Well Testing AnalysisFall 2005Mazher Ibrahim

Basis for Grade:Homework20%Examinations (3)45%Final Examination25%Class Participation/Pop Quizzes 10%total = 100%Grade Cutoffs: (Percentages)A: < 90 B: 89.99 to 80 C: 79.99 to 70 D: 69.99 to 60 F: < 59.99

Introduction to Well Testing

ObjectivesList the more common objectives of well testing.Describe the diffusivity equation by explainingits purpose and applications assumptions made in its derivation and how it is derivedits form for one-dimensional radial flow.List, define, give the units for, and specify typical sources for each of the variables that influence responses in a well test.Compute the total compressibility for different reservoir systems (undersaturated oil, saturated oil, gas).

What Is A Well Test?A tool for reservoir evaluation and characterizationInvestigates a much larger volume of the reservoir than cores or logsProvides estimates of permeability under in-situ conditionsnear-wellbore conditionsdistances to boundariesaverage pressure

How Is A Well Test Conducted?Well is allowed to produce normallySensor is lowered into wellProductionremains constantPressure stabilizes

How Is A Well Test Conducted?q = 0Sensor is lowered into wellWell is shut inProduction drops to 0Pressurerises

Fundamental ConceptsApplications and objectives of well testingDevelopment of the diffusivity equationDefinitions and sources for data used in well testing

Types and Purposes of Well TestsPressure transient testsWe generate and measure pressure changes with timeDeliverability testsWell controlled production(Production Analysis)Use of production data for goals usually achieved by well testing

Production data analysisReservoir properties (permeability, skin factor, fracture half-length, etc).Reservoir pore volume (estimated using long-term production performance).Estimated ultimate recovery (EUR)movable fluid volumes.

Well Test Applications

Well Test Objectives Define reservoir limitsEstimate average drainage area pressureDiagnose productivity problemsCharacterize reservoirEvaluate stimulation treatment effectiveness

Single-, Multiwell TestsWell is allowed to produce normallySensor is lowered into well

Single-, Multiwell TestsWell is shut in, pressure is measured

Single-, Multiwell TestsWell is shut inSensor is lowered into offset well. . . pressure is measured at offset well(s)

Kinds of Well TestsProduce well at constant ratePlot pressure responseLower sensor into well

Kinds of Well TestsProduce well at constant rateShut in wellLower sensor into wellPlot pressure response

Kinds of Well TestsInject fluid into well at constant ratePlot pressure response

Kinds of Well TestsInject fluid into well at constant rateMeasure pressure responseShut in well

Multiwell Tests. . . measure pressure response at offset well(s)Produce one well at constant rate . . .

Multiwell Tests. . . measure pressure response at offset well(s)Alternately produce and shut in one well . . .

PTA: Single-Well Testsone well in which the pressure response is measured following a rate change. pressure buildup test shut in after controlled productiondrawdown or flow test (specific drawdown tests: are called reservoir limits testspressure falloff test similar to a pressure buildup test, except it is, conducted on an injection wellinjectivity test Inject into the well at measured rate and measure pressure as it increases with time analogous to pressure drawdown testing.

PTA: Multiwell TestsFlow rate is changed in one well Pressure response is measured in one or more other wellsDirectional variations of reservoir properties (orientation of natural fractures)Presence or lack of communication between two points in the reservoirRatio of the porosity-compressibility products of the matrix and fracture systems

Multiwell tests:Interference tests The active well is produced at a measured, constant rate throughout the test(Other wells in the field must be shut in so that any observed pressure response can be attributed to the active well only.)Pulse testsThe active well produces and then, is shut in, returned to production and shut in againRepeated but with production or shut-in periods rarely exceeding more than a few hoursProduces a pressure response in the observation wells which usually can be interpreted unambiguously (even when other wells in the field continue to produce)

Deliverability tests (DT)production capabilities of a well under specific reservoir conditionsprimarily for gas wellsabsolute openflow (AOF) potentialinflow performance relationship (IPR) or gas backpressure curve

DT: Flow-After-Flow Tests (referred to as gas backpressure or four-point tests) producing the well at a series of different stabilized flow rates measuring the stabilized bottomhole flowing pressure at the sandface typically, with a sequence of increasing flow rates

DT: Single-Point Testslow-permeability formationsflowing the well at a single rate until the bottomhole flowing pressure is stabilizedrequired by many regulatory agenciesrequires prior knowledge of the well's deliverability behavior (from previous testing or from correlations with other wells producing in the same field under similar conditions)

DT: Isochronal TestsSpecifically, the isochronal test is a series of single-point tests developed to estimate stabilized deliverability characteristics without actually flowing the well for the time required to achieve stabilized conditionsThe isochronal test is conducted by alternately producing the well, then shutting in the well and allowing it to build up to the average reservoir pressure prior to the beginning of the next production period.

IssuesDevelopment Wells vs. Exploration WellsProducing Wells vs. Injection WellsShallow Wells vs. Deep WellsStimulated Wells vs. Unstimulated WellsEffects of Reservoir PropertiesLow Permeability vs. High Permeability FormationsSingle Zones vs. Multiple ZonesSafety and Environmental ConsiderationsSweet Gas vs. Sour and Corrosive GasesOther environmental Concerns

Production data analysisReservoir properties (permeability, skin factor, fracture half-length, etc).Reservoir pore volume (estimated using long-term production performance).Estimated ultimate recovery (EUR)movable fluid volumes.

End of Class

The Diffusivity EquationDescribes the flow of a slightly compressible fluid having constant viscosityin a porous mediumat constant temperatureDerived from basic relationships ofcontinuityflow equation (Darcys law)equation-of-state

The Continuity Equation

Flow Equation (Darcys Law)or, in differential form,

Equation of State for a Slightly Compressible Liquid

The Diffusivity EquationOne-dimensional, radial form:

Formation Volume Factor

ViscosityA fluids resistance to flowGasolinelow viscosityVaselinehigh viscosity

Fluid Compressibility

Porosity

Permeability

Pore Compressibility

Net Pay Thicknessh = h1 + h2 + h3

Net Pay ThicknessVertical well, horizontal formationDeviated well, horizontal formationDeviated well, slanted formationVertical well, slanted formation

Saturations

Wellbore Radius

Total Compressibility

Modeling Radial Flow

Instructional ObjectivesState the Ei-function solution to the diffusivity equation, and list all the assumptions on which it is based. State practical rules for determining the numerical values of the Ei-function.Given formation and fluid properties, be able to calculate the radius of investigation at a given time and the time necessary to reach a given radius of investigation.Describe the effects of reservoir properties on the radius of investigation.

Radial Flow Reservoir Model

Ei-Function Solutionto the Diffusivity Equation

Ei-Function Graph

Short-Time Approximation for Ei-Function Solution(large radius or small time)

Long-Time Approximation to Ei-Function Solution(small radius or large time)

Pressure Profile During DrawdownDistance from center of wellbore, ftPressure, psi20001000110100010010000riririri

Pressure Profile During Buildupriririri

Radius of Investigation EquationsRadius of investigation for a given time t:Time required to reach a given radius of investigation ri:

Characterizing Damage and Stimulation

Instructional ObjectivesList factors that cause skin damage or geometric skin factor.Calculate skin factor for a given additional pressure drop due to damage; conversely, calculate additional pressure drop for a given skin factor.Calculate flow efficiency given the skin factor, wellbore pressure, and average drainage area pressure.Express skin factor as an apparent wellbore radius; conversely, express apparent wellbore radius as a skin factor.Express a given skin factor as an equivalent fracture halflength (for an infinite-conductivity fracture); conversely, express fracture half-length as an equivalent skin factor.

Drilling Fluid Damage Mud filtrate invasionFines may clog pore throats, reducing effective permeabilityFiltrate may cause clays to swell, causing damage

Production Damage

Injection Damagedirty waterincompatible water

Reservoir ModelSkin Effect

Reservoir Pressure Profile

Skin and Pressure Drop

Skin and Pressure Drop

Skin Factor and Properties of the Altered Zone

Skin Factor and Properties of the Altered Zone

Effective Wellbore Radius

Minimum Skin Factor

Minimum Skin FactorExample

Converging Flow to PerforationsGeometric Skin

Partial PenetrationGeometric Skin

Incompletely Perforated IntervalGeometric Skin

Partial PenetrationApparent Skin FactorGeometric Skin

Deviated WellboreGeometric Skin

Deviated Wellbore Apparent Skin Factor

Well With Hydraulic FractureGeometric Skin

Completion SkinkRkd

Gravel Pack Skin

Productivity Index

Flow Efficiency

Flow Efficiency and Rate

Semilog AnalysisFor Oil Wells

Instructional ObjectivesAnalyze a constant-rate drawdown test using semilog analysis.Analyze a buildup test following a constant-rate flow period using the Horner method.

Ei-Function Solution0.001100-x246-Ei(-x)

Reservoir Pressure ProfilePositive (damage) skin (s = +5)Negative skin (s = -2)Pressure, psiUnsteady-state pressure (s=0)

Incorporating Skin into the Ei-Function SolutionFor r > raFor r = rw

Log Approximation to the Ei-Functiony = mx + bUse |m| in computations from this point forward

Estimating Permeability and Skin

Drawdown Test GraphPowers of 10Plot pressure vs. time

Exampleq = 250 STB/Dpi = 4,412 psiah = 46 ft = 12%rw = 0.365 ftB = 1.136 RB/STBct = 17 x 10-6 psi-1 m = 0.8 cp

Exampleq = 250 STB/Dpi = 4,412 psiah = 46 ft = 12%rw = 0.365 ftB = 1.136 RB/STBct = 17 x 10-6 psi-1 m = 0.8 cp

ExampleExtrapolate to get p1hrp1hr 3,540 psim 100Plot data points from field datap10hr 3,440 psislope = p10 hr-p1 hr -100

Exampleq = 250 STB/Dpi = 4,412 psiah = 46 ft = 12%rw = 0.365 ftB = 1.136 RB/STBct = 17 x 10-6 psi-1 m = 0.8 cp

p1hr 3,540 psim 100

Problems with Drawdown TestsIt is difficult to produce a well at a strictly constant rateEven small variations in rate distort the pressure response

Alternative to Drawdown TestsThere is one rate that is easy to maintain a flow rate of zero.A buildup test is conducted by shutting in a producing well and measuring the resulting pressure response.

Buildup Test - Rate HistorySum after shut-in of 0.Rate after shut-in of -qRate during production of +q.

Buildup Pressure ResponsePressure normally declines during production...but rises during the injection (buildup) period...yielding a pressure curve that is the sum of the two rate curves:

Buildup Test - Superpositiony = mx + b

Buildup Straight-Line AnalogyHorner time ratio

Buildup Test Graph

Estimating Skin Factor From a Buildup Test

Horner Pseudoproducing Time

Semilog AnalysisFor Gas Wells

Instructional Objectives1.Identify range of validity of pressure, pressure-squared, and adjusted pressure analysis methods 2.Estimate pressure drop due to nonDarcy flow3.Analyze flow and buildup tests using semilog analysis

OutlineFlow Equations For Gas WellsPseudopressurePressure-SquaredPressureAdjusted PressureNon-Darcy FlowExample

Diffusivity Equation - LiquidsContinuity EquationEquation of State For Slightly Compressible LiquidsDarcys Law

Real Gas LawpV=znRT

Real Gas Pseudopressure

Gas Flow EquationReal Gas PseudopressureContinuity EquationReal Gas Law Equation of StateDarcys Law

Gas Flow Equation Pressure-SquaredContinuity EquationReal Gas Law Equation of StateDarcys LawThe term z Is Constant

Pressure-Squared Ranges 00.1602,0004,0006,0008,00010,000Pressure, psiamu*z, psi/cpTf = 200 FFairly constant at rates

Gas Flow Equation: PressureContinuity EquationReal Gas Law Equation of StateDarcys LawIf p/z is constant,

Pressure: Range Of ApplicationTf = 200FFairly constant at rates >3,000 psi

Gas - Dependent VariablesPressure-Squared - Valid Only For Low Pressures (< 2000 psi)Pressure - Valid Only For High Pressures (> 3000 psi)Real Gas Pseudopressure - Valid For All Pressure Ranges

Gas Flow Equation:Real Gas PseudopressureContinuity EquationReal Gas Law Equation of StateDarcys LawStrong Variation With Pressure

Real Gas Pseudotime

Adjusted Variables

Using Horner Time RatioWith Adjusted Time

Non-Darcy FlowFlow equations developed so far assume Darcy flowFor gas wells, velocity near wellbore is high enough that Darcys law failsNon-Darcy behavior can often be modeled as rate-dependent skin

Apparent Skin Factor

Estimating Non-Darcy CoefficientFrom Multiple Testss = 3.4D = 5.1x104D/Mscf

Estimating Non-Darcy CoefficientFrom Turbulence ParameterOften, only one test is availableIf so, we can estimate D from

Estimating Turbulence ParameterIf is not known, it can be estimated from

Wellbore Storage

ObjectivesDefine wellbore unloadingDefine afterflowCalculate wellbore storage (WBS) coefficient for wellbore filled with a singlephase fluidCalculate WBS coefficient for rising liquid level

Fluid-Filled Wellbore - UnloadingBottomhole Rate0Ei-function solution assumes constant reservoir rateMass balanceequation resolvesproblems

Fluid-Filled Wellbore - AfterflowBottomhole RateSurface RateBottomhole flow continues after shut-in

Rising Liquid LevelBottomhole Rate

Wellbore Storage

Wellbore Storage Definition

Type Curve Analysis

Objectives1.Identify wellbore storage and middle time regions on type curve.2.Identify pressure response for a well with high, zero, or negative skin.3.Calculate equivalent time.4.Calculate wellbore storage coefficient, permeability, and skin factor from type curve match.

Dimensionless Variables

Radial Flow With WBS And Skin

Gringarten Type CurveConstant rate productionVertical wellInfinite-acting homogeneous reservoirSingle-phase, slightly compressible liquidInfinitesimal skin factorConstant wellbore storage coefficient

Gringarten Type Curve

Gringarten Type CurveSimilarities of curves makematching difficult

Pressure Derivative

Derivative Type CurvetD/CD0.01100100,000tD/PDDifferences in curve shapes make matching easier

Pressure + Derivative Type CurvestD/CD0.01100100,000PDCombining curves gives each stem value two distinctive shapes

Pressure/Derivative Type CurvetD/CD0.01100100,000PDWBSTransitionRadial FlowEarly Time RegionMiddle Time Region

Pressure + Derivative Type CurvetD/CD0.01100100,000PD

Equivalent Time For PBU Tests

Equivalent Time For PBU Tests

Equivalent Time For PBU Tests

Equivalent Time For PBU Tests

Properties Of Equivalent Time

Adjusted Variables For Gas Wells

Field Data Plot

Overlay Field Data on Type Curve

Move Field Data Toward Horizontal

Move Field Data Toward MatchBegin to move toward unit slope line

Move Field Data Toward Stems

Move Field Data Toward StemsExtrapolate curve as necessaryp/pD kTeq/tD CDCalculate s from matching stem valueLets say s=7x109Assume Dp = 262Assume pD = 10Assume teq = 0.0546Assume tD/CD = 1

Use Reservoir, Well Properties q = 50B = 1.325m = 0.609h = 15f = 0.183ct = 1.76 x 10-5rw2 = 0.25CD = 1703

Calculate k From Pressure Match

Calculate CD From Time Match

Calculate s From CDe2s

Manual Log-Log Analysis

Instructional ObjectivesTo be able to manually estimate permeability and skin factor from the log-log diagnostic plot without using type curves

Estimating Permeability and Skin Factor from the Diagnostic Plot

Estimating Permeability and Skin Factor

Exampleq= 50 STB/Dpwf= 2095 psiah= 15 ftf= 18.3%B= 1.36 RB/STBct= 17.9 x 10-6 psi-1m= 0.563 cp rw= 0.25 ft

Estimate (tp)r, tr, and Dpr

Estimate Permeability

Estimate Skin Factor

Flow Regimes and the Diagnostic Plot

Objectives1.Identify early, middle, and late time regions on a diagnostic plot.2.Identify characteristic shapes of flow regimes on a diagnostic plot.3.List factors that affect pressure response in early time.4.List boundaries that affect pressure response in late time.

The Diagnostic PlotPressure change (Dp) Pressure derivative (Dp )

The Diagnostic PlotElapsed time (Dt ), hrsPressure change (Dp ) and derivative (Dp ), psiMiddle-timeregionLate-timeregionEarly-timeregionUnit-slope line(wellbore storage)

The Diagnostic PlotElapsed time (Dt ), hrsPressure change (Dp ) and derivative (Dp ), psiMiddle-timeregionLate-timeregionEarly-timeregionPartial penetration, phase redistribution, fracture conductivity Homogenous reservoir horizontal derivative(best estimate of k )

The Diagnostic PlotElapsed time (Dt ), hrsPressure change (Dp ) and derivative (Dp ), psiMiddle-timeregionLate-timeregionEarly-timeregionPartial penetration, phase redistribution, fracture conductivity Infinite-acting behavior Boundary effects

Flow RegimesCommon characteristic shapes of derivativeVolumetricRadialLinearBilinearSphericalDifferent flow patterns may appear at different times in a single testFlow regimes follow sequence within model

Volumetric Behavior

Volumetric BehaviorPseudosteady-State Flow

Volumetric BehaviorDerivative

Volumetric BehaviorElapsed time (Dt ), hrsPressure change (Dp ) and derivative (Dp ), psiPressure derivativePressure change during recharge or pseudosteadystate flow

Volumetric Behavior

Radial Flow

Radial Flow

Radial Flow

Radial FlowVertical Well

Radial FlowDerivative

Radial Flow

Radial Flow

Spherical Flow

Spherical Flow

Spherical FlowSmall part ofzone perforated

Spherical FlowCertain wireline testing tools

Spherical FlowSpherical Probe (RFT)

Spherical FlowDerivative

Spherical Flow

Spherical Flow

Linear Flow

Linear Flow

Linear Flow

Linear Flow

Linear Flow

Linear FlowDerivative

Linear FlowPressure change in undamaged fractured wellPressure change in fractured/damaged or horizontal well

Bilinear Flow

Bilinear FlowHydraulic Fracture

Bilinear FlowDerivative

Bilinear FlowPressure in fractured, undamaged wellPressure in fractured, damaged well

Diagnostic Plot

Estimating Average Reservoir Pressure

Estimating Reservoir PressureMiddle Time Region MethodsMatthews-Brons-Hazebroek MethodRamey-Cobb MethodLate Time Region MethodsModified Muskat MethodArps-Smith Method

Middle-Time Region MethodsBased on extrapolation and correction of MTR pressure trendAdvantageUse only pressure data in the middle-time regionDisadvantagesNeed accurate fluid property estimatesNeed to know drainage area shape, size, well location within drainage areaMay be somewhat computationally involved

Matthews-Brons-HazebroekProducing time prior to shut-in, tp = 482 hrPorosity, f = 0.15 Viscosity, m = 0.25 cpTotal compressibility, ct = 1.615 x 10-5Drainage area, A = 1500 x 3000 ft (a 2x1 reservoir)

Curves for Square Drainage Area

Curves for 2x1 Rectangle

Curves for 4x1 Rectangle

Matthews-Brons-Hazebroekp*=2689.4m=26.7Step 1: Plot pressure vs. Horner time ratioStep 2: Extrapolate slope m to find p*

Step 3: Calculate dimensionless producing timeMatthews-Brons-Hazebroek

Step 4: On appropriate MBH curve, find pMBHDMatthews-Brons-Hazebroek

Matthews-Brons-Hazebroek

Matthews-Brons-HazebroekPlot pws vs (tp+t)/t on semilog coordinatesExtrapolate to (tp+t)/t=1 to find p*Calculate the dimensionless producing time tpADUsing the appropriate MBH chart for the drainage area shape and well location, find pMBHDIf tp >> tpss, more accurate results may be obtained by using tpss in place of tp in calculating the Horner time ratio and tpAD

AdvantagesApplies to wide variety of drainage area shapes, well locationsUses only data in the middle-time regionCan be used with both short and long producing timesDisadvantagesRequires drainage area size, shape, well locationRequires accurate fluid property dataMatthews-Brons-Hazebroek

Reservoir Shapes

Reservoir Shapes

Reservoir Shapes

Reservoir ShapesDietz shape factor CA = 31.62Dietz shape factor CA = 31.6Dietz shape factor CA = 19.17Dietz shape factor CA = 21.9Dietz shape factor CA = 0.098Dietz shape factor CA = 27.1

Ramey-CobbStep 1: Plot pressure vs. Horner time ratioStep 2: Calculate dimensionless producing time

Ramey-CobbStep 3: Find the Dietz shape factor CA for the drainage area shape and well location

Ramey-Cobb

Ramey-CobbPlot pws vs (tp+t)/t on semilog coordinatesCalculate the dimensionless producing time tpADFind the Dietz shape factor CA for the drainage area shape and well locationCalculate HTRavgExtrapolate middle-time region on Horner plot to HTRavg

Ramey-CobbAdvantagesApplies to wide variety of drainage area shapes, well locationsUses only data in the middle time regionDisadvantagesRequires drainage area size, shape, well locationRequires accurate fluid property dataRequires producing time long enough to reach pseudosteady state

Late-Time Region MethodsBased on extrapolation of post-middle-time region pressure trend to infinite shut-in timeAdvantagesNo need for accurate fluid property estimatesNo need to know drainage area shape, size, well location within drainage areaTend to be very simpleDisadvantageRequire post-middle-time-region pressure transient data

Late-Time Region Data

Late-Time Region Data

Modified Muskat MethodShut-in pressure

Modified Muskat MethodStep 1: Assume a value for average pressure

Time, minutesModified Muskat Method

AdvantagesVery simple to applyDisadvantagesSomewhat subjective: Which data points should I try to straighten?More sensitive to estimates that are too low than to estimates that are too highNot easily automatedModified Muskat Method

RecommendationsDont try to straighten data until there has been a clear deviation from the middle-time regionOnce middle-time region has ended, try to straighten all dataExpect best reliability for wells reasonably centered in drainage areasModified Muskat Method

Arps-Smith Method

Arps-Smith MethodStep 1: Assume a value for average pressure, accepting theory based on empirical observation

Arps-Smith MethodStep 2: Plot dpws/dt vs pws on Cartesian scalePavg = 5575 psiStep 3: Fit a straight line through the data points

Arps-Smith MethodOptional: Estimate the productivity index in STB/D/psi from the slope b and the wellbore storage coefficient C

Arps-Smith MethodAdvantagesSimple to applyEasily automatedDisadvantagesRequires data in late-time region, after all boundaries have been feltRequires numerical differentiation of pressure with respect to time

HydraulicallyFractured Wells

Hydraulically Fractured WellsFlow RegimesDepth of InvestigationFracture DamageStraight Line AnalysisBilinear Flow AnalysisLinear Flow AnalysisSemilog AnalysisType Curve Analysis

Ideal Hydraulic FractureReservoir sand(permeability=kr ) Fracture halflength, LfHydraulic fracture(permeability =kf )Wellbore

Dimensionless Variables for Fractured Wells

Flow Regimes in FracturesFracture flowLinearBilinearFormation flowLinearEllipticalPseudoradial

Fracture Linear FlowTransient moves down fracture lengthTransient has not moved into reservoirTransient has not reached end of fracture

Fracture Linear FlowTimePressure(Too early for practical application)

Fracture Linear FlowTimePressure

Bilinear FlowPressure transient moves down fracture, into formation

Bilinear Flow

Bilinear Flow

Bilinear Flow(Time depends on dimensionless flow, fracture conductivity)

Bilinear Flow(Time depends on dimensionless flow, fracture conductivity)

Bilinear FlowData can yield fracture conductivity wkf if kf is known.

Bilinear FlowData cannot yield Lf, but may identify lower bound .

Formation Linear FlowTransient moves linearly into wellboreNegligible pressure drop down fractureFlow from beyond ends of fracture not yet significant

Formation Linear Flow

Elliptical Flow

Pseudoradial Flow

Pseudoradial Flow

Depth Of Investigation

Depth Of InvestigationFor linear flow, pseudosteady-state flow exists out to a distance b at a dimensionless time given byDepth of investigation for a linear system at time t

Depth of InvestigationDepth of investigation along minor axisDepth of investigation along major axisArea of investigation

Hydraulic Fracture With Choked Fracture DamagekLfLs

Choked Fracture Skin Factor

Hydraulic Fracture With Fracture Face DamagekwfkfksLf

Fracture Face Skin Factor

Bilinear Flow AnalysisProcedureIdentify the bilinear flow regime using the diagnostic plotGraph pwf vs. t1/4 or pws vs tBe1/4Find the slope mB and the intercept p0 of the best straight lineCalculate the fracture conductivity wkf from the slope and the fracture skin factor sf from the intercept

Bilinear Equivalent Time

Bilinear Flow AnalysisEquationsBuildupDrawdown

Bilinear Flow Analysisp0=2642.4 psim=63.8 psi/hr1/4pwf=2628.6 psips

Limitations of Bilinear Flow AnalysisApplicable only to wells with low-conductivity fractures (Cr < 100)Bilinear flow may be hidden by wellbore storageRequires independent estimate of kGives estimate of wkf and sfCannot be used to estimate Lf

Linear Flow AnalysisProcedureIdentify the linear flow regime using the diagnostic plotGraph pwf vs. t1/2 or pws vs tLe1/2Find the slope mL and the intercept p0 of the best straight lineCalculate the fracture half-length Lf from the slope and the fracture skin factor sf from the intercept

Linear Equivalent Time

Linear Flow AnalysisEquationsBuildupDrawdown

Linear Flow Analysispa0=2266.0 psim=211 psi/hr1/2pawf=1656.2 psips

Limitations of Linear Flow AnalysisApplicable only to wells with high-conductivity fractures (Cr > 100)Wellbore storage may hide linear flow periodLong transition period between end of linear flow (tLfD < 0.016) and beginning of pseudoradial flow (tLfD > 3)Requires independent estimate of kGives estimate of Lf and sfCannot be used to estimate wkf

Pseudoradial Flow AnalysisProcedureIdentify the pseudoradial flow regime using the diagnostic plotGraph pwf vs. log(t) or pws vs log(te)Find the slope m and the intercept p1hr of the best straight lineCalculate the formation permeability k from the slope and the total skin factor s from the interceptEstimate fracture half-length from total skin factor

Pseudoradial Flow AnalysisEquationsBuildupDrawdown

Pseudoradial Flow Analysisp1hr=2121 psim=120 psi/cycle

Apparent Wellbore Radius

Estimating Lf From Skin Factor1.Calculate rwa from rwa = rwe-s2.Estimate Lf from Lf = 2rwa3.Estimate fracture conductivity wkf4.Calculate FcD from FcD = wkf/kLf5.Find Lf/rwa from graph or equation6.Estimate Lf from Lf = (Lf/rwa)*rwa7.Repeat steps 4 through 6 until convergence (Warning: may not converge)

Limitations of Pseudoradial Flow AnalysisBoundaries of reservoir may be encountered before pseudoradial flow developsLong transition period between linear flow and pseudoradial flowPseudoradial flow cannot be achieved for practical test times in low permeability reservoirs with long fracturesGives estimate of k and stDoes not give direct estimate of Lf, wkf, or sf

Dimensionless Variables For Fractured Wells

Type-Curve Analysis: Fractured Wells, Unknown k1.Graph field data pressure change and pressure derivatives2.Match field data to type curve3.Find match point and matching stem4.Calculate Lf from time match point5.Calculate k from pressure match point6.Interpret matching stem value (wkf, sf, or C)

Interpreting Match Points,Unknown Permeability

Type Curve Analysis: Fractured Wells, Known k1.Graph field data pressure change and pressure derivatives2.Calculate pressure match point from k3. Match field data to type curve, using calculated pressure match point4.Find match point and matching stem5.Calculate Lf from time match point6.Interpret matching stem value (wkf, sf, or C)

Interpreting Match PointsKnown Permeability

Cinco Type Curve

Cinco Type Curve:Interpreting Cr Stem

Choked Fracture Type Curve

Choked Fracture Type Curve:Interpreting sf Stem

Barker-Ramey Type Curve

Barker-Ramey Type CurveInterpreting CLfD Stem

Limitations of Type Curve AnalysisType curves are usually based on solutions for drawdown - what about buildup tests?Shut-in timeEquivalent time (radial, linear, bilinear)Superposition type curvesType curves may ignore important behaviorVariable WBSBoundariesNon-Darcy flowNeed independent estimate of permeability for best results

Pressure Transient Analysis for Horizontal Wells

Horizontal Well AnalysisDescribes unconventional and complex reservoirs Defines effectiveness of completion technique optionsDistinguishes between poor reservoir and damaged wellbore Differentiates between completion success and in-situ reservoir quality

Complications in AnalysisThree-dimensional flow geometry, no radial symmetrySeveral flow regimes contribute data Significant wellbore storage effects, difficult interpretation Both vertical and horizontal dimensions affect flow geometry

Steps to Evaluating DataIdentify specific flow regimes in test dataApply proper analytical and graphical procedures Evaluate uniqueness and sensitivity of results to assumed properties

Step 1: Identify Flow RegimesFive major and distinct regimes possiblemay or may not even occur may or may not be obscured by wellbore storage effects, end effects, or transition effects

Step 2: Apply ProceduresEstimate important reservoir properties Determine parameter groups from equations Expect complex iterative processes requiring use of a computer

Step 3: Evaluate ResultsExpect nonunique resultsSimulate test to confirm that the analysis is consistent with test data Use simulator to determine whether other sets of formation properties will also lead to a fit of the data

Horizontal Well Flow RegimesFive possible flow regimes(1) early radial (2) hemiradial(3) early linear(4) late pseudoradial(5) late linearAny flow regime may be absent from a plot of test data because of geometry, wellbore storage or other factors.

Well and Reservoir Geometryh

Well and Reservoir Geometryhz0

Flow RegimesRadial

Flow RegimesHemiradial

Flow RegimesEarly Linear

Flow RegimesEarly LinearFlow effects not seen at ends of wellbore

Flow RegimesLate Pseudoradial

Flow RegimesLate Linear

Flow Regimes/Drawdown p 'Log (Dp) or Log (p)WellborestorageEarlyRadialFlowEarlyLinearFlowPseudoradialFlowLateLinearFlowLog (time)

Required Permeabilities

Flow

Regime

Result

of

Analysis

Permeabilities Required for Limit Calculations

Permeabilities Required to Calculate Skin

Early Radial

End - kz and ky

and kx/kz

Hemiradial

End - kz and ky

and kx/kz

Early Linear

Start - kz

End - ky

kx and kz

Late Pseudoradial

Start - ky

End - ky and kx

kx, ky and kz

Late Linear

Start - ky and kz

End - kx

kx and kz

Note: We can use

in our analysis. In some cases, for simplicity, we assume kx = ky = kh. This assumption may reduce analysis accuracy.

_903942730.unknown

_903942733.unknown

_914228399.unknown

_914228476.unknown

_903942731.unknown

_903942727.unknown

_903942728.unknown

_903942725.unknown

Determines kh and kz Determines properties useful in horizontal test design (using an analytical or finite-difference simulator)Identifies likely flow regimes Estimates required test duration Identifies probable ambiguitiesPretesting a Vertical Section

Required Distances

Flow

Regime

Result

of

Calculation

Distances Required for Limit Calculations

Distances Required to Calculate Skin

Early Radial

Lw

End - dz and Lw

Hemiradial

Lw

End - dz and Lw

Early Linear

Lw and h

Start - Dz

End - Lw

Lw and h

Late Pseudoradial

h

Start - Lw

End - dy, Lw, and dx

Lw, h and dz

Late Linear

b and h

Start - Dy, Lw, and Dz

End - dx

b, h and dz

Early Radial Flow RegimeMay be masked by wellbore storage effects

End of Early Radial Flow

Early Radial Flow Pressure

Early Radial Flow/DrawdownSemilog plotTime Dp 1000.147 33

Early Radial Flow/DrawdownSemilog plotTime Dp 1000.147 33

Skin in Early Radial Flow

Early Radial Flow Buildup PlotCorrect only if (tp+t) and t appear simultaneously or if tp >> t.Semilog plot Horner Time Ratio Dp 101,00047 33

Early Radial Flow Buildup PlotSemilog plotTime Dp 1000.147 33

Early Radial Flow Buildup PlotSemilog plotTime Dp 1000.147 33

Early Radial Flow/Buildup

Start of Hemiradial FlowBegins after closest vertical boundary (at distance dz from wellbore) affects dataand before farthest boundary (at Dz from wellbore) affects the data.

Start of Hemiradial FlowBegins after closest vertical boundary (at distance dz from wellbore) affects data and before furthest boundary (at Dz from wellbore) affects the data.

End of Hemiradial FlowEnds when furthest boundary (at distance Dz from wellbore) affects the data . . .

End of Hemiradial Flow. . . or when effects are felt at ends of wellbore, whichever comes first.

Hemiradial Flow/DrawdownSemilog plotTime Dp 1000.147 33

Hemiradial Flow/DrawdownSemilog plotTime Dp 1000.147 33

Hemiradial Flow/Drawdown

Early Linear Flow RegimeStart

Early Linear Flow RegimeEnd

Early Linear Flow/DrawdownDpTime1/211418Cartesian plot

Early Linear Flow/Drawdown

Early Linear Flow/DrawdownConvergence skinFlow converges from total cross-section of reservoir radially into small area of wellbore

Early Linear Flow/Buildup

Early Linear Flow/Buildup

Late Pseudoradial FlowStart

Late Pseudoradial FlowStart

Late Pseudoradial Flow

Late Pseudoradial Flow(whichever is reached first)

Pseudoradial Flow/Drawdown

Pseudoradial Flow/ Drawdown

Pseudoradial Flow/Buildup

Late Linear FlowLate Linear

Late Linear FlowLate Linear

Late Linear Flow

Late Linear Flow(whichever is reached last)

Late Linear FlowEnd

Late Linear/DrawdownEstimate kx

Late Linear FlowCalculate total skin, st, including partial penetration skin, sp

Late Linear FlowCalculate total skin, st, including partial penetration skin, sp

Late Linear FlowCalculate total skin, st, including partial penetration skin, sp

Late Linear Flow/BuildupPressure is plotted vs.

Late Linear Flow/Buildupor From the slope, miv we can calculate kx:

Late Linear Flow/Buildup

Late Linear Flow/BuildupCalculate total skin, st, from

and skin due to altered permeability, sa, from

Summary of Analysis ProceduresCalculate kxEarly linear flow regime data: from effective wellbore length, LwLate linear flow regime: from reservoir length, b, parallel to wellbore Effective wellbore length, Lw, can be calculated from data in the early linear flow regime if kx has been calculated.

Summary of Analysis ProceduresCalculate kxEarly linear flow regime data: from effective wellbore length, LwLate linear flow regime: from reservoir length, b, parallel to wellbore. Length of the boundary, b, parallel to wellbore can be calculated from data in late linear flow regime if kx is known.

Summary of Analysis ProceduresCalculate kxIf data such as Lw or b are unknown or if flow regimes are missing, analysis is iterative at best and will result in nonunique results.Calculate kz from data in early radial or hemiradial flow regimesCalculate ky from pseudoradial flow regime

Summary of Analysis ProceduresCalculate kxCalculate kz from data in early radial or hemiradial flow regimesCalculate ky from pseudoradial flow regime

We can assume kx = ky = kh and often simplify analysis, but validity is questionable.

Summary of Analysis ProceduresCalculate kxCalculate kz from data in early radial or hemiradial flow regimesCalculate ky from pseudoradial flow regimeCheck on expected durations of flow regimes using tentative results from the analysis to minimize ambiguity in results

Pressure Transient Analysis for Horizontal WellsUsing the Techniques

Drawdown Diagnostic Plotp 'Log (Dp) or Log (p)Log (time)Wellbore storageunit-slope lineRadial flow horizontal derivativeLinear flow halfslope line

Drawdown Diagnostic PlotLog (Dp) or Log (p)WellborestorageEarlyRadialflowEarlyLinearFlowPseudoradialFlowLateLinearFlowLog (time)Shapes may not appear in buildup tests(better chance if tp>>Dtmax)

Field Example: Well AHorizontal exploration wellVertical tectonic fracturePermeability probably results from fracture

Ld, ft

2,470

Lw, ft

-

rw, ft

0.25

(, %

5

h, ft

150

q, STB/D

104

Bo, RB/STB

1.40

(, cp

0.45

tp, hours

238

Well A: Diagnostic Plot110010t, hrLog (Dp or p )10,000100010010Radial flow?p '

Well A: Horner PlotTest time too short to detect lower boundary, linear flow, or anisotropyk = 0.011s = 2.9

Well A: Buildup History Match110010t, hrLog (Dp or p )10,000100010010Radial flowp 'k = 0.027s = 11.5k = 0.011s = 2.9(from Horner plot)

Field Example: Well BWell in west Texas carbonateExpected isotropic k caused by fracturing, dissolution

Ld, ft

2,000

Lw, ft

-

rw, ft

0.30

(, %

17

h, ft

75

q, STB/D

200

Bo, RB/STB

1.60

(, cp

1.80

tp, hours

1,320

Well B: Diagnostic Plot

Well B: Horner Plotk = 0.14tErf = 165 hrk = 0.15k = 0.14

Well B: Buildup History Matchk = 0.15k = 0.14k = 0.15

Well B: Tandem-Root Plotm = 39.6h = 75 ftNearest boundary = 29 ft

Field Example CHorizontal wellHigh-k sandstoneExtensive underlying aquifer

Ld, ft

1,400

Lw, ft

484

rw, ft

0.41

(, %

17

h, ft

54

q, STB/D

2,760

Bo, RB/STB

1.10

(, cp

4.88

tp, hours

36

Well C: Diagnostic Plot100010010.1Dp, psia or pt, hrNo apparent wellbore storageRadial, hemiradial,or elliptical flowDecline caused by underlying aquifer

Well C: Type-Curve Match100010010.1Dp, psia or pt, hr

Well C: Horner Plotk = 53k ~ 48(confirms validity of earlier findings of nowellbore storage)

Well C: Regression Match100010010.1Dp, psia or pt, hrGeometric average of horizontal, vertical k ~ 48

Horizontal Well Test ConfigurationMeasurements usually made above horizontal wellbore Tools may be too rigid to pass through curveConventional tools can be used in horizontal well tests

Horizontal Well Test Configuration

Horizontal Well Test Configuration

Factors That Affect Transient ResponseHorizontal permeability (normal and parallel to well trajectory)Vertical permeabilityDrilling damageCompletion damageProducing interval that may be effectively much less than drilled length Variations in standoff along length of well

Obstacles to InterpretationMultiple parameters frequently yield inconclusive test analysis resultsWellbore storage obscures effects of transient behaviorMiddle- and late-time response behavior may require several hours, days, or months to appear in transient data

Ensuring Interpretable DataEstimate horizontal and vertical k from tests in pilot hole before kicking off to horizontal borehole segment Estimate standoff from directional drilling survey Determine producing part of wellbore from production log flow surveyFlow wells in developed reservoirs long enough to equilibrate pressures along the wellbore and minimize crossflow

Effects of Errors in Input Data

Presentation OutlineIntroductionSources of Error in Input DataEffects of Error on Results of Welltest InterpretationExamplesSummary

Problem 1Well A estimates from PBU testPermeability, 10 mdSkin factor, 0Distance to boundary, 250 ftAnalysis assumed net pay 25 feet

Problem 2Seismic interpretation indicates boundary 300 ft from Well BPBU test interpretation indicates nearest boundary 900 ft awayWhat could have caused this much error in the distance estimate?

Sources of Input DataLog interpretationFluid propertiesReservoir and well properties

Data From Log InterpretationPorosityWater saturationNet pay thickness

Causes of Error in Log InterpretationFailure to calibrate the logging toolFailure to make necessary environmental correctionsFailure to calibrate the log-derived properties against core measurementsFailure to select appropriate cutoffs for net pay estimation

Error in Log Interpretation DataParameterDeviationWithoutWith correctioncorrectionPorosity15 % 5 % Water saturation 40 % 10 % Net pay50 % 15 %

Fluid Properties DataFormation volume factorCompressibilityViscosity

Error in Fluid Properties DataParameterDeviationBg from composition 1.1% to 5.8% Bg from composition 1.3 % to 7.3%(as much as 27% if impurities are ignored)cgNegligible at low pressure mg 2% to 4%, gg < 1 up to 20% low, gg > 1.5From Gas Properties Correlations

Error in Fluid Properties DataParameterDeviationBo, p > pb 10%Bo, p pb 5%co, p > pb Up to 50% low at high pressureBest near pbco, p pb 10%, p > 500 psi 20%, p < 500 psimo Order of magnitude onlyFrom Oil Properties Correlations

Other Input DataFlow rateWellbore radiusFormation compressibilityTotal compressibility

Error in Well and Reservoir DataParameterErrorFlow rateFailure to record rate before BU testInaccuracy in estimates, averagesWellbore radius Poor choice of measurementFormation compressibilityEstimation errorsTotal compressibilityVariations in fluid saturationsAbnormally pressured reservoirOil compressibilityFrom Measurement or Calculations

Total Compressibility

Effects of ErrorsVertical wellSingle-phase flowHomogeneous reservoirBoundaryNo-flow, linear constant pressure, closedTestDrawdown, buildup, injection, or fall-offDuration long enough to identify boundary

Errors in ViscosityIf input = 2 true Then:kcalc = 2 ktrueNothing else will be affected

Errors in PorosityIf input = 2 true, Then:scalc = strue+ 0.5ln(2)Lx calc = Lx true/sqrt(2)A calc = Atrue/2

Errors in Water SaturationCause errors in calculating total compressibility

Errors in CompressibilityIf ct input = 2 ct true Then:scalc = strue+ 0.5ln(2)Lx calc = Lx true/sqrt(2)A calc = Atrue/2

Errors in Net PayIf hinput = 2 htrueThen:kcalc = ktrue/2scalc = strue+ 0.5ln(2)Lx calc = Lx true/sqrt(2)A calc = Atrue/2

Errors in Flow RateIf qinput = 2 qtrue Then:kcalc = 2 ktruescalc = strue- 0.5ln(2)Lx calc = sqrt(2) Lx trueA calc = 2 Atrue

Errors in Formation Volume FactorIf Binput = 2 Btrue Then:kcalc = 2 ktruescalc = strue- 0.5ln(2)Lx calc = sqrt(2) Lx trueA calc = 2 Atrue

Errors in Wellbore RadiusIf rw input = 2 rw trueThen:scalc = strue+ ln(2)

Solution to Problem 1Well A estimatesPermeability, 10 mdSkin factor, 0Boundary, 250 ftAssumed net pay 25 ftNet pay50 ftPermeability, 5 mdSkin factor, 0.35Boundary, 177 ft

Solution To Problem 2Seismic interpretation indicates boundary 300 ft from Well BPBU test interpretation indicates nearest boundary 900 ft awayBoundary could be a factor of 3 too far away

SummaryPermeability is most affected by errors in viscosity, net pay, and flow rateDistances to boundaries and drainage area are most affected by errors in compressibilitySkin factor is not affected to a large degree by any input variable

Bounded Reservoir Behavior

CautionsRecognizing may be as important as analyzingMany reservoir models may produce similar pressure responsesInterpretation model must be consistent with geological and geophysical interpretations

CharacteristicsBoundaries control pressure response following middle-time regionEquivalent time functions apply rigorously only to situations where eitherProducing and shut-in times both lie within middle-time regionShut-in time is much less than producing timeBoundaries affect pressure responses of drawdown and buildup tests differently

Shapes of curvesDurations of flow regimes explain shape of drawdown pressure responses Shape of buildup derivative type curve depends on how the derivative is calculated and plottedShut-in timeEquivalent timeSuperposition time

Superposition in spaceRadial flow pattern

Superposition in spaceEqual distances from no-flow boundary

Superposition in spaceProducing wellImage wellImage well

Producing wellSuperposition in space

Superposition in space

Infinite-acting reservoir

Infinite-acting reservoirDrawdown Type Curve

Buildup ResponseDerivative with respect to shut-in timeInfinite-acting reservoirShape depends on duration of production time prior to shut-in Dimensionless shut-in time

Buildup ResponseDerivative with respect to equivalent timeInfinite-acting reservoir

Buildup ResponseDerivative taken with respect to equivalent time, plotted against shut-in time0.010.11101001E+031E+041E+051E+061E+071E+081E+09Dimensionless time functionDimensionless pressuretpD=105,106,107,108DrawdownInfinite-acting reservoirLargest time on plot is not limited to producing or shut-in time

Linear no-flow boundaryProducing well(If so, far away.)

Drawdown Type CurveLinear no-flow boundaryHemiradial flowChange in derivative from 0.5 to 1Change occurs over about 12/3 log cycles

Buildup ResponseDerivative with respect to shut-in timeLinear no-flow boundaryThe longer the equivalent time before shut-in, the longer the coincidence between buildup and drawdownDimensionless shut-in time

Buildup ResponseDerivative with respect to equivalent timeLinear no-flow boundaryDerivative doubles over only a tiny fraction of a log cycle for very short producing times prior to shut-in

Buildup ResponseDerivative with respect to equivalent time, plotted against shut-in timeLinear no-flow boundarySimilar to drawdown response

Linear constant-p boundaryProducing well

Drawdown Type CurveLinear constant-p boundary

Buildup ResponseDerivative with respect to shut-in timeDimensionless shut-in timeLinear constant-p boundarySlope steeper than drawdown slope for very short producing times before shut-in

Buildup Response Derivative with respect to equivalent timeLinear constant-p boundary

Buildup ResponseDerivative with respect to equivalent time shut-in timeLinear constant-p boundary

Channel reservoirProducing well(Effects of ends not felt )

Drawdown Type CurveChannel reservoir

Buildup ResponseDerivative with respect to shut-in timeDimensionless shut-in timeChannel reservoirDerivative reaches a slope of -1/2 if shut-in time is much larger than producing time

Buildup ResponseDerivative with respect to equivalent time, plotted against dimensionless timeChannel reservoir

Buildup ResponseDerivative with respect to equivalent time, plotted against shut-in timeChannel reservoirDerivative curve shape resembles drawdown curve shape

Intersecting sealing faultsWedge reservoirProducing well

Drawdown Type CurveIntersecting sealing faultsDerivative levels off at (360/) x (derivative of infinite-acting response)The narrower the angle, the longer to reach new horizontal

Buildup ResponseDerivative with respect to shut-in timeDimensionless shut-in timeIntersecting sealing faultsDramatic difference in curves when shut-in is greater than producing time prior to shut-in

Buildup ResponseDerivative with respect to equivalent timeIntersecting sealing faultsDerivative shape same as drawdown response only when producing period reaches fractional flow regime

Buildup ResponseDerivative with respedt to equivalent time, plotted against shut-in timeIntersecting sealing faultsDerivative, drawdown curves similar

Closed circular boundaryProducing well

Drawdown Type CurveClosed circular boundaryBoth slopes approach unit slope at late times(pseudosteady-state flow)Reservoir limits test yields pore volume of intervalUnit slope may be seenearlier if two zones withdifferent permeabilityare present

Buildup ResponseDerivative with respect to shut-in timeDimensionless shut-in timeClosed circular boundarytpD=106,107,108Derivative falls rapidly for all combinations of plotting functions

Buildup ResponseDerivative with respect to equivalent timeClosed circular boundarytpD=107,108Slope drops sharply for very small values of producing time before shut-in

Buildup ResponseDerivative with respect to equivalent time, plotted against shut-in timeClosed circular boundarytpD= 106, 107, 108Derivative, drawdown type curves differ fundamentally

Circular constant-p boundaryProducing wellPossibly strong aquifer supporting pressure equally from all directions

Drawdown Type CurveCircular constant-p boundary

Buildup ResponseDerivative with respect to shut-in timeDimensionless shut-in timeCircular constant-p boundaryCurve can be identical to drawdown plot just seen

Buildup ResponseDerivative with respect to equivalent timeCircular constant-p boundaryDerivative falls off rapidly

Buildup Response Derivative with respect to equivalent time, plotted against shut-in timeCircular constant-p boundaryResults in somewhat-changed curve on the plot

Radially composite reservoirProducing wellSignificant difference in permeability near, farther from wellk1k2

Drawdown Type CurveVarying M1/M2Radially composite reservoirResponses resemble other tests

Drawdown Type CurveVarying S1/S2Radially composite reservoirIf s1/s2> 1, plot looks like closed circular drainage areaIf S1/S2

Final commentsAssuming a well is in an arbitrary point in a closed, rectangular reservoir can lead to apparent fit of test for many different reservoirs

Arbitrary well position

CautionsMake sure the model is consistent with known geology before using the modelTwo most dangerous models (because they can fit so many tests inappropriately)Composite reservoirWell at arbitrary point in closed reservoir

Final commentsAssuming a well is in an arbitrary point in a closed, rectangular reservoir can lead to a poor fit of test for many different reservoirs

Buildup Testing and the Diagnostic Plot

ObjectivesBecome familiar with time plotting functions used with diagnostic plots for buildup testsBecome aware of the very different shapes in the diagnostic plots of buildup and drawdown tests as buildup tests approach stabilization

Time-Plotting FunctionsShut-in TimeHorner Pseudoproducing TimeMultirate Equivalent TimeSuperposition Time Function

Variable Rate Historytqq1q2qn-1qnt1t2tn-2ttn-1t0

Horner Pseudoproducing TimeCumulative produced oilFinal rate before shut-inExpressed another way...

Horner Pseudoproducing TimeCumulative produced oilFinal rate before shut-inGood results as long as last producing time is at least 10x maximum shut-in time.

Multirate Equivalent Time

Superposition Time FunctionPressure derivative for buildup calculated aspressure derivative with respect to superposition time function; plotted vs. shut-in timeSome literature recommends . . .

Superposition Time Function(previous equation, rearranged)

Superposition Time Function(previous equation, rearranged again using properties of natural logarithm)

Superposition Time Function

Superposition Time FunctionSuperposition time function is simply the log of a constant plus the log of the equivalent time.

Derivitive with respect to multirate equivalent time = derivitive with respect to superposition time

Superposition Time FunctionPressure derivative for buildup calculated aspressure derivative with respect to superposition time function; plotted vs. shut-in timeSome literature recommends . . . Pressure derivative for buildup calculated aspressure derivative with respect to equivalent time functionSome literature recommends . . .

Superposition Time FunctionSince the derivatives with respect to multirate equivalent time and superposition time are equal,

ConclusionsHorner pseudoproducing time is adequate when producing time is 10 times greater than the maximum shut-in time

ConclusionsDerivatives with respect to time for the superposition time function and radial equivalent time are identical. They can be plotted vs. shut-in time, superposition time, or equivalent time

ConclusionsSome literature or software documentation may specify the method of taking or plotting the derivative, but any of these will work for these situation.

Radial Flow

Approaching StabilizationStabilization is the stage where pressure has built up completely and is no longer changing.

Stabilization In Radial SystemProducing times must be at least 10x maximum shut-in time

Linear Flow

Stabilization in Linear System

Volumetric Behavior

Stabilization in Volumetric SystemAll boundaries have been feltDrawdown responsefeels boundary later than build-up response

ConclusionsShapes of the buildup and drawdown diagnostic plots are fundamentally different as the reservoir approaches stabilization.Dont expect to see the same shape on a diagnostic plot for a build up test as for a drawdown test.

Integrated Well Test Interpretation

Integrating Test Interpretation

Interpreting Integrated DataImportance of Model SelectionIntegrating Other DataGeological DataGeophysical DataPetrophysical DataEngineering DataValidating the Reservoir ModelCommon Errors and Misconceptions

Similar Model ResponsesWell in a WedgeComposite Reservoir

Multiple Knobs ConfuseMobility ratio M1/M2Storativity ratio S1/S2Distance to boundary RDistance to wall D1Well in a BoxWD2LD1Composite ReservoirM2,S2M1,S1Distance to wall D2Reservoir length LReservoir width W

Models Simplify GeologyInterpretation model must be consistent with (not identical to) geological modelHave we oversimplified the geology?

Responses Differ With Test TypeClosed Reservoir - DD TCConst Pres Boundary - DD TCClosed Reservoir - BU TCConst Pres Boundary - BU TC

Importance Of Model SelectionMost major errors caused by use of wrong model instead of wrong methodMeaningless estimatesMisleading estimatesTwo aspects of model selectionSelecting reservoir geometryIdentifying features of pressure response

Geology Offers InsightsDepositional environmentReservoir sizeShapeOrientationReservoir heterogeneityLayeringNatural fracturesDiagenesisTypes of boundariesFaultsSealingPartially sealingFluid contactsGas/oilOil/water

Geophysics and PetrophysicsStructureFaultsLocation SizeReservoir compartmentsShapeOrientationNet pay thicknessPorosityFluid saturationsFluid contactsLithologyLayeringEvidence of natural fractures

Engineering DataDrilling datadaily reportsProduction and flow test dataStimulation treatment resultsFracture design half-length, conductivityFracture treating pressure analysis resultsProblems during treatmentdaily reportsData from offset wellsPossible interferenceproduction records Well test results

Reality Checks Validate ModelWellbore storage coefficientSkin factorCore permeabilityPressure response during flow periodProductivity indexAverage reservoir pressureRadius of investigationDistances to boundariesIndependent estimates of model parameters

Wellbore Storage CoefficientWBS coefficient from test should be within order of magnitude of estimatePhase segregation can cause smaller WBSWBS coefficient >100x estimated value may indicate reservoir storage instead of WBS

Skin FactorLikely estimates by completion typeNatural completion0Acid treatment-1 to -3Fracture treatment-3 to -6Gravel pack+5 to +10Frac pack-2 to +2Local field experience may suggest more appropriate valuesSkin factor < -6 very unlikely

Core PermeabilityIn-situ permeability from well testCore permeability to airHighoverburden and saturationLownatural fracturesTotal kh from core adjusted to in-situ value less than kh from well testFracturesMissing coreMost useful when entire interval cored

Production Period PressureMust be consistent with shut-in pressure responseMust ensure consistencyInterpret flow periods independentlyPredict flow period pressures from results of buildupMatch flow and buildup periods simultaneously

Productivity IndexCorrect model should give consistent values

Average Reservoir PressureCompare average reservoir pressure from test interpretationMaterial balanceAnalytical simulationNumerical simulationResults should be similar if same reservoir model is used

Radius of InvestigationEstimate radius of investigationBeginning of middle-time regionEnd of middle-time regionUnrealistically large ri may indicate selected MTR is incorrectVery small ri may indicate wrong MTR or test not measuring reservoir characteristics

Distance to BoundariesReservoir sizeProduction dataGeological dataGeophysical dataDistances to boundariesGeological dataGeophysical dataGeoscience professionals should develop common interpretation model

Independent ParametersDual porosity from fracture width, spacingStorativity ratio Interporosity flow coefficient

Independent ParametersComposite reservoir parameters for waterflood-injection wellRadius of waterflooded zoneMobility ratio (k/)1/(k/)2Storativity ratio (ct)1/ (ct)2Dual porosity from fracture width, spacing

Independent ParametersComposite reservoir parameters for waterflood-injection wellDual porosity from fracture width, spacingFracture properties from treatment designFracture half-length lfFracture conductivity wkf

Common Errors/MisconceptionsMost-often-misused modelsWell between two sealing faultsWell in a radially composite reservoirWell in a rectangular reservoirCommon misconceptionsUnit-slope line indicates wellbore storagePeak in derivative indicates radial flowStrong aquifer acts as constant-pressure boundary

Well Between Two Sealing FaultsAngle between faultsDistance from well to 1st faultDistance from well to 2nd fault

Radially Composite ReservoirMobility ratio M1/M2Storativity ratio S1/S2Distance to boundary R

Rectangular ReservoirDistance to wall D1Distance to wall D2Reservoir length LReservoir width W

Unit-slope line always indicates wellbore storageUnit-slope line may be caused byPseudosteady-state flow (drawdown test only)Recharge of high-permeability zone (either drawdown or buildup test)

Peak in derivative implies radial flowPeak in derivative may be caused by a flow restriction for any flow regime

Strong aquifer acts as constant pressure boundaryMobility of water must be much higher than that of reservoir fluid to act as constant pressure boundaryMaybe, maybe not for oilNever for gas

A well test is conducted byChanging production rate at a wellMeasuring resulting pressure response at the same well or another well

A well test is conducted byChanging production rate at a wellMeasuring resulting pressure response at the same well or another well

ExplorationIs this zone economic?How large is this reservoir?Reservoir engineeringWhat is the average reservoir pressure?How do I describe this reservoir in order to estimate reserves?forecast future performance?optimize production?Production engineeringIs this well damaged?How effective was this stimulation treatment?Why is this well not performing as well as expected?Define reservoir limitsDistances to boundariesDrainage areaEstimate average drainage area pressureCharacterize reservoir PermeabilitySkin factorDual porosity or layered behaviorDiagnose productivity problemsPermeabilitySkin factorEvaluate stimulation treatment effectivenessSkin factorFracture conductivityFracture half-lengthA well test is conducted byChanging production rate at a wellMeasuring resulting pressure response at the same well or another well

A well test is conducted byChanging production rate at a wellMeasuring resulting pressure response at the same well or another well

A well test is conducted byChanging production rate at a wellMeasuring resulting pressure response at the same well or another well

A well test is conducted byChanging production rate at a wellMeasuring resulting pressure response at the same well or another well

A well test is conducted byChanging production rate at a wellMeasuring resulting pressure response at the same well or another well

A well test is conducted byChanging production rate at a wellMeasuring resulting pressure response at the same well or another well

A well test is conducted byChanging production rate at a wellMeasuring resulting pressure response at the same well or another well

A well test is conducted byChanging production rate at a wellMeasuring resulting pressure response at the same well or another well

A well test is conducted byChanging production rate at a wellMeasuring resulting pressure response at the same well or another well

Obtained by combiningContinuity equationEquation of state for slightly compressible liquidsFlow equation - Darcys law

The continuity equation is a restatement of the conservation of matter. That is, the rate of accumulation of fluid within a volume element is given by the rate at which the fluid flows into the volume minus the rate at which the fluid flows out of the volume.

NomenclatureA =Cross-sectional area open to flow, ft2m =Rate of accumulation of mass within the volume, lbm/secv =Fluid velocity, ft/sec =Density of fluid, lbm/ft3

NomenclatureA =Cross sectional area open to flow, cm2k =Permeability, darciesL =Length of flow path, cmp =Pressure, atmp =Pressure difference between upstream and downstream sides, atmq =Flow rate, cm3/secux =Flow velocity, cm/secx =Spatial coordinate, cm =Viscosity, cp

This equation describes the change in density with pressure for a liquid with small and constant compressibility.

Nomenclaturec =Compressibility, psi-1p =Pressure, psi =Density of fluid, lbm/ft3

The diffusivity equation is obtained by combining-The continuity equation-The equation of state for a slightly compressible liquid-Darcys lawOther transient flow equations may be obtained by combining different equations of state and different flow equations-Gas flow equation-Multiphase flow equationThe formation volume factor is the volume of fluid at reservoir conditions necessary to produce a unit volume of fluid at surface conditions.Symbol Bo, Bg, BwUnits res bbl/STB, res bbl/ MscfSource Lab measurements, correlationsRange and typical valuesOil1 2 res bbl/STB, Black oil2 4 res bbl/STB, Volatile oilWater1 1.1 res bbl/STBGas0.5 res bbl/Mscf, at 9000 psi5 res bbl/Mscf, at 680 psi30 res bbl/Mscf, at 115 psi

Compressibility is the fractional change in volume due to a unit change in pressure.Symbol co, cg, cwUnits psi-1, microsips (1 microsip = 1x10-6 psi-1)Source Lab measurements, correlationsTypical ValuesOil15x10-6 psi-1, undersaturated oil180x10-6 psi-1, saturated oilWater4x10-6 psi-1Gas1/p, Ideal gas60x10-6 psi-1, at 9000 psi1.5x10-3 psi-1, at 680 psi9x10-3 psi-1, at 115 psi

Porosity is the ratio of volume of pore space to bulk volume of rock.Symbol - Units Equations - fractionReports - % (or fraction)SourceLogs, coresRange or Typical Value30%, unconsolidated well-sorted sandstone20%, clean, well-sorted consolidated sandstone8%, low permeability reservoir rock0.5%, natural fracture porosityPermeability is the measure of capacity of rock to transmit fluid.Symbolk UnitsDarcy or millidarcy (md or mD)SourceWell tests, core analysisRange0.001 md - 10,000 md

Pore volume compressibility is the fractional change in porosity due to unit change in pressure.Symbol cfUnits psi-1, microsipsSource Lab measurement, correlation, guessRange or Typical Value4x10-6 psi-1, well-consolidated sandstone30x10-6 psi-1, unconsolidated sandstone4 to 50 x 10-6 psi-1 consolidated limestones

The net pay thickness is the total thickness of all productive layers in communication with the well.Net pay includes any rock that has sufficient vertical permeability to allow fluid to move to a layer from which it may be produced.Thickness is measured perpendicular to bed boundaries.Symbol h Units ftSource logsRange or Typical ValueMay be as small as 5 ft or even lessMay be as large as 1,000 ft or moreSaturation is the fraction of pore volume occupied by a particular fluid.Symbol So, Sw, SgUnits fraction or %Source logsRange or Typical Value15 to 25% connate water saturation in well-sorted, coarse sandstones40 to 60% connate water saturation in poorly sorted, fine-grained, shaly, low-permeability reservoir rock

Wellbore radius is the size of wellbore.SymbolrwUnitsfeetSourceBit diameter/2Caliper logRange or Typical Value2 to 8 in.

The total compressibility is the sum of pore compressibility and saturation weighted fluid compressibilities.Symbol ctUnitspsi-1, microsipsSourceCalculatedRange or Typical ValueSee exercises

Upon completion of this section, the student should be able to:1.Given formation and fluid properties, be able to calculate the radius of investigation achieved at a given time or the time necessary to reach a given radius of investigation.2.Describe, without looking at the equation, how each of the following parameters affects the time required to reach a given radius of investigation: permeability, compressibility, viscosity, porosity, net pay thickness, flow rate.

AssumptionsSingle-phase liquid with constant , c, BFormation with constant , hWell completed over entire sand thicknessInfinite reservoir containing only one wellUniform pressure in reservoir prior to productionConstant production rate q beginning at time t=0Homogeneous reservoir

The Ei-function solution to the diffusivity equation assumes line source well (finite size of wellbore can be neglected).This solution is valid only for r > rw .It predicts the pressure response in the reservoir as a function of both time t and distance from the center of the wellbore r.The argument of the Ei-function, x, is given by:

Short times or large distances large xLong times or short distances small x

For short times, x > 10, pressure response predicted by the Ei-function is negligible.For long times, x < 0.01, pressure response may be calculated using the logarithmic approximation to the Ei-function.For intermediate times, 0.01 < x < 10, the full Ei-function must be used to calculate the pressure response.

At any given point in the reservoir, at sufficiently early times, the pressure response is essentially negligible.This approximation applies wheneverAt any given point in the reservoir, at sufficiently late times, the pressure response is approximately logarithmic in time.This approximation applies whenever

.

Consider the pressure profile in an infinite-acting reservoir during drawdown. At t = 0 the pressure is uniform throughout the reservoir.At t = 0.01 hours only a small region within 10 ft of the wellbore has shown the effects of the transient.Sometime later, at t =1 hour, the pressure transient has moved into a larger region, perhaps 100 ft from the wellbore.Still later, at t =100 hours, the pressure transient has moved even further from the wellbore.As production continues, the pressure transient continues to move through the reservoir until it has reached all of the boundaries of the reservoir.Consider what happens when we shut in the well from the previous slide for a buildup.At the instant of shutin, t=0, the pressure begins to build up in the well. However, this rise in pressure does not affect the entire reservoir at once.At t = 0.01 hours, the pressure buildup has affected only that part of the reservoir within about 10 ft of the wellbore. A pressure gradient still exists in the bulk of the reservoir. This means that fluid continues to flow in most of the reservoir, even during buildup.At t =1 hour, the pressure has built up in a larger area, within about 100 ft of the wellbore.As the shutin period continues, the region within which the pressure has built up grows until the entire reservoir is at uniform pressure.AssumptionsRadial flowInfinite-acting reservoirHomogeneous reservoirEffect of reservoir properties Increasing porosity, viscosity, or total compressibility increases the time required to reach a given radius of investigation.Increasing permeability decreases the time required to reach a given radius of investigation.Changing the rate has no effect on the radius of investigation.

Upon completion of this section, the student should be able to:1.List and describe 5 factors that cause skin damage.2.List and describe 3 factors that cause a geometric skin factor.3.Given formation and fluid properties, be able to calculate skin factor from additional pressure drop due to damage and vice-versa.4.Calculate the flow efficiency given the skin factor, the wellbore pressure, and the average drainage area pressure.5.Express skin factor as an apparent wellbore radius and vice-versa, given the actual wellbore radius.6.Convert skin factor to an equivalent fracture half-length for an infinite-conductivity fracture and vice-versa.

Mud filtrate invasion reduces effective permeability near wellbore.Mud filtrate may cause formation clays to swell, causing damage.In an oil reservoir, pressure near well may be below bubblepoint, allowing free gas which reduces effective permeability to oil near wellbore. In a retrograde gas condensate reservoir, pressure near well may be below dewpoint, allowing an immobile condensate ring to build up, which reduces effective permeability to gas near wellbore.Injected water may not be clean - fines may plug formation.Injected water may not be compatible with formation water - may cause precipitates to form and plug formation.Injected water may not be compatible with clay minerals in formation; fresh water can destabilize some clays, causing movement of fines and plugging of formation.Consider an undisturbed formation of thickness h and permeability k.

Assume that something (drilling the well, producing fluid from the well, injecting fluid into the well) changes the permeability near the wellbore. One simple model of this effect is to assume that this altered zone has uniform permeability ka and radius ra, and that the rest of the reservoir is undisturbed.

For generality, we allow the permeability in the altered zone to be either smaller or larger than the permeability in the undisturbed formation.

We define the skin factor in terms of the additional pressure drop due to damage.As defined, the skin factor is dimensionless -- it has no units.Nomenclaturek=mdh=ftq=STB/DB =bbl/STBps =psi=cpThe skin factor equation may be rearranged to give the additional pressure drop caused by a given skin factor.The skin factor may be calculated from the properties of the altered zone.If ka < k (damage), skin is positive.If ka > k (stimulation), skin is negative.If ka = k, skin is 0.The equation on the previous slide can be rearranged to solve for the permeability in the altered zone. If we know the reservoir permeability and the skin factor and can estimate the depth of the altered zone, we can estimate the permeability of the altered zone using this equation.

If the permeability in the altered zone ka is much larger than the formation permeability k, then the wellbore will act like a well having an apparent wellbore radius rwa.The apparent wellbore radius may be calculated from the actual wellbore radius and the skin factor.The minimum skin factor possible (most negative skin factor) would occur when the apparent wellbore radius rwa is equal to the drainage radius re of the well.

For a circular drainage area of 40 acres (re = 745 feet) and a wellbore radius of 0.5 feet, this gives a minimum skin factor (maximum stimulation) of -7.3.

When a cased wellbore is perforated, the fluid must converge to one of the perforations to enter the wellbore. If the shot spacing is too large, this converging flow results in a positive apparent skin factor. This effect increases as the vertical permeability decreases, and decreases as the shot density increases.

When a well is completed through only a portion of the net pay interval, the fluid must converge to flow through a smaller completed interval. This converging flow also results in a positive apparent skin factor. This effect increases as the vertical permeability decreases and decreases as the perforated interval as a fraction of the total interval increases.

When a well penetrates the formation at an angle other than 90 degrees, there is more surface area in contact with the formation. This results in a negative apparent skin factor. This effect decreases as the vertical permeability decreases, and increases as the angle from the vertical increases.

Often to improve productivity in low-permeability formations, or to penetrate near-wellbore damage or for sand control in higher permeability formations, a well may be hydraulically fractured. This creates a high-conductivity path between the wellbore and the reservoir. If the fracture conductivity is high enough relative to the formation permeability and the length of the fracture, there will be virtually no pressure drop down the fracture. This distributes the pressure drop due to influx into the wellbore over a much larger area, resulting in a negative skin factor.

For a well that has been perforated, there is an additional pressure drop across a zone surrounding the perforations. This pressure drop may be calculated using the radial flow form of Darcys law.NOTE: This expression does not include the effects of non-Darcy flow, which may be extremely important, especially in high-rate gas wells!Nomenclature:sp- geometric skin due to converging flow to perforationssd - damage skin due to drilling fluid invasionsdp - perforation damage skinkd - permeability of damaged zone around wellbore, mdkdp - permeability of damaged zone around perforation tunnels, mdkR - reservoir permeability, mdLp - length of perforation tunnel, ftn - number of perforationsh - formation thickness, ftrd - radius of damaged zone around wellbore, ftrdp - radius of damaged zone around perforation tunnel, ftrp - radius of perforation tunnel, ftrw - wellbore radius, ftWhen a well is gravel packed, there is a pressure drop through the gravel pack within the perforation.

NOTE: This expression does not include the effects of non-Darcy flow, which may be extremely important, especially in high-rate gas wells!

Nomenclaturesgp - skin factor due to Darcy flow through gravel packh - net pay thicknesskgp - permeability of gravel pack gravel, mdkR - reservoir permeability, mdLg - length of flow path through gravel pack, ftn - number of perforations openrp - radius of perforation tunnel, ftThe productivity index is often used to predict how changes in average pressure or flowing bottomhole pressure pwf will affect the flow rate q.The productivity index is affected byReservoir quality (permeability)Skin factor

We can express the degree of damage on stimulation with the flow efficiency.For a well with neither damage nor stimulation, Ef = 1.For a damaged well, Ef < 1For a stimulated well, Ef > 1

We can use the flow efficiency to calculate the effects of changes in skin factor on the production rate corresponding to a given pressure drawdown.

qnew =Flow rate after change in skin factorqold =Flow rate before change in skin factorEfnew =Flow efficiency after change in skin factorEfold =Flow efficiency before change in skin factor

Upon completion of this section, the student should be able to:1.Analyze a constant-rate drawdown test using semilog analysis.a.Identify the data that correspond to the middle time region on the diagnostic plot.b.Calculate permeability and skin factor from a semilog graph.2.Analyze a buildup test following a constant-rate flow period using the Horner method.a.Calculate the Horner pseudo-producing time for variable rate production.b.List the conditions that must be satisfied for the Horner pseudo-producing time to be applicable without referring to the text. c.Identify the data that correspond to the middle time region on the diagnostic plot.d.Calculate permeability, skin factor, and initial pressure from a Horner graph for a buildup test in a well in an infinite-acting reservoir.Note that the skin factor affects the pressure response only within the altered zone. The pressure profile at points beyond the radius of the altered zone is not affected by the skin factor.We have already seen that the additional pressure drop due to skin at the wellbore can be calculated from the flow rate and fluid and rock properties.We can modify the Ei-function solution to apply for 2 cases: 1) at the wellbore, and (2) beyond the altered zone.Neither of these expressions is valid within the altered zone.Neither of these expressions is valid until after the logarithmic approximation to the Ei-function becomes applicable throughout the altered zone.When < 0.01, we may use the logarithmic approximation to the Ei-function.

This expression may be written in the same form as the equation of a straight line.A graph of pwf vs. log10(t) should fall on a straight line.Slope m allows us to estimate permeability.Intercept b (which is usually referred to as p1hr), allows us to estimate skin factor s.

Slope m

(t1, pwf1), (t2, pwf2) are any two points on the straight line portion of the graph.Normally, t1 and t2 are chosen to be powers of 10.For best accuracy, pick points several log cycles apart.The point p1hr is the pressure on the best straight line through the data at a time of 1 hr. It may be necessary to extrapolate the straight line to a time of 1 hr to read p1hr.In a real test, all, some, or none of the data points may fall on a straight line of the correct slope.This correct semilog straight line corresponds to the data identified as the middle-time region on the diagnostic plot.

The data summarized below were recorded during a pressure drawdown test from an oil well. Estimate the effective permeability to oil and the skin factor using the graphical analysis technique for a constant-rate flow test. q= 250 STB/Dpi= 4,412 psiah= 46 ft= 12%rw= 0.365 ftB= 1.136 RB/STBct= 17 x 10-6 psi-1m= 0.8 cp

The data summarized below were recorded during a pressure drawdown test from an oil well. Estimate the effective permeability to oil and the skin factor using the graphical analysis technique for a constant-rate flow test. q= 250 STB/Dpi= 4,412 psiah= 46 ft= 12%rw= 0.365 ftB= 1.136 RB/STBct= 17 x 10-6 psi-1m= 0.8 cp

The data summarized below were recorded during a pressure drawdown test from an oil well. Estimate the effective permeability to oil and the skin factor using the graphical analysis technique for a constant-rate flow test. q= 250 STB/Dpi= 4,412 psiah= 46 ft= 12%rw= 0.365 ftB= 1.136 RB/STBct= 17 x 10-6 psi-1m= 0.8 cp

Consider the rate history for an idealized buildup test. A well is produced at rate q for a time tp, then is shut in for a buildup test. The rate history can be represented as the algebraic sum of two different constant rate flow periods one at rate q, beginning at t = 0, and another at rate -q, beginning at t = 0.

The pressure response for the rate history shown on the previous slide can also be obtained by adding the pressure responses from each of the two rate flow histories.

The second term on the RHS of this equation gives the pressure change due to production at constant rate q beginning at t = 0.

The third term on the RHS of this equation gives the pressure change due to injection at constant rate q beginning at t = tp, or t = 0.

This equation can be simplified by canceling terms within the square brackets, as shown on the next slide.The quantity is called the Horner time ratio.

A graph of pws vs. should fall on a straight line

Slope m of the resulting straight line allows us to estimate permeability

Intercept b at = 0 or = 1 gives us the initial pressure pi.

The slope m is obtained from

( , pws1) and ( , pws2) are any two points on the straight line portion of the graph.Normally and are chosen to be powers of 10.For best accuracy, pick points several log cycles apart (Why?)In a real test, all, some, or none of the data may fall on a straight line of the correct slope. This correct semilog straight line corresponds to the data identified as the middle-time region on the diagnostic plot.Note that the HTR on the x-axis decreases from left to right, so that time increases from left to right. You may also see Horner plots drawn with HTR increasing from left to right. When this is the case, be aware that time increases from right to left.

To estimate the skin factor from a buildup test, we have to know the flowing bottomhole pressure at the instant of shutin, pwf.

The point p1hr is the pressure on the best straight line through the data at a time of 1 hour. It may be necessary to extrapolate the straight line to a Horner time ratio corresponding to a time of 1 hour in order to read p1hr.

Wells are almost never produced at exactly constant rate prior to shut-in.Variations in rate prior to shut-in can be accounted for in many cases by the use of the Horner pseudoproducing time approximation. When this approximation is used, the buildup analysis is performed by treating the well as if it had produced at rate qlast for a time tp.This approximation applies when the well produced at rate qlast for a period at least 10x as long as the duration of the shut-in period.If the last rate qlast is lower than the average production rate, the Horner pseudoproducing time will be longer than the actual elapsed production time.The Horner pseudoproducing time preserves material balance. That is, a well producing at a constant rate qlast for a time tp will produce exactly the same amount of fluid as the actual variable rate history.Upon completion of this section, the student should be able to:1.Identify the range of validity of each of the following analysis variables: pressure, pressure-squared, pseudopressure, and adjusted pressure.2.Estimate pressure drop due to non-Darcy flow.3.Analyze flow and buildup tests for gas wells using semilog analysis and any of the following analysis variables: pressure, pressure-squared, pseudopressure, and adjusted pressure.The PDE describing flow of a slightly compressible liquid of constant viscosity in a homogeneous porous medium is the diffusivity equation. This equation is derived from 3 principles - the cont