UNDERSATURATED OIL-GAS SIMULATION IMPES SOLUTION SIG4042 Reservoir Simulation

UNDERSATURATED OIL-GAS SIMULATION

IMPES SOLUTION

SIG4042 Reservoir Simulation

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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 »


Solution by IMPES Method »

Boundary Conditions »


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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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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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

Qq

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

Qq

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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Continue

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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Continue

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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Continue

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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Continue

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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Nomenclature

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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With link to an web-site with contact information, taks and publications

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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/

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UNDERSATURATED OIL-GAS SIMULATION IMPES SOLUTION SIG4042 Reservoir Simulation