undersaturated oil-gas simulation impes solution sig4042 reservoir simulation
TRANSCRIPT
UNDERSATURATED OIL-GAS SIMULATION
IMPES SOLUTION
SIG4042 Reservoir Simulation
FAQReferenc
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Oil and Gas Flow Equations
Questions
Black Oil Fluid Properties
Discretization of Flow Equations
Boundary Conditions
Solution by IMPES Method
Nomenclature
HomeContents
Discretization of Flow Equations »
Black Oil Fluid Properties
Solution by IMPES Method »
Boundary Conditions »
Oil and Gas Flow Equations
Questions
• Introducion
• Constant gas injection rate
• Injection at constant bottom hole pressure
• Constant oil production rate
• Production at constant reservoir voidage rate
• Production at constant bottom hole pressure
• Flow terms
• Oil storage term
• Gas storage term
• Discretized form of oil and gas equations
Handouts(pdf file)
• Assumptions
• IMPES oil pressure solution
• IMPES bubble point pressure solution
• Aplicability of IMPES method
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Oil and Gas Flow EquationsIf the oil is under-saturated, there is no free gas in the pores. Thus, we really
have one phase flow (in absence of water). However, if the amount of gas in solution varies, such as in the case of gas injection into an under-saturated reservoir, we need to keep track of the bubble point pressure, or saturation pressure, of the oil in order to account for the gas as well as in order to be able to compute the oil properties. Again, the oil density is given by:
Continue
o oS gSRso
Bo
For an under-saturated oil, the oil pressure is per definition higher than the bubble point pressure:
Thus, the formation volume factor depends on both oil pressure and bubble point pressure (or saturation pressure):
Po Pb
p
>and there is no free gas in the reservoir:
So = 0
Bo = f(Po,Pbp)Rso = f(Pbp)
Continue
The solution gas-oil ratio, on the other hand, is now a function of bubble point pressure only, and not on oil pressure:
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Oil and Gas Flow EquationsAgain, we will, for mass balance purposes, separate the oil density into two parts,
one that remains liquid at the surface and one that becomes gas:
Continue
The oil mass balance is written so that its continuity equation includes the liquid part only, while the gas mass balance includes the solution gas in the reservoir, and thus all free gas at the surface:
As in the case of saturated oil-gas systems, the solution gas will flow with the rest of the oil in the reservoir, at oil relative permeability, viscosity and pressure.
o oS gSRso
Bo
oS
Bo
gSRso
Bo
oL oG
oLooL tu
x
oGooG tu
x
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Oil and Gas Flow Equations
Although we have single phase flow, we will keep the relative permeability in the equations in case it is not equal to one. Substituting Darcy's equations and the liquid oil density and the solution gas density definitions into the continuity equations, and including production/injection terms in the equations, we obtain the following two flow equations:
A free gas rate term is included in the gas equation for gas injection purposes.
oo
o
oo
ro
Btq
x
P
B
kk
x
osoosog
o
oo
roso B
Rt
qRqx
P
B
kkR
x
The Darcy equation for the single phase in the reservoir is thus the one for oil:
x
Pkku o
o
roo
Continue
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Black Oil Fluid Properties
The Black Oil fluid properties for under-saturated oilunder-saturated oil:
In the figures, the solid lines represent saturated conditions, while the dotted lines represent under-saturated behavior, with the bubble point pressure being defined by the intersection of the dotted line and the saturated line. Thus, the bubble point pressure depends on the amount of gas present in the system. The more gas, the higher the bubble point pressure.
Po
o
o vs. Po
Pbp
Rso
Rso vs. Pbp
Po
Bo
Bo vs. Po
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Discretization of Flow EquationsDiscretization of flow equations
The discretization procedure for the flow terms of the oil-gas equations is very much similar to the one for saturated oil-gas equations.
Flow terms
The left side of the oil equation is identical to the one for a saturated system:
)()( 11 21
21 ioioixoioioixo
i
o
oo
ro PPTPPTx
P
B
kk
x
ioioixosoioioixoso
i
o
oo
roso PPTRPPTR
x
P
B
kkR
x
11 21
21
For the gas equation, we only have the solution gas terms, which may be approximated just as for the solution gas term of the saturated equation:
Continue
Continue
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Discretization of Flow EquationsDefinitions of the terms in the equation:
i
i
i
ii
io
xoi
kx
kx
x
T
1
1
21
21
2
o kro
oBo
The upstream mobilities are selected as:
Solution gas-oil ratios:
i
i
i
ii
io
xoi
kx
kx
x
T
1
1
21
21
2
ioioio
ioioio
ioPPif
PPif
1
11
21
ioioio
ioioio
ioPPif
PPif
1
11
21
Where:
ioioiso
ioioiso
isoPPifR
PPifRR
1
11
21
ioioiso
ioioiso
isoPPifR
PPifRR
1
11
21
Continue
Continue
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Discretization of Flow EquationsOil storage term
We expand the right hand side of the oil equation as follows, keeping in mind that the formation volume factor is a function of oil pressure as well as bubble point pressure:
For the last term, we use the standard approximation:
Continue
where Pbp now is the variable bubble point pressure, a new unknown to be solved for together with oil pressure.
Continue
t
P
P
B
t
P
P
B
t
P
dP
d
BBtbp
bp
oo
o
oo
ooo
/1/11
The two first terms above are identical to the two first terms of the saturated oil equation, with the exception that the derivative of the inverse formation volume factor is now a partial derivative, and that the the gas saturation is now zero. Thus, we can write the approximation of these two terms directly:
tioio
io
o
o
ri
i
o
o
oo
oo
PPP
B
B
c
tt
P
P
B
t
P
dP
d
B
/1/11
tibpibp
ibp
oi
i
bp
bp
o PPP
B
tt
P
P
B
/1/1
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Discretization of Flow EquationsGas storage term
We expand the right hand side of the gas equation as follows:
Continue
Since the first term is equal to the right hand side of the oil equation, multiplied by the solution gas-oil ratio, we can write directly:
Continue
The second term becomes:
t
P
dP
dR
BBtR
B
R
tbp
bp
so
ooso
o
so
tibpibp
ibp
oisotioio
io
o
o
riso
ioso PP
P
B
t
RPP
P
B
B
c
t
R
BtR
/1/1
tibpibp
ibp
so
io
i
i
bp
bp
so
oPP
dP
dR
tBt
P
dP
dR
B
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Discretization of Flow EquationsThe oil and gas equations in discretized form may be written, including the well terms:
where, for the oil equation: More
t
ibpibpibpotioioipoo
oiioioixoioioixo
PPCPPC
qPPTPPT
1121
21
t
ibpibpibpgtioioipog
giiosoioioixosoioioixoso
PPCPPC
qqRPPTRPPTR
1121
21
io
o
o
riipoo
P
B
B
c
tC
)/1(
ibp
oiipoo
P
B
tC
)/1(
and for the gas equation:
io
o
o
risoipog
P
B
B
c
t
RC
)/1(
ibp
so
obp
oso
iibpg
dP
dR
BP
BR
tC
1)/1(
The derivative terms to be computed numerically for each time step based on the input table to the model, now are:
, andio
o
P
B
)/1(
ibp
o
P
B
)/1(
ibp
so
dP
dR
Ni ,...,1
Ni ,...,1
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Boundary ConditionsBoundary conditions - introduction
The boundary conditions for undersaturated oil-gas systems are somewhat simpler than those for saturated oil-gas systems since only one phase is produced in the reservoir. Normally, we inject gas in a grid block at constant surface rate or at constant bottom hole pressure, and produce oil (and solution gas) from a grid block at constant bottom hole pressure, or at constant surface oil rate. Again, we may sometimes want to specify constant reservoir voidage rate, where either the amount of injection of gas is to match a specified rate of oil production at reservoir conditions, so that average reservoir pressure remains constant, or the reservoir production rate is to match a specified gas injection rate.
Continue
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Boundary ConditionsConstant gas injection rate
Again, a gas rate term is already included in the gas equation. Thus, for a constant surface gas injection rate of Qgi (negative) in a well in grid block i:
It is normally assumed that the injected gas immediately goes into solution with the oil in the injection gridblock. The bottomhole pressure in the well will therefore be a function of the oil mobility around the well, since there is not any free gas present. Thus, the bottom hole presure may be determined from the same expression for the well as for saturated oil-gas flow:
and since the gas mobility term is zero:
Continuei
gigi xA
w
e
ii
rr
hkWC
ln
2
i
e
xyr
)(' ibhiogioigi
oi
i
igi PP
B
B
xA
WCq
)(' ibhiooigi
oi
i
igi PP
B
B
xA
WCq
The well constant is as before given by:
and:
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Boundary ConditionsInjection at constant bottom hole pressure
Using the previous well equation:
The terms must be included in the appropriate coefficients in the pressure solution. At the end of the time step, the above equation may be used to compute the actual gas injection rate for the step.
)(' ibhiooigi
oi
i
igi PP
B
B
xA
WCq
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Boundary ConditionsConstant oil production rate
For the oil equation, this condition is handled as for the saturated case. Thus, for a constant surface oil production rate of Qoi (positive) in a well in grid block i:
Oil production will always be accompanied by solution gas production, and this is accounted for in the gas equation by the term q’
oiRsoi.
At the end of a time step, after having solved the equations, the bottom hole production pressure for the production well may be calculated using the well equation for oil:
Continue
As for oil-water systems, production wells in oil-gas systems are normally constrained by a minimum bottom hole pressure. If this is reached, the well should be converted to a constant bottom hole pressure well.
The gas-oil ratio, GOR, at the surface, is for undersaturated systems equal to the solution gas-oil ratio:
i
oioi xA
Continue
ibhiooiioi PPWCQ
isoi RGOR
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Boundary ConditionsProduction at constant reservoir voidage rate
If the total production of fluids from a well in block i, at reservoir conditions, is to match the reservoir injection volume so that the reservoir pressure remains approximately constant, the following relationship must be obeyed:
Thus:
Continue
The bottom hole pressure of the production well may be computed at the end of the time step from:
Continue
oiinjg
injo
injginjginjgoioi B
BQBQBQ
i
oiinjginjg
oioi xA
BQ
Bq
1
q oi WCi
Axi
oi(Poi Pbhi )
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Boundary ConditionsProduction at constant bottom hole pressure
For a production well in grid block i with constant bottom hole pressure, Pbhi, we have an oil rate of:
Continue
or
)( ibhioioiio PPWCQ
The terms appearing in the rate expressions will again have to be included in the appropriate matrix coefficients when solving for pressures. At the end of each time step, actual oil rate and GOR are computed.
)( ibhiooii
ioi PP
xA
WCq
)( ibhiooii
iisooiiso PP
xA
WCRqR
The solution gas rate is:
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Solution by IMPES MethodSolution by IMPES method
The procedure for IMPES solution is similar to the previous cases. To be correct, the method should now be labeled IMPEBPP, for Implicit pressure, Explicit Bubble Point Pressure. We make the same type of assumptions in regard the coefficients in the equations:
ContinueHaving made these approximations, the equations become:
Ni ,...,1
Ni ,...,1
tso
txo RT ,
tpog
tpoo CC ,
tbpg
tbpo CC ,
tibpibp
tibpo
tioio
tipoooiioio
tixoioio
tixo PPCPPCqPPTPPT 11
21
21
t
ibpibptibpg
tioio
tipog
giiosoioiot
ixosoioiot
ixoso
PPCPPC
qqRPPTRPPTR
1121
21
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Solution by IMPES MethodIMPES oil pressure solution
The oil pressure equation for the saturated oil-gas becomes:
Ni ,...,1
tibpo
tibpg
i C
C
t
ioiotipogi
tipoo
ioit
sogioi
ioiot
ixosoitixoioio
t
ixosoitixo
PPCCqRqq
PPTRTPPTRT
1121
21
21
21
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tibpo
tibpg
i C
C
Solution by IMPES MethodIMPES pressure solution (cont)
We then rewrite the oil pressure equation as:
iiiiiii dPcPbPa ooo 11
Ni ,...,1 tixosoi
tixoi TRTa
21
21 tixosoi
tixoi TRTc
21
21
tipog
t
ixosot
ixosoi
tipoo
tixo
tixoi
CTRTR
CTTb
21
21
21
21
iosogioi
tio
tipogi
tipooi qRqqPCCd )(
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Modifications for boundary conditions
As before, all rate specified well conditions are included in the rate terms qoi’, qgi’ and Rsoiqoi’ , and are already appropriately included in the di term above.
For injection of gas at bottom hole pressure specified well conditionsinjection of gas at bottom hole pressure specified well conditions, the following expression apply:
In a block with a well of this type, the following matrix coefficients are modified:
)(' ibhiooigi
oi
i
igi PP
B
B
xA
WCq
toit
gi
toi
i
itipog
tixg
tixgi
tipo
tixo
tixoi
B
B
xA
WCCTT
CTTb
21
21
21
21 )(
ibhtoit
gi
toi
i
iiibh
toi
i
itio
tipogi
tipooi P
B
B
xA
WCP
xA
WCPCCd
)(
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For production at bottom hole pressure specified well production at bottom hole pressure specified well conditionsconditions, we have the following expressions:
In a block with a well of this type, the following matrix coefficients are modified:
q oi WCi
Axi
oi(Poi Pbhi )and
Again, the pressure equation may now be solved for oil pressures by using Gaussian elimination.
)( ibhiooiiso
i
ioiiso PPR
xA
WCqR
toiiso
i
itipog
t
ixosot
ixosoi
toi
i
itipoo
tixo
tixoi
RxA
WCCTRTR
xA
WCCTTb
21
21
21
21 )(
ibhtoi
tiso
i
iiibh
toi
i
itio
tipogi
tipooi PR
xA
WCP
xA
WCPCCd
)(
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IMPES bubble point pressure solution
Having obtained the oil pressures above, we need to solve for bubble point pressures using either the oil equation or the gas equation. In the following we will use the oil equation for this purpose:
Since bubble point pressure only appears as an unknown in the last term on the right side of the oil equation, we may solve for it explicitly:
For grid blocks having pressure specified oil production wells, we make appropriate modifications, as discussed previously:
Continue
tibpibp
tibpo
tioio
tipoooiioio
tixoioio
tixo PPCPPCqPPTPPT 11
21
21
tioio
tipoooiioio
tixoioio
tixo
tibpo
tibpibp PPCqPPTPPT
CPP 11
21
21
1
tioio
tipoo
tibhio
toi
gi
oi
i
iioio
tixoioio
tixo
tibpo
tibpibp PPCPP
B
B
xA
WCPPTPPT
CPP 11
21
21
1
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IMPES bubble point pressure solution (cont)
Having obtained oil pressures and water saturations for a given time step, well rates or bottom hole pressures may be computed, if needed, from, from the following expression for an injection well:
For a production well:
The surface gas-oil ratio is of course equal to the solution gas-oil ratio:Continue
q oi WCi
Axi
oi(Poi Pbhi )
)( ibhiooiiso
i
ioiiso PPR
xA
WCqR
and
)(' ibhiooigi
oi
i
igi PP
B
B
xA
WCq
isoi RGOR
Required adjustments in well rates and well pressures, if constrained by upper or lower limits, are made at the end of each time step, before all coefficients are updated before proceeding to the next time step.
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Solution by IMPES MethodApplicability of IMPES method
Applicability of the IMPES method for undersaturated oil-gas systems is fairly much as for the previously discussed systems. However, due to the fact that changes in bubble point pressure may be rapid, relatively small time step sizes may be required.
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Questions1) What are the criteria for undersaturated flow? What are the
functional dependencies of Rso and Bo? Next2) What are the primary unknowns when solving the undersaturated
equations?
3) Write the equations for undersatrated flow.
4) Make sketches of Black Oil fluid properties for undersaturated system.
5) Write the two flow equations for oil and gas on discretized forms for a
undersaturated system in terms ot transmissibilities, storage
coefficients and pressure differences.
All questions are taken from former exams in Reservoir Simulation
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B - formation volume factor
C - storage coefficient
c - compressibility, 1/Pa
GOR - gas-oil ratio
k - permeability, m2
kr - relative permeability, m2
L - lenght, m
N - number of grid blocks
P - pressure, Pa
Pbh - bottom hole pressure, Pa
Pbp - bubble point pressure, Pa
Pc - capillary pressure, Pa
Q, q - flow rate, Sm3/d
Rso - solution gas-oil ratio
r - radius, m
S - saturation
T - transmissibility, Sm3/Pa·s
t - time, s
u - Darcy velocity, m/s
WC - well constant
x - distance, m
x, y, z - spatial coordinate
x - lenght of grid block, m
t - time step, s
- porosity
- mobility, m2/Pa·s
- viscosity, Pa·s
- density, kg/m3
Subscripts:
d - drainage
e - end of reservoir
f - fluid
G, g - gas
i - block number
inj - injection
ir - irreducible
L, l - liquid
o - oil
S - surface (standard) conditions
r - residual
r - rock
t - total
w - water
w - well
Nomenclature
Back to presentation
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General information
About the author
Title: Undersaturated Oil-Gas Simulation - IMPES Solution
Teacher(s): Jon Kleppe
Assistant(s): Szczepan Polak
Abstract: Criteria for undersaturated oil conditions. Black Oil fluid properties for undersaturated oil. IMPES solution for undersaturated system.
Keywords: undersaturated oil-gas simulation, black oil fluid properties, IMPES, bubble point pressure
Topic discipline: Reservoir Engineering -> Reservoir Simulation
Level: 4
Prerequisites: Good knowledge of reservoir engineering
Learning goals: Learn basic principles of Reservoir Simulation
Size in megabytes: 1.0
Software requirements: -
Estimated time to complete: 45 minutes
Copyright information: The author has copyright to the module
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FAQ
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References
Aziz, K. and Settari, A.: Petroleum Reservoir Simulation, Applied Science Publishers LTD, London (1979)
Mattax, C.C. and Kyte, R.L.: Reservoir Simulation, Monograph Series, SPE, Richardson, TX (1990)
Skjæveland, S.M. and Kleppe J.: Recent Advances in Improved Oil Recovery Methods for North Sea Sandstone Reservoirs, SPOR Monograph, Norvegian Petroleum Directoriate, Stavanger 1992
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Summary
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About the AuthorName
Jon Kleppe
Position
Professor at Department of Petroleum Engineering and Applied Geophysics at NTNU
Address: NTNU S.P. Andersensvei 15A 7491 Trondheim
E-mail: [email protected]
Phone: +47 73 59 49 33
Web: http://iptibm3.ipt.ntnu.no/
~kleppe/