Material Balance Material Balance EquationEquation

For Oil ReservoirsFor Oil Reservoirs

Oil Material Balance EquationOil Material Balance Equation

( )[ ] ( ) ( )

( )we

wc

fwcw

gi

g

oi

gssioiooiiwpgspop

BW

pS

CSCm

BB

m

BBRRBB

BNBWBRRBN

+

+++

+

+=++

111

NomenclatureNomenclature

Np = cumulative oil produced at reservoir pressure p, STB Ni = initial oil in place, STB

m = oii

gii

BNBG

Rs = ratio of gas in place to oil in place under standard conditions,SCF/STB Gi = initial gas in place, SCF We = cumulative water influx, STB Wp = cumulative water produced, STB Rp = cumulative producing gas oil ratio, SCF/STB = Gp/Np Gp = cumulative gas produced, SCF

NomenclatureNomenclature

p = (pi-p), psia, where pi is the initial pressure Rs = solution gas ratio, SCF/STB Bo = oil formation volume factor, bbl/STB

Bg = gas formation volume factor, bbl/SCF = .005zTp

Bw = water formation volume factor, bbl/STB Cw = compressibility of water, psi-1 Cf = compressibility of formation, psi-1 Swc = connate water saturation, fraction subscript i = initial conditions (For example, Boi = initial formation volume factor, bbl/STB)

Oil Material Balance EquationOil Material Balance Equation

In words, the oil material balance equation In words, the oil material balance equation says that any production we obtain is due says that any production we obtain is due to change in volume of reservoir rock and to change in volume of reservoir rock and fluids and/or displacement by encroaching fluids and/or displacement by encroaching water from an adjoining aquifer.water from an adjoining aquifer.

Left Hand SideLeft Hand Side

Production from the reservoir Production from the reservoir at reservoir at reservoir conditionsconditionszz We measure these termsWe measure these terms

NNppBBoo Oil produced at reservoir conditions, (res. Oil produced at reservoir conditions, (res. bbl)bbl)zz Note that Note that BBoo includes changes in oil volume due to includes changes in oil volume due to

gas going into solution.

( )[ ] wpgspop BWBRRBN ++

gas going into solution.

Left hand SideLeft hand Side

NNpp((RRpp RRss))BBgg Amount of Amount of freefree gas gas produced at reservoir conditions.produced at reservoir conditions.zz Note Note RRpp is total produced gas (free + is total produced gas (free +

dissolved) per barrel; dissolved) per barrel; RRpp is gas dissolved at is gas dissolved at reservoir conditions that is produced.reservoir conditions that is produced.

zz The effect of oil volume changes due to The effect of oil volume changes due to solution gas, solution gas, RRss is included in is included in BBoo

WWppBBww Amount of water produced (or Amount of water produced (or injected)injected)

Right Hand SideRight Hand Side

Accounts for expansion of oil and free gas Accounts for expansion of oil and free gas in the reservoir, as well as influx of water in the reservoir, as well as influx of water and change in reservoir pore volume.and change in reservoir pore volume.

Expansion of oil + dissolved gas

( ) ( )( )gssioioi BRRBBN +

Right Hand SideRight Hand Side

zz Expansion of gas in the gasExpansion of gas in the gas--capcap

zz Expansion of rock and connate waterExpansion of rock and connate water

zz Natural Water influxNatural Water influx

giigigi

goii BGBGB

BmBN =

1

( ) ( ) pS

CSCBGBNp

SCSC

mBNwc

fwcwgiioii

wc

fwcwoii

++=

++

111

weBW

Problem (from Craft and Problem (from Craft and Hawkins)Hawkins)

Using the letter symbols for reservoir engineering, Using the letter symbols for reservoir engineering, write expressions for the following terms in a write expressions for the following terms in a volumetric undersaturated reservoirvolumetric undersaturated reservoirzz The initial oil in place in stockThe initial oil in place in stock--tank barrelstank barrelszz The fractional recovery after producing The fractional recovery after producing NNpp STBSTBzz The volume occupied by the remaining oil after The volume occupied by the remaining oil after

producing producing NNpp STBSTBzz The The scfscf of gas producedof gas producedzz The The scfscf of initial gasof initial gaszz The The scfscf of gas in solution in the remaining oil.of gas in solution in the remaining oil.zz The The scfscf of free gas in the reservoir after producing of free gas in the reservoir after producing NNpp

STBSTBzz The reservoir volume occupied by the escaped gasThe reservoir volume occupied by the escaped gas

Initial Reservoir ConditionsInitial Reservoir Conditions

Reservoir Conditions

Oil

Gas

Ni STB

NiRsi scf

Gi scf

Surface Conditions

{ Original Reservoir Oil = NiBoi res bbl.

Original Reservoir Gas = GiBgi res bbl. = m NiBoi res bbl

Initial Reservoir ConditionsInitial Reservoir Conditions Reservoir Pore Volume = Reservoir Pore Volume = VVpp bblbbl Pore Volume occupied by water = Pore Volume occupied by water = VVppSSwcwc bblbbl Hydrocarbon Pore Volume (HCPV) = Hydrocarbon Pore Volume (HCPV) = VVpp (1 (1 SSwcwc) bbl) bbl But HCPV = But HCPV = NNiiBBoioi + m + m NNiiBBoioi = (1+m) = (1+m) NNiiBBoioi bblbbl So So VVpp = (1+m) = (1+m) NNiiBBoioi / (1 / (1 SSwcwc) bbl) bbl Free gas volume = Free gas volume = mNmNiiBBoioi bblbbl So initial gas saturation = So initial gas saturation = mNmNiiBBoioi/ / VVpp

= = mNmNiiBBoioi (1 (1 SSwcwc) /((1+m) ) /((1+m) NNiiBBoioi ) ) Initial oil saturation = Initial oil saturation = NNiiBBoioi/ / VVpp

= = NNiiBBoioi (1 (1 SSwcwc) /((1+m) ) /((1+m) NNiiBBoioi ) )

After Production of Oil and GasAfter Production of Oil and Gas

Reservoir Conditions

Oil

Gas

Ni STB

NiRsi scf

Gi scf

Surface Conditions

Np STB

NpRp scf

(Ni - Np)Bo bblcontaining

(Ni - Np)Rs scf gas

(Gi+NiRsi - NpRp-(Ni - Np)Rs)Bg

bbl gas

Free Gas in Reservoir = (mNiBoi/Bgi+NiRsi - NpRp- (Ni Np)Rs)Bg bbl.

After Production of Oil and GasAfter Production of Oil and Gas

Gas Saturation = Free Gas Volume Gas Saturation = Free Gas Volume Pore Pore VolumeVolumezz ((mNmNiiBBoioi/B/Bgigi+N+NiiRRsisi -- NNppRRpp -- (N(Nii NNpp)R)Rss)B)Bgg [[(1+m) (1+m) NNiiBBoioi

/ (1 / (1 SSwcwc) ]) ]

Oil Saturation = Oil Volume Oil Saturation = Oil Volume Pore Pore VolumeVolumezz (N(Nii NNpp)B)Boo (1+m) (1+m) NNiiBBoioi / (1 / (1 SSwcwc))

ExampleExampleFor a solution gas drive reservoir, calculate the original oil in place if the following information is given. Assume water and rock compressibility are negligible.

p = 2,000 psia Bo = 1.22 bbl/STB Rs = 350 SCF/STB z = 0.80 Rsi = 600 SCF/STB Boi = 1.3 bbl/STB Np = 20.0 MMSTB T = 150o F Rp = 900 SCF/STB

SolutionSolution

Solution Gas drive reservoir implies:Solution Gas drive reservoir implies:zz Negligible water influx.Negligible water influx.zz No initial gas cap.No initial gas cap.

( )[ ]( ) ( )( )gssioioi

wpgspop

BRRBBNBWBRRBN

+=++

SolutionSolution

Calculate Calculate BBgg

( )( )bbl/SCF10625.7

000,24601508.005.

005.

4=

+=

=p

zTBg

SolutionSolution

Calculate reservoir productionCalculate reservoir production

Calculate PVT dependent terms on RHSCalculate PVT dependent terms on RHS

( )[ ] ( )[ ]bbl res. 10279.3

010625.735090022.110207

46

=++=++ wpgspop BWBRRBN

( ) ( ) ( ) ( )bbl/STB res. 1106.

10625.73506003.122.1 4

=+=+ gssioio BRRBB

SolutionSolution

Solve for initial oil in placeSolve for initial oil in place

( )[ ]( ) ( ) gssioio

wpgspopi BRRBB

BWBRRBNN +

++=

MMSTB2961106.

10279.3 7 ==iN

Condensed NotationCondensed Notation

The material balance equation is lengthy; The material balance equation is lengthy; cumbersome to work with.cumbersome to work with.

Introduce shorthand notation to facilitate Introduce shorthand notation to facilitate manipulation.manipulation.

Total Production of oil, water and gas:Total Production of oil, water and gas:

( )[ ] wpgspop BWBRRBNF ++=

Condensed NotationCondensed Notation Reservoir fluid expansion terms (on a Reservoir fluid expansion terms (on a

per STB basis)per STB basis)zz Expansion of oil and dissolved gasExpansion of oil and dissolved gas

zz Expansion of gasExpansion of gas--cap gascap gas

zz Expansion of rock and connate waterExpansion of rock and connate water

( ) ( ) gssioioo BRRBBE +=

= 1

gi

goig B

BmBmE

( ) ( ) pS

BmEmwc

oiwf

+=+1

11 ,cSc fwcw +

NoteNote

The fluid expansion terms The fluid expansion terms EEoo, , EEgg, and , and EEf,wf,ware composed only of fluid PVT properties are composed only of fluid PVT properties and connate water saturation.and connate water saturation.zz At reservoir conditions (Temperature At reservoir conditions (Temperature

constant), they are functions of reservoir constant), they are functions of reservoir pressure only.pressure only.

Material Balance EquationMaterial Balance Equation

In terms of our shorthand notationIn terms of our shorthand notation

Simplifications:Simplifications:zz Solution Gas Drive Reservoir: WSolution Gas Drive Reservoir: Wee = 0, m = 0= 0, m = 0

F = Ni Eo + mEg + 1+ m( )Ef , w( )+ WeBw

( )wfoi EENF ,+=

Solution Gas DriveSolution Gas Drive

Production (Production (FF) is measured.) is measured. EEoo and and EEf,wf,w are determined from PVT and are determined from PVT and

rock properties.rock properties.Material balance is a straight line equationMaterial balance is a straight line equation

zz Plot of Plot of FF versus versus EEoo + + EEf,wf,w is a straight line with is a straight line with intercept 0 and slope intercept 0 and slope NNii

Solution Gas DriveSolution Gas Drive

F

Eo(0,0)

Slope = Ni

Gas Cap Drive Gas Cap Drive No Water No Water InfluxInflux

In this case, the material balance equation can In this case, the material balance equation can be simplified tobe simplified to

Straight line form:

( )( )wfgoi EmmEENF ,1+++=Straight line form:( ) ( )( )

( ) ( )( )wfo wfgiiwfowfgwfoi

EEEE

mNNEE

F

EEmEENF

,

,

,

,,

+++=+

+++=

Gas Cap drive PlotGas Cap drive Plot

(0,0)

( )wfo EEF

,+

( )( )wfo wfg EEEE

,

,

++

} iN

Slope = mNi

Gas ReservoirsGas Reservoirs

Material BalanceMaterial Balance

ProblemProblem

Suppose we had a Suppose we had a tank of gas buried tank of gas buried undergroundundergroundzz Fixed known Fixed known

temperaturetemperaturezz Pressure knownPressure knownzz Tank Volume unknown

GaspiTGiTank Volume unknown

Problem (ContProblem (Contd)d)

Suppose we remove Suppose we remove GGpp scfscf gasgaszz Pressure falls to a new Pressure falls to a new

measured valuemeasured valuezz Temperature constantTemperature constant

Can we determine the Can we determine the original original scfscf of gas in of gas in the tank?

GaspT

Gi - Gp

Gp

the tank?

SolutionSolution From real gas law:From real gas law:

Original number of moles of gas in the tank, Original number of moles of gas in the tank, nnii

Solving for original volume of the tankSolving for original volume of the tank

znRTpV =

RG

RG

RTzVpn ii

i

ii 0283.0520

7.14 ===

giii

ii BGp

TzGV == 0283.0

Solution (ContSolution (Contd)d)

Number of moles removed from tankNumber of moles removed from tank

Number of moles left in the tankNumber of moles left in the tankR

Gn pr 0283.0=

( )R

GGn pi

= 0283.0left

Solution (ContSolution (Contd)d) Gas left occupies the entire tank volume, soGas left occupies the entire tank volume, so

Tank volumeTank volume

( )RT

RGG

zpV

RTznpV

pi

=

=0283.0

left

( ) ( )pigpi GGBGGpzTV == 0283.0

Gas Material Balance Gas Material Balance Volumetric ReservoirVolumetric Reservoir

We have two expressions for tank volumeWe have two expressions for tank volumezz Must be equalMust be equal

( )

=

=

==

i

ii

g

giip

igipig

zp

zp

GBB

GG

GBGGBV

11

Straight line plotStraight line plot

For a volumetric gas reservoir, a plot of For a volumetric gas reservoir, a plot of GGppversus p/z will be a straight line of slope versus p/z will be a straight line of slope ((GGii/(p/(pii/z/zii)) and intercept )) and intercept GGii

In practice, people plot p/z versus In practice, people plot p/z versus GGpp and and extrapolate to p/z = 0extrapolate to p/z = 0

p/z versus p/z versus GGpp

pG

zp

i

iz

p

iG0

0

ProblemProblem

Suppose when we Suppose when we remove remove GGpp scfscf gas, gas, WWeeBBww res. bbl of res. bbl of water encroachedwater encroachedzz Pressure falls to a new Pressure falls to a new

measured valuemeasured valuezz Temperature constantTemperature constant

Can we determine the Can we determine the original original scfscf of gas in of gas in the tank?

GaspT

Gi - Gp

Gp

WeBwthe tank?

SolutionSolution

Original volume of gas in tankOriginal volume of gas in tank

Final Volume of gas in tankFinal Volume of gas in tank

Material Balance with water influxMaterial Balance with water influx

gii BGV =

( ) ( )pigpiwe GGBGGpzTBWV == 0283.0

( )pigwegii GGBBWBG =

p/z versus p/z versus GGpp with water influxwith water influx

pG

zp

i

iz

p

iG0

0

Effect of water influx

Production of GasProduction of Gas

Processes that determine gas production:Processes that determine gas production:zz Expansion of gasExpansion of gaszz Water InfluxWater Influxzz Expansion of rock and connate waterExpansion of rock and connate water

In most cases expansion of rock and In most cases expansion of rock and connate water is small compared to gas connate water is small compared to gas expansion.expansion.

In In abnormallyabnormally pressured gas reservoirs, pressured gas reservoirs, this term may be significant.this term may be significant.

Gas Material BalanceGas Material Balance

General formGeneral form

Gas formation volume factorGas formation volume factor

( ) wewc

fwcwgigigiwpgp BWpS

cScBBBGBWBG +

++=+

1

/scfft 0283.0 3p

zTBg =

Gas Material Balance Gas Material Balance -- pp//zz

It is customary to express the gas material It is customary to express the gas material balance in terms of balance in terms of pp//zz

If there is no water influx and formation If there is no water influx and formation and rock compressibility are negligibleand rock compressibility are negligible

Plot of Plot of pp//zz vs. vs. GGpp is a straight lineis a straight linezz Intercepts the xIntercepts the x--axis at axis at GGi

( )piii

i GGzG

pzp =

i

Abnormally Pressured Abnormally Pressured ReservoirsReservoirs

Normal pressure gradients for gas Normal pressure gradients for gas reservoirs are in the range of 0.4reservoirs are in the range of 0.4--0.5 0.5 psiapsia/ft of depth/ft of depth

Abnormally pressured reservoirs have Abnormally pressured reservoirs have gradients of 0.7gradients of 0.7--1.0 1.0 psiapsia/ft of depth/ft of depthzz > 300 abnormally pressured gas reservoirs > 300 abnormally pressured gas reservoirs

offshore Gulf Coast; gradients > 0.65 with offshore Gulf Coast; gradients > 0.65 with depths over 10000 ft.depths over 10000 ft.

P/z for Abnormally Pressured P/z for Abnormally Pressured ReservoirsReservoirs

Material Balance EquationOil Material Balance EquationNomenclatureNomenclatureOil Material Balance EquationLeft Hand SideLeft hand SideRight Hand SideRight Hand SideProblem (from Craft and Hawkins)Initial Reservoir ConditionsInitial Reservoir ConditionsAfter Production of Oil and GasAfter Production of Oil and GasExampleSolutionSolutionSolutionSolutionCondensed NotationCondensed NotationNoteMaterial Balance EquationSolution Gas DriveSolution Gas DriveGas Cap Drive No Water InfluxGas Cap drive PlotGas ReservoirsProblemProblem (Contd)SolutionSolution (Contd)Solution (Contd)Gas Material Balance Volumetric ReservoirStraight line plotp/z versus GpProblemSolutionp/z versus Gp with water influxProduction of GasGas Material BalanceGas Material Balance - p/zAbnormally Pressured ReservoirsP/z for Abnormally Pressured Reservoirs