050919 TBlasingame SPE DL Pres (1 Hour Version)

8/6/2019 050919 TBlasingame SPE DL Pres (1 Hour Version)

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Slide 1/53T.A. Blasingame Performance-Based Reservoir Characterization State-of-the-Technology

SPE DISTINGUISHED LECTURER SERIES

is funded principallythrough a grant of the

SPE FOUNDATIONThe Society gratefully acknowledges

those companies that support the programby allowing their professionalsto participate as Lecturers.

And special thanks to The American Institute of Mining, Metallurgical,and Petroleum Engineers (AIME) for their contribution to the program.

Society of Petroleum EngineersDistinguished Lecturer 2005-06 Lecture Season


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Society of Petroleum EngineersDistinguished Lecturer 2005-06 Lecture Season

Performance-Based Reservoir

Characterization

State-of-the-Technology(with all annotation)

Thomas A. Blasingame, Texas A&M U.

Department of Petroleum EngineeringTexas A&M University

College Station, TX 77843-3116+1.979.845.2292 [email protected]


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Prelude: Performance-Based Reservoir CharacterizationQ. What is "Performance-Based Reservoir Characterization?"A. In the context of this lecture, "Performance-Based Reservoir Charac-

terization" is the diagnosis, analysis, and interpretation of production

datataken from an individual well

and the integration of the resultsof this analysis with other forms of data (e.g., petrophysical data, re-sults of well test analysis, geological data, etc.).

Discussion: PreludeProduction analysis(PA) versus pressure transient analysis(PTA)? (same)Weakness of PA? (data quality/quantity)

Strength of PA? (continuous monitoring, PTA becomes a subset of PA)

Diagnosis/Interpretation:

Generally pretty good,mostly an issue of pro-duction data quality.

Analysis:

Modern tools for pro-duction data analysis areexcellent analytical,semi-analytical, andnumerical modelling areessentially without flaw.

Integration:

Evolving major issuesare data quality and thescale(s) of the results.


4/57


Guidelines: Performance-Based Reservoir CharacterizationsREVIEWproduction data for consistency (allocations, accuracy, etc.).REVIEWwell history, particularly recompletions/stimulations.

GATHER/CORRELATEavailable petrophysical data (core, logs, etc.).PERFORMsimplified analysis of production data (Arps, EUR, etc.).

REVIEWmeasured rate/pressures (quality check).PERFORMmodel-based analysis of production (and well test) data.

INTEGRATEresults at different scales to establish correlation(s).

Prelude: Guidelines for Production Analysis


5/57


Darcy's Law:

Oil Material Balance Eq.: (p>pb)

Oil Diffusivity Eq.: (p>pb)

Oil Pseudosteady-State Flow Eq.: (p>pb)

Primer: Reservoir Engineering

Discussion: Primer Reservoir EngineeringDiffusivity equation? (mass continuity (balance) + Darcy's Law)Oil pseudosteady-state equation? (material balance + diffusivity equation)

Implications? (origin of virtually all reservoir engineering relations)

Q. What do I need to know for reservoir engineering ... ?A. All you need is ... Darcy's law and material balance.

poi

o

ti N

B

B

Ncpp

1

o

wA

oowf qs

r

ACekh

Bpp2

14ln21141.2

r

prB

kh

.q

2141

1)(

t

p

k

c

r

p

rr

p t

12

2


6/57


Orientation: PA TheoryQ. What is the theory to represent boundary-dominated flow behavior?A. Combine material balance and pseudosteady-state flow equations.

Discussion: Orientation PA Theory "Production Analysis Relation" implies? (plot: log[(pi-pwf)/qo] vs. log[Np/qo])Use of "log-log" plot(i.e., log[(pi-pwf)/qo] vs. log[Np/qo])? (diagnosis/analysis)

Does material balance time work? (yes

very robust formulation)

The quantity Np/qo (or

Gp/qg) is known as "ma-terial balance time" isrigorous for boundary-dominated flow, and is avery good approxima-tion for transient flow.

Oil Material Balance Eq. (MBE): (p>pb)

Oil Pseudosteady-State Flow Eq. (PFE): (p>pb)

Production Analysis Relation:

oi

o

tpss,oppss,oi

B

B

NcmNmpp

1where

sr

A

Cekh

B

bqbppwA

oopss,oopss,owf 2

14

ln2

1

141.2where

o

p

pss,opss,oo

wfi

q

N

mbq

pp

)(

(Oil analytical form)

g

ppss,gpss,g

g

wfi

q

Gmb

q

pmpm

)()(

(Gas approximate form)

dpz

pp

ppm

gbase2

1)(


7/57Slide 7/53T.A. Blasingame Performance-Based Reservoir Characterization State-of-the-Technology

Orientation: Production Analysis(PA)

Discussion: Orientation Production Analysis(PA)What are the major issues? (pressure and rate data quality/quantity)What are the benefits? (est. reservoir properties/volume, rate prediction)

Where are the tools? (commercial vendor products)

Production Analysis(PA) isanalogous to raising children it is problematic everycase is unique(and poten-

tially very frustrating). BUT,the PA experience is alsopotentially very rewarding ...you always learn somethingin the process.



Orientation: Production Analysis(PA) DataQ. What is Production Analysis(PA)?A. Combined analysis of rate and flowing bottomhole pressure data.

Discussion: Orientation Production Analysis(PA) DataPA is a passive technology can it yield high resolution results? (yes)What is/are the key data issue(s)? (qand pwf data accurate andcorrelated)

Erratic q and pwf data

how to analyze? (boundary-dominated flow theory)

Rate Data:Flowrates are measuredon a per-well basis formost gas wells oilflowrates are often al-located(this is a majorissue).

Pressure Data:

Measured bottomholepressure data are es-sentially non-existentsurface pressure dataare often available forgas wells, flowing(sur-face) pressure data for

oil wells are rare atbest.



Orientation: PA Data Analysis Example(Gas)Q. Does material balance time work? (prove with an example)A. Typical low productivity gas well example (mid-Continent US).

Discussion: Orientation PA Data Analysis Example(Gas)What are the limitations of this approach? (poor rate data, poor sampling)What is the "best" data frequency? (minimum preferred frequency = daily)

What is the effect of pressure? (pressure data critical

but in practice ...)

Transient Behavior:

Appears to reflect be-havior of a vertical wellwith a finite-conducti-vity vertical fracture.

BDF/PSSBehavior:Clear indication of(approximate) materialbalance behavior.

Material balance behav-

ior is independent ofreservoir shape!

(1))(

g

ppss,g

bdfg q

Gm

q

pm

(1/4))(

g

ppss,g

transg q

Gm

q

pm



Orientation: PA ObjectivesQ. What are the objectives of Production Analysis(PA)?A. Estimate reservoir properties and volume, and predict performance.

Discussion: Orientation PA Objectives In THEORY any differences in PA and PTA? (none, could use same tools) In PRACTICE differences in PA and PTA? (PTA data is higher resolution)

In FUTURE

what will happen with PA and PTA? (applications will merge)

Transient Radial Behavior: (PTA/PA) Pseudosteady-State Behavior: (PA)

r(t)

)ln(

)ln()(

cr,trns

cr,trnscr,trnswfi

mptd

dp

dt

dt'p

tmbppp

tmptd

dp

dt

dt'p

tmbppp

cr,pss

cr,psscr,psswfi

)ln(

)(

Common Characteristics:

Constant rate(q).Constant reservoir properties.Constant fluid properties.

Differences:

Volume ONLY from pseudo-

steady-state data.Reservoir properties ONLY

from transient flow data.

re

0

erdr

dp

(closed outerboundary)

(infinite-actingradial flow behavior)

...),(...)( s,kfb,kfm cr,trnscr,trns ...),(...)( s,kfb,Nfm cr,psscr,pss



Orientation: PA Data RequirementsQ. What are the data requirements for Production Analysis(PA)?A. Rate and pressure data, reservoir and fluid properties, well history.

Discussion: Orientation PA Data Requirements Issues with pressure? (measured infrequently at surface) Issues with rate? (gas usually good, oil problematic, water poor)

Issues with well completion? (review history, keep completion simple)

Required Data:

Good pressure data!Consistent rate!Well history.Fluid properties.

Issues:

Effect of well comple-tion(see example).

Conversion of ptfpwf (will remain amajor issue, considerpwf measurement).

Always review well

history prior to per-forming PA.



History: Production Analysis(PA)

Discussion: History Production Analysis(PA)PA up to 1970? (estimate reserves, maximum rate, and future rates)PA 1970-1990? (estimate reservoir properties simple models [pwf=con])PA 1990-2000? (estimate reservoir properties general qand pwf profiles)

PA 2000-? (interactive diagnosis, analysis-by-modelling, and forecasting)

Past:

Estimate reserves.Simple rate forecasts.

Simple estimates ofreservoir properties.Present:

Full variable-rate/vari-able pressure capabil-ity analysis andmodelling(rigorous).

Robust analysismethods and verygood software.

Future:

Measured pwf data.Larger data volumes.Will still need tools for

poor(i.e., field) qual-ity data.



This relationship is rigor-ous for low pressure gasreservoirs(it can be deriv-ed from the p2-form of thegas diffusivity equation(n=1) for laminar flow).

Q. What is "deliverability" analysis, and why did it evolve?A. Analysis of rate-pressure data, evolved toestimate maximum rate.

History: Production Analysis(PA) q(pwf)

Discussion: History Production Analysis(PA) q(pwf)Purpose of a q(pwf) relation? (maximum rate, inflow performance)Typical applications? ("deliverability" tests, darcy/non-darcy flow)

Limitations? (rate a function of pressure (not time), laminar flow)

From:Rawlins,E.L.andM.A.

Schellhardt:Backpressure

DataonNaturalGasWellsand

TheirApplicationTo

ProductionPractices,Monograph7,U.S.BureauofMines,

Washington,DC,(1936).

Well Deliverability:

The original purposeof well performanceanalysis was to quan-tify well deliverability.

The empirically de-rived"deliverability"

equation is given by:n

wfgppCq )(22

a. Wellbore Diagram: Ancient,but still accurate.

b. "Deliverability" Plot: (pav2-pwf

2)versus qg (log-log format) used to estimate qg(pwf=0) (anindicator of well performance

and recovery).



Q. What is "rate-time" analysis, and what was its original purpose?A. Analysis of rate-time data(graphically), originally used for taxation.

Discussion: History Production Analysis(PA) q(t) (Case 1)Theory for q(t)=qiexp(-Dit)? (... log(qo) vs. t straight-line data confirms model)Theory for q(t)=qi/[(1+bDit)

(1/b)]? (... other plots would be required to confirm)

Theory for "Averaged and Extrapolated" trend given by Cutler? (unknown)

Concept:

Create an extrapolationof the rate-time profile toyield "estimated ultimaterecovery"(EUR) (i.e., toprovide a reserves esti-mate).

Limitation(s):

Early rate-time modelshad no direct basis intheory but ironically,these models have beenproven in modern times,and are regularly usedas standards for re-serves estimation(i.e.,

the exponential andhyperbolic rate-timerelations).

History: Production Analysis(PA) q(t) (Case 1)

From:Cutler,W.W.,Jr.:EstimationofUndergroundOil

Res

ervesbyOil-WellProductionCurves,Departmentoft

he

Inte

rior,U.S.BureauOfMines

Bulletin228(1924).



Q. "Non-exponential" q(t) behavior is it real?A. See calibrated analysis using exponential and hyperbolic models.

History: Production Analysis(PA) q(t) (Case 2)

From:Cutler,W.W.,Jr.:EstimationofUndergroundOil

R

eservesbyOil-WellProduc

tionCurves,Departmento

fthe

In

terior,U.S.BureauOfMine

sBulletin228(1924).

Discussion: History Production Analysis(PA) q(t) (Case 2)q(t) behavior? ("higher" resolution than qo(Np))qo(Np) behavior? ("confirms" q(t) behavior also extrapolate Np(qo0)) Issue1?(exponential and hyperbolic relations show early linear qo vs. log(t))

Issue2? (in concept apply to single well; in practice field/pool data)

a. "Rate-Time" Plot: qo versus t(or qg versus

t) hyperbolic behavior dominant, notethat "early" exponential is required.

b. "Rate-Cumulative" Plot: qo versus Np (or qg

versus Gp) apparent hyperbolic behavior,analysis calibrated with q(t) data.



Oil Material Balance Eq. (MBE): (p>pb)

Oil Pseudosteady-State Flow Eq. (PFE): (p>pb)

Steps:1. Differentiate oil MBEand oil PFEwith respect to time.2. Assume: pwf = constant [i.e., d(pwf)/dt= 0].3. Equate results 1st order o.d.e.4. Separate/integrate.5. Exponentiate result final form:

Q. Can the "exponential" rate-time relation(q(t)=qiexp(-Dit)) be derived?A. Yes, see steps below slightly compressible liquid, pwf=constant.

Discussion: History Production Analysis(PA) q(t)=qiexp(-Dit) Is the q(t)=qiexp(-Dit) relation rigorous? (oil (p>pb) yes, gas ?)Other formulations? (rate-cumulative: q(t)=qi DiNp)

Applications? (estimate reserves (Np,max = Np(q

0)), q(t) predicton)

The q(t)=qiexp(-Dit) form

is correct for boundary-dominated flow behavior slightly compressibleliquid, pwf=constant.oi

o

tpss,oppss,oi

B

B

NcmNmpp

1where

sr

A

Cekh

B

bqbppwA

oo

pss,oopss,owf 2

14

ln2

1

141.2where

History: Production Analysis(PA) q(t)=qiexp(-Dit)

pss,o

pss,oiii

bmDtDqq )exp(



Q. What is an Estimated Ultimate Recovery(or EUR) plot?A. Plot q(t) vs. Np, extrapolate to zero rate using a straight line ((EUR)exp).

Discussion: History Production Analysis(PA) (EUR)exp Is the estimated ultimate recovery(EUR)exp supported by theory? (yes) Is the(EUR)exp conservative or liberal? ((EUR)exp is always conservative)

Other issues? (use of other EURmodels (e.g., hyperbolic) requires caution)

History: Production Analysis(PA) (EUR)exp

Fro

m:Fetkovich,M.J.:"Declin

eCurveAnalysisUsingType

Cur

ves,"

JPT(June1980)1065

-1077.

Theory: (exp model)

The q(t)=qiexp(-Dit)model(i.e., slightly com-pressible liquid, pwf=constant) is integratedto yield:

Application: (EUR)expPlot q(t) versus Np, ex-trapolate trend using astraight-line model toNp(q=0) this givesestimated ultimate re-

covery(EUR)exp.

pii NDqq



Q. Origin and purpose of the EUR versus qo,1yr correlation?A. Potential value as a correlation, but must quantify theory(N, k, s, etc).

Discussion: History Production Analysis(PA) EURvs. qo,1yr Origin? (1919 production data correlation (>85 years old)!) Theory? (Constant pwf (liquid) boundary-dominated flow conditions)

Rationale? (Correlate reserves versus production (or reservoir properties))

"Ancient" Technique:

The proposed correla-tion of EURvs. qo,1yr wasused to estimate oil re-serves from initial pro-duction performancedata.

Modern Application:

Approach is based intheory EUR = f[k, s, xf,... and contactedfluidsin-place(i.e., N or G)].

Could be used as a "re-servoir characterization"tool to classify well per-

formance.

History: Production Analysis(PA) EURvs. qo,1yr

Fro

m:ManualforTheOilandGasIndustryUnderThe

Rev

enueActof1918,Treasury

Department

UnitedStates

Inte

rnalRevenueService(1919).

Hi P d i A l i (PA) A



Arps' observations:b=0 Reservoir is highly undersaturated (p>pb).b=0 Gravity drainage and no free surface.b=0.5 Gravity drainage with free surface.

b=0.667

Soln. gas-drive reservoir ( vs. Np

linear).b=0.333 Soln. gas-drive reservoir ( vs. Np linear).

Q. Theory for Arps' relations?A. Arps derived the exponential and hyperbolic relations from loss ratio.

Discussion: History Production Analysis(PA) Arps "Theory" for the Arps' relations? (loss ratio (exp) and its derivative (hyp))Validity of the Arps' observations? (only qualitative (except for p>pb case))

Graphical analysis using the hyperbolic relation? (only using Fetkovich TC)

History: Production Analysis(PA) Arps

Fro

m:Arps,J.J:"AnalysisofDeclineCurves,"

Trans.,AIME

(1945)160,228-247.

Case Rate Relation Cumulative Relation

)exp( tDqq ii

)1()(1

b/i

i

tbD

qq

)(1 tD

qq

i

i

Exponential: (b=0)

Hyperbolic: (0



Q. What is the "Fetkovich" Decline Type Curve, and how is it used?A. A composite of analytical(pwf=con) and empirical(Arps) solutions

used as a "type curve"(data overlay) to estimate reservoir properties.

Discussion: History Production Analysis(PA) FetkovichAn original purpose of the Fetkovich TC? (graphical solution of Arps Eqs.)Use of "transient" stems? (estimate reservoir properties kand s)

Use of "depletion" stems? (estimate reservoir volume, predict rate)

History: Production Analysis(PA) Fetkovich

Fro

m:Fetkovich,M.J.:"Declin

eCurveAnalysisUsingType

Curves,"

JPT(June1980)1065

-1077.

Transient Stems: (left)Infinite-acting radial

flow model(pwf= con).q(t) is concave up.

Depletion Stems: (right)

Bounded circular re-servoir(pwf= con).

q(t) is concave down.b=0: pwf = con.b=1: qo = con. (qo/p).b>1: transient flow or

external drive energy.

Reservoir Properties:

k

y-axis match.N x&y-axis matches.s reD match.

swwa

wa

e

wfiDd

wa

e

w

ewt

Dd

err

r

rB.

ppkh

tqq

r

r

r

rrc

kt.t

2

1ln2141

)(

)(

2

1ln1

2

1

1006330

22

Variables for the Fetkovich Decline Type Curve

Hi P d i A l i (PA) C



Q. What is the "Carter" Decline Type Curve, and how is it used?A. A numerically-generatedgasrate solution(pwf=con) used as a

"type curve"(data overlay) to estimate reservoir properties.

Discussion: History Production Analysis(PA) CarterGenisis of the Carter TC? ("correction" of Fetkovich gas flow solutions)Use of "transient" stems? (estimate reservoir properties kand s)

Use of "depletion" stems? (estimate reservoir volume, predict rate)

History: Production Analysis(PA) Carter

Fro

m:Carter,R.D.:"TypeCurvesforFiniteRadialand

line

arGasFlowSystems:Con

stantTerminalPressure

Cas

e,"

SPEJ(October1985)7

19-728.

Transient Stems: (left)Numerical flow model

(pwf= con).q(t) is concave up.

Depletion Stems: (right)

q(t) is concave down.b=0: pwf = con.

b=1: qo = con. (qo/p).b>1: transient flow or

external drive energy.: numerical gas flow

cases( =f(pwf/(pi)).Reservoir Properties:

k y-axis match.G x&y-axis matches.s reD match.

swwa

wa

egigi

wfiDd

wa

e

w

ewtigi

Dd

err

r

rB.

ppkh

tqq

r

r

r

rrc

kt.t

2

1ln2141

)(

)(

2

1ln1

2

1

1006330

22

Variables for the Carter Decline Type Curve

M d PA O i t ti



Modern PA: OrientationVariable Rate/Pressure:

Material balance time(equi-valent constant rate timefunction).

Derived by applying the

superposition formulationto the pseudosteady-statesolution.

Auxiliary Plotting Functions:

Time-averaged rate func-tion(rate integral function).

Derivative, integral, and in-

tegral-derivative functions.Semi-Analytical Solutions:Ansah, et al gas solutions/

Buba method.Advanced methods for

hyperbolic rate analysis.

Adv. Decline Type Curves:

"Palacio" type curve."Doublet" type curve.Cumulative production TC.

"Modern production analysis (PA) is a mature and functionalprocess but we have to recognize that these methods maymake us overconfident in our abilities to diagnose, interpret,and analyze a particular field data case."

Blasingame

Discussion: Modern PA OrientationMajor advances in PA? (variable q/variable pwf analysis methods)

(auxiliary plotting functions to improve analysis)

(semi-analytical solutions

advances from the Arps' relations)

M d PA M t i l B l Ti (Th )


23/57


Modern PA: Material Balance Time(Theory)Q. Concept of material balance time?A. Constant-rate equivalent function(variable-rate constant rate case).

Discussion: Modern PA Material Balance Time(Theory)Value of material balance time? ("deconvolution" constant rate response)p/qo versus Np/qo (pressure drop form)? (PTA analog (constant rate case))

qo/p versus Np/qo (rate decline form)? (decline curve analysis format)

Pressure Drop Form:

tmbq

ppss,opss,o

o

Flowrate Form:

tmbp

q

pss,opss,o

o

1

(intercept)

oi

o

t

pss,o

B

B

Nc

m1

s

r

A

Cekh

Bb

wA

oopss,o 2

14ln

2

1141.2

(slope)

(material balance time)o

p

q

N

t

M d PA M t i l B l Ti (A li ti )


24/57


Modern PA: Material Balance Time(Application)Q. Application of material balance time?A. Requires pressure and rate histories (accurate rates essential).

Discussion: Modern PA Material Balance Time(Application)Effectiveness on raw data? (note example excellent correlation)p/qo versus Np/qo data? (transient flow ?, BDFB yes!)

qo/

p versus Np/qo (rate decline form)? (transient flow

?, BDFB

yes!)

Theory:

Material balance timeand the general boun-dary-dominated flowrelation:

Application:

Plotp/qo versus Np/qoand qo/p versus Np/qoto correct for rate ef-

fects

and to establishboundary-dominatedflow behavior(BDFB).

o

p

q

Nt

tmbq

ppss,opss,o

o

tmbp

q

pss,opss,o

o

1

M d r PA A ili Pl tti F ti (Th )


25/57


Normalized Pressure Drop: "Pressure Derivative"

"Pressure Integral"

"Pressure Integral-Derivative"

Normalized Flowrate: "Rate Derivative"

"Rate Integral"

"Rate Integral-Derivative"

Modern PA: Auxiliary Plotting Functions(Theory)Q. What are the "auxiliary" plotting functions for PA?A. Time(or material balance time)-averaged pressure drop and flowrate

functions "integral" and "integral-derivative" forms are used.

Discussion: Modern PA Auxiliary Plotting Functions(Theory)Theory for auxiliary plotting functions? (just calculus ...)Purpose of auxiliary plotting functions? (more resolution for PA data)

Advice? (use BOTHpressure drop and flowrate functions (separate plots))

tdq

pt

tq

p

oio

0

1

ioidoq

ptd

dtq

p

tdp

qt

tp

q o

i

o

0

1

i

o

id

op

qtd

dtp

q

p

q

td

dt

p

q o

d

o

Theory:The auxiliary func-tions(specifically theintegral and integralderivative functions)are designed to pro-vide "smooth" datafunctions whichshould improve inter-pretation and analy-sis.

odoq

p

td

dt

q

p

Application:

"Pressure drop"functions "Nor-malized PI" plot .

Flowrate" functions "Blasingame"plot .

Modern PA Auxiliary Plotting Fcns (Application 1)


26/57


Modern PA: Auxiliary Plotting Fcns(Application1)Q. Application1 What is the "Normalized PI" plot?A. Pressure drop auxiliary functions versus material balance time.

Discussion: Modern PA Aux. Plotting Fcns("Normalized PI" plot)p/qo functions versus Np/qo? (PTA form: transient flow ?, BDFB yes!)Validity of(p/qo)d function? (exceptional case downhole pressure data)

(

p/qo)i and(

p/qo)id functions? (smooth and representative)

Theory:

(p/qo) fcns versus Np/qo:

Application:

Analysis approach same

as PTA

look for char-acteristic behavior.

o

p

q

Nt

tdq

pt

tq

p

oio

01

ioidoq

p

td

dt

q

p

odoq

p

td

dt

q

p

Modern PA: Auxiliary Plotting Fcns (Application 2)


27/57


Modern PA: Auxiliary Plotting Fcns(Application2)Q. Application2 What is the "Blasingame" plot?A. Rate auxiliary functions versus material balance time.

Discussion: Modern PA Auxiliary Plotting Fcns("Blasingame" plot)qo/p versus Np/qo? (rate decline form: transient flow ?, BDFB yes!)Validity of(qo/p)d function? ((again) good downhole pressure data)

(qo/

p)i and(qo/

p)id functions? (well-behaved and representative)

Theory:

(qo/p) fcns versus Np/qo:

Application:

Transient flow behavior iscomplex, but transitionand BDFB unique.

o

p

q

Nt

tdp

qt

tp

qo

io

01

i

o

id

o

p

q

td

dt

p

q

p

q

td

dt

p

q o

d

o

Modern PA: Ansah et al Gas Solution/Buba Method


28/57


Modern PA: Ansah, et al Gas Solution/Buba MethodQ. What is the Ansah, et al gas solution/Buba method?A. A gas-specific solution for qg(Gp).

Discussion: Modern PA Ansah, et al Gas Solution/Buba MethodValidity of the Ansah, et al gas solution? (general criteria: pi < 6000 psia)Application of the Buba method? (assortment of plotting functions)

Advantages? (rigorous solution for gas flow cases (pwf = constant))

Theory:

Rate-Cumulative Result:

Gzp

zp

qD

GG

DGDqq

ii

wfwf

gii

pi

pigig

2

2

/

/

1

2

where

2

1

G

G

z

p

z

p

zp

zpCq

p

i

i

wf

wfg

1

22

Modern PA: Hyperbolic EUR Methods (Type Curves)


29/57


Modern PA: Hyperbolic EUR Methods(Type Curves)Q. Straight-line or characteristic behavior for Arps hyperbolic relation?A. Rate-cumulative ratios used to define "type curve" behavior.

Discussion: Modern PA Hyperbolic EUR Methods(Type Curves)Example gas case from Fetkovich(SPE13169)? ("typical" gas case) log[(qg/qgi)] versuslog[1-(Gp/G)]? (straight-line characteristic behavior)

Cartesian(qg/qgi) versus(Gp/G)? (note that 0.4


30/57


Liquid: (pwf = constant)

Gas: (pwf = constant)

Hyperbolic:

Harmonic:

Modern PA: EUR Methods(Schematic Behavior)Q. Collection of modern EUR methods?A. Comparison of methods illustrates behavior.

Discussion: Modern PA EUR MethodsLiquid(oil) flow behavior? (straight-line rate-cumulative behavior)Gas flow behavior?(rigorous) (rate-quadratic cumulative behavior)

General hyperbolic relation? (straight-line log[qg] versus log[1-(Gp/G)])

i

gi

D

qGpGiDgiqgq

i

gi

bpgig

Db

qG

G

Gqq

)(1

1 )1(

1

p

gi

igig G

q

Dqq exp

Gzp

zp

qD

GG

DGDqq

ii

wfwf

gii

pi

pigig

2

2

/

/1

2

2

1

Modern PA: Palacio Type Curve (Constant p case)


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Auxiliary functions:

Rate integral:

Rate integral-derivative:

dtq

t

tq oi,o

0

1

i,oid,o qdt

dtq

Modern PA: Palacio Type Curve(Constant pwf case)Q. Enhancement of the functions on the original Fetkovich type curve?A. "Palacio" type curve uses auxiliary rate functions(better resolution).

Discussion: Modern PA Palacio Type Curve(Constant pwf case) Auxiliary functions: Rate integral, integral derivative. (smoothness/detail) Rate integral: qo,i =(1/t) Int(qo, 0, t). (time-averaged rate function)

Rate integral derivative: qo,id = t|d/dt[qo,i]|. (derivative-like rate function)

Modern PA: Doublet Type Curve (Constant Rate Eq )


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Auxiliary functions:

Material BalanceTime:

Rate integral:

Rate integral-derivative:

tdp

qt

tp

q o

i

o

0

1

i

o

id

o

p

q

td

dt

p

q

dtq

t

qt o

o 0

1

Discussion: Modern PA Doublet Type Curve(constant rate eq.) Material Balance Time: tmb =(1/q) Int(qo, 0, t). (material bal. "deconvolution") Auxiliary Functions: Defined using qo andp. (accounts for qo andp = f(t))

Validity: Variable-rate/variable pressure drop cases. (rigorous for pssflow)

Modern PA: Doublet Type Curve(Constant Rate Eq.)Q. Type curve solution for variable-rate/variable pressure case?A. "Doublet" type curve uses material balance time function.

Modern PA: Rate Cumulative Type Curves


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Modern PA: Rate-Cumulative Type CurvesQ. Rate-cumulative type curves ... how can these help?A. "Gross" comparison of rate-cumulative behavior.

Discussion: Modern PA Rate-Cumulative Type CurvesValue? (probably most value as a screening tool)Pitfalls? (VERY difficult to distinguish individual trends hyperbolic)

Implementation? (recommend use as a validation tool with Arps' analysis)

Modern PA: Normalized PI Plot


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Modern PA: Normalized PI PlotQ. What is the "Normalized PI" plot(and how is it used)?A. This is pressure transient analysis (PTA) analog analysis plot.

Discussion: Modern PA Normalized productivity index(PI) plotHow is the Normalized PI plot used? (diagnose characteristic flow regimes)Transient flow behavior? (radial flow, fractured wells, horizontal wells ... )

Boundary-dominated flow behavior? (unique "unit-slope" behavior)

Plotting Functions:

(

p/qo)id versus Np/qo:(Pressure Integral-Derivative)

o

p

q

Nt td

q

pt

tq

p

oio

0

1

ioido q

p

td

dt

q

p

Orientation:

(0) IARF k(1/4) Fin. Con. FcD(1/2) Inf. Con. xf(1/2) Horizontal Lh

(1) BDF N


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Modern PA: Analysis-by-Modelling


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n

j j,DDcp,Dj,wfi ttqppB

kh

.q 1 1)()(2141

1

n

jj,DDcr,sDjji ttpqq

kh

B.pp

111 )()(2141

Variable-rate case:

Variable pressure drop case:

Approach:

Assemble time-pressure-rate(TPR) data.Perform quality control(particularly on pressure data).Establish initial reservoir model using normalized PI/Blasingame plots.Generate: rate data + model pwf(t); pressure data + model q(t)Generate: FORECAST of production and/or pressure performance.

Modern PA: Analysis-by-ModellingQ. What is "Analysis-by-Modelling?"A. Interactive(dynamic) analysis using a specified reservoir model.

Discussion: Modern PA "Analysis-by-Modelling Issues? (superposition is rigorous qand pwf data must be accurate)Typical case? (pressure data quality is less-than-optimal interpretation)

Unexpected benefit(s)? (modelling validates/disputes qand pwf data quality)

Issues:

These superpositionrelations are rigorous i.e., for EXACT in-puts, we will obtainEXACT results.

The superposition rela-tions must be modifiedfor the gas case OR,

the gas(or multiphase)case(s) can be gen-erated NUMERICALLY.

Constant Rate Solution

Constant Pressure Solution

Modern PA: Analysis-by-Modelling (Oil Case) (1/3)


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Modern PA: Analysis-by-Modelling(Oil Case) (1/3)

Discussion: Modern PA Oil CaseOrigin of data? (DAILY data: rate surface, pressure downhole gauge)Likelihood of a successful analysis? (VERY HIGH)

Rate decline at late times? (damage, gas-blocking, ?)

Points to Consider:

DAILY data(pressure is

measured continuous-ly, with a single valuereported per day).

Rates appear to be ac-curate and well-corre-lated with pressure.



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Discussion: Modern PA "Normalized PI" and "Blasingame" plotsTransient flow data? (Transition regime is apparent, NOTtransient flow)Boundary-dominated flow? (very strong agreement all data functions)

Influence of late-time effect?(damage?) (not significant compared to BDF)

a. "Normalized PI" Plot: (

p/qo) functionsversus Np/qo extraordinary agreementbetween data and reservoir model.

b. "Blasingame" Plot: (qo/

p) functions versusNp/qo excellent data-model performance(note that even derivative function is valid).

Q. Behavior of "normalized PI" and "Blasingame" plots for this case?A. Both plots perform EXTREMELY well driven by data quality.



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Discussion: Modern PA Final model/data matchRate match? (EXCELLENT near perfect match until late times)Pressure match? (virtually perfect)

Accounting for "damage?" (use PTA, or "cheat"

use variable skin effect)

Q. Performance of final model match? (i.e., rate and pressure functions)A. Extraordinary agreement of model and data again, data quality ...

Comment:

Excellent data qualityyields results similar topressure transientanalysis(PTA).

This case strong ad-vocates continuousrate and pressuremeasurement the

data are worth the cost.Downhole pressure

measurement must beconsidered if the eco-nomics support theimplementation.

There is no substitute

for vigilance in dataacquisition.

Modern PA: Analysis-by-Modelling (Gas Case) (1/4)


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Modern PA: Analysis by Modelling(Gas Case) (1/4)

Discussion: Modern PA Gas CaseOrigin of data? (DAILY data: rate surface, pressure surface)Likelihood of a successful analysis?(high to very high, data well-correlated)

Integration of PTA data? (designed for production, modelling for shut-ins)

Points to Consider:

Using both PA and PTAcan yield compliment-ary analyses.

Accurate rate and pres-sure data are required.

Q. Analysis-by-modelling for gas cases?A. Same general procedure, must use pseudopressure/pseudotime, or

numerical(or semi-analytical) gas solutions.



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Discussion: Modern PA "Normalized PI" and "Blasingame" plotsTransient flow data? (confirms fractured well behavior, moderate FcD)Boundary-dominated flow?("convergence" (Blasingame plot) confirms BDF)

What is required to achieve similar results? (accurate qand pwf data)

a. "NormalizedPI

" Plot: (pp/qg

) functionsversus Gp/qg excellent agreement indata and model functions.

b. "Blasingame" Plot: (qg/p

p) functions versus

Gp/qg excellent agreement note that theqDd functions converge (confirms BDF).

Q. Behavior of "normalized PI" and "Blasingame" plots for this case?A. Both plots perform EXTREMELY well driven by data quality.



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Discussion: Modern PA PA model/data matchRate match? (EXCELLENT near perfect match until late times)Pressure match? (virtually perfect including shut-ins)

Accounting for "damage?" (use PTA, or "cheat"

use variable skin effect)

Q. How well can a gas case be modeled?A. Depends on the data can be excellent...

Comment:

Modelling capturesindividual shut-ins,for both rate andpressure data.

Such performanceREQUIRES veryaccurate rate andpressure data.

This case is an ex-ception(low per-meability reservoirpermits use of dailydata). High per-meability reservoirsrequire continuousrate and pressuremeasurements.



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Discussion: Modern PA PTA model/data matchPTA match? (VERY GOOD note that production pwf data also matched)Comparison of PA and PTA results? (minor differences, due to data quality)

etc.? (note that pressure history is matched for PTA (entire history))

Q. Analysis of individual pressure transient tests?A. Should be straightforward vigilance in data acquisition is required.

Comment:

Excellent match on log-logplot(pressure drop funct-ions) and very good matchof the entire productionpressure history.

Daily rate and pressuredata are sufficient for thislow permeability reservoir

case.

Summary: Production Analysis (PA)


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Summary: Production Analysis(PA)

PA Present:

Strong roots in theory.Very good software tools in

the market.Fair to good data acqui-

sition practices.PA Future:

Improved data acquisitionsystems, particularlypermanent bottomholepressure measurement.

Integration of analytical/numerical tools numeri-cal models may govern thefuture of production dataanalysis.

"Event" analysis(PTA) willfade and "continuous" data

analysis will dominate.

"The future looks familiar ... We should be able to performcontinuous reservoir monitoring using production analysis(PA). The theory and tools are well-established the trick is

vigilant data acquisition and quality control"Blasingame

Discussion: Summary Production Analysis(PA)PA Theory? (variable q/variable pwf analysis methods proven)PA Tools? (software products are robust and easy to use)PA Future?(better data acquisition, numerical models, continuous analysis)

Summary: Production Analysis (PA) Issues


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Discussion: Summary Production Analysis(PA) Issues Pressure data are critical! (ptf pwf conversion hard to model/correlate) Consistent rate measurements! (rates frequency/accuracy is an issue) Review the well history! (note that several completion changes can occur)

Summary: Production Analysis(PA) IssuesQ. Major issues for PA?A. Rate data, pressure data, and well completion history.

Pressure:

Measure consistently.Check for relevance

(i.e., is the pressuremeasured in the cor-rect location?).

Pressure conversionsto bottomhole condi-

tions are always ap-proximate(at best).Flowrates:

Daily rates are best (asa practical minimum).

Be sure to check ratesfor accuracy.

Summary: Production Analysis (PA) Analysis


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Discussion: Summary Production Analysis(PA) Analysis Interpretation of production data should be obvious. (if complex, beware) DATA should provide analysis. (do not impose an analysis) Keep the reservoir/well model simple as possible. (simple = appropriate)

Summary: Production Analysis(PA) AnalysisQ. Analysis approach for PA?A. Not like PTA in PA data quality determines "analyzability" of data.

Interpretation:Boundary-dominated

flow data should "self-interpret."

Transient flow datashould also "self-inter-pret" however, amajor issue is thequantity(i.e., frequen-cy) of transient flowdata.

Modelling:

Simple is always best.Do not use approxi-

mations in modelling

(if at all possible).

Summary: Production Analysis (PA) Pressures


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Discussion: Summary Production Analysis(PA) Pressures Reality: All pressure data are suspect. (scrutinize ALL pressure data) Fantasy: Pressure at bottomhole conditions. (ptf pwf conversion issue)

Future: pwf (bottomhole measurements) common. (rate becomes issue)

Summary: Production Analysis(PA) PressuresQ. Pitfalls of pressure data as these relate to PA?A. Put simply pressure data(quality/quantity) remain weakest link.

Due Diligence:

Correlate pwf versus q this should givesome indication of"correlation."

No correlation noanalysis/interpretation.

Advice:

Pressure is the weaklink but ... try toperform an analysisregardless, at leastvalidate that pressuredata are relevant ornot.

Reconstruct a pwf(t)

profile by any reason-able means.

Summary: Production Analysis (PA) Flowrates


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Discussion: Summary Production Analysis(PA) Flowrates Reality: Rates are generally better than pressures. (allocations are an issue) Fantasy: Rates measured at high frequency. (rate-pressure mismatch)

Future: Downhole flowrate measurement. (continuous monitoring/analysis)

Summary: Production Analysis(PA) FlowratesQ. Pitfalls of flowrate data as these relate to PA?A. Flowrate data beware of allocations/frequency of measurements.

Due Diligence:

Allocated flowratesalways have issues.

Advice:

Flowrate data aregenerally better thanpressure data youshould be able toobtain an analysis.

Current Assessment: Production Analysis(PA)


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Cu e t ssess e t oduct o a ys s ( )

"Technology ... is a queer thing. It bringsyou great gifts in one hand, and it stabsyou in the back with the other."

C.P. Snow (1905-1980)

Discussion: Current Assessment Production Analysis(PA)Analysis? (Very good tools for PA and PTA (incremental improvements))Modelling?(Analytical modelling is sufficient, numerical modelling evolving)Data Issues?(Improvement in data handling (good/bad data, large volumes))Data Acquisition? (Lack of innovation in testing/monitoring methodologies)

Integration? (Breakthrough to integrate PA/PTA results with other data)

Current Assessment: PA Reality Check


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Reservoir Volume-Averaging:

Pressure transient analysis (PTA).Production data analysis (PA).Reservoir simulation.

Advanced Solutions:

Same view of the reservoir just more

"knobs" (i.e., reservoir/well parameters).Time-pressure-rate data will always "see" apressure/volume-averaged reservoir system.

Challenges:

Technical data acquisition/management.Operational "reservoir management"

mentality to ensure use of data.

y

Fro

m:SimulatorParameterAssignmentandtheProblem

of

ScalinginReservoirEngin

eering

Halderson(1986).

Q. What is the "reality check" with regard to PA?A. It is just time-pressure-rate data, do not expect a miracle.

Discussion: Current Assessment PA Reality Check Reservoir scale issues? (petrophysical data, PTA, PA, seismic data, etc.) Results from PTA/PA? ("reservoir-scale" flow character)

Key to success? (high precision/frequency reservoir performance data)

Current Assessment: Future Work in PTA/PA


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From: Simulator Parameter Assignment andthe Problem of Scaling in ReservoirEngineering Halderson (1986).

From: How Heterogeneity AffectsOil Recovery Weber (1986).

Crossplot: ko (PA and PTA) vs. klog mean Santa Barbara Field (Venezuela)..

Discussion: Current Assessment Future Work in PTA/PA

Additional reservoir/well models. (elliptical flow, moving boundary, ...)Full incorporation of PVT character. (volatile oil, gas condensate, ...)Reservoir scaling for PTA/PA. (scaling of petrophysical data?)Handling poor quality rate/pressure data. (major issue at present)Continuously measured pwf data. (this is coming ... high frequency rates?)Multiple well analysis(integration). (analytical (material balance) or models)

Coupling of analysis/interpretation(3D/3P models). (over-determined case)

Q. What are the prospects for PTA/PA?A. Most important issue is data quality/frequency characterization.

Society of Petroleum Engineers


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y gDistinguished Lecturer 2005-06 Lecture Season

Performance-Based Reservoir CharacterizationState-of-the-Technology

End of Presentation

Thomas A. Blasingame, Texas A&M U.

Department of Petroleum EngineeringTexas A&M UniversityCollege Station, TX 77843-3116

+1.979.845.2292 [email protected]

Society of Petroleum Engineers


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SPE DISTINGUISHED LECTURER SERIESis funded principally

through a grant of the

SPE FOUNDATIONThe Society gratefully acknowledges

those companies that support the program

by allowing their professionalsto participate as Lecturers.

And special thanks to The American Institute of Mining, Metallurgical,and Petroleum Engineers (AIME) for their contribution to the program.

y gDistinguished Lecturer 2005-06 Lecture Season

Selected References: Production Data Analysis


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y

(Appendix)

References Production Data Analysis:

1. Lewis, J.O., and Beal, C.H.: "Some New Methods for Estimating the Future Production of Oil Wells," Trans. AIME (1918) 59, 492-525.2. Manual for The Oil and Gas Industry Under The Revenue Act of 1918, Treasury Department United States Internal Revenue Service (1919).3. Cutler, W.W., Jr.: Estimation of Underground Oil Reserves by Oil-Well Production Curves, Department of the Interior, U.S. Bureau Of Mines Bulletin 228 (1924).4. Roeser, H.M.: "Determining the Constants of Oil-Production Decline Curves," Trans AIME (1925) 71, 1315-1321.5. Larkey, C.S.: "Mathematical Determination of Production Decline Curves," Trans AIME (1925) 71, 1322-23.6. Johnson, R.H. and Bollens, A.L.: "The Loss Ratio Method of Extrapolating Oil Well Decline Curves," Trans. AIME (1927) 77, 771.7. Lindsly, B.E., and Berwald, W.B.: Effect of Vacuum on Oil Wells, U.S. Department of Commerce, Bureau of Mines, Bulletin 322 (1930).8. Allen, R.E.: "Control of California Oil Curtailment," Trans AIME (1931) 92, 47-66.9. Rawlins, E. L. and M. A. Schellhardt: Backpressure Data on Natural Gas Wells and Their Application To Production Practices, Monograph 7, U.S. Bureau of Mines, Washington, DC, (1936).

10. Arps J.J.: "Analysis of Decline Curves," Trans. AIME (1945) 160, 228-247.11. Gurley, J.: "A Productivity and Economic Projection Method Ohio Clinton Sand Gas Wells," JPT (Nov. 1963) 1183-1188.12. Slider, H.C.: "A Simplified Method of Hyperbolic Decline Curve Analysis," JPT (Mar. 1968) 235-236.13. Stewart, P.R.: "Low-Permeability Gas Well Performance at Constant Pressure," JPT (Sept. 1970) 1149-1156.14. Gentry, R.W.: "Decline-Curve Analysis," JPT (Jan. 1972) 38-41.15. Gringarten, A.C.: "Reservoir Limits Testing for Fractured Wells," paper SPE 7452 presented at the 1978 SPE Annual Technical Conference and Exhibition, Houston, TX., 1-3 October 1978.16. Fetkovich, M.J.: "Decline Curve Analysis Using Type Curves," JPT (March 1980) 1065-1077.17. Nind, T.E.W.: Principles of Oil Well Production, McGraw-Hill Book Company, 2nd Ed., 1981, 31-44.18. Carter, R.D.: "Type Curves for Finite Radial and linear Gas Flow Systems: Constant Terminal Pressure Case," SPEJ (October 1985) 719-728.19. Fraim, M.L., Lee, W.J., and Gatens, J.M., III: "Advanced Decline Curve Analysis Using Normalized-Time and Type Curves for Vertically Fractured Wells," paper SPE 15524 presented at the 1986

SPE Annual Technical Conference and Exhibition, New Orleans, LA, 05-08 October 1986.20. Fetkovich, M.J., Vienot, M.E., Bradley and M.D., Kiesow, U.G.: "Decline Curve Analysis Using Type Curves Case Histories," SPEFE (Dec. 1987) 637-656.21. McCray, T.L.: Reservoir Analysis Using Production Decline Data and Adjusted Time, M.S. Thesis, Texas A&M U., College Station, TX (1990).22. Palacio, J.C. and Blasingame, T.A.: "Decline Curve Analysis Using Type Curves Analysis of Gas Well Production Data," paper SPE 25909 presented at the 1993 Joint Rocky Mountain

Regional/Low Permeability Reservoirs Symposium, Denver, CO, 26-28 April 1993.23. Doublet, L.E., Pande, P.K., McCollum, T.J., and Blasingame, T.A.: "Decline Curve Analysis Using Type Curves Analysis of Oil Well Production Data Using Material Balance Time: Application

to Field Cases," paper SPE 28688 presented at the 1994 Petroleum Conference and Exhibition of Mexico held in Veracruz, Mexico, 10-13 October 1994.24. Shih, M.-Y. and Blasingame, T.A.: "Decline Curve Analysis Using Type Curves: Horizontal Wells," paper SPE 29572 presented at the 1995 Joint Rocky Mountain Regional/Low Permeability

Reservoirs Symposium, Denver, CO, 20-22 March, 1995.25. Doublet, L.E. and Blasingame, T.A.: "Evaluation of Injection Well Performance Using Decline Type Curves," paper SPE 35205 presented at the 1996 SPE Permian Basin Oil and Gas Recovery

Conference, Midland, TX, 27-29 March 1996.26. Crafton, J. W.: "Oil and Gas Well Evaluation Using the Reciprocal Productivity Index Method," paper SPE 37409 presented at the 1 997 SPE Production Operations Symposium, held in Oklahoma

City, Oklahoma, 09-11 March 1997.27. Agarwal, R.G., Gardner, D.C., Kleinsteiber, S.W., and Fussell, D.D.: "Analyzing Well Production Data Using Combined Type Curve and Decline Curve Analysis Concepts," paper SPE 49222

prepared for presentation at the 1998 SPE ATCE, New Orleans, LA, 27-30 September 1998.28. Marhaendrajana, T. and Blasingame, T.A.: "Decline Curve Analysis Using Type Curves Evaluation of Well Performance Behavior in a Multiwell Reservoir System," paper SPE 71514 presented

at the 2001 Annual SPE Technical Conference and Exhibition, New Orleans, 30 September-03 October 2001.29. Araya, A. and Ozkan, E.: "An Account of Decline-Type-Curve Analysis of Vertical, Fractured, and Horizontal Well Production Data," paper SPE 77690 presented at the SPE Annual Technical

Conference and Exhibition held in San Antonio, Texas, 29 September-02 October 2002.

30. Pratikno, H., Rushing, J.A., and Blasingame, T.A.: "Decline Curve Analysis Using Type Curves: Fractured Wells," paper SPE 84287 presented at the 2003 Annual SPE Technical Conference andExhibition, Denver, CO., 05-08 October 2003.

31. Mattar, L. and Anderson, D.: "A Systematic and Comprehensive Methodology for Advanced Analysis of Production Data," paper SPE 84472 presented at the SPE Annual Technical Conferenceand Exhibition held in Denver, Colorado, U.S.A., 05-08 October 2003.

32. Anderson, D. and Mattar, L.: "Practical Diagnostics Using Production Data and Flowing Pressures," paper SPE 89939 presented at the SPE Annual Technical Conference and Exhibition held inHouston, Texas, U.S.A., 26-29 September 2004.

33. Fuentes-Cruz, G., Camacho-Velzquez, R., and Vsquez-Cruz, M. : "Pressure Transient and Decline Curve Behaviors for Partially Penetrating Wells Completed in Naturally Fractured-VuggyReservoirs, paper SPE 92116 presented at the 2004 SPE International Petroleum Conference in Mexico, Puebla, Mexico, 08-09 November 2004.

Selected Vendors: PA Software


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Slide 55/53T.A. Blasingame Performance-Based Reservoir Characterization State-of-the-Technology (Appendix)

Vendor A: FeketeProduct: RTAContact: Ed Ferguson ([email protected])

web: www.fekete.comVendor B: Kappa EngineeringProduct: TopazeContact: Kevin Siggery ([email protected])web: www.kappaeng.com

Vendor C: Weatherford/eProduction SolutionsProduct: PanSystemContact: Carol Marini ([email protected])web: www.ep-solutions.com

Rules for Life: Blasingame


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Simple Rules for Life:

Righty-tighty lefty-loosey

solves most problems in life.You can learn a lot from a

Chihuahua (no fear, all love).Don't mow the grass until

the city tells you to.

Leadership is not defined byvision, strength, integrity, orcourage JUST PASSION.If you have to herd cats, then

be a rat.

If you have to hold yourselfout as an example, thenmake sure to be a bad one(Mark Twain).

Important Rules for Life:

Always work harder than

those you work for.Never own anything that

eats while you sleep.Never own anything that

needs repainting.

g

"An empty stomach is not a good politicaladvisor moral know your convictions,but take care of yourself as well"

Mahatma Gandhi (+ Blasingame)

(Appendix)

Rules for Mathematics: Blasingame


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Mathematics:

(for reservoir engineering)

1.Cheat(approximate, trun-cate, etc.). (except formaterial balance)

2.Take the Derivative.

3.Take the Integral.

4.Take the Laplace Transform.

5.Go back to Rule 1 (Cheat).

Favorite Numbers:0, 1, .

(What else do we need?)

g

Documents

050919 TBlasingame SPE DL Pres (1 Hour Version)