Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk
BASIC WELL TEST DESIGN AND ANALYSIS (5 Days) FOR
BP EXPLORATION OPERATING COMPANY
COURSE NOTES
May 2001
BY
DOMINIQUE BOURDET AND PIERS JOHNSON
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk CONTENTS Page 1.
INTRODUCTION .......................................................................................................................................................... 7
2.
FUNDAMENTALS OF WELL TESTING................................................................................................................. 9
2.1 2.2 2.3 3. 3.1 3.2 3.3 4. 4.1 5. 5.1 5.2 5.3
DESCRIPTION OF A WELL TEST..................................................................................................................................... 9 TERMINOLOGY AND DEFINITIONS ................................................................................................................................ 9 INPUT DATA REQUIRED FOR AND INFORMATION GAINED FROM WELL TESTING ........................................................ 11 AN OVERVIEW OF PRESSURE TRANSIENT ANALYSIS.............................................................................. 15 BASIC EQUATIONS....................................................................................................................................................... 15 WELL TEST INTERPRETATION METHODOLOGY ......................................................................................................... 16 TYPES OF WELL TESTS .............................................................................................................................................. 17 TYPICAL TEST PROCEDURE................................................................................................................................ 21 OPERATIONAL TIPS ..................................................................................................................................................... 21 FLOW STATES............................................................................................................................................................ 23 STEADY STATE............................................................................................................................................................ 23 PSEUDO-STEADY STATE ............................................................................................................................................. 23 TRANSIENT STATE ...................................................................................................................................................... 23
6.
DARCY’S LAW............................................................................................................................................................ 24
7.
THE DIFFUSIVITY EQUATION AND ITS SOLUTIONS .................................................................................. 26
7.1 7.2 7.3 7.4 7.5 7.6 8. 8.1 8.2 8.3 9.
HYPOTHESES ............................................................................................................................................................... 26 DARCY'S LAW.............................................................................................................................................................. 26 PRINCIPLE OF CONSERVATION OF MASS (CONTINUITY EQUATION)............................................................................ 26 EQUATION OF STATE OF A CONSTANT COMPRESSIBILITY FLUID ................................................................................ 26 DIFFUSIVITY EQUATION .............................................................................................................................................. 27 DIFFUSIVITY EQUATION IN DIMENSIONLESS TERMS ................................................................................................... 27 RESERVOIR AND FLUID ASSUMPTIONS FOR SOLVING THE DIFFUSIVITY EQUATION .............. 29 THE LINE SOURCE SOLUTION ...................................................................................................................................... 29 SEMI-LOG APPROXIMATION : RADIAL FLOW REGIME ................................................................................................. 29 RADIUS OF INVESTIGATION......................................................................................................................................... 30 WELLBORE STORAGE............................................................................................................................................ 31
9.1 9.2 9.3
DEFINITION ................................................................................................................................................................. 31 CARTESIAN PLOT ANALYSIS........................................................................................................................................ 32 ESTIMATING WELLBORE STORAGE FROM COMPLETION ............................................................................................. 33
10.
SKIN ............................................................................................................................................................................... 35
10.1 10.2 10.3 11. 11.1 11.2 12.
DEFINITION.................................................................................................................................................................. 35 RADIAL FLOW REGIME ................................................................................................................................................ 36 SEMI-LOG EQUATION................................................................................................................................................... 37 SUPERPOSITION THEORY..................................................................................................................................... 39 MULTI RATE THEORY : TIME SUPERPOSITION ............................................................................................................. 39 IMAGE WELL THEORY TO MODEL BOUNDARIES : SPACE SUPERPOSITION ................................................................... 42 LOG LOG ANALYSIS................................................................................................................................................ 44
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk 12.1 12.2 12.3 13. 13.1 13.2 14. 14.1 14.2 14.3 14.4 15. 15.1 15.2 15.3 15.4 15.5 16. 16.1 16.2 16.3 16.4 16.5 16.6 16.7 17. 17.1 17.2 18. 18.1 18.2 18.3 18.4 19. 19.1 19.2 19.3 19.4 19.5 19.6 20. 20.1 20.2 21.
LOG-LOG SCALE .......................................................................................................................................................... 44 PRESSURE CURVES ANALYSIS : EXAMPLE OF "WELL WITH WELLBORE STORAGE AND SKIN, HOMOGENEOUS RESERVOIR" ................................................................................................................................................................ 45 BUILD-UP ANALYSIS : BUILD-UP TYPE CURVE ............................................................................................................ 48 SEMI LOG ANALYSIS............................................................................................................................................... 50 M.D.H. ANALYSIS : ∆P VS LOG∆T .............................................................................................................................. 50 HORNER ANALYSIS...................................................................................................................................................... 50 DERIVATIVE ANALYSIS......................................................................................................................................... 52 DEFINITION.................................................................................................................................................................. 52 DERIVATIVE TYPE-CURVE : EXAMPLE OF "WELL WITH WELLBORE STORAGE AND SKIN, HOMOGENEOUS RESERVOIR" ................................................................................................................................................................ 52 DATA DIFFERENTIATION ............................................................................................................................................. 55 THE ANALYSIS SCALES ................................................................................................................................................ 56 PRINCIPAL FLOW REGIMES ................................................................................................................................ 58 RADIAL FLOW ............................................................................................................................................................. 58 LINEAR FLOW.............................................................................................................................................................. 59 BI-LINEAR FLOW ........................................................................................................................................................ 63 SPHERICAL FLOW ........................................................................................................................................................ 66 PSEUDO STEADY STATE............................................................................................................................................... 67 BOUNDARY THEORY .............................................................................................................................................. 68 ONE SEALING FAULT ................................................................................................................................................... 68 TWO PARALLEL SEALING FAULTS ............................................................................................................................... 70 TWO INTERSECTING SEALING FAULTS ........................................................................................................................ 77 CLOSED SYSTEM.......................................................................................................................................................... 81 CONSTANT PRESSURE BOUNDARIES............................................................................................................................ 88 SEMI PERMEABLE BOUNDARY..................................................................................................................................... 92 PREDICTING DERIVATIVE SHAPES ............................................................................................................................... 94 RESERVOIR PRESSURE .......................................................................................................................................... 96 DEFINITIONS ................................................................................................................................................................ 96 APPLICATIONS OF RESERVOIR PRESSURE .................................................................................................................... 97 MOBILITY CHANGE THEORY ........................................................................................................................... 101 DEFINITIONS .............................................................................................................................................................. 101 RADIAL COMPOSITE BEHAVIOR................................................................................................................................. 102 LINEAR COMPOSITE BEHAVIOR ................................................................................................................................. 107 MULTICOMPOSITE SYSTEMS ..................................................................................................................................... 109 PARTIAL PENETRATION THEORY .................................................................................................................. 110 DEFINITION................................................................................................................................................................ 110 CHARACTERISTIC FLOW REGIMES ............................................................................................................................. 110 LOG-LOG ANALYSIS .................................................................................................................................................. 111 SEMI-LOG ANALYSIS ................................................................................................................................................. 112 GEOMETRICAL SKIN SPP ........................................................................................................................................... 112 SPHERICAL FLOW ANALYSIS ..................................................................................................................................... 113 HYDRAULIC FRACTURE THEORY................................................................................................................... 115 INFINITE CONDUCTIVITY OR UNIFORM FLUX VERTICAL FRACTURE ......................................................................... 115 FINITE CONDUCTIVITY VERTICAL FRACTURE ........................................................................................................... 117 HORIZONTAL WELL THEORY .......................................................................................................................... 119
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk 21.1 21.2 21.3 21.4 21.5 21.6 21.7 21.8 22. 22.1 22.2 22.3 23. 23.1 23.2 23.3 23.4 23.5 24. 24.1 24.2 24.3 24.4 25. 25.1 25.2 25.3 25.4 25.5 25.6 26. 26.1 26.2 26.3 26.4 26.5 26.6 26.7 27. 27.1 27.2 27.3 27.4 27.5 27.6 27.7 27.8
DEFINITION................................................................................................................................................................ 119 CHARACTERISTIC FLOW REGIMES ............................................................................................................................. 120 LOG-LOG ANALYSIS .................................................................................................................................................. 120 VERTICAL RADIAL FLOW SEMI-LOG ANALYSIS ......................................................................................................... 122 LINEAR FLOW ANALYSIS ........................................................................................................................................... 123 HORIZONTAL PSEUDO-RADIAL FLOW SEMI-LOG ANALYSIS ...................................................................................... 124 DISCUSSION OF THE HORIZONTAL WELL MODEL ...................................................................................................... 126 OTHER HORIZONTAL WELL MODELS ......................................................................................................................... 132 SKIN FACTORS ........................................................................................................................................................ 136 THE DIFFERENT SKIN FACTORS ................................................................................................................................. 136 GEOMETRICAL SKIN .................................................................................................................................................. 136 ANISOTROPY PSEUDO-SKIN....................................................................................................................................... 137 PERFORMING A TEST DESIGN.......................................................................................................................... 138 INTRODUCTION.......................................................................................................................................................... 138 HARDWARE ............................................................................................................................................................... 139 GAUGES ..................................................................................................................................................................... 139 PRESSURE RESPONSE ................................................................................................................................................ 140 AN EXAMPLE TEST DESIGN....................................................................................................................................... 142 GAS WELL TESTING .............................................................................................................................................. 145 GAS PROPERTIES ....................................................................................................................................................... 145 TRANSIENT ANALYSIS OF GAS WELL TESTS .............................................................................................................. 146 DELIVERABILITY TESTS............................................................................................................................................. 150 ODEH-JONES ANALYSIS ............................................................................................................................................ 155 FISSURED RESERVOIRS....................................................................................................................................... 159 PRESSURE PROFILE .................................................................................................................................................... 159 DEFINITIONS .............................................................................................................................................................. 161 DOUBLE POROSITY BEHAVIOR, RESTRICTED INTERPOROSITY FLOW (PSEUDO-STEADY STATE INTERPOROSITY FLOW). ....................................................................................................................................................................... 164 DOUBLE POROSITY BEHAVIOR, UNRESTRICTED INTERPOROSITY FLOW (TRANSIENT INTERPOROSITY FLOW)......... 178 MATRIX SKIN............................................................................................................................................................. 184 EXAMPLES OF COMPLEX HETEROGENEOUS RESPONSES ........................................................................................... 186 FACTORS COMPLICATING WELL TEST ANALYSIS.................................................................................. 188 RATE HISTORY DEFINITION ....................................................................................................................................... 189 ERROR OF START OF THE PERIOD .............................................................................................................................. 190 TIME ERROR CORRECTION........................................................................................................................................ 193 CHANGING WELLBORE STORAGE............................................................................................................................. 194 TWO PHASES LIQUID LEVEL ...................................................................................................................................... 195 PRESSURE GAUGE DRIFT ........................................................................................................................................... 197 PRESSURE GAUGE NOISE ........................................................................................................................................... 199 WELL TESTING HARDWARE ............................................................................................................................. 200 SURFACE TEST EQUIPMENT ...................................................................................................................................... 200 SUBSEA EQUIPMENT ................................................................................................................................................. 210 PRESSURE MEASUREMENT ....................................................................................................................................... 211 DOWNHOLE EQUIPMENT ........................................................................................................................................... 212 QUALITY CONTROL CHECKS .................................................................................................................................... 215 SAMPLING ................................................................................................................................................................. 216 SAFETY ...................................................................................................................................................................... 218 ENVIRONMENTAL ISSUES .......................................................................................................................................... 220
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk 28. 28.1 28.2 28.3 28.4 29. 29.1 29.2 29.3 29.4
INTERPRETATION PROCEDURE, REPORTS AND PRESENTATION OF RESULTS.......................... 222 METHODOLOGY ........................................................................................................................................................ 223 THE DIAGNOSIS : TYPICAL PRESSURE AND DERIVATIVE SHAPES .............................................................................. 225 SUMMARY OF USUAL LOG-LOG RESPONSES .............................................................................................................. 227 CONSISTENCY CHECK WITH THE TEST HISTORY SIMULATION .................................................................................. 231 APPENDICES............................................................................................................................................................. 237 NOMENCLATURE ....................................................................................................................................................... 237 LIQUID TO GAS CONVERSION CHART ........................................................................................................................ 241 FLARING FLOW CHART ............................................................................................................................................. 242 TESTING FLOW CHART .............................................................................................................................................. 243
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk OVERVIEW
This intensive course teaches participants how to design and analyse pressure transient tests, for both oil and gas, commonly used by the petroleum industry. Participants are taught how to properly design tests to achieve specific objectives and how to use both classical and interactive computer analysis methods (PIE) to analyse data. The course covers both pressure transient theory and the practical aspects of well testing, including the equipment employed, using both lecture and problem sessions.
OBJECTIVES
Participants will: • • • • •
Be able to design and analyse well tests with classical and software tools Gain experience with real test data sets Learn about the equipment used for well testing Become aware of the implications of accurate planning Appreciate how the theory can affect the practical aspects of testing and interpretation.
Acknowledgements are due to Mike Wilson of Well Test Solutions Ltd., for his support in the preparation of these notes and his permission to include examples generated in PIE, the well test interpretation software package.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk
1. INTRODUCTION The pressure behaviour of a well can be easily measured and is extremely useful in analysing and predicting reservoir performance or diagnosing the condition of a well. Various instruments to measure flowing and static pressures in oil and gas wells have been in use since the 1920's. The recording devices that have been used include mechanical, (the Borden tube which records via a stylus mark on a blackened metal sheet), sonic (echometers which measure liquid levels) and electronic instrumentation (which measure pressure and temperature). Continuous recording instruments, such as the Amerada gauge, have been available since the early 1930's. Today the preferred instrument is the electronic (memory) gauge. One of the earliest applications of bottom-hole pressure measurements in wells was the determination of static or average reservoir pressures from the bottom-hole pressure measured after a well had been shut-in for between 24 and 72 hours. Although this static measurement indicated the average formation pressure in the permeable and productive reservoir it soon became apparent to engineers that static pressure measurements depended considerably on the time for which the well had been shut-in. Thus, the lower the permeability, the longer the time required for pressure stabilisation in the well. This lead to the realisation that when a well was shut-in, the rate of the pressure build-up would be a reflection of the reservoir permeability around the well. Since a well test and subsequent pressure transient analysis is the most powerful tool available to the reservoir engineer for determining reservoir characteristics and planning production schedules, the subject of well test analysis has attracted considerable attention. Petroleum Engineering literature alone includes more than 500 published technical papers on this subject whilst the field of ground water hydrology also contains a similar number of publications on pump test analysis. A well test is the only method available to the reservoir engineer for examining the dynamic response in the reservoir and considerable information can be gained from a well test. Therefore, well testing is a subject which should be considered seriously. After static pressure measurements, the most common methods of transient (time dependant) pressure analysis required that data points be selected such that they fell on a well-defined straight line on either semi-logarithmic or cartesian graph paper. The well test analyst must then insure that the proper straight line has been chosen if more than one line can be drawn through the plotted data. This aspect of interpretation of well test data requires the input of a reservoir engineer. Equally important is the design of a well test to ensure that the duration and format of the test is such that it produces good quality data for analysis.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk With the advent of powerful desktop computers and software, analysis of transient well test pressure data took another step forward with the introduction of type curve analysis. The computer can perform large numbers of calculations necessary to generate a type curve which is specific to the reservoir itself and also takes into account many different flow periods which not all straight line analysis did. This eliminates generalisations but still requires interpretation of the data set which the reservoir engineer must perform. This is considered to be the best and most efficient method of well test analysis currently available. The results obtained from transient pressure analysis are used to set up numerical simulation models for predicting future production and to assist in making estimates of the Hydrocarbons Originally in place. Both explicitly compelling reasons to carry out well tests. It should also be noted that all units within these course notes where not clearly stated are “oilfield” units which means psi, feet, Barrels etc rather than S.I. Units.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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2. FUNDAMENTALS OF WELL TESTING 2.1 Description of a Well Test A Well Test is the examination of the transient behaviour of a porous reservoir as the result of a temporary change in production conditions performed while measuring all the relevant parameters. It is usually performed over a relatively short period of time in comparison to the producing life of a field. 2.2 Terminology and Definitions Drawdown/Injection period The drawdown can be both the part of the test when the well is flowing (fluid extracted from the reservoir) and a value represented by the difference in pressure between the initial static reservoir pressure and the flowing bottom hole pressure. The injection period is the opposite of a drawdown in that fluid is injected into the reservoir creating an increase of pressure in the near well bore area and the reservoir. Build up/Fall off The build up can be both the part of the test when the well is shut in (not flowing) and a value represented by the difference in the pressure measured at any time during the build up and the final flowing pressure. A fall off is the time after the ending of an injection period where the pressure falls. In both cases the flow rate is known to be zero. Perforating This is the activity of making holes in the casing and/or reservoir so that there is communication between the reservoir and the well bore. This is usually achieved by detonating explosives when the perforating guns are known to be at the correct perforating depth. If the pressure in the wellbore prior to perforating is less than the reservoir pressure, this is known as underbalanced perforating. This can have the effect of improving well productivity by allowing the perforations to flow immediately after perforating reducing the skin (see below for definition). When the pressure in the well is greater than in the reservoir, this is overbalanced perforating. Underbalanced perforating is preferred whenever practical and safe and is widely used today. It can also be useful to replace the drilling mud with specialised completions fluids (brine, diesel, base oil for example) to improve well productivity on perforating. Well Bore Storage
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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Wellbore storage is an early transient phenomenon whose effect decays in time and occurs every time a rate change takes place. Usually the well rate is controlled at the surface by means of a valve or choke. When the valve or choke is opened, wellbore fluids are initially produced at the wellhead while production at the perforations remains zero. During this early time period the wellbore is said to be unloading or decompressing. Eventually, the perforations will also start to produce and in time equal the production at the wellhead. During constant production, wellbore storage effects are negligible. At the point of shut-in, wellbore storage is referred to as afterflow. Whilst the rate at the wellhead is zero, production at the perforations continues, gradually decaying to zero. Well bore storage is expressed in units of volume per unit of pressure, barrels per psi, with the nomenclature of C. Skin This is a dimensionless value attributed to the near well bore damage or stimulation. Reservoir permeability in the near wellbore area is frequently altered as a result of drilling, production, or stimulation of the well. For example, the invasion of drilling fluids, or migration of fines during production tends to lower the permeability in the near wellbore region. The well is subsequently referred to as damaged under these circumstances and this is represented by a positive skin value. Conversely, stimulation treatments such as acidizing or fracturing may create an increase in the near wellbore permeability relative to the overall reservoir permeability. This is summarised below; Skin > 0 Damaged Skin = 0 Neutral Skin < 0 Stimulated (not usually less than -5) The value is dimensionless and is represented by the letter S. Permeability Permeability is a measure of the ability of a porous rock to transport a fluid through it and is measured, usually, in Darcys, D, or millidarcies, mD. Its units are L2. Porosity Porosity is the amount of void (space) in a porous rock measured as a percentage of the whole rock. Mobility
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk Mobility is the combination of the ability of a rock to transport a certain fluid through it given a fluid’s viscosity and is permeability divided by viscosity, k/µ. Partial Penetration This describes a well completion where only part of the total permeable formation is perforated or communicating with the well. Hydraulic Fractures These are, usually, purposefully induced cracks in the formation rock which allows a much greater wellbore contact with the formation and should not be confused with Natural fractures explained below. Natural Fractures (or Double Porosity) This is where naturally occurring fractures in the formation rock exist meaning there are two different porosities, one for the fractures and one for the rock between the fractures. In order for this effect to be observed in transient test analysis, the difference in the two porosities must be significant, generally an order of magnitude. Radius of Investigation. When a change of flow rate occurs in a well, ie, initial flow, a change in the reservoir pressure from its initial undisturbed state is created. Over time this pressure disturbance propagates further away from the wellbore. The radius of investigation is defined as the distance that a significant pressure disturbance has propagated away from the well. The mathematical description is given in section 8.3. 2.3 Input data required for and Information gained from well testing The fundamental role of the reservoir engineer is to produce oil and gas reservoirs having determined the characteristics of the in-place fluids and the reservoir. In general, the characteristics are listed below and it should be noted that whilst the first four parameters are used to estimate the amount of hydrocarbons in place the reservoir permeability establishes the ability of the fluids in place to flow through the reservoir: a)
Net pay.
b)
Porosity.
c)
Reservoir description;
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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-
Lateral extent, Shape, Heterogeneities, Boundaries.
d)
Average reservoir pressure (static pressure) and temperature.
e)
Formation permeability given net pay.
In practice net pay, porosity and the reservoir description are derived from the geological interpretation of core data, log data and geophysical surveys. Pressure transient tests are generally conducted to determine the average reservoir pressure, the effectiveness of the well completion (determining a value of skin - see below), the well productivity/deliverability, distances to boundaries and permeability of the reservoir volume drained by the well. It should also be noted that pressure transient tests can be designed to estimate the size and shape of the drainage volume. Other reasons for testing may be to determine the nature of the produced fluids, production problems, to clean up the well for production, the maximum possible flow rate, and connectivity in the reservoir. For any well production (or injection) usually takes place via the drilled wellbore hence the conditions prevailing around the wellbore are of particular interest particularly if the sand face may be damaged during drilling or as a result of production operations. Quantitative information gained from a welltest, therefore, enables the reservoir engineer to determine whether or not low productivity in a well is due to damage, low formation permeability, or a low driving force for moving fluid to the well. Similarly if a well has been stimulated to remove formation damage the success of the operation can be evaluated hence the reservoir engineer can make practical decisions regarding future well stimulation treatments and/or operating practices. Thus it can be said that a pressure transient test is a fluid flow experiment used to determine one or more of the reservoir characteristics and properties mentioned above. Input Data Required To perform a test analysis a considerable amount of information is required. The following list summarises this to perform an interpretation of a single pressure test. a) through g) are mandatory while h) and i) are optional. a)
The well production rates as a function of time.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk b)
The bottom-hole pressure as a function of time.
c)
The wellbore configuration; -
d)
A documented test history including; -
e)
a complete sequence of events. any operational problems.
Well data -
f)
completion report. string diagram. gauge depths.
drilled v true depths if well not vertical. direction and length of horizontal or inclined sections relative to reservoir boundaries or discontinuities. Well bore radius (drill bit size not casing size)
Rock and fluid properties Net pay (formation thickness, h) Porosity Water Saturation Oil viscosity Water compressibility* Oil compressibility* Rock compressibility*
g)
Geological interpretation characteristics.
of
the
reservoir
extent
and
likely
General information obtained from Geologists and Geophysicists h)
Production logs (optional but recommended with large h) To determine Producing intervals.
i)
Gradient surveys (optional but recommended) to confirm; -
Static fluids levels and fluid contacts/interfaces Static fluid densities.
formation
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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* Individual compressibilities can be combined and represented by ‘Total Compressibility’ of the reservoir. Whenever possible, the above data should be obtained prior to performing the test and used to run a test design to examine the theoretical response for the planned test. Failing to plan the test is planning to fail. For exploration well testing where test parameters may not be readily available, Robert C Earlougher’s SPE Monograph Advances in Well Test Analysis has an appendix in which many of the above parameters can be derived from correlations. Why perform well tests? A large amount of information can be gained from performing well tests. A non exhaustive list of this information follows; Well Tests (Transient Pressure Tests) are performed to Determine: Nature of produced fluids. Rates of produced fluids. Reservoir and well characteristics. Location of reservoir limits (or not). Well deliverability (drawdown/rates). Sand production (if any). Maximum rate. Hydrocarbons in place. Well connectivity (interference tests). Reservoir layers. Production characteristics and any production problems. The long term productivity from a short term test operation. When well tests can derive such a large amount of information it should be evident that Well Testing is a valuable tool to the reservoir engineer. Therefore:
PERFORM WELL TESTS WHENEVER POSSIBLE
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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3. AN OVERVIEW OF PRESSURE TRANSIENT ANALYSIS 3.1 Basic equations Transient pressure analysis models are solutions to the diffusivity equation obtained by a combination of Darcy’s law and various hypotheses for the fluid in the reservoir. When it is assumed that the fluid density remains constant and that the flow is through a horizontal linear porous medium then Darcy’s law can be expressed as follows;
q=−
kA ∆ p l µ
(Eq. 3-1)
where, q k A µ l ∆P
is the flow rate is the permeability of the porous medium, is the area available to flow, is the viscosity of the fluid flowing through the porous medium, is the length of porous medium through which the flow is transported. is the pressure drop over the length l
For radial flow, which is flow in a hydrocarbon reservoir into a well bore, Equation 3.1 becomes; q = 2π
kh dp µ dr
(Eq. 3-2)
which, after separating the variables, integrating with respect to r from well bore, rw to the effective radius of investigation, re, introducing an equation for skin by van Everdingen and expressing the equation in field units the equation becomes; pe − p wf = 1412 .
qBµ re ln + S kh rw
(Eq. 3-3)
In summary, for radial flow into a well bore, the pressure difference (drawdown) is directly proportional to the following parameters; Flow rate
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk Viscosity Formation volume factor Permeability formation thickness well bore radius (limited effect because of the ln term) skin The application of Darcy’s Law and the different solutions to the diffusivity equation will be explained in more detail later in this text. References: L.P Dake, Fundamentals of Reservoir Engineering, Chapter 4, sections 4.1 to 4.7
3.2 Well Test Interpretation Methodology Well test interpretation can be simplified if it is considered as a special pattern recognition problem. This concept is illustrated by the following schematic;
INPUT(I)
SYSTEM (S)
OUTPUT (O)
Thus, the problem is to define the system when knowing the output for a given input. In a well test, a known constant signal, I (a constant production rate) is applied to a system, S (the well and reservoir) and the response of that system, O, (the change in bottom hole pressure) is measured. The aim of well test interpretation is, therefore, to identify and characterise the system knowing only the input and output signals. This is called the INVERSE PROBLEM where; O / I ----> S The solution to this problem involves the selection of a well defined theoretical model where output for the same input signal is as close as possible to that of the actual reservoir. Hence, the construction of the model response involves the DIRECT PROBLEM where; I * S ----> O
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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Interpretation therefore relies on theoretical models which are assumed to have characteristics which are representative of those of the actual well and reservoir. The solution to the inverse problem is not unique but the number of possibilities decreases as the number of output responses increases and measurements become more accurate. The aim of well testing is to evaluate the well and reservoir system under dynamic conditions. The interpretation of measured data combined with reservoir engineering expertise results in the production of a reservoir model. The reservoir model subsequently enables the prediction of field production in terms of production rate and fluid recovery. Together with economic considerations, the reservoir model can be used to formulate a development strategy. Having determined the development strategy, the production facilities and completion can then be optimised accordingly.
3.3 Types of Well Tests Throughout this course a number of different tests will be considered and discussed. The following sections illustrate the wide variety of tests available to the reservoir engineer. Classification by Completion Production Test The well is completed as a production well in cased hole hence the well completion is permanent, even though it may be pulled out of the well on completion of the test. Production tests are generally carried out on initially completing development wells to define reservoir parameters and then routinely run as part of the reservoir management strategy. DST (Drill Stem Test) In a DST, the well is completed temporarily with down-hole control provided by conventional DST tools. This type of completion can be used in cased or in open hole, and is usually associated with exploration and appraisal wells where the aim of the test is to determine as many reservoir parameters as practically possible or to confirm the observations of previous tests on other exploration wells. Drill Stem tests are so called because originally the drill string (drill pipe) was used as the production string. This is now considered unsafe in all parts of the world and production tubing is employed. High Pressure High Temperature Tests
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk These tests are classified separately since the pressures and temperatures encountered give rise to special considerations. However, permanent production equipment is usually used and in all other ways the test will be similar to a production test. Classification by Test Procedure Pressure build-up Pressure build-up tests are performed with the well is shut in and not flowing. In all cases a build up is recommended. The well must be flowing, ideally at a constant rate, before the well is shut-in and the rise in bottom-hole pressure recorded. Almost without exception, a build up will produce better quality pressure data than a drawdown. Besides the issue of data quality which in itself is important, the other compelling reason a build up is used is that it is the only time in a well test when the flow rate is categorically known without any margin of error. It is always zero. Pressure drawdown In a pressure drawdown test the well is shut-in prior to commencing the test so that pressure can equalise throughout the formation. Having run pressure measuring equipment into the wellbore the flowing pressure is recorded as the well is produced at a nominal constant rate. Data quality can be erratic but with good data and a constant rate, drawdowns can be used to derive drainage areas as well as to derive reservoir properties by normal type curve analysis. Injection Injection tests are usually only performed to determine maximum sustainable injection rates for reservoir injection projects or stimulation operations. In practice injection tests are normally followed by fall-off tests. Note that drawdown and injectivity tests are not popular as it is frequently not possible to hold the rate at a constant level throughout the test period. These tests can be analysed, as a stand alone test or as part of a subsequent production test/DST, in a similar way to any other test, by making the production rate negative. Pressure fall-off Pressure fall-off tests, which are similar to build-up tests in that the production rate is zero, follow injection tests. Ideally the injection rate is stabilised and held constant for a predetermined duration prior to ceasing injection. The decline in the bottom-hole pressure is then recorded as the reservoir pressure stabilises.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk Step-rate Step rate tests involve flowing the well at different rates. The main advantage of these tests is that the well is not shut-in between rates saving time. Flowing the well at different rates helps establish production characteristics from low to high rates for future reference and to determine rate dependant phenomena. (See gas well testing below) Gas well testing In gas wells special tests are used to determine the rate dependent skin factor and the Absolute Open Flow (AOF) potential of the wells tested. Some of the most common tests encountered are;
i)
Flow after flow (FAF) and multi-rate tests.
ii)
Isochronal tests.
iii)
Modified Isochronal tests.
Whilst the flow after flow test can be conducted without shutting-in the well thus saving time, the isochronal and modified isochronal test sequences require the well to be shut-in between different flow rates. In an isochronal test the well is shut-in after each flow period until the previous shut-in pressure is reached, (ie, all flow and build-up periods may be different). For the modified isochronal test all flow and build-up periods are equal in duration except for the final stabilised flow. Generally, it has been found that isochronal tests (of both types) produce better results and more build ups for analysis and a modified isochronal test takes less time and is therefore popular with financial managers. Interference and Pulse Tests Interference and Pulse tests are used primarily to define reservoir rather than individual well characteristics. They are commonly used to evaluate communication between wells completed in the same formations and are therefore excellent for determining average reservoir perrmeabilities between wells. A typical test would involve the measurement of the pressure response at a shut-in observation well or wells due to a rate change (interference test) or a series of rate changes (pulse test) at another well. In vertical interference tests the test is designed to evaluate communication between different formations in the same well, where the same formation is encountered at different depths or is vertically displaced in a faulted reservoir. Classification by Operational phase
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk Exploration The initial exploration phase of drilling a well into a structure about which nothing is known other than from geological and seismic data (sometimes referred to as a wildcat) carries the most risk of not being tested. If hydrocarbons are identified from mud logs and electric logs and a test is performed, these are usually the simplest and shortest. The primary aim will be to confirm the presence and nature of hydrocarbons, take a some samples of the produced fluids, obtain an indication of the flow rates and to determine a value of permeability and skin. Historically, wildcat wells have had approximately a one in ten chance of finding commercial hydrocarbons. This ratio is lower in known hydrocarbon provinces, such as the North Sea or the Gulf of Mexico, perhaps one in eight. Usually the transient solution to the diffusivity equation is employed for this type of testing where the limits of the reservoir are not encountered by the “pressure transient”. Appraisal Once the presence of hydrocarbons has been confirmed from an exploration well test, the structure is appraised by drilling further wells to examine the extent of the hydrocarbons. Appraisal well tests will usually be designed specifically to clarify any structure/reservoir uncertainty such as the limit of the reservoir or changes in fluid phases across the structure (as may happen in a large gas condensate field). Development At the field development phase, the well may not even be tested or it may be completed, cleaned up and suspended for future production. Only if something unexpected is found from drilling or logging is the well fully tested.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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4. TYPICAL TEST PROCEDURE For any test, drawdown or build-up the following procedures should be incorporated into the test programme; a) Prepare the well. This can mean running casing, changing the completion fluid from mud to brine, which can allow the annular pressure operated downhole tools to operate more reliably, or place a plug or cement in the well. b) Run a bottom-hole pressure gauge and a completion in the well. Ideally the gauges should be located opposite or below the perforations and below any fluid contacts in the wellbore. c) Produce the well (ideally at a constant rate or rates) for some period of time, tp, (hours). d) During the flow period measure the relevant parameters, determine produced fluid properties and take PVT samples. Take some samples early in the test in case of subsequent problems. e) Shut the well in and monitor the bottom-hole pressure response if surface read-out is available. When using downhole shut-in tools monitor the wellhead pressure (which should not be bled down to zero) response to ensure that the well is shut-in and remains shut-in throughout the test. f) Make well safe to allow retrieval of gauges and completion if applicable. g) Retrieve gauges and validate data before terminating operations. Repeat test if data quality is poor or corrupt. h) Proceed with interpretation. 4.1 Operational Tips Pressure test everything on site. Ensure sufficient fixed chokes are available, usually from 16/64” to 1” every 4/64 and always flow for long periods of time after clean up on a fixed choke. Examine adjustable choke tip and zero before each test. Ensure sufficient orifice plates are available for the Daniel meter.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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Calibrate the measuring instruments on the separator; liquid meters with pumps and tank, Barton with Dead Weight Tester and calibrated air supply. Verify the separator’s safety valve certification and all equipment certification if applicable. Ensure remote ignition system of burners works if offshore. Ensure burner booms are hung and rigged correctly to avoid problems during adverse weather conditions. Record high and low tides throughout the test. Record every pressure cycle of the annulus if using annular operated downhole tools. Do not move or hit anything connected to the well during the build up as long as all is well. Check policy on opening the well in the dark and encountering H2S prior to commencing operations. Make friends with the Chef. You may miss regular meals when testing so it is useful to have an ally in the galley!
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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5. FLOW STATES For pressure transient analysis the governing flow equation, referred to as the diffusivity equation, expressed in cartesian coordinates is: 2 2 ∂p k y ∂2 p kx ∂ p kz ∂ p + + = φ ct 2 2 2 µ ∂x µ ∂y µ ∂z ∂t
(Eq. 5-1)
Only the following flow states, as solutions to the above equation, are considered in common well test interpretation theory and in this material. 5.1 Steady State The time derivative is assumed to be equal to zero, ie: ∂p = 0 ∂t
(Eq. 5-2)
This means the pressure in the reservoir never changes with time and occurs when a constant pressure boundary exists at some distance from the well. 5.2 Pseudo-steady State The time derivative in Equation 5.1 is a constant, ie: ∂p = constant ∂t
(Eq. 5-3)
This means that the pressure is changing constantly with time and the value of the constant depends on the reservoir. In other words, all boundaries have been encountered and the reservoir is depleting. 5.3 Transient State The time derivative is expressed as a function of time and space, ie:
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk ∂p = f (x, y,z,t) ∂t
(Eq. 5-4)
This state is most frequently encountered in exploration pressure transient testing. However, as the radius of investigation increases with time Pseudo-steady state may be observed if the radius of investigation reaches the outer boundaries.
6. DARCY’S LAW Darcy's law expresses the rate through a sample of porous medium as a function of the pressure drop between the two ends of the sample. q A dp / dl Figure .6-16-2 : Rate through a sample. q A
=V =
k dp µ dl
(Eq. 6-1)
With : q : volumetric rate A : cross sectional area of the sample V : flow velocity k : permeability of the porous medium µ : viscosity of the fluid The flow velocity V is proportional to the mobility k/µ and to the pressure gradient dp/dl.
In case of radial flow, the Darcy's law is expressed : q 2πrh
=V =
k dp µ dr
(Eq. 6-2)
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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re
q
q rw
Figure 6-3 : Radial flow. For steady state flow condition, the pressure difference between the external and the internal cylinders is : pe − pw =
qµ 2 π kh
ln
re rw
The relationship is used in the definition of the dimensionless pressure.
(Eq. 6-3)
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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7. THE DIFFUSIVITY EQUATION AND ITS SOLUTIONS
7.1 Hypotheses • Constant properties : k, µ, φ and the system compressibility. • Pressure gradients are low. • The formation is not compressible and saturated with fluid. 7.2 Darcy's law →
V=
→
k µ
(Eq. 7-1)
grad p
7.3 Principle of conservation of mass (continuity equation) The difference between the mass flow rate in, and the mass flow rate out the element, defines the amount of mass change in the element during the time dt. →
div ρV = − φ
The density ρ =
m v
∂ρ ∂t
(Eq. 7-2)
is used.
7.4 Equation of state of a constant compressibility fluid The compressibility, defined as the relative change of fluid volume, is expressed with the density ρ : c=−
1 ∂v v ∂p
=
1 ∂ρ ρ ∂p
(Eq. 7-3)
With a constant compressibility, the fluid equation of state is : ρ = ρ0e
ct p − p0
(Eq. 7-4)
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk For a liquid flow in a porous medium, the total system compressibility ct is attributed to an equivalent fluid : ct = co So + cw S w + c f
(Eq. 7-5)
7.5 Diffusivity equation Combining the three equations: k → div ρ grad µ
∂ρ ∂p = φ ρ ct p = φ ∂t ∂t
(Eq. 7-6)
In radial coordinates, and with the condition of low pressure gradients defined with the
( )
∂p approximation ∂r
2
≅ 0 it comes,
∂ p ∂ r → φµ ct ∂ p 1 ∂r div grad p = = ∇2 p = r ∂r k ∂t
The ratio
k φµ ct
(Eq. 7-7)
is called hydraulic diffusivity.
7.6 Diffusivity equation in dimensionless terms (U.S. oil field system of units)
The dimensionless pressure is given by;
pD =
kh 141. 2 qB µ
∆p
with the dimensionless time given as;
(Eq. 7-8)
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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tD =
0. 000264 k φµct rw2
∆t
(Eq. 7-9)
and the dimensionless radius rD =
r rw
(Eq. 7-10)
The diffusivity equation in dimensionless terms becomes:
∂ pD ∂ rD ∂ pD 1 ∂ rD = ∇ 2 pD = ∂ rD ∂tD rD
(Eq. 7-11)
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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8. RESERVOIR AND FLUID ASSUMPTIONS FOR SOLVING THE DIFFUSIVITY EQUATION 8.1 The line source solution • Initial condition : the reservoir is at initial pressure. pD = 0 at tD < 0
(Eq. 8-1)
• Well condition : the rate is constant, the well is a "line source". ∂ pD = −1 Lim rD ∂ r D r→0
(Eq. 8-2)
• Outer condition : the reservoir is infinite. Lim pD = 0 r→∞
(Eq. 8-3)
The solution is called Exponential Integral. p D ( t D , rD ) =−
∞
Ei( − x ) =− ∫ x
1 rD2 Ei − 2 4t D
e −u du u
8.2 Semi-log approximation : radial flow regime The Exponential Integral can be approximated with a log. For x < 0.01, Ei( x ) =− ln(γ x )
(Eq. 8-4)
(Eq. 8-5)
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk with γ = 1.78 : Euler's constant p D (t D , rD ) =
[ (
)
1 ln t D rD2 + 0.809 2
p(r,t)
ri1
pi
]
(Eq. 8-6)
log(r) ri2
t1 pw1
t2
pw2
Figure 8-1 : Pressure profile versus the log of the distance to the well.
8.3 Radius of investigation The radius of investigation ri is in general defined with one of the two relationships; t D riD2 =
1 4
or t D riD2 =
1 γ2
.
(Eq. 8-7)
This gives respectively, ri = 0. 032 kt φµct
(Eq. 8-8)
ri = 0. 029 kt φµct
(Eq. 8-9)
and
Time is given in hours, porosity as a decimal and radius of investigation is in feet. Note also that the radius of investigation is independent of the rate.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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9. WELLBORE STORAGE 9.1 Definition The production at surface is due to the expansion of the fluid in the wellbore. The reservoir contribution is negligible. " ! !
!
rw
r
pi
pw
Figure 9-1 : Wellbore storage effect. Pressure profile.
During constant production, wellbore storage effects are negligible. At the point of shut-in, wellbore storage is referred to as afterflow. Whilst the rate at the wellhead is zero, production at the perforations continues gradually decaying to zero. Wellbore storage is defined as the difference between the sandface and wellhead rates and is assumed to be proportional to the wellbore storage constant, C, ie:
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
Rate, q
Pressure, p
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q surface q sand face Time, t
Figure 9-2 : Wellbore storage effect. Sand face and surface rates. q sf − q =
24 C dp w B dt
(Eq. 9-1)
where qsf = the flow rate at the sand face q = the flow rate at the surface B = the formation volume factor dpw = the change in downhole pressure after the rate change dt = the time after rate change
9.2 Cartesian plot analysis Wellbore storage effects dominate the pressure transient response at early time independent of the reservoir characteristics. It is possible to estimate a value of well bore storage, C, during this early time period from or a cartesian plot. Assuming that a well has just been put on production then the sand face rate, qsf, will initially be zero until it stabilises at the constant surface rate, q. Substituting these conditions into equation 9.1 gives: ∆p =
qB∆t 24 C
(Eq. 9-2)
Equation 9.2 above indicates that the wellbore pressure response is linear with time at early time, after a rate change when wellbore storage effects are dominant. A cartesian plot of the early time data will exhibit a straight line which passes through the origin.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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Dp
mWBS
0 0
Dt
Figure 9-3 : Well Bore Storage plot (Cartesian) For the above cartesian plot of ∆t versus ∆p and equation 9.2 the slope of the gradient of the early time data, mWBS,which falls on a straight line will be given as follows; mWBS =
qB 24 C
(Eq. 9-3)
from which the well bore storage constant can be derived with knowledge of the other parameters.
9.3 Estimating wellbore storage from completion An estimate of well bore storage can be made from the volume of the completion and the compressibility of the fluid as follows; C = Vw c
(Eq. 9-5)
where; C is the well bore storage constant Vw is the well bore volume and c is the compressibility of the fluid in the well bore. This method is not as accurate as deriving the well bore storage from the plots described above since the compressibility of the fluid in the well bore will change with pressure
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk (which changes with time), particularly with gas. It does, however, allow the engineer to make a first estimate. Typically, values of well bore storage are less than 1 bbl/psi for both oil and gas and for a downhole shut in, which reduces the compressible well bore volume significantly, values can be as low as 10-3 bbl/psi or less. It is desirable to minimise the well bore storage to eliminate the risk of masking, or hiding, reservoir effects and also to reduce the amount of time required to carry out the test. In order to minimise these effects, the gauge measuring the downhole pressure should be placed as near to the reservoir as possible and a downhole shut in employed. The above theory is applicable for wells which are filled completely. When the well bore is only partially filled and the liquid level is changing, the well bore storage constant C, is given by the following equation; C=
Vu ρ g 144 g c
(Eq. 9-4)
where Vu is the well bore volume per unit length in barrels ρ is the fluid density g is the acceleration due to gravity ft/sec2and gc is a units conversion factor 32.17 lbmft/lbfsec2
Reference: SPE Monograph Advances in Well Test Analysis, Robert C Earlougher.Page 10
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2001
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10. SKIN 10.1 Definition
S=
kh 141. 2 qB µ
∆pSkin
(Eq. 10-1)
Damaged well (S > 0) : poor contact between the well and the reservoir (mud-cake, insufficient perforation density, partial penetration) or invaded zone Stimulated well (S < 0) : surface of contact between the well and the reservoir increased (fracture) or stimulated zone Steady state flow in the circular zone :
k
rs rw
ks
pw, S − pw, S =0 =
141. 2 qB µ kS h
ln
rS rw
−
141. 2 qB µ kh
ln
rS rw
(Eq. 10-2)
The skin is expressed :
S=
k r kh pw , S − pw , S = 0 = − 1 ln S 1412 . qBµ k S rw
(
)
(Eq. 10-3)
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk 10.2 Radial flow regime " ! ! ! " "
!
#
#
# #
rw
r
pi
S=0
pw
Figure 10-1 : Radial flow regime. Pressure profile. Zero skin. rw
r
pi S>0
pw(S=0) pw(S>0)
Dp(skin)
Figure 10-2 : Radial flow regime. Pressure profile. Damaged well, positive skin factor.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk rw
r
rwe
pi pw(S<0)
S<0 pw(S=0)
Figure 10-3 : Radial flow regime. Pressure profile. Stimulated well, negative skin factor.
10.3 Semi-log equation
∆p = 162.6
qBµ k − 3.23 + 0.87 S log ∆t + log 2 φ µ ct rw kh
(Eq. 10-4)
The skin, S, can be calculated from the following equation
(
)
pi − p wf (1hr ) k − log + S = 1151 . 3 . 23 2 m φµ cr w
(Eq. 10-5)
where; Pi is the initial reservoir pressure (or final flowing pressure for a build up) Pwf(1hr) is the pressure determined from a semi log plot of (log) time versus pressure after 1 hour (zero on the log scale) m is the slope of the plot mentioned above during the transient period k is the permeability φ is the reservoir porosity µ is the fluid viscosity and rw is the well bore radius
2001
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Dp slope m Dp(1hr)
0 Log(Dt)
Figure 10-4 Radial flow regime. Specialized analysis on semi-log scale.
Another concept frequently used in pressure transient analysis is the concept of the equivalent wellbore radius. This is the equivalent radius assuming a skin factor of zero, ie, well undamaged, that gives the same pressure drop as a well with skin. The equivalent wellbore radius rwe, as a function of skin can be estimated from: rwe = rw e − s
(Eq. 10-6)
where, If S > 0, rwe < rw and If S < 0, rwe < rw So, where the skin factor is positive, implying damage, the equivalent wellbore radius will be less than the actual drilled wellbore radius (the fluid flows an additional distance thereby increasing the pressure drop). Conversely, for stimulated wellbores the equivalent radius will be larger. To summarise, the value of the skin factor quantifies the communication between virgin formation and the wellbore.
Reference: L.P.Dake, see index under Mechanical Skin factor, determination of
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2001
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11. SUPERPOSITION THEORY 11.1 Multi rate theory : time superposition Example of a shut-in after a single rate drawdown
5000
Dp drawdown 4000
Dp build-up
Dp drawdown
p, psi
3000 0
100
200
300
400
500
time, hr
Figure 11-1 : History drawdown - shut-in.
The superposition principle : an injection period is superposed onto the drawdown period.
5000
p(tp) p(Dt)
4000
p(tp+Dt)
p, psi
3000
Drawdown Injection 2000 0
100
200
300 time, hr
400
500
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk Figure 11-2 : History drawdown - injection.
[p
D
( ∆t ) D ]BU
(
= pD ( ∆t ) D − pD t p + ∆t
)
D
( )
+ pD t p
(Eq. 11-1)
D
Semi-log scale superposition function The superposition time is used for semi-log analysis :
[∆p( ∆t )]
BU
= 162.6
t p ∆t qBµ k + log − + S 3 . 23 0 . 87 log φ µ ct rw2 kh t p + ∆t
(Eq. 11-2)
With the superposition time, the correction compresses the ∆t scale. 1500 2S
CDe tp
1000
Dp, psi
type curve
compression
drawdown and compressed build-up 500
0 1.0E-02
build-up
1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
Dt & (tp.Dt)/(tp+Dt), hours
Figure 11-3 : Drawdown and build-up type curves on semi-log scale. With the Horner method, the superposition time is simplified as : pws = pi − 162. 6
qB µ kh
log
t p + ∆t ∆t
(Eq. 11-3)
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk Multi- rate superposition At time ∆t of flow period # n, the multi-rate dimensionless pressure is :
[
pD ( ∆t ) D
]
n −1
MR
qi − qi −1 pD (tn −1 − ti ) D − pD ( tn −1 + ∆t − ti ) D + pD ( ∆t ) D q q − 1 − n n i =1
[
=∑
]
(Eq. 11-4)
4000
3900
p, psi 3800
Dt
3700
0
t1
50
100
t2
150
200
250
t3 t4
300
350
400
450
500
time, HR.
Figure 11-4 : Multi- rate history.
The multirate superposition time is expressed : pws ( ∆t ) = pi − 162.6
Bµ kh
n −1
∑ (q
i
i =1
− qi −1 ) log( t n + ∆t − ti ) +( qn − qn −1 ) log( ∆t ) (Eq. 11-5)
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www.opc.co.uk 11.2 Image well theory to model boundaries : space superposition One sealing fault example " ! !
" ! "
#
rw
# # L
r
pi pw
Figure 11-5 : One sealing fault. Pressure profile. The fault is not reached. Infinite reservoir behavior. rw
L
r
pi
pw
Figure 11-6 : One sealing fault. Pressure profile. The fault is reached, but it is not seen at the well. Infinite reservoir behavior.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
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www.opc.co.uk rw
r
L
pi
p infinite pw
Figure 11-7 : One sealing fault. Pressure profile. The fault is reached, and it is seen at the well. Boundary effect.
Flow regimes : • Radial flow • Semi-radial flow
This sequence of regime can be simulated by introducing the interference effect of an image well. This well, at distance 2L from the active well, produces at same rate q. The median line is no-flow.
L Well (q)
A second semi-log straight line with a slope double (2m)
L Image (q)
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2001
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Dp 2m m
0 Log(Dt)
Figure 11-8 One sealing fault. Semi-radial flow regime.
12. LOG LOG ANALYSIS 12.1 Log-log scale
102
101
∆P, psi
100
10-1 10-3 (3.6 sec)
10-2 (36 sec)
10-1 (6 mn)
100
101
102
∆t, hr Figure 12-1 : Log-log scale.
On the log-log plot, the well pressure response is compared to a set of dimensionless theoretical curves.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk p D = A ∆p , t D = B ∆t ,
{ A= f ( kh,...)}
{B = g( k , C, S ...)}
(Eq. 12-1)
The shape of the response curve is characteristic : the product of one of the variables by a constant term is changed into a displacement on the logarithmic axes. log pD = log A + log ∆p (Eq. 12-2) log t D = log B + log ∆t The log-log scale expands the response at early time.
12.2 Pressure curves analysis : example of "Well with wellbore storage and skin, homogeneous reservoir" Dimensionless terms Dimensionless pressure kh
pD =
141. 2 qB µ
∆p
(Eq. 12-3)
∆t
(Eq. 12-4)
Dimensionless time tD =
0. 000264 k φµct rw2
Dimensionless wellbore storage coefficient CD =
0.8936C φct hrw2
(Eq. 12-5)
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk 1.0E+02 1.00E+60
pD
1.0E+01
slope 1 1 1.00E-01
1.0E+00
1.0E-01 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
tD/CD
Figure 12-2 : Pressure type-curve : Well with wellbore storage and skin, homogeneous reservoir. Log-log scale. CD.e(2S) = 1060, 1030, 1015, 106, 103, 10, 1, 0.1.
Dimensionless time group tD CD
= 0. 000295
kh ∆t µ C
(Eq. 12-6)
Dimensionless curve group CD e 2 S =
0.8936C 2 S e φ ct hrw2
(Eq. 12-7)
Log-log matching procedure The log-log data plot ∆p, ∆t is superimposed on a set of dimensionless type-curves pD, tD / CD. At early time, well bore storage create a unit slope straight line. In log-log co-ordinates equation 12.2 becomes: log ∆p = log
qB + log ∆t 24 C
(Eq. 12-8)
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk The early time unit slope straight line is matched on the "wellbore storage" asymptote but the final choice of the CD e2S curve is frequently not unique.
1.0E+03
1.0E+02 Dp
Slope=1
1.0E+01
1 1.0E+00 1.0E-03
1.0E-02
1.0E-01
1.0E+00
1.0E+01
Dt
Figure 12-3 : Build-up example. Log-log plot
Results of log-log analysis
1.0E+02
1.0E+01
pD
1
1.0E+00
1.0E-01 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
tD/CD
Figure 12-4 : Build-up example. Log-log match.
Pressure match : the permeability thickness product
1.0E+02
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
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kh = 1412 . qBµ ( PM )
(Eq. 12-9)
Time match : the wellbore storage coefficient C = 0.000295
kh 1 µ TM
(Eq. 12-10)
Curve match : the skin S = 0.5 ln
CD e 2 S Match CD
(Eq. 12-11)
It is also important to determine from the log-log match when the end of well bore storage occurs. In the past a “rule of thumb” existed which stated that well bore storage finished after 1 ½ log cycles on the time scale of a log log plot. There was no scientific base to this and was sometimes completely wrong. It is suggested that the end of well bore storage occurs, for a build up, once the sand face flow rate falls to less than 1% of the original surface flow rate. This convention is adopted in PIE, a transient test analysis software package, and is used to mark the plot (shown in figure 23.1, as ENDWBS).
12.3 Build-up analysis : build-up type curve Build-up data should be matched against build-up type curves, generated with the time superposition. When ∆t>>tp , the pressure curve is compressed on the ∆p axis.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
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www.opc.co.uk 1.0E+04
tp 2S
CDe
type curve
Dp, psi
1.0E+03
drawdown
build-up
1.0E+02
1.0E+01 1.0E-02
1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
Dt, hours
Figure 12-5 : Drawdown and build-up type curves (tp = 2 hr).
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
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13. SEMI LOG ANALYSIS 13.1 M.D.H. analysis : ∆p vs Log∆t Semi-log straight line of slope m :
∆p = 162.6
qBµ k − 3.23 + 0.87 S log ∆t + log 2 φ µ ct rw kh
(Eq. 13-1)
Results: kh = 162. 6
qB µ
(Eq. 13-2)
m
∆p k S = 1151 . 1 hr − log + 3.23 2 φµ ct rw m
(Eq. 13-3)
13.2 Horner analysis
pws = pi − 162. 6
qB µ kh
log
t p + ∆t ∆t
(Eq. 13-4)
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk 4000.00
p*
m
p, psi
3750.00
3500.00
3250.00
3000.00 1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
Log [(tp+Dt)/Dt]
Figure 13-1 : Semi-log Horner plot
Horner analysis : The slope m, The pressure at 1 hour on the straight line The extrapolated pressure p*. Results : kh = 162.6
qBµ m
∆p tp +1 k S = 1151 . 1 hr − log + log + 3 . 23 φµ ct rw2 tp m
(Eq. 13-5)
(Eq. 13-6)
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
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14. DERIVATIVE ANALYSIS 14.1 Definition The natural logarithm is used. ∆p ' =
dp dp = ∆t d ln ∆t dt
(Eq. 14-1)
For a shut-in after a single drawdown period (the Horner method is applicable), the derivative is generated with respect to the modified Horner time given in the superposition Equation 11-2 : ∆p ' =
t p + ∆t dp dp = ∆t t p ∆t tp dt d ln t p + ∆t
(Eq. 14-2)
For a complex rate history, the multirate superposition time is used. In all cases, the derivative is plotted on log-log coordinates versus the elapsed time ∆t since the beginning of the shut-in period.
14.2 Derivative type-curve : example of "Well with wellbore storage and skin, homogeneous reservoir" Radial flow
Log(Dp) Log(Dp')
Dp'=constant
Log(Dt)
Figure 14-1 : Pressure and derivative responses on log-log scale. Radial flow.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
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∆p = 162.6
qBµ k − 3.23 + 0.87 S log ∆t + log 2 φ µ ct rw kh
(Eq. 14-3)
The derivative is a constant. ∆p ' = 70. 6
qB µ kh
(Eq. 14-4)
In dimensionless terms, dp D = 0.5 d ln( t D C D )
(Eq. 14-5)
Wellbore storage ∆p =
q ∆t 24 C
(Eq. 14-6)
∆p ' =
q ∆t 24 C
(Eq. 14-7)
slope 1
Log(Dp) Log(Dp')
Log(Dt)
Figure 14-2 : Pressure and derivative responses on log-log scale. Wellbore storage
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Derivative analysis of previous example
1.0E+03
Dp', psi
1.0E+02
Slope=1 0.5 line 1.0E+01
1.0E+00 1.0E-03
1.0E-02
1.0E-01
1.0E+00
1.0E+01
1.0E+02
Dt, hours
Figure 14-3 : Derivative of build-up example Figure 12.3. Log-log scale.
1.0E+02 1.00E+60
pD'
1.0E+01
1.0E+00
slope 1
0.5
1.00E-01 1.0E-01 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
tD/CD
Figure 14-4 : "Well with wellbore storage and skin, homogeneous reservoir" Derivative of type-curve Figure 12.2, log-log scale. CD.e(2S) = 1060, 1030, 1015, 106, 103, 10, 1, 0.1.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
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www.opc.co.uk 1.0E+02
pD'
1.0E+01
1.0E+00
1.0E-01 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
tD/CD
Figure 14-5 : Derivative match of example Figure 12.4.
14.3 Data differentiation The algorithm uses three points, one point before (left = 1) and one after (right = 2) the point i of interest. It estimates the left and right slopes, and attributes their weighted mean to the point i. On a p vs. x semi-log plot,
∆p ∆p ∆x2 + ∆x1 ∆x 2 dp ∆x 1 = dx ∆x1 + ∆x2
( 14-8)
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
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www.opc.co.uk L (2) . . . . ... . . . .. . . . . . . . .. (1). Dx1
p
Dp1
Dp2
Dx2
Superposition x Figure 14-6 : Differentiation of a set of pressure data.
It is recommended to start by using consecutive points. If the resulting derivative curve is too noisy, smoothing is applied by increasing the distance ∆x between the point i and points 1 and 2. The smoothing is defined as a distance L, expressed on the time axis scale. The points 1 and 2 are the first at distance ∆x1,2>L. The smoothing coefficient L is increased until the derivative response is smooth enough but no more, over smoothing the data introduces distortions. With this smoothing method, L is usually no more than 0.2 or 0.3.
At the end of the period, point i becomes closer to last recorded point than the distance L. Smoothing is not possible any more to the right side, the end effect is reached. This effect can introduce distortions at the end of the derivative response.
14.4 The analysis scales The log-log analysis is made with a simultaneous plot of the pressure and derivative curves of the interpretation period. Time and pressure match are defined with the derivative response, adjusting the curve match on pressure and derivative data the CD e(2S) group is identified by.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
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pD & pD'
1.0E+02
1.0E+01
slope 1 0.5
1.0E+00
1.0E-01 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
tD/CD
Figure 14-7 : Pressure and derivative type-curve for a well with wellbore storage and skin, homogeneous reservoir.
The double log-log match is confirmed with a match of the pressure type-curve on semi-log scale to adjust accurately the skin factor and the initial pressure. A simulation of the complete test history is presented on linear scale in order to control the rates, any changes in the well behavior, the average pressure etc.
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15. PRINCIPAL FLOW REGIMES In pressure transient analysis the reservoir engineer is interested in analysing the pressure behaviour at the wells resulting from well and reservoir effects. Individual wells may or may not penetrate the formation fully, be fractured or damaged and these factors will affect the pressure behaviour at the well in different ways. The flow regimes are, without exception, relatively near well bore effects, since ultimately in an “infinte” reservoir the flow regime tends to radial flow. Thus, it becomes necessary to understand the various flow regimes which can exist near wellbores and these are considered below. 15.1 Radial Flow If it is assumed that a well extends through the productive zone of the reservoir, the flow lines at all elevations around the well would be radial. For a producing well the flow lines would converge whilst for an injection well they would diverge. A schematic diagram of the flow lines, showing the lines of equal pressure (isopotentials), is presented in Figure 16.1 below. Thus if flow is radial everywhere in the reservoir, the pressure at the well for a constant rate drawdown can be expressed by the following equation: ∆p = ( pi - p w ) = 162.6
qµB log t + constant kh
(Eq. 15-1)
where, ∆p pi pw q B µ k h t
is the pressure drop in psi, is the initial reservoir in psi, is the flowing pressure at time t in psi, is the flow rate in STB/Day, is the formation volume factor RB/STB, is the viscosity of the reservoir fluid in cp, is the reservoir permeability in mD, is the reservoir thickness in feet and is the producing time in hours.
Inspection of Equation 15.1 indicates that a plot of pw v log t will be a straight line.
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SECTION
PLAN Figure 15-1 Geometry of flow lines in Radial Flow
15.2 Linear Flow Linear flow occurs at early time in hydraulic fractures as depicted in Figure 15.2. It can also occur in a horizontal well during transition from vertical radial flow to pseudo radial flow
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
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www.opc.co.uk or in a well situated between two parallel no-flow boundaries at later times. In the latter case flow is radial until the radius of investigation reaches the parallel boundaries after which the flow becomes linear. The pressure response during linear flow is proportional to the square root of time. qB µ ∆p = ( pi - p w ) = 4.064 x f h φ ct k
1/ 2
t 1/ 2 + constant (Eq. 15-2)
where, pi pw q B xf µ φ ct k t
is the initial reservoir in psi, is the flowing pressure at time t in psi, is the flow rate in STB/Day, is the formation volume factor RB/STB, is the fracture half-length (or half-length of horizontal drain or half distance between faults) in feet, is the viscosity of the reservoir fluid in cp, is the reservoir porosity, is the total system compressibility in 1/psi, is the reservoir permeability in mD and, is the producing time in hours.
With reference to Equation 15.2 a plot of pw v t1/2 will give a straight line. Infinite conductivity fracture response
xf
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Figure 15-2 : Infinite conductivity fracture. Geometry of the flow lines. Linear and radial flow regimes.
On a plot of the pressure versus the square root of time, the response follows a straight line intercepting the origin.
Dp slope mLF
0 SQRT(Dt)
0
Figure 15-3 Infinite conductivity fracture. Pressure versus the square root of time.
∆p = 4 . 06
qB hx f
∆p' = 2.03
qB hx f
µ φ ct k
∆t
µ ∆t φ ct k
(Eq. 15-3)
(Eq. 15-4)
On log-log scale, the pressure and derivative follow a half unit slope straight line, the derivative is half the pressure.
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slope 1/2 Log(Dp) Log(Dp')
Log(Dt)
Figure 15-4 : Pressure and derivative responses on log-log scale. Infinite conductivity fracture.
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15.3 Bi-Linear Flow Bi-linear flow occurs in finite conductivity hydraulic fractures in very early time. This regime is termed bi-linear because fluid stored in the fracture flows towards the well in a linear pattern parallel to the fracture face. At the same time flow in the formation tends to be linear and perpendicular to the fracture face as illustrated by Figure 15.5.
w
Figure 15-5 : Finite conductivity fracture. Geometry of the flow lines during the bilinear flow regime.
During a drawdown, the bi-linear regime is usually followed by a linear flow regime when the pressure losses inside the fracture become negligible. At later time the bi-linear regime may be followed by a radial, (pseudo-radial) flow regime. In practice this flow regime is rarely observed in the pressure response. The drawdown pressure response during bi-linear flow is: ∆p = ( pi - p w ) = 44.1
qµB 1/ 4 + constant 1/ 4 t h( k f w ) ( φµ ct k ) 1/ 2
where, pi pw q B µ h kfw φ ct k
is the initial reservoir in psi, is the flowing pressure at time t in psi, is the flow rate in STB/Day, is the formation volume factor RB/STB, is the viscosity of the reservoir fluid in cp, is the reservoir thickness in feet, is the fracture conductivity in mDft, is the reservoir porosity, is the total system compressibility in 1/psi and is the reservoir permeability in mD and
(Eq. 15-5)
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
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is the producing time in hours.
From Equation 15.5 it can be seen that a plot of pw versus t1/4 will be a straight line.
Dp
slope mBLF
0 0
4thRT(Dt)
Figure 15-6 Finite conductivity fracture. Pressure versus the fourth root of time.
On log-log scale, the pressure and derivative follow a quarter unit slope straight line, the derivative is 1/4 the pressure. ∆p = 44.11
qB µ h kf
. ∆p' = 1103
w4
4
φµ ct k
∆t
qBµ h k f w 4 φ µ ct k
4
(Eq. 15-6)
∆t
(Eq. 15-7)
slope 1/4 Log(Dp) Log(Dp')
Log(Dt)
Figure 15-7 : Pressure and derivative responses on log-log scale.
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15.4 Spherical Flow Spherical flow is most frequently encountered in partially penetrating wells as shown in Figure 15.8. At early time, flow into a partially penetrating well is radial over the perforated interval thickness. At later time, however, the flow becomes spherical if the perforated interval is small compared to the total formation thickness. At very late time the flow becomes radial again with respect to the total formation thickness.
h
hw zw
Figure 15-8 : Well in partial penetration. Geometry of the flow lines. Radial, spherical and radial flow regimes.
During spherical flow the pressure response is:
∆p = ( pi - p w ) = 2452.91
qµB ( φ ct )1/ 2 kh k
1/ 2 z
t
-1/ 2
+ constant (Eq. 15-8)
where pi pw q B µ φ ct kh kz t
is the initial reservoir in psi, is the flowing pressure at time t in psi, is the flow rate in STB/Day, is the formation volume factor RB/STB, is the viscosity of the reservoir fluid in cp, is the reservoir porosity, is the total system compressibility in 1/psi and is the horizontal reservoir permeability in mD, is the vertical reservoir permeability in mD and is the producing time in hours.
From Equation 15.8 it can be seen that a plot of pw versus t-1/2 gives a straight line.
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Dp
slope mSPH
0 1/SQRT(Dt)
0
Figure 15-9 Well in partial penetration. Pressure versus 1/ the square root of time. On log-log scale, the derivative follow a negative half unit slope straight line, the pressure is not characteristic.
∆p = 70. 6
qB µ k S rS
∆p ' = 1226. 4
− 2452. 9
qB µ φµ ct
(Eq. 15-9)
k S3 2 ∆t
qB µ φµ ct
(Eq. 15-10)
k S3 2 ∆t
Log(Dp)
slope -1/2 Log(Dp')
Log(Dt)
Figure 15-10 : Pressure and derivative responses on log-log scale. Well in partial penetration.
15.5 Pseudo steady state At late time during drawdwon, the response follows a straight line on linear scale :
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∆p = 0.234
qB qBµ A ∆t + 162.6 log 2 − log(C A ) + 0.351 + 0.87 S φ ct hA kh rw
(Eq. 15-11)
Dp slope m* 0 0
Dt
Figure 15-11 Closed reservoir. Linear scale.
On log-log scale, the pressure and derivative follow the same unit slope straight line. ∆p' = 0.234
qB ∆t φ ct hA
(Eq. 15-12)
Log(Dp)
slope 1 Log(Dp')
Log(Dt)
Figure 15-12 : Pressure and derivative responses on log-log scale. Closed system. (Drawdown).
16. BOUNDARY THEORY 16.1 One sealing fault Characteristic flow regime
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
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www.opc.co.uk 1. Radial flow 2. Semi-radial flow
Figure 16-1 Flow regimes for a well near a boundary
With a boundary at a fixed distance from the well, in early time, the typical radial flow regime is observed with a corresponding flattening of the derivative on a log log plot. During later time, the boundary causes the flow regime to change to an approximate semi radial flow with a corresponding upturn in the derivative on the log log plot which eventually flattens out again at a value twice the radial flow value. This is illustrated for different boundary distances on the log log plot below in figure 16.2. Log-log analysis 1.0E+02
pD & pD'
1.0E+01
LD=100 1.0E+00
300 1.0E-01 1.0E+00
1.0E+01
1.0E+02
1.0E+03
1000
1.0E+04
3000
1.0E+05
tD/CD
Figure 16-2 : Responses for a well with wellbore storage and skin in a homogeneous reservoir limited by one sealing fault. Log-log scale. Several distances. CD = 100, S = 5, LD = 100, 300, 1000, 3000.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
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www.opc.co.uk Semi-log analysis The time of intercept ∆tx between the two semi-log straight lines can be used to estimate the distance between the well and the sealing fault : L = 0. 01217
k ∆t x
(Eq. 16-1)
φµ ct
20.0
slope 2m
pD
15.0
10.0
LD =100 300 1000 3000
slope m
5.0
0.0 1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
tD/CD
Figure 16-3 : Semi-log plot of previous examples Figure 16.2.
16.2 Two parallel sealing faults Definition
L2 Well L1
Characteristic flow regimes 1. Radial flow
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Log-log analysis 1.0E+02
pD & pD'
°A
°B
1.0E+01
B
slope 1/2
1.0E+00
0.5 1.0E-01 1.0E-01
1.0E+00
1.0E+01
A
1.0E+02
1.0E+03
1.0E+04
1.0E+05
tD/CD
Figure 16-4 : Responses for a well with wellbore storage in a homogeneous reservoir limited by two parallel sealing faults. Log-log scale. One channel width, two well locations. CD = 3000, S = 0, L1D = L2D = 3000 (curve a) and L1D = 1000, L2D = 5000 (curve b).
1.0E+02
L1D=L2D=500 1000
pD & pD'
1.0E+01
2500 5000 500
1.0E+00
1.0E-01 1.0E-01
1.0E+00
1.0E+01
1.0E+02
tD/CD
1.0E+03
1.0E+04
1.0E+05
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www.opc.co.uk Figure 16-5 : Responses for a well with wellbore storage and skin near two parallel sealing faults. Homogeneous reservoir. Log-log scale. The well is located midway between the two boundaries, several distances between the two faults are considered. CD = 300, S = 0 L1D = L2D = 500, 1000, 2500 and 5000.
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Semi-log analysis On semi-log scale, only one straight line is present. During the late time linear flow, the responses deviate in a curve above the radial flow line. 40.0
500 30.0
pD
1000 20.0
2500 10.0
5000
slope m
0.0 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
tD/CD
Figure 16-6 : Semi-log plot of previous example Figure 16.5.
20.0
pD
15.0
B
10.0
slope m
A
5.0
0.0 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
tD/CD
Figure 16-7 : Semi-log plot of previous example Figure 16.4.
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Linear flow analysis 40.0
500 30.0
pD
1000 slope mch 20.0
2500 5000
10.0
0.0 0.0E+00
5.0E+01
1.0E+02
1.5E+02
2.0E+02
2.5E+02
3.0E+02
3.5E+02
SQRT(tD/CD)
Figure 16-8 : Square root of time plot of previous example Figure 16.5.
The pressure change ∆p is plotted versus the square root of the elapsed time ∆t. The slope mLF and the intercept ∆pint of the linear flow straight line are used to estimate the channel width and the well location.
mLF = 8133 .
µ qB h( L1 + L2 ) kφ ct
L1 + L2 = 8.133
σ=
kh 141. 2 qB µ
qB
µ
hmLF
k φ ct
∆pint − S
L + L2 L1 1 exp( −σ ) = arcsin 1 L1 + L2 π 2π rw
(Eq. 16-2)
(Eq. 16-3)
(Eq. 16-4)
(Eq. 16-5)
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www.opc.co.uk Build-up analysis 1.0E+02
°C
°D
pD & pD'
1.0E+01
D 1.0E+00
C 1.0E-01 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
tD/CD
Figure 16-9 : Build-up responses for a well with wellbore storage in a homogeneous reservoir limited by two parallel sealing faults. Log-log scale. One channel width, two well locations. The two dotted derivative curves correspond to the drawdown solution. The build-up responses (solid lines) are generated for (tp/C)D = 2000. CD = 3000, S = 0, L1D = L2D = 5000 (curve a) and L1D = 2000, L2D = 8000 (curve b).
9.0
D 8.0
C
slope m
pD
7.0
6.0
5.0
4.0
3.0 1.0E+00
1.0E+01
1.0E+02
1.0E+03
(tpD + tD) / tD
Figure 16-10 : Semi-log plot of previous example Figure 16.9 (Horner time).
The extrapolation of the Horner straight line does not correspond to the infinite shut-in time pressure.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
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www.opc.co.uk For an infinite channel, when both the drawdown and the shut-in periods are in linear flow regime, the superposition function is expressed as (tp+∆t) 1/2 - (∆t) 1/2. The extrapolation of the linear flow straight line to infinite shut-in time, at (tp+∆t) 1/2 - (∆t) 1/2 = 0, is used to estimate the initial reservoir pressure.
9.0
D 8.0
C
slope mch
pD
7.0
6.0
5.0
4.0
3.0 0.0E+00
1.0E+01
2.0E+01
3.0E+01
4.0E+01
5.0E+01
SQRT (tpD+tD/CD) - SQRT (tD/CD)
Figure 16-11 : Square root of time plot of previous example Figure 16.9. pD versus [(tpD+tD)/CD]1/2 - [tD/CD]1/2.
16.3 Two intersecting sealing faults Definition
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L2
θ
Well
θw
L1
The angle of intersection θ between the faults is smaller than 180°, the wedge is otherwise of infinite extension. LD is the dimensionless distance between the well and the faults intercept and the well location in the wedge is defined with θw. The distances L1 and L2 between the well and the sealing faults are expressed as : L1 = LD rw sinθ w
L2 = LD rw sin(θ − θ w )
(Eq. 16-6) (Eq. 16-7)
Characteristic flow regimes 1. Radial flow 2. Fraction of radial flow
Log-log analysis If for example the angle between the faults is 60° (π/3), the wedge is 1/6 of the infinite plane (2π), and the derivative stabilizes at 3.
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°B °A pD & pD'
1.0E+01
180° / θ=3 B 1.0E+00
A
0.5 1.0E-01 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
tD/CD
Figure 16-12 : Responses for a well with wellbore storage in a homogeneous reservoir limited by two intersecting sealing faults. Log-log scale. CD = 3000, S = 0, LD = 5000, θ = 60°, θw = 30° (curve a) and θw = 10° (curve b).
θ = 360°
∆p1st stab.
(Eq. 16-8)
∆p2nd stab.
1.0E+02 10°
pD & pD'
10° 1.0E+01
20° 180° 45° 90° 135° 180°
1.0E+00
1.0E-01 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
tD/CD
Figure 16-13 : Responses for a well with wellbore storage in a homogeneous reservoir limited by two intersecting sealing faults. Log-log scale.
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www.opc.co.uk Several angles of intersection θ, the well is on the bisector θw = 0.5 θ, the distance to the two faults is constant L1D = L2D = 1000, the distance LD to the fault intercept changes. CD = 1000, S = 0, θ = 10°, LD = 11473; θ = 20°, LD = 5759; θ = 45°, LD = 2613; θ = 90°, LD = 1414; θ = 135°, LD = 1082; θ = 180°, LD = 1000.
Semi-log analysis On a complete response, two semi-log straight lines can be identified. The first, of slope m, describes the infinite acting regime. The second, with a slope of (360/θ)m, defines the fraction of radial flow limited by the wedge. 60.0
10° slope (360° /θ) m
50.0
20°
pD
40.0
30.0
45°
20.0
90° 135°
slope m
180°
10.0
0.0 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
tD/CD
Figure 16-14 : Semi-log plot of previous example Figure 16.13. θ = 360°
m1st line m2nd line
(Eq. 16-9)
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk 20.0
B slope 6 m
pD
15.0
A
10.0
slope m 5.0
0.0 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
tD/CD
Figure 16-15 : Semi-log plot of previous example Figure 16.12.
16.4 Closed system Definition Only the rectangular reservoir shape is considered. The well is at dimensionless distances L1D, L2D, L3D, and L4D from the four sealing boundaries, the dimensionless area of the closed reservoir is expressed as: A = ( L1D + L3 D )( L2 D + L4 D ) rw2
(Eq. 16-10)
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk 5050
pi pressure, psi
5000
4950
p-
4900
pseudo steady state m*
4850
4800 0
1000
2000
3000
4000
5000
6000
7000
8000
time, hours
Figure 16-16 : Drawdown and build-up pressure response. Closed system. The pseudo steady state regime The well, at initial reservoir pressure pi, is produced until all reservoir boundaries are reached. At the end of the drawdown, the pseudo steady state regime is shown by a linear pressure trend. The well is then closed for a shut-in period, the pressure builds up until the average reservoir pressure p- is reached, and then the curve flattens. The difference (p -p-), i
between the initial pressure and the final stabilized pressure defines the depletion. 1.0E+02
slope 1
°B °A pD & pD'
1.0E+01
A&B B 1.0E+00
0.5 A 1.0E-01 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
tD/CD
Figure 16-17 : Drawdown and build-up responses for a well with wellbore storage in a closed square homogeneous reservoir. Log-log scale. The two dotted derivative curves correspond to the drawdown solution. The build-up responses (solid lines) are generated for (tp/C)D = 1000.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk CD = 25000, S = 0 .Curve A: L1D = L2D = L3D = L4D = 30000, curve B: L1D = L2D = 6000, L3D = L4D = 54000. Drawdown log-log analysis
pD & pD'
1.0E+02
1.0E+01
1.0E+07
A/rw2=1.0E+06 1.0E+00
1.0E+08
1.0E-01 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
tD/CD
Figure 16-18 : Drawdown responses for a well with wellbore storage in a closed square homogeneous reservoir. Log-log scale. Three reservoir sizes, the well is centered or near one of the boundaries. CD = 100, S = 0, A/rw2 = 106, 107, 108 (L1D = 200)
pD & pD'
1.0E+02
slope 1
°
°
D
C
1.0E+01
D C slope 1/2
1.0E+00
0.5 1.0E-01 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
tD/CD
Figure 16-19 : Pressure and derivative drawdown responses for a well with wellbore storage in a closed channel homogeneous reservoir. Log-log scale. CD = 1000, S = 0 curve a: L1D = L3D = 20000, L2D = L4D = 2000 curve b: L1D = L2D = L3D = 2000, L4D = 38000 .
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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Drawdown semi-log straight lines analysis 20.0
A/rw2=1.0E+06
1.0E+07
1.0E+08
15.0
pD
slope 2m 10.0
slope m 5.0
0.0 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
tD/CD
Figure 16-20 : Semi-log plot of Figure 16.18.
1.0E+04
1.0E+05
1.0E+06
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk Drawdown linear and semi-linear flow analysis 50.0
D 40.0
slope 2 mch
pD
30.0
20.0
C slope mch
10.0
0.0 0.0E+00
2.0E+02
4.0E+02
6.0E+02
8.0E+02
SQRT(tD/CD)
Figure 16-21 : Linear flow analysis plot of Figure 16.19.
The slope for the infinite channel behavior (Curve a) is expressed in Equation 16.2. For the limited channel (b) the slope of the linear flow straight line is double :
mSLF = 16.27
qB µ h( L2 + L4 ) kφ ct
(Eq. 16-11)
Drawdown pseudo-steady state analysis 50.0
A/rw2=1.0E+06 40.0
1.0E+07
slope m*
pD
30.0
20.0
1.0E+08
10.0
0.0 0.0E+00
2.0E+05
4.0E+05
tD/CD
6.0E+05
8.0E+05
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk Figure 16-22 : Pseudo steady state flow analysis plot Figure 16.18.
During pseudo-steady state regime, the drawdown dimensionless pressure is expressed as : p D = 2 π t DA + 0.5 ln
A rw2
+ 0.5 ln
2. 2458 CA
+S
(Eq. 16-12)
The dimensionless time tDA is defined with respect to the drainage area : t DA =
0. 000264 k φµct A
∆t
(Eq. 16-13)
The "shape factor" CA characterizes the geometry of the reservoir and the well location. With real data, the pressure during pseudo steady state flow regime is expressed :
∆p = 0.234
qB qBµ A 2.2458 ∆t + 162.6 + 0.87 S log 2 + log kh rw CA φ ct hA
(Eq. 16-14)
the slope m* of the pseudo-steady state straight line provides the reservoir connected pore volume : φhA = 0. 234
qB
(Eq. 16-15)
ct m *
When kh and S are known from semi-log analysis, the shape factor CA is estimated from the intercept ∆pint of the pseudo-steady state straight line :
C A = 2.2458∗10
Build-up analysis
A ∆pINT − log 2 − 0.87 S 162.6qBµ kh rw
(Eq. 16-16)
2001
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
www.opc.co.uk Pressure and derivative build-up analysis Figure 16.23 presents three build-up examples for a well in a closed rectangle. The rectangular reservoir configuration is described in the Shape Factors Tables with CA = 0.5813 and the start of pseudo steady state is defined at tDA = 2 (or tD/CD = 54600). The well is closed for build-up just before or during the pure pseudo steady state flow, the three production times are tpDA = 0.6, 2 and 10.
1.0E+02
° pD & pD'
1.0E+01
tpDA = 0.6 1.0E+00
tpDA = 2, 10 1.0E-01 1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
tD/CD
Figure 16-23 : Build-up responses for a well with wellbore storage and skin in a closed rectangle homogeneous reservoir. Log-log scale. The well is close to one boundary. Three production times are considered. CD = 292, S = 0, L1D = 500, L2D = 1000, L3D = 3500, L4D = 1000 tpD/CD (tpDA) = 16400 (0.6), 54600 (2), 273000 (10) When all reservoir boundaries have been reached during drawdown, the shape of the subsequent build-up is independent of tp. At late times, the stabilized dimensionless pressure p-D is expressed as : A rw2 p D = 1151 . log + 0.35 + S CA
Semi-log analysis of build-up
(Eq. 16-17)
2001
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
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pD8.0
slope m pD
6.0
4.0
0.0 1.0E+00
1.0E+01
1.0E+02
1.0E+03
10
2
tpDA = 0.6
2.0
1.0E+04
1.0E+05
1.0E+06
(tpD + tD) / tD
Figure 16-24 : Semi-log plot of Figure 16.23. When tp>>∆t, the Horner time can be simplified with tp+∆t ≅ tp :
log
t p + ∆t ∆t
= log t p − log ∆t
(Eq. 16-18)
For different production time tp in a depleted reservoir, the Horner straight lines of slope m are parallel. The Horner plot Figure 16-24 is presented in dimensionless terms. The straight line extrapolated pressure p* changes with t and, later, the curves flatten to reach p = 8.62 of D
p
D
Equation 16.17. For examples B and C, p*D > p D , but not for exampe A. With real pressure, the average pressure p decreases when tp increases.
16.5 Constant pressure boundaries Definition
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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Gas
Oil Water
L Well (q)
L Image (-q)
Figure 16-25 Representation of Constant Pressure Boundary
2001
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
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Log-log analysis The dimensionless stabilized pressure is defined as : pD = ln(2 LD ) + S
(Eq. 16-19)
1.0E+02
pD & pD'
1.0E+01
3000
1000
300
1.0E+00
LD=100
1.0E-01 1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
tD/CD
Figure 16-26 : Responses for a well with wellbore storage and skin near one constant pressure linear boundary in a homogeneous reservoir. Log-log scale. Several distances. CD = 100, S = 5, LD = 100, 300, 1000, 3000.
1.0E+02 o
pD & pD'
1.0E+01
sealing fault : 1 1.0E+00
0.5 1.0E-01
constant pressure
1.0E-02 1.0E-01
1.0E+00
1.0E+01
1.0E+02
tD/CD
1.0E+03
1.0E+04
1.0E+05
2001
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
www.opc.co.uk Figure 16-27 : Pressure and derivative responses for a well with wellbore storage and skin near two perpendicular faults in a homogeneous reservoir. Log-log scale. The closest fault is sealing, the second at constant pressure. CD = 100, S = 0, θ= 90°, θw = 20°, LD = 1000.
Semi-log analysis 15.0
3000 1000 300
slope m 10.0
pD
LD =100
5.0
0.0 1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
tD/CD
Figure 16-28 : Semi-log plot of Figure 16.26. The time of intercept ∆tx between the semi-log straight line and the constant pressure line is used, as for a sealing fault, to estimate the distance of the boundary :
L = 0. 01217
k ∆t x φµ ct
(Eq. 16-20)
The difference of pressure between the start of the period and the final stabilized pressure, [p- - p(∆t=0)], can also be used to estimate L :
L = 0.5rw e
kh p− p ( ∆t = 0 ) − S 1412 . qBµ
(Eq. 16-21)
2001
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
www.opc.co.uk 16.6 Semi permeable boundary Definition lf
kf
The partially communicating fault, at distance L from the well, has a thickness lf and a permeability kf. The dimensionless fault transmissibility ratio α is expressed as : α=
k f lf
(Eq. 16-22)
k L
Log-log analysis
pD & pD'
1.0E+02
1.0E+01
1.0E+00
0.5
0.5 1.0E-01 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
tD/CD
Figure 16-29 : Pressure and derivative drawdown response for a well with wellbore storage near a semi-permeable linear boundary. Homogeneous reservoir. Log-log scale. CD = 104, S = 0, LD = 5000, α = 0.05
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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pD & pD'
1.0E+02
1.0E+01
α =1,
0.1,
0.01,
0.001
1 1.0E+00
0.5 1.0E-01 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
tD/CD
Figure 16-30 : Responses for a well with wellbore storage and skin near a semipermeable linear boundary. Log-log scale. Several transmissibility ratios. CD = 100, S = 5, LD = 300, α = 1, 0.1, 0.01, 0.001.
Semi-log analysis 20.0
slope 2m 15.0
α = 0.001 0.01 0.1 1 slope m
pD
slope m 10.0
5.0
0.0 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
tD/CD
Figure 16-31 : Semi-log plot of Figure 16.30.
1.0E+04
1.0E+05
1.0E+06
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2001
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16.7 Predicting derivative shapes 4
3 5 1
2
Figure 16-32 : Closed reservoir example.
Example of a drawdown in a closed system, the shape of the reservoir is assumed to be a trapezoid. After wellbore storage, the response shows : 1 - the infinite radial flow regime (derivative on 0.5), 2 - one sealing fault (derivative on 1), 3 - the wedge response (derivative on π/θ), 4 - linear flow (derivative straight line of slope 1/2), 5 - pseudo steady state (straight line of slope 1).
5 3
4
Log(Dp') 2 1
Log(Dt)
Figure 16-33 : Derivative response for a well in a closed trapezoid.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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17. RESERVOIR PRESSURE 17.1 Definitions
6000.
Figure 17.1 illustrates the pressures p*, pi and p which can be derived from a typical semi log build-up plot. It is interesting to note that they are all different in value. A brief review of these three pressures will be made before considering the applications and methods of determining average reservoir pressures.
4000.
Pbar SLOPE
P*
3000.
P PSI
5000.
Pi
Homogeneous Reservoir
1000.
2000.
slope of the line = -.13335 PSI/cycle extrapolated pressure = 4739. PSI R(inv) at 1.750 hr = 351. FEET R(inv) at 2.611 hr = 429. FEET prod. time=48.00 hr at rate=5000.000 STB/D skin = -1.68 permeability = 36.6 MD Perm-Thickness = 3660. MD-FEET 5.3 HR
0.
5000.
.48 HR
10000.
.048 HR
15000.
SUPERPOSITION
Figure 17-1 : Plot showing different reservoir pressures pi - Initial reservoir pressure This is a definitive term describing the virgin reservoir conditions prior to any fluid withdrawal. The initial reservoir pressure can be derived from RFT/FMT measurements while logging the well prior to testing. This pressure value can also be determined from extrapolation of a semi log or superposition plot to infinite shut-in time if the amount of fluid withdrawn for the reservoir prior to the survey is negligible compared with the hydrocarbons in-place and if the reservoir is infinite-acting at the time of the shut-in, (ie, the reservoir boundaries have not been seen). Generation of a type curve which accurately matches a set of pressure data from a test will also yield a value of pi providing the reservoir system is infinite. Volumetric average reservoir pressure, p
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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This is the pressure that will eventually be reached if a well is shut-in long enough for the reservoir to reach static equilibrium. It will change continuously with fluid withdrawal or production, if no water influx is assumed. Note that the average reservoir pressure is not a value that can be derived by direct extrapolation of any plot of the build-up data and applies to a closed system. The semi-log extrapolated or "False" pressure, p* This is the pressure obtained by extrapolating the correct semi-log straight line portion of a plot to infinite shut-in time, ie, when: t + ∆t t + ∆t = 1 or log = 0 ∆t ∆t
(Eq. 17-1)
As discussed earlier, p* can be equal to pi if production has been relatively minimal and the reservoir is still infinite-acting at the time of shut-in, eg, during a DST. If this is not the case, p* has no physical significance. A large amount of error can be introduced by assuming that p* is equal to p . As will be demonstrated later, the difference between p* and p can be quite large and is a function of the reservoir shape, well configuration and producing time. The difference between p* and p increases with producing time for most reservoir shapes. It should also be noted that the true p* is derived from extrapolation of the semi log straight line used to calculate the permeability thickness product, kh, or permeability k. In a well near a boundary, there may be two (or more) straight lines on the semi log plot. p* is obtained from extrapolation of the first (earlier time) straight line. This p* may be less than p and possibly even less that pws or the measured build-up pressures.
17.2 Applications of reservoir pressure An important task of the reservoir engineer is to determine the quantity of fluids in place. Undoubtedly the porosity, φ, thickness, h, drainage area, A, and water saturation, Sw, play an important role. In addition to these parameters, the average reservoir pressure assists in determining the quantity of fluids in place and recoverable.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk For a well producing from a closed (bounded) system the average reservoir pressure is the maximum pressure which will be recorded if the well is shut-in for an infinite time assuming no water influx. In developed fields, the field average reservoir pressure used to be determined by estimating individual drainage areas, shapes, and hydrocarbon volumes and then weighting the observed individual well 's p . A preferable method currently is to use type curve analysis since this describes the reservoir characteristics more accurately. As the reservoir is produced, the relationship of p with fluid withdrawal volumes provides important evidence regarding the reservoir drive mechanism, ie, depletion drive (originally saturated or undersaturated oil), water drive, limited water influx, etc. Determination of individual well productivity, which indicates how many barrels per day or mmscfd can be withdrawn for a given sand face pressure drawdown, requires that p (or the reservoir boundary pressure, pe) be known. A volumetrically averaged pressure is required for wells within a field to help determine if the entire reservoir is being adequately drained. Average reservoir pressure is an essential piece of data for history matching reservoir performance with computer simulation models. An understanding of the average reservoir pressure concept is required to properly use individual build-up derived pressures and relate them on an appropriate radial basis. The manner in which different areas of a reservoir deplete (or re-pressure with secondary or enhanced recovery schemes) can help characterise the extent of sand lenses, location and effectiveness of faults continuity between different areas of a field, partially transmissible boundaries and other reservoir heterogeneities. Drainage Volumes of Wells Each producing well in a bounded reservoir will drain a certain volume surrounding that well. Initially, (during the early transient period), all wells will drain from an equal volumetric share of the reservoir (if thickness, porosity, saturations and compressibility are constant) since the radius of drainage is a function of the dimensionless producing time, tDp, not the production rate. During the early transient period, therefore, the individual well flow rates are proportional (for equal drawdowns) to kh/µ. Once the reservoir is producing under pseudo steady-state, ie, dp/dt is nearly constant throughout the reservoir, each well will drain a volume of the reservoir proportional to its producing rate
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk Generally, it is sufficiently accurate to assume that drainage areas are symmetrical and divide them according to a convenient parameter such as well spacing. This becomes acceptable if the p 's so-derived are used in material balance calculations where, once again, pressure differences (changing p's) rather than absolute p 's are used. It is important, however, to understand the drainage area concept so that in cases of large production imbalances, irregular well spacing, heterogeneous pay thickness, etc., or in applications where an accurate absolute pressure is required, that appropriate consideration be given to non-symmetrical drainage. From a cartesian plot of ∆p v ∆t it is possible to determine the drainage area by identifying the data points corresponding to pseudo-steady state flow. Provided that the data points lie on a straight line the following expression may be used: ∆p = 0.234
qB ∆t + ∆ p INT φ ct hA
(Eq. 17-2)
If Equation 17.2 is re-written in the form: ∆p = m * ∆t + ∆ p INT
(Eq. 17-3)
Then the slope m* corresponding to pseudo-steady state flow is given by:
m* =
0.234qB φ ct hA
(Eq. 17-4)
The drainage area can be calculated by rearranging Equation 17.4 to obtain:
A =
0.234qB φ ct hm*
(Eq. 17-5)
Note that this analysis will only be applicable to drawdowns of very long duration, ie, those exceeding 3 months. The Matthews-Brons-Hazebroek (MBH) Method Matthews-Brons-Hazebroek superimposed the effects of many image wells and calculated build-up curves for a variety of different reservoir shapes and relative well locations. The superposition of the image wells utilised the Ei function which necessarily assumes that the
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk drawdown and build-up are both following infinite-acting behaviour rather than assuming the drawdown had reached pseudo-steady state as was done by Miller, Dyes and Hutchinson. This method is largely redundant with today’s interpretation technology since a type curve in a bounded system can be generated which represents, albeit theoretically, the exact pressure response for the given system including the fluid withdrawal. The initial pressure is therefore found for the actual reservoir system. The Miller-Dyes-Hutchinson (MDH) Method Miller, Dyes and Hutchinson presented a technique that could be used to estimate the average reservoir pressure for a well producing at pseudo-steady state prior to shut-in, in the centre of a circular (or nearly geometrically regular) drainage area. For the same reason presented above in the MBH method, this method is largely redundant and its use is not recommended.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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18. MOBILITY CHANGE THEORY 18.1 Definitions (k/µ)2, (φct)2 (k/µ)1, (φct)1
(k/µ)2, (φct)2 (k/µ)1, (φct)1
Radial composite Linear composite Figure 18-1 : Model for composite reservoirs. With the radial composite model, the well is at the center of a circular zone of radius r. With the linear composite model, the interface is at a distance L. The well is located in the region "1". The parameters of the second region are defined with a subscript "2". Mobility & storativity ratios
M=
F=
( k µ )1 ( k µ )2
(φ ct )1 (φ ct )2
(Eq. 18-1)
(Eq. 18-2)
Dimensionless variables The dimensionless variables (including the skin) are expressed with reference to the region "1" parameters.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk k1h
pD =
tD CD
141. 2 qB µ 1
= 0. 000295
CD =
S=
141. 2 qB µ 1
LD =
(Eq. 18-3)
k1h ∆t µ1 C
0.8936C (φ ct )1 hrw2
k1h
rD =
∆p
(Eq. 18-4)
(Eq. 18-5)
∆pSKIN
(Eq. 18-6)
r rw
(Eq. 18-7)
L rw
18.2 Radial composite behavior Influence of heterogeneous parameters M and F
(Eq. 18-8)
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk 1.0E+02
M = 10
pD & pD'
1.0E+01
M = 10 M=2
1.0E+00
M = 0.5
0.5 1.0E-01
M = 0.1 0.5 M
1.0E-02 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
tD/CD
Figure 18-2 : Radial composite responses, well with wellbore storage and skin, changing mobility and constant storativity. Log-log scale. The two dotted curves correspond to the closed and the constant pressure circle solutions. CD = 100, S = 3, rD = 700, M = 10, 2, 0.5, 0.1, F =1.
25.0
M = 10
20.0
M=2
15.0
pD
M = 0.5 slope m
10.0
M = 0.1 slopes m M
5.0
0.0 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
tD/CD
Figure 18-3 : Semi-log plot of Figure 18.2.
1.0E+04
1.0E+05
1.0E+06
2001
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
www.opc.co.uk 1.0E+02
F = 10 1.0E+01
pD & pD'
F = 0.1 F = 10 0.5
1.0E+00
0.5 F = 0.1
1.0E-01 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
tD/CD
Figure 18-4 : Radial composite responses, well with wellbore storage and skin, constant mobility and changing storativity. Log-log scale. CD = 100, S = 3, rD = 700, M = 1, and F =10, 2, 0.5, 0.1
15.0
F = 10
slopes m
slope m
10.0
pD
F = 0.1
5.0
0.0 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
tD/CD
Figure 18-5 : Semi-log plot of Figure 18.4.
Log-log analysis The permeability thickness product k1h of the inner region is estimated from the pressure match, and C from the time match : k1h = 1412 . qBµ 1 ( PM )
(Eq. 18-9)
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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C = 0.000295
k1h 1 µ 1 TM
(Eq. 18-10)
At early time, the homogeneous (CD e2S)1 curve defines the skin factor S. The mobility ratio M is estimated from the two derivative stabilizations. M=
∆p2nd stab. ∆p1st stab.
(Eq. 18-11)
Semi-log analysis The first semi-log straight line defines the mobility of the inner zone, and the wellbore skin factor S.
∆p = 162.6
qBµ 1 k1 log ∆t + log − 3.23 + 0.87 S 2 k1h (φµ ct )1 rw
(Eq. 18-12)
The second line, for the outer zone, defines M and the global skin SRC.
∆p = 162.6
qBµ 2 k2h
k2 log ∆t + log − 3.23 + 0.87 S RC 2 (φµ ct )2 rw
(Eq. 18-13)
The skin SRC includes two components : the wellbore skin factor S and a radial composite geometrical skin effect, function of the mobility ratio M and the radius rD of the circular interface : S RC =
1 1 S + − 1 ln rD M M
(Eq. 18-14)
When the mobility near the wellbore is higher than in the outer zone (M>1), the geometrical skin is negative.
Build-up analysis
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk 1.0E+02
build-up drawdown pD & pD'
1.0E+01
1.5 0.5
1.0E+00
1.0E-01 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
tD/CD
Figure 18-6 : Build-up radial composite response, well with wellbore storage and skin, changing mobility and constant storativity. Log-log scale. The dotted pressure and derivative curves correspond to the drawdown solution. CD = 11500, S = 5, rD = 2000, M = 3, F = 1.
With a strong reduction of mobility (M>>10), drawdown and build-up responses can show the behavior of a closed depleted system, before the influence of the outer region is seen. 1.0E+02
50
pD & pD'
build-up drawdown 1.0E+01
0.5
1.0E+00
tp
1.0E-01 1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
1.0E+07
tD/CD
Figure 18-7 : Drawdown and build-up responses for a well with wellbore storage and skin in a radial composite reservoir. The dotted pressure and derivative curves correspond to the drawdown solution. CD = 1000, Sw = 0, rD = 10000, M =100, F =1 and tp/CD=3200.
2001
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
www.opc.co.uk 18.3 Linear composite behavior Influence of heterogeneous parameters M and F The second homogeneous behavior is defined with the average properties of the two regions : 1 k k = 0 . 5 1 + µ APPARENT M µ 1
(Eq. 18-15)
1.0E+02
M = 10
pD & pD'
1.0E+01
M = 10 0.5
1.0E+00
1.0E-01
M = 0.1 1.0E-02 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
tD/CD
Figure 18-8 : Linear composite responses, well with wellbore storage and skin, changing mobility and constant storativity. Log-log scale. The two dotted curves correspond to the sealing fault and the constant pressure fault solutions. CD = 100, S = 3, LD = 700, M = 10, 2, 0.5, 0.1, F = 1.
2001
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
www.opc.co.uk 15.0
M = 10
slope m
M = 0.1
pD
10.0
5.0
0.0 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
tD/CD
Figure 18-9 : Semi-log plot of Figure 18-8. Log-log analysis The two derivative stabilizations are used to estimate the mobility ratio M : M=
∆p2nd stab. 2 ∆p1st stab. − ∆p2nd stab.
(Eq. 18-16)
1.0E+02
Radial
pD & pD'
1.0E+01
Linear
Radial
1.0E+00
Linear
0.5 1.0E-01 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
tD/CD
Figure 18-10 : Comparison of radial and linear interfaces. Well with wellbore storage and skin in composite reservoirs. Log-log scale. CD = 200, S = 0 Linear composite : M = 5, F = 1, LD = 300 Radial composite : M = 1.667, F = 1, rD = 300
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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18.4 Multicomposite systems Three inner regions with abrupt change of mobility 1.0E+01
pD & pD'
1.0E+00
rD= 1000, M=0.1 rD= 2500, M=0.15 rD=50000, M= 0.5
0.5 033 rD= 1000, M=0.1
0.1 1.0E-01
0.05
rD= 2500, M=0.15 rD=50000, M= 0.5
1.0E-02 1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
tD/CD
Figure 18-11 Pressure and derivative responses for a well with wellbore storage and skin in a 4 regions radial composite reservoir. CD = 5440, Sw = 0, F =1. r1D = 1000, k/µ2 = 1.5 k/µ1, r2D = 2500, k/µ3 = 5 k/µ1, r3D = 50,000, k/µ4 = 10 k/µ1. The dashed curves correspond to radial composite responses with only one zone (rD = 1000, M = 0.1, rD = 2500, M = 0.15, rD = 50,000, M = 0.5).
Two inner regions with a linear change of mobility
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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pD & pD'
1.0E+01
1.0E+00
0.5 rD=10000 rD=1000
1.0E-01
1.0E-02 1.0E+00
1.0E+01
1.0E+02
0.05
1.0E+03
1.0E+04
1.0E+05
1.0E+06
1.0E+07
tD/CD
Figure 18-12 Pressure and derivative responses for a well with wellbore storage and skin in a radial composite reservoir, linear change of transmissivity. CD = 1000, Sw = 0, F =1. From r1D = 1000 to r2D = 10,000, M decreases linearly from 1 to 0.1. The dashed curves correspond to radial composite responses (M=0.1, rD = 1000, rD = 10,000).
19. PARTIAL PENETRATION THEORY 19.1 Definition
h
hw zw
Figure 19-1 : Well in partial penetration. Geometry of the flow lines. hw : open interval of thickness zw : distance of the center of the open interval to the lower reservoir boundary kH : horizontal permeability kV : vertical permeability
19.2 Characteristic flow regimes
2001
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
www.opc.co.uk 1. Radial flow over the open interval : first derivative stabilization (at 0.5 h/hw) and first semi-log straight line. Results : permeability-thickness product for the open interval, kHhw, and skin of the well, Sw. 2. Spherical flow : -1/2 slope derivative straight line. Results : permeability anisotropy kH/kV and location of the open interval in the reservoir thickness. 3. Radial flow over the entire reservoir thickness : second derivative stabilization and second semi-log straight line. Results : permeability-thickness product for the total reservoir, kHh, and the total skin of the well, St.
The total skin combines the wellbore skin Sw and an additional geometrical skin Spp due to partial penetration : • Spp is large when the penetration ratio hw/h or the vertical permeability kV are low (high anisotropy kH/kV). • For damaged well, the product (h/hw)Sw can be larger than 100. St =
h S w + S pp hw
(Eq. 19- 1)
A skin above 30 or 50 is indicative of a partial penetration effect. 19.3 Log-log analysis 1.0E+02
c b a pD & pD'
1.0E+01
slope -1/2 first stabilization 1.0E+00
0.5 kH/kV = 10 (a),
1.0E-01 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
100 (b), 1.0E+04
1000 (c) 1.0E+05
1.0E+06
tD/CD
Figure 19-2 : Responses for a well in partial penetration with wellbore storage and skin. Log-log scale. hw/h = 1/5, CD = 33, S=0, kH / kV = 10 (curve a), 100 (curve b) and 1000 (curve c).
2001
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The kHh product is estimated from the pressure match. The wellbore skin Sw and the penetration ratio hw/h are estimated from the first radial flow (derivative plateau at 0.5 h/hw): hw ∆p2nd stab. m2nd line = = ∆p1st stab. h m1st line
(Eq. 19- 2)
When the vertical permeability kV is low (high kH/kV), the start of the spherical flow regime is delayed (-1/2 derivative slope moved to the right). The permeability anisotropy kH/kV and location of the open interval are estimated from the spherical flow -1/2 slope match.
19.4 Semi-log analysis 40.0
c
slope m
b
30.0
pD
a Spp 20.0
10.0
0.0 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
tD/CD
Figure 19-3 : Semi-log plot of Figure 19.2. Influence of kH / kV on Spp (Sw=0).
The final semi-log straight line defines kHh and St. When a first semi-log straight line is seen (radial flow over the open interval), it defines the permeability-thickness kHhw (penetration ratio hw/h with Eq. 19-2), and the wellbore skin Sw.
19.5 Geometrical skin Spp When the penetration ratio hw h and the dimensionless reservoir thickness-anisotropy group (h rw ) k H k V are not very small, Spp can be expressed :
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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h π h S pp = − 1 ln hw 2 rw
hw kH h h + ln h k V hw 2+ w h
( z + hw 4)(h − z + hw 4) ( z − hw 4)(h − z − hw 4)
(Eq. 19- 3)
With hw h = 0.1 and kH/kV = 1000, Spp = 65 whereas with hw h = 0.5 and kH/kV = 10, Spp = 6 only.
19.6 Spherical flow analysis Plot of ∆p versus 1 ∆t . The straight line is frequently not well defined and the analysis is difficult : on example (c), the spherical flow regime is established between tD/CD=104 and 106. The straight line is very compressed, it ends before 1
∆t =0.01.
35.0
c 30.0
pD
b
25.0
a
20.0 0
0.01
0.02
0.03
0.04
0.05
1/SQRT(tD/CD)
Figure 19-4 : Spherical flow analysis of responses Figure 19.2. One over square root of time plot.
When the open interval is in the middle of the formation, the slope mSPH of the spherical flow straight line:
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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∆p = 70. 6
qB µ k S rS
− 2452. 9
qB µ φµ ct k S3 2 ∆t
(Eq. 19- 4)
The spherical permeability is expressed as : φµ ct k s = 2452.9qBµ mSPH
23
(Eq. 19- 5)
The permeability anisotropy is: kH kH = kV k s
3
(Eq. 19- 6)
If the open interval is close to the top or bottom sealing boundary, flow is semi-spherical and the slope mSPH must be divided by two in Equation 19-4.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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20. HYDRAULIC FRACTURE THEORY 20.1 Infinite conductivity or uniform flux vertical fracture Characteristic flow regimes 1. Linear flow: 1/2 slope straight line. Results : fracture half length xf. 2. Pseudo radial flow : derivative stabilization at 0.5. Results : permeability-thickness product kh and the geometrical skin S. Log-log analysis Dimensionless terms : t Df =
0. 000264 k
CDf =
φµct x 2f
∆t
(Eq. 20-1)
0.8936C
(Eq. 20-2)
φ ct hx 2f
1.0E+01
b a pD & pD'
1.0E+00
0.5 line
b slope 1/2
a
1.0E-01
1.0E-02 1.0E-04
1.0E-03
1.0E-02
1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
tDf
Figure 20-1 : Responses for a well intercepting a high conductivity fracture. Loglog scale. Infinite conductivity (curve a) and uniform flux (curve b). No wellbore storage effect CD = 0.
2001
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On Figure 20-1, CD = 0. The two models are slightly different during the transition between linear flow and radial flow. With the uniform flux model, the transition is shorter and the pressure curve is higher. Match results :
xf =
0.000264 k φµ ct (TM )
(Eq. 20-3)
With the infinite conductivity fracture, the geometrical skin effect is defined as : x f = 2 rw e − S
(Eq. 20-4)
And, for the uniform flux solution, x f = 2. 7 rw e − S
(Eq. 20-5)
Linear flow analysis 1.40
b
mLF
1.20
a 1.00
pD
0.80 0.60 0.40 0.20 0.00 0.0E+00
2.0E-01
4.0E-01
6.0E-01
8.0E-01
SQRT(tDf)
Figure 20-2 : Square root of time plot of Figure 20-1. Early time analysis.
1.0E+00
2001
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
www.opc.co.uk Plot of the pressure versus the square root of time : the response follows a straight line intercepting the origin. The slope mLF is :
∆p = 4. 06
qB
µ
hx f
φ ct k
∆t
(Eq. 20-6)
The half fracture length xf is estimated from the slope :
x f = 4.06
µ qB φ ct k hmLF
(Eq. 20-7)
Fractured well with wellbore storage 1.0E+01
pD & pD'
1.0E+00
1.0E-01
CD = 0 1.0E-02
1000 1.0E-03 1.0E-04
10000
1.0E-03
1.0E-02
1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
tDf
Figure 20-3 : Responses for a fractured well with wellbore storage Infinite conductivity fracture. Log-log scale. CD = 0, 103, 104.
20.2 Finite conductivity vertical fracture Characteristic flow regimes
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk 1. Bi-linear flow : 1/4 slope straight line. Results : fracture conductivity kfw. 2. Linear flow : 1/2 slope straight line. Results : fracture half length xf. 3. Pseudo radial flow : derivative stabilization at 0.5. Results : permeability-thickness product kh and the geometrical skin S. Log-log analysis The dimensionless fracture conductivity kfDwfD is defined as : k fD w fD =
k f wf
(Eq. 20-8)
kx f
For large fracture conductivity kfDwD, the bilinear flow regime is short lived and the 1/4 slope pressure and derivative straight lines are moved downwards. The behavior tends to a high conductivity fracture response.
1.0E+03
slope 1/2
1.0E+02
pD & pD'
0.5 line 1.0E+01
slope 1/4
1.0E+00
1.0E-01
1.0E-02 1.0E-04
1.0E-02
1.0E+00
1.0E+02
1.0E+04
1.0E+06
tDf
Figure 20-4 : Response for a well intercepting a finite conductivity fracture. Log-log scale. No wellbore storage effect CDf = 0, kfDwfD = 100.
The kh product is estimated from the pressure match and the fracture half length xf from the time match (Eq. 20-1). The fracture conductivity kfw is estimated from the match on the bilinear flow 1/4 slope.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk Bi-linear flow analysis On a plot of the pressure versus the fourth root of time, the straight line intercepts the origin. The slope mBLF is : ∆p = 44.11
qB µ
4
h k f w 4 φµ ct k
∆t
(Eq. 20-9)
The fracture conductivity kfwf is estimated with 1 qBµ k f w = 1944.8 φµ ct k hmBLF
2
(Eq. 20-10)
21. HORIZONTAL WELL THEORY 21.1 Definition kV
kH kH h
L
L
zw
Figure 21-1 : Horizontal well geometry. L : effective half length of the horizontal well zw : distance between the drain hole and the bottom sealing boundary kH : horizontal permeability kV : vertical permeability
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk 21.2 Characteristic flow regimes
Vertical radial flow
Linear flow
Horizontal radial flow
Figure 21-2 : Horizontal well flow regimes. 1. Vertical radial flow : a first derivative plateau at 0.25(h L) k H kV . Results : the permeability anisotropy kH/kV and the wellbore skin Sw or the vertical radial flow total skin STV. 2. Linear flow between the upper and lower boundaries : 1/2 slope derivative straight line. Results : effective half length L and well location zw of the horizontal drain. 3. Radial flow over the entire reservoir thickness : second derivative stabilization at 0.5. Results : reservoir permeability-thickness product kHh, and the total skin STH.
21.3 Log-log analysis 1.0E+01
1.0E+00
pD & pD'
0.5 line
1.0E-01
slope 1/2
first stabilization 1.0E-02 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
tD/CD
Figure 21-3 : Response for a horizontal well with wellbore storage and skin in a
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk reservoir with sealing upper and lower boundaries. Log-log scale. With long drain holes, the 1/2 derivative slope is moved to the right. When the vertical permeability is increased, the first derivative stabilization is moved down. The kHh product is estimated from the pressure match. The effective half length L and well location zw are estimated from the intermediate time 1/2 slope match. The vertical radial flow total skin STV and the permeability anisotropy kH/kV are estimated from the first radial flow in the vertical plane (permeability thickness 2 kV k H L and derivative plateau at 0.25(h L) k H kV ).
Influence of L The examples presented Figures 21-4 to 21-22 are generated with h = 100 ft and rw = 0.25 ft.
pD & p'D
1.0E+01
1.0E+00
1.0E-01 1.0E+00
L/h=2.5
1.0E+01
1.0E+02
1.0E+03
1.0E+04
5
1.0E+05
10
1.0E+06
tD/CD
Figure 21-4 : Influence of L on pressure and derivative log-log curves. SQRT (kV kH)*L constant, (∆p1st stab.)D= 0.223. CD = 100, Sw=0, Lw/h=2.5, kV/kH=0.2; Lw/h=5, kV/kH=0.05; Lw/h=10, kV/kH=0.0125.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk 1.0E+02
pD & p'D
1.0E+01
1.0E+00
L/h=2.5 1.0E-01 1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
5
10
1.0E+05
1.0E+06
tD/CD
Figure 21-5 : Influence of L on pressure and derivative log-log curves. SQRT (kV kH)*L constant, (∆p1st stab.)D= 1. CD = 100, Sw=0, Lw/h=2.5, kV/kH=0.01; Lw/h=5, kV/kH=0.0025; Lw/h=10, kV/kH=0.000625.
Influence of zw 1.0E+01
pD & p'D
1.0E+00
1.0E-01
zw/h=0.125
1.0E-02 1.0E-01
1.0E+00
1.0E+01
0.25
1.0E+02
0.5
1.0E+03
1.0E+04
1.0E+05
tD/CD
Figure 21-6 : Influence of zw on pressure and derivative log-log curves. CD = 1000, Sw=2, Lw/h=15, kV/kH=0.02, zw/h=0.5, 0.25, 0.125.
21.4 Vertical radial flow semi-log analysis
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk ∆p =
kV k H ∆t 162.6 qBµ − 3.23 log 2 kV k H L φ µ ct rw2 k 1 k +0.87 S w − 2 log 4 V + 4 H kV 2 kH
(Eq. 21- 1)
The skin STV measured during the vertical radial flow is expressed with the wellbore skin Sw and the anisotropy skin Sani : S TV = S w + S ani = S w − ln
4 k k +4 k V H H kV
(Eq. 21- 2)
2
Sometimes, the vertical radial flow skin is expressed as S'TV, defined with reference to the equivalent fully penetrating vertical well : ' S TV =
h 2L
kH S TV kV
(Eq. 21- 3)
3.0
pD
2.0
1.0
horizontal radial flow vertical radial flow
0.0 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
tD/CD
Figure 21-7 : Semi-log plot of Figure 21-3.
21.5 Linear flow analysis
Linear flow analysis
1.0E+04
1.0E+05
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk ∆p =
8128 . qB 1412 . qBµ 1412 . qBµ µ ∆t Sw + Sz + kH h 2Lh φ ct k H 2 kV k H L
(Eq. 21- 4)
During the linear flow regime, the flow lines are distorted vertically before reaching the horizontal well, producing a partial penetration skin Sz. S z = −1151 .
π r kH h log w 1 + kV L h
kV π z w sin k H h
(Eq. 21- 5)
21.6 Horizontal pseudo-radial flow semi-log analysis ∆p = 162.6
k ∆t qBµ log H 2 − 3.23 + 0.87 S TH k H h φ µ ct rw
(Eq. 21- 6)
STH measured during the horizontal radial flow combines S'TV of Equation 3.11 and the geometrical skin SG of the horizontal well (function of the logarithm of the well effective length and a partial penetration skin SzT , close to the linear flow skin Sz ) : S TH =
h 2L
kH S w + SG kV
(Eq. 21- 7)
L + S zT rw
(Eq. 21- 8)
S G = 0.81 − ln
S zT = 1151 .
π r kH h k π z log w 1 + V sin w kV L k H h h 2 k H h 2 1 zw zw − 0.5 − + kV L2 3 h h 2
(Eq. 21- 9)
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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4.0
zw/h=0.125 0.25 0.5
2.0
slope mVRF slopes mHRF 1.0
0.0 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
tD/CD
Figure 21-8 : Semi-log plot of Figure 21-6.
2 kV/kH = 1
kV/kH = 0.1
kV/kH = 0.01
kV/kH = 0.001
0
-2
S
pD
3.0
-4 kV/kH = infinity -6 zw/h = 0.5 _____ zw/h = 0.1 - - - - -8 h/rw=1000
-10 1.0E+02
1.0E+03
1.0E+04
1.0E+05
L/rw
Figure 21-9 : Semi-log plot of the geometrical skin SG versus L/rw. Influence of kV/kH. h/rw =1000, zw/h=0.5, 0.1.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk 2 h/rw = 2000
h/rw = 4000
h/rw = 1000 0
h/rw = 500
S
-2
-4 kV/kH = infinity -6 zW /h = 0.5 _____ zW /h = 0.1 - - - - -8 kV/kH=0.1
-10 1.0E+02
1.0E+03
1.0E+04
1.0E+05
L/rw
Figure 21-10 : Semi-log plot of the geometrical skin SG versus L/rw. Influence of h/rw. kV/kH =0.1, zw/h=0.5, 0.1.
21.7 Discussion of the horizontal well model Several well conditions can produce a pressure gradient in the reservoir, parallel to the wellbore. The vertical radial flow regime is then distorted, and the derivative response deviates from the usual stabilization at 0.25(h L) k H kV ). During horizontal radial flow, the geometrical skin can be larger or smaller than SG of Equation 21-8 and 21-9.
Non-uniform mechanical skin
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk 1.0E+01
pD & p'D
1.0E+00
Sw i = 4,
1.0E-01
4, 4, 4
8, 5.3, 2.6, 0 0, 8,
8, 0
8, 0,
0, 8
1.0E-02 1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
tD/CD
Figure 21-11 : Influence of non-uniform skin on pressure and derivative curves. CD = 100, L =1000 ft, h =100 ft, rw =0.25 ft, zw/h =0.5, kV/kH=0.1. The well is divided in 4 segments of 500 ft with skins of Swi=4, 4, 4, 4 (uniform damage), Swi=8, 5.33, 2.66, 0 (skin decreasing along the well length), Swi=0, 8, 8, 0 (damage in the central section), Swi=8, 0, 0, 8 (damage at the two ends).
The two ends of the well are more sensitive to skin damage.
Finite conductivity horizontal well When the pressure gradients in the wellbore are comparable to pressure gradients in the reservoir, the flow is three-dimensional and the derivative is displaced upwards during the early time response. During horizontal radial flow, the total skin STH is less negative.
Partially open horizontal well When only some sections of the well are opened to flow, the response first corresponds to a horizontal well with the total length of the producing segments. Later, each segment acts like a horizontal well, and several horizontal radial flow regimes are established until interference effects between the producing sections are felt. Then, the final horizontal radial flow regime is reached for the complete drain hole. The more distributed the producing sections, the more negative the total skin STH.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk 1.0E+01
pD & p'D
1.0E+00
0.5 0.5/2 0.5/4
1.0E-01
w ell
c urv e
1.0E-02 1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
1.0E+07
tD/CD
Figure 21-12 : Influence of number of open segments on pressure and derivative log-log curves. Total half-length 2000 ft, effective half-length 500 ft. CD = 100, Leff =500 ft, h =100 ft, rw =0.25 ft, zw/h =0.5, kV/kH=0.1. 1, 2 and 4 segments, Swi=0.
When the producing segments are uniformly distributed along the drain hole, the total skin STH can be very negative even with a low penetration ratio. On the examples Figure 21-13, with penetration ratios of 100, 50, 25 and 12.5%, STH is respectively –7.9, -7.4, -6.6 and –5.1.
pD & p'D
1.0E+01
1.0E+00
12.5%
1.0E-01
25% 50% 100% 1.0E-02 1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
1.0E+07
tD/CD
Figure 21-13 : Influence of the penetration ratio on pressure and derivative log-log curves. Four segments equally spaced. Total half-length 2000 ft, penetration ratio 12.5, 25, 50 and 100%. CD = 100, Leff =250, 500, 1000 and 2000 ft, Swi=0, h =100 ft, rw =0.25 ft, zw/h=0.5, kV/kH=0.1.
Non-rectilinear horizontal well
2001
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
www.opc.co.uk During the vertical radial flow, the upper and lower sealing boundaries can be reached at different times when the well is not strictly horizontal. The transition between vertical radial flow and linear flow is then distorted. 1.0E+01
pD & p'D
1.0E+00
1.0E-01
1.0E-02 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
1.0E+07
tD /C D
Figure 21-14 : Non-rectilinear horizontal wells. Pressure and derivative curves. CD = 100, L = 2000 ft (500+1000+500), Swi=0, h =100 ft, rw =0.25 ft, kV/kH=0.1, zw/h =0.95, 0.5, 0.95 and 0.5, 0.95, 0.5 (average 0.725).
Changes in vertical permeability On Figure 21-15, a thin reduced permeability interval is introduced in the main layer. When a homogeneous layer of total thickness is used for analysis, the vertical permeability overestimated and effective well length is too small. 1.0E+01
thre e la ye rs
pD & p'D
1.0E+00
1.0E-01
o ne layer = h 1+h2 +h 3 h3 1.0E-02 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
1.0E+07
tD/CD
Figure 21-15 : Horizontal well in a reservoir 3 layers with crossflow. Pressure and derivative log-log curves. CD = 100, L = 1000 ft, Sw=0, h =100 ft (h1=45ft, h2=5ft, h3=50ft), k1=k3=100k2, rw =0.25 ft, (kV/kH)i=0.1, zw/h = 0.25 (well centered in h3).
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk One layer (h1+h2+h3) : k= (k1h1+ k2h2+ k3h3) / (h1+h2+h3), L = 550 ft, Sw=-0.2, kV/kH=0.4, zw/h = 0. 5 (well centered in h1+h2+h3). One layer (h3) : k= k3, L = 1000 ft, Sw=0, kV/kH=0.1, zw/h = 0. 5 (well centered in h3).
Anisotropic horizontal permeability In anisotropic reservoirs, horizontal well responses are also sentitive to the well orientation.
kz
ky kz ky 2L
kx ky L2
kx ky h
Figure 21-16 : Effective permeability during the three characteristic flow regimes towards a horizontal well.
The final horizontal radial flow regime defines the average horizontal permeability k H = k x k y . During the linear flow regime, only the permeability ky normal the the well orientation is acting. At early time, the average permeability during the vertical radial flow is k z k y .
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk 1.0E+01
pD & pD'
1.0E+00
ky L
2
kx ky h
1.0E-01
kz ky 2L 1.0E-02 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
tD/CD
Figure 21-17 : Influence of the permeability anisotropy during the three characteristic flow regimes.
When the isotropic horizontal permeability model is used for analysis, the apparent effective half-length is : La = 4 k y k x L
(Eq. 21- 10)
(the vertical permeability kz is unchanged).
ky ky kx
kx
Figure 21-18 : Horizontal well normal to the maximum permeability direction : apparent effective length increased.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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ky ky kx
kx
Figure 21-19 : Horizontal well in the direction of maximum permeability : apparent effective length decreased.
Presence of a gas cap or bottom water drive When the constant pressure boundary is reached at the end of the vertical radial flow regime (or hemi radial in the examples Figure 21-20), the pressure stabilizes and the derivative drops. It the thickness of the gas zone is not large enough, the derivative stabilizes at late time to describe the total oil + gas mobility thickness. 1.0E+01
p D & p 'D
1.0E+00
no gas cap 1.0E-01
h g as = 20 ft h g as
1.0E-02
10 0 ft
h o il 1.0E-03 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
tD /C D
Figure 21-20 : Horizontal well in a reservoir with gas cap and sealing bottom boundary. Pressure and derivative log-log curves. CD = 100, L = 1000 ft, Sw=2, h =100 ft, rw =0.25 ft, (kV/kH)=0.1, zw/h = 0.2 (well close to the bottom boundary). Gas cap : hgas= 0.20, 1.0, 5.0 h, µgas=0.01 µoil, ct gas=10 ct oil.
21.8 Other horizontal well models
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk Multilateral horizontal well As for partially penetrating horizontal wells, the different branches of multilateral wells start to produce independently until interference effects between the branches distort the response. At later time, pseudo radial flow towards the multilateral horizontal well develops.
pD & p'D
1.0E+01
1.0E+00
tw o branc hes
1.0E-01
one branc h
f our branc hes
1.0E-02 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
1.0E+07
tD/CD
Figure 21-21 : Multilateral horizontal wells. Pressure and derivative curves. CD = 100, L = 1000 ft (500+500 or 250+250+250+250), Swi=0, h =100 ft, rw=0.25 ft, kV/kH=0.1, zw/h = 0.5.
In the case of intersecting multilateral horizontal wells, increasing the number of branches does not improve the productivity. With the radial symmetric examples of Figure 21-21, total skin STH of the horizontal well is STH =-6.8 (one branch) and respectively –6.6 and –6.2 with two and four branches.
When the distance between the two producing segments is large enough, the response becomes independent of the orientation of the branches. The responses Figure 21-22 tend to be equivalent to the example with two segments of Figure 21-12. The total skin STH is more negative when the distance between the branches is increased. For the two multilateral horizontal wells of Figure 21-22, STH =-7.1 (and STH =-6.8 with one branch).
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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1.0E+01
pD & p'D
: one branch 1.0E+00
1.0E-01 : tw o branches at 90°
: tw o branches parallel 1.0E-02 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
1.0E+07
tD/CD
Figure 21-22 Multilateral horizontal wells. Pressure and derivative curves. CD = 100, L = 1000 ft (500+500), Swi=0, h =100 ft, rw=0.25 ft, kV/kH=0.1, zw/h = 0.5. The distance between the 2 parallel branches is 2000ft, on the second example the intersection point is at 1000ft from the start of the 2 segments.
Fractured horizontal well Two configurations are considered : longitudinal and transverse fractures. At early time, the different fractures produce independently until interference effects are felt. With longitudinal fractures, bi-linear and linear flow regimes can be observed, possibly followed by horizontal radial flow around the different fractures. For a single fracture of half-length yf, the slope mBLF and mLF are expressed : mBLF = 44.11
mLF = 4.06
yf
qB h yf
qBµ k f w 4 φ µ ct k H
( 21-11)
µ k H φ ct
( 21-12)
With transverse fractures, the flow is first linear in the formation and radial in the fracture, it changes into linear flow, and later into the horizontal radial flow regime around the fracture segments. The radial linear flow regime yields a semi-log straight line whose slope is function of the fracture conductivity. For a single transverse fracture of radius rf, the slope mRLF and mLF are:
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk mRLF = 813 .
mLF = 517 .
qBµ kk w
qB h rf
( 21-13)
µ k H φ ct
( 21-14)
Once the interference effect between the different fractures is fully developed, the final pseudo radial flow regime towards the fractured horizontal well establishes. As for partially open horizontal wells, the time of start of the final regime is a function of the distance between the outermost fractures.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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22. SKIN FACTORS 22.1 The different skin factors Name Sw SG Sani SRC S2φ D.q
Description Infinitesimal skin at the wellbore.
Type Positive or negative Geometrical skin due to the stream line curvature (fractured, partial Positive or penetration, slanted or horizontal wells). negative Skin factor due to the anisotropy of the reservoir permeability. Negative Skin factor due to a change of reservoir mobility near the wellbore Positive or (permeability or fluid property, radial composite behavior). negative Skin factor due to the fissures in a double porosity reservoir. Negative Turbulent or inertial effects on gas wells. Positive
22.2 Geometrical skin Well A
Well B
Well C
Figure 22-1 Configuration of wells A, B and C. A = fully penetrating vertical well, B = well in partial penetration, C = horizontal well.
2001
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
www.opc.co.uk 1.0E+02
B S>0
1.0E+01
pD & pD'
S<0 C
S>0 1.0E+00
B S<0
C
1.0E+03
1.0E+04
1.0E-01 A : vertical well B : partial penetration C : horizontal well 1.0E-02 1.0E-02
1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+05
1.0E+06
tD/CD
Figure 22-2 Pressure and derivative response of wells A, B and C. Log-log scale. 30.0
B
pD & pD'
A : vertical well B : partial penetration C : horizontal well 20.0
S>0
10.0
A S<0 C
0.0 1.0E-02
1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
tD/CD
Figure 22-3 Semi-log plot ofFigure 22-2 examples.
22.3 Anisotropy pseudo-skin An equivalent transformed isotropic reservoir model of average radial permeability is used, by a transformation of variables in the two main directions of permeability kmax and kmin. With k = k max k min
(Eq. 22- 1)
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk x' = x
k = x 4 min k max k max
(Eq. 22- 2)
y' = y
k = y 4 max k min k min
(Eq. 22- 3)
k
k
The wellbore is changed into an ellipse whose area is the same as in the original system, but the perimeter is increased. The elliptical well behaves like a cylindrical hole whose apparent radius is the average of the major and minor axes, and produces an apparent negative skin : rwa =
[
1 rw 4 k min k max + 4 k max k min 2
S ani = − ln
]
(Eq. 22- 4)
4k 4 min k max + k max k min
2
+ k max k = − ln min 2 k
(Eq. 22- 5)
Sani is in general low but, for horizontal wells, when kV/kH <<1, Sani =-1 may be observed.
23. PERFORMING A TEST DESIGN 23.1 Introduction A test design is the examination of one or more theoretical pressure responses in the reservoir given (or by estimating) all other parameters. In order to obtain the best results, optimum data and achieve objectives it is important to carry out a test design before performing the actual test. Indeed it can also determine if the original objectives can actually be achieved. In an ideal world this should be carried out by the engineer who will actually supervise the test, or have a direct influence on it, since it also gives the engineer an idea of what the optimum operating conditions will be to achieve the objectives. Consideration should also be given to the hardware to be used to ensure that safe operating conditions will be encountered. This is relatively simple since in many parts of the world the operating conditions can easily be estimated from local wells drilled in the area and the equipment is a choice of two or maybe three options off the shelf. In some exceptional circumstances, where the test is planned to be unconventional, equipment may have to be designed specifically for the test. There is also the issue of obtaining approval from
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk relevant authorities to carry out a test. Enough time should be allowed to gather the information required and submit this to the authorities to ensure that the approval will be forthcoming in a timely manner. 23.2 Hardware Equipment for well testing usually comes as standard. More details of the equipment can be found in section 26. However, the most important considerations are the pressure and temperature to be encountered, with temperature nudging ahead of pressure in order of importance, from the reservoir to the burners. This information can be obtained from offset wells ahead of drilling the well and from the drilling and electric logs while the well is being drilled. The logging stage is sometimes too late to change the equipment so if in doubt, select the higher rated equipment. To date the majority of wells have surface shut in well head pressures of less than 10,000 psi but this may change in the future. The reason the temperature is more important to know and consider than the pressure is that the properties of materials, particularly metals, change more significantly with temperature than with pressure. This issue becomes particularly significant in high pressure high/temperature wells and there are guidelines issued by the Institute of Petroleum called Well Control during the Drilling and Testing of High Pressure Offshore Wells, Model code of safe practice, Part 17 which addresses these type of tests (and the drilling of the wells) in more detail. Consideration should also be given to the tubing properties and the annular fluid when using annular pressure operated tools since it is possible to collapse the tubing downhole when applying surface annular pressure. Also consider the materials’ properties at reservoir temperatures.
23.3 Gauges The gauges measure the downhole pressure (the output response of the system) and are therefore of the utmost importance. We have established that it is best to observe the pressure during a period when the well is not flowing and some of the analysis is carried out using the change in pressure from the time of shut in. Depending on the reservoir characteristics, the change in pressure can be very small after a long or short period of shut in. So small in fact that the change in pressure cannot be measured by even the best of gauges. It is therefore vital that a test design is performed, to examine the pressure change at the end of the build up to ensure that gauges are chosen that can measure the response. If not, the following options exist to resolve the problem;
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk a. Flow the well for longer b. Create a bigger drawdown c. Obtain a better gauge As a guide, 0.1 psi per hour will be measured by a quartz gauge but this will change with time as the technology improves. Consult with the gauge provider for gauge characteristics. And finally on the subject of gauges, put them as close as possible to the perforations/reservoir to minimise the risk of phase segregation or pressure gradient changes and use several for redundancy. 23.4 Pressure Response During any flow period, well bore storage must have finished before any pressure data can be used for reservoir analysis. So this must be the first check; that well bore storage is over. As stated in section 9, this can be said to be when the sand face flow rate becomes less than 1% different to the surface flow rate. The plots below shown in Figure 23-1, show two build ups, one with a long build up for analysis and the other with a build up that limits the amount of information that can be gained from any analysis. In the lower plot, it can be seen that some information can be derived from the analysis. 1996/01/02-0000 : OIL
10 -1 10 -2 10 -3
UNIT SLP
10 -4
DP + DERIVATIVE (PSI/STB/D)
ENDWBS
10 -3
10-2
10 -1
10 0
101
DT (HR)
End of Well Bore Storage not reached
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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1996/01/04-0000 : OIL
10 0 10 -1
STABIL
10 -2
DP + DERIVATIVE (PSI/STB/D)
ENDWBS
10 -3
UNIT SLP
10 -2
10-1
10 0
10 1
102
DT (HR)
End of WBS Reached
Figure 23-1 Considering Well Bore Storage
Another consideration for the test design is to ensure that the flow and build up will be long enough to determine the existence of one or more boundaries or for determination of hydrocarbons in place. As has been explained in previous sections of this document, a single linear boundary manifests itself in the derivative as an upturn and eventual flattening of the derivative with the value of the second “stabilisation” being twice the first. To examine hydrocarbons in place, a known volume is extracted while measuring a corresponding drop in pressure which must be measurable with the gauge to be employed. These checks are performed in a similar way to checking the end of well bore storage. To summarise; 1. Check the gauge can read the pressure change at the end of the build up period. 2. Ensure well bore storage ends well before the data does. 3. Ensure flow periods will be long enough to achieve objectives.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk 23.5 An Example Test design This example is illustrative of how a test can be planned (designed) to achieve objectives. In this example, which in reality may be considered a little tenuous but illustrates the procedure well, it is believed that there are some boundaries at a distance of approximately 300 feet. One of the objectives of the test is to determine if the boundaries are intersecting (pinched out) or parallel (a channel). The question which needs to be answered is how long must the test be to confirm which of the two possibilities actually exists? It will be assumed that the flow rate (oil) will be 2,500 BOPD (estimated from another well previously drilled into a similar reservoir) and the initial reservoir pressure is 5000 psi (from drilling and logging). The reservoir type assumed will be homogeneous with a vertical well drilled into it. The static data is as follows; Oil formation volume factor Viscosity (cp) Porosity (%) Water Saturation (%) Formation thickness, h (feet) Oil compressibility (1/psi) Water compressibility (1/psi) Formation compressibility (1/psi) Well bore radius (feet)
1.5 2.1 15 25 350 1e-5 1e-7 1e-6 0.354
The well and reservoir data needed for the test design is as follows; well. storage (Bbls/psi) 0.01 skin 0.0 permeability (mD) 100 One boundary (Feet) 300 Other boundary (Feet) 300 Angle of boundary intersection, θ 60 (in the case of intersecting boundaries only) (Degrees) Using the above parameters and a process of trial and error it is possible to determine optimum flow period durations in order to be able to distinguish one reservoir type from the other. As a starting point, a drawdown of 12 hours as a minimum can be used and increased accordingly to be sure of achieving objectives. For the purposes of a test design the build up can be very long and modified later to be of a practicable duration. A starting point for the duration of the build up could be 150 hours or more. After creating the theoretical pressure response for the above conditions and parameters (24 hour flow and 150 hour build up) the build up response is shown below in Figure 23-2.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk Examination of the plot reveals that the build up should be in the order of 50 - 60 hours to be sure of determining which reservoir response is occurring. However, examination of the digital pressure data shows that the pressure change at the end of the build up is less than the recommended 0.1 psi per hour and therefore liable to inaccuracies in the gauge measurement. Increasing the drawdown to 48 hours resolves this problem and the derivative plot appears to be virtually unchanged, shown in Figure 23-3. This process can be adapted to any reservoir investigations including the end of well bore storage, the time to reach “stabilisation” and distances to boundaries. The difference in the two responses is clear. For a reservoir with parallel boundaries (channel) the later time response on the derivative is a positive half slope which is indicative of linear flow. This will continue until such time as linear flow is interrupted. In the case of intersecting boundaries, the derivative will eventually display a second stabilisation (a flattening of the derivative) which will occur at 2π/θ times the value of the first stabilisation. (refer to chapter 16 - Boundary theory) The build up needs to be long enough to determine which response is occurring.
10 -2
Intersecting Boundaries
Parallell Boundaries 10 -3
DP + DERIVATIVE (PSI/STB/D)
1998/01/02-1200 : OIL
10 -2
10 -1
10 0
10 1
10 2
DT (HR)
Figure 23-2: Derivative plot: 12 hour flow,150 hour build up
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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10 -2
Intersecting boundaries
Parallell boundaries 10 -3
DP + DERIVATIVE (PSI/STB/D)
1998/01/03-1200 : OIL
10 -2
10 -1
10 0
10 1 DT (HR)
Figure 23-3: Derivative plot: 48 hour flow, 150 hour build up
10 2
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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24. GAS WELL TESTING Two different types of test are used with gas wells. Transient analysis provides a description of the producing system, as for oil wells. With deliverability testing, the theoretical rate at which the well would flow if the sandface was at atmospheric pressure, "the Absolute Open Flow Potential" AOFP, is estimated. 24.1 Gas properties Gas compressibility and viscosity The viscosity µ and the compressibility of gas cg change with the pressure. cg =
1 p
−
1 ∂Z
( 24-1)
Z ∂p
Z is the real gas deviation factor. For an ideal gas Z=1, and the compressibility is cg=1/p. Pseudo-pressure The pseudo-pressure m(p), also called "real gas potential", is defined : m( p) = 2
p
p
∫ µ( p)Z ( p) dp
( 24-2)
p0
The pressure p is expressed in absolute unit, m(p) has the unit of (pressure)2 / viscosity, psia2 / cp with the usual system of units. The reference pressure p0 is an arbitrary constant, smaller than the lower test pressure. The complete pressure data is converted into pseudo-pressure m(p) before analysis. The change of pseudo-pressure, expressed as m(p)-m(p[∆t=0]), is independent of the reference pressure p0. Pseudo-time The pseudo-time tps is sometimes used as a complement of m(p).
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk t
t ps = ∫ 0
1 dt µ ( p)ct ( p)
( 24-3)
The pressure must be known during the complete flow rate sequence in order to estimate µ and ct before calculation of the superposition with the pseudo time tps.
24.2 Transient analysis of gas well tests Simplified pseudo-pressure for manual analysis On Figure 24-1, µZ is plotted versus p for a typical natural gas at constant temperature : - When the pressure is less than 2000 psia, the product µZ is almost constant and m(p) simplifies into : m( p) =
2 µZ
p
∫ p0
pdp =
p 2 − p02 µ i Zi
( 24-4)
On low pressure gas wells, it is possible to analyze the test in terms of pressure squared p2. - When the pressure is higher than 3000 psia, the product µZ tends to be proportional to p and p/µZ can be considered as a constant. The pseudo-pressure m(p) becomes : p
2p 2 pi m( p) = dp = ( p − p0 ) ∫ µ Z p0 µ i Zi
( 24-5)
On high pressure wells, the gas behaves like a slightly compressible fluid, and the pressure data can be used directly for analysis. - Between 2000 psia and 3000 psia, no simplification is available and m(p) must be used.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk 0.04 0.04 0.03
Mu*Z, cp
0.03 0.02 0.02 0.01 0.01 0.00 0
1000
2000
3000
4000
5000
6000
7000
8000
Pressure, psia
Figure 24-1 : Isothermal variation of µZ with pressure.
Dimensionless parameters Nomenclature In field units, the standard pressure is psc =14.7psia and the temperature is Tsc = 520°R (60°F, all temperatures are expressed in absolute units). The gas rate is expressed in standard condition as qsc in Mscf/D (103scft/D ). When the pseudo-pressure is used, the dimensionless terms are defined with respect to the gas properties at initial condition (subscript i). With the pressure and pressure squared approaches, the properties are defined at the arithmetic average pressure of the test (symbol -). Dimensionless pressure m(p): kh Tsc ( m( pi ) − m( p)) Tq sc psc kh = 7.03∗10−4 ( m( pi ) − m( p)) Tq sc
pD = 1987 . ∗10−5
( 24-6)
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk p2:
(
kh Tsc 2 pi − p 2 µ ZTq sc psc kh pi2 − p 2 = 7.03∗10−4 µ ZTq sc
pD = 1987 . ∗10−5
(
)
)
( 24-7)
p: pD = 3.976∗10 −5
kh p Tsc ( pi − p) µ ZTq sc psc
( 24-8)
kh p = 1406 . ∗10 ( pi − p) µ ZTq sc −3
Dimensionless time m(p): tD =
0. 000263k
tD =
0. 000263k
φµi cti rw
2
∆t
( 24-9)
∆t
( 24-10)
p2 and p: φµ ct rw
2
Dimensionless wellbore storage As for oil wells, the wellbore storage coefficient is expressed in Bbl/psi. m(p): CD =
0.8936C
CD =
0.8936C
φ cti hrw2
( 24-11)
p2 and p: φ ct hrw2
( 24-12)
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk Dimensionless time group tD/CD m(p): tD CD
= 0. 000295
kh ∆t
( 24-13)
µi C
p2 and p: tD CD
= 0. 000295
kh ∆t
( 24-14)
µ C
Skin On gas wells, the skin coefficient S' is expressed with a rate dependent term, also called turbulent effect or non-Darcy skin. S ' = S + Dq sc
( 24-15)
In a multirate sequence, the analysis is made with respect to the rate change (qn - qn-1), and the skin is estimated from the change of ∆pskin between the flow periods n and n-1. S' is expressed :
S' =
q n ( S + Dq n ) − q n −1 ( S + Dq n −1 ) q n − q n −1
= S + D(q n + q n −1 )
( 24-16)
During shut-in periods (qn = 0) and during a period immediately after shut-in (qn-1 = 0), the actual flow rate is used in Equation 24-15.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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S'=S+D(qn+qn-1)
12.0
10.0
slope = D
intercept = S
8.0
6.0 0
1000
2000
3000
4000
5000
6000
7000
8000
qn+qn-1, Mscf/D
Figure 24-2 : Variation of the pseudo-skin with the rate qn + qn-1.
24.3 Deliverability tests Deliverability equations Empirical approach (Fetkovitch, or "C & n")
pi2-pwf2, psia2
1.0E+09
1.0E+08
slope = 1/n
pwf=14.7psia
1.0E+07
AOF=9000Mscf/D 1.0E+06 1.0E+03
1.0E+04
1.0E+05
q, Mscf/D
Figure 24-3: Deliverability plot for a back pressure test. Log-log scale, pressure squared method.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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(
2 qsc = C pi2 − pwf
)
n
( 24-17)
The stabilized flowing pressure pwf is expressed in absolute units. The coefficients C and n are two constant terms. n can vary from 1 in the case of laminar flow, to 0.5 when the flow is fully turbulent.
Theoretical approach (LIT, or Houpeurt's, or Jone's, or "a & b") In a closed system, the difference between the pseudo-steady state flowing pressure pwf and the following shut-in average pressure p- is expressed as :
() ( )
m p − m pwf
T A rw2 T = 1637 log + 0.35 + 0.87 S qsc + 1422 Dq sc2 kh CA kh
( 24-18)
4.0E+04
Dm(p)/q, psia2*D/cp/Mscf
stabilized 3.5E+04
transient slope = b 3.0E+04
intercept = a 2.5E+04
2.0E+04
1.5E+04 0.0E+00
2.0E+03
4.0E+03
6.0E+03
8.0E+03
1.0E+04
q, Mscf/D
Figure 24-4 Deliverability plot for an isochronal or a modified isochronal test. Linear scale, pseudo-pressure method.
With a circular reservoir of radius re, CA = 31.62 and ∆m(p) is simplified :
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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() ( )
m p − m pwf = 1637
T T 0.472re + 0.87 S q sc + 1422 Dq sc2 2 log kh rw kh
( 24-19)
Before the pseudo-steady state regime, the response follows the semi-log approximation and ∆m(p) is :
() ( )
m p − m pwf = 1637
T k∆t T + 3.23 + 0.87 S q sc + 1422 Dq sc2 log 2 φµ i cti rw kh kh
( 24-20)
The two ∆m(p) deliverability relationships can be expressed as a(t) qsc + b q2sc. During the infinite acting regime, a(t) is an increasing function of the time whereas "a" is constant when pseudo-steady state is reached. The coefficient "b" is the same in the two equations. The Absolute Open Flow Potential is :
q sc , AOF =
(
− a + a 2 + 4b m( p) − m(14.7) 2b
)
( 24-21)
Back pressure test (Flow after flow test) The well is produced to stabilized pressure at three or four increasing rates and the different flow periods have the same duration.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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pi Pressure, psia
7000
pwf1
pwf2
pwf3 pwf4
6950
6900
6850
6800 0
100
200
300
400
500
600
700
800
900
1000
Time, hr
Figure 24-5 Pressure and rate history for a back pressure test.
Dm(p)/q, psia2*D/cp/Mscf
3.5E+04
3.0E+04
slope = b
2.5E+04
intercept = a 2.0E+04 0.0E+00
2.0E+03
4.0E+03
6.0E+03
8.0E+03
q, Mscf/D
Figure 24-6 Deliverability plot for a back pressure test. Linear scale, pseudo-pressure method.
Isochronal test The well is produced at three or four increasing rates and a shut-in period is introduced between each flow. The drawdowns, of same duration tp, are stopped during the infinite acting regime. The intermediate build-ups last until the initial pressure pi is reached. A final flow period is extended to reach stabilized flowing pressure.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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7050
pi Pressure, psia
7000
pwf1
6950
pwf2
pwf5
6900
pwf3 pwf4
6850
6800 0
100
200
300
400
500
600
700
800
Time, hr
Figure 24-7 Pressure and rate history for an isochronal test. 1.0E+08
(pi2 or pws2) - pwf2, psia2
stabilized p=14.7psia transient slope = 1/n
1.0E+07
1.0E+06
AOF=8000Mscf/D 1.0E+05 1.0E+03
1.0E+04
1.0E+05
q, Mscf/D
Figure 24-8 Deliverability plot for an isochronal or a modified isochronal test. Log-log scale, pressure squared method.
Modified isochronal test
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk The intermediate shut-in periods have the same duration tp as the drawdowns, and the last flow is extended until the stabilized pressure is reached.
7000
pws1 pws2
pws3
pi
pws4
Pressure, psia
pwf1 6800
pwf2 pwf5
pwf3 6600
pwf4
6400
6200 0
100
200
300
400
500
600
Time, hr
Figure 24-9 Pressure and rate history for a modified isochronal test.
24.4 Odeh-Jones Analysis Introduction The Odeh-Jones plot is used to plot all data from all flow-periods in the current period for analysis. This plot can be used to compare the data in several build-up's and draw-down's. Drawing a straight-line on this plot is used to determine a value of permeability and skin that is global for the entire period for analysis and not just for a single flow-period. The Odeh-Jones plot is particularly useful for the analysis of tests with simultaneous sandface rate and pressure measurements. The Odeh-Jones time transform is a numerical evaluation of the general convolution integral:
t
Pw = Sq (t ) +
∫ q (τ )
τ =0
dPd dt
t −τ
dτ
( 24-22)
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2001
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This integral uses the rate-history together with the line-source solution to define the timetransform of the data. Note that the line-source solution,’Pd’', and skin-factor,’S’', include coefficients to convert them to 'psi' when multiplied by a rate. A typical application of the Odeh-Jones plot is for gas-well A.O.F. (absolute open flow) test. Instead of assuming pseudo-steady-state flow as in a typical L.I.T. analysis, the Odeh-Jones plot allows all the pressure and rate-data to be 'convolved' to analyse the transient behaviour in all the flow-periods of the test. The resulting Odeh-Jones analysis yields permeability, skin-factor, and the turbulence coefficient. This is the recommended method for a gas-well test analysis to derive the turbulence factor. The Odeh Jones Transform An Odeh-Jones plot, plots the transformed time, 'X', and transformed pressure, 'Y', defined by the equations: n
∑ (q Xk =
Yk =
j =1
j
− q j −1 ) log( tk − t j ) qn
( pi − pk ) qn
( 24-23)
( 24-24)
where 'n' ranges over all of the rates and 'k' ranges over all the pressure points in the period for analysis. The value of 'Pi' is the extrapolated pressure entered on the small table displayed prior to creating the plot. It is recommended that a value of extrapolated pressure is obtained from the Superposition plot analysis of a build-up. Otherwise, a reasonable value of initial-pressure should be used, perhaps from an RFT or MDT, or as a last resort, the highest pressure value in the test data. Note that if this value is wrong, the plot will show interesting curves and displacements due to odd behaviour in the 'Y' transform. If the value of extrapolated pressure is not available, then try several values of 'Pi' for the 'Y' transform so the best straight-lines are seen on the plot. PIE, a well test interpretation software package which incorporates this technique, uses a convention of ALWAYS having the 'X' and 'Y' transform value INCREASE with increasing time. The problem with superposition functions is that they increase or decrease with time depending on the flow rate changes. To avoid this, a 'rotation' of the data is made so there is
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk a consistent behaviour. If this was not done, the plot for several flow periods with storage, skin, and reservoir heterogeneities would be impossible to understand. Clearly, there is a problem for the 'X' transform when a flow-period has a zero rate. PIE follows a convention of dividing by the preceding rate, '', in order to avoid division by zero. This fix-up results in all build-up pressures plotting in the lower left quadrant of the plot. Note that this convention means the value of skin factor for a build-up cannot be calculated. Even though this is a limitation to the transform, it is useful to plot build-up data because this data usually shows the radial-flow behaviour most clearly. When analyzing a gas-well or multi-phase test, the objective is to place a line of constant permeability through the data for each producing flow-period in order to obtain an estimate of turbulence coefficient and mechanical skin factor. Select each flow-period in turn and use the FIXSL function to set a straight-line through the data. With several lines on the plot, PIE will compute the turbulence factor and mechanical skin from the least-squares best-fit of the skin-factors vs. rates. An example Odeh-Jones plot for a gas-well test is shown in Figure 24-10.
Odeh-Jones Analysis When analyzing a gas-well or multi-phase test, the objective is to place a line of constant permeability through the data for each producing flow-period in order to obtain an estimate of turbulence coefficient and mechanical skin factor. Select each flow-period in turn and use the FIXSL function to set a straight-line through the data. With several lines on the plot, PIE will compute the turbulence factor and mechanical skin from the least-squares best-fit of the skin-factors vs. rates. An example Odeh-Jones plot for a gas-well test is shown in the figure below;
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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.040
ODEH-JONES PLOT Reference Pressure= 5801.520 PSI FP.003 Slope= .009171 Int.= .051497 MPSI2/CP/MSCF/D FP.003 K= 3.92 MD S= .855 FP.001 K= 3.92 MD S= -.891 FP.004 K= 3.92 MD S= .555 Turb=.00082 1/MSCF/D Smech= -1.47
.004 .003
ODEH.004 .001
.020
SLOPE
0.
ODEH.001
-.020
.005
-.060
-.040
(PI-P(T))/Q (MPSI2/CP/MSCF/D)
.060
1999/01/03-0143 : GAS (PSEUDO-PRESSURE)
-5.
-4.
-3.
-2.
-1.
0.
1.
SUPERPOSITION/Q
Bloom-hardy Brap #1 Figure 24-10 Odeh-Jones plot showing analysis and results
The Odeh-Jones plot is very similar to the Superposition plot, therefore, all of the diagnostic characteristics seen in the Superposition plot apply to the Odeh-Jones plot as well. For example, a linear no-flow reservoir boundary shows up on the Superposition plot as a doubling of slope. This characteristic is also seen on the Odeh-Jones plot. Use the OdehJones plot as a diagnostic plot to help decide which model would best match the test behaviour.
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2001
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25. FISSURED RESERVOIRS In fissured reservoirs, the fissure network and the matrix blocks react at a different time, and the pressure response deviates from the standard homogeneous behavior. 25.1 Pressure profile
"
! ! ! &
$ ! %
%
' #
rw
pm
r
pi
pw
pf
pi = pm > pf
Figure 25-1 : Double porosity behavior. Pressure profile. Fissure system homogeneous regime.
First, the matrix blocks production is negligible. The fissure system homogeneous behavior is seen.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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"
! ! ! &
" ! % '
#
' #
rw
r
pi pm pw
pf
pi > pm > pf
Figure 25-2 : Double porosity behavior. Pressure profile. Transition regime.
When the matrix blocks start to produce into the fissures, the pressure deviates from the homogeneous behavior to follow a transition regime.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk "
! ! !
$ ! # '
"
%
# '
%
rw
r
pi
pw
pi > pf = pm
Figure 25-3 : Double porosity behavior. Pressure profile. Total system homogeneous regime (fissures + matrix). When the pressure equalizes between fissures and matrix blocks, the homogeneous behavior of the total system (fissure and matrix) is reached.
25.2 Definitions Permeability The fluid flows to the well through the fissure system only and the radial permeability of the matrix system does not contribute to the mobility (km = 0). The permeability thickness product kh estimated by the interpretation is used to define an equivalent bulk permeability of the fissure network, over the complete thickness h : kh = k f h f
Porosity
(Eq. 25-1)
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www.opc.co.uk φf and φm : ratio of pore volume in the fissures (or in the matrix), to the total volume of the fissures (of the matrix). Vf and Vm : ratio of the total volume of the fissures (or matrix) to the reservoir volume (Vf + Vm = 1). φ = φ f V f + φ mVm
(Eq. 25-2)
In practice, φf and Vm are close to 1. The average porosity of Equation 25.2 can be simplified as : φ = Vf + φm
(Eq. 25-3)
Storativity ratio ω
ω=
(φ Vct ) f (φ Vct ) f = (φ Vct ) f + (φ Vct )m (φ Vct ) f +m
(Eq. 25-4)
Interporosity flow parameter λ λ = α rw2
km kf
(Eq. 25-5)
α is related to the geometry of the fissure network, defined with the number n of families of fissure planes. For n = 3, the matrix blocks are cubes (or spheres) and, for n = 1, they are slab. α=
n ( n + 2) rm2
(Eq. 25-6)
rm is the characteristic size of the matrix blocks. It is defined as the ratio of the volume V of the matrix blocks, to the surface area A of the blocks : rm = nV A
(Eq. 25-7)
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk When a skin effect (Sm in dimensionless term) is present at the surface of the matrix blocks, the matrix to fissure flow is called restricted interporosity flow. Sm =
k m hd rm k d
(Eq. 25-8)
km rm
hd kd n=3, cubes
n=1, slabs
Figure 25-1 Matrix skin. Slab and sphere matrix blocks.
The analysis with the restricted interporosity flow model (pseudo-steady state interporosity flow) provides the effective interporosity flow parameter λeff :
λ eff = n
rw2 k d rm hd k f
(Eq. 25-9)
λeff is independent of the matrix block permeability km.
Dimensionless variables pD =
kh ∆p 1412 . qBµ
(Eq. 25-10)
2001
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
www.opc.co.uk tD kh ∆t = 0.000295 CD µ C
CDf =
0.8936C (φ Vct ) f hrw2
CDf + m =
0.8936C (φ Vct ) f +m hrw2
(Eq. 25-11)
(Eq. 25-12)
(Eq. 25-13)
The storativity ratio ω correlates the two definitions of dimensionless wellbore storage : CDf + m = ω CDf
(Eq. 25-14)
25.3 Double porosity behavior, restricted interporosity flow (pseudo-steady state interporosity flow). Log-log analysis Pressure type curves Three component curves : 1. - (CDe2S)f at early time, during fissure flow. 2. - λeff e-2S during transition regime, between the two homogeneous behaviors. 3. - (CDe2S)f+m at late time, when total system behavior is reached. A double porosity response goes from a high value (CDe2S)f when the storativity corresponds to fissures, to a lower value (CDe2S)f+m when total system is acting.
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2001
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1.0E+02 2S
CD e
λ eff e
-2S
pD
1.0E+01
1.0E+00
1.0E-01 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
tD/CD
Figure 25-2 Pressure type-curve for a well with wellbore storage and skin in a double porosity reservoir, pseudo steady state interporosity flow. CDe2S = 1030, 1010, 103, 5, 0.1, 5.10-3. λeffe-2S = 10-30, 10-10, 10-6, 10-2, 0.5.
Typical responses The limit "approximate start of the semi-log straight line" shows that the wellbore storage stops during the fissure regime with example A. With example B, wellbore storage lasts until the transition regime, and the fissure (CDe2S)f curve is masked. 1.0E+02 o o o
approximate start of the semi-log straight line
A B
pD
1.0E+01
1.0E+00
1.0E-01 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
tD/CD
Figure 25-3: Pressure examples for a well with wellbore storage and skin in a double porosity reservoir, pseudo steady state interporosity flow.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
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www.opc.co.uk CDe2S = 1030, 1010, 105, 104, 1, 0.1, 5.10-3. λeffe-2S = 10-30, 10-7, 3.10-4, 10-2. o = A : (CDe2S)f = 1, (CDe2S)f+m = 0.1, ω = 0.1, λeffe-2S = 3.10-4. ∝ = B : (CDe2S)f = 105, (CDe2S)f+m = 104, ω = 0.1, λeffe-2S = 10-7.
On semi-log scale, two parallel straight lines are present with example A. With example B, only the total system straight line is seen. 10.0
B 8.0
slope m 6.0
pD
slope m
4.0
A 2.0
slope m
0.0 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
tD/CD
Figure 25-4: Semi-log plot of Figure 25-3 examples.
1.0E+02 o o o
pD & pD'
A B
1.0E+01
1.0E+00
1.0E-01 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
tD/CD
Figure 25-5: Pressure and derivative examples of Figure 25-3 for a well with wellbore storage and skin in a double porosity reservoir, pseudo steady state interporosity flow. CDe2S = 1030, 1010, 105, 104 (only pD), 1, 0.1, 5x10-3 (only pD).
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
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www.opc.co.uk λeffe-2S =10-30, 10-7, 3x10-4, 10-2. λeffCDf+m/ω(1-ω) =10-2, 3x10-4. λeffCDf+m/(1-ω) = 103, 3x10-5.
With the derivative, example A shows two stabilizations on 0.5. The derivative of example B stabilizes on 0.5 only during the total system homogeneous regime. On the derivative type-curve, the transition is described with two curves, labeled
(λ
eff
CD f + m
) [ω (1 − ω )] (decreasing derivative) and (λ
eff
CD f + m
)
(1 − ω ) .
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk Match results kh = 1412 . qBµ ( PM )
C = 0.000295
S = 0.5 ln
(C
kh 1 µ TM
De
2S
)
f +m
CDf + m
(C e ) ω= (C e )
(Eq. 25-15)
(Eq. 25-16)
(Eq. 25-17)
2S
D
f +m
2S
D
(
(Eq. 25-18)
f
)
λ eff = λ eff e −2 S e 2 S
(Eq. 25-19)
Pressure and derivative response With the restricted interporosity flow model, the derivative exhibits a valley shaped transition between the two stabilizations on 0.5, when the three characteristic regimes are developed.
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www.opc.co.uk 1.0E+02
pD & pD'
1.0E+01
1.0E+00
1.0E-01 1.0E-01
0.5 line
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
tD/CD
Figure 25-6 Pressure and derivative response for a well with wellbore storage in double porosity reservoir, pseudo-steady state interporosity flow. CDf+m = 103, S = 0, ω = 0.1, λeff= 6.10-8 (CDe2Sf =104, λeffe-2S= 6.10-8 and CDe2Sf+m = 103)
Influence of the heterogeneous parameters ω and λeff Influence of ω With small ω values, the transition regime from CDe2Sf to CDe2Sf+m is long. On the derivative responses, the transition valley drops when ω is reduced. On semi-log scale, the first straight line is displaced upwards and the horizontal transition between the two parallel lines is longer.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
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www.opc.co.uk 1.0E+02
ω = 1.0E-03
pD & pD'
1.0E+01
1.0E+00
ω = 1.0E-01
0.5
1.0E-01
1.0E-02
ω = 1.0E-03
1.0E-03 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
1.0E+07
1.0E+08
tD/CD
Figure 25-7 Double porosity reservoir, pseudo-steady state interporosity flow. Influence of ω. Log-log scale. CDf+m =1, S =0, λeff=10-7 and ω =10-1, 10-2 and 10-3
10.0
slope m 8.0
ω = 1.0E-03
slope m
pD
6.0
4.0
ω = 1.0E-01 2.0
0.0 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
tD/CD
Figure 25-8 Semi-log plot of Figure 25-7.
1.0E+05
1.0E+06
1.0E+07
1.0E+08
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk Influence of λeff The interporosity flow parameter defines the time of end of the transition regime. The smaller is λeff, the later the start of total system flow. On the pressure curves, the transition regime occurs at a higher amplitude and, on the derivative responses, the transition valley is displaced towards late times. 1.0E+02
λ = 1.0E-08
pD & pD'
1.0E+01
1.0E+00
0.5
1.0E-01
λ = 1.0E-06 1.0E-02 1.0E01
1.0E+00
1.0E+01
1.0E+02
λ = 1.0E-08 1.0E+03
1.0E+04
1.0E+05
1.0E+06
1.0E+07
tD/CD
Figure 25-9 Double porosity reservoir, pseudo-steady state interporosity flow. Influence of λeff. Log-log scale. CDf+m =100, S =0, ω =0.02 and λeff=10-6, 10-7 and 10-8
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
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www.opc.co.uk 12.0
λ = 1.0E-08 8.0
pD
slope m
slope m
λ = 1.0E-06
4.0
0.0 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
tD/CD
Figure 25-10 Semi-log plot of Figure 25-9.
1.0E+05
1.0E+06
1.0E+07
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2001
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Analysis of the semi-log straight lines 10.0
8.0
Double porosity
pD
6.0
4.0
slope m
Homogeneous
2.0
0.0 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
tD/CD
Figure 25-11 Semi-log plot of homogeneous and double porosity responses. CD = CDf+m = 100, S = 0, ω = 0.01 and λeff= 10-6
During fissure flow, if the first semi-log line is present,
∆p = 162.6
qBµ k − + S log ∆t + log 3 . 23 0 . 87 kh (φ Vct ) f µ rw2
(Eq. 25-20)
The second line, for the total system regime is :
∆p = 162.6
qBµ k log ∆t + log − 3 . 23 + 0 . 87 S kh (φ Vct ) f +m µ rw2
(Eq. 25-21)
The vertical distance δp between the two lines gives ω : ω = 10−δp m
(Eq. 25-22)
When only the first semi-log straight line for fissure regime is present, if the total storativity is used instead of that of the fissure system, the calculation of the skin gives an over estimated value Sf :
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk S f = S + 0.5 ln
1 ω
(Eq. 25-23)
2001
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
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Build-up analysis Log-log pressure build-up analysis When the production time tp is small, the three characteristic regimes of a double porosity response are not always fully developed on build-up pressure curves. Whatever long are the three build-up examples of Figure 25-12, only example C exhibits a clear double porosity response. The build-up curve A does not show a double porosity behavior, but only the build-up response of the fissures. For example B, the build-up curve flattens at the same ∆p level as the λeffe-2S transition, there is no evidence of total system flow regime.
1.0E+01
drawdown
fiss. & tot. system
C B A
pD
double porosity 1.0E+00
1.0E-01 1.0E-01
1.0E+00
1.0E+01
tpC
tpB
tpA
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
tD/CD
Figure 25-12 Drawdown and build-up pressure responses for a well with wellbore storage and skin in double porosity reservoir, pseudo-steady state interporosity flow. Log-log scale. CDf+m = 0.1, S = 0, ω = 0.1, λeff= 3.10-4 (CDe2Sf =1, λeffe-2S= 3.10-4 and CDe2Sf+m = 0.1). tpD/CD = 100 (A), 9.103 (B), 3.105 (C).
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk 8.0
tpC
drawdown build-up
6.0
tpB C
pD
tpA
B
4.0
A 2.0
0.0 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
tD/CD
Figure 25-13 Semi-log plot of drawdown and build-up pressure responses of Figure 25-12.
Horner & superposition analysis In example C, the initial pressure pi is obtained by extrapolation of the second straight line, the first one extrapolates to pi + m ln (1/ω). If the drawdown stops during the transition, (example B), the extrapolated pressure p* is between pi and pi + m ln (1/ω), depending upon tp.
0.0
(p-pi)D
slope m
p*>pi
-2.0
A p*=pi -4.0
B
-6.0 1.0E+00
C 1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
(tp + Dt) / Dt
Figure 25-14 Horner plot of the three Build-ups of Figure 25-12. A (tpD/CD = 100), B (tpD/CD = 9.103) and C (tpD/CD = 3.105).
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk
Derivative build-up analysis 1.0E+00
pD'
C 1.0E-01
A
B build-up drawdown 1.0E-02 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
tD/CD
Figure 25-15 Drawdown and build-up derivative responses of Figure 25-12.
2001
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
www.opc.co.uk 25.4 Double porosity interporosity flow)
behavior,
unrestricted
interporosity
flow
(transient
Log-log analysis Pressure type-curve Two pressure curves : 1. - β' at early time, during transition regime before the homogeneous behavior of the total system 2. - (CDe2S)f+m later, when the homogeneous total system flow is reached The two families of curves have the same shape: the β ' transition curves are equivalent to CDe2S curves whose pressure and time are divided by a factor of two. β' is defined as :
β '= δ '
(C
D
e2 S
λe
)
−2 S
f +m
(Eq. 25-24)
The constant δ' is related to the geometry of the matrix system. For slab matrix blocks δ '=1.89, and for sphere matrix blocks δ ' = 1.05.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk 1.0E+02 2S
CD e β'
pD
1.0E+01
1.0E+00
1.0E-01 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
tD/CD
Figure 25-16 Pressure type-curve for a well with wellbore storage and skin in a double porosity reservoir, transient interporosity flow. CDe2S = 1030, 1010, 103, 5, 0.1, 5.10-3. β' = 1030, 1010, 103, 5, 0.1.
Typical responses A long transition on a β' curve is seen on example A. With example B, the wellbore storage is large, and the transition is shorter on the tD/CD time scale. 1.0E+02 o o o
A B
pD
1.0E+01
1.0E+00
1.0E-01 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
tD/CD
Figure 25-17: Pressure examples for a well with wellbore storage and skin in a double porosity reservoir, transient interporosity flow, slab matrix blocks. CDe2S = 1030, 1010, 6.103, 10, 0.1. β' = 1030, 1010, 106, 5. o = A : (CDe2S)f+m = 10, ω = 0.001, β' = 106, λe-2S = 1.8914*10-5.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk ∝ = B : (CDe2S)f+m = 6.103, ω = 0.001, β' = 1010, λe-2S = 1.1348*10-6.
On semi-log scale, example A shows a first straight line of slope m/2 during transition, before the total system straight line of slope m. With example B, only the total system straight line is present. 10.0
8.0
B slope m
A
pD
6.0
slope m/2
4.0
slope m 2.0
0.0 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
tD/CD
Figure 25-18: Semi-log plot of Figure 25-17 examples. 1.0E+02 o o o
pD & pD'
A B
1.0E+01
1.0E+00
1.0E-01 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
tD/CD
Figure 25-19: Pressure and derivative examples of Figure 25-17. CDe2S = 1030, 1010, 6.103, 10, 0.1. β' = 1030, 1010, 106, 5. λCDf+m (1-ω)2 = 3.10-2, 3.10-3, 3.10-4, 3.10-5.
1.0E+05
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk With the derivative, example A shows a first stabilization on 0.25 before the final stabilization on 0.5 for the total system homogeneous regime. The derivative of example B exhibits only a small valley before the stabilization on 0.5. The end of transition, and the start of the total system homogeneous regime, is described by a ( λ CD )
(1 − ω )2
derivative curve.
Match results On a double porosity response with unrestricted interporosity flow, after the wellbore storage hump the derivative exhibits a first stabilization on 0.25 before the final stabilization on 0.5.
pD & pD'
1.0E+02
1.0E+01
0.5 line
1.0E+00
0.25 line 1.0E-01 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
tD/CD
Figure 25-20 Pressure and derivative response for a well with wellbore storage and skin in double porosity reservoir, transient interporosity flow, slab matrix blocks. Log-log scale. CDf+m =10, S = 5, ω = 0.01, λ = 10-5. (CDe2Sf = 2.2 107, λe-2S= 4.5 10-10 and CDe2Sf+m = 2.2 105)
λ =δ'
(C
De
2S
β 'e
)
f +m −2 S
(Eq. 25-25)
ω is difficult to access with the transient interporosity flow model. Slab and sphere matrix blocks With the two types matrix geometry, the pressure curves look identical but the derivatives are slightly different. At late transition time, the change from 0.25 to the 0.5 level is steeper on the curve generated for slab matrix blocks.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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pD & pD'
1.0E+01
1.0E+00
0.5 Sphere Slab
0.25 1.0E-01 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
tD/CD
Figure 25-21 Double porosity reservoir, transient interporosity flow, slab and sphere matrix blocks. Log-log scale. CDe2Sf+m=1, β'=104 and ω=10-2. Slab: λe-2S = 1.89 10-4, Sphere: λe-2S = 1.05 10-4.
Influence of the heterogeneous parameters ω and λ Influence of ω
pD & pD'
1.0E+02
ω = 1.0E-03
1.0E+01
1.0E+00
0.5
ω = 1.0E-01 ω = 1.0E-03
1.0E-01 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
1.0E+07
1.0E+08
tD/CD
Figure 25-22 Double porosity reservoir, transient interporosity flow, slab matrix blocks. Influence of ω on pressure and derivative curves. CDf+m =1, S =0, λ =10-7 and ω =10-1, 10-2 and 10-3
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk 10.0
8.0
slope m/2
slope m
pD
6.0
4.0
ω = 1.0E-03
ω = 1.0E-01
2.0
0.0 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
1.0E+07
1.0E+08
tD/CD
Figure 25-23 Semi-log plot of Figure 25-22.
Influence de λ 1.0E+02
pD & pD'
λ = 1.0E-08 1.0E+01
λ = 1.0E-06
1.0E+00
1.0E-01 1.0E-01
0.5 λ = 1.0E-08
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
1.0E+07
1.0E+08
tD/CD
Figure 25-24 Double porosity reservoir, transient interporosity flow, slab matrix blocks. Influence of λ on pressure and derivative curves. CDf+m =100, S =0, ω =0.02 and λ =10-6, 10-7 and 10-8
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk 12.0
λ = 1.0E-08
pD
8.0
λ = 1.0E-06
slope m/2 slope m 4.0
0.0 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
1.0E+07
1.0E+08
tD/CD
Figure 25-25 Semi-log plot of Figure 25-24.
Build-up analysis 1.0E+00
pD & pD'
C A
B
1.0E-01
build-up drawdown 1.0E-02 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
tD/CD
Figure 25-26 Drawdown and build-up derivative responses, double porosity reservoir, unrestricted interporosity flow, slab matrix blocks. CDf+m = 0.1, S = 0, ω = 0.1, λ = 3.10-4. tpD/CD = 100 (A), 9.103 (B), 3.105 (C).
25.5 Matrix skin
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk 1.0E+01
Sm = 0 0.5
pD & pD'
1.0E+00
Sm = 0 0.25 0.1
1.0E-01
1 Sm = 100
10 1.0E-02 1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
tD/CD
1.0E+05
1.0E+06
1.0E+07
Figure 25-27: Double porosity reservoir, transient interporosity flow, slab matrix blocks with interporosity skin. CDf+m = 1, S = 0, ω = 0.01, λ = 10-5. Sm = 0, 0.1, 1, 10, 100.
pD'
1.0E+00
1.0E-01 ooo
restricted
Sm = 1 1.0E-02 1.0E+01
1.0E+02
1.0E+03
1.0E+04
Sm = 10 1.0E+05
Sm = 100 1.0E+06
1.0E+07
tD/CD
Figure 25-28: Comparison of Figure 25-27 derivative responses with the restricted interporosity flow model. λ eff = 2.500x10-6 (Sm = 1), λ eff = 3.323x10-7 (Sm = 10), λ eff = 3.333x10-8 (Sm = 100). 1.0E+01
Sm = 0
pD & pD'
1.0E+00
0.5 Sm = 0 0.25
1.0E-01
0.1 1 10
1.0E-02 1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
tD/CD
1.0E+05
Sm = 100 1.0E+06
1.0E+07
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk Figure 25-29: Double porosity reservoir, transient interporosity flow, sphere matrix blocks with interporosity skin. CDf+m = 1, S = 0, ω = 0.01, λ = 10-5. Sm = 0, 0.1, 1, 10, 100.
pD'
1.0E+00
1.0E-01
ooo
restricted
Sm = 1 1.0E-02 1.0E+01
1.0E+02
1.0E+03
1.0E+04
Sm = 10 1.0E+05
Sm = 100
1.0E+06
1.0E+07
tD/CD
Figure 25-30: Comparison of Figure 25-29 derivative responses with the restricted interporosity flow model. λ eff = 1.66x10-6 (Sm = 1), λ eff = 1.96x10-7 (Sm = 10), λ eff = 2.00x10-8 (Sm = 100).
1.0E+01
unrestricted sphere unrestricted slab
pD & pD'
1.0E+00
1.0E-01
restricted
1.0E-02 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
tD/CD
Figure 25-31: Log-log plot of pressure and derivative responses for a well with wellbore storage and skin in double porosity reservoir, restricted and unrestricted interporosity flow, slab and sphere matrix blocks. CDf+m = 1, S = 3, ω = 0.02, λ = 10 -4. CDe2Sf+m=403, λe-2S = 2.48*10-7. Slab: β' = 3.07*10 9, Sphere: β' = 1.71*10 9
25.6 Examples of complex heterogeneous responses
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk One sealing fault in double porosity reservoirs: 1.0E+02
o
pD & pD'
1.0E+01
start of the sealing fault
fissure regime 1.0E-01 1.0E-01
1.0E+00
1.0E+01
1.0E+02
1
1
0.5
1.0E+00
transition 1.0E+03
1.0E+04
1.0E+05
total system 1.0E+06
tD/CD
Figure 25-32 : One sealing fault in double porosity reservoir, pseudo-steady state interporosity flow. Log-log scale. CD = 104, S = 0, LD = 5000, ω = 0.2, λ = 10-9
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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Triple porosity solution The model considers two sizes of matrix blocks. The blocks are uniformly distributed in the reservoir.
pD & pD'
1.0E+01
total system
fissure + group 1
fissure regime 1.0E+00
0.5 1.0E-01 1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
1.0E+07
tD/CD
Figure 25-33 : Triple porosity response, pseudo-steady state interporosity flow. Log-log scale. CDf+m = 1, S = 0, ω = 0.01, λeff1 =10-5, δ1 =0.1, λeff2 =5x10-7, δ2 =0.9.
9.0
slopes m
fissure regime
6.0 pD
total system fissure + groupe 1
3.0
0.0 1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
tD/CD
Figure 25-34: Semi-log plot of Figure 25-33 example.
26. FACTORS COMPLICATING WELL TEST ANALYSIS
1.0E+07
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk 26.1 Rate history definition Two approaches can be used in order to simplify the rate history : 1. An equivalent production time is defined as the ratio of the cumulative production divided by the last rate (called equivalent Horner time). On the test example, tp=120. 2. When there is a shut-in period in the rate history, if the bottom hole pressure is almost at initial pressure, it can be assumed that the rate history prior this shut-in is negligible.On the test example, tp=20.
pressure, psi
4000 3900 3800 3700
tp=120 tp=20
3600 3500 0
50
100
150
200
250
300
350
400
450
500
time, hr
Figure 26-1 : Example of a two drawdowns test sequence. Linear scale.
Dp & Dp', psi
1.0E+03
1.0E+02
tp=20
1.0E+01
tp=120
1.0E+00 1.0E-02
1.0E-01
1.0E+00
1.0E+01
1.0E+02
Dt, hr
Figure 26-2 : Log-log plot of the final build-up. The derivative is generated with three different rate histories.
1.0E+03
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk In practice, if the duration of the analysed period is ∆t, it is possible to simplify the rate history for any rate changes that occurred at more than 2∆t before the start of the period. All rate variations immediately before the analysed test period must be used in the superposition time.
26.2 Error of start of the period 3830
e
pressure, psi
3810
d 3790
a b
3770
c 3750 169.7
169.8
169.9
170.0
170.1
170.2
170.3
time, hr
Figure 26-3 : Example of Figure 26-1 at time of shut-in. Time and pressure errors. - Shut-in time error : curve a = 0.1 hr before and curve b = 0.1 hr after the actual shutin time. - Shut-in pressure error : curve c = 10 psi below and curve d = 10 psi above the last flowing pressure. - Error in time and pressure : curve e.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk
Dp & Dp', psi
1.0E+03
1.0E+02
1.0E+01
a 1.0E+00 1.0E-02
1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
Dt, hr
Figure 26-4 : Case a = shut-in time too early.
Dp & Dp', psi
1.0E+03
1.0E+02
1.0E+01
b 1.0E+00 1.0E-02
1.0E-01
1.0E+00
1.0E+01
Dt, hr
Figure 26-5 : Case b = shut-in time too late.
1.0E+02
1.0E+03
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk
Dp & Dp', psi
1.0E+03
1.0E+02
1.0E+01
c 1.0E+00 1.0E-02
1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
Dt, hr
Figure 26-6 : Case c = last flowing pressure too low.
Dp & Dp', psi
1.0E+03
1.0E+02
1.0E+01
d 1.0E+00 1.0E-02
1.0E-01
1.0E+00
1.0E+01
1.0E+02
Dt, hr
Figure 26-7 : Case d = last flowing pressure too high.
1.0E+03
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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Dp & Dp', psi
1.0E+03
1.0E+02
1.0E+01
e 1.0E+00 1.0E-02
1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
Dt, hr
Figure 26-8 : Case e = shut-in time too late, last flowing pressure is taken in the build-up data, during the wellbore storage regime.
26.3 Time Error Correction
correction for zero time = -3.000 seconds slope of the line = 1043. PSI/hr Volume = .3581E+04 Reservoir-BBLS Storage Coefficient (calculated) = .0358 BBLS/PSI End of Well-bore Storage dt = 13.27972 hr
150. 100. 50. 0.
DP (PSI)
200.
250.
1980/12/12-0600 : OIL
SLOPE -.10
0.
.10
.20 DT
.30
(HR )
Figure 26-9 : PIE Well Bore Storage plot(Cartesian)
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk As stated before, pure wellbore storage yields a straight line on a cartesian plot passing through the origin. If the straight line does not pass through the origin, or the points in pure wellbore storage on a log-log plot do not lie on a unit slope straight line, or the pressure points cross over the derivative, there could be a time error as illustrated in Figure 9.3 Time errors are usually due to an error in recording the exact shut-in time, a pressure reading not being recorded at the exact shut in time due to infrequent sampling rates, or the length of time involved to fully open the well, ie, difficulty in defining a start time. If a cartesian plot showing a straight line can be made from the early time data, it may be corrected for the time error. Some software packages (PIE for instance) will automatically make this correction for the user once the well bore storage has been identified on the plots. However, it should be noted that time errors are early time phenomena relating to the well bore and do not ultimately effect the reservoir response. There is more on this subject within chapter 25.
26.4 Changing Wellbore Storage There are instances in which various types of wellbore storage phenomena can combine (for example, fluid expansion followed by a falling liquid level in a fall-off test). In this event the line of unit slop may be distorted and data can depart from the line of unit slope. During this period absolutely nothing may be learned of the reservoir flow capacity, formation diffusivity or skin effect. These effects are known as changing well bore storage effects (see section below) and should be planned to be avoided. During drawdown, the wellhead pressure may drop below the bubble point of the oil, hence, in the tubing, there will be both free gas and oil. Since the wellbore storage constant is proportional to the compressibility of the fluid in the tubing a large wellbore storage constant will initially be observed if the well is shut-in (due to large compressibility of gas).
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk 1988/07/08-1229 : GAS (PSEUDO-PRESSURE)
ENDWBS
10 -1
** Simulation Data ** well. storage = .00228 Skin(mech) = 1.72 permeability = .470 Turbulence = 0. Perm-Thickness = 30.8 Initial Press. = 11680. Skin(mech)+DQ = 1.72 Smoothing Coef = 0.,0.
BBLS/PSI MD 1/MSCF/D MD-FEET PSI
10 -2
PD=1/2
10 -3
DP + DERIVATIVE (MPSI2/CP/MSCF/D)
Homogeneous Reservoir
10 -3
10 -2
10 -1
10 0
10 1
DT (HR)
Figure 26-10 : Changing Well Bore Storage Later, as the pressure builds-up, the wellhead pressure will pass the bubble point hence gas will re-dissolve in the fluid thereby giving a smaller wellbore storage constant. Changing wellbore storage can yield a log-log pressure behaviour which shows a slope greater than unity. This is often seen as a `humping' of the early time log-log data and is particularly evident in gas wells. This is illustrated above in plot 9.5. A type curve is overlaid on the data illustrating for this well in the reservoir the well bore storage in early time is not the same as that which matches the reservoir in later time. It has changed.
26.5 Two phases liquid level In diphasic wells (oil + water, or gas + condensate), a phase redistribution in the wellbore can produce a characteristic humping effect.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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diphasic flow
changing liquid level
end of phase segregation effect
Figure 26-11 : Changing liquid level after phase segregation.
When, after shut-in, water falls at the bottom of the well for example, the weight of the column between the pressure gauge and the formation is not constant as long as the water level rises and the gauge pressure is not parallel to the formation pressure. In some cases the build-up pressure can show a temporary decreasing trend after some shut-in time. During this time interval, the derivative becomes negative.
4000.00
pressure, psi
3500.00
Humping Pressure difference after phase segregation
3000.00
2500.00
Pressure difference before phase segregation
2000.00 18.00
28.00
time, hr
Figure 26-12 : Example of build-up response distorted by phase segregation. Humping effect.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk If the interface between the two phases stabilizes or reaches the depth of the pressure gauge, the pressure difference between gauge and formation returns to a constant, and the remaining build-up data can be properly analyzed. When phase redistribution is expected, the pressure gauge should be as close as possible to the perforated interval (or even below).
Dp & Dp', psi
1.00E+04
1.00E+03
1.00E+02
1.00E+01 1.00E-03
1.00E-02
1.00E-01
1.00E+00
1.00E+01
1.00E+02
Dt, hr
Figure 26-13 : Log-log plot of the build-up example of phase segregation.
26.6 Pressure gauge drift
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk 300.0
Dp, psi
200.0
100.0
0.0 0.0
100.0
200.0
300.0
Dt, hr
Figure 26-14 : Final build-up of Figure 26-1. Drift of ± 0.05 psi/hr. Linear scale.
Dp & Dp', psi
1.0E+03
1.0E+02
1.0E+01
1.0E+00 1.0E-02
1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
Dt, hr
Figure 26-15 : Log-log plot of the build-up example. Drift of ± 0.05 psi/hr.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk
26.7 Pressure gauge noise
250.0
Dp, psi
200.0
150.0
100.0
50.0
0.0 0.0
100.0
200.0
300.0
Dt, hr
Figure 26-16 : Final build-up of Figure 26-1. Noise of +1 psi every 2 points. Linear scale.
Dp & Dp', psi
1.0E+03
1.0E+02
1.0E+01
1.0E+00 1.0E-02
1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
Dt, hr
Figure 26-17 : Log-log plot of the build-up example. Noise of + 1 psi every 2 points. Three points derivative algorithm. No smoothing.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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27. WELL TESTING HARDWARE Well testing hardware can be broken down into three main areas; Surface well test equipment, subsea well test equipment (only applicable on offshore floating drilling units) and downhole/botttomhole equipment. The practical aspects of well testing can have a significant impact upon the quality of the test data and thus on the validity of the test results. When designing the mechanical aspects of a completion, future needs must be considered in the initial design. The following discussion on equipment will be orientated toward testing requirements. 27.1 Surface Test Equipment Surface Test Tree, Flowhead or Wellhead The Wellhead, flowhead or test tree is usually a hollow cross (sometimes manufactured out of a large block of metal) with a valve on each of the arms of the cross. The master valve is a full opening valve at the bottom of the cross which can stop flow from the well. When flowing the well this remains open. The swab valve is a full opening valve, similar to the master valve, which is placed at the top of the flow cross. The swab valve allows wireline instruments to be introduced into a flowing well without shutting a well-in to rig up a lubricator. Unless logging or there is wireline in the hole, the swab valve is usually closed while flowing the well. On one side of the cross is the flow wing valve which can also stop the flow from the well and this is sometimes hydraulically operated and connected to the emergency shut down system so that this valve closes when the emergency shutdown system is activated. Any wireline in the hole is not cut. On the other side of the cross, is the kill wing valve which is connected to a kill pump which could kill the well in case of emergency. This is usually closed during testing and open for pressure testing and killing the well or reverse circulating. There is also often a check (or non return valve) in the kill wing valve preventing from the well/flowhead into the kill line to the pump. The flowhead should have a rated working pressure greater than the flowing, shut-in, or injection pressure of the well. Wellhead damage can easily result in losing control of the well. For this reason, downhole safety valves are frequently used to prevent blowouts should a wellhead be damaged. Operations are made easier if the flowhead is able to support the weight of the completion and the lower part of the flowhead can swivel independently from the upper part of the flowhead for packer setting. A further (lower master) valve is sometimes added beneath the swivel as an extra isolation point in case the swivel leaks.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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Coflexip Hoses/Chicksans These link the wellhead to the choke manifold or the rig high pressure piping leading to the choke manifold. Today, coflexip hoses are mainly used and in some cases chicksans are used. A coflexip hose is a metal armoured flexible thermoplastic hose capable of holding pressures of up to 15,000 psi with a temperature rating of between –4 and 266 degF. Chicksans are solid metal pipe with a rotational high pressure link allowing movement. These are rarely used today but originally were always used. The flexibility is particularly important when testing on floating drilling units. Pipework Frequently, the drilling unit on which the test is performed, has fixed piping specifically for linking well testing equipment together. This can make the rig up easier but can contain debris if used infreqently or not hold the required pressure. Other sections of pipe are taken to the location as part of the test kit and used to link together all of the test equipment. It therefore must be pressure tested once it is rigged up. The pipe is linked together using screw “Weco” (brand name) unions which contain a plastic/rubber seal. This seal must be rated to the correct pressure and service (C02, H2S) Sand Trap A sand trap is usually installed between the flowhead and the choke manifold. Typically it consists of two vertical pressure cylinders containing a wire mesh of a certain size which can be changed to suit the operations. There are two cylinders to allow for one to be “on stream” while the other is bypassed and emptied. They can be used to measure the quantity of sand produced. Sand production can also be monitored by the use of a sand detection probe placed in the well flow stream and works by sand grains impacting on the probe. Data Header The data header is specially manufactured short length of pipe installed just upstream from the choke manifold. It contains screw tapped holes for attaching pressure gauges sampling points and other measuring devices (sand production monitoring device for instance). This is also the point at which the “bubble hose” is attached for monitoring the early stages of a test in low flowrate wells. Choke Manifold
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk The choke manifold is a pressure reduction and flow control device and the well isolation point (the well is usually shut in using the choke valves). The manifold will have a minimum of four valves and usually a fixed and variable choke. The variable choke is used in the early stages of a test to vary the flow rate until the well is cleaned up and relatively stable. The variable choke is a tapering cone which is screwed in or out of a cylindrical hardened (usually Tungsten Carbide) orifice to vary the size. Thereafter, a fixed choke is used of a known, accurately measured size. The fixed choke is similar to the cylindrical hardened orifice of the variable choke but without the cylindrical cone sitting in it. The choke size is measured in 1/64” and fixed chokes are usually found in steps sizes of 4/64” from 16/64” to 64/64”. It is useful to check a complete set is available prior to testing. Some choke manifolds will have a bypass. For the choke manifold to control the well (as opposed to other parts of the production system) the choke should operate under critical flow conditions. This is when the upstream pressure is more than twice the downstream pressure. A positive choke should be used at the wellhead to control the well's flow rate during a drawdown test having cleaned the well up on an adjustable choke. Several different chokes sizes should be available so that the desired flow rates can be obtained. Whenever possible, chokes should be changed without shutting in the well. If a wellhead or choke has twin valves, which in the case of a choke is usual, this is possible. Because of wear and erosion, adjustable chokes do not provide the desired accuracy to use them in testing for the measurement of produced field volumes. Furthermore, the adjustable choke should be checked at the end of each flow period for erosion and the zero setting. Steam Heat Exchanger or Heater This device is used to heat the well effluent. This could be to reduce the risk of hydrates forming by raising the temperature or to lower the viscosity of oil. It consists of a series of looped pipes separated by a choke to allow for heating prior to pressure reduction. There is a strong preference to use a separate steam generation unit which can pipe steam to the heat exchanger water bath. This removes the risk of ignition of the well effluent. This is an indirect heater where the steam is piped through a water bath, the temperature of the water is subsequently raised and the hot water heats the well effluent. Separator The separator is the “heart” of the surface well testing equipment. It serves the purpose of separating the oil/condensate, gas and water and allowing measurement of their respective flow rates. They can be vertical or horizontal and rated to different pressures (usually 600 or 1440 psi) allowing for different flow rates as indicated below;
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk Pressure rating 600 psi 1440 psi
Liquid flow rate BOPD 10,500 14400
Gas flow rate MMscf/d 28 60
The above rates are maxima for one phase only and cannot be achieved at the same time. The separation is achieved by retaining the fluid in the separator and allowing relative densities to separate the fluids. The retention time will determine the effectiveness of the separation and is regulated by the pressure and level of liquids in the separator. The pressure is controlled by an automatic control valve regulated by the operator. The level is controlled by a float the position of which is regulated by the operator. The float should normally be removed for transportation since it can fall off. Liquid production rate measurement Liquid production rates are usually measured by a positive-displacement meter which consists of two primary elements; a stationary case, and a moving element. The moving element isolates a fixed volume of fluid in the case for each cycle of operation. The mobile element can be vanes (turbine), cam arrangement with a disc, or a piston. Positive displacement and turbine meters must meet certain requirements as to proper installation and operation. The minimum recommended capacity for a positive displacement meter is 20% of the maximum flow rate. The meters must be installed so that gas breakout does not occur in the meter. Gas flow through turbine meters or positive displacement meters must also be avoided if the meters are to measure volumes accurately. Meters must also be proved (tested) on a regular basis. The proving of a meter establishes a meter factor which is used in calculating the volume for the period. The meter reading must also be corrected for temperature, pressure and compressibility. The calculation of actual volume through a positive displacement meter is given by the following equation:
Actual Volume =
(Gross Volume)(MF)( F tL ) 1 - (P - VP)(F)
Gross Volume = Meter reading. MF
=
Meter factor from the last proving.
FTl
=
Volume correction factor (temperature correction).
P
=
Operating pressure, psig.
(Eq 27-1)
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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VP
=
Vapour pressure at operating temperature, psig.
F
=
Compressibility factor.
Practically, all of the above is combined into a meter and shrinkage factor measured from comparing the meter reading to the actual volume recorded in a tank after the gas has left the oil. Tank Gauging is another accurate way of measuring a liquid flow rate. The time to fill a fixed, known volume of a tank is used to estimate the production rate. The volume can be determined by calculating or having calibrated the volume per unit length (or height) of the tank. Gas production rate measurement To measure the gas production, it is necessary to have a continuous flow rate recorder. In most situations a Barton meter (or its equivalent) is used. The Barton recorder is a two pen recorder that marks on a circular chart. The chart is driven by a clock mechanism. One pen records the meter run pressure upstream of a calibrated orifice. The other pen records the differential pressure (pressure loss) across the orifice. A continuous pressure recorder on the flowline can be useful in interpreting test data. It can also prove to be an invaluable asset if the bottom hole instrument should fail. With a continuous record of the surface pressure, the test will not be a complete loss. If a continuous pressure recorder is used on the flowline, it should be proved by using a dead weight tester. Continuously calculating computers for orifice meters are also available and can prove valuable in establishing the well flow rate prior to or during the test. Orifice Meters are used in conjunction with the Barton recorder to measure the gas flow rate. The orifice meter consists of a flat plate orifice, of a known size, with the absolute pressure upstream of the plate and differential pressure across it being measured. The orifice plate is installed in a section of pipe, known as the meter run with a known, fixed ID. The gas flow rate is calculated using the equation:
Q=0.024 C' hwpf where:
(Eq 27-2)
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk hw= Differential across the orifice measured in inches of water. Pf =Flowing pressure in psi. Q=Flow Rate in MCFD C'=Orifice constant The orifice constant used in most calculations is: C' = Fb*Fpb*Ftb*Fg*Ftf*Fr*Y*Fpv*Fm*F1*Fa
(Eq 27-3)
where the orifice factors are obtained from tables published by the Service Companies. The factors are: Fb = Basic orifice factor. Fb is a function of the experimental constant for different orifices and the following assumptions: Standard temperature=60°F = 520°R Gas gravity=1.00 (air) Flowing temperature=60°F = 520°R Fpb=Pressure base factor which corrects C' for the case when standard pressure is not 14.73 psia. It is calculated from: Fpb = 14.73/pb where pb is the standard pressure used. Ftb=Temperature base factor which corrects C' for the case when the standard temperature is not 60°F and is calculated from: Ftb = Tb/520 where Tb =standard temperature in °R Fg =Specific gravity force which corrects the orifice coefficient for the specific gravity of the gas where. Fg = (1/G).5 = √1/G and G = Specific gravity of the gas.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk Ftf=Flowing temperature factor which corrects for the case when the flowing temperature is not 60°F. Ftf = √ (520/(460 + Tf)) where Tf = Flowing temperature in °F. Fr=Reynolds number factor which corrects the orifice coefficient for variation in Reynolds number. Fr = 1 + b/√hwpf and b is obtained from charts. The charts assume that gas viscosity is essentially constant. Y=Expansion factor to correct for the change in gas density as the pressure changes across the orifice. The specific heat ratio, cp/cv is assumed constant at 1.3. Fpv=Supercompressiblity factor to correct for the compressibility variation of the gas from an ideal gas where: Fpv = _1/z Fm=Manometer factor which corrects for errors in a mercury actuated meter at higher pressures. It is only used when mercury type meters are recording the pressure differentials.
F1=Gauge location factor which corrects for meter locations when the meter is other than at 45° latitude and sea level. Fa=Thermal expansion factor to correct for expansion (or contraction) of the meter plate when the temperature varies appreciably from the conditions under which the orifice was bored. It is used when temperature of the meter run is above 120°F or below 0°F. Since standard tables based on experimental data are used to obtain Fb, the physical set-up of the metering system becomes important. A meter run should be sized so that it can handle the anticipated maximum and minimum flow rates. The smallest meter run should not have an ID smaller than 3 inches. To determine the meter run size needed, calculate the orifice size from the following equation: Orifice size =
C′ 250
(Eq 27-4)
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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where C' is calculated from Equation 26.1 after setting hw to pf (numerical values not units). The Meter run should then be sized from: Meter run = Orifice size x 1.5 Measurement of the pressure drop across the orifice is generally made with a bellows assembly. The bellows is connected to the differential pen of a two pen recorder. The flowing pressure is measured by a Bourdon tube which is connected to the second pen of the two pen recorder. The two pen recorder marks a chart which is later integrated to determine the amount of gas produced. To integrate the chart, several items must be known, ie, orifice size, tap type and the temperature of the flowing stream. Recently another method of measuring the pressure drop has seen use. The pressure is measured by transducers which transmit an electrical signal. In most cases the signal is fed into a microcomputer. The computer then calculates the flow rate using Equation 11.1 after the orifice size has been programmed into the computer and pressure and temperature measurements input by the transducers. Tanks A gauge tank is a simple atmospheric tank for collecting and measuring the liquids. A surge tank is a low pressure vessel serving the same purpose. During testing operations, produced fluids must be either stored in appropriate gauge tanks or disposed of in a proper manner. Produced gas can either be sold to the pipeline or flared using burners and booms. Oil can be temporarily stored until it is sold or burned. For small amounts of oil it is often more convenient to burn the oil. Water should be stored until it can be disposed of in an environmentally acceptable manner. There must be sufficient storage capacity on location at the start of a test to handle the anticipated fluid that is produced during the test.
Transfer pumps These are used to pump out the dead oil from the tanks. These are usually electrically driven but for safety can be air driven. Gas and oil diverter manifolds This is a collection of valves allowing flow to be diverted from one burner to the other. A gas diverter manifold is a simple two valve manifold. In the case of the oil manifold it also
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk incorporates more valves to allow the tanks to pumped out and diverted to the burners. An oil manifold will consist of five valves. Burners It is necessary to dispose of oil and gas offshore in an environmentally friendly manner. The issue of the protecting the environment has become more significant in the 1990’s and BP has been leading the way in this field. A burner is employed which is attached to the end of a boom with piping along it to the burners themselves. The burner consists of one or more heads which incorporate a nozzle which mixes the oil with air to atomise it and allow the oil to burn more easily. The air and oil mixture is then sprayed with a water ring to cool it and reduce the smoke effects. It is recommended that the air is supplied from a separate supply from the rig which are usually two or more (for back up) air compressors. Depending on the oil flow rate, it is usually necessary to cool the drilling rig with a water screen which also reduces the effects of radiated heat. An manually operated ignition system requires a methane supply and electricity to ignite a spark plug. The gas is simply piped to the end of boom and burns with the oil. Two booms and burners are required to cater for changes in wind direction and to avoid the drilling rig being engulfed in smoke. In summary, to burn oil offshore the following is required; 1. 2. 3. 4. 5.
A point to attach the burner boom An abundant supply of water Air compressors A methane supply (containers) Electricity for the ignitions system.
Even with the optimum operating conditions, a moderate wind, pure oil, good burner nozzles and an abundant air supply and water, there is still a risk of some oil carrying over into the sea. This is highly undesirable and not good for the environment. There is a slow move towards so called “Green Testing” whereby the oil is stored in tanks offshore or piped to a collection vessel. There are also specialist vessels with all the test equipment on board which hold the produced oil on board in tanks similar to a FPSO but on a smaller scale. This has the added benefit of offering the opportunity to sell the oil at the end of the test to offset some of the costs. Onshore, a pit is dug a safe distance from the operations and the liquid hydrocarbons simply burnt. Gas is diverted to a similar area and burnt. Consideration should be given to the heat generated which could damage the surrounding area.
Vent lines Vent lines are tied into the burners or run overboard.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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Emergency Shut Down System The well and facilities are protected by an automatic shut down system which can be manually operated as well. It will automatically shut down when an input signal is received and this can be the following; ( High pressure upstream and downstream of the choke ( Low pressure upstream and downstream of the choke (ruptured pipe) ( High liquid level in the separator ( High pressure in separator ( High liquid level in tank ( Flamable gas detection ( H2S detection ( Manual shut down
Data Recorder Usually electronic data recorders are used today for collecting all the relevant test data. This includes, the fluid rates, pressures throughout the production test system and temperatures. The information is presented in real time and can be useful in observing trends and detecting problems. Hand recording of all of the data must be carried out too in case of electronic disaster. Lubricators This is necessary to safely enter a wellbore with a wireline string. The lubricator is simply a series of steel tubes long enough to hold the wireline tools. The lubricator has a stuffing box and flow tubes at the top end, which is a sealing device which pumps viscous oil around the wireline and holds the well head pressure with the help of packing elements. There is a valve on top of the flowhead at the bottom of the lubricator. The valve is closed until the tools are inside and the lubricator is screwed together. The valve is then opened and the tools lowered into the well. The lubricator assembly is made up on the top of the swab valve (if the well is so equipped). The packing elements prevent pressure and fluid escaping from the wellbore. A stuffing box with the flow tubes generally has a working pressure of up to 10,000 psi. For electric wireline a control head with a grease injector must be used while for slick line only a stuffing box is generally employed. The type of lubricator should be coordinated with the service company providing the wireline services to ensure compatibility with the flowhead and for pressure rating.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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27.2 Subsea Equipment Lubricator valve A lubricator is a surface controlled safety valve which is used as an isolation valve in the landing string and can also act as a second barrier from the formation after the downhole valve. This allows the entry of long wireline strings into the well bore without the need for a long lubricator. Sub Sea Test Tree This is vital for safe testing operations offshore using a floating drilling rig. The valve is called different trade names and the most commonly used are E-Z Valve and SSTT. The valve functions from the surface with control hoses which are attached to the valve itself. The valve acts as a seal for pressure from below with a rotating ball valve and a backup flapper valve which will only hold pressure from below. The ball valve will cut wire, sometimes with extra assistance required but the flapper valve will not. It is possible to pump through the valve from above thus allowing the opportunity to kill the well if necessary. The main purpose of the valve is to allow the landing string to become safely disconnected from the main tubing string without the need to kill the well allowing the floating rig to move off location and leave a live well in a safe condition. The reasons for movement of the rig are usually for severe weather warnings (typhoons or hurricanes) or could be for any reason which threatens the safety of the rig in its testing location. The “Dummy Run” When testing offshore from a floating structure, the test string is fixed in the BOP on the seabed at the hanger. The section of the test string from the fluted hanger to the rig floor is called the landing string. Since the rig moves up and down around this section of tubing/pipe there must be enough sticking up above the rig floor to cater for tides and heave in bad weather. The annulus is made by closing the rams (usually the pipe rams and/or annular preventer) around a specially prepared short section of pipe called a slick joint above which the sub surface test tree is placed. On the lower end of the slick joint is a fluted hanger which fits into the hanger in the BOP. If the Sub Surface Test Tree has to be employed, it is important to be able to close the top rams or the annular preventer above the lower section of the test tree left in the BOP stack. While there will usually be a diagram of the BOP Stack on the rig, it is wise not to rely on it and to actually physically check the selected rams close on the slick joint and not on the Test Tree. This is done by performing a “dummy run”.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk To perform a “dummy run”, the slick joint is painted white. This is then attached to the landing string with no Test Tree and with the fluted hanger attached below. The landing string is then run until the fluted hanger lands off in the BOP hanger. The selected rams are closecd, left for a few moments closed and then opened again. While this is done the actual length of landing string run can be measured at the rig floor so that suitable length “pup joints” can be selected to obtain the optimum stick up. Once the landing string has been pulled out of the hole again, the painted slick joint is examined to ensure that the rams closed somewhere in the middle of it and not where the Test Tree would have been. To summarise, the “dummy run” achieves two objectives; 1. To confirm that the position of the rams is such that they will close on the slick joint and not the Test Tree. 2. To allow pup joints to be selected to give the optimum stick up on the rig floor. 27.3 Pressure Measurement Downhole Gauges Bottom hole pressure is usually obtained by running a bottom hole pressure (BHP) gauge to a predetermined depth on wireline or by running a number of gauges (frequently four) in the downhole test assembly. Normally, the gauges are set at the bottom of the open perforations. However, in all cases it is important to position the gauge as near as practically possible to the perforations or formation. The majority of the gauges are either "hung off" downhole or are suspended downhole on a wireline. To hang off a gauge, an instrument hanger, wireline mandrel and a seating nipple are used. All of the wireline run equipment must be able to pass through the tubing string and be able to be held in place, when "hung off" by the appropriate completion equipment. Therefore, the equipment run in with the tubing will dictate the future feasibility of testing a specific well. When using gauges, which are almost always now electronic, the following should be noted; i) It is necessary to allow the gauges to reach thermal and mechanical equilibrium. This is especially important during static surveys. ii) Proper handling of any hysteresis effects. Hysteresis, if present, will be known if the instrument is properly calibrated.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk iii) Proper calibration of gauges is critical, if believable and accurate pressures are to be obtained. With surface readout the downhole pressure can be obtained in real time and analysis can be performed during the test. The test can then be terminated as soon as enough data is gathered and the well returned to production, suspended or abandoned. Furthermore, operational problems can be detected and the test modified or curtailed as required. Several items should be considered when planning a test using surface readout instruments; i) The power supply available must have a voltage and frequency compatible with the instrument. ii) If the equipment is battery powered, extra batteries should be on location and the batteries should be tested prior to being run. iii) The surface equipment may require a protected environment (air conditioning) iv) Surface readout gauges are not recommended for use in H2S or other corrosive environments. The gauges are carried in a gauge carrier and can be arranged to record pressure from within the tubing or the annulus pressure. Dead Weight Tester A dead weight tester should be used to monitor the surface flowing pressures as a back-up to any data acquisition system. Surface pressures should be recorded periodically during the test with earlier time readings being made more frequently. 27.4 Downhole Equipment The practical aspects of well testing have a significant impact upon the quality of the data acquired and therefore on the validity of the test results. Landing Nipples The industry has various types of landing nipples available to be used in the tubing string. The two main types of nipples are selective and non-selective. Selective nipples allow any number of nipples of the same ID to be located in the tubing string. The nipple generally has a top no-go which is an enlarged ID that the running tool passes down through. Once
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk the desired nipple has been located, the wireline tools are moved upward into the nipple. The dogs on the locking mandrel will engage the no-go. The mandrel is then released from the wireline. For a bottom hole shut-in tool, the packing of the mandrel seals in the polish bore of the nipple. When using an instrument hanger, a mandrel without packing is used so that the well fluids can mover through the tubing. Other downhole shut in tools are self contained and usually annular pressure operated. There are some drawbacks to running bottom hole pressure tests using the hanging-off procedure. If the well produces sand or was fracced either sand can settle in the nipple and cause the packing seals on the mandrel to leak, (thereby nullifying the desired bottom hole shut-in) or sand will fall off the side of the tubing and land on the mandrel causing it to stick (A stuck mandrel can be difficult if not impossible to retrieve by wireline. Furthermore the additional wireline work necessary to run, set, and later retrieve the bottom hole pressure tools can sometimes damage these sensitive tools. If the wellbore does not have landing nipples, there are other alternatives in obtaining bottom hole pressure data. One method is to leave the instruments suspended by the wireline. The main disadvantages of this are the increased cost of a wireline unit on location and deterioration and failure of the wireline if it is exposed to a hostile environment (ie H2S) over a long period of time. There is also a safety consideration in leaving the wireline in the well. If a well has to be shut-in, the only valve that can be closed is the wing valve since closing the master valves will cut the wireline resulting in a fishing job. It is, therefore, recommended that a soft set running tool and a shock absorber be used when "hanging off" bottom hole pressure instruments. The softest tools are designed to release without a jarring action. The shock absorber suspends the recording instruments below the landing mandrel. The springs of the shock absorber are designed to minimise impact on the bottom hole instruments during the running and retrieving operations.
Wireline Reentry guide This is a specially strengthened slant ended short piece of pipe to guide wireline tools back into the test string. Packers A packer is a tool which is designed to isolate one section of the wellbore from another. It also acts as a support for some or all of the test string above it. There are two types of packers available, permanent and retrievable. Both types are designed to isolate sections of a wellbore by using slips and elastic packing elements. Permanent packers are set using an
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk electric wireline or by applying pressure to expand certain parts of it and to allow the slips to grip the casing and set the packer. A setting tool sets off a charge which causes the slips to set and the packing element to expand and seal. The permanent packer has a polished bore into which a tubing seal element is positioned. A permanent packer allows tubing movement and removal of tubing. If tubing subs and a nipple are run on the bottom of the packer, a plug can be used to shut-in the well while the tubing is removed without killing the production zone. Permanent packers hold pressure from above or below the packer. Permanent packers are only permanent in the sense that when the tubing is removed, the packer remains in the hole. Permanent packers can be removed by milling over the packer slips and fishing the tool. Retrievable packers are designed to be removed from the well or hole when the tubing is removed. Retrievable packers can be set in a number of ways. The most common methods are mechanical and hydraulic. Mechanical setting varies with the packer but usually a combination of rotating the tubing and setting down some of the tubing weight on the packer is employed. Hydraulic systems generally involve exceeding a predetermined pressure to shear pre-set shear pins which allow the packer to set by the hydraulic pressure acting on pistons. Wireline set packers are similar to the permanent packer and vary as to whether they hold pressure from either above or below, or both. These packers are difficult to drill or mill up if left in the hole but can be run with tubing subs and nipples below the packer. Consideration should be given to whether or not the packer will be subjected to a greater pressure from below than above, as is experienced during stimulation pumping. Some retrievable packers have slips that only prevent movement downwards and will therefore come unseated when pressured from below. Tester Valve The tester valve is a valve that is opened and closed for flow and shut-in periods. The valve is actuated by either a reciprocating motion, rotation or more usually today by annular pressure. In the case of a rotating ball valve, the valve opens to the full bore size, usually 2 ¼”, the valve is designed to cut wire (so beware of this fact when performing wireline operations) and to open with a differential pressure across it. More recent innovations for the operations of down-hole tester valves include pre-programming of the tool by electronics and operating it by sending lower pressure pulses to it via the annulus. Recent tester valves also allow circulation for the spotting of acid or nitrogen. Reversing Valves
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2001
www.opc.co.uk These are “one shot” or re-usable valves establishing communication between the annulus and the tubing. At the end of a test it is useful to be able remove the oil from the tubing and this is achieved by reverse circulating (pumping down the annulus and up the tubing). Two are usually employed for reduntancy. Safety Joint This is a short length of pipe used to back off the string from above the packer to retrieve the gauges in the event the packer gets stuck in the well. Hydraulic jar This is used in conjunction with the safety joint to create a “hammer” action to assist in retrieving stuck pipe. 27.5 Quality Control Checks Correct field interpretation of data is very important because once the pressure record is retrieved, the decision must be made to either accept the accuracy of the data or repeat the test. A similar judgement regarding test quality must also be made by the reservoir engineer when subsequently evaluating the data. It is therefore necessary to determine whether or not the test was satisfactory from a mechanical standpoint, and whether the accuracies of the recorded pressures are adequate for subsequent evaluation. It is therefore suggested that the guidelines discussed earlier are followed. Occasionally, a well may indicate depletion and a limited reservoir based on a DST. Such an observation is made when the extrapolated final shut-in has a pressure lower than the pi from the initial shut-in. Before making such a statement, the DST data should be examined closely. Earlougher states that depletion effects should not even be considered unless the difference between pisi and pfsi is more than 5% of pisi. Earlougher also writes that a second test should be conducted prior to concluding that a reservoir is limited. Barriers, reservoir heterogeneity, limited flow times, and gauge inaccuracies may also contribute to a lower value of pfsi. For the reasons noted above, a conclusion that the reservoir has experienced depletion should be examined closely. Very limited reservoirs may indeed see a pressure drawdown during a DST. The value of a reservoir that shows depletion after a DST may also be limited. This fact should be discovered by the DST analyst - not by an uneconomic full field development.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk 27.6 Sampling The purpose of obtaining a reservoir fluid sample is to collect a sample which is representative of the reservoir fluid. A reservoir fluid sample is analyzed in the laboratory to determine fluid characteristics. The results of the laboratory analysis are used to manage the reservoir production. If the samples obtained are not characteristic of the reservoir fluids, then effective management of the reservoir can not be attained. The proper sampling of reservoir fluids is thus quite important hence the planning and taking of samples will be discussed in the following sections. Sample Type Reservoir fluids can be sampled by either subsurface means or by collecting a sample on the surface. Representative reservoir fluid samples may be collected using a subsurface sampler when the flowing bottom hole pressure is greater than the bubble point for oil or dew point for gas-condensate. For a well that makes free water or when a large volume of sample is required, subsurface samplers are not good choices. Surface samples can be collected in two ways; Recombination. Surface samples are taken at the wellhead or at separators. Planning the Sampling Programme The first step in planning a sampling programme is to determine if a sample of original reservoir fluid is required or is the sample simply that of the fluid below the bubble point or dew point. The former is the more difficult task and will be covered in this discussion. For the latter case, a recombination sample is recommended. Type of Reservoir The sampling programme and technique are dependent upon the type of reservoir that is being studied. There are four basic types of reservoirs: 1. 2. 3. 4.
Oil Volatile oil Gas Condensate Gas
The first three types of reservoirs require that a sample be obtained before the bottom hole pressure drops below the bubble point/dew point pressure. Once the bottom hole pressure falls below the bubble point/dew point, the original fluid characteristics will not
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk be observed. Thus, the original samples must be obtained early in a reservoir's life. For a gas reservoir the fluid remains constant throughout the life of the reservoir so a sample may be obtained at any time. Reservoir Characteristics The following characteristics should be considered when planning the sampling programme. If a group of wells is separated from another group because of structural and/or stratigraphic changes, each group of wells should be sampled. Thick reservoirs often have variations in fluid compositions and should be sampled at several depth intervals. If the reservoir appears to be homogeneous, but GOR and/or oil gravity vary over the field, each area of similar GOR and gravity should be sampled. Condition of the reservoir fluid near the wellbore must be considered (avoid gas coning, high flow rates creating high gas saturation near the wellbore, near wellbore pressure is below the bubble point, etc.). Producing Characteristics Important producing characteristics of a well which should be recorded at the time of taking a sample are: Current water production, if any. GOR and oil gravity typical of surrounding wells and reservoir. Well productivity index:- high or low. Well flows naturally. Presence of water-oil, gas-oil, gas-water contacts. Conditioning the Well The aim of sampling a well is to obtain a sample which is representative of the reservoir fluid. In the wellbore and near wellbore vicinity, the fluid is often not representative of the reservoir fluid. This is because the greatest pressure drop occurs close to the wellbore and this often results in some alterations of the otherwise typical reservoir fluid.
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2001
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Shutting in a well may restore the well to reservoir pressure, but it may not restore the fluid to its original composition. Conditioning a well is designed to restore the fluid composition to original reservoir conditions by displacing the altered fluid with representative reservoir fluid. Conditioning generally requires flowing a well at much lower than normal rates. Conditioning the different types of well is reviewed below. Flowing Oil Wells A flowing oil well is conditioned by producing the well at successively lower rates thus decreasing the pressure drop near the wellbore. The well is flowed at each rate until the GOR stabilises. The flow rate reduction continues until the GOR remains constant for two successive flow rates. A constant GOR indicates that the oil is above the bubble point. If the GOR decreases after a rate reduction, this can be indicative of free gas saturation near the wellbore. If GOR increases after a rate reduction, it can be indicative of the well producing from a gas zone and an oil zone. Such a well should probably not be used for sampling. Pumping Oil Wells The same procedure for flowing oil wells is followed except that the rate reduction is obtained by reducing the pump rate. Most wells on pump will be below the bubble point and are not satisfactory for obtaining a bottom hole reservoir fluid sample in the hopes of evaluating conditions above the bubble point. Gas Condensate Wells Gas condensate wells are conditioned in the same fashion as a flowing oil well. Volatile-Oil Wells Conditioning a volatile oil well is as for a flowing oil well. However, in a volatile oil well, GOR will usually remain stable between flow rates even though the fluid composition is changing. If a reservoir is believed to be a volatile oil reservoir, then reservoir fluid samples should be obtained as soon as possible after completion. It may also be prudent to consider having anti foaming chemicals on the location to prevent (or minimise) foaming in the separator.
27.7 Safety The safe conduct of any test is of the utmost importance. The safety considerations covered are general recommendations and are not intended to be an exhaustive list but some ideas
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk for consideration. Each test will be different and so adjustments to the safety items will be necessary for each test while there are always some facts which will never change. Before conducting any test, a check with regulatory agencies for any specific safety requirements should be made.
Ensure that all equipment is certified by an external certifying body such as Det Norsk Veritas or similar. Make sure that the wellhead, flanges, fittings and other equipment rated working pressure is greater than the expected pressures and is compatible Make sure that there is no leakage at or through any pipe, valve or connection. In other words pressure test everything prior to testing. Ensure there is a safe method of igniting the flare. If well fluids contain H2S, ensure that all personnel are aware of potential H2S problems and safety equipment is available. Comply with any company and/or government safety requirements. Use common sense. Restrict the use of all hot work. It is recommended that all hot work ceases during flowing of the well and is limited to safe areas once the well is live. Keep personnel outside the accommodation to a minimum during flowing. Establish criteria for opening the well in the dark and for shutting the well down if H2S is encountered unexpectedly. Hold regular safety meetings, especially prior to opening the well for the first time to ensure that all personnel are aware of what will be happening. Do not lift objects over wireline while running in or out of the hole or over pressured pipes or vessels. Smoking is prohibited outside of the accommodation. Make sure the emergency shut down buttons function and everyone knows where they are. Arrange for someone to check the pipes for leaks throughout the flow period.
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2001
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Ensure that downhole safety valves are open but able to function. Do not leave wireline in hole without government or company approval. Ensure that a tool joint is not in the pipe rams when the packer(s) is set and verify that the pipe ram rubbers are the proper size for the pipe. Note also that BOP's should be tested prior to beginning a DST. Know the formation strength at the casing seat from an earlier pressure test or from offset information. Make sure that the choke manifold, and/or meter run, is rated for the maximum surface pressure anticipated. Insure that flare line(s) is properly staked and that the flare can be ignited safely. Have the wellhead rated for greater than the maximum anticipated pressure. Check the opening and closing of the valves on the wellhead. Keep the annulus full tripping in and out of the hole and monitor the annulus fluid level during the test. Keep track of volumes and pump strokes into the annulus. It is recommended that any DST be run during a period of good visibility, preferably daylight. Visibility is necessary when observing the annulus and fluid flow, if any, through the choke manifold. Comply with any company and regulatory authority requirements. 27.8 Environmental issues BP are leading the way in minimising the impact on the environment from burning hydrocarbons. The following is taken from internal BPX correspondence. In line with the BPX stated goal of no damage to the environment , John Browne’s (BP’s Chairman) commitment that our goal is to only flare in emergencies, and to reduce emissions to zero as soon as practicable, the following is an attempt to give guidance on the environmental considerations during Well Testing. Given that the technology to allow us to meet these goals is not yet available and there are cases where testing is considered advantageous to optimum development of any given reservoir, it is important that the environmental and business implications of testing are fully considered. It is not the intent to ‘prohibit’ testing within BPX but to ensure that consistent decisions are made in line with our corporate aspirations These guidelines apply to both operated and partner operated activities. It is acknowledged that partner operations can be more difficult however we still require assurance that we are doing all in our power to influence our partners
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2001
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1. The need to test and preplanning Prior to any well test, either conventional or extended, it is important that a full review of the need to test the well is conducted by an appropriate level of peer review. Once this has been agreed the duration of the test must be minimised to reduce the volume of emissions which will be caused by any well testing . Prior to any testing the Business unit must evaluate all emissions and clearly have a program of environmental protection in place. “ In all cases where testing takes place the most efficient burners available must be used. Prior to well testing the business unit must ensure to the satisfaction of the peer review team that measures have been put in place to ensure cleanest possible burn and to reduce potential start up problems. At completion of the operation a full environmental report will be prepared 2.Gas Testing Currently all testing involves flaring or in certain cases cold venting where the option of gas disposal does not exist.(eg. Export or re-injection to a nearby well) For short duration tests flaring or cold venting will be allowed, for longer (more than 1 week) tests all practical efforts must be made to dispose of the gas either by export or via re-injection. In cases where this is deemed impracticable by the business unit EXCO approval must be received prior to testing. 3.Oil testing For short term tests flaring is allowed on remote sites where liquid export is impracticable. Offshore, drop out must be minimised and vessels with anti pollution capability available on standby. A peer review of operational procedures and equipment must be performed. Onshore, bunding must be used to capture any unburnt liquid hydrocarbons which must then be disposed of in an appropriate manner. For extended well tests flaring liquids is unacceptable . Associated gas will fall within the same criteria as gas in section 2. In all cases testing must be immediately suspended if equipment in use is not operating to the standards assumed in preplanning of the test. As we move forward we will be investigating new testing technology including gas to liquid conversion, and business unit support will be sought to help drive these technologies through to delivery. Reference should be made to the appendix which has two flow charts referring to the planning phases of the testing operations and considerations to the environment.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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28. INTERPRETATION PROCEDURE, REPORTS AND PRESENTATION OF RESULTS The procedure to perform a well test interpretation and the style and content of any well test interpretation report will vary from location to location. This is in no way intended as a strict format to follow but is included as a guide since it may be useful in the absence of any other guidelines.
An outline for the method to perform the analysis follows; 1. Gather all available data together and review it. 2. Determine other data that is required but which is not available and make estimates in consultation with other specialists with knowledge of the well and reservoir. The data required is listed in section 2.3. 3. Load pressure, rate and static data into the chosen software package. 4. Compare gauges’ data (assuming more than one) and select one which has the longest (not necessarily the most) amount of reliable useful data. 5. Examine the plots and confirm that the data can be analysed and that it is not all well bore storage. 6. Perform a diagnostic analysis to determine initial values of permeability, skin and boundary distances to ensure the solutions are within the expected range (if known). 7. Select a type curve model 8. Attempt to obtain a suitable match on all three plots; log-log/derivative, semi log/superposition, data plot. 9. Repeat steps 7 and 8 for as many reservoir models as are suitable. 10. Produce a report which may include all the plots with both diagnostic and type curve analysis. It may be a good idea to include in the report a summary of the results at the beginning, a description of the parameters used in the analysis, their origin and any assumptions which were used to derive them. It is also useful to include somewhere a sequence of events for the test, a summary of the actual data used in the analysis so that someone else could repeat the interpretation at a later date, a log of the reservoir interval and a completion diagram.
The reports last a long time and ideally should contain everything that another engineer would need to repeat the interpretation several years later since it will usually become separated from other data.
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2001
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28.1 Methodology Well test analysis is a three steps process : 1 2 3
Identification of the interpretation model. The derivative plot is the primary identification tool. Calculation of the interpretation model. The log-log pressure and derivative plot is used to make the first estimates. Verification of the interpretation model. The simulation is adjusted on the three usual plots : log-log, test history and superposition.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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Log-log analysis
Model selection (derivative)
Estimate parameters : kh, C, heterogeneities , boundaries (derivative) and S (pressure)
Simul
Test history simulation
#1 . . . . . . #n
• Adjust initial pressure pi • Check the data (variable skin, consistent rate history) • Check the model response on a larger time interval
Superposition simulation
Adjust parameters (pi, S, C...)
Next model End
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2001
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28.2 The diagnosis : typical pressure and derivative shapes
Flow regime identification GEOMETRY
LOG-LOG shape
slope
TIME RANGE Early
Intermediate
Late
Radial
No 0
Double porosity restricted
Homogeneous Semi infinite behavior reservoir
Linear
1/2 1/2
Infinite conductivty fracture
Horizontal well
Two sealing boundaries
Finite conductivity fault
Double porosity unrestricted with linear flow
Bi-linear
1/4 1/4
Spherical
No -1/2
Pseudo Steady State
1 1
Finite conductivity fracture
Well in partial penetration
Wellbore storage
Layered no Closed crossflow reservoir with (drawdown) boundaries Conductive fault
Steady State
0 -1 (−∞)
Constant pressure boundary
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2001
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Changes of properties during radial flow Mobility decreases : Sealing boundaries, composite reservoirs, horizontal well with a long drain hole. Dp
Log(Dp')
Log(Dt)
Figure 28-1 The mobility decreases (kh ↓).
Log(Dt)
Log-log and semi-log scales.
Mobility increases : Composite reservoirs, constant pressure boundaries, layered systems, wells in partial penetration. Dp
Log(Dp')
Log(Dt)
Figure 28-2 The mobility increases (kh ↑).
Log(Dt)
Log-log and semi-log scales.
Storativity increases : Double porosity reservoirs, layered and composite reservoirs.
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
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Log(Dp')
Log(Dt)
Log(Dt)
Figure 28-3 The storativity increases (φ ct h ↑). Log-log and semi-log scales.
Storativity decreases : Composite systems. Dp
Log(Dp')
Log(Dt)
Log(Dt)
Figure 28-4 The storativity decreases (φ ct h ↓). Log-log and semi-log scales.
28.3 Summary of usual log-log responses Well models Wellbore storage and Skin p'
Wellbore storage, C Radial, kh and S
C p&
1 2
1 S
kh ∆t
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2001
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Linear, xf Radial, kh and ST
p&
1 2
p'
Infinite conductivity fracture 1/2 xf
kh, ST
∆t
Bi-linear, kf wf Linear, xf Radial, kh and ST
xf
p&
1 2 3
p'
Finite conductivity fracture kh, ST
1/2
kf w f
1/4 ∆t
Radial, hw and Sw Spherical (mobility ↑), kV Radial, kh and ST
-1/2
p&
1 2 3
p'
Partial penetration
kV
kh, ST
hw, Sw ∆t
Radial vertical, kV and Sw Linear (mobility ↓), L Radial, kh and ST
p&
1 2 3
p'
Horizontal well
1/2 L kV , Sw ∆t
Reservoir models
kh, ST
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3
Radial fissures, k Transition (storativity ↑), ω and λ Radial fissures + matrix, kh and S
p'
1 2
restricted ω
kh, S
p&
Double porosity, interporosity flow
λ ∆t
1 2
p'
unrestricted
Transition, λ Radial fissures + matrix, kh and S
p&
Double porosity, interporosity flow
λ
kh, S
∆t
Radial inner, k1h and Sw Transition (mobility ↑ or ↓), r Radial outer, k2h and ST k1h > k2h;
p&
1 2 3
p'
Radial composite
k2h, ST
k1h, Sw
k1h < k2h
r
∆t
Radial inner, k1h and Sw Transition (mobility ↑or ↓), L Radial total, (k1h+k2h)/2 and ST k1h > k2h; k1h < k2h
p&
1 2 3
p'
Radial Composite
(k1h+k2h)/2, ST
k1h, Sw L
∆t
1 2 3
No crossflow Transition (storativity ↑), ω, κ and λ (kV) Radial, kh1+kh2 and ST
p'
skin ω,κ
p&
Double permeability, same S1=S2
kh, ST λ
∆t
2001
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1 2 3
Radial, k2h2 and S2 Transition (mobility ↑), λ (kV) Radial, kh1+kh2 and ST
p'
partial k2h2, S2
p&
Double permeability, penetration S1= ∞
kh, ST λ
∆t
Boundary models
1 2 3
Radial, kh and S Transition (mobility ↓), L Hemi-radial
p&
p'
Sealing fault
kh, S L
∆t
1 2 3
Radial, kh and S Linear, L1+L2 : Off-centered Radial, kh and S Hemi-radial, L1 Linear, L1+L2
p&
1 2
p'
Channel 1/2
L1
L1+L2 kh, S ∆t
Radial, kh and S Linear, L1+L2 Transition (mobility ↓), L3 Semi-linear
1/2
p&
1 2 3 4
p'
Channel closed at one end
1/2 L3
L1+L2
kh, S ∆t
1 2 3 4
Radial, kh and S Linear, L1+L2 Fraction of radial, θ : Off-centered Radial, kh and S Hemi-radial, L1 Linear, L1+L2 Fraction of radial, θ
θ
p&
1 2 3
p'
Intersecting faults
L1
1/2 L1+L2
kh, S ∆t
2001
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p'
Closed system centered
1
Radial, kh and S Pseudo steady state, A : Build-up Radial, kh and S
2
Average pressure, p and A
1
p&
1 2
p
A
kh, S
∆t
Closed channel 1
3
Average pressure, p and A
p
p'
1 2
Radial, kh and S Linear, L1+L2 Pseudo steady state, A : Build-up Radial, kh and S Linear, L1+L2
1/2
p&
1 2 3
L1+L2
A
kh, S ∆t
Closed with intersecting faults
4
Average pressure, p and A
1 p
p'
1 2 3
Radial, kh and S Linear, L1+L2 Fraction of radial, θ Pseudo steady state, A : Build-up Radial, kh and S Linear, L1+L2 Fraction of radial, θ
p&
1 2 3 4
θ
A
1/2 L1+L2
kh, S ∆t
1 2
Radial, kh and S Transition (mobility ↑), L One boundary Multiple boundaries
p&
p'
Constant pressure boundaries -1
kh, S
∆t
28.4 Consistency check with the test history simulation
L
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2001
www.opc.co.uk In the following examples, the initial pressure is 5000 psi. The interpretation model, defined from log-log analysis of the short shut-in period, may be inconsistent when applied to the complete rate history.
Increase of derivative response after the last build-up point (second sealing boundary) The log-log derivative plot suggests the presence of a sealing fault.
Dp & Dp', psi
1.0E+00
1.0E-01
1.0E-02
1.0E-03 1.0E-03
1.0E-02
1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
Dt, hr
Figure 28-5 Log-log plot of the final build-up. Homogeneous reservoir with a sealing fault.
The sealing fault model is not applicable on the extended production history.
pressure, psi
5000
pi=4914 psi
4800
4600
4400 0
200
400
600
800
time, hr
Figure 28-6 Test history simulation. Linear scale. Homogeneous reservoir with a sealing fault.
1000
10 95 90 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 50 0 1200
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Dp & Dp', psi
1.0E+00
1.0E-01
1.0E-02
1.0E-03 1.0E-03
1.0E-02
1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
Dt, hr
Figure 28-7 Log-log plot of the final build-up. Homogeneous reservoir with two parallel sealing faults.
pi=5000 psi
pressure, psi
5000
4800
4600
4400 0
200
400
600
800
1000
100 950 900 850 800 750 700 650 600 550 500 450 400 350 300 250 200 150 100 500 0 1200
time, hr
Figure 28-8 Test history simulation. Linear scale. Homogeneous reservoir with two parallel sealing faults.
When a second sealing fault, parallel to the first, is introduced farther away in the reservoir, the extended production history match is correct.
Decrease of derivative response after the last build-up point (Layered semi infinite reservoir)
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The log-log derivative plot suggests the presence of two parallel sealing faults.
Dp & Dp', psi
1.0E+00
1.0E-01
1.0E-02 1.0E-03
1.0E-02
1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
Dt, hr
Figure 28-9 Log-log plot of the final build-up. Homogeneous reservoir with two parallel sealing faults.
With the parallel sealing faults model the initial pressure before the production history is too high.
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10000 9500 9000 8500 8000 7500 7000 6500 6000 5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0
5500
pi=5443 psi
pressure, psi
5000
4500
4000
3500
3000 0
200
400
600
800
1000
time, hr
Figure 28-10 Test history simulation. Linear scale. Homogeneous reservoir with two parallel sealing faults.
The reservoir is a two layer no crossflow, one layer is closed. At late time, the derivative stabilizes to describe the radial flow regime in the infinite layer. The hump at intermediate time corresponds to the storage of the limited zone.
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Dp & Dp', psi
1.0E+00
1.0E-01
1.0E-02 1.0E-03
1.0E-02
1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
Dt, hr
Figure 28-11 Log-log plot of the final build-up. Two layers reservoir, one infinite and one closed layer.
10000 9500 9000 8500 8000 7500 7000 6500 6000 5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0
5000
pi=5000 psi pressure, psi
4500
4000
3500
3000 0
200
400
600
800
time, hr
Figure 28-12 Test history simulation. Linear scale. Two layers reservoir, one infinite and one closed layer.
1000
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29. APPENDICES 29.1 Nomenclature
A B cg co ct ct− C CA D e Ei k kf kH km ks kV h hw L m m(p) m* M n p pf pi pm pw p* p− q r ri rm rw S
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Surface, sq ft Formation volume factor, RB/STB Gas compressibility, psi-1 Oil compressibility, psi-1 Total compressibility, psi-1 Total compressibility at the average pressure of the test, psi-1 Wellbore storage coefficient, Bbl/psi Shape factor Diffusivity ratio (outer zone / inner zone), or turbulent flow coefficient Exponential (2.7182 . . .) Exponential integral Permeability, md Fracture or fissures permeability, md Horizontal permeability, md Matrix blocks permeability, md Spherical permeability, md Vertical permeability, md Thickness, ft Perforated thickness, ft Distance, or length of an horizontal well, ft Straight line slope (semi-log or other) Pseudo pressure or gas potential, psia2/cp Slope of the pseudo steady state straight line, psi/hr Mobility ratio (outer zone / inner zone) Number of fissure plane directions, or turbulent flow coefficient Pressure, psi Fissure pressure, psi Initial pressure, psi Matrix blocks pressure, psi Well pressure, psi Extrapolated pressure, psi Reservoir average pressure, or during the test, psi Flow rate, STB/D or Mscf/D (103scft/D) Radius, ft Radius of investigation, ft Matrix blocks size, ft Wellbore radius, ft Skin coefficient, or saturation
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk Spp St Sw
= Geometrical skin of partial penetration = Total skin = Skin over the perforated thickness
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk
t tp v V xf w Z Z− α β δ ∆ γ φ φf φm κ λ µ µ− θ θw σ ω ρ
= = = = = = = = = = = = = = = = = = = = = = = = =
Time, hr Horner production time, hr Volume, cu ft Volume ratio (fissures or matrix), or flow velocity Half fracture length, ft Fracture width, ft Distance to the lower reservoir limit, ft, or real gas deviation factor Real gas deviation factor at the average pressure of the test Geometric coefficient in λ , or transmissibility ratio of a semi-permeable fault Transition curve of a double porosity transient interporosity flow Constant of a β curve Difference Euler's constant (1.78 . . . ) Porosity, fraction Fissures porosity, fraction Matrix blocks porosity, fraction Mobility ratio Interporosity (or layer) flow coefficient Viscosity, cp Viscosity at the average pressure of the test, cp Angle between two intersecting faults Well location between two intersecting faults Geometrical coefficient of the location of a well in a channel Storativity ratio Density, lb/cu ft
= = = = = = = = = = =
Absolute Open Flow Potential Bi-linear flow (slope m) Build-up Dimensionless External Fracture, fissures or formation Horizontal Initial or investigation Intersection of straight line Linear flow (slope m) Matrix
Subscripts AOF BLF BU D e f H i int LF m
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk o p pp ps PSS s sc S SLF SPH t V w wf ws WBS 1 2
= = = = = = = = = = = = = = = = = =
Oil Production (time) Partial penetration Pseudo (time) Pseudo permanent Spherical Standard conditions Skin Semi linear flow (slope m) Spherical flow (slope m) Total Vertical Well, or water Flowing well Shut-in well Wellbore storage regime (slope m) Inner zone, or high permeability layer(s) Outer zone, or low permeability layer(s)
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk 29.2 Liquid to gas conversion chart
2001
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
www.opc.co.uk 29.3 Flaring Flow Chart
BPX WELL TESTING ENVIRONMENTAL GUIDELINES Asset Team Decides Data or Clean-Up Requirements
BPX Goal Formation of Peer Review • Appropriate Level • Skills Brought • Demonstrably Independent Of the Asset
Should be Early Enough to Influence Design
• Is PVT critical, can you use an MDT • Is Deliverability critical, what about if the well is obviously good • Distance to faults and other late time effects will they really change anything. • DST’s can reduce uncertainty not remove it
Data Acq. or Clean-Up Plan
• How clean do you need to be • What completion fluid, what perf debris • Why can’t the production facilities handle the fluids • What would it take to enable the production facilities to take the fluid
Must Not Be Flared
Objective Of Peer Review • Justification Of Emissions
• The technology to achieve this along with Testing and Clean Up is not yet available but we must minimise the impact.
– Will it change any decisions ?
• Minimisation of Emissions – Have all appropriate measures been taken ? – Is the best equipment being used ?
• Feasibility Of Ops Plan
Peer Review
Internal Challenges on Environmental Aspects
Oil
Assoc. Gas
Oh No A Decision Shr ub
Yes
– Will it work Operationally – Will it get the Data required
Business Unit Leader
Environmental Report Prog Vs Act Emissions Sheen Monitoring Spills Chemicals Used / Discharged Equipment Performance
Is the Test Longer than 1 Week ?
Gas Exco Approval Required Inform Tech. Directors
Lessons Learned
No Yes
No
Detailed Programme
• Zero Damage to the Environment • Flare only in Emergencies • Reduce Emissions to Zero
Environmentally or Politically Particularly Sensitive
Statutory Approval To Flare
If the Equipment Performance is unsatisfactory and causing Pollution or Excess Emissions
Asset Assurance Procedures Evaluation of Emissions Environmental Impact Programme of Environmental Protection Contingency Plans HAZOP/HAZID/ Risk Assessment
Test or Clean-Up
Peer Review Technical Integrity
Suspend Immediately
Fundamentals of Well Test Design and Analysis, D. Bourdet and P. Johnson
2001
www.opc.co.uk 29.4 Testing flow chart
Well testing and environment Guidelines flowchart
Complete environemtal report
Burn gas
Burner selection and burner trial
Peer Review of operational procedures and equipment
ExCo approval to burn gas > 1 week
Gas Team decided test required
Peer Review on test need
Environmental assessment and control program
Technical Director endorsement
Oil Complete environemtal report
Burn oil and gas short tests only
Burner selection and burner trial
Peer Review of operational procedures and equipment