feb22_mar1_mbe
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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
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