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Heriot-Watt UniversityINSTITUTE OF PETROLEUM ENGINEERING

HeriotHeriot--Watt UniversityWatt UniversityINSTITUTE OF PETROLEUM ENGINEERING

MATERIAL BALANCE EQUATIONMATERIAL BALANCE EQUATIONAPPLICATIONAPPLICATION

Adrian C ToddAdrian C Todd

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Material Balance ApplicationMaterial Balance Application

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

zz 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

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

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Laurie Laurie DakeDake quote from the quote from the PractisePractise of of Reservoir EngineeringReservoir Engineering--Elsevier.Elsevier.

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the material balance to oilfields, the belief that it is now material balance to oilfields, the belief that it is now superceded by the application of modern numerical simulation.superceded by the application of modern numerical simulation.

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

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

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

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

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

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

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

zz OdehOdeh & & HavlenaHavlena (1963) rearranged MB equation into (1963) rearranged MB equation into different linear forms.different linear forms.

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

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

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Material Balance as Straight LineMaterial Balance as Straight Line

zz 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.

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

zz 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 EquationThe material balance equation can be written as

( )[ ] injwpgspop WBWBRRBN ++( ) ( )[ ]gssioio BRRBBN +=

+ 1

BB

mNBgi

goi

( ) ( )( ) ewc

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...WS1

pcScNBm1E ewc

fswoifw +

++=The expansion of the free gas

STB/bbl...1BB

mNBEgi

goig

=

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The material balance in this simplified form can be written

efwgo WNENmENEF +++=

Using this equation Havlena and Odehmanipulated 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 EE

GNEF +=o

g

o EE

GNEF +=

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

zz Covered in Chapter 17Covered in Chapter 17

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Depletion drive or other?Depletion drive or other?zz 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, there is other energy support, egeg. Water drive. Water drive

( )o fw eF N E E W ...bbl= + +Divide by Eo +Efw

e

o fw o ew

WF N ...STBE 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

zz 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.

zz 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.

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

.( ) ( ) ewc

fwcwgigigwpgp WpS1

cScGBBBGBWBG +++=+

Havlena and Odeh approach gives:

( )( ) scf/rcf....p

S1cScBE

scf/rcf...BBEft.cu.res....BWBGF

wc

fwcwgifw

gigg

wpgp

+=

=+=

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MB Approach to Gas ReservoirsMB Approach to Gas ReservoirsShort hand MB equation for gas reservoirs( ) efwg WEEGF ++=With gas reservoirs the pore and water compressibility can

be ignored

eg WGEF +=

g

e

g EWG

EF +=

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MB Approach to Gas ReservoirsMB Approach to Gas Reservoirs

g

e

g EWG

EF += Plot F/Eg vs. Gp, time or p

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MB Approach to Gas ReservoirsMB Approach to Gas Reservoirs

zz Plot gives initial gas Plot gives initial gas in placein place

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

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

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

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

determine gas in placedetermine gas in place

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

f wc wcp i g e w g

wc

c c SG G GB GB p W B / B1 S

+= f wc wc

p i g e w gwc

c c SG G GB GB p W B / B1 S

+= zz We is the net water influx (includes We is the net water influx (includes WpWp))

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

p gi e w

g gi

G B W B1 1G B GB

=

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p/z approachp/z approach

zz 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 p 1z z G

= Well known p/z plot

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zz 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

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

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

zz Depletion drive gas reservoirs will exhibit straight p/z plot weDepletion drive gas reservoirs will exhibit straight p/z plot well ll 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 effectzz Because of the high Because of the high

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

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

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

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

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Material Balance Equation Applied to Material Balance Equation Applied to Oil Reservoirs Oil Reservoirs Depletion DriveDepletion Drivezz 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

zz Above the Bubble PointAbove the Bubble Pointzz Production due to compressibility of the total system.Production due to compressibility of the total system.

zz Although appears complex MB equation isAlthough appears complex MB equation is

zz vv =C x V x =C x V x ppzz 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.zz 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 + = + zz No gas capNo gas cap

zz Aquifer small in volume We = Aquifer small in volume We = WpWp =0=0

zz RsRs==RsiRsi==RpRp all gas at surface dissolved in oil in all gas at surface dissolved in oil in reservoirreservoir

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Solution gas drive above bubble point.Solution gas drive above bubble point.zz Oil compressibility Oil compressibility -- ( )o oi

ooi

B Bc

B p=

zz 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+ = +

o o w wc fp o oi

wc

p o oi e

c S c S cN B NB p1 S

orN B NB c p

+ +=

=

So + Swc = 1

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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 p1 S

orN B NB c p

+ +=

=

( )e o o w wc fwc

1c c S c S c1 S

= + + ( )e o o w wc fwc

1c c S c S c1 S

= + +ce is the effective saturation weighted compressibility of the reservoir system

Recovery at bubble point p oie

ob

N B c pN B

=

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Solution Gas DriveSolution Gas Drivezz Reservoir pressure drops below bubble point solution gas Reservoir pressure drops below bubble point solution gas

drive effective.drive effective.

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

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

zz However also very inefficient.However also very inefficient.

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

zz 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 gasInstantaneous producing gas--oil ratio equation.oil ratio equation.

Saturation equationSaturation equation

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Instantaneous GasInstantaneous Gas-- Oil RatioOil Ratiozz Instantaneous GasInstantaneous Gas-- Oil Ratio, R, is the ratio of 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.

zz 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

zz 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 GasInstantaneous Gas-- Oil RatioOil Ratiozz Where:Where:

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

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

zz BBgg =gas formation volume factor, =gas formation volume factor, bblsbbls/SCF/SCF

zz BBoo = oil formation volume factor, = oil formation volume factor, bblsbbls/STB/STB

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

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

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

go s

g

gg o s

g

qFree Gas= Solution Gas=Q R

Bq

Total gas production rate: Q Q RB

= +

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Instantaneous GasInstantaneous Gas-- Oil RatioOil Ratio

oo

o

go s

g

o o

o

o oo s

go

g

qOil producing rate is: QB

qQ R

BCombining equations gives: R

q / Bq

q BSince: Q R RqBB

=

+=

= = +

oo

o

go s

g

o o

o

o oo s

go

g

qOil producing rate is: QB

qQ R

BCombining equations gives: R

q / Bq

q BSince: Q R RqBB

=

+=

= = +

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Instantaneous GasInstantaneous Gas-- Oil RatioOil Ratio

eg eog o

g e w o e w

2 k h p 2 k h pq and qln r / r ln r / r = =

Therefore in previous equation:

eg

g g e ws

eo

o o e w

2 k h pB ln r / r

R R2 k h pB ln r / r

= +

o eg os

g eo g

B kR R

B k= +

Instantaneous GasInstantaneous Gas-- Oil Oil Ratio EquationRatio Equation

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Instantaneous GasInstantaneous Gas-- Oil RatioOil Ratioo eg o

sg eo g

B kR R

B k= +

zz 1. Above 1. Above PbPb, no free gas. , no free gas. Keg is zero, R=Keg is zero, R=RsRs==RsiRsi..

zz 2. Short time when gas 2. Short time when gas saturation below critical saturation below critical value, keg still zero but value, keg still zero but R=R=RsRs

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Instantaneous GasInstantaneous Gas-- Oil RatioOil Ratiozz Instantaneous GOR is not the same as cumulative Instantaneous GOR is not the same as cumulative

GOR.GOR.

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

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

zz Two Two GORGORss related as follows.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 Equationzz 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.

ooil volume remainingS

total pore volume= ( )( )p oo ob wc

N N BS

NB / 1 S=

o wc

p b

S - oil saturation at any time, S - connate water sat'nN - oil in place at bubble point, N -cumulative oil production below P .

Equation can be rearranged as:

( )p oo wob

N BS 1 1 SN B

= cThe Oil Saturation

Equation

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History MatchingHistory Matchingzz History matching History matching if your model cannot predict the past its if your model cannot predict the past its

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

zz Instantaneous GOR can be used to history match relative Instantaneous GOR can be used to history match relative permeabilitiespermeabilities..

zz Rearranged takes the form.Rearranged takes the form. ( )eg o gseo g o

k BR R

k B=

zz Production data provides R and Production data provides R and NpNp as a function of pressure.as a function of pressure.

zz RsRs, B and , B and values from PVT report.values from PVT report.zz NpNp values provide So from oil saturation equation.values provide So from oil saturation equation.

zz Can generate therefore kCan generate therefore kegeg//kkeoeo vs. Svs. Soo

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Solution Gas Drive CharacteristicsSolution Gas Drive Characteristics

zz Rapid pressure declineRapid pressure decline

zz Water free productionWater free production

zz Rapidly increasing gasRapidly increasing gas--oil ratiooil ratio

zz Low ultimate oil recoveryLow ultimate oil recovery

zz Prediction methodsPrediction methods SchilthuisSchilthuis, , TarnerTarner and Tracy & and Tracy & TarnerTarner

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Solution Gas DriveSolution Gas Drive--TarnerTarnerss MethodMethodzz Similar approach to Similar approach to SchilthuisSchilthuis procedureprocedure

zz Above Above PbPb use effective compressibility use effective compressibility equation equation

p oie

ob

N B c pN B

=

zz 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 DriveSolution Gas Drive--TarnerTarnerss MethodMethod

zz Assemble dataAssemble data

zz Production dataProduction data

zz Field data and rockField data and rock

zz Field dataField data Formation volume factorsFormation volume factors

Gas solubilityGas solubility

Gas compressibilityGas compressibility

Gas and oil viscosities

zz Rock dataRock data

zz Laboratory relative Laboratory relative permeabilitiespermeabilities

zz Past production dataPast production data Oil productionOil production

Gas productionGas production

Water productionWater production

New water influxNew water influxGas and oil viscosities

All presented as a function of pressure

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Solution Gas DriveSolution Gas Drive--TarnerTarnerss MethodMethodzz TarnerTarnerss method uses MB equation rearranged to calculate method uses MB equation rearranged to calculate

gas production gas production GpGp..

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

zz Step1 Step1

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

zz 2. Select a future pressure and assume a value of 2. Select a future pressure and assume a value of NpNp at at that pressure. Sometimes express that pressure. Sometimes express NNpp as a function of N.as a function of N.

zz 3. Solve MB 3. Solve MB eqneqn. For . For NNppRRpp, , ieie. . GGpp..

( )( ) ( )o si s g ob p o s gp p p

g

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

B+ = =

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Solution Gas DriveSolution Gas Drive--TarnerTarnerss MethodMethod

( )p oo wcob

N BS 1 1 SN B

=

zz 4. Using assumed 4. Using assumed NpNp solve oil saturation equation for solve oil saturation equation for So. This enables keg/So. This enables keg/keokeo to be determined.to be determined.

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

sg eo g

B kR R

B k= +

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

i i 1p1

R R N2

++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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Solution Gas DriveSolution Gas Drive--TarnerTarnerss MethodMethodzz 6. Total gas produced from MB 6. Total gas produced from MB eqneqn. and IGOR . and IGOR eqneqn. .

Compared and assumed value of Compared and assumed value of NNpp adjusted and steps 2 adjusted and steps 2 to 6 repeated until MB and IGOR values for to 6 repeated until MB and IGOR values for GGpp match.match.

zz Step 2Step 2

zz 1 Second pressure selected and new 1 Second pressure selected and new NNpp assummedassummed..

zz 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 g2 p2 p2 p1 p1 p1 p1

g

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

B+ = =

zz 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.

zz 4. With assumed value of Np2 from 4. With assumed value of Np2 from satsatnn eqn.determineeqn.determine So.So.

zz 5. Calculate IGOR5. Calculate IGOR

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Solution Gas DriveSolution Gas Drive--TarnerTarnerss MethodMethodzz 6. Calculate gas produced during second step6. Calculate gas produced during second step

( ) ( )i 1 i 2 p2 p1 2R R N N G2+ ++ =zz 7. G7. G22 from MB compared with Gfrom 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.

zz By plotting these two values By plotting these two values vsvs NpNp a convergence point a convergence point can be determined.can be determined.

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

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TracyTracys Form of s Form of TarnerTarnerss MethodMethodzz Tracy took MB equation and generated a shorthand versionTracy took MB equation and generated a shorthand version

( ) ( )( ) ( )p o s g p g e po 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 po 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

+ = + + ( )

( ) ( )o s gn o oi si s g oi g gi giB R B

B B R R B mB B B / B

= + +

( ) ( )gg o oi si s g oi g gi giB

B B R R B mB B B / B = + +

( ) ( )w o oi si s g oi g gi gi1

B B R R B mB B B / B = + +

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TracyTracys Form of s Form of TarnerTarnerss MethodMethodzz For simplicity assume no gas cap. Then:For simplicity assume no gas cap. Then:

( )( )

o s gn

o oi si s g

B R BB B R R B

= +

( )g

go oi si s g

BB B R R B

= +

( )w o oi si s g1

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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TracyTracys Form of s Form of TarnerTarnerss MethodMethodzz 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= + zz If we assume no water encroachment or productionIf we assume no water encroachment or production

p n p gN N G= + zz Tracy considered two pressure conditions Tracy considered two pressure conditions PPjj & & PPkk and and

the oil production the oil production NNpp during this pressure interval.during this pressure interval.zz Tracy estimates producing GOR Tracy estimates producing GOR RkRk at the lower at the lower

pressure rather than pressure rather than NpNp..zz For For kkthth pressure.pressure.

pk nk pk gkN N G= +

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TracyTracys Form of s Form of TarnerTarnerss MethodMethodzz If N=1 then equation takes on fractional recovery formIf N=1 then equation takes on fractional recovery form

pk nk pk1 N G= +zz alsoalso ( ) ( )pj pk nk pj pk gk1 N N G G= + + +

zz andand ( ) ( )'pj pk nk pj avg pk gk1 N N G R N= + + + zz wherewhere

'j k'

avg

R RR

2+=

zz rearrangingrearranging

( )'pj nk pj gk pk nk avg gk1 N G N R= + + +

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TracyTracys Form of s Form of TarnerTarnerss MethodMethod

( )'pj nk pj gk pk nk avg gk1 N G N R= + + + zz Solving for Solving for NpkNpk

pj nk pj gkpk '

nk avg gk

1 N GN

R = +

zz Only unknown is Only unknown is RRavgavg all the rest from PVT data or all the rest from PVT data or calculated at previous stepcalculated at previous step

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

o eg ok s

g eo g

B kR R

B k= +

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TracyTracys Form of s Form of TarnerTarnerss MethodMethod

zz SSoo obtained from oil saturation equationobtained from oil saturation equation

( )p oo wcob

N BS 1 1 SN B

= Tracys Procedure

zz Set pressure step below Set pressure step below PbPb..

zz 1. Estimate 1. Estimate RRkk

zz At bubble point = At bubble point = RRsisi

zz From extrapolation of trendFrom extrapolation of trend

zz 2. Estimate 2. Estimate RRavgavg 'j k'avg

R RR

2+=

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Tracys Procedure continued.zz 3. Determine PVT functions 3. Determine PVT functions nn and and gg..zz 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

zz 5. Using 5. Using NNpp determine Sdetermine Soo using saturation using saturation eqn.andeqn.and thereby kthereby kegeg//kkeoeo from Sfrom Soo vs. kvs. kegeg//kkeoeo data.data.

zz 6. Calculate 6. Calculate RRkk from IGOR equation.from IGOR equation.

zz 7. Compare 7. Compare RRkk with with RRkk. According to tolerance for . According to tolerance for RRkk = = RRkk. If not repeat steps 1 to 6. If not repeat steps 1 to 6

c ACTODD

Tracys Procedure continued.

zz 8. Estimate 8. Estimate GGpp and and GGpp. . GGpp ==NNpp x R x R avgavgzz Set next pressure step and repeat steps 1 to 8.Set next pressure step and repeat steps 1 to 8.

c ACTODD

Gas Cap Drive ReservoirsGas Cap Drive Reservoirs

zz TarnerTarnerss method can also be used for gas cap method can also be used for gas cap drive reservoirsdrive reservoirs

c ACTODD

Average Reservoir PressureAverage Reservoir Pressure

zz MB equation sometimes considered as a MB equation sometimes considered as a tank modeltank model..

zz 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 decline gives confidence in using MB eqneqn..

zz DakeDake suggests if equilibrium is not achieved then can still suggests if equilibrium is not achieved then can still use MB use MB eqneqn..

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

c ACTODD

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=

Average Reservoir PressureAverage Reservoir Pressure

c ACTODD

Average Reservoir PressureAverage Reservoir Pressurez 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 pdV dp

q cV cV pdt dt

= = = =

j j 'j j j

dV cV pdV dp

q cV cV pdt dt

= = = =

For constant compressibility 'j j j

'j j j

j'

j jj

V q / p

p q / pp

q / p

=

c ACTODD

Average Reservoir PressureAverage Reservoir Pressurezz Material balance often applied at regular intervals .Material balance often applied at regular intervals .

zz Change in underground Change in underground withdrawl,UWwithdrawl,UWjj can be used can be used over a pressure drop over a pressure drop ppjj..

zz Then:Then:j j j

j

j jj

p UW / pp

UW / p

=

zz DakeDake suggested that the MB approach be used prior 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

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

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

zz 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.

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

MATERIAL BALANCE EQUATIONAPPLICATIONMaterial Balance ApplicationLaurie Dake quote from the Practise of Reservoir Engineering-Elsevier.Material Balance as an Equation of a Straight LineMaterial Balance as Straight LineMaterial Balance EquationNo Water Drive and No Gas CapGas Drive Reservoirs, No Water Drive and Known Gas CapGas Drive Reservoirs, No Water Drive and N & G unknownWater Drive ReservoirsDepletion drive or other?Depletion drive or other?Gas Field Application of MB EquarionBeware of the p/z plot.MB Approach to Gas ReservoirsMB Approach to Gas ReservoirsMB Approach to Gas ReservoirsMB Approach to Gas Reservoirsp/z approachp/z approachp/z approachp/z approach rate effectMaterial Balance Equation Applied to Oil Reservoirs Depletion DriveSolution gas drive above bubble point.Solution gas drive above bubble point.Solution gas drive above bubble point.Solution Gas DriveInstantaneous Gas- Oil RatioInstantaneous Gas- Oil RatioInstantaneous Gas- Oil RatioInstantaneous Gas- Oil RatioInstantaneous Gas- Oil RatioInstantaneous Gas- Oil RatioOil Saturation EquationHistory MatchingSolution Gas Drive CharacteristicsSolution Gas Drive-Tarners MethodSolution Gas Drive-Tarners MethodSolution Gas Drive-Tarners MethodSolution Gas Drive-Tarners MethodSolution Gas Drive-Tarners MethodSolution Gas Drive-Tarners MethodTracys Form of Tarners MethodTracys Form of Tarners MethodTracys Form of Tarners MethodTracys Form of Tarners MethodTracys Form of Tarners MethodTracys Form of Tarners MethodTracys Procedure continued.Tracys Procedure continued.Gas Cap Drive ReservoirsAverage Reservoir PressureAverage Reservoir PressureAverage Reservoir PressureAverage Reservoir PressurePredictions as a function of time

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