PRODUCTION ENGINEERING FUNDAMENTALS I 212.3 Inflow Performance of Gil Wells2.3.1/ntroductionThe proper design of any artificiallift system requires an accurate knowledge of the fluid rates that can be producedfrom the reservoir through the given well. Present and also future production rates are needed to accomplish thefollowing basic tasks of production engineering:• selection of the right type of lift• detailed design of production equipment• estimation of future well performanceThe production engineer, therefore, must have a clear understanding of the effects governing fluid inflow into awell. Lack of information may lead to over-design of production equipment or, in contrast, equipment limitations mayrestrict attainable liquid rates. Both of these conditions have an undesirable impact on the economy of artificialliftingand can cause improper decisions as well.A well and a productive formation are interconnected at the sandface, the cylindrical surface where the reservoiris opened. As long as the well is kept shut in, sandface pressure equals reservo ir pressure and thus no inflow occurs tome well. It is easy to see that, in analogy to flow in surface pipes, fluids in the reservoir flow only between points havingdifferent pressures. Thus, a well starts to produce when the pressure at its sandface is decreased below reservoir pressure.Auid particles in the vicinity of the well then move in the direction of pressure decrease and, after an initial period, aStabilized rate develops. This rate is controlled mainly by the pressure prevailing at the sandface, but is also affected byiI. multitude of parameters such as reservoir properties (rock permeability, pay thickness, etc.), fIuidproperties (viscosity, density, etc.) and well completion effects (perforations, well damage). These latter parametersbeing constant for a given well, at least for a considerable length of time, the only means of controlling production ratesisthe control of bottomhole pressures. The proper description of well behavior, therefore, requires that the relationshipbetween bottomhole pressures and the corresponding production rates be established. The resulting function is calledthe wells inflow performance relationship (IPR) and is usually obtained by running well tests.This section discusses the various procedures available for the description of well inflow performance. Firstly, themost basic terms with relevant descriptions are given.23.2 Basic conceptsDarcys LawThe equation describing filtration in porous media was originally proposed by Darcy and can be written in anyrnnsistent set of units as:2.34This formula states that the rate of liquid flow,q, per cross-sectional area, A, of a given permeable media is directlypuportional to permeability, k, the pressure gradient, dp/dl, and is inversely proportional to liquid viscosity. Themegative sign is included because flow takes place in the direction of decreasing pressure gradients. Darcys equationiiiII55UITlesa steady state, linear flow of a single-phase fluid in a homogeneous porous media saturated with the same fluidoAlmough these conditions are seIdom met, all practical methods are based on Darcys work.
22 I GAS LIFr MANUALDrainage Radius, reConsider a welI producing a stable fluid rate from a homogeneous formation. Fluid particles from alI directionsaround the welI flow toward the sandface. ln idealized conditions, the drainage area, i.e. the area where fluid is movingto the welI, can be considered a circle. At the outer boundary of this circle, no flow occurs and undisturbed reservoirconditions prevail. Drainage radius, fe, is the radius of this circle and represents the greatest distance of the given welIsinfluence on the reservoir under steady-state conditions.Average Reservoir Pressure, PRThe formation pressure outside the drainage area of a welI equals the undisturbed reservoir pressure, PR, which canusualIy be considered a steady value over longer periods of time. This is the same pressure as the bottomhole pressuremeasured in a shut-in welI, as seen in Figure 2-3.PressureSondfoceWell shut -inOrawdown == SBHP -FBHPFBHPDistonceFormationFig. 2-3 Pressure distribution around a well in the formation.Flowing Bottomhole Pressure (FBHP),PwfFigure 2-3 shows the pressure distribution in the reservoir around a producing welI. In shut-in conditions, theaverage reservoir pressure, PR prevails around the welIbore and its value can be measured in the welI as SBHP.After flowhasstarted, bottomhole pressure is decreased and pressure distribution at intermediate times is represented by thedashed !ines. At steady-state conditions, the welI produces a stabilized liquid rate and its bottomhole pressure attains astable value, PwI- The solid line on Figure 2-3 shows the pressure distribution under these conditions.Pressure DrawdownThe difference between static and FBHP is calIed pressure drawdown. This drawdown causes the flow of formationfluids into the welI and has the greatest impact on the production rate of a given welI.
PRODUCTION ENGINEERING FUNDAMENTALS I 232.3.3 The productivity index conceptThe simplest approach to describe the infIow performance of oil weIls is the use of the productivity index (PI)concept. It was developed using the foIlowing simplifying assumptions:• fIow is radial around the weIl,• a single-phase liquid is fIowing,• permeability distribution in the formation is homogeneous, and• the formation is fuIly saturated with the given liquidoFor the previous conditions, Oarcys equation (Equation 2.34) can be solved for the production rate:O.00708kh (PR - Pwf)q = r ),uB ln (r:2.35where: q = liquid rate, STB/dk = effective permeability, mOh = pay thickness, ft/l = liquid viscosity, cPB = liquid volume factor, bbI/STBre = drainage radius of weIl, ftrw = radius of weIlbore, ftMost parameters on the right-hand side are constant, which permits coIlecting them into a single coefficient caIled PI:2.36This equation states that liquid infIow into a weIl is directly proportional to pressure drawdown. lt plots as astraight line on apressure vs. rate diagram, as shown in Figure 2-4. The endpoints of the PI line are the average reservoirpressure, PR, at a fIow rate of zero and the maximum potential rate at a bottomhole fIowing pressure of zero. Thismaximum rate is the weIls absolute open fIow potential (AOFP) and represents the fIow rate that would occur if fIowingbottomhole pressure could be reduced to zero. In practice, it is not possible to achieve this rate, and it is only used tocompare the deliverability of different weIls.The use of the PI concept is quite straightforward. If the averagereservoir pressure and the PI are known, use of Equation 2.36 gives the fIowIate for any FBHP. The weIls PI can either be caIculated from reservoirparameters, or measured by taking fIow rates at various FBHPs.SBHPa..Ia:lLLLíquid RateAOFPFig. 2~ Well performance with the PI concept.
24 I GAS Lwr MANUALExample 2-9. A well was tested at Pwf = 1,400 psi (9.7 MPa) pressure and produced q = 100 bpd(15.9 m3/d) of oil. Shut-in bottom pressure was Pws = 2,000 psi (13.8 MPa). What is the wells PI and what isthe oil production rate at Pwf = 600 psi (4.14 MPa).SolutionSolving Equation 2.36 for PI and substituting the test data:PI = q / ( PR - Pwf) = tOO / (2,000 - 1,400) = 0.17 bopd/psi (0.0039 m3lkPald)The rate at 600 psi (4.14 MPa) is found from Equation 2.36:q = PI ( Pws - Pwf ) = 0.17 ( 2,000 - 600 ) = 238 bopd (37.8 m3/d).2.3.4/nflow performance relationships2.304.1 Introduction. In most wells on artificial lift, bottomhole pressures below bubblepoint pressure areexperienced. Thus, there is a gas phase present in the reservo ir near the wellbore, and the assumptions that were used todevelop the PI equation are no longer valido This effect was observed by noting that the PI was not a constant as suggestedby Equation 2.36. Test data from such wells indicate a downward curving line, instead of the straight line shown in Figure2-4.The main cause of a curved shape of inflow performance is the liberation of solution gas due to the decreasedpressure in the vicinity of the wellbore. This effect creates an increasing gas saturation profile toward the well andsimultaneously decreases the effective permeability to liquido Liquid rate is accordingly decreased in comparison tosingle-phase conditions and the well produces less liquid than indicated by a straight-line PI. Therefore, the constantPI concept cannot be used for wells producing below the bubblepoint pressure. Such wells are characterized by theirIPR curves, to be discussed in the following section.2.3 04.2 Vogels IPR correlation. Vogel used a numerical reservoir simulator to study the inflow performanceof welIs depleting solution gas drive reservoirs. He considered cases below bubblepoint pressure and varied pressuredrawdowns, fluid, and rock properties. After running several combinations on the computer, Vogel found that all thecalculated IPR curves exhibited the same general shape. This shape is best approximated by a dimensionless equation:qqmax = 1-0.2 ;wf _ 0.8( Pwf )2R PR2.37where: q = production rate at bottomhole pressure Pwf, STB/dqmax = maximum production rate, STB/dPR = average reservoir pressure, psiEquation 2.37 is graphically depicted in Figure 2-5.Although Vogels method was originally developed for solution gas drive reservoirs, the use of his equation isgenerally accepted for other drive mechanisms as welI . It was found to give reliable resuIts for almost any well witha bottomhole pressure below the bubblepoint of the crude.In order to use Vogels method, reservoir pressure needs to be known along with a single stabilized rate and thecorresponding FBHP. With these data, it is possible to construct the wells IPR curve by the procedure discussed in thefollowing example problem.
PRODUCTION ENGINEERING FUNDAMENTALS I 25FBHP, Fraction of Reservior Pressure••••••••••••••••••••••••......••••••••.....•.•.••........."""......••"~"-"-"-I...~."",,0.90.80.70.188.8.131.52.20.1oO 0.2 0.4 0.6 0.8Production Rate, Fraction of MaximumFig. 2-5 Vogels dimensionless inflow performance curve.Example 2-10. Using data of the previous example nnd the wells AOFP and construct its IPR curve, byassuming multiphase flow in the reservoir.SoIutionSubstituting the test data into Equation 2.37:1001 cfrnax = 1 - 0.2 (1,400/2,000) - 0.8 (1,400/2,000)2 = 0.468.From the previous equation the AOFP of the well:cfrnax= 10010.468 = 213.7bopd (34 m3/d)Now nnd one point on the IPR curve, where Pwf = 1,800 psi (12.42 MPa) using Figure 2-5.Pwjl PR = 1,800/2,000 = 0.9From Figure 2-5:cf 1 cfrnax = 0.17, and q = 213.70.17 = 36.3 bopd (5.8 m3/d).The remaining points of the IPR curve are evaluated the same way.
26 I GAS LIFT MANUALFigure 2-6 shows the calculated IPR curve along with a straight line valid for PI = 0.17 (0.0039 m3/kPald),as found in Example 2-9. Calculated parameters for the two cases are listed as folIows:VogelConstant PIMax. Rate213.7340Rate at 600 psi185.5238.~ 2000cD 1800•....:J 1600(/) (/)Q)1400•.... c... 1200Q)"61000..c E 800o - 600ÕCO400Ol c 200.~ oOLi: O50100150200250300350Fig.2-6 Comparison of IPR curves for Examples 2-7 and 2-8.Comparison of the preceding results indicates that considerable errors can occur if the constant PI methodis used for conditions below the bubblepoint pressure.184.108.40.206 Fetkovichs method. Fetkovich demonstrated that oil welIs producing below the bubblepoint pressureand gas welIs exhibit similar inflow performance curves . The general gas welI deliverability equation can thus alsobe applied to oil wells:q = C(p~- p~l2.38Coefficients C and n in this formula are usualIy found by curve-fitting of multipoint welI test data. Evaluation ofwelI tests and especialIy isochronal tests is the main application for Fetkovichs method.2.4 Single-phase Flow2.4.1/ntroductionIn all phases of oil production, several different kinds of fluid flow problems are encountered. These involve verticalor inclined flow of a single-phase fluid or of a multiphase mixture in welI tubing, as welI as horizontal or inclined flowin flowlines. A special hydraulic problem is the calculation of the pressure exerted by the static gas column present ina welIs annulus. AlI the problems mentioned require that the engineer be able to calculate the main parameters of theparticular flow, especialIy the pressure drop involved.In this section, basic theories and practical methods for solving single-phase pipe flow problems are covered thatrelate to the design and analysis of gas lifted welIs. As alI topics discussed have a common background in hydraulictheory, a general treatment of the basic hydraulic concepts is given first. This includes detailed definitions of andrelevant equations for commonly used parameters of pipe flow.