bwtasec1

Well Test Analysis

Halliburton 1 - 1 Section 1 2000, Halliburton

Section 1Table of ContentsTable of Contents ..........................................................................................................................................1Introduction ...................................................................................................................................................3Purpose of this Chapter .................................................................................................................................4Transient Pressure Response.........................................................................................................................4Mathematical Basis for Pressure Analysis Methods .....................................................................................5Flow of Oil (Constant Compressibility Liquid) ............................................................................................6Basic Concepts and Terms ............................................................................................................................7

Flow of Gas ...............................................................................................................................................7Total Compressibility ................................................................................................................................8Types of Flow Regimes .............................................................................................................................8Skin Effect ...............................................................................................................................................11Flow Efficiency .......................................................................................................................................12Wellbore Storage .....................................................................................................................................12Principle of Superposition .......................................................................................................................14Effect of Boundaries ................................................................................................................................15Radius of Investigation............................................................................................................................16Phase Redistribution ................................................................................................................................18

Production Well Testing: Types of Tests and Techniques of Analysis.......................................................19Introduction .............................................................................................................................................19Test Types................................................................................................................................................19

Pressure Drawdown Test .....................................................................................................................19Pressure Buildup Test ..........................................................................................................................19Drill Stem Testing (DST).....................................................................................................................20Multi-rate Testing ................................................................................................................................20Multi-well Testing................................................................................................................................20

Planning the Test .....................................................................................................................................20Test Objectives ........................................................................................................................................20Single Well Test Benefits ........................................................................................................................21

Leaks Near or in the Wellbore or Reservoir ........................................................................................21Stimulation Treatments ........................................................................................................................21Step-out Locations ...............................................................................................................................21Time Decay of Performance ................................................................................................................21Critical Flow Rates ..............................................................................................................................21Detecting Impediments ........................................................................................................................21

Multi-well Test Benefits ..........................................................................................................................22Communication....................................................................................................................................22Competitive Production .......................................................................................................................22Detecting Undrilled Reserves ..............................................................................................................22Infill Drilling........................................................................................................................................22Reserves in a Naturally Fractured Reservoir .......................................................................................22

Summary of Well Testing Benefits .........................................................................................................22Establishing Test Procedures...................................................................................................................22Reservoir Limits Tests.............................................................................................................................22Quantitative Analysis Methods of Pressure Transient Tests ...................................................................23

Type Curves .........................................................................................................................................23How to Use Type Curve Matching ......................................................................................................24

Well Test Analysis


Characteristics of Type Curves ............................................................................................................251. Interference Type Curve (line source solutionFig. 1.19) .................................................................... 252. Pressure Drawdown Type Curve (unfractured reservoir)........................................................................ 253. Fractured Reservoir Type Curves............................................................................................................ 26

Characteristics of the fractured well type curves .................................................................................27Semi-log Methods................................................................................................................................30

The Mathematical Basis of the Semi-log Methods ......................................................................................... 30Applying Semi-log Method to Drawdown/Fall-off Tests .............................................................................. 34Applying Semi-log Method to Pressure Buildup Tests................................................................................... 34MDH Plot (Miller-Dyes-Hutchinson plot) ...................................................................................................... 36Type Curve Matching and Semi-log Method for Gas Reservoirs ................................................................... 37Reservoir Limit Test ....................................................................................................................................... 39Derivative Approach ....................................................................................................................................... 39Dual Porosity Reservoirs................................................................................................................................. 39Warren and Root Model.................................................................................................................................. 41Kazemi Model................................................................................................................................................. 41Definitions....................................................................................................................................................... 41

Derivative Plots for a Heterogeneous System .........................................................................................43Gas Well Deliverability Testing ..............................................................................................................43Backpressure Tests ..................................................................................................................................45

Rate-After-Rate Tests...................................................................................................................................... 45Isochronal Tests .............................................................................................................................................. 45Modified Isochronal Tests............................................................................................................................... 45

Pressure Transients and Backpressure Tests ...........................................................................................47Backpressure Plots ...............................................................................................................................48Effect of permeability and perforated interval .....................................................................................49

Nomenclature ..............................................................................................................................................51Greek & Other Symbols ..........................................................................................................................51

References ...................................................................................................................................................52General Bibliography ..................................................................................................................................54

DST Chart Interpretation and Analysis ...................................................................................................54Production Well Testing ..........................................................................................................................54

Well Test Analysis


IntroductionThe original idea of analyzing pressure versustime data from a producing or shut-in well toobtain information on the producing stratum firstappeared in hydrology. Hydrologists were mainlyinterested in the behavior of underground waterflow in large aquifers. Shortly after, Theis1published his pioneering work on fluid flow inporous media, which included his point-sourcesolution, Muscat2 studied a problem more suitedto oil reservoirsthe eventual static pressurebehavior of a shut-in well in a closed oilreservoir. When compared to the initial reservoirpressure, this new static pressure can be usedto measure the quantity of hydrocarbonsproduced up to the time of the test.The need for a reliable quantitative analysis wasdriven by economics. Accurate formationpermeability determination is needed to betterforecast reservoir production and, in turn, betterforecast cash flow. Determination of initialreservoir pressure and drainage boundariesgives us the information needed to determinethe amount of hydrocarbon in place.Determination of well condition (skin factor)enables us to predict how much expenditure isneeded to improve productivity and the expectedreturn of this workover.Since Muscat, literally thousand of papers havebeen published on the analysis of pressuretests. Several new pressure tests were devisedto determine specific reservoir parameter. Thisexplosion in literature was basically due to theease in which pressure behavior could bemeasured and the enormous value of theparameters calculated from these tests. Amongother useful information, pressure tests can beused to estimate how efficiently a well is completed the need for and the success of a

stimulation treatment the general type of well treatment desired the degree of connectivity of one well to

anotherModern well test analysis started when Horner3and Miller et al.4 presented their famous paperswhere semi-log straight lines were introduced asthe first technique to analyze pressure tests.Within a few years, other fundamental

developments were introduced when VanEverdingen and Hurst5 and Moore et al.6introduced the concept of wellbore storage.Shortly after, Hurst7 and Van Everdingen8introduced the concept of wellbore skin factor.Finally, Matthews et al.9 introduced an analyticalapproach to calculate average reservoirpressure from extrapolated pressure. Thesolution was a function of reservoir area, shape,and location of the well in the reservoir.The period described above may be consideredthe first phase of modern well test analysis. Thesecond phase of modern well test analysis maybe called type curve analysis," as compared tothe "semi-log straight line" phase describedabove. This phase was pioneered by Ramey.Numerous publications by Ramey10-13 and hisstudents, initially at Texas A&M and then atStanford University, made this technologyavailable and popular.The third phase of well test analysis is the"derivative." This phase was initiated by Tiaband Kumar.14,15 However, the technology wasmade popular through a series of papers byBourdet.16-18 The derivative technology requireshigher precision pressure measurements thatwere not available earlier. The derivativetechnology has significantly improved modelrecognition, which has led to a surge in modeldevelopment. For the first time it was possible todetermine various models with some degree ofcertainty.The fourth phase of well test analysis was thedevelopment of "computer-aided analysis,"which is essentially a nonlinear optimizationtechnique to match the observed data againstexisting models.19-21 This development, althoughit made analysis significantly easier and allowedengineers to consider excessively complicatedsituations, could be considered a double-edgedsword. Many times, because of the complexity ofthe models considered, an analysis may not beunique. The best way to overcome thisuniqueness problem is to consider informationfrom other sources, such as seismic, logging,etc.

The last phase is use of "knowledge-basedsystem/Neural network"22 to determine thepossible models that match the data. Whichmodel is the most realistic should probably bedetermined from other information as previouslydiscussed.

Well Test Analysis


Original models for well testing were basicallyconcerned with homogenous, isotropic systemsproducing under radial flow conditions. Now,well testing has been expanded tremendously toconsider a variety of complex models thatenhanced the applicability and usefulness of thetechnology. These models include but are notrestricted to Dual porosity models including the sugar

cube or layered models representingnaturally fractured systems

Uniform flux model that approximates the

behavior of some the naturally fracturedreservoirs

Hydraulically fractured reservoir with infinite

or finite conductivity Radially composite reservoirs Hydraulically fractured radially composite

reservoir Layered reservoir Horizontal well model Fractured horizontal well model, both with

transverse or longitudinal fractures

Purpose of this ChapterThis chapter will show you how to use welltesting results to learn more about the reservoir.By learning how to interpret test results, you canmake wise decisions about the future of thewell-reservoir system. You will also learn how toplan and conduct a test to meet your specificneeds.Production well tests are usually conducted aftercompletion of the well. Usually, before aproduction well test, you already have somequalitative information about reservoirparameters from core analysis, well logging, anddrillstem testing (DST). This means that theextent of sampling in the reservoir has beenrelatively small. In production testing, yousample a much greater portion of the reservoirthan in the other tests mentioned (Fig. 1.1).DSTs and production well tests determineseveral reservoir parameters under dynamic(flowing) conditions; as a result, the larger theextent of the sampling, the more properly

weighted the parameters will be and the closerthey should be to the parameters that will applyafter the well is on production. This will allow youto make better completion decisions andeconomical projections for the well the wellunder considerations.

Transient PressureResponseTransient pressure response refers to thepressure response as a function of time thatresults from a change in a wells production rate.This change could be as an increase or

decrease in rate. It could take the form ofshutting in a producing well (buildup test). Itcould be also a result of putting a previouslyclosed-in well on production (drawdown test).More elaborate tests can be designed involvingone or more wells and one or more ratechanges. A limited number of these tests will becovered in this document: specifically,drawdown, buildup, and interference tests.

ProductionT esting DSTLog

Core

Fig. 1.1Relative extent of the reservoirsampled by cores, logs, drillstem tests, andproduction well tests under normal conditions

Well Test Analysis


Operations, advantages, and limitations of thesethree tests will be discussed in a later section.A well test analysis involves qualitatively and/orquantitatively determining the system propertiesfrom the measured response. This problem isusually referred to as inverse problem, in directcontrast to the direct problem. In direct problem,the system is well defined and the systemresponse to an imposed change may becalculated. Design of a well test is an example ofthis direct problem.The reverse problem, analysis of a well test, issignificantly more complicated. Except for thesimplest system, the response may not beuniquely defined by a model. An analyst willhave to determine the most likely system thatmatches the observed behavior. This shouldnot be done based on pressure-time behavioralone. Other sources such as seismic, drillinglog, and well logging are extremely important inproducing a realistic picture of the reservoir. Thisis especially true when analyzing some of thecomplicated cases involving a combination ofthe models listed above.

In summary, pressure tests may be used for anyof the following reasons: Determine formation permeability Determine condition of the well by

determining skin factor Determine average formation pressure,

which may be used in material balancecalculations to determine recoverablereserves

Detection of a permeability barrier in the

vicinity of wellbore Determine the size of a reservoir Porosity of the reservoir Help in drilling and completion decision

makingIn addition, pressure tests may be used beforeor after a stimulation treatment to Select stimulation candidates Determine degree of success or failure of a

stimulation treatment Properly design a stimulation treatment

Mathematical Basis forPressure AnalysisMethodsExcept for flow near a high gas flow rate well,fluid flow in a reservoir is usually laminar.Extensive analysis of this type of flow in porousmedia has been made. Coupled with knowledgeof physical properties of fluids and rock, fluidflow theory forms the principal basis of for thestudy of fluid flow in porous media. In thissection, the mathematical principles of thistheory are briefly discussed and presented. Formore details, please refer to SPE Monograph #1by Matthews and Russel23 or SPE Monograph#5 by Earlougher.24

We will begin the mathematical description ofpressure analysis by showing a material balanceover a differential element of reservoir volume.We will end up with the governing partialdifferential equation describing how fluid flows inthe system. Because of the very low fluid flowvelocity, conservation of energy may be ignored.

In other words, the inertia effect is not a factor influid flow in porous media. Except in a fewcases, this assumption is usually valid. Onlyconservation of mass is used to develop thegoverning differential equations:[ ] [ ] [ ]ratestorageMassoutrateMassinrateMass =

1.1) ........(............................................................For development of the governing differentialequation, the above equation is applied to aninfinitesimally small element.Neglecting gravitational forces, Darcys equationfor laminar flow in porous media is

)1.2......(..............................dsdpk

=

Darcys equation may be used to express themass rate terms in Eq. 1.3. For a radial flowsystem of constant thickness, taking the limits ofEq. 1.3 as the element under considerationapproaches zero volume, Eq. 1.3 becomes

( ) )3.1(....................1

trprk

rr

=

Well Test Analysis


Equation 1.3 is the continuity equation underradial flow conditions. The general threedimensional form for fluid flow in a reservoir isgiven in Eq. 1.4

( ) )4.1..(.................... t

pkru

=

!where:

zk

yj

xi

+

+

=

Both Eqs. 1.3 and 1.4 assume single phase flowin a porous media. Equations describingmultiphase flow have been developed in thefashion given above. However, that

development will result in a separate differentialequation for each phase. In addition, equationsdescribing the interaction between the variousphases will also have to be developed. Theinteraction equations are the mass conservationof each phase, evolution of one phase intoanother or solution of one phase into another. Inaddition, the pressure of each phase is relatedto other phases through the capillary pressurerelationship. Clearly, the multi-phase system isconsiderably more complex than a single-phasesystem. For simplification, the rest of thischapter will be devoted to single-phase flowonly.

Flow of Oil (ConstantCompressibility Liquid)Because the overwhelming majority of liquidreservoirs are isothermal (fluid flow at a constanttemperature), density will be treated as afunction of pressure only. The isothermalcompressibility (hereafter calledcompressibility) is defined in the followingequation of state:

)5.1....(..........11TT pp

C

=

=

If compressibility (C) is constant, Eq. 1.5 can besolved using separation of variables, yieldingthe following equation:

)6.1..(..............................)( scppcsce = This is the equation of state for a constantcompressibility liquid. It applies reasonably wellfor many liquids such as oil and water.Equations 1.3 (radial flow) and 1.6 can besimultaneously solved either analytically ornumerically to get the pressure distribution in aradial flow system when a constantcompressibility liquid is flowing through theporous medium. If the rock and fluid propertiesare assumed to be constant, then substitutingEq. 1.6 for density ( ) into Eq. 1.3 will yield thefollowing governing partial differential equation.

)7.1.(....................1tp

kc

r

pr

rr

=

Equation 1.7 describes how a specific fluidwould flow in porous media. To generalize Eq.1.7, the following dimensionless form forpressure and time and radius are used:

)8.1.(..............................2.141 qpkhpD =

)9.1........(....................0002637.0 2wt

Drc

ktt =

)10.1...(........................................w

Dr

rr =

Substituting Eqs. 1.8 and 1.9 into Eq. 1.7produces the following dimensionless flowequation:

)11.1.........(..........122

D

D

D

D

DD

D

tp

r

prr

p

=

+

Equation 1.11 is used to construct the generalsolution of fluid flow through porous media. Thissolution is in terms of the dimensionlessparameters pD, tD and rD. In other words thesolution would be independent of reservoir andfluid properties. Thus using this general solution,one could easily construct the solution for aspecific reservoir and fluid parameters. In thiscase the differential Eq. 1.11, may be solvedonly once and produced as a type curve. Thisillustrates the importance of the use ofdimensionless parameters.The assumptions implicitly used in constructingthe Eq. 1.11 are listed as follows.1. Homogenous and isotropic permeability2. Constant porosity

Well Test Analysis


3. Constant thickness4. Small and constant compressibility5. Constant temperature6. Constant viscosity

7. Radial flow in reservoir8. Laminar flow in reservoir9. Negligible gravity force10. Small pressure gradient

Basic Concepts andTermsIn this section, some of the basic conceptsencountered in welltest analysis are explained.

Flow of GasAlthough many excellent equations of state areavailable for gases, the one based upon the lawof corresponding states has achieved a wideacceptance in petroleum reservoir engineering.This is because it can be readily applied to multi-component gases.

)12.1..(........................................znRTpv =Using Eq. 1.12, the compressibility of real gas isexpressed as:

)13.1.....(..............................11dpdz

zpCg =

For an ideal gas, the compressibility factor, z, isunity and dz is 0, and the compressibility issimply the reciprocal of pressure.Substituting Eq. 1.13 into Eq. 1.14 results in thefollowing equation:

)14.1...(..........12

2

tp

kzcp

z

r

rr

=

Solution of Eqs. 1.14 would yield the pressuredistribution in a gas reservoir. Remember thatbecause both viscosity and compressibilityfactor are functions of pressure, Eq. 1.14 in itsexisting form is nonlinear, which makes itsignificantly more difficult to solve. In addition,as will be explained later, superposition may notbe applied to non-linear equations. Therefore,linearizing the gas flow equation is highlydesirable.There are two approaches to linearize Eq. 1.14.First, at pressures below 1,000 psi, the viscosity-compressibility product may be approximated asfollows:

zz =

where and z are evaluated at averagereservoir pressure ( )p .Based on this approximation, Eq. 1.14 would beidentical to Eq. 1.7, except that the solution is interms of p2 instead of p. This simply means thatthe techniques developed for liquid reservoirsmay be applied to gas reservoirs provided thatp2 is used in various plots instead of p. Of

300

200

100

0500 1,000 1,500 2,0000

Fig. 1.3: m(p)-p curve for the gas in Figure 1.2.

Gas Gravity = 0.66Reduced T emperature = 1.6

0.05

0.04

0.03

0.02

0.01

0

z

z= ConstantZ,

cp

2,000 4,000 6,000 8,000 10,0000

p, psiaFig. 1.2Variation of p and Z with pressure.

z, cp

Constant=z

Well Test Analysis


course, conversion factors and analysisequations would have to be modified tocorrespond to the new plotting parameter.

At pressures above 5,000 psi, the viscosity-compressibility product is a linear function ofpressure (Fig. 1.2). Using this approximation,the resulting governing differential equationwould be in terms of pressure.Thus, if pressure is low, p2 may be used, and ifpressure is high, use of p in equations would beacceptable. In the middle range neither one isapplicable.A general approach to linearize Eq. 1.14 is touse the transformation suggested by Al-Hussainy, et al.25

( ) )15.1.........(....................20=p

dpz

ppm

Thus, the governing partial differential equationof gas flow in a porous medium becomes:

( ) ( ) )16.1......(1tpm

kc

r

pmr

rr

=

The function m(p) is usually called real gaspotential or gas pseudo-pressure. Therelationship between the pseudo pressurefunction and pressure is given in Fig. 1.3.The similarity between Eqs. 1.7 (equation for oiland water flow) and 1.16 (equation for gas flow)indicates that the solution for Eq. 1.7 can beused as solution for the pseudo-pressuredistribution in a gas reservoir. Further, thisimplies that all techniques developed foranalyzing the transient pressure behavior of anoil well can be applied to a gas well if m(p) issubstituted for pressure.Equation 1.16 may be transformed to adimensionless form using dimensionlesspressure and time similar to the ones defined inliquid case. Because of dependency of gasproperties on pressure, the dimensionlesspressure and time are defined as follows:

)17.1.........(....................424.1

)(qTpmkhpD

=

)18.1.....(....................0002637.0 2wii

Drc

ktt =Substituting the pD and tD definitions into Eq.1.16 yields the dimensionless form of fluid flowequation given earlier in Eq. 1.10. Thus,

regardless of the type of fluid, the same generalequation may be used to produce the pressurebehavior with time. Only when we convert thesolution into a dimensional form, the effect of thetype of fluid would appear through the use of thedimensionless definition of pressure and time.

Total CompressibilityThe term ct

, total compressibility, is generallydefined as

)19.1........(..........fggwwoot cscscscc +++=The solution presented in the type curves lateron in this chapter assumes that a single-phasefluid flows and occupies the pore space.However, the above equation also allows fornon-flowing phases, such as connate water, tobe considered. Eqs. 1.19 can be simplified forboth oil and gas reservoirs. In oil reservoirs, ifthe formation pressure is higher than the bubblepressure for the crude oil, the free-gas phasewould not exist and Eq. 1.19 becomes

)20.1.........(....................fwwoot cscscc ++=On the other hand, in gas reservoirs the oilphase does not exist, simplifying Eq. 1.19 to

)21.1........(....................fwwggt cscscc ++=But, because Cf and Cw Sw are usually muchsmaller then CgSg, Eq. 1.21 may beapproximated by Eq. 1.22:

)22.1......(........................................ggt scc

Types of Flow RegimesAlthough we will concentrate on radial flowregime, there are other regimes that may existdepending on reservoir and well condition.Figure 1.4

illustrates some of these flowregimes. Other flow regimes exist underspecialized conditions, such as finiteconductivity fractures and horizontal wells.

Radial - This flow pattern occurs in wells

located in infinitely large reservoirs. It canalso occur in a finite reservoir, provided theeffects of the reservoir boundaries onpressure behavior have not been felt yet. Inradial flow, the stream lines are convergingtowards the wellbore. The density of streamlines per unit area increases as they getcloser to the wellbore. In other words the

Well Test Analysis


fluid velocity increases as it gets closer tothe wellbore, causing a higher pressure dropnear the wellbore. This will cause alogarithmic distribution of pressure versusdistance away from the wellbore. In extremecase, the convergence of the stream lineswould cause flow to become turbulent,causing an extra pressure drop that wouldappear in the analysis as an extra skinfactor. The long term approximation of thisflow regime is logarithmic is )log(tp .

Linear - This flow pattern occurs in

laboratory experiments when fluid is injectedat one end of a cylindrical core and it flowsin parallel stream lines through the uniformlypermeable sample. It also occurs at earlytime of production around infinitelyconductive fractures, when fluid flows inparallel stream lines into the fracture, thenflows in similar pattern in the fracture to thewellbore. The long-term approximation ofthis flow regime is tp .

Spherical - In this flow regime, the stream

lines are converging towards the center of asphere. The iso-potential lines are sphericalin shape.

Hemispherical -This flow regime may occur

if hydrocarbon is produced through a probe,like and RFT (Repeat Formation Tester) orSFT. This flow regime is similar to sphericalflow except that the sphere is cut in half.Equations controlling both flow regimes areessentially the same. The long termapproximation for both spherical and

hemispherical flow is t

p 1 .

Elliptical - This flow pattern occurs in a

fractured reservoir after the initial linear flowhas occurred in an infinitely conductivefracture. This looks fairly similar to the radialflow regime, except that the iso-potentiallines are elliptical shaped instead of circular.This flow regime is usually difficult toanalyze, however it may be noticed fromFig. 1.4 that as the outer ellipses becomeless elongated. In other words after areasonably long time the ellipses may beapproximated by circles, and the flow istermed pseudo-radial flow. At that point, theradial flow equations are applied to this flowregime with a negligible error. Thus the longterm approximation for this flow regime islogarithmic, )log(tp .

Well Test Analysis


FLOW LINESFLOW LINES

WELL

FLOW LINES

FLOW LINESFLOW LINES

WELL

WELL

WELL

LINEAR

RADIAL

ISOPOTENTIALLINES

ISOPOTENTIALLINES

FLOW LINES

FLOW LINESISOPOTENTIALLINES

ISOPOTENTIALLINES

WELL

WELL

WELL

FRACTURE

SPHERICAL

ELLIPTICAL

FLOW LINES

WELLE.

D.

C.

B.

A.

Top View Side View

Fig. 1.4Flow regimes.

Well Test Analysis


Skin EffectUsually, permeability of a formation is found tobe reduced near the wellbore as a result ofdrilling and completion practices. Drilling fluidinvasion of the formation, dispersion of clay,presence of a mud cake, and cement tend toreduce formation permeability around thewellbore. This same effect can be produced by adecrease in the area of flow near the wellbore.Thus, partial well penetration and limitedperforation or plugging of perforations wouldalso give the impression of a damagedformation. Conversely, an inclined well orformation increases the area of flow near thewellbore, giving the impression of a stimulatedwell (higher permeability around the wellbore).The zone of reduced (or higher) permeabilityhas been called a skin, and the resulting effecton permeability is called skin factor. Skin factorcan be used as a relative index to determine theefficiency of drilling and completion practices. Itis positive for a damaged well, negative for astimulated well, and zero for an unchanged well.(See Fig. 1.5). Acidized wells usually show anegative skin. Hydraulically fractured wells,though not affecting the formation permeability,often show values of s (skin factor) ranging to aslow as -7.Hawkins26 derived the following expressionrelating the skin factor to wellbore radius,damage radius and permeabilities of bothreservoir and damaged area.

).......(....................ln 2311

=

w

d

d r

r

kk

s

This expression indicates that if the area aroundthe wellbore has a lower permeability than theoriginal reservoir permeability, that is a damagedwell, skin factor would be greater than zero(positive value). If the permeability around thewellbore is higher than the reservoirpermeability, that is, stimulated well, skin factorwould be lower than zero, (negative value). Skinfactor of zero indicates no change inpermeability around the wellbore.If the permeability around the wellbore isinfinitely higher than the original reservoirpermeability (a larger well radius), Eq. 1.23 maybe written in one of the following forms:

)24.1....(................................ln'

=

w

w

r

rs

)25.1......(......................................' sww err ='

wr is usually referred to as effective wellboreradius. If s is a negative value, the effectivewellbore radius would be larger than r

w. If s is a

positive value, the effective wellbore radiuswould be smaller than r

w. Effective wellbore

radius is a term that was developed to describethe radius of an equivalent well with skin factorof zero. One can easily see from Eq. 1.25 that ifs is positive the effective wellbore radius, 'wr , issmaller than r

w. Thus the damaged well under

consideration is equivalent to a well with zeroskin but smaller radius r

w. Thus both real and

equivalent wells would have the sameproductivity under the same pressure drop.The concept of effective wellbore radius hasbeen used widely in solving the differentialequation for fluid flow under a negative skincondition. Without this equivalency the equationwould become highly unstable under stimulatedconditions. This is because the skin factor formsa pressure source/sink condition.An equation similar to Eq. 1.25 had beendeveloped for fractured wells. In this case anequivalent skin factor and wellbore radius maybe related to the length of a vertical hydraulicfracture with infinite conductivity through Eq.1.26.

)26.1...(.....................................2 swf erx =Thus, if r

w = 3 in. and s is -5, the half-length of

an infinitely conductive fracture, xf

, is 74.2 ft.Equation 1.27 relates the fracture half-length ofa natural unpropped fracture to an equivalentskin factor and wellbore radius.

)27.1(..........7183.2 swswf ereerx ==Where e is the natural logarithm base.

W ellbore

S tatic P ressu re

Skin or Zone ofDam age

F low ing P ressure

Pressure D ropAcross Skin

P ressure in Form ation

skin

=

Fig. 1.5Pressure distribution in a reservoir with a skin.

Well Test Analysis


Thus, the fracture half-length for the sameconditions as previously shown (r

w = 3 in. and

s = -5) is 100.86 ft.As mentioned earlier, the skin factor (s) acts asa pressure sink/source. This extra pressure dropdue to presence of skin may be calculated usingEq. 1.28.

)28.1.(..............................2

=

khq

spskin

In field units, Eq. 1.28 becomes

)29.1...(....................2.141

=

khq

spskin

If the skin is positive, the flowing pressure pwf ,

will actually be lower than that of an undamagedwell by the amount of pskin. The above equationmay be used to quickly yield the effect ofreducing skin factor on flowing pressure.

Flow EfficiencyAnother relative index for determining theefficiency of drilling and completion on a well isprovided by flow efficiency. This is defined asthe ratio of actual productivity index of a well toits productivity index had the skin factor beenreduced to zero.

)30.1......(..........ideal

actual

JJ

efficiency Flow =

Where:

)31.1....(..............................wf

actual ppqJ

=

)32.1....(..........)( skinwfideal pppqJ+

=

pressure reservoir averagep =Thus the final expression of flow efficiency isgiven below:

)33.1......(wf

skinwfpp

pppefficiency Flow

=

If the well is unchanged, the flow efficiency isunity. If the well is stimulated, the flow efficiencywill be higher than unity. Damaged wells have aflow efficiency less than unity. However, the flowefficiency of a damaged well can not be less

than zero. If the well does not flow (it is plugged)the flow efficiency will become zero.Sometimes the term damage ratio (DR) is usedto describe the well conditions. Damage ratio isthe reciprocal of flow efficiency.

Wellbore StorageWellbore storage, also called after-flow, after-production, after-injection, and wellboreunloading or loading, has long been recognizedas an important parameter affecting short-timetransient pressure behavior. Pressure transienttheory presented so far assumed that the shut-inof a well in a buildup test (or production rate in adrawdown test) occurs at the sand-face.However, in many tests, the well is shut in at thesurface causing the wellbore volume (wellborestorage) to affect the early time pressureresponse. When this wellbore storage issignificant, it must be considered duringtransient test data or meaningless data may beanalyzed.28

Wellbore storage causes the sand-face flow rateto change slower than the surface flow rate.Wellbore storage is the ability of the wellbore tostore fluid per unit of pressure change. Zerowellbore storage means that the flow condition isimposed at the sand-face. Figure 1.6schematically shows the ratio of sand-face (qsf)to surface rate (q) when a shut-in well starts toproduce at a surface rate q. For wellborestorage greater than zero, the majority of flowrate will come out of the wellbore storage. Theformation contribution to total flow rate willinitially be very small. However the ratio willincrease with time until it reaches 1, meaningthat all fluid produced at the surface is comingout of the formation. The larger the wellborestorage, the longer it would take for stabilizationto occur. On the other hand when the wellborestorage coefficient is negligible, qsf /q = 1 at alltimes.Depending on the system, one of two methodsis used to calculate the wellbore storagecoefficient. In a compressible system thefollowing equation could be readily derived todescribe storage:

(13.34)....vcpvC tcoefficien storagewellbore ==(13.35)

6155..............

.

hcAC tcoefficien storagewellbore wb=

Well Test Analysis


Thus, in this case, the storage ability of the welldepends on the wellbore volume.In case of declining or rising liquid level, theequation describing wellbore storage may beexpressed as follows:

)36.1(..........6155 g.

z144g C tcoefficien storagewellbore c

=

Wellbore storage obscures the effect offormation on pressure response. When wellborestorage is dominant, the pressure versus timeplot on a log-log scale is linear (unit slopestraight line) as shown in Fig. 1.7. Nomeaningful information can be obtained till thisperiod of wellbore storage dominance hasexpired. This period may be described using thefollowing equation:

)37.1.....(........................................0417.0C

tqp =Because Eq. 1.37 is only dependent on wellboreand fluid parameters, no information could beobtained as long as the observed behavior iscontrolled by this equation.Ways have been devised to minimize the effectof wellbore storage on pressure behavior. Themost obvious way is to reduce the volume of thewell by using a packer to eliminate the annulusvolume. One may reduce this volume further byintroducing a downhole closure mechanism toeliminate the majority of the tubing volumeAs we have seen earlier, dimensionlessdefinition of parameters may be used togeneralize a specific equation. In case ofwellbore storage, the dimensionless form is asfollows:

)38.1.......(........................................2

615.52

wtD

rhcCC =

Sometimes the wellbore storage coefficient maychange during transient testing. For example,consider a fall-off test in a water injection wellwith a high wellhead pressure during testinjection. When the well is shut-in, initially thepressure at well head would be high but woulddecrease to atmospheric pressure andeventually would go on vacuum if the staticformation pressure is below hydrostaticpressure. At this point the change in pressurewould be due to falling liquid level instead ofdecompression of liquid. As a result, thewellbore storage coefficient, which is for fluiddecompression, changes to the higher falling-liquid-level coefficient. This second storagecoefficient easily could be a hundred to athousand times greater than the first. The

reverse situation can occur as well; with high,rising-liquid-level storage at the beginning ofinjection changing to fluid-compression storageas the wellhead pressure begins to increase.Figure 1.7

illustrates behavior when wellborestorage changes in value.

Fig. 1.6Effect of wellbore storage on sand-face flow rate, C3 > C2 > C1.

C3

00

1

tD

C1C2

q sf/q

t1Log (t)

Log

( p)

Exponential Integral Solution

t2

C1

C2

Fig. 1.7Theoretical pressure response for bothincreasing and decreasing wellbore storage: C2 >C1. (Adaptation of data from Earlougher, Kersch,and Ramey)

Well Test Analysis


Principle of SuperpositionPrinciple of superposition is a mathematicalprinciple that applies to linear differentialequations with linear boundary conditions. Inessence it states that a complex problem maybe broken into a group of simpler problems. Theaddition of the solutions of the simpler problemswill yield the solution of the complex problem.This principle may be applied to account for

changes in rates (superposition in time) or toconsider the effect of boundaries (superpositionin space) or to consider the addition of new wellsto a reservoir (superposition in both time andspace).The most widely used application ofsuperposition principle is the well-known Hornerplot used in analysis of buildup tests. The rateversus time of a buildup test may berepresented by Fig. 1.8.A. This figure may beconsidered as the summation of Figures 1.8.B

Fig. 1.8Schematic of actual and equivalent systems.

A.Actual System

Equivalent System

q

q

-q

Timetp tp + t

Rate

Rate

Rate

B.

C.

Well Test Analysis


and 1.8.C. Principle of superposition states thatthe summations of the solutions of problems1.8.B and 1.8.C is a solution to problem 1.8.A.Following this procedure, Horner equation maybe derived from the basic drawdown equation.Considering skin factor, the long term solution ofthe problem in Fig. 1.8.B is the fairly simpledrawdown equation,

( ) )39.1..(303.222275.3loglog 21

+

++=s

rc

kttmp

wtp

Where

)40.1(..................................................6.162kh

qm

=

The solution for the problem in Figure 1.8.C issimilar, however one will have to remember thatrate is injection rather than production.

( ) )41.1...(303.222275.3loglog 22

+

++=s

rc

kttmp

wtp

Adding the two solutions produces Hornersequation.

)42.1(........................................log

+

=

t

ttmp pt

Thus, if pt or pws (shut-in pressure) is plottedvs.

+

t

tt plog , the long term data will form a

straight line with a slope m. Formationpermeability may be calculated from the slope,m. Skin factor may be calculated using thefollowing equation:

)43.1.....(2275.3log1515.1 21

+

=

wt

wfhr

rc

km

pps Application of the above two equations inanalysis will be illustrated in the examples givenin a later section. Application of principle ofsuperposition to account for boundary effect isdiscussed in the following section.

Effect of BoundariesIn this section, we will study how the presence ofboundaries affects well pressure. To effectivelydo that, we will demonstrate how an engineermay account for the presence of barriers usingthe principle of superposition.The effect of boundaries may be calculatedusing the method of images. If a producing wellhas a nearby infinite single sealing barrier,

method of images may be used to replace thewell-barrier system with a two well system.Basically, the barrier acts as a mirror and maybe replaced with an imaginary well identical tothe real well in physical properties andappearance. Figure 1.9 is a schematic of boththe actual and equivalent systems.Because of the distance between the two wells,the effect of the imaginary well will beinsignificant to start with and observed pressurewill be as if the real well is located in an infinitereservoir. At any time, using principle ofsuperposition, the following equation gives thetotal pressure drop at the real well:

+=303.22

1688log6.162

2

s

rc

ktkh

qppwt

wfi

( ) )44.1....(..........29486.702

ktLcEi

khq t

At early time, the second term of Eq. 1.44 isnegligible compared to the first one and theequation may then be simplified into thefollowing expression:

+=

303.22

21688log6.162 s

wr

tc

ktkh

qwfpip

1.45).........(3026.22

1688loglog 2

++=s

rc

ktm

wt

Fig. 1.9Well near a no-flow boundary.

Sealing

Producing

Actual Image

L

L L

Actual System

Equivalent System

Well Test Analysis


After a sufficiently long time Eq. 1.44 will beapproximated by the following expression:

+=

303.22

21688log6.162 s

wr

tc

ktkh

qwfpip

( ) )46.1.........(..........21688log6.162 2

+Lc

ktkh

q

t

This equation may be written in the simpler form:

( ) ...303.221688loglog2.325

++=s

Lrck

tkh

qppwt

wfi

...(1.47)Equation 1.47 may be written in the followingform:

( ) ...........303.221688loglog2

++=s

Lrck

tmppwt

wfi ...(1.48)Equations 1.47 and 1.48 indicate that whenflowing pressure pwf is plotted versus time, thelate time behavior will have two straight lines.The slope of the second one is double the slopeof the first. The first straight line reflects theperiod before the boundary is felt. The secondstraight line indicates the period when the effectof the barrier on the well is felt. An example ofsuch a test is given in Fig 1.10. The samebehavior would also appear during a builduptest.

When a well is near multiple barriers, thesituation becomes a lot more complex. If a wellis in between two parallel sealing boundaries,the system may be replaced by an infinitenumber of wells as shown in Fig. 1.11.Practically speaking, only three or four levels ofwells may be needed to emulate the presence ofthe two barriers. A well in a closed rectanglesystem may be represented in a similar fashion.

Radius of InvestigationIf a well is opened to flow, shut in, or theproduction rate is changed, a pressure gradientbetween the wellbore and the reservoir willresult. This pressure gradient (or transient) thenpropagates away from the wellbore at a speedthat depends on rock properties and in-situ fluidproperties. Rock properties include permeability,porosity, and thickness. Fluid properties includeviscosity, compressibility, and fluid saturation.Contrary to popular belief, propagation speed of

the pressure transient does not depend on fluidproduction rate.To find the radius of the pressure transient awayfrom the wellbore, use these equations forapproximate radial flow.

M1

M2

Slope)1 to2(

21

2=

MM

BoundaryEffect

Fig. 1.10 Test on a wellnear a boundary.MDH: P versus Log(dt)

pi

Image Wells

ProducingWell

A

I I I I IIA123 1 2 3

Actual System

Equivalent System

Sealing Boundaries

Fig. 1.11Well between two no-flow boundaries.

Well Test Analysis


Gas well:

( ) )49.1.......(....................948iggg

gi

cs

tkr

=

Oil well:

)50.1....(..............................948 toi

oi

c

tkr

=

where sg is gas saturation and ( )iggc is theproduct of gas viscosity and gas compressibilityat initial reservoir pressure.The above equations correspond to the distancetraveled by a pressure pulse created by aninstantaneous source, hence its independenceof rate. Some researchers have expressedconcern about using Eqs. 1.44 and 1.45 todetermine radius of investigation. An equationthat includes flow rate and gauge resolution has

been developed.27 The following equationdescribes this approach for a gas reservoir.

( ) ( )

=

+

=

kttrcEiqq

Tkhs

kn

itn

kkk

f

y

1

2

11

19.379267.459

2828.705

...(1.51)Where Tf is formation temperature and Sy is thegauge resolution.

Clearly, radius of investigation depends on thegauge resolution. Fig. 1.12 shows anexponential decline in both gas in place (GIP)and radius investigated versus a decreasinggauge resolution (increasing values of Sy). Inthis well case, the use of an electronic quartzgauge with .01 psi resolution would more thandouble the volume investigated by a gauge withonly .1 psi resolution.

Radius of Investigation (ri) & Gas in Place (GIP)vs

Gauge Resolution (Sy)

0

200

400

600

800

1000

1200

0.00

10.

137

0.27

30.

409

0.54

50.

681

0.81

70.

953

1.08

91.

225

1.36

11.

497

1.63

31.

769

1.90

52.

041

2.17

72.

313

2.44

9

Gauge Resolution (psi)

Rad

ius

of In

vest

igat

ion

(ft)

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

Gas

in P

lace

(MMS

CF)

GIP

ri

Fig. 1.12 Radius of Investigation and Gas in Place vs Gauge Resolution (Sy )

Sy ri GIP(psi) (ft) (MMSCF)

.001 1042 8224

.01 813 5007

.1 559 23671.0 229 3972.5 94 67

Reservoir Parametersnet pay = 100 ftwellbore radius = 0.333 ftpermeability = 10 mdgas viscosity = 0.0249 cpporosity = 10%sw = 10%gas rate = 1000MSCF/Dflow time = 24 hoursz factor = .99972Bg = 6.6477E-4 RB/SCFct = 1.1865E-4 1/psiskin = 0initial pressure = 5000 psiformation temp = 200 deg F

Well Test Analysis


Raghavan,28 on the other hand, suggestscalculating the distance to sealing fault that itseffect on pressure is about to appear. In thiscase one may call this the radius of area devoidof boundaries. This approach should beapplicable to pressure buildup tests, too. Thisdefinition may be tied to production rate andgauge resolution.Other authors have stated that what is not seenduring a drawdown would not be seen in abuildup test. This may be easily visualized in atest where more than one flow regime may beencountered, such as effect of boundary, dualporosity system, fractured reservoirs, etc. Forexample, if the drawdown test was too short toobserve effect of a boundary, buildup could notbe used to calculate distance to the boundaryregardless how long it may be. In a veryinteresting article, Earlougher and Kazemi29stated that radius investigation during adrawdown test should be at least four times thedistance to a sealing fault to observe doubling ofslope during a buildup test.Another example is a buildup test on a dualporosity system. In a dual porosity system, onewould observe wellbore storage effect followedby effect of fractured system, a transition period,

and eventually the total system including thematrix. Figure 1.13 shows the buildup responsefor various producing times as well as theresponse from a long drawdown test. It is clearfrom Fig. 1.13 that as the producing time getslonger, the behavior of the buildup test willapproach that of a drawdown test. It is also clearthat a buildup test would not reveal a flowregime that had not been investigated by thepreceding drawdown. For example if thedrawdown test was terminated before the matrixsystem was investigated, the buildup test couldreveal reliable information on that portion of thereservoir.

Phase RedistributionDuring a buildup test, pressure is expected tomonotonously increase with time. However insome instances wellbore effect may cause asharper than expected increase in pressurefollowed by a decrease in pressure, as given inthe field example illustrated in Fig. 1.14. Thispeculiar phenomenon is due to phase (gas andliquid) redistribution inside the wellbore. Whengas bubbles rise inside a liquid column it causes

S = 0CD = 100 = 0.1 = 0.001

tD/CD

10

10

10

10

10

1010 10 10 10 10 10 10 10 10

1

0

-1

-2

-3

-4

-2 -1 0 1 2 3 4 5 6

106

TPD = 10

PD

105104103

102

Fig. 1.13Log-log plot of PD vs. tD/CD for buildups for a naturally fractured formation.

Well Test Analysis


pressure to increase by an amount equal to thedifference in specific weights of the phasestimes the column height. Stegemeier andMatthews30 performed analytical andexperimental research and were the first toexplain this phase redistribution phenomenon.In actual field cases, the increase (hump) inpressure does not usually reach the pressuredefined above, because as pressure inside thewellbore increases quickly, fluid flow wouldreverse, i.e. flow into the formation. Thus it isexpected that as permeability gets higher, themagnitude of the hump would be smaller.Formulating the phase redistribution as asecondary wellbore storage, Fair31 presented asolution and type curves for wells exhibiting thisphenomenon.

Production Well Testing:Types of Tests andTechniques of Analysis

IntroductionThis section will cover techniques currently inuse for analyzing data from pressureinterference, drawdown, and buildup tests in oiland gas wells. We will provide the mathematicalbasis for such analysis techniques as type curvematching and semi-log. A straight-forward, step-by-step procedure is provided to apply the actualtechniques and find such factors as formationpermeability, skin, reservoir conductivity, storagecoefficient, porosity, and others. Exampleproblems and their solutions are included in theback of this section.

Test Types

Pressure Drawdown Test

This test is conducted by producing a new wellor a well that has been shut-down for a long time

at a constant rate. During the test, thebottomhole pressure is monitored. The behaviorof this pressure will be affected by wellborestorage at early time and by skin factorthroughout the test. Therefore, analysis of thistest by semi-log or type-curve matching methodscan theoretically determine skin, storagecoefficient, permeability and porosity. Practicallyspeaking, however, only permeability and skincan be determined with high accuracy.

Pressure Buildup Test

This is the most common test and probably theeasiest to conduct. In this test, the well isproduced at a constant rate (or almost constantrate). The well is then shut-in and the pressurerise with time is recorded. The pressureresponse is affected by both skin factor andwellbore storage. Analysis of this test yieldsreservoir and fracture parameters. Semi-logmethods and type curve matching can be usedfor analysis. The main reason behind thepopularity of this test is the expectedsmoothness of the data due to the constant rateduring the buildup period (zero rate).

SurfaceTubing Pressure

( TP)

BottomholePressure(40 HP)

300

psia

300

psia

300

psia

148 HP ( TP)

1200

1100

1000

900

800

700

600

500

400

300

200

100

Pre

ss

ure

C

han

ges,

ps

i

Time After Closing In, min

100 20000

Fig. 1.14Pressure buildup example of phaseredistribution, South Texas well.

Well Test Analysis


Drill Stem Testing (DST)

DST is a temporary completion of a well to allowtesting of a particular zone. DST allows anoperator to determine key reservoir parameterssuch as formation permeability and formationinitial reservoir pressure. It will also giveindication of possible problems with the reservoirsuch as depletion, sand tendency, and thepresence of a nearby barrier. Following a DST,an operator may choose to either permanentlycomplete the well or plug & abandon the welldepending on the wells economic feasibility. ADST usually consists of four periods: two flowsand two shut-ins.

Multi-rate Testing

The simplest multirate test is a buildup test. Inthe case of a buildup test the second period hada zero flow rate. Other multirate tests are fairlyeasy to conduct and analyze as long as bothrate and pressure are accurately monitored forall periods. Multirate tests includes pulsetesting, isochronal testing, and modifiedisochronal testing. This type of tests yieldsinformation similar to what could be obtainedfrom a drawdown test.

Multi-well Testing

To obtain information that may characterize thereservoir in various directions, one may have toconduct a multi-well test. In such a test, the flowrate at a producing well is varied while thepressure is monitored at one or moreobservation wells. Analysis of these pressuredata yields information that could not have beenobtained from a single well test. For example,directional permeability, direction of naturalfractures or hydraulic fractures may be obtained.

The simplest form of a multi-well test is aninterference test in which only one observationwell is employed. Because of the distance (tensor hundreds of feet) between the producer andobservation wells, it is expected that fairly smallchanges in pressure at the observation wellswould be monitored. The use of electronicgauges may be necessary to achieve a reliableanalysis.

Planning the TestThe first step in conducting a production welltest is planning. Too often, inadequate planningleads to trouble and costly mistakes. Naturally,knowing about the well-reservoir system ishelpful for planning a test; collect as muchpretest information as possible. If available, youcan gather pretest in formation from: Seismic data (geology) Drilling operations information Core samples logs any other previous production tests.Neglecting this information usually increasestesting costs. For example, consider a fault or afluid interface close to the wellbore where abuildup test is being used to determine reservoircharacteristics. Because of the closeness of thefault, it would be advised to minimize wellborestorage to obtain useful data. This may beaccomplished by shutting the well in near thesand face instead of at the surface. By doingthis, you would separate the time between theend of wellbore effects and the beginning ofboundary effects. A downhole shut-in tool maybe necessary if phase segregation is occurringin the wellbore when a fault or fluid interface isnear the wellbore.Pretest information usually allows you toproperly select a bottomhole pressure gauge forresolution and range. It usually provides theneeded reservoir parameters so you can designthe most efficient test that would achieve the testobjectives.By knowing the history and the future plans for awell-reservoir system, you can set realistic testobjectives.

Test ObjectivesOperators conduct production well tests todetermine some or all reservoir and wellborecharacteristics, to predict individual wellperformance, or both. Well testing is mostbeneficial when used for exploration. Testing todiscover new reserves or preventing dry holesare testings two main purposes. Often,operators use production well tests to provethere are enough hydrocarbons in place tojustify the cost of building a pipeline to the well.Production well tests can be time consuming,but well worth the effort if data is gatheredcorrectly.

Well Test Analysis


Single Well Test BenefitsIn addition to determining reservoir parameters,production tests on individual wells offer severalother potential benefits.

Leaks Near or in the Wellbore orReservoir

One benefit of a well test in either a wildcat orproducer, is that it helps detect leaks. Wellboreleaks in the very early part of the test, andreservoir leaks very late in the test are importantto know for gas storage and other projects (Fig.1.15).

Stimulation Treatments

From a short-term test, you can usually decidewhether it is economical to stimulate a well andhow to stimulate it. Running a short-term testbefore and after the stimulation treatment allowsyou to determine whether the treatment waseffective.

Step-out Locations

By running a long-term test, you can provewhether a reservoir exists at a step-out drillinglocation. You can also determine the direction ofthe step-out by analyzing geological or seismicsources.

Time Decay of Performance

You can also predict the time decay of wellperformance from well test results. For gaswells, the shift of the back-pressure line iscommonly used. For oil wells, the projecteddecay of the productivity index is used. Youmust know the initial reserves and the drainageshape of the well before you can predict timedecay of well performance.

Critical Flow Rates

During well testing, you can also determine thecritical flow rates where coning or waterfingering begins (Fig. 1.16 and 1.17).

Detecting Impediments

You can detect impediments such as sealingfaults, leaky or unsealed faults, or rock or fluiddiscontinuities by running well production tests.By knowing this information, you can morecompletely describe the reservoir and makereliable economical projections for the well.

F1

Water

F3

F2Oil

Fig. 1.15Leaks near or in the wellbore orreservoir.

F4

WaterF3F2

Oil

F1

Fig. 1.16Fingering of water into thewellbore.

F4

Water

F3

F1

Oil

Fig. 1.17Water coning into the wellbore.

Well Test Analysis


Multi-well Test BenefitsWhen more than one well is involved, you canrun an interference or pulse test to obtain otherbeneficial information, provided the propertesting procedures are used.

Communication

Interference tests are commonly run todetermine if two or more wells arecommunicating through the zones from whichthey are producing. This is important forsecondary recovery processes and is necessarybefore determining fracture orientation anddirectional reservoir parameters.

Competitive ProductionIf two operators have adjoining wells that areproducing from the same zone, interferencetesting ensures that neither is producinghydrocarbons from the other operators lease.

Detecting Undrilled ReservesIn many reservoirs, if you carefully gather andanalyze field data, undrilled reserves can bedetected.

Infill Drilling

In tight reservoirs, optimum well spacing isimportant to efficiently drain the reservoir formaximum return on your investment. Productionwell testing from this can help you determine ifinfill drilling will be profitable.

Reserves in a Naturally FracturedReservoir

The only reasonable way to determinehydrocarbon volume in a naturally fracturedreservoir is from multiple-well testing.

Summary of Well TestingBenefitsWithout a properly designed testing program,you could waste a large amount of money. Byevaluating well testing data and other well-reservoir system data, you can avoid conductingunnecessary stimulation treatments. If

stimulation is necessary, you can make it aseconomical as possible. A well designed testingprogram can also help you establish productionrates to recover maximum hydrocarbons and tojustify the cost of a pipeline to the well. Afterestablishing the goals of the test, you can selectthe most economical, effective type of test forthe goal you have set.

Establishing Test ProceduresThe way you conduct the test depends on thetest objectives, the characteristics of the well-reservoir system, the way you will analyze thetest data, and more than likely, a governmentalagencys requirements. Transient well testingprimarily involves four types of tests: Drawdown/buildup tests Injection/falloff tests Interference/pulse tests Drillstem tests (DSTs)Since drawdown/buildup tests are the mostcommon, we will discuss them in detail in thissection. Keep in mind, though, that you conductinjection/falloff tests and DSTs similarly.Interference/pulse tests are used less often andare not within the scope of this section. Recentpublications on interference testing consider theeffects of anisotropic reservoirs. From the welltest analysts point of view, you should not selecta drawdown test alone, unless the well cannotbe shut in for operational or economicalreasons. Buildup test data is normally the mainsource of data used for determining the well-reservoir description. Of course, a buildupcannot occur unless well drawdown occurs first.Analysis techniques allow you to analyzedrawdown and buildup data simultaneously.

Reservoir Limits TestsIf the single-rate drawdown test lasts longenough for the pressure transient to be affectedwellbore, the test becomes a reservoir limitstest. Figure 1.10 shows a reservoir limit testwhere the rate of pressure decline is twice asfast because of the impediment effect on thepressure as compared to the rate of pressuredecline for an infinite-acting reservoir.Figure 1.18

shows a reservoir limit testperformed on a well in the middle of a reservoir.

Well Test Analysis


This figure above shows that a plot of pressureversus time would eventually become linear,where the slope of the straight line is a functionof reservoir area. This period of linear pressuredecline with time is the period where pseudo-steady conditions prevail. Pseudo-steadycondition starts at

.

10637.21.04

Acktx

tt

DA

==

Quantitative Analysis Methodsof Pressure Transient TestsType Curve Matching is a useful technique foranalyzing data from drawdown, buildup andinterference tests. To understand type curvematching, however, one must first understandthe basic characteristics of type curves.

Type Curves

As discussed earlier, the followingdimensionless groups are used in the petroleumengineering literature for an oil reservoir.

( ) )52.1..(..............................2.141 q

ppkhp iD

=

)53.1(..............................10637.2 24

wtD

rc

ktxt

=

)54.1........(........................................w

Dr

rr =

If Eqs. 1.52 to 1.54 are substituted into thegoverning differential equation, the followinggeneral dimensionless equation would result.

)55.1..(....................122

D

D

D

D

DD

D

tp

r

prr

p

=

+

The same dimensionless groups are used for agas reservoir except that the dimensionlesspressure is defined as:

( ) )56.1......(..............................417.1 Tq

pmkhpD

=

where

m(p)=change in gas pseudo-pressure, MMpsi2/cpq = production rate in MMCF/DT = reservoir temperature in R.The solution of Eq. 1.55 is general. Thepressure behavior of any reservoir that satisfiesassumptions 1 to 10 on Page 5 should resemblethis general solution. However, the reservoirdata will shift in both the x and y directions by acertain factor that depends on reservoir and fluidproperties. If both the general solution of Eq.

Fig. 1.18Cartesian plot of the drawdown test data showing pseudo steady state.

Flow

ing

Pres

sure

, pw

f, ps

ia

Flowing Time, t (hrs)

pi

Well Test Analysis


1.55 and the field data are plotted on a log-loggraph, the shift, or transformation, from oneplotted curve to the other will be linear. The log-

log plot of the general solution is usually calledtype curve.Type Curve Matching is a technique designedto find the amount of shift between the field datacurve and the general solution curve along thep (pressure) and the t (time) axes. Bysubstituting the shift amount along the p axisand the shift amount along the t axis into Eq(s).1.52-54, the unknown reservoir properties maybe calculated. These parameters may includeformation permeability, porosity and skin factor,and others.

How to Use Type Curve Matching

When a drawdown or a pressure buildup test istoo short for the semi-log straight line todevelop, the data cannot be analyzed usingsemi-log methods. The general type curvematching method that will be described here canbe applied to any system with known pD vs. tD.Type curve matching may be used fordrawdown, buildup and interference testing.The type-curve matching techniques have beendescribed in many ways; the method outlinednext is probably the simplest, and will beillustrated for Fig. 1.19. This figure is the typecurve for pressure interference tests using theline source solution.

10 1 10 2 10 3

10 0

10 1

10 2

10 3

T im e, days

Cha

nge

in

Pr

ess

ure

, ps

i

Fig. 1-20Field data for type curve matching

104

10

1

1

10-1

10 103 10410210-1

105 106 107 108

PD

tD/rD2

10-2

Fig. 1.19Line source solution.

( )22

,

0002637.02.141

rc

ktr

t

ppq

khp

tD

D

triD

=

=

Well Test Analysis


1. From reservoir, well, and test conditionschoose the type curve that matches thetest, well and reservoir condition; this willusually be a log-log plot of pD vs. tD. Noticethat on Fig 1.20, the observed test data areplotted as p vs. test time, t, using thesame size scale as the base type curve.

2. Calculate p, or change in pressure withtime, for pressure buildup data for theinterference test:

)57.1.......(....................).........(tppp wfi = .In general, for any kind of test,

( ) )58.1.......(....................0 tptpp ww == .Note that p is always calculated as apositive number. Also, the time parameteris the running test time, t.

3. Plot observed test data by placing tracingpaper over the desired type curve, tracingthe major grid lines and marking theobserved p (psi) and, t (hours), as givenin Fig. 1.20. Note: Any pressure or timeunits can be used as long as theconversion factors in Eq. 1.52 and 1.53 aremodified accordingly. Plotting p vs. tusing the type curve grid showing throughthe tracing paper as a guide will guaranteethat the data plot and the type curve havethe same scale.

4. Slide the tracing paper with plotted dataalong the type curve, keeping the grids oneach graph parallel, until the data pointsmatch the type curve. After the match iscompleted, pick a convenient match point,such as an intersection of major grid lines.Finally, record values of that point on thedata plot [(p)

m , (t)

m] and the corresponding

values beneath that point on the type curvegrid [(pD)m , (tD)m]. The ordinate (pD) of thistype curve is dimensionless pressure,

)59.1.......(....................2.141 qphkpD = .

5. By substituting the match point values fromStep 4 and rearranging this equation, weestimate formation permeability:

( )( ) )60.1(....................2.141

m

mD

pp

hqk

= .

6. Similarly use the definition of the abscissaon the type curve,

)61.1.....(....................10637.2 24

2wtD

D

rc

ktxr

t

= .

7. Using the time-scale match-point data andthe permeability just determined, toestimate the reservoir porosity:

( ) )62.1......(10637.22

2

4

mD

D

m

wt

r

t

t

rc

kx

=

.

Characteristics of Type CurvesSeveral type curves exist in the literature. Thetests for these type curves have already beendiscussed. They are illustrated in the exampleproblems in the last section.1. Interference Type Curve (line source

solutionFig. 1.19) This type curve describes the effect a

producing well has on an observation well. The distance between the observation and

producing well should be much larger thanthe wellbore radius of the producing well.

If pressure is plotted versus time on a semi-log scale, the data corresponding to (tD/rD2)> 25 will yield a straight line. The slope (m)of this straight line is

)63.1..(..............................6.162hkq

m

= .

2. Pressure Drawdown Type Curve (unfracturedreservoir)

Figure 1.21 is designed for pressure

drawdown test of a single well. The effectsthat skin factor and wellbore storage haveon reservoir performance are considered.Notice, however, that the shapes of thevarious curves on these graphs are similar,making it fairly difficult to achieve a uniquetype curve match.

All the curves begin as a straight line withunit slope. The unit slope straight lineindicates the presence of wellbore storage.As the dimensionless wellbore storage CDbecomes smaller, the duration of the unitslope straight line gets shorter.

Well Test Analysis


As the skin factor gets higher, the curvesflatten out more abruptly.

If pressure is plotted vs. time on semi-logscale, the data corresponding to

( ) )64.1.......(....................5.360 sCt DD +will fall on the straight line with a slope m.

)65.1..(..............................6.162hkq

m

=

This type curve is also used for matchingbuildup tests. However, it should beremembered that in order to use pressurebuildup data for type curve matching one ofthe following conditions should be satisfied.(1) That the drawdown test has lasted long

enough such that the change inpressure with time is small (pseudo-steady state). (Figure 1.22)

(2) That the pressure buildup data havebeen corrected for the change inpressure with time using the principle ofsuperposition.

Although only an estimate is involved in thefirst condition, if the condition is satisfied,as it is the case in many buildup tests, apressure buildup test should yield fairlyaccurate results. The second condition is

rigorous and should always yield anaccurate analysis. Pressure buildup datamust be corrected if the producing time isshort. Otherwise, type-curve matching willgive erroneous results. Specializedtechnique have been developed foranalysis of such cases, however they areoutside the scope of this chapter.

3. Fractured Reservoir Type CurvesIn this section, we will only consider a wellintercepting a vertical fracture with fractureheight equal to the formation height. This typecurve is designed for a drawdown test, however,it can also be used for pressure buildup tests.Figures 1.22 and 23 are schematic diagrams ofa well intercepting a vertical fracture. Threequantities are usually mentioned in descriptionsof fractured reservoirs.a. Distance from wellbore to the outer

boundary, xe , as illustrated in Fig. 1.22b. Fracture conductivity.Dimensionless fracture conductivity, CfD , isdefined in the following equation:

)66.1....(........................................f

ffD kL

wkC =

where:

10-1

CD = 0

C D

= 10

2

C D = 10

3

C D

= 10

4

C D

= 10

5

102

10

1

PD

s = 10s = 20

s = 5s = 0

s = -5

102 103 104 105 106 107tD

( )

2

2

26146.5

0002637.02.141

wtD

wtD

wfiD

hrcCC

rc

ktt

ppq

khP

=

=

=

Fig. 1.21PD vs. tD for a well with storage and skin effect (radial flow).

Well Test Analysis


kf is the fracture permeability in millidarcies, w isthe width of the fracture in feet, and Lf is thefracture half-length in feet.In terms of conductivity, three types of fracturesexist:

Infinite conductivity fractureIn this type, it is assumed that there is nopressure drop inside the fracture. Figure1.25

gives the type curve for this type offracture.

Uniform flux fractureThis specific solution assumes that fluidflow into the fracture is uniform along thefracture face, which implies a fairly smallpressure drop along the fracture. This typecurve, Fig. 1.26, was initially used toapproximate the behavior of dual porositysystems. In modern analysis specializedmodels for dual porosity have beendeveloped that replace this type curve.

Finite conductivity fractureIn this case there is an appreciablepressure drop inside the fracture. The flowregimes that may arise in this case may becomplicated and require more care inanalysis in pressure transient tests. Figure1.27

gives the type curve for a wellintersecting a finite conductivity fracture.

Characteristics of the fractured welltype curves

The type curves of infinite conductivity anduniform flux fracture start with straight lineswhose slopes are one-half. This slopeindicates a linear flow regime in theformation. This straight line characterizesthe infinite conductivity in Fig. 1.25 and theuniform flux type curve in Fig. 1.26.

The finite conductivity fracture goes throughmore complicated flow regimes illustrated inFig. 1.28.

The finite conductivity does nothave the half-slope straight linecharacteristic of infinite conductivityfractures because of the pressure drop inthe fracture is significant in comparison tothe pressure drop in the formation. Thiscauses the system to first go through alinear flow regime (linear in the fracture)

dp1dp2

Pi

Pressure

Pws Line

Pwf Line

p2

tTime

+ q

Rate

tp t

Time

+ qRate

tp tTime

- q

p1

tp

Fig. 1.22Buildup schematic.

Fig. 1.23Schematic diagram for a wellintercepting a vertical fracture.

Wellbore

Lfxe

No FlowBoundary

Well Test Analysis


followed by a bi-linear flow regime (thefracture and formation).

The linear flow of the fracture is due toexpansion of fluid in the fracture and isusually too short in duration to be of anypractical use in evaluating the fracture.

The bi-linear regime of finite conductivityfractures is characterized by a quarterslope straight line. The duration of the bi-linear flow period depends on thedimensionless fracture conductivity. (seeFig. 1.29)

After a transition period, the fracturedreservoir data forms a semi-log straight linewhose slope is given in the following

equation.

)67.1........(....................6.162 kh

qm

=

The start of the semi-log straight linedepends on dimensionless fractureconductivity as shown in Fig. 1.29.

The dimensionless time on these twographs is defined a little differently than onprevious graphs. It is

).68.1(....................10637.2 t 24

Dfft Lc

kt

=

The fracture half-length may be calculatedfrom type-curve matching. The skin factorfor these may be related to the fracture

Fracturekr

xfw

Wellbore

Fig. 1.24Schematic of a fractured well.

Dimensionless Production Time, tD

xe/xf = 1 10/7 2 10/3

5

10

Dim

ensi

onle

ss W

ellb

ore

Pre

ssu

re D

rop,

p w

D

2Lf

2xe

Drainage AreaA = (2xe)2

10-2 10-1 100 101 102 103

100

102

10-1

101

Fig. 1.25Propped hydraulic fractures. (Infinite conductivity fractures where p=0)

Well Test Analysis


half-length as follows.for infinite conductivity fractures

(1.69)...............................2 swf erL =for uniform flux fractures

(1.70)........................718.2 swf erL =

for finite conductivity fractures

(1.71)............................... swf enrL =

where n is a factor that depends on fractureconductivity. Figure 1.31 gives this factoras a function of conductivity.

xe/x f = 1 10/7 2 10/3 5

10

Dimensionless Production Time, tD

Dim

en

sio

nle

ss W

ellb

ore

Pre

ssu

re D

rop,

p wD

10 -2 10 -1 10 0 10 1 10 2 10 3

10 0

10 1

10 2

10 -1

Drainage AreaA = (2xe)2

2Lf

2xe

20

Fig. 1.26Natural, unpropped fractures. (Uniform Flux fractures where p is small)

10-4 10-3 10-1 110-5 10-210-3

1

10-2

10

10-1

Dim

ensi

onle

ss P

ress

ure

, pw

D

Dimensionless Time, tDL

0.1

0.51

510

50100500

DimensionlessFractureConductivity(CfD)

f

( )( )[ ]

( )

f

ffD

fitxD

wD

wD

wD

kLwL

C

Lckt

t

qTpmkh

p

zTqpkh

p

qpkh

p

f

=

=

=

=

=

2

4

2

10634.2

Gas1424

Gas1424

Oil2.141

Fig. 1.27Log-log type curves for finite capacity vertical fractures (constant well rate).

Well Test Analysis


Semi-log Methods

It is sometimes difficult to find a unique matchwhen attempting to use the type curve matchtechnique. This difficulty especially occurs whenusing it for wells with wellbore storage and skin.However, analysis of the solution of thegoverning differential equation has shown that inthe transient pressure region, pressure isdirectly proportional to a logarithmic function oftime. Thus, if pressure is plotted versus time ona semi-log graph, a straight line will result. Theslope of the straight line is a unique function offluid and rock properties as well as theproduction rate.

The Mathematical Basis of the Semi-log Methods

A solution of the governing partial differentialequation for an oil reservoir is:

1.72) (..................4

6.70 2

+=t

rEikhqpp wiwf

where

1.73) (....................................10637.24

tc

k

=

Linear Fracture Flow

Bilinear Flow

Linear Formation Flow

Eliptical or Transitional Flow

Pseudo Radial Flow

Fig. 1.28Fracture flow regimes.

Fracture Tip

yD Reservoir

FractureWell

yD= w / 2LfxD(0.0) 1

Dimensionless Fracture Conductivity, CfD

Tim

e a

t the

End

of t

he B

ilinear

Flow

Per

iod,

t eb

D

10-1

10-2

10-3

10-4

10-51 1010-1 102

Fig. 1.29Dimensionless time at the end of thebilinear flow period, constant pressure production.

Well Test Analysis

Hal

bwtasec1

Documents

Transcript of bwtasec1