Material Balance Course

Reservoir Engineering I

Material Balance

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Arron Singhe

OUTLINE INTRODUCTION MATERIAL BALANCE EQUATION Tarner’s Formulation DRIVE INDICES

APPLICATIONS Undersaturated Oil Reservoir Gas Reservoirs

WATER INFLUX

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GRAPHICAL METHOD Havlena-Odeh Method

INTRODUCTION Material Balance is  Powerful method to estimate OOIP, OGIP  Estimate aquifer influx

Disadvantage  Require production from reservoir  Requires accurate pressure monitoring and production/injection measurements  Requires accurate PVT data

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Advantage  Independent from volumetric methods  Used to verify volumetric results

MATERIAL BALANCE EQUATION

 Given the reservoir pore volume under reservoir conditions 

Initial pressure, pi and reservoir temperature Ti

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 The tank represents the total pore volume of the reservoir

MATERIAL BALANCE EQUATION

 Fit all hydrocarbons of the reservoir into the tank 

0-dimensional representation

 The size of the potential gas cap is measured relative to the size of the oil volume.  m is a volume ratio

p = pi

NBoi

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GF Bgi  m  NBoi

MATERIAL BALANCE EQUATION

 Expansion of the reservoir fluids due to decrease in reservoir pressure 

p < pi

 At the lower pressure p, the gas cap expands, the aquifer expands and the oil volume changes At pi:

at p:

GF Bgi  mNBoi

NBo  N ( Rsi  Rs ) Bg

NBoi We

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GF Bg

MATERIAL BALANCE EQUATION

 When trying to fit the expanded fluid volumes back into the reservoir

N p ( Rp  Rs ) Bg

Produced Volume N p Bo

GF Bg

W p Bw

GF Bgi  mNBoi NBo  N ( Rsi  Rs ) Bg NBoi We

Initial Volume

Expanded Volume

Reservoir Content

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

tank, some fluid volume does not fit in The superfluous volumes are the produced volumes of oil, gas and water

MATHEMATICAL FORMULATION Initial volume:

NBoi  GF Bgi  NBoi  mNBoi Expanded volume at pressure p:

mNBoi

Bg Bgi

 NBo  NBg  Rsi  Rs   We





N p Bo  N p  ( Rp  Rs ) Bg  Wp Bw  N p  Bo  Bg ( Rp  Rs )  Wp Bw Material Balance Equation: [Expanded volume] – [initial volume] = [produced volume]

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Produced volume at pressure p:

MATERIAL BALANCE EQUATION

m  N  Boi 

Bg Bgi

 N  Bo  N  Bg  ( Rsi  Rs )  We  N  Boi  m  N  Boi

 N p  [ Bo  Bg  ( R p  Rs )]  W p  Bw

N 





N p Bo  Bg  ( R p  Rs )  (We  W p Bw )  Bg   mBoi   1  Bg  ( Rsi  Rs )  ( Boi  Bo ) B   gi 

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Tarner‘s formula (1944):

EXPLANATION OF TERMS IN EQUATION

All terms in reservoir volume at pressure p produced oil & gas



N p  Bo  Bg  ( R p  Rs )  (We  W p Bw )  Bg  mBoi    1  Bg  ( Rsi  Rs )  ( Boi  Bo ) B  gi  

expansion of

expansion of

shrinkage of

gas cap

dissolved gas

reservoir oil

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N



net water influx

DRIVE INDICES 1   1 1 N  Bo  Boi   Rsi  Rs   mNBoi   B  Bg   gi Bg B  N p  o  R p  Rs   Bg 

 1  We  Wp Bw   B g  1

Drive mechanisms  The solution gas drive Is  two phase expansion (dissolution of gas and shrinkage of the oil)



Ig

expansion of the gas cap

 The water drive

Iw

 expansion of the aquifer

Is+ Ig+ Iw= 1

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 The gas drive

SOLUTION GAS DRIVE INDEX - Is

 1  We  Wp Bw   B g  1

1  N  Bo  Boi   Rsi  Rs   Bg  Is   Bo  N p   R p  Rs   Bg 

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1   1 1 N  Bo  Boi   Rsi  Rs   mNBoi   B  Bg   gi Bg  Bo  N p   R p  Rs   Bg 

GAS DRIVE INDEX - Ig

 1  1  mNBoi   B  B g   gi Ig   Bo  N p   R p  Rs   Bg 

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1   1 1 1 N  Bo  Boi   Rsi  Rs   mNBoi     We  W p Bw  B B  B  Bg   gi g  g 1  Bo  N p   R p  Rs   Bg 

WATER DRIVE INDEX - Iw

1   1 1 1 N  Bo  Boi   Rsi  Rs   mNBoi     We  W p Bw  B B  B  Bg   gi g  g 1  Bo  N p   R p  Rs   Bg 

 Bo  N p   R p  Rs   Bg 

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Iw 

1  We  W p Bw  Bg

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DRIVE INDICES VARY WITH TIME

DRIVE INDICES SUMMARY Solution Gas Drive Index

Is

 1 N Bg   

  Bo i   Rsi  Rs     Bo  Np  R p  Rs  B    g 

Bo

Ig

 1 1   mNBo i   B Bg  gi    B  N p  o  R p  Rs     Bg 

Water Drive Index

Iw 

1 We  W p Bw  Bg  Bo  Np  R p  Rs  B    g 

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Gas Cap Drive Index

MATERIAL BALANCE EQUATION

 N

Solving the MB-equation





N P  Bo  Bg  ( R p  Rs )  (We  W p Bw )  Bg  mBoi    1  Bg  ( Rsi  Rs )  ( Boi  Bo ) B  gi  





for water influx, We, yields



   Bg   N  mBoi    1  Bg  ( Rsi  Rs )  ( Boi  Bo )   B   gi  

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We  N P  Bo  Bg  ( R p  Rs )  W p Bw

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EXAMPLE 7.1: Water Influx and Drive Indices

UNDERSATURATED OIL RESERVOIRS mNBoi 

Bg Bgi

 NBo  NBg  ( Rsi  Rs )  We  NBoi  mNBoi 

N p  [ Bo  Bg  ( R p  Rs )]  W p Bw

 

In undersaturated reservoirs, there is no gas cap (m = 0) Hence:

NBo  NBg  ( Rsi  Rs )  We  NBoi  N p [ Bo  Bg  ( Rp  Rs )]  Wp Bw



The Rs function is constant above the bubble point pressure Therefore:



Or:

N  Bo  Boi   We  N p Bo  Wp Bw

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

NBo  We  NBoi  N p Bo  Wp Bw

UNDERSATURATED OIL RESERVOIRS



The compressibility of the connate water and rock have an important role in undersaturated reservoirs:

V  Vw  V p  cwVw p  cV p p  cw NBo i

S wi 1 p  c NBo i p 1  S wi 1  S wi

 c  S wi cw  NBo i    1 S wi 

If entered into the following equation

N  Bo  Boi   We  N p Bo  Wp Bw 

yields

Boi c  S wicw p   N   Bo  Boi    We  N p Bo  W p Bw 1  S wi  

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

    p 

GAS RESERVOIRS The compressibility of the connate water and rock can be neglected

  

The reservoir initial volume: The expanded reservoir volume: The produced volume:



Then, the material balance equation is:

GBgi GBg+We GpBg+Wp

GBg  We  GBgi  G p Bg  Wp GBg  Bgi   G p Bg  Wp  We

G

G p Bg Bg  Bgi



We  W p Bg  Bgi

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

GAS RESERVOIRS G

G p Bg Bg  Bgi



We  W p Bg  Bgi

 Bg 

ZTP0 Z C T pT0 z0 p

if C and T are constant, then : Z    We  W p  p G  Gp   Z   Z  Bg  Bgi       p   p i

If there is no water influx, then We and Wp are zero:

 p   p   Gp    C1  C2G p      1  Z Z G    i  



Then, a plot of Gp versus p/Z will show a linear function!

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

GAS RESERVOIRS Plot of cumulative Gas Production (Gp) versus p/Z will show linear function, if no water influx.

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

MATERIAL BALANCE FORECAST METHOD (p/Z) ZT Z C p p G p Bg  G ( Bg  Bgi ) Bg  0.0283

 Z Zi  Z   G pC  G C  C p pi   p Z Z Z Gp  G G i p p pi p

 G

Z Z  G i P Pi

Zi  G pi G p  G Z p    p pi G p  G Z Zi  G Gp pi  p 1   Z Zi  G

  

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G

MATERIAL BALANCE – GRAPHICAL METHOD (HAVLENA/ODEH) In 1963 Havlena and Odeh showed that material balance data can be combined to an equation of a straight line 





Because the material balance equation is linear

This allows the determination of two parameters 

E.g.: OOIP and water influx coefficient



Until their publication, it has been used only to determine one single parameter

Assumptions:    

The reservoir may have an initial gas and an initial oil phase The gas is allowed to dissolve in the liquid phase Water is allowed to invade the reservoir from the aquifer during production The water and the rock are compressible

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

MATERIAL BALANCE – GRAPHICAL METHOD (HAVLENA/ODEH)



Introduce:

 o  Bo  Boi  Bg ( Rsi  Rs ) w 

Boi c  S wi cw p 1  S wi

 ow   o   w  g  Bg  Bgi C w  We



Substituting into the Tarner‘s formula

N ow  G g  C w  QF

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QF  N p [ Bo  Bg ( R p  Rs )]  W p  (WI  GI Bg )

OIL RESERVOIR WITHOUT WATER INFLUX

MATERIAL BALANCE EQUATION

N ow  G g  C w  QF

DRIVE INDICES APPLICATIONS

Oil reservoir without gas cap

QF  N o

GRAPHICAL METHOD

WATER INFLUX



Co-ordinate system o vs. QF; the slope of the line is N

Oil reservoir with gas cap (m known)   Bo QF  N  o  m  B  g  

 

  g  i

   

Co-ordinate system o +m(Bo/Bg)i g vs. QF; slope is N If it does not yield a straight line, the m is incorrect

Gas reservoir

QF  G g



G p Bg  G g

Co-ordinate system g vs. GpBg; straight line with slope G

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INTRODUCTION

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RESERVOIRS WITHOUT WATER INFLUX

RESERVOIR WITH WATER INFLUX Oil reservoir without gas cap

QF

 ow

 

 N C

w  ow

Co-ordinate system QF/ow vs. We/ ow provides a straight line, the intersection with the axis y gives N. If the line is not straight, then the water influx is incorrect

C N We/ ow

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QF/ow

RESERVOIR WITH WATER INFLUX Oil reservoir with gas cap

  i  i

 N 

o

We  Bo  m B  g

  i  i



Co-ordinate system QF/[o+m(Boi/Bgi) g] vs. We/ /[o+m(Boi/Bgi) g]

 

Straight line, the intersection with the axis y gives N. If the line is not straight, then the water influx (or m) is incorrect

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o

QF  Bo  m B  g

RESERVOIR WITH WATER INFLUX Gas reservoir:

G p Bg  W p

g

Coordinate system (GpBg+Wp)/g vs. We /g Straight line, the intersection with the axis y gives G. If the line isn’t straight- the water influx is incorrect

(GpBg+Wp)/g C G We/ g

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 

w  GC g

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MATERIAL BALANCE – GRAPHICAL METHOD SUMMARY

WATER INFLUX

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The water influx depends on:  Size of the aquifer  Permeability of the aquifer  Compressibility of the rock  Compressibility of the water Very seldom verified  no wells are drilled to explore the aquifer

INTRODUCTION

RESERVOIR BOUNDARY CONDITIONS

MATERIAL BALANCE EQUATION DRIVE INDICES APPLICATIONS GRAPHICAL METHOD

Finite Closed System (Pseudo-steady-state) (3)  If the whole amount of water flowing into the reservoir is due to expansion of the aquifer, then the exterior boundary is closed  Finite closed aquifer Closed System, Constant Pressure (Steady-state) (1)  The pressure at the exterior boundary is constant  Finite aquifer with constant pressure at the exterior boundary Infinite System (Non-steady-state) (2)  No effects “felt” at the exterior boundary  „infinite“aquifer

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WATER INFLUX

NON-STEADY-STATE WATER INFLUX

Originally by Van EVERDINGEN and HURST (1949) In 1971, FETKOVICH presented a simplified approach  easier to use! Especially for numerical computations  Iterative solution process

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Basic equations are based on  a “productivity index” of the aquifer, Jw  a maximum possible water influx, Wei  and the pressure difference between aquifer and reservoir

FETKOVICH METHOD Calculation for individual time intervals For a time interval, the flow rate is given by:



qw  J w P j  P wf ( j 1)



Where pj= average aquifer pressure, pwf=average inner (reservoir) pressure

The influx, the cumulative influx and average aquifer pressure are given by

We ( j 1)  qw t j 1

We ( j 1)   Wen n 1

P j 1

Pi  We ( j 1)  Pi Wei

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



INTRODUCTION

EXAMPLE 7.2 – FETKOVICH AQUIFER MODEL

MATERIAL BALANCE EQUATION DRIVE INDICES APPLICATIONS GRAPHICAL METHOD

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WATER INFLUX