1

( ) ( ) ( )

( ) BW + BR - RN + BN =

BG + BW + W +

pS - 1

SC + C GB + NB + B - BG + B - BN

wpgsoipptp

IgIIwIe

twi

wiwfgitigigtit ∆

Material Balance Calculations

A general material balance equation that can be applied to all reservoir types was

first developed by Schilthuis in 1936. Although it is a tank model equation, it can

provide great insight for the practicing reservoir engineer. It is written from start of

production to any time (t) as follows: oil zone oil expansion + gas zone gas expansion + oil zone and gas zone pore volume and connate water expansion + water influx + water injected + gas injected = oil produced + gas produced + water produced

Where: N initial oil in place, STB Np cumulative oil produced, STB G initial gas in place, SCF GI cumulative gas injected into reservoir, SCF Gp cumulative gas produced, SCF We water influx into reservoir, bbl WI cumulative water injected into reservoir, STB Wp cumulative water produced, STB Bti initial two-phase formation volume factor, bbl/STB = Boi Boi initial oil formation volume factor, bbl/STB Bgi initial gas formation volume factor, bbl/SCF Bt two-phase formation volume factor, bbl/STB = Bo + (Rsoi - Rso) Bg Bo oil formation volume factor, bbl/STB Bg gas formation volume factor, bbl/SCF Bw water formation volume factor, bbl/STB BIg injected gas formation volume factor, bbl/SCF BIw injected water formation volume factor, bbl/STB Rsoi initial solution gas-oil ratio, SCF/STB Rso solution gas-oil ratio, SCF/STB Rp cumulative produced gas-oil ratio, SCF/STB Cf formation compressibility, psia-1 Cw water isothermal compressibility, psia-1 Swi initial water saturation ∆pt reservoir pressure drop, psia = pi - p(t) p(t) current reservoir pressure, psia

(1)

2

( ) ( ) ( )

( ) BW + BR - RN + BN =

BG + BW + W +

pS - 1

SC + C GB + NB + B - BG + B - BN

wpgsoipptp

IgIIwIe

twi

wiwfgitigigtit ∆

BN B G =

volumeoil initial volumecap gas initial

ti

gi = m

( ) ( ) ( )

( ) BW + BR - RN + BN =

BG + BW + W +

pS - 1

SC + C BNm + NB + B - B BBNm + B - BN

wpgsoipptp

IgIIwIe

twi

wiwftitigig

gi

titit ∆

Material Balance Equation as a Straight Line

Normally, when using the material balance equation, each pressure and the

corresponding production data is considered as being a separate point from other pressure

values. From each separate point, a calculation is made and the results of these

calculations are averaged.

However, a method is required to make use of all data points with the

requirement that these points must yield solutions to the material balance equation that

behave linearly to obtain values of the independent variable. The straight-line method

begins with the material balance written as:

Defining the ratio of the initial gas cap volume to the initial oil volume as:

and plugging into the equation yields:

Let:

3

( )[ ] B G - B W - B W + B R - R + B N =F

p S - 1

S C + C =E

B - B =E

B - B =E

IgIIwIwpgsoiptp

twi

wiwfwf,

gigg

tito

∆

( )

( ) W + E B m + 1 + E BBm + E N =

W + E B m + 1 N + E BBNm + E N =F

ewf,tiggi

tio

ewf,tiggi

tio

( )

E B m + 1 + E

BBm + E = D wf,tig

gi

tio

DW + N =

DF e

Thus we obtain:

Let:

Dividing through by D, we get:

Which is written as y = b + x. This would suggest that a plot of F/D as the y coordinate

and We/D as the x coordinate would yield a straight line with slope equal to 1 and

intercept equal to N.

(2)

4

Drive Indexes from the Material Balance Equation

The three major driving mechanisms are: 1) depletion drive (oil zone oil

expansion), 2) segregation drive (gas zone gas expansion), and 3) water drive (water

zone water influx). To determine the relative magnitude of each of these driving

mechanisms, the compressibility term in the material balance equation is neglected and

the equation is rearranged as follows:

Dividing through by the right hand side of the equation yields:

The terms on the left hand side of equation (3) represent the depletion drive index (DDI),

the segregation drive index (SDI), and the water drive index (WDI) respectively. Thus,

using Pirson's abbreviations, we write:

DDI + SDI + WDI = 1

( ) ( ) ( ) ( )[ ] B R - R + B N = BW - W + B - BG + B - B N gsoiptpwpegigtit

( )( )[ ]

( )( )[ ]

( )( )[ ] 1 =

B R - R + B NBW - W +

B R - R + B N

B - BG + B R - R + B N

B - BN

gsoiptp

wpe

gsoiptp

gig

gsoiptp

tit

(3)

5

Tracy Material Balance

Tracy started with the Schilthuis form of the material balance equation:

Since:

Plugging into the equation, rearranging, and solving for N yields:

Let:

( ) ( ) ( ) wpgsoipptpegigtit BWB R - R N + B N = WB - BG + B - B N ++

( )B R - R + B =B

B =B

R N =GBBNm =G

gsosoiot

oiti

ppp

gi

ti

[ ]

( ) ( )

−−

B - B BBm + B R - R + B - B

BWWB G + B R - B N = N

giggi

oigsosoioio

wpegpgsoop )(

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

Φ

Φ

Φ

B - B BBm + B R - R + B - B

=

B - B BBm + B R - R + B - B

B =

B - B BBm + B R - R + B - B

B R - B =

giggi

oigsosoioio

W

giggi

oigsosoioio

gG

giggi

oigsosoioio

gsooN

1

6

Phi vs Reservoir Pressure

0.01

0.1

1

10

100

0 500 1000 1500 2000

Reservoir Pressure, psia

Pres

sure

Fac

tor

phiGphiW

PhiN vs Reservoir Pressure

0

5

10

15

20

25

30

35

40

0 500 1000 1500 2000

Reservoir Pressure, psia

Pres

sure

Fac

tor

phiN

Thus we have:

If there is no water influx or water production, the equation is written as:

ΦΦ GpNp G + N = N

Phi factors can be calculated at all desired pressures using data from a reservoir fluid

analysis. Then a table or plot of these factors can be used to calculate oil in place or

predict future performance.

Phi factors are infinite at the bubble point and decline rapidly as pressure declines below

the bubble point. Characteristic shapes of these pressure functions are shown next.

( )Φ−+ΦΦ WewpGpNp WBW G + N = N

7

Tracy Method for Predicting Future Performance

For oil reservoirs above the bubble-point pressure, the Tracy prediction method is

not needed. It is normally started at the bubble-point pressure or at pressures below. To

use this method for predicting future performance, it is necessary to choose the future

pressures at which performance is desired. This means that we need to select the pressure

step to be used. At each selected pressure, cumulative oil, cumulative gas, and producing

gas-oil ratio will be calculated. So the goal is to determine a table of Np, Gp, and Rp

versus future reservoir static pressure such as:

n p Np Gp Rp

0

1

2

3

4

p0 = pi = pb

p1

p1

p3

p4

0

Np1

Np2

Np3

Np4

0

Gp1

Gp2

Gp3

Gp4

Rsb

R1

R2

R3

R4

Oil is produced from volumetric, undersaturated reservoirs by the expansion of reservoir

fluids. Down to the bubble-point pressure, the production is caused by liquid (oil and

connate water) expansion and rock compressibility. Below the bubble-point pressure, the

expansion of the connate water and the rock compressibility are negligible, and as the oil

phase contracts owing to release of gas from solution, production is a result of expansion

of the gas phase. As production proceeds, pressure drops and the gas and oil viscosities

and volume factors continually change.

Tracy method for predicting the performance of internal gas drive reservoir uses

the material balance equation for a volumetric undersaturated oil reservoir which is

written as:

( ) ( ) ( ) wpgsoipptpegigtit BWB R - R N + B N = WB - BG + B - B N ++

8

Since:

Plugging into the equation, rearranging, and solving for N yields:

Let:

Thus we have:

( )B R - R + B =B

B =B

R N =GBBNm =G

gsosoiot

oiti

ppp

gi

ti

[ ]

( ) ( )

−−

B - B BBm + B R - R + B - B

BWWB G + B R - B N = N

giggi

oigsosoioio

wpegpgsoop )(

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

Φ

Φ

Φ

B - B BBm + B R - R + B - B

=

B - B BBm + B R - R + B - B

B =

B - B BBm + B R - R + B - B

B R - B =

giggi

oigsosoioio

W

giggi

oigsosoioio

gG

giggi

oigsosoioio

gsooN

1

9

If there is no water influx or water production, the equation is written as:

ΦΦ GpNp G + N = N

Phi factors can be calculated at all desired pressures using data from a reservoir fluid

analysis. Then a table or plot of these factors can be used to calculate oil in place or

predict future performance.

At time level j, the above equation is written as:

Since:

Thus:

Rearranging and solving for ∆Np we get:

Where Npj-1, Gp

j-1 are the cumulative oil and gas production at the old time level j-1

respectively.

( )Φ−+ΦΦ WewpGpNp WBW G + N = N

( ) ( )Φ∆Φ∆ jGp

1-jp

jNp

1-jp G + G + N + N = N

( ) N 2

R + R = R N = R N = G p

jp

1-jpave

ppppp ∆

∆∆∆

( ) ( ) N R + + G + N =

N R + G + N + N = N

pjG

avep

jN

jG

1-jp

jN

1-jp

jGp

avep

jG

1-jp

jNp

jN

1-jp

∆ΦΦΦΦ

Φ∆ΦΦ∆Φ

ΦΦ

ΦΦ∆jG

avep

jN

jG

1-jp

jN

1-jp

p R + G - N - N = N (4)

10

The tarner method for predicting reservoir performance by internal gas drive will

be presented in a form proposed by Tracy as follows:

1) Calculate ΦN and ΦG at p = pi - ∆p using:

2) Assume Rpj = Rso

j

3) Calculate Rpave using:

4) Calculate ∆Np using equation (4):

5) Calculate Np = Npj-1 + ∆Np

( ) ( )

( ) ( )

Φ

Φ

B - B BBm + B R - R + B - B

B =

B - B BBm + B R - R + B - B

B R - B =

giggi

oigsosoioio

gG

giggi

oigsosoioio

gsooN

2

R + R = Rjp

1-jpave

p

ΦΦ

ΦΦ∆G

avepN

G1-j

pN1-j

pp R +

G - N - N = N

11

6) Calculate So and SL as follows:

7) Evaluate ko at Sw and kg at SL

8) Calculate Rpj using the following equation:

9) Calculate the difference between the assumed and the calculated Rpj value. If these

two values agree within some tolerance, then the calculated ∆Np is correct. On the other

hand, if they don't agree then the calculated value should be used as the new guess and

steps 3 through 9 are repeated.

( )

S + S = S

BB

NN - 1 S - 1 = S

owL

oi

opwo

BB

kk + R = R

g

o

g

o

o

gso

jp µ

µ