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Material Balance

Derivation of material balance

General form of material balance

Linear form of material balance equation

Uses and limitations of material balance

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Cross Section of a Combination Drive

Reservoir

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Use of Material Balance Equations

Determining hydrocarbon in place under various

conditions Determining water influx

Predicting reservoir pressure

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Material Balance

During production, from time 0 to time t:

Change in Oil volume + Change in Free Gas Volume

+ Change in Water Volume + Change in Void Space

Volume = 0

During production, overall volume change is zero.

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What do we need for the calculation?

Production, reservoir, and lab data:

Initial reservoir pressure, average reservoirpressure at time t

Produced oil, gas, water

Amount of water that enters into the reservoirFormation volume factors

Gas oil ratios Initial ratio of gas and oil volume

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Symbols Used in Material Balance

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Symbols Used in Material Balance

N: Initial reservoir oil, STB (t0)

Np: Cumulative produced oil, STB (t)

G: Initial reservoir free gas, SCF (t0)

Gf: Volume of free gas in the reservoir, SCF (t)W: Initial reservoir water, bbl (t0)

Wp: Cumulative produced water, STB (t)

We: Water influx into reservoir, bbl (t)

Vf: Initial void space, bbl (t0)m: initial reservoir free gas volume / initial reservoir oil volume (t0)

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Boi: Initial oil formation volume factor, bbl/STB (t0)

Bo: oil formation volume factor, bbl/STB (t)

Bgi: Initial gas formation volume factor, bbl/SCF (t0)

Bg: Gas formation volume factor, bbl/SCF (t)

Bw: Water formation volume factor, bbl/STB (t)

cw: Water isothermal compressibility, psi

-1

p: change in average reservoir pressure, psia (t)

Swi: Initial water saturation (t0)

Vf: Initial void space, bbl (t0)

Rsoi: Initial solution gas-oil ratio, SCF/STB (t0)

Rp : Cumulative produced gas-oil ratio, SCF/STB (t)Rso: Solution gas-oil ratio, SCF/STB (t)

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Oil Volume Change

Change in oil volume = NBoi - (N-Np)Bo

Initial reservoir oil volume: NBoi

Oil volume at time t and pressure p: (N-Np)Bo

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Free Gas Volume Changegi

oi

GB

m NB

gi oiGB NmB

f

SCF initial gas (free + dissolved) - SCF gas produced - SCF in solution

G [ ] [( ) ]

Reservoir free gas volume at time t=

( )

oisoi p p p so

gi

oisoi p p p so g

gi

NmB NR N R N N RB

NmB NR N R N N R BB

Ratio of initial free gas to initial oil volume:Initial free gas volume =

( )oioi soi p p p so ggi

NmB NmB NR N R N N R BB

SCF free gas at t:

Change in free gas volume:

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Water Volume Change

Change in water volume:

W (W - BwWp + We+ Wcwp) = BwWp We - Wcwp

Initial reservoir water volume (t0): W (res bbl)

Cumulative water produced at t = Wp (STB)Reservoir volume of cumulative produced water(t) = BwWp

Volume of encroached water at t: We (res bbl)Change in volume of water due to change in

pressure: Wcwp (res bbl)Reservoir water volume at t: W - BwWp + We+ Wcwp

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Rock Volume Change

f f f V V c p

[ ] f f f f f f V V V c p V c p

Change in void space volume =

Initial void space volume = Vf

Volume at time t =

Change in rock volume =

f fV c p

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Total Change in Water and Rock Volume

f fWith W=V , and V ,1

oi oiwi

wi

NB NmBS

S

e w p w f f W B W Wc p V c p

We have total change in water and rock volume:

[ ]( )1

(1 )

1

oi oi

e w p w wi f wi

w wi f

e w p oi

wi

NB NmB

W B W c S c pS

c S cW B W m NB p

S

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Total Change in Oil and Free Gas Volume

[ ]oi g

oi o p o oi soi g p p g

gi

g so p g so

NmB B NB NB N B NmB NR B N R B

B

NB R N B R

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Material Balance

During production, from time 0 to time t:

Change in Oil volume + Change in Free Gas Volume

==

-(Change in Water Volume + Change in Void SpaceVolume)

During production, overall volume change is zero.

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p

[ ]

(1 )1

Adding and subtracting the term N ,

[ ]

oi g

oi o p o oi soi g p p g

gi

w wi f

g so p g so e w p oi

wi

g soi

oi g

oi o p o oi soi g p p g g so p

gi

NmB B NB NB N B NmB NR B N R B

B

c S c NB R N B R W B W m NB p

S

B R NmB B

NB NB N B NmB NR B N R B NB R N BB

p pN N (1 )1

Then grouping items:

[ ( ) ] [ ( ) ]

( ) [ ] (1 )1

Wri

g so

w wi f

g soi g soi e w p oi

wi

oi oi o soi so g p o soi so g

oi g w wi f

p soi g p e w p oi

gi wi

R

c S c B R B R W B W m NB pS

NB NmB N B R R B N B R R B

NmB B c S c R R B N W B W m NB p

B S

oi o soi so g tte B and [B +(R -R )B ]=B , we have:tiB

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tiN(B ) [ ( ) ] (1 )

(1 )1

g

t p t p oi g ti

gi

w wi f

e w p ti

wi

B B N B R R B NmB

B

c S cW B W m NB p

S

tN(B ) ( ) (1 )1

[ ( ) ]

w wi f titi g gi ti e

gi wi

p t p soi g w p

c S cNmB B B B m NB p W

B S

N B R R B B W

Rearrange to become:

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tN(B ) ( ) (1 )1

[ ( ) ]

w wi f titi g gi ti e

gi wi

p t p soi g w p

c S cNmB B B B m NB p W

B S

N B R R B B W

Expansion of oilor gas zones

Change in voidspace volume

Water influx

Oil and gasproduction

Water production

General Material Balance Equation

For saturated oil reservoir with an associated gas cap.

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tN(B ) [ ( ) ]1

w wi f

ti ti e p t p soi g w p

wi

c S c B NB p W N B R R B B W

S

p tiN and NmB p p gi R G GB

tN(B ) ( ) ( )1

( )

w wi f

ti g gi ti gi e

wi

p t p soi g w p

c S c B G B B NB GB p W

S

N B G NR B B W

Unsaturated oil reservoir: m = 0

Simplifications

Gas reservoir:

( )1

w wi f

g gi gi e p g w p

wi

c S cG B B GB p W G B B W

S

No initial oil, N = 0, Np = 0:

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Drive Mechanisms

Segregation

Drive (Gas cap)

Water Drive

Depletion Drive(Volumetric reservoir)

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tN(B ) ( ) (1 ) 1

[ ( ) ]

w wi f ti

ti g gi ti egi wi

p t p soi g w p

c S cNmB

B B B m NB p W B S

N B R R B B W

t

N(B ) ( ) ( ) [ ( ) ]ti

ti g gi e w p p t p soi ggi

NmBB B B W B W N B R R B

B

Drive Mechanisms

With three drives, compressibility term is negligibleRearrange

t

( )( )N(B )

1[ ( ) ] [ ( ) ] [ ( ) ]

ti g gi

gi e w pti

p t p soi g p t p soi g p t p soi g

NmBB B

B W B W B

N B R R B N B R R B N B R R B

Divide each term by the right hand side term

Depletion Drive Index Segregation Drive Index Water Drive Index

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Assumption of Material Balance Equations

The reservoir may have an initial gas phase andliquid phase. The gas is dissolved in the liquid phase and is inequilibrium.

Oil can become volatile and enter into the gasphase.

Water is allowed to invade the reservoir from the

aquifer during production. Water and rock are compressible.

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Limitations of Material Balance Equations

Sources of error:

Assumptions:- Gas and oil may not be at equilibrium

Data:

- gas libration processes simulate those thatoccur in reservoir

- Average pressure (how it was averaged, extent

of spatial variation)

- Error in estimates of gas and water production.

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Havlena and Odeh method: linearization

t

I, I,

[ ( ) ]

= N[(B ) ( ) (1 ) ]1

Here W cumulative water injection, G cumulative gas injection, formation volume factor of injected

p t p soi g w p I I Ig

w wi f titi g gi ti e

gi wi

Ig

N B R R B B W W G B

c S cmB B B B m NB p W

B S

B

gas

F: net production from reservoir

Eo:expansion of oil E

f,w

: expansion offormation and waterEg:

expansion of gas

,(1 ) tio ti f w g egi

NmBF NE N m B E E W B

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Havlena and Odeh method: linearization

,(1 )

ti

o ti f w g egi

NmB

F NE N m B E E W B

When no original gas cap, no water influx, and negligible

formation and water compressibilities:

oF NE