General Principles of Bottomhole Pressure Tests

1:23:09 PM - 1 -

GENERAL PRINCIPLES

OF

BOTTOM-HOLE PRESSURE TESTS

BHP BALANCE

ACHIEVED OBJECTIVE

BHP TEST

Field Operators

Test Analyst

Proposal

Writer

Prof. Mike Obi Onyekonwu Petroleum Engineering Department University of Port Harcourt, Nigeria.

ACHIEVED OBJECTIVE

2

PREFACE Bottom-hole pressure test is one of the most economical methods of obtaining information required for reservoir management. The test involves measuring the bottom-hole pressure at the sandface under specified rate conditions. The bottom-hole pressure contractors (wireline operators) are responsible for running the test while the reservoir engineers analyze the tests. For the past twelve years, we have been analyzing pressure tests for companies in Nigeria. The analysis involves using sophisticated software and models to obtain reservoir parameters that will not only match obtained pressure, but that are in agreement with known information about the reservoir. From our experience, we observed that more than fifty percent of the errors in BHP analysis could be due to ignorance on the part of the field operators responsible for measurements. Therefore, there is need for everyone involved with BHP test, to understand the purpose of the test and factors that can affect the test. This is what we call the total concept approach to BHP tests and analysis. The total concept approach requires that all parties involved in BHP operations - BHP

proposal writer, BHP contractor, production staff, reservoir engineers and users of

BHP test data - need to understand the principles of BHP tests. This book provides required general information for all the parties involved in BHP test and analysis. Chapter 1 of the book is on general introduction and definitions while Chapter 2 is on field practices. The field practices cover gauges, running procedure and procedure for data quality check. Chapter 3 is on basis for BHP analysis. Subjects such as units, flow phases and flow regimes are discussed. Chapter 4 is on analysis models while Chapter 5 dwells on skin factor and concepts. Chapter 6 is on analysis methods. The conventional and type-curve methods are covered. Chapter 7 is on sensitivity on factors affecting BHP tests. Factors such as leaks, rate variation and gauge problems are considered. Chapter 8 is on field cases. Both good and bad tests are presented. The bad tests could be a result of man-made problems and problems resulting from malfunction of equipment. For the bad ones, solutions are proffered on how to avoid future mistakes. Chapter 9, on the theory of welltesting, is for the academic minded group that is not only interested in how, but why. The fundamental equation, diffusivity equation, is derived and solutions presented for different conditions. Concepts like superposition are also discussed. Finally, there is analogy between liquid and gas flow equations. Chapter 10 is on bottom-hole pressure analysis in horizontal wells. Several equations are presented and procedures for using the equations outlined. Finally some field cases on horizontal wells are presented. Materials for this book were taken from manuals that we have used successfully in teaching three different short courses – “Practical Aspects of Bottom-Hole Pressure

Tests, General Principles of Bottom-Hole Pressure Tests and Theory of Welltesting.” These courses were organized for participants with different backgrounds. We are therefore certain that anyone who needs some knowledge on bottom-hole pressure tests will find this book useful. In this era of multidisciplinary approach to problems, this book will give reservoir and non-reservoir engineers a good working knowledge of bottom-hole pressure tests.

Mike Onyekonwu

3

ACKNOWLEDGEMENT For me, writing a book is simply a way of expressing my understanding of a subject.

We therefore owe our teachers, students and collegues a lot in contributing to our

knowledge on the subject. We are grateful to Late Professor H. J. Ramey (Jr) of

Stanford University, California – the father of Well Testing – for teaching us a

difficult subject in a simple way. We are also grateful to Steve Rice and Jim Willetts

of Shell Petroleum Development Company who challenged us and helped us acquire

much needed practical experience on this subject. Tom McAllister who took over

from Steve and Jim has continued to encourage us. We appreciate his contributions to

our continued growth. We are also particularly grateful to petroleum and production

staff of Shell Petroleum Development Company, Warri, for their support and use of

our services.

I also thank my good friend, Dr G. A. Okpobiri, who was my co-consultant when we

taught the first Well Testing course. I appreciate the immense contributions made by

Laser Engineering Consultants staff and particularly Mr. Obi Ekeh and Mr. Cornel

Udoh for being part of the analysis team. Mrs. Margaret Anele and Mr. Sam Jumbo

of Laser Engineering did a good job at typesetting the book. I am equally indebted to

my students both in the oil industry and University of Port Harcourt for allowing us

use them as the guinea pigs. Their contributions are indeed invaluable.

I also thank Professor G. K. Falade who taught us the first course on “Well Testing”

at University of Ibadan. I am equally indebted to Professor Chi Ikoku for creating a

nice working environment at the University of Port Harcourt and therefore made it

possible for me to continue practicing engineering. Finally, I thank my dear wife,

Tina, for always standing by me. May God reward everyone. This is actually our

book.

Mike Onyekonwu

4

ABOUT THE AUTHOR

Include photgraph!

Dr Mike Onyekonwu has a B.Sc. (First Class Honours) in Petroleum Engineering

from University of Ibadan, Nigeria. He also has an MS and Ph.D. degrees in

Petroleum Engineering from Stanford University.

Dr Onyekonwu is a Senior Lecturer and former Head of Petroleum Engineering

Department, University of Port Harcourt. He is a member of University Senate. Dr

Onyekonwu is the founder and Managing Consultant of Laser Engineering

Consultants, Nigeria. Dr Onyekonwu worked for Shell Petroleum Development

Company Nigeria and Stanford University Petroleum Research Institute, California.

Dr Onyekonwu is a registered engineer and a member of different professional

bodies. He consults for Shell, Mobil, Elf, NNPC, Agip and other oil operating and

service companies. His area of specialization includes welltest analysis, reservoir

simulation, recovery methods and computer applications.

5

TABLE OF CONTENTS

PAGE

PREFACE ............................................................................................................................ ii COURSE ADVERTISEMENT ............................................................................................ iii

1 INTRODUCTION .......................................................................................... 1 1.1 Objectives of BHP Survey .. .............................................................................. 1 1.2 Uses of BHP Derived Information..................................................................... 1 1.3 Common Types of BHP Tests ........................................................................... 2 1.4 Ideal Conditions and Information Derived from Test........................................ 3 1.5 Importance of Sticking to Ideal Condition for Test ........................................... 4 1.6 Uses of Information Derived from BHP Tests................................................... 6 1.7 Well Test Equipment ......................................................................................... 7 1.8 Electronic Gauges and Problems ...................................................................... 11 1.9 Flowing Gradient/Buildup/Static Gradient(Fg/Bu/Sg) Survey.......................... 11 1.10 Flowing Gradient/Buildup/Static Gradient Survey Proposal............................. 19 1.11 Useful Hints on Proper Testing of Wells ........................................................... 23 1.12 Practical Hints.................................................................................................... 23 1.13 Gauge Quality Check Procedure........................................................................ 27 1.14 Roles of Field Staff In BHP Survey................................................................... 28 1.14.1 Roles of Production Staff...................................................................... 32 1.14.2 Roles of BHP Contractor Staff ............................................................. 33

2. BASIS OF ANALYZING BOTTOM HOLE PRESSURE TESTS ............................ 34 2.1 Flow Phases ...................................................................................................... 34 2.2 Features of Different Phases ............................................................................. 35

3. ANALYSIS OF BOTTOM HOLE PRESSURE TESTS ........................................... 46

4. EFFECT OF CERTAIN FACTORS ON ANALYSIS OF SIMULATED DATA ..... 64 4.1 Analysis of Ideal BHP Data .............................................................................. 64 4.2 Effect of Gauge Accuracy and Datum Correction ............................................ 73 4.3 Effect of Noise .................................................................................................. 78 4.4 Effect of Gauge Sensitivity ............................................................................... 98 4.5 Effect of Rate Variation .................................................................................... 98 4.6 Effects of Leaks ................................................................................................ 112 4.7 Effect of Interference/Leak ............................................................................... 112

5. FIELD CASES ........................................................................................................... 117 5.1 Good Test .......................................................................................................... 117 5.2 Effect of Gauge Movement ............................................................................... 117 5.3 Effect of Gauges Off-Depth .............................................................................. 125 5.4 Effect of Reporting Wrong Rates ..................................................................... 125 5.5 Effect of Ineffective Shut-in/Well not flowing before Shut-in .......................... 131 5.6 Effect of Leak ................................................................................................... 132 5.7 Effect of Gauge Oscillations/Sensitivity Problems ........................................... 135 5.8 Effect of Gas Phase Segregation ....................................................................... 140 5.9 Effect of Liquid Interface Movement ............................................................... 144 5.10 Effect of Gaslift ................................................................................................ 144 5.11 Effect of Short Buildup or Flow Period ............................................................ 144

6. CLASS DISCUSSION ............................................................................................... 154

7. FLOWING AND STATIC GRADIENT SURVEYS ................................................. 165 8. SKIN FACTOR 8.1 What is Skin Factor............................................................................................ 172

6

8.2 Causes of Skin ................................................................................................... 174 8.3 Classification of Pseudoskins ............................................................................ 174 8.4 Calculation of Pseudoskins ................................................................................ 176 8.5 Relationship Between Total Skin and Pseudoskins ........................................... 180 8.6 Pressure Change Due to Skin............................................................................. 181 8.7 Relationship Between Skin and WIQI............................................................... 181 Exercises ............................................................................................................ 182

APPENDIX A: TYPICAL PROPOSAL .............................................................................. 184

APPENDIX B: PROPOSAL WITH WRONG INSTRUCTION ......................................... 192

APPENDIX C: MISCELLANEOUS MATERIAL...............................................................

7

1. INTRODUCTION

Oil well tests are made for numerous reasons and the type of test required depends on the objective of the test. Common well tests include: (a) Potential test (b) Gas-oil ratio test (c) Productivity test (d) Bottom-hole pressure test Potential test involves measurement of the amount of oil and gas a well produces during a given period (normally 24 hours or less) under certain conditions fixed by regulatory bodies. The information obtained from these tests is used in assigning a producing allowable of the well. The gas-oil ratio test is made to determine the volume of gas produced per barrel of oil so as to ascertain whether or not a well is producing gas in excess of permissible limit. Bottom-hole pressure test involves measurement of sandface pressure and flowrate variation with time. Such tests are quite economical to run and they yield valuable information about the reservoir characteristics and well characteristics. Hence, bottom-hole pressure tests are usually referred (Earlougher, 1977; 1982) to as welltests. Productivity tests are made on oil wells and include both the potential test and the bottom-hole pressure (BHP) test. The purpose of this test is to determine the effects of different flow rates on the pressure within the producing zone of the well and thereby establish producing characteristics of the producing formation. In this manner, the maximum potential rate of flow can be calculated without risking possible damage to the well which might occur if the well were produced at its maximum possible flow rate. In this book, the term welltest will be used for bottom-hole pressure test unless otherwise stated. In this chapter, the purpose of well testing, types of well tests and well test equipment are discussed. In addition, other practical aspects of BHP tests such as test procedure and equipment problems are discussed.

1.1 OBJECTIVES OF BHP SURVEY

Bottom-hole pressure tests are conducted to obtain data that can be used for the following purposes:

• Determine Well Parameters - Skin - Productivity Index - Wellbore storage constant - Fluid distribution in wellbore - Flowing pressures in wellbore - Static gradients

• Determine Reservoir Parameters - Average pressure in the drainage area - Permeability - Distance to boundaries

8

- Vertical/Horizontal permeability - Gas/oil contacts

• Determine Dynamic Influence of other Wells/Aquifer

• Assess Changes Since Previous Survey - Changes in datum pressure - Changes to damage skin - Changes in drainage area (from a drawdown test) - Confirm boundaries 1.2 USES OF BHP DERIVED INFORMATION

Results obtained from BHP tests are used for the following purposes:

* Reservoir Surveillance * Determination of Stimulation Candidates * Gaslift Optimisation * Input for Reservoir Simulation * Material Balance Calculation Examples of benefits from BHP test compiled by a client are given in Table 1.1. The benefits were realised by using results from BHP tests for good well and reservoir surveillance. 1.3 COMMON TYPES OF BOTTOM-HOLE PRESSURE (BHP) TESTS

Common types of bottom-hole pressure tests include the following: (a) Drawdown test (b) Injectivity test (c) Buildup test (d) Falloff test (e) Interference/pulse tests (f) Others The definitions of the tests and the rate and pressure profiles during the test are as follows:

Table 1.1: Benefits from BHP Surveys

ACTIVITIES SAVINGS ($ million)

Well Surveillance

Stimulation (abort 5 jobs, contribute to finding 5 more) Gaslift Optimization (10% improvement of target at $1/bbl)

0.8 1.0

Reservoir Surveillance

Sand F4.0/F4.1X Production (3 Mbopd) 1.0 Sand-X Block (new well cancelled) 3.0 Dump Creek (10% of the 6 fewer wells required) 3.0 Well -11 (sidetrack raise trajectory) 1.0 Sand D5.0X Development (horizontal well changed to recompletion) 3.0 (10% of 8 well campaign) 4.0 Total 16.8

9

1. Drawdown Test: Involves measuring the variation of sandface pressure with time while the well is flowing. For a drawdown test, the well must have been shut in to attain average pressure before production commences for the test. The rate and pressure profiles during drawdown test are in Fig 1.1.

Pwf

0 timetime

q

0

Fig. 1.1: Rate and Pressure Profiles During Drawdown Test

2. Injectivity Test: This is the counterpart of a drawdown test and involves

measuring the variation of sandface pressure with time while fluid is being injected into the well. The rate and pressure profiles during drawdown test are in Fig 1.2.

q Pwf

0 time

Injection (-ve q)

time

Fig. 1.2: Rate and Pressure Profiles During Injectivity Test

3. Buildup Test: Involves measuring the variation of sandface pressure with time while well is shut-in. The well must have flowed before shut-in. Figure 1.3 shows the rate and pressure profiles during the flow and buildup periods. Buildup tests are more common and will be the main subject of our discussion.

Pw

0 time 0 time

q

Shut-in Time

0

Drawdown

Buildup

Drawdown

Buildup

Fig. 1.3: Rate and Pressure Profiles During Drawdown and Buildup

4. Falloff Test: This is the counterpart of buildup test and it involves measuring the

variation of sandface pressure with time while well is shut-in. In this case, some fluid must have been injected into the well before shutting. Figure 1.4 shows the rate and pressure profiles during the injection and falloff periods.

10

q Pw

0 time Injection

time

Fig. 1.4: Rate and Pressure Profiles During Injectivity and Falloff Test

5. Interference Test: Unlike the first four tests (drawdown, injectivity, buildup,

falloff) which are tests involving only one well (single well tests), the interference test involves the use of more than one well (multiple well test). During interference tests, pressure changes due to production or injection or shut-in at an active well is monitored at an observation well. The active well and the observation well are shown in Fig 1.5. Only one active well is required, but there could be more than one observation well.

Interference tests are primarily used to establish sand continuity between the active and observation wells. In situation where more than one observation well is used, interference test can be used to find (Ramey and ) maximum and minimum permeability and their directions.

Active Well Observation Well

Sand continuityGauge

q > 0

q = 0

q < 0

q = 0

Fig. 1.5: Active and Observation Wells in an Interference Test

1.4 IDEAL CONDITIONS AND INFORMATION DERIVED FROM TEST

If possible, BHP tests should be run under the stated ideal conditions as this makes interpreting such tests easy. The ideal conditions for running different BHP tests and information that can be obtained from the tests are given in Table 1.2 Table 1.2: Types of Well Tests and Derivable Information

Type of Test Ideal Conditions for Test Information Derived from Test

Drawdown 1. Constant rate production 2. Well shut-in long enough

before test to attain uniform pressure in reservoir.

1. Permeability 2. Skin factor 3. Reservoir drainage volume 4. Flow efficiency 5. Distance to linear no-flow

barrier

Buildup 1. Constant rate production before shut-in.

1. Permeability 2. Skin factor

Injectivity Falloff

Shut-in

11

3. Flow efficiency 4. Average Pressure 5. Distance to linear no-flow

barrier

Interference 1. Constant rate production or injection at the active well.

1. Permeability 2. Storativity 3. Anisotropic permeability

values and orientation

4. Sand continuity

1.5 IMPORTANCE OF STICKING TO IDEAL CONDITION FOR TEST

In this section, we shall discuss the importance of sticking to the ideal condition for any test. Two factors considered are constant rate production and not shutting well for long period before a drawdown test. (1) Constant Rate Production: Rate variation makes tests difficult to analyse because effect of rate changes last until well is shut-in and builds up to average pressure. Rate changes are modelled using the concept superposition illustrated in Fig. 1.6. Figure 1.6 shows that a rate which occurred at where at time, t, will continue to affect

pressure response until time, t + ∆t. In a layman’s language, wells do not forget rate changes that occurred in them unless they are shut in to build up to average pressure.

q1

q2

q1

- q1

q2+

t

≡

t + ∆t∆t ∆t

Fig. 1.6: Effect Rate Variation

Causes of Rate Variation

Some tests like the potential tests are designed such that the rates in the wells are varied. Such rate variations create no problem during analysis because the rates are measured and therefore can be considered during analysis. Situations that create problems include cases where the rates are varied and not measured. Such situations may occur under the following conditions: (a) Partly closing wing value to lower tools (b) Slow shutting of well at end of flowtest (c) Not allowing for rate stabilization. Surface indications of well stabilization

include (i) Constant wellhead flowing pressure (ii) Constant gas production rate (iii) Constant fluid production rate.

12

(2) Long shut-in Requirement: The rate and pressure profiles for wells shut-in for long and short times are shown in Fig. 1.7 and Fig. 1.8. Short Shut-in Period

q2 = 0

q3

q1

Rate

Time

Time

Pressure

Fig. 1.7: Rate and Pressure Variation During Short Shut-in Period

Long Shut-in Period

q2 = 0

q3q1

Rate

Time

Time

Pressure

Fig. 1.8: Rate and Pressure Variation During Long Shut-in Period

For the case of short shut-in period, the well did not build up to the average pressure before the drawdown test was started. In this case, analysis of the test will involve using three rates. However, for the case of long shut-in time before the drawdown, the well reached the average pressure during the buildup. Hence, analysis of the drawdown will simply involve a single rate. A single rate test is usually simpler to analyse than a three rate test. 1.6. FLOWING GRADIENT (FG) AND STATIC GRADIENT (SG) SURVEYS

We regard the flowing gradient and static gradient surveys as auxiliary surveys that complement bottom-hole pressure tests. These tests and their uses are described. Flowing Gradient (FG): The flowing gradient survey involves measuring flowing pressure at different depths in the well while the well is flowing. Results from this test are used for gaslift optimization. Figures 1.9 shows cases where the flowing

13

pressures measured along the traverse of the well reveal examples of optimized and non-optimized gaslifting.

Depth

Fig. 1.9: Optimized and non optimized gaslift.

In the non-optimized case, we may have a “U” tube effect in which injected gaslift gas is simply re-circulated giving rise to lower flowing pressure gradient in the upper part of the tubing. This is shown in Fig.1.10.

Gas Gas

Fig. 1.10: Non-Optimized Gaslift.

Flowing gradient surveys also provide flowing pressures which can be used to determine the appropriate correlation for modelling flow along the wellbore. Such models are used for gaslift optimization. In all cases during the flowing gradient survey, the depth where pressure was measured is important. Static Gradient Survey: In this case, we measure pressure at different depths in the well while the well is shut in. This implies that it can be run in well that has not been flowing. Usually, before a static gradient survey is run, the well must have been shut in for sufficient time to allow the pressure to stabilize. At every static gradient stop along well, the gauges must be left for a minimum of 15 minutes so that pressures will be steady. The static gradient survey is used to determine the fluid distribution in the wellbore. This information is required for pressure correction and locating the depth for the operating gaslift valves. In a well that is closed in, the static gradient survey is a good source of pressure data that can be used in calculating the datum pressure with no oil deferred. The basis for determining fluid gradients using static gradient survey is that fluid gradients depend on the density of the fluid. Therefore, pressure gradient in the gas zone is small because gas has the smallest density of the wellbore fluids. Figure 1.11 shows fluid gradients determined using results from static gradient survey in a wellbore that contains gas, oil and water.

Pressure Pressure

Depth

Optimized

Non-optimized

14

Water (0.433 psi/ft)

Pressure, psi

Gas (≈ 0.07 psi/ft)

Oil (≈ 0.35 psi/ft)Depth, ft

Fig. 1.11: Wellbore Fluid Distribution Determined with Static Gradient Survey

With the static gradient survey, we can determine the gas-oil contact which is important in selecting the depth of the gaslift operating valve. An operating valve that is above the gas-oil contact as shown in Fig. 1.10, will result to a non-optimized gaslift operation. To ensure that correct wellbore fluid contacts are determined using static gradient survey, a minimum of two static survey stops must be taken in each phase as the gauge moves through the fluid phases. Generally, it is recommended that in gaslifted wells, there should be two stops around the region where the gaslift mandrels are installed. This helps determine fluid contacts (if any) around the gaslift mandrels. In bottom-hole pressure (BHP) tests, it is required that we measure pressure at the sandface (mid-perforation). However, in some situations it is not possible for the gauge to be lowered to the sandface. In such situations, the static gradient survey provides the fluid gradient required for obtaining the pressure at the mid perforation. The equation for calculating the pressure at mid perforation is as follows:

Pmid perf. = Pgauge + (Fluid Gradient x ∆z) 1.1

Where Pmid perf = Pressure at the mid perforation Pgauge = Pressure recorded by gauge at the last stop

∆z = Distance between mid perforation and last gauge stop

Fluid Gradient = Wellbore fluid gradient in the interval ∆z A graphical interpretation of Eq. 1.1 is shown in Fig. 1.12 for a case where the last gauge depth is the “XN” nipple. From this, it is obvious that we need to be careful in reporting the gauge depth. An error of 5 ft with water near the perforations (gradient of 0.433 psi/ft) means a 2.1 psi error and this is more than the absolute accuracy of the crystal gauges.

15

Depth

Pressure Fig.1.12: Extrapolating Pressure to top of Perforation

1.7. COMPLETE BOTTOM-HOLE PRESSURE TESTS PROFILES

A complete buildup or drawdown survey requires that both flowing and static gradient surveys should also be taken. Typical pressure profiles for such tests are as follows:

Flowing Gradient/Buildup/Static Gradient (FG/BU/SG): Typical pressure profile for this test is shown in Fig. 1.13.

Time Fig 1.13: Pressure Profile for FG/BU/SG Survey

The sequence of events performed during the test that resulted to the pressure profile shown in Fig 1.13 are as follows: Event Description A – B Gauge in lubricator reading atmospheric pressure as there is no

communication yet with the well B – C Increasing pressure due to running in hole C – D Flowing gradient stops D – E Running hole to final survey depth E – F Flowtest with gauge at final survey depth F – G Buildup period G – H Static Stops near the final survey depth H – I Pulling gauge out of hole I – J Static stops in the upper part of tubing for liquid level detection J – K Pulling gauge out of hole to lubricator

XN

Water

Oil

Extrapolation

to mid perf

Increasing pressure

A B

C

D

E F

G

H

I

J

K

Flowing Gradient Buildup Static Gradient

16

Although events in Fig 1.13 are typical, but there may be variations. For example, at the end of buildup, the gauge may be pulled out about 200 ft and then moved down about 200 ft to original survey depth. The profile for this case is shown in Fig. 1.14 with events GH and HI representing the pull out and run back respectively. This could be used for checking the accuracy of depth measurement as pressure at the same depth in a well that has been shut in for sufficiently long time must always be the same. Another variation is that of where well is shut in while the gauge is run in hole. With the gauge on the bottom, the well is then opened for a flowtest and then shut-in again for a buildup. A typical profile for this case is shown in Fig.1.15.

Fig 1.14: Pressure Profile for another FG/BU/SG Survey

Fig 1.15: Pressure Profile for Buildup Survey

Drawdown

Buildup

Gradient stops

Time

Increasing pressure

Running in hole

Pulling out of hole

Increasing pressure

A B

C

D

E F

G

I

J

Flowing Gradient Buildup Static Gradient

Time

H

17

If this test is properly run, there will be the advantage of obtaining both buildup and drawdown data that can be analyzed. Also as the well is shut in while the gauge is run in hole, the problem of lowering gauge especially in high flowrate wells will be eliminated. The problem with this type of test is that the duration of the shut-in while the gauge is run in hole may not be adequate for the drawdown and buildup tests to be easily analyzed without using superposition. That is, the duration may be too short for the system to attain average pressure, a condition required prior to good drawdown test. Static Gradient/Drawdown/Flowing Gradient (SG/DD/FG): This is the complete test sequence in a situation where the well is just programmed for a drawdown. Typical pressure profile for the tests is shown in Fig. 1.16.

Fig 1.16: Pressure Profile for SG/DD/FG Survey

1.8 USES OF INFORMATION DERIVED FROM BHP TESTS

Most petroleum engineers already know how parameters derived from BHP tests are used. To our readers who are non-petroleum engineers, this section will help them understand the importance of some parameters derived from BHP tests. A. Permeability k: This is a measures of the ability of a formation allow fluid flow

through it. Permeability is one of the parameters required for rate prediction as shown in the following equations used for rate prediction:

Linear: q (STB/D) = 1.127 x 10 3− KA p

B L

∆µ

Radial: q (STB/D) = 7.08 x 10 Kh p

Inr

r

-3

e

w

∆

Bµ

B. Skin Factor: A measure of the efficiency of drilling and completion practices

used. The skin factor can be used in calculating additional pressure drop around the wellbore caused by drilling and completion practices. Skin factor is discussed in detail in Chapter ????

Static Gradient Drawdown Flowing Gradient

Time

Increasing pressure

18

The pressure drop may be caused by the following: 1. Alteration of permeability around the wellbore caused by invarion of drilling

fluid, dispersion of clay, mud cake and cement, acidizing etc. In the case of lower permeability around the wellbore, the skin in this case can be likened to the extra fuel spent in driving through a bad road.

2. Partial well completion as shown in Fig 1.17.

Fully completed Partial compleion

Fig. 1.17: Flow Streamlines in Fully and Partially Completed Wells

In the case of a partially completed well, the skin could be likened to the extra energy lost at the door when many people want to go out (assuming there is fire outbreak in the room) of a room at the same time. The pressure drop due to skin is wasted because it does not contribute to the useful drawdown. The skin simply causes an additional pressure drop at the well as shown in Fig. 1.18.

Pw f if no sk in

Pw f if there is sk in

∆∆∆∆p s

rw re

usefu l

d rawdown

Fig. 1.18: Pressure Drop due to Skin

Pressure drop due to skin, ∆pskin , and efficiency are related as follows:

Flow Efficiency (FE) = P

P

- P - p

- Pwf skin

wf

∆

Note that the skin and permeability are determined by the amount of pressure rise and the rate at which pressure rises with time. This is shown in Fig. 1.19. This implies that if pressure does not rise, there will be no pressure rise and skin and permeability cannot be obtained from the test.

19

Time

PressureSkin

permeability

P*

Fig. 1.19: Pressure Rise During Buildup

C. Reservoir Drainage Volume: This is the volume of the reservoir drained by test well. Drainage volume is required in choosing adequate well spacing and reservoir management. Note that wells drain reservoir volumes in proportion to their rate. This is shown in Fig 1.20.

2 q.

q.

q.

Fig. 1.20: Relationship between Drainage Area and Rate.

D. Porosity φφφφ: A measure of void spaces in the reservoir. Porosity can be obtained from interference test. Volumetric calculation of initial oil-in-place requires porosity as an input parameter. This is shown in the equation:

N = 7758 A Soiφ

Boi.

E. Average Pressure: This is a measure of depletion as the amount of fluid in reservoir is related to average pressure in the reservoir. Average pressure is used in material balance calculations.

20

2.0 FIELD PRACTICES Over the years, emphasis has been on developing sophisticated models for BHP analysis. However, Fig 2.1 show that achieving the objective of BHP test requires that the parties involved – the proposal (programme) writer, field staff (BHP contractors and production staff), test analyst and test result users – must play their roles correctly.

Fig. 2.1: BHP Balance

In this chapter, we shall discuss basic BHP equipment, procedures for running BHP tests and proposals. We shall also highlight roles of production staff and BHP contractor during BHP surveys. 2.1 WELL TEST EQUIPMENT

The main equipment used for bottom-hole pressure surveys is as follows: (1) Pressure recorders (2) Lubricator (Wireline BOP) (3) Wireline unit (4) Christmas tree with hydraulically operated value Figure 2.2 taken from Dake (1978) is a schematic of the equipment while Figure 2.3 shows detail of some of the equipment. Most of the equipment are also used in wireline operations and will not be discussed in this test. We shall only discuss the pressure gauges, which are the most important equipment for bottom-hole pressure tests. Fig 2.2 and fig 2.3 here???

Leave a page for Figure 2.3: Wireline Surface Equipment

(Example of an Arrangement) 2.1.1. Pressure Gauges

Different types of gauges are used for measuring bottom-hole pressure. The sensitivity and the accuracy of the gauges vary. The accuracy of a gauge is principally concerned with systematic errors, often attributed to the calibration of the gauge. For example, if the accuracy of a gauge is 5 psi and the gauge reads 1000 psi, this implies that the correct readings lie in the range (1000 - 5) psi to (1000 + 5) psi.

BHP BALANCE

Field Operators

Proposal Writer

Test Analyst And Users

ACHIEVED

OBJECTIVE

21

The sensitivity or resolution of a gauge is described as the smallest pressure change

that can be reliably measured by the gauges. Table 2.1 shows bottom-hole pressure

measured in a Niger Delta well with an insensitive gauge. The constant pressure

readings after shut-in time of 3 minutes is not necessarily due to stabilization, but the

gauge could not “discern” the pressure changes with time.

Table 2.1: Buildup Data from Niger Delta Well

Shut-in Time, min Shut-in Pressure, psi

1 3488

2 3531

3 3539

5 3539

10 3539

20 3539

30 3539

40 3539

50 3539

60 3539

90 3539

120 3539

The types of gauges, their principle of operation, accuracy and sensitivity are given in Table 2.2. Table 2.2: Types of Gauges and Operation Principles

Type of Gauge Principle of Operation Accuracy Sensitivity

Amerada Strain Gauge Quartz Crystal (Electronic)

Bourdon tube (Mechanical) Change in resistivity Change in frequency

± 0.2% FSD

± 0.05% FSD

± 0.035% R

0.05% FSD 0.0025% FSD 0.0001% FSD

FSD = Full Scale Deflection, e.g. 5000 or 10000 psi R = Reading, i.e. the measured pressure

The implications of information in Table 2.2 for a 5,000 psi and 10,000 psi rated

gauges are shown in Table 2.3.

Table 2.3: Sensitivity and Accuracy of 5000 psi and 10000 psi Rated Gauges

Type of Gauge FSD = 5000 psi FSD = 10,000 psi

Accuracy Sensitivity Accuracy Sensitivity

Amerada 10 psi 2.5 psi 20 psi 5.0 psi Strain Gauge 2.5 psi 0.125 psi 5.0 psi 0.25 psi Quartz Crystal (Electronic)

1.75 psi 0.005 psi 3.5 psi 0.01 psi

For the electronic gauge, the calculated sensitivity is the maximum because we assumed that the measured pressure is equal to the Full Scale Deflection (FSD). Some deduction from Tables 2.2 and 2.3 are as follows:

22

1. The electronic gauges are more sensitive and accurate than the strain gauge

while the strain gauge is more sensitive and accurate than the Amerada gauge. 2. If a 5000 psi rated gauge can do the job, do not use a 10,000 psi rated gauge

because the 10000 psi rated gauge has lower accuracy and sensitivity. This deduction is more relevant when Amerada gauges are used.

Ideally, we recommend that electronic gauges be used. However, it costs more than other gauges, but it is worth it. 2.1.1.1 Amerada Gauges

Until 1994 about 80% of bottom-hole pressure tests in Nigeria were run with Amerada gauges. Now many companies do not use Amerada gauges. However, for historical reasons, we need to discuss the Amerada gauge because it clearly shows the components of any gauge: a clock, pressure sensor and recorder. Figure 2.4. taken from Dake (1978) is a schematic of the Amerada gauge. The continuous trace of pressure versus time is made by the contact of a stylus with a chart that has been specially treated on one side to permit the stylus movement to be permanently recorded. The chart is held in a cylindrical chart holder, which in turn is connected to a clock that drives the holder in the vertical direction. The stylus is connected to a bourdon tube and is constrained to record pressures in the perpendicular direction to the movement of the chart holder. The combined movement is such that, on removing the chart from the holder after the survey, a continuous trace of pressure versus time is obtained as shown in Fig. 2.4b, for a typical pressure buildup survey. Fig 2.4 here ?????? 2.1.1.2 Electronic Gauges and Problems

For BHP surveys, the electronic gauges have now replaced the Amerada gauges. The electronic gauges are more sensitive and accurate, but they are also more delicate and require regular calibrations. Figures 2.5 to 2.12 show pressures measured with electronic gauges and problems associated with the measurements. The problems include gauge “shifts,” vibrations, synchronization, failure, etc. Discussion of some of the measurements and problems are as follows:

a) Good Measurement: Figure 2.5 shows good pressure measurements from the lower and upper gauges taken with electronic gauges during a buildup test in a well in the Niger Delta region. The bottom graph in Fig. 2.5 is the pressure difference between the lower and upper gauges. The implications of the pressure difference will be discussed in the section on quality control of measured pressure.

23

Fig. 2.5: Good Measurements with Electronic Lower and Upper Gauges

b) Pressure Shift: Pressure shifts are drastic changes in pressure not cause by pressure changes in the environment of measurement. Figure 2.6 shows an example of pressure shift in the lower gauge at about 20.7 hours. Pressure shifts are common with electronic gauges and are mainly caused by change in the sampling frequency of the gauges.

In most cases, the pressure shifts are less than 2 psi, but such pressure shifts cause distortion of the pressure derivative and this could lead to misinterpretation of the test.

To minimize the effect of pressure shifts, it is recommended that sampling frequencies of should not be changed during the critical periods of the test. For tests in high permeability formation, the critical periods are 30 minutes prior to a rate change and 1hour after the rate change. Generally, the sampling frequency of the gauge should not be changed during the transient state phase.

24

Fig. 2.6: Pressure Shift in the Lower Gauge

c) Pressure Vibration: This is erratic variation in pressure readings, which are

caused by known and unknown factors. Figure 2.7 shows two situations where pressure variations occurred during buildup tests. In the situation shown in Fig. 2.7, the variations occurred in the two gauges and we suspect that the variations were caused by vibration or local phenomena in the wellbore. We believe that it is not a reservoir response.

Pressure vibration could completely mar a test if it occurred during the critical periods of the test. This will be discussed in Chapter ? In some of the wells, pressure vibration always occurred irrespective of the time of test and gauges used for the test. This implies that some unknown factors may be responsible for the vibration.

Fig 2.7 here ????

Fig. 2.7: Pressure Vibration in the Lower and Upper Gauges

d) Gauge Sensitivity: Although electronic gauges are sensitive, but we have seen cases where electronic gauges could not precisely “discern” the pressure correctly. Figure 2.8 shows a case where the upper gauge had sensitivity problem while Fig 2.9 shows case where both gauges had the problem. Sensitivity problem is purely a gauge problem and could be remedied by calibration.

Put Fig 2.8 and 2.9 here here ????? d) Gauge Synchronization: It is recommended that two gauges be run in tandem

during any test. Originally, this procedure was adopted because the second gauge will serve as a backup if one of the gauges failed. In addition to this, pressure

25

readings with two gauges are used for quality check and also detecting wellbore phenomenon such as phase segregation effects.

Gauge synchronization simply means that clocks used in both gauges must give same readings every time. Figure 2.10 shows pressure readings when the clocks are not synchronized. Same events are recorded to have occurred at different times by the gauges. Pressures in Fig. 2.10 are from a surging well and the “peaks” and “lows” of the surge occur at different times. This is due to the non-synchronization of the times and this will affect the results of the quality check. Unsynchronized pressure readings could be corrected with welltest analysis software.

Fig. 2.10: Unsynchronized Pressure Readings

e) General Gauge Failure: A gauge is considered to have failed if the pressure recordings are not realistic. In addition to the enumerated gauge problems, there are other problems that could result in gauge failure. Figure 2.11 shows an example where the upper gauge failed and therefore gave pressures that are higher than that of the lower gauge. Figure 2.12 shows case where one of the temperature elements of the gauges failed and measured temperatures are erratic.

In many cases, gauge failure can be remedied by re-calibrating the gauges. The BHP contractors are therefore advised to perform a quality check on the BHP data after each test to ascertain whether the gauges can be used for the next test. Even in situations where the gauges have performed well, it is generally required that gauges be re-calibrated after every six months.

26

Fig. 2.11: Gauge Failure – Upper Gauge Giving Higher Pressures than the Lower

Gauge

Fig. 2.12: Temperature Element Failure

2.2 GAUGE QUALITY CHECK PROCEDURE

Checking the quality of gauge measurements is one of the best things that have happened in welltesting. Gauge quality check (QC) now enables us to do the following:

a) Display the entire readings of the gauges and determine which of the gauges obtained a more reliable pressure data. Some of the gauge anomalies have been shown.

b) Determine whether gauge readings are consistent. c) Determine wellbore phenomena such as phase segregation, fluid interface

movement, etc.

27

Quality check can easily be performed on data obtained with digital instruments such as quartz gauges or strain gauges because the measured values are now known precisely. This is not so for values read from Amerada chart as such values may depend on the person that read the Amerada chart. 2.2.1 Quality Check Procedure: There are software that can be used for quality check. The procedures for the QC are summarized as follows: a) Load pressure data from both gauges. b) Synchronize the data if they are not synchronized. c) Using the zoom option, enlarge the different sections of the test so that you can

see fine details. We are able to see abrupt pressure changes of about 0.1 psi caused by changing the sampling frequency of the gauges! d) Take readings of lower and upper gauges corresponding to specific events. The events that you may choose in a buildup are a flow period, early buildup, mid buildup, late buildup and a static stop.

e) Plot the pressure difference between the lower gauge and the upper gauge. f) Interpret pressure difference plot

g) Repeat procedure for temperature data from both gauges. Some of the pressure anomalies may be caused by temperature changes.

h) Fill the quality check report sheet. Table 2.4 is a sample copy of QC report sheet. Table 2.4 here ????? Inferences that could be made from the pressure difference plot for two gauges that were placed 4 ft apart are as follows: 1. The pressure differences will reflect the fluid in the 4-ft. column between the

gauges. 2. If the 4-ft column was filled with gas, the pressure difference will be about 0.4 psi

and if the column between the gauges was filled with oil, the pressure difference between the gauges will be about 1.5 psi

3. Ideally the maximum pressure difference will be 4 ft x 0.433 psi/ft (water gradient) = 1.732 psi.

Getting pressure differences as high as 3 psi in a situation where the gauges are 4 ft apart shows that something is wrong with one of the gauges or both gauges. Note that some companies run their gauges 6-ft apart. This implies that the expected maximum pressure difference will be about 2.6 psi.

2.2.2. Quality Check Plots: Some quality check graphs are discussed in this section. The graphs show effects of phase segregation, liquid interface movement, leak and unrealistic pressure readings. Gas Phase Segregation: Figures 2.13 and 2.14 show pressure and temperature differences from two tests. In Fig.2.13, the pressure difference between the upper and lower gauge was constant and this implies that there was no gas segregation. In Fig. 2.14, there is evidence of gas segregation. The pressure difference was initially low and later increased as gas bubbled out leaving a denser fluid. Detecting this phenomenon helps while analyzing tests because non-reservoir responses will not be interpreted. Fig 2.13 and 2.14 here?????

28

Liquid Interface Movement: Figure 2.15 shows quality check in which fine details are revealed. In this case, there was an initial buildup followed by a pressure drop and then a final buildup. Movement of liquid interface in the wellbore caused the pressure drop. This is discussed in detail under field cases. Such phenomenon can cause misinterpretation of test. Fig 2.15 here? Leak: Part of quality check of pressure data involves enlarging different sections of the data to view fine detail. Such fine detail easily reveals things like leaks. Figure 2.16 is a quality check plot showing the effect of leak. Fig 2.16 here? Unrealistic Pressure Differences: Figure 2.17 is a quality check plot showing pressure difference of about 10 psi. This is unrealistic for gauges that are just 4 ft apart. With this detection, we now proceed to determine the source of the error before choosing the data to be analyzed.

Fig 2.17 here?

2.3 SAMPLE BHP SURVEY PROPOSAL

The proposal writer is the first man in the relay race to achieving the desired BHP objective. This implies that without proper education, the proposal writer can spoil the race. In most companies, writing BHP proposals is not considered as a job that requires skill. Hence, trainee engineers are assigned the job of writing BHP proposals using already existing templates. This may be a problem if they just change the test dates without going through well files to see changes that have occurred in the wells since the last test. They may also forget to review the success of the last test, which is a guide to the changes required for the new test to succeed. A typical FG/BU/SG survey proposal is in Appendix A. Some of the information in the proposal and reasons for including such information are discussed. We shall discuss Page A.1 to Page A.8. Page A.1: Contains the following information: (a) Objectives of test, (b) Types of tests required (c) Depth reference data (DFE, CHH). The objectives of a tests and types of tests required are related. For example, if one of the objectives is to optimize gaslift, a flowing gradient survey must be included as one of the test. Also, for us to calculate false pressure, P*, skin and permeability, both buildup and static gradient surveys must be included. Always match the type of survey with the objectives of the survey. Different depth references are used in the field. The drillers use the derrick floor as his reference and therefore his depths are stated as xx ftDFE. This means xx ft below derrick floor elevation (DFE). The BHP contractor, in most cases, gets to the well when the derrick floor is no more there. Hence, his depths are measured with respect to the casing head housing. That is xx ftCHH, which means xx ft below the casing head housing.

29

The status diagram depths used in determining survey depths are stated with respect to the DFE. Therefore the depth references must be supplied in the proposal. The depth reference (DFE and DFE - CHH) shows the basis for calculating depths with respect to the casing head. The BHP contractor should ensure that stated depth references agree with what is in the status diagram. Note that both the driller and the BHP contractor use along hole (ah) depth, which is different from vertical depths used by reservoir engineers. The vertical depths are measured with respect to the sub-sea (SS) level. The different depths and their interpretations are as follows: XX ahftDFE: XX ft along hole below the derrick floor YY ahftCHH: YY ft along hole below the casing head housing ZZ ftSS: ZZ vertical ft below the sub-sea level Page A.2: Main information on this page of the proposal is as follows: (a) Production rate (b) Perforated Interval (c) Location of sleeve and nipple The rate supplied in the proposal is the flowrate at the time of writing the proposal. The BHP contractor should always check with the flowstation staff to confirm that the stated rate is the current rate. With this, we can know the current state of the well. There is no point running a flowing gradient and buildup surveys in a well that is not flowing. Even in flowing wells, the contractor should check the rate stated in the proposal and ascertains that the gauges can be run in hole while the well is flowing. The proposal states on Page A.4 that if the gauges cannot be run hole at the current rate, the rate should be adjusted (i.e. change bean size) 24 hours before test. Changing bean size requires that the BHP contractor contacts the production staff. We expect that the BHP contractor will be guided by his experience The rate stated in the proposal is just for information and does not mean that during the flowtest, the bean size has to be adjusted to obtain the rate in the proposal. The flowtest preceeding the buildup provides the rate required for analysis of test. Therefore the rate must be measured during the flowtest. Measuring the flowrate correctly in a BHP test is as important as the pressure measurements. We therefore recognize the important role that flowstation staff plays in giving us accurate rate data. Perforated interval is an important data because it gives the interval of interest in the survey. Also, we are interested in the pressure at the middle of the perforation. In tests run in the long string, the gauges can be lowered, in most cases, to the top of the perforation. This is not possible for tests run in the short strings because of the “Amerada” stops, which will not allow gauges pass through them. In all cases where the gauges can be lowered to the top of the perforation, the BHP contractor is advised to do so because doing so will minimize phase segregation effects and also errors in correcting to the top of the perforation. In situations where the gauges cannot be lowered to the perforations, static gradient stops should be taken at short intervals so that type of fluid at bottom can be determined. We need this information for correcting pressure to the top of the perforation.

30

The BHP contractor should note the positions of the nipple and sleeves, as they are needed for depth control. Depths in the proposal must be verified with the status diagram. Also, the speed at which the wireline is lowered should be reduced in areas where there are restrictions in the well to avoid hitting the gauges against objects. Gauges are sensitive and can easily be damaged. Page A.3: This page gives information on the following: (a) Gauge specification (b) Sampling rate (c) Depth Control procedure The BHP contractor should ensure that their gauges meet stated specifications. Usually, we recommend that two gauges be used in any survey. This must be adhered to because readings from both gauges can be used to detect wellbore phenomenon and also for data quality check. Also, in a situation where one gauge fails, we can still rely on data from the other gauge. Using the sampling rate given in the proposal in important. Many changes in sampling rate during a survey is not welcome because we have observed that changing sampling frequency could cause pressure shifts that are as much as 3 psi. Such pressure shifts cause discontinuities that may make analysis of such test difficult and hence obtained results may be unrealistic. We wish to encourage BHP contractors to consider depth control as a very important issue because the depth at which pressure was measured is as important as the pressure measurement. The contractors are advised to follow the recommended procedure and document locations of sleeve and nipple if they are different from what is in the status diagram. If the BHP analyst knows the measured depths he can correct the recorded pressure to the reference depth. Page A-4: This page contains information on the following: (a) Well conditioning (b) Gauge Quality Control Well conditioning is required in some cases to ensure that well produced at stable conditions before shut-in. Normally, no time should be spent on well conditioning if rate was not changed while the gauges were run in hole or during the flowtest. This is the reason why all required rate changes should occur at least 24 hours before test commences. Quality checks on gauge measurements by BHP contractors are required so that anomalies can be detected early and faulty gauges removed. This implies that BHP contractors should purchase quality check software. In addition, operating companies should also perform their own quality check. Page A-5: This page shows the flowing gradient stops and the duration of the stops. There must be at least two stops around the gaslift mandrels if the flowing pressures will be used in vertical lift performance studies and determining correct gaslift valve locations. Page A-6: This page contains information on the following (1) The duration of the flowtest.

31

(2) The duration of the build up test. (3) Data to be gathered during the flowtest. The duration of the flowtest does not include whatever time it takes to run in hole and perform the flowing gradient survey. The gauges must be at the final survey during the flowtest. If the gauges are moved, the flowtest should be repeated. Measurements to be taken during the flowtest include the THP, THT, GOR and flowrate. These parameters should be measured every 15 minutes during the flowtest. In Page A-6, there is also instruction on the need to secure the well with the gauges at fixed location in cases where the well will be shut-in overnight. This is to ensure that there is no slippage or tampering. Page A-7: This gives information on the location of the static gradient stops and the duration of the stops. It is important to note the following: (a) The distance between stops should be smaller around the final survey depths.

This is important because we need to accurately determine the fluid gradient required for pressure correction.

(b) There must be, at least two stop around the gaslift mandrels so that the gas/oil contact (if any) can be determined accurately.

(c) Measurement of the static oil gradient is useful for estimating the oil density in the reservoir.

Class Exercise and Discussion

A sample of the FG/BU/SG survey proposal that has a problem is enclosed in Appendix B for discussion. The problem with this proposal is that gauges were moved after the flowtest. This was because of confusing instruction in the proposal. 2.4 PROCEDURE FOR RUNNING FG/BU/SG TEST

A summary of the procedure for running FG/BU/SG survey is as follows:

1. Perform dummy run to determine hold-up-depth (HUD). Locate accessory closest to the interval of interest (eg. XN-Nipple) and mark the wireline. Why?.

2. Run in hole both gauges to specified depths while well is flowing. Use wireline mark as depth control.

3. Suspend gauges at final survey depth and allow flow to continue for the specified period which is usually between 2 to 6 hours. Obtain the flowrate figures from the flowstation staff.

4. Shut in well for buildup survey. In case of gas-lifted wells, shut off gas supply prior to shutting in.

5. At the end of the specified buildup period, start the static gradient survey, which involves measuring static pressures at specified depth.

2.5 USEFUL HINTS ON PROPER TESTING OF WELLS

(a) The gauges must be in good conditions to record accurate pressure. (b) Depths where the pressure measurements were taken must be known. (c) Correct stable flowrate must be known during flowing gradient survey and prior

to shutting in. (d) Sufficient gas must be available for reliable stable flow of gaslifted intervals. (e) Survey programme must be correct, understood and followed.

32

2.6 PRACTICAL HINTS 1. Report events such as leaks, gauge movements that occur during the tests. The golden rule is that it is better to report than to cover up! There is no blame. It just means that we can interpret the test with the actual information rather than being puzzled by an inconsistency. Figure 2.18 and Table 2.4 show example of truthful reporting while Fig. 2.19 and Table 2.5 show strange pressure response which does not agree with test sequence.

2. Report activities in nearby wells if you are aware of them. 3. Once you have started the flowtest prior to shutting in, do not move the gauges even if you just realized that you are not at the correct depth. Just record the correct depth of the gauges when the flowtest started.

4. Do not change rate during the test. Any rate adjustment should occur 24 hours before test.

5. Avoid activities that will cause unnecessary vibration of gauges. 6. Ensure that test well is correctly hooked on to the test separator during the flowtest. Fig 2.18 and table 2..4/ 2.19 and 2.5 ????????? ask 2.7 ROLES OF FIELD STAFF IN BHP SURVEY

Field staff involved in BHP survey is production staff and BHP contractor staff. The roles played by both are vital for good data to be obtained. Also, both production and BHP contractor staff need to understand the principles involved in BHP test and analysis. This will help them understand the importance of their roles. The test and analysis principles are illustrated in Fig. 2.20.

Fig. 2.20: Test and Analysis Principles

The test principle involves allowing some known rate changes to occur in the reservoir and measuring the resulting pressure changes. Note that some characteristics of the reservoir such as the permeability are not known. Analysis principle involves applying the same rate changes to a mathematical model whose characteristics are known and observing the resultant pressure changes. The model characteristics can be changed until the pressure changes from the model become equivalent to pressure changes obtained when the same input (rate changes) were applied to the reservoir. We now conclude that the model characteristics are equivalent to the reservoir characteristics. The implications of the test and analysis principles are as follows: 1.) Rate changes (input) are needed to create pressure changes (output).

Reservoir k? s?

Pressure change

(Output)

Same

Rate change

Rate change

(Input)

(Input) (Output)

Pressure change

Test Principle

Analysis Principle

Model

k, s, etc known

33

2.) Correct rate (input) applied to the reservoir must be known. 3.) Unrecorded rate changes will render test useless in some cases. 4.) Correct pressure changes caused by rate changes must be measured. 5.) Factors such as leak, gauge movement, etc. cause pressure changes not

associated with rate changes. Based on deductions from test and analysis principles, the roles of production staff and BHP contractors are as follows:

Roles of Production Staff

Production staff benefits from results obtained by analyzing BHP tests because such tests are used for good well and reservoir management. We are sure everyone is happy when wells are producing at optimum rate. The production staff can contribute very much to the success of any BHP survey by doing the following: 1.) Ensuring that a full flowtset is conducted prior to shutting in the well for

buildup. Note that the BS&W and GOR are also required during test analysis. 2.) Ensuring that the correct well is hooked on for test. 3.) Reporting everything that could introduce errors in recorded production rate

(e.g. zero rate tests, changes to gaslift availability, surging). 4.) Telling the BHP contractor the correct status of well before survey starts. 5.) Taking more interest in BHP surveys. After all, we are all partners in

progress.

Roles of BHP Contractor Staff

The success of BHP survey depends largely on the field staff that runs the test. A few things the field staff should be aware of are summarized as follows: 1.) A well that is not flowing or a well that cannot be shut in will not produce the

required pressure change needed for a good buildup test. 2.) Moving gauges when they are required to be at a certain position will produce

pressure changes that will distort the correct pressure changes induced by applied rate changes.

3.) Due to the relationship between depth and pressure, good depth control is needed so that we can associate pressure with correct depths.

4.) Leaks are unwelcome events because they affect pressure changes. 5.) Good gauges are always needed for correct pressure measurements. 6.) Mistakes give rise to wrong interpretations and wasted resources. 7.) Cooperation with flowstation staff is necessary. 8.) Accurate reporting of all deviations from programme. If the test analyst has

accurate information, he may be able to compensate for deviations.

34

3. BASIS OF ANALYZING BOTTOM HOLE

TESTS In this section, we shall discuss some concepts that form the basis for analyzing bottom-hole pressure tests. The concepts include the following: Graphical presentation of Data Units and conversions Flow phases and identification Flow geometry Typical models

3.1 Graphical Presentation of Data

Graphs are one of the preferred methods for presenting information and yet they are often poorly made. In welltesting, many graphs of bottom-hole pressure versus time are used to deduce desired information. In this section, we shall discuss how to make good graphs and common types of graphs used in welltest analysis. 3.1.1 Making Good Graphs

To make a good graph involves the following: (a) Correct Labelling: The axis of the graph must be correctly labelled with units. For example, time (hr); pressure ( psi); Dimensionless pressure, etc. (b) Proper Scaling: Choosing the scale of a graph is important. The scales should be chosen such that unnecessary subdivision of the grids in the graph paper is avoided. It is convenient to choose the grid spacing such that each unit represents 1, 2, 5, 10, etc. (c) Data Points: Data points on any graph should be made conspicuous using

symbols such as ∅, ∆, ∇, etc. The practice of representing data points with dots should be discontinued because once a line passes through such points, the position of the point may be obscure. We are more interested in the points and not in the line. In addition, each data set should be represented using the same type of symbol. Different data sets should be represented with different symbols. (d) Title of Graph: Every graph must have a title, which briefly explains what the graph is all about. It is now common practice to put the title at the bottom of the graph. In some cases, some information may be put on the graph in the form of legend or something to make the graph understandable. 3.1.2 Types of Graphs The three common graphs used in well testing are the Cartesian, semilog and log-log graphs. The Cartesian, semilog and log-log graphs are included as Specimen A, B and C respectively. Further discussion on these graphs follows: Leave 3 pages for the specimens ????

38

(a) Cartesian Graph Paper: This is used for graphing data set (x,y) of the form

y = a + m x 3.1 Equation 3.1 is simply the equation of a straight line where “a” is the intercept while “m” is the gradient (slope). The intercept is the value of y when x = 0 while the

gradient is calculated as change in y (∆y) divided by change in x (∆x). The units of the intercept a, and gradient m are

a = intercept (units of y) and m = gradient unitsof y

unitsof x.

(b) Semilog Graph Paper: This is used for graphing data set (x,y) of the form y = a + m log x 3.2 The advantage of using the semilog graph for plotting data set that satisfy Eq. 3.2 is that on the semilog paper, the abscissa (x-axis) is already in log scales. Hence the data points can just be plotted without finding the logarithm of x. From Eq. 3.2, it can be shown that the value of “a” corresponds to the value of y when x is equal to one. This follows from the fact that log 1 is zero. The gradient m, in Eq. 3.2 is defined as m = change in y per cycle of x Mathematically,

12

12

loglog xx

yym

−−

=

Generally, x2 and x1 are chosen so that they are one cycle apart. In that case, the

denominator is unity.

The major grids on a log scale are scaled to the powers of ten. That is, 10-2, 10-1, 10o,

101, 102, etc. The interval between 10x and 10x+ 1, where x is an integer is known as a cycle. Note that a data set that satisfied Eq. 3.2 may still be graphed on a Cartesian graph to obtain a straight line. In that case, the transformation X = log x 3.3 is required. Hence the final equation becomes: y = a + m X 3.4 (c) Log-log Graph Paper: This type of graph is used for data set of the form: log y = a + m log x 3.5 The gradient m, in Eq. 3.5 is given in cycles of y per cycle of x. The gradient may be calculated using the equation:

39

m = 12

12

xlog - xlog

y log - y log 3.6

The points (x1, y1) and (x2, y2) are taken arbitrarily from the straight line. The log-log

graph paper is already divided into log scales and the data points are graphed directly without finding the logarithm of anything. By suitable transformation, the data set that satisfy Eq. 3.6 may be graphed on a Cartesian graph to get a straight line but the slope of the straight line will be different from that obtained from Eq 3.5. 3.2 SYMBOLS, UNITS AND CONVERSIONS

In this section, we shall discuss symbols, units and unit conversions.

a) Symbols and Units The three unit systems used in welltesting are the CGS, Oilfield and SI units. Although the Oilfield units are more common, but the preferred unit is the SI unit. Some of the welltest parameters, symbols and units are shown in Table 3.1. For more symbols and units, refer to SPE Metric Standard (1982). We assumed that you are already familiar with basic definitions of reservoir properties.

Table 3.1: Symbols and Units

Parameter Symbol CGS Units Oilfield Units Practical S.I. Units

Liquid Flow Rate q cc/sec STB/day m3/day Permeability k Darcy millidarcy

(mD) millidarcy (mD)

Time t seconds (s) hours (hr) seconds (s) Liquid Viscosity µ centipoise centipoise millipascal-second

Compressibility c atm-1 psi-1 kpa-1 Wellbore Storage Cs cc/atm bbl/psi m3/kpa

Porosity φ fraction of bulk volume same as CGS same as CGS

Saturation S fraction of pore volume same as CGS same as CGS Pressure p atm psi kpa Thickness h cm ft m Skin Effect s dimensionless dimensionless dimensionless

b) Conversion Factors

Some basic conversion factors used in this welltesting are as follows: 1 atmosphere (atm) = 14.7 psi = 1.01325 x 105 kpa

1 cp = 1 x 10-6 kpa-s

1 barrel (bbl) = 1.589873 x 10-1 m3 1 bb1/day = 1.840131 x 10-6 m3/s = 1.840131 cc/s More conversion factors are given in Table 3.2 taken from Earlougher (1977).

Leave two pages for conversion factors ??????

TABLE 3.2-CONVERSION FACTORS USEFUL IN WELL TEST ANALYSIS.

40

SI conversions are in boldface type. All quantities are current to SI standards as of 1974. An asterisk (*) after the sixth decimal indicated the conversion factor is exact and all following digits are zero. All other conversion factors have been rounded. The notation E + 03 is used in place of 103, and so on.

43

c) Unit Conversion

The Society of Petroleum Engineers (SPE) recommends the use of S.I. units. Articles that appear in SPE journals are now written using both the SI units and the Oilfield units. The Oilfield units are also referred to as English units. Conversion from one set of units to another is easy. Two cases will be discussed here. The first case considers conversion of a quantity expressed in one unit to another unit. The second case considers conversion of an equation with parameters expressed in some units to an equivalent equation with parameters given in different units. In both cases, what is required is a conversion factor. Consideration for given cases follows:

Case 1: Conversion of a given quantity from one unit to another

Example 1 (a) ft → inches

3ft = 3ft x 12 inches

ft1

↑ ↑ old conversion units factor (first) = 36 inches

Example 2

bb1 → ft3

5 bb1 = 5 bb1 x 5.614583

b b l ft3 ≅ ( 5.61 x 5) ft3

Note that in both cases, if the units in the numerator and denominator are cancelled, the resultant unit (unit to remain uncancelled) is the new unit.

Case 2: Conversion of units in an equation

Example 1

q = K A p

L

∆µ

3.7

Equation 3.7 is the steady state form of Darcy’s law for a linear system. The parameters in Eq. 3.7 are in Darcy’s units. That is:

(cm) L (cp)

(atm) p (cm)A (D)k =

s

cc q )(

µ∆

3.8

Equation 3.8 can be converted to Oilfield units. That is:

44

q )(s

cc ⇑ q (

STB

day)

k (D) ⇑ k (mD)

A (cm2) ⇑ A (ft2)

∆p (atm) ⇑ ∆p (psi)

µ (cp) ⇑ µ (cp)

L (cm ⇑ L (ft) Note that while the flowrate in Darcy’s units is expressed at reservoir conditions, in Oilfield units, it is expressed at stock tank conditions. Conversion of Eq. 3.8 to Oilfield units is done by replacing all terms in the original equation (Eq 3.8) with other terms and combining to get the overall conversion factor. The terms in Eq. 3.8 and their replacements are as follows: Parameters in Original Equation Replacements

k (D) ⇒ K (mD) x 1 (D)

1000 (mD)

A (cm2) ⇒ A (ft2) x 30.48 (cm

(

2 )

1 ft )2

∆p (atm) ⇒ ∆p (psi) x 1

147

( )

. ( )

atm

psi

µ (cp) ⇒ µ (cp)

L (cm) ⇒ L (ft) x 30.48 (cm)

1 (ft)

q(cc/s) ⇒ q (STB/day) Bobbl

STB x

1 day

(24x3600)sec

158987.3 cm

1 (bbl)

3

x

Substituting,

q (STB

day x B (

bbl

STB x

1 day

(24 x 3600) secso) ) x 0.158973m

bbl x

100 cm

m

3 3 3

3

= k(mD) x D

1000mD x A(ft ) x

(30.48) cm

ft

) x 30.48cm

ft

22 2

2

µ(cp) L (ft

x p (psi) x (atm)

∆ 14 7. psi 3.9

45

Evaluating, Eq. 3.9 becomes

qB STB

day x 1.84013 = 0.002073469

k (mD) A (ft p (psi)

(cp) L (ft)

2

o

) ∆µ

3.10

and therefore,

q (STBday

x 10 K(mD) A(ft p (psi)

(cp) B (bbl)

(STB) L(ft)

-32

o

) .)

= 1 127∆

µ. 3.11

The conversion factor 1.127 x 10-3 should be familiar to those who have taken basic reservoir engineering courses.

Example 2

Dimensionless pressure is defined in Darcy’s unit as:

PD = µπ

q

P] - [P 2 ikh 3.12

The term PD in Eq. 3.12 may for now be considered to be just a dimensionless number

(pressure). Equation 3.12 is in Darcy’s unit. That is:

PD (dimensionless) =

)((S)

(cc)q

(atm) p (cm)h (D) 2

cp

k

µ

π ∆ 3.13

Conversion of Eq. 3.13 to Oilfield units is done as follows:

PD (dimensionless) = 2πk(mD) x ( ).481

1000

30D

mDh ft x

cm

ftx

x

∆p psi psi

qSTB

dayxB

bbl

STBx

day

x s

x

o

atm

14.7

( ) ( )( )sec24 3600

01589873 106 3. x cm

bblx cpµ 3.14

PD (dimensionless) = 2π x 0.0011268 K mD h ft p psi

qSTB

dayB

bbl

STBcpo

( ( )

( )

∆

µ

= kh p

qBo

∆1412. µ

3.15

46

This implies that dimensionless pressure, defined in Darcy’s unit as

PD = 2π

µkh p

q

∆, 3.12

is given in oilfield units as

PD = kh p

qBo

∆1412. µ

3.16

Similarly, it can be shown that dimensionless time tD is defined in Darcy’s units as:

tD = kt

c rt w∅µ 2 3.17

is given in Oilfield units as

tD = 0 000264

2

. kt

c rt w∅µ 3.18

The time, t, in Eq. 3.18 is in hours. If it is in days, the conversion factor will be 0.00634. Note that in converting from Darcy’s to Oilfield units, nothing should be done to the dimensionless parameters PD. Why? They are dimensionless.

In Case 2 where the parameters in an equation are converted from one set of units to another, if the units in the numerator and denominator are cancelled, the old units will remain. This is the difference between Case 1 and Case 2. d) Rule of Thumb for unit conversion Welltest equations usually contain dimensionless groups. The ability to recognize these groups and their equivalent in the different unit systems forms the basis of the rule of thumb used in unit conversion. Table 3.3 shows the groups and their equivalent in different unit systems.

Table 3.3: Definitions of Dimensionless Groups in Different Unit Systems

Parameter CGS Oilfield SI

Dimensionless Time, tD

ktc rt w∅µ 2 0.000264

2

kt

c rt w∅µ

2

610557.3

wt rc

ktx

µ∅

−

Dimensionless Time Based on Area, tDA

ktc At∅µ 0.000264 kt

c At∅µ Ac

ktx

tµ∅

−610557.3

Dimensionless Pressure, PD 2πµ

kh p

q

∆

kh p∆141.2qBµ

µqBx

pkh310866.1

∆

Dimensionless Distance rrw

rrw

rrw

The use of the rule of thumb is illustrated with examples.

47

Example 1

Convert the equation

q = 2π

µ

kh p

r

re

w

∆

ln

from Darcy’s unit to oilfield units. Solution

Rearrange the equation into recognizable dimensionless groups as follows

ln r

r

kh p

qe

w=

2πµ

∆

↑ ↑ (Form of Dimensionless Distance) (Dimension Pressure) Therefore, the equation in oilfield unit is

ln r

r

kh p

qBe

w=

∆141.2 µ

Rearranging

q = 7.08 10 3x kh p

Br

re

w

− ∆

µ ln

Example 2

Convert the following equation in Darcy’s unit to Oilfield unit.

P = pi + )4

(4

2

kt

rcE

kh

q t

i

µπµ ∅

−

Solution

Rearranging to bring out the dimensionless group,

2 x [ ]2

4

2πµ

kh p p

qE

c r

kti

i

t−= − −

∅µ

Recognizable dimensionless groups and their oilfield equivalents are as follows

[ ] [ ]2

141.2qB

πµ µ

kh p p

q

kh p pi i−→

−

∅

→∅µ µc r

kt

c r

ktt t

2 2

0.000264

48

Substituting, the oilfield equivalent of the equation is

[ ]2

141.2qB 4x0.000264

2kh p pE

c r

kti

it−

= − −∅µ

µ

making “p” the subject gives

p = pi + 70.6qB

0.001056

2µkh

Ec r

ktit−

∅µ

3.3 Dimensionless Forms Many dimensionless parameters are defined in petroleum engineering. The dimensionless parameters make it possible to cast fluid flow equations into dimensionless forms. The advantages and disadvantages of the dimensionless forms are given in this section. Also, the definitions of some of the dimensionless parameters are given. a). Advantages of Dimensionless Forms

(i) Ease of comparing solutions (ii) Makes it possible for results to be generalized. For example, the pressure at any point in a single well reservoir produced at constant rate q, is written in a general form using dimensionless pressure as:

Pi - P(r, t) = 1412.

, , ,qB

kh

µ [P (t r C geometry ) + s]D D D D 3.19

Equation 3.19 is in Oilfield units and the dimensionless pressure, PD, is a function of

other dimensionless parameters, rD, tD and CD.

(iii) With pressure expressed in dimensionless form, It becomes easier to apply superposition concept to handle varying flowrates and pressure drop in multiwell systems. (iv) Dimensionless form aid in presentation of results in a more compact form. Also, the results are invariant in form, irrespective of the unit system used.

b) Disadvantages of Dimensionless Forms

(i) With dimensionless form, the engineer may loose a sense of magnitude of the quantity. For example, a time of 24 hours may correspond to dimensionless time of 300 for a tight reservoir and a dimensionless time of 1000 for a highly permeable sand. This follows from the way dimensionless parameters are defined. 3.3.1 Definition of Dimensionless Parameters In Darcy Units

Dimensionless Time

tD = kt

c rt w

2∅µ 3.20

49

Dimensionless Time Based on Area

tDA

t

= kt

c A∅µ 3.21

Dimensionless Radius

rD = r

rw 3.22

Dimensionless Pressure

PD = 2 kh

q (P - P (r, t) )i

πµ

3.23

PD = P - P (r, t)

P - P

i

i wf

3.24

The form of dimensionless pressure, PD, used in any problem depends on the type of boundary condition used at the well. Use Eq 3.23 if well produces at constant pressure and Eq 3.24 for well producing at constant rate.

Dimensionless Cumulative Production

qD = q (t)

2 kh (P - Pi wf

µπ )

QD

o

tDW

= q dtD DW∫ 3.25

Other Dimensionless Parameter

kt

c r rt D∅µ ∅µ r =

kt

c r r

= t

2

t w

2

w

2

D

2 2 3.26

3.4 FLOW PHASES

Just like a man goes through phases in life, a well undergoing a test also passes through phases with each phase revealing some information about the well or reservoir. It is therefore necessary that we understand the phases and how to identify them to avoid problem of obtaining required information from a wrong phase. In this section, we shall describe the phase and methods of diagonizing the phases. 3.4.1. Flow Phases in Drawdown and Buildup Tests Figure 3.1 shows the phases that a well undergoes during a drawdown or buildup test. In case, there is a wllbore storage phase, transient phase and late-time phase. The late-time phase depends on the nature of the boundary and test (drawdown or buildup).

50

Wellbore Storage

A B C D E

Transient State Phase Late-Time Phase

Increasing Flow Time

Figure 3.1: Flow Phases in Buildup or Drawdown Test. Some deductions from Fig 3.1 are as follows: 1. The wellbore storage phase is an independent phase and can occur concurrently

with the transient state. This happened in Interval BC. 2. The wellbore storage phase with no concurrent transient phase is known as the

“strong wellbore storage”. This is represented by Interval AB. 3. If wellbore storage and transient state occur concurrently, the pressure-time data

acquired will be “polluted” and cannot be analyzed using conventional method of analysis.

4. The wellbore storage phase may last so long that all the transient state phase may be “polluted”. Tests must be designed so that this type of “pollution” does not occur as the most valuable information from such tests are obtained from the unpolluted transient state phase.

5. The transient state phase without concurrent wellbore storage phase is the “good transient state phase”. This is represented by Interval CD.

6. The transient state phase and late-time phase are dependent. The transient state must end before the boundary effect phase is reached.

7. For the buildup test, The late-time phase represents the stabilization phase. During this period, the well builds up to average pressure if interference from other wells is minimal.

8. For drawdown test, the late-time phase represents the pseudo-steady state phase if the reservoir boundary is closed to inflow (bounded reservoir). On the other hand, if the reservoir boundary is open to inflow, the late-time phase will represent the steady state phase.

9. Information derived from the pressure time data obtained during the different phases is shown in Table 3.4.

Table 3.4: Information from Different Phases

Phases Derived Information

1. Strong Wellbore Storage Wellbore Storage Constant, Cs

2. Good Transient State (1) Permeability, k (2) Skin Factor, s

3. Pseudo-Steady State Drainage Volume, Vd

4. Stabilization Phase Average Pressure

3.4.2 Definition of the Phases The function of the pressure gauge is simply to measure pressure irrespective of the flow phase occurring in the well. Depending on the information we need from the test, we need to ascertain that the acquired pressure-time data used in the analysis correspond to the phase of interest.

51

In this section, we shall define the phases and the characteristic features of pressure-time data obtained during the phases. Wellbore Storage Phase: This occurs early in the life of the test well. The pressure changes that occur during this phase are caused by fluid stored in the wellbore or stored fluid produced from the wellbore. This is caused by the fact that wells are opened or shut at the surface during tests and also reservoir fluids are compressible. Figure 3.2 shows a test well that has just been shut in. Note that the production rate

(q) is zero at the surface, but the reservoir is still producing (qsf ≠ 0) and the produced fluid is stored in the well. The duration in which the surface rate is zero and the sandface rate is not zero is the wellbore storage phase for a buildup test.

q = 0 q = 0

qsf = 0qsf ‡ 0

Wellbore Storage

phaseEnd of wellbore Storage

phaseIncreasing Shut-in time

Figure 3.2: Surface and Reservoir Production During Shut-in. Figure 3.3 shows wellbore storage phase in a drawdown test. In this case, on opening the well for a drawdown test, the initially produced fluid is the fluid stored in the wellbore. Thai is, q = qwb. Wellbore storage phase ends in this case when the total production is equal to the fluid produced by the reservoir. That is, q = qsf.

52

q = qwb + qsfq = qsf

qsf = q

qwb = 0

qsf = 0

or

qsf < q

Wellbore Storage

phaseEnd of wellbore Storage

phase

Increasing production time0

qwb

Figure 3.3: Surface, Wellbore, and Reservoir Production in an Opened Well.

In Figure 3.3, qwb is the production from fluid stored in the well. Note that if qsf = 0 for drawdown or qsf = q in a shut-in well, we have strong a wellbore storage phase. The manner in which q, qsf and qwb vary during buildup and drawdown tests are shown in Figure 3.4.

q = qsf + qwb

q

0 time ts ∆tete

qwb

qsf

qsf

Drawdown Buildup

Wellborestorage

Wellborestorage

Figure 3.4: Rate Variation During Buildup and Drawdown Tests. In Figure 3.4, te = end of wellbore storage phase for drawdown test ts = shut-in time

∆te = End of wellbore storage phase for buildup test.

53

The wellbore storage phase is a nuisance because it “pollutes” the transient phase from which we can get useful information about our reservoir. The duration of the well storage phase must be reduced or eliminated if it is possible. To reduce the duration of wellbore storage phase will require knowledge of the factors that can affect the phase. The factors are mainly compressibilty of the wellbore fluid and wellbore volume. (a) The compressibility of the fluid in the well: The higher the compressibility of the

fluid, the longer the duration of the wellbore storage phase. The compressibility of the wellbore fluid depends on the gas-oil ratio (GOR). Wells with high GOR has high wellbore fluid compressibility . Can the GOR be reduced?

(b) The volume of the well that communicate with the tubing: This is shown by the

volume of the shaded region in Fig. 3.5. Figure 3.5a represents a case with a packer while Fig 3.5b represents a case without a packer or a non-sealing packer. The duration of the wellbore storage phase increase with the increase in volume communicating with the tubing.

(c) Production Rate: The production rate of the well also affects the duration of the

wellbore storage phase. The higher the production rate, the smaller the duration of the wellbore storage phase.

(a) Sealing packer (b) No packer or non sealing packer

Figure 3.5: Volume communicating withTtubing

Exercise: Explain how the following will affect the duration of the wellbore storage phase: (i) Test well was shut at the flow station because the wing valve was faulty.

Draw the phase box diagram showing situation where this could make it difficult to calculate permeability and skin.

(ii) Test well was shut in with a downhole shut-in tool. Also, show the phase box diagram in this case.

The practical implications of these are as follows:

54

(a) Do not shut the well at the flowstation as that will increase the volume communicating with the tubing and thus increase the duration of the wellbore storage phase.

(b) In wells with high gas-oil ratio (GOR) the duration of the wellbore storage phase is long because of the high compressibility of the fluid in the wellbore.

(c) Use a downhole shut-in tool in situations with unusually long wellbore storage duration.

(d) In buildup tests, if there are leaks, the wellbore storage phase may not end because the sandface production, qsf, will not be zero.

(e) For wells producing less than 500 STB/D, the duration of wellbore storage phase should cause some concern.

Transient State Phase: This is the most important phase because important reservoir parameters (permeability, skin, etc.) are deduced from pressure-time data obtained during this phase. The useful part of the pressure-time obtained during this phase is the part not “polluted” by the wellbore storage phase. Due to the usefulness of this phase, the following guidelines must be followed during tests. (1) Design and run test so that not all parts of the transient state phase will be

“polluted” by the well bore storage phase. (2) Test duration must be such that the transient state phase must be reached before

the test is stopped. The transient state phase occur when the pressure changes at the wells are not influenced by the nature of the boundary. For example, if you drop a little stone into a bowl containing water, concentric waves will move outwards as shown in Figure 3.6.

Figure 3.6: Waves illustrating Transient State Period. The waves will continue to move outwards until they hit the side of the bowl. The waves get distorted and become less orderly. The period during which the waves have not hit the boundary can be likened to transient state phase because the effect of the boundary has not been felt. The mathematicians describe this phase as period when the rate of pressure change with time is neither zero or constant. All systems go through the transient state irrespective of the nature of the boundaries. The duration of the transient state is affected by many factors which includes the following: (a) Permeability of the Formation: The higher the permeability, the shorter the

duration of the transient state phase (waves move faster). For Niger Delta

55

formation with permeabilities greater than 1000 mD, the transient state phase has a short duration. This is a problem because the short transient state duration could easily be marred by the wellbore storage phase.

(b) Location of Test Well: The location of the test well with respect to reservoir

boundary affects the duration of the transient state. Wells that are closer to the boundary will have a shorter transient state period compared to wells that are farther from the boundary. Figure 3.7 shows two cases.

Longer TS phase Shorter TS phase Figure 3.7: Effect of Well Location on Duration of Transient State Phase.

Late-Time Phase: The nature of the late-time phase depends on nature of test. For buildup test, the well will build up to average pressure at late-time if there is just one well in the reservoir. In case where we have many wells, the well will build up and then drops due to interference effect. In a situation where the drainage area of the test well is large, the pressure may build up to average pressure before dropping. The pressure profile in such case is shown in Fig 3.8.

Fig. 3.8: Effect of Interference

For drawdown, the nature of the late-time phase depends on the type of the outer boundary condition of the reservoir. If the boundary is open to inflow (water influx), a steady state will be attained during the late-time phase. During steady state, pressure in the system will no longer be changing with time. If the reservoir boundary is closed to flow, a pseudo-steady state will be attained at late time. During the pseudo-steady state period, pressure in the system will be changing, but the rate at which the pressure will be changing everywhere in the system will be constant. The value of the is related to the drainage volume of the test well and this is the basis of reservoir limit test. Figure 3.9 shows the transient state, steady state and pseudo-steady state phases during a drawdown test

Shut in Pressure

Shut in Time

No Interference Interference

56

.

For steady state to be attained there must be an adjoining aquifer providing the source of water influx and the aquifer permeability must be large to permit ease of flow of water into the reservoir. Reservoirs in highly faulted environment are more likely to be sealed off and less likely to be in contact with such an aquifer. 3.5 Flow Geometry

Depending on the reservoir and nature of perforations in well, flow could occur linearly, radially, spherically or elliptically. In some cases, the flow geometry could change with time from one form to another. In this section, we shall discuss equations governing the different flow geometries and parameters that may be deduced from the flow geometry. 3.5.1 Linear Flow

The plan and elevation in situations where linear flow occur is shown in Figure 3.10 Plan Elevation

Fig. 3.10: Linear Flow Geometry

The linear flow described here is the one-directional flow that occurs in Cartesian co-ordinates. The practical situations where linear flow could in a reservoir are as follows: a. Single Vertical Fracture Intersecting Well

This case is illustrated in Figure 3.11. The permeability in the fracture zone is much greater than the permeability of the formation (infinite conductivity). Therefore, fluid flows linearly into the fracture and the wellbore. Such flow occurs at early time and will explained later.

Transient

Steady State

Pseudo-Steady State

Log (Flowing Time)

Pw

Fig 3.9: Transient, Steady and Pseudo-Steady States

57

Fig 3.11: Vertically Fractured Well

Linear flow is also observed at early time in situations where the flow per unit area of

fracture is constant (uniform flux fracture).

b. Horizontal Well

At some period, linear flow could occur in horizontal wells. The flow streamlines in a horizontal well during linear flow is shown in Figure 3.12. ?????????????? Cut and paste

Fig. 3.12: Linear Flow in Horizontal Wells

c. Reservoirs with Strong Partial Influx

In a reservoir where a segment is open to strong water influx, flow will be dominant in the direction of the influx. Depending on the size of segment, the flow to the wellbore could be linear. In situations where linear flow occur, the flow could be modelled with the equation:

∂ ρ∂

φµ ∂ρ∂

2

2x

ctk t

=

3.27 Equation 3.27 is also a diffusivity equation. Hence, all assumptions relating to diffusivity equation will hold. The solution to Equation 3.27 is of the form

∆p = (At)½ --------------------------------------------------------------------------- 3.28 Taking log of both sides,

Log ∆p = ½ log t + B ------------------------------------------------------------- 3.29

In Equations 3.28 and 3.29, ∆p is the pressure change while A and B are constants that depend on fluid and rock properties.

Reservoir Boundary

Well

Fracture

xf

58

The implication of Equation 3.29 is that a graph of ∆p versus t (time) on a log-log paper will give a straight line with slope ½. This is distinguishing feature of all forms of linear flow irrespective of where they occur. A similar relationship existing between dimensionless pressure and dimensionless time for a well intersecting a single vertical fracture is given as

DxfD tP π=

3.28 where tDxf is dimensionless time based on fracture half length and is defined in Oilfield units as

2

000264.0

ft

Dxfxc

ktt

φµ=

3.29 Equation 3.28 also shows that a graph of PD versus tDxf will give a slope of 0.5.

3.5.2 Bilinear Flow

This is a form of linear flow observed in some wells with single vertical features. In this case, the permeability in the fracture is not much greater than the permeability in the formation (finite conductivity fracture). Hence, there is linear flow to the fracture and another linear flow from the fracture to the wellbore. This is illustrated in Figure 3.13.

Plan showing Well and Fracture Elevation showing the Well

Fig. 3.13: Bilinear Flow Geometry

For the bilinear flow, the relationship between pressure change and time is

∆p = A t¼ ---------------------------------------------------------------------------- 3.30 Taking log,

Log ∆p = ¼ log t + B ------------------------------------------------------------ 3.31

59

Equation 3.31 implies that a graph of ∆p versus t (time) on a log-log graph has a unique slope of ¼.

3.5.3 Radial Flow

This is the most common flow geometry and occurs when the flowing fluid surrounds the wellbore and flow streamlines come from distances that are large compared to the size of the wellbore. Streamlines converge towards a central point in each plane. Figure 3.14 shows the plan and elevation during radial flow.

Plan showing Well and Streamlines Elevation showing Well and Strealines

Fig 3.14: Radial Flow

Irrespective of what it is called (pseudo-radial, late-time radial, early-time radial, etc.) the relationship between pressure and time in all forms of radial flow is

∆p = A log t + B ----------------------------------------------------------------- 3.32

In Figure 3.14, ∆p is pressure change, (Pi – p), t is time (or some form of time function), and A and B are constants that depend on the reservoir characteristics.

The implication of Equation 3.32 is that a graph of ∆p (or pressure) versus log t gives a straight line. This is shown in Figure 3.15 for a drawdown test.

Fig 3.15: Semilog Straight Line due to Radial Flow

Radial flow could occur virtually in all system including horizontal well. In a reservoir with a single vertical fracture, a form of radial flow, pseudo-radial flow, occurs at the end of the linear flow. Figure 3.16 shows the pseudo-radial flow occurring in a system with a single vertical fracture.

0.1 1 10 100 1000

2000

2500

3000

Pressure

Time

60

Fig 3.16: Pseudo-radial Flow occurring in well with a Single Vertical Fracture

In Fig 3.16, the streamlines come from far distances and the fracture behaves as a point source. A form of radial flow (pseudo-radial) also occurs in horizontal wells and this is shown in Figure 3.17. ???????????? cut and paste

Fig 3.17: Pseudo-radial Flow occurring in Horizontal Well

A form of radial flow, hemiradial flow, occurs when the well is close to a boundary as shown in Fig. 3.18. Equation 3.32 also holds for a hemiradial flow with constants, A and B defined appropriately.

Fig 3.18: Hemiradial Flow

3.5.4 Spherical Flow

For this case, flow occurs from all directions towards the wellbore. This is shown in Figure 3.19.

Plan Elevation

Fig 3.19: Spherical Flow

Boundary

61

Spherical flow could occur in thick reservoirs if the perforated interval is small. Moran and Finklea (1962) used spherical flow equations for analysing pressure transient data. Raghavan (1975) found an expression for vertical permeability in a partially penetrating well using spherical flow equations. The spherical flow equations at early and long times are of the forms: Early Time

∆p = At1/2-------------------------------------------------------- 3.33 Long Time

∆p = 1 - 112Bt

-------------------------------------------------

3.34 In Equations 3.33 and 3.34, A and B are constants. Graphical implications of Equations 3.33 and 3.34 are obvious. Onyekonwu and Horne (1983) published detail on the equations. Equation 3.34 is generally recognized as the spherical flow equation and hence a plot of pressure versus the inverse of the square root of time yields a straight line. 3.6 Flow Geometry and Phases

In this section, we illustrate using box diagrams the phases and geometries that occur during transient in some well and reservoir systems. a) System with Wellbore Storage and Skin in Homogeneous Reservoir

Fig. 3.20: Radial Flow in a Homogeneous Reservoir

b) System with Well on a Single Vertical Fracture

Fig. 3.21: Flow Geometry and Phases in a Well on a Single Vertical Fracture

Wellbore Storage Phase

Radial Flow


Increasing Time



Increasing Time

Linear Flow Pseudo-Radial Flow

62

An implication of Fig 3.21 is that linear flow can easily be marred by wellbore storage phase. c) Horizontal Well in a Homogeneous System

Fig. 3.22: Flow Geometry and Phases in a Well on a Horizontal Well

Depending on the conditions, horizontal wells could exhibit other flow regimes as explained by Du and Stewart (1992) and Kuchuk (1995). Figure 3.22 shows the main flow regimes published by Lichtenberger (1994). Note that there is usually a transitional flow between the flow geometry.

3.7 Distinguishing the Phases and Geometry The pressure gauges simply measure pressure and time irrespective of the flow phase and geometry. However, information derived from the pressure-time data depends on the flow phase and geometry. This implies that we must be able to distinguish pressure-time data gathered during each phase and geometry. Until 1980, pressure plots were solely relied on for diagnosing flow phases and geometry. This is simply based on equation relating pressure and time. Bourdet et al (1983) introduced the concept of pressure derivative, which has been found to be more unique and reliable in diagnosing flow phases and geometry. The pressure derivative is defined as

))(( tfdIn

dpP =′ 3.35

where f(t) is a time function which may be defined as follows: f(t) = t (drawdown test)

f(t) = ∆t (buildup test)

f(t) = (tp + ∆t)/ ∆t (buildup test) The derivative of pressure drop could also be used and in that case, the pressure term

in Eq. 3.35 is replaced with ∆p.



Increasing Time

Linear Flow Pseudo-Radial Flow Radial Flow

63

As pressure values during test are obtained at discrete times, the pressure derivative is obtained numerically. Horne (1990) published an algorithm that can be used in obtaining the derivative as follows:

ii t

pt

t

p

∂∂

=

∂∂ln

3.36

= ( )

( ) ( )kijiiji

jikii

tttt

ptt

−++

+− ∆

/ln/ln

/ln+

( )( ) ( )kiiiji

iikiji

tttt

pttt

−+

−+ ∆

/ln/ln

/ln 2

- ( )

( ) ( )kijikii

kiiji

tttt

ptt

−+−

−+ ∆

/ln/ln

/ln 3.37

In Eq. 3.36 and 3.37, the time function is assumed to be time. The constraints on the time in Eq. 3.37 are as follows:

In ti+j - In ti ≥ 0.2 and In ti - In t i-k ≥ 0.2 The value of 0.2 (known as differentiation interval) could be replaced by smaller of larger values (usually between 0.1 and 0.5), with consequent differences in the smoothing of the noise in the pressure data. Higher values yield a more smoothened derivative. Pressure derivatives characteristic shapes published by Gringarten (1987) are shown in Fig. 3.23. ????? leave 3 inches

Fig 3.23: Characteristic Shapes of Pressure Derivatives

Table 3.5 summarizes the feature and the characteristics of the derivative

Table 3.5: Features and Characteristics of the Derivatives

Features Characteristics of Derivative

Wellbore storage and skin Hump with derivative attaining a maximum

Stimulated or fractured well No hump, No maximum

Heterogeneous behaviour A “valley”

Fault Upward turn

Closed boundary (drawdown) Upward turn

Closed boundary (buildup) Downward turn

Constant pressure (due to influx or gas-cap)

Downward turn

Infinite-acting radial flow (IARF) “flat” (zero gradient)

Linear Flow Gradient with slope of 0.5

Spherical flow Gradient with slope of - 0.5

Strong wellbore storage Gradient with slope of 1 (unity)

Hemiradial flow “flat” (zero gradient at a higher level)

Table 3.5 will form the basis for using pressure derivative for diagnosis. 3.7.1 Diagnostic Plots for Distinguishing Flow Phases

In this section, we shall show the diagnostic features of the phases using pressure (or pressure drop) and pressure derivative. Discussion now follows:

64

Diagnosing Wellbore Storage Phase: Wellbore storage is distinguished on a log ∆p versus log t (or log ∆t) plot. The following features help us determine pressure response obtained during this phase. (1) Pressure responses obtained when wellbore storage is strong lie on a unit slope

line. This corresponds to time ending at t* on Figure 3.24. This follows from the fact that

log(∆p) = log t + C. 3.38

For buildup, t* is replaced by ∆t*. (2) Pressure responses obtained when wellbore storage is not strong will have a slope

that is in the range of 0 < m < 1. The duration of this part of wellbore storage phase is in time range of 10t* < tewb< 50t* (1 to 1.5 cycle rule) where t* is the time when strong storage effect ended. The times are shown in Figure 3.24.

(3) A typical log-log plot is shown in Figure 3.25.

Wellbore Storage

Transient Late Time

timet* t esl

(Figure not to scale)

tewb

Figure 3.24: Phases and Duration of Wellbore Storage Phase.

Figure 3.25: Log-log Showing Wellbore Storage Influenced Data

0.1 1 10 100

Time, t or Shut-in Time, ∆t 1

10

100

∆p

45o

t* tewb

65

Parameters in Figure 3.25 are defined as follows:

∆p = Pi - pwf (Drawdown)

∆p = Pws - pwf (tp) (Buildup) t = flowing time (drawdown)

∆t = shut-in time (Buildup) Pi = initial pressure pwf = flowing pressure tp = total flowing time pwf(tp) = flowing pressure at shut-in time Pws = shut-in pressure. The wellbore storage phase is more easily distinguished on a log-log plot of the pressure derivative. On this plot the wellbore storage phase forms clearly defined “hump” as shown in Fig.3.26.

Derivative

W ellbo re S to rage Hump

45

T ime

Fig. 3 .26: D erivative P lot - D iagnostic P lot for W ellbore S torage Phase

Since the advent of the derivative plot, I have always relied on it when detecting pressure responses affected by wellbore storage phase. In most cases, both the pressure and derivative plots are displayed in one graph. This is shown in Figure 3.27.

Derivative

and P ressure

W ellbo re S to rage H ump

45

T ime

Fig. 3 .27 : Log-log P lo t o f P ressu re and Deriva tive .

Pressure

D erivat ive

Diagnosing Transient State Radial FlowPhase: Pressure responses obtained during the good transient state radial flow phase or infinite-acting radial flow phase (IARF) will fall on a straight line when pressure is plotted against log of time (semilog plot). The time in this case is defined as follows: time = flowing time for drawdown

66

time = shut-in time for buildup (MDH Plot)

time = t t

t

p +∆

∆ for buildup (Horner Plot)

This follows from the fact that during the good transient, p = A + m[ log t(ime) + C]. 3.39 where A and C are constants. Also, m is a constant that depends on fluid and rock properties. The semilog plots are shown in Figure 3.28, 3.29 and 3.30.

Figure 3.28: Semilog Plot (Drawdown Test)

Fig 3.30: MDH Plot

Log (time)

Pw

tew

Good Transient

Log t

Pws

Good TraTransie

Log (t +∆t/∆t) 1

Pws

Stabilizing at average pressure

Fig. 3.29: Horner

Plot

67

The good transient state phase is the transient state not influenced by wellbore storage phase. It occurs in the time range tewb < t < tesl. The term tesl is the time when transient ends and it is shown in Figure 3.24.

Many factors affect the semilog straight line obtained during the transient state phase. Figure 3.31 taken from Mathews and Russell (1967) show how these factors affect the semilog straight line. ????? ask for Fig 3.31. The good transient state phase can also be clearly discerned on a derivative plot. The derivative plot in this case falls on a horizontal line (“flat”) after the end of the wellbore storage hump. This is shown in Figures 3.32.

Fig 3.32: Flow Phases Discerned on Pressure Derivative

Transient state is more uniquely identified on a dimensionless pressure-time plot. A graph of dimensionless pressure versus log of dimensionless time has a unique slope with a value of 1.151. This is shown in Fig 3.33.

Fig 3.33: Dimensionless Plot Showing Transient State

Derivative

Log ∆p

Log (time)

m= 1.151

Log tD

PD Wellbore Storage

Good

Wellbore Storage

Transient State Phase

Pressure

Unit slope line

68

The unique slope follows from the fact that PD = 1.151 [log tD + C] 3.40 Equations that can be used for diagnosing other flow geometries during the transient state have been discussed under flow geometries. Diagnosing Late Time Phase: The diagnostic plot for the late time phase depends on the nature of the outer boundary and the type of test. We shall not go in detail, but a summary of the different plots is as follows: Drawdown Test (Closed Boundaries): For drawdown test in wells with closed outer boundaries, the system attains pseudo-steady state at late time. Cartesian plots of flowing pressure versus time during the late time phase yields a straight line. The slope of the straight line is related to the drainage volume of the well. Figure 3.34 shows the diagnostic plot.

Fig. 3.34: Cartesian Plot Showing Pseudo-Steady State

The basis of the plot shown in Fig 3.34 follows from the equation that holds during pseudo-steady state: Pwf = A + mt 3.41 Where A and m are constants. The value of m is related to the drainage volume and this forms the basis of reservoir limit test. The dimensionless plot is a more unique plot for detecting pseudo-steady state because a graph of dimensionless pressure versus dimensionless time based on area

has a slope of 2π during pseudo-steady state. Figure 3.35 shows the dimensionless plot.

Time

Wellbore storage and transient state Pseudo-steady

Flowing

Slope = m

69

Fig. 3.35: Dimensionless Plot Showing Pseudo-Steady State

The basis of Fig 3.35 is Eq 3.42 which is as follows:

PD = 2πtDA + C 3.42 Where C is a constant. Drawdown Test (Open Boundaries): For a drawdown test with boundaries open to flow, the system will attain steady state (pressure remains constant) if the influx through the boundaries is sufficient to stop further pressure decline. A semilog graph for systems with closed boundaries and constant pressure boundaries is shown in Fig 3.36.

Fig. 3.36: Semilog Plot of Drawdown Data with Different Boundary

Conditions

Slope = 2π

Pseudo-steady state

PD

Dimensionless Time based on Area, tDA

Pseudo-steady state

Wellbore storage

Transient Late-time

Log (time)

Steady state

Pw

70

Buildup Test: For a well in a closed system that is shut-in, the pressure builds up to average pressure during the late time phase. If there are many wells in the system, pressure in the test well will build up and later drop due to interference effect. If there is a large gas-cap, the system attains a constant pressure due to the gas-cap. The derivative plots for the different late-time conditions are discussed in Chapter 4.

71

4. ANALYSIS MODELS

Pressure – time data obtained from BHP tests are normally analyzed using mathematical models. Figure 4.1 shows both the test and analysis principles.

Reservoir

k? s?

Pressure change

(Output)

Model

k, s, etc known

Same

Rate change

Rate change

(Input)

(Input) (Output)

Pressure change

Test Principle

Analysis Principle

Fig. 4.1: Test and Analysis Principles

The test principle involves perturbing the reservoir by applying some input (usually rate changes) and measuring the resulting pressure. The analysis principle involves perturbing a mathematical model using similar input applied to the reservoir and comparing the resulting pressure with actual pressures obtained during the test. The mathematical model is fine-tuned until the actual pressures obtained during the test agree with pressures obtained with the mathematical model. It is then inferred that properties of the mathematical model are similar to that of the reservoir. The uniqueness of the result is not the subject of discussion now.

The choice of mathematical model is not arbitrary because pressure and pressure derivatives obtained during the test contain “signatures” that reveal the nature of the type of model to be used in the analysis. Therefore understanding the models will help in relating to the reservoirs.

Each model used in test analysis may consist of 3 sub-models: well model, reservoir model and boundary model. Options in the sub-models are shown in Table 4.1. The term reservoir model is used here to represent the behaviour of the reservoir during transient state phase. Any well model can be used with any reservoir model and any boundary model to make up the analysis model. However, in some cases, the mathematical models for the chosen sub-models may not be available or physically feasible. Available choices show that a number of mathematical models are available. The boundary model will not be used if the late-time was not reached.

Although it is not possible to describe all analysis models, but the following models deserve some mention: a. Wellbore storage and skin well in homogeneous reservoir b. Wellbore storage and skin well on single vertical fracture c. Wellbore storage and skin well in a double porosity reservoir. Discussion on these follows:

72

Table 4.1: Analysis Models

Well Models Reservoir Models Boundary Models • Wellbore storage and skin • Homogeneous • Closed

• Changing Wellbore storage • Single Fracture • Constant Pressure

• Limited entry well • Double Porosity • Fault

• Horizontal well, etc. • Composite • Leaky Fault

• Well on a Fracture

4.1 WELLBORE STORAGE AND SKIN WELL IN HOMOGENEOUS

RESERVOIR.

A homogeneous reservoir is one whose properties (permeability and porosity) are invariant in the direction of flow. This is the most common model and many high permeability formations (eg. Niger Delta) are homogenous. Typical profiles for a well with wellbore storage and skin in a homogenous reservoir are shown in Figures 4.2.

Figure 4.2: Log-Log and Semilog Profiles from Well with Wellbore Storage and Skin

in Homogenous Reservoir

Note that no boundary model was used because test ended during the infinite-acting radial flow (I.A.R.F) phase.

Parameters that can be deduced with this model are as follows

Cs = wellbore storage constant

s = skin factor

73

k = permeability

4.2 WELL ON A SINGLE VERTICAL FRACTURE

It is not unlikely that a well is located on a single vertical or horizontal fracture as shown in Fig 4.3. The fracture, which was caused by faulting or fracturing, becomes a fast track for fluids getting to the wellbore.

Parameters that can be deduced with this model include the following: xf = fracture half length

k = permeability Cs = wellbore storage constant

s = skin factor

Three forms of this model depending on the flow in the fracture exist. They are as follows:

(a) Infinite conductivity fracture (b) Finite conductivity fracture (c) Uniform flux fracture

Fig. 4.3a: Horizontal Fracture

74

Fig. 4.3b: Vertically Fractured Wells

Gringarten et al (1974) published solutions for these models. Features of the models are discussed.

4.2.1 Infinite Conductivity Fracture

This represents the case where the fracture permeability, kf, is much greater than the permeability of the matrix, k (kf >> k). The fracture is considered to have an infinite permeability and therefore there is no pressure drop during flow in the fracture. The pressure profile in this case is shown in Figure 4.4.

This reservoir goes through a linear flow, followed by a pseudo-radial flow before the boundary effect. However, no pseudo-radial flow will appear if xf/xe = 1. This is shown in Figure 4.5.

4.2.2 Uniform Flux Fracture

In this case, fluid enters the fracture at uniform flowrate per unit area of fracture face so that there is a pressure drop in the fracture. Features of uniform flux fracture are similar to the infinite conductivity case shown in Figures 4.4 and 4.5.

4.2.3 Finite conductivity fracture

In this case, fluid flows within the fracture and there is a pressure drop along the length of the fracture. Features of this case are shown in Figure 4.6.

75

Fig. 4.4: Pressure and Derivative for Well on A Single Vertical Infinite Conductivity Fracture.

Fig. 4.5: Effect of xf/xe on Well on a Single Vertical Infinite Fracture

76

Fig 4.6: Log – Log Graph of Data from Well with Finite Conductivity Fracture

4.3 DOUBLE POROSITY RESERVOIR

The double porosity reservoir is simply a fissured (naturally fractured) reservoir and is shown in Figure 4.7.

Fig. 4.7: Fissured Reservoir

77

The main feature of this reservoir is that the pore space is divided into two distinct media: the matrix, with high storativity and low permeability, and the fissures with high permeability and low storativity. Flow between the fissure and matrix can occur under pseudo steady state on transient state. However, the former is more common.

In addition to fissured reservoirs, double-porosity models can also represent layered reservoirs in which one layer has a permeability that is much higher than the other. This is shown in Fig 4.8. Fluid essentially reaches the wellbore through the layer with higher permeability.

Fig. 4.8: Layered Reservoir

Layered reservoirs are also modeled with double permeability model (Bourdet, 1985), but the features of double permeability model are similar to the double porosity model.

Warren and Root (1963), de Swaan (1976), Bourdet and Gringarten (1980) and Gringarten (1984) published double porosity solutions. Informations that may be deduced with the double porosity model are as follows:

k = permeability

s = skin factor

Cs = wellbore storage constant

ω = ratio of the storativity in the most permeable medium to that of the total reservoir

k2 >> k1

k1

78

λ = Inter porosity flow coefficient

Some parameters in the equations are defined as follows:

km = permeability of matrix or least permeable layer

kf = permeability of fracture or most permeable layer

V = ratio of the total volume of one medium (matrix or fissure) to the bulk volume

α = characteristics of the geometry of the interporosity flow.

Figure 4.9 shows features of a double porosity reservoir with pseudo-steady state interporosity flow. Features in Fig 4.9 are explained as follows:

1. At early time, only the fissures are detected. A homogeneous response corresponding to the fissure storativity and permeability may be observed.

( )( ) ( ) =

VC

VC + VC

t fissure (f)

t (f) t (m)

∅

∅ ∅fissure matrix

= r kkw2

m

f

α

79

Fig. 4.9 Features of Double Porosity Reservoir

2. When the interporosity flow starts, a transition period develops. This is seen as an inflection in the pressure response and a “valley” in the derivative.

3. At the end of the transition, the reservoir acts as a homogeneous medium, with the total storativity and fissure permeability.

A few things to note while analyzing test with double porosity models are as follows:

(a) Wellbore storage may mask all indications of heterogeneity.

(b) The depth of the transition valley is a function of ω. When ω decreases (low fissure storativity) the valley is more pronounced and the transition starts early.

(c) The time the transition ends is independent on ω.

(d) The time when transition occurs is a function of λ. When λ increases (higher km/kf ), the transition occurs earlier. The time the transition ends is proportional to 1/λ only.

(e) The value of ω can be less than or equal to one. The double porosity system

degenerate to single porosity system when ω = 1.

(f) The values of λ are usually small (≈ 10-3 to 10-10). If λ is larger than 10-3, the level of heterogeneity is insufficient for dual porosity effects to be important. The system then acts as a single porosity reservoir.

(g) If inter porosity flow is transient, the “valley” is less evident. 4.3.1 Practical Hints on Double Porosity Models

The double-porosity model can either be used for a fissured reservoir on a multi layer reservoir with high permeability contrast between the layers. As a result, it is not possible, from the shape of the pressure versus time curve alone, to distinguish between the two possibilities. The following practical hints will be of help in distinguishing the systems. Common features are also highlighted.

(a) If well is damaged, an increase in C, after an acid job and the resulting high value of wellbore storage constant are characteristic of fissured formation. This is because when the well is damaged, most of the fissures intersecting the wellbore are plugged and do not contribute to wellbore volume. On the other hand, there is no significant change in the wellbore storage constant following an acid job in a multi layer reservoir.

(b) Double-porosity reservoirs have skin value for non-damaged well that is lower than zero. In reality, double-porosity reservoirs exhibit pseudo skins, as created by hydraulic fractures. A skin of –3 is normal for non-damaged wells in formations with double-porosity behaviour. Acidized wells may have skins as low as –7, whereas a

80

zero skin usually indicates a damaged well. A very high wellbore-storage constant and a very negative skin should suggest a fissured reservoir, even if the well exhibits homogeneous behaviour.

(c) The parameters ω and λ may change with time for the same well depending on the

characteristics of the reservoir fluid. The reason is that ω and λ both depend on fluid properties, not just on rock characteristics. The parameters ω and will definitely change as pressure falls below bubble point.

4.4 BOUNDARY MODELS

It is impossible to cover all possible boundary models. Appendix C taken from Middle East Well Evaluation Review shows features of different analysis models including different boundary models.

Although it is not absolutely correct to use models derived during drawdown for buildup analysis, but for practical purposes, it is accepted. Pressure and pressure derivative obtained during buildup show similar features seen in drawdown tests. Figure 4.10 published by Economides (1988) for different models shows this.

4.5 MODEL SELECTION

Two key steps in the process for estimating reservoir properties from pressure/production data are as follows:

1. Selection of an appropriate reservoir model.

2. Estimation of parameters with the chosen model.

The selection of an appropriate model requires selecting appropriate set of material and energy balances for the physical processes involved, as well as the fluid properties and reservoir geometry. The problem in choosing the most appropriate reservoir model is that several different models may apparently satisfy the available information about the reservoir. That is, the models may be consistent with available geologic and petrophysical information and seem to provide more or less equivalent matches of the measured pressure/production data.

81

Fig. 4.10: Different Models used for Buildup Analysis

Watson et al (1988) suggested a method of model selection, which is summarized as follows:

1. Select candidate models that are consistent with all available information about the reservoir. A pool of candidates may be formed as a hierarchy of models as shown in Figure 4.11. The number of independent parameters to be estimated from the models is also shown.

2. Using a parameter estimation (automatic history matching) method, estimate the independent parameters.

3. Using the calculated independent parameters, calculate the expected pressure and production data.

82

4. Compare the calculated data with actually obtained data. The correct model is the one that minimizes the difference between calculated and actual data in the least square sense. Weighting factors can be included.

Although the model can be chosen at the end of Procedure 4, there is still need to find out whether a simpler model can be used. This is because pressure and production data are not known with certainty. Also with a simpler model, fewer numbers of unknowns are calculated. A model that has too many parameters for the given set of data will often result in parameter estimates that have large errors associated with them. The reason for this is that the estimation process using models with too many parameters tends to be poorly conditioned in that many different set of parameter values tend to give essentially equivalent fits to the data. Consequently small measurement errors may result in large errors in parameter estimates.

In deciding whether a simpler model can be used, Watson et al (1988) suggest using an F-test to find out whether the estimated parameters are very different from known values of such parameters for the simpler model. For example, if at end of Procedure 4, a

double-porosity model is chosen, calculated values of ω and λ are compared with 1 and 0 which correspond to a simpler single porosity model. A hypothesis test is done for chosen level of significance.

Fig 4.11: Model Hierarchy

Single Porosity With Skin

Dual Porosity

Dual Porosity with

Skin

Dual Porosity Infinite Acting

Single Porosity, Infinite Acting

Single Porosity, Infinite Acting,

With Skin

Dual Porosity

Infinite Acting

With Skin

83

EXERCISES

1. State the uses of pressure derivative plots. 2. Assuming reservoir is infinite, sketch the pressure derivative for the following

models: i ) Wellbore and skin model in homogeneous reservoir ii) Fractured model with well intersecting a single vertical fracture

iii) Double porosity model

3. Using buildup pressure and pressure derivative, show the distinguishing features between the following systems:

i) Well producing near a single fault and a well producing between two parallel faults. ii) Well producing in a reservoir with gas-cap and a well producing in a well without

gas-cap. iii) Well producing in a truncated channel and a well in a channel that is not truncated. iv) Well producing from an infinite system and well producing in a rectangle with

boundaries close to flow.

4. Categorise the following into well, reservoir and boundary models:

Wellbore storage and skin, Single fracture, leaky fault, horizontal well, double

porosity, composite, constant pressure.

84

5. SKIN FACTOR, sT The skin factor is useful in evaluating the well condition, but it is usually applied wrongly. In this chapter, we shall discuss the basic things about skin and how to use the skin factor properly.

What is Skin Factor? Factor used in BHP analysis to account for extra-ordinary pressure changes caused by flow around the wellbore Example of Normal (Ordinary) Pressure Change Around Wellbore

Open Hole Completion of Entire Interval (no restriction)

h t

Fig.5.1: Unrestricted Flow at Wellbore

p pqB

rr

x khw e

e

w= − −

µ ln

.7 08 10 3 (steady State) 5.1

p p

qBrr

x khw w

e

w= −−

−

_ln .

.

µ 0 75

7 08 10 3 (pseudo-steady state) 5.2

Example of Extra-Ordinary Pressure Change Around Wellbore

Partially Completed Well

h thp

Fig.5.2: Restricted Flow at Wellbore

Due to the “stampede” at the wellbore, the equation for bottom-hole flowing pressure becomes

85

p p

qBrr s

x khw e

e

wc

= −+

−

µ ln

.7 08 10 3 (steady state) 5.3

p p

qBrr s

x khw w

e

wc

= −− +

−

_ln .

.

µ 0 75

7 08 10 3 (pseudo-steady state) 5.4

where sc is the skin factor due to partial completion. Generally,

p p

qBrr s

x khw e

e

wT

= −+

−

µ ln

.7 08 10 3 (steady state) 5.5

p p

qBrr s

x khw w

e

wT

= −− +

−

_ln .

.

µ 0 75

7 08 10 3 (pseudosteady state) 5.6

where sT is the total skin due to all factors that cause extra-ordinary pressure drop around the wellbore. The total skin is calculated from BHP data. Note: 1) Given the same drawdown and positive skin, well with normal pressure around

wellbore will produce more than well with skin. 2) If the average pressure in the drainage areas are the same, both wells can produce at

the same rate if the flowing pressure in well with skin is lower by an amount equal to

∆pqB

khs

qB

x khss T T= = −

1412

7 08 10 3

.

.

µ µ. 5.7

This is shown in the Fig. 8. 3.

Pwf if no skin

Pwf if there is skin

∆∆∆∆ps

rw re

useful

drawdown

Fig.5.3: Effect of Total Skin on Wellbore Pressure (van Everdingen Model)

86

5.2 Causes of Skin

a) Invasion of region around wellbore by drilling fluid and fines b) Dispersion of clay c) Presence of mud cake and cement d) Stimulation e) Sand Consolidation f) Presence of high gas saturation around wellbore g) Partial well completion I) Limited Perforation j) Slanting of wells k) Gravel Pack l) Sand in wellbore if flow occurs through it m) Non-Darcy effects (caused by turbulence especially in gas wells) n) etc. 5.3 Classification of Pseudoskins

Factors that cause skin are classified under different pseudoskins as follows: 1) Pseudoskin due to Damage, sd: Includes Factors a to f and other factors that result to

alteration (relative to the native permeability of the formation) of permeability around the wellbore.

2) Pseudoskin due to Completion (also called Mechanical Skin) sc: Includes Factor g. 3) Pseudoskin due Perforation, sp: Includes Factor i. 4) Pseudoskin due to Slanted, ssw: Includes Factor j 5) etc. Each pseudoskin creates its own extra-ordinary pressure around the wellbore. This is shown in the figure for cases where the pseudoskins are positive.

∆∆∆∆pT

rw re

useful

drawdown

∆∆∆∆psc

∆∆∆∆psd

∆∆∆∆psppwf

p

Fig.5.4: Effect of Pseudoskins on Wellbore Pressure (van Everdingen Model)

Note:

1) The total pressure drop ∆pT caused by the total skin is the sum of pressure drop caused by individual pseudoskins

2) Stimulation can only reduce pressure loss caused by pseudoskin due to damage

87

3) Total skin may not be a good yardstick for determining stimulation candidates. It is better to used pseudoskin due to damage, but preferably (personal experience) use ratio defined as follows:

Rp

P p

sd

wf

=−

∆_ 5.8

where ∆psd is the pressure loss caused by skin due to damage and P pwf

_

− is the

drawdown. Table 5.1 . shows the pressure loss due to skin and calculated R ratios. Table 5.1: Skin and Pressure Losses from Wells

WELL TEST NO

1 2 3 4 5 6 7

Total Skin 18.4 3.5 26.4 192 6.1 196 63.6

∆pT, psi 57.86 174.2 216.6 723 9.9 119 367

Damage Skin 3.34 9.1 16.6 45 -17.6 66 3.15

∆psd, psi 30.72 135.2 204 677 -17.5 115 190

R 0.34 0.71 0.75 0.91 - 0.95 0.47

Inferences from this table are as follows: 1) Using the magnitude of total skin for ranking the priority for stimulation, the wells

will be ranked as follows: 6, 4, 7, 3, 1, 5, 2. Note that Well 5 does not require stimulation because the skin due to damage is negative.

2) Using the damage skin or R ratio, the wells will be ranked as follows: 6, 4, 3, 2, 1,7,

5. The added advantage of using R ratio is that it takes values from 0 to 1 and from experience we do not recommend stimulation if R ratio is less than 0.6. Tables 2 and 3 comparing pre-stimulation and post-stimulation surveys from two wells illustrate this.

Table 5.2: Results of Pre-Stimulation and Post-Stimulation Tests (Well A)

PARAMETER PRE-

STIMULATION

POST-

STIMULATION

Rate, STB/day 568 570 Drawdown, psi 68.16 168.72 Total skin,s 9.82 4.85

∆p due to total skin, psi 32.45 59.7

Mechanical skin 2.2 2.2 Damage skin 7.57 1.3

∆pdamageskin, psi 25.02 32.3

R -

∆pdamageskin/Drawdown

0.367 0.191

Oil PI, STB/day/psi 2.37 3.08

88

Table 5.3 : Results from Pre- and Post-Stimulation Analysis (Well B)

Parameter Pre-Stimulation Post-Stimulation

Rate, STB/day 66 539

Total skin, s 46.74 8.58

Mechanical skin 0.2 0.2

Damage skin 46.57 8.4

Drawdown, psi 365.63 212.18

∆pdamageskin, psi 259.53 102.2

R -

∆pdamageskin/Drawdown

0.71 0.48

Oil PI 0.15 2.14

From the R ratios calculated from pre-stimulation, Well A was not a clear stimulation candidate (R < 0.6) while Well B was a good candidate (R > 0.6). The post-stimulation oil gains also confirm this. 5.4 Calculations of Pseudoskins

a) Pseudoskin due to Damage, sd

rw r1 re

kk 1

Fig.8.5: Damaged and Undamaged Zones

sk

k

rrdw

= −

1

11 ln 5.9

if k > k1, sd > 0 (damage) and if k < k1, sd < 0 (stimulation) Some implications of Equation. 5.9 that will give you a sense of magnitude is given in Table 5.4.

89

Table 5.4: Effect of Permeability Alteration on Pseudoskin Due to Damage

k1 (% of k) Depth of Damage, r1, in (rw = 6 in)

8 24 60 120

90 0.03 0.15 0.25 0.33

50 0.29 1.39 2.30 3.00

10 2.59 12.47 20.7 26.96

5 5.46 26.34 43.7 56.9

Note that in high permeability formation, permeability reduction can be large if fines block the pore throat of the porous media. The problem with the equation for calculating pseudoskin due to damage is that depth of invasion and permeability of the damage zone are not known. Hence, we now estimate the damage skin from the total skin. This will be discussed. An equation for determining the pseudoskin due to consolidation is given as:

sr r

h

k k

k

r

rdc

c w

p

h c

c

c

w

= −−

−

1 0 2. ln 5.10

Comparing Equations 8.9 and 8.10, rc = r1, kh = k and kc = k. The term hp is the perforated interval b) Pseudoskin due to Partial Completion (Mechanical Skin), sc: This depends on

i) ratio of completed interval hp to formation thickness, hT (most important parameter) ii) location of completion relative to total thickness iii) ratio of vertical to horizontal permeability

hp hT

Fig8 6: Effect of Partial Completion on Flow

Table 5.5 gives a sense of magnitude of how sc varies with hp/hT. Table 5.5: Effect of Penetration Ratio on Mechanical Skin

hp/hT 0.1 0.2 0.3

sc 36.9 16.4 9.6

90

There are many sources for obtaining sc, but Odeh’s equation is preferred. Photocopy containing his article and equation is enclosed. Saidikowski’s (SPE 8204 1979 SPE Conference) equation is simpler and gives reasonable answer for practical purposes. The equation is

sh

h

h

r

k

kc

T

p

T

w

H

v

= −

−

1 2ln 5.11

c) Pseudoskin due to Slanting of Well, ssw: This is always negative as slanting of wells will generally enhance productivity because more wellbore length will be exposed to flow. This is shown in Figure . The limiting case is a horizontal well.

Length available

for fluid entry

Fig. 5.7: Length Available to Flow - Vertical and Slanted Well

The pseudoskin due to slanting of well depends on the angle of slant, α , and ratio of total thickness to wellbore radius, hT/rw.. It can be approximated using the equation published by Cinco, Miller and Ramey. The equation is

sh

rsw

T

w

= −

−

α α41 56 100

2 06 1 865. .

log 5.12

and it is valid for 0 < α < 75o and h T/rw > 40 (i.e. about hT > 12 ft). Typical values of ssw for different hT/rw and angles of slant are given in Table 5.6. Table 5.6: Pseudo Skin Due to Slanting of Well

hT/ rw

hT (ft) rw = 0.4 ft

ANGLE (Degrees)

20 30 40 50 60 75

50 20 -0.18 -0.43 -0.79 -1.26 -1.85 -2.95

100 40 -0.23 -0.53 -0.95 -1.51 -2.19 -3.47

200 80 -0.27 -0.62 -1.11 -1.75 -2.53 -3.99

300 120 -0.30 -0.67 -1.21 -1.89 -2.73 -4.29

400 160 -0.32 -0.71 -1.27 -1.99 -2.88 -4.51

91

Note: If k k defineh

r

h

r

k

kH v

T

w

T

w

H

v

≠ ≡,

A graph for estimating pseudoskin due to slanting of well is enclosed. d) Pseudoskin due to Perforation, sp: This depends on many parameters, but more importantly on perforation penetration and perforation density. Figure illustrates how this contributes to skin.

Fig.5.8: Flow Convergence to Perforation

The skin due to perforation can be positive (for shallow penetration) or negative (for deep penetration). The enclosed chart by Harris will be used to have a sense of magnitude. The best value is about -3 and the worst value is about 10. In calculating the pseudoskin due to perforation, most models (including Harris) neglect the pressure drop due to flow in the perforation tunnels. This pressure drop could be substantial and contribute to skin if tunnel is filled with sand or gravel. This is because there could be turbulent flow within the tunnels. Deductions from studies that consider pressure losses within the perforation tunnels are as follows: i) There is optimum perforation length for minimum skin. This is shown in Fig 5.7. The productivity ratio used in the figure is defined as

PR

rr

rr s

w

wp

=+

ln

ln

1

1

5.13

Based on this, the skin caused by perforation tunnel being filled with gravel can be between 13 to 30. ii) Minimum skin due to perforation is smallest for highest shot density. iii) Minimum skin depends on gravel type, viscosity, rate openhole/casing diameter, etc. 8.5 Relationship between Total Skin, sT and Pseudoskins

1) Full Penetration of Entire Interval: s T = sc + sd + sp + ssw + .. 5.14 2) Partial Penetration a) Shallow Damage Skin (Damage restricted to perforated interval)

92

[ ]s sh

hs s s sT c

T

p

d p sw i= + + + + 5.15

b) Deep Damage Skin (Damage covers entire total skin)

[ ]s s sh

hs s sT c d

T

p

p sw i= + + + + 5.16

If total skin is known (from BHP test result) and other pseudoskins calculated, the pseudoskin due to damage can be deduced from these equations. 8.6 Pressure Change Due to Skin

The pressure change caused by skin can be calculated as follows: 1) Total Skin, sT:

∆pqB

khss

T

T=1412. µ

5.17

2) Pseudoskin Due to Partial Penetration (Mechanical Skin), sc:

∆pqB

khssc

T

c=1412. µ

5.18

3) Pseudoskin Due to Damage,sd: a) Shallow Damage Skin:

∆pqB

khssd

p

d=1412. µ

5.19

b) Deep Damage Skin:

∆pqB

khssd

T

d=1412. µ

5.20

4) Other Pseudoskins, si:

∆pqB

khssi

p

i=1412. µ

5.21

Note that 7.08 x 10-3 is the inverse of 141.2.

93

8.7 Relationship Between Skin and Well Inflow Quality Indicator (WIQI)

WIQIPI

PI

actual

ideal

= 5.22

where:

PIx kh

Br

rs

actual

e

w

T

=− +

−7 08 10

0 75

3.

ln .µ 5.23

and

PIx kh

Br

rs

ideal

e

w

c

=− +

−7 08 10

0 75

3.

ln .µ 5.24

The mechanical skin is included in defining the ideal productivity index (PI) because it is “permanent” and can only changed if the completion is changed. From Equations 5.22 to 5.24, we have

WIQI

r

rs

r

rs

p p

p

e

w

c

e

e

T

wf sd

wf

=− +

− +≈

−ln .

ln .

0 75

0 75

∆ ∆∆

5.25

Exercises

1) A well was completed with a 7 in diameter casing diameter as shown in the Fig. 5.9

50 ft

150 ft

Fig. 5.9: Partially Completed Well

The casing has a perforation density of 4 shots/ft and 120o phasing. It is estimated that the formation penetration is 15 inches and total skin determined from BHP analysis is 25. Considering only the pseudoskin due to partial completion and skin due to perforation, Calculate the following: a) Skin due to damage assuming damage is shallow. b) Skin due to damage assuming damage is deep. c) Pressure change due to total skin. d) Pressure change due to damage skin due to damage assuming damage is shallow e) Pressure change due to damage skin due to damage assuming damage is deep f) Is there any difference between answers from d and e.

94

(Use k = 1000 md, q = 1000 STB/D, B = 1.10 rb/STB, µ = 0.6cp) 2) Repeat Exercise 1 for case where well is slanted at angle 60o. 3. Five pages of a report on a test considered unreliable are enclosed. The calculated

parameters were not entered in the provided slots, but just written down. Even though the results are considered unreliable, but someone still wants to know the skin due to damage, pressure loss due to it and WIQI. Your job is to calculate required parameters. All data required can be found in the enclosed report.

Skin Correction: In some analysis, it is difficult to match the correct flowing pressure before shut-in. This is usually due to rate or time problem. The case where the real flowing pressure was not matched is shown in the following simulation plot.

P

real data

simulated profile

(Analysis results)

∆∆∆∆p = analysis pwf - actual pwf

Time

flowrate

In a situation like this, the analysis skin was calculated with the following equation:

sP p t

m

k

c r

hr wfa

t w

=− =

−

+

11510

3231

2.( )

log .∆

φµ

where:

pwfa(∆t=0) = flowing pressure just before shut-in (used for analysis)

m = semilog straight line slope=162 6.

/qB

khpsi cycle

µ.

As the analysis flowing pressure is greater than the actual flowing the actual skin will be

s + ∆s where

∆∆ ∆

sp t p t

m

wfa wfr== − =

1151

0 0.

( ) ( )

and

pwfr(∆t=0) = real flowing pressure just before shut-in

??????????????????

95

6. ANAYSIS OF BOTTOM-HOLE PRESSURE TESTS

There are three methods of analyzing bottom-hole pressure tests: conventional, type-curve and regression (automated type-curve) methods. To the “layman” the methods can be likened to three ways of getting an overall work dress for a worker. The conventional method is like going to a tailor who takes measurements at three points of the body and with just three measurements, the overall is made. Note that with the exception of the three measurements, other dimensions used in making the dress are extrapolations or interpolations. Also, if the tailor uses the dimension of the waist to be that of the shoulder, the dress will not fit. This underscores the importance of where the measurements were taken. The type-curve method can be likened to just walking into a shop and choosing from already made dresses. The problem here is that the exact size may not be available and one has to manage with the next lower of higher size. In high permeability-high skin Niger Delta formations, we have observed that available type-curves do not match the pressure data. The regression method is simply an automated type-curve method where the exact size that takes into account all dimensions is made. The problem here is that if you are not standing properly while the measurements are taken, the wrong overall will invariably be made. This implies that wrong pressure data will create problems during analysis with regression method if such data are not removed. The advantage of the regression method is that the correct-sized dress will be made and therefore there is no problem with the high permeability-high skin Niger Delta formations. In this section we shall discuss methods of analyzing drawdown, buildup and interference tests. Emphasis will be on the conventional and type-curve methods. However, we shall show that in an ideal situation, the three analysis methods give comparable results. 6.1 Conventional Method

Bottom-hole pressure tests can be analyzed using the conventional method. The conventional analysis is based on locating the straight lines that are characteristic of the different flow phases. The desired parameters are then calculated from the slopes of the straight lines. This implies that if you put the wrong straight lines, you will get the wrong answers. The procedure for analyzing drawdown and buildup test using conventional method is as follows: 6.1.1 Drawdown Test

The transient state equation during a drawdown test is given as follows:

96

+−

+−= s

rc

kt

kh

qBPP

wt

iwf 87.023.3loglog6.162

2φµµ

6.1

Equation 6.1 forms the basis of calculating permeability and skin from drawdown test using data not influenced by wellbore storage effects. The procedure for the analysis is oultined.

Procedure for Analyzing Drawdown Test 1. Locate pressure influenced by wellbore storage effect and calculate wellbore storage constant. How?

(a) Make a graph of (Pi -pwf) versus t on a log-log paper (b) Locate data with strong wellbore storage effect on the unit slope line.

(c) Calculate the wellbore storage constant, Cs, with ∆p= (Pi -Pwf) and t taken from a point on the unit slope line. Use the following equation.

Cs = qBt bbl

psi24∆ (rb/psi) 6.2

(d) Locate the data not strongly influenced by wellbore storage effect. Use the

gentle slope rule or the 10t* to 50t* rule.

2. Locate data obtained during the transient state phase and calculate permeability and skin. How?

a) Plot pwf versus log t. Use a semilog paper with time graphed on the log scale axis.

b) With the knowledge of the time when wellbore storage effect has died down

completely, put the correct straight line that represents the good transient state

behaviour.

c) Determine the slope of the semilog straight line, m, and calculate the permeability

using the equation

k (mD) = mh

qBu6.162− 6.2

Note that the slope, m, is negative and therefore permeability, k, is positive.

d) Calculate the total skin using the following equation:

s = 1.1513( )

+−−

232.3log2

1

wt

ihr

ruc

k

m

PP

φ 6.3

97

where: P1hr is the pressure taken from the straight line portion when flowing time is equal to one. Exercise 6.1:. The log-log and semilog plot of the pressure-time obtained during a drawdown test are shown in Fig 6.1 and Fig 6.2 (leave space for these figures ???? ) Data from the test are shown in Table 6.1 ??? (ask me). Draw the necessary straight lines and calculate Cs, k and s. Other information required for the analysis are as follows:

rw = 0.5 ft h = 50ft Bo = 1.125 ct = 20 x 10-6 psi-1 µ = 0.6 cp φ = 0.25. q = 1000 STB/D

Table 6.1: Drawdown Data

Time, hr Flowing Pressure, psi Time, hr Flowing Pressure, psi

.00000000E+00 3183.763 .10176000E+01 3123.157

.10000000E-03 3183.245 .12576000E+01 3122.951

.80000000E-03 3179.758 .14976000E+01 3122.781

.20000000E-02 3174.302 .17376000E+01 3122.638

.48000000E-02 3163.727 .19776000E+01 3122.513

.96000000E-02 3150.896 .22176000E+01 3122.402

.12000000E-01 3146.317 .24576000E+01 3122.303

.18240000E-01 3138.186 .28176000E+01 3122.172

.21600000E-01 3135.356 .31776000E+01 3122.056

.27840000E-01 3131.839 .35376000E+01 3121.953

.32400000E-01 3130.196 .38976000E+01 3121.860

.39600000E-01 3128.530 .42576000E+01 3121.776

.44400000E-01 3127.814 .46176000E+01 3121.698

.55680000E-01 3126.796 .49776000E+01 3121.626

.60000000E-01 3126.553 .53376000E+01 3121.559

.88800010E-01 3125.731 .56976000E+01 3121.496

.11040000E+00 3125.446 .60576000E+01 3121.438

.13440000E+00 3125.224 .64176000E+01 3121.383

.17760000E+00 3124.928 .67776000E+01 3121.330

.24960000E+00 3124.572 .71376000E+01 3121.281

.37440000E+00 3124.154 .74976000E+01 3121.234

.53760000E+00 3123.788 .79536000E+01 3121.177

.77760000E+00 3123.421

Although you have gone through the exercise manually, there are computer programmes that do the graphing and calculations in a twinkle of an eye. I used the programme in

98

analyzing this test and the results obtained are as follows: Cs = 0.93 x 10-2, k = 991.2, and s = 24.72.

In a situation where the drawdown test is run long enough, the volume drained by the test well can be determined. The equation that forms the basis of this analysis is as follows:

+++−= s

Cr

A

kh

qBt

hAc

qBPP

Awt

iwf 22458.2

lnln2.141234.0

2

µφ

6.4

The implications of this equation are as follows: (a) A graph of Pwf versus t gives a straight with a slope,

thAc

qBm

φ234.0

−= (psi/hr) 6.5

(b) From the slope, the drainage volume, φhA, is calculated. Exercise 6.2: Table (6.2 ??? ask) shows pressure data from a drawdown test in a well in the center of a square which was tested long enough to reach pseudo-steady state. Using the conventional method and data in Table 6.3, analyze the test and calculate the following: a) Formation permeability b) Skin factor c) Wellbore storage constant d) Drainage area table 6.2 here ?? Table 6.3: Parameters used for Analysis

Parameter Value

Wellbore radius, rw, ft 0.5

Total compressibility, ct, psi-1 20 x 10-6

Porosity, φ 0.25

Oil formation volume factor, Bo, rb/STB 1.125

Oil viscosity, µ, cp 0.6

Production rate, q, STB/D 1000 Formation thickness, h, ft 50

Solution: Plots from the analysis and results are shown in Fig 6.3. ????? ask 6.1.2 Buildup Test

Two forms of transient state equation can be used in the analysis of buildup test. The first equation was developed by Horner (1951) by assuming that the test well produced

99

for a while (late-time state was not reached) before the well is shut in. The second form of the equation was developed by Miller, Dyes and Hutchinson (1950) by assuming that the test well produced for a long time (late-time state was reached) before the well was shut in. Based on the assumptions used in developing the equations, their applications were limited. For example, the Horner equation was used for new wells while Miller, Dyes and Hutchinson (MDH) equation was used for old wells. However, Ramey and Cobb (1971) have shown that the Horner equation is superior even when it is used for old wells. On the other hand Agarwal (1980) has shown that MDH equation can be used for new wells by using an equivalent shut-in time defined as follows:

∆+

∆=∆

p

eq

t

t

tt

1

6.6

Parameters in Eq. 6.6 are defined as follows:

∆teq = equivalent shut-in time to be used for analysis

∆t = actual shut-in time tp = production time before shut in

Discussions on the Horner and MDH equations and procedure for using them follow:

Horner Equation The Horner equations for infinite and developed systems are given as follows:

∆

∆+−=

t

tt

kh

qBPP

p

iws log6.162 µ

for infinite reservoir 6.7

and

∆

∆+−=

t

tt

kh

qBPP

p

ws log6.162* µ

for finite or developed reservoir 6.8

Note that as shut-in time, ∆t, increases, the Horner time {(tp + ∆t)/ ∆t} decreases. At infinite shut-in time, Horner time is unity (1).

MDH Equation The MDH equation is given as follows:

100

+

+∆+=

−

A

rC

rc

kt

kh

qBPP wA

wt

ws

2

2log

000264.0loglog

6.162

φµµ

6.9

Unknown parameters in Eqs. 6.7 to 6.9 are defined as follows:

−

P = Average pressure P* = false pressure Pi = initial pressure CA = shape factor or geometric factor A = drainage area The shape factor depends on the location of the well with respect to the boundary. Typical shape factors are in Table 6.4. Table 6.4: Reservoir Geometries and Shape Factors

Geometry and Well location Shape Factor

Well in center of circle

31.62

Well in center of square

31.6

2 Well in the center of 2 to 1 rectangle

21.8369

1

2 Well in a quadrant of 2 to 1 rectangle

2.0769

Shape factors for other reservoir geometries and well location are given by Earlougher (1977). Irrespective of the equation used for analysis, the procedure for calculating wellbore storage constant and skin is the same. We shall therefore present a general procedure for analyzing buildup test and highlight the variations resulting from the use of any of the equations (MDH or Horner)

1

101

Procedure for Analyzing Buildup Test

Calculation of Wellbore Storage Constant 1. Locate pressure influenced by wellbore storage effect. How?

(a) Make a graph of ∆p = ( )[ ]p p tws wf p− versus ∆t on a log-log paper.

(b) Locate data with strong wellbore storage effect on the unit slope line.

(c) Calculate the wellore storage constant, Cs, using ∆p and ∆t from a point on the unit slope line and the following equation:

Cs = qB t

p

∆∆24

rb/psi 6.10

(d) Locate the data not strongly influenced by wellbore storage effect. Use the

gentle slope rule or the 10∆t* (1 cycle) to 50∆t* 1.5 cycle) rule.

Calculation of Permeabilty MDH Plot

2. Plot pws versus log ∆t. Use a semilog paper with time graphed on the log scale axis. 3. With the knowledge of the time when wellbore storage effect has died down

completely, put the correct straight line that represents the good transient state behaviour.

4. Determine the slope of the semilog straight line, m, and calculate the permeability using the equation.

k (mD) = 162 6. qBu

mh 6.11

At the end of this step, move to Step 8 to calculate skin factor.

Horner Plot

5. Plot pws versus

∆

∆+

t

tt plog . Use a semilog paper with Horner time graphed on the

log scale axis. The graph can be made with Horner time increasing from left to right to left. The later is usually preferred as it gives a graph with shut-in time increasing from left to right.

6. With the knowledge of the time when wellbore storage effect has died down completely, put the correct straight line that represents the good transient state behaviour.

7. Determine the slope of the semilog straight line, m, and calculate the permeability using the Eq. 6.11.

8. Extrapolate the straight line to Horner time equal to 1 (infinte shut-in time) and read off P* or Pi. They are needed for estimating the average pressure in a well that was not shut in long enough to reach average pressure.

102

Calculation of Skin

9. The correct equation is given as

+

++

−

−= 23.3

1loglog

)(1513.1

2

1

p

p

wt

pwfhr

t

t

rc

k

m

tPPs

φµ 6.12

However, if production time before shut-in, tp >> 1, Eq. 6.12 degenerates to

+

−

−= 23.3log

)(1513.1

2

1

wt

pwfhr

rc

k

m

tPPs

φµ 6.13

In Eqs. 6.12 and 6.13, P1hr is the pressure taken from the straight-line portion of the

semilog plot at shut-in time, ∆t = 1 hour. Note that shut-in tome of 1 hr corresponds to Horner time of (tp + 1). Note on Production Time before Shut-in, tp

For a well that produced at constant rate (unusual) before shut in, tp, is the actual production time before shut in. This is illustrated in Fig 6.4. Rate Time Fig.: 6.4: Actual Production Time Before Shut in Equal to tp

If a well did not produce at constant rate (usually the case) before shut in, tp can be calculated from cumulative production provided that a stable flow rate was attained during the flowtest prior to shutting in. The equation for calculating tp is given as follows:

s

p

pq

Nt

∆= 6.14

where ∆Np = Cumulative production since last shut in qs = stabilized rate prior to shutting in This case is illustrated in Fig 6.5.

q

t = tp

103

Fig.: 6.5: Actual Production Time Before Shut inis not Equal to tp

Experience has shown that for wells that produced for a long time (reached pseudo-steady state or steady state) before shut in, tp, can be replaced by a time that is at least the time to reach pseudo-steady state, tpss. The Horner time will simply be defined as as [(tpss

+ ∆t)/ ∆t]. For high permeability formation obtained in the Niger Delta, we normally use 1000 hours.

Exercise 6.3: Figures 6.6 to 6.8 log-log and semilog plots of pressure-time data obtained during a buildup test. Draw the necessary straight lines and calculate Cs, k and s. Other information required for the analysis are as follows:

rw = 0.5ft h = 50ft Bo = 1.125 Ct = 20 x 10-6 psi-1 µ = 0.6 cp and φ = 0.25. Q = 1000 STB/D tp = 1000 hr.

Answers: k = 1000 mD, s = 25 and Cs = 0.009 rb/psi

Calculation of Average Pressure For wells that were not shut-in long enough to average pressure, the average pressure can be obtained if the transient state was attained. An example of such a well is shown in Figure 6.9.

qs

Rate

Time t ≠ tp

Good Transient

Log (t +∆t/∆t) 1

Pws

P*

104

The average pressure is obtained based on the fact that P* is related to the average pressure by the equation

)(303.2

* PDADMBH tPm

PP −=−

6.15

where −

P = Average pressure P* = False pressure PDMBH = Dimensionless pressure defined by Mathews, Brons and Hazebroek (1954) m = Slope of the semilog straight line tpDA = Dimensionless (based on area) production time before shut-in

Ac

kt

t

p

φµ000264.0

= Oilfield units.

The PDMBH values depend on geometry of reservoir, well location and dimensionless production time based on area. The values for some cases can be obtained from Fig 6.10. Leave a page for this figure ????

Fig 6.10: Mathews, Brons and Hazebroek Dimensionless Pressure, PDMBH

Note that Horner plot and Mathews, Brons and Hazebroek dimensionless pressure, PDMBH Were used to obtain the average pressure. The Dietz (1965) method which involves the used of MDH plot in obtaining average pressure will not be discussed. The values of PDMBH can also be calculated at certain periods using the following

equations:

1. Early production time before shut-in (tp < tpss)

PDMBH

= 4πtDA 6.16

2. Long Production time before shut-in (tp ≥ t pass ) P

DMBH = In CAtpDA 6.17

Fig. 6.9: Horner Plot

105

Confirm the correctness of these equations with Fig 6.10. Some facts about average pressure are as follows:

(a) For infinite system, p = p* = pi

(b) If cumulative production is not sufficient to reduce the average pressure significantly, p* is still a good estimate of average pressure.

(c) Average pressure decreases with increase in production time before shut in. This is shown in Fig 6.11

leave space here???? ask

Fig 6.11: Relationship between Average Pressure and False Pressure

(d) The average pressure calculated from a test well is the average pressure in the drainage area of the test well. The reservoir average pressure is then calculated as follows:

reservoirP =

p qii=1

N

i

tq

∑ 6.18

where

pi = average pressure from Well i.

qi = production rate from Well i.

qt = total production from all the wells in the reservoir.

In situation where all the wells were not tested, the average pressure in the untested well

can be estimate using the p - q relationship from the tested well. You have to interpolate

or extrapolate. 6.1.3 Interference Test

The equation that forms the basis of analyzing interference is as follows:

−

+−= 23.3loglog

6.162),(

2rc

kt

kh

qBPtrP

t

i φµµ

6.19

106

Equation 6.19 predicts the transient state pressure at an observation point that is r distance from the active well. This equation does not include wellbore storage and skin in either the active or the observation well. The procedure for using Eq 6.19 for interference test analysis is as follows: (a) Graph P(r,t) (pressure recorded in the observation well) versus time, t, on a

semilog graph. (b) The transient state data will fall on a straight line. Permeability can be calculated

with straight line slope, m, with the equation:

k (mD) = 162 6. qBu

mh 6.20

(c) Porosity or storativity can then be calculated from the equation

23.3log 1 +−

=

m

PP

c

k hri

tφµ 6.21

In Eq 6.21, P1hr is the pressure at time, t = 1 hr, taken from the straight-line portion of the curve. 6.2 Type-Curve Method

Gringarten (1987) defined a type curve as a graphic representation of the theoretical

solution of the fluid flow equation used in representing the test well and the reservoir

being tested. For a constant-pressure test, the theoretical solution is presented in the form

the change in production rate with time. This is usually used in decline curve type

curves. For a constant-rate test, the theoretical solution is presented as change in pressure

at the bottom of the well as a function of time. This form is normally used in bottom-hole

pressure test analysis. Other types of response are also used, such as the time derivative

of the bottom-hole pressure.

Type curves are derived from solutions to the flow equations under specific initial and

boundary conditions. For the sake of generality, type curves are usually presented in

dimensionless terms, such as a dimensionless pressure vs. a dimensionless time. A given

interpretation model may yield a single type curve or one or more families of type curves,

depending on the complexity of the model.

107

Type-curve analysis involves finding a type curve that “matches” the actual response of

the well and the reservoir during the test. The reservoir and well parameters, such as

permeability and skin, can then be calculated from the dimensionless parameters defining

that type curve.

The match can be found graphically, by physically superposing a graph of the actual test

data with a similar graph of the type curve(s) and searching for the type curve that

provides the best fit. Alternatively, an automatic fitting technique involving a linear or

nonlinear regression can be used.

Figure 6.12 taken from Gringarten (1987) gives an example of a graphic type-curve

match. The graph of the data is positioned over the graph of the type curves, with the

axes kept parallel, so that the test data match one of the type curves. Reservoir parameters

are calculated from the value of the dimensionless parameter defining the type curve

being matched and from the x and y axis shifts.

Fig 6.12 here ask ????

Fig 6.12: Graphical Type-Curve Matching Process

There are many kinds of type-curves, but we shall only discuss the Theis (1935) type-

curve used for analyzing interference test and Bourdet et al (1983) type-curve used for

analyzing buildup and drawdown tests. The type curves are shown in Fig. 6.13 and Fig

6.14. The Theis type-curve is also known as line source solution.

Leave a page for this fig 6.13

Fig 6.13: Theis Type-Curve

Leave a page for this fig 6.14

Fig 6.14: Bourdet Type-Curve

108

6.2.1 Basis of Type-Curve Analysis

The basis of type-curve analysis will be illustrated using the interference type case and

application of type-curve will be extended to buildup tests.

Interference Test: The basis of type-curve analysis is the relationship between

dimensionless parameters and the non-dimensionless parameters used in interference test

type-curve. The relationships are shown for dimensionless pressure and time as follows:

PD = kh[ p]

141.29B

∆µ 6.22

t

r r

D

D

2 2 =

0.000264kt

c tφµ 6.23

Taking log of Eqs 6.22 and 6.23 gives

log PD = log ∆p + log µqB

kh

2.141 6.24

log t

r r

D

D

2 2 = log t + log

0.000264k

c tφµ 6.25

Note that log(kh/141.2qBµ) and log(0.000264k/φµctr2) are constants and therefore

regarded as shifts.

The implications of Eqs 6.24 and 6.25 are as follows:

(1) On a log-log paper, PD is directly related to ∆p and t

r

D

D

2 is also directly related to t.

The parameters of interest in the analysis constitute the “shifts” in the relationships.

109

(2) A graph of ∆p versus t on a log-log paper will look exactly like that of PD versus

t

r

D

D

2 on a similar log paper.

(3) ∆p - t plot can then be overlain on the Pr

D

D

- t D2 type curve. This is known as type-

curve matching process. Normally, the field data are plotted on transparent paper so

that the match can easily be made. Details on the process of type-curve matching

will be demonstrated in the class

On matching the curves, any point on the type-curve ( PD and tD/rD2) and the

corresponding point on the field data (∆p and t ) is known as a match point. With values

of the match point, the desired parameters can be calculated with the following equations.

K = 1412. qB

h p m

µ .

PD

∆

6.26

φµ

Cr

t

D m

= 0.000264k

r

t

t2

D / 2

6.27

Noe that Equations 6.26 and 6.27 are simply rearranged forms of dimensionless

parameters defined by Eqs. 6.22 and 6.23.

Buildup Test (Pressure Match): Using Bourdet’s type-curve, a match could be obtained

using the pressure type-curve or the pressure derivative type-curve. Both type-curves are

in Fig. 6.14. For pressure match, dimensionless parameters defined by Bourdet et al

(1983) for analysis of buildup test are as follows:

PD = Kh

141.2qB p

µ∆ 6.28

t

C

D

D

= 0.000295 Kh

t

Cµ∆

6.29

110

Chr

D

w

= 0.8936 C

C tφ 2 6.30

The procedure for using the type-cure is as follows:

1. Graph ∆p = Pws – Pwf (tp) versus ∆t on a transparent paper with same log cycle

dimensions as the type-curve.

2. Match the real graphed data with a type-curve

3. Obtain the match point values (∆p, ∆t, PD tD/CD) and the CDe2s of the representative

type-curve

4. With the match point values, calculate

a k from Eq. 6.28

b C from Eq. 6.29

c CD from Eq. 6.30

d s using CD and CDe2s of the representative type-curve.

Buildup Test (Derivative Match): For this case, we give you the opportunity to outline the procedure. 6.3 Comparison of Analysis Methods

Under ideal conditions, all analysis method will yield close results. The ideal conditions includes good data acquisition, test run long enough to reach required phases and also accurate identification of the phases. Table 6.5 show results obtained by analyzing drawdown test using conventional and type-curve methods. Data for this test are in Table 6.1. In this case, both analysis methods gave close results. Table: 6.5 DrawDown Analysis Results

Parameter Conventional Method Type-Curve Method

Permeability, k, md 991.2 1000 Skin, s 24.72 25 Wellbore Storage, Cs, rb/psi 0.93 x 10-2 0.9 x 10-2

Some comments on the analysis are as follows: 1. Both analysis methods gave close results, but the type-curve method is more accurate.

We know about this because the analyzed pressure data were generated with k = 1000md, s = 25 and Cs = 0.9 x 10-2 rb/psi. This implies that the type-curve method gave exact answers.

111

2. The test was run long enough as the transient state is clearly shown on the derivative plot.

Table 6.6 shows results obtained by analyzing drawdown test using conventional and type-curve methods. The derivative for this analysis is shown in Fig 6.15. Both analysis methods did not give close results because the transient state is not apparent. We cannot say which resulta are correct because data are actual field data.

Table 6.6: Buildup Analysis Results

Parameter Conventional Method Type-Curve Method

Permeability, k, md 65 293 Skin, s 12 75

The test well whose data are shown in Fig 6.15 was producing 129 (STB/day) and the gas-oil ratio (GOR) was 882 SCF/STB. With such a low production and high GOR, what do you expect?. Wellbore storage effect will last for unusually long period and the transient state phase will be masked.

Leave space for fig 6.15 ask

Fig 6.15: Derivative from Test Data

Note that with the type-curve method, we do not look for straight lines corresponding to the different phase. Rather, we do a curve-fit to match the pressure responses with the ideal responses for the entire phases. When an acceptable curve-fit is obtained, the desired reservoir parameters can be calculated.

Problem Set 6

1 Pressure-time data from a buildup test are given in Table 6.7. The well and reservoir data are as follows:

rw = 0.5 ft, h = 50 ft, q = 1000 STB/day, ct = 20 x 10-6 psi-1, φ = 0.25, µo = 0.6 cp, Bo

= 1.125 rb/STB, tp = 1000 hr A. Using the Horner plot determine the following: a) effective permeability to oil b) total skin factor c) wellbore storage constant d) average pressure assuming that well is in the centre of a square with a drainage area

of 80 acres

112

Table 6.7: Buildup Data

∆∆∆∆t, hr pws, psi ∆∆∆∆t, hr p, psi

0 3183.76 0.0888 3241.80 0.0001 3184.28 0.1776 3242.60 0.0008 3187.77 0.3774 3243.37 0.0020 3193.22 0.5376 3243.74 0.0048 3203.80 0.7776 3244.10 0.0120 3221.21 1.0176 3244.37 0.0278 3235.69 1.2576 3244.57 0.0557 3240.73 1.9776 3245.01

2) Confirm results obtained in Question 1(a), (b) and (c) using the MDH method.

3) The pressure drop, ∆p, and the derivative of the pressure drop, ∆p’, from a buildup test are given in Table 6.8. The well and reservoir data are as follows:

rw = 0.5 ft, h = 50 ft, q = 1000 STB/day, ct = 20 x 10-6 psi-1, φ = 0.25, µo = 0.6 cp, Bo

= 1.125 rb/STB, tp = 1000 hr. A. Using the Horner plot determine the following: a) effective permeability to oil b) total skin factor c) wellbore storage constant B. Repeat the analysis using type-curve analysis and pressure derivative data Table 6.8: Buildup Data

∆t, hr ∆p, psi ∆p’ ∆t, hr ∆p, psi ∆p’ 0 0 - 0.0888 58.032 1.668

0.0001 0.5180 - 0.1776 58.835 1.200 0.0008 4.005 4.646 0.3774 59.609 1.195 0.0020 9.4610 9.087 0.5376 59.975 1.169 0.0048 20.036 15.461 0.7776 60.34 0.987 0.0120 37.446 18.078 1.0176 60.605 0.979 0.0278 51.923 11.773 1.2576 60.811 - 0.0557 56.967 3.984

4). An interference test was run in shallow water sand. The active well produced 466 STB/D and pressure was measured in the observation well, which was 99 ft from the

active well. Estimated rock and fluid properties are as follows: µw = 1.0 cp, Bw = 1.0

bbl/STD, h = 9 ft, rw = 3 in, ct = 27.4 x 10-6 psi-1 Match points obtained from type -

curve matching with the pressure response are as follows:

t = 128 minutes, t

rD

D2 = 10 , ∆p = 5.1 psi, and PD = 1.0

Using supplied information calculate permeability and porosity.

113

5) An interference test is run between two oil wells. One is produced at 477 STB/D and the pressure response is measured at a nearby (263ft) well with a high precision guage. Pressure and reservoir data are given below. Using the semilog analysis method and type curve method, find the effective permeability to oil, md, and the effective in place porosity, fraction of bulk volume. h = 15ft rw = 0.275ft

µo = 0.89cp ct = 13.5 x 10-6 psi-1 Bo = 1.10 RB/STB

Table 6.9: Pressure Data at Observation Well

t, hrs ∆p, psi 0 0 25 57.5 37.5 66.3 50 73.8 62.5 78.8 75 82.5 87.5 86.6 125 95.0

114

7. SENSITIVITY ON FACTORS AFFECTING BHP

ANALYSIS

In this chapter, we shall discuss the effect of certain factors on analysis of bottom-hole pressure (BHP) tests. Factors considered include leaks, rate variation, sampling frequency, interference effect, wrong datum correction, and gauge sensitivity and accuracy. The effects of these factors were simulated in ideal BHP data and results from analysis of such data compared with the ideal case. We present the ideal data case and results from analysis of the data. We shall use the buildup test, which is the commom test, in illustrating these factors affect results obtained from BHP analysis. The ideal BHP test data are used as the standard because fluid and reservoir parameters used in generating such tests are known. 7.1. Analysis of Ideal BHP Data

The ideal BHP data were obtained by simulating the buildup test using parameters in Table 7.1. The simulation is also called test design. Generated pressure-time data are in Table 7.2. Table 7.1: Parameters for Test Design

Parameter Design Value



Formation thickness, h, ft 50

Porosity, φ 0.25



Production rate, q, STB/D 1000 Production time before Shut-in, tp, hr 1000

Permeability, k, md 1000 Total skin, s, 25 Wellbore storage constant, Cs 0.009

The ideal buildup data were analyzed using the conventional and type-curve methods. The analysis procedure is as follows: A Conventional Analysis Method

(i) Graph on log-log paper, ∆p versus ∆t and calculate Cs (see Figure 7.1)

(ii) Graph on semilog paper, Pws versus t t

t

p + ∆

∆ (Horner plot). Calculate k and s (see

Fig. 7.2) (iii) Use values calculated in Steps (i) to (ii) to generate pressure profiles. Plot the

profiles with solid lines and compare with ideal data. This comparison is shown in Figs. 7.3. What can you conclude from the comparison?

115

Table 7.2: Designed Buildup Data Shut-in Time, hr Pressure, psi Shut-in Time, hr Pressure, psi ************************************************************************ 0.00000 3183.763 1.01760 3244.368 0.00010 3184.281 1.25760 3244.574 0.00080 3187.768 1.49760 3244.743 0.00200 3193.224 1.73760 3244.887 0.00480 3203.799 1.97760 3245.011 0.00960 3216.630 2.21760 3245.122 0.01200 3221.209 2.45760 3245.220 0.01820 3229.340 2.81760 3245.352 0.02160 3232.170 3.17760 3245.467 0.02780 3235.686 3.53760 3245.569 0.03240 3237.330 3.89760 3245.662 0.03960 3238.996 4.25760 3245.746 0.04440 3239.712 4.61760 3245.824 0.05570 3240.730 4.97760 3245.895 0.06000 3240.973 5.33760 3245.962 0.08880 3241.795 5.69760 3246.024 0.11040 3242.080 6.05760 3246.082 0.13440 3242.302 6.41760 3246.137 0.17760 3242.598 6.77760 3246.189 0.24960 3242.953 7.13760 3246.238 0.37440 3243.372 7.49760 3246.285 0.53760 3243.738 7.95360 3246.341 0.77760 3244.104

B. Type curve Analysis Method We used the automatic type curve analysis method. Results obtained were used in generating simulated profiles. The simulated profiles are compared with the ideal data and shown in Fig. 7.4. Table 7.3 is a summary of calculated results and the expected results. Table 7.3 Buildup Analysis Results

Parameter Calculated Results Corrected Results

Conventional Type Curve

K, md 1000 1000.29 1000 S 25.01 25.01 25 Cs rb/psi 1.03 x 10-2 0.8996 x 10-2 0.9 x 10-2 P* 3251 3251 3251

From Table 7.3, we conclude that for the ideal case, both analysis methods will give the expected results. We now investigate the effects of some factors on the results.

116

7.2 Effect of Gauge Accuracy and Datum Correction. The effect of gauge accuracy was investigated by introducing consistent errors in the ideal BHP data. Two cases were considered. In the first case we assumed that the gauge readings were consistently lower by 15 psi. This case may also represent a case where the pressures were measured at depth that is 40 ft shallower than the datum.

Datum

Measurement depth

Gauge

Top of perforation

Z/2

h

Figure 7.1: Gauge position and Datum. As expected, the pressure profiles obtained from the analysis of this first case are similar to what we obtained with the idea data. The only difference is the shift in pressure level. This implies that P* and average pressure obtained in this analysis will be lower by 15 psi. Table 7.4 shows a summary of the results. Table 7.4: Effect Gauge Accuracy and Datum Correction-Case 1



K, md 1000 1005 1000 S 25.01 25.01 25 Cs rb/psi 0.962 x 10-2 0.901 x 10-2 0.9 x 10-2 P* 3236 3236 3251

In the second part of this investigation, we introduced 5psi, 7psi and 10psi errors over certain ranges. In real situation, this may be likened to wrong calibration over certain pressure ranges. Figures 7.5 and 7.6 show the plots of pressure data for this case. Table 4.5 shows the results obtained from the analysis of this case. Table 7.5: Effect of Gauge Accuracy -Case 2


Conventional Type curve

K, md 1002 995.8 1000 S 24.02 24.02 25 Cs rb/psi 0.94 x 10-2 0.93 x 10-2 0.9 x 10-2 P* 3229 3229

117

The “staggering” of the pressure profile in this case is embarrassing, but we are pleased that the calculated results are close to the correct results. This does not mean that good gauge calibration is not important. Who knows the magnitude of the error you will introduce by not calibrating your gauge properly. 7.3 Effect of Noise

By noise, we mean erratic pressure perturbations caused by gauge malfunction or erratic flow in the wellbore. The effect of noise was simulated by randomly introducing errors in the pressure data. Figures 7.7 to 7.8 show the pressure data after introducing the errors. Table 4.6 gives a summary of the results for this case. Table 7.6: Effect Gauge of Noise



K, md 1003 995 1000 S 35.42 35.42 25 Cs rb/psi 0.32 x 10-2 0.32 x 10-2 0.9 x 10-2

Some comments on the results are as follows: 1 Noise may cause overestimation or under-estimation of parameters. 2 Due to noise, the derivative plot may exhibit features, which may be wrongly misinterpreted as boundary effect 7.4 Effect of Gauge Sensitivity. We simulated the effect of gauge sensitivity by setting the pressure readings in some regions to be the same. This implies that the gauge could not “discern” the difference in pressure in that region. Figure 7.9 shows data for this case while Table 7.7 shows the summary of the results from the analysis. Table 7.7: Effect Gauge of Sensitivity



K, md 1003 987 1000 S 24.9 24.9 25 Cs rb/psi 1.42 x 10-2 1.4 x 10-2 0.9 x 10-2

7.5 Effect of Rate Variation We simulated the bottom-hole pressure using design rate shown in Table 7.8. We also analyzed the test using the rate shown in Table 7.8. Both the design and analysis rates

118

are also shown in Fig. 7.10. Table 7.8 and Figure 7.10 represent a case where a well flowed at 1000 STB/D for 999 hours and then just an hour prior to shut in the rate was reduced to 500 STB/D. During the analysis, a rate of 500 STB/D was used and this implies that the complete rate history was not used. The rate variation simulated here could be caused by reducing the choke size to drop the tool. A tool whose size could obstruct flow reasonably could also cause it. Table 7.8: Rate Schedule for Design and Analysis

Time, hrs Design Rate, STB/D Time, hr Analysis Rate, STB/D

0 1000 0 500 999 500 1000 0 1000 0

Design Rate Analysis Rate

1000

1000

1000999

500

500

timetime

00

qq

Figure 7.10: Design Rate and Analysis Rate. The pressure profiles in this case are shown in Fig. 7.11 and Fig 7.12. The figures and analysis look reasonably, but the results shown in Table 7.9 are not correct. Table 7.9 also shows results obtained by analyzing the test using the correct rate history. Figures 7.13 and 7.14 show the match obtained using the wrong rate history and correct rate history respectively. Table 7.9: Effect of Rate Variation

Parameter

Calculated Results

(Wrong Rate)

Calculated Results

(Correct Rate)

Correct Results

Conventional Type Curve Conventional Type

Curve

k, md 743.9 736.9 958.6 1000 1000 s 17.24 17.24 23.68 25 25 Cs, rb/psi 1.01 x10-2 1.01 x10-2 0.92 x 10-2 0.91 x 10-2 0.9 x 10-2

Figure 7.11 to 7.14 here ??? From these results, we make the following inferences:

119

1 Analyzing a bottom-hole pressure test with incorrect rate schedule will give wrong answers. Even the Horner’s approximation for handling multirate test as a single rate test did not give satisfactory results. 2 Avoid things that will cause unnecessary rate changes during test. If rate changes

occur, record it properly and let the one who will analyze the test be aware of it. 3 Rate variation could lead to the beginning of another wellbore storage phase and

this could be misinterpreted as a boundary. This is shown in Fig 7.15 for actual data.

Fig 7.15 ????? here 7.6 Effect of Leaks Leaks may result in the gauge measuring pressure that may be lower than the actual pressures especially when the leaking system is at higher pressure. For a buildup test,

leaks may result to continuous flow and hence the sandface rate not being zero (qsf ≠ 0). In this section, we shall consider leaks caused by the sandface rate not equal to zero. The following may cause such leaks: (a) Leaking lubricator (b) Leaking packer, which may result in communication between two sand intervals. We simulated the bottom-hole pressure test with a leak of 1% of the original rate during the buildup stage and analyzed the test with the assumption that there was no leak (qsf = 0) during the buildup period. The results of the analysis are shown in Table 7.10. Table 7.10: Effect of Leak

Parameter Calculated Results Correct Results


k, md 944 964 1000 s 22.94 23.84 25 Cs, rb/psi 1.13 x 10-2 0.96 x 10-2 0.9 x 10-2

Concluding, leaks of any form are not welcome. Both permeability and skin may be underestimated or overestimated. 7.7 Effect of Interference /Leak Producing wells that are around the test well can cause interference. These wells cause some pressure drop at the test well because the wells are draining the same reservoir and each well behaves as if it does not know that the other well is present. We simulated this effect by deducting the pressures due to interference from the ideal pressure data. The resulting data could also represent a case with leaks.

120

Figures 7.16 shows the log-log plot for this case and results obtained are in Table 7.11. Table 7.11: Effect of Interference



K, md 1043 1316.7 1000 S 26.18 34.6 25 Cs rb/psi 1.016 x 10-2 0.917 x 10-2 0.9 x 10-2

Deductions from the analysis are as follows: (1) From the derivative plot, the effect of interference could be misinterpreted as constant

pressure boundary effect. (2) Both analysis methods gave wrong answers, but the type-curve results are worse.

This is because during the type curve analysis, every data point is given equal weighting while the conventional analysis considers only the segment of interest.

Concluding, we should not allow leaks and if there are serious interference effects, the effects must be accounted for before analyzing your data. For the real tests, all these factors may be present during one test and results obtained from analyzing the test will be wrong. We therefore advice that you avoid as much as possible any of the factors that may adversely affect BHP tests.

121

8. FIELD CASES In this chapter, we shall present some of the tests we have analyzed and problems encountered. We shall first present analysis of a test that was properly run to show that tests can actually be run properly. 8.1 Good Test

A good test requires the following: (a) The gauges must be calibrated properly so that gauge readings will be reliable. (b) Accessories must be tagged for effective depth control. (c) Test programme must be correct, understood and followed. Figures 8.1 and 8.2 show the readings of the upper and lower gauges for a good test. Table 8.1 is a worksheet that shows the temperature and pressure readings at different times during the test. From the worksheet we infer that the readings are consistent as the expected pressure differences between the gauges agree with the calculated pressure differences based on the fluid gradient between the gauges. The same deduction could be made from the temperature differences. ???

Fig. 8.1: Good Measurements with Upper and Lower Gauges During Buildup Period

?????? Fig. 8.2: Good Measurements with Upper and Lower Gauges During Entire Test Period

Table 8.1: Worksheet For Gauge Quality Check General Information Field: Niger Well: 1 Reservoir: E 1.0 Top Gauge:WX1 Depth 7713 ft/CHH Bottom Gauge: WX2 Depth 7717 ft/CHH Test Date: 01/01/95 Gross (STB/D): 340 BSW (%): 0 GOR (SCF/STB): 1236

Quality Check Analysis

Differential Pressure Analysis

Event Time (hr)

Plower (psi)

Pupper (psi)

∆p (psi) Fluid Pressure Gradient (psi/ft)

Calculated

∆p (psi) Difference in

∆p (psi)

Flowing Period

14.5 3082.69 3081.54 1.15 oil 0.30 1.2 -0.12

Early Buildup

14.91 3240.73 3239.14 1.59 oil 0.30 1.2 -0.92

Mid Buildup

20.46 3246.29 3244.15 1.59 oil 0.30 1.2 -0.86

Late Buildup

26.43 3245.73 3244.15 1.58 oil 0.30 1.2 -0.88

Static Stop

27.82 3005.19 3005.24 0.95 oil 0.30 1.2 -1.17

Average Difference in calculated ∆p (psi): - 0.722

122

Differential Temperature Analysis Event Time

(hr) Tlower

(°C)

Tupper

(°C) ∆T (°C) Fluid Temp.

Gradient

(°C/m)

Calculated

∆T (°C)

Difference

in ∆T (°C)

Flowing Period

14.5 159.28 158.45 0.83 oil 0.01 0.04 0.136

Early Buildup

14.91 159.85 159.05 0.80 oil 0.01 0.04 0.045

Mid Buildup

20.46 159.77 158.74 0.83 oil 0.01 0.04 0.066

Late Buildup

26.43 159.82 158.99 0.83 oil 0.01 0.04 0.034

Static Stop

27.82 156.15 155.39 0.76 oil 0.01 0.04 0.103

Average Difference in calculated ∆T ( C) : 0.083 Comments: The pressure gauges gave consistent readings. The average pressure offsets of -0.722 psi can

be accepted as the pressure rise during the buildup was about 1565 psi. The pressure difference plot shows

no evidence of phase segregation. This is clearly shown on the pressure difference plot in Fig 8.2.

Figures 8.3 to 8.5 show the plots used for the analysis. The derivative plot shows the wellbore storage phase, infinite-acting radial flow phase and late-time data influenced by the constant pressure boundary. Note that in this analysis, the pressure profiles simulated with the calculated results (solid lines) match the real data very well. In this test there were no leaks and fluid segregation effects. ???????

Fig. 8.3: Log-Log Plot of Data from Good Test

???????

Fig. 8.4: Semi-Log Plot of Data from Good Test

??????? Fig. 8.5: Simulation Plot of Data from Good Test 8.2: Effect of Gauge Movement

From experience, moving gauges during flowtest prior to shutting in or during buildup renders tests useless. This is also true even if the test analyst knows that the gauges were moved. This implies that once you start the flowtest, do not move the gauges even if you have just realized that gauges are not at specified depth. Just state in your report the exact location of the gauges. On the other hand, if you must move the gauges (say to the perforations to minimize fluid interface movement) you must start the flowtest all over again on reaching the new depth.

123

Figure 8.6 shows an example in which gauges were moved prior to shutting in the well. The gauge was moved further down after the flowtest. In this case, the flow period at the new depth was not sufficient. ????

Fig 8.6: Gauge Movement Prior to Shutting in

The problem with analyzing this test is that two different bottom hole flowing pressure before shut-in could be used in the analysis. The first bottom-hole flowing pressure is the one obtained prior to the gauge movement. Analysis plots obtained by using this bottom-hole pressure are shown in Figures 8.7 and 8.8. Obtained results are shown in Fig 8.8. A second bottom-hole pressure before shut in that could be used in the analysis is the one prior to shutting in. That is, after the gauge was moved. The analysis plot for this case is shown in Fig 8.9. The second option looks more reasonable, but may be difficult to recognize. It may be regarded as an artefact as pressure has started building up. We recognize it because the BHP contractor was instructed to move the gauge after flowtest – a wrong instruction.

An important issue in moving gauges prior to shutting in is that the correct shut-in pressure may not be used for the analysis. This is an important parameter in the calculation of skin and also in type-curve matching. Also, gauge movements can yield a “dome” that may be misinterpreted as wellbore storage phase as shown in Figure 8.8. This is because if you are running in the gauges at constant speed, pressure will vary linearly with time. During the wellbore storage phase, pressure also varies linearly with time. Therefore, do not complicate the analysis with unnecessary movement of the gauges. 8.3 Effect of Gauges off Depth

We are not just interested in measured pressure, but we also want to know the depth where the pressure was measured. If depth is off by 50ft from datum in a reservoir that produces dry oil of gravity 0.35psi/ft, the error in datum pressure will be 17.5psi. This may seem small, but many reservoirs may not have that much depletion in one year. This implies that a 1-year error has been introduced in reserve calculations! Depth control is important and that is why we insist on tagging accessories, marking wire, etc., but many operators still do not understand the importance. Poor depth control is a source of error in calculating wellbore fluid gradient. The equation for calculating the fluid gradient is as follows:

12

12

xx

PPGradientFluid

−−

= 8.1

where P1 = Pressure measurement at Poin1 x1 = Vertical depth at Point 1 (along hole depth corrected for deviation) P2 = Pressure measurement at Poin2

124

x2 = Vertical depth at Point 2 (along hole depth corrected for deviation)

Figure 8.10 shows an example where we detected poor depth control. Both the upper and lower gauge readings are shown. After correcting for deviation, the calculated static gradients in Column 4 gave unrealistic results. For example, in some cases, a lighter fluid is found below a denser fluid. This is not physically possible and was likely caused by depth error. ?????? Leave a page

Fig. 8.10: Incorrect Static Gradient Due to Poor Depth Control

The magnitude of the depth error can be estimated by introducing shifts in the depths until a satisfactory fluid gradient trend is obtained. Column 5 shows the shifts and the final fluid gradients are in Column 8. The shifts are considered as the suspected depth errors. This should be brought to the notice of the BHP contractor so that he will be careful with his depth measurements. Also, leaks could cause faulty static gradients. This is shown in Fig 8.11. In this case, there is no remedy as both the numerator and denominator in Eq 8.1 may be affected. ?????? Leave a page

Fig. 8.11: Incorrect Static Gradient Due to Leak

We have observed some contradiction between the claimed gauge position and resulting effect on pressure. For example, with the gauge at mid perforation we do not expect any effect on pressure due to liquid interface movement.

8.4 Effect of Reporting Wrong Rates

Flowrate data is as important as the pressure data. Without the flowrate data, permeability, the most important parameter, cannot be calculated from BHP test. The relationship between flowrate and permeability is given in Oilfield units by the following equation:

KqB

mh=162 6. µ

8.2

where: K = permeability, md q = flowrate, STB/day m = slope obtained from semilog plot of BHP data, psi/cycle B = formation volume factor rb/STB

125

µ = Viscosity, cp h = drainage thickness, ft A consequence of using wrong rate is that permeabilities calculated from different tests on the same well will be inconsistent. This is not expected unless there is a process altering the permeability in the drainage area. Figure 8.12, shows the expected feature. The slope of the semilog plot is constant and therefore permeability is constant, but average pressure in the drainage area changes time. ????? Here

Fig 8.12: Consistency in Permeability and Analysis Rates

Flowrate is also used in computer programs used in BHP analysis for calculating wellbore storage constant as shown for a buildup by the following equation:

CqB t

ps =

∆∆24

8.3

where: Cs = wellbore storage constant.

∆p = Pws-Pwf(tp) from unit slope of log-log plot of wellbore storage phase.

∆t = shut-in time corresponding to ∆p. The wellbore storage constant can also be estimated with equations that do not involve rate but depends on storage mechanism (fluid expansion, falling liquid, etc.), fluid compressibility in the wellbore, wellbore and connected volume. The wellbore storage constant calculated with these equations should agree within reasonable limits with wellbore storage constant calculated with Equation 8.3. In situations where they do not agree, it is likely that the wrong rate was used in the analysis. For example, in one of our analysis, the wellbore storage constant calculated using Equation 8.2 was 0.00176 STB/psi while estimated wellbore storage constant depending on whether tubing communicates with casing or not lies between 0.00556 to 0.043 STB/psi. This implies that the rate supplied for the analysis is off by a factor of at least three. In another case, we calculated a wellbore storage constant of 0.0071 STB/psi using Equation 8.3 while the estimated wellbore storage constant is 0.00724 STB/psi. In this case, we believe that the rate used for the analysis was correct. This could be used as a quality check for rate measurement if measured pressures are reliable. To overcome the problem of reporting the wrong rate, we recommend the following: 1. The BHP contractor should be interested in obtaining correct rate data as much as he

is interested in obtaining good pressure data. 2. The flowstation supervisors should cooperate with the BHP contractor and both

should ensure that the test well is correctly hooked on to the test separator so that correct flowrate data will be obtained.

126

3. We are aware that the flowstation supervisor has scheduled dates for testing each well, but he should utilize the opportunity created by the BHP to perform a complete test on the test well. That is, obtain rate, BSW, GLR, etc.

8.5 Effect of Ineffective shut-in / Well not flowing before Shut-in

If there are no speed limits, the state of many roads can be determined by how fast vehicles move on the roads. In the same manner, important reservoir parameters can be determined from BHP test from the rate at which pressure drops (as in drawdown) or rises (as in buildup) with time. This implies that during a good buildup test, pressure must rise. This will occur if the well was flowing and then effectively shut in. The correct pressure rise during a buildup test can be achieved by flowing the well until stabilization and then completely shutting the well to flow. The implications of the last sentence are as follows:

1. During a buildup, the well must be flowed until stabilization. The stabilization condition is removed if well is surging. However, in that case, well must be flowed for at least 6 hours.

2. If well cannot be completely shut-in due to faulty valve or wax or any other reason, the buildup part of the test should be aborted.

Table 8.2 shows the production data recorded by a BHP contractor before the well was shut for buildup. It is apparent from the flowrate data that the well was not flowing prior to shutting in. The little production he observed when he opened the well could have come entirely from the wellbore and not the reservoir. Do not forget that fluid in the wellbore is compressed and this could cause the wellbore fluid to be produced on opening the well initially.

Table 8.2: Production Data Prior to Shut-in

OIL

TIME Meter Reading(bbl) Obser. Vol. (bbl)

12:45:00 205.70 67 13:00:00 210.00 4.30 13:15:00 248.40 38.40 14:00:00 336.70 2.90 14:15:00 355.00 18.30 14:30:00 372.50 17.50 15:30:00 391.70 0.00 15:45:00 391.80 0.10 16:30:00 391.90 0.00 16:45:00 391.90 0.00 17:00:00 391.90 0.00 17:15:00 391.90 0.00 18:15:00 391.90 0.00 18:30:00 391.90 0.00

The well was shut-in at 18:30:00

127

Figure 8.13 is a pressure plot of a well that could not be shut in or the well was not flowing prior to shutting in. In this case, there was no pressure rise and the test is useless if the objective of the test is to calculate permeability and skin. ???

Fig 8.13: Ineffective Shutting in of Well

It is easy to know whether a well flowed or could not be shut in during buildup test. In this case, the flowing gradients (misnomer for a dead well) and the static gradients are same. This is shown in Fig 8.14. Figure 8.15 is the static and flowing gradients for a well that was flowing and properly shut in during the buildup. Note that for this case, the static pressure at a given depth is higher than the flowing gradient at the same depth. ????? Leave a page

Fig 8.14: Static and Flowing Gradients for a Well that is not Flowing

Fig 8.15: Static and Flowing Gradients for a Live Well

In addition to our interest in ensuring that wells are effectively shut during buildup, we are equally interested in your shutting the well as fast as possible. Pressure profiles in Fig. 8.16 show a well that was slowly shut in. Slow shutting in of wells makes tests difficult or impossible to analyze. ??????

Fig 8.16: Slow Shutting in of Well

8.6 Effect of Leak Using the road analogy, a leak could be likened to a detour on the road. Such a detour will make it difficult to ascertain the condition of the road from the speed of the vehicles. Leaks make it impossible to analyze BHP tests because leaks cannot be quantified. Also, when there are leaks, calculated fluid gradients are wrong as shown in Fig 8.11.

During BHP tests, leaks could occur through (a) leaking lubricator. (b) communication between long string and short string or between a string and casing.

This could occur through gas lift ports or non-sealing packers. (c) not completely shutting the well, etc. At early shut-in time (wellbore storage phase), the effect of leak may not be detected from pressure data because during that period, the rate at which pressure rises is usually greater than the rate at which it leaks. Therefore, a net pressure rise is observed. But, during the infinite-acting radial flow phase in formation with high permeability as in the Niger Delta, the rate at which the pressure rises will in most cases be lower than the rate at which pressure leaks. A net pressure drop is recorded and such a test cannot be

128

analyzed. Figure 8.17 shows pressure plots from a test with leak. Note that pressure difference between lower and upper gauge may be constant if there is leak. ?????

Fig 8.17: Pressure Drop Caused by Leak

Figures 8.18 and 8.19 show the semilog and log-log for a case where there was an intial leak through the lubricator. The leak was stopped by pumping the lubricator before pressure started to rise again. It is always better to avoid such problems by using tested lubricators. ?????? page 107

Fig 8.18: Semilog Plot of Data with Initial Leak from Lubricator

??????

Fig 8.19: Log-log Plot of Data with Initial Leak from Lubricator

In conclusion, make sure your lubricator is not leaking and that wells are completely shut in. Leaks due to communication between strings and casing can be detected by recording the surface pressures of all the strings and casing using a surface chart pressure recorder. ??? Figures 8.20 and 8.21 show the pressure chart using in recordind surface pressure for two cases. Figure 8.20 shows that there was communication between the test string (Well 15S) and casing during the test. Pressure from this string merged with pressure in the casing after a while. Fig 8.21 is an example of a case with no communication between test string and casing. The pressures from both strings and casing followed their own trend. We recommend the use of these charts. ??? Leave a page

Fig. 8.20: Surface Pressures of Short String, Long String and Casing (Case with Communication)

??? Leave a page

Fig. 8.21: Surface Pressures of Short String, Long String and Casing (Case with no Communication)

??????????? 8.7 Effect of Gauge Oscillations/Sensitivity Problems

We have observed strange oscillations in pressure data measured with electronic gauges. In most cases, the oscillations occur in both gauges as shown in Fig. 8.22. Definitely, such oscillations are neither due to well nor reservoir responses. We believe that the oscillations may be due to the procedure used while running the test. Example of procedures that could produce such oscillations include “jarring” the gauges and allowing the gauges hit objects in the well. The oscillations could also be caused by electronic problems that we do not understand. ??????

129

Fig 8.22: Gauge Oscillations

Another factor that could cause “milder” oscillations is poor gauge sensitivity or temperature effects. This is shown in Fig. 8.23. The gauge gives readings that are about the correct value. An example of the effect of such oscillation on pressure data graphed on Horner plot is shown on Fig. 8.24. These oscillations cause discontinuities on the pressure derivative as shown in Fig. 8.25. The implication of this is that the different flow phases may not be clearly discerned.

?????

Fig 8.23: Gauge Oscillations Due to Sensitivity Problem

????

Fig 8.24: Effect of Gauge Oscillations on Horner Plot

????

Fig 8.25: Effect of Gauge Oscillations on Pressure and Derivative on Log-log Plot

Depending on when and how the oscillations occur, oscillations can make test analysis impossible. Also, in cases where BHP tests with oscillations in data are analyzed, the oscillations cause a large band of uncertainty for the calculated parameters. 8.8 Effect of Gas Phase Segregation Liquid phase is usually produced with dissolved gas. Hence when a well is shut-in, the dissolved gas comes out of solution and moves to the top of the wellbore creating an unusual pressure rise (or “hump”) in the wellbore. This pressure “hump” is illustrated in Fig. 8.26 and Fig 8.27. ????

Fig 8.26: Gas Phase Segregation “Hump” on a Semilog Plot

???? Fig 8.27: Gas Phase Segregation “Hump” on a Simulation Plot

Gas phase segregation is more common in wells with high gas-oil ratio(GOR) and can be identified using the following: (a) Pressure difference (i.e. lower gauge pressure minus upper gauge pressure) between

the gauges increases with time because as gas comes out of solution, liquid between the gauges become denser.

130

(b) The pressure changes due to phase segregation cause a “dip” in the pressure derivative. The dip in the pressure derivative may be erroneously interpreted as double porosity feature. Figure 8.28 illustrates this.

Fig: 8.28: Pressure Derivatives Showing Dips Due to Phase Segregation and Double

Porosity

Figure 8.29 shows an example of a real case of the effect of gas phase segregation on pressure derivative.

??????

Fig 8.29: Effect of Gas Phase Segregation on Pressure Derivative

(c) On semilog plot of the pressure data, phase segregation effects cause a “hump”. This

is shown on Fig 8.26. Generally, gas phase segregation is unlikely to prevent correct interpretation of test because the effect dies down quickly. 8.9 Effect of Liquid Interface Movement Quite like gas, oil and water will separate in a well that is shut in. Depending on the position of the gauges and amount of water produced, recorded pressure data can be affected by oil/water interface movement. The oil/water interface can fall below the gauges or rise above the gauges. A schematic from Lingen (1995) showing the effect on pressure for case of rising and falling contacts are in Figs. 8.30 and 8.31. Figure 8.30 shows a case where the oil/water contact was initially below the gauges and later rose above the gauges. In this case, pressure builds up, drops and then rises again. This is likely to occur in wells producing with BS&W of about 45% or more if the gauges are not close to the perforation. Figure 8.31 shows a case where the oil/water contact was initially above the gauges and letter dropped below the gauges. The pressure rises normally and then more rapidly before the normal pressure rise continues. This is more likely to occur in wells with gas with the gas pushing down the oil/water contact.

?????

Log ∆t

Log ∆p’

Dip due to Phase

Segregation

Dip due to Double Porosity

Log ∆t

131

Fig 8.30: Effect of Rising Oil/Water Contact

?????

Fig 8.31: Effect of Falling Oil/Water Contact

These effects can be explained by the fact that the density of the fluid in the well and between the gauges affect the rate of pressure change during buildup. Unlike the case of gas phase segregation, pressure distortion caused by liquid interface movement can last for a much longer time and therefore can make the test uninterpretable. Figures 8.32 and 8.8.33 show examples of a real case of rising liquid interface. Figure 8.32 is on a Cartesian plot while Fig. 8.33 is the same phenomenum on a semilog log plot. Liquid interface movement is an unwanted phenomenon. To eliminate or minimize it, the gauges must be at the perforations. This is especially important if the BS&W is high (BS&W > 30% - not a magic number).

????

Fig 8.32: Liquid Interface Movement on a Cartesian Plot

????

Fig 8.33: Liquid Interface Movement on a Semilog Plot

8.10 Effect of Gaslift

Distortions on pressure data caused by gaslift effects are unpredictable. Intermittent opening and closing of gaslift valves during the buildup cause the distortions. The distortions could be worse if the gas supply to the well was not turned off. Distortions caused by gaslift malfunction make flow phase indetification difficult. Hence analysis will yield wrong results. Figures 8.34 and 8.35 show two interpretations of a test with gaslift distortion. The log-log and semilog plots used in the interpretation are shown. Figures 8.34a and 8.34b show one form of interpretation and Fig 8.35a and 8.35b show another form of interpretation of the same test. Results from both interpretations are also shown. Note that the gaslift effects makes it difficult to simulate the correct pressure profiles and result obtained from analyzing such test may depend on the assumed profile. ????????

Fig 8.34 a and b

Fig 8.35 a and b

8.11 Effect of Short Buildup or Flow Period

We have discussed the different flow phases that a well that is shut-in goes through. We also discussed the information that may be derived from each phase. For example,

132

accurate values of skin and permeability are derived from the infinite-acting radial flow phase. In this section, we present results showing the effect of test duration. In one case, the test was run long enough to reach the infinite acting radial flow (IARF) phase while in another case, the some test data were removed to simulate a case where the test was not run enough to reach (IARF) phase. The two cases were analyzed and the results compared. In the first case where the (IARF) phase was reached, analysis was easy. Both conventional and automated type-curve methods could be used. Figure 8.36a and b show the log-log and semilog plots used in the analysis. Figures 8.37a and b show plots used in the analysis in the case where the (IARF) phase was not reached and the the skin constrained. That is, the skin was not varied during the autmatic type-curve matching. Figures 8.38a and b show plots used in the analysis in the case where the (IARF) phase was not reached and the skin not constrained. That is, the skin was varied during the autmatic type-curve matching. Results obtained from analyzing the different cases are shown in Table 8.3. Table 8.3: Effect of Buildup Period on Calculated Results.

Parameter Calculated Results

Long Shut-in Short Shut-in

Case 1 (contrained skin)

Case 2 (uncontrained skin)

Permeability, md 1840 708 2090 Skin 32.4 9.09 37.8

Cs, STB/psi 0.0248 0.0237 0.025

Figure 8.36 to 8.38 here ??????

Deductions from the analysis are as follows: 1. Tests run long enough to reach a clearly defined infinite-acting radial flow phase will

yield correct values of permeability and skin. 2. For a test that is not run long enough (e.g. infinite-acting radial flow phase not

reached) the calculated permeability and skin will be wrong irrespective of the analysis method.

3. In these cases where the wellbore storage phase was observed, the calculated wellbore storage constants are close.

4. The semi-log plots show that if the infinite-acting radial flow phase is not reached test cannot be analyzed using the conventional method.

In addition to our being interested in the duration of the buildup period, we are also interested in the duration of the flow period prior to shutting in of well. A flow period of about 2 to 6 hours (good for Niger Delta high permeability formations) with the gauges at the survey depth where the well will be shut is necessary for the following reasons:

133

1. To ensure that there is a flow period that can be used for matching simulated profile with real data.

2. To ensure that the correct flowing pressure at shut-in is used. This affects calculated skin and productivity index.

8.12 Class Discussion

It is not possible to present and discuss every type of problem encountered in BHP test and analysis. In some cases, we are not able to figure out what caused the problem. In such cases, we give probable cause of the problem. In this section, we shall present some cases for discussion. Every one will participate in trying to find out what happened during the tests or the characteristics shown by the test data. Case 1: Figures 8.39a and b are the log-log and semilog plots of a test we regard as a mystery test. The mystery if the sudden rise in pressure of about 300 psi that occurred at about 3.00 am while the well was shut for overnight. The gauge depth was less than 150 ft from the maximum well depth. Now explain what caused the problem. ???

Fig 8.39: Log-log and Semilog Plots of the Mystery Test

Case 2: Figures 8.40a and b are the log-log and semilog plots of a test with a problem. Explain what the problem is. ?????

Fig 8.40: Log-log and Semilog Plots of a Test

Case 3: Figures 8.41a and b are the log-log and semilog plots of a test with a problem. Explain what the problem is. ?????


Case 4: Figures 8.42a and b are the log-log and semilog plots of a test with a problem. Explain what the problem is. Varying kh ?, Composite system? Etc. ?????


Case 5 : Figures 8.43 shows the log-log plot of a test with ran with an Amerada gauge. Explain the trend.

134

????? Fig 8. 43: Log-log of a Test ran with Amerada Gauge

Case 6 : Figures 8.44 shows the Cartesian plot of a test with a final shut-in pressure that is lower than the flowing pressure. Suggest the cause of the problem. ?????

Fig 8.44: Cartesian Plot of a Test with a Problem

Case 7 : Figures 8.45a and b show the semilog plots from tests in two adjacent wells draining the same reservoir. The permeability calculated from the semilog plots differ by a factor of about 10. Explain the big unrealistic difference.

Fig 8.45: Semilog Plots of Tests in Adjacent Wells Draining the same Reservoir

One deduction from this case is the need for system approach to welltest analysis. Tests from wells in the same reservoir must be compared to determine unrealistic results. This comparison will help show the local (well) and global (reservoir) effects.

135

9. THEORETICAL CONCEPTS We have given a lot of equations without showing how they were obtained. Readers who are mathematically inclined will like to know the origin of the equations. Therefore, for completeness, we decided to include a chapter on theoretical concepts of bottom-hole pressure test analysis. In this chapter, we shall derive some of the equations in Darcy units, outline method of prediction pressure for test design and throw a little light into the concept of superposition which forms the basis for generating multi-rate flow equations. 9.1 DERIVATION OF FLUID FLOW EQUATIONS To understand and correctly analyze well test data require:

(a) Understanding of physical processes involved. For example, the fluid flow processes the effect of reservoir geometries and heterogeneties.

(b) Selection or production of proper mathematical equations or models for analysis of results.

To satisfy conditions (a) and (b) will require understanding the mathematics of fluid flow in porous medium. Discussion on the derivation of some of the fluid flow equations follow. The basic concepts involved in the derivation of fluid flow equations are: (a) Conservation of mass equation (b) Transport rate equation (e.g. Darcy’s law) (c) Equation of state The use of these basic concepts in derivation of liquid and gas flow equations in porous medium is discussed. Flow in cylindrical coordinates is considered with flow in angular and z-directions neglected. The equations are given as follows: 9.1.1 Liquid Flow Equations

A. Conservation of Mass Equation: Mass rate in - Mass rate out = Mass rate storage 9.1 B. Transport Rate Equation

q = - k A

dp

drµ 9.2

By conserving mass in an elemental control volume shown in Fig. 9.1 and substituting the transport rate equation, the following equations can be obtained:

136

= r

p

rhk2 r

−

∂∂ρ

µπ

rr r

p

rhk2∆+

−

∂∂

ρµ

π + 2 r r h

t ( )π

∂∂

ρ∅∆ 9.3

Expanding the first term on the right hand side of Eq. 9.3 about “r” using Taylor’s series expansion gives:

1

r

r {rkp

p

r} =

t ( )

∂∂ µ

∂∂

∂∂

ρ∅ 9.4

Figure 9.1: Schematics of the Reservoir and Control Volume.

Equations 9.1 to 9.4 apply to both liquid and gas. For each fluid, the density or pressure term in Eq. 9.4 may be replaced by the correct expression in terms of pressure or density. For slight but constant compressibility liquid,

ρ ρ = sc ec[P - psc] 9.5

Substituting for pressure in Eq. 9.4, the final equation in terms of density is:

∂∂

∂∂

∂∂

2

+ 1

r

p =

(c + c

k

p

t

rp

r r2

∅µ ) 9.6

In terms of pressure, Eq. 9.4 is:

∂∂

∂∂

∂∂

∂∂

2r +

1

r

p + c(

p

r =

(c + c

p

t

p

r r k2

2))

∅µ 9.7

If pressure gradient is small everywhere in the system, Eq. 9.7 becomes:

∂∂

∂∂

µ ∂∂

2

+ 1

r

p =

C

k

p

t

tp

r r2

∅ 9.8

r

r+∆

r r+∆

137

Equation 9.8 is the diffusivity equation. Some deductions about the form of the derived equation are:

1. The partial differential equation in terms of density is linear (Eq. 9.6).

2. The partial differential equation in terms of pressure (Eq. 9.7) is non-linear but if the pressure gradient is small, the resulting equation (Eq. 9.8) becomes linear.

The assumptions inherent in the derivation of the diffusivity equation are as follows: (a) The formation is homogeneous, horizontal and of uniform thickness. (b) Flow is radial, with gravity and capillary effects negligible. (c) The fluid is considered to be of slight and constant compressibility. (d) The pressure gradient in the reservoir is considered to be small. (e) No reaction between fluid and formation matrix occurs. (f) Reservoir is at isothermal condition. (g) Flow is laminar. 9.1.2 Gas Flow Equations Equations 9.1 to 9.4 also hold for gases. In addition, the equation of state for gases is:

ρ = PM

ZRT 9.9

For real gases, isothermal compressibility, c, is defined as:

c = 1

PT

- 1

z dz

dp

9.10

For ideal gases,

c = 1

P 9.11

Combining Eq. 9.4, Eqs. 9.9 and 9.10, the different forms of gas flow equations will be:

A. Pressure-Squared Form (p2 Form)

∇∅

. [kpM

ZRT ] =

cpM

p

tµ∂∂

∆p RT

9.13

Differentiating Eq. 9.13 and assuming that permeability is constant gives:

1

Z -

1

Z

Z)

dp ( p =

c

kZ

p

t

2

2

22

µ µµ ∂

∂∇ ∇

∅p

d (2 2 2) 9.14

Hence,

138

∇ ∇∅µ2 2p -

d

dp [ ln ( Z)] ( p =

c

k

p

dt

2

2

22

µ∂

) 9.15

Under certain conditions, Eq. 9.15 simplifies to :

dt

p

k

c = p

222 ∂φµ

∇ 9.16

Equation 9.16, which is the commonly used p2 - form of gas flow equation follows from Eq. 9.15 and assuming that the pressure gradient is small everywhere in the system. For ideal gases, Eq. 9.16 holds but the assumption of small pressure gradient is not necessary. This is because for ideal gases, the gas compressibility factor z is unity and ideal gas viscosity is not dependent on pressure. The viscosity of ideal gas only depends on temperature. Hence, the second term on the left-hand side of Eq. 9.15 will always be zero.

B. Pseudo -Gas Pressure Form (m (p) Form)

Equation 9.13 still holds. That is

∇ ∇∅

. [ 1

z p =

c

kZ

p

t

22

µ∂∂

] 9.13

The pseudo-gas pressure is defined as:

∫p

Pp z

2pdp = m(p)

µ 9.17

Thus,

∆ ∆m (p) = 1

z p2

µ 9.18

Substituting Eq. 9.18 into Eq. 9.13 gives:

∆2 m(p) = 9.19

Equation 9.19 holds for both real gas and ideal gas with no assumptions on the magnitude

of pressure gradient term. Equation 9.19 appears linear but it is not as µ and z are dependent on pressure. The resemblance between the gas flow equation (Eq. 9.19) and liquid flow equation (Eq. 9.8) form the basis of adapting liquid flow solution to gas flow problems. Summary of the equations of state and resulting flow equations are give in Table 9.1.

139

Table 9.1: Summary of Flow Equations

Continuity Equation: )( t

}r

p

rk {

r

1∅=

∂∂ ρ

∂∂ρ

µ∂∂

r

Equation of State

Resulting Flow Equation

Liquid

]P-c[P

scsce = ρρ

∇∅µ2 p =

ct

k

p

t

∂∂

Gas

ρ = PM

ZRT

∇ ∇∅

2 p - d

d p [ln Z] ( =

ct

k

p2

2

2

2

µµ ∂

∂p

t)

∇∅µ2 m(p) =

c

k

t m(p)∂

∂

9.1.3 Relationship between m(p), p2 and p

The real gas pseudo-pressure is defined as:

ψ = ∫p

Pb z

2pdp = m(p)

µ 9.17

At low pressure (p < 2000 psi)

µz = µi zi = constant 9.20a

Hence,

m (p) = P

i

2

µ z i

9.20b

for Pb = 0

Substituting Eq. 9.20 into Eq. 9.19, it can be shown that the m(p) form of the gas

equation is exactly the same as the p2 form at low pressure. At high pressure (p > 3000 psi)

140

p

z iµ µ =

P

z

i

i

9.21

Hence

m(p) = 2 p

z Pi

iµ i

9.22

for Pb = O

Substituting Eq. 9.22 for m(p) into Eq. 9.19, it can be shown that the m(p) form of the equation degenerates to the p-form. That is, the gas flow equation will be same as the liquid flow equation (diffusivity equation). The relationships in Eqs. 9.20a and 9.21 are shown in Figure 9.2. Note that in Figure 9.2,

the assumptions that µz is constant at low pressure and µzP

is constant at high pressure

may be acceptable for practical purposes.

141

Figure 9.2: Variation of Gas Deviation Factor-Viscosity Product for a Real Gas (after

Wattenberger and Ramey, 1968)

Calculation of Real Gas Potential, m(p)

The m(p) can be calculated graphically or read from Tables. The graphical method

requires that p, µ, z be given and m(p) calculated from area under a curve. This is illustrated below for a case where the trapezoid method is used in calculating the area under the curve. Table 9.2: Graphical Method of Calculating m(p)

P µµµµ z 2P/(µµµµz) Area m(p)

P0 µ0 z0 2P/(µz)0 = Y0

P1 µ1 z1 2P/(µz)1 = Y1 A1 = 0.5(Y0 +Y1)( P1-P0) A1

142

P2 µ2 z2 2P/(µz)2 = Y2 A2 = 0.5(Y1+Y2)( P2-P1) A1 + A2

P3 µ3 z3 2P/(µz)3 = Y3 A3 = 0.5(Y2 +Y3)( P3-P2) A1 + A2+ A3

Pn µn zn 2P/(µz)n = Yn An = 0.5(Yn-1 +Yn)( Pn-Pn-1) A1 + A2+ A3+…+ An

The graphical method is illustrated in Figure 9.3.

Figure 9.3: Graphical Method of Obtaining m(p)

Tables 9.3 and 9.4 can be used in obtaining m(p). Note that the data in the tables were generated using some assumed typical gas properties. Both tables give reduced pseudo-pressure defined as:

2

1

2 c

rP

µΨ=Ψ 9.22a

where

ψ = m(p)

µ1 = initial viscosity Pc = critical pressure From Eq 9.22a, m(p) can be calculated. The differences between values in both tables are simply due to the use of different bases. Table 9.3 uses reduced pressure of 0.2 as base while Table 9.4 uses reduced pressure of 0 as base. However, the values can be reconciled with the knowledge that

2p/(µz)

p

m(p)

Pb

143

∫−∫=∫2.0

002.0

yyypp

9.22b

Note that at low pressure, m(p) can simply be calculated with Eq 9.20b.

144

Table 9.3: Reduced Pseudo-Pressure Integral (ΨΨΨΨr) With 0.2 as Base

Pseudo-

Reduce

d

Pressur

e

Values of integral for Pseudo-Reduced Temperature Tpr of

Ppr 1.05 1.15 1.30 1.50 1.75 2.00 2.50 3.00 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50 5.75 6.00 6.25 6.50 6.75 7.00 7.25 7.50 7.75 8.00 8.25 8.50 8.75 9.00 9.25 9.50 9.75

10.00 10.50 11.00 11.50 12.00 12.50 13.00 13.50 14.00 14.50 14.00

0.0257 0.0623 0.1102 0.1698 0.2418 0.3264 0.4236 0.5326 0.6546 0.7903 0.9484 1.1444 1.3671 1.5828 1.7924 1.9959 2.1926 2.3821 2.5649 2.7424 2.9147 3.0825 3.2464 3.4066 3.5633 3.7169 3.8679 4.0165 4.3788 4.7278 5.0653 5.3938 5.7144 6.0276 6.3347 6.6368

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

0.0229 0.0553 0.0971 0.1485 0.2105 0.2835 0.3678 0.4631 0.5691 0.6855 0.8126 0.9503 1.0980 1.2546 1.4191 1.5883 1.7595 1.9321 2.1071 2.2841 2.4619 2.6399 2.8172 2.9937 3.4683 3.3403 3.5094 3.6766 4.0876 4.4874 4.8766 5.2579 5.6367 6.0088 6.3897 6.7235 7.0706 7.4124 7.7495 8.0821 8.4099 8.7330 9.0520 9.3670 9.6786 9.9876

10.2936 10.5963 11.1935

- - - - - - - - - - - - - - - - -

0.0198 0.0477 0.0839 0.1283 0.1810 0.2419 0.3111 0.3889 0.4755 0.5707 0.6734 0.7838 0.9020 1.0277 1.1606 1.3001 1.4457 1.5966 1.7526 1.9138 2.0791 2.2473 2.4186 2.5935 2.7710 2.9504 3.1320 3.3153 3.7771 4.2400 4.7052 5.1693 5.6277 6.0822 6.5308 6.9714 7.4044 7.8304 8.2497 8.6632 9.0711 9.4731 9.8703

10.2635 10.6531 11.0398 11.4223 11.7998 12.1731 12.5433 12.9102 13.2735 13.6340 13.9925 14.3483 14.7011 15.3996 16.0892 16.7703 17.4427 18.1069 18.7642 19.4147 20.0588 20.9676 21.3318

0.0170 0.0409 0.0716 0.1091 0.1532 0.2037 0.2608 0.3246 0.3954 0.4734 0.5579 0.6484 0.7449 0.8473 0.9558 1.0703 1.1906 1.3464 1.4474 1.5838 1.7253 1.8712 2.0214 2.1758 2.3341 2.4957 2.6612 2.8308 3.2685 3.7223 4.1897 4.6678 5.1539 5.6459 6.1412 6.6377 7.1355 7.6343 8.1338 8.6336 9.1326 9.6297

10.1249 10.6185 11.1091 11.5957 12.0794 12.5615 13.0416 13.5194 13.9939 14.4644 14.9322 15.3980 15.8609 16.3205 17.2313 18.1318 19.0212 19.8976 20.7640 21.6238 22.4762 23.3216 24.1596 24.9921

0.0145 0.0348 0.0609 0.0927 0.1303 0.1734 0.2221 0.2763 0.3358 0.4004 0.4702 0.5452 0.6255 0.7114 0.8025 0.8983 0.9988 1.1042 1.2146 1.3293 1.4498 1.5744 1.7034 1.8370 1.9751 2.1169 2.2626 2.4123 2.8038 3.2178 3.6504 4.0997 4.5638 5.0406 5.5280 6.0234 6.5252 7.0320 7.5449 8.0622 8.5836 9.1085 9.6364

10.1665 10.6973 11.2279 11.7587 12.2897 12.8211 13.3532 13.8858 14.4187 14.9513 15.4834 16.0146 16.5447 17.6030 18.6590 19.7090 20.7507 21.7858 22.8166 23.8434 24.8616 25.8642 26.8596

0.0126 0.0303 0.0530 0.0807 0.1132 0.1505 0.1927 0.2397 0.2915 0.3483 0.4098 0.4758 0.5461 0.6209 0.7001 0.7840 0.8724 0.9653 1.0624 1.1636 1.2687 1.3777 1.4904 1.6068 1.7268 1.8504 1.9778 2.1094 2.4534 2.8178 3.2016 3.6049 4.0268 4.4663 4.9203 5.3860 5.8621 6.3472 6.8412 7.3442 7.8551 8.3739 8.8993 9.4298 9.9647

10.5034 11.0452 11.5897 12.1377 12.6897 13.2440 13.7993 14.3558 14.9128 15.4700 16.0274 17.1463 18.2662 19.3931 20.5120 21.6135 22.7156 23.8144 24.9057 25.9948 27.0862

0.0100 0.0241 0.0421 0.0640 0.0898 0.1194 0.1529 0.1902 0.2312 0.2761 0.3248 0.3773 0.4335 0.4932 0.5566 0.6235 0.6940 0.7679 0.8454 0.9264 1.0111 1.0994 1.1912 1.2862 1.3846 1.4864 1.5915 1.6998 1.9849 2.2896 2.6119 2.9516 3.3077 3.6788 4.0649 4.4664 4.8825 5.3130 5.7575 6.2150 6.6844 7.1643 7.6544 8.1543 8.6633 9.1808 9.7064

10.2398 10.7812 11.3308 11.8872 12.4497 13.0182 13.5926 14.1700 14.7499 15.9178 17.0928 18.2738 19.4614 20.6575 21.8627 23.0724 24.2820 25.4964 26.7197

0.0083 0.0200 0.0349 0.0532 0.0747 0.0993 0.1271 0.1580 0.1920 0.2292 0.2695 0.3129 0.3594 0.4090 0.4616 0.5173 0.5760 0.6378 0.7025 0.7700 0.8401 0.9144 0.9907 1.0700 1.1522 1.2373 1.3232 1.4159 1.6550 1.9144 2.1841 2.4731 2.7782 3.0994 3.4357 3.7865 4.1511 4.5286 4.9194 5.3241 5.7413 6.1699 6.6104 7.0633 7.5283 8.0049 8.4921 8.9884 9.4932

10.0062 10.5281 11.0583 11.5962 12.1421 12.6952 13.2545 14.3923 15.5560 16.7372 17.9315 19.1388 20.3556 21.5858 22.8246 24.0719 25.3268

145

Table 9.4: Reduced Pseudo-Pressure Integral (ΨΨΨΨr) as a Function of Tr and Pr.

146

9.2 SOLUTION OF LIQUID FLOW EQUATIONS Solutions of the liquid flow equation (diffusivity equation) are discussed in this section. Three cases - steady state, pseudo-steady state and the unsteady state (transient state) - are considered. Conditions under which they are attained are also given. 9.2.1 Steady State Solution

Steady state condition is attained when the rate of change of pressure is zero in the entire reservoir. That is,

0 = t

p

∂∂

9.23

The condition expressed in Eq. 9.23 can be attained in a reservoir undergoing depletion if there is influx of some material (e.g. water) at the external boundary to recharge the reservoir and hence prevent the pressure from dropping as a result of the production at the well. This implies that for a reservoir to attain steady state condition, its boundary must be open to flow. However, the entire boundary does not necessarily need to be open to flow. As long as the water influx from any part of the boundary is strong enough to keep the pressure in the reservoir from changing, the system will attain steady state condition. The diffusivity equation, inner and outer boundary conditions for the steady state case are given as: Diffusivity Equation

0 = )r

pr (

r r

1

r

p r

1+

2

2

∂∂

∂∂

∂∂

=∂∂

r

p 9.24

Inner Boundary Condition

r dp

dr = = constant

r

q

khw

µπ2

9.25

Outer Boundary Condition

p = pe at r = re 9.26

The solution to Eq. 9.24 subject to the inner boundary condition is:

p - p = q

2 kh ln

r

rw

w

µπ

9.27

Applying the condition at the outer boundary (r = re, p = pe), Eq. 9.27 becomes:

P - P = q

2 kh ln

r

re w

e

w

µπ

9.28

147

Rearranging, Eq. 9.28 gives

q = 2 kh

[Pe - P

w

re

rw

πµ

]

ln

9.29

The volumetric average pressure for any system is given as follows:

p =

pdV

V

o

VT

T

∫ 9.30

where

V = (r - r2

w

2π )h∅ 9.31

And

V hT = (r - r2

e

w

2π ) ∅ 9.32

From Eq. 9.13,

dV = 2πrh∅dr 9.33 We can use Eqs. 9.30 to 9.33 to determine the expression for average pressure for a system at steady state. Substituting Eqs. 9.32 and 9.33 into Eq. 9.30 and integrating gives:

pr re e

= p + q

2 kh

r

- r 1n r - 1n r -

r

- r 1n r -

1

2 w

e

2

w

2 w ww

2

w

2 w

µπ 2 2

9.34

For re >> rw, Eq. 9.34 becomes

]2

1 -

r[1n

kh2

q +p = e

w

wrp

πµ

9.35

Rearranging Eq. 9.35,

q

rw

= 2 kh [p - p ]

[1n r

- 1

2 ]

w

e

π

µ 9.36

148

=

]r 0.6061n [

] p - p[kh 2

e

w

wrµ

π 9.37

Combining Eqs. 9.28 and 9.36 gives:

p e - p = (p - p - (p - p = q

4 khe w w) )

µπ

9.38

Equation 9.38 shows the relationship between the pressure at the external boundary and the average pressure for a system that attained steady state. Equation 9.37 is not a very useful equation because of the difficulty in obtaining the average pressure. When a well that attains steady state is shut in, the pressure does not build up to the average pressure. The pressure builds up to the initial pressure because of influx from contiguous aquifers. And for a system that attains steady state, pe = pi.

9.2.2 Pseudo-Steady State Solution A reservoir attains pseudo-steady state if the rate of pressure decline with time is a constant. At that state, the mass rate of production is equal to mass rate of depletion. Further discussion on pseudo-steady state follows. Pseudo-steady state is also referred to as semi-steady state, quasi-steady state and in some cases, wrongly referred to as steady state.

The conditions necessary for this state to be attained are as follows: (a) Reservoir outer boundary must be closed to flow.

(b) Well producing from finite drainage volume due to long production time as shown in Figure 9.4. The finite drainage volume will be in proportion to the production rate of that particular well in the drainage area. That is, a well producing at a rate of 2q will drain twice the drainage volume of a well producing at rate q.

Figure 9.4: Reservoir Depletion under Pseudo-Steady State

Mathematically, at pseudo-steady state condition,

∂∂p

t =

dp

dt = constant 9.39

2q q

q

149

Ideally, only reservoirs with closed outer boundaries should attain pseudo-steady state but partially closed reservoir with very limited influx may attain a state, which for practical purpose may be considered to be pseudo-steady state. 9.2.2.1 Pseudo-Steady State Equation Equations for pseudo-steady state are given in this section. First, the value of the constant

(∂∂p

t = constant) is determined. Secondly, the solution to the pseudo-steady state

equation is given and lastly, the expressions for the volumetric average pressure and other related equations are derived.

A. Derivation of Value of Constant When pseudo-steady state condition is attained, Mass production rate = rate of mass depletion That is,

qρw = − V d

dt ( )T ρ∅ 9.40

= − ∅ V c dp

dtT tρ

Substituting for the total volume gives:

q ρ π ρw e

2

w

2

t = - [r - r ] h c dp

dt∅ 9.41

Therefore,

dp

dt =

- q

(r - r ) h c

e

2

w

2

t

w

∅πρρ

From the equation of state for a constant compressibility fluid

ρρw = e [p - p ]-c

w 9.42

Substituting Eq. 9.42 into Eq. 9.41 gives

dp

dt =

- q

[r - r ] h c e

e

2

w

2

t

-c [ p - pw

∅π] 9.43

150

For liquid of small compressibility,

ec [ −

≅p - pw 1

]

Hence

dp

dt =

- q

[r - r ] h c =

p

te

2

w

2

t∅π∂∂

9.44

= −qVc t

= constant 9.45

Equation 9.45 is the basis of reservoir limit test. This equation implies that if the pressure-time data obtained during pseudo-state is graphed on a Cartesian paper, a straight line whose slope is related to the well’s drainage volume is obtained. Equation 9.45 can be derived from a volume balance (a degenerate form of mass balance, see Craft and Hawkins, page 286), but such derivation does not explicitly show that an assumption was made about the fluid compressibility.

B. Pseudo-Steady State Solution The governing diffusivity equation is given by Eq. 9.8. At pseudo-steady state, the rate of pressure change is given by Eq. 9.44. Substituting Eq. 9.44 into Eq. 9.8, the diffusivity equation at pseudo-steady state becomes:

1

r

r (r p

r =

- q

kh (r - re

2

w

2

∂∂

∂∂

µπ

)) 9.46

Using the inner boundary condition given by Eq. 9.25 and the fact that the pressure at any point, r, in the reservoir is p, the solution to Eq. 9.46 is

( )( ))

P - P = q

2 kh (r - r 1

2 r - r + r ln r / rw

e

2

w

2 w

2 2

e

2

w

µπ

9.47


P - P = q

2 kh [ln r / r -

1

2 rw w

2µπ

/ ]re2 9.48

When r = re, p = pe, then Eq. 9.48 becomes

P - P = q

2 kh [ln

r

r -

1

2e w

e

w

µπ

] 9.49

151

Hence

q = 2 kh [p - p

[ln r

r -

1

2

e w

e

w

π

µ

]

]

9.51

C. Volumetric Average Pressure Using the same principles as in steady-state case, the volumetric average pressure when pseudo-steady state condition is attained is:

p = p + q

2 kh [r - r [r

2 - r ln rw

e

2

w

2

w

2

e

2

w

µπ ]

] -

+

q

kh [r - r {r ln r -

r

4 ln +

r

4 (r

e

2

w

2 e

2

ee

2

w

2

e

4µπ ]

( ) )}2

2 2 41

2

1

2

1

8r r re w w− − −

9.52


p = p + q

2 kh [ln

r

r-3

4w

e

w

µπ

] 9.53

Hence

q = 2 kh [p - p

[ln r

r

w

e

w

π

µ

]

]−3

4

= 2 kh [p - p

ln 0.472 r

r

w

e

w

π

µ

] 9.54

Some deductions from the pseudo-steady state equation are:

1. The wellbore pressure pw, average pressure p , and the pressure at the external

boundary Pe in Eq. 9.54 are dependent on time. This is not explicit in Eq. 9.54.

2. When a system with closed outer boundaries is shut-in, the pressure builds up to the

average pressure. Hence Eq. 9.54 is a useful equation. 3. Note the similarities between Eq. 9.37 derived for steady state case and Eq. 9.51

derived for the pseudo-steady state case. The equations are summarized in Table 9.5.

152

4. The term 0.472re in Eq. 9.54 is usually replaced with rd called the drainage radius. This is a misnomer because it gives the impression that just a part of the reservoir is being drained. Actually, the whole reservoir is drained.

Table 9:5: Summary of Stabilized Flow Equations in Darcy Units

Steady State Pseudo-Steady State (re >>>>>>>>r) General relationship between p and r

p pw− = q

2 kh ln

r

rw

µπ

p pr

w

e

− = q

2 kh [ln

r

r -

r

w

2µπ 2 2

]

Flow equations when p = pe and r = re

p = q

2 kh ln

r

r

e

w

e wp−µ

π

p = q

2 kh ln

r -

1

2

ee w

w

pr

−

µπ

Flow equations in terms of reservoir average pressure

p - p = q

2 kh ln

r

r -

1

2w

e

w

µπ

p - p = q

2 kh ln

r

r -

3

4w

e

w

µπ

D. Other Useful Pseudo-Steady State Equations By combining the pseudo-steady state equations, the relationship between the reservoir average pressure, the initial pressure Pi and the pressure at the external boundary Pe

during pseudo-steady state are obtained as follows:

Pi - p = qt

[r - r h c

e

2

w

2 t∅π ] 9.55a

pi – p = (pi - p ) + ( p - pw) - (p - pw)

= q

2 kh [2

r

r

kt

c r + ln

r

r-3

4 +

1

2 (r

r]w

e t w

2

eµπ

∅µ

2

2)e 9.55b

(pe - p ) = (pe - pw) - ( p - pw) = q

8 kh

µπ

9.56

153

pi - pe = (pi - p ) - (pe - p ) = q

2 kh [

2kt

c r -

1

4t e

2

µπ ∅µ

] 9.57

9.2.3 Unsteady State (Transient State) Solution

During the transient state, the rate of pressure change in the reservoir has a value which is neither zero nor a constant. Transient state behavior occurs in every system when the boundary effects are not yet felt. For example, a reservoir that is infinite (let us for now assume that there is something like that) will always be in transient state. Also, reservoirs that are finite (the boundary may or may not be closed to flow) will undergo transient behavior at early time when the boundary effects or interference due to production from and/or injection into other wells are not yet felt. The early time when the transient state behavior persists is generally known as the infinite- acting stage. In this section, a simple analytical solution of the transient state equation is given. The

solution may be obtained by Boltzman transformation discussed in Matthew and Russel

(1967). The governing equation and associated boundary and initial conditions are:

Diffusivity Equation

∂∂

∂∂

∂∂

2

+ 1

r

p

r =

c

k

p

t

tp

r 2

∅µ 9.58

Inner Boundary Condition (Constant Production)

kh2

q = ]

r

p[

0

r

πµ

∂∂

→w

r 9.59

Outer Boundary Condition

Lim p = pi as r → ∞ 9.60

Initial Condition

P = Pi, when t = O 9.61 Equation 9.59 specifies that the well has a vanishing wellbore radius. That is the wellbore is considered to be a line and hence the solution to the diffusivity equation obtained with such inner boundary condition is called the line source solution. If a finite wellbore inner boundary condition is considered, a solution can be obtained by Laplace transformation.

154

Although the outer boundary condition (Eq. 9.60) indicates that the system is infinite but this is just a mathematical representation of systems where the boundary effects or effects due to interference have not been felt. Solution to diffusivity equation subject to the given boundary and initial condition is:

p(r, t) = pi + q

kh

r

kt

µπ4 4

2

E - (c

it∅µ

) 9.62

For x < 0.0025,

Ei (- x) ≈ ln (γx) = ln x + 0.5772 9.63

The factor, γ, is the Euler’s constant and it is equal to 1.781. Ei(- x) is called the

exponential integral function and is defined as

Ei(- x) = - e

u du

-u

x

∞

∫ 9.64

Applying the approximation (generally called log approximation) given as Eq. 9.62, for

∅µc r

kt

t

2

4 < 0.0025. That is (

kt

c t∅µ r > 100)

2,

the transient state solution becomes

p(r, t) = pi + q

4 kh ln

( c rt

2µπ

γ∅µ )

4kt 9.65

At the well

p(rw, t) = Pi + q

4 kh ln

( c rt

2µπ

γ∅µ )

4kt 9.66

The above solution is the line source solution also called the exponential integral solution or Theis solution. Equation 9.65 makes it possible to predict the pressure at any point r, in the reservoir due to constant rate production at the well. This equation forms the basis of interference test. Equation 9.66 makes it possible to predict the pressure at the well due to constant rate production at the well. This equation forms the basis of drawdown tests The interest in the transient state solution is due to the fact that every well, at early time, goes through the transient state. That is, the well behaves like a single well in an infinite reservoir. At later times, the effects of other wells, reservoir boundaries, aquifer influence cause it to deviate from the infinite acting behaviour. Figure 9.5 is a semilog graph of

155

bottom-hole flowing pressure versus time showing that every system irrespective of the boundary condition initially goes through the transient state phase.

Figure 9.5: Plot of Pwf versus log t for Reservoirs with Different Outer Boundary

Conditions

Determination of Ei Function

For practical purposes, the log approximation to the Ei function holds for

kt

c t∅µ r 2 > 5 with about 2 percent error.

When the log approximation does not hold, the function can be read from Table 9.6.

Table 9.6: Values of the Ei Functions

156

9.3 SUMMARY OF TRANSIENT, PSEUDO-STEADY AND STEADY STATE

FLOW EQUATIONS The equations describing flow during the transient, pseudo-steady (semi-steady) and steady state periods have been given. A brief review of the different types of flow is given in the paper published by Matthew’s (1986). Further discussion on transient, pseudo-steady, and steady state flows is given by Ramey (1975), Matthew and Russel (1967) and Earlougher (1977). A summary of the governing equations during the flow periods is given in Table 9.7. The equations are given in both Oilfield and Darcy’s Units. In the equations in Table 9.7, the skin effect is neglected. Table 9.7: Summary of General Flow Equations

Flow Type: Transient

Darcy Units: P = P +

q

4 kh E

- ct r

kti i

2µπ

∅µ

4

Oil Field Units:

∅kt 0.00264 x 4

rct - E 70.6 + P =

2

ii

µµkh

qBP

Practical SI Units

∅kt 0.00264 x 4

rct - E 10 x 9.33 + P =

2

i

2

i

µµkh

qBP

157

Flow Type:

Pseudo-Steady State

Darcy Units: P = P +

q

2 kh

r

r -

3

4w

e

w

µπ

ln

Oil Field Units: P = P + 141.2

qB

kh

r

r -

3

4w

e

w

µln

Practical SI Units

4

3 -

r

rln

kh

qB10 x 1.866 + P =

w

e3

w

µP

Flow Type:

Steady State

Darcy Units:

P = P + q

2 kh ln

r

rw

w

µπ

Oil Field Units: P = P + 141.2

qB

kh

r

r w

w

µln

Practical SI Units

r

rln

kh

qB 10 x 1.866 + P =

w

3

w

µP

9.4 LIQUID FLOW SOLUTIONS IN DIMENSIONLESS FORMS

In this section, the dimensionless forms of the diffusivity equation, steady state, pseudosteady state and transient state solutions are given Diffusivity Equation:

∂∂

∂∂

∂∂

2

D

D

D

D

D

+ 1

r

p

r =

p

t

P

r

D

D

2 9.66

Steady State Solution ∂∂p

t

D

D

= O

q = 2 kh

[P - P

r

r

e w

e

w

πµ

]

ln

9.67

and Pe = Pi

Then in dimensionless form

158

P = 2 kh

q

Dw

[Pi - Pwπ

µ] = ln

re

rw 9.68

Generally,

r

r

q

pPkhP ei

D ln][2

=−

=µ

π 9.69

Pseudo-Steady State Solution, ( )∂∂p

t

D

D

= constant

P - P = q

2 kh 2

r

r

kt

c r + ln

r

r-3

4 +

1

ri

w

e

2

t w

2

eµπ µ

∅

2

2

re 9.70

In dimensionless form,

P = 2 kh

q (P - PDw i w

πµ

) = 2t

r + ln r -

3

4 +

1

2 rD

De

2 De De

2 9.71

Unsteady State (Transient State) Solution (Line Source Solution)

P = P + q

4 kh E

(- cri i

2µπ

∅µ )

4kt 9.72

In dimensionless form

P = - 1

2 E

(-rD i

D

2

4

)

tD 9.73

For r

t

D

D

2

4 < 0.0025, the log approximation holds. That is,

P = -1

2 ln

( rD

D

2γ )

4tD 9.74

+==

303.2

80907.0tlog1.15] 0.80907 +

r

t[1n

2

1

2

D

2

D

D

Dr 9.75

Figure 9.6 is a schematic of dimensionless time for an infinite system, closed outer boundary system and constant pressure outer boundary system.

159

Figure 9.6: Dimensionless Pressure and Time for Different Reservoir Outer Boundary Conditions

9.5 COMPARISON BETWEEN LINE SOURCE SOLUTION AND FINITE

WELLBORE SOLUTION

The line source solution also called the exponential integral solution is very easy to obtain using the Boltzman transformation technique. The finite wellbore solution can be obtained in the Laplace space using Laplace transform technique. Solution in real space may be obtained numerically using Stefhest algorithm. The comparison between the exponential integral solution and finite wellbore solution is shown in Fig. 9.7. Deductions that may be made from Fig. 9.7 are:

1. When t

r

D

D2 ≥ 25 (for all rD) the exponential integral solution and the finite wellbore

solution are same

2. Also at locations for which rD ≥ 20 both solutions are also the same when t

r

D

D2 ≥ 0.5.

160

Figure 9.7: Line Source Solution and Finite Wellbore Solution

9.5 GENERALIZED FLUID FLOW EQUATIONS

So far, we have considered a single well draining a circular reservoir. The dimensionless pressures for such case have been given for different conditions. The drainage area in many reservoirs may not be circular. Hence, the reservoir geometry must be accounted for. This is done by the introduction of Dietz shape factors, which allows a general form of inflow equation to be developed for a wide range of geometries of drainage area and positions of the well within the boundary. The pressure drop at any point in a single-well reservoir being produced at constant rate q, is then described with the general solution:

( ){ }P - p = q

2 khPD t geometryi D µ

π, , ,r CD D

9.76 The dimensionless pressure in Eq. 9.76 is dependent on dimensionless time,

dimensionless radius, wellbore storage dimensionless constant, and reservoir geometry.

The skin factor is added to the dimensionless pressure only when calculating the pressure

at the well. That is

( ){ }P - p = q

2 kh P t geometry + Si w D D

µπ , , ,r CD D

9.77 Practically speaking, dimensionless pressure is just a number that may be given by an equation, read from a table or a graph for different systems. The equations for calculating dimensionless pressure for some systems are given.

161

A. Infinite (Infinite Acting) System without Wellbore Storage and Skin

PD i

D2

D

= -12 E

r

4t

−

9.78

5 > r

tFor

2

D

D

Pr

DD

= 1

2ln

t + 0.80907 D

-

-

2

9.79 Equations 9.78 and 9.79 should be familiar.

B. Closed System (General Case)

For all closed systems undergoing transient flow, the dimensionless pressure is given within 1 percent as:

PwD = 1

2 [ln t + ln

A

r + 0.80907 ]DA

w2

9.80 where 0.000025 < tDA < t`DA and t`DA is the time to use infinite system solution with

less than 1 percent error. Values of t`DA for different reservoir geometries and well

locations are given in the last column of Table 9.8 taken from Earlougher (1975). Does Eq. 9.80 imply that the transient state equation depends on drainage area? No. The area term in Eq. 9.80 cancels out if tDA is expressed as a function of tD and A. Equation 9.80

is exactly the same as Eq. 9.79 for rD= 1 (wellbore).

162

Table 9.8: Shape Factors for Various Closed Single - Well Drainage Areas

164

At long times, every closed system attain pseudo-steady state and the dimensionless pressure is given as:

PwD = 2 t + 1

2 ln

A

r +

1

2 ln

2.2458

CDA

w A2

π

9.81

Equation 9.81 applies for different systems when tDA is greater than the time under the

column with heading, “exact for tDA >” given in Table 9.8. With a maximum of 1%

error, the time when pseudo-state will start can be read from the column with the heading “less than 1 percent error for tDA >.” This makes it possible to use the pseudo-state

equation to calculate pressure in a period that ideally is considered a transition period (transition from transient to pseudo-steady state). The CA in Eq. 9.81 is the geometric or shape factor, which is also given in Table 9.8 for different drainage systems and well locations. From Eq. 9.81, a Cartesian graph of dimensionless pressure versus dimensionless time based on area gives a straight line with

slope of 2π when pseudo-steady state condition is reached.

C. Infinite System with Single Vertical Fracture (No Storage and Skin)

(i) Uniform Flux Fracture: For uniform flux fracture, fluid enters the fracture at a uniform flowrate per unit area of fracture face so that there is a pressure drop in the fracture. Uniform flux fractures are closer to natural fractures. The dimensionless pressure for uniform fracture is:

P = t erf 1

1 -

1

2 E -

1

4twD Dxf i

Dxf

π

2tDxf

9.82

where dimensionless time is based on half-fracture length defined as

t = t r

xDxf D

w

f

2

The function erf is the error function defined as:

erf (x) = 2

π e du-u2

o

x

∫

9.83 For tDxf > 10, Eq. 9.82 becomes

165

PwD = 1

2 [ln t + 2.80907]Dxf

9.84 With less than 1 percent error and for tDxf < 0.1,

PD = tDxfπ

9.85 Taking the log of Eq. 9.85 gives

log PD = 1

2 log [t + ]Dxf π

9.86 Equation 9.86 implies that a log-log plot of PD versus tDxf gives a straight line with a

slope of 0.5. This implies that at short times, flow into fracture is linear. Linear flow is characterized by a slope of 0.5 on log-log graph of PD versus tDxf or (∆p versus t). (ii) Infinite Conductivity Fracture: In this case, the fracture is considered to have infinite permeability and therefore uniform pressure in the fracture. The dimensionless pressure for infinite conductivity fracture systems is given by Earlougher (1977) as:

PD = 1

2 t [erf(

0.134

t + erf (

0.866

t ]Dxf

Dxf Dxf

π ) )

-0.067Dxf

i

Dxf t

(-0.750) E 0.433 - )

t

0.018-( iE

9.87 But when tDxf > 10,

PD = 1

2 [ln t + 2.20]Dxf

9.88 Also, with less than 1 percent error and for tDxf < 0.01

PD = tDxfπ

9.89 Equation 9.89 indicates that at short times, flow into infinite conductivity fracture is also linear.

166

D. Closed System with Single Vertical Fracture (No Storage and Skin)

Equations given for the infinite system with single vertical fracture apply but when the system attains pseudo-steady state.

Px

x CD

e

f A

= 2 t + 1

2+

1

2DAπ ln ln

.

22 2458

9.90 where xe is half-length of the side of the system parallel to the fracture.

9.6 WELLTEST DESIGN In new fields or special circumstances, it is important that tests be designed before they

are run. The objectives of such test design are as follows:

1. To determine the duration of the flow phases 2. To determine the expected pressure changes. 3. To ensure that correct equipment (especially gauges) are used.

During test design, values of well and reservoir parameters are assumed while equations are used to calculate the required parameters. This will be illustrated with an example.

Example: Table 9.9 shows assumed and estimated well and reservoir parameters. Using the values in Table 9.9 and assuming well in the center of a 2:1 rectangle and 40 acre drainage area calculate the following:

(a) The duration of the wellbore storage phase. (b) Duration of the transient state period (c) The time when pseudo-steady state will start (d) Draw the box diagram showing the duration (e) The maximum pressure change at the end of strong wellbore storage effect (f) The pressure change at the end of transient (g) The pressure change at the beginning of pseudo-steady state Table 9.9: Parameters for Test Design

Parameter Value



Porosity, φ 0.25



Production rate, q, STB/D 1000 Formation thickness, h, ft 50 Permeability, md 1000 Skin Factor 25 Wellbore Storage Constant, rb/psi 0.009

167

Solution A. For drawdown, wellbore storage ends at time

tkh

ws = (200000 + 12000S)Cs

/ µ = 0.054 hrs

B. For well in the center of 2:1 rectangle, transient ends at a time read from Table 9.8 as tDATs = 0.025. Converting tDATs to ordinary time gives

tA

ktTs DATs

= ct∅µ

0000264.. = 0.495hrs

C. For this system pseudo-steady state starts at time read from Table 9.8 as tDAPSS

=

0.3. In absolute time, this is

t A

K=

c tt

DApss

φµ0000264.

= 5.94 hrs

D. Box Diagram Showing Duration for Drawdown Test O t(hr) E. Assuming wellbore storage effect died down completely at 50t*, therefore

t * = tws

50

tr

D

w

* = 0.000264 kt *

c = 380.16

tφµ 2

Cc hr

D

t w

= 0.8936 C

= 128.68s

φ 2

0.054

0.495

Transient phase

Wellbore storage phase Wellbore storage phase

Pseudo-steady state

5.94

168

Dimensionless pressure at end of strong wellbore storage is

PC

D

D

= t *D

The pressure change now becomes

[ ]∆pB

= 141.2q

kh P + SD

µ = 1.9062 [2.95 + 25] = 53.28 psi

F. Dimensionless time at end of transient is

tc r

D

t w

= 0.000264kt

= 1.74 x 10Ts 5

φµ 2

Dimensionless pressure at end of transient is

PD i = -1

4t12

D

Ε

or

[ ]P In tD = 12 D + 080907. if log approximation holds.

Log approximation holds, for t

r

D

D

2 > 5 , therefore, PD = 6.44

Pressure drop at the end of transient is

[ ]∆pB

= 141.2q

kh P + SD

µ = 1.9062 [25 + 6.44] = 59.93 psi

G. Dimensionless pressure at the beginning of pseudo-steady state is

PD pss = 2 t + In

A

r + In

2.2458

C = 8.626DA

1

2

w

2

1

2

A

π

Pressure drop at the beginning of pseudo-steady state is then

[ ]S + kh

141.2qB = DPp

µ∆ = 1.9062 [8.626 + 25] = 64.10 psi

9.7 SUPERPOSITION

So far, we have considered the well to be produced at constant rate and the effect due to only one well in a given reservoir system. In this section, we shall use superposition concept to account for production at different rates and also the effect of having more than one well in a given reservoir system.

169

Simply put, superposition generally means that any sum of solutions to a linear differential equation is also a solution. In Petroleum Engineering, this implies that the total pressure drop at any point in a reservoir that contains more than one well is the sum of the pressure drop due to each well acting as if it is in the reservoir alone. Hence superposition concept can be used for generation of changing flowrate or pressure boundary condition in pressure buildup analysis and water-drive material balance methods. Also superposition can be used for determining the effect of many wells in a reservoir. In this section, the application of superposition will be illustrated with examples. Superposition in both space and time will be considered. Discussion on these follows. 9.7.1 Superposition in Space Considering a reservoir containing two wells as shown in Figure 9.8, the pressure drop due to both wells at point (x,y) in the reservoir is given using superposition in space as: Pressure drop at x, y = Pressure drop at x, y due to Well 1 +Pressure drop at x, y due to Well 2 9.91

Figure 9.8: Superposition in Space

Using the notations in Figure 9.8, Eq. 9.91 implies: Pi - P = (P - P + (P - Px, y, t i a, t i b, t) )

9.92

Well 2

P(x, y)

b a

d

Well 2

q 2 q 1

170

Generally, (P - P = q

2 kh P (r ti r, t D D D) , )

µπ

9.93 Therefore,

) t,(b P 2

q + ) t,(a P

2

q = t)y, (x, P - DDDDDD

21

khkhPi π

µπ

µ

9.94 At Well 1, Eq 9.94 becomes

) t,(d P 2

q + ])(t [P

2

q = P - DDD1DDwf1

21

khs

khPi π

µπ

µ+ 9.95

The second term on the right hand side of equation 9.95 is the interference effect due to

Well 2 at Well 1.

At Well 2, Eq 9.94 becomes

])(t [P 2

q + )t(d P

2

q = P - 2DDD D,Dwf2

21 skhkh

Pi +π

µπ

µ 9.96

The first term on the right hand side of equation 9.95 is the interference effect due to Well

1 at Well 2.

For q1 = q2 = q, Eq. 9.94 becomes

2πkhqu

tD P - P (x, y, t) = P (a + P (b t( )i D D D D D, ) , ) 9.97

At Well No. 1, Eq. 9.97 is:

2πkhqu

P - P = P (1, t + S + P (d t( )i wf1 D D 1 D D D) , ) 9.98

and at Well No. 2, Eq. 9.97 is:

2πkhqu

P - P = P (d , t + P (1 t + S( )i wf2 D D D D D 2) , ) 9.99

Note that at the wells, the skin factor terms are added. Also, for superposition in space, the wells may have different production rates but for all wells, production must commence at the same time. Signs are changed (q is negative) for injectors.

171

Implications of Superposition in Space

1. The pressure distribution for one well producing at constant rate q located at a distance d/2 from a linear flow barrier, or one well in a semi-infinite reservoir can be simulated by super-imposing solutions of two wells. This is shown in Figure 9.9. The two wells must either be injectors or producers.

Figure 9.9: Simulation of No-Flow Boundary

2. Similarly, a single well producing near a constant pressure barrier can be simulated with two wells. In this case, one of the wells must be an injector while the other must be a producer. This is shown in Figure 9.10.

Figure 9.10: Simulation of Constant Pressure Boundary

3. Single well producing in closed square can be generated by considering an infinite array of wells as shown in Figure 9.11.

d/2 d/2 d/2

= =

172

Figure 9.11: Simulation of a Well in a Closed Square

For Figure 9.11, if we want to compute the pressure at the well in the center of a closed

square with dimension A rw/ = 2000 and at time tDA = 0.1, the procedure is as

follows: (i) Draw the closed square with the well at center. Then, draw a few of the infinite arrays of wells around the well of interest. A few of the infinite array of wells is required because the further the wells are from region of interest, the lesser their contribution to the total pressure effect. The array of wells is shown in Figure 9.12. (ii) Of the array of wells around the well of interest, locate wells that are of equal distance from the well of interest. This was done by assigning the same code number to wells at equal distance from well of interest as shown in Figure 9.12.

Figure 9.12: Schematic of Array of Wells.

= =

5 5

4

3

4

5

1

4

2

4

4 3

2

3 4

1

4

5

4

4

2

1

4

3

173

(iii) Calculate the distance from the well of interest to array of wells surrounding it. The

square has an area of “A” and each side of the square has a length of A . Also the

following relationship holds

t

r

D

D2 =

kt

cr

2∅µ 9.100

The above relation is important because the dimensionless pressure due to each well is a function of tD/rD

2. Hence the contribution of each well is easily found from the

exponential integral solution. Note that each well will act as if it is in the reservoir alone. Table 9.10 shows the calculated distances and dimensionless pressures. Table 9.10: Calculated Distances, Time and Dimensionless Pressures

Assigned

No.

No. of Wells

Distance From Well to Well of

Interest

t rD D/ 2

=kt

cr2∅µ

0.1) = (

)/( 2

DA

DD

t

For

rtP

1 4 A tDA .012458

2 4 2A tDA

2

.000574

3 4 2 A tDA

4

.0000025

4 8 5A tDA

5

0

(iv) Applying the superposition principle,

2

2 4 5

2

2 2 2

kh

q P - P = PD (t + S + 4 PD

t = t

4 P t

= t

+ 4 P t

= t

+ 8 P = t

[ i wf DD

DA

DD DA

DD DA

DDA

µ] )

r

r r

t

r

D

D D

D

D

+

9.101

The first term on the right hand side of Eq. 9.101 is the contribution of the well of interest. This contribution is calculated as follows:

tD = kt

crw2∅µ =

kt

cA .

A

rD2∅µ = t .

A

rDA

D2 9.102

For the case of interest, it is given that

174

A rw/ 2 = 4 x 10 and t = 0.1 6DA

Therefore

P (t = P (0.1 x 4 x 10D D D6) ) = 6.8542

Neglecting the skin factor S, and substituting values in Eq. 9.101 gives PD = 6.8542 + 4 x 0.012458 + 4 x 0.000574 + 4 x 0.0000025 + 0 = 6.90634

This compares favourably with a value of 6.9063 calculated by Earlougher et al (1968). The same procedure may be used to obtain dimensionless pressures at any point in the well. 9.7.2 Superposition in Time

Superposition in time is used for handling flow rate that varies with time at a given well. A schematic of how this is done is for a two-rate case is shown in Figure 9.13.

Figure 9.13: Superposition in Time (2-Rate Case)

In words, the pressure drop due to a well producing in an infinite reservoir at rate q1 for

time “t”, and then for rate q2 for a period of “∆t” is equivalent to the sum of pressure drop

due to the well producing at rate q1 for time “t + ∆t” and the same well producing at rate

(q2 - q

1 ) for a period of “∆t”.

That is:

[ ] )DDD12

D1

i t,r P 2

)q-(q + )(,r P

kh2

q = ) t (r, P - P (

D∆∆+

khtt D π

µπ

µ 9.103

q2-q1

∆t t +∆t t ∆t

q2

q1

q q q

+ =

175

At the well, rD =1, P(r,t) = Pwf and the skin term will be added to give

(P - P =q

2 kh P [1, (t + t) +

(q - q

2 kh P (1, t +

q

2 kh Si wf

1 2 1 2D D D D) ]

))

µπ

µπ

µπ

∆ ∆ 9.104

If q2 is zero, Eq. 9.104 becomes

{ }][][)( DDDD1

wsi t 1, P - t) +(t ,1P kh2

q = P - P ∆∆

πµ

9.105

Note that the skin term has disappeared and we have changed notation from pwf (pressure well flowing) to pws (pressure well shut-in). If we assume transient and log approximation,

]80907.0)[ln(5.0])(,1[ +∆+=∆+ DDD ttttP 9.106

and

]80907.0[ln5.0],1[ +∆=∆ DDD ttP 9.107

Substituting Eq 9.106 and 9.107 into Eq 9.105 gives

∆∆+t

ttln

kh2

q = P - P 1

wsi )(π

µ 9.108

Equation 9.108 is the basis of the buildup test. We have derived the equations for a two-rate case. Figure 9.14 shows how superposition is done for a three-rate case. You should derive the equations for this case as an exercise.

=

q

t1 + ∆t2 + ∆t2 t1 ∆t2 ∆t

q1

q2

q3

176

Figure 9.14: Superposition in Time (3-Rate Case)

EXERCISES

1. Estimate the pressures at the well located in the center of a 2:1 rectangle after the well has produced 800 STB/day of dry oil for 15 minutes and 15 days. Other data are:

Pi = 3265 psi, ko = 900 md, µo = 6.2 cp, Bo = 1.02 RB/STB, h = 47 ft

∅ = 0.17, ct = 2 x 10-5 psi-1 rw = 0.50 ft A = 40 acres, s = 7

2. The dimensionless pressure at pseudo-steady state well in the center of a circular reservoir is:

Pr r r

ie w e

- P = q

2 kh 2

r

kt

c + ln

r -

3

4 +

1

2 r

ww

t rw2

e wµπ µ

∅

2 2

And the generalized dimensionless pressure for any closed system with any geometry is

where CA is the geometric factor. Assuming re >>rw, find the geometric factor for a well

in the center of a circular reservoir by relating both equations. 3. Ramey and Cobb (1971) have shown that for a well situated at the center of a regular shaped drainage area, (for instance, a circle square or hexagon) the transient period is of extremely short duration. Under these circumstances, it is possible to equate the transient flow equation and pseudo-steady state equation to determine the time at

+

+

q2 – q1

q3 – q2

177

which pseudo-steady state condition starts. Find an expression for tDA at which this

occurs for a well in the center of a circle? 4. Neglecting skin, the pseudo-steady state equation for flow in circular reservoir is:

4

3-

rln

kh2

q = P - e

wf

wrP

πµ

The generalized flow equation at pseudo-steady state is

2

wA

wfr C

4Aln

2

1

kh2

q = P -

γπµ

P

where CA is the shape factor. By comparing the two equations, estimate the shape factor for a well in the centre of a circular reservoir.

178

10. BOTTOM-HOLE PRESSURE TESTS IN

HORIZONTAL WELLS The objectives of bottom-hole pressure tests in vertical and horizontal wells are similar. Actually, a horizontal well can be viewed as a vertical well with infinite conductivity vertical fracture or a highly stimulated vertical well. However, BHP tests in horizontal wells are more difficult to analyze for the following reasons. 1. The horizontal wells are not perfectly horizontal as assumed by the analysis models. The wells may be snake-like as shown in Fig. 10.1.

Fig. 10.1 Snake-Like Nature of some Horizontal Wells

The consequence of the snake-like nature of horizontal wells is that waves emanating from different sections of the horizontal well at the same time may hit different boundaries. This complicates pressure response. 2. The total drilled length may not all contribute to the producing length. Some drilled length may intersect non-productive interval. The unknown non-producing length is erroneously accounted for as skin factor. 3. There are many possible flow regimes which depending on the reservoir and well conditions may or may not be properly discerned from pressure tests. In this chapter, we shall present basic information about horizontal wells and how to identify the flow regimes obtained during horizontal welltests. We shall also present flow equations and how to analyze horizontal welltests.

Vertical Depth

Horizontal Distance

179

10.1 Introduction A horizontal well is a well drilled parallel to the reservoir bedding plane while a vertical well is drilled perpendicular to the reservoir plane. This is shown in Fig 10.2. With horizontal well, we can enhance reservoir contact by well and therefore hence enhance well productivity. This implies that the drainage area of the horizontal well is bigger than that of a vertical well. Joshi (1988) showed two methods of calculating the drainage area of horizontal well based on knowledge of the drainage area of a vertical well. Joshi’s procedures are presented.

Fig. 10.2: Vertical and Horizontal Wells.

Drainage Area of Horizontal Well

If the drainage area of a vertical well is:

A = π rev2 10.1

Where rev is shown in Fig 10.3.

Fig. 10.3: Drainage Area of Vertical Well

The two method for the calculating the drainage area of horizontal well are as follows:

Horizontal Well Vertical Well

rev

180

Method 1: This assumes that each end of the horizontal well drains a semi-circle drainage area while the length drains a rectangular drainage area as shown in Fig 10.4.

rev

Fig. 10.4: Rectangular/Semi-Circle Drainage Area

Assuming that h ≥ rev,

HWDA = ( ){ }πrev2 + L 2rev 10.2

Where HwDA = Drainage area of the horizontal well L = Productive length of the horizontal well

Method 2: We assume an elliptical drainage area in the horizontal plane, with each end of a well as a foci of drainage ellipse as shown in Fig 10.5.

Fig. 10.5: Elliptical Drainage Area

In this case, the drainage area is,

HWDA = πab 10.3 where

L

L

181

a = half major axis of ellipse = L

2 + rev

b = half minor axis = rev

The calculated drainage areas (using different methods) are not the same. Therefore, the average of the two can be used as the effective drainage area.

Applications of Horizontal Wells

(a) Some naturally fractured reservoirs (b) Reservoir with gas/water coning problems (c) Thin reservoirs (d) Reservoirs with high vertical permeability (e) In EOR projects with injectivity problems (f) In fields (e.g. offshore) requiring limited wells due to cost or environmental

problems.

Advantages of Horizontal Wells

(a) Horizontal well productivities are 2 to 5 times greater than that of unstimulated vertical well. Actually, the performance of horizontal wells depends on the effective length of the horizontal section in the formation. Higher productivity may result to early payout.

(b) Horizontal well may intersect several fractures or compactments and help drain them effectively

(c) Reduce coning tendencies (d) As injectors can improve sweep efficiency in EOR projects

Disadvantages of Horizontal Wells

(a) Ineffective in thick (500ft to 600ft) low permeability reservoirs. (b) Cannot easily drain different layers (c) Technological limitations (d) Cost more (1.4 to 2 times) than cost of drilling a vertical well.

Dimensionless Parameters used in Horizontal Wells

Dimensionless parameters are also used in horizontal wells. Figure 10.6 shows dimensions and coordinates in horizontal wells.

182

Fig. 10.6: Dimensions in a Horizontal Wells

Table 10.1 shows the different dimensionless parameters in Darcy unit.

Table 10.1: Dimensionless Parameters used in Horizontal Wells

Dimensionless Parameters Equations

Dimensionless Pressure in terms of h and kr PDh = k h p

Br

s o o

∆141.2q µ

Dimensionless Pressure in terms of L and k kr z PDL = k h L p

Br z

s o o

∆141.2q µ

Dimensionless Time in terms of L/2 tD = 0.0002637k

2)2r

t

t

cφ µ (L /

Dimensionless Time in terms of h and kz tDz = 0.0002637k

2z

o t

t

c hφµ

Dimensionless Time in terms of ye and ky tDy = 0.0002637k

2

y

o t e

t

c yφ µ

Dimensionless Time in terms of h and ye tDhy = 0.0002637ky

o t

t

c hyeφ µ

Dimensionless Time in terms of and rw tDrw = 0.0002637

2

k k t

c rr z

o t wφ µ

h

x

z

y

zw

L/2

183

Derivative in terms of pDL ( )pdp

d tDLDL

D

'

ln=

Derivative in terms of pDh ( )pdp

d tDhDh

D

'

ln=

Dimensionless x, y, z Coordinate xD = 2(x-xw)/L yD = 2(y-yw)/L zD = z/h

Dimensionless Wellbore Location

xwD = 2xw/xe ywD = 2yw/ye zwD = zw/h

Dimensionless x, y-direction boundary width xeD = 2xe/L yeDL = 2ye/L

Dimensionless Wellbore Length LD =

L k

kz

r2h

Dimensionless Wellbore Radius rwD = 2rw/L

10.2. Flow Regimes during BHP Tests In Horizontal Wells

During bottom-hole pressure tests in a horizontal well, the followoing flow regimes could be discerned: linear, radial, hemiradial, and pseudo-radial. Figure 10.7 is a box diagram showing some of the flow regimes and the order in which they occurred.

Some parameters in Table 5.1 are defined as follows:

Zw = vertical distance measured from bottom of payzone to the well

Xw, Yw, Zw = Well location co-ordinates

Xe, Ye = reservoir boundaries in x and y directions

L = Length of horizontal well

kr = kh = permeability in horizontal place

= k kx y

kz = kv = vertical permeability

184

Fig. 10.7: Typical Flow Regimes in a Horizontal Well

The pressure and pressure derivative for the case shown in Fig 10.7 is shown in Fig 10.8 taken from Lichtenberger (1994).

Fig. 10.8: Pressure and Derivative for a Typical Horizontal Well BHP Test

Details on the flow regimes are as follows: Early Radial Flow: Figure 10.9 shows the early time radial flow period in a vertical plane, which develops, when the well is put initially on production. The well acts as though it is a vertical well turned sideways in a laterally infinite reservoir with thickness, L. This flow period ends when the effect of the top or bottom boundary is felt or when flow across the well tip affects pressure response. This flow regime may not develop (Kuchuk , 1995) if the anisotropic ratio, kH/kV is large.

Transient State

Wellbore Storage

Radial Linear Pseudoradial Boundary Effect

185

Fig. 10.9: Early Radial Flow in a Horizontal Well

Many authors (Ozkan et al, 1989; Goode and Thambynayagam, 1987; Odeh and Babu, 1990; Du and Stewart, 1992; Lichtenberger, 1994 and Kuchuk, 1995) published equations for identifying the different flow regimes. Inferences from their publication show that the early radial flow can be identified using the pressure derivative or semilog plot if it is not marred by wellbore storage effects. The pressure derivative gives a zero slope as shown in Fig 10.8 while a graph of pressure versus log of time yields a straight line. The basis for the straight line is the equation:

Pi - pwf =

−+−

As

rc

kk

Lkk

qB

wt

vy

yv

87.023.3 t

log

6.1622φµ

µ 10.4

where s = skin due to damage/stimulation. If it is positive, it is denoted as Sm, mechanical damage due to drilling and completion. Also, A is a constant given by Lichtenberger (1994) as

A = 2.303 log ½ k

k

y

v

4 + k

k

v

y

4

10.5

The equation implies that a graph of Pwf versus log t gives a straight line with slope

m1 = 162 6. qB

k kv y

µ L

10.6

From this, the equivalent permeability in the vertical plane, k kv y , can be calculated.

186

k km L

v y = qB162 6

1

. µ 10.7

The skin equation for the first radial flow period is

S = 1.151( )Pi

r

- P 1 hr

m - log

k k

c + A + 3.23

1

y v

t w2φµ

10.8

Note:

1. In arealy isotropic reservoir, kx = ky = kh

2. If effective reservoir permeability k kv y is known, the given equation can be used

in determining the effective producing length. 3. The early-time flow regime can be short and may be completely marred by

wellbore storage effect. Use of downhole shut-in tool is useful here. Time to end of wellbore storage is given by Lichtenberger (1994) as:

µ/

)2404000(

Lkk

Cst

vh

mEus

+= 10.9

where

tEus = time for end of wellbore storage effects, hours sm = skin factor C = wellbore storage constant, rb/psi L = effective producing length of well, ft

µ = viscosity of oil, cp

kh = horizontal permeability = yxkk

kv,kx,ky = permeability in the vertical, x and y directions respectively.

Early radial flow ends when either of the following will occur: a. Effect of bottom or top boundary is felt. b. Flow across well tips affects pressure response Mathematically, different authors gave time to the end of early radial flow, te1, as follows: Goode and Thambynayam (1987)

te1 = v

wz

k

d t

095.0095.2 c r 190 φµ−

10.10

where dz is the distance of the well to the closest boundary (top or bottom) Odeh and Babu (1990) and Licthenberger (1994)

187

te1 = min

d C

k

C

k

z2

t

v

t

y

1800

125 2

φµ

φµL

10.11

The first equation represents the time when the effect of the boundary (top or bottom) will distort early radial flow while the second equation represents time when flow across the tip of the well will distort early radial. Lichtenberger assumed areal isotropy in his equations. DU and Stewart (1992)

te1 = min

C d

k

C d

k

t z2

v

t x2

x

947

947

φµ

φµ

10.12

where dx is nearest distance from well point to the boundary normal to the well length axis. Other parameters are as defined earlier. Although the equations by different authors are not exactly the same, but they are similar and therefore can be used as a guide. Hemiradial Flow : When the wellbore is closer to any of the no-flow boundaries, hemi-radial (or hemi-elliptical) flow may develop. This produces slope doubling on the semilog plot and p1 (pressure derivatives) will plateau at twice the radial flow value. The flow equation during the hemi-radial flow period is given by Kuchuk (1995) and Lichetenberger (1994) as

Pwf = Pi - 2 x162 6

3 230 87

22

.log .

.qB

k k

k k

c r

SA

H v

H v

t w

µφµ L

t

− + −

10.13

where

A = log

w

z

r

d

v

H

k

k + 1

dz = distance to the nearest boundary (top or bottom)

Lichtenberger (1994) gave the time for the end of the hemiradial flow as

188

tEhrf = 1800 d c

k

z2

t

v

φµ 10.14

Implications of the Eq 10.13 are as follows: a. A plot of pwf versus log t is a straight line with slope m

b. k kmL

H v = 2 qB162 6. µ

10.15

c. The skin equation is

S = 2.30 ( )

w

zi

r

d

r

P

V

H

2

wt

vH

k

k+1log + 3.23+

c

kklog -

m

hr 1P -

φµ 10.16

Intermediate Linear Flow: This flow regime may develop after the effects of upper and lower boundaries are felt at the well. Figure 10.10 shows the streamlines during the intermediate-time linear flow. This flow regime develops if the well length is sufficiently long compared with reservoir thickness and there is no constant pressure boundary.

Fig. 10.10: Intermediate Linear Flow

The flow equation during the linear flow is given as follows:

Pi - Pwf = ( )8128. qB

Lh

t

c kS S

t y

z

µφµ

µ +

141.2 qB

L k ky v

+ 10.17

where SZ is the pseudo-skin factor caused by partial penetration in the vertical direction and is given by different authors as:

189

Odeh and Babu (1990)

SZ = lnh

rw

−

°

−

+ 0.25 ln

k

k

Z

h

y

v

wln sin .180

1838 10.18

Lichtenberger (1994)

SZ

+≈

h

d sin 1

h zw ππ

y

v

k

kr 10.19

Kuchuk (1995)

SZ = ln

h

d sin

k

k + 1

h

w

H

vw ππr 10.20

Implications of the flow equations are as follows:

a. A graph of (Pi -Pwf) versus t is a straight line

b. Slope of line

m2 = 8128. qB

Lh c kt y

µφ

10.21

Therefore

L2 ky = 8128

2

2. qB

hm ct

µφ

10.22

c. ∆p ( )t oy v

zL k k

S S= + = 141.2 qBµ

10.23

where ∆pt=0 is the pressure drop at time equals zero. The skin due to damage, S, can therefore be calculated as SZ is known (calculated from Equation 10.18, 10.19 or 10.20)

Time to end of early linear flow is given by different authors as follows:

Goode and Thambynayagam (1987)

te2 = 20 8. c L

k

t2

x

φµ 10.24

Du and Stewart (1992)

190

te2 = 16 c L

k

t2

x

φµ 10.25

Odeh and Babu (1990) and Lichtenberger (1994)

te2 = 160 c L

k

t2

x

φµ 10.26

Odeh and Babu (1990) also gave the time to the start of early linear flow period as

tS2 = 180 D c

k

z2

t

v

φµ 10.27

where Dz (= h-dz) is the maximum distance between the well and the z-boundaries (top or bottom boundary)

Pseudoradial Flow: In sufficiently large reservoir, pseudo radial flow will develop eventually as the dimensions of the drainage areain the horizontal plane becomes much larger that the effective well length. Figure 10.11 is a schematic showing the streamlines during the pseudoradial flow. This is similar to what happens in a horizontal with a vertical fracture.

Fig 10.11: Pseudoradial Flow

The flow equation during the pseudoradial flow period is given as follows:

Pi – Pwf = ( )SSkkL

qBA

Lc

tk

kk

qBZ

vyt

x

yx

++

−

µφµ

µ 2.141log

h

6.1622

10.28

where “A” is a contant given by many authors as follows:

A = 2.023 (Goode and Thambynayagam, 1987) A = 2.5267 (Kuchuk, 1995) A = 1.76 (Odeh and Babu, 1990) A = 1.83 (Lichtenberger, 1994

191

Implications of Eq 10.28 are as follows: a. Graph of Pwf versus log t is a straight line b. Slope,

m3 = 162 6. qB

k hx

µky

10.29

Therefore

k kqB

m hx y =

162 6

3

. µ

The skin equation is

S = ( )1151 1

32

.log

L

h

k

k

Pi P hr

m

k

c LA Sv

x

x

t

Z

−− +

−φµ

Note:

a. Pseudo radial develops if L>>h (hD ≤ 2.5) b. If top or bottom boundary is maintained at constant pressure, no pseudo-radial flow

period will occur. Instead, there is steady state flow at late time. Time for beginning of pseudoradial flow, ts3, is given by different authors as follows:

tS3 = 1230 L c

k

2t

x

φµ (Goode and Thambynayagam

tS3 = 1480 L c

k

2t

x

φµ (Odeh and Babu, Lichtenberger used 1500)

tS3 = 2841 c L

k

t2

x

φµ (Du and Stewart)

Lichtenberger (1994) gave the time for the end of the pseudoradial flow as follows:

tEprf = min

( )

1650

2000 4

2

2

φµ

φµ

C D

k

C L D

k

t x

H

t w y

H

/ +

10.30

where Dx and Dy are the lateral distances of the reservoir in the x and y directions respectively.

Late Linear Flow Period: After, the pseudoradial flow, it is possible that a late-time linear flow period develops. The flow equation for this phase is

192

Pi - Pwf = ( )8128

2

. qB

x

t

k cS

eh y t

x

µφ

µ

+

141.2qB

L k k + S + S

y v

z 10.31

where 2xe = width of reservoir Sx = pseudo skin due to partial penetration in the x direction. Implications of Eq 10.31 are obvious.

Skin in Horizontal Well

The skin factor will serve the same purpose in horizontal well as it does in vertical wells. The dominant pseudoskins in horizontal wells are the pseudoskin due to damage and pseudoskin due to convergence in the z-direction. The pseudoskin due to damage is dominant because of more fluid losses resulting from larger area contacted by the well. The pressure loss due to skin is defined with respect to the formation thickness in vertical wells and well length in horizontal wells. This is shown in the following equations: Pressure loss due to skin in vertical wells:

∆pqB

khs vertical S=

1412. µ 10.32

Pressure loss due to skin in horizontal wells:

∆pqB

kLs Horizontal

S =

1412. µ 10.33

From Eqs 10.32 and 10.33, we infer that the pressure loss due to skin in horizontal well is much smaller than the pressure loss due to skin in vertical wells because the horizontal well length is usually longer than the formation thickness. The small pressure loss due to skin in horizontal wells does not imply that skin has small effect on horizontal well productivity because the drawdown in horizontal wells is also small. The remedial factor, R, used in vertical wells should also be used in horizontal wells to quantify the effect of skin.

Problem Set

A 2100ft long well is completed in a 100 ft thick formation with closed top and bottom boundaries. The estimated average horizontal permeability from several vertical well test is 1500md while the vertical permeability is 300md. The horizontal well has a diameter of 8½ in and is located 30ft from top of the sand. Other parameters are as follows:

φ = 20%, µ = 0.65cp, ct = 20 x 10-6psi-1 and s = 5. Also, assume dx = L/2

193

(a) Assuming that the early radial flow will not be distorted by wellbore storage effects, determine the time when the early radial flow will end. Also, determine when wellbore storage effect will end if wellbore storage constant, c = 0.025 rb/psi.

(b) Calculate the time to start and end of the early linear flow period for the horizontal well whose parameters have been given

(c) For given reservoir and well data, calculate time required to start a pseudo-radial flow.

(d) Using the following additional data calculate the pressure change in a horizontal well. S = 25, q = 4000 STB/D and Bo = 1.05 rb/STB

Tabulated Solution to Problem Set

Authors

End of early radial flow tel x10-3 (hrs)

End of well-bore storage effect tews x10-5 (hrs)

Start of early linear flow ts2 x10-3 (hrs)

End of early

linear flow te2(hrs)

Start of Pseudo

Radial Flow ts3 (hrs)

Skin due to partial pent. In vert. dir.

Sz

∆P @ the start of Pseudo Radial flow

∆P (psi) Goode et al 2.560 N/A N/A 0.159 9.402 N/A N/A

Odeh and Babu 14.040 N/A 7.644 1.223 11.313 3.996 12.11

Du and Stewart 7.387 N/A N/A 0.122 21.717 N/A N/A

Lichtenberger N/A 5.998 N/A N/A 11.466 2.15 x 10-4 10.833

N/A implies author did not give required equation

10.3 Detecting Flow Regimes Using Pressure Derivative

The pressure derivative is the best diagnostic tool for detecting flow regimes. This follows from the characteristic slopes of the derivatives obtained for different flow regimes on a log-log plot. Figure 10.12 shows the characteristic slope for the first radial, hemiradial and pseudoradial flow regimes in a horizontal well. Note that the derivatives have zero slopes at different levels. For linear flow, the derivative has a slope of 0.5.

Log

Log (time)

Fig 10.12: Prssure Derivatives for the Different Radial Flow Regimes

Pressure Derivative

Early Radial

Hemiradial

pseudoradial

194

Figure 10.13 taken from Kucuk (1995) shows the pressure derivative for cases with well and reservoir parameters shown in Table 10.2.

Fig 10.13 : Derivatives for Cases shown in Table 10.2

Table 10.2: Reservoir Parameters for Examples Shown in Fig 10.13

Example h, ft kH, md KV, md Lw, ft zw, ft rwD

1 100 100 10 500 20 0.00146 2 100 100 1 500 20 0.00389 3 100 100 5 500 5 0.00194 4 40 100 5 500 20 0.00197 5 200 200 1 500 20 0.00530

+

=

V

H

w

wwD

k

k

L

rr 1

2

Deductions from the derivatives are as follows: (a) The first radial period can be seen in all cases. (b) In Example 3, the well is close to the boundary (5ft) and therefore, hemiradial flow

occurred after a short duration early radial flow. (c) In Example 4, a linear flow regime manifested because the well length is much

greater than formation thickness (d) In all cases, the pseudoradial flow developed Du and Stewart (1992) quantified the effect of parameters on the flow regimes. In their work, they defined dimensionless parameters as follows:

PDL = 2π

µk k

pz r L

q.∆ 10.34

195

PDL1 =

dp

d ln tDL

DZ

(Pressure Derivative) 10.35

tDZ = kz t

c ht2φµ 10.36

Values of PDL1 for different flow regimes are as follows:

PDL1 = 0.5 vertical radial flow

PDL1 = 1 vertical hemi-radial flow

PDL1 = LD pseudo radial flow

Du and Stewart (1992) concluded that parameters zWD (dimensionless well location in the z-direction) and LD (dimensionless wellbore length) have the dominant effect on flow regimes obtained in horizontal wells. These parameters are defined as follows:

zwD = zw/h and H

vD

k

k

h

LL

2=

Figure 10.14 shows effect of LD on flow regimes for infinite reservoir with no flow top and bottom. Horizontal well is in the center of the formation (ZWD = 0.5).

Fig. 10.14: Effect of LD in Homogeneous Laterally Infinite Reservoir With Sealed Top and Bottom

Inferences from Fig. 10.14 are as follows:

(i) For LD ≥ 3 the flow regimes are: Vertical radial flow (VRF: PDL1 = 0.5) + transition

+ early linear flow opposite the completed section (ELF: Gradient = ½ ) + transient

reservoir pseudo radial flow (PRF: PDL1 = LD).

196

(ii) For LD < 3, the flow regimes are: VRF + vertical spherical flow (VSF) Gradient = - ½ + transition + PRF. The smaller LD, the longer the duration of VSF.

(iii) The smaller the LD, the shorter the duration of VRF and longer the length of PRF.

(iv) For LD ≤ 0.1, no VRF at all. For a 100ft thick formation and Kz / Kr = 0.2, this implies a minimum well length of 45 ft.

Figure 10.15 shows effect of ZWD. From Fig 10.15, we infer that when ZWD ≤ 0.1 (well close to one of the boundaries), there is a hemi-radial flow (HRF), between VRF and ELF with a transition in between.

Figure 10.16 shows the effect of ZWD in a situation with gas cap. The dimensionless well length, LD, is large.

Fig. 10.15: Effect of ZwD in Homogeneous Laterally Infinite Reservoir With Sealed Top

and Bottom

197

Fig. 10.16: Effect of ZwD in Homogeneous Laterally Infinite Reservoir with Bottom

Constant Pressure Boundary and Top No Flow Boundary

Inferences from the graph are as follows: (i) When ZWD < 0.5 (well closer to the bottom no-flow boundary) the flow regimes are

as follow: VRF + transition + HRF + transition + constant pressure effect (rollover).

(ii) When ZWD ≥ 0.5 (well nearer the constant pressure boundary) the flow regimes are VRF + transition + rollover (constant pressure effect).

(iii)The nearer the well to the constant pressure boundary, the stronger the constant pressure boundary effect.

The effect of the boundaries is similar to what obtains in a vertical well. A no - flow boundary causes stabilization at higher level with respect to the infinite reservoir case. Figure 10.17 shows the effect of lateral boundaries, which are parallel to the direction of well length.

Fig. 10.17: Effect of Lateral Boundaries in Reservoir with Sealed Top and Bottom

10.4 Field Cases

In this section, we shall discuss some field examples of bottom-hole pressure tests. The objective is to see what may actually obtain in real life. Table 10.3 shows basic parameters or the different tests.

Table 10.3: Parameters for Different Field Cases

Parameters Values for Different Cases 1 2 3 4

198

Formation thickness, ft 73 95 19 123

Well length, ft 1984 1387 232 1330

Well location, ft 7 7.4 8 16.1

Oil rate STB/D 3948 4144 685 4951

Porosity, % 29 29 24 29

Oil Viscosity, cp 1.97 1.97 0.34 2.23

Formation volume factor, rb/STB 1.128 1.120 1.539 1.298

Wellbore radus, ft 0.4 0.4 0.4 0.3

Calculated Permeability, md 15363 16320 1070 3820

Calculated Skin 35 23 0 -1.4

Drawdown, psi 15 10 9 34

Discussions on the field cases follow:

Case 1: Figure 10.18 shows the pressure and pressure derivative for this case.

Fig. 10.18: Pressure and Pressure Derivative for Case 1

The pressure derivative shows wellbore storage, early vertical radial flow, early linear flow, hemiradial flow and “rollover” due to a constant pressure boundary. The hemiradial flow was inferred because of the nearness of the well to the bottom boundary

and the fact that ∆p′ (VRF) ≈ 2∆p′ (HRF). The constant pressure effect was caused by gas-cap. Figure 10.18 shows that mathematical models give us good insight unto pressure and flow regime obtained during actual BHP tests.

199

Case 2: Figure 10.19 shows pressure data for Case 2. Case 2 and Case 1 are from the same reservoir in western Niger Delta. The pressure and pressure derivative for the two cases exhibit similar characteristics. The exception is the scatter in Case 2 data during the wellbore storage phase. The scatter was due to gauge shift. Case 3: Data for this case were obtained from the eastern Niger Delta reservoir. Figure 10.20 shows the pressure and derivative for this case. The identified flow regimes are as follows: wellbore storage phase + early vertical radial flow + vertical spherical flow + pseudo radial flow + “rollover” due to lateral constant pressure boundary.


200


The vertical spherical flow regime resulted because the well length is small and therefore

LD < 3 (actually LD ≈ 2). This is in agreement with the finding of Du and Stewart (1992).

Case 4: Data for this case were obtained from the first horizontal well in eastern Niger Delta reservoir. Figure 10.21 shows the pressure and derivative for this case.


We believe that the distortions in pressure and derivative were caused by the “snakelike” nature of the horizontal part of the well. The distortions made it difficult to clearly discern the flow regimes.

201

In field situations, there could be problem resulting because the gauge may not get to the horizontal part. 10.4 Analysis Procedure

Analysis of bottom-hole pressure test in horizontal wells, requires the following a Identifying boundaries and main features such as faults, fractures, etc. from flow

regimes analysis. b Estimating well/reservoir parameters and refining the model that is obtained from

flow regime analysis. The graphical type curve procedure is practically impossible for the analysis of horizontal welltest data because of the many unknowns (kH, kV, s, C, Lw, h, dz, ) even in the case of a single-layer reservoir. Thus, along with the flow regime analysis, non-linear least-square techniques are usually used to estimate reservoir parameters. In applying these methods, one seeks not merely a model that fits a given set of output data (pressure, flowrate, and/or their derivatives) but also knowledge of what features in that model are satisfied by the data.

A flowchart showing recommended procedure for test analysis in horizontal well is shown in Fig 10.22.

Fig. 10.22: Flow chart of Procedure for Test Analysis in Horizontal Well.

Regimes

Clear

Diagnose Flow Regimes

No Any

Early radial? Test cannot

be Analyzed

Analyze Tests using the 3 methods

Yes

Analyze test with

Regression

Time

constraints ? No Results not

accepted

Yes Simulate profile and compare

Accept results with best confidence

General Principles of Bottomhole Pressure Tests

Documents

Transcript of General Principles of Bottomhole Pressure Tests