Pressure Transient Analysis

CHAPTER 2.PRESSURE TRANSIENT ANALYSIS IN DRAWDOWN AND BUILDUP EPS Training 40 CHAPTER 2. PRESSURE TRANSIENT ANALYSIS IN DRAWDOWN AND BUILDUP......................................39 2.1. Background to Transient Pressure Analysis .........................................................................................41 2.1.1. Introduction .............................................................................................................................41 2.1.2. Development of Pressure Testing .......................................................................................42 2.1.3. Exploration Well Testing........................................................................................................43 2.2. Radial Flow Theory....................................................................................................................................49 2.2.1. The Basic Flow Equations .....................................................................................................49 2.2.2. Fluid of Constant Compressibility........................................................................................50 2.2.3. Further Development of the Accumulation Term...............................................................51 2.2.4. Linearisation of the Radial Flow Equation...........................................................................52 2.2.5. Initial and Boundary Conditions...........................................................................................53 2.2.6. Dimensionless Form of the Diffusivity Equation ...............................................................55 2.2.7. The Line Source Analytical Solution in an Infinite Reservoir..........................................58 2.2.8. Well-bore Damage and Improvement Effects .....................................................................60 2.2.9. Analytical Solution for the Case of a Bounded Circular Reservoir .................................62 2.2.10. Analytical Solution for a Constant Pressure Outer Boundary .......................................65 2.2.11. SPE Field Units ......................................................................................................................66 2.2.12. The Depth of Investigation and Radius of Drainage.......................................................66 2.2.13. The Dynamics of Reservoir Pressure Response..............................................................71 2.3. Pressure Drawdown Testing ....................................................................................................................72 2.3.1. Introduction .............................................................................................................................72 2.3.2. Pressure Drawdown Analysis in Infinite-Acting Reservoirs ...........................................73 2.4. The Principle of Superposition.................................................................................................................76 2.4.1. Introduction .............................................................................................................................76 2.4.2. Multiple-Well Situations........................................................................................................76 2.4.3. Variable Rate Situations .........................................................................................................77 2.5. Pressure Build-Up Testing........................................................................................................................78 2.5.1. Introduction .............................................................................................................................78 2.5.2. Pressure Build-Up Test Analysis during the Infinite-Acting Period ..............................79 2.5.3. After Production......................................................................................................................81 2.5.4. Determination of Reservoir Parameters................................................................................82 2.5.5. Peaceman Probe Radius .........................................................................................................86 EPS Training 41 2.1. Background to Transient Pressure Analysis 2.1.1. Introduction One of the greatest problems facing the petroleum engineer is that of characterising the physical nature of the subterranean reservoir from which the crude oil or gas is produced. The significance which can be put on the results of sophisticated numerical simulations of reservoir performance is entirely dependent on the quality of reservoir description inherent in the model. The difficulty in obtaining a reliable description stems from the large scale and heterogeneous nature of the reservoir and the very limited number of points, i.e. wells, at which observations can be made. In the case of an offshore reservoir this difficulty is compounded by the fact that the well spacing is much larger than that typical of onshore operation. There are several ways by which it is possible to gain information about the reservoir characteristics; the most important are: (a) Seismic and associated geological studies (b) Information obtained during t he well drilling programme; this comprises: (1) the analysis of cuttings and cores (2) the interpretation of various logs (c) Wireline formation testing (1) virgin reservoir (exploration and appraisal wells) (2) produced reservoir (new development wells) (d) Transient pressure testing of wells (including production logging) (e) Analysis of reservoir performance, e.g. through history matching a simulator A consistent description of the reservoir can only be generated by collating and assessing all the available information from these different sources and synthesising a coherent physical model of the system which minimises inconsistencies in the data. Note that in the reservoir development stage only items (a), (b) and (c) are applicable and important engineering decisions will be made on the basis of rather sparse and sometimes conflicting evidence. It is the subject of transient pressure testing of wells which will be dealt with here. The pressure behaviour of an oil or gas well is both a readily measurable and a very useful entity. In the context of pressure testing it is the pressure at the bottom of the well adjacent to the producing formation, referred to as the bottom-hole pressure (BHP), which is of significance. The pressure analysis of wells essentially concerns the dynamic relation between the producing rate, the BHP and the reservoir pressure in the vicinity of the well under consideration. Knowledge of this relation from field tests, combined with a realistic model for fluid flow in the reservoir rock surrounding the well, allows parameters of the flow system such as permeability to be established by inference. In fact pressure transient analysis is simply a parameter estimation technique in which one or more of the parameters of a differential equation are computed from a measured solution. Parameter estimation is the inverse of simulation and is a well known procedure in systems engineering. The relation between producing rate, the BHP and the pressure at the well head, referred to as the top-hole pressure (THP), is known as the vertical lift characteristic of the well and is an entirely separate issue which is not treated here. Although the topic is referred to as the pressure testing of wells it is important to appreciate that it is information about the reservoir which is obtained from such tests. Moreover the information gained is not restricted in scope to the immediate vicinity of the well-bore as is the case for data from logs. One of the advantages of transient pressure testing is that it yields average values of permeability taken over substantial volumes of rock. Since the theory of pressure testing is basically that of unsteady-state, radial flow in the vicinity of the well bore the reservoir engineering aspects of radial flow will also be covered. EPS Training 42 2.1.2. Development of Pressure Testing The technological progress of transient pressure testing is very much linked to the development of sub- surface measuring instruments for recording bottom-hole pressures. Instruments for measuring maximum pressures in wells were developed and applied in the United States in the 1920's. However the utility of early bottom-hole pressure instruments was greatly increased by the development of continuously recording instruments such as the Amerada gauge; this of course allowed transient pressure changes to be monitored and laid the foundation for the modern techniques. In the Amerada and similar systems the whole assembly comprising the pressure element (Bourdon tube), mechanical recording system (based on scribing with a needle on a tin chart), clock and thermometer are lowered down-hole in a bomb for the period of the test. Although the Amerada type gauges are still used such mechanical systems have inherent limitations with regard to accuracy and reproducibility. There is nothing more annoying than to run an expensive pressure test (in terms of lost oil production or rig time) and then discover, on retrieving the bomb by wireline, that the internal clock had stopped during the test. However in deep wells, where exceptionally high temperatures are encountered, the only viable pressure recording instrument may be the purely mechanical Amerada system and one or two service companies specialise in calibrating gauges and reading Amerada charts with high precision. The weak link of electronic systems is the batteries used to power the circuitry and memory devices. There is no doubt that the introduction of the bottom-hole recording system was the spur to the development of modern, transient well testing as a practical and useful engineering technique. The measurement of wellhead pressure alone requires that the downhole pressure be deduced using some form of vertical flow model and there is simply far too much uncertainty in this process. Some success in the testing of gas wells has been achieved using wellhead pressure measurement only, since in single phase conditions it is possible to make a much more reliable prediction of the vertical flow characteristics and a commercial device called the "Spider", which simultaneously measures surface flow-rate and pressure and uses a computer program to deduce bottomhole pressure, is available. The development of technology which would allow the accurate measurement of downhole flow-rate, of comparable quality to the extremely accurate detection of pressure now routinely available, would constitute a further major advance in the subject. At least two service companies are developing bottom-hole venturi meters, which have the advantage of no moving parts, and may help to facilitate a good in-situ rate measurement especially in single phase flow. At the present time downhole flow measurement is limited to the use of conventional production logging tools based on spinner devices and the accuracy in multiphase conditions is problematic to say the least. The last two decades have seen a further technological breakthrough with the introduction of very precise surface recording pressure tools. These were first introduced by the Hewlett-Packard company and are based on a quartz crystal pressure transducer with accurate temperature compensation; the signal can be transmitted to the surface where it is displayed and recorded on a digital data logging system. These instruments have been pioneered in the North Sea and, despite their high cost, have become something of an offshore industry standard. In deep wells with highly permeable formations it is necessary to accurately measure fairly small changes in the down-hole pressure at a high pressure level; this demands instrumentation of correspondingly high sensitivity. In the early days the usual type of measurement was the determination of the so-called 'static' pressure; this was done by lowering a pressure-measuring device to the bottom of a well which had been closed-in for a period of time, say 72 hours. These static measurements sufficed to indicate the pressure in permeable, high- productivity reservoirs. However petroleum engineers soon recognised that in most formations the static pressure measurements were very much a function of closed-in time. The lower the permeability, the longer the time required for the pressure in a well to equalise at the prevailing reservoir pressure. Thus it was realised very early that the rapidity with which pressure build-up occurred when a well was closed in was a reflection of the permeability of the reservoir rock around that well. This qualitative observation was an important step in developing an understanding of well pressure behaviour and led to the other basic type of measurement called transient pressure testing. In this technique, which is the basis of modern well testing, the pressure variation with time is recorded after the oil flow-rate of the well is changed. A stimulus for developing a quantitative interpretation of pressure data came with the introduction of the material balance method of calculating the original oil in place (OOIP) in a reservoir. This procedure requires knowledge of the static reservoir pressure and, rather than closing in the wells for long periods of time, the question naturally arose as to whether flowing transient pressure measurements could be extrapolated in some way to give a reliable estimate of the pressure that would exist in the reservoir if all fluid motion ceased. EPS Training 43 The first attempt to present an extrapolation theory and to relate the change in pressure with time to the parameters of the reservoir was presented in 1937 by Muskat (1) . He deduced mathematically a method for extrapolating the measured well pressure to a true static pressure. At the time Muskat stated that his theory only had a qualitative application and, in a sense, this was true since the analysis did not take into account the important aspect of fluid compressibility. However it should be pointed out that in 1935 a French hydrologist, Charles Theis (2) , had presented the theory of pressure buildup analysis now associated with the name of Horner in the petroleum literature. The first comprehensive treatment of pressure behaviour in oil wells to include the effects of compressibility was that of Miller, Dyes and Hutchinson (3) in 1950. The following year Horner (4) presented a somewhat different treatment. These two papers still furnish the fundamental basis for the modern theory and analysis of oil well pressure behaviour. Subsequent works have brought a multitude of refinements and a deeper understanding of the subject. The procedure for extrapolation to static reservoir pressure was perfected by Matthews, Brons and Hazebroek (5) while Ramey (6) has extended the theory of pressure behaviour to cover gas reservoirs. Recent developments have been concerned with the detection of faults and fractures within the reservoir and with the investigation of the degree of continuity between wells. This latter aspect is known as interference testing and the state-of-the-art in this area has been radically improved by the introduction of the new, highly sensitive pressure transducers. The excellent monograph by Matthews and Russell of Shell Oil entitled 'Pressure Build-up and Flow Tests in Wells' and published by the SPE in 1967 gives a thorough review of the subject as it stood at that time. 2.1.3. Exploration Well Testing In broad terms the subject of well testing may be divided into two broad categories, namely, the testing of exploration or appraisal wells and the testing of development wells. For offshore reservoirs the exploration and appraisal wells are tested from a semi-submersible rig and the methodology involves down-hole shut-in with some form of annulus pressure operated testing string as illustrated in Figure 2.1.1. The three principal elements of the test string are: - the packer which is usually set by weight on the tail pipe - the testing valve which is operated by annulus pressure - the bottom-hole pressure transducer which records the response 1 Muskat, M.:"Use of Data on the Build-Up of Bottom-Hole Pressures",Trans AIME(1937), 123, 44-48 2 Theis, C.V.:"The Relation between the Lowering of the Piezometric Surface and the Rate and Duration of Discharge of a Well Using Ground Water Storage", Trans AGU (1935), 519-524 3 Miller, C.C, Dyes, A.B. and Hutchinson, C.A.:"The Estimation of Permeability and Reservoir Pressure from Bottom-Hole Pressure Build-Up Characteristics", Trans AIME (1950), 189, 91-104 4 Horner, D.R.:"Pressure Build-Up in Wells", Third World Petroleum Congress, The Hague (1951), Sec II, 503-523 5 Matthews, C.S., Brons, F. and Hazebroek,P.:"A Method for Determination of Average Pressure in a Bounded Reservoir", Trans AIME (1954), 201, 182-191 6 Al-Hussainy, R., Ramey, H.J. and Crawford, P.B.:"The Flow of Real Gases through Porous Media", JPT (May 1966), 624-636 EPS Training 44 Downhole Memory Surface Recording Casing Tubing Testing Valve (operated by annulus pressure) Packer (set by weight on string) Pressure Transducer Tailpipe Figure 2.1.1 Drillstem Testing Assembly Well Test Surface Hardware Choke Gas Oil Surface Choke provides Rate Control Orifice Plate Flow Measurement q o Q Test Rate Limited by Separator Capacity Fig 2.1.1b Test Separator A well test conducted with such a system is commonly referred to as a drill stem test (DST) although this nomenclature is somewhat out of date since in modern practice the test string is not set on drill pipe but on special tubing. Historically drill stem tests were carried out in open-hole conditions but again in modern practice the well will have been cased, cemented and perforated before the DST is performed. The important point about a DST is that the testing valve is downhole and that it is the first test on a new well and hence the pressure will normally already have been determined by a wireline formation tester (WFT) survey prior to casing. In the exploration - appraisal well situation the reservoir is unproduced and all layers of the tested interval are likely to be at the same potential i.e. pressure corrected to datum level. In a typical test the well is flowed and then shut in by opening and closing the testing valve and the downhole pressure response recorded by the transducer. The flowrate is normally measured at the surface where the produced fluids are passed through a separator and the flow-rates of oil, water and gas are individually registered as illustrated in Figure 2.1.1b. Very recently service companies have introduced a downhole venturi flowmeter which complements the downhole pressure measurement with downhole flow measurement. With a modern quartz pressure transducer it is possible to measure pressure to approximately five decimal places of accuracy. However the rate measurement of the flow-rate of the single-phase fluids from the separator is not nearly so precise; in fact the error is of the order of t10% which is very disappointing indeed. Given this poor quality of rate surface measurement it is really only possible to determine permeability within this error range i.e. t10%. The error in rate measurement also crucially affects the validity of a recombination sample of the single-phase streams which has been one of the principal sampling techniques. In many exploration well tests the capacity of the test separator is much lower than the potential flow-rate from the well in a production system. Therefore the surface choke is used to limit the test rate to a value which can be handled by the test separator. The oil-field choke in critical (two-phase) flow is not an ideal control device since the rate still depends on the upstream pressure. A recent innovation is to utilise a muliphase surface flowmeter and send a feed-back signal to a process engineering pressure control valve (PCV) which will give much better rate control than a manually variable choke. Test design is based on achieving essentially constant rate in the flow periods of a test. The first quartz pressure transducer introduced by Hewlett-Packard was a surface read-out device and this was welcomed because the pressure data could be analysed in real time as the test proceeded. However in most DSTs currently being carried out the pressure data is recorded downhole using electronic memory devices which are retrieved at the end of the test; this is cheaper and intrinsically more safe than the surface read-out option which requires a cable connection to bypass the testing valve. The determination of reservoir parameters depends on both pressure and rate being accurately measured. As the testing string is introduced into the well, the mud or completion fluid is displaced and on setting the packer the pressure in the sealed volume beneath attains reservoir pressure. When the valve is opened the pressure immediately changes to the pressure in the tubing above. In order to limit the pressure differential on the formation a cushion or liquid column is placed in the tubing as it is being run in. This cushion may be diesel oil or a heavy aqueous solution. In some cases the whole tubing may be filled with liquid which means that the well rate is under choke control from the start of the first flow period. When a partial cushion is used this is displaced by the formation fluid as the well starts to flow once the testing valve is opened. Hence there is a period of rising liquid level until liquid reaches the surface. During this period the rate of gas issuing from the well used to be assessed by placing a hose in a bucket of water and observing the "strength of the blow". However modern safety regulations usually prohibit such ad hoc procedures. In some tests the reservoir pressure is not sufficient for fluids to reach the surface and the DST data is confined to a period of rising liquid level followed by a shut-in; this is known as a slug test and special methods, which account for the rapidly decreasing rate EPS Training 45 as the back pressure due to the rising liquid column increases, are required for interpretation. In shallow reservoirs with competent formations the well may be tested dry i.e. with no cushion. Some operators require that the duration of the initial, short flow period be such that, say, four times the tubing volume of reservoir fluid be produced at the surface; this is to ensure complete displacement of the cushion from the well. However if a heavy cushion has been employed even this precaution may not guarantee displacement. The objectives of testing exploration or appraisal wells can be summarised as: 1. Determine the nature of the formation fluids 2. Measure well productivity 3. Measure reservoir temperature and pressure 4. Obtain samples for laboratory PVT analysis 5. Obtain information on reservoir description (permeability, heterogeneity ) 6. Estimate completion efficiency i.e. skin factor However it must be stressed that all these objectives must be subordinate to the issue of safety. A schematic of a typical drillstem test is shown in Figure 2.1.2 in which a dual flow period and dual shutin are present. The purpose of the initial short flow period - typically only five minutes - is to relieve any supercharging (excess pressure due to mud filtrate invasion in low permeability formations) present and establish communication with the reservoir. The very short flow period ensures that negligible depletion of the reservoir will occur even if a small compartment is being investigated The well is then shutin for the first buildup to determine the initial reservoir pressure, denoted p i , and the duration of this first buildup is long enough for the pressure to essentially stabilise so that no, or very little, extrapolation is required to fix p i . The well is then flowed for a long period - the second drawdown - and shut-in for the second or final buildup. As shown in Figure 2.1.2 the rate may be brought up in steps; this may be to investigate any sanding tendency of the formation and an acoustic sand monitor can be installed to register the concentration of sand particles in the flowing stream. The main objective of the second flow period is to achieve a sustained period of constant rate production prior to the final buildup and the last flowing pressure before shutin, denoted p wf (t p ), must be accurately known to allow proper interpretation of the ensuing buildup. If the data obtained in the (second) drawdown period is analysed the initial pressure, p i , is known. The design of a well test principally revolves round the decision as to how long the major flow period should last and the question of depth of investigation of the pressure disturbance is usually the main issue from a reservoir engineering point of view. However in the context of offshore operations, where the expense of rig time is an important issue, there is an obvious incentive to curtail the duration of the test. The final decision must reflect a compromise between cost and the value of additional information gained from prolonging the test time. The issue of depth of investigation will be treated at length later in this chapter. The duration of the final buildup should be approximately 1½ times that of the preceding flow period. In an exploration or appraisal well test the choice of flow-rate is governed by several factors. For an oil well, in a prolific reservoir, the limiting consideration is often the capacity of the separator which will be rated, say, at 5000 or 10,000 bbl/day. In a gas well test on a semisubmersible rig the capacity of the flare system may well decide the maximum rate. The Initial Flow I ni t i al Shut i n Afterflow Final Shutin Final Flow Time Prod. Rate Initial Res. Pressure Drawdown Buildup BHP Time Figure 2.1.2 Dual Flow - Dual Shutin Test Fig 2.1.3 4 k h π q B s µ Transient Well Testing Buildup Analysis - Horner (Theis) Plot Semilog Analysis l n t + t p ∆ ∆t Buildup Affected by Wellbore Storage Intercept gives skin factor S Affected by Boundaries ETR MTR LTR slope = − p* p ws Nomenclature due to D. Pozzi (EPR) EPS Training 46 highest recorded gas well test rate offshore probably occurred in the testing of a Troll field well in Norwegian waters where 160 million SCFD was achieved generating a drawdown of about 2 psi in the extremely permeable and thick formation. The buildup took place over a period of about two minutes. In the testing of very high pressure and temperature gas condensate wells in the central area of the North Sea basin, the limiting rate is controlled by the wellhead temperature which cannot be allowed to exceed the rating of elastomers used in the BOP stack. The well must be flowed at a low enough rate that heat loss to the surrounding rock, at geothermal temperature, sufficiently reduces the temperature of the flowing stream as it progresses up the tubing. The surface choke is, of course, used to limit the well rate to an acceptable value. In tight formations the rate is controlled by the reservoir deliverability and in extreme cases no transient test is possible because a constant stable rate cannot be sustained; this is a problem with prefracture tests in very low permeability reservoirs. An important part of the design of a well test is to use a production engineering nodal analysis software package to simulate the flow behaviour of the proposed system viz. tubing , choke and formation based on estimates of the likely permeabilities which may be encountered; this exercise also requires estimates of the PVT properties of the produced fluid. The final buildup is usually analysed using the Horner plot illustrated diagrammatically in Figure 2.1.3; here the bottomhole shut-in pressure, p ws , recorded by the transducer is plotted against a logarithmic time function i.e.: ln t p + ∆t ∆t . . . . . (1.1) where t p is the production time and ∆t is time measured from the moment where the well was shut in. The theory of this interpretation method will be given later. The slope and intercept of the middle time region of the semilog graph yield, respectively, the average permeability of the formation, k , and the skin factor, S . The physical meaning of the skin factor has been discussed at length in Chapter 1 and is a combination of formation damage and perforation effects as illustrated in Figure 2.1.4. altered zone p w p wf ∆p s Unaltered Permeability k r w r s r e wellbore radius altered zone radius external radius ∆p = I ncrement al Ski n Pressure Drop s Fig 2.1.4a k s Near Wellbore Altered Zone Fig 2.1.4b n = Number of perforations per foot l = Length of penetration = Phase angle s p θ r w r s k s l p Altered Zone k Combined Skin Damage + Perforation Perforated Completion S f n l k r k s p s s · , , , , , θ e j When an exploration or appraisal well is drilled the mud system will not be optimised with respect to formation damage; for example water based muds are often employed for the appraisal wells whereas the eventual development wells will be drilled with oil based mud which minimises alteration in water sensitive formations. In addition exploration and appraisal wells are often drilled with a substantial overbalance which again will result in significant mud filtrate loss promoting damage. The most important feature of transient well testing is that the interpretation allows the independent assessment of formation average permeability, k, and the skin factor, S. Thus even if the skin effect in the exploration/appraisal well is high the productivity index of the future development wells can be predicted using a formula of the form: J sss = 2πkh Bµ ¸ ¸ , _ ln r e r w − 3 4 + S . . (1.2) where the permeability is taken from the appraisal well test and the skin factor is an estimate of what can be achieved with optimised drilling and perforation in the future development wells. For example, in the case of tubing conveyed perforation (TCP) in an underbalanced condition, it is possible to obtain a negative skin of EPS Training 47 the order of -1.5. The drainage area radius, r e , used in this formula is related to the well spacing but it should be noted that J sss is quite insensitive to the value of r e since it enters a logarithmic term. The important point is that the dynamics of the transient pressure response allows a discrimination between near wellbore effects - the skin - and the formation permeability. A simple determination of steady-state or semi-steady-state productivity index using the defining equation: q s = J sss (p − − p wf ) . . . . (1.3) only requires measurement of the well flow rate, q s , and the corresponding flowing bottomhole pressure, p wf , assuming the reservoir pressure, p − , is known. However the contributions to the resultant J sss due to permeability and skin cannot be resolved without transient information i.e. a pressure buildup. The development of pressure transient testing and, in particular, buildup analysis has been primarily motivated by this feature of dynamic discrimination between the effects of intrinsic formation permeability and near wellbore alteration and perforation on well productivity. In the routine testing of development wells which have been flowing for a considerable time the main objectives of the buildup is to determine the current well skin factor and reservoir pressure; this is referred to as reservoir monitoring. Again the analysis will usually be carried out on a Horner plot as shown on Figure 2.1.5 but using a synthetic flowing time denoted t sia and the slope may be forced to the known permeability since this will have been determined in the first test on the well when it was drilled. Forcing the permeability to a known value means that the skin factor is evaluated on a common basis and it is possible to make a comparison between the original and present skin factor i.e. monitor any deterioration in well performance due to scale deposition, fines migration, asphaltene precipitation or any other such mechanism. The adjustment of the extrapolation of the straight line, p*, to the reservoir pressure is known as the MBH correction and requires knowledge of any no-flow boundaries present in the vicinity of the well. Again the theory of this approach will be given in subsequent chapters. The Horner (semilog) buildup plot is the vehicle for such an interpretation. In the case of development wells the valve used to shut-in the system for a buildup is located at the wellhead and the possibility of wellbore storage effects, due to the capacity of the compressible fluid mixture in the well at the moment it is closed, must be expected. The mechanism of wellbore storage and methods of modelling the phenomenon will be treated in Chapter 3. If a well is located in a closed drainage area, such as a fault block with perfectly sealing boundaries, and it is flowed at constant rate three principle flow regimes are encountered in an extended drawdown test; these are illustrated in Figure 2.1.6 which shows a Cartesian plot of flowing bottom-hole pressure versus time. The period in which the propagating pressure disturbance has not yet encountered any boundaries is known as the infinite-acting or transient flow regime. It is this data which yields a straight line on the semilog plot, i.e. the pressure is varying with the log of time, and it is also referred to as the middle time region (MTR). Once the pressure behaviour of the well is influenced by boundaries the late time region (LTR) is entered and in a closed system produced at constant rate a state of semi -steady-state (SSS) depletion as described in Chapter 1 is eventually attained. In this flow regime the bottom-hole flowing pressure, p wf , varies E T R MT R LTR sl ope p* q µ 4 kh π m = − l n t + t si a ∆ ∆t p p ws ∆p MBH Fig 2.1.5 Horner Plot 0 Determination of Average Pressure WELL PRESSURE STARTS TO BE AFFECTED BY BOUNDARIES MTR LTR TRANSIENT I.A. FLOW TRANSITION LATE TRANSIENT SEMI-STEADY-STATE FLOW p i p wf 0 TIME SCHEMATIC PLOT OF PRESSURE DECLINE AT PRODUCING WELL CONSTANT RATE WELL BOUNDED RESERVOIR Figure 2.1.6 dp d t q B c r h s t e ·− φ π 2 Cartesian Plot Flow Regimes MTR LTR ln t + t p ∆ ∆ depletion buildup drawdown Horner Plot p ws p* p** Closed "tank" of pore volume, V Figure 2.1.7 p i 0 Time, t q Detection of Depletion EPS Training 48 linearly with time as shown in Figure 2.1.6. A well test in which the flowing period is sufficiently long for this flow regime to be attained is termed a reservoir limit test since the size i.e. pore volume of the closed drainage area may be found from the slope of the Cartesian plot in the SSS regime. The interval of transition between the end of infinite-acting flow and the beginning of semi -steady-state depletion is known as the late transient period. This transition is very short when the well is at the centre of an approximately square or circular drainage area but becomes significant when the well is asymmetrically located or the reservoir compartment is rectangular or triangular. Well test interpretation becomes very difficult when the infinite-acting regime is masked, say, by wellbore storage effects. These flow regimes refer to constant rate drawdown (CRD); when a buildup follows a period of constant rate production it is referred to as a constant rate buildup (CRB). One of the main objectives of a drillstem test is the identification of any depletion as illustrated in Figure 2.1.7. Here the pressure attained in the final buildup is less than the initial pressure, p i , indicating a closed system of finite volume. The detection of depletion depends on the correct extrapolation of the buildup to the final stabilised pressure. The prescription that the shut-in time, ∆t max , should be 1½ times the flowing time, t p , has the objective of reducing the uncertainty in the extrapolation process so that it is feasible to detect any significant depletion. In recent times the objectives of well testing have developed from the straightforward determination of an average permeability and skin factor, to be used in the expression for productivity index, to sophisticated approaches aimed at defining the parameters of more complex reservoir models. The progression from the basic methodology to fuller reservoir description has been possible because of the improvements in pressure gauge resolution allowing the use of derivative techniques and the use of interactive software packages based on type curve matching and non-linear regression. For what might be termed category I well test interpretation a simple model of a homogeneous reservoir containing a vertical well with skin is sufficient; this canonical model is illustrated in Figure 2.1.8. Interpretation using a homogeneous model implies that an average permeability will be obtained when data from a heterogeneous system is processed. The form of average e.g. arithmetic or geometric depends on the nature of the Skin r e k Homogeneous Finite Reservoir d d Well Image No Flow Boundary k 1 k 2 r 1 Composite Infinite Reservoir d 2 d 1 Single Linear Fault Multiple Faults µ 2 µ 1 θ Some Well Test Models Fig 2.1.8 T(t=0) x x Heated Bar Thermocouples in thermowells Transient Heat Conduction Equation Inner B.C. i.e. . . . specified gradient (2 kind BC) nd Linear Flow t Penetration Depth Initial Temp. T Face Temp. Dynamic Temperature Distribution Fig 2.1.9 ∂ ∂ ρ ∂ ∂ T t k C T x p · 2 2 q A k T x x ·− · ∂ ∂ 0 ∂ ∂ T x q kA x· ·− 0 Constant heat flux EPS Training 49 heterogeneity in the system. The integration of core analysis data with well test interpretation results essentially revolves around the definition of such averages and the modern approach to geostatistics is giving new insights into this activity. In a layered system for example the arithmetic average is the appropriate method of treating core data. In what might be classified as category II well test interpretation the methodology is extended to include the determination of reservoir description parameters related to: • heterogeneity • boundaries • layering • anisotropy and more complex models are required some of which are also illustrated in Figure 2.1.8. In the 19 th century the mathematical theory of transient heat conduction was developed based on Fourier’s equation. A temperature transient propagating in a metallic bar with constant rate heat injection at one end is illustrated diagrammatically in Figure 2.1.9 2.2. Radial Flow Theory 2.2.1. The Basic Flow Equations The mathematical foundation of transient pressure analysis is the theory of unsteady-state, single-phase, one-dimensional radial flow in a porous medium. This is based on the following assumptions concerning flow in the vicinity of the well-bore. (1) The formation is homogeneous and isotropic with respect to both the porosity and permeability which are also considered to be constants and thus independent of pressure. (2) The producing well is completed across the entire formation thickness thus ensuring fully radial flow. (3) The formation is completely saturated with a single fluid and is uniformly thick. In certain circumstances it is necessary to partly relax these assumptions and allow for the effects of such phenomena as non-uniform or two-phase flow or the porosity being a function of pressure. When these effects are significant, allowance will be made by applying appropriate correction factors to the basic theory. The basic homogeneous, radial system is illustrated in Figure 2.2.1 where the following nomenclature applies: h = formation thickness k = average permeability r = general radius r e = external radius of drainage area r w = wellbore radius p = pressure q = in-situ volumetric flow-rate t = time u r = superficial velocity φ = porosity µ · viscosity ρ = fluid density The equations of motion governing the flow follow from the principles of the conservation of mass and momentum; the assumption is also made that the flow is isothermal. For the present purpose the momentum equation takes the form of D'Arcy's law and the effect of turbulent or non-D'Arcy flow, which is usually only of importance in gas reservoirs, will be taken up later. Thus the basic flow equations, in radial co- ordinates, expressing the symmetry of the flow system, are: h r e r w q Well in the Centre of a Ci rcul ar Reservoi r Radial Flow Fig 2.2.1 Model Reservoir EPS Training 50 Continuity equation: 1 r ∂(rρu r ) ∂r = − φ ∂ρ ∂t . . . (2.1) Darcy's Law: u r = − k µ ∂p ∂r . . . (2.2) D'Arcy's law is employed as a quasi-steady approximation to the general momentum equation, i.e. a form in which the momentum accumulation term has been discarded. This is perfectly valid provided information is not sought concerning pressure disturbances moving at the velocity of sound in the fluid. All pressure variations predicted by the model are associated with local pressure gradients due to laminar fluid flow. Although the fluid compressibility will be introduced into the model sonic effects are still neglected. In pressure analysis the flow is assumed to be horizontal and hence the pressure, p, appears in D'Arcy's law rather than the flow potential; thus gravity forces are neglected. The superficial radial velocity, u r , may be eliminated between (2.1) and (2.2) giving: 1 r ∂ ¸ ¸ , _ rρk µ ∂p ∂r ∂r = φ ∂ρ ∂t . . . (2.3) This is the basic partial differential equation (PDE) describing the unsteady-state flow of a single-phase fluid in a porous medium, provided the flow is laminar. Equation (2.3) represents a general form for the combination of the continuity equation and D'Arcy's law or the momentum equation. The PDE is non-linear because the density, ρ , and the viscosity, µ , are pressure dependent. As it stands the equation involves two dependent variables, pressure and density, and another equation is necessary to yield a determinate system; this of course is the equation of state relating these t wo quantities, ρ = ρ(p) , in an isothermal system. 2.2.2. Fluid of Constant Compressibility The occurrence of pressure transients only takes place because the reservoir fluid is compressible to some extent and local accumulation or depletion of fluid in the reservoir occurs. Although liquid compressibility effects can safely be neglected in most fluid flow situations, this is not true in an oil reservoir where very large volumes of liquid at high pressure are involved. In the case of single phase liquids the thermodynamic equation of state is adequately represented by the model of a fluid of constant compressibility. The definition of the compressibility, c , of a fluid is: c = − 1 V ¸ ¸ , _ ∂V ∂p T,m . . . (2.4) i.e. the relative change in the volume of the fluid per unit change in pressure; it has the dimensions of reciprocal pressure. Equation (2.4) may be put in terms of fluid density as follows; for a constant mass of fluid, m: ρ = m V or V = m ρ hence: c = − 1 V ∂V ∂p = − ρ m ∂ ¹ ' ¹ ¹ ; ¹ m ρ ∂p · − ρ . ¸ ¸ , _ − 1 ρ 2 ∂ρ ∂p i.e. c = 1 ρ ∂ρ ∂p . . . . (2.5) EPS Training 51 Not surprisingly the compressibility is also the relative change in density per unit change in pressure; naturally if the volume decreases the density increases. For a fluid of constant compressibility equation (2.5) may be integrated to yield: ρ · ρ i e c(p − p i ) . . . . (2.6) where ρ i is the liquid density at some reference pressure, p i . Differentiating equation (2.5) with respect to time gives: cρ ∂p ∂t = ∂ρ ∂t . . . . (2.7) which on substitution in (2.3) results in: 1 r . ∂ ¸ ] 1 kρr µ ∂p ∂r ∂r = φcρ ∂p ∂t . . (2.8) which has achieved a partial elimination of the density from (2.3). The assumption of constant compressibility is always valid for liquids, provided of course no gas comes out of solution, and is sometimes approximately valid for gases at high pressures. The theory of the unsteady-state flow of real gases will be considered later. 2.2.3. Further Development of the Accumulation Term In the derivation of the continuity equation the accumulation term represents the time rate of change of the mass of fluid per unit superficial volume. When the porosity is not constant and varies with formation pressure this term must be written: ∂(φρ) ∂t which can be expanded as follows: ∂(φρ) ∂t = φ ∂ρ ∂t + ρ ∂φ ∂t = φc l ρ ∂p ∂t + ρ ∂φ ∂p . ∂p ∂t . . (2.9) where the liquid compressibility has been denoted c l . On defining the compressibility of the formation as: c f = 1 φ ∂φ ∂p . . . . (2.10) this may be written: ∂(φρ) ∂t = φρ ( ) c l + c f ∂p ∂t . . . (2.11) Although in reality the φ on the right-hand side of equation (2.11) is still a function of pressure, as a first approximation it can be treated as a constant evaluated at some representative average formation pressure. In this form equation (2.11) adequately allows for the small effect of rock compressibility in all but the most exceptional circumstances. A further refinement of equation (2.11) is warranted; in an undersaturated reservoir two liquids are in fact present - oil and immobile connate water - both of which are compressible. Hence the liquid compressibility, c l , is given by the sum of two contributions: c l = S wc c w + S o c o . . . (2.12) EPS Training 52 where c w and c o are the compressibilities of water and oil and S wc and S o are the respective saturations. Thus the compressibility, c , in the basic equation (2.8) is identified with the total system compressibility, c t ,defined as: c t = c l + c f = S wc c w + S o c o + c f . . (2.13) Note that the permeability, k , in equation (2.8) is not the absolute permeability of the porous medium but the permeability to oil at the connate water saturation, i.e. k = k o (S wc ) . . . the end-point permeability Since the flow model assumes horizontal flow the permeability also refers to this direction, i.e. it is the radial permeability. When the total compressibility, c t , is employed the density, ρ , refers to the mass per unit pore volume of oil and connate water. 2.2.4. Linearisation of the Radial Flow Equation The partial differential equation describing the unsteady-state, radial flow of a fluid of constant compressibility has just been derived as: 1 r . ∂ ¸ ] 1 kρr µ ∂p ∂r ∂r = φc t ρ ∂p ∂t . . . (2.14) This form still involves two dependent variables, p and ρ , and remains non-linear. In order to transform (2.14) into a linear PDE in a single dependent variable, i.e. pressure, further simplifying assumptions are necessary. Such a transformation is desirable because linear PDE's derived from (2.14) have analytical solutions which are easily manipulated and circumvent the necessity of employing cumbersome numerical methods. The latter are especially time-consuming when the inverse problem of non-linear parameter estimation is involved. The additional assumptions necessary for the linearisation of (2.14) are certainly no more severe than those already made concerning formation homogeneity etc. For a liquid the permeability, k , and the viscosity, µ , are independent of pressure and equation (2.14) may be written: 1 r . ∂{rρ ∂p ∂r } ∂r = φc t ρµ k . ∂p ∂t . . (2.15) The derivative term on the left-hand side of (2.15) may be expanded using the chain rule for differentiation; thus: 1 r ¸ ¸ , _ rρ ∂ 2 p ∂r 2 + ρ ∂p ∂r + r ∂p ∂r . ∂ρ ∂r = φc t ρµ k . ∂p ∂t . (2.16) The compressibility is defined as: c t = 1 ρ ∂ρ ∂p . . . . (2.17) and differentiating this equation with respect to 'r' results in: ∂ρ ∂r = c t ρ ∂p ∂r . . . . (2.18) Substitution of (2.18) in (2.16) gives: EPS Training 53 1 r ¸ ¸ , _ rρ ∂ 2 p ∂r 2 + ρ ∂p ∂r + rρc t ¸ ¸ , _ ∂p ∂r 2 = φc t ρµ k . ∂p ∂t (2.19) The assumption is now made that the pressure gradient ∂p/∂r is small and hence (∂p/∂r) 2 can safely be neglected. Under this assumption, and cancelling ρ throughout, equation (2.19) becomes: ∂ 2 p ∂r 2 + 1 r ∂p ∂r = φc t µ k . ∂p ∂t . . . (2.20a) or, alternatively: 1 r ∂ ¸ ¸ , _ r ∂p ∂r ∂r = φc t µ k . ∂p ∂t . . . (2.20b) For the flow of liquids the above assumptions are quite reasonable and have frequently been applied. However this simple linearisation by deletion must be treated with caution and can only be applied when the following condition is satisfied: c t p << 1 . . . . . (2.21) For example when the c t p product is 0.1 the solutions of (2.19) and (2.20) differ by about 5%. This condition makes it necessary to modify the assumption concerning compressibility such that it is not just constant but both small and constant. When dealing with reservoir systems which have a high compressibility, so that (2.21) is not valid, it is necessary to linearise (2.21) or its equivalent using some form of mathematical transformation. Such an approach will be required when dealing with a gas reservoir, since in this case the compressibility of the gas alone, assuming it to be ideal, is the reciprocal of the pressure; hence to a first approximation the c t p product is unity and condition (2.21) is certainly not satisfied. 2.2.5. Initial and Boundary Conditions The linearised radial flow equation : ∂p ∂t = k φµc t . 1 r . ∂ ¸ ¸ , _ r ∂p ∂r ∂r is a second-order, linear, parabolic PDE having pressure as the dependent variable and radial position and time as the independent variables. Similar second-order parabolic equations of this form are frequently encountered in the theory of diffusional transport processes, e.g. unsteady-state heat conduction, and all equations of this type are referred to as 'the diffusivity equation'. Hence the quantity k/(φµc t ) is known as the hydraulic diffusivity, α , and has dimensions L 2 T -1 . However it should be emphasised that there is no diffusional aspect to D'Arcy's law and its similarity to Fourier's law of heat conduction, for example, is purely mathematical. Fortunately, because of its importance in transport phenomena, almost every conceivable analytical solution to the diffusivity equation is available in the literature. In order to determine the solution of a second-order PDE of the parabolic type an initial condition and two boundary conditions must be specified. The fundamental solutions of interest in the development of pressure analysis methods are those for the case of flow into a centrally located well at a constant volumetric rate of production, q. Later on it will be demonstrated how the principle of superposition may be used to yield solutions for arbitrary rate histories. The geometry of the system is again shown in Figure 2.2.1; the reservoir is assumed to be cylindrical in shape with thickness, h , and ext ernal radius, r e . The well is of radius r w and in all cases of interest r w << r e . (a) Initial Condition EPS Training 54 Before any flow into the well takes place the whole cylindrical element is assumed to be at the uniform initial reservoir pressure, p i . Using the notation p(r,t) to represent the pressure as a general function of radial position, r , and time, t , the initial condition may be written succinctly as: p(r, o-) = p i , r w ≤ r ≤ r e . . . (2.22) Here o- signifies any time prior to t = 0. (b) Boundary Condition at the Well Bore - Constant Rate Inner Boundary Condition At time t = 0 the well is set in production at a constant volumetric flow-rate, q s , and maintained at this level from then on. This oil flow-rate, q s , is measured at stock tank conditions and hence the in situ reservoir flow rate is given by q = q s B where B is the oil formation volume factor. Although this latter quantity is a weak function of pressure it is treated as a constant, evaluated at the initial reservoir pressure. Applying D'Arcy's law just inside the formation at the sand face gives: − u r r=r w = q s B 2πr w h = k µ ∂p ∂r r=r w or, on re-arranging: ∂p ∂r r=r w = q s Bµ 2πkh . 1 r w , t ≥ o . . . (2.23) This imposes a constraint on the formation pressure gradient at the well-bore and is known as a boundary condition of the second kind. For all ordinary purposes the finite wellbore radius boundary condition (2.23) may be replaced by the alternative form: Lim r→o ¸ ¸ , _ r ∂p ∂r = q s Bµ 2πkh , t > o . . (2.24) which is known as the 'line source' approximation to the original condition. This has been shown to yield identical results (from a practical standpoint) to those obtained from solution of the problem with the exact, less-tractable version. The line source approximation assumes the well-bore radius to be vanishingly small and is acceptable because the dimension of the well-bore is very small compared to the radius of the volume drained by the well. When fractured wells are treated as cylindrical wells of large effective radius the full finite wellbore radius solution must be employed; this is really the only occasion when the line source approximation is not valid. (c) External Boundary Condition Three distinct forms of external boundary condition arise, each one pertaining to a different physical situation. Naturally each specific boundary condition yields its own unique solution to the PDE. The three basic cases of interest are: (1) Infinite Reservoir In this case the well is assumed to be situated in a porous medium of infinite radial extent. This condition is also valid for a reservoir of finite size provided any pressure disturbance generated at the well never reaches the outer boundary of the reservoir which remains at the initial pressure, p i , throughout the period of interest. This condition may be written: p(r, t) = p i as r→∞ , t > o . . (2.25) (2) Bounded Reservoir EPS Training 55 In this case the well is assumed to be located in the centre of a cylindrical reservoir of radius r e with no flow across the exterior boundary. The no-flow condition implies zero superficial velocity at the outer boundary and hence the local pressure gradient must also be zero, i.e. ∂p ∂r r=r e = 0 , t > 0 . . . . (2.26) (3) Constant Pressure Outer Boundary Here the well is situated in the centre of a cylindrical area with a constant pressure, equal to the initial pressure, p i , maintained along the outer boundary. This condition takes the form: p(r e , t) = p i , t > 0 . . . . (2.27) The specific application of each of these cases will become apparent later. 2.2.6. Dimensionless Form of the Diffusivity Equation The linearised partial differential equation of compressible fluid flow in a porous medium, embodying the principle of mass conservation and D'Arcy's law, takes the form: ∂p ∂t = α 1 r . ∂ ¸ ¸ , _ r ∂p ∂r ∂r . . . . (2.28) where α = k φµc t = hydraulic diffusivity. A typical set of initial and boundary conditions (infinite reservoir case) are: I.C. at t < 0 : p = p i for all r (2.29) B.C.1 at r = r w : ∂p ∂r = q s Bµ 2πkh . 1 r w for all t ≥ o (2.30) B.C.2 at r = ∞ : p = p i for all t ≥ o (2.31) The solution to the problem is a mapping p(r,t) representing the pressure as a function of position and time; naturally the solution p(r,t) must satisfy both the differential equation and the prescribed initial and boundary conditions which in combination are referred to as the differential system. The constants appearing in this system, e.g. α , p i and the quantity q s Bµ /(2πkh) are known as the system parameters and these must be specified before a particular solution can be obtained. However, it is much preferable first to obtain a general analytical solution which is quite independent of any specific values given to the parameters. Once this is available any particular solution, corresponding to a certain set of parameter values, is readily obtainable. The general solution is achieved by transforming the differential system into dimensionless form. This is a standard procedure in engineering mathematics and should always be carried out where possible. When the solution in terms of pressure, p(r,t) , has been established the remaining variables, if required, follow from D'Arcy's law and the equation of state, i.e. u r (r, t) = − k µ ∂{p(r,t)} ∂r . . . . (2.32) EPS Training 56 ρ(r, t) = ρ i exp[ ] c t (p (r, t) − p i ) . (2.33) In order to understand the philosophy behind the procedure for rendering the system dimensionless it is helpful to consider briefly the nature of solutions to the diffusivity equation. Some typical pressure profiles in a cylindrical reservoir at various times following the well being set in production are shown in Figure 2.2.2; initially the reservoir was everywhere at pressure p i . At early time the pressure disturbance, due to fluid expansion and motion, is localised in a region around the central well but this progressively propagates itself further out into the reservoir. The difference in pressure between the initial value, p i , and that in the well (r = r w ) is known as the drawdown and increases with the depth of penetration of the pressure disturbance into the formation. Note that, at the well-bore, the gradient of the pressure profile is identical at all times, which is indicative of the constant flow-rate boundary condition. The time taken for the leading edge of the pressure disturbance to reach a radial position, r, is given approximately by : t r = 0.1 r 2 α . . . . (2.34) and the quantity: t w = r 2 w α . . . . (2.35) can be regarded as a characteristic time constant of the system. For a reservoir of finite size, the infinite reservoir solution will be valid for all times 't' such that: t < 0.1 t e = 0.1 r 2 e α i.e. t De = t D r 2 De = kt φµc t r 2 e < 0.1 . . (2.36) which is the condition that the pressure disturbance has not yet reached the external boundary to any significant extent. Thus choosing r w as a characteristic dimension of the system and t w as a characteristic time it is possible to define the dimensionless independent variables r D and t D as follows: r D = r r w . . . . . (2.37) t D = t t w = αt r 2 w ln r t p( r , t ) p i Well- Bore Tr ansi ent Devel opment of t he For mat i on Pr essur e Di st r i but i on Fig 2.2.2 EPS Training 57 or t D = kt ϕµ c t r 2 w . . . . . (2.38) When considering the dependent variable, i.e. pressure, p , it is convenient to form dimensionless ratios by dividing by the quantity q s Bµ/(2πkh) suggested by the inner boundary condition. Also, in order to ensure that the dimensionless boundary conditions assume as compact a form as possible, it is preferable to formulate the system in terms of pressure deviation from the initial value. Hence the dimensionless pressure (difference) is defined as: p D = p i − p q s Bµ 2πkh = 2πkh q s Bµ (p i − p) . . . (2.39) This is equivalent to measuring pressure relative to a datum of p i . Note that 'q s ' is positive for production and a positive p D represents drawdown. Making the substitutions: t = t D r 2 w α r = r D r w p = p i − q s Bµ 2πkh p D into equations (2.28), (2.29), (2.30) and (2.31) the dimensionless differential system becomes: ∂p D ∂t D = 1 r D ∂ ¸ ¸ , _ r D ∂p D ∂r D ∂r D . . . (2.40) I.C. at t D < 0 p D = 0 for all r D (2.41) B.C.1 at r D = 1, ∂p D ∂r D = − 1 for all t D > 0 (2.42) B.C.2 at r D = ∞, p D = 0 for all t D > 0 (2.43) The differential system has now been put into fundamental form amenable to solution by the standard mathematical methods for linear parabolic PDEs of the diffusivity type. The general analytical solution to the system can be obtained without any reference to the physical situation it represents. In this systems engineering approach the mathematical analysis, i.e. the mechanics of deriving the solution, whether by analytical or numerical methods, is completely divorced from the engineering aspects concerned with setting up equations which adequately describe the physical situation and interpreting the solution once it is available. Indeed it may be preferable to rewrite the system using the conventional nomenclature of the mathematical literature, e.g. ∂x ∂t = 1 z ∂{z ∂x ∂z } ∂z z > 1 t < 0 , x = 0 all z (2.44) z = 1 , ∂x ∂z = − 1 all t > 0 z = ∞ , x = 0 all t > 0 EPS Training 58 where x is the dependent variable and 't' and 'z' are the independent time and space variables respectively. In this way the problem is posed as a purely mathematical one. The analytical solution of the dimensionless differential system is denoted p D (r D , t D ), or equivalently x(z, t), and represents the general solution to the basic system for all values of the parameters and dimensions. A particular solution is easily obtained from the general solution using equation (2.39), i.e. p(r, t) = p i − q s Bµ 2πkh p D ¸ ¸ , _ r r w , αt r 2 w . . (2.45) 2.2.7. The Line Source Analytical Solution in an Infinite Reservoir For this case the dimensionless differential system takes the form: ∂p D ∂t D = 1 r D ∂ ¸ ¸ , _ r D ∂p D ∂r D ∂r D r D > 0 . (2.46) t D < 0, p D = 0 all r D Lim r D → 0 ¸ ¸ , _ r D ∂p D ∂r D = − 1 all t D > 0 r D = ∞, p D = 0 all t D > 0 The analytical solution of this problem is given in many textbooks concerned with transport phenomena, e.g. Carslaw and Jaeger (7) . In 1949 Hurst and van Everdingen (8) presented a comprehensive description of the solution of the radial diffusivity equation applicable to reservoir engineering problems. Their solutions were obtained by applying the Laplace transformation to the diffusivity equation primarily in order to solve the problem of water influx into a radial reservoir from a similarly shaped aquifer of both finite and infinite extent. The flow of reservoir fluids into the well-bore has precisely the same nature as the flow of water into a radial reservoir but on a different scale. A more concise analytical solution of the present problem, not involving Laplace transformations has been quoted by Matthews and Russell. Only the result of these analyses will be given here and the references can be consulted for further details if required. The analytical solution of the differential system (2.46) is: p D (r D , t D ) = − 1 2 Εi ¸ ¸ , _ − r 2 D 4t D . . . (2.47) where −Ei(−x) is a standard mathematical function called the exponential integral which is defined by: Εi(-x) = − ⌡ ⌠ x ∞ e -u u du . . . . (2.48a) 7 Carslaw, H.S. and Jaeger, J.C.:"The Conduction of Heat in Solids", 1959, Clarendon Press, Oxford (2 nd Edition) 8 van Everdingen, A.F. and Hurst, W.:"The Application of the Laplace Transformation to Flow Problems in Reservoirs", Trans AIME (1949), 186, 305-324 EPS Training 59 Εi(x) = ⌡ ⌠ x ∞ e -u u du . . . (2.48b) and hence Ei(− x) = − Ei(x). The Ei function is tabulated in mathematical data books and is shown, plotted on a log-log scale, in Figure 2.2.3; Ei(x) is large for small values of x and vice- versa. Fortunately when x is small, i.e. x < 0.01, the exponential integral is very closely approximated by a simple logarithmic function given below; thus for x < 0.01: −Ei(−x) = Ei(x) = − ln(γ x) . . . (2.49) where γ is Eulers constant and has a value of 1.781 or e 0.5772 . Hence the general analytical solution may be written in the simpler form: p D (r D , t D ) = − 1 2 ln ¸ ¸ , _ γ r 2 D 4t D . . (2.50) or, on re-arranging and entering the numerical value of γ : p D (r D , t D ) = 1 2 ¸ ] 1 ln (t D /r 2 D ) + 0.80907 . (2.51) The latter equation is, of course, only valid when: r 2 D 4t D < 0.01 Fortunately this is nearly always the case in connection with pressure testing when the wellbore pressure at r D = 1 is required and hence (2.51) is the most useful form of the analytical solution. The solution in terms of the actual variables and system parameters follows from the equation: p(r, t) = p i − q s Bµ 2πkh p D ¸ ¸ , _ r r w , αt r 2 w . . (2.52) Noting that: r 2 D t D = r 2 αt = φ µ c t r 2 kt . . . (2.53) 10 4 10 5 10 6 10 7 10 8 10 9 10 1 10 -1 10 -2 10 -1 1 10 10 2 10 3 10 4 p D t /r D D 2 Figure 2.2.3 Single Well in an Infinite Reservoir (No Skin) Exponential Integral Solution EPS Training 60 the exact solution becomes: p(r, t) = p i − q s Bµ 2πkh ¸ ¸ , _ − 1 2 Εi ¸ ¸ , _ − φµc t r 2 4kt . . (2.54) and the approximate version: p(r, t) = p i − q s Bµ 2πkh . 1 2 ¸ ¸ , _ ln kt φµc t r 2 + 0.80907 . (2.55) The condition that the latter is valid may be written: kt φµc t r 2 > 25 Indeed when kt/(φµc t r 2 ) is equal to 5 the error in using the log approximation is still only about 2%. Note that for values of t D /r D 2 less than 5 the line-source solution based on the Ei function starts to deviate from the solution to the radial diffusivity corresponding to the proper boundary condition (2.24). The preceding equations give the pressure as a general function of both time and position in the reservoir. However in most applications the important item of interest is the pressure at the well-bore; this is the observable quantity in well tests. The dimensionless well-bore pressure, i.e. that corresponding to r D = 1 is given by: p D (1, t D ) = 1 2 [ln t D + 0.80907] . . (2.56) It is convenient to represent the pressure just inside the homogeneous formation adjacent to the well-bore by the symbol p w and hence from equation (2.55): p w (t) = p(r w , t) = p i − q s Bµ 2πkh . 1 2 ¸ ] 1 1 1 ln kt φµc t r 2 w + 0.80907 (2.57) 2.2.8. Well-bore Damage and Improvement Effects In the preceding treatment only the dynamic pressure behaviour within the homogeneous formation has been analysed. It is now necessary to relate the pressure in the well-bore itself, i.e. the bottom-hole fluid pressure as measured by a transducer, to that in the adjacent formation, p w . The bottom-hole fluid pressure at the mid-point of the producing interval in a flowing well is denoted p wf . In general the two quantities p w and p wf are not identical because of the method of completing the well. The phenomenon of skin effect and well-bore damage or improvement has already been introduced in Chapter 1 and discussed at length in the treatment of steady-state flow. One way of quantifying such damage to operating wells represents the well-bore condition by a steady- state pressure drop at the well-face in addition to the normal pressure profile in the formation: this is illustrated in Figure 2.2.4. The incremental pressure drop called the 'skin effect' is presumed to occur over an infinitesimally thin 'skin zone' in which the permeability impairment is confined. Accordingly, the bottom-hole flowing pressure is given by: ∆p s p w p wf "SKIN" t PRESSURE PROFILE I N THE FORMATI ON r w NEGATIVE SKIN FACTOR i.e. STIMULATION −∆ p s p wf pw k > k a PRESSURE PROFILE IN FORMATION RIGOROUS SKIN CONCEPT PROFILE t STIMULATED ZONE Figure 2.2.4 Dimensionless Skin S p q B kh s s · ∆ µ π 2 Positive Skin Factor i.e. Damage EPS Training 61 p wf (t) = p w (t) + ∆p s . . . (2.58) where the skin pressure drop, ∆p s , is a function of the instantaneous well flow-rate, q s B , the fluid viscosity, µ , and the characteristics of the altered zone, i.e. its average radial thickness and permeability. The important assumption is made that, since the skin is so thin there is insignificant accumulation or depletion of fluid in this region and hence quasi-steady-state conditions exist. For all practical purposes the skin pressure drop reacts immediately to any changes in production rate, e.g. if the flow stops ∆p s disappears without delay. Of course even at constant flow-rate long term changes in ∆p s can occur due to progressive plugging of the well-face, hence the term 'quasi-steady-state'. The dimensionless skin pressure drop is denoted by S and is defined by the equation: S = ∆p s q s Bµ 2πkh . . . . (2.59) Since the pressure drop over the damaged zone, assuming steady-state laminar D'Arcy flow, should be proportional to the product q s Bµ the dimensionless skin factor, S, only depends on the nature of the impairment. The dimensionless form of equation (2.58) may now be written: p D (1-, t D ) = p D (1, t D ) + S . . (2.60) where the notation r D = 1- implies pressure in the well-bore. Hence the dimensionless working equations for transient flow with a well-bore skin effect take the form: p D (1-, t D ) = p wfD = 1 2 [ln t D + 0.80908 + 2S] (2.61) and the corresponding equation in actual variables and parameters becomes: p wf (t) = p i − q s Bµ 2πkh . 1 2 ¸ ] 1 1 1 ln kt φµc t r 2 w + 0.80908 + 2S (2.62) This formulation is the basis for constant rate drawdown analysis on a semilog graph illustrated in Figure 2.2.5 in which the permeability is obtained from the slope and the skin factor from the intercept. EPS Training 62 CRD p i 0 q p wf TIME, t 0 CARTESIAN PLOT INTERCEPT SLOPE, m ln t 0 p wf SEMILOG PLOT Figure 2.2.5 = − qµ 4 kh π p (t=1) wf Rate Schedule Ideal (CSFR) Drawdown 2.2.9. Analytical Solution for the Case of a Bounded Circular Reservoir Of course no real reservoir is infinite in extent and the solution of the preceding section is only valid while the pressure transient is confined within the limits of a particular cylindrical volume. As soon as the pressure at the outer boundary starts to deviate from the initial value one of the external boundary conditions becomes operative. Usually the alternative form most consistent with physical reality is the no flow constraint (2.26). Occasionally the mathematical boundary may coincide with a physical barrier, i.e. the extremity of the reservoir. However, a much more common situation arises when several producing wells, placed more or less symmetrically, are distributed over the reservoir. In this case no flow boundaries arise because of the reservoir drainage patterns which develop; deviation from the transient, infinite reservoir solution occurs when the expanding, radially symmetric pressure disturbances from adjacent wells first come in contact. The concept of drainage volumes will be taken up in detail later. In the meantime an individual well will be assumed to be located in the centre of a cylindrically shaped drainage area of uniform thickness, h , and external radius, r e , with no flow across the external boundary. The dimensionless differential system now takes the form: ∂p D ∂t D = 1 r D ∂ ¸ ¸ , _ r D ∂p D ∂r D ∂r D 1 ≥ r D ≥ r De (2.63) t D < 0 , p D = 0 all r D r D = 1 , ∂p D ∂r D = −1 all t D > 0 EPS Training 63 r D = r De , ∂p D ∂r D = 0 all t D > 0 where r De = r e r w . The derivation of the analytical solution, using the Laplace transform technique, to this differential system is also given by Matthews and Russell and the result is: p D (r D , t D ) = 2 r 2 De −1 ¸ ¸ , _ r 2 D 4 + t e − r 2 De lnr D r 2 De − 1 − (3r 4 De − 4r 4 De lnr De − 2r 2 De − 1) 4(r 2 De − 1) + π ∑ m=1 ∞ exp (−α 2 m t D ) J 2 1 (α m r De ) A α m [J 1 2 (α m r De ) − J 1 2 (α m )] . . . (2.64) where A = J 1 (α m )Y 0 (α m r D ) − Y 1 (α m )J 0 (α m r D ) and α m are the roots of: J 1 (α m r De )Y 1 (α m ) − J 1 (α m )Y 1 (α m r De ) = 0 . (2.65) For the dimensionless pressure at the well-bore, r D = 1 , and for the case where r e >> r w , i.e. r De >> 1 , equation (2.64) simplifies considerably and can be written: p D (1, t D ) = 2t D r 2 De + ln r De − 3 4 + 2 ∑ m=1 ∞ exp(−α 2 m t D ) J 2 1 (α m r De ) α m 2 ¸ ] 1 J 2 1 (α m r De ) − J 2 1 (α m ) . (2.66) The α m values in equations (2.64) or (2.66) take on monotonically increasing values as 'm' increases, i.e. α 1 < α 2 . . . . . . < α m . . . . Thus for a given value of t D the exponentials decrease monotonically. Also the Bessel function portion of the terms in the series becomes less as 'm' increases. Hence as t D becomes large, the terms for large 'm' become progressively smaller and the summation is rapidly convergent. Indeed for sufficiently large t D all the terms of the series are negligibly small; for practical purposes this occurs for t D /r De 2 greater than about 0.3, i.e. if the following condition is satisfied: t De = t D r 2 De = kt φµc t r 2 e > 0.3 . . (2.67) the dimensionless pressure is given by equation (2.64) without the series summation and hence becomes a linear function of t D . In this case ∂p D /∂t D is a constant, irrespective of position, which implies that the EPS Training 64 dimensionless pressure is changing at the same rate everywhere in the system and that the pressure profiles are therefore not altering in shape as time proceeds. This situation is described as the semi -steady-state or pseudo-steady-state flow period and the dimensionless pressure at the well-bore during this period is given by: p D (1, t D ) = 2t D r 2 De + ln r De − 3 4 . . (2.68) The concept of semi -steady-state is an important one and is treated at length subsequently. It has already been indicated that for values of the parameter t D /r De 2 less than 0.1 the pressure disturbance has not yet reached the outer boundary at r e and the solutions to systems (2.46) and (2.63) are indistinguishable provided the line-source approximation is valid. Thus when the dual condition t De < 0.1 t D r 2 D > 5 . . (2.69) is satisfied the bounded reservoir solution (2.64) may be replaced by the mathematically much simpler line- source, infinite-reservoir solution (2.47). Naturally when t D /r D 2 > 25 the logarithmic approximation to the exponential integral, i.e. equation (2.51), is preferable. The pressure behaviour in this period is not affected by the external boundary and is essentially the same as in an infinite reservoir; this is described as transient flow or the infinite acting period. The interim time during which neither the infinite acting nor the semi-steady-state asymptotes are applicable to the pressure behaviour of the bounded reservoir is known as the late transient period and occurs when: 0.1 < t De < 0.3 . . . (2.70) Thus the pressure behaviour domain is divided into three distinct regimes viz. transient, late transient and semi-steady-state each corresponding to a specific physical state of the reservoir. Condition (2.36), upon which conditions (2.69) and (2.70) are partially founded, was arrived at by considering the error in the predicted pressure at the external boundary. If it is only the pressure at the well-bore which is of concern the infinite acting solution is acceptable for a bounded, circular reservoir over a longer period of time. Indeed up to values of t De of 0.3 the deviation between the values of well-bore pressure predicted by equations (2.56) and (2.66) is less than 1%. Hence, in a circular drainage area with a central well, for all practical purposes regarding the well-bore pressure there are only two flow regimes - the infinite acting and semi- steady-state periods - with the demarcation occurring at t De equal to 0.3. Thus in summary: Circular, Bounded Reservoir, Well-bore Pressure (a) t De < 0.3 : Transient flow p D (1, t D ) = 1 2 [ln t D + 0.80907] . . (2.71) (b) t De < 0.3 : Semi-steady-state flow p D (1, t D ) = 2t D r 2 De + ln r De - 3 4 . . (2.68) This is very convenient since the complicated late transient solution involving Bessel functions need not be employed. Alternatively in terms of actual variables and parameters these become: (a) t < 0.3φµc t t r 2 e k : Transient flow EPS Training 65 p w (t) = p i − q s Bµ 2πkh . 1 2 ¸ ] 1 1 1 ln kt φµc t r 2 w + 0.80907 . (2.57) (b) t > 0.3φµc t r 2 e k : Semi-steady-state flow p w (t) = p i − q s Bµ 2πkh ¸ ] 1 1 1 2kt φµc t r 2 e + ln r e r w − 3 4 . (2.72) 2.2.10. Analytical Solution for a Constant Pressure Outer Boundary In this event the condition at the external limit takes the form: r D = r De , p D = 0 all t D . . (2.73) and inflow through the outer boundary takes place at a rate determined by the local pressure gradient and D'Arcy's law, i.e. ¹ ¹ ∂p ∂r r=r e = q e Bµ 2πkhr e . . . . (2.74) where q e is the inflow from a sink surrounding the cylindrical volume. Using the definitions of dimensionless quantities equation (2.74) may be written: q e = q ¹ ¹ ∂p D ∂r D r D =r De . . . . (2.75) For the moment only the analytical solution for the dimensionless well-bore pressure will be given, i.e. p D (1, t D ) = ln r De − 2 ∑ m=1 ∝ exp(− β 2 m t D ) J 2 o (β m r De ) β 2 m ¸ ] 1 J 2 1 (β m ) − J 2 o (β m r De ) . (2.76) where the β m are the roots of: J 1 (β m ) Y 0 (β m r De ) − Y 1 (β m ) J 0 (β m r De ) = 0 (2.77) Again for sufficiently long times the summation is negligible and equation (2.76) reduces to: p D (1, t D ) = ln r De . . . (2.78) or, in terms of actual variables p w = p i − q s Bµ 2πkh ln r e r w . . . (2.79) In this circumstance the inflow at the external boundary and the production rate are equal, i.e. q e = q s and a true steady-state exists within the cylindrical volume. Note that in equation (2.79) p i represents the pressure at the outer boundary and is synonymous with p e . EPS Training 66 2.2.11. SPE Field Units In the preceding treatment the equations have been presented in fundamental form applicable in any system of consistent units e.g. strict metric S.I. where the basic units are: Production rate, q s : m 3 /s Formation thickness, h : m Permeability, k : m 2 Viscosity, µ : Ns/m 2 Pressure, p : Pa Radius, r : m Time, t : s Compressibility, c t : Pa -1 Table 2.1 S.I. Metric Units However most engineers prefer to work in a more practical system of units, for example SPE field units which are defined as: q s : STbbl/day h : ft k : md µ : cp p : psia r : ft t : hr c t : psia -1 Table 2.2 SPE Field Units In terms of SPE field units the definitions of dimensionless pressure and time take the form: p D = 2πkh∆p 887.2q s Bµ t D = 0.0002637kt φµc t r 2 w and any equation can be converted into field units using the appropriate definit ions of p D and t D . 2.2.12. The Depth of Investigation and Radius of Drainage These analytical solutions to the diffusivity equation predict the dynamic response of the pressure in a model reservoir after it is put on production at a constant rate, q. The pressure response to a variable (time- dependent) production rate will be considered subsequently. The distributed pressure behaviour of the reservoir is conveniently represented on a plot of pressure drop versus radial distance from t he well-bore at particular times, all on a dimensionless basis; such a diagram is shown in Figure 2.2.6. In order to compute this information it is necessary to evaluate the full exponential integral solution using an algorithm of the form: 0 ≤ x ≤ 1 Ei(x) = − ln x + a 0 + a 1 x + a 2 x 2 + a 3 x 3 + a 4 x 4 + a 5 x 5 + ε(x) where a 0 = − 0.57721566 a 3 = 0.05519968 a 1 = 0.99999193 a 4 = − 0.00976004 ε(x) < 2×10 -7 a 2 = − 0.24991055 a 5 = 0.00107857 1 ≤ x ≤ ∞ 10 4 10 5 5x10 5 10 6 2x10 6 t = 3x 10 D 6 0 1 2 3 4 5 6 7 8 9 1 200 400 600 800 1000 r D p D r =10 De 3 Figure 2.2.6 Dimensionless Pressure Distributions in Radial Flow SSS I A Well in a Closed Reservoir EPS Training 67 Ei(x) = 1 xe x . x 4 + a 1 x 3 + a 2 x 2 + a 3 x + a 4 x 4 + b 1 x 3 + b 2 x 2 + b 3 x + b 4 + ε(x) where a 1 = 8.5733287401 b 1 = 9.5733223454 a 2 = 18.059016973 b 2 = 25.6329561486 ε(x) < 2×10 -8 a 3 = 8.6347608925 b 3 = 21.0996530827 a 4 = 0.2677737343 b 4 = 3.9584969228 The Ei function has the form shown in Figure 2.2.7; as the argument, x, becomes large the function asymptotically approaches zero. Note that the region in which the log approximation is valid occurs at very small values of the argument viz. x < 0.01. The exponential function is mainly required for the interpretation of well to well interference tests where the radius, r, is the distance between observation and active well. When the well is put on flow at time t = 0 the oil production rate, q , is initially sustained by the expansion of fluid immediately around the well-bore. However this expansion is, of course, accompanied by a reduction in pressure and a local pressure gradient is established in the reservoir. Thus fluid from the next adjacent annular zone flows toward the well-bore at a rate governed by D'Arcy's law and the process of fluid expansion- pressure decline is extended further into the reservoir. In this way a progressively increasing zone of pressure drawdown propagates out from the active well until it reaches the external boundary. The propagation of this pressure disturbance is analogous to the temperature transient in a cylindrical block of material subjected to a constant heat flux at the face of a central cavity. Since the well is produced at a constant rate the pressure gradient at the well-bore, r = r w , is the same at all times. The shape of the instantaneous pressure profile at any time during the transient period is still much influenced by the radial nature of the flow, i.e. rapidly increasing gradient as the well-bore is approached, and this results in the well-bore pressure exhibiting the strongest dynamic response to a well flow-rate change. In most transient well tests this is precisely the quantity which is observable by measurement using down-hole pressure recording instruments. In Figure 2.2.6 the dimensionless pressure profiles at t D = 10 3 , 10 4 and 10 5 have been constructed using the exponential integral solution i.e. equation (2.47); the dimensionless pressure, p D , is plotted as a function of dimensionless radius, r D , for various values of dimensionless time, t D . It is not until t D /r 2 De = 0.1, equivalent to t D = 10 5 in this reservoir, that the external boundary pressure starts to decline and the exponential integral solution is invalid. The profiles for t D = 5×10 5 and greater have been computed from the semi -steady-state solution and it is apparent how the pressure decline takes place uniformly without change in the shape of the distribution. For dimensionless times between t D = 10 5 and 5×10 5 strictly speaking the bounded reservoir (late transient) solution should be employed but, as previously stated, there is very little error in omitting this regime; the graphical presentation of the pressure behaviour shows no signs of discontinuity between infinite- acting and semi-steady-state behaviour. Remember, however, that this is only true for a well in the centre of a circular drainage area. During the infinite-acting period a region of pressure disturbance propagates out from the well at which a flow-rate change has occurred. This penetration aspect is quite characteristic of systems whose behaviour is described by the second-order, parabolic diffusivity equation and is well-known in heat conduction and mass transfer. The analogy between diffusional t ransport processes and compressible fluid flow in a porous medium derives from the similar mathematical nature of underlying phenomenological description. For example, the basis of unsteady-state heat conduction is: (a) local flux equation (Fourier's law) q A = − k dT dr HENCE E(x) IS DENOTED -Ei (-x) 1 1 2 3 0.4 1.0 1.6 E (x) 1 -Ei(-x) 0 x Fig 2.2.7 Ei x e u u x ej · − − − ∞ z Exponential Integral Function EPS Training 68 (b) relation between energy density and temperature (state equation) de = ρ C p dT (c) law of energy conservation whereas compressible fluid flow is described by: (a) local flux equation (D'Arcy's law) q A = − k µ dp dr (b) relation between mass density and pressure (state equation) dρ = ρ c dp (c) law of mass conservation Although compressible fluid flow in porous media is not diffusional in nature the governing equations have the same mathematical form. Because of the importance of the diffusivity equation in transport phenomena, analytical or numerical solutions exist for almost every conceivable geometry and boundary condition. The velocity with which the pressure disturbance moves out through the reservoir is determined by the system hydraulic diffusivity, α = k/(φµc t ). The leading edge of the pressure front, defined loosely as the location where the pressure is say 1% different from the initial value, is shown in Figure 2.2.8. The dimensionless pressure drop at a distance, r D , from the wellbore and at time, t D , is given by the exponential integral solution: p D = 1 2 Ei ¸ ¸ , _ r 2 D 4t D If the argument of the Ei function is arbitrarily set to unity, i.e. r 2 D 4t D = 1 , the value of p D is given by: p D (r Di ) = 1 2 Ei(1) = 0.2194 2 = 0.11 This implies that at any time, t D , the dimensionless pressure drop is less than 0.1 at a distance r Di from the wellbore where: r Di = r i r w = 4t D = 4kt φµc t r 2 w . . . (2.80) Thus the depth of penetration in radial geometry is given by: r i = 4α t = 4kt φµc t . . . (2.81) 10 3 5x10 3 t=10 D 4 1 100 200 r D PRESSURE DISTURBANCE FRONT Fig 2.2.8 Radius of Influence p = 0. 1 D EPS Training 69 Although this definition of depth of investigation or radius of influence is borrowed from transient heat conduction, where it is termed depth of penetration, the concept is extremely useful in well test analysis and is a measure of the extent of the reservoir which has been influenced by the pressure disturbance. Equation (2.81) may also be written in the alternative dimensionless form: t Di = t D r 2 Di = kt φµc t r 2 i = 0.25 . . (2.82) The definition of the radius of investigation is somewhat arbitrary since the pressure profile is in fact asymptotic to p i and no sharp front exists. Hence the value of the constant in (2.82), for example, depends on the level of deviation from p i which is chosen to be significant. The value of 0.25 has been accepted as defining the reservoir volume which has contributed to the dynamic pressure response measured at the well- bore. Taking the constant as 0.1, as suggested previously, would seriously overestimate the region which had any real influence on pressure at the well-bore. In SPE field units equation (2.81) becomes: r i = 4×0.0002637kt φµc t = 0.00105kt φµc t = 0.032 kt φµc t (2.83) In practical terms, since the porosity, φ , and the compressibility, c t , do not vary very much the dynamic pressure response is largely determined by the reservoir permeability, k, and the oil viscosity, µ . The implication of this statement is that, if the response to a flow-rate change is measured and the oil viscosity is known, it should be possible to infer the value of the reservoir permeability. This is one of the main objectives of transient well testing. The concept of depth of investigation is particularly useful in estimating how far from a well information is being obtained at a particular time in a transient well test. In addition, it is of vital importance in estimating the duration of a proposed well test to evaluate the entire drainage area of a well. The nature of the transient pressure response is further illuminated by considering the behaviour of a well to well interference test illustrated in Figure 2.2.9 where the pressure is measured in an inactive observation well some distance, r, from the flowing well. The pressure at the observation well, again modelled by the exponential integral solution, is also plotted as a function of time in Figure 2.2.9. Since sonic phenomena are disregarded in the diffusivity equation the analytical solution predicts that the pressure at the observation well starts to change immediately the active well is set in flow. In reality, of course, pressure disturbances cannot exceed the speed of sound in the porous medium. However the pressure propagation process essentially follows a diffusional model and after some time the dimensionless pressure change at the observation well, p D , exceeds the arbitrary value of 0.1; at this point the depth of investigation is deemed to have reached the observation well. It is readily apparent from this physical situation that the actual moment at which a pressure change at the observation well becomes detectable is related to the resolution of the pressure transducer; once the pressure deviation from p i becomes larger than the gauge resolution, δp, it is resolvable. In Chapter 4 an alternative definition of depth of investigation, based on pressure transducer resolution, will be given. A related concept to depth of investigation is that of radius of drainage which is illustrated in Figure 2.2.10 where the dimensionless pressure profile at some time, t D , is plotted versus ln r D . Over a large range of r D the pressure profile exhibits a straight line on this plot indicating that quasi-steady-state conditions exist in the vicinity q ACTIVE WELL OBSERVATION WELL MINIMUM OBSERVABLE p DEPENDS ON GAUGE RESOLUTION ∆ r D "ARBITRARY" CRITERION Ei SOLUTION p i p wo OBS WELL PRESSURE 0 t Fig 2.2.9 p Ei r t D D D · F H G I K J 1 2 4 2 p p kh q D · · ∆ 2 01 π µ . Depth of Investigation 1 2 3 4 5 6 p D 0 1 2 3 4 5 6 7 ln r D Radius of Drainage r D Fig 2.2.10 I.-A. Transient Pressure Profile at t = 10 D 5 Steady-State Pressure Profile for Same p(1,t ) D D r Dd r D i r = classical depth of investigation D i EPS Training 70 of the wellbore i.e. the pressure distribution follows the steady-state model: (p(r,t) − p wf (t))2πkh q s Bµ = ln r r w . . . (2.84) The radius of drainage is defined through the equivalent annulus having the same overall steady-state pressure drop as the dynamic case; thus: p D (1,t D ) = (p i − p wf (t))2πkh q s Bµ = 1 2 ln 4t D γ = ln r Dd = ln r d r w . (2.85) i.e. 1 2 ln 4α t γ r 2 w = ln r d r w ∴ r d = 1 γ 4α t = 0.7493 4α t which corresponds to r d = 0.75r i . Thus the radius of drainage, r d , is somewhat smaller than the classical depth of investigation, r i . This definition of radius of drainage corresponds to finding the intercept, r Dd , of the straight line portion of the dynamic pressure profile shown in Figure 2.2.10. The important fact emerges that the flow conditions in the vicinity of the wellbore conform to quasi-steady-state implying that the local flow-rate, q r , is indistinguishable from the fixed sand-face rate, q , over quite a range of radii; this implies that there is negligible expansion of fluid occurring in the near wellbore region. This observation is crucial in the understanding of multiphase flow, for example, and the behaviour of radial composite systems. The idea of radius of drainage is closely related to the concept of transient productivity index denoted J t and defined by the equation: q s = J t (p i − p wf (t)) . . . . (2.86) Comparing this definition with the transient flow expression including a skin effect: p D (t D ) = (p i − p wf (t))2πkh q s Bµ = 1 2 ¸ ¸ , _ ln 4kt γφµc t r 2 w + 2S (2.87) J t is given by: J t = 2πkh Bµ ¸ ¸ , _ 1 2 ln 4kt γφµc t r 2 w + S = 2πkh Bµ ¸ ¸ , _ ln r d r w + S . (2.88) Thus the transient productivity index is just another way of expressing the p D function and serves to demonstrate that the transient PI, as defined above, decreases with time as the pressure disturbance propagates out into the reservoir. Note that the concept of transient PI is very useful when the pressure drop or drawdown is held approximately constant and the rate declines as a consequence of the sandface pressure gradient decreasing as the disturbance moves deeper into the system. Note that J t is defined in terms of the initial pressure, p i , since the notion of average pressure really has no meaning in an infinite-acting system. EPS Training 71 2.2.13. The Dynamics of Reservoir Pressure Response The theoretical well-bore pressure response for the same reservoir (r De = 1000) is shown in more detail in Figure 2.2.11 where the dimensionless pressure drop, p wD , is plotted against dimensionless time, t D , on both logarithmic and linear scales. During the infinite-acting period when the logarithmic approximation to the t = 0.3 De r = 1000 De 4 2 0 t = 10 D t = 25 D p wD ln t D 1 10 10 2 10 3 10 4 10 5 10 6 0 2 x 10 3 4 x 10 3 6 x 10 3 8 x 10 3 10 x 10 3 0 1 2 3 4 5 p wD PRODUCTION SHUT-IN 0 q 0 t D 0 2 4 6 8 10 12 14 6 8 10 Cartesian Graph t D t D Semilog Plot Pressure Drawdown at the Wellbore Fig 2.2.11 Ei function is valid the dimensionless pressure drop (for t he case of zero skin) is given by: p wD = 1 2 [ ln t D + 0.80908] . . . (2.56) and this is clearly shown as a straight line on the semi -log plot (Figure 2.2.11(a) ). Equation (2.56) is applicable when t D > 25 and t De < 0.3 as indicated. For t D between 10 and 25 the proper exponential integral solution should be used and for t D < 10 the line source solution (dotted line on Figure 2.2.11(a) ) must be replaced by one based on a finite well-bore radius shown in Figure 2.2.12. Note that it is not valid to extrapolate the linear portion of the semi-log plot back to early times. For values of t De greater than 0.3 the semi-steady-state solution: p wD = 2t D r 2 De + ln r De − 3 4 . . . (2.68) is applicable and the well-bore pressure declines more rapidly than the prediction of the infinite-acting model. Hence the theoretical pressure response falls below the extrapolation of the linear portion of the semi -log plot as indicated on Figure 2.2.12(a). Figure 2.2.12(b) shows a direct plot of well-bore drawdown versus time on a dimensionless basis and demonstrates the very rapid initial rate of change of pressure when the well is put on production. Figure 2.1.6 delineates the three flow regimes and illustrates that in the SSS regime the plot of well-bore pressure versus time is linear. The addition of the skin effect to the semi -steady-state flow equation results in: p D (1-, t D ) = p wfD = 2t D r 2 De + ln r De − 3 4 + S . (2.89) and the corresponding equation in actual variables and parameters becomes for semi -steady-state flow: EPS Training 72 p wf (t) = p i − q s Bµ 2πkh ¸ ] 1 1 1 2kt φµc t r 2 e + ln r e r w − 3 4 + S (2.90) where p wfD = p D (1-, t D ) = p i − p wf q s Bµ 2πkh The pressure distribution in the reservoir during the SSS period is illustrated in Figure 2.2.13 where the stabilised shape of the pressure profiles at successive times is apparent. r D = 1.0 2 . 0 1 . 2 2 0 EXPONENTIAL INTEGRAL SOLUTION 10 -2 10 -1 1 10 10 2 10 3 10 -2 10 -1 1 10 p D t /r D 2 D Finite Wellbore Radius Solution Finite Wellbore Radius (FWR) Solution Fig 2.2.12 p p e p wf Stabilised Pressure Distribution SSS Depletion Wel l i n Centre of a Cl osed Ci rcul ar Reservoi r r e r w Fig 2.2.13 Pore Vol ume = hA φ q s 2.3. Pressure Drawdown Testing 2.3.1. Introduction The analytical solution to the diffusivity equation for a uniform pressure initial condition and a constant flow-rate inner boundary condition has led to an expression for the dynamic well-bore pressure behaviour of a model reservoir having homogeneous formation permeability and instantaneous skin effect. The objective of a well test is to measure the dynamic response of an actual reservoir under these same conditions and determine unknown reservoir parameters by inference. The two most important such parameters are the permeability thickness product, kh , and the skin factor, S. The productivity of a well can only be predicted if these quantities are known. The problem of well testing is essentially one of parameter estimation in which the unknown properties are adjusted until the theoretical solution or ideal model matches the measured system behaviour. In linear systems this can often be achieved directly without a search process. Often the first significant transient event at an oil well is the initial production period that results in a pressure drawdown at the formation face. Provided the production rate can be controlled at a constant value, the physical situation corresponds to the model conditions and thus it seems logical to investigate what can be learned about the well and the reservoir from pressure drawdown data. In this section constant-rate drawdown testing in the infinite-acting and semi-steady-state flow regimes will be considered. Although drawdown testing is not limited to the initial productive period of a well, that may be the ideal time to obtain drawdown data. Figure 2.3.1 schematically illustrates the production and pressure history during a drawdown test. Ideally the well is shut-in until it reaches static reservoir pressure before the test. This requirement is met in new reservoirs; it is less often met in old reservoirs. The drawdown test is run by producing the well at a constant flow-rate while continuously recording bottom-hole pressure. While most reservoir information obtained from a drawdown test can also be obtained from a pressure build-up test, there is an economic advantage to drawdown testing since the well is produced during the test. The main technical advantage of drawdown TIME, t Pressure Drawdown Testing RATE q 0 0 SHUT-I N PRODUCING TIME, t p = p ws i 0 Fig 2.3.1 Bottom Hole Pressure p wf EPS Training 73 testing is the possibility of estimating reservoir volume. The major disadvantage is the difficulty of maintaining a constant production rate and the fact that the skin factor, S, may change due to the well cleaning up. 2.3.2. Pressure Drawdown Analysis in Infinite-Acting Reservoirs The bottom-hole pressure at an active well producing at a constant rate in an infinite-acting reservoir is given by equation (3.1): p wfD = 1 2 [ ] lnt D + 0.80908 + 2S . (3.1) or in terms of actual variables: p wf = p i − q s Bµ 2πkh . 1 2 ¸ ] 1 1 1 ln kt φµc t r 2 w + 0.80908 + 2S . (3.2) i.e. p wf = p i − q s Bµ 2πkh . 1 2 ¸ ] 1 1 1 lnt + ln k φµc t r 2 w + 0.80908 + 2S (3.3) if the reservoir is at p i initially; here t is the time from the start of production. Theoretically a plot of measured flowing bottom-hole pressure versus the natural logarithm of flowing time (commonly called the semilog plot) should be a straight line of slope m and intercept p t=1 - when lnt = 0, t = 1. Hence the analysis of drawdown data consists of making a plot of p wf against lnt giving: p wf = m lnt + p t=1 . . . (3.4) Such a graph is shown in Figure 2.3.2; from equation (3.3) the slope is given by: m = − q s Bµ 4πkh . . . . (3.5) and the intercept corresponding to lnt equal to 0 by: p t=1 = p i + m ¸ ] 1 1 1 ln k φµc t r 2 w + 0.80908 + 2S . (3.6) Once the slope of the straight line portion of the semilog plot, m, has been determined the permeability thickness product, kh , can be calculated from equation (3.5). This presumes, of course, that the oil production rate, q s , has been measured in the test and that the oil formation volume factor, B, and viscosity, µ , are known from laboratory PVT studies. If the formation thickness (net pay), h , is known from log evaluation the formation permeability, k , can be obtained. Equation (3.6) may be rearranged as an explicit expression for the skin factor, i.e. S = 1 2 ¸ ] 1 1 1 p t=1 − p i m − ln k φµc t r 2 w − 0.80908 . . (3.7) Hence if the initial reservoir pressure, p i , the porosity, φ , the total compressibility, c t , and the well-bore radius, r w , are known the skin factor can be calculated from the slope and intercept of the plot using equation EPS Training 74 (3.7). Note that both the numerator and the denominator in the first term in the brackets in equation (3.7) are intrinsically negative. It is apparent from Figure 2.3.2 that the data points corresponding to early times do not coincide with the fitted linear portion of the semilog plot. Indeed theory suggests early points should lie below the straight line whereas the initial measured data falls considerably above it. This deviation from ideal behaviour is due to well-bore storage and damage effects. The theoretical model envisages a step change in the oil flow-rate at the well-face at time t=0. However in practice it is impossible to achieve such an instantaneous change and the well-face flow changes from zero to the final value over a finite time interval. Even if the surface oil rate could be rapidly set at the desired constant value the compressibility of the large amount of fluid in the well-bore will sustain the initial production and the actual well-face flow will lag significantly behind the surface flow. This phenomenon is known as well-bore storage and will be treated in detail in the next chapter. The influence of well-bore storage is compounded by the presence of a skin effect. In principle the bottom- hole pressure should decline by an amount ∆p s immediately the well is put on production. However, because of the lag in the build-up of well-face flow the skin pressure drop does not reach its full value instantaneously. In fact there is also some capacity associated with the skin zone itself and this further contributes to a delay in the establishment of ∆p s . Thus the deviation of the early data points from the idealised model is a result of the combination of well-bore storage and skin effects. Before determining the slope and the intercept of the fitted straight line it is essential to exclude all points affected by these phenomena. A rational method for doing this based on a log-log plot will be given in Chapter 3 on well-bore storage. Note that the intercept, p t=1 , must be determined by extrapolation of the linear trend on the semilog graph. Once the data points influenced by well- bore storage and damage have been eliminated from consideration, the most convenient way of determining the best slope and intercept is to use a least-squares linear regression routine. However it is essential that the plotted data be first examined visually for anomalous points. The preceding equations can be used with either of the sets of consistent units. However, within the oil industry, there is still a preference for using field units and the working equations are easily transformed to accommodate this. The field units version of equation (3.3) is: p wf = p i − 887.217q s Bµ 2πkh.2 ¸ ] 1 1 1 ln 0.0002637kt φµc t r 2 w + 0.80908 + 2S . . . (3.8) or on rearranging: p wf = p i − 70.6q s Bµ kh ¸ ] 1 1 1 lnt + ln k φµc t r 2 w − 7.43173 + 2S . . . (3.9) Hence, in field units, a plot of p wf (psi) versus lnt (t:hr) gives a straight line of slope, m , and intercept, p t=1 where: m = − 70.6 q s Bµ kh . . . . (3.10) and 0 p t=1 ln t NOTE : ln t = 0 corresponds to t = 1 Fig 2.3.2 Deviation from straight line caused by damage and wellbore storage effects slope, m = − 4 k h π q B s µ Drawdown Semilog Plot Bottom Hole Pressure p wf EPS Training 75 p t=1 = p i + m ¸ ] 1 1 1 ln k φµc t r 2 w − 7.43173 + 2S . (3.11) which on solving for the skin factor, S, becomes: S = 1 2 ¸ ] 1 1 1 p t=1 − p i m − ln k φµc t r 2 w + 7.43173 . (3.12) The preceding treatment is based on the usage of the natural logarithm, lnt , and advocates plotting p wf against lnt on ordinary linear graph paper; this approach is more amenable to analysis using least-squares regression procedures. The natural log is fundamental to the basic theory and the working equations are simpler if it is retained. However, it is also possible to plot p wf versus t directly on semilog graph paper and this approach has been much used in the past. Hence it is necessary to derive the equations on the basis of log to the base ten on which logarithmic graph paper is founded. The dimensionless drawdown equation is: p wfD = 1 2 ln 4t D γ + S . . . . (3.13) which on converting to log 10 becomes: p wfD = 2.3026 2 ¸ ] 1 1 log 4t D γ + 0.86859 S . . (3.14) Changing to actual variables in field units this becomes: p wf = p i − 887.217q s Bµ 2πkh . 2.3026 2 ¸ ] 1 1 1 log 0.0002637×4kt φµc t r 2 w γ + 0.86859 S . . . . . (3.15) and the working equations in field units and log 10 become: m = − 162.6 q s Bµ kh psi/log cycle . . . (3.16) p t=1 = p i + m ¸ ] 1 1 1 log k φµc t r 2 w − 3.2275 + 0.86859 S . . . . (3.17) S = 1.1513 ¸ ] 1 1 1 p t=1 − p i m − log k φµc t r 2 w + 3.2275 . (3.18) Again in equation (3.18) p t=1 must be from the semilog straight line. If pressure data measured at 1 hour do not fall on that line, the line must be extrapolated to 1 hour and the extrapolated value of p t=1 must be used in equation (3.18). This procedure is necessary to avoid calculating an incorrect skin by using a well-bore storage influenced pressure. If the drawdown test is long enough, the bottom-hole flowing pressure will eventually deviate from the semilog straight line and make the transition from infinite-acting to semi-steady-state behaviour. EPS Training 76 Although a properly run drawdown test yields considerable information about the reservoir the test may be hard to control since it is a flowing test. If a constant rate cannot be maintained within a reasonable tolerance the analysis technique presented in this section cannot be used. Variable rate procedures are considered in Chapter 6. Another practical problem is that of measuring accurately the small pressure drops encountered, at the fairly high absolute pressures involved. 2.4. The Principle of Superposition 2.4.1. Introduction The analytical solutions developed for the diffusivity equation in section 2 were all for the idealised case of a single well operating at a constant rate from time zero onward. Since real reservoir systems usually have several wells operating at varying rates, a more general approach is needed to study problems associated with transient well testing. Fortunately, because the diffusivity equation is linear, variable-rate, multiple well problems can be handled by applying the principle of superposition. The constant rate inner boundary condition takes the form: r = r w , ∂p ∂r = q s Bµ 2πkhr w in which the well production rate, q s , remains unchanged throughout the whole production history. Except under special test conditions this will not normally be the case and flowing well rates may change because of demand variation or altered reservoir conditions. Some well tests, by their very nature, involve variable rate and production histories, e.g. short drill stem tests or offshore exploratory well tests. In the general case the well production rate is time dependent and should be written q(t). The inner boundary condition now takes the form: r = r w , ∂p ∂r = q s (t)Bµ 2πkhr w . . . (4.1) and the dynamic pressure behaviour of the reservoir under such a time dependent boundary condition is obviously much more complex than in the simple constant rate case. The superposition principle states that adding solutions to a linear differential equation results in a new solution to that differential equation, but for different boundary conditions. Superposition can be applied to include more than one well, to change rates and to impose physical boundaries. Superposition is easily applied to infinite systems but for bounded systems it must be used with more care - not because the principle is different but because the basic solutions frequently do not give the necessary information for correct superposition. 2.4.2. Multiple-Well Situations The simplest illustration of superposition is that of determining the pressure drop at some point in a field in which two sinks are located. Consider the three well infinite system shown in Figure 2.4.1. At time t = 0 well 1 starts producing at rate q 1 and well 2 starts producing at rate q 2 . It is desired to determine the pressure at the shut-in observation point, well 3. To do this the pressure change at well 3 caused by well 1 is added to the pressure change at well 3 caused by well 2, i.e. ∆p 3 = ∆p 3,1 + ∆p 3,2 . . . (4.2) • o Well 1 q 1 Well 2 q 2 r 1 r 2 Well 3 Observation Well Active Well Three Well System Figure 2.4.1 Principle of Superposition EPS Training 77 In order to utilise equation (4.2) expressions for the individual pressure drops of the form: ∆p i,j = q i B j µ 2πkh p D (t D , r Dj ) . . . (4.3) must be employed with the appropriate well flow-rate, q j B j , and dimensionless distance, r Dj , to the observation point, i , for each sink, j . Thus equation (4.2) may be written: ∆p 3 = µ 2πkh (q 1 B 1 p D (t D , r D1 ) + q 2 B 2 p D (t D ,r D2 )) . . (4.4) which can be extended to an arbitrary number of wells: ∆p(t,r) = µ 2πkh ∑ j=1 n q j B j p D (t D , r Dj ) . (4.5) Note that equation (4.5) adds pressure changes (or dimensionless pressures). If the point of interest is an operating well the skin factor must be added to the dimensionless pressure for that well only. If the wells do not all start producing at the same time t D in equation (4.5) should be replaced by t Dj , the dimensionless production time for each individual well. There is no restriction on the number or location of sinks (or sources) and it is perfectly valid to consider two or more sinks as coincident, i.e. at the same point in space in order to generate a variable rate history at this point. 2.4.3. Variable Rate Situations To illustrate the application of the principle of superposition to varying flow-rates, consider a single well system with the production rate schedule shown in Figure 2.4.2. q 2 q t T 1 q - q 2 1 Two-Rate Flow Schedule Fig 2.4.2 Injection well rate q q 2 1 − Production well rate q 1 Superposition of Rates Total Response Extrapolated Pressure Principle of Superposition Injection Well at Rate q - q 2 1 Well at Rate q1 0 T 1 t 0 p p i w − ∆p DD ∆p DD Fig 2.4.3 The oil flow-rate is q 1 from t = 0 to t = t 1 , and q 2 thereafter. To perform the superposition calculation the single well may be visualised as two wells located at the same point, with one producing at rate q 1 , from t = 0 to t and the second (imaginary) well producing at rate (q 2 - q 1 ), starting at t 1 and continuing for a time period (t - t 1 ). The net superposed rate after time t 1 would be q 1 + (q 2 - q 1 ) = q 2 as desired. As in the previous example delta-p's are added for these conditions. The general form of the equation for N rates, with changes at t j , j = 1, 2 . . N, is: ∆p = µ 2πkh ∑ j=1 N {(q j B j − q j-1 B j-1 )(p wD (t − t j-1 ) D + S)} . (4.6) where (t-t j ) D is the dimensionless time calculated at time (t-t j ). For the rate schedule of Figure 2.4.2, N=2, only two terms of the summation are needed and equation (4.6) becomes: EPS Training 78 ∆p = Bµ 2πkh {q 1 (p wD (t D ) + S) + (q 2 − q 1 )(p wD (t D − t D1 ) + S)} . . . . (4.7) Figure 2.4.3 illustrates the calculation i.e. superposition. The lower dashed curve (including the first portion of the solid curve) is the pressure change caused by rate q 1 , alone. The topmost curve is the pressure change caused by the rate q 2 - q 1 , after t 1 ; that ∆p is negative because (q 2 - q 1 ) < 0. The sum of the dashed curve and topmost curve is the pressure response for the two-rate schedule. 2.5. Pressure Build-Up Testing 2.5.1. Introduction The most widely used form of transient well testing technique is pressure build-up analysis. This type of testing was first introduced by groundwater hydrologists but it has been extensively used in the petroleum industry. Pressure build-up testing entails shutting in a producing well and recording the closed-in bottom- hole pressure as a function of time. The most common and simplest analysis techniques require that the well produce at a constant rate, either from start-up or long enough to establish a stabilised pressure distribution before shut-in. If possible the flowing bottom-hole pressure prior to shut-in should also be recorded; indeed it is essential if an estimate of skin is required. Figure 2.5.1 schematically shows flow-rate and bottom-hole pressure behaviour for an ideal pressure build-up test. Here t p is the production time and ∆t is the running shut-in time. The pressure is measured immediately before shut-in and is recorded as a function of time during the shut-in period. The resulting pressure build-up curve is analysed for reservoir properties and well-bore condition. As in all transient well tests, knowledge of surface and subsurface mechanical conditions is important in build-up test data interpretation. Therefore it is recommended that tubing and casing sizes, choke size, well depth, packer locations etc be determined before data interpretation starts. Short -time pressure observations are usually necessary for complete delineation of well- bore storage effects. Data may be needed at intervals as short as 15 seconds for the first few minutes of some build-up tests. As the test progresses, the data collection interval can be extended. Stabilising the well at a constant rate before testing is an important part of a pressure build-up test. If stabilisation is overlooked or is impossible, conventional analysis techniques may provide erroneous information about the formation. Thus it is important to determine the degree and adequacy of the stabilisation; one way is to check the length of the pre-shut-in constant rate period against the time required to reach semi -steady state flow, i.e. q De = 0.3 for a central well. For wells with significantly varying rates before shut-in, build-up test analysis is still possible using the variable-rate methods discussed later. If the well produces at a constant rate, q , for a time,t p , and pressures are subsequently recorded for closed- in times, ∆t , then the bottom-hole pressure at any time after the well has been shut-in can be obtained from a superposed solution based on: q acting for time (t p + ∆t) + (o − q) acting for time ∆t The build-up test rate schedule is the simplest form of two-rate test (q 2 = 0) and the two-rate superposition formula: ∆p = Bµ 2πkh { q 1 [ ] p wD (t D ) + S + (q 2 − q 1 ) [ ] p wD (t D − t 1D ) + S } R A T E t p ∆ ∆ t t FLOWING SHUT-IN t p B H P p ws p ( t=0) wf ∆ Figure 2.5.1 Schematic Flow-Rate and Pressure Behaviour for an Ideal Buildup q EPS Training 79 . . . (5.1) will predict the pressure behaviour during the second zero rate period; on putting: q 1 = q s ; q 2 − q 1 = −q s ; t D = t pD + ∆t D ; t D − t 1D = ∆t D this becomes: ∆p = q s Bµ 2πkh { p wD (t pD + ∆t D ) − p wD (∆t D ) } i.e. p ws (∆t) = p i − q s Bµ 2πkh { p wD (t pD + ∆t D ) − p wD (∆t D ) } (5.2) where p wD is the applicable dimensionless pressure function; note that the shut-in pressure, p ws , is not affected by the skin factor in an ideal build-up test. Equation (5.2) provides the theoretical basis for the analysis of pressure build-up tests. 2.5.2. Pressure Build-Up Test Analysis during the Infinite-Acting Period There are several ways for analysing the results of a build-up test the most popular being the Horner method which is based on the supposition that the reservoir is infinite in extent and a negligible amount of fluid is removed from the system during the production period prior to closure. In practice this case corresponds to an initial well test conducted in a virgin reservoir. During an infinite-acting period both dimensionless pressure terms in equation (5.2) are replaced by the logarithmic approximation to the exponential integral, i.e. p wD (t D ) = 1 2 (lnt D + 0.80908) and equation (5.2) becomes: p ws (∆t) = p i − q s Bµ 2πkh . 1 2 (ln(t pD + ∆t D ) − ln ∆t D ) or p ws (∆t) = p i − q s Bµ 2πkh ln t p + ∆t ∆t . . . (5.3) This is the Horner build-up equation which predicts a linear relationship between p ws and ln((t p + ∆t)/∆t) ; note that the ratio is independent of the units of the time quantities. This equation also indicates that, for an infinite-acting system, the bottom-hole shut-in pressure will eventually build up to the initial reservoir pressure, p i . Equation (5.3) describes a straight line of the form: p ws = m ln t p + ∆t ∆t + p* . . . (5.4) with slope m = − q s Bµ 4πkh . . . . (5.5) and intercept, p* = p i . . . . (5.6) In this case the intercept, p*, corresponds to ln((t p + ∆t)/∆t) equal to zero or (t p + ∆t)/∆t = 1 which implies ∆t >> t p - a shut-in time very long compared to the production time. From such a plot the reservoir average permeability can be obtained as: EPS Training 80 k = − q s Bµ 4πmh . . . . (5.7) provided all the quantities on the right-hand side of (5.7) are known. In oilfield units equation (5.3) becomes: p ws (∆t) = p i − 70.6q s Bµ kh ln t p + ∆t ∆t . . (5.8) or p ws (∆t) = p i − 162.6q s Bµ kh log t p + ∆t ∆t . (5.9) The most important feature of the semilog plot is that the reservoir kh product can be determined from the slope of the build-up. This is a much better value than can be obtained by trying to average core measured permeabilities over the entire producing interval. The determination of the permeability is the most significant result to be obtained from an initial test on a reservoir simply because, under these circumstances, the initial pressure could be obtained from a spot measurement prior to opening the well in the first place. Once the well has been producing for a significant period of time this statement is no longer true because the well will now have produced an amount of fluid which may not be insignificant in comparison to the initial oil in place. Therefore it can no longer be assumed that the pressure will build up to its initial value even for an infinite closed-in time, but rather to some lower value p which will be representative of the average pressure within the drainage volume of the well. This corresponds to pressure build-up in a bounded reservoir which will be considered later. It is instructive to examine the process of pressure build-up within the reservoir itself as well as at the well- bore. Defining the dimensionless reservoir pressure after shut-in as: p sD = p sD (r D , ∆t D ) = p i − p s (r, ∆t) q s Bµ 2πkh . . (5.10) the superposition theorem states that: p sD = p D (r D , t pD + ∆t D ) − p D (r D , ∆t D ) . (5.11) However at locations far removed from the well-bore the logarithmic approximation is not valid and the dimensionless pressure functions on the right-hand side of (5.11) must be represented by the proper exponential integral solution for an infinite-acting system. The dimensionless shut-in pressure distribution for an infinite reservoir which has produced for a dimensionless production time of t pD = 10 4 is shown in Figure 2.5.2; this diagram was drawn in accordance with equation (5.11) using Ei functions for the p D terms. When the well is shut-in the inner boundary condition becomes one of no flow, i.e. zero pressure gradient at the well- bore. This is clearly seen in Figure 2.5.2 from the pressure profiles at various values of the dimensionless shut- in time, t D all these have zero slope at r D = 1. p ws ln t + t p ∆ ∆t p* Devi ati on from Strai ght Line caused by Afterflow and Skin 0 slope, m = − 4 π k h q B s µ Semilog (Horner) Plot for a Buildup Fig 2.5.1b r D 0 5 1 200 q 0 t p D ∆ tD t = 10 pD 4 Pressure Build-Up in a Reservoir Figure 2.5.2 t = 10 D 4 ∆ 10 50 200 2 10 3 p sD EPS Training 81 This is in marked contrast to the pressure profiles during flowing periods which have very steep slopes. Fluid continues flowing toward the well after shut-in but since it cannot emerge from the reservoir this fluid accumulates which it can only do by compressing and the reservoir pressure rises. This process continues until the pressure is everywhere equalised. However it is apparent from Figure 2.5.2 that the final stages of pressure recovery are slow because reservoir pressure gradients are small and fluid flow is difficult. The behaviour of the closed-in well-bore dimensionless pressure is shown in more detail in Figures 2.5.3 and 2.5.4. 0 4 0 7 l n ∆ ∆ t D t D t + pD 0 5 p sD ∆ t D 10 10 3 × × 10 3 5 1 0 3 t = 10 p D 4 Pressure Build-Up at Wellbore Semilog Cartesian Dimensionless Response (Horner) Fig 2.5.3 p sD t + t p ∆ ∆t 13 5 2 6 ∆t D t = 10 pD 4 ∆t < 10 D Ei Function not Represented by Log Approximation Fig 2.5.4 p D Dimensionless Build-up Semilog (Horner) Plot Figure 2.5.3 demonstrates theoretically the linearity of the Horner plot for dimensionless shut-in times, ∆t D , greater than 10. However for values of ∆t D less 10 it is no longer valid to use the logarithmic approximation to the exponential integral solution. Thus for very early shut-in times: p wsD = p D (tp D + ∆t D ) − p D (∆t D ) ≠ 1 2 ln t p + ∆t ∆t In fact: p wsD = 1 2 (ln(t pD + ∆t D ) + 0.80908) + 1 2 Εi ¸ ¸ , _ −1 4∆t D . (5.12) Figure 2.5.4 shows the behaviour of p wsD at very early shut-in times evaluation using (5.12) and it can be seen that the exact solution deviates from the extrapolation of the linear portion; clearly such an extrapolation is not valid. In theory then one would expect early build-up points to lie above the straight line. In practice they often fall below the line and the reason for this is again the effect of well-bore storage and skin. 2.5.3. After Production The preceding theory of pressure build-up analysis is based on the supposition that the oil flow-rate at the well-face can be instantaneously reduced from the constant rate, q , to zero. Often the well is shut-in at the surface and pressure increases in the well-bore after shut-in require the influx of sufficient fluid fromthe formation to compress the contents of the whole well-bore. Note however that, in drill stem tests or offshore exploration well tests on semi-submersible rigs or drill ships, the closing-in valve is at bottom-hole. The duration and magnitude of this effect, called afterflow in pressure build-up testing, depends on the flow-rate before shut-in, the capacity of the well-bore and the fluid compressibility. The effect is minimised when the well production rate is high and hence in highly productive wells afterflow is not a serious problem. In general it is advisable always to avoid analysis of pressure build-up data for times when bottom-hole pressures are affected by after production. Thus even though the well is shut-in during pressure build-up testing, the afterflow caused by well-bore storage has a significant influence on pressure build-up data. Figure 2.5.5 shows that the pressure points fall below the semilog straight line while well-bore storage is important. Two effects cause this less than ideal rate of pressure build-up. Firstly pressure build-up in the formation itself is less rapid because of the continuing production from the well-face. Secondly the skin pressure drop, which should disappear instantaneously at the moment of closure, declines in response to the afterflow. EPS Training 82 t + t p ∆ ∆t Effect of Afterflow on a Horner Plot Data Affected by Wel l bore St orage Correct Semilog Straight Line p ws 0 p* slope m Fig 2.5.5 ETR MTR 1 1000 0.001 100 t ∆p (psi) Unit Slope Dat a of Cor r ect Semi l og Sl ope (hr) Fig 2.5.6 MTR Log - Log Diagnostic Plot for Afterflow The duration of these effects may be estimated by making the log-log data plot described in Chapter 3 on well-bore storage. For pressure build-up testing, plot log [p ws - p wf (∆t = 0)] versus log ∆t. When well-bore storage dominates that plot will have a unit slope straight line; as the semilog straight line is approached the log-log plot bends over to a gently curving line with a low slope as shown in Figure 2.5.6. In all pressure build- up analyses where afterflow may be of importance the log-log data plot should be made before the straight line is chosen on the semilog data plot, since it is often possible to draw a semi -log straight line through well-bore storage dominated data. This phenomenon occurs because well-head shut-in does not correspond to sand- face shut-in. As the sand-face flow does drop off to zero, the pressure increases rapidly to approach the theoretically predicted level. The semilog data plot is steep and nearly linear during this period, and may be analysed incorrectly seriously underestimating the formation permeability. 2.5.4. Determination of Reservoir Parameters The basic method analysing pressure build-up data in an infinite-acting reservoir is the Horner plot of p ws versus ln((t p + ∆t)/∆t). It is important to remember that in this context "infinite-acting" refers to both the production period before shut-in and to the ensuing closed-in period; thus it is a dual condition requiring that neither the pressure disturbance from the initial production nor that from the rate change at shut-in reaches the external boundary. It is recommended that the data points be plotted on ordinary graph paper as a plot of p ws versus ln( (t p + ∆t)/∆t) with bottom-hole shut-in pressure as ordinate and the natural log term as abscissa. The data points influenced by well-bore storage are eliminated from consideration and a straight line is fitted to those remaining by a linear least-squares regression. This process yields the slope, m , and the intercept, p*, of equation (5.4). Although it is not valid to extrapolate the straight line portion of the plot to very long shut-in times since the term p D (∆t D ) in the superposition is then no longer represented by the logarithmic approximation, nevertheless, in an essentially infinite-acting reservoir, i.e. one with a short initial flow period, t p , it is often assumed that the extrapolated pressure (intercept), p*, is synonymous with the initial pressure, p i , and the mean drainage area pressure, p - . This is permissible provided the amount of fluid withdrawn from the reservoir during the flow period is very small compared to the oil in place, i.e. negligible depletion has occurred. In this situation then p − = p i = p* . . . (5.13) and the average drainage area pressure can be determined simply by extrapolating the linear portion of the build-up. The permeability-thickness product, kh , is calculated in the usual way from the measured slope, m , of the fitted straight line using the formula: kh = − q s Bµ 4πm . . . . (5.14) or, if field units are being used: EPS Training 83 kh = − 70.6q s Bµ m (natural log) . (5.15) kh = − 162.6q s Bµ m (log base 10) . (5.16) Again if the net pay, h , is known the average permeability, k , of the area investigated can be calculated. The superposition process showed that the ideal pressure behaviour (no well bore storage) after shut-in is not influenced by the skin effect, S. Hence build-up data alone cannot be used to determine formation damage; only the flowing pressure prior to shut-in is affected by the skin. For an infinite-acting reservoir the flowing pressure just prior to shut-in is given by: p wf (∆t = 0) = p i − q s Bµ 4πkh ¸ ¸ , _ ln kt p φµc t r 2 w + 0.80908 + 2S (5.17) and, again for an infinite-acting system, the initial reservoir pressure, p i , can be replaced by the extrapolated pressure, p*; hence equation (5.17) may be written: p wf (∆t = 0) = p* + m ¸ ¸ , _ ln kt p φµc t r 2 w + 0.80908 + 2S (5.18) where − q s Bµ 4πkh has been replaced by the measured slope of the build-up Horner plot, m . Solving equation (5.18) for the skin factor, S , results in: S = 1 2 ¸ ¸ , _ p wf (∆t = 0) − p* m − ln kt p φµc t r 2 w − 0.80908 (5.19) where the slope, m , is an intrinsically negative quantity. The equivalent formulae for the skin factor when using field units are: (a) natural log basis S = 1 2 ¸ ¸ , _ p wf (∆t = 0) − p* m − ln kt p φµc t r 2 w + 7.43173 (5.20) where m = − 70.6q s Bµ kh and (b) log base 10 S = 1.1513 ¸ ] 1 1 1 p wf (∆t = 0) − p* m − log 10 kt p φµc t r 2 w + 3.2275 (5.21) where m = − 162.6q s Bµ kh When semilog graph paper with a limited number of cycles is being employed it may be inconvenient to extrapolate the straight line portion of the Horner plot sufficiently far to obtain p* directly. However it is EPS Training 84 always possible to determine the pressure on the straight line 1 hour after shut-in; this is denoted p ws (∆t = 1) or more simply p 1hr . The shut-in pressure, p ws , is given by: p ws = p* + m log t p + ∆t ∆t . . . (5.22) and hence: p ws (∆t = 1) = p 1hr = p* + m log (t p + 1) . (5.23) i.e. p* = p 1hr − m log (t p + 1) . . . (5.24) Substituting this expression for p* into equation (5.21) gives: S = 1.1513 ¸ ] 1 1 1 p wf (∆t = 0) − p 1hr m + log t p + 1 t p − log k φµc t r 2 w + 3.2275 . . . (5.25) in which the term log ((t p + 1)/t p ) is frequently quite negligible. This formula for the skin factor often appears in the literature and the procedure for finding p 1hr is shown in Figure 2.5.7. One advantage of a modern electronic pressure transducer, with a high sampling rate, is that the last flowing pressure, p wf (∆t = 0) , can be accurately determined as illustrated in Figure 2.5.8; this is important for the calculation of the skin. Note that the clock time corresponding to ∆t = 0 (the point at which the valve actually closes) can also be accurately bracketted. Precise estimates of pressure and time at ∆t = 0 are necessary for the log-log diagnostic plots discussed in the next chapter. The analysis of a buildup by the CRB method can be carried out even when the rate is varying during the drawdown period as shown in Figure 2.5.8. The equivalent constant rate drawdown time is defined as: t p = Q q . . . . . (5.26) Here Q is the cumulative volume produced over the whole flow period and q is the last, stabilised rate. This approach should not be used when there is a strongly declining rate in the flowing period, as in a slug (rising liquid level) test; in this case full superposition is necessary as described in Chapter 6. Log t + t p ∆ ∆t x x x x x x x x x x x p* p 1hr slope m 0 Determination of p on the Horner Plot 1 hr Fig 2.5.7 MTR Straight Line ∆t = 1 hr S p p m t t k c r wf hr p p t w · − + + − + L N M M O Q P P 11513 1 3 2275 1 2 . . log log φµ p ws EPS Training 85 Determine and very accurately p ( t=0 t( t=0) w f ∆ ∆ ) + t( t=0) ∆ p ( t=0) wf ∆ End of Drawdown Buildup ∆t Stabilise flow-rate before shutin q Q = cumulative volume Flow- Rate ∆t Shutin Afterflow ∆ − ∆ p = p p ( t=0) BU ws wf ∆ − ∆ t = t t( t=0) t = p Q q Test Precautions Fig 2.5.8 In Figure 2.5.9 a typical chart from the original Amerada gauge is depicted and it can be seen that during the flowing periods the bottom-hole pressure is actually increasing. This is the rising liquid level phenomenon apparent in many old DST’s and Horner analysis of the final buildup is not really recommended because of the implied rate variation. In 1976 the Hewlett-Packard company introduced the first quartz crystal pressure transducer and this proved essential for the satisfactory conduct of well tests in the high permeability North Sea basin. In Figure 2.5.10 a Horner plot of a pressure buildup in a Piper (Occidental) well is shown where the total pressure change in the buildup (∆p BU ) is less than 5 psi; it is immediately apparent why a high resolution pressure gauge is necessary in this application. Time P r e s s u r e Flow period Reservoir Disturbance Shut-in period Reservoir Recovery Horner Plot Large m Small m Same p r p ws Log t + t p ∆ ∆t Small Reservoir Disturbance Low flow-rate Small viscosity High permeability Large Reservoir Disturbance High flow-rate Viscous fluid Low permeability Permeability of Reservoir Rock from a DST After Matthews and Russell Large m Small m Scribed Tin Chart from Amerada Gauge Fig 2.5.9 EPS Training 86 2.5.5. Peaceman Probe Radius In Figure 2.5.2 the pressure redistribution in the reservoir during a buildup process has been generated using the exponential integral solution and superposition; this diagram clearly illustrates how the buildup is a relaxation of the pressure profile existing at the moment of shut-in back to equilibrium i.e. uniform pressure. The buildup test measures the wellbore pressure, p ws , as a function of shut-in time, ∆t , and it is of interest to show how the buildup record can be used to infer the spatial pressure distribution present in the reservoir at the moment of shut-in. Thus a transformation which takes p ws (∆t) and generates p r (t p ,r), as shown in Figure 2.5.11, is required. 3424 3423 3422 p ws 8 7 6 5 4 Horner Plot Early Piper Well (HP Gauge) slope m = 0.7465 psi − kh = 1.067 * 10 md.ft S = 3.08 6 q = 11750 bbl/d B = 1.28 = 0.75 cp r = 0.362 ft = 0.237 c = 1.234 * 10 psi s w t µ φ -5 - 1 (psia) Fig 2.5.10 l n t t t p + ∆ ∆ 3425 0 t p ∆t Pressure Build-Up in a Reservoir ∆ t 1 ∆ t 2 ∆t 3 ∆ t 4 ∆t 5 p ( t , r ) r p p1 p ( t , r ) r p p 2 p ( t , r ) r p p 3 p( t , r ) r p p 4 p ( t , r ) r p p 5 p r r r w Reservoir pressure distribution at moment of shut-i n, p(t) r p Peaceman Probe Radius Concept Fig 2.5.11 p p w r r r w · · The basis of the method can be examined for the case of a homogeneous system where the flowing pressure at radius, r d , is predicted by the exponential integral expression: p wf,D( ) r D , t pD = 1 2 Ei ¸ ¸ , _ r 2 D 4 t pD . . . (5.27) The shut-in pressure at time ∆t D is given by: p ws,D( ) ∆t D = 1 2 ln t p + ∆t ∆t . . . (5.28) If the exponential integral can be replaced by the log approximation viz.: p wf,D( ) r D , t pD = 1 2 ln 4 t pD γ r 2 D . . . (5.29) then equating the pressures in (5.29) and (5.28) gives the correspondence result: 4 t pD γ r 2 D ≡ t p + ∆t ∆t i.e. r d ≡ 4 k ∆t e γ φ µ c t (5.30) This calculation was first performed by Peaceman ( 9 ) who termed the equation for r d given above the probe radius formu la. 9 Peaceman, D.W.:“Interpretation of Well-Block Pressure in Numerical Simulation”, JPT (June 1976), 183-194 EPS Training 87 The ability to generate the pressure distribution at fixed time, t p , allows the average pressure in the near wellbore region to be computed. For example suppose a simulator block contains a well and it is desired to relate the simulator block pressure to buildup surveys. If the simulator block has an area, A ∼ , then the equivalent radius, r ∼ , is defined as: r ∼ = A ∼ π . . . . (5.31) The average pressure in this region can be obtained by integration of the reconstituted pressure profile i.e. p ∼ = ⌡ ⌠ r w r ∼ 2π p r r dr A ∼ 2.5.6. Transient Productivity Index, J t It is quite rare for the semi-steady-state regime to be attained in an appraisal well test and the drawdown period will usually fall into the infinite-acting, transient period. Supposing the major flow period is terminated at time, t p , as illustrated in Figure 2.5.12 where the last flowing pressure is denoted p wf (t p ). The buildup, when plotted on a Horner plot, will extrapolate to the initial reservoir pressure, p i , as indicated by the small vertical bar on the Fig 2.5.12 Cartesian plot. It is often the practice to then define a transient productivity index (P.I.) as: J t = q s p i − p wf (t p ) . . . . (5.32) It is apparent from the diagram that this quantity is highly time dependent especially when the flowing time, t p , is short i.e. the flush production is high. In a well test the surface chokes will be set to maintain the flow-rate at a value, q s , compatible with the test separator capacity. If the well is flowed for a long time a semi-steady-state condition will be reached and the semi -steady-state P.I. of the form: q s = J sss¸ ¸ , _ p − − p wf . . . . (5.33a) can be determined. In this case J sss will be described by an equation of the form: J sss = 2πkh Bµ ¸ ¸ , _ ln r e r w − 3 4 + S . . (5.33b) p wf p i Time, t t p2 t p1 t p1 t p3 t p3 t p4 Transient Productivity Index, J t p (t ) wf p3 p ws or J is strongly time dependent t p wf J q p p t t s i wf p · − di Fig 2.5.12 EPS Training 88 where r e is the equivalent circular radius of the well confining compartment. The important point is that the transient P.I. defined by (5.32) is much higher than the stabilised value of (5.33). From the theory of constant rate drawdown the transient P.I. is given by: J t = 2πkh Bµ ¸ ¸ , _ 1 2 ln 4kt γφµc t r 2 w + S . . . (5.34) The difference between these two quite distinct forms of the productivity index is not properly appreciated by many production engineers. EPS Training 89 Fig 2.1.1 Drillstem Testing Assembly ............................................................................................................................43 Fig 2.1.2 Dual Flow Dual Shutin Test.............................................................................................................................45 Fig 2.1.3 Buildup Analysis on the Horner (Theis) Plot ...............................................................................................46 Fig 2.1.4 Physical Interpretation of the Skin Factor.....................................................................................................46 Fig 2.1.5 Use of the Horner Plot in Reservoir Monitoring ..........................................................................................47 Fig 2.1.6 Principle Flow Regimes in Constant Rate Drawdown..................................................................................47 Fig 2.1.7 Detection of Depletion in a Drillstem Test....................................................................................................48 Fig 2.1.8 Some Well Test Models ...................................................................................................................................48 Fig 2.1.9 Transient Heat Conduction in a Linear System............................................................................................49 Fig 2.2.1 Prototype Radial Reservoir of Fixed Thickness............................................................................................53 Fig 2.2.2 Transient Pressure Profiles in Radial Geometry............................................................................................56 Fig 2.2.3 Exponential Integral Function for a Single Well in an Infinite Reservoir..................................................59 Fig 2.2.4 Dimensionless Skin Factor, S..........................................................................................................................60 Fig 2.2.5 Ideal Constant Sandface Rate Drawdown .....................................................................................................61 Fig 2.2.6 Dimensionless Pressure Distributions in Radial Flow.................................................................................66 Fig 2.2.7 Exponential Integral Function .........................................................................................................................67 Fig 2.2.8 Depth of Investigation .....................................................................................................................................68 Fig 2.2.9 Well to Well Interference Test........................................................................................................................69 Fig 2.2.10 Radius of Drainage..........................................................................................................................................69 Fig 2.2.11 Theoretical Dimensionless Wellbore Pressure Response........................................................................71 Fig 2.2.12 Finite Wellbore Radius Solution...................................................................................................................71 Fig 2.2.13 Stabilised Pressure Distribution in SSS.......................................................................................................72 Fig 2.3.1 Diagrammatic Production and Pressure History for a Drawdown .............................................................72 Fig 2.3.2 Drawdown Semilog Plot ...................................................................................................................................73 Fig 2.4.1 Three Well Infinite System..............................................................................................................................76 Fig 2.4.2 Two-Rate Schedule ...........................................................................................................................................77 Fig 2.4.3 Superposition of Pressure Changes ...............................................................................................................78 Fig 2.5.1 Schematic Flow-Rate and Pressure Behaviour for an Ideal Buildup .........................................................78 Fig 2.5.2 Dimensionless Buildup Pressure Distributions............................................................................................80 Fig 2.5.3 Dimensionless Buildup Semilog Graph..........................................................................................................81 Fig 2.5.5 Effect of Wellbore Storage on a Buildup.......................................................................................................81 Fig 2.5.6 Log-Log Diagnostic Plot for a Buildup..........................................................................................................82 Fig 2.5.7 Procedure for Finding the MTR Pressure at 1 Hour ....................................................................................84 Fig 2.5.8 Test Precautions................................................................................................................................................84 Fig 2.5.9 Scribed Tin Chart from an Amerada Gauge...................................................................................................85 Fig 2.5.10 Horner Plot from an Early Piper Well............................................................................................................85 Fig 2.5.11 Peaceman Probe Radius Concept .................................................................................................................86 Fig 2.5.12 Transient Productivity Index, J t .....................................................................................................................87 EPS Training 90 Fig 2.1.1...ptaintf1...............................................................................................................................................................43 Fig 2.1.2...ptaintf2...............................................................................................................................................................45 Fig 2.1.3...ptaintf3...............................................................................................................................................................46 Fig 2.1.4...ptaintf4...............................................................................................................................................................46 Fig 2.1.5...ptaintf5...............................................................................................................................................................47 Fig 2.1.6...ptaintf6...............................................................................................................................................................47 Fig 2.1.7...ptaintf7...............................................................................................................................................................48 Fig 2.1.8...ptaintf8...............................................................................................................................................................48 Fig 2.1.9...ptaintf9...............................................................................................................................................................49 Fig 2.2.1...ptabfi1................................................................................................................................................................53 Fig 2.2.2...ptabfi2................................................................................................................................................................56 Fig 2.2.3...ptabfi3................................................................................................................................................................59 Fig 2.2.4...ptabfi7................................................................................................................................................................60 Fig 2.2.5...ptabfi8................................................................................................................................................................61 Fig 2.2.6...ptabfi4................................................................................................................................................................66 Fig 2.2.7...ptabfi9................................................................................................................................................................67 Fig 2.2.8...ptabfi5................................................................................................................................................................68 Fig 2.2.9...ptabfi10..............................................................................................................................................................69 Fig 2.2.10...ptabfi11............................................................................................................................................................69 Fig 2.2.11...ptabfi6..............................................................................................................................................................71 Fig 2.2.12...ptabfi12............................................................................................................................................................71 Fig 2.2.13...ptabfi13............................................................................................................................................................72 Fig 2.3.1...ptapdtf1..............................................................................................................................................................72 Fig 2.3.2...ptapdtf2..............................................................................................................................................................73 Fig 2.4.1...ptaposf1.............................................................................................................................................................76 Fig 2.4.2...ptaposf2.............................................................................................................................................................77 Fig 2.4.3...ptaposf3.............................................................................................................................................................78 Fig 2.5.1...ptapbtf1..............................................................................................................................................................78 Fig 2.5.2...ptapbtf2..............................................................................................................................................................80 Fig 2.5.3...ptapbtf3..............................................................................................................................................................81 Fig 2.5.5...ptapbtf5..............................................................................................................................................................81 Fig 2.5.6...ptapbtf6..............................................................................................................................................................82 Fig 2.5.7...ptabfi14..............................................................................................................................................................84 Fig 2.5.8...ptabfi15..............................................................................................................................................................84 Fig 2.5.9...ptabfi16..............................................................................................................................................................85 Fig 2.5.10...ptabfi17............................................................................................................................................................85 Fig 2.5.11…ptapbtf8...........................................................................................................................................................86 Fig 2.5.12…ptapbtf9...........................................................................................................................................................87