Transcription of 1 Well Testing Analysis - Elsevier.com
1 1 Well y Reser voir Characteristics1 Flow Equations1 Well Testing1 Cur ves1 Derivative Method1 and Pulse Tests1 Well Testing1/1331/2 WELL Testing Primary Reservoir CharacteristicsFlow in porous media is a ver y complex phenomenon andcannot be described as explicitly as flow through pipes orconduits. It is rather easy to measure the length and diam-eter of a pipe and compute its flow capacity as a function ofpressure; however, in porous media flow is different in thatthere are no clear-cut flow paths which lend themselves Analysis of fluid flow in porous media has evolvedthroughout the years along two fronts: the experimental andthe analytical.
2 Physicists, engineers, hydrologists, and thelike have examined experimentally the behavior of variousfluids as they flow through porous media ranging from sandpacks to fused Pyrex glass. On the basis of their analyses,they have attempted to formulate laws and correlations thatcan then be utilized to make analytical predictions for main objective of this chapter is to present the math-ematical relationships that are designed to describe the flowbehavior of the reser voir fluids. The mathematical forms ofthese relationships will var y depending upon the characteris-tics of the reser voir. These primar y reser voir characteristicsthat must be considered include: types of fluids in the reser voir; flow regimes; reser voir geometr y; number of flowing fluids in the reser Types of fluidsThe isothermal compressibility coefficient is essentially thecontrolling factor in identifying the type of the reser voir general, reser voir fluids are classified into three groups:(1) incompressible fluids;(2) slightly compressible fluids.
3 (3) compressible isothermal compressibility coefficientcis describedmathematically by the following two equivalent expressions:In terms of fluid volume:c= 1V V p[ ]In terms of fluid density:c=1 p[ ]whereV=fluid volume =fluid densityp=pressure, psi 1c=isothermal compressibility coefficient, 1 Incompressible fluidsAn incompressible fluid is defined as the fluid whose volumeor density does not change with pressure. That is V p=0and p=0 Incompressible fluids do not exist; however, this behaviormay be assumed in some cases to simplify the derivationand the final form of many flow compressible fluidsThese slightly compressible fluids exhibit small changesin volume, or density, with changes in pressure.
4 Knowing thevolumeVrefof a slightly compressible liquid at a reference(initial) pressurepref, the changes in the volumetric behaviorof this fluid as a function of pressurepcan be mathematicallydescribed by integrating Equation , to give: c pprefdp= VVrefdVVexp[c(pref p)]=VVrefV=Vrefexp[c(pref p)][ ]where:p=pressure, psiaV=volume at pressurep,ft3pref=initial (reference) pressure, psiaVref=fluid volume at initial (reference) pressure, psiaThe exponential exmay be represented by a series expan-sion as:ex=1+x+x22!+x23!+ +xnn![ ]Because the exponentx(which represents the termc(pref p)) is ver y small, the exterm can be approximatedby truncating Equation to:ex=1+x[ ]Combining Equation with gives:V=Vref[1+c(pref p)][ ]A similar derivation is applied to Equation , to give: = ref[1 c(pref p)][ ]where.
5 V=volume at pressurep =density at pressurepVref=volume at initial (reference) pressurepref ref=density at initial (reference) pressureprefIt should be pointed out that crude oil and water systems fitinto this categor fluidsThese are fluids that experience large changes in volume as afunction of pressure. All gases are considered compressiblefluids. The truncation of the series expansion as given byEquation is not valid in this categor y and the completeexpansion as given by Equation is isothermal compressibility of any compressible fluidis described by the following expression :cg=1p 1Z Z p T[ ]Figures and show schematic illustrations of the vol-ume and density changes as a function of pressure for thethree types of Flow regimesThere are basically three types of flow regimes that must berecognized in order to describe the fluid flow behavior andreser voir pressure distribution as a function of time.
6 Thesethree flow regimes are:(1) steady-state flow;(2) unsteady-state flow;(3) pseudosteady-state Testing ANALYSIS1/3 PressureCompressibleSlightly CompressibleIncompressibleVolumeFigure volume CompressibleCompressibleFluid Density0 Figure density versus pressure for different fluid flowThe flow regime is identified as a steady-state flow if the pres-sure at ever y location in the reser voir remains constant, ,does not change with time. Mathematically, this condition isexpressed as: p t i=0[ ]This equation states that the rate of change of pressurepwithrespect to timetat any locationiis zero.
7 In reser voirs, thesteady-state flow condition can only occur when the reser voiris completely recharged and supported by strong aquifer orpressure maintenance flowUnsteady-state flow (frequently called transient flow) isdefined as the fluid flowing condition at which the rate ofchange of pressure with respect to time at any position inthe reser voir is not zero or constant. This definition suggeststhat the pressure derivative with respect to time is essentiallya function of both positioniand timet, thus: p t =f i,t [ ]Pseudosteady-state flowWhen the pressure at different locations in the reser voiris declining linearly as a function of time, , at a con-stant declining rate, the flowing condition is characterizedas pseudosteady-state flow.
8 Mathematically, this definitionstates that the rate of change of pressure with respect totime at ever y position is constant, or: p t i=constant[ ]It should be pointed out that pseudosteady-state flow is com-monly referred to as semisteady-state flow and quasisteady-state shows a schematic comparison of the pressuredeclines as a function of time of the three flow Testing ANALYSISTimeUnsteady-State FlowLocation iSemisteady-State FlowSteady-State FlowPressureFigure ViewPlan ViewFlow LinesFigure radial flow into a Reservoir geometryThe shape of a reser voir has a significant effect on its flowbehavior.
9 Most reser voirs have irregular boundaries anda rigorous mathematical description of their geometr y isoften possible only with the use of numerical , for many engineering purposes, the actual flowgeometr y may be represented by one of the following flowgeometries: radial flow; linear flow; spherical and hemispherical flowIn the absence of severe reser voir heterogeneities, flow intoor away from a wellbore will follow radial flow lines a substan-tial distance from the wellbore. Because fluids move towardthe well from all directions and coverage at the wellbore,the term radial flow is used to characterize the flow of fluidinto the wellbore.
10 Figure shows idealized flow lines andisopotential lines for a radial flow flowLinear flow occurs when flow paths are parallel and the fluidflows in a single direction. In addition, the cross-sectionalWELL Testing ANALYSIS1/5p1p2 AFigure ViewWellPlan ViewFractureFigure linear flow into vertical ViewFlow LinesFigure flow due to limited ViewFlow LinesWellboreFigure flow in a partially of FlowDistancePressurexp1p2 Figure versus distance in a linear to flow must be constant. Figure shows an ideal-ized linear flow system. A common application of linear flowequations is the fluid flow into vertical hydraulic fractures asillustrated in Figure and hemispherical flowDepending upon the type of wellbore completion config-uration, it is possible to have spherical or hemisphericalflow near the wellbore.
