1969).

72. Shiu, K. C. and Beggs, H. D.: "Predicting Temperatures in Flowing Oil Wells," J. Energy Res. Tech. (March 1980) Trans., ASME.

73. Karcher, B. J., Giger, F. M., and Combe, J.: "Some Practical Formulas to Predict Horizontal Well Behavior", paper SPE 15430 (October 1986).

74. Ansari, A. M. et al.: "A Comprehensive Mechanistic Model for Upward Two-Phase Flow in Wellbores," paper SPE 20630 (May 1994) 143-152.



75. Arirachakaran, S. et al.: "Intelligent Utilization of a Unified Flow Pattern Prediction Model in Production System Optimization," paper SPE 22869 (October 1991) 503-516.

76. Xiao, J. J., Shoham, O., and Brill, J. P.: "A Comprehensive Mechanistic Model for Two-Phase Flow in Pipelines," paper SPE 20631 (September 1990) 167-180.

77. Scott, S. L. and Kouba, G. E.: "Advances in Slug Flow Characterization for Horizontal and Slightly Inclined Pipelines," SPE 20628 (September 1990) 125-140.

78. Beggs, H. D. et al.: "Design Criteria for Selecting Velocity Type Subsurface Safety Valves," ASME (February 1980).

79. Eaton, B. A., Andrews, D. E., and Knowles, C. R.: "The Prediction of Flow Patterns, Liquid Holdup and Pressure Losses Occurring During Continuous Two-Phase Flow in Horizontal Pipelines," JPT (June 1967) 815-828.


Index

A a - term Jones equation, 32 Alves et al. unified model, 105 Ansari Mechanistic, 89 API 14B, 95 Ashford & Pierce, 100 Aziz, et al, 88

B b - term Jones equation, 32 Babu and Odeh, 62 back pressure 4 pt test, 34 back pressure equation, 34 Baxendell & Thomas, 87 Beggs & Brill, 88, 91 Beggs, Brill, & Minami, 91 Bernoulli, 95 beta factor gravel pack, 81

C C - term back pressure equation, 34 chokes, 93 collapsed perforation, 76 compacted zone perforation, 70 completion component, 69 completion effects, 10 completion models

collapsed perforation, 76 gravel pack, 79 gravel pack collapsed perforation, 83 gravel pack open hole, 82 gravel pack stable perforation, 83 gravel-pack open perforation, 83 open hole completion, 69 open perforation completion, 70

constant productivity index, 20 constant productivity index (PI), 20 Cooke equation, 82 Coulter & Bardon, 105 critical flow, 94 crushed zone perforation, 70 Cullender & Smith, 90

D D - Ramey D term, 26 damaged zone, 73 Darcy equation, 24 datafile inflow, 43 deviated well, 107

differential graph, 11 dimensionless fracture conductivity, 39 dimensionless pressure drop in the fracture and

reservoir, 40 dimensionless time, 40 dimensionless wellbore tunnel length, 47 directional survey, 107 downhole network, 111

calculation, 113 drainage area and shape factor, 27 drainage ellipsoid, 52 drainage shape factor, 57 Dukler, 91 Duns & Ros, 88, 90

E Economides, 52 elevation, 86 erosion, 101 erosional velocity, 102

F Fancher & Brown, 87 Firoozabadi and Katz equation, 82 Flow Assurance, 106 flow rate, 40 flow through restrictions, 93 flow velocity, 41 flowing temperature distribution, 104 flowline, 85 four point test

back pressure equation, 34 Jones equation, 33

fracture penetration ratio, 39 fractured well, 38 fractured well non-Darcy flow, 41 Fundamental Flow, 90 Fundamental Flow Adjusted, 90

G general, 5 Giger, 49 Goode and Thambynaya, 66 gradient, 8 gravel pack beta turbulence factor, 81 gravel pack completion, 79 gravel pack open perforation, 83 gravel pack stable perforation, 83 gravel-pack collapsed perf, 83 gravel-pack open hole completion, 82

Index PERFORM Technical Reference Manual


Gray, 90

H Hagedorn & Brown, 87, 90 Hagedorn and Brown, 86 heat transfer, 104

coefficients, 106 correlations, 105

horizontal flow oil well, 91

horizontal well, 43

I inflow, 2 inflow performance

back pressure equation and 4 pt test, 34 constant productivity index, 20 Darcy equation, 24 datafile, 43 drainage area and shape factor, 27 fractured well analysis, 38 Jones 'a' and 'b' term and 4-pt. test, 33 Jones equation, 30 transient equation, 36 Vogel equation, 21

inflow performance relationship (IPR), 19 integration constant, 47 IPR (inflow performance relationship), 19

J Jones 'a' and 'b' user-entered, 33 Jones equation, 30 Jones four-point test, 33 Joshi, 54

K kickoff point, 44 Kuchuk, 59

L linear temperature gradient, 104 link, 111

M Modified Gray, 90 MONA, 88, 91 MONA Modified, 88, 91 Mukherjee & Brill, 88, 91 multilateral, 111 multilayer, 111 multiphase flow calculations

vertical, 89 multiphase flow correlations, 86

Muskat, 49

N n - term back pressure equation, 34 nodal analysis, 2 node, 2, 111

fundamental requirements of, 2 non-Darcy fractured well flow, 41 non-Darcy turbulent term, 26

O open hole completion, 69 open perforation completion, 70 Orkiszewski, 88 outflow, 2

P perforated completion, 70 perforation crushed zone, 70 perforation interval, 13 perforation shot density, 12 Perkins, 93, 97 PI (productivity index), 20 Poettmann & Carpenter, 87 productivity index (PI), 20

constant, 20

R Ramey, 105 Ramey D term, 26 references, 115 Renard and Dupuy, 57 reservoir, 19 reservoir skin, 9 restrictions, 93 Reynold's number, 41 Ros & Gray, 90

S Sagar et al. simplified model, 105 Saucier beta equation, 82 Scale, 106 separator pressure, 16 shape factor, 27, 28 Shiu & Beggs, 105 skin, 9

physical, 26 skin effect, 26 slip, 86 static reservoir pressure, 43 subcritical flow, 94 Sylvester & Yao Mechanistic, 89 system analysis

PERFORM Technical Reference Manual Index


applying, 7 defined, 1

T temperature survey, 105 Tenneco beta equation for resin packs, 82 time to pseudosteady state, 37 transient flow equation, 36 transient time to pseudosteady state, 37 tubing flow component, 85 tubing size effect, 14 turbulence beta factor, 41 turbulence coefficient beta, 81 Turner, 103

U unloading, 103

V vertical flow

gas well, 90 oil well, 86

vertical flow component, 85 Vogel equation, 21

combination, 22

W wellbore damaged zone, 73 wellbore deviation, 107 wellbore flowing bottomhole pressure

defined, 2 wellhead pressure, 16

X Xiao Mechanistic, 91

Documents

Manual Perform