Advanced Production Engineering - Ptpr 465

Course Module Advanced Production Engineering – PTPR 465

Introduction to System NODAL Analysis PTPR 6093 Module 1

Revised: September 2007

MacPhail School of Energy

redefine yourself www.sait.ca

Introduction to System NODAL Analysis Rationale Why is it important for you to learn this material? Production Systems NODAL Analysis is a key analytical technique applied to hydrocarbon production systems. It provides an accurate quantitative understanding of the interrelationships among all the components of the hydrocarbon production system. It allows for the isolation of any single component or groups of components for detailed analysis of the effect of possible changes to increase production rate from the well. It can be used to optimize production from existing wells or to plan critical components of production systems for new wells.

Learning Outcome When you complete this module you will be able to …. Understand the concept of System NODAL Analysis for production optimization and how it is used to determine the inter-relationships between system components and the production rate from a hydrocarbon system.

Learning Objectives Here is what you will be able to do when you complete each objective. 1. Understand the concept of System NODAL Analysis and how it is used to optimize production from oil and gas wells. 2. Identify the various components that affect pressure drop, and hence production, from the reservoir to the stock tank, pipeline or gas plant. 3. Learn how to pick critical nodes for analysis that most affect production rates and which types of calculations and correlations are appropriate.

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OBJECTIVE ONE When you complete this objective you will be able to… Understand the concept of System NODAL Analysis and how it is used to optimize production from oil and gas wells.

Learning Activity Complete each of the Learning Activities listed below. 1. Read and study the learning material in this objective. 2. Do the exercise provided. 3. Do the assignment provided. 4. Research outside sources (that is, the Internet, library materials, reference texts, and so on) for additional information.

Learning Material CONCEPT OF SYSTEM NODAL ANALYSIS NODAL analysis is a technique used to analyze the components of the hydrocarbon production system and to determine the production rate the system can deliver to the sales or custody meter. It is an effective way to analyze an existing well and recommend changes to optimize production or to plan for new wells. The hydrocarbon production system starts at the reservoir and ends at the sales point, sales line, stock tank or custody meter (Figure 1).

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Figure 1 Simple Production System (Courtesy of Schlumberger)

Each component of the production system between the reservoir and the sales point contributes its own characteristic loss of available pressure (Figure 2) and has its own unique response to any change in flow variables in terms of the resulting pressure loss. This unique response of each component is described by different flow equations or correlations. The hydrocarbon production system is composed of interacting components. In other words, the pressure for each component depends on the pressure immediately adjacent to it either on the upstream side in the inflow system or on the downstream side in the outflow system.

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Figure 2 Location of Various Nodes (Courtesy of Schlumberger)

NODAL analysis is done at a division point, called a node, in the production system (Figure 2). A solution node is selected depending on which component or segment of the production system it is desired to evaluate. If a solution node is selected between the two ends of the production system, the solution node pressure is calculated from both directions starting at the known, fixed end-point pressures. The pressure changes from the starting point of the production system are evaluated and their response curves are added together until the solution node is reached on the inflow side and an inflow relationship is constructed. Similarly, the pressure changes from the end point are evaluated until the solution node is reached from the other side and the response curves for each component or segment are added together to form an outflow relationship at the solution node. The intersection of the inflow and outflow response curves at the solution node represents the system operating point which is the solution to the analysis of the total production system. It is not necessary to re-calculate the inflow system curve if changes are made only in the outflow system curve to investigate their effect on total system flow rate. Likewise, it is not necessary to recalculate the outflow system curve if changes are made in the inflow system to investigate their effect on total system flow rate. 4 PTPR 6093

A node is described as a functional node if a pressure differential exists across it and the pressure or flow rate response can be described by a flow equation or correlation. Lab, manufacturer or field-based measurements can also sometimes be a source of pressure and flow data for functional nodes or components. An example of a functional node is a surface choke. In the hydrocarbon production system there are two pressures which are not a function of changes in flow rate over the short term. One is average reservoir pressure at one end of the production system and the other is the separator or sales line or stock tank pressure at the other end of the system. While average reservoir pressure can change over time, it normally doesn’t change significantly over the short-term. Any production systems that discharge to atmospheric tanks obviously operate at constant end pressure. Any production systems that discharge into pipelines normally do not affect the pipeline pressure with relatively small changes in flow rates. However, large changes in flow rates can sometimes change pressures in production systems that discharge into pipelines although the changes tend to be small. Regardless, any solution to the total production system analysis must be started from one or both of the known, and assumed constant, conditions at either end of the production system. If a solution node is picked at either end of the production system then all the pressure losses (or gains) from the other end are evaluated and summed up for each component or node until the solution node is reached.

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OBJECTIVE TWO When you complete this objective you will be able to… Identify the various components that affect pressure drop, and hence production, from the reservoir to the stock tank, pipeline or gas plant.

Learning Material PRODUCTION SYSTEM COMPONENTS A diagram of the production system for a flowing well and some common nodes are shown in Figure 3.

Figure 3 Common Nodes for a Simple Production System (Courtesy of Schlumberger)

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A partial list of applications that can be analyzed utilizing production system NODAL analysis techniques includes: •

Maximizing rate in competitive reservoirs

•

Stimulation evaluation

•

Gravel pack design

•

Perforation density

•

Artificial lift design

•

Abnormal flow restrictions

•

Tubing size

•

Subsurface safety valve design

•

Choke size

•

Flow line size and length

•

Compression evaluation

Figure 4 illustrates the pressure differentials in some of the above listed components of the production system.

Figure 4 Possible Pressure Losses in Production System Courtesy of Schlumberger

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Each pressure loss indicated in Figure 4 corresponds to a different component of the production system. Each component generally requires a unique way of calculating its pressure loss. For example, the tubing pressure loss often must be calculated using multi-phase flow correlations that differ from correlations used for multi-phase flow in horizontal flow lines. Furthermore, there are often several correlations available that have been developed by different authors to cover one component of a production system or one range of flow rates or one type of fluid or one range of pressures better than others. Thus, care must be taken to use the appropriate correlation to calculate pressure loss that is best matched to the specific conditions of the production system under evaluation. Subsequent modules in this course will provide more information on the applicability of various correlations for multi-phase flow which is the most common complication in production system analysis.

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OBJECTIVE THREE When you complete this objective you will be able to… Learn how to pick critical nodes for analysis that most affect production rates and which types of calculations and correlations are appropriate

Learning Material CRITICAL NODE SOLUTIONS Nodes are selected to provide information on the performance of specific system components and to determine whether there is a benefit to changing operating parameters or if there is an economic opportunity to modify a system component. Figure 5 shows example intake and outflow curves at Node 3 which is the well head. The intake curve is the response curve for all the system components upstream of the well head and includes the reservoir, the perforations, any restrictions in the tubing, the well bore tubing itself and any surface choke. The outflow curve is the response curve for the surface flow line and the separator. The intersection of the intake and outflow curves will be the system operating point. The system operating point is dictated by the pressures at either end of the system and the pressure loss characteristics of every component on each side of the solution node as summed in the respective intake or outflow curves.

Figure 5 Solution to Production System at Node 3 (Courtesy of Schlumberger)

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Figure 5 indicates that if the horizontal system performance curve can be flattened and/or lowered then the production from the well should increase. The intersection point of the intake and outflow curves will move to the right thereby indicating a higher production rate. Cleaning any restrictions out of the flow line by “pigging” it will reduce the pressure loss in the flow line thereby flattening the outflow curve which results in increased production. Replacing the flow line with a larger diameter flow line, “twinning” the flow line, making it shorter or reducing the end point pressure will also flatten the outflow curve and result in increased production. Figure 6 shows an example of the effect on well production of the flattening of the surface flow line curve as a result of an increase in its diameter from 2-inch to 3-inch.

Figure 6 Effect of Change in Flow Line Diameter at Node 3 (Courtesy of Schlumberger)

The procedure for picking a critical node for analysis involves selecting the node that provides the most useful information about the effect of a change in system pressure on flow rate. If it is desired to evaluate the benefits of a change to a surface flow line or a change in separator pressure compared to a change in well tubing to a larger diameter or to install artificial lift, then a node at the well head is the appropriate choice. This is because a node at the well head isolates the flow line from the rest of the production system so that the effect of changes in the horizontal flow line or the vertical tubing becomes obvious. 10 PTPR 6093

On the other hand if surface equipment can not be easily changed and well head pressure is fixed, then an appropriate critical node may be at Node 6 (Figure 3) which is the bottom hole node. Information about the outflow curve at Node 6 provides information about the tubing and tubing attachment performance in addition to the IPR (Inflow Performance Relationship), or well inflow performance curve, which represents the inflow curve to Node 6. In this case the effect of a larger (or smaller) tubing string diameter on the system intake performance (outflow curve at Node 6) or the effect of improved completion efficiency on the IPR curve (inflow curve at Node 6) can be evaluated for effect on pressures and production rate.

Figure 7 Production System Analysis at Node 3 (Courtesy of Schlumberger)

In Figure 7, if well head pressure is fixed, then the system intake performance curve (outflow curve) represents the performance of the completion and the tubing and all its attachments. If well head pressure is not fixed, then the system intake performance curve (outflow curve) is for the whole production system as indicated. If well head pressure is fixed, increasing tubing diameter or removing tubing restrictions will lower tubing intake pressure by reducing friction losses and flatten the intake performance curve (outflow curve) resulting in increased inflow from the reservoir. 11 PTPR 6093

Similarly, if the Inflow Performance Relationship can be improved through stimulation or better perforation techniques, the IPR curve will be lifted and/or flattened and its intersection with the system intake performance curve will shift right indicating an increase in production rate for the production system. FUNCTIONAL NODE SOLUTIONS When a functional node such as a surface choke is selected as the solution node, then the node itself creates a pressure drop that is a function of pressure and/or flow rate. Where a functional node is selected as a solution node, the intake and outflow curves are determined as normal for the case where the solution node is not a functional node. The pressure differential, or loss, across the functional node, in this case a surface choke, is then determined for various choke settings which govern the flow rate through the choke. The pressure loss is then plotted between the intake and outflow curves as shown in Figure 8 and the flow rate for each differential pressure is noted. Then a single curve of choke pressure loss versus flow rate, Figure 9, represents the total production system performance curve for a choked well. Using data for the specific choke, choke pressure loss can be related to bean size which is the orifice size of the choke.

Figure 8 Surface Choke Evaluation (Courtesy of Schlumberger)

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Figure 9 Total System Performance Curve for Surface Choke (Courtesy of Schlumberger)

The remaining modules for this course will detail the procedures and calculations to determine the flow rate and pressure drops of the various components of the hydrocarbon production system.

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Exercise One 1. A production system for a flowing oil well has the following conditions: Psep, 100 psi Flowline, 2-in. dia., 3000 ft long Water-oil ratio, 0 Perforation depth, 5000 ft Gas-oil ratio = 400, scf/bbl Tubing size, 2 ⅜ in Pr, 2200 psi IPR (productivity index assumed constant) = 1.0/bopd/psi a) Using the following table (from Reference 1) of flow rates and pressure losses calculated for you, and using node 8 as the solution node with reference to Figure 3, determine the flow rate of this production system at the given reservoir pressure. Table 1 (Courtesy of Schlumberger)

b) Determine the flow rate of this production system at a reservoir pressure of 1500 psi. Use the following procedure: i. Select flow rates to construct the outflow system curve (system intake performance curve) at node 8. This example assumes 200, 400, 600, 800, 1000 and 1500 bopd. ii. For each rate, starting at Psep = 100 psi, calculate and add all the pressure losses until reaching node 8 (this is done for you in this example). iii. Calculate and plot the resulting total system pressure loss in the outflow system versus flow rate data as the system performance curve (outflow curve at node 8). iv. Plot the reservoir pressure (being constant, it will be a horizontal line) on the system performance curve. v. Read the intersection of the two curves to determine the predicted flow rate and flowing bottom hole pressure. vi. Repeat step iv. for a reservoir pressure of 1500 psi. 14 PTPR 6093

Exercise One Solutions Adding up the total system pressure losses completes the table as follows:

Plotting the system outflow curve and reservoir pressure gives an intersection at 900 bopd for the system operating point if average reservoir pressure is 2200 psi and 700 bopd if average reservoir pressure is 1800 psi.

Selecting the solution node at the reservoir allows evaluation of the effect of a decline in average reservoir pressure on this production system.

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Module Self-Test 1. NODAL analysis is: a. a technique used to analyze the components of the hydrocarbon production system b. an effective way to analyze an existing well and recommend changes to optimize production c. used to determine the production rate the system can deliver to the sales or custody meter d. all of the above 2. A node is described as a functional node if a. b. c. d.

it interacts with other components of the production system the pressure loss is constant as the flow rate varies the flow rate is constant as the pressure loss varies a pressure differential exists across it that can be calculated by some means

3. Each node in a production system except for the end nodes a. has different flow equations or correlations to describe the pressure loss through each node b. contributes the same pressure loss to the system c. divides the production system into groups of upstream and downstream components d. all of the above 4. A solution node can be described as a. b. c. d.

a node selected at one end of the production system a division of the hydrocarbon production system the point where the inflow and outflow response curves intersect all of the above

5. The inflow response curve is a. the sum of the pressure responses on the outflow side of a bottom hole node b. the sum of the pressure responses of all the components on the inflow side of the functional node c. the sum of the pressure responses of all the components on the inflow side of the solution node d. none of the above 16 PTPR 6093

6. The outflow response curve is a. the sum of the pressure responses on the inflow side of the sales point or custody meter/shipping tank b. the sum of the pressure responses of all the components on the outflow side of the solution node c. the sum of the pressure responses of all the components on the outflow side of a functional node d. all of the above 7. The intersection of the inflow and outflow response curves at a solution node represents a. the operating point of the production system for the given end conditions b. the solution to the analysis of the total production system c. the flow rate the production system will produce at for the given end conditions d. all of the above 8. When using a functional node as a solution node to analyze a production system, a. the pressure losses and component responses on either side of the functional node are evaluated as normal b. the intersection of the intake and outflow curves provides the system operating point only if the choke is wide open c. the intersection of the intake and outflow curves must be modified by the pressure loss through the functional node d. all of the above 9. The Inflow Performance Relationship (IPR) a. is the well inflow performance curve b. represents the inflow curve to a node at the bottom hole c. describes the variation in production rate with bottom hole pressure d. all of the above 10. The most common complication affecting production system analysis is a. b. c. d.

selecting one end of the production system as the solution node selecting a functional node as a solution node selecting the appropriate multi-phase flow correlation determining the intersection of the inflow and outflow curves

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Module Self-Test Answers 1. d 2. d 3. c 4. d 5. c 6. b 7. d 8. d 9. d 10. c

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Assignment 1. What tools are available to determine the pressure differential across a functional node? 2. Which pressures in the production system are not functions of flow rate? 3. List all the components of a typical hydrocarbon production system.

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References 1. Brown, K.E., Mach, J. and Proano, E., A NODAL Approach for Applying Systems Analysis to the Flowing and Artificial Lift Oil or Gas Well, JohnsonMacco/Schlumberger training material, undated. 2. Brown, K.E., Mach, J. and Proano, E., Systems Analysis as Applied to Producing Wells, Congreso Pan Americano de Ingenieria del Petroleo, Mexico City, Mexico, March 19-23, 1979.

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Appendix Table 2 Conversion Factor Table

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Course Module

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Course Module Advanced Production Engineering – PTPR 465

Liquid Deliverability (Inflow Performance Relationship or IPR) PTPR 6094 Module 2

Revised: October 2007

MacPhail School of Energy

redefine yourself www.sait.ca

Liquid Deliverability (Inflow Performance Relationship or IPR) Rationale Why is it important for you to learn this material? One of the most common questions addressed by those involved in hydrocarbon production is “What rate will the well produce at?” This module will provide you with the basic tools, techniques and terminology and the understanding to enable you to find an answer to that question. This module will focus on liquid (oil) deliverability of producing wells with reservoir pressures either above or below the bubble point and for wells that are either damaged or stimulated.

Learning Outcome When you complete this module you will be able to …. Calculate the relationship between the liquid (or oil) flow rate from the reservoir to the well and the pressure drop between the reservoir and the bottom of the well.

Learning Objectives Here is what you will be able to do when you complete each objective. 1. Calculate and use the Productivity Index (PI) for pressures above the bubble point. 2. Calculate and use the Vogel Liquid IPR equation for zero skin (and pressures below the bubble point) and link it with the Productivity Index. 3. Calculate and use the Vogel/Standing Liquid IPR equations for near well bore damage or stimulation (non-zero skin). 4. Calculate and use the Vogel/Standing Liquid IPR equations for reservoir pressures above and below the bubble point.

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OBJECTIVE ONE When you complete this objective you will be able to… Calculate and use the Productivity Index (PI) for pressures above the bubble point.

Learning Activity Complete each of the Learning Activities listed below. 1. Read and study the learning material in this objective. 2. Do the exercise provided. 3. Do the assignment provided. 4. Research outside sources (that is, the Internet, library materials, reference texts, and so on) for additional information.

Learning Material TYPES OF RESERVOIR DRIVE MECHANISMS Before we discuss and learn to use the Productivity Index (PI) and then the Inflow Performance Relationship (IPR) for oil wells, some background on the basic types of reservoir drive mechanisms is required. This background will assist in understanding the derivation, nature and applicability of the Productivity Index and Inflow Performance Relationship equations and curves. The three main naturally occurring types of oil reservoir drives are solution gas drive, water drive and gas cap drive. Solution Gas Drive Solution gas drive is also referred to as depletion drive, volumetric performance or internal drive. In a solution gas drive reservoir, the reservoir pressure is below the bubble point of the reservoir liquid, either at discovery or as a result of subsequent pressure depletion by production. Pressure depletion below the bubble point causes the liberation and expansion of gas that was in solution in the oil. The liberated gas takes the form of tiny bubbles in the oil phase which expand under pressure depletion, thereby mobilizing the oil. This provides the main “internal” drive to mobilize the oil in solution gas reservoirs that are below the bubble point.

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The expression “bubble point” pressure originates from tests of reservoir fluid samples. The pressure of a reservoir fluid that has the same composition as the insitu reservoir fluid is slowly reduced while maintaining the sample at reservoir temperature. The observed pressure when the first bubbles appear within the reservoir fluid sample is the bubble point pressure for that fluid sample. Continued production from a solution gas drive reservoir eventually causes an increase in the gas phase within the reservoir sufficient to reduce the permeability to oil significantly. Furthermore, solution gas drive reservoirs are subject to relatively rapid pressure decline when enough solution gas has been liberated and produced. This has the effect of depleting the energy of the solution gas drive mechanism of the reservoir. Water Drive In a water drive reservoir, sometimes referred to as water encroachment or hydraulic control or active aquifer, there is at least some degree of pressure support as the oil and gas are withdrawn. This is due to encroachment by a mobile water leg, or aquifer, which underlies, and is in contact with, the oil leg. In many cases, the water leg is so mobile, or active, that it completely replaces any liquid or gas volumes produced from the reservoir. In such cases, the 100% pressure support provided by the underlying aquifer maintains the reservoir at its original pressure and its production at original rates. Gas Cap Drive In gas cap expansion drive reservoirs, also called gravity drainage or segregation drive, the expansion of an overlying gas zone provides the energy to help maintain production. In high permeability gas cap drive reservoirs, gas liberated from the oil as a result of pressure depletion can move into the gas cap to further help maintain reservoir pressure and production rates. The larger the size of the gas cap in relation to the oil leg and the higher the vertical permeability, the greater the pressure support provided by the gas cap. Oil recovery from gas cap expansion drive reservoirs tends to be intermediate between solution gas drive and water drive reservoirs. Note that some reservoirs called, surprisingly, combination drive reservoirs, can have one or more of the above natural drive mechanisms operating at the same time.

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PRODUCTIVITY INDEX The inflow performance of a well is the ability of the reservoir to give up fluids through a specific well bore. In the simplest case, a plot of bottom hole producing pressure versus flow rate is a straight line (Figure 1). This is the ideal case for a water drive reservoir with 100% pressure support or for an oil reservoir above its bubble point. A plot of bottom hole producing pressure versus flow rate will curve downward (Figure 1) for solution gas drive reservoirs below the bubble point.

Figure 1 Sample Inflow Performance Curves

The Productivity Index, denoted as J, is a commonly used measure of a well’s ability to produce. It is defined as the production rate divided by the pressure drawdown at the mid-point of the producing interval.

Productivity Index, J =

q m3 /d/kPa (pe - p w )

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For oil wells producing little water, the productivity index is usually calculated using only the oil rate. For some wells such as those in water drive reservoirs with strong aquifer support or those with sufficient pressure support from injection to maintain reservoir pressure, the productivity index remains constant over a wide range of pressure drawdowns. For wells in solution gas drive reservoirs above the bubble point, the productivity index remains constant above the bubble point With reference to Figure 1, the example water drive reservoir IPR curve is straight and therefore has a constant slope. That slope is the Productivity Index (PI) which is constant for the sample well in a water drive reservoir. For other wells, such as those in solution gas drive reservoirs below the bubble point, the IPR curve is exactly that (Figure 1) and the PI is no longer constant as pressure drawdown and flow rate increase. When a well’s IPR is curved, the well’s PI is defined by the slope of a tangent to the curve at a point corresponding to the bottom hole pressure. Obviously, with reference to the sample curve for a solution gas drive reservoir in Figure 1, the PI of a well in a solution gas drive reservoir continues to decrease with increasing drawdown and production rate. Reservoir pressure declines with time in many reservoirs and a well’s PI will also therefore decline with time. Water drive reservoirs with 100% aquifer pressure support or reservoirs with 100% pressure support from water injection will have constant reservoir pressure and have well PIs that do not decline with time. In solution gas drive reservoirs, the PI of all the wells decreases over time due to an increase in oil viscosity as a result of the reduction in reservoir pressure. As gas in solution; that is, the light ends; is released out of the oil and into the reservoir, the viscosity of the remaining oil increases. This causes oil productivity, and hence the PI, to decrease. Not all of the liberated gas is produced out of the reservoir. Some of it remains in the reservoir and causes the formation of a gas phase. The PI of all the wells will therefore also decrease over time as increased gas saturation in the reservoir decreases the permeability of the reservoir rock to oil. Finally, with reference to Figure 1, note that if the producing bottom hole pressure is maintained close to zero, then the maximum available drawdown is being used and the maximum rate is obtained. The well is then considered “pumped-off”. Similar to the Productivity Index, an Injectivity Index can be defined as follows for injection wells: Injectivity Index, I = PTPR 6094

q m3 /d/kPa (p w - pe )

5

Obviously, for injection wells, the Injectivity Index requires the difference between bottom hole injection pressure and reservoir pressure to be used in the denominator. The Injectivity Index applies to injection wells used in pressure maintenance schemes or in secondary recovery (for example, waterfloods) as well as to salt water disposal wells.

Exercise One 1. A well was tested with the following results: Reservoir pressure, 9500 kPa Bottom hole producing pressure, 5200 kPa Oil rate, 15 m3/d Water rate, 59 m3/d Assuming a reservoir with strong aquifer support (the ideal case of a constant PI), calculate the PI for this well. 2. Using the calculated PI, what is the expected total rate of this well, assuming a constant PI, if bottom hole pressure can be reduced to 2200 kPa with improved artificial lift?

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OBJECTIVE TWO When you complete this objective you will be able to…

Calculate and use the Vogel Liquid IPR equation for zero skin (and pressures below the bubble point) and link it with the Productivity Index.

Learning Material VOGEL’S LIQUID IPR EQUATION

Vogel published a method in 1968 (Reference 1) to determine the inflow performance relationship (IPR) curve for a well producing by solution gas drive below the bubble point. He assumed radial, uniform, two-phase flow of oil and gas with constant water saturation and no gravity segregation. He also assumed a skin factor of zero. His work resulted in a reference curve which allows construction of an IPR curve from only one test on a well. The equation of Vogel’s curve is;

qo q o,max

⎡p ⎤ ⎡p ⎤ = 1 - 0.20 ⎢ wf ⎥ - 0.80 ⎢ wf ⎥ ⎣ pR ⎦ ⎣ pR ⎦

2

For comparison, the inflow performance curve for a constant IPR (straight line) is; qo q o,max

⎡p ⎤ = 1 - ⎢ wf ⎥ ⎣ pR ⎦

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Vogel’s reference curve is shown in Figure 2.

V OGEL'S IPR FOR S OLU T ION -GAS D R IV E R E SE R V OIR S 1.00 0.90 0.80 0.70 0.60 0.50 . 0.40 0.30 0.20 0.10 0.00 0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

D IME N SION LES S R AT E (q/ q m a x ), Fra ction of Ma ximum

Figure 2 Vogel’s Inflow Performance Relationship for Solution Gas Drive Reservoirs

In his original work, Vogel also showed there was a progressive deterioration in the IPRs for a solution gas drive reservoir as depletion proceeds (Figure 3). Note the rapid deterioration in productivity indicated by the low recovery factors (ultimate recovery factor of only 14%) shown in Figure 3 for the example well in a solution gas drive reservoir. This type of performance is all too typical for solution gas drive reservoirs.

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Figure 3 Calculated IPRs for a Solution gas Drive Reservoir © Journal of Petroleum Tech. This material has been copied under license from Access Copyright. Resale or further copying of this material is strictly prohibited.

Vogel’s reference IPR curve is strictly valid only for solution gas drive reservoirs producing below the bubble point of the reservoir fluid. Because it was constructed by computer modeling of a variety of solution gas reservoirs with different fluid and rock properties, expect a little error when applying it to a specific reservoir. That said, Vogel’s reference IPR has been applied to solution gas reservoirs producing water with reasonable accuracy - provided water cuts are less than 50%. Note that when evaluating artificial lift systems, which are typically necessary in wells producing significant amounts of water with the oil, the productivity index is calculated using total liquid rate (oil + water). 9 PTPR 6094

Exercise Two 1. A well was tested with the following results Reservoir pressure, 9500 kPa Reservoir fluid bubble point, 9500 kPa Bottom hole producing pressure, 5200 kPa Oil rate, 74 m3/d Assuming a solution gas drive reservoir with no aquifer support or gas cap; (a) Determine the maximum oil rate for this well using Vogel’s IPR reference curve for solution gas drive reservoirs. (b) Calculate the maximum oil rate for this well using Vogel’s IPR equation for solution gas drive reservoirs. (c) Determine the predicted oil rate at a bottom hole pressure of 2200 kPa using Vogel’s IPR equation for solution gas drive reservoirs.

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OBJECTIVE THREE When you complete this objective you will be able to…

Calculate and use the Vogel/Standing Liquid IPR equations for near well bore damage or stimulation (non-zero skin).

Learning Material STANDING’S EXTENSION OF VOGEL’S IPR FOR NON-ZERO SKIN

As mentioned, Vogel’s initial work assumed that the well was neither damaged nor stimulated, that is, a skin factor of zero applied. Standing (Reference 3) created a chart (Figure 5) to aid in the calculation of production rate of a damaged or stimulated well as a function of flow efficiency. In order to be able to use that chart, an understanding of the concept of flow efficiency is required. Flow efficiency is a measure of the condition of a well. Flow efficiency, FE, is defined as follows: FE =

p - p'wf undamaged/unstimulated drawdown = R actual drawdown p R - p wf

Since p'wf = pwf + ∆pskin, (referring to Figure 4) then: FE =

p R - p wf - Δpskin p R - p wf

A well that is neither damaged nor stimulated would have a flow efficiency of 1.0. Damaged wells would have a flow efficiency of less than 1 while stimulated wells would have flow efficiency greater than 1. Inspection of Figure 4 will make the above nomenclature clear. Also note in Figure 4, for future reference, that at radius equal to 0.472 re, where re is the no flow boundary of the reservoir, the pressure is equal to the average reservoir pressure. In addition, the effective drainage radius stabilizes at the value of 0.472 re.

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Figure 4 Reservoir Pressure Profile for Damaged Well © Journal of Petroleum Tech. This material has been copied under license from Access Copyright. Resale or further copying of this material is strictly prohibited.

In order to determine ∆pskin, the following relationship defined by van Everdingen (Reference 4) is used;

Δpskin = S

qμ 2πkh

The value for the skin factor, S, is normally obtained from a pressure transient analysis of a well test which is also the normal source of a value for the reservoir permeability, k. Note that a method to determine FE from two flow tests is given in Reference 2, pages 273-275. Figure 5 can be used to determine the maximum rate, qo(max): • • •

for a well that is damaged (FE < 1) for a well that is stimulated (FE > 1) for a well that has had the damage removed (FE = 1.0)

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Figure 5 Standing’s IPR Curves for Wells Producing by Solution Gas Drive © Journal of Petroleum Tech. This material has been copied under license from Access Copyright. Resale or further copying of this material is strictly prohibited.

Figure 5 can also be used to determine: •

The flow rate at any bottom hole pressure for different values of FE

•

IPR curves showing flow rate versus bottom hole pressure for damaged or improved wells

In order to use Figure 5 correctly, it is usually necessary to first determine qo(max) at an FE = 1.0. Also, note that it is not recommended to extrapolate the curves in Figure 5 beyond the ratio of 1.0, for producing rate divided by the maximum producing rate at a FE = 1.0, as errors become unacceptable. The best way to learn to use Figure 5 is to work through some exercises. 13 PTPR 6094

Exercise Three 1. Given the following data for a damaged well: Oil rate, 10 m3/d Average reservoir pressure, 4000 kPa Producing bottom hole pressure, 3000 kPa Flow efficiency, 0.7 a) Determine the maximum possible rate for this well at FE = 0.7. b) Determine the oil rate at a producing bottom hole pressure of 2000 kPa for this well. 2. Given the following data for a damaged well Oil rate, 10 m3/d Average reservoir pressure, 4000 kPa Producing bottom hole pressure, 3000 kPa Flow efficiency, 0.7 a) Determine the maximum possible rate for this well if it is stimulated to a FE = 1.2. b) Determine the oil rate at a producing bottom hole pressure of 1200 psi and a FE = 1.2 for this well. 3. Given the following data for a stimulated well Oil rate, 20 m3/d Average reservoir pressure, 4000 kPa Producing bottom hole pressure, 1200 kPa Flow efficiency, 0.7 a) Determine the bottom hole pressure at which this well will produce 20 m3/d if FE = 1.0.

14 PTPR 6094

OBJECTIVE FOUR When you complete this objective you will be able to…

Calculate and use the Vogel/Standing Liquid IPR equations for reservoir pressures above and below the bubble point.

Learning Material EXTENSION OF VOGELS’S IPR CURVE FOR FLOW ABOVE AND BELOW THE BUBBLE POINT

Vogel’s initial work assumed that the well was producing from a reservoir that had a pressure below the bubble point pressure of the reservoir fluid. There is a method that makes it possible to use Vogel’s reference IPR curve to analyze wells that have production tests either above or below the bubble point. Obviously, this requires knowledge of what the reservoir fluid bubble point pressure is and what the average reservoir pressure is. This information is normally obtained from actual tests (PVT and reservoir pressure). Note that a reservoir pressure that is higher than the bubble point pressure of the reservoir fluid is the definition of an undersaturated oil reservoir. An undersaturated reservoir is one in which the reservoir fluid has the capacity to dissolve more gas in solution. Undersaturated reservoirs are typically a result of an oil pool subjected to increased burial depth after the accumulation of hydrocarbons has occurred within it. This can result in the reservoir pressure being higher than the bubble point of the original hydrocarbon mixture. Note that these reservoirs are sometimes brought closer to surface later as a result of erosion or other geologic conditions. This can result in such reservoirs also being overpressured for their current depth. Undersaturated reservoirs produce as a result of liquid expansion drive only until the bubble point is reached unless they are subject to a water drive. By definition, undersaturated reservoirs will not have a gas cap as any free gas will have been dissolved into the oil.

15 PTPR 6094

It is preferable to have a test sample of original reservoir fluid taken at bottom hole conditions before any production has been taken from the reservoir. If that is not possible, the next best option is to take samples of produced liquid and gas at surface, as soon after production starts as possible, and recombine them in the same proportion as the producing gas-oil ratio to attempt to recreate the original reservoir fluid. The basic technique to analyze wells that have production tests above or below the bubble point involves constructing a composite well IPR curve that is straight above the bubble point and curved, as usual per Vogel’s reference IPR, below the bubble point. Production Test Data above the Bubble Point (after Reference 4)

Figure 4 Flow Tests Above and Below Bubble Point (Courtesy of Schlumberger)

If well flow test data (q1, pwf1) is available above the known bubble point (pb), the procedure to calculate qo(max) and to estimate the flow rate, q, at any bottom hole pressure, pwf2, below the bubble point is as follows: •

Calculate the constant PI above the bubble point using: PI =

q1 p R - p wf1

16 PTPR 6094

•

Calculate the flow rate, qb, at the bubble point pressure using the PI above the bubble point: qb = PI (p R - p wf1 )

•

Calculate qo(max): q o (max) = q b +

•

J (p b ) 1.8

To estimate the flow rate, q, at a pressure below the bubble point, pwf2 begin by inspecting Figure 4 and applying Vogel’s IPR below the bubble point as follows: ⎡p ⎤ q2 p = 1 - 0.20 wf2 - 0.80 ⎢ wf2 ⎥ qx pb ⎣ pb ⎦

•

2

Substituting qx = qo, max - qb , and q2 = q – qb, and re-arranging gives: q = q b + (q o,max

2 ⎧⎪ ⎡ p wf2 ⎤ ⎫⎪ p wf2 - q b ) ⎨1 - 0.20 - 0.80 ⎢ ⎥ ⎬ p R ⎣ p R ⎦ ⎭⎪ ⎩⎪

Note: The expression for qo(max) above is derived by differentiating Vogel’s IPR reference curve equation with respect to pwf. Production Test Data below the Bubble Point (after Reference 4)

If well test data (q, pwf) is available below the known bubble point, pb, the procedure to calculate qo(max) and to estimate the flow rate, q, at any bottom hole pressure, pwf2, below the bubble point is as follows: •

Start with the equation to solve for q derived in section 1: q = q b + (q o,max

•

2 ⎡ p wf2 ⎤ ⎪⎫ p wf2 ⎪⎧ - q b ) ⎨1 - 0.20 - 0.80 ⎢ ⎥ ⎬ pR ⎣ p R ⎦ ⎪⎭ ⎪⎩

Derive an equation for the PI at the bubble point, pb:

LET; ⎡p ⎤ p A = 1 - 0.20 wf2 - 0.80 ⎢ wf2 ⎥ pb ⎣ pb ⎦

2

17 PTPR 6094

Then; q = q b + (q o,max - q b ) A

(1)

qb = PI (p R = p b )

(2)

PI(p b ) 1.8

q o(max) = q b + •

Substituting (3) and (2) into (1) and re-arranging to solve for PI gives: PI =

•

qb pb ⎡ ⎤ ⎢⎣ p R - p B + 1.8 (A) ⎥⎦

With PI known, qo(max) can be found using qb = PI (pR – pb) and;

q o(max) = q b + •

(3)

PI (p b ) 1.8

With qo(max) known, the flow rate, q, at a bottom hole pressure of pwf2 can be determined as follows: q = q b + (q o,max

2 ⎧⎪ ⎡ p wf2 ⎤ ⎫⎪ p wf2 - q b ) ⎨1 - 0.20 - 0.80 ⎢ ⎥ ⎬ pR ⎣ p R ⎦ ⎭⎪ ⎪⎩

Exercise Four 1. Given the following data for an oil well Oil rate, 30 m3/d Reservoir pressure, 16 000 kPa Producing bottom hole pressure, 12 000 kPa Reservoir fluid bubble point, 8000 kPa a. Determine the maximum flow rate possible for this well. b. Determine the expected oil rate for a producing bottom hole pressure of 5000 kPa.

18 PTPR 6094

Exercise One Answers 1. PI = J =

qo + qw 15 + 59 = = 0.017 m3 /d /kPa pR - pw 9500 - 5200

2. PI = J =

qo + qw pR - pw

Re-arranging to solve for q o + q w q o + q w = J(p R - p w ) = 0.017(9500 - 2200) = 124 m3 /d

Exercise Two Answers 1 a. Refering To Figure 2; p 5200 At wf = = 0.55, pR 9500 q q max

= 0.65

Therefore, q max =

q 74 = = 114 m3 /d 0.65 0.65

b. Using the Equation For Vogel's Reference IPR Curve: q q max

⎡p ⎤ ⎡p ⎤ = 1 - 0.20 ⎢ w ⎥ - 0.80 ⎢ w ⎥ ⎣ pR ⎦ ⎣ pR ⎦

2

= 1 - 0.20 [ 0.55] - 0.80 [ 0.55]

2

= 0.65 Therefore, q 74 q max = = = 114 m3 /d 0.65 0.65 c. At p wf = 2200 kPa, p wf 2200 = = 0.23 pR 9500 Using Vogel's reference IPR curve p q = 0.91 At wf = 0.23 q max pR Therefore: q = 0.91q max = 0.91(124) = 113 m3 /d PTPR 6094

19

Exercise Three Answers 1. a. Given q o = 10 m3 /d p wf = 3000 kPa

p R = 4000 kPa FE = 0.7

Determine q o(max) At FE = 0.7 First, find q o(max) for FE = 1.0 Using

p wf 3000 = 0.75 = pR 4000

From Fig. 2 for FE = 0.7,

qo

= 0.281

q o(max)

Therefore, at FE = 1.0, qo 10 = = 35.6 m3 /d 0.281 0.281 Next, find q o for FE = 0.7 q o(max) -

From Fig. 2, for FE = 0.7, and for qo q o(max)

p wf =0 pR

= 0.87

Therefore a = 0.87(35.6) = 31.0 m3 /d b. Determine q o For p WF = 2000 kPa At FE = 0.7 p wf 2000 = = 0.5 pR 4000 From Fig. 2, for FE = 0.7,

qo q

= 0.523

)FE = 1.0

o(max

Therefore, q o = 0.523(q o(max) FE=1.0 ) = 0.523(35.6) = 18.6 m3 /d

20 PTPR 6094

2. a. Given q o = 10 m3 /d

p R = 4000 kPa

p wf = 3000 kPa

FE = 0.7

Determine q o(max) At FE = 1.2 First, find q o(max) for FE = 1.0 p wf 3000 = = 0.75 pR 4000 From Fig. 2, for FE = 0.7 qo = 0.281 q o(max) 10 = 35.6 m3 /d 0.281 For FE = 1.2

Therefore, q o(max) Fe=1.0 = Next, find q o(max) From Fig. 2, for

p wf = 0 and FE = 1.2 pR

qo q o(max) FE=1.0 Therefore q o = 0.96(35.6) = 34.2 m3 /d b. Find q o for p wf - 2000 kPa and FE - 1.2 p wf 2000 = = 0.5 pR 4000 From Fig. 2, using FE = 1.2 curve qo = 0.788 q o(max)Fe=1.0 q o = 0.788(35.6) = 28.1 m3 /d 3. Given q o = 20 m3 /d p wf = 1200 kPa

p R = 4000 kPa FE = 0.7

Determine the bottom hole pressure at which this well makes 20m3 /d if FE = 1.0. Of the total drawdown of 2800 kPa, 1960 kPa occurs accross the formation. Therefore, for p wf = 400 - 1960 = 2040 kPa, This well will make 20 m3 /d if FE = 1.0.

21 PTPR 6094

Exercise Four Answers 1. a. Given q = 30 m3 /d

p wf = 12 000 kPa

p R = 16 000 kPa

p b = 8000 kPa

Determine q o(max) First, determine PI above bubble point PI =

q 30 = = 0.0075 m3 /d /kPa p R - p wf 16 000 - 12 000

Next, determine q b q b = PI(p R - p b ) = 0.0075(16 000 - 8000) = 60 m3 /d Then, find q o(max) J pb 0.0075 g 8000 = 60 + 1.8 1.8 3 = 93 m /d b. Determine q at Pwf = 5000 kPa q o(max) = q b +

use q = q b + (q o(max)

2 ⎧⎪ ⎡ Pwf ⎤ ⎡ Pwf ⎤ ⎫⎪ - q b ) ⎨1 - 0.20 ⎢ ⎥ - 0.80 ⎢ ⎥ ⎬ ⎣ PR ⎦ ⎣ PR ⎦ ⎭⎪ ⎪⎩

2 ⎪⎧ ⎡ 5000 ⎤ ⎡ 5000 ⎤ ⎫⎪ q = 60 + (93 - 60) ⎨1 - 0.2 ⎢ 0.80 ⎢⎣16 000 ⎥⎦ ⎬ ⎣16 000 ⎥⎦ ⎪⎩ ⎭⎪ 3 = 92.5 m /d

22 PTPR 6094

Module Self-Test Questions 1. The three main naturally-occurring types of oil reservoir drives are: a. b. c. d.

Depletion drive, volumetric performance and internal drive Depletion drive, volumetric performance and gas cap drive Solution gas drive, water drive, gravity drainage drive Solution gas drive, internal drive and gas cap drive

2. In a segregation drive reservoir, the expansion of tiny bubbles of liberated gas that was in solution mobilizes the oil and provides the main source of energy. a. True b. False 3. In a reservoir that is producing by solution gas drive, the reservoir pressure is below the bubble point of the reservoir fluid. a. True b. False 4. In a solution gas drive reservoir, the expansion of tiny bubbles of liberated gas that was in solution maintains reservoir pressure. a. True b. False 5. The inflow performance of a well in a solution gas drive reservoir: a. Is a plot of producing bottom hole pressure versus flow rate b. Is the ability of a reservoir to give up fluids at that specific well c. Is represented by a plot of producing bottom hole pressure versus flow that always curves downward d. All of the above 6. The productivity index of a well a. Is constant above the bubble point in solution gas drive reservoirs b. Is constant for any oil reservoir that has 100% pressure support from an underlying aquifer c. Curves below the bubble point in solution gas drive reservoirs d. All of the above 7. The PI of a well a. Is calculated using the oil rate only for wells producing little water b. Is calculated using the sum of the oil and water rates for wells producing up to about 50% water c. Is not recommended for wells producing more than a 50% water cut d. All of the above PTPR 6094

23

8. The most direct measure of the condition of a well is a. b. c. d.

J PI FE IPR

9. Vogel’s initial work on the reference IPR for solution gas wells a. Assumed the well was producing from a reservoir that was below its bubble point b. Assumed the well was neither stimulated nor damaged c. Showed there was a progressive deterioration in IPR for solution gas drive reservoirs as depletion proceeded d. All of the above 10. Vogel’s reference IPR a. Is strictly valid only for solution gas drive reservoirs above the bubble point b. Is the best measure of the condition of a well c. Has been successfully applied to solution gas reservoirs producing more than 50% water cut though not strictly valid d. None of the above 11. The value for a well’s skin factor is normally obtained a. b. c. d.

From an analysis of pressure build-up data From van Everdingen’s equation for ∆pskin From a production test of a well All of the above

12. A well that has a skin factor of zero has a flow efficiency of 1.0. a. True b. False

24 PTPR 6094

Module Self-Test Answers 1. c 2. b 3. a 4. a 5. d 6. d 7. d 8. c 9. d 10. d 11. a 12. a

25 PTPR 6094

Assignment 1. Name three main types of naturally-occurring reservoir drives, one other type of naturally-occurring reservoir drive type and two types of man-made reservoir drives. In a sentence or two for each type of reservoir, describe what makes that type of reservoir unique and different from the others. 2. Describe what a PVT test is, how it is carried out and what information is sought from it. 3. Define bubble point pressure; explain its impact on reservoir liquid deliverability and its significance in constructing IPR curves. 4. Define undersaturated reservoir and describe how its drive mechanism differs from a solution gas drive reservoir.

26 PTPR 6094

References 1. Vogel, J.V., Inflow Performance Relationships for Solution Gas Drive Wells, J. Pet. Tech., January 1968. 2. Brown, K.E., Beggs, H.D., The Technology of Artificial Lift Methods, PennWell Publishing Co., 1977. 3. Standing, M.B., Inflow performance Relationships for Damaged Wells Producing by Solution Gas Drive, J. Pet. Tech., November, 1970. 4. Production and Reservoir Systems Analysis, Johnston-Macco/Schlumberger, undated training material.

27 PTPR 6094

Appendix Table 1 Conversion Factor Table

28 PTPR 6094

Course Module

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Course Module Advanced Production Engineering – PTPR 465

Gas Deliverability -Simplified Method (Inflow Performance Relationship or IPR) PTPR 6095 Module 3

Revised: October 2007

MacPhail School of Energy

redefine yourself www.sait.ca

Gas Deliverability – Simplified Method (Inflow Performance Relationship or IPR) Rationale Why is it important for you to learn this material? “Well test information is second only to production data in importance for the prudent management of oil or gas reservoirs. As such, well testing is an integral part of the overall production and depletion strategy of a reservoir. The lowest cost and the greatest benefit are realized when an appropriate number of high quality tests are run throughout the producing life of the reservoir.” AEUB Directive 040, Pressure and Deliverability Testing Oil and Gas Wells, December 15, 2006. Knowledge of gas well testing is a critical part of conventional oil and gas production engineering. Many subsequent decisions on exploration, development, and capital rely on competent gas well deliverability tests, their interpretation and use.

Learning Outcome When you complete this module you will be able to …. Understand and calculate the relationship between the gas flow rate and the pressure drop from the reservoir to the bottom of the well using the Simplified Method of Gas Deliverability Analysis (laminar flow) and the pseudo-pressure function for gas.

Learning Objectives Here is what you will be able to do when you complete each objective. 1. Understand and apply the Darcy radial inflow equation for gas. 2. Calculate Gas IPRs with the Simplified Gas Analysis Method for stabilized (non-transient) conditions. 3. Understand and calculate the gas pseudo-pressure.

1 PTPR 6095

OBJECTIVE ONE When you complete this objective you will be able to… Understand and apply the Darcy radial inflow equation for gas.

Learning Activity Complete each of the Learning Activities listed below. 1. Read and study the learning material in this objective. 2. Do the exercise provided. 3. Do the assignment provided. 4. Research outside sources (that is, the Internet, library materials, reference texts, and so on) for additional information.

Learning Material INTRODUCTION – WELL TESTING According to EUB Directive 040, “Bottom hole deliverability relationships are required for all producing gas wells prior to or during the first three consecutive calendar months of sales.” There are exceptions – refer to Directive 040 for details. In addition, initial subsurface pressures are required as follows: “Gas Wells – on all productive wells, within the first three months of production (one well per pool per section).” Again, there are exceptions. Almost every well file for a gas well has at least one deliverability test and either one measurement of static bottom hole pressure or a pressure build-up test to measure reservoir pressure. It is important to understand some of the theory behind gas well testing in order to be able to correctly interpret and use those deliverability tests. Decisions such as whether a well is economic to tie-in, workover, have compression installed, etc. all depend upon familiarity with gas well deliverability tests. In addition, as covered in Module 1 of this course, the technique for nodal analysis of the total hydrocarbon production system relies on an accurate inflow performance relationship from the reservoir to the bottom hole of a well. Hydrocarbon systems nodal analysis will be further covered in Module 7. 2 PTPR 6095

RADIAL GAS FLOW EQUATION BASED ON DARCY’S LAW The Darcy radial flow equation for gas applies only to laminar flow in nontransient (stabilized) conditions. The following common versions of the Darcy radial flow equation can be used to estimate gas flow rates from a reservoir under stabilized conditions where turbulence can be neglected. a) For steady-state flow in oilfield units, qg =

0.703 k g h (p e 2 - p wf 2 ) μ g z T [ln (re /rw ) + s]

mscfd

b) For steady-state flow in metric units, qg =

0.000763 k g h (p e 2 - p wf 2 ) μ g z T [ln (re /rw ) + s]

103 m3 /d

c) For pseudo steady-state flow in oilfield units, qg =

0.703 k g h (p R 2 - p wf 2 ) μ g z T [ln 0.472 (re /rw ) + s]

mscfd

d) For pseudo steady-state flow in metric units, qg = Where: q kg h pe pR pwf μg re rw z s

= = = = = = = = = = =

0.000763 k g h (p R 2 - p wf 2 ) μ g z T [ln 0.472 (re /rw ) + s]

103 m3 /d

gas rate, mscfd or 103m3/d effective permeability to gas, md net pay, ft or m reservoir pressure at drainage radius, psia or kPa(a) average reservoir pressure, psia or kPa(a) producing bottom hole pressure, psia or kPa(a) average gas viscosity, cp or mPa•s drainage radius, ft or m nominal bore hole radius, ft or m real gas deviation factor, dimensionless total of skins other than sd (true formation damage), dimensionless

3 PTPR 6095

The following must be noted when applying the above equations: 1. A well is considered to be in steady-state if it is infinite acting; that is, the radius of investigation has not encountered a no-flow boundary. Wells with 100% pressure support from an underlying aquifer or from offset injection would also be considered to be in steady-state as the pressure at the external boundary does not change with production. 2. A well is considered to be in pseudo steady-state if the radius of investigation has encountered a no-flow boundary. Examples of no-flow boundaries include the drainage boundary of an offset producer or the productive edge of the reservoir. In these cases, the reservoir pressure declines at what is assumed to be a constant rate. 3. The steady state equation requires knowledge of the pressure at the external boundary of the reservoir, pe. The pseudo steady-state equation requires knowledge of the average reservoir pressure, pR. The average reservoir pressure is defined as the volumetrically weighted pressure in the reservoir. Usually, pR is obtained from a pressure build-up test. If not known, pR may be calculated from a stabilized flowing bottom hole test, where the stabilized flow rate and flowing bottom hole pressure are known, using the stabilized bottom hole deliverability equation. 4. The above equations require the use of the average values for μ and z between pe and pwf or pR and pwf. 5. The above equations are strictly correct only for radial flow for a well in the center of a circular reservoir. For all other geometric cases, the above equations can be modified to incorporate the appropriate Dietz shape factor (References 7 and 8) for the specific well/reservoir geometry. 6. The time at which a well transitions to pseudo steady-state, tpss, which is also the time that the radius of investigation reaches the external boundary of the reservoir, was given by Earlougher (Reference 9) in oilfield units as: t pss =

φμC t A t DA 0.000264 k

Where: A is the drainage area Ct is the system compressibility tDA is a variable whose value depends on the geometric shape of the drainage area Values for tDA can be found in Reference 9. 7. The above equations are strictly correct only for laminar flow. 4 PTPR 6095

“BACKPRESSURE” EQUATION FOR SIMPLIFIED ANALYSIS OF GAS WELL TESTS It will be shown in Module 4 that the familiar and commonly used “backpressure” equation used in the simplified method of gas well deliverability analysis, to be reviewed in detail later in this module, is an approximation of the Forcheimer equation, introduced in the next section, with the rate-dependent turbulence neglected. The form of the backpressure deliverability equation is: qg = C(Δp2)n Or, rearranging, 1/ n

⎛ qg ⎞ Δp = ⎜ ⎟ ⎝C⎠ 2

Where: C is a constant called the flow coefficient ∆p2 is the difference between reservoir pressure squared and producing bottom hole pressure squared n, the flow exponent, has a value between 0.5 and 1.0 If laminar flow is assumed or exists, n has a value of 1.0 and ∆p2 = qg/C. For well tests that show the effect of turbulence, n can range down to 0.5 for fully turbulent flow. In the simplified method of gas well deliverability test analysis, it is assumed that n is either 1.0 or that n is constant and not a function of flow rate. LAMINAR- INERTIAL-TURBULENT FLOW (LIT ANALYSIS) DELIVERABILITY EQUATIONS BASED ON THE FORCHEIMER EQUATION FOR TURBULENT FLOW

The following Forcheimer-type equations that take turbulent flow into account will be covered in more detail in Module 4 of this course. a) In oilfield units, for steady-state flow; qg =

0.703 k g h (p 2 - p wf 2 ) μ g z T [ln (rd /rw ) + s + Dq]

mscfd

5 PTPR 6095

In metric units, for steady-state flow; qg =

0.000763 k g h (p 2 - p wf 2 ) μ g z T [ln (rd /rw ) + s + Dq]

103 m3 /d

In the above Forcheimer equations, D is the non-Darcy (turbulent) flow coefficient and Dq is termed the turbulence skin effect (or pseudo-skin factor). Note that in the Forcheimer equation, rd is the effective drainage radius. It is time dependent until rd = 0.472 re; thereafter, rd/rw = 1.5 (tD) 0.5. The value of tD, dimensionless time, is given in oilfield units by; tD =

0.000264 k g t φ(μ g C t )i rw 2

Rearranged, Forcheimer-based equations for deliverability analysis of gas well tests that have turbulence accounted for have the following form: Δp 2 = aq g + bq g 2

Where: the constant a accounts for the laminar component of gas flow the constant b accounts for the turbulent component of gas flow DELIVERABILITY EQUATIONS FOR TURBULENT FLOW USING GAS PSEUDO-PRESSURE

The use of the gas pseudo-pressure in the Forcheimer equation will be covered in detail in Module 4. It is simply the Forcheimer (LIT) flow equation above with the pressure terms replaced by the gas pseudo-pressure as follows. Δψ = aq g + bq g 2

Where: ψ is the gas pseudo-pressure function It takes into account the variation of gas viscosity and gas deviation factor when pressure gradients in the reservoir have a significant effect on the solution to the flow equation. Gas viscosity and gas deviation factor are functions of pressure and in reservoirs with significant pressure gradients, using values averaged between reservoir pressure and producing bottom hole pressure does not provide acceptable accuracy. Discussion and calculation of the gas pseudo-pressure will be done later in this module. 6 PTPR 6095

Exercise One 1) An apparently very good gas well in Oklahoma was tested to stabilization as follows: Choke size, in

Flow Rate, mscfd

0 ¼ 20/64 3/8 7/16

0 4023 5974 8240 9228

Bottom hole pressure at MPP, psia 4068 4042 4030 4017 4012

Other data for this well: Net sand thickness, 54 ft Average gas viscosity, 0.028 cP Reservoir temperature, 260°F Assumed re, 1300 ft (160 ac) Average Z factor, 0.931 Well bore diameter, 0.532 ft (casing size) It is assumed the well is undamaged and unstimulated and that steady state conditions apply. a) Calculate the average permeability of the reservoir using all four flow rates. b) Determine the bottom hole absolute open flow (AOF) using the Darcy radial flow equation and the average permeability.

7 PTPR 6095

OBJECTIVE TWO When you complete this objective you will be able to…

Calculate Gas IPRs with the Simplified Gas Analysis Method for stabilized (nontransient) conditions.

Learning Material GAS WELL DELIVERABILTY - SIMPLIFIED METHOD

The following equation can be used to estimate the flow rate from the reservoir into the bottom of a gas well: q g = C(p R 2 - p wf 2 ) n

Where: qg = gas flow rate at standard conditions, m3/d or mscfd C = deliverability coefficient pR = average reservoir pressure obtained by shut-in of the well to complete stabilization, kPa(a) or psia pwf = flowing bottom hole pressure, kPa(a) or psia n = deliverability exponent which is the inverse of the slope of the stabilized deliverability line This equation, called the “backpressure” equation, was discovered empirically and published by Rawlins and Schellhardt in 1936 (Reference 1). “Empirically” means that a large amount of actual gas well flow test data was analyzed and the above relationship between drawdown and flow rate was observed to represent the actual test data satisfactorily. It is important to note that the well tests analyzed by Rawlins and Schellhardt were for gas wells flowed until stabilized rates were achieved. In addition, the wells were typically shallower, lower pressure wells. In the simplified method of gas well deliverability analysis, a plot of (pR2 - pwf2) versus flow rate, q, on log-log paper (Figure 1) is assumed to give a line of constant slope. For gas wells with laminar gas flow at all times, that constant slope is equal to one giving a value for the flow exponent, n, of one. Many, if not most, gas wells actually have some degree of turbulent flow. For those wells, a plot of (pR2 - pwf2) versus flow rate, q, on log-log paper (Figure 2) gives a line of slope somewhere between 1/n = 2 for wells with fully turbulent flow and 1/n = 1 for wells with fully laminar flow. 8 PTPR 6095

In other words, an intermediate value of n, somewhere between 0.5 and 1.0 represents flow that is somewhere between fully turbulent and fully laminar. Most gas well deliverability tests result in an intermediate value of n. Relatively few wells have flow exponents of either 0.5 or 1.0. It is not necessary that most of the reservoir within the drainage radius of the well be in turbulent flow to result in a flow exponent having a value that approaches 0.5. The main determinant of laminar versus turbulent flow is the product of kh (in md-m or md-ft) where k is the permeability to gas and h is the thickness of the reservoir. A relatively high value of kh would tend to indicate more laminar flow while a relatively low value of kh would tend to indicate more turbulent flow.

Figure 1 Simplified (Stabilized) Gas Deliverability Plot – Metric Units

The position of the deliverability line along the x-axis depends upon the value of the flow coefficient C.

9 PTPR 6095

In the simplified method of gas well deliverability the following assumptions are made: a) Average reservoir pressure (or static well head pressure) is determined when well pressure has completely stabilized. b) The coefficient C and the exponent n are assumed to be constant. c) Extrapolation of the deliverability curve beyond the range of the flow rates used to construct the curve is standard practice and is assumed sufficiently correct. Thus, one of the limitations of the simplified gas well deliverability analysis is that it assumes a constant value for the turbulent flow exponent n. This constant value is also used to extrapolate the flow test data to determine the absolute open flow (AOF) potential of the reservoir (or well). In Figure 2, it can be seen that for a typical well producing at higher rates, the exponent n actually changes as a function of gas flow rate.

Figure 2 Actual Gas Deliverability Plot – Oilfield Units © Prentice-Hall. This material has been copied under license from Access Copyright. Resale or further copying of this material is strictly prohibited.

10 PTPR 6095

Note in Figure 2: •

At low flow rates, the slope is 1 and n = 1 indicating laminar flow as would be expected.

•

At higher flow rates, the slope starts to increase and even approach 2 (n = 0.5) in some cases indicating fully turbulent flow. For this well, the maximum observed slope was 1.48 (n = 0.675).

•

The absolute open flow potential (AOF) of the reservoir, not the well head AOF at surface, is 58,000 mscfd for the sample well. It is determined at the extrapolated maximum drawdown of the well which occurs when pwf is 14.7 psia (0 psig). Note that in this case, the extrapolation is a straight line which does not account for a likely further change in n.

•

If the permeability to gas of this reservoir were higher or if the pay were thicker or if the gas viscosity were lower, and so on, the flow coefficient C would be higher and the deliverability curve would shift to the right. This would indicate a higher reservoir absolute open flow potential and deliverability.

The flow coefficient C is obviously a function of permeability, reservoir thickness, gas viscosity, reservoir temperature, gas deviation factor, well bore radius, external drainage radius and skin factor. Mathematically, in oilfield units, C = 0.703 kg h/[μ z T ln(re/rw) + s]. In the simplified analysis, these variables are assumed to be constant not just from average reservoir pressure to a flowing bottom hole (or well head) pressure of zero but also over the life of the well. The Rawlins and Schellhardt conventional deliverability equation required that the well be flowed to a stabilized condition. This could take a very long time particularly in low permeability reservoirs,. Another issue was waste and loss of revenue due to flaring of gas if a well had to be tested for long periods before it could be tied-in. To speed up gas well testing and reduce waste of gas and loss of revenue, several methods of flow rate analysis were developed culminating with the modified isochronal well test. The modified isochronal well test procedure requires that a well be shut-in for exactly as long as it was produced for each of several flow rates. Usually, there are four such transient flow and shut-in periods. A best fit line through these points defines the slope of the deliverability curve. A final flow rate extended to stabilization determines the position of the deliverability curve on the log-log deliverability plot. Note that in the modified isochronal test, the preceding shut-in pressure is used to calculate the drawdown rather than the stabilized reservoir pressure except for the final extended flow period. 11 PTPR 6095

A sample isochronal well test plot is shown in Figure 3. Note the best fit line drawn through the four transient data points to determine the slope of the deliverability line and the stabilized flow data point used to determine the position of the deliverability line.

Figure 3 Transient and Stabilized Gas Deliverability (Reference 3)

Note that there are actually two related gas well deliverability equations in common use. When bottom hole data is used, the deliverability equation represents the capacity of the reservoir to produce gas. In other words, it is the inflow performance relationship at the bottom hole. It can also be used to determine the Absolute Open Flow (AOF) potential of the reservoir. The bottom hole AOF is often used by regulatory bodies to set maximum allowable production rates for gas wells. When well head data is used, the well head deliverability equation represents the actual sales gas production rate of the well and the effects of gathering line “backpressure”. Gathering line back-pressure is simply the pressure in the gathering line that the well must overcome at surface to produce gas. 12 PTPR 6095

The well head deliverability equation is as follows: q g = C' (p ts 2 - p tf 2 ) n'

Where: qg = gas flow rate at standard conditions, m3/d or mscfd C' = well head deliverability coefficient pts = well head static pressure obtained by shut-in of the well to complete stabilization, kPa(g) or psig ptf = flowing tubing head (well head) pressure, kPa(g) or psig n' = well head deliverability exponent which is the inverse of the slope of the stabilized well head deliverability line In this module and in Module 4, we will focus on the bottom hole gas deliverability equation and the inflow performance relationship. Once the gas deliverability relationship is determined from testing, or by other means, an IPR curve of flowing pressure versus flow rate can be constructed as shown in Figure 4. Gas Well IPR

Well Head Pressure, kPag

30000

25000

20000

15000

10000 IPR Curve, n=0.8

5000

0 0

10

20

30

40

50

60

70 3

80

90

100

110

120

130

140

150

3

Gas Rate, 10 m /d

Figure 4 Sample Well Head Curve for Gas Well – Simplified Analysis

Figure 4 is a well head performance curve. A similar IPR curve can be constructed at bottom hole conditions to represent reservoir deliverability. Note that in Figure 4, the well head IPR curve is valid at only one static shut-in well head pressure. A series of similar curves at different static shut-in well head pressures would be required to predict future well performance as depletion reduces static shut-in well head pressure. In the simplified method of analysis, the deliverability coefficient C and the deliverability exponent n are assumed to remain constant with depletion. PTPR 6095

13

Exercise Two 1) A gas well is producing 7.1 103m3/d at a drawdown of 2750 kPa. Static (or stabilized) reservoir pressure is 11 000 kPa(a). a) Assuming simplified analysis and 100% laminar flow; calculate the bottom hole AOF of this well using the backpressure equation. b) What drawdown is required to double the flow rate? c) If the deliverability exponent is 0.855, determine the deliverability coefficient and the bottom hole AOF. 2) Determine the bottom hole deliverability relationship using the Simplified Method for a gas well in Oklahoma that was tested to stabilization as follows: Choke size, in

Flow Rate, mscfd

0 ¼ 20/64 3/8 7/16

0 4023 5974 8240 9228

Bottom hole pressure at MPP, psia 4068 4042 4030 4017 4012

Other data for this well: Net sand thickness, 54 ft Average gas viscosity, 0.028 cP Reservoir temperature, 260°F Assumed re, 1300 ft (160 ac) Average Z factor, 0.931 Well bore diameter, 0.532 ft (casing size) a) Plot (pe2 – pwf2) in psi2 versus flow rate in mscfd on a log-log chart in Excel. b) Do a best fit of the data points using the Excel trend line tool and extrapolate with a straight line on the log-log plot to determine bottom hole AOF. c) Use the Excel equation for the trend line to calculate the deliverability coefficient C and the deliverability exponent n for this well. d) Write the completed bottom hole deliverability equation. Was the simplified method of analysis valid for this well? Justify your answer.

14 PTPR 6095

OBJECTIVE THREE When you complete this objective you will be able to…

Understand and calculate the Gas Pseudo-Pressure

Learning Material GAS PSEUDO-PRESSURE FUNCTION (after Reference 3)

The main limitation of the Forcheimer LIT equation (turbulent flow) deliverability analysis is that it assumes that pressure gradients are small and that as a result, gas viscosity and gas compressibility are constant. In fact, pressure gradients, particularly in tighter reservoirs, can be large. To overcome this problem, a pseudo-pressure function, m (p), denoted by the symbol ψ (psi), is used. A plot of ψ versus pressure, called a ψ – p plot, is constructed for a specific gas and reservoir temperature. Using this plot, pressure can be converted to pseudo-pressure and vice-versa. Wherever a value for pressure, p, appears in the gas deliverability equation being used, it is replaced with the appropriate value of the pseudo-pressure, ψ. The pseudo-pressure function, ψ, describes the variation of gas viscosity, μ, and the real gas deviation factor (sometimes termed the gas compressibility), Z, with pressure. The pseudo-pressure function was defined by Al-Hussainy in 1965 (Reference 4) as:

ψ=2

∫

p

p0

(p/μZ) dP

The value under the integral can be solved numerically by plotting 2 p/μZ versus p and determining the area under the curve from any convenient reference pressure p0 to a pressure p. The area under the curve is the value of ψ corresponding to a pressure p. Whenever the pressure difference, and therefore the pseudo-pressure difference, is required in a calculation and not a pressure alone, then the actual reference pressure, p0, used is immaterial. Note that different authors in the literature use different values for the base pressure when evaluating ψ for different gases such as sour gases.

15 PTPR 6095

When required, it is easy to convert ψ from one reference pressure to another using the following relationship: (ψ)pp0 = (ψ)0p - (ψ)0p

0

The ERCB provided a table (Table 1) of values for ψ as a function of pseudoreduced temperature, Tr, and pseudo-reduced pressure Pr of sweet gases. It uses the viscosity correlations of Carr, Kobayashi and Burrows (Reference 5) and the compressibility factor correlation of Standing and Katz (Reference 6) and uses zero as the reference pressure.

16 PTPR 6095

Table 1 Values of Reduced Pseudo-Pressure ψr as a Function of Pr and Tr

For sour gases the pseudo-pressure, ψ, is obtained by numerical integration or using the tables of Zana and Thomas (Reference 7).

17 PTPR 6095

Exercise Three (after Reference 3) 1. Given the following properties for a sweet gas: Pressure, P kPa

Real Gas Deviation Factor, Z

0 3000 6000 9000 12000 15000

1.000 0.952 0.908 0.871 0.847 0.834

and, Pc = 4581 kPa,

Tc = 198 K,

Gas Viscosity, μ

μ Pa⋅s 12.0 12.8 13.9 15.0 16.3

μi = 11.6 μPa⋅s

Note: 1 cp = 1000 μPa⋅s = 1 mPa⋅s

a) Calculate the pseudo-pressure using simple numerical integration and plot the results on a plot of y versus pressure. Procedure (construct a table to contain the result for each step for every pressure): i. Calculate 2(p/μZ) at each pressure p. ii. Calculate the mean of two successive values of 2(p/μZ) for each pressure p. iii. Calculate the pressure difference, ∆p, for two successive values of p. iv. Calculate the product of 2(p/μZ)∆p. v. Calculate ψ by adding the value from iv) to the previous value for ψ. b) Calculate ψ using Table 1 for each pressure and compare with the results for simple numerical integration. Procedure (construct a table to contain the result for each step for every pressure): i. Calculate reduced temperature Tr. Calculate reduced pressure, pr, at each pressure p. ii. Obtain a value of ψr by interpolating between values of pr and Tr. iii. Calculate the value for ψ from the product of ψr and 2pc2 μi at each pressure. 18 PTPR 6095

Exercise One Answers 1. a. Use the Darcy radial flow equation in oil field units for steady-state conditions. Rearranging to solve for permeability, kg: q g M g ZT [ lm(re/rw) + s ]

kg =

0.703 n(pe 2 -p wf 2 )

qg

p wf

kg

mscfd 0

psia 4068

md --

4023 5974 8240 9228

4042 4030 4017 4012

72.3 73.8 75.9 77.6 74.9 md

Exercise Two Answers 1) a. Deliverability (Back Pressure) Equation for simplified analysis is: q = C(p R 2 - p wf 2 ) n First, solve for the flow constant C: C= =

(p R

2

q - p wf 2 ) n

7100 (11 0002 - 82502 )1.0

= 0.000134 m3 /d/ kPa 2

Then, calculate the AOF: q = 0.000134(11 0002 – 02)1.0 = 16.2 103m3/d

19 PTPR 6095

b. What drawdown is required to double the flow rate? q = 14.2 103 m3 /d = 0.000134 (p R 2 - p wf 2 ) p R 2 - p wf 2 = 105.97 • 106 kPa 2 p wf = 11 0002 - 105.97 • 106 = 3877 kPa(a) Drawdown = 11 000 kPa(a) - 3877 kPa(a) = 7123 kPa c. Determine AOF if n = 0.855 First, determine C at q = 14.2 103 m3 /d C= =

q (p R - p wf 2 ) n 2

14 200 (11 0002 - 3877 2 )0.855

= 0.00195 m3 /d /kPa 2n AOF = 0.00195(11 0002 - 02 )0.855 = 15.9 103 m3 /d

20 PTPR 6095

2. a. An Excel plot of Δp2 versus q is shown below. Exercise Two, Problem 2) a. Simplified Analysis

1.00E+08

0.9931

y = 55.051x

2

2

(pR -pwf ), psi

2

1.00E+07

1.00E+06

1.00E+05 1000

10000

100000

1000000

Flow rate, mscfd

b. Extrapolation of the data on the log-log plot to a Δp2of 16 549 000 indicates an AOF of ≅ 330,000 mscfd. c. The Excel generated best fit trend line for the data is given by: y = 55.051 x0.9931 Δp2 = 55.051 q0.9931 Rearranging: 1.007

⎛ Δp 2 ⎞ q= ⎜ ⎟ ⎝ 55.051 ⎠

1.007

⎛ 1 ⎞ Where C = ⎜ ⎟ ⎝ 55.051 ⎠

= 0.0177

n = 1.007 21 PTPR 6095

Exercise Three Answers 1. a. By simple numerical integration: (i) p

0 3000

Z

0.952

(ii)

(iii)

⎛ P⎞ ⎛ P⎞ z⎜ ⎟ Δp z⎜ ⎟ μz ⎝ ⎠ ⎝ μz ⎠ μPags 103 kPa/mPags 103 kPa/mPags kPa μ

12.0

0.0

-

525.2

-

262.6

3000

(iv)

(v)

⎛ P⎞ z ⎜ ⎟ Δp ⎝ μz ⎠ 109 kP 2 a/mPags

ψ 109 kP 2 a/mPags

-

0.0

0.788

0.788

6000

0.908

12.8

1037.5

778.9

3000

2.337

3.125

9000

0.871

13.9

1486.8

1259.7

3000

3.779

6.904

12000 0.847

15.0

1889.0

1687.9

3000

5.064

11.968

15000 0.834

16.3

2206.8

2047.9

3000

6.144

18.112

b. From Table 1, by interpolation: (i)

(ii)

p

pr

ψr

0 3000

0 0.65

0 0.215 0.863 1.901 3.311 4.986

(iii) ⎛p 2 2⎜ r ⎝ μi

⎞ ⎟ ψr = ψ ⎠ 0.0 0.779 3.121 6.878 11.979 18.040

22 PTPR 6095

2. The plot of the pseudo-pressure function ψ, versus pressure as calculated by simple numerical integration is shown below:

Exercise Three, Problem 1) b.

15

9

2

Pseudo-pressure, ψ, 10 kPa /m•Pa-s

20

10

5

0 0

5000

10000

15000

20000

Pressure, kPaa

23 PTPR 6095

Module Self-Test 1. Well deliverability test information is obtained because a. It minimizes costs and maximizes benefits throughout the life of a reservoir b. It is necessary for the prudent management of reservoirs c. It is required by Alberta’s EUB d. It is an integral part of the production and depletion strategy for a reservoir e. All of the above 2. Bottom hole deliverability relationships are required by the EUB for all producing gas wells prior to or during the first three consecutive calendar years of sales. a. True b. False 3. Gas well deliverability equations based on Darcy’s law are best used to estimate flow rates from gas wells with turbulent flow. a. True b. False 4. A well is infinite-acting if the radius of investigation has not yet “felt” any boundary in the reservoir. a. True b. False 5. The effective drainage radius of a well a. Eventually stabilizes at a value equal to 47.2 percent of the external radius of the reservoir and does not change after that b. Is the same as the external radius of a radial, cylindrical reservoir c. Is time dependent until it reaches the external boundary of the reservoir. d. None of the above 6. The Rawlins-Schellhardt gas well deliverability equation a. Is an approximation of the Forcheimer gas deliverability equation for turbulent flow b. Was developed empirically based on analysis of actual deliverability tests c. Assumes that the flow exponent is constant d. Assumes that the flow exponent is equal to one e. All of the above 24 PTPR 6095

7. The simplified method of gas well test analysis is a. good for wells tested in very tight reservoirs but not in high permeability reservoirs. b. not very good for wells tested in very tight reservoirs but good in high permeability reservoirs. c. not very good for testing wells in very tight reservoirs or in high permeability reservoirs. d. None of the above 8. A well head deliverability equation has the same form and the same values for the deliverability coefficient and flow exponent as a bottom hole deliverability equation a. True b. False 9. A modified isochronal well test usually involves four short-term tests of equal flow and shut-in length and an extended flow test to stabilization. a. True b. False 10. Gas viscosity and gas compressibility factor a. Vary less for wells with high pressure gradients in the reservoir than for wells with low pressure gradients b. Are a function of temperature, gas composition, pressure and pressure gradients in the reservoir c. Are constant for wells with small pressure gradients in the reservoir d. All of the above 11. The appropriately-defined gas pseudo-pressure function ψ, ψ2, or ∆ψ2, can be used in place of a pressure term p, p2 or ∆p2 in a gas deliverability equation. a. True b. False

25 PTPR 6095

Module Self-Test Answers 1. e 2. b 3. b 4. a 5. a 6. e 7. c 8. b 9. a 10. b 11. a

26 PTPR 6095

Assignment 1. Obtain copies of the EUB Gas Well Deliverability Test Summary report forms for both Simplified Analysis and LIT Analysis. Fill in the Simplified Analysis form as completely as possible using the data for Exercise One Part 2) and fill in the form’s Simplified Analysis calculation area (convert the oil field unit’s data to metric). 2. List the assumptions inherent in the Darcy radial gas flow equation. 3. List the assumptions inherent in the Simplified Analysis of gas well deliverability tests. 4. List the assumptions inherent in the Forcheimer turbulent gas flow equation. 5. Which assumptions are not part of the use of pseudo-pressure in the Forcheimer turbulent gas flow equation?

27 PTPR 6095

References 1. Rawlins, E.L., Schellhardt, M.A., Backpressure Data on Natural Gas Wells and their Application to Production Operations, U.S. Bureau of Mines, Monograph 7, 1936. 2. Craft, B.C., Hawkins, M.F., Applied Petroleum Reservoir Engineering, Prentice-Hall, Inc., 1959. 3. ERCB Guide G-3, Gas Well Testing, 4th Ed, 1979. 4. Al-Hussainy, R., The Flow of Real Gases Through Porous Media, M.Sc. Thesis, Texas A&M. Univ., 1965. 5. Carr, N.L., Kobayashi, R., Burrows, D.B., Viscosity of Hydrocarbon Gases under Pressure, Trans. AIME, 201, 264-272, 1954. 6. Standing, M.B., Katz, D.L., Density of Natural Gases, Trans. AIME, 146, 140, 1942. 7. Dietz, D.N., Determination of Average Reservoir Pressure from Build-up Surveys, JPT, pp. 955-959, Aug. 1965. 8. Economides, M.J., Hill, A.D. and Ehlig-Economides, C., Petroleum Production Systems, Prentice-Hall, 1993. 9. Earlougher, R.C., Advances in Well Test Analysis, SPE, 1977.

28 PTPR 6095

Appendix Table 2 Conversion Factor Table

29 PTPR 6095

30 PTPR 6095

Course Module

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Course Module Advanced Production Engineering – PTPR 465

Gas Deliverability - LIT Analysis (Inflow Performance Relationship or IPR) PTPR 6096 Module 4

Revised: October 2007

MacPhail School of Energy

redefine yourself www.sait.ca

Gas Deliverability – LIT Analysis (Inflow Performance Relationship or IPR) Rationale Why is it important for you to learn this material? Deliverability testing of many wells - particularly high-pressure wells, high-rate wells, wells in low permeability reservoirs and wells that can not be tested to stabilized rates, require the application of techniques that account for turbulent flow, for variation in gas properties in reservoirs having significant pressure gradients and for flow tests conducted under transient conditions.

Learning Outcome When you complete this module you will be able to …. Understand and calculate the relationship between the gas flow rate from the reservoir to the well and the pressure drop between the reservoir and the bottom of the well assuming either laminar or turbulent flow.

Learning Objectives Here is what you will be able to do when you complete each objective. 1. Calculate Gas IPRs with the Laminar-Inertial-Turbulent (LIT) Method (both laminar and turbulent) using stabilized (non-transient) conditions. 2. Understand and calculate the effect of formation damage or stimulation (skin) on the laminar Simplified Gas Analysis Method and the LIT Method (both laminar and turbulent). 3. Apply the transient Modified Isochronal Well Testing technique to both the Simplified Gas Analysis and LIT Methods.

1 PTPR 6096

OBJECTIVE ONE When you complete this objective you will be able to… Calculate Gas IPRs with the Laminar-Inertial-Turbulent (LIT) Method (both laminar and turbulent) using stabilized (non-transient) conditions.

Learning Activity Complete each of the Learning Activities listed below. 1. Read and study the learning material in this objective. 2. Do the exercise provided. 3. Do the assignment provided. 4. Research outside sources (that is, the Internet, library materials, reference texts, and so on) for additional information.

Learning Material OVERVIEW OF WELL DELIVERABILTY TESTS Conventional Back Pressure Test As previously discussed in Module 3, the conventional back pressure test requires flowing a gas well until pressure stabilization is achieved for each rate (Figure 1).

Figure 1 Conventional Backpressure Deliverability Test Plot © Amoco Production Company. This material has been copied under license from Access Copyright. Resale or further copying of this material is strictly prohibited.

2 PTPR 6096

Stabilization occurs when the radius of investigation (the well pressure transient) has reached an external boundary of the reservoir. When that happens, average reservoir pressure, and pressure at every point in the reservoir, decreases at a constant (stabilized) rate. Conventional back pressure testing is usually not practical for reservoirs with low permeability because of the time required to reach pressure stabilization for each rate. Isochronal Test An isochronal test of a gas well is conducted by flowing it at several different flow rates, with four being a typical number, each of equal duration (Figure 2 below).

Figure 2 Isochronal Deliverability Test Plot © Amoco Production Company. This material has been copied under license from Access Copyright. Resale or further copying of this material is strictly prohibited.

The length of each flow period is usually much less than the time required for stabilization. The well is shut-in long enough between each flow period for the shut-in pressure to approach static conditions. The isochronal test takes advantage of the fact that the radius of investigation is a function of time and not flow rate. However, an extended flow rate long enough to reach pressure stabilization is required to determine the position of the deliverability curve on the log-log plot of ∆p2 versus qg. In tight reservoirs the length of time required to reach pressure stabilization between flow periods could also make an isochronal test impractical. 3 PTPR 6096

Modified Isochronal Test An isochronal test in which the shut-in between flow periods is of the same duration as each flow period is called a modified isochronal test (Figure 3). It also requires an extended flow period to determine the position of the stabilized deliverability line. There are typically four flow and shut-in periods. These equal length periods are often four hours each though that can vary depending on the reservoir.

Figure 3 Modified Isochronal Deliverability Test Plot © Amoco Production Company. This material has been copied under license from Access Copyright. Resale or further copying of this material is strictly prohibited.

Some authors believe the theoretical validity of the modified isochronal test is not as good as the isochronal test and therefore prefer the isochronal test. However, the modified isochronal test requires the shortest time to conduct the test which results in its widespread application nonetheless. Single Point Test A single point test consists only of an extended flow rate. It requires an estimate of the degree of turbulent flow in the formation. This estimate is often based on information provided by other wells in the same formation or calculated from reservoir and fluid properties.

4 PTPR 6096

Simplified or LIT Analysis? The two common methods of gas well deliverability analysis are the Simplified Method that was covered in Module 3 and the Laminar-Inertial-Turbulent (LIT) Method to be covered in detail in this module. The LIT method is the most rigorous and technically sound method but it requires more calculation effort and time. As a result, it is normally used to analyze those gas well tests where turbulence is significant and where a straight-line extrapolation to determine the AOF would result in significant error. The laminar-inertial-turbulent terminology arises from the concept that between laminar flow at low Reynolds numbers and turbulent flow at high Reynolds numbers there is a transition period dominated by inertial effects as flow changes from laminar streamline flow to turbulent (chaotic) flow. Pressure, Pressure Squared or Pseudo-Pressure? In the past, many analysts have used the following guidelines to determine whether ∆p, ∆p2, or ∆ψ terms are used in gas inflow equations; Static Reservoir Pressure 0 > 2500 psi; 0-17,500 kPa(a) 2500 > 3500 psi; 17,500-24,500 kPa(a) > 3500 psi; >24,500 kPa(a)

Pressure Function p2 m(p) or ψ p

The above guidelines should be used with a good deal of caution as they are not rigorous and may not apply in many cases - particularly in view of the following. The Alberta EUB has an extensive technical discussion in Reference 2, page 2136, regarding which pressure term will provide the best accuracy for any particular well. In summary, the EUB’s finding is that if ψ is a linear function of p then the pressure approach becomes identical to the ψ approach. If ψ is a linear function of p2 then the p2 approach becomes identical to the ψ approach. Obviously if ψ is not a linear function of either p or p2 then the ψ approach must be best. Furthermore, the EUB uses a chart presented in Reference 3 to state that “… it essentially negates the generalization that the p2 approach is valid at low pressure and the p approach applies at high pressures.” If in doubt and much is at stake, the pseudo-pressure approach works for all pressure ranges although, as shown in Module 3, it requires much more computation. 5 PTPR 6096

Transient or Steady State? In their useful Guide G-3 (Reference 2), the EUB, and others, refer to the flow during steady-state conditions as transient flow. This is because during transient flow the well transient (radius of investigation) has not yet reached the external no-flow boundary of the reservoir and is not influenced by either boundary effects or well bore storage effects. As a result, the transient period provides the maximum amount of information about the reservoir. However, the use of the term transient to describe the steady state flow period is cause for some confusion as the two words are somewhat opposite in meaning. Stabilized or Unstabilized? Deliverability testing a well to stabilization does not mean flowing it at constant rate until the bottom hole pressure stops dropping but until the bottom hole pressure, and the pressure everywhere else in the reservoir, declines at a constant rate. This happens when the radius of investigation of the pressure disturbance from opening the well to flow (the pressure transient) has reached the external boundary of the reservoir. At that time the well is said to be stabilized sometimes referred to as depletion mode. Flow testing a well so that none of the pressure transients have enough time to reach the external boundary of the reservoir results in the technique of transient deliverability testing. This means that transient deliverability testing is conducted within the period the well is infinite-acting, that is, in steady state conditions. This technique, to be covered in detail later in this module, minimizes test time and costs as well as waste of gas if it can not be done into a flow line. Obviously a calculation of the time to reach pseudo steady-state flow, to be covered later in this module, is required to confirm that the radius of investigation does not reach, or is affected by, the external boundaries of the reservoir. GAS WELL DELIVERABILTY - LIT ANALYSIS The following Forcheimer-type equations that take turbulent flow into account were introduced in Module 3 of this course. 1. In oilfield units, for steady state flow in a gas reservoir with constant pressure outer boundaries open to flow (for example, reservoirs with 100% aquifer pressure support): qg =

6

0.703 k g h (p e 2 - p wf 2 ) μ g z T [ln (re /rw ) + s + Dq]

mscfd

(1)

2. In metric units, for steady state flow in a gas reservoir with constant pressure outer boundaries open to flow: PTPR 6096

qg =

0.000763 k g h (p e 2 - p wf 2 ) μ g z T [ln (re /rw ) + s + Dq]

103 m3 /d

(2)

3. In oilfield units, for pseudo steady-state flow in a gas reservoir with no-flow boundaries where the pressure everywhere in the reservoir declines at the same rate (for example, a volumetric reservoir): qg =

0.703 k g h (p R 2 - p wf 2 ) μ g z T [ln 0.472 (re /rw ) + s + Dq]

mscfd

(3)

4. In metric units, for pseudo steady-state flow in a gas reservoir with no-flow boundaries where the pressure everywhere in the reservoir declines at the same rate: qg =

0.000763 k g h (p R 2 - p wf 2 ) μ g z T [ln 0.472 (re /rw ) + s + Dq]

103 m3 /d

(4)

In the above Forcheimer-type equations, D is the non-Darcy (turbulent) flow coefficient, Dq is termed the turbulence skin effect (or pseudo-skin factor) and s is the sum of all the skins other than the rate-dependent (turbulent) skin. It was shown in Reference 6 that the effective drainage radius in pseudo steadystate flow (reservoirs with no-flow boundaries) initially increases with time but then stabilizes at a value given by: rd = 0.472 re The metric equation for pseudo steady-state (stabilized) flow can be rearranged as follows; μg z T q g ⎡⎣ln 0.472 (re /rw ) + s + Dq g ⎤⎦ (pR2 - pwf2) = 0.000763 k h g

=

If we define,

A =

and,

B =

1310 μ g z T kg h

1310 μ g z T kg h

⎡⎣ ln 0.472 (re /rw ) + sq g + Dq g 2 ⎤⎦

[ln 0.472 (re

/rw ) + s ]

1310 μ g z T D kg h 7

PTPR 6096

then we get, ∆p2 = (pR2 - pwf2) = Aqg + Bqg2

(5)

Equation (5) represents the LIT method for analysis of stabilized gas well deliverability tests. Note that equation (5) can also be written in terms of pseudopressure as follows: ∆ψ = (ψR - ψwf) = Aqg + Bqg2

(6)

Equation (6) represents the (LIT)ψ method for analysis of stabilized gas well deliverability tests. In both equations, the A-term is the pressure-squared or pseudo-pressure drop due to laminar and well bore effects while the B-term is the pressure-squared or pseudo-pressure drop due to inertial-turbulent flow effects. Note that the inertial-turbulent flow effect increases as the well bore is approached in a radial system. This is due to the increase in flow velocity (v = q/A) as the flow area (A = πr2) decreases with the square of the distance (r). If we ignore the turbulence term Bqg2, or assume it is negligible, we get qg = C (pR2 - pwf2) where

C = 1/A n = 1

This, of course, is the conventional backpressure deliverability equation obtained empirically by Rawlins and Schellhardt (Reference 1) that was covered in Module 3. Rearranging equation (5) above, we get (p R 2 - p wf 2 ) qg

= A + Bqg

which has the form y = mx + b, the equation of a straight line of slope m with intercept b. This indicates that a plot of (pR2 - pwf2)/qg versus qg will give a straight line of slope B (the value of the turbulent flow term) with intercept A (the value of the laminar flow term).

8

If the value of kg h is known, for example from a build-up plot, then the turbulent flow term B can be solved to determine the turbulence coefficient D. PTPR 6096

Similarly, the laminar flow term A can be solved to determine the non-rate dependent skin s if kg h is known. In the Forcheimer equations, rd is the effective drainage radius. It is time dependent until rd = 0.472 re; thereafter, rd/rw = 1.5 (tD)0.5. The value of tD, dimensionless time, is given in oilfield units by: tD =

0.000264 k g t φ(μ g c t )i rw 2

Note in equation (5) above that the A-term which is multiplied by qg will increase more slowly than the B-term which is multiplied by qg2. This implies that turbulence is more of an issue in high kg h reservoirs that will have higher flow rates than in low kg h reservoirs that will have lower flow rates.

9 PTPR 6096

Exercise One 1. A gas well was tested as follows; Stabilized rate, mscfd 0.0 9424 15628 20273

Bottom hole flowing pressure, psia 408.2 394.0 378.7 362.7

Other data for this well: Permeability to gas, 1405 md Reservoir thickness, 115 ft Gas viscosity, 0.02 cP Real gas deviation factor, 0.91 Reservoir temperature, 120°F Well spacing, four/section Well bore diameter, 7-in. Gas gravity, 0.60 Perforated interval, 115 ft Non-turbulent skin, zero a) Plot the backpressure data for this well on a log-log deliverability chart. b) Calculate the laminar flow term for this well assuming non-turbulent skin is zero. Plot a laminar deliverability line on the deliverability chart. c) Determine the inertial-turbulent flow term for this well and plot the inertial-turbulent deliverability line on the deliverability plot. d) Sum the laminar and inertial-turbulent lines and plot the result on the loglog chart. e) Calculate the turbulence coefficient D by correlation and compare to the value of D calculated in 1. c). f) Write the complete LIT deliverability equation for this well. g) Determine the bottom hole AOF for this well. h) Calculate one or two extra points on the deliverability curve to better define it and create a bottom hole Inflow Performance Relationship (IPR) curve for this well. Can this curve be used throughout the life of this well?

10 PTPR 6096

OBJECTIVE TWO When you complete this objective you will be able to…

Understand and calculate the effect of formation damage or stimulation (skin) on the laminar Simplified Gas Analysis Method and the LIT Method (both laminar and turbulent).

Learning Material SIMPLIFIED METHOD SKIN CALCULATION

The conventional backpressure deliverability equation is: qg = C (pR2 - pwf2)n where n = 1.0 for 100% laminar flow n = 0.5 for 100% turbulent flow with intermediate values of n representing the relative amount of laminar and turbulent flow. The simplified method assumes stabilized flow that is made up of some combination of laminar and turbulent components. Ignoring the turbulent flow component and assuming 100% laminar flow, n = 1.0 and the conventional backpressure equation can be rearranged as follows: (p R 2 - p wf 2 ) =C qg where (in metric units), C =

and (in oilfield units),

C=

kg h

1310 (μ g z)avg T [ ln 0.472 (re /rw ) + s ] kg h

1424 (μ g z)avg T [ ln 0.472 (re /rw ) + s ]

If flow is laminar and the deliverability equation flow coefficient is determined from a stabilized well test, then it is possible to calculate the non-turbulent skin factor, s, if the product of kg h is known. Normally the product of kg h (in md-m or md-ft), known as the flow capacity of the reservoir, is obtained from a build-up test. 11 PTPR 6096

When kg h is small, the assumption of laminar flow is good and the value of s obtained by this method is reasonably accurate. LIT METHOD SKIN CALCULATION

In Exercise One above, the information that the non-turbulent skin, s, was zero was provided for the problem well. The non-turbulent skin, s, shows up only in the laminar flow term of the deliverability curve while the turbulence coefficient, D, shows up in the turbulent flow term of the deliverability curve. If the value of the non-turbulent skin, s, is not known then it is necessary to solve a pair of simultaneous equations in order to determine the two unknowns, s and D, which both figure in the calculation of (pR2 - pwf2). The pair of equations to be solved is: (pR2 - pwf12) = Aqg1 + Bqg12 and

(pR2 - pwf22) = Aqg2 + Bqg22

Once values for A and B are determined, then the equations for A and B can be solved for the non-turbulent skin factor, s, and the turbulent flow factor, D. If it turns out that the total apparent skin is constant at two different gas well flow test rates, then non-Darcy (turbulent) flow is not a factor at the higher of the two flow rates. Note that turbulent flow could become a factor at a higher flow rate. The best way to illustrate this is to work through Exercise Two.

12 PTPR 6096

Exercise Two 1. A gas well was tested as follows; Stabilized rate, mscfd 0.0 9424 15628 20273

Bottom hole flowing pressure, psia 408.2 380.8 350.1 316.3

Other data for this well: Permeability to gas, 1405 md Reservoir thickness, 115 ft Gas viscosity, 0.02 cP Real gas deviation factor, 0.91 Reservoir temperature, 120°F Well spacing, four/section Well bore diameter, 7-in. Gas gravity, 0.60 Perforated interval, 115 ft Note that the laminar and turbulent skin factors for this well are not known. a) Plot the backpressure data for this well on a log-log deliverability chart. b) Using the second and third flow rates, write a pair of simultaneous deliverability equations and solve for the laminar flow term A and the turbulent flow term B. c) Solve the laminar flow term A for the skin factor s and the turbulent flow term B for the turbulence factor D. d) Plot the laminar deliverability line and the inertial-turbulent deliverability line on the log-log deliverability chart. Determine the slopes of the laminar deliverability line and the inertial-turbulent deliverability line. e) Sum the laminar and inertial-turbulent deliverability lines and plot the calculated LIT deliverability line on the log-log plot. f) Write the complete deliverability equation. g) Calculate the bottom hole AOF for this well. h) Plot the IPR for this well with a skin factor on the same chart as the well in Exercise One which has a skin factor of zero. Note the effect of skin on deliverability and AOF. 13 PTPR 6096

OBJECTIVE THREE When you complete this objective you will be able to…

Apply the transient Modified Isochronal Well Testing technique to both the Simplified Gas Analysis and LIT Methods.

Learning Material FLOW REGIMES OVERVIEW

During a well flow test that is conducted at a constant flow rate, three flow regimes are apparent (Figure 4).

Figure 4 Flow Regimes for Constant Pressure Drawdown © Prentice Hall. This material has been copied under license from Access Copyright. Resale or further copying of this material is strictly prohibited.

14 PTPR 6096

Early Time Flow (Well Bore Storage)

The early-time flow period is strongly affected by well bore storage effects and linear flow through any fractures. It is usually short. When a well is opened, flow is initially from the well bore and any connected fractures (natural or induced). Flow from the reservoir increases from zero until all the flow is from the reservoir. This occurs at a time tws given by the following (for wells with zero skin) in metric units: t ws =

2653 μ Vws C ws kh

Vws is the well bore storage volume and Cws is the total compressibility of the fluids in the well bore storage volume. The above equation also gives the time when the transient (steady state) period starts. For wells with non-zero skin, References 8 and 9 provide dimensionless type curves showing the combined effect of well bore storage and skin on the time of the end of the well bore storage (early time flow) period and the start of the transient (pseudo steady-state) flow period. Transient Period (Steady State Flow)

The transient period starts after the early-time flow period (there may be a period of transition) and ends when the reservoir boundary begins to have a significant effect. During this period, the well is infinite-acting because the well pressure transient sees a constant pressure equal to the pressure at the external boundary of the reservoir until it reaches that boundary. During transient (steady state) flow, there is flow across the constant pressure “boundary” equal to the flow from the well bore. Transient flow ends when the well transient (that is, radius of investigation) reaches the external boundary, at re, of the closed reservoir at a time given by: t ss =

0.25 re 2 φ μ c λk

15 PTPR 6096

During the transient (steady state) period, the “reach” of the well transient, or radius of investigation, is given by the following: rinv

⎡λ kt⎤ =2 ⎢ ⎥ ⎣φ μ c⎦

0.5

This equation, a re-arrangement of the preceding one, is valid only for rinv ≤ re. Note the difference between the radius of investigation, rinv, and the effective drainage radius, rd. It was shown that the effective drainage radius for a well in stabilized (pseudo steady-state) flow stabilizes at a value given by: rd = 0.472 re The effective drainage radius is a mathematical definition required to force the steady state equation to represent the pseudo steady-state solution and does not have a physical counterpart. Stabilized Period (Pseudo Steady-State)

The steady state period ends when the pressure transient encounters the external boundary of the reservoir. At this time the reservoir pressure at the boundary, and at all points in the reservoir, start to decrease at the same constant rate – hence the name pseudo steady-state. Depending on the shape of the reservoir’s external boundary and the location of the well relative to that boundary, there may be a long transition period between the transient (or steady-state) flow regime and the stabilized (or pseudo steadystate) flow regime. Not all reservoirs are circular with a well in the center by any means! Boundary effects ultimately dominate the pressure behaviour at a well.

16 PTPR 6096

TRANSIENT DELIVERABILTY

To this point the deliverability relationships presented - the Rawlins and Schellhardt conventional back pressure deliverability equation and the Forcheimer versions of the deliverability equation for turbulent flow represented by equations (5) and (6) - are all at stabilized (pseudo steady-state) conditions, that is, where rinv ≥ re. The transient flow deliverability equation presented below is not derived empirically or from the Forcheimer equation for turbulent flow. It is an approximation to the analytical solution of the classic diffusivity equation as applied to flow in reservoirs with appropriate inner boundary conditions. The mathematics of deriving the deliverability equation for transient (steady state) conditions will not be covered here. Detail on the derivation of this equation can be found in chapters 2 and 4 of Reference 4. The deliverability equation for transient (steady state) flow conditions, was written using ∆p2 as follows in oil field units by Fetkovich (Reference 7): (p R 2 - p wf 2 ) = Where pR pwf μg z T s

β

λ k h φ rw qg t tD and,

= = = = = = = = = = = = = = =

tD =

1424 (μg z)avg T kh

⎡⎣{0.5 (ln t D + 0.809) + s} q g + Dq g 2 ⎤⎦

(7)

static reservoir pressure, psia flowing bottom hole pressure, psia gas viscosity, cps gas deviation factor, dimensionless reservoir temperature, °Rankin non-turbulent skin, dimensionless turbulence factor, ft-1 gas gravity effective permeability to gas, md net pay, ft porosity, fraction well bore radius, ft gas flow rate, mscfd time, days dimensionless time 6.33 x 10-3 k P t φ μ g rw 2

17 PTPR 6096

Fetkovich provided a definition of the turbulence coefficient D as: 2.226 x 10-15 β k λ D = h rw μg If we define,

At =

and,

Bt =

1424 ( μ g z)avg T kg h

[{0.5(ln t D

(8)

+ 0.809) + s}]

1424 ( μ g z)avg T D kg h

then, by substituting in Equation 7 we get, Δ p t 2 = (p R 2 - p wf 2 ) = A t q g + Bt q g 2

(9)

Equation (9) represents the LIT method for analysis of transient (pseudo steadystate) gas well deliverability tests. However, Equation (9) is strictly correct only in terms of pseudo-pressure as follows: Δ ψ t = (ψ R - ψ wf ) = A t q g + Bt q g 2

(10)

Equation (10) represents the LIT-ψ method for analysis of transient (pseudo steady-state) gas well deliverability tests. By combining the definitions for B and D above, we get:

rearranging,

3.17 x 10-12 z T λ Bt = β h 2 rw

(11)

Bt h 2 rw 3.17 x 10-12 z T λ

(12)

β=

A value for β is determined using equation (12) with the value of Bt determined from solving a pair of simultaneous equations for values of At and Bt. Then a value for D can be calculated. The turbulent flow skin effect is then given by Dq.

18 PTPR 6096

Exercise Three 1. A modified isochronal test was conducted on a gas well. Average reservoir pressure was 1948 psia. Given the following flow test data: Flow No. 1 2 3 4 Extended

pwf, psia 1784 1680 1546 1355 1233

pws, psia 1948 1927 1911 1887 1948

qg, mscfd 4500 5600 6850 8250 8000

a) Determine C and n for this well for stabilized conditions. b) Determine the stabilized AOF under stabilized and transient conditions using graphical and equation methods as outlined below. Procedure: Graphical Method – Stabilized Conditions a. Plot the data on a log-log deliverability chart and draw a best fit line through the four transient data points. b. Determine n by calculating the slope of the transient curve. To do this, read the flow rate change over one log cycle of ∆ p2 and divide by one (log 10x – log x is always 1). c. Draw a stabilized line with the same slope as the transient line through the stabilized test point. d. Calculate C by using the stabilized deliverability equation. e. Determine stabilized AOF by extrapolating the stabilized deliverability line and by calculating it using the stabilized deliverability equation. Do the values agree? Procedure: Equation Method – Transient Conditions f. Since values for s and D are not known, plot values of (pR2 - pwf2)/qg versus qg to determine At and Bt graphically. g. Write the transient deliverability equation and use it to calculate the transient AOF for this well. h. Calculate a stabilized IPR using the stabilized deliverability equation. Plot the stabilized IPR curve and the transient (test data) IPR curve on a Cartesian plot. 19 PTPR 6096

Exercise One Answers 1. a. Construct the deliverability plot of

(p

2 R

− p wf 2 ) versus q g on Log-Log chart

(p

p wf

2 R

-p wf 2 )

qg

psi 2

mscfd

480.2 394.0

0 11391

0 9424

378.7

23214

15628

362.7

35076

20273

psia

b. Calculate the laminar deliverability line using the L-I-T (for cheimer) equation for pseodo-steady state flow in oilfield units:

(p

2 R

-p wf 2 ) = A q g +Bqg 2

where A=

⎛ rd 1424μgZT ⎡ ⎢ln 0.472 ⎜ kgh ⎝ rw ⎣

⎞ ⎤ ⎟ + s⎥ ⎠ ⎦

solving A=

1424i0.02i0.91i58o ⎡ ⎛ 1489 ⎞ ⎤ ln 0.472 ⎜ ⎟ + o⎥ ⎢ 1405i115 ⎝ 0.292 ⎠ ⎦ ⎣

= 0.72436

(p

2 R

-p wf 2 )

11391 23214 35076

Aq g 6826 11320 14685

20 PTPR 6096

c. Calculate the inertial-turbulent deliverability line using hte L-T-T (fercheimer) equation for pseudo-steady state flow in oilfield units:

(p

2 R

-p wf 2 ) =A q g +Bqg 2

where B=1424

kgZT D kgh

Solving 1424i0.02i0.91i580 D 1405i115 = 0.093033 D To find D, use flow rate # 3 data 35076=0.72436 (20273) + 0.093033 D (20273) B=

D= 5.3329 i 10-4 B= 0.093033 i5.329i10-4

and

= 4.9577i10-5 (p R 2 -p wf 2 ) 11391 23214 35076

Bq g 2 4406 12117 20391

d. Calculate the turbulence coefficient D by correlation and compare to the value of D from the test data.using the correlation: D=

6g10-5 γk -0.1 h μ rw h 2 pert

Solving, D=

6g10-5 g0.60g(1405)-0.1 g115 0.02g0.292g(115) 2

= 2.597 g10-5

e. Add the laminar and inertial turbulent deliverability lines to determine the calculated LIT line

(p

2

-p wf 2 )

Aq g

Bq g 2

A q g + Bq g 2

11391

6826

4406

11233

23214

11320

12113

23438

35076

14685

20391

35076

R

PTPR 6096

21

f. The LIT deliverability equation for this well is, in oilfiend units:

(p

2 R

-p wf 2 ) = 0.72436q g + 4.9614i10−5 q g 2

g. There are several ways to determine the bottom hole 4of for this well: (i) Extrapolate by "eyeball" the deliverability curve on the log-log plot (quick, but least accuarate) (ii) use the forcheimer flow equation to calculate q g for p wf =0 (iii) Re-arrange the deliverability equation and solve for q g with p wf = 0 Using the quadratic formula we will use method (iii) Re-arranging the LIT equation in 1)f above to the form ax 2 +bx+c=0 gives 4.9614i10-6 q g 2 + 0.72436q g − 166627 = 0 The solution to equations of the above form is: y=

-b± b 2 -4ac 2a

( 0.72436 )

−0.72436 ±

qg =

2

− 4(4.9614i10−5 i−166727)

2(4.9614i10−5 )

-0.72436 ± 5.7957 9.9228i10-5 Only the + root yields a positive answer therefore -0.72436+5.7957 qg = 9.9228i10-5 = 51 100 mscfd = 4 of =

Exercise Two Answers 1. a. Plot (p R 2 -p wf 2 )versus q g on a log-log delverability plot. p wf

(p R 2 -p wf 2 )

qg

psi 2

mscfd

408.2

0

0

380.8

21632

9424

350.1

44083

15628

316.3

66069

20273

psia

22 PTPR 6096

Exercise Two, Problem 1) L-I-T Analysis

Test data 1.00E+06

(pR -pwf ), psi

2

1.00E+05

2

2

1

1.00E+04

1.00E+03 1000

10000

100000

1000000

Flow rate, mscfd

23 PTPR 6096

b. Use (p R 2 -p wf 2 ) = A qg + Bqg 2 #3

(408.22 − 350.12 ) = 15626 A + (15628) 2 B

#4

(408.22 − 316.32 ) = 20273 A + (20273)2 B

#3

44083 = 15628A+244.23i106 B

#4

66609 = 20273A+410.99i106 B 2252.6 = 4645 A + 166.76 i106 B A = 4.850-35900 B

#4-#3

44083 = 15628 (4.850-35900B) +244.23i106 B

substituting

44083 = 75796 - 561.05i106 B + 244.23i106 B 3i6.82B = 31713 B = 1.001i10-4 A = 1.256

substituting

c. Laminar flow term in oilfield units A=

1424μ gZT ⎡ ⎛ rd ⎞ ⎤ ln 0.472 ⎜ ⎟ + s ⎥ ⎢ kgh ⎝ rn ⎠ ⎦ ⎣

1.256=

1424i0.02i0.91i580 ⎡ ⎛ 2980 ⎞ ⎤ ln 0.472 ⎜ ⎟ + s⎥ ⎢ 1405i115 ⎝ 0.292 ⎠ ⎦ ⎣

1.256 − 8.48 0.09303 = 5.02

s=

The turbulent flow term is given by: B=

1424 μg ZT D kgh

1.001 i10-4 = 0.09303D D= 1.076i10-3

24 PTPR 6096

d. Plot the laminar deliverability line given by (p R 2 -p wf 2 ) = Aq or (p R 2 -p wf 2 ) psi 2

q2 =

(PR 2 -Pwf 2 ) A q Aq

mscfd

0

0

21632

17223

11837

44083

35098

19629

66609

53033

25463

Plot the inertial- turbulent deliverability line given by: (p R 2 -p wf 2 ) = Bq g 2 (p R 2 -p wf 2 )

q

Bq 2

21632

17223

8816

44083

35098

24243

66609

53033

40797

psi 2

d. See laminar and inertial-turbulent deliverability lines plotted below. Slope (1/n) of laminar deliverability line is 1 (n=1). Slope (1/n) of inertial-turbulent line is 2 (n=0.5).

25 PTPR 6096

Exercise Two, Problem 1) d. L-I-T Analysis

Test data Inertial-Turbulent Line Laminar Line

1.00E+06

(pR -pwf ), psi

2

1.00E+05

2

2

1

1.00E+04

1.00E+03 1000

10000

100000

1000000

Flow rate, mscfd

e. Sum laminar and inertial-turbulent deliverability lines and plot on log-log chart: (p R 2 -p wf 2 )

Aq

Bq 2

Aq +Bq 2

21632 44083

11837 19629

8816 24243

20652 43872

66609

25463

40797

66529

inertial turbulent Δp excess laminar Δp 2 at about 12 000 mscfd 2

26 PTPR 6096

Exercise Two, Problem 1) e. L-I-T Analysis

Test data Inertial-Turbulent Line Laminar Line Calculated L-I-T Line

1.00E+06

(pR -pwf ), psi

2

1.00E+05

2

2

1

1.00E+04

1.00E+03 1000

10000

100000

1000000

Flow rate, mscfd

27 PTPR 6096

f. The deliverability equation is (408.22 -Pwf 2 )= 1.256 q g + 1.00i10−4 q g 2 g. Re-arranging the deliverability equation to the form: ax 2 +bx+c=0 1.001i10-4 q g 2 + 1.256q g − 166627 = 0

Gives

for Pwf=0 and q g =Aof The solution to the quadratic equation is: −b ± b 2 − 4ac x= 2a substituting, Aof=

-1.256 ±

(1.256 )

2

− 4i1.00i10−4 i−1.66627

2i1.001i10-4

= 34900 mscfd h. Well IPR curves with zero non-turbulent skin and with +5 non-turbulent skin are plotted below. Exer. Two, Prob 1) h. 500

IPR Curve, Skin = +5

400

pwf, psia

IPR Curve, Skin = 0 300

200

100

0 0

10000

20000

30000

40000

50000

q g, m scfd

28 PTPR 6096

60000

Exercise Three Answers 1. a. Plot of data on log-log deliverability chart with best fit line 1 log 6000 − log1600 = scope log106 − log105 3.78-3.20 = 1 = 0.58 c. See log-log deliverability chart below: b. n=

Exercise Three, Problem 1) a., 1) c. L-I-T Analysis

(pR 2-pwf2), psi2

1.00E+07

1 1.00E+06

Transient Test Data Stabilized Test Data

1.00E+05 1000

10000

100000

Flow rate, mscfd

29 PTPR 6096

d. For the stabilized line, q = c( p R 2 - p wf 2 ) n and q

C=

(p

2 R

- p wf )

n

From the stabilized line At (p R 2 - p wf 2 ) = 1i106 psi 2 q = 5000 mscfd Therefore 5000 (1i106 )0.54

c=

= 2.877 mscfd/psi 2 AOF = 2.877 (19482 -0)0.54

e.

= 10,273 mscfd By extrapolation AOF

10,200 mscfd

f. To determine A t and Bt ,calculate

(p

2 ws

− p wf 2 )

qg

and plot versus q g , on a cartesian plot. The intercept of the straight

line fitted to the transient data is A t and the slope is Bt . Test

Flow

no

Rate

p ws

mscfd psia 0 1948 1 4500 1984 2 5600 1927 3 6850 1911 4 8250 1887 EyL. 8000 1948 Intercept = A t = 49.233

p wf

psia 1948 1284 1680 1546 1355 1233

p ws 2 -p wf 2

psia 2 0 612048 840929 1261805 1724744 2274415

p ws 2 -p wf 2 qg psia2/mscfd 136.01 159.09 184.21 209.66 284.30

Slope = Bt = 0.0195

30 PTPR 6096

g. The transient deliverability equation is

(p

2 R

-p wf 2 ) =19.666q g +0.0267q g 2

Re-arranging, 0.0195 q g 2 + 49.233q g − ( p R 2 -p wf 2 ) = 0 Which has the form ax 2 +bx +c=0 That can be solved using the quadratic formula −b ± b 2 − 4ac x= 2a At p wf = 0 p R = 1978 0.0195 q g 2 + 49.233q g − 3794704 = 0 and q g =

-49.233 ±

( 49.233)

2

− 4i0.0195i−3.794709

2i0.0195

= 12 745 mscfd This is the AOF during the transient period

31 PTPR 6096

Prob. 1. e. and 1. g. Plot showing stabilized and transient AOFs shown below. Exercise Three, Problem 1) e., 1) g. L-I-T Analysis

(pR 2-pwf2), psi2

1.00E+07

1 1.00E+06

Transient Test Data Transient AOF Stabilized Test data Stabilized AOF 1.00E+05 1000

10000

100000

Flow rate, mscfd

h. Calculate stabilized IPR curve using stabilized deliverability equation: q g =1.656 (19482 -p wf 2 )0.58 p wf

32

psia 1948 1800 1600 1300 900 600 300 0

qg mscfd 0 5553 5652 7700 9430 10230 10689 10839 PTPR 6096

Exercise Three, Problem 1) h. Transient and Stabilized IPR Curves 2000

Flowing Bottom Hole Pressure, psia

1800 1600 1400 1200 1000 800 600 400

Transient IPR from Test Data

200

Calculated Stabilized IPR

0 0

2000

4000

6000

8000

10000

12000

Gas Rate, mscfd

33 PTPR 6096

Module Self-Test 1. The conventional backpressure test requires each flow be done a. b. c. d.

until pressure has stabilized and does not decline any further until pressure is observed to decline at a constant rate for several equal time intervals followed by a shut-in to stabilized pressure for equal time intervals each of which is followed by a shut-in of equal time

2. When a pressure transient reaches an external reservoir boundary, all pressures in the reservoir begin to decrease at a constant rate. a. True b. False 3. The modified isochronal test is the most theoretically rigorous test and can be completed in the shortest amount of time. a. True b. False 4. The Laminar-Inertial-Turbulent method of gas well deliverability test analysis is the best method to use for wells where turbulence dominates well inflow. a. True b. False 5. Under some conditions, the p2 approach and the p approach are just as accurate as the ψ (pseudo-pressure) approach to well test analysis. a. True b. False 6. The transient flow period a. is also called the steady state period b. is characterized by a well transient that has reached the external boundary of the reservoir c. means the well is infinite acting d. all of the above 7. When the radius of the well pressure transient, the radius of investigation and the drainage radius all have values equal to the radius of a circular reservoir, pseudo steady-state flow begins. a. True b. False 34 PTPR 6096

8. If the flow capacity of a reservoir is small, then laminar flow becomes more likely. a. True b. False 9. Which of the following is not true? a. Well bore storage flow is followed by transient flow which is followed by steady state flow. b. Early time flow is followed by steady state flow which is followed by pseudo steady-state flow. c. Early time flow is followed by transient flow which is followed by pseudo steady-state flow. d. There may be a transition period between well bore storage flow and transient flow as well as between transient flow and pseudo steady-state flow. 10. If a well is in pseudo steady-state flow, the external boundary of the reservoir is a no-flow boundary a. True b. False 11. On the log-log plot of an LIT analysis of a deliverability test for a gas well with significant turbulence, a. the slope of the laminar deliverability line is always 1 and the slope of the inertial-turbulent deliverability line is always 2 b. the laminar and inertial-turbulent deliverability lines are always straight but the overall well deliverability line is curved c. the overall well deliverability curve is the sum of the laminar and inertialturbulent deliverability lines. d) all of the above 12. For a gas well tested at different rates during the transient flow period the non-Darcy skin will not change even if turbulence is significant. a. True b. False 13. For wells that are not in the center of circular reservoirs, flow equations must be modified with appropriate shape factors. a. True b. False 35 PTPR 6096

Module Self-Test Answers 1. b 2. a 3. b 4. a 5. a 6. a 7. b 8. a 9. a 10. a 11. d 12. b 13. a

36 PTPR 6096

Assignment 1. Identify the main types of well deliverability tests and describe how each is unique. 2. Identify the main types of flow conditions in a reservoir and describe the distinguishing characteristics of each. 3. Discuss the differences between a stabilized deliverability line and a transient deliverability line. 4. Discuss the difference between a laminar deliverability line, an inertialturbulent deliverability line and an overall well deliverability line. 5. Discuss the different values that the deliverability exponent, n, can have depending on whether flow is laminar or turbulent and depending on what position on an LIT deliverability line it is measured at.

37 PTPR 6096

References 1. Rawlins, E.L., Schellhardt, M.A., Backpressure Data on Natural Gas Wells and their Application to Production Operations, U.S. Bureau of Mines, Monograph 7, 1936. 2. ERCB Guide G-3, Gas Well Testing, 4th Ed, 1979. 3. Aziz, K., Mattar, L., Ko, S. and Brar, G.S., Use of Pressure, PressureSquared or Pseudo-Pressure in the Analysis of Gas Well Data, J. Can. Pet. Tech., 1975. 4. Economides, M.J., Hill, A.D. and Ehlig-Economides, C., Petroleum Production Systems, Prentice-Hall, 1993. 5. Systems Nodal Analysis, Engineering Reference I-23, Amoco Production Company, 1984. 6.

Aronofsky, J.S., Jenkins, R., A Simplified Analysis of Unsteady Radial Gas Flow, Trans. AIME, 201, 149-154, (1954).

7.

Fetkovich, M.J., Multipoint Testing of Gas Wells, SPE Well Test Analysis Course, March 17, 1975.

8.

Ramey, H.J.. Jr., Short-Time Well Test Data Interpretation in the Presence of Skin Effect and Well Bore Storage, J. Pet. Tech.., 23, 14093-1505. 1970).

9.

Agarwal, R.G., Al-Hussainy, R., Ramey, H.J., Jr., An Investigation of Wellbore Storage and Skin Effect in Unsteady Liquid Flow, Soc. Pet. Eng. J., 10, 279-290. (1970).

38 PTPR 6096

Appendix Table 1 Conversion Factor Table

39 PTPR 6096

40 PTPR 6096

Course Module

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Course Module Advanced Production Engineering - PTPR 465

Flow in Tubulars (Outflow Performance Relationship) PTPR 6097 Module 5

Revised: November 2007

MacPhail School of Energy

redefine yourself www.sait.ca

Flow in Tubulars (Outflow Performance Relationship) Rationale Why is it important for you to learn this material? Well bore tubulars are one of the key components of the Outflow Performance Relationship that determines production rate together with the Inflow Performance Relationship. Understanding the factors that affect flow in tubulars and knowledge of the computation tools to determine pressure drops in them are critical to being able to optimize the hydrocarbon production system.

Learning Outcome When you complete this module you will be able to …. Calculate the relationship between the pressure drop up a well bore tubular and the flow rate.

Learning Objectives Here is what you will be able to do when you complete each objective. 1. Understand and calculate the relationship (including friction factor) between flow rate and pressure drop for single-phase flow. 2. Understand the difficulties caused by two- and three-phase flow when calculating the relationship between flow rate and pressure drop. 3. Estimate the flow rate and pressure drop relationship for two- and three-phase flow using pressure traverse curves.

1 PTPR 6097

OBJECTIVE ONE When you complete this objective you will be able to… Understand and calculate the relationship (including friction factor) between flow rate and pressure drop for single-phase flow.

Learning Activity Complete each of the Learning Activities listed below. 1. Read and study the learning material in this objective. 2. Do the exercise provided. 3. Do the assignment provided. 4. Research outside sources (that is, the Internet, library materials, reference texts, and so on) for additional information.

Learning Material INTRODUCTION TO FLOW IN TUBULARS Well bore tubulars are one of the main components of the hydrocarbon production system. They usually consist of one or more strings of tubing run inside the well bore casing pipe. Flow of reservoir fluids is up the tubular(s) or up the annulus between the tubular(s) and the casing, or sometimes both. Packers are sometimes used to isolate some of the available flow paths up the well bore while production equipment or tubing attachments can assist, or interfere, with flow. Some tubing attachments (for example, sliding sleeves) permit changing the flow path between a tubular and the annulus with a simple operation on the wire line. Although some wells produce single-phase oil, gas or water, most wells have twophase flow, (gas and oil, gas and water, oil and water) or three-phase flow (oil, gas and water). Depending on many factors such as fluid type and properties, pressures and flow rates and mechanical properties such as tubular diameters and lengths, friction factors and inclination, the characteristics of multi-phase fluid flow can be very complex and can vary widely. Determining the pressure drop and flow rate sometimes requires the ability to determine the specific flow regime and always requires knowledge of the appropriate specific multi-phase correlation to estimate flow rate and pressure drop with reasonable accuracy. 2

PTPR 6097

This module will focus on an understanding of single and multi-phase flow and the use of published pressure traverses to estimate the performance of the tubulars component of the hydrocarbon production system. Single-Phase Incompressible Flow As illustrated in Figure 1, the single-phase flow of an incompressible fluid can be characterized as either laminar or turbulent depending on the value of a dimensionless quantity called the Reynolds number. Laminar flow, also called streamline flow, is characteristic of lower fluid velocities. It has only a longitudinal component of velocity with no radial component. Turbulent flow on the other hand has randomly fluctuating flow velocity in all directions, that is, eddies. Turbulent flow, sometimes called plug flow, has a flatter profile of fluid velocity across the pipe diameter - hence the name plug flow – than does laminar flow. The flatter velocity profile across a pipe section containing turbulent flow is due to radial mixing. Laminar flow has a larger variation in fluid velocity across the pipe diameter with maximum velocity at the center of the pipe and a minimum velocity near the well of the pipe due to a lack of radial mixing. For either type of flow, there is always a thin layer of fluid at the pipe wall, the “boundary layer” or laminar sub-layer, which is always moving in laminar flow.

Figure 1 Laminar and Turbulent Flow in Pipe © Crane Co. This material has been copied under license from Access Copyright. Resale or further copying of this material is strictly prohibited.

There is a narrow zone between laminar and turbulent flow, called the critical zone, where flow can be either laminar or turbulent depending on factors such as changes in pipe section, changes in direction of flow and hindrances to flow such as valves. This zone starts at the Reynolds lower critical number of about 2000 and generally extends up to the Reynolds upper critical number of about 4000. Above the Reynolds upper critical number there is a transition zone to fully turbulent flow. 3 PTPR 6097

The Reynolds number, NRE, is a dimensionless combination of fluid variables that can be thought of as the ratio of inertial forces, that is, those caused by acceleration or deceleration of fluids, to the forces of viscous shear. For single-phase incompressible flow in a circular pipe the dimensionless Reynolds number is given by: NRE = where v D ρ μ

= = = =

vDρ μ

average velocity, m/s diameter, m density, kgm/m3 viscosity, Pa-s (kgm/m-s)

Discussion of Units of Dynamic Viscosity Dynamic viscosity, μ, is defined as shear stress on a fluid, τ, divided by the resulting rate of change of the fluid’s velocity with distance, dv/dx. The resulting units of viscosity are N-sec m-2. Since by definition 1 newton, N, is the force required to accelerate 1 kgm by 1 m sec-1 per sec, then: μ = N-sec m-2 = kgm m sec-2 sec m-2 = kgm m-1 sec-1 By modern definition: 1 Pa-s = 1 kgm m-1 sec-1 where Pa-s is a Pascal-second. In the past, 1 Poiseuille was the name given to 1 kgm m-1 sec-1 and 1 Poise was the name given to 1 gmm cm-1 sec-1. It should be apparent that 1 Poise is 1/10th of a Poiseuille and that 1/100th of a Poise, or 1 cP, is 1/1000th of a Poiseuille, or, today, 1/1000th of a Pa-sec. It is easy to be confused about units of dynamic viscosity. Fortunately, this module does not require discussion of kinematic viscosity and its conversion to dynamic viscosity! Convention in the oil patch is to use cP (1/1000th of a Pa-s) for both gas and liquid viscosities. Liquid viscosities are generally a few cP or higher while gas viscosities are a small fraction of a cP. Obviously, when using any flow equations, attention to the correct units is critical to obtaining the correct answer.

4

PTPR 6097

Definitions of Reynolds Numbers for the Oilpatch By substituting q/A = v and A = πD2/4 for circular pipe in the expression for the Reynolds number, we get the general expression of the Reynolds number for flow in a circular pipe: NRE =

4qρ π Dμ

Then, by converting seconds to days and m to mm we get the dimensionless expression for the Reynolds number for single-phase incompressible flow in a circular pipe in metric units of m3/d, kgm/m3, mm, and Pa·s: NRE =

0.0147 q ρ Dμ

In oilfield units of bbl/d, lbm/ft3, in. and cP, the dimensionless expression for the Reynolds number for single-phase incompressible flow in a circular pipe is: NRE =

1.48 q ρ Dμ

Pressure Drop in Single-Phase Flow To determine the pressure drop in pipe, a mechanical energy balance approach is used. The total pressure drop, ∆p, is the sum of all gains or losses due to any changes in potential energy, ∆pPE, kinetic energy, ∆pKE, and flowing friction, ∆pF In other words:

∆p = ∆pPE + ∆pKE + ∆pF

Integration of the differential form of the above mechanical energy balance equation, which assumes incompressible flow and no work energy added or removed from the fluid, gives the following expression for the total pressure drop in a pipe of total length L: ∆p = p1 – p2 =

2f f ρ u 2 L g ρ ρ ∆z + ∆u2 + gc D 2g c gc

Eq. 1

where g/gc = a term to convert oilfield (and imperial engineering system) units between force and mass and to adjust equations when gravitational force differs significantly from the value at mean sea level (at sea level, g = 32.2 ft sec-2; gc, the constant of proportionality = 32.2 ft lbm lbf-1 sec-2; therefore, g/gc = 1lbf/lbm at sea level ρ = density of incompressible fluid, lbm/ft3 ∆z = change in elevation, ft ∆u = change in flow velocity, ft/sec ff = Fanning friction factor, dimensionless L = length of pipe between points 1 and 2, ft D = internal pipe diameter, ft 5 PTPR 6097

To calculate the change in pressure due to change in elevation (potential energy, PE) for flow in a straight pipe of length L and inclination angle Ø, as measured from the horizontal, where flow is upward (point 1 lower than point 2): ∆pPE =

g ρ L sin Ø gc

Note that for horizontal flow, the sin of 0 deg is 0 (no potential energy change). For vertical flow, the sin of 90 deg is 1. To calculate the change in pressure due to a change in the velocity of the fluid (kinetic energy, KE) as a result of a change in pipe diameter between points 1 and 2: 8ρq ∆pKE = 2 (D2-4 – D1-4) π gc To calculate the frictional pressure drop (always positive) the Fanning equation is used: 2f f ρ u 2 L ∆pF = gc D For laminar flow, the Fanning friction factor is a simple function of the Reynolds number: ff, lam = 16/NRE

6

PTPR 6097

For turbulent flow, the Fanning friction factor depends on both the Reynolds number and the pipe roughness. The Moody friction factor diagram (Figure 2) relates relative surface roughness, defined as ε/D, where ε is the height of the protrusions from the nominal pipe internal diameter, D. Note that the Moody friction factor, f, is 64/NRE.

Figure 2 Moody Friction Factor Diagram © Crane Co. This material has been copied under license from Access Copyright. Resale or further copying of this material is strictly prohibited.

Figure 3 gives relative roughness values for different types and sizes of clean pipe. Note in Figure 3 the increase in relative roughness, and therefore friction factor, with a decrease in diameter for each type of pipe. It is common for the roughness of pipe to increase over time due to scale, corrosion and other deposits. It is possible to calculate friction factors using the following correlation for smooth wall pipe developed by Drew, Koo and McAdams (Reference 10); f = 0.0056 + 0.5NRE -0.32

(3000 < NRE < 1,000,000)

For fully rough pipe, the correlation of Nikuradze (Reference 11) can be used; f--2 = 1.74 – 2 log (2

/D)

7 PTPR 6097

Figure 3 Relative Roughness of Pipe © Crane Co. This material has been copied under license from Access Copyright. Resale or further copying of this material is strictly prohibited

Reference 2 (Ex. 7-2, 7-3 and 7-4) provides examples of the use of the above equations to calculate the pressure drops for single-phase incompressible flow.

8

PTPR 6097

Single-Phase Compressible Flow

In a gas well, because the fluid is compressible, fluid density and fluid velocity are not constant along the tubular. The mechanical energy balance for gas in a tubular can be written without the kinetic energy term as the diameter of tubulars in a well bore does not usually change significantly and because the change in kinetic energy due to changes in temperature and pressure are small compared to changes in potential energy and losses due to friction. Thus, for gas flow in well bore tubulars with no work energy mechanically extracted or added from the gas stream: ∆p = ∆pPE + ∆pF In differential form and slightly re-arranged, Eq. 1is: 2f ρ u 2 L dρ g = dz + f ρ gc gc D Substituting dz = sin Ø dL allows combining the two terms on the right as follows: 2f ρ u 2 ⎞ dρ ⎛ g =⎜ sin Ø + f ⎟ dL ρ ⎝ gc gc D ⎠

Re-arranging the real gas law to calculate density and using = MWg/28.97 in terms of fluid density gives an expression for gas density: ρ=

28.97 γ g p ZRT

The gas velocity, u, is given by u = q/A = q/(πd2/4) as follows: u=

⎛ T ⎞ ⎛ psc ⎞ 4 q sc Z ⎜ ⎟⎜ ⎟ 2 πD ⎝ Tsc ⎠ ⎝ p ⎠

By making the substitutions for ρ and u in the differential form of the mechanical energy equation and re-arranging the result, ready for integration, is:

ZRT dp + 28.97γ g p

2 ⎛ T ⎞ ⎛ psc ⎞ ⎤ ⎪⎫ 32 f f ⎡ ⎪⎧ g sin Ø + ⎨ ⎢q sc Z ⎜ ⎟⎜ ⎟ ⎥ ⎬ dL = 0 g c π 2 D5 ⎣ ⎝ Tsc ⎠ ⎝ p ⎠ ⎦ ⎪⎭ ⎪⎩ g c

Eq. 2 9

PTPR 6097

If average values for the temperature and for the real gas deviation (compressibility) factor are used, Eq. 2 can be integrated to give: 2

⎛ q sc Zavg Tavg psc ⎞ s 32f f p =e p + ⎜ ⎟ ( e - 1) 2 5 Tsc g c π D sin Ø ⎝ ⎠ 2 2

s

2 1

Eq. 3

where:

s=

-(2)(28.97)γg (g/g c ) sin Ø L Zavg R Tavg

Eq. 4

In order to obtain the friction factor, the Reynolds number for compressible flow must first be calculated as follows: N RE =

4(28.97) γ q psc π D μ avg R Tsc

Eq. 5

Combining the constants and conversion factors in equations (3), (4) and (5) gives, in oilfield units: 2

⎛ Zavg Tavg ⎞ s p = e p + 2.685 × 10 ⎜ ⎟ ( e - 1) ⎝ D5 sin Ø ⎠ 2 2

s=

s

2 1

-3

-0.0375 γ g sin Ø L Zavg Tavg

N RE = 20.09

γg q D μ avg

Eq. 6

Eq. 7

Eq. 8

Equations (6), (7) and (8) are the working equations in oilfield units for compressible flow in circular pipe.

10

PTPR 6097

Exercise One 1) Given: Fluid velocity, 0.0914 m/s Fluid specific gravity, 0.82 Pipe ID, 101.6 mm Fluid viscosity, 3 cP a) Calculate the Reynolds number. b) Prove in your calculation that the units cancel so the result is a dimensionless number. c) Is the flow laminar or turbulent? 2) Given: Flow rate, 403 bopd Oil density, 51.07 lb/ft3 Pipe ID, 4 in. Fluid viscosity, 3 cP a) Calculate the Reynolds number. b) Calculate the average fluid velocity in ft/sec. c) Illustrate with a hand movement the average fluid velocity in 4 in. pipe of a well producing 400 bopd. 3) For new oil field well bore tubing with an internal diameter of 1.995 in. at a Reynolds number of 150,000: a) Determine the friction factor using Figures 2 and 3. b) If ε/D is 0.0009, determine the friction factor using the correlation of Nikuradze.

11 PTPR 6097

OBJECTIVE TWO When you complete this objective you will be able to…

Understand the difficulties caused by two- and three-phase flow when calculating the relationship between flow rate and pressure drop.

Learning Material INTRODUCTION TO MULTI-PHASE FLOW

Multi-phase flow occurs whenever there is simultaneous flow of free gas and liquid. Some applications of vertical multi-phase flow analysis include: •

Flowing wells

•

Annular vertical flow

•

Dewatering gas wells/siphon strings

•

Artificial lift design

•

Gas lift design

•

Design of tubulars in deviated wells

Many correlations have been developed to determine the pressure drop in vertical, inclined and horizontal multi-phase flow (horizontal multi-phase flow will be covered in Module 6). Care has to be taken to ensure the best multi-phase flow correlation is used for the particular conditions. Since the multi-phase flow correlations were developed largely based on field or lab tests, many provide reasonable accuracy only over limited ranges of such factors as flow pattern, pipe diameter, fluid velocity, fluid viscosity and gas-liquid ratio, among others. Flow Regimes

Figure 4 shows the main vertical multi-phase flow regimes, or patterns, as categorized by Orkiszewski (Reference 2). Note that many multi-phase flow correlations utilize their own categorization of multi-phase flow regimes. Flow regimes for horizontal flow (Module 6) differ substantially from flow regimes for vertical or inclined multi-phase flow. During bubble flow, small gas bubbles form and move at different velocities within a continuous liquid phase.

12

PTPR 6097

Figure 4 Vertical Multi-Phase Flow Patterns © Amoco Production Co. This material has been copied under license from Access Copyright. Resale or further copying of this material is strictly prohibited.

In slug flow, the gas phase is more pronounced. The bubbles have enlarged and coalesced to form plugs or slugs. The liquid phase is still continuous but the gas bubbles move at a higher velocity than the liquid. The change from a continuous liquid phase to a continuous gas phase occurs in the slug-annular transition flow pattern. In this flow pattern, gas phase effects dominate. Liquid may be entrained in the gas bubbles which may join together. In the annular-mist flow pattern, the gas phase is continuous. Liquid is entrained as small droplets in the gas phase and also coats the pipe wall with a film. Liquid Holdup

An important concept when dealing with multi-phase flow is liquid holdup. In upward multi-phase flow, the lightest phase, because of density, viscosity and/or surface tension effects, will tend to move faster than the heavier phases. This results in the pipe having a greater quantity of the heavier phase(s) than is entering the pipe. Holdup, y, is simply defined, for the denser phase β, as yβ = Vβ /V where yβ is a fraction and Vβ is the volume occupied by the denser phase in a given volume of pipe, V. The holdup of the lighter phase, α, is given by yα = 1-yβ. Correlations for liquid holdup are important components of most multi-phase flow correlations. 13 PTPR 6097

Multi-Phase Flow Equations

The mechanical energy balance equation for compressible flow can be applied to multi-phase flow with adjustments to account for the presence of two phases (gas and liquid) with often two liquids (oil and water) being present. The variables in the mechanical energy balance that must be adjusted for gasliquid flow include: ρ = ρm, density of gas-liquid mixture v = vm, velocity of mixture f = fm, friction factor for flow of gas-liquid mixture

and the Reynolds number for flow of a gas-liquid mixture: (N RE ) m =

vm D ρm μm

In order to calculate the density of the gas-liquid mixture, it is necessary to know the fraction of the pipe that contains liquid. This is known as liquid holdup, denoted as HL. The expression for mixture density is therefore: ρm = ρL (HL) + ρg (1 - HL)

Because of slippage, the tendency for gas to flow faster than liquid, a holdup correlation is required to accurately assess the density of the mixture. If two liquids are present in the liquid phase, properties of the liquid mixture must be determined as follows: Liquid phase density, ρL = ρo Fo + ρw Fw

where Fo and Fw are oil and water fractions Liquid phase viscosity, μL = μo Fo + μwFw

Other fluid properties of the two-phase mixture can be determined as follows: Gas-liquid mixture viscosity, μm = μL (HL) + μg (1 - HL)

14

PTPR 6097

Figure 5 illustrates the change with gas flow rate in the relative contribution of ∆pPE and ∆pF toward total ∆p (with = ∆pKE assumed negligible) in multi-phase flow.

Figure 5 Relative Contributions to Total Pressure Drop in Multi-Phase Flow © Amoco Production Co. This material has been copied under license from Access Copyright. Resale or further copying of this material is strictly prohibited.

Multi-Phase Flow Correlations

One of the best and most popular multi-phase flow correlations is that of Hagedorn and Brown. It was developed using data from a 1500 ft deep instrumented well with 1¼ in. to 2 ⅞ in. tubing. It uses a correlation, rather than actual measurements, for liquid holdup. Another well-known correlation is that of Beggs and Brill. It is based on horizontal flow regimes with corrections to liquid holdup for non-horizontal situations and is applicable to any pipe inclination, including horizontal and downward flow. It was developed using air and water flowing in 1 in. and 1.5 in. acrylic pipe 90 ft long that could be inclined at any angle. Test rates were less than 300 mscfd and 1000 bbl/d while pressures were between 35 and 95 psia. Other good multi-phase correlations include those of Dun and Ros, Orkiszewski and Govier and Aziz. There are many others - some good, some not so good and most with limited applicability in terms of rates, pressures, fluids and flow regimes. 15 PTPR 6097

Special Cases of Multi-Phase Flow Heading

Heading is the main cause of irregular behaviour during the latter stages of a well’s flowing life when rates have fallen off significantly. It can also be found in new oil wells that have low gas-liquid ratios. A well that has reached the heading stage will flow longer with communication between the annulus and tubing than if a packer isolating them is installed. This is because the accumulation of gas in the annulus cyclically “blows around” and acts to gas lift the liquid out of the well bore at a high rate. When the lift cycle is complete and most of the liquid in the annulus and tubing has been “blown out”, the bottom hole pressure is at its lowest possible value and inflow of oil and gas resumes. Some of the gas bypasses the tubing string and builds up again in the annulus until the cycle repeats. This method of production is inefficient as it produces out the last liquid volumes from the well bore with an excess of gas. Using packers will reduce or eliminate heading in a well but the well will normally die sooner than if a packer was installed. More efficient production systems to control heading in a well and use it to advantage include regulators activated by casing pressure to control flow out of the tubing, gas valves in the annulus and plunger lift in the tubing. Directional Wells

Directional wells often have a vertical or slanted straight section near surface, a section where angle is built to the final angle and a more or less straight final section at the final angle to reach the downhole target depth. Such wells can be analyzed for vertical and/or inclined multi-phase flow if the well trajectory is divided into sections having reasonably constant angles. Another way to tackle multi-phase flow in directional wells was presented by Ney (Reference 8) and Fuentes (Reference 9). In summary, a vertical multi-phase flow correlation is used assuming a length of tubing equal to the true vertical depth (TVD) of the well and a horizontal multi-phase flow correlation is used assuming a length of tubing equal to the difference between the actual length of the tubulars and a length equal to the TVD.

16

PTPR 6097

Annular Flow

Sometimes well flow is up the annular space between the tubing and casing. An example of this is a dual completion (two separate reservoirs producing) having a packer and only one tubing string. For the case of annular flow, the hydraulic radius concept allows the use of multiphase flow correlations. Hydraulic radius is defined as the cross-sectional flow area divided by the wetted perimeter. When applied to an annulus, the hydraulic radius, rh, can be shown to be equal to (Di – Do)/4. Since for circular pipe, rh = D/4 then for an annulus, the hydraulic diameter, Dh = Di –Do. If the hydraulic diameter of the annulus is used, then any multi-phase flow correlation for circular pipe can be used.

17 PTPR 6097

OBJECTIVE THREE When you complete this objective you will be able to…

Estimate the flow rate and pressure drop relationship for two- and three-phase flow using pressure traverse curves.

Learning Material PRESSURE TRAVERSE CURVES

A pressure traverse, or gradient curve, is a plot of the flowing pressure in a tubular versus depth. Usually, the curves start at atmospheric pressure so that the pressure traverse indicates the total flowing pressure loss, ∆p, at any depth. There are two types of pressure traverse curves. Field pressure traverse curves are generated by computer using the best vertical multi-phase correlation for a particular field and formation. These are then used by field engineers and technicians to help analyze producing wells and solve production problems. General pressure traverses are also prepared by computer but can utilize any of several multi-phase correlations each of which was developed from a different sample of wells, fields, fluids and tubulars. General pressure traverse curves may or may not work well for any particular field. Whether field or general pressure traverse curves are used, either should be proved against actual field data to ensure their accuracy is acceptable for that application. Pressure traverses are calculated by a trial and error procedure. The tubular is divided into a number of increments, usually by length. The unknown pressure at the end of the first increment is estimated and fluid properties are evaluated at average conditions of pressure in the increment. The pressure at the end of the increment is then calculated using the appropriate multi-phase flow correlation and compared to the estimate. If the two figures do not agree well enough, the fluid properties are recalculated using a new estimate of pressure to determine average pressure until the estimated and calculated pressures agree sufficiently. This trial and error procedure is repeated for each of the remaining increments until the end of the tubulars is reached. The shorter the increments the better the accuracy as the fluid properties will vary less between the ends of the segment If temperature is not sufficiently constant along the length of the tubulars being evaluated, an estimate for temperature will also have to be made for the end of the increment and used, along with an estimate for pressure, to determine average temperature and pressure in the increment in order to determine fluid properties.

18

Figure 6 is an example of a general pressure traverse curve for three-phase flow. References 5, 6 and 7 provide additional sources of prepared. general pressure traverse curves. PTPR 6097

Some applications of pressure traverse curves include: 1. Estimation of flowing bottom hole pressure when surface pressure is known. 2. Estimation of flowing pressure drop in tubulars when either surface or bottom hole pressure is known. 3. Estimation of maximum surface pressure for a desired flow rate if a bottom hole IPR curve is available for the well. 4. Estimation of the minimum production rate that will prevent a well from loading up and dying when surface pressure is fixed. This is done by making a plot of flowing bottom hole pressure versus production rate. The plotted curve will show a minimum value of flowing bottom hole pressure, at which the well will not flow on its own but load up and die.

Exercise Two 1. A well is producing at a rate of 1200 stbbl/d. Determine the flowing bottom hole pressure and the flowing pressure drop in the tubing given the following conditions: Gas-liquid ratio, 600 scf/stbbl Oil API, 35° Water specific gravity, 1.07 Water cut, 50% Tubing size, 1.991-in i.d. Tubing length, 9400 ft Separator pressure, 400 psi Procedure: a) Find the flowing pressure traverse curve which most closely approximates the given conditions (sometimes, though not in this case, it may be necessary to interpolate between two or more sets of pressure traverse curves). b) Determine the equivalent depth corresponding to the well head pressure (this represents the surface) and add the tubing length (the sum represents the equivalent well bottom). c) Intersect the appropriate GLR curve at the depth representing the equivalent well bottom and read the flowing bottom hole pressure. d) Calculate the flowing pressure drop in the tubing. 19 PTPR 6097

Figure 6 Sample General Pressure Traverse Curves © Pennwell Publishing Co. This material has been copied under license from Access Copyright. Resale or further copying of this material is strictly prohibited.

20

PTPR 6097

Exercise One Solutions 1. a)

Given: Fluid velocity, 0.0914 m/s Fluid specific gravity, 0.82 Pipe ID, 101.6 mm Fluid viscosity, 3 cP N RE = Use

=

vdρ μ 0.0914 m s -1 × 0.1016 m × 820 kg m m -3 0.003 kg m m -1s -1

= 2,538 b) Units in the above equation cancel out so the answer is dimensionless c) Flow is in the critical zone where it is changing from laminar to turbulent and may be either or both. 2. a) Given: Flow rate, 403 bopd Oil density, 51.07 lb/ft3 Pipe ID, 4 in. Fluid viscosity, 3 cP qρ dμ 403 × 51.07 = 1.48 × 4×3 = 2,538

N RE = 1.48

Use

b) Calculate the average fluid velocity in ft/sec

V=

q A 403

Use

=

= 0.3

1 d 5.615 ft 3 bbl × × 86,400 sec d bbl π⎛ 1 ft ⎞ ⎜ 4 in . × ⎟ 4⎝ 12 in . ⎠

2

ft sec 21

PTPR 6097

c) A fluid velocity of 0.3 ft/sec or 18 ft/min is not very fast! 3. a) First, determine the relative roughness, ε/D , for commercial steel pipe from figure 1.995 in = 50.67 mm ε/D = 0.0009 From figure at ε/D = 0.0009 and NRE= 150,000. The friction factor is 0.021. b) Using the correlation of Nikuradze for fully rough pipe: with ε/D =0.0009; f -2 =

1

= 1.74-2 log(2 ε/D) f = 7.229 f = 0.019

Exercise Two Solutions 1. a) Using Figure 3, equivalent depth of separator (surface) pressure of 400 psi is 2200 ft. b) Adding the equivalent depth of 2200 ft for surface pressure to the tubular length of 9400 ft gives an equivalent well bottom at 11, 600 ft. c) Moving horizontally from the well bottom depth of 11, 600 ft to the pressure traverse curve for a GLR of 600 scf/stbbl yields a flowing bottom hole pressure of 2920 psi The flowing pressure drop in the tubing is 2520 psi.

22

PTPR 6097

Module Self-Test Directions: • • • •

Answer the following questions. Compare your answers to the enclosed answer key. If you disagree with any of the answers, review learning activities and/or check with your instructor. If no problems arise, continue on to the next objective or next examination.

Flow in Tubulars

1. A laminar boundary layer always exists next to the pipe wall whether flow is laminar or turbulent. a) True b) False 2. The Reynolds number a) indicates whether flow will be laminar, turbulent or in between. b) is a dimensionless quantity in both metric and oilfield units. c) is a function of fluid properties, fluid velocity and a length that depends on the particular cross-section of the flow conduit. d) all of the above. 3. One hundred cP (centipoise) are equal to 1 Pa·s. a) True b) False 4. In a typical flowing oil well of 7000 ft depth producing 400 bopd up 2 ⅞-in. tubing, which would you expect to be the dominant pressure loss? a) ∆pPE b) ∆pKE c) ∆pF 5. The units of all the pressure loss terms in the mechanical energy balance equation are lbf ft-2 in the lbm, lbf, ft, sec. system of units. a) True b) False

23 PTPR 6097

6. For incompressible flow, a) the Fanning friction factor is a function of the Reynolds number under laminar conditions. b) the Fanning friction depends on both the Reynolds number and the pipe roughness under turbulent flow. c) The Fanning friction factor is indeterminate or undefined in the critical zone between laminar and turbulent flow. d) all of the above. 7. In a flowing gas well. which of the following does not change along the tubular? a) b) c) d) e)

flowing pressure fluid density fluid viscosity temperature none of the above

8. There are many multi-phase flow correlations, each of which were developed using different samples of well types, fluid properties, tubular diameters and flow regimes. a) True b) False 9. Which of the following is not true? a) Hydraulic radius is cross-sectional flow area divided by wetted perimeter. b) Heading describes irregular flow late in the life of a flowing well when rates have declined. c) In the bubble flow regime, small bubbles of liquid form and move at different velocities in a continuous gas phase. d) In the annular-mist flow pattern, the gas phase is continuous and the liquid is entrained as small droplets in the gas phase. 10. Liquid holdup means that the heavier liquid phase in the tubulars holds up the lighter gas phase so that the gas phase moves slower than the heavier liquid phase. a) True b) False 11. Any set of pressure traverse curves being contemplated for use in a well analysis should be proven against actual field data. a) True b) False 24

PTPR 6097

Module Self-Test Answers 1. a 2. d 3. b 4. a 5. a 6. d 7. e 8. a 9. c 10. b 11. a

25 PTPR 6097

Assignment 1) Describe the velocity profiles for laminar and turbulent flow across a section of circular pipe. 2) Compare the flow regimes as presented by at least two investigators of multiphase flow correlations other than Govier and Aziz. 3) List at least four different multi-phase correlations for vertical flow and describe, with sources, the conditions the correlations are reputed to be best for in the opinion of the reviewer.

26

PTPR 6097

References 1. Flow of Fluids, Technical Paper No. 410M, Crane Co., 1977. 2. Orkiszewski, J., Predicting Two-Phase Pressure Drops in Vertical Pipe, JPT, June 1967. 3. Systems Nodal Analysis, Engineering Reference I-23, Amoco Production Co., Feb.1 1984. 4. Brown, K.E., Beggs, H.D., The Technology of Artificial Lift Methods, PennWell Publ. Co., 1977. 5. Handbook of Gas-Lift, U.S. Industries Petr. Equip. Div., 1960. 6. Brown, K.E., Gas-Lift Theory and Practice, Petr. Publ. Co., 1965. 7. Gradient Curves for Well Analysis and Design, CIM Special Vol. 20 (SI Units). 8. Ney, C., A Laboratory Investigation of Holdup and Pressure Loss in Directional Multiphase Flow, M.S. Thesis, University of Tulsa, 1968. 9. Fuentes, A.J., A Study of the Multiphase Flow Phenomena in the Directional Well, M.S. Thesis, University of Tulsa, 1968. 10. Drew, T.B., Koo, E.C., McAdams, W.H., Trans. Am. Inst. Chem. Engrs., 28, 1930. 11. Nikuradze, J., Forschungsheft, p. 301, 1933.

27 PTPR 6097

Appendix Table 1 Conversion Factor Table

28

PTPR 6097

Course Module

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Printed in Canada on Recycled Paper

Course Module Advanced Production Engineering - PTPR 465

Flow in Surface Pipelines and Facilities (Outflow Performance Relationship) PTPR 6098 Module 6

Revised: November 2007

MacPhail School of Energy

redefine yourself www.sait.ca

Flow in Surface Pipelines and Facilities (Outflow Performance Relationship) Rationale Why is it important for you to learn this material? The surface flow lines and facilities are the second key component (the first being the well bore tubulars) of the Outflow Performance Relationship that together with the Inflow Performance Relationship determines production rate. Understanding the factors that affect flow in surface pipelines and facilities and knowledge of the computation tools to determine pressure drops in them are critical to being able to optimize the hydrocarbon production system.

Learning Outcome When you complete this module you will be able to …. Calculate the relationship between pressure drop in the surface flow lines and equipment and the flow rate.

Learning Objectives Here is what you will be able to do when you complete each objective. 1. Understand and calculate the flow rate-pressure drop relationship for singlephase flow (including friction factor). 2. Understand the difficulties caused by multi-phase flow on calculating the flow rate-pressure drop relationship. 3. Estimate the flow rate-pressure drop relationship for two- and three- phase flow using pressure traverse curves.

1 PTPR 6098

OBJECTIVE ONE When you complete this objective you will be able to… Understand and calculate the flow rate-pressure drop relationship for single-phase flow (including friction factor).

Learning Activity Complete each of the Learning Activities listed below. 1. Read and study the learning material in this objective. 2. Do the exercise provided. 3. Do the assignment provided. 4. Research outside sources (that is, Internet, library materials, reference texts, and so on) for additional information.

Learning Material INTRODUCTION TO FLOW IN SURFACE LINES AND FACILITIES The surface flow lines and facilities are the second key component of the Outflow Performance Relationship in the hydrocarbon production system. As with vertical well bore tubulars, flow in surface flow lines may be single-phase gas or liquid, or two-phase (gas and oil or water, or oil and water), or three phase (gas, oil and water). Module 5 focused on vertical single- and multi-phase flow and touched on inclined and directional multi-phase flow. This module will focus on horizontal single- and multi-phase flow and will touch on inclined multi-phase flow as encountered in hilly terrain. As usual, the knowledge and ability to convert back and forth from oil field to metric units and to ensure equations are consistent in units is expected and required.

2

PTPR 6098

Single-Phase Incompressible Flow The review of laminar and turbulent flow provided in Module 5, including the equations to calculate the Reynolds number and the means to determine friction factors, are also applicable to horizontal flow in surface lines and equipment. To determine the pressure drop in horizontal pipe, the mechanical energy balance approach is used as usual. The total pressure drop, ∆p, is the sum of all gains or losses due to any changes in potential energy, ∆pPE, kinetic energy, ∆pKE, and flowing friction, ∆pF, as follows: ∆p = ∆pPE + ∆pKE + ∆pF Integration of the differential form of the above mechanical energy balance equation, with these assumptions: 1. 2. 3. 4.

incompressible flow no work energy added or removed from the fluid no significant changes in elevation due to topography (∆pPE = 0) no change in pipe diameter (∆pKE = 0)

gives the following expression for the total frictional pressure drop in a pipe of total length L: 2 ff ρ u 2 L ∆p = p1 – p2 = gc D where:

gc ρ u ff L D

= = = = = =

32.17 ft lbm lbf-1 sec-2 density of incompressible fluid, lbm/ft3 flow velocity, ft/sec Fanning friction factor, dimensionless length of pipe between points 1 and 2, ft internal pipe diameter, ft

Note that if the slope of the laminar line on a friction factor diagram is 16/NRE, the friction factor presented is the Fanning friction factor, denoted as ff in this course. If it is 64/NRE then it is the Moody (or Darcy-Weisbach) friction factor denoted here as f. In other words, ff = f/4

3 PTPR 6098

Single-Phase Compressible Flow (after Reference 1) For single phase compressible flow (gas), fluid density and fluid velocity are not constant along a horizontal surface flow line. The mechanical energy balance for gas in a surface flow line which is very nearly horizontal can be written without the potential energy term. However, the kinetic energy term may be significant for some cases of compressible flow. Thus, for gas flow in a horizontal surface flow line: ∆p = ∆pKE + ∆pF In differential form, slightly re-arranged, this equation is: 2f u 2 dL Dp u = du + f ρ gc gc D

Eq. 1

Re-arranging the real gas law to calculate density and using γg = MWg/28.97, where γg is gas gravity, gives an expression for gas density: ρ=

28.97 γ g p ZRT

The gas velocity, u, at the temperature and pressure under flowing conditions is given by u = q/A = q/(πd2/4) as follows: u=

4q sc Z ⎛ T ⎞ ⎛ psc ⎞ ⎜ ⎟ ⎜ ⎟ πD 2 ⎝ Tsc ⎠ ⎝ p ⎠

The differential form of the kinetic energy term for a gas is therefore: ⎛ 4 q sc Z psc ⎞ u du = ⎜ ⎟ 2 p ⎠ ⎝ πD

2

dp p3

By making the substitutions for ρ and u du in equation (1) and re-arranging, the result, after integration, assuming average values for the temperature and for the real gas deviation (compressibility) factor Z, is: 2

p ⎞ 32 28.97 γ g Zavg Tavg ⎛ q sc psc ⎞ ⎛ 2f f L p -p = 2 + 1n 1 ⎟ Eq. 2 ⎜ ⎟ ⎜ 4 p2 ⎠ π R gc D ⎝ Tsc ⎠ ⎝ D 2 1

4

2 2

PTPR 6098

Combining the constants and conversion factors in equation (2) gives, in oilfield units: 4.195 × 10-7 γ g Zavg Tavg q sc2 ⎛ 24 f f L p ⎞ p -p = + 1n 1 ⎟ ⎜ 4 p2 ⎠ D ⎝ D 2 1

2 2

Eq. 3

where: p1, p2 = inlet and outlet pressures, psi

γg = Zavg = Tavg = qsc = D = ff = L =

gas gravity average real gas deviation factor average gas temperature, oR flow rate in mscfd pipe internal diameter, in. Fanning friction factor surface line length, ft

Equation (3) is the working equation in oilfield units for gas flow in horizontal surface lines having no significant changes in elevation. The friction factor can be determined from a Moody friction factor chart with the Reynolds number, in oil field units, calculated as follows:

N RE = 20.09

γ g q sc D μ avg

5 PTPR 6098

Exercise One 1. Given: Produced water injection pump discharge pressure, 450 psig Length of cement-lined injection line, 5500 ft Absolute roughness of cement, 0.005 ft ID of cement liner, 2.4 in. Produced water specific gravity, 1.05 Produced water viscosity, 1.1 cP a) Calculate the relative roughness. b) Calculate the Reynolds number. Is the flow laminar or turbulent? c) Determine the Moody and Fanning friction factors using Fig. 2, Module 5 (Moody diagram). d) Determine the pressure drop in the injection pipe line. 2. Given: Minimum compressor suction pressure 20 psia to avoid overloading driver Maximum gas rate at maximum available horsepower, 4,400 mscfd Length of gas flow line from well head to compressor, 4000 ft Gas specific gravity, 0.7 Pipe ID, 3-in. Fluid viscosity, 0.013 cP a) Determine the friction factor assuming fully turbulent flow in a new, clean steel flow line. b) Calculate the required well head pressure to flow the desired gas flow rate assuming Zavg is 1.0 for this gas at low pressure and the kinetic energy term is negligible.

6

PTPR 6098

OBJECTIVE TWO When you complete this objective you will be able to…

Understand the difficulties caused by multi-phase flow on calculating the flow rate-pressure drop relationship.

Learning Material MULTI-PHASE FLOW IN SURFACE LINES

Two-phase flow occurs whenever there is simultaneous flow of free gas and liquid. Multi-phase flow occurs whenever there is simultaneous flow of free gas, oil and water. The main applications of horizontal multi-phase flow analysis in the oil patch are: 1. Sizing of transmission lines and pipelines for various mixtures of gas, oil and water. 2. Sizing of production flow lines from well heads to batteries. 3. Determining the pressure loss in production flow lines in order to calculate well production rates. 4. Determination of flow regime/pattern to optimize sizing of surface vessels such as separators and slug catchers. Horizontal Multi-Phase Flow Correlations

Many correlations have been developed to determine the pressure drop in vertical, inclined and horizontal multi-phase flow (vertical multi-phase flow was covered in Module 5). Care has to be taken to ensure the most appropriate multi-phase flow correlation is used for the particular conditions. Since the multi-phase flow correlations were developed largely based on field or lab tests, many provide reasonable accuracy only over limited ranges of such factors as flow pattern, pipe diameter, fluid velocity, fluid viscosity and gas-liquid ratio, among others. Table 1 below summarizes the better horizontal multi-phase flow correlations according to Reference 4.

7 PTPR 6098

Table 1 Horizontal Multi-Phase Flow Correlations Summary © PennWell Publishing Co. This material has been copied under license from Access Copyright. Resale or further copying of this material is strictly prohibited.

Horizontal Flow Regimes

The well-known correlation of Beggs and Brill is based on horizontal flow regimes with corrections to liquid hold-up for non-horizontal situations. It is applicable to any pipe inclination, including horizontal and downward flow. This gives it an advantage over other correlations. It was developed using air and water flowing in 1-inch and 1.5-inch acrylic pipe 90 ft long that could be inclined at any angle. Test rates were less than 300 mscfd and 1000 bbl/d while pressures were between 35 and 95 psia. Figure 1 shows the main horizontal multi-phase flow regimes and patterns as categorized by Beggs and Brill (Reference 2). Note that the flow regimes for horizontal flow differ substantially from the flow regimes for vertical or inclined multi-phase flow (see Module 5). In the horizontal segregated flow regime, the two phases are more or less separate. There are three flow patterns in the segregated regime. In the stratified flow pattern, the liquid is moving along the bottom of the pipe while gas occupies the upper part. There is a well defined smooth interface between the phases. At higher gas rates, the interface is no longer smooth but wavy. This is called stratified wavy or ripple flow. At high gas and liquid rates, an annular flow pattern results with liquid surrounding a central core of gas that contains entrained liquid droplets. 8

PTPR 6098

Plug flow, in which elongated gas bubbles flow along the top of the pipe, and slug flow, in which large slugs of liquid alternate with high-velocity gas bubbles that fill most of the pipe, are the two patterns that comprise the intermittent flow regime. Bubble flow and mist flow are the two patterns in the distributive flow regime. In horizontal flow, bubbles will flow in the upper part of the pipe and not be equally distributed as in vertical flow. The mist flow pattern occurs when high gas rates accompany low liquid rates. The liquid becomes entrained as droplets in the gas. It should be noted that if the flow pipe is inclined upward only a few degrees, stratified or wave flow can not exist because the liquid is held back by gravity and slug flow results instead. Conversely, if the pipe is inclined downward only a few degrees, stratified flow predominates because of gravity and slug flow will not occur where predicted.

Figure 1 Horizontal Multi-Phase Flow Patterns per Beggs and Brill © Journal of Petroleum Technology. This material has been copied under license from Access Copyright. Resale or further copying of this material is strictly prohibited. PTPR 6098

9

Flow Regime Mapping

There are many different correlations to predict the flow regime and pattern. The Beggs and Brill correlation uses a plot of the mixture Froude number, NFR, defined below, versus the liquid input fraction, γl, also defined below, to determine the flow regime: N FR =

and where:

γ1 = um us usg g D

= = = = =

u 2m gD

u sl u sl + u sg

usl + usg superficial liquid velocity, ft/sec superficial gas velocity, ft/sec 32.17 ft/sec2 internal pipe diameter, ft

Superficial velocity is simply the volumetric flow rate at flowing conditions divided by the pipe cross-sectional area.

Figure 2 Horizontal Flow Regime Map, Beggs and Brill Correlation © Journal of Petroleum Technology. This material has been copied under license from Access Copyright. Resale or further copying of this material is strictly prohibited.

10

PTPR 6098

Another widely used correlation to determine flow regime and pattern is that of Baker, Figure 3, as modified by Scott (Reference 3). Note that transitions between zones are not abrupt as indicated by the wide boundaries on the Baker flow regime map. The dimensionless quantities plotted on the axes of the Baker flow regime map are essentially gas velocity versus liquid velocity and are defined below: Gg = gas flow rate, lbm/hr-ft2 Gl = liquid flow rate, lbm/hr-ft2 γ = [(ρg/0.075)(ρl/62.4)]0.5 Ø = (73/σl)[μl(62.4/ρl)2]0.333

Figure 3 Horizontal Flow Regime Map, Baker Correlation © Academic Press. This material has been copied under license from Access Copyright. Resale or further copying of this material is strictly prohibited.

11 PTPR 6098

Selected Topics in Horizontal Multi-Phase Flow Inclined Multi-Phase Flow

Inclined multi-phase flow in surface flow lines is caused by hilly terrain or by the flow line crossing river valleys. Short flow lines contained on a lease tend to be very nearly horizontal while longer ones, especially those tying in wells to a central battery tend not to be strictly horizontal and in some cases traverse steep hills and valleys. It should be noted that most investigators recognize some recovery on the down slope side of the pressure loss suffered on the upslope side. However, most multiphase correlations assume no recovery on the down slope side. As mentioned previously, the Beggs and Brill correlation is applicable to vertical, horizontal and inclined multi-phase flow. They derived the following equation for multi-phase flow from the general mechanical energy balance equation: ⎡ ( g/g c )( ρ m sin Ø ) + {2 f TP ρ m u 2m / ( g c D )} ⎤ ⎥ p1 – p2 = z1 – z2 ⎢ 1 - ⎡⎣ρ m v m vsg / ( g c p ) ⎤⎦ ⎢⎣ ⎥⎦ where: (g/gc) (ρm sin Ø) = the potential energy term 2 fTP ρm u 2 /(gc D) = the friction loss term m

ρm vm vsg /(gc p) = the kinetic energy term The Beggs and Brill correlation requires determining whether flow is segregated, transition, intermittent or distributed (see Figure 2) in order to calculate the liquid hold-up (see pp. 162-163, Reference 1). The Beggs and Brill correlation also requires calculation of a two-phase friction factor, fTP. Another widely used correlation to predict liquid hold-up in inclined flow pipes is that of Flanigan. In studying a 16-inch pipeline with multi-phase flow he noted the following: 1. For lower gas velocities, the largest portion of the total pressure drop occurred in the up slope sections. 2. The elevation component of the pressure drop is directly proportional to the sum of the elevation increases in the pipeline. 3. The slope of the inclined sections is not important, only the total rise. 4. The difference in elevation between the start and end of the pipeline is immaterial. 5. The pressure drop in the up slope section is inversely proportional to the gas velocity.

12

PTPR 6098

Flanigan treated the up slope sections as though the pressure drop in them was proportional to their vertical height. The equation he used to calculate the pressure drop solely due to elevation rises is: ∆p = where:

∆p ρL HF ∑hi

= = = =

ρ L g H F Σh i g c 144

pressure drop due to elevation increase, psi liquid density, lbm/ft3 elevation factor, dimensionless sum of elevation increases in the flow direction, ft

Flanigan’s correlation for HF, with vsg being superficial gas velocity in ft/sec at conditions of average flowing pressure and temperature, is shown in Figure 4.

Figure 4 Flanigan Correlation for Inclined Multi-Phase Flow © PennWell Publishing Co. This material has been copied under license from Access Copyright. Resale or further copying of this material is strictly prohibited.

13 PTPR 6098

Single-Phase Flow through Chokes

A choke is simply a mechanical restriction with an interchangeable insert, called the bean, available in various size openings measured in 64ths of an inch. A common application of chokes is to control flow during gas well tests. Some reasons to control flow during well production include: •

Protect surface equipment from erosion

•

Prevent gas or water coning

•

Prevent sand production resulting from excessive rate

•

Produce reservoir at most efficient rate

•

Maintain rate at well allowable

When the ratio of the upstream and downstream pressures across a choke exceeds the critical pressure ratio, pc, the choke is in critical flow. This means the fluid has been accelerated to the speed of sound for that fluid through the reduced diameter of the choke bean. The flow rate through the choke is then no longer a function of downstream pressure because any downstream pressure disturbances can not travel faster than the sonic velocity for that fluid. In other words, the relationship between flow rate and well head pressure is controlled by the choke when it is in critical flow and not by whatever the downstream pressure happens to be as a result of any change in flow rate or for any other reason. The critical pressure ratio, pc, for single-phase gas flow through a choke is given by: (p2/p1)c = [2/(γ + 1)] γ/(γ-1) where:

γ = heat capacity ratio of the gas

For air and other diatomic gases γ is approximately 1.4 and the critical pressure ratio is 0.53. Thus the rule of thumb that flow through a choke in single-phase gas flow is critical when the downstream pressure is less than half the upstream pressure. Single-phase flow of an incompressible fluid through a choke or orifice nozzle is described by: q = C A (2 gc 144 ∆p/ρ)0.5 where:

14

C A gc ∆p Ρ

= = = = =

flow coefficient cross-sectional area of the choke 32.17 ft lbm lbf -1sec-2 pressure drop across choke or orifice nozzle, psi fluid density, lbm/ft3 PTPR 6098

In oilfield units of bbl/d, choke size in inches, pressure in psi and density in lbm/ft3, the above equation becomes: q = 22,800 C D2 (∆p/ρ)0.5 For single phase flow of a compressible fluid (that is, gases), a net expansion factor, Y, is added to the above equation: q = Y C A (2 gc 144 ∆p/ρ)0.5 The flow coefficient, C, and the net expansion factor, Y, can be obtained from Figures 5 and 6 as follows:

Figure 5 Orifice Flow Coefficient C © Crane Co. This material has been copied under license from Access Copyright. Resale or further copying of this material is strictly prohibited.

15 PTPR 6098

Figure 6 Net Expansion Factor Y © Crane Co. This material has been copied under license from Access Copyright. Resale or further copying of this material is strictly prohibited.

Multi-Phase Flow through a Choke

Often, the flowing fluid is a multi-phase mixture of gas and liquid. Most correlations for multi-phase flow through a choke are valid only for critical flow. One of the better multi-phase methods of analyzing flow for a well flowing gas with liquid through a well head choke is that of Gilbert (Reference 7). It can be shown theoretically, with some assumptions, that: pwh = C R0.5 q / S2 where:

16

pwh R q S C

= = = = =

flowing tubing head pressure, psia gas/liquid ratio, mscf/bbl liquid rate, bbl/d bean size, 64ths in. constant

PTPR 6098

Gilbert showed from analysis of data from a specific field that the following equation would serve as a good first approximation if pwh is in psig: pwh = 435 R0.546 q / S1.89 This is the same equation as equation 10-75 given in the course text, Reference 1 but in different form. Further details and capabilities on Gilbert’s method and on other multi-phase flow correlations for choke flow can be found in Reference 4. Emulsified Flow

Whenever there is three-phase flow of gas, oil and water, there is the possibility of an emulsion forming. An emulsion is a dispersal of oil droplets in the water phase, or vice-versa, aided by a surfactant. This is not a well understood phenomena and generally can not be predicted. The viscosity of emulsions is high and so, therefore, is the attendant pressure drop. If an emulsion is formed, one treatment option is adding chemical demulsifiers. Since each emulsion is unique, the treatment must be determined with lab testing of actual emulsion samples. Equivalent Lengths of Pipe Fittings

The turbulence caused by any fittings such as elbows, tees, valves, sudden contractions or enlargements, results in additional pressure drop in surface lines. This pressure drop is accounted for by using an equivalent length for each fitting that is added to the surface line length in order to calculate the total flowing pressure drop. An excellent discussion of equivalent lengths of fittings can be found in Reference 6 and sample equivalent lengths can be found in Reference 1.

17 PTPR 6098

Exercise Two 1. Determine the flow pattern using the Baker flow pattern map (Figure 3) given: • 1500 bbl/d of oil 3 o Density, 800 kg/m o Viscosity, 4 cP • flowing with 1000 mscfd of gas o Specific gravity, 0.71 o Viscosity, 0.018 cP o d Zavg, 0.91 • In a 3.5-in. ID pipeline at 120°F and 800 psi • Oil/gas interfacial tension, 30 dynes/cm Procedure: a) b) c) d)

Calculate the gas mass flux Gg. Calculate the liquid mass flux GL. Calculate the x- and y-axis values for the Baker flow pattern map. Determine the flow pattern using Figure 3.

2. Given: Flow line ID, 3.0 in. Flow line length, 2,000 ft Flow rate, 2,000 bwpd Specific gravity of water, 1.07 Gas/liquid ratio, 100 scf/bbl Gas gravity, 0.65 Gas Z-factor, 0.95 Water/gas interfacial tension, 66.7 dynes/cm Gas viscosity, 0.015 cP Water viscosity, 1.0 cP Average flowing temperature, 120°F Average flowing pressure, 750 psi The following hill heights: Hill 1 2 3 4 5

Height 20 15 160 100 55

a) Determine the multi-phase pressure drop in the flow line due to hilly terrain using Flanigan’s correlation. b) How would you determine the total multi-phase pressure drop in the flow line? 18

PTPR 6098

3. Given: Well head pressure, 800 psia Gas/liquid ratio, 1000 scf/bbl Oil gravity, 40°API Choke size, 14/64ths Downstream pressure, 100 psia a) Determine if the choke is in critical flow. b) Calculate the flow rate through the choke using Gilbert’s multi-phase flow correlation.

19 PTPR 6098

OBJECTIVE THREE When you complete this objective you will be able to…

Estimate the flow rate-pressure drop relationship for two- and three-phase flow using pressure traverse curves.

Learning Activities HORIZONTAL PRESSURE TRAVERSE CURVES

Pressure traverses, or gradient curves, are available for horizontal multi-phase flow just as they are for vertical multi-phase flow. A horizontal pressure gradient curve is a plot of the flowing pressure in a horizontal flow line versus length. Figure 7 shows a sample set of horizontal general pressure traverse curves. Just as with vertical flow, field pressure traverse curves can be generated by computer using the best horizontal multi-phase correlation for a particular field and formation. General pressure traverses are also prepared by computer but can utilize any of several multi-phase correlations each of which was developed from a different sample of wells, fields, fluids and tubulars. General pressure traverse curves may or may not work well for any particular field. Whether field or general pressure traverse curves are used, either should be proved against actual field data to ensure their accuracy is acceptable for that application. References 4 and 5 are sources of published general pressure traverses for horizontal multi-phase flow. They are prepared from the Eaton correlation, as is the set of curves in Figure 7, and are good except for very low pressure, low rate and low gas-oil ratios and when liquid viscosity is greater than 10 cP. As with vertical pressure traverses, horizontal pressure traverses are calculated in short increments in order to maximize accuracy. Using increments that are too long, or not enough increments, results in average fluid properties that depart too much from end conditions in such increments. These errors accumulate from the first increment of the pressure traverse to the last increment. Some applications of horizontal pressure traverse curves include: 1. Estimation of flowing well head pressure to determine well productivity. 2. Determination of optimum size of surface flow line.

20

PTPR 6098

Figure 7 Horizontal Multi-Phase Flow General Pressure Traverse © PennWell Publishing Co. This material has been copied under license from Access Copyright. Resale or further copying of this material is strictly prohibited.

21 PTPR 6098

Exercise Three 1. A well is tied into a separator operating at 100 psi with a 5000 ft flow line of 2-in. ID. What well head pressure is required to move 1000 bblpd at 95% water cut and 1,000 scf/bbl gas/liquid ratio with a gas gravity of 0.65? Procedure: a) Find the horizontal flowing pressure gradient curve which most closely approximates the given conditions (sometimes, though not in this case, it may be necessary to interpolate between two or more sets of pressure traverse curves). b) Determine the equivalent length corresponding to the separator pressure. Add this length to the length of the flow line to get the total equivalent flow line length. c) Intersect the appropriate GLR curve at the length representing the total equivalent flow line length and read the required flowing well head pressure.

22

PTPR 6098

Exercise One Solutions ε

0.005 ft 1 ft 1. a) Relative roughness D 2.4 in. × 12 in. = 0.025 =

b) Reynolds number; N Re =

1.48 qρ DM

1.48 × 1500 bbld -1 × 1.05 × 62.4 16 lb m ft -3 2.4 in. × 1.1 cP = 55,100 (Turbulent) =

c) Friction factor, ft, using Figure 2, Module 5 (Moody diagram) f= 0.053 since scope of laminar line is 64/NRF. Figure 2, Module 5 gives the Moody friction factor, f. f The Fanning friction factor, f f = 4 ff =

0.053 = 0.0133 4

d) Δp = P1 - P2 =

μ=

q = A

z ff ρ μ 2 L gc D

1500 bbld -1 ×

1d 5.615 ft 3 × 86,400 sec bbl

π ⎛ 2.4 in. × 1 ft ⎞ ⎜ ⎟ 4⎝ 12 in. ⎠

2

0.0975 ft 3 sec-1 0.314 ft 2 = 3.105 ft sec-1 =

2 × 0.0133 × 1.05 × 62.416 lb m ft -3 × (3.105) 2 ft 2 sec-2 × 5500 ft Δp = 32.174 ft·lb m lb-1f sec-2 × 0.2 ft = 14360 lb f ft -2 ×

ft 2 144 in.2

= 99.7 lbf in.-2 = 99.7 psi

23 PTPR 6098

2. a) From Figure 3, Module5, relative roughness

ε

= 0.0006 for complete D turbulence and the friction factor, f = 0.0175 (f is Moody friction factor) f Fanning friction factor, ff = = 0.00438 4

b) 4.195 × 10-7 × γ g × Zavg Tavg q sc 2 ⎛ ft L P ⎞ P -P = + ln 1 ⎟ ⎜ 24 4 D P2 ⎠ D ⎝ 4.195 × 10-7 × 0.7 × 1.0 × 560 × (4400) 2 ⎛ 24 × 0.00438 × 4000 ⎞ = ⎜ ⎟ 3 (3) 4 ⎝ ⎠ 3184(140.16) = 81 = 5510 psi 2 2 1

2 2

P1 = 5510 + (2o )2 = 77 psia

Exercise Two Solutions 1. a) Calculate gas mass flux Gg Gg = u sg ρ g lb m hr -1 ft -2

Gas superficial velocity at pipeline conditions ⎛ T ⎞ ⎛ Psc ⎞ q 4 = q sc Z ⎜ ⎟⎜ ⎟ 2 A πD ⎝ Tsc ⎠ ⎝ p ⎠ 4 1d ⎛ 580 ⎞ ⎛ 14.7 ⎞ = 1 × 106 × 0.91⎜ ⎟⎜ ⎟× 2 ⎝ 460 ⎠ ⎝ 814.7 ⎠ 86,400 sec ⎛ 3.5 ⎞ π⎜ ⎟ ⎝ 12 ⎠ = 2.95 ft sec-1 28.97 γ g P ρg = ZRT 28.97 lb m (lb-mol)-1 × 0.71 × 814.7 lbf in.-2 = 0.91 × 10.73 lbf in.-2 ft 3 (lb mol)-1 °R -1 × 580°R

u sg =

= 2.37 1b m ft -3 G g = 2.95 ft sec-1 × 2.37 lb m ft -3 ×

3600 sec hr

= 25,139 lb m hr -1 ft -2

24

PTPR 6098

b) Calculate the liquid flux G L = USL ρ L Liquid superficial velocity

u SL =

q = A

1500 bbl d -1 × 5.65 ft 3 bbl × π ⎛ 3.5 ⎞ ⎜ ⎟ 4 ⎝ 12 ⎠

1d 86,400 sec

2

= 1.45 ft sec-1 G L = 1.46 ft sec-1 × 0.8 × 62.4 lb m ft -3 ×

3600 sec hr

= 262,200 lb m hr -1 ft -2

d) Calculate the y- axis parameter λ

⎡⎛ ρ g ⎞ ⎛ ρ L ⎞ ⎤ λ = ⎢⎜ ⎟⎜ ⎟⎥ ⎢⎣⎝ 0.075 ⎠ ⎝ 62.4 ⎠ ⎥⎦

0.5

⎡⎛ 2.37 ⎞ ⎛ 49.9 ⎞ ⎤ = ⎢⎜ ⎟⎜ ⎟⎥ ⎣⎝ 0.075 ⎠ ⎝ 62.4 ⎠ ⎦ = 5.03

0.5

Calculate the y-axis parameter ∅

73 ⎡ ⎛ 62.4 ⎞ ∅= ⎢4 ⎜ ⎟ 30 ⎢⎣ ⎝ 49.9 ⎠

2 1/3

⎤ ⎥ ⎥⎦

= 5.20 y-axis value for baker map is: Gg λ

=

25,139 = 5000 5.03

x-axis value is G g λ∅ Gg

=

262,200 × 5.03 × 5.20 = 273 25,139

d) Predicted flow pattern is slug flow. 25 PTPR 6098

2. a) Flanigan’s correlation for pressure drop for inclined flow is: Δp =

ρ L gH F Σh i g c 144

Σh i = 20 + 15 + 160 + 100 + 55 = 350 ft HF =

1 or from Figure 4 1 + 0.3264 v1.006 sg

vsg = superficial gas velocity = =

qg A ⎛ Tavg A q sc Zavg ⎜ 2 πD ⎝ Tsc

=

4

× 1 ft ⎞ ⎛ π ⎜ 3 in. ⎟ 12 in. ⎠ ⎝ = 5.89 ft sec -1

2

⎞ ⎛ Pse ⎟ ⎜⎜ ⎠ ⎝ Pavg

×

⎞ ⎟⎟ ⎠

100 scf 2000 bbl 5.615 ft 3 × × × 0.95 × bbl d bbl

1d ⎛ 560°R ⎞ ⎛ 14.7 psia ⎞ ⎟ ⎜ ⎟⎜ ⎝ 460°R ⎠ ⎝ 764.7 psia ⎠ 86,400 sec

From Figure 4 at Vsg= 5.89 ft sec-1 H F = 0.35 and

Δp = =

ρ L g H F Σh i g c 144 1.07 × 62.4 lb m ft -3 × 32 ft sec-2 × 0.35 × 350 ft 32.174 ft lb m lb f -1sec-2 × 144 in.2 ft -2

= 56.8 lbf in.-2 = 56.8 psi

b) Use a horizontal multi-phase correlation such as Beggs and Brill (with sin ∅ = 0 for ∅ = 0 for horizontal flow) to determine the frictional and kinetic energy losses and add them to the pressure loss due to hilly terrain. 3. a) P2 100 = = 0.125 P1 800

Since

26

P2 <0.5, flow is critical P1

PTPR 6098

b) Using Gilberts correlation: 435 R 0.546 q Pwh = S1.89 P S1.89 q = wh 0.546 435 R 800 × (14)1.89 = 435(1)0.546 = 270 bbl/d

Exercise Three Solutions 1. a) Use Figure 7 which is for conditions very close to the problem conditions. b) Equivalent length of 100 ps separator pressure is 200 ft. Total equivalent length of flowline is 5200 ft. c) The required flowing well head pressure at a length of 5200 ft and a GLR of 1000 scf/bbl is 545 psi.

27 PTPR 6098

Module Self-Test Directions: • • • •

Answer the following questions. Compare your answers to the enclosed answer key. If you disagree with any of the answers, review learning activities and/or check with your instructor. If no problems arise, continue on to the next objective or next examination.

1. On a Moody diagram, the friction factor is a function of Reynolds number and the relative roughness of the pipe being evaluated. a) True b) False 2. The mechanical energy balance approach to determine pressure drop in horizontal flow: a) indicates whether flow will be laminar, turbulent or inbetween. b) is the sum of all gains or losses in potential energy, kinetic energy and friction if no work is added to the system. c) is a function of fluid properties, fluid velocity and length that depends on the particular cross-section of the flow conduit. d) all of the above. 3. The Darcy-Weisbach friction factor is four times the Moody friction factor. a) True b) False 4. In single-phase compressible flow in a horizontal flow line: a) Fluid velocity and density are not constant along the length b) Fluid velocity, density and pressure are not constant along the length c) Fluid velocity, density, temperature and pressure are not constant along the length d) Fluid velocity, density, temperature, pressure and mass flow rate are not constant along the length 5. In single-phase flow of a compressible fluid the velocity changes when the pressure changes. a) True b) False

28

PTPR 6098

6. The Beggs and Brill multi-phase flow correlation for vertical flow is based on the flow patterns of horizontal multi-phase flow. a) True b) False 7. Which is true? a) Short flow lines tend to be very nearly horizontal and tend to vary little in fluid temperature along their length. b) Long flow lines tend to encounter more variations in terrain and have a significant change in fluid temperature along their length. c) Long transmission lines tend to encounter many variations in terrain but have little change in temperature along most of their length. d) All of the above. 8. In inclined multi-phase flow, liquid hold-up is higher in the up slope sections at low gas velocities. a) True b) False 9. In inclined multi-phase flow, liquid hold-up in the up slope sections can be reduced by increasing gas velocities. a) True b) False 10. For sections of inclined multi-phase flow in a long flow line: a) The slope of the inclined sections is important, not the total rise. b) The largest portion of the total pressure drop occurs in the down slope sections. c) The pressure drop is proportional to gas velocity in the up slope sections. d) The difference in elevation between the start and end of the flow line is immaterial. 11. When the ratio of the upstream and downstream pressures across a choke is greater than about 0.5, the choke is said to be in critical flow. a) True b) False 12. The flow rate from a gas well in critical flow through a choke can be adjusted by reducing the pressure downstream of the choke. a) True b) False 29 PTPR 6098

Module Self-Test Answers 1. a 2. b 3. b 4. c 5. a 6. a 7. d 8. a 9. a 10. d 11. b 12. b

30

PTPR 6098

Assignment 1. Describe the effect of changes in line size, flow rate, viscosity, gas/liquid ratio, and water/oil ratio on the pressure drop during horizontal multi-phase flow. 2. List at least four different mappings of flow regime and describe what appears to be the main difference distinguishing each mapping from the others. 3. List at least six different multi-phase correlations for horizontal flow and provide at least one feature of each that distinguishes it from the others.

31 PTPR 6098

References 1. Economides, M.J., Hill, A.D. and Ehlig-Economides, C., Petroleum Production Systems, Prentice-Hall, 1993. 2. Beggs, H.D. and Brill, J.P., A Study of Two-Phase Flow in Inclined Pipes, J.Pet. Tech., (May 1973), 607. 3. Scott, D.S., Properties of Cocurrent Gas-Liquid Flow, Advances in Chem. Eng., Vol. 4, Academic Press, New York, pp. 200-278, 1963. 4. Brown, K.E., Beggs, H.D., The Technology of Artificial Lift Methods, PennWell Publ. Co., 1977. 5. Brown, K.E., Gas-Lift Theory and Practice, Petr. Publ. Co., 1965. 6. Flow of Fluids, Technical Paper No. 410M, Crane Co., 1977. 7. Gilbert, W.E., Flowing and Gas-Lift Well Performance Drilling and Production Practice, 1954, API, P.143.

32

PTPR 6098

Appendix Table 2 Conversion Factor Table

33 PTPR 6098

Course Module

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Course Module Advanced Production Engineering - PTPR 465

Production Optimization Using Graphical NODAL Analysis PTPR 6099 Module 7

Revised: December 2007

MacPhail School of Energy

redefine yourself www.sait.ca

Production Optimization Using Graphical NODAL Analysis Rationale Why is it important for you to learn this material? In order to optimize production, it is necessary to be able to graphically analyze the hydrocarbon production system using System NODAL Analysis to determine the effect of the variables that affect well production rate.

Learning Outcome When you complete this module you will be able to …. Calculate the best production possible from a well given enforced constraints.

Learning Objectives Here is what you will be able to do when you complete each objective. 1. Combine the Inflow and Outflow Performance Relationships for a reservoir and well to determine the production rate for the system. 2. Calculate the effect of changing constraints on either the reservoir or the well on the production rate (production optimization).

1 PTPR 6099

INTRODUCTION This module brings together all you have learned so far about the reservoir and the other components of the hydrocarbon production system and about the technique of System NODAL Analysis. You will learn how to do the traditional graphical solution method of System NODAL Analysis to determine the production rate of a hydrocarbon production system. You will also learn to identify and determine the effects of the key parameters that affect production rate so that production can be efficiently and economically optimized for a particular well.

2 PTPR 6099

OBJECTIVE ONE When you complete this objective you will be able to… Combine the Inflow and Outflow Performance Relationships for a reservoir and well to determine the production rate for the system.

Learning Activity Complete each of the Learning Activities listed below. 1. Read and study the learning material in this objective. 2. Do the exercise provided. 3. Do the assignment provided. 4. Research outside sources (that is, Internet, library materials, reference texts, and so on) for additional information.

Learning Material PRODUCTION OPTIMIZATION USING GRAPHICAL SYSTEM NODAL ANALYSIS As discussed in Module 1, if it is desired to determine the production rate from a well, a common selection of solution node for the production system is at bottom hole. At that node, the intersection of the well’s IPR curve and the intake curve for the remainder of the production system (comprised of the tubulars, any choke, a flow line, separator, and so on) provides the system operating point. At bottom hole, the system operating point is the production rate and producing bottom hole pressure. If the well head pressure is fixed, as is often the case, the intersection of the well’s IPR curve and the vertical lift performance curve (the intake curve for the tubulars only) provides the system operating point (see Figure 1). Fixed well head pressure is common in wells with short flow lines to separators. It is to be noted for vertical flow that as the gas/liquid ratio, GLR, increases there may be a point in the vertical lift performance curve where the producing bottom hole pressure reaches a minimum. To that minimum point, an increasing GLR reduces the hydrostatic pressure drop in the vertical tubulars because of the difference in density between gas and liquid and because the proportion of gas increases with increasing GLR. However, as the gas/liquid ratio continues to increase beyond that point, flowing friction starts to increase faster than the decrease in hydrostatic pressure and bottom hole pressure starts to increase with increasing flow rate. 3 PTPR 6099

The tubing intake curve in Figure 1 illustrates what a well-defined minimum in tubing intake pressure looks like. This means the producing bottom hole pressure is also at a minimum at that point.

Figure 1 Solution to System Production Rate with Fixed Well Head Pressure (after Reference 1)

Figure 2 is a sample graphical analysis of the entire hydrocarbon production system for a producing well. The fixed system pressures that do not vary with flow rate are the end conditions of static reservoir pressure and stock tank pressure (which is usually atmospheric pressure). Obviously the graphical representation of the fixed end pressures is a straight line on a pressure versus flow rate chart such as Figure 2. The other system pressure that is usually considered not to change with flow rate is separator pressure. A typical horizontal separator is shown in Figure 3. It provides mechanical means such as gravitational segregation, baffles, contact plates, and level control to separate gas from liquid and the residence time, as a result of the vessel’s volume, to do it efficiently. There is little in it to restrict flow. Two- or three-phase fluid enters at one end and gas and liquids are removed at the other. Separators designed to separate gas, oil and water are called threephase separators and will therefore have three outlets. If the production system discharges into a gas or liquid pipeline or a facility such as a battery or gas plant, it must be determined whether or not the end pressure will remain constant or is affected by any changes in the flow rate from the production system under analysis into the pipeline or facility. 4 PTPR 6099

Figure 2 Graphical Representation of the Total Hydrocarbon Production System for a Well (Reference 3)

When the performance curves of all the available system components are plotted on a chart such as Figure 2, it is possible to see at a glance which components produce the largest pressure drop at the system’s operating point which is at the well’s production rate, qL. This provides a guide to prioritizing the components to evaluate for the effect of any change in their operating conditions on production rate.

Figure 3 Typical Horizontal Separator (Reference 4)

5 PTPR 6099

For example, in the graphical representation of the sample well shown in Figure 2, the largest pressure drop of any component at the well’s production rate occurs in the tubing (∆p tubing). The tubing intake curve (that is, the outflow performance curve at the bottom hole) represents the pressure losses in the tubing, the choke (if there is one), the flow line and the separator. Because together they contribute the largest pressure drop in the sample production system, the tubing string, the choke, the flow line and separator would likely be the first components evaluated for any possible increase in production rate from changes in their operating conditions. For example, if the choke bean size can be increased, or the choke removed altogether, the choke pressure drop can be reduced or eliminated and the tubing intake curve moves downward shifting its intersection with the bottom hole IPR curve to the right. This indicates increased production is possible with the choke bean size increased or with the choke itself removed. In Figure 2, the pressure drop in the flow line is also relatively significant. Some methods to reducing pressure drop in a horizontal flow line are: •

using larger diameter pipe

•

twinning an existing line

•

removing any significant impediments to flow in the line such as a hill by re-routing the pipeline right-of-way around it

•

pulling a smooth plastic liner through rough or corroded pipe

•

regularly “pigging’ the line to remove wax, scale, sand or, for gas flow, liquids in low spots

•

injecting diluent or heating the liquid to reduce friction if flowing viscous crude

Reducing the pressure drop in the flow line moves the flow line performance curve down which in turn moves the tubing intake curve down. As before, this downward shift in the tubing intake curve shifts its intersection with the bottom hole IPR curve to the right indicating the potential for increased production. If a separator can be run at a significantly different pressure, the result of a change in separator pressure on the pressure drops in the flow line and tubing may need to be evaluated. The indicated pressure drop between the sand face IPR and the bottom hole IPR in Figure 2 also indicates a significant loss in production rate. This is because of the relatively flat IPR curves. This indicates that for the sample well in Figure 2, completion efficiency is worthy of an evaluation for any possible improvement. For each tubing size there is a critical gas rate to lift well bore liquids. This rate is in the range of 200 to 800 mscfd depending on the tubing size and other factors. 6 PTPR 6099

Exercise One 1) Given: Oil well depth, 7500 ft Gas/oil ratio, 400 scf/bbl Productivity index, 1.2 Static reservoir pressure, 2800 psi Fixed well head pressure, 200 psi Oil cut, 100% a. Determine the difference in producing rate between 2.5-inch and 3-inch tubing for this well. Procedure: i. On a Cartesian chart of pressure versus flow rate, plot a straight line PI of 1.2. ii. Using vertical pressure traverse plots (C-104, C-105, C-106 and C112, C-113 from Reference 1), calculate several points of a VLP curve for each tubing size and plot the resulting curves to determine the difference in producing rate. b. If oil netback for this well is $50/bbl, what is the incremental net cash flow for this well with 3-inch ID tubing? c. If this well is currently producing with 2.5-inch ID tubing and the cost of a workover to change the tubing to 3-inch ID is $45,000, would you recommend the workover? 2) Determine the production rate from a well given: Tubing length, 5000 ft Tubing size, 2.5-inch ID Flow line size, 2.5-inch Flow line length, 4400 ft Static reservoir pressure, 2600 psi PI = 1.5 Oil gravity, 35 oAPI Water-cut, 90% Gas/liquid ratio 200 scf/bbl Separator pressure, 140 psi.

7 PTPR 6099

Procedure: i. On a Cartesian chart of pressure versus flow rate, plot a straight line IPR with a PI of 1.5. ii. Calculate the well head pressures using horizontal pressure traverse curves D.107, D.108, and D.109 from Reference 1 and plot them on the chart. iii. Using the same flow rates as in step ii., and the resulting well head pressures, calculate the tubing intake (bottom hole) pressures using vertical pressure traverse curves C.109, C.110 and C.111 from Reference 1. iv. Plot the tubing intake curve on the chart and determine the system operating point; that is, the well production rate, from its intersection with the IPR curve. 3) For the well in 2), if static reservoir pressure is 2600 psi, bubble point pressure of the reservoir fluid is 2600 psi and a test for the well showed 700 bblpd at a producing bottom hole pressure of 1800 psi; a. Determine the production rate. b. Explain why it is permissible to use the tubing intake curve as calculated for Problem 2) above. Procedure: i. Use the Vogel relationship to calculate qL, max using the test data and to calculate several points of the IPR curve. Plot the IPR curve on a Cartesian chart. ii. Use the tubing intake curve calculated in Problem 2) to determine the operating point (production rate and producing bottom hole pressure) of this system.

8 PTPR 6099

OBJECTIVE TWO When you complete this objective you will be able to… Understand and calculate the effect of changing constraints on either the reservoir or the well on the production (production optimization).

Learning Material Recall from Module 1 that System NODAL Analysis can be done at any one of many available nodes between components, or at a functional node, in the hydrocarbon production system. The node or functional node that is selected depends upon which component or functional node is desired to be analyzed for its effect on production. The node selected should isolate the component to be analyzed as much as possible. Recall also that the inflow or outflow performance curves at a solution node are not affected by changes in the other. Only the system operating point changes when a change is made in either the inflow or outflow curve. For example, in order to evaluate the effect of changes in a flow line such as larger diameter pipe or twinning an existing line or removing significant restrictions to flow in the line such as a hill by re-routing the pipeline right-ofway, an appropriate solution node selection is likely to be at the well head. Once an inflow curve (IPR and VLP curves) is established for the flow line at the well head, it will serve no matter what changes are evaluated or made downstream from the well head in the flow line performance (outflow) curve. In the case of functional nodes such as chokes, inflow and outflow curves are both required. Once the inflow and outflow curves are established, the system operating point is determined by the available range of changes in pressure or flow rate that the functional node can provide.

9 PTPR 6099

A table of some common solution nodes, performance curves analyzed, system constraints and expected results follows. Table 1 Common Solution Nodes NODE

Bottom hole

PERFORMANCE CURVES ANALYZED IPR and Vertical Lift Performance

Bottom hole

IPR and Tubing Intake System

Fixed system delivery/sales pressure

*Submersible pump Well head

IPR, Pump and Vertical Lift Performance Inflow and Outflow System

Fixed well head pressure

*Choke

Inflow, Outflow and Choke Well head IPR, Compressor curve and pipeline curve Inflow System and Compressor curve

Pressure drop through choke Fixed system delivery/sales pressure

*Compressor, well head *Compressor, field

CONSTRAINT(S)

MAIN RESULT(S)

Fixed well head pressure

Production rate (reservoir/completion capability), tubing size,. Production rate, Reservoir capability.

Fixed delivery/sales pressure

Fixed system delivery/sales pressure

Submersible pump capacity and head, tubing size. Flow line size, separator pressure. Choke size and well production rate. Compressor size, horsepower, minimum inlet pressure Compressor size, horsepower, minimum inlet pressure

*denotes functional node

When all possible components of a particular production system (see Figure 2) have been evaluated and all practical changes that are economic have been implemented, then it can be said that production from the system has been optimized and the well is producing at its maximum practical economic rate.

10 PTPR 6099

EFFECT OF PRODUCING GAS/LIQUID RATIO For very low GLR production, hydrostatic pressure is dominant and the Vertical Lift Performance (VLP) curve tends to be relatively straight and flat. For very high GLR production, including gas wells, friction dominates as flow rate increases and the VLP curve is relatively curved and steep. Figure 4 shows bottom hole pressure as a function of flow rate for the same size tubing for production at low and high GLR.

Figure 4 Effect of Gas/Liquid Ratio on Vertical Lift Performance Curves

The effect of GLR on bottom hole pressure is easily seen on all vertical pressure traverse charts which show pressure as a function of both depth and GLR. The higher the GLR, the lower the bottom hole pressure for a given tubing size and flow rate. Sometimes gas is injected into wells in order to provide assist in lifting liquids. Figure 4 indicates that for high GLR wells there exists a point where injecting too much gas actually increases bottom hole pressure. This reduces liquid production from the well. PTPR 6099

11

EFFECT OF TUBING SIZE In order to determine the effect of tubing size on production rate, typical VLP curves are calculated and plotted for different tubing sizes in Figure 5. The intersection of the VLP curve for each tubing size with the well’s IPR curve determines the system production rate as shown for the example well in Figure 2.

Figure 5 Solutions to System Production Rate with Various Tubing Sizes (after Reference 2)

The VLP curve for each tubing size can be obtained from published vertical pressure traverse curves by plotting the well bottom pressure versus flow rate for each tubing size or it can be calculated for the specific well. If the tubing string selected is too large there is the possibility of premature loading of the well with liquids, which will reduce production, and the subsequent expense of lost production along with the cost of a workover to replace the tubing with a smaller string. Those costs and the higher cost of a larger tubing string must be weighed against a higher initial production rate for a larger tubing string.

12

Occasionally, tapered tubing strings are required if a smaller diameter liner has been run into the production casing. A tapered tubing string consists of a length of smaller diameter tubing run below a larger diameter length of tubing. In this case, a logical selection for the solution node is the point where the tubing changes diameter. The inflow system then consists of the reservoir and the lower string of tubing while the outflow system consists of the upper string of tubing to either a fixed well head pressure or to a flow line with a fixed endpoint pressure. PTPR 6099

EFFECT OF INCREASING WATER CUT In many reservoirs, particularly those with underlying water or those with water injection, a well may initially produce clean oil (<1% water), or oil at a relatively low water cut, and then at some point, either gradually or rapidly, start to produce increasing amounts of water. The effect of increasing water cut on well production is shown in Figure 10 below. The increase in water cut results in a higher flowing bottom hole pressure due to the increase in hydrostatic pressure. As a result, total liquid production from the well decreases. Frequently, the increased bottom hole pressure, along with the frictional pressure drop and well head pressure, is enough to cause the well to flow erratically or to stop flowing. The installation of artificial lift is then required to produce the well.

Figure 6 Solutions to System Production Rate with Increasing Water Cut (after Reference 1)

The VLP curves at different water cuts are determined in the usual way from published vertical pressure traverse curves or calculated by hand or computer. PTPR 6099

13

EFFECT OF FLOW LINE SIZE AND SEPARATOR PRESSURE The effect of a change in the flow line size from 2.5-inch ID to 3.5-inch ID and for a change in separator pressure from 100 psi to 50 psi for the 2.5-inch ID flow line is shown for a specific well and conditions in Figure 7.

Figure 7 Solutions to System Production Rate with Larger Flow Line Diameter and with Reduced Separator Pressure (after Reference 2)

The solution node selected is the well head where the inflow curve represents the well IPR and VLP curves while the outflow curve represents the flow line curve with the separator pressure as the fixed constraint at the end of the flow line. The specific well conditions (Reference 2) are: • • • • •

14

reservoir pressure = reservoir fluid bubble point pressure = 2400 psi well test data, 710 stbd at 2000 psi well head pressure oil cut, 100 % GLR, 800 scf/bbl flow line length, 3000 ft

The Vogel relationship can be used to calculate the inflow relationship at bottom hole and vertical pressure traverse curves are used to calculate the VLP curve to determine the flowing pressure at the well head. Horizontal pressure traverse curves can be used to calculate the flowing pressure at the well head node for each flow line size and separator pressure evaluated. PTPR 6099

For this well and this specific set of conditions, a larger diameter flow line has a much larger effect on production rate than a reduction in separator pressure. EFFECT OF RESERVOIR PRESSURE The effect of depletion indicated by declining static reservoir pressure over time is shown by the IPR curves and their intersection with the tubing intake curve in Figure 8. The effect on production rate is substantial for relatively small declines in static reservoir pressure due to the shape of the IPR curve.

Figure 8 Effect of Declining Reservoir Pressure on Well Production (after Reference 3)

Static reservoir pressure as a result of depletion can be determined by mathematical material balance methods (Reference 5, pp167-202), by Standing’s extension of Vogel’s work to predict future IPR curves (Reference 1, pp. 22-23), by graphical methods for gas wells (Reference 5, pp 203-204) and by graphical methods for oil wells (Reference 1, pp 25-31). One method for calculating a future IPR curve for a gas well is that of Fetkovich (SPE 4529) as follows: qg = Ci (pr/pi) (p2r – p2wf)n where:

qg Ci pr pi n pi

= = = = = =

gas production rate at a future time initial deliverability coefficient average reservoir pressure at a future time initial average reservoir pressure initial deliverability exponent initial average reservoir pressure

The above equation to calculate future gas well IPRs is a simple modification to the “backpressure” deliverability equation. PTPR 6099

15

Fetkovich (Reference 6) also found empirically that a similar “backpressure” deliverability curve was also applicable to flowing oil wells as follows: qo = J’o (pR2 – pwf2)n and that future oil IPRs could be calculated in the same simple way as future IPRs for gas as follows: qo = J’oi (pr/pi) (pr2 – pwf2)n Note that the “deliverability” exponent n for flowing oil wells also ranges between 0.5 and 1. The “deliverability” coefficient for oil, J’o, is a form of productivity index analogous to the gas well deliverability coefficient C. Both n and J’o for oil wells are determined from tests just as n and C are for gas wells. EFFECT OF SKIN FACTOR The effect of a positive skin factor on well production is indicated by the intersection of the IPR curves for flow efficiencies of 1.0 (undamaged) and 0.54 with a sample tubing intake curve in Figure 9.

Figure 9 Effect of Skin Factor on Well Production (after Reference 3)

As with declining reservoir pressure, the effect of flow efficiency on a well’s production is substantial because of the shape of the IPR curve and is greatest at maximum drawdown.

16 PTPR 6099

EFFECT OF WELL HEAD COMPRESSION A well in Northern Alberta was tested resulting in the following stabilized deliverability data: Table 2 Flow Test

1

2

3

4

Flowing pressure, psi

1950

1789

1475

900

Flow rate, mscfd

5000

10000

15000

20000

It is desired to determine the additional production possible from two different well head compressors, A and B, capable of lowering well head pressure to 750 and 500 psi, respectively, from the current line pressure of 1000 psi. To analyze the effect of a well head compressor that is installed on the well lease, select the well head as the solution node. The stabilized well head deliverability line (inflow performance relationship at the well head solution node) is shown in Figure 10. Example 1 10000000

pts2 - ptf2, psi 2

Stabilized well head data

1000000

100000 1000

10000

100000

Flow Rate, mscfd

Figure 10 Stabilized Well Head Deliverability Line for Example 1 (after Reference 2)

17 PTPR 6099

From Figure 11 it can be seen that well head deliverability at the current line pressure of 1000 psi is 19,200 mscfd and that the incremental production expected from the well head compressors is 1,500 mscfd and 2,300 mscfd respectively.

Example 1

Flowing Well Head Pressure, psi

2500

2000 Stabilized well performance curve 1500 No compression

1000

Compressor A Compressor B

500

0 0

5000

10000

15000

20000

25000

Flow rate, mscfd

Figure 11 Stabilized Well Head Performance Curve for Example 1 (after Reference 2)

An economic evaluation is required to determine whether the well head compression modeled is economic in this case and, if so, which of the two available compressors provides the better economics. If it is desired to do a detailed compression valuation, the compressor’s performance curve would be compared to the well’s performance curve. Some of the variables that will affect the compressor’s performance curve include gas temperature and composition, compression ratio and the number of stages required, and minimum compressor inlet pressure based on available horsepower and compressor design. Service factors to be considered would include type of service (sweet or sour gas) and whether reciprocating or rotary compression would be best. Economic factors to be considered include a forecast of incremental production from compression, installation and maintenance costs and whether terms of purchase, rent or lease to own provide the best economics.

18

Sometimes new shallow, low pressure gas wells producing into an older highpressure gas gathering system require compression from day one. In other cases, compression is required later in the life of a gas well due to depletion of reservoir pressure, if gas prices have increased or if gathering system pressures have increased due to the tie-in of additional wells. PTPR 6099

DISCUSSION OF LIQUID LOADING IN GAS WELLS In a volumetric or depletion-drive type natural gas reservoir, the energy (reservoir pressure) available to lift produced fluids to surface declines with production of reserves. At some point, fluid flow rates decline enough that all of the liquids produced with the gas are no longer lifted to surface. The sources of these well bore liquids are the liquids condensed from the gas (usually mostly condensate) due to reduction in pressure and free liquids (usually mostly water) produced into the well bore with the gas. These liquids accumulate in the well bore over time and result in increased hydrostatic backpressure on the formation. This results in a further decrease in the energy available to lift fluids to surface. If this continues, enough liquid accumulates in the well bore to counterbalance the available reservoir pressure and the well will cease to flow, or die. This process is called gas well loading-up. Various means are available to deal with gas well loading up including: •

Blowing down the tubing to atmosphere (the increased flow rate clears the liquids from the well bore – not a good environmental or resource conservation practice)

•

Installing plunger lift (uses buildup of reservoir energy beneath a mechanical plunger in the tubing to lift liquids – a cyclical process that must be carefully fine-tuned to each specific well)

•

Installing a smaller diameter tubing string (the higher velocity to lift liquids comes at a price of increased friction pressure loss)

•

Dropping soap sticks in the annulus to “foam” the liquid and reduce the hydrostatic head on the formation ( a home-brew remedy much loved by field personnel as a last ditch effort to resuscitate a dead well)

•

Injecting gas into the annulus to “reverse circulate” well bore liquids out up the tubing

•

Installing artificial lift and occasionally pumping out the accumulated liquids

•

Running a portable small diameter coiled tubing string temporarily into the production tubing and injecting gas to lift out the well bore liquids (this needs to be repeated regularly and tends to be pricey)

19 PTPR 6099

Exercise Two 1. An oil well’s production is represented by a Fetkovich “backpressure” equation in oil field units as follows: qo = 0.00025 (4 x 106 – pwf2)1.0 Determine the production rate for this well at average reservoir pressures of 2000, 1600 and 1200 psia given: • Fixed well head pressure, 120 psi • GOR, 500 scf/bbl • Oil cut, 100% • Tubing size, 2-inch ID • Well depth, 7000 ft Procedure: i. Assume a series of pwf values and calculate qo to determine an IPR curve for each reservoir pressure. ii. Use a vertical multi-phase pressure traverse to calculate the VLP curve for the tubing. iii. Determine the oil production rate at each future reservoir pressure from the intersections of the tubing intake curve and the IPR curves.

20 PTPR 6099

Exercise One Answers 1. a) i. For PI=1.2: Pwf , Psi 2800

2000

1200

750

q 0 , Stbd

960

1920

2460

0

See chart below:

Module 7 - Exercise One Problem 1

Bottom Hole Pressure, p wf, psi

3000 PI = 1.2

2500 2000 1500 1000 500 0 0

500

1000

1500

2000

2500

Oil rate, stbd

ii.

Tubing ID, in Well Head Pressure, psi

2.5 200

3.0 200

q0

Pwf

q0

Pwf

stbd 800

psi 1700

stbd 1000

psi 1610

1000

1750

2000

1770

1500

1940

Incremental production with 3-in ID tubing is 180 stbd PTPR 6099

21

Module 7 - Exercise One Problem 1 3000 Bottom Hole Pressure, p wf, psi

IPR, PI = 1.2 VLP, 2.5-in id tubing

2500

VLP, 3-in id tubing 2000 1500 1000 500 0 0

500

1000

1500

2000

2500

Oil rate, stbd

b) At an oil netback of $50/bbl incremental income with 3-inch ID tubing is $9000/ day. c) If the cost of a workover to replace 2.5 inch ID tubing with 3-inch tubing is $45 000, pay out is 5 days (plus 2 days deferred revenue due to downtime). Therefore, this tubing change-out is highly attractive economically. 2. Calculate and plot the straight line IPR as for A P.I. = 1.5 as follows: 2600 2000 Pwf, psi qL, stbd 0 72 Using appropriate horizontal pressure traverse curse (0.107, 0.108, 0.109) from reference 1, calculate the wellhead pressures: qL, stbd ptf, psi

800 186

1000 205

1500 250

Calculate the tubing intake (bottom hole) pressures using vertical pressure traverse charts 0.109, 0.110, 0.111 from reference 1 qL, stbbl Ptf, psi Tubing intake, psi

800 186 1540

1000 205 1585

1500 250 1750

22 PTPR 6099

Module 7 - Exercise One Problem 2 3000 IPR, PI = 1.5 2.5-in id Flow Line Performance 2.5-in Tubing Intake Curve

Pressure, psi

2500 2000

Tubing intake curve

1500 1000

Tubing pressure drop

500 Flow line pressure drop 0 0

500

1000

1500

2000

2500

Liquid Rate, stbd

From the intersection of the tubing intake performance curve and the IPR curve, the system operating point is 1160 stbd at a bottom hole pressure of 1620 psi. 3. a)

Since flow is below the bubble point, assume Vogel relationship is valid for 90% water cut and use it to determine the well’s IPR as follows: Vogel:

qL q L, max

⎛P ⎞ ⎛P ⎞ = 1 - 0.2 ⎜ wf ⎟ -0.8 ⎜ wf ⎟ ⎝ PR ⎠ ⎝ PR ⎠

2

Detemine q L, max : q L, max =

7.00 ⎛ 1800 ⎞ ⎛ 1800 ⎞ 1 - 0.2 ⎜ ⎟ -0.8 ⎜ ⎟ ⎝ 2600 ⎠ ⎝ 2600 ⎠

2

= 1464 blpd Calculate several points on the IPR curve: Pwf , psi

2600 0

2000 546

1400 967

800 1263

200 1435

0 1464

23 PTPR 6099

b) The tubing intake curve is unaffected by the change in the IPR curve as the solution node is the bottom hole and fluid properties have not changed. This will produce 1550 bblpd at a bottom hole pressure of 870 psi. Module 7 - Exercise One Problem 3 3000 IPR, Vogel 2.5-in id Flow Line Performance 2.5-in Tubing Intake Curve

Pressure, psi

2500 2000

Tubing intake curve

1500 Tubing pressure drop

1000 500

Flow line performance curve

0

Flow line pressure drop

0

500

1000

1500

2000

2500

Liquid Rate, stbd

Exercise Two Answers 1. Assume a series of Pwf values and calculate a qo to determine an IPR curve for each reservoir pressure as follows: 1.0 ⎛P ⎞ qo = J oi ⎜ r ⎟ ( Pr 2 − Pwf 2 ) ⎝ Pi ⎠ ⎛ P ⎞ = 0.00025 ⎜ r ⎟ ( Pr 2 − Pwf 2 ) ⎝ 2000 ⎠

24 PTPR 6099

Table 3 PR, psia

2000

1600

1200

pwf

qo

pwf

qo

pwf

qo

psia

stbpd

psia

stbpd

psia

stbpd

2000 1600 1200 800 600 400 200 0

0 360 640 840 910 960 990 1000

1600 1400 1200 800 600 400 200 0

0 120 224 384 440 480 504 512

1200 800 600 400 200 0

0 120 162 192 210 216

Using vertical pressure traverse charts From reference 5 qo Pwf

200 1170

400 1220

600 1300

Exercise Two, Problem 1 2500 IPR, 2000 psia IPR, 1600 psia IPR, 1200 psia VLP Curve, 2-in i.d. tubing

Pressure, psia

2000

1500

1000

500

0 0

100

200

300

400

500

600

700

800

900 1000 1100

Oil Rate, stbpd

Oil rates are 590, 230 and 10 stbpf at 2000, 1600 and 1200 psia average reservoir pressures, respectively.

25 PTPR 6099

Assignment 1) Research choke manufacturers and describe the main types, features and basic operation of chokes used to control gas flow. 2) Research plunger lift systems. Briefly describe the main mechanical components, a typical production cycle and how such systems increase production. 3) Research the Fetkovich “back-pressure” deliverability equation for oil and describe how a value for the “deliverability” coefficient j’o is determined.

26 PTPR 6099

Module Self-Test 1. Well head pressure can be safely assumed to be fixed when calculating a VLP curve. a) True b) False 2. There may be a minimum pressure between the ends of a tubing intake curve on a plot of pressure versus flow rate for multi-phase flow when: a) a well produces at maximum rate b) hydrostatic pressure decreases with increasing flow rate c) friction pressure starts to increase faster than hydrostatic pressure is decreasing d) hydrostatic pressure is decreasing faster than friction pressure is decreasing 3. The hydrocarbon production system’s operating point is often determined by the intersection of the bottom hole IPR curve and the tubing intake curve. a) True b) False 4. In a hydrocarbon production system, the fixed pressures that are considered not to change with flow rate are: a) b) c) d)

Static reservoir pressure, well head pressure and stock tank pressure Tubing intake pressure, stock tank pressure and separator pressure Static reservoir pressure, separator pressure and stock tank pressure Tubing intake pressure, well head pressure and separator pressure

5. If a hydrocarbon production system discharges into a pipeline or a gas plant it can be safely assumed that the system discharge pressure will not vary with system flow rate. a) True b) False 6. Inflow and outflow performance curves at a solution node are not affected by a change in the other. a) True b) False 27 PTPR 6099

7. Which is true? a) If a tubing string diameter is too large, a well may load up with liquids and cease flowing efficiently or altogether. b) Tapered tubing strings are installed to decrease pressure drop in the tubing. c) Decreasing water cut may cause a well to stop flowing efficiently or cease flowing altogether. d) All of the above. 8. If a tubing string diameter is too small, a well may load up with liquids and cease flowing efficiently or altogether. a) True b) False 9. A relatively small decline in reservoir pressure can result in a relatively large decrease in production from a flowing well. a) True b) False 10. Gas is sometimes injected into a well to: a) increase the GLR and reduce the flowing friction pressure in the tubing; b) provide assistance to lift liquids by reducing the flowing friction pressure in the tubing; c) provide assistance to lift liquids by reducing the hydrostatic pressure in the tubing; d) all of the above

28 PTPR 6099

Module Self-Test Answers 1. b 2. c 3. a 4. c 5. b 6. a 7. a 8. a 9. a 10. c

29 PTPR 6099

References 1. Brown, K.E., Beggs, H.D., The Technology of Artificial Lift Methods, PennWell Publ. Co., 1977. 2. Systems NODAL Analysis, Engineering Reference I-23, Amoco Production Co., Feb.1 1984. 3. Brown, K.E., Mach, J. and Proano, E., Systems Analysis as Applied to Producing Wells” Johnston-Macco/Schlumberger training material, undated. 4. Fundamentals of Petroleum, Petroleum Extension Service, University of Texas, Austin, 2nd Ed., 1981. 5. Economides, M.J., Hill, A.D. and Ehlig-Economides, C., Petroleum Production Systems, Prentice-Hall, 1993. 6. Fetkovich, M.J., The Isochronal Testing of Oil Wells SPE 4529, 1973.

30 PTPR 6099

Appendix Conversion Factor Table

31 PTPR 6099

Course Module

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Course Module Advanced Production Engineering – PTPR 465

Hydraulic Fracturing and Production PTPR 6100 Module 8

Revised: October 2007

MacPhail School of Energy

redefine yourself www.sait.ca

Hydraulic Fracturing and Production Rationale Why is it important for you to learn this material? Hydraulic fracture stimulation to overcome near-well bore damage is often used to increase well production rates. An increase in production rate can lead to an increase in reserves if there is a competitive drainage situation or if there is an increase in the well’s economic life. An increase in well production rate can also improve economics by accelerating cash flow from the well (time value of money or net present value). Hydraulic fracture stimulation is also commonly used in tight formations to provide production rates large enough to be economic. This module will show you how to analyze performance and condition of a vertical well, estimate production after a propped hydraulic fracture stimulation and optimize propped hydraulic fracture stimulation design.

Learning Outcome When you complete this module you will be able to …. Estimate production of a vertical well after a hydraulic fracture stimulation with the Darcy radial flow equation using equivalent well bore diameter and understand the parameters that affect hydraulic fracture stimulation effectiveness.

Learning Objectives Here is what you will be able to do when you complete each objective. 1. Understand and use several versions of the Darcy radial flow equation modified for analysis of damaged and stimulated wells. 2. Describe the difference between true formation skin (damage or improvement amenable to stimulation or damage), other skins not amenable to stimulation and pseudo-skin. 3. Explain and use the concepts of well bore radius equivalent to skin factor and of dimensionless fracture conductivity when estimating well productivity after a propped hydraulic fracture stimulation. 4. Learn how to optimize the design of a hydraulic fracture stimulation of a vertical well.

1 PTPR 6100

Introduction The ultimate goal of this module is to provide you with an understanding of the factors which affect the productivity of hydraulically fractured vertical wells. We will use a slightly modified version of the standard Darcy steady-state radial flow equation to illustrate the effect of those factors. The equation is modified such that it separates the true formation damage (or stimulation) skin from all the other skins which can affect productivity. This is done by replacing the nominal well bore radius in the radial flow equation with a well bore radius that is equivalent to true formation damage (or stimulation) only. This allows the effect of a propped vertical fracture to be incorporated directly into the equation to gauge the effect of different hydraulic fracture designs on well productivity. Thus, easy optimization of hydraulic fracture design parameters is possible. The remaining skins which can affect well productivity are handled by a skin factor term as normal. This works because the remaining skins are not amenable to normal stimulation or damage and will remain constant under different conditions of true formation damage or stimulation.

2 PTPR 6100

OBJECTIVE ONE When you complete this objective you will be able to… Understand and use several versions of the Darcy radial flow equation modified for analysis of damaged and stimulated wells.

Learning Activity Complete each of the Learning Activities listed below. 1. Read and study the learning material in this objective. 2. Do the exercise provided. 3. Do the assignment provided. 4. Research outside sources (that is, the Internet, library materials, reference texts, and so on) for additional information.

Learning Material DARCY STEADY-STATE RADIAL FLOW EQUATIONS The standard form of the Darcy steady-state radial flow equation in oilfield units for a compressible liquid such as oil is: qo =

7.08 k o h (p e - p wfs ) bopd Bo u o [ ln (re /rw ) + S']

However, a slightly modified form of the Darcy steady-state radial flow equation will be used in this module, and in Module 9, to better illustrate the effect of a propped hydraulic fracture stimulation of a well with formation damage. This modified form will allow a better understanding of the various down hole skin factors which can affect well productivity. The two types of skin factors are those which are amenable to stimulation and those which are not. It does this by showing the magnitude of the two types of skins, when present; to better gauge their effect on well productivity. It also reduces confusion when estimating production from wells with formation damage which are not vertical or have only some of the pay penetrated or have excessive losses through the perforations or have non-laminar inflow.

3 PTPR 6100

The radial flow equation is modified by the following two changes: 1. The nominal well bore radius, rw, is replaced with an effective, or equivalent, well bore radius, r'w, defined here as that equivalent well bore radius which is solely due to true formation damage (or improvement) amenable to improvement (or damage) by fracturing, acidizing or other chemical and mechanical treatments to overcome near-well bore formation damage. 2. The total apparent skin factor, S', is replaced by the sum of the remaining skin factors, s, which are not amenable to improvement or damage by stimulation activities but are the result of drilling and completion practices or non-laminar (turbulent) or phase flow in the reservoir. The remaining four skin factors not amenable to stimulation treatments down the well bore include: • • • •

inclination of the well bore through the pay zone partial penetration or partial completion of the pay zone perforation entry losses non-laminar (turbulent) flow or phase flow in the reservoir

Non-laminar (turbulent) flow is usually called a pseudo-skin because, while it results in a pressure drop from the reservoir to the well bore - in effect a less permeable “skin” around the well bore as do all the other skins - it is not amenable to any improvement by any means in the near-well bore area. This is because it is a rate-dependent skin. Of course, excessive perforation entry losses can be remedied by re-perforating to higher shot density and/or to bigger hole sizes. This is in fact often done prior to hydraulic fracture stimulation particularly in older wells or in newer wells which were carelessly shot or shot in a manner to minimize cost. These wells were often shot with small diameter hole sizes and/or at low shot density. In order to minimize friction pressure loss through the perforations when pumping a viscous gel or other fluid at high rate during a fracture stimulation treatment in order to carry sand at high concentration into the formation, such wells are routinely reperforated with big hole charges and increased shot density. For the purposes of this module, all the skins other than the true formation skin, sd, are assumed to be zero. This is usually the case for a typical vertical oil well with all the pay perforated and with no unusual perforation losses. Module 9 will show you how to calculate the remaining skins including the rate-dependent and phase flow pseudo-skins and to determine their effect on production rates. The modified Darcy steady-state radial flow equation for a compressible liquid (oil) in a bounded drainage area can be written in oilfield units as:

4

qo =

7.08 k o h (p e - p wfs ) bopd Bo u o [ ln (re /r'w ) + s ] PTPR 6100

and in metric units as: qo =

0.5352 k o h (pe - p wfs ) 3 m /d Bo u o [ ln (re /r'w ) + s ]

Similarly, the Darcy steady-state radial flow equation for gases in a bounded drainage area can be written in oilfield units as: qg =

0.703 k g h (p e 2 - p wfs 2 ) μ g z T [ ln (re /r'w ) + s ]

mscfd

and in metric units as: qg = where:

qo qg ko kg h pe

= = = = = = pwfs = Bo = μo = μg = re = rw = r'w = z = s =

0.000763 k g h (p e 2 - p wfs 2 ) μ g z T [ ln (re /r'w ) + s ]

103 m3 /d

oil rate, bopd or m3/d gas rate, mscfd or E3m3/d effective permeability to oil, darcys or μ m2 effective permeability to gas, darcys or μ m2 net pay, ft or m reservoir pressure at drainage radius, psia or kPa(a) producing bottom hole pressure, psia or kPa(a) oil formation volume factor, res vol/st vol oil viscosity at producing bottom hole pressure, cp or mPa•s gas viscosity at producing bottom hole pressure, cp or mPa•s drainage radius, ft or m nominal bore hole radius, ft or m equivalent or effective well bore radius, ft or m real gas deviation factor, dimensionless total of skins other than sd, dimensionless

The foregoing Darcy steady-state radial flow equation for a compressible liquid (oil) can easily be applied to an incompressible fluid (water) by assuming a water formation volume factor of 1.0. While some gas can be in solution in formation water, this simplifying assumption provides adequate accuracy for production engineering estimating purposes.

5 PTPR 6100

For gas wells, note the following guidelines for the correct pressure function to use in any gas flow equations: Static Reservoir Pressure 0 > 2500 psi; 0-17,500 kPa(a) 2500 > 3500 psi; 17,500-24,500 kPa(a) > 3500 psi; >24,500 kPa(a)

Pressure Function p2 m(p) or ψ p

The pressure function guidelines show that it is not always necessary to use the real gas pseudo-pressure function, m(p) or ψ, to obtain practical engineering accuracy when using the Darcy radial flow equations. This is because there is always a degree of inaccuracy in the real life value of most of the other variables in the equations and because real gas properties do not vary significantly within the ranges of the guidelines. For a more rigorous approach to the use of the real gas pseudo-pressure function and its applicability, the Alberta EUB Guide G-3, Gas Well Testing - Theory and Practice, 4th Ed., 1979 is a recommended reference. The steady state equation is applicable when reservoir pressure at the drainage radius is constant and not changing due to depletion of reserves or (offset) injection. Thus, it can apply to reservoirs with fully active aquifers. It is also applicable before the well drainage radius has reached a no-flow boundary in the reservoir. In other words, when a new well is still infinite acting and has not yet “felt’ the reservoir no-flow boundaries, it is considered to be in steady-state. The only requirement is that the readjustment time to any change in the system from the well to the reservoir is small. This is usually, but not always, the case. Pseudo-steady state occurs when the pressure at a well’s drainage radius is not constant but changing at a constant rate due to depletion of reserves or offset production or unbalanced injection, this is called. In this state, a well’s drainage radius has extended out to the reservoir’s no-flow boundaries and it “feels” the changing pressure at the boundary. A slightly different form of the Darcy radial flow equation using average reservoir pressure applies in the pseudo-steady state. Many wells that are hydraulic fracture stimulated are new wells in new pools and are therefore infinite-acting. During this period, the well’s drainage radius is timedependent. However, the standard Darcy radial flow equation is not very sensitive to the value for the well’s drainage radius. Table 1 shows the difference in calculated oil rate using different typical values for well drainage radius. The difference between assuming 40-acre and 640-acre drainage areas is only 15%.

6 PTPR 6100

Table 1 Effect of Well Drainage Radius on Calculated Production Rate © Prentice-Hall. This material has been copied under license from Access Copyright. Resale or further copying of this material is strictly prohibited.

Furthermore, the modified Darcy radial flow equation used in this module to illustrate the effect of formation damage and stimulation is even less sensitive to well drainage radius. This is because in most cases, as shown in Table 1, the well’s drainage radius spans only about one order of magnitude while the equivalent well bore radius can vary by many orders of magnitude between a damaged well and a stimulated well. In order to convert between true formation skin and equivalent well bore radius, the following relationship defining effective or equivalent well bore radius is used (Reference 7):

r' = rw e(-sd ) or, re-arranging: s d = ln (rw /r'w )

7 PTPR 6100

Exercise One 1. A new oil well has been determined to have: An effective permeability to oil of 100 mD A net perforated pay thickness of 1.5 m A reservoir pressure of 9000 kPa(a) An oil formation volume factor of 1.073 An oil viscosity at bottom hole temperature and pressure of 3.0 cP A drainage radius of 227 m A nominal bore hole diameter of 200 mm The well is vertical, all the pay is perforated, there are no significant perforation losses and inflow is laminar. a) If the bottom hole flowing pressure will be 2000 kPa(a), estimate the production rate from this well using the modified steady-state radial flow equation assuming no formation skin. b) If the well is subsequently damaged to a true formation skin of +5 as a result of (perhaps clumsy or incompetent) operations to equip the well bore, i. Calculate the damaged well’s equivalent well bore radius. ii. Calculate the damaged well’s productivity. c) If it is expected that the well can then be hydraulically fracture stimulated to a true formation skin of -3 based on similar offset wells, i. Calculate the stimulated well’s equivalent well bore radius. ii. Calculate the stimulated well’s productivity.

8 PTPR 6100

OBJECTIVE TWO When you complete this objective you will be able to…

Describe the difference between true formation skin (damage or improvement amenable to stimulation or damage), other skins not amenable to stimulation and pseudo-skin.

Learning Material SKIN FACTORS

The modified Darcy steady-state radial flow equations previously shown in this module include two types of skin. The first type of skin is true formation skin, denoted as sd, which is amenable to stimulation or damage. Its magnitude is indicated by a positive number for damage and by a negative number for improvement or stimulation. Negative true formation skins of magnitude larger than -5 are rarely achieved with stimulation, however, positive true formation skins with magnitudes of +10 or greater are common. In other words, it is easier to damage a well to the point where it produces little or nothing than it is to stimulate a well much beyond what amounts to a two- to four-fold improvement in production rate. The second type of skin includes all the remaining skins, the sum of which is denoted as s. The common property of all the remaining skin factors is that they are not amenable to improvement (by stimulation) or damage (by other means) of the formation in the near well bore area. These remaining skin factors are well bore angle of inclination through the pay interval, partial penetration or partial completion of the pay interval, perforation entry losses and losses due to nonlaminar (turbulent) or phase flow in the reservoir. Their magnitude is also denoted by positive (damage) or negative (improvement) numbers in the same as true formation skin. Productivity loss due to non-laminar (turbulent) or phase flow in the reservoir is termed a pseudo-skin because, although it can be quantified as a pressure loss and have an affect on production like the other skin factors, it is not physically connected to the well bore but to the reservoir and is rate (or phase) dependent. As such it can not be treated from, or be changed by, operations from the well bore as it occurs in the reservoir.

9 PTPR 6100

The relationships between the different skin factors are summarized below: The total apparent skin factor, S', obtained from normal pressure transient test analysis or from production performance matching, is the sum of the following true formation damage skin factor and other skin factors: S' = sd + sa+pc+ sperf + sDq or:

S' = sd + s

where:

sd = true formation damage (or improvement) skin factor amenable to, or a result of, stimulation s = sum of remaining skin factors including pseudo-skin = sa+pc + sperf + sDq

where: sa+pc = inclination angle and partial completion skin factor sperf = perforation entry loss skin factor sDq = turbulent or phase flow pseudo-skin factor By separating out the true formation damage skin factor and the total of the remaining skin factors including the pseudo-skin factor, it is easier to determine what the major problem(s) is (are) in an underperforming well and what the best approach is to remedy it (them). Note that the effective, or equivalent, well bore radius will be larger than the nominal well bore radius for an improved/stimulated well and will be smaller than the nominal well bore radius for a damaged well. The nominal well bore radius is the bore hole diameter/2 or the diameter/2 of the drill bit used to penetrate the pay zone. Sometimes, for formations which are not competent such as poorly consolidated sands or “sloughing” shales, well bore radius can be obtained from a caliper survey. For the case of a well with a total apparent skin of zero, (neither damaged nor improved) the effective or equivalent well bore radius will be equal to the nominal well bore radius. The total apparent skin S' can be obtained from pressure transient test analysis or from production performance matching. Production performance matching allows you to estimate the total apparent skin of a well if values for all the other variables in the Darcy radial flow equation are known or can be guessed at with sufficient “engineering accuracy”.

10 PTPR 6100

The individual remaining skin factors take into account the effect on production of any bore hole angle of penetration different than 90° (deviated/inclined) through the pay zone, partial penetration of the pay interval (TD above bottom of pay or only part of the pay perforated), any significant pressure drop through the perforations in the steel casing/cement sheath and any losses due to non-laminar, or turbulent, flow or phase flow in the reservoir. Their estimation will be discussed in detail in Module 9. Note that the Darcy radial flow equation applies only to laminar flow in the reservoir. This is generally the case except for very high rate gas wells or highly restricted entry into the well bore. If there is any significant pressure loss into the well bore due to non-laminar (turbulent) or phase flow, it must be accounted for separately by means of a pseudo-skin factor.

Exercise Two 1. A vertical oil well drilled with a 168 mm diameter bit is estimated to have: An effective permeability to oil of 20 mD based on nearby wells in the same formation A net perforated pay thickness of 2 m A reservoir pressure of 6000 kPa(a) An oil formation volume factor of 1.04 An oil viscosity at bottom hole temperature and pressure of 2.0 10 3 μm2 A drainage radius of 227 m At a producing bottom hole pressure of 1500 kPa(a) it makes 3 m3/d of oil. There are no significant perforation losses, all the pay is perforated, inflow is laminar and the well is infinite acting. a) What is the nominal well bore radius, rw, for this well? b) What is the estimated equivalent well bore radius for this well? c) What is estimated true formation skin factor for this well?

11 PTPR 6100

OBJECTIVE THREE When you complete this objective you will be able to…

Explain and use the concepts of well bore radius equivalent to skin factor and of dimensionless fracture conductivity when estimating well productivity after a propped hydraulic fracture stimulation.

Learning Material DIMENSIONLESS FRACTURE CONDUCTIVITY

Dimensionless fracture conductivity is defined as: FCD =

where:

FCD kf wf k xf

= = = = =

kf w k xf

dimensionless fracture conductivity retained fracture permeability in darcies, D average propped fracture width in millimetres, mm effective reservoir permeability in millidarcies, mD propped fracture half length in metres, m

Using the units indicated in the above definitions will result in the correct numerical value for FCD without having to convert kf to millidarcies or wf to metres. In oilfield units, using millidarcies and feet throughout results in the correct numerical values for FCD. The numerical value will be the same in either system because it is a dimensionless value. Fracture conductivity, Cf, is sometimes referred to and is defined as follows: Cf = kf wf where:

Cf = fracture conductivity, md-m (same as darcy-m) or md-ft kf = retained fracture permeability in d or md wf = average propped fracture width, mm or ft

Fracture conductivity is a measure of the flow capacity of a propped hydraulic fracture. Dimensionless fracture conductivity is a measure of the flow capacity of a propped fracture relative to the formation.

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Fracture conductivity is directly proportional to retained fracture permeability and to fracture width. Retained fracture permeability reflects the final characteristics of a specific proppant in-situ, that is, after the treatment is over and the well starts to produce. It is significantly different than the permeability of the bulk proppant as measured at surface conditions. Retained fracture permeability is influenced by: 1. The type of frac fluid used to carry the proppant into the fracture and how much of its residue remains between the proppant grains after the frac fluid gel “breaks”. 2. The concentration of proppant, in kg/m3, actually achieved in the hydraulic fracture. 3. Whether any fines have been generated during handling and pumping and whether any crushing of the proppant occurs after closure or “healing” of the induced hydraulic fracture once the treatment pumps are stopped. 4. Whether any embedment of the proppant in the fracture faces occurs after the fracture closes once the treatment pumps are stopped. Average propped fracture width and propped frac half length depend upon the amount of proppant used, reservoir rock stresses and mechanical properties, assumptions or knowledge of fracture geometry, and fracture treatment parameters such as fracture fluid viscosity, pump rate, fluid volumes, and schedule of fluid and proppant stages. A symmetrical, vertical penny-shaped fracture as shown in Figure 1 can be assumed as a good first approximation when estimating performance of a well stimulated with a propped hydraulic fracture.

13 PTPR 6100

Figure 1 Hydraulic Fracture Orientation When Overburden Stress is the Dominant Tri-axial Rock Stress (Courtesy of Schlumberger)

An induced hydraulic fracture will proceed in the direction of the maximum horizontal stress, that is, the reservoir rock will split apart against the minimum horizontal stress. In Alberta, in a wide belt near the foothills, the maximum horizontal stress “frozen” into the subsurface is a result of the compression forces from the ancient collision of tectonic plates which gave rise to the Rocky Mountain chain which runs approximately NW/SE. In that wide belt, the maximum horizontal stress can be expected to point towards the mountains and you can expect typical induced hydraulic fracture orientation to be approximately NE/SW. Studies of well bore “rugosity”, or the degree and direction of departure from a circular bore hole to one that is “oval”, within a wide belt near the Alberta foothills, confirms the direction of maximum horizontal stress. Well bores that tend to “spall” in this area tend to have a slightly larger diameter in an approximate NE/SW direction than in the NW/SE direction. The “spalling” of material from the well bore face is increased in the direction of the maximum horizontal stress. 14 PTPR 6100

Because overburden stress is generally the dominant stress below 300-600 m, fractures are vertical below those depths. At shallower depths, fractures may be horizontal. We will concern ourselves only with vertical propped hydraulic fractures in this module. While hydraulic fracture treatments can create long fractures in the reservoir rock, only the length of the fracture that remains propped open with a significant sand pack after the treatment is over counts as effective propped frac length. Stimulation service companies often have sophisticated 3-D rock-mechanics software which models and predicts fracture size and shape. These may or may not provide a more accurate estimate for propped fracture half-length depending upon how much is known about in-situ rock properties (often very little) and reservoir fluid properties. The sophisticated 3-D rock-mechanics software for fracture simulation requires specific and detailed information on rock and fluid properties. This information is rarely all available and is usually costly to obtain. Values for the following rock and fluid properties are generally required as input for a 3-D fracture simulator: •

Poisson’s ratio

•

Static Young’s modulus

•

Closure stress

•

Pore pressure

•

Permeability

•

Bulk compressibility

•

Fracture toughness

•

Porosity

•

Water saturation

15 PTPR 6100

Exercise Three 1. A fractured oil well has: A retained fracture permeability of 580 darcies Fracture width of 2 mm Reservoir permeability to oil of 100 mD A frac half-length of 17.275 m a) Calculate the dimensionless fracture conductivity using the above values and units. b) Calculate the dimensionless fracture conductivity converting the numbers in a) to mD and ft as necessary. c) Calculate the dimensionless fracture conductivity using units of darcies and metres.

16 PTPR 6100

OBJECTIVE FOUR When you complete this objective you will be able to…

Understand and use several versions of the Darcy radial flow equation modified for analysis of damaged and stimulated wells.

Learning Material EQUIVALENT OR EFFECTIVE WELL BORE RADIUS

“The production behaviour of a homogeneous cylindrical reservoir producing a single fluid through a centrally located vertical fracture of limited lateral extent (and high FCD) can be represented by an equivalent radial flow reservoir of equal volume. The effective well bore radius of this equivalent reservoir is equal to onefourth of the total effective fracture length (or one-half of the effective fracture half-length, xf). The behaviour of vertically fractured reservoirs can be interpreted in terms of simple radial flow reservoirs of large well bore.” (Reference 1) In other words, it can be shown that for a high dimensionless conductivity fracture, the effective well bore radius can be given by: r'w =

where:

xf 2

r’w = effective or equivalent well bore radius, m xf = effective fracture half-length, m

Note that in a well stimulated by a hydraulic fracture, the well behaves as if the effective, or equivalent, well bore radius is much larger than the nominal well bore radius. If the effective fracture half-length of a hydraulic fracture stimulation can be determined, then the effective well bore radius, and hence productivity, of the stimulated well can be easily determined. The effective fracture half-length, xf, is approximately, but always a little less than, the propped fracture half-length. This is because there is usually deceasing sand concentration in the propped fracture as the tip of the fracture is approached. This is because of sand dilution effects and because proppant is introduced into the well bore in stages during treatment with the early stages at low proppant concentration and the last stages at high proppant concentration in the frac fluid.

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It can be assumed, within typical engineering accuracy, that the effective propped fracture half-length is the same as the propped fracture half-length. It can be estimated using simple geometry if the amount of sand placed in the reservoir, the bulk density of the sand and the average frac width are known as follows:

⎛ SP x 1000 ⎞ xf = ⎜ ⎟ ⎝ π ρs w f ⎠ where:

0.5

SP = sand placed, tonnes ρs = sand bulk density, g/cm3 wf = fracture average propped width, mm

The above assumes a symmetrical penny-shaped frac that intersects a vertical well bore. Typically, in Alberta, you will find propped fracture half-lengths for most fracs in conventional formations to be in the range of 10 to 100 m. It should be emphasized that the above relation only holds true for wells with dimensionless fracture conductivity, FCD, larger than about 30. For propped fractures with an FCD less than about 1, the following relationship has been found to hold: r'w = 0.28

Cf k

For most values of dimensionless fracture conductivity, the equivalent well bore radius may also be determined graphically by using Figure 2, which is based on the work in Reference 3. Alternatively, a mathematical relationship from a best-fit curve to the data in Figure 21 may be used, particularly for the region where 1< FCD < 30.

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EQUIVALENT WELLBORE RADIUS vs DIMENSIONLESS FRACTURE CONDUCTIVITY 1

r'w / xf

FCD > 30, r'w = x f / 2 and well performance is x f limited 0.1

FCD < 1, r'w = 0.28 Cf / k and well performance is Cf limited 0.01 0.1

1

10

100

FCD Figure 2 Equivalent Well Bore Radius vs. Dimensionless Fracture Conductivity

There are some important observations to be made from Figure 2 above. 1. For low FCD fractures, post-fracture productivity is dominated by fracture conductivity (fracture flow capacity). Low FCD fractures typically are those in high permeability reservoirs. In these reservoirs, productivity of wells stimulated with propped fractures is insensitive to fracture length. 2. For high FCD fractures, post-fracture productivity is dominated by effective fracture half-length. High FCD fractures typically are those in low permeability formations. In these reservoirs, fracture conductivity is relatively unimportant because the reservoir is incapable of delivering high fluid rates into the propped hydraulic fracture. Thus, to stimulate a high permeability (high productivity) formation, the goal in hydraulic fracture design and placement is to create a high conductivity fracture with a high FCD. In this case fracture length does not control well productivity and is not important. It is only necessary to get fracture length past any true formation damage which is typically very close to the well-bore and does not extend very far into the reservoir.

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Conversely, to stimulate a low permeability (tight) formation, the goal in hydraulic fracture design and placement is to create a long propped fracture. In this case, fracture conductivity does not control well productivity because the reservoir will never have the capability to deliver high fluid rates into the propped fracture. Compensation for a tight reservoir is made by providing a large fracture face area to conduct fluid into the propped fracture. Because the formation is tight relative to the conductivity of the propped fracture, fracture conductivity and FCD will be high relative to the formation. Since the pay height will be more or less constant, increasing fracture length is thus the primary means to improve well productivity in tight reservoirs. Sample values for retained fracture permeability, assuming 60% of proppant permeability is retained in-situ (retained permeability factor) and typical values of sand proppant bulk density are shown below for common sand sizes. Table 2 Sample Retained Fracture Permeability Values Sand Size Mesh 20/40 16/30 12/20 8/10

Proppant In-situ Permeability Bulk Density at 0.6 Retained Factor Darcies g/cm3 200 1.59 390 1.57 580 1.63 770 1.60

Values of proppant in-situ permeability should be obtained from fracturing service providers or obtained empirically from treatment results. Different types of frac fluids, gel quality and treatment parameters will result in different values for the retained permeability factor. Also, the type, quality or source of proppants may result in different values of bulk density than shown above. A final consideration is the crushing of proppant that is subjected to stresses beyond its capability to withstand due to high formation stresses. Generally, stresses on proppants increase with depth and rock hardness. Beyond a certain depth sand is no longer a suitable proppant due to excessive crushing and fines generation. The crushing will produce fines that will reduce in-situ proppant pack porosity and permeability. Frac width will depend partly on rock properties such as horizontal stress profile and proppant embedment. Sometimes, embedment of proppant into softer formations when the hydraulically-induced fracture closes after the pumps stop can reduce fracture width and, hence, fracture conductivity. Frac fluid properties such as viscosity and leak-off from the fracture during treatment and treatment parameters such as pump rate and sand/frac fluid pumping schedule, among other factors, can also affect achieved frac width. 20 PTPR 6100

There is a technique called “tip-screen-out” (TSO) which is often applied to high permeability formations to maximize fracture width by “ballooning” the fracture. This technique is designed to dramatically increase propped fracture width without increasing its length and packing it full of high permeability proppant. This maximizes fracture conductivity and, in high permeability formations, provides a higher value for FCD. Detailed discussion of this technique is beyond the scope of this course. Just be aware that it exists and where it applies best. Frac height growth is controlled by in-situ horizontal stress profiles and fracture treatment mechanics. High horizontal stress contrasts in formations tend to act as barriers to fracturing “out-of zone”. Because overburden stress, the dominant stress below 300-600 m depth, increases with depth, fracture growth out of zone (frac height growth) tends to be upward as in Figure 3.

Figure 3 3-D Frac Simulator Result Showing, Frac Width, Frac Shape and Half-length (Courtesy of Schlumberger)

The sample 3-D frac simulator output shown in Figure 3 is typical for an Alberta well. Note the maximum propped fracture width of only 4 mm at the well bore and the effective propped fracture half-length of about 40 m. This was a 24 ton frac. Note in particular the preferential height growth of the propped fracture into formations above the pay zone. This is common due to the way rock stresses normally increase with depth and where no significant geological barrier to fracture height growth exists. Obviously, longer propped fractures can be obtained for a given treatment size if the fracture can be contained entirely within the pay zone. Now you have all the tools needed to estimate the productivity of a well for different treatment size (tonnes of sand), sand size, retained frac fluid/proppant pack permeability and frac width. PTPR 6100

21

With the tools presented in this module you can estimate productivity of stimulated wells with reasonable accuracy with a minimum of expensive data gathering. Note that the best estimates are obtained when any assumptions or guesses can be fine-tuned from the results of similar nearby wells producing from the same formation, from the results of pressure transient analysis and from stimulation service company data for the same formation in the area. The procedure to estimate productivity of a hydraulically fractured vertical well is as follows; 1. Obtain values for all variables in the appropriate Darcy radial flow equation except equivalent well bore radius. 2. Obtain a value for effective propped fracture half-length. 3. Calculate dimensionless fracture conductivity. 4. Determine equivalent well bore radius. 5. Calculate expected well production rate using the Darcy radial flow equation. By repeating the calculations for various tonnages and types of proppant, for insitu retained fracture permeability (which depends upon the type of frac gel used and the sand concentration achieved in the fracture) and achieved fracture width, different hydraulic fracture designs can be identified and production forecasts can be generated. By determining the cost of the different hydraulic fracture designs, a net present value can be calculated for each and the hydraulic fracture design with the best practical economic return can be found.

22 PTPR 6100

Exercise Four 1. A new oil well drilled with a 200 mm diameter bit will be stimulated with a hydraulic fracture using 20 tonnes of 20/40 sand. Expected average fracture width, 3 mm Reservoir permeability to oil, 40 md Net pay, 3 m Reservoir pressure, 5000 kPa(a) Bottom hole pump intake pressure, 1000 kPa(a) Oil formation volume factor, 1.05 Oil viscosity at bottom hole conditions, 2.5 Cp Drainage radius, 227 m Assuming a vertical well with all the pay perforated, no significant perforation losses and laminar flow: a) Calculate the effective fracture half-length for this well. b) Calculate the dimensionless fracture conductivity. c) Determine the equivalent well bore radius using Figure 3. d) Calculate the expected production rate. e) Determine the expected true formation skin after the stimulation.

23 PTPR 6100

Exercise One Answers 1. a. Use Darcy radial flow equation for oil modified for equivalant wee bore radius qo =

0.5352 k o h ( pe -p wfs ) Bo μ [ lm(re /rw )+s ]

Substituting qo =

0.5352i0.100i1.5 ( 9000-2000 ) 1.073i3.0 [ lm(227/0.100)+0]

= 22.6 m3 /d b. i Determine Equivalent well bore radius of true formation skin r1w = e-Sd Substituting, r1w = 0.100 e-S = 0.000674 m ii Substituting in Darcy Radial flow equation: qo =

0.5352i0.100i1.5 ( 9000-2000 )

1.073i30 [ lm (227/0.000674)+0]

= 13.7 m3 /d c. i Determine equivalent well bore radius of true formation skin r1w = e-Sd Substituting r1w = 0.100 e

-(-3)

= 2.0086 m ii Substituting in Darcy radial flow equation qo =

0.5352i0.100i1.5i(9000-2000) 1.073i3.0 [ lm (227 / 2.0086) + 0]

= 36.9 m3 /d

24 PTPR 6100

Exercise Two Answers 168 mm = 84 mm = 0.084 m 2 b. Re-arranging the darcy radial flow equation to solve for equivalent well bore radius 0.5352 k o h(pe -p wfs ) ln(re /rw ) = q o Bo μ o

1. a. rw =

Substituting ln(227/rw ) =

0.5352i0.02i2.0(6000-1500) 3.0i1.04i 2.0

= 15.439 227/rw = e15.439 rw =0.00004477 m c. Re-arranging the equation to convert equivalent well bore radius to skin Sd = f m (rw /rw ) Substituting Sd = ln(0.004/0.00004477) = 7.54

Exercise Three Answers 1. a. Using the equating calculator dimensionless fracture conductivity FCD =

kf wf kx f

substituting FCD =

580 Di2 mm 100 mDi17.275 m

= 0.671 b. FCD =

580 000 mDi0.0065644 100 mDi56.67 ft

= 0.671 c. FCD =

580Di0.002 m 0.100Di17.275 m

= 0.671

25 PTPR 6100

Exercise Four Answers 1. a . Use the equation to calculate effective fracture half length: ⎛ SPi1000 ⎞ xf = ⎜ ⎟ ⎝ πps w f ⎠ Subsituting ⎛ 20i1000 ⎞ xf = ⎜ ⎟ ⎝ π i1.59i3 ⎠ = 36.5m

0.5

0.5

kfwf k xf 200i3 = 40i36.5 = 0.411 c. From the figure 3 for FCD =0.411 b. FCD =

r 1w = 0.093 xf and rw = 0.093i36.5 m = 3.39 m d. Using the modified darcy radial flow equation for steady state: qo =

0.5352k o h(pe-pwf) Bo μo ⎡⎣ln(re/rw1 ) + s ⎤⎦

Substituting qo =

0.5352i0.040i3.0 ( 5000 − 1000 )

⎡ ⎛ 227 ⎞ ⎤ 1.05i2.5 ⎢ln ⎜ ⎟ + 0⎥ ⎣ ⎝ 3.39 ⎠ ⎦ = 23.3 m3 /d

e. Using the equation to convert equivalent well bore radius to skin factor rw = rw c-sd re-arranging ⎛ rw ⎞ sd = ln ⎜ 1 ⎟ ⎝ rw ⎠ subsituting ⎛ 0.100 ⎞ sd = l n ⎜ ⎟ = −1.22 ⎝ 3.39 ⎠ 26 PTPR 6100

Module Self-Test 1. Total apparent skin is the sum of all the skins less any pseudo-skins. a. True b. False 2. A well with a total apparent skin factor of zero a. b. c. d.

is damaged is stimulated is neither damaged nor stimulated could be somewhat stimulated (negative true formation skin) but have an offsetting non-laminar pseudo-skin

3. The effective propped fracture half-length is usually a little more than the propped facture half-length. a. True b. False 4. If a well completed in a conventional lower permeability formation is stimulated with an average hydraulic fracture treatment you would expect high dimensionless fracture conductivity. a. True b. False 5. Faced with a tight (low permeability) reservoir, would you be looking at a. a fracture stimulation that provides a short effective fracture half-length and high fracture conductivity? b. a fracture stimulation that provides a long effective fracture half-length and lower fracture conductivity? c. a fracture stimulation that provides a short effective fracture half-length and lower fracture conductivity? d. a fracture stimulation that provides a long effective fracture half-length and high fracture conductivity? 6. Retained in-situ propped fracture permeability a. b. c. d.

is a fraction of the permeability of that proppant as measured at surface partially depends on the mesh size of the sand used partially depends on how much frac fluid residue remains all of the above 27

PTPR 6100

7. Fracture conductivity is proportional to a. b. c. d.

fracture length tonnes of sand placed in the formation reservoir permeability none of the above

8. In a damaged well with a true formation skin of +4, the equivalent well bore radius is smaller than the nominal well bore radius. a. True b. False 9. Propped hydraulic fractures are often not contained within the pay zone. a. True b. False 10. Induced hydraulic fractures in formations less than less than 300-600 m tend to be horizontal a. True b. False 11. If a propped hydraulic fracture has broken out of the pay zone, you would expect the tendency is to preferentially grow down. a. True b. False 12. In a wide belt close to the foothills in Alberta and paralleling the Rocky Mountain chain, in what azimuthal direction would you expect an induced hydraulic fracture to extend in a well deeper than 200-600 m? a. b. c. d.

Approximately horizontally. Approximately NW/SE. Approximately NE/SW. Approximately vertically.

28 PTPR 6100

Module Self-Test Answers 1. b 2. d 3. a 4. b 5. b 6. d 7. d 8. b 9. a 10. a 11. b 12. c

29 PTPR 6100

Assignment 1. What are three reasons that wells are stimulated with propped hydraulic fractures? 2. Assuming a symmetrical penny-shaped frac, what factors determine the effective propped fracture half-length? 3. Explain the concept of equivalent well bore radius. 4. What is the difference between steady-state and pseudo-steady state conditions in a reservoir surrounding a producing well? 5. List six skin factors which are not true formation skin. 6. Obviously, it is possible to stimulate a damaged well. Describe how it is possible to damage a new well that has just been successfully stimulated with a propped hydraulic fracture and is waiting to be equipped for production? Under what condition(s) would the use of a high-strength proppant be considered?

30 PTPR 6100

References 1. Prats, M., Hazebroek, P., Strickler, W.R.; Effect of Vertical Fractures on Reservoir Behavior - Compressible Fluid Case, SPEJ (June 1962), pp. 87-94. 2. Cinco-Ley, H., Ramey Jr., H.J., et al; Behaviour of Wells with LowConductivity Vertical Fractures, SPE 16776, Sept. 1987. 3. Smith, M.B., Hannah, R.R., High Permeability Fracturing: The Evolution of a Technology, JPT, July 1996, pp. 628-33. 4. Economides, M.J., Hill, A.D., Ehlig-Economides, C., Petroleum Production Systems, Prentice-Hall, 1994. 5. Economides. M.J., Nolte, K.G., Reservoir Stimulation, 2nd Ed. Prentice-Hall, 1989. 6. Marcinew, R, MacFarlane, C., Hydraulic Fracture Optimization, Schlumberger, undated presentation ca. 1999. 7. Mathews, C.S., Russell, D.G., Pressure Build-up and Flow Tests in Wells, AIME, Monograph Vol. 1, SPE-AIME, 1967.

31 PTPR 6100

Appendix Table 2 Conversion Factor Table

32 PTPR 6100

Course Module

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Course Module Advanced Production Engineering - PTPR 465

Skin Factors Other than True Formation Damage PTPR 6101 Module 9

Revised: November 2007

MacPhail School of Energy

redefine yourself www.sait.ca

Skin Factors Other than True Formation Damage Rationale Why is it important for you to learn this material? In order to better understand all the variables that can affect well productivity, it is necessary to understand and be able to evaluate skin factors other than the true formation damage skin factor (covered in Module 8). The material in this module will enable you to understand and estimate the magnitude of the skin effects that are due to an inclined well bore through the pay zone, partial completion in the pay zone, perforation friction losses and nonlaminar (turbulent) flow in the reservoir. This knowledge will help with decisions on how best to develop and exploit those reservoirs which have, or may require, a well other than the standard vertical well or those reservoirs which have high flow rates.

Learning Outcome When you complete this module you will be able to …. Understand and calculate the skin factors other than true formation damage, that is, inclination, partial penetration, turbulent flow and phase flow that affect well productivity.

Learning Objectives Here is what you will be able to do when you complete each objective. 1. Calculate the effect on skin of partial completion in the reservoir and of deviation of the well from vertical. 2. Calculate the effect on skin of perforations through the casing. 3. Calculate the effect on skin of the pseudo-skin factors, especially turbulent flow.

1 PTPR 6101

INTRODUCTION Because of surface constraints such as dwellings and other structures, topography, surface water, and the increasing demand to minimize environmental impact and as a result of improvements in drilling technology to control well trajectory, it has become more common to drill deviated or directional wells. Usually, these wells penetrate the formation at an angle other than vertical. A skin factor is used to quantify the effect on well productivity of penetrating the formation on an angle other than vertical. Occasionally, a well reaches total depth before all the pay is penetrated due to miscalculation, operational problems or lack of knowledge about a reservoir. Sometimes, only part of the pay is penetrated or perforated to avoid a water leg or to avoid coning gas in an oil well or to avoid coning oil in a gas well. Sometimes, for a variety of other reasons, only a fraction of the available pay is perforated. A skin factor can be calculated to account for partial penetration of the reservoir or for a limited perforation interval within the pay interval. Sometimes, wells are perforated with small diameter holes and/or with holes at low shot density and/or with charges providing only a shallow penetration depth. This is more common with older wells but is sometimes found in newer wells completed on “shoe-string” budgets or having poor completion designs. A perforation pressure loss skin factor can be calculated to evaluate these situations for their effect on well productivity or to optimize the re-perforating of old wells or the perforating of a new well. Finally, there may be present, most commonly in high rate gas wells, a skin factor that accounts for non-laminar, that is, turbulent, flow in the reservoir. This skin is termed a pseudo-skin as it is, unlike all the other skin factors, entirely rate dependent as well as being dependent on reservoir, not well bore, parameters. In condensate or gas reservoirs, a phase-flow or multi-phase flow skin factor may affect well productivity. This is also a pseudo-skin as it is rate dependent and occurs in the reservoir. This module will show you how to analyze the magnitude and effect on production of the skin factors associated with the above drilling and completion practices and reservoir fluid effects.

2

PTPR 6101

OBJECTIVE ONE When you complete this objective you will be able to… Calculate the effect on skin of partial completion in the reservoir and of deviation of the well from vertical.

Learning Activity Complete each of the Learning Activities listed below. 1. Read and study the learning material in this objective. 2. Do the exercise provided. 3. Do the assignment provided. 4. Research outside sources (that is, Internet, library materials, reference texts, and so on) for additional information.

Learning Material INCLINATION OF THE WELL BORE THROUGH THE PAY ZONE As indicated in the introduction, the drilling of non-vertical wells has become much more common in the last 10 to 15 years. This has been spurred by new technology in small diameter and steerable drilling equipment developed for horizontal wells. Horizontal wells require precise control of trajectories to intercept what are often thin pay zones deep in the subsurface and at large horizontal displacements from the surface location. A horizontal reach of 1000 m or more is not unusual for horizontal wells. In addition, more stringent environmental regulations and surface land considerations often dictate that a bottom hole target can only be obtained from a surface location not directly over it. Economics also can play a part as some reservoirs can be more efficiently exploited, with less disturbance of surface land and with better centralization and economy of surface facilities, by drilling multiple directional wells from a single surface pad. Note that the terms slant, inclined, deviated and directional are often, and sometimes incorrectly, used interchangeably to describe non-vertical wells. The term slant well should be reserved for wells that start on an angle at surface. Often, that angle is maintained to the desired bottom hole target resulting in a straight well bore. Typically, these straight slant wells are to be pumped using rod strings which benefit greatly from a straight well bore. 3 PTPR 6101

It is possible to rod pump slant wells that have a moderate deviation from being straight. However, greater than normal tubing and rod wear, as well as maintenance costs, inevitably crop up in such cases. Other times, slant wells are required to reach out to shallower bottom hole targets that do not provide enough depth to build the required curvature of a directionally drilled well. In this case, a slant well can be used to start a directional well and extend the horizontal reach to targets at shallower depths. Slant rigs with a superstructure that can be inclined at the necessary angle are used to drill all slant wells. Slant pump jacks are also available for such wells which, of course, will have slanted well heads installed whether they are rod pumped or not. Inclined and deviated mean about the same thing; that is, a well drilled at an angle other than vertical through the formation. Typically, such wells start out vertical at surface and then build to a planned trajectory that will bring the well bore into the pay zone at the desired bottom hole co-ordinates. It is sometimes possible to drill such wells with an “S-shaped” trajectory to reduce or eliminate the inclination or deviation from vertical of the well bore when it penetrates the pay zone. This may be useful if the well requires a hydraulic fracture stimulation. This topic will be covered in some detail later in this module. Directional wells are wells deliberately drilled to a specific azimuth and which must be precisely controlled to that azimuth in order to hit a target at a precise depth such as a small pinnacle reef, a narrow channel, the boundary/corner of a drilling spacing unit or a small drilling spacing unit. As you will soon discover, the skin factor for an inclined well bore through the pay zone is always negative due to increased exposure of the well bore to the formation. It is appears highly uncommon to justify drilling inclined or deviated wells for this reason but perhaps this should be evaluated more often for its economic justification and practicality especially where hydraulic fracture stimulation is not practical. PARTIAL COMPLETION WITHIN THE PAY ZONE Occasionally, a well reaches total depth before all the pay is penetrated due to drilling miscalculation or operational problems, perforating problems or costs, lack of knowledge about a reservoir, or, deliberately, to avoid a water leg or, sometimes, an oil leg that the gas owners have no rights to. In the latter two cases, there would typically be some stand-off above the top of the liquid leg to the first perforation to achieve isolation. A skin factor, always positive due to a reduced well exposure to the reservoir, can be calculated to account for these situations.

4

PTPR 6101

CALCULATION OF INCLINATION AND PARTIAL COMPLETION SKIN Cinco-Ley et al (Reference 1) provided the tables that follow of skin factors for both inclination of the well bore and partial penetration of the reservoir. In order to use these tables, values for the following variables must be known: θ rw h hw zw

= = = = =

wellbore inclination through pay zone borehole radius, m formation (pay) thickness, m perforated interval length, m elevation of midpoint of perforated interval, m (above bottom of pay)

Reference to Figure 1 will explain these variables further.

Figure 1 Inclination and Partial Completion Skin Variables © Society of Petroleum Engineers (SPE). This material has been copied under license from Access Copyright. Resale or further copying of this material is strictly prohibited.

The following tables from Reference 1 show partial completion skin factors, sp, inclination skin factors, sθ and composite, sθ + p, skin factors for wells that are, respectively, partially completed, inclined, or both.

5 PTPR 6101

Table 1 Inclination and Partial Completion Skin for hD = 100 © Society of Petroleum Engineers (SPE). This material has been copied under license from Access Copyright. Resale or further copying of this material is strictly prohibited.

Table 2 Inclination and Partial Completion Skin for hD = 1000 © Society of Petroleum Engineers (SPE). This material has been copied under license from Access Copyright. Resale or further copying of this material is strictly prohibited

6

PTPR 6101

In order to use Tables 1 and 2, the following dimensionless quantities must be calculated:

h dimensionless formation (pay) thickness rw z = w dimensionless elevation of midpoint of perforated interval rw (above bottom of pay) z = w dimensionless perforated interval length rw

hD = zwD

hwD h w D cosθ nD

= dimensionless well penetration ratio

7 PTPR 6101

Several observations can be drawn from inspection of Tables 1 and 2: 1. The skin factor due to inclination, sθ, is always negative indicating an improvement over a vertical well in the same reservoir. In fact, for higher inclinations through the reservoir, sθ becomes very significantly negative. 2. The skin factor due to partial completion or penetration, sp, is always positive and is usually larger than sθ. In fact, it can often mask or dwarf the skin effect due to true formation damage or stimulation. Referring to module 8, the true formation skin factor ranges from about -5 to +10 or more. FRACTURING INCLINED WELLS

There are complex and not very well understood interactions between tri-axial rock stresses, inclined and/or directional well bores through the reservoir and fracture mechanics. As a result, the hydraulic fracture stimulation of inclined well bores tends to be more difficult, provides less improvement in productivity and has a higher chance of execution failure than hydraulic fracture stimulation of vertical wells. Premature “sanding-off”, which is the failure to pump all of the intended sand tonnage into the hydraulically induced fracture, is a frequent result of a failed stimulation attempt. If a slant, inclined or directional well bore is not aligned with the azimuth of the maximum horizontal rock stress in the target reservoir rock, higher formation breakdown and treating pressures are observed. In addition, multiple fractures can be initiated, drastically reducing effective propped fracture length. In the worst case, only a small fraction of the sand can be injected into the induced hydraulic fractures due to reduced fracture width. Ideally, wells that begin as slant, inclined, deviated or directional wells are drilled with a final trajectory that brings the well bore to vertical through the pay zone. In this case, the wells are no longer inclined or deviated in the target producing formation. This will eliminate the problems encountered hydraulically fracture stimulating wells that are non-vertical. Often, slant, inclined, deviated or directional wells well can not be drilled vertically through the target formation due to cost, lack of enough depth to “build” to vertical or because a straight well bore is desired for artificial lift. In this case, the next best option for wells that require stimulation with a propped hydraulic fracture is to penetrate the formation in the direction of the maximum horizontal rock stress. In this case, perforating is carried out at 180° phasing and oriented to the top and bottom of the inclined casing. This ensures the fracture is initiated along the entire length of the well bore and starts in the direction it will proceed in. 8

PTPR 6101

In wells drilled at an azimuth other than the maximum horizontal rock stress in the target formation, more than one fracture may initiate along the perforated interval. These multiple fractures can cut across the well bore and have only a restricted access to the well bore as a result. Any fractures initiated along the inclined well bore will tend to turn vertical causing narrow and restrictive “pinchpoints”.

Exercise One 1. Determine the partial completion, inclination and composite skin factors for a well with the following characteristics: Well bore deviation from vertical through the pay zone, 30 deg Bore hole diameter, 0.174 m Pay interval thickness, 8.7 m Length of perforated interval, 1.0 m Elevation of mid-point of perforations above bottom of pay interval, 6 m Interpolate arithmetically or graphically and round off your answers to one decimal point.

9 PTPR 6101

OBJECTIVE TWO When you complete this objective you will be able to…

Calculate the effect on skin of perforations through the casing.

Learning Material SKIN DUE TO PERFORATIONS

Often, inflow into a well bore is controlled more by completion efficiency than by reservoir characteristics or artificial lift capability. The three most common types of completion available for both vertical and inclined wells are open-hole, perforated casing and cement and gravel pack. Less commonly, different types of uncemented slotted liners can also be used in either type of well. In an open hole completion, the casing is set at the top of the formation to be produced and is cemented in place. Then the formation is drilled out with a smaller bit than the main hole and left open to the well bore with no casing or cement across it. Typically, carbonate formations are the best candidates for an open hole completion as carbonate rock is often very competent compared to many sandstones and most shales. In the open hole case, no casing, cement or perforations are required. Sometimes, however, open hole sections are shot in an attempt to bypass near-well bore damage. When a formation contains unconsolidated sands that demonstrate a tendency to produce into the well bore a gravel pack completion can be used. An uncemented perforated or slotted liner is set across the formation and the annulus between the liner and the formation is packed with “gravel”. This gravel is usually sand that is larger than the unconsolidated formation sand to prevent the formation sand moving into the well bore. Sometimes a larger annulus is deliberately washed out behind the perforated or slotted liner to make room for a thicker gravel pack. If the liner was perforated, the gravel would also pack into any perforating tunnels that extended into the formation. Gravel pack completion is typically used in the high rate gas wells completed into the soft unconsolidated sands found under the Gulf of Mexico. In Alberta, gravel packs for the typical low oil rate wells that produce sand are not economic. In such cases, sand production is usually handled by lift equipment that can tolerate some sand but maintenance costs are usually high.

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PTPR 6101

In this module, we will concern ourselves primarily with the friction losses characteristic of perforated casing and cement which is by far the most common type of completion. ESTIMATION OF PERFORATION SKIN

Typically, in Alberta’s oil patch, decisions on how to perforate a well are too often not based on science but based on what everyone else is doing in the area, ”… how things have always been done”, and what the service company representative recommends – often to his profit. Often, there is not much more science involved than picking out of the perforating company’s catalog the largest diameter perforating gun that will fit into the well bore or the guns and charges providing the biggest hole diameter, highest density of holes and deepest penetration that you can afford. Usually, larger charges and higher shot density require a larger perforating gun diameter which is usually limited by casing size for casing guns and tubing size for through-tubing guns. The optimum phasing of perforations around the well bore will not be discussed here as it is beyond the scope of this course. Suffice it to say that certain phasings of holes around the well bore are believed to be optimum for the hydraulic fracture stimulation of vertical wells. Optimal phasing of perforations in inclined well bores aligned with the maximum horizontal stress was discussed earlier in this module. In this section, you will be introduced to the procedure developed by Karakas and Tariq (Reference 2), as outlined in References 3 and 4, to estimate the magnitude of perforation skin, sp, as a function of phasing, shot density and well bore size.

11 PTPR 6101

They divided the perforation skin effect into three components as follows to allow for an analytical solution for the calculation of the skin effect: sp = sH + sV + swb where:

sH = the plane flow effect sV = the vertical converging effect swb = the well bore effect

Figure 2 Variables for Perforation Skin Calculation © Prentice-Hall. This material has been copied under license from Access Copyright. Resale or further copying of this material is strictly prohibited.

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PTPR 6101

Table 3 Table of Constants for Perforation Skin Calculation © Prentice-Hall. This material has been copied under license from Access Copyright. Resale or further copying of this material is strictly prohibited.

To calculate the plane flow effect, sH: sH =

ln rw r'w (θ)

where: r'w (θ) = lperf/4 for θ = 0° (or 360°) = aθ (rw + lperf) for θ >0° To calculate the vertical converging effect, sV; sV = 10a hD b-1 rD b where:

a b rD hD

= = = =

a1 log rD + a2 b1 rD + b2 rperf (1 + (kV/kH)0.5 )/ 2hperf hperf (kH/kV)0.5 / lperf

To calculate the well bore effect, swb; swb = c1ec2 rwD

where:

a = a1 log rD + a2 rwD = rw / (lperf + rw )

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Exercise Two 1. A well is perforated at 13 jspf at 90° phasing with charges providing 0.5-in. diameter holes and 16-in. depth of penetration. Formation horizontal and vertical permeability are both 20 mD and well bore diameter is 8-in. Convert all measurements to feet in the equations. a) b) c) d) e)

14

What is the plane flow effect, sH, for this well? What is vertical converging effect, sV, for this well? What is the well bore effect, swb, for this well? What is the perforation skin effect for this well? If the vertical permeability is only 5 mD, what is the perforation skin effect for this well?

PTPR 6101

OBJECTIVE THREE When you complete this objective you will be able to…

Calculate the effect on skin of the pseudo-skin factors, especially turbulent flow.

Learning Material PSEUDO-SKIN EFFECTS

The two main pseudo-skin effects are the turbulent flow pseudo-skin effect and the phase flow pseudo-skin effect. The turbulent flow pseudo-skin effect occurs primarily in high-rate wells, usually gas, and is called the turbulent flow skin effect. Its presence can be seen in a production test on a high rate gas well as in Figure 3.

Figure 3 Determination of Rate-Dependent Pseudo-Skin © Prentice-Hall. This material has been copied under license from Access Copyright. Resale or further copying of this material is strictly prohibited.

Referring to Figure 3, the total apparent skin in a well subject to turbulent flow is seen to increase at a constant rate directly proportional to flow rate. The slope or rate of this increase is the turbulent flow coefficient, D. Recall from Module 8 that the total apparent skin, S', is equal to the sum of the true formation skin, sd, and the sum of the remaining skin factors, including the rate-dependent pseudo-skin effect. In other words: S' = s + Dq where:

s = the sum of all the skin factors excluding the rate-dependent skin effect Dq = the rate-dependent skin effect 15

PTPR 6101

Obviously, none of the other skin factors are subject to the effect of turbulence. This is because they all pre-suppose laminar flow by definition. The turbulent flow coefficient D is a constant that can be estimated from an empirical correlation such as the one suggested by Reference 2: D = where:

γ k h μ rw

hperf

= = = = = =

6 x 10-5 γ k -0.1 h μ rw h 2 perf

gas gravity near-well bore permeability, mD reservoir (net pay) thickness, ft gas viscosity, cP well bore radius, ft perforated interval length, ft

The turbulent flow coefficient D can also be determined by gathering test data at two flow rates and applying the appropriate flow equation to each. This allows determination of the value of the total apparent skin factor S'. It will be seen to differ at the two different flow rates if turbulent flow is significant. Then the two unknowns, s (the sum of all the skin factors other than the rate dependent pseudoskin) and D, the turbulent flow coefficient of the rate-dependent skin, can be determined by simultaneously solving a pair of flow equations – one for each flow rate. The Darcy radial flow equation for steady state flow of gas modified for nonDarcy effects is as follows in oilfield units; qg =

0.703 k g h (p 2 avg - p 2 wfs ) μg, avg z avg T [ln(re /rw ) + s + Dq g ]

mscfd

The above equation can be re-arranged and then solved simultaneously for s and Dqg if data from two separate tests is available. PHASE FLOW SKIN EFFECT

The most common example of phase flow skin effect results from the condensation of gas liquids in the reservoir surrounding a gas well. This is due to the reduction in pressure in the zone near the well bore with production. The resulting liquid condensate reduces the relative permeability to gas flow and causes an additional resistance to flow which is treated as a skin.

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PTPR 6101

Fetkovich (1973) in Reference 4 gave an approximate solution for the magnitude of the condensate skin effect, sc, in metric units as follows: sc = where:

(k - k skin ) ln ( 0.0024 q2st μ Z R'c t / h2 φ π k p Sc r2w) 2 k skin

sc = pseudo-skin due to condensate drop-out in the reservoir k = reservoir permeability, mD kskin = effective permeability to gas in the condensate saturated area of the reservoir qst = flow rate, 103m3/d μ = gas viscosity, cP R'c = m3 of gas accumulation in the reservoir per m3 of total (recombined) gas produced per kPa t = temperature, K h = reservoir thickness, m φ = reservoir porosity, fraction pavg = average reservoir pressure, kPa(a) Sc = condensate liquid saturation to reach mobility, fraction of pore volume rw = well bore radius, m

It can be observed that the phase flow skin factor is primarily dependent upon the reduction in permeability in the reservoir surrounding the well bore and, secondarily, dependent on flow rate. Referring back, it can also be observed that the skin factors for inclination, limited perforation interval and perforation friction depend primarily on geometry and are completely independent of flow rate. There are other ways to estimate the phase flow pseudo-skin effect such as substitution of single-phase flow parameters in flow equations with effective parameters for multi-phase flow. For example, effective total production rate would be the sum of the gas and condensate flow rates. The most accurate way, however, is to use reservoir simulation to model multi-phase flow.

17 PTPR 6101

Exercise Three 1. An oil well was flowed at a rate of 7099 mscfd for 36 hrs with a flowing bottom hole pressure of 1399.6 psia at that time. It was shut-in and the bottom hole pressure recovered to 2030.5 psia. It was flowed again at 9938 mscfd for 24 hrs at which time the flowing bottom hole pressure was 1595.4 psia. The bottom hole pressure again recovered to 2030.5 psia after shut-in. a) Estimate the radius of investigation for each flow period assuming the well is in the centre of a circular reservoir. b) Calculate the turbulence coefficient, D, the skin due to turbulent flow and the normal skin, s, for this well assuming steady-state (infiniteacting) behaviour. c) Estimate the turbulence coefficient using the empirical correlation from Reference 2 and compare the result to b above.

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PTPR 6101

Exercise Three Answers 1. Given: θ = 30o

rw = 0.087

h w =1.0 m

zw = 6 m

Calculate:h D = z wD =

h = 8.7 m

h 8.7 m = =100 rw 0.087 m zw 6m = = 69 rw 0.087 m

h wD cosθ 11.5 i 0.866 = = 0.100 hD 100 z wD 69 = = 0.69 hD 100 From Table 1: For θ w = 30o ,

h wD cos θ = 0.100 hD

At

z wD = 0.8, Sθ + p = 14.185 hD

At

z wD = 0.6, Sθ + p = 13.636 hD

Interpolating (arithmetically or graphically) z At wD = 0.69, sθ + p 13.9 hD

19 PTPR 6101

Exercise Three Answers θ = 90o

1. Given: 4 jspf, lperf = 16

rp = 0.25 in.

rw = 4 in.

k H = 20 mD

k v = 20 mD a. Calculate

sH = =

lm rw ln rw = 1 r w (θ) a θ ( rw + lperf )

ln (0.33) = -0.91 0.726(0.33 + 1.33)

b. Calculate s V = 10a h D b-1 rD b Where h D = h perf (k H /k v )0.5 /lperf = 0.25 (1)0.5 /1.33 = 0.188 rD = rperf (1+(k v /k H )0.5 )/2h perf = 0.0208(1 + (1)0.5 )2 i 0.25=0.0833 b = b1 rD + b 2 = 1.5674 i 0.0833 + 1.6935 = 1.824 a = a1log rb + a 2 = -1.905 i log 0.0833 + 0.1038 = 2.160 Therefore s V = 10a h D b-1 rD b = 102.16 i 0.1880.824 i 0.08331.824 = 144.5 i 0.252 i 0.01075 = 0.39 c. Calculate s wb = c1ec2 rwD Where

rWD = rw /(lperf + rw ) = 0.33/(1.33 + 0.33) = 0.197

Therefore, s wb =0.0019 e

6.155g0.197

= 0.0019 i 3.36 = 0.06 d. Calculate s p = sθ + s v + s wb = -0.91 + 0.39 + 0.06 = -0.46

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PTPR 6101

e. Given Kv = 5 mD h D = h perf (k H /k v )0.5 / lperf = 0.25 (4)0.5 /1.33 = 0.376 rD = rperf (1 + (k v /k H )0.5 ) / 2h perf = 0.0208 (1 + (0.25)0.5 )/2 i 0.25 = 0.0624 b = b1 rD + b 2 = 1.5674 i 0.0624 + 1.6935 = 1.79 a = a1 log rD + a 2 = -1.905 log 0.0624 + 0.1038 = 3.00 Therefore, s V = 10a h D b-1 rD b = 103.00 i 0.3760.79 i 0.06241.79 = 1000 i 0.462 i 0.00697 = 3.22

Exercise Three Answers 1 a. Calculate rinv1 and rinv2 using: ⎛ 2.637 i 10-4 k t ⎞ rinv =2 ⎜ ⎟ φM c t ⎝ ⎠

0.5

For rinv ≤ re Given total fluid compressibility, c t , as follows: c t1 = 5.167 i 10−4 psi

-1

c t2 = 6.862 i 10−4 psi -1 rinv1

⎛ 2.637 i 10-4 i 20 i 24 ⎞ = 2⎜ −4 ⎟ ⎝ 0.15 i 0.0158 i 5.167 i 10 ⎠ = 643 ft

0.5

⎛ ⎞ 2.637 i 10−4 i20 i 36 rinv2 = 2 ⎜ −4 ⎟ ⎝ 0.15 i 0.0146 i 6.862 i 10 ⎠ = 711 ft

0.5

21 PTPR 6101

b. Given: kg = 0.020 D T = 579.6°R

h = 39.37 ft rw = 0.410 ft

p R = 2030.5 psia re = 745 z1 = 0.841

μ1 = 0.0158

q 2 = 9938 mscfd p wf 2 = 1399.6 psia z 2 = 0.856

μ 2 = 0.0146

q1 = 7099 mscfd p wf 1 = 1595.4 psia 2

Assume: Steady state (infinite acting) p approach and re-arranging Darcy radia flow equation: ⎡ ⎛ rinv ⎞ ⎤ 0.703k g h (p R 2 − p wf 2 ) q g ⎢ln ⎜ ⎟ + s + Dq ⎥ = μg z T ⎣ ⎝ rw ⎠ ⎦ For flow (1): 7099 ⎡⎣7.357 + s + Dq ⎤⎦ = 0.0719(4.123 i 106 - 2.544 i 106 ) 52.227 i 103 + 70995 + 50.396 i 106 D = 113.530 i 103 70995 + 50.396 i 106 D = 61.303 i 103 5 + 7099 = 8.64 For flow (2): 9938 ⎡⎣7.458 + 5 + Dq ⎤⎦ = 0.0764(4.123 i 106 - 1.959 i 106 ) 74.118 i 103 + 99385 + 98.764 i 106 D = 165.33 i 103 99385 + 98.764 i 106 D = 91.212 i 103 5 + 9938 D = 9.18 Subtracting (2) from (1) 839 D = 0.54 D = 1.90 i 10-4 /mscfd s = 7.3 c. Using the correlation in references:

D=

6 i 10-5 γk -0.1 h μ rw h 2 perf

for flow (1) D1 =

6 i 10-5 i 0.611 i (20)-0.1g39.37 0.0158 i 0.410 i (39.37)2

= 1.07 i 10-4 /mscfd mw 17.7 use γ = = = 0.611 mw air 28.97 Compare to D = 1.90 i 10-4 /mscfd using the Darcy radial flow equation.

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PTPR 6101

Module Self-Test Directions: • • • •

Answer the following questions. Compare your answers to the enclosed answer key. If you disagree with any of the answers, review learning activities and/or check with your instructor. If no problems arise, continue on to the next objective or next examination.

1. A slant well can be used to a. provide a straight well bore for rod pumping b. provide numerous wells from a single surface lease c. provide a longer horizontal reach to a shallower formation than a deviated well started as a vertical well d. all of the above 2. Which of the following skin factors are always negative? a. b. c. d.

Partial penetration into the formation Inclined well bore through the formation Limited perforation interval None of the above

3. Which of the following skin factors are never negative? a. b. c. d.

Formation damage Turbulent flow Inclined well bore through the formation Both a and b

4. Which of the following is true of the skin factor due to partial completion or partial penetration? a. It is sometimes negative b. It is usually smaller in magnitude than the skin factor due to well bore inclination c. It is always dwarfed by the skin effect due to formation damage or stimulation d. None of the above 5. At increasing inclination of a well bore through the formation, a. b. c. d.

it becomes easier to successfully stimulate by hydraulic fracture true formation damage increases perforation friction increases the well bore inclination skin factor becomes more negative 23

PTPR 6101

6. Ideally, inclined or deviated wells are: a. turned to penetrate the formation vertically so it is easier to rod pump them b. drilled in the direction of minimum horizontal stress so it is easier to hydraulically fracture stimulate them c. turned to penetrate the formation vertically so it is easier to hydraulically fracture stimulate them d. drilled as vertical wells to start so it is easier to rod pump them 7. Wells can be drilled open hole if a. formation sand is too unconsolidated and liable to collapse when running casing b. a gravel pack completion would be too difficult and expensive c. it is less expensive when it is not necessary to perforate the pay zone d. the rock is competent and does not require casing and cement 8. Which of the following perforating gun specifications is the most critical to the successful execution of a hydraulic fracture stimulation carried out at high rate with a viscous gel carrying high sand concentration? a. b. c. d.

Perforating hole diameter Perforating gun diameter Depth of penetration (length of perforation tunnel) Phasing of shots

9. Which is true in regard to perforation skin effect? a. b. c. d.

Large perforation phasing angles reduce perforation skin Long perforation tunnels increase perforation skin High density of perforations increases perforation skin Low vertical permeability increases perforation skin

10. The magnitude of the turbulent flow skin effect a. b. c. d.

is higher in highly deviated gas wells is affected by the diameter of the perforations in the casing is a constant in high rate gas wells is directly proportional to flow rate in high rate gas wells

11. The phase-flow skin effect is a. primarily a function of reduced permeability in the reservoir due to gas liquid condensation b. primarily a function of liquid flow rate in the reservoir c. primarily a function of reservoir permeability d. primarily a function of gas flow rate in the reservoir 24

PTPR 6101

Module Self-Test Answers 1. d 2. b 3. b 4. d 5. d 6. c 7. d 8. a 9. d 10. d 11. a

25 PTPR 6101

Assignment 1. Provide as many reasons as you can why it is frequently necessary to drill wells that are not vertical. 2. Discuss the effect of inclination and of azimuth on the hydraulic fracture stimulation of non-vertical wells and why a vertical well is the best type of well to drill if the formation must be hydraulically fracture stimulated for economic production. 3. Explain how to determine the turbulent flow skin factor from well testing. 4. Describe the cause of the phase-flow skin effect in some gas reservoirs.

26

PTPR 6101

References 1. Cinco-Ley, H., Ramey, H.J., Miller, F.G., Pseudoskin Factors for Partially Penetrating Directionally Drilled Wells, SPE Paper 5589, 1975. 2. Economides, M.J., Hill, A.D., Ehlig-Economides, C., Petroleum Production Systems, Prentice-Hall, 1994. 3. Economides. M.J., Nolte, K.G., Reservoir Stimulation, 2nd Ed. Prentice-Hall, 1989. 4. Fetkovich, M.J., Decline Curve Analysis Using Type Curves, SPE Paper 4629, (1973). 5. ERCB Guide G-3, Gas Well Testing, Fourth Ed., Metric, 1979.

27 PTPR 6101

Appendix Table 4 Conversion Factor Table

28

PTPR 6101

Course Module

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