TPG4150 Reservoir Recovery Techniques 2004
Material Balance Equations
1
Material Balance Equations
To illustrate the simplest possible model we can have for analysis of reservoir behavior, we will start with
derivation of so-called Material Balance Equations. This type of model excludes fluid flow inside the reservoir,
and considers fluid and rock expansion/compression effects only, in addition, of course, to fluid injection and
production. First, let us define the symbols used in the material balance equations:
Symbols used in material balance equations
Bg
Formation volume factor for gas (res.vol./st.vol.)
Bo
Formation volume factor for oil (res.vol./st.vol.)
Bw
Formation volume factor for water (res.vol./st.vol.)
Cr
Pore compressibility (pressure-1)
Water compressibility (pressure-1)
Cw
!P
Gi
Gp
P2 ! P1
Cumulative gas injected (st.vol.)
Cumulative gas produced (st.vol.)
m
N
Np
Initial gas cap size (res.vol. of gas cap)/(res.vol. of oil zone)
Original oil in place (st.vol.)
Cumulative oil produced (st.vol.)
P
Rp
Pressure
Cumulative producing gas-oil ratio (st.vol./st.vol) = G p / N p
Rso
Solution gas-oil ratio (st.vol. gas/st.vol. oil)
Sg
Gas saturation
So
Sw
T
Vb
Oil saturation
Vp
Pore volume (res.vol.)
We
Cumulative aquifer influx (st.vol.)
Wi
Wp
!
Cumulative water injected (st.vol.)
!
Water saturation
Temperature
Bulk volume (res.vol.)
Cumulative water produced (st.vol.)
Density (mass/vol.)
Porosity
Then, the Black Oil fluid phase behavior is illustrated by the following figures:
Fluid phase behavior parameters (Black Oil)
Bo
R so
P
Bg
P
Department of Petroleum Engineering and Applied Geophysics
Norwegian University of Science and Technology
Bw
P
P
Professor Jon Kleppe
August 17, 2004
TPG4150 Reservoir Recovery Techniques 2004
Material Balance Equations
2
!o S + ! g SR s o
Oil density:
!o =
Water compressibility:
C w = !(
Water volume change:
B w 2 = Bw 1 e "cw #P ! Bw 1 (1 " c w # P)
Bo
1 "Vw
)(
)
V w "P T
Finally, we need to quantify the behavior of the pores during pressure change in the reservoir. The rock
compressibility used in the following is the pore compressibility, and assumes that the bulk volume of the rock
itself does not change.
Pore volume behavior
Rock compressibility:
1 "!
Cr = ( )( ) T
! "P
Porosity change:
! w 2 = !w 1 e cr #P " !w 1 (1 + cr #P)
The material balance equations are based on simple mass balances of the fluids in the reservoir, and may in
words be formulated as follows:
Principle of material conservation
!Amount of fluids present% ! Amount of % !Amount of fluids remaining %
#
# #
# #
#
" in the reservoir initially & ( "fluids produced & = " in the reservoir finally &
#
# # (st. vol.) # #
#
(st. vol.)
(st. vol.)
$
' $
' $
'
We will define our reservoir system in terms of a simple block diagram, with an initial reservoir stage before
production/injection starts, and a final stage at which time we would like to determine pressure and/or
production.
Block diagram of reservoir
oil production: Np
gas production: RpNp
water production: Wp
Final stage (2)
Initial stage (1)
Gas
Gas
Oil
Oil
Water
Water
gas injection: Gi
water injection: Wi
aquifer influx: We
Department of Petroleum Engineering and Applied Geophysics
Norwegian University of Science and Technology
Professor Jon Kleppe
August 17, 2004
TPG4150 Reservoir Recovery Techniques 2004
Material Balance Equations
3
The two stages on the block diagram are reflected in the fluid phase behavior plots as follows:
Initial and final fluid conditions
Bg
Bw
2
|
2
|
1
|
1
|
P
P
Bo
R so
2
|
1
|
2
|
1
|
P
P
Note: If a gas cap is present initially, then the initial pressure is
equal to the bubble point pressure
Now, we will apply the above material balance equation to the three fluids involved, oil, gas and water:
Equation 1: Oil material balance
! Oil present %
Oil %
#in the reservoir # !
#
# #
#
( " produced & =
"
&
initially
#
# # (st. vol.) #
'
#$ (st. vol.) #' $
! Oil remaining %
#in the reservoir #
#
#
"
&
finally
#
#
#$ (st. vol.) #'
or
N - Np = V p2 S o 2 /Bo 2
yielding
So 2 =
( N ! N p ) Bo 2
Vp2
Equation 2: Water material balance
! Water present %
#in the reservoir # ! Water % ! Water % ! Aquifer %
#
# #
# #
# #
#
"
& ( " produced & + "injected & + " influx & =
initially
#
# # (st. vol.) # #(st. vol.)# #(st. vol.)#
' $
' $
'
#$ (st. vol.) #' $
!Water remaining %
# in the reservoir #
#
#
"
&
finally
#
#
#$
#'
(st. vol.)
or
V p1 S w 1 /B w1 - Wp + Wi + We = Vp 2 S w 2 /Bw 2
yielding
(
+B
" S
%" 1 %
S w 2 = *(1 + m)NBo 1$ w 1 ' $
' + Wi + We ! Wp - w 2
# 1! Sw1& # Bw1&
*)
-, V p 2
(
Department of Petroleum Engineering and Applied Geophysics
Norwegian University of Science and Technology
)
Professor Jon Kleppe
August 17, 2004
TPG4150 Reservoir Recovery Techniques 2004
Material Balance Equations
4
Equation 3: Gas material balance
Free gas
! Solution gas % !
%
# present in the # # present in the # ! Gas % ! Gas %
#
# #
# #
# #
#
"reservoir initially& + "reservoir initially& ( " produced & + "injected &
#
# #
# # (st. vol.) # #(st. vol.)#
' $
'
#$
#' #$
#' $
(st. vol.)
(st. vol.)
Free gas %
! Solution gas % !
# present in the # # present in the #
#
# #
#
="
&+"
&
reservoir
finally
reservoir
finally
#
# #
#
#$
(st. vol.) #' #$
(st. vol.) #'
or
NRs o1 + mNBo1 /Bg 2 - R p N p = (N-N p )R s o2 + Vp 2 S g 2 /B g 2
yielding
)+ #
-+ B
B &
g2
Sg 2 = * N % (Rso1 " Rso2 ) + m( o1 )( " N p (R p " Rso2 ) + G i . (
)
Bg 2 ('
+, %$
+/ V p 2
In addition to these three fluid balances, we have the following relationships for fluid saturations and pore
volume change:
!
Equation 4: Sum of saturations
S o + S w + S g = 1.0
Equation 5: Pore volume change
V p 2 =Vp 1( 1+cr !P)
Department of Petroleum Engineering and Applied Geophysics
Norwegian University of Science and Technology
Professor Jon Kleppe
August 17, 2004
TPG4150 Reservoir Recovery Techniques 2004
Material Balance Equations
5
By combining the 5 equations above, and grouping terms, we obtain the material balance relationships, as shown
below:
THE COMPLETE BLACK OIL MATERIAL BALANCE EQUATION:
(
)
F = N E o + mE g + E f ,w + (Wi + We )Bw 2 + Gi Bg 2
where
production terms are
[
(
) ]
F = N p Bo 2 + R p ! Rso2 Bg 2 + W p Bw 2
oil and solution gas expansion terms are
E o = (Bo 2 ! B o 1) + (Rso1 ! Rso2 )B g 2
gas cap expansion terms are
" Bg2 %
E g = B o 1$$
! 1''
# Bg1 &
and rock and water compression/expansion terms are
E f ,w = !(1 + m)Bo 1
Cr + Cw S w 1
"P
1! Sw1
Department of Petroleum Engineering and Applied Geophysics
Norwegian University of Science and Technology
Professor Jon Kleppe
August 17, 2004