chap16- mbal appl

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MATERIAL BALANCE EQUATIONMATERIAL BALANCE EQUATIONAPPLICATIONAPPLICATION

Adrian C ToddAdrian C Todd

Heriot-Watt UniversityHeriot-Watt University

INSTITUTE OF PETROLEUM ENGINEERING

Heriot-Watt UniversityHeriot-Watt University

INSTITUTE OF PETROLEUM ENGINEERING

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Material Balance ApplicationMaterial Balance Application

No one universal solution to the MB equation.No one universal solution to the MB equation.

Recently the computing power behind modern Recently the computing power behind modern reservoir situation has cast a shadow of reservoir situation has cast a shadow of confidence in the material balance approachconfidence in the material balance approach

To quote the late Professor Laurie Dake a To quote the late Professor Laurie Dake a proponent of the MB equation.proponent of the MB equation.

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Laurie Dake quote from the Practise of Laurie Dake quote from the Practise of Reservoir Engineering-Elsevier.Reservoir Engineering-Elsevier.

““It seems no longer fashionable to apply the concept of the material It seems no longer fashionable to apply the concept of the material balance to oilfields, the belief that it is now superceded by the application balance to oilfields, the belief that it is now superceded by the application of modern numerical simulation.of modern numerical simulation.

Acceptance of this idea is a tragedy and has robbed engineers of their Acceptance of this idea is a tragedy and has robbed engineers of their most powerful tool for investigating reservoirs and understanding their most powerful tool for investigating reservoirs and understanding their performance rather than imposing their wills upon them, as is often the performance rather than imposing their wills upon them, as is often the case when applying numerical simulation directly in history matching…..case when applying numerical simulation directly in history matching…..

There should be no competition between MB and simulation instead they There should be no competition between MB and simulation instead they must be supportive of one another: the former defining the system which must be supportive of one another: the former defining the system which is used as input to the modelis used as input to the model

Material balance is excellent at history matching production performance Material balance is excellent at history matching production performance but has considerable disadvantages when it comes to prediction, which is but has considerable disadvantages when it comes to prediction, which is the domain of numerical simulation.”the domain of numerical simulation.”

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Material Balance as an Equation of a Material Balance as an Equation of a Straight LineStraight Line

Material balance not a difficult concept.Material balance not a difficult concept.

Difficult in applying it to real reservoirsDifficult in applying it to real reservoirs

There is often inadequate understanding of drive mechanisms.There is often inadequate understanding of drive mechanisms.

Odeh & Havlena (1963) rearranged MB equation into different linear Odeh & Havlena (1963) rearranged MB equation into different linear forms.forms.

Their method requires the plotting of a variable group against Their method requires the plotting of a variable group against another variable group selected depending on the drive mechanism.another variable group selected depending on the drive mechanism.

If linear relationship does not exist, then this deviation suggests that If linear relationship does not exist, then this deviation suggests that reservoir is not performing as anticipated and other mechanisms are reservoir is not performing as anticipated and other mechanisms are involved.involved.

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Material Balance as Straight LineMaterial Balance as Straight Line

Once linearity has been achieved, based on Once linearity has been achieved, based on matching pressure and production data then a matching pressure and production data then a mathematical model has been achieved.mathematical model has been achieved.

The technique is referred to as The technique is referred to as history history matchingmatching..

The application of the model to the future The application of the model to the future enables predictions of the future reservoir enables predictions of the future reservoir performance.performance.

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Material Balance EquationMaterial Balance Equation

The material balance equation can be written as

injwpgspop WBWBRRBN gssioio BRRBBN

1

B

BmNB

gi

goi

e

wc

fswoi WS1

pcScNBm1

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Wp, Winj and We are sometimes not included

Havlena and Odeh simplified equation to:-

efwgo WNENmENEF

Left hand side are production terms in reservoir volumes

gspop BRRBNF

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The right hand side includes oil and its originally dissolved gas, Eo, where

B....bbl/STBRRBBE gssioioo

The expansion of the pores and connate water, Efw.

STB/bbl...W

S1

pcScNBm1E e

wc

fswoifw

The expansion of the free gas

STB/bbl...1B

BmNBE

gi

goig

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The material balance in this simplified form can be written

efwgo WNENmENEF

Using this equation Havlena and Odeh manipulated the equation for different

drive types to produce a linear equation

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No Water Drive and No Gas CapNo Water Drive and No Gas Cap

efwgo WNENmENEF A plot of F vs. Eo should produce a straight line through the origin.Slope of line gives oil in place.

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Gas Drive Reservoirs, No Water Drive Gas Drive Reservoirs, No Water Drive and Known Gas Capand Known Gas Cap go mEENF Plot of F vs. (Eo + mEg) should produce a straight line slope N.

If m is not known then m can be adjusted to generate linear form at correct value for m.

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Gas Drive Reservoirs, No Water Drive Gas Drive Reservoirs, No Water Drive and N & G unknownand N & G unknown

efwgo WNENmENEF

o

g

o E

EGN

E

Fo

g

o E

EGN

E

F

Plot of F/Eo vs. Eo/Eg should be linear with a slope of G=mN and intercept N.

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Water Drive ReservoirsWater Drive Reservoirs

Covered in Chapter 17Covered in Chapter 17

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Depletion drive or other?Depletion drive or other? Material Balance can be used in short hand Material Balance can be used in short hand

form to get an indication of whether field is form to get an indication of whether field is depleting volumetrically ( depletion drive ) or depleting volumetrically ( depletion drive ) or there is other energy support, eg. Water drivethere is other energy support, eg. Water drive

o fw eF N E E W ...bbl

Divide by Eo +Efw

e

o fw o ew

WFN ...STB

E E E E

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Depletion drive or other?Depletion drive or other? Two unkowns, N & We. Dake suggests plot of F/(Eo+Efw)

vs. Np, or time or pressure drop

We = 0, no aquifer

Energy from oil and dissolved gas.Intercept oil in place

Pressure support probably from infinite aquifer.Could be abnormal

compaction Finite aquifer, less support later.

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Gas Field Application of MB Gas Field Application of MB EquarionEquarion

In earlier chapter introduced p/z plot for a gas In earlier chapter introduced p/z plot for a gas reservoir without water drive.reservoir without water drive.

Many have warned about the application of Many have warned about the application of this approach since it neglects another this approach since it neglects another possible energy support.possible energy support.

Plots of Gp vs. p or p/z can give wrong Plots of Gp vs. p or p/z can give wrong indications of gas in place. Under estimate indications of gas in place. Under estimate when Gp vs. p and over estimate when water when Gp vs. p and over estimate when water drive ignored.drive ignored.

Beware of the “p/z plot”.

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Beware of the “p/z plot”.

Craft & Hawkins

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MB Approach to Gas ReservoirsMB Approach to Gas ReservoirsFluid production = gas expansion + water expansion & pore

compaction and water influx

”. e

wc

fwcwgigigwpgp Wp

S1

cScGBBBGBWBG

Havlena and Odeh approach gives:

scf/rcf....pS1

cScBE

scf/rcf...BBE

ft.cu.res....BWBGF

wc

fwcwgifw

gigg

wpgp

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MB Approach to Gas ReservoirsMB Approach to Gas Reservoirs

Short hand MB equation for gas reservoirs

efwg WEEGF With gas reservoirs the pore and water compressibility can

be ignored

eg WGEF

g

e

g E

WG

E

F

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g

e

g E

WG

E

F Plot F/Eg vs. Gp, time or p

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Plot gives initial gas Plot gives initial gas in placein place

Advancing water only Advancing water only evident when gas evident when gas water contact arriveswater contact arrives

Mobility ratio of water Mobility ratio of water displacing gas as low displacing gas as low as 0.1as 0.1

Gas moving 100 Gas moving 100 times faster than times faster than waterwater

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p/z approachp/z approach Long established in gas reservoir engineering to Long established in gas reservoir engineering to

determine gas in placedetermine gas in place

Gas produced = gas initially in place – gas remaining Gas produced = gas initially in place – gas remaining in reservoirin reservoir

f wc wcp i g e w g

wc

c c SG G GB GB p W B / B

1 S

f wc wcp i g e w g

wc

c c SG G GB GB p W B / B

1 S

We is the net water influx (includes Wp)We is the net water influx (includes Wp)

Compressibility terms small for water & poresCompressibility terms small for water & pores

p gi e w

g gi

G B W B1 1

G B GB

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p/z approachp/z approach

Replacing gas formation factor with z/p givesReplacing gas formation factor with z/p gives

p

i

e w gii

G1

Gp pW B / Bz z

1G

WeBw/GBgi water invaded volume

Higher this term the higher the pressure and

vice versa

With no water drive becomes

pi

i

Gp p1

z z G

Well known p/z plot

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The equation enables gas in place to be The equation enables gas in place to be determined when p/z=0determined when p/z=0

p/z approachp/z approach

If any pressure If any pressure support curve will support curve will deviate from deviate from linear.linear.

In early time In early time periods pressure periods pressure support may not support may not be felt.be felt.

Depletion drive gas reservoirs will exhibit straight p/z plot well Depletion drive gas reservoirs will exhibit straight p/z plot well established. A straight line plot however does not prove established. A straight line plot however does not prove existence of depletion drive.existence of depletion drive.

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p/z approach rate effectp/z approach rate effect Because of the high mobility of Because of the high mobility of

gas then if gas extracted at a gas then if gas extracted at a high rate then pressure decline high rate then pressure decline faster since water mobility faster since water mobility cannot keep up.cannot keep up.

If however gas extraction rate If however gas extraction rate low then water drive will give low then water drive will give pressure support.pressure support.

This effect can distort p/z plot This effect can distort p/z plot for water drive reservoirs.for water drive reservoirs.

Varying rates are common in Varying rates are common in relation to winter and summer relation to winter and summer rates.rates.

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Material Balance Equation Applied to Material Balance Equation Applied to Oil Reservoirs – Depletion DriveOil Reservoirs – Depletion Drive Solution gas drive has two stages of depletionSolution gas drive has two stages of depletion

– First stage above bubble point pressureFirst stage above bubble point pressure

– Second stage below bubble point pressureSecond stage below bubble point pressure

Above the Bubble PointAbove the Bubble Point

Production due to compressibility of the total system.Production due to compressibility of the total system.

Although appears complex MB equation isAlthough appears complex MB equation is

v =C x V x v =C x V x pp Production = Expansion of reservoir fluidsProduction = Expansion of reservoir fluids

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Solution gas drive above bubble point.Solution gas drive above bubble point. MB equation above bubble point simplifies to:-MB equation above bubble point simplifies to:-

o oi w wc fp o oi

oi wc

B B c S cN B NB p

B 1 S

o oi w wc fp o oi

oi wc

B B c S cN B NB p

B 1 S

No gas capNo gas cap

Aquifer small in volume We = Wp =0Aquifer small in volume We = Wp =0

Rs=Rsi=Rp all gas at surface dissolved in oil in Rs=Rsi=Rp all gas at surface dissolved in oil in reservoirreservoir

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Oil compressibility -Oil compressibility -

Solution gas drive above bubble point.Solution gas drive above bubble point.

o oio

oi

B Bc

B p

Replacing oil term in MB equation givesReplacing oil term in MB equation gives

o oio

oi

B Bc

B p

w wc f

p o oi owc

c S cN B NB c p

1 S

w wc fp o oi o

wc

c S cN B NB c p

1 S

So + Swc = 1o o w wc f

p o oiwc

p o oi e

c S c S cN B NB p

1 S

or

N B NB c p

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Solution gas drive above bubble point.Solution gas drive above bubble point.o o w wc f

p o oiwc

p o oi e

c S c S cN B NB p

1 S

or

N B NB c p

e o o w wc fwc

1c c S c S c

1 S

e o o w wc fwc

1c c S c S c

1 S

ce is the effective saturation weighted compressibility of the reservoir system

Recovery at bubble point p oie

ob

N Bc p

N B

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Solution Gas DriveSolution Gas Drive Reservoir pressure drops below bubble point solution gas drive Reservoir pressure drops below bubble point solution gas drive

effective.effective.

More complex as gas comes out of solution.More complex as gas comes out of solution.

Most common reservoir drive mechanism.Most common reservoir drive mechanism.

However also very inefficient.However also very inefficient.

Often associated with other drive mechanisms.Often associated with other drive mechanisms.

In order to use MB equation to predict production versus In order to use MB equation to predict production versus pressure need other independent equations.pressure need other independent equations.

– Instantaneous producing gas-oil ratio equation.Instantaneous producing gas-oil ratio equation.

– Saturation equationSaturation equation

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Instantaneous Gas- Oil RatioInstantaneous Gas- Oil Ratio Instantaneous Gas- Oil Ratio, R, is the ratio of gas Instantaneous Gas- Oil Ratio, R, is the ratio of gas

production to oil production at a particular point in production to oil production at a particular point in production time, at a particular reservoir pressure.production time, at a particular reservoir pressure.

Instantaneous producing GOR is:Instantaneous producing GOR is:

Gas producing rate, SCF/dayR=

Oil producing rate, STB/day

Gas producing rate, SCF/dayR=

Oil producing rate, STB/day

Gas production comes from gas in solution in reservoir Gas production comes from gas in solution in reservoir and from free gas in reservoir which has come out of and from free gas in reservoir which has come out of solution.solution.

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Instantaneous Gas- Oil RatioInstantaneous Gas- Oil Ratio Where:Where:

qqgg = free gas flow rate, res.bbls/day = free gas flow rate, res.bbls/day

qqoo = oil producing rate, res.bbls/day = oil producing rate, res.bbls/day

BBgg =gas formation volume factor, bbls/SCF =gas formation volume factor, bbls/SCF

BBoo = oil formation volume factor, bbls/STB = oil formation volume factor, bbls/STB

QQoo = oil flow rate,STB/day = oil flow rate,STB/day

QQgg = total gas producing rate, SCF/day = total gas producing rate, SCF/day

RRss = gas solubility, SCF/STB = gas solubility, SCF/STBg

o sg

gg o s

g

qFree Gas= Solution Gas=Q R

B

qTotal gas production rate: Q Q R

B

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Instantaneous Gas- Oil RatioInstantaneous Gas- Oil Ratio

oo

o

go s

g

o o

o

o oo s

go

g

qOil producing rate is: Q

B

qQ R

BCombining equations gives: R

q / B

qq B

Since: Q R RqBB

oo

o

go s

g

o o

o

o oo s

go

g

qOil producing rate is: Q

B

qQ R

BCombining equations gives: R

q / B

q

q BSince: Q R R

qBB

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eg eog o

g e w o e w

2 k h p 2 k h pq and q

ln r / r ln r / r

Therefore in previous equation:

eg

g g e ws

eo

o o e w

2 k h p

B ln r / rR R

2 k h pB ln r / r

o eg os

g eo g

B kR R

B k

Instantaneous Gas- Oil Instantaneous Gas- Oil

Ratio EquationRatio Equation

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o eg os

g eo g

B kR R

B k

1. Above Pb, no free gas. Keg is 1. Above Pb, no free gas. Keg is zero, R=Rs=Rsi.zero, R=Rs=Rsi.

2. Short time when gas saturation 2. Short time when gas saturation below critical value, keg still zero below critical value, keg still zero but R=Rs<Rsibut R=Rs<Rsi

2-3. Gas reached critical 2-3. Gas reached critical saturation, keg increases as keo saturation, keg increases as keo decreases. Gas very mobile decreases. Gas very mobile compared to oil. Free gas compared to oil. Free gas produced from oil still in reservoir.produced from oil still in reservoir.

3. Maximum GOR value3. Maximum GOR value

4. Bg is increasing with decreasing 4. Bg is increasing with decreasing pressure. pressure.

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Instantaneous GOR is not the same as cumulative Instantaneous GOR is not the same as cumulative GOR.GOR.

Instantaneous GORInstantaneous GOR,R, is ratio at particular moment in ,R, is ratio at particular moment in time.time.

Cumulative GORCumulative GOR, Rp, is ratio of total oil and gas , Rp, is ratio of total oil and gas produced up to a particular moment.produced up to a particular moment.

Two GOR’s related as follows.Two GOR’s related as follows.


pNi pi

p p pp0

i pi

R NR RdN R

N

where R is the average GOR over period that N produced.

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Oil Saturation EquationOil Saturation Equation Oil saturation equation provides an average Oil saturation equation provides an average

oil saturation for a reservoir at any time.oil saturation for a reservoir at any time.

o

oil volume remainingS

total pore volume

p o

oob wc

N N BS

NB / 1 S

o wc

p b

S - oil saturation at any time, S - connate water sat'n

N - oil in place at bubble point, N -cumulative oil production below P .

Equation can be rearranged as:

p oo wc

ob

N BS 1 1 S

N B

The Oil Saturation

Equation

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History MatchingHistory Matching History matching – if your model cannot predict the past its History matching – if your model cannot predict the past its

value in predicting the future is in question.value in predicting the future is in question.

Instantaneous GOR can be used to history match relative Instantaneous GOR can be used to history match relative permeabilities.permeabilities.

Rearranged takes the form.Rearranged takes the form. eg o gs

eo g o

k BR R

k B

Production data provides R and Np as a function of pressure.Production data provides R and Np as a function of pressure.

Rs, B and Rs, B and values from PVT report. values from PVT report.

Np values provide So from oil saturation equation.Np values provide So from oil saturation equation.

Can generate therefore kCan generate therefore kegeg/k/keoeo vs. S vs. Soo

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Solution Gas Drive CharacteristicsSolution Gas Drive Characteristics

Rapid pressure declineRapid pressure decline

Water free productionWater free production

Rapidly increasing gas-oil ratioRapidly increasing gas-oil ratio

Low ultimate oil recoveryLow ultimate oil recovery

Prediction methodsPrediction methods

– Schilthuis, Tarner and Tracy & TarnerSchilthuis, Tarner and Tracy & Tarner

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Solution Gas Drive-Tarner’s MethodSolution Gas Drive-Tarner’s Method Similar approach to Schilthuis procedureSimilar approach to Schilthuis procedure

Above Pb use effective compressibility Above Pb use effective compressibility equation equation

p oie

ob

N Bc p

N B

Below bubble point pressure use MB, Below bubble point pressure use MB, Instantaneous GOR and Oil Saturation equationsInstantaneous GOR and Oil Saturation equations

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Solution Gas Drive-Tarner’s MethodSolution Gas Drive-Tarner’s Method

Assemble dataAssemble data

Production dataProduction data

Field data and rockField data and rock

Field dataField data

– Formation volume factorsFormation volume factors

– Gas solubilityGas solubility

– Gas compressibilityGas compressibility

– Gas and oil viscositiesGas and oil viscosities

Rock dataRock data

Laboratory relative Laboratory relative permeabilitiespermeabilities

Past production dataPast production data

– Oil productionOil production

– Gas productionGas production

– Water productionWater production

– New water influxNew water influx

All presented as a function of pressure

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Tarner’s method uses MB equation rearranged to calculate Tarner’s method uses MB equation rearranged to calculate gas production Gp.gas production Gp.

Procedure is a trial & error approach using independently Procedure is a trial & error approach using independently MB and Instantaneous GOR eqns.MB and Instantaneous GOR eqns.


Step1 Step1

1. Start at bubble point pressure1. Start at bubble point pressure

2. Select a future pressure and assume a value of Np at 2. Select a future pressure and assume a value of Np at that pressure. Sometimes express Nthat pressure. Sometimes express Npp as a function of N. as a function of N.

3. Solve MB eqn. For N3. Solve MB eqn. For NppRRpp, ie. G, ie. Gpp..

o si s g ob p o s g

p p pg

N B R R B B N B R BN R G

B

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p oo wc

ob

N BS 1 1 S

N B

4. Using assumed Np solve oil saturation equation for 4. Using assumed Np solve oil saturation equation for So. This enables keg/keo to be determined.So. This enables keg/keo to be determined.

5. Calculate instantaneous GOR.5. Calculate instantaneous GOR.o eg o

sg eo g

B kR R

B k

6. Calculate gas produced during pressure drop over 6. Calculate gas produced during pressure drop over

period.period.i i 1

p1

R RN

2

Ri = instantaneous GOR at start of periodRi+1 = instantaneous GOR at end of periodNp1= cumulative oil produced at end of period

Assumption R vs Np linearTherefore use small

pressure drops

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6. Total gas produced from MB eqn. and IGOR eqn. 6. Total gas produced from MB eqn. and IGOR eqn. Compared and assumed value of NCompared and assumed value of Npp adjusted and steps 2 adjusted and steps 2 to 6 repeated until MB and IGOR values for Gto 6 repeated until MB and IGOR values for Gpp match. match.

Step 2Step 2

1 Second pressure selected and new N1 Second pressure selected and new Npp assummed. assummed.

2. Solve MB for N2. Solve MB for Np2p2. This is cumulative gas at end of . This is cumulative gas at end of second pressure.second pressure.


o si s g ob p2 o s g

2 p2 p2 p1 p1 p1 p1g

N B R R B B N B R BG N R N R N R

B

3. Calculate gas produced during 23. Calculate gas produced during 2ndnd step by removing from step by removing from cumulative gas from step 1.cumulative gas from step 1.

4. With assumed value of Np2 from sat’n eqn.determine So.4. With assumed value of Np2 from sat’n eqn.determine So.

5. Calculate IGOR5. Calculate IGOR

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6. Calculate gas produced during second step6. Calculate gas produced during second step


i 1 i 2p2 p1 2

R RN N G

2

7. G7. G22 from MB compared with G from MB compared with G2 2 from IGOR and new from IGOR and new assumed value of Nassumed value of Np2p2 until convergence achieved. until convergence achieved.

By plotting these two values vs Np a convergence point By plotting these two values vs Np a convergence point can be determined.can be determined.

Further steps as for step 2.Further steps as for step 2.

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Tracy’s Form of Tarner’s MethodTracy’s Form of Tarner’s Method Tracy took MB equation and generated a shorthand versionTracy took MB equation and generated a shorthand version

p o s g p g e p

o oi si s g oi g gi gi

N B R B G B W WN

B B R R B mB B B / B

p o s g p g e p


N B R B G B W WN


o s g

n


B R B


g

g


B


w


1


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For simplicity assume no gas cap. Then:For simplicity assume no gas cap. Then:

Tracy’s Form of Tarner’s MethodTracy’s Form of Tarner’s Method

o s g

no oi si s g

B R B

B B R R B

g

go oi si s g

B

B B R R B

wo oi si s g

1

B B R R B

These functions are only dependent on reservoir pressure and oil properties.

They can all be obtained from PVT data.

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Using this shorthand system MB equation can be Using this shorthand system MB equation can be written:written:


p n p g e p wN N G W W

If we assume no water encroachment or productionIf we assume no water encroachment or production

p n p gN N G Tracy considered two pressure conditions PTracy considered two pressure conditions Pj j & P& Pkk and and

the oil production the oil production NNpp during this pressure interval. during this pressure interval.

Tracy estimates producing GOR Rk at the lower Tracy estimates producing GOR Rk at the lower pressure rather than pressure rather than Np.Np.

For kFor kthth pressure. pressure.pk nk pk gkN N G

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If N=1 then equation takes on fractional recovery formIf N=1 then equation takes on fractional recovery form


pk nk pk1 N G

alsoalso

pj pk nk pj pk gk1 N N G N

pj pk nk pj pk gk1 N N G N

andand 'pj pk nk pj avg pk gk1 N N G R N

wherewhere'

j k'avg

R RR

2

rearrangingrearranging

'pj nk pj gk pk nk avg gk1 N G N R

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'pj nk pj gk pk nk avg gk1 N G N R

Solving for Solving for NpkNpkpj nk pj gk

pk 'nk avg gk

1 N GN

R

Only unknown is R’Only unknown is R’avgavg all the rest from PVT data or all the rest from PVT data or calculated at previous stepcalculated at previous step

RRkk can also be estimated from liquid saturation known can also be estimated from liquid saturation known using IGOR eqn.using IGOR eqn.

o eg ok s

g eo g

B kR R

B k

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SSoo obtained from oil saturation equation obtained from oil saturation equation


p oo wc

ob

N BS 1 1 S

N B

Tracy’s Procedure

Set pressure step below Pb.Set pressure step below Pb. 1. Estimate R’1. Estimate R’kk

At bubble point = RAt bubble point = Rsisi

From extrapolation of trendFrom extrapolation of trend 2. Estimate R’2. Estimate R’avgavg '

j k'avg

R RR

2

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Tracy’s Procedure continued. 3. Determine PVT functions 3. Determine PVT functions nn and and g.g.

4. Determine 4. Determine NNpkpk and and NNpp..

pj nk pj gkpk '

nk avg gk

1 N GN

R

At first pressure step, pj = pb, Npj =0, Gpj =0

5. Using 5. Using NNpp determine S determine Soo using saturation using saturation

eqn.and thereby keqn.and thereby kegeg/k/keoeo from S from Soo vs. k vs. kegeg/k/keoeo data. data.

6. Calculate R6. Calculate Rkk from IGOR equation. from IGOR equation.

7. Compare R7. Compare Rkk with R’ with R’kk. According to tolerance for . According to tolerance for

RRkk = R’ = R’kk. If not repeat steps 1 to 6. If not repeat steps 1 to 6

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8. Estimate 8. Estimate GGpp and and GGpp. . GGpp = =NNpp x R avg x R avg

Set next pressure step and repeat steps 1 to 8.Set next pressure step and repeat steps 1 to 8.

Tracy’s Procedure continued.

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Gas Cap Drive ReservoirsGas Cap Drive Reservoirs

Tarner’s method can also be used for gas cap Tarner’s method can also be used for gas cap drive reservoirsdrive reservoirs

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Average Reservoir PressureAverage Reservoir Pressure

MB equation sometimes considered as a ‘tank model’.MB equation sometimes considered as a ‘tank model’.

If there is uniform pressure decline in all wells then this If there is uniform pressure decline in all wells then this decline gives confidence in using MB eqn.decline gives confidence in using MB eqn.

Dake suggests if equilibrium is not achieved then can still Dake suggests if equilibrium is not achieved then can still use MB eqn.use MB eqn.

He suggests an average pressure.He suggests an average pressure.

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Equilibrium

Non Equilibrium

Well positions and drainage boundaries

In figure wells have their own pressure declines.Dake presents a volume weighting for each drainage area.

Pj, Vj and qj are the pressure, volume and reservoir rate for the area j.The volume weighted average pressure is therefore.

j j jj j

p p V / V


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Dake suggests an alternative method based on production rate, based on the time derivative of the compressibility equation


j j 'j j j

dV cV p

dV dpq cV cV p

dt dt

j j '

j j j

dV cV p

dV dpq cV cV p

dt dt

For constant compressibility 'j j j

'j j j

j

'j j

j

V q / p

p q / p

pq / p

c ACTODD

Material balance often applied at regular intervals .Material balance often applied at regular intervals .

Change in underground withdrawl,UWChange in underground withdrawl,UWjj can be used can be used

over a pressure drop over a pressure drop ppjj..

Then:Then:


j j jj

j jj

p UW / p

pUW / p

Dake suggested that the MB approach be used prior Dake suggested that the MB approach be used prior to numerical simulation approach .to numerical simulation approach .

c ACTODD

Predictions as a function of timePredictions as a function of time

None of the terms in the MB equation include None of the terms in the MB equation include time.time.

Only a pressure volume solution .Only a pressure volume solution .

Need to use another method which uses time Need to use another method which uses time to work alongside MB solution.to work alongside MB solution.

Productivity of wells for example.Productivity of wells for example.

chap16- mbal appl

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