sc re chap18- immiscible displ

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Immiscible DisplacementImmiscible Displacement

Adrian C ToddAdrian C Todd

Heriot-Watt UniversityHeriot-Watt University

INSTITUTE OF PETROLEUM ENGINEERING

Heriot-Watt UniversityHeriot-Watt University

INSTITUTE OF PETROLEUM ENGINEERING

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Immiscible displacementImmiscible displacement Context: water displacing oil.Context: water displacing oil.

Water drive a very effective recovery mechanism.Water drive a very effective recovery mechanism.

Water injection a very common oilfield practise.Water injection a very common oilfield practise.

Modelling water drive is often carried out within Modelling water drive is often carried out within reservoir simulators.reservoir simulators.

Objective: to understand the mechanisms rather Objective: to understand the mechanisms rather than blindly using the ‘black box’.than blindly using the ‘black box’.

Useful texts; Dake & Chierici.Useful texts; Dake & Chierici.

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Reasons for Water InjectionReasons for Water Injection Main intervention method.Main intervention method.

High recovery & available fluids.High recovery & available fluids.

Historically considered as a secondary recovery process.Historically considered as a secondary recovery process.

Using water to sweep out previously unrecovered oil.Using water to sweep out previously unrecovered oil.

Todays perspective is keeping reservoir pressure above Todays perspective is keeping reservoir pressure above bubble point.bubble point.

Pressure maintenance. Voidage replacementPressure maintenance. Voidage replacement

Prevents solution gas drive.Prevents solution gas drive.

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Technical challenges of water InjectionTechnical challenges of water Injection

Water is produced.Water is produced.

Environmental. A disposal challenge of Environmental. A disposal challenge of contaminated water.contaminated water.

Reinjection of produced water.Reinjection of produced water.

Chemical incompatibility -scaleChemical incompatibility -scale

CorrosionCorrosion

BacteriaBacteria

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Benefits of water InjectionBenefits of water Injection- Zone Isolation- Zone Isolation In some cases not all of the reservoir may receive aquifer In some cases not all of the reservoir may receive aquifer

support.support.

Aquifer support may be uncertainAquifer support may be uncertain

Only after production support becomes evident.Only after production support becomes evident.

TimeRate

RateTime

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High PermeabilityHigh Permeability

kA dPq

dl

High injection / production ratesHigh injection / production rates

Wide well spacingWide well spacing

Rapid fluid movementRapid fluid movement

Accelerated developmentAccelerated development

Qi Qp

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Oil viscosityOil viscosity

Low oil viscosity leads to Low oil viscosity leads to good mobility ratio.good mobility ratio.

'rw w'ro o

k /Mobility Ratio, M

k /

Krw’Kro’

M < 1 gives piston like displacement

1-Sor

Sw

Swc

x

OilWaterKrw’

Kro’

Oil

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Undersaturated Undersaturated Reservoirs Reservoirs

Pressure surveys Pressure surveys can give an can give an excellent insight excellent insight into communication into communication in the reservoir.in the reservoir.

Enables layered Enables layered nature so important nature so important in waterflooding to in waterflooding to be defined.be defined.

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Overpressured ReservoirsOverpressured Reservoirs

Can let the pressure fall significantly Can let the pressure fall significantly to get a ‘feel’ of the reservoir.to get a ‘feel’ of the reservoir.

P

Time

5500psi

3500psi

4500psi

Pressure

De

pth

1000-2000psiN. Viking Graben-N.Sea

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Reservoir DepthReservoir Depth In offshore environments In offshore environments

facilities very costly.facilities very costly.

In waterflooding injection In waterflooding injection wells at the extremes of the wells at the extremes of the formation.formation.

Well slots have to be Well slots have to be capable of reaching these capable of reaching these limits.limits.

Horizontal wells now enable Horizontal wells now enable shallower accumulations to shallower accumulations to be reachedbe reached

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Facility DesignFacility Design

Issues:Issues:

Where to inject.Where to inject.

Ability to inject.Ability to inject.

When and how much water is produced.When and how much water is produced.

How to handle it.How to handle it.

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Thermal FracturingThermal Fracturing Experience over recent years has Experience over recent years has

shown that water injection injectivity shown that water injection injectivity has been better than expected.has been better than expected.

The cold water has cooled the The cold water has cooled the formation such that injection formation such that injection pressures have been able to pressures have been able to overcome fracture pressure, which overcome fracture pressure, which reduces when cooled.reduces when cooled.

Thermal fracturing.Thermal fracturing.

Provides a higher injection area Provides a higher injection area than for a radial injection geometry than for a radial injection geometry

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Water HandlingWater Handling Water handling- one of the Water handling- one of the

major technical challenges major technical challenges of the oil industry.of the oil industry.

Environmental pressures Environmental pressures increasing in relation to oil increasing in relation to oil emissions.emissions.

Reservoir engineers need Reservoir engineers need to predict water production to predict water production to enable design.to enable design.

When and how much.When and how much.

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Basic Waterdrive TheoryBasic Waterdrive Theory Method applicable to gas injection as well as water injection.Method applicable to gas injection as well as water injection.

Immiscible processImmiscible process

No mixing of the respective injection and displaced fluids.No mixing of the respective injection and displaced fluids.

Miscible processMiscible process where injected phases mix with displaced phase where injected phases mix with displaced phase by mass transfer of components.by mass transfer of components.

In our calculations we are combining properties, measurements and In our calculations we are combining properties, measurements and application over a huge range of physical scales.application over a huge range of physical scales.

This is the case in oil recovery predictions.This is the case in oil recovery predictions.

Important not to make ‘unrealsitic’ jumps in application of dataImportant not to make ‘unrealsitic’ jumps in application of data

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In water-oil displacement we are dealing with a process which takes place at In water-oil displacement we are dealing with a process which takes place at a range of scales.a range of scales.

Pore or microscopic scalePore or microscopic scale..

Isolation and movement of fluids is dependant on, IFT, wettability, viscosity, Isolation and movement of fluids is dependant on, IFT, wettability, viscosity, pore size and shape.pore size and shape.

Larger, macroscopic scaleLarger, macroscopic scale..

Behaviour at laboratory level scale, e.g. core plug scale. Permeability, relative Behaviour at laboratory level scale, e.g. core plug scale. Permeability, relative permeability and capillary pressure.permeability and capillary pressure.

Field Scale, or behavioural scaleField Scale, or behavioural scale..

Quantum leap of scale. Heterogeneous formations.Quantum leap of scale. Heterogeneous formations.

Vertical segregation over large thickness.Vertical segregation over large thickness.

Basic Waterdrive TheoryBasic Waterdrive Theory

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Levels of viewing Levels of viewing displacementdisplacement

Field scale

Electron microscope

scale

Core scale

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Water Oil Displacement at Microscopic and Water Oil Displacement at Microscopic and Macroscopic levelsMacroscopic levels

Trapped oil in swept area held by capillary Trapped oil in swept area held by capillary and IFT forces.and IFT forces.

Residual saturation SorResidual saturation Sor

10-40% of the pore space10-40% of the pore space

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Water Oil Water Oil Displacement at Displacement at Microscopic levelMicroscopic level

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Water Oil Displacement at Reservoir LevelWater Oil Displacement at Reservoir Level

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Factors that Dictate Flooding EfficiencyFactors that Dictate Flooding Efficiency

Mobility RatioMobility Ratio

HeterogeneityHeterogeneity

GravityGravity

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Relative PermeabilityRelative Permeability Darcy’s law for 100% single Darcy’s law for 100% single

phase saturation.phase saturation.

Relative permeability used to Relative permeability used to relate absolute permeability ( relate absolute permeability ( single phase ) to the effective single phase ) to the effective permeability when more than permeability when more than on phase present.on phase present.

Q k pu

A l

ewrw

kk

k

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Relative Relative PermeabilityPermeability Darcy’s law for linear Darcy’s law for linear

flow for two fluids in flow for two fluids in an inclined an inclined configuration.configuration.

ro oo o

o

rw ww w

w

kk A pq gSin

x

kk A pq gSin

x

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Relative Relative PermeabilityPermeability

End point relative End point relative permeabilities, krw’ & kro’.permeabilities, krw’ & kro’.

Limiting saturations of the Limiting saturations of the respective phasesrespective phases

Relative permeabilities from Relative permeabilities from 1D core floods.1D core floods.

Either viscous displacement or Either viscous displacement or steady state co-injection of steady state co-injection of fluids.fluids.

Unlikely to be representative Unlikely to be representative of reservoirof reservoir

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Relative PermeabilityRelative Permeability

SwcSorA

L

'rw w'ro o

k /Mobility Ratio, M

k /

Express volumes in the core plug as pore

volumes, PV.

PV AL

or wcMovable oil volume, MOV. PV 1 S S

End point rel. perm. values represent maximum velocity of the water flow compared to the maximum velocity of the oil.

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Fractional FlowFractional Flow For flow in core plug or a reservoir. Ratio of flow of water at any For flow in core plug or a reservoir. Ratio of flow of water at any

point is termed:point is termed:

fractional flowfractional flow..

ww

w o

qf

q q

o t wq q q

Subtracting Darcy equations for water and oil, P atmos.

c o wP p p

x x x

w o where

w o t o cw 6

rw ro ro

q P gSinq A

kk kk kk x 1.0133 10

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Fractional FlowFractional Flow

Values now inserted into fractional flow equation

ww

w o

qf

q q

ro c

6t o

wro w

rw o

kk A P gSin1

q x 1.0133 10f

k1

k

Dake simplified this to; wro w

rw o

1 Gf

k1

k

4 ro

t o

kk A gSinG 4.886 10

q

G = positive gravity number

qt/A=velocity term

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Impact of parameters on fractional flowImpact of parameters on fractional flow

Angle of dipAngle of dip

Water injected downdip gravity term is positive-Water injected downdip gravity term is positive-reduces fractional flow.reduces fractional flow.

If gas injection downdip gravity term positive If gas injection downdip gravity term positive increasing fractional flow of gas.increasing fractional flow of gas.

If dip angle zero, horizontal flow, gravity term is zeroIf dip angle zero, horizontal flow, gravity term is zero

4 ro

t o

kk A gSinG 4.886 10

q

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Impact of parameters on fractional flow- Impact of parameters on fractional flow- Capillary pressureCapillary pressure

Capillary pressure increases fractional flow of water

c c w

w

P P S

x S x

ro c6

t ow

ro w

rw o

kk A P gSin1

q x 1.0133 10f

k1

k

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Impact of parameters on fractional flow- Impact of parameters on fractional flow- velocityvelocity

For edge water drive velocity much higher than For edge water drive velocity much higher than bottom water drive- e.g. v=0.2ft/dbottom water drive- e.g. v=0.2ft/d

G=0.22G=0.22

For bottom water much lower velocity v= 0.004 ft/dFor bottom water much lower velocity v= 0.004 ft/d

G=10.29G=10.29

Leads to piston displacement for bottom water Leads to piston displacement for bottom water drive.drive.

Problem here however is well bore coningProblem here however is well bore coning

4 ro

t o

kk A gSinG 4.886 10

q

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Fractional FlowFractional Flow

If angle of dip and If angle of dip and capillary pressure capillary pressure effects ignored. effects ignored.

Fractional flow Fractional flow equation becomesequation becomes

ww ro

o rw

1f

k1

k

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Relationship between capillary pressure,relative Relationship between capillary pressure,relative permeability and fractional flowpermeability and fractional flow

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Relationship Relationship between capillary between capillary pressure, relative pressure, relative permeability and permeability and fractional flowfractional flow

At A the well only produces oil

At B,45%saturation both oil & water produced with water cut of 50%.

At C advancing water isolated irreducible oil saturation

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Displacement TheoriesDisplacement Theories Assumptions:Assumptions:

Displacement is incompressibleDisplacement is incompressible..

qqtt=q=qoo+q+qww=q=qii

qqtt=total flow rate in reservoir.=total flow rate in reservoir.

qqoo=oil flow rate in reservoir.=oil flow rate in reservoir.

qqww=water flow rate in reservoir.=water flow rate in reservoir.

qqii=water injection flow rate in reservoir.=water injection flow rate in reservoir.

Diffuse FlowDiffuse Flow

The saturations at any point are uniformly distributed The saturations at any point are uniformly distributed over the thickness.over the thickness.

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Diffuse FlowDiffuse Flow Enables a 1 dimensional simple analysis.Enables a 1 dimensional simple analysis.

A simple core flood this assumption not A simple core flood this assumption not unreasonable.unreasonable.

Can be encountered in a reservoir injection Can be encountered in a reservoir injection where rates are high and vertical equilibrium where rates are high and vertical equilibrium not able to be established.not able to be established.

For low injection rates where the thickness is For low injection rates where the thickness is small compared to transition zone.small compared to transition zone.

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Buckley-Leverett TheoryBuckley-Leverett Theory

Established in 1942 for displacement calculations.Established in 1942 for displacement calculations.

Theory for linear one dimensional displacement.Theory for linear one dimensional displacement.

Total flow rate is constant. (Incompressible ).Total flow rate is constant. (Incompressible ).

Theory determines the velocity of a plane of constant Theory determines the velocity of a plane of constant water saturation moving through a linear system.water saturation moving through a linear system.

e.g. a core.e.g. a core.

Well founded on principle of conservation of massWell founded on principle of conservation of mass

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Mass flow rate in - mass flow rate out = rate of increase of mass in volume

w w w w w wx x dxq q A dx S

t

or w w w w w w w wx xq q q dx A dx S

x t

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w w w wq A Sx t

For incompressible flow , wconstant

w w

t x

q SA

x t

Differential of water saturation is

w ww t

x

S SdS dx dt

x t

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Buckley-Leverett TheoryBuckley-Leverett TheoryWe are examining the advancement of a particular saturation.

Sw is constant therefore dSw=0, therefore

w w

Swx

S S dx

t x dt

Alsow w w

t w t

q q S

x S t

Combining in w w

t x

q SA

x t

givesw

Swt

q xA

S t

For incompressible flow qt is constant.

Water flow rate is total flow times fractional flow, qw=qt x fw.

t wSw

Sw w Sw

q fdxv

dt A S

VSw is the velocity of the plane of saturation,Sw

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This is the Buckley-Leverett equation.This is the Buckley-Leverett equation.

Also called the Equation of Characteristics.Also called the Equation of Characteristics.

It indicates the velocity of a plane of saturation It indicates the velocity of a plane of saturation moving through the linear system.moving through the linear system.

It enables the calculation of Sw as a function of time It enables the calculation of Sw as a function of time and distance.and distance.

It indicates the dependance on the derivative of the It indicates the dependance on the derivative of the fractional flow curve.fractional flow curve.

t wSw

Sw w Sw

q fdxv

dt A S

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Equation of CharacteristicsEquation of Characteristics

Clierici has given a thorough analysis of the Clierici has given a thorough analysis of the displacement process for three fractional flow displacement process for three fractional flow curve types.curve types.

Important to understand the initial boundary Important to understand the initial boundary conditionsconditions

L

w wiS S for o<x L, t o w orS 1 S for x=0, t o

w wiS S for o<x L, t o w orS 1 S for x=0, t o

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Equation of CharacteristicsEquation of Characteristics

If the initial conditions at t=0 are applied to the general equation

t wSw

Sw w Sw

q fdxv

dt A S

and the equation is then integrated a general solution of the displacement process is obtained for the calculation of Sw in

terms of x and t.Velocity x time =distance.

t ww 0 wt

w Sw

q dfx S x S t

A dS

This equation describes a series of straight lines, the characteristics, with an initial ordinate value of 0 wx S

and slope oft w

w Sw

q df

A dS

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Viscous OilsViscous Oils Viscosity of displaced phase much greater Viscosity of displaced phase much greater

than injected water phase.than injected water phase.

Fractional flow curve has a concave Fractional flow curve has a concave downward slope.downward slope.

Its gradient fw’ increases from Sw=1-Sor. To a Its gradient fw’ increases from Sw=1-Sor. To a maximum at Sw=Swi+maximum at Sw=Swi+SwiSwi

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Viscous Oils

Cierici

c ACTODD

Velocity of Sw increases from its value at Sw=1-Sor to its maximum Velocity of Sw increases from its value at Sw=1-Sor to its maximum at Sw just greater than Swi.at Sw just greater than Swi.

Heavy oil produces water very early and steadily increases until too Heavy oil produces water very early and steadily increases until too high levels of water.high levels of water.

Viscous OilsViscous Oils

Cierici

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Very light oilsVery light oils

Low relative viscosity to injected water.Low relative viscosity to injected water.

Large gravitational effects.Large gravitational effects.

Low velocityLow velocity

Concave upward fractional flow curve.Concave upward fractional flow curve.

fw’ curve decreasing from value at Sw=1-Sor fw’ curve decreasing from value at Sw=1-Sor to a minimum at Sw just greater than Swi.to a minimum at Sw just greater than Swi.

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Very light oils

Cierici

upwards

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Fastest saturation 1-Sor quickly overtakes other saturations.Fastest saturation 1-Sor quickly overtakes other saturations.

Shock front developed.Shock front developed.

Until shock front arrives water free oil productionUntil shock front arrives water free oil production

Very light oilsVery light oils

Cierici

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Typical Medium Density OilsTypical Medium Density Oils Displacement velocities not unlike field Displacement velocities not unlike field

valuesvalues

S shaped fractional flow curveS shaped fractional flow curve

Two curvaturesTwo curvatures

Derivative curve, fw’. Slope Derivative curve, fw’. Slope increases from its starting value increases from its starting value Sw=1-Sor and then decreases.Sw=1-Sor and then decreases.

Cierici

c ACTODD

Typical Medium Density OilsTypical Medium Density Oils Development of the saturation would be a steady increase in Development of the saturation would be a steady increase in

the velocity of the increasing saturation.the velocity of the increasing saturation.

This would reach a maximum at Swf whereSwi<Swf<(1-Sor)This would reach a maximum at Swf whereSwi<Swf<(1-Sor)

Behind this velocities decrease with increasing Sw. Behind this velocities decrease with increasing Sw.

Cierici

c ACTODD

Typical Medium Density OilsTypical Medium Density Oils A shock front is developed at value of Swf.A shock front is developed at value of Swf.

The saturations greater than this moving at a lower velocity.The saturations greater than this moving at a lower velocity.

Behind this shock front a steady increase in the saturations Behind this shock front a steady increase in the saturations moving at decreasing velocitymoving at decreasing velocity

Cierici

c ACTODD

Water free oil is produced until breakthrough at Water free oil is produced until breakthrough at saturation saturation SSwfwf and fractional flow of and fractional flow of ffwbtwbt..

Saturation then climbs until irreducible oil saturation Saturation then climbs until irreducible oil saturation value, when only water is produced.value, when only water is produced.

Typical Medium Density OilsTypical Medium Density Oils

Cierici

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Velocity of shock frontVelocity of shock front Material balance across Material balance across

the shock front.the shock front.

R ahead L behind.R ahead L behind.

Velocity of front vf=dxf/dt

w,L w,R w,L w,R fq dt q dt A S S dx

Sincetw wq q f

w,L w,Rtf

f

w,L w,F

f fqdxv

dt A S S

Since t tq Au

w,L w,Rtff

w,L w,F

f fudxv

dt S S

Rankine-Hugoniot

condition for velocity of shock fronts

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Velocity of shock frontVelocity of shock front

Limiting conditionsLimiting conditions

w,L or w,R wi

w,L w,R

S 1 S S S

f 1 f 0

Therefore

w,L w,R

or wiw,L w,R

f f 1tan

1 S SS S

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From Buckley Leverett equation1 Sor

t wSw

1 Sor w 1 Sor

u fdxv

dt S

Combining equations

w,L w,Rt w t t t1 Sor

1 Sor w or wiw,L w,R1 Sor

f fu f u u udx 1v tan

dt S 1 S SS S

If we apply the Rankine-Hugonoit condition to the medium oil viscosity condition, the S shaped fw curve

w,L wf w,R wi

w,L w wf w,R

S S S S

f f S f 0

w,L w,Rt w t t wf twf

wf w wf wiw,L w,Rwf

f fu f u u f udxv tan

dt S S SS S

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Graphical ProcedureGraphical Procedure Examination of these equations provides a convenient Examination of these equations provides a convenient

procedure to determine conditions at the shock frontprocedure to determine conditions at the shock front

w wf

w wf wiwf

df ftan

dS S S

w wf

w wf wiwf

df ftan

dS S S

Tangent drawn to the fractional flow curve from Swi,0, meets the curve at

shock front conditions.

Time for breakthough to producer

wf wibt

f t w swf

S SL Lt

v u f

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Welge AnalysisWelge Analysis Welge provided a method to obtain average saturation Welge provided a method to obtain average saturation

behind shock front.behind shock front.

Useful for oil recovery calculations.Useful for oil recovery calculations.

Water injected at a rate of qw

Time for breakthrough, tbt wf wi

btf w w swf

S SL ALt

v q f

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Welge Welge AnalysisAnalysis

Before water arrives at producer, the volume of

oil produced is equivalent to volume of water injected, Wi=qw x t.

At breakthrough volume of oil produced, Np is also the difference between the initial oil volume, (AL(1-Swi). Less

that remaining in terms of the average water saturation, Sw, (AL(1-Swavg)

pbt w btN q t

pbt w wiN AL S S

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wf wipbt w bt w wi

w swf

S SN q t AL AL S S

f

Therefore wf wiw wi

wf

S SS S

f

From a previous equation w wf

w wf wiwf

df ftan

dS S S

Therefore w wi

w

w Swf

1S S

dfdS

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Combining equations gives

wf wi w wf wwf wbt wf

w wi w wi w Swf

S S S S df1 f 1 S S

S S S S dS

Rearranging gives wfwbt wf

w

w Swf

1 fS S

dfdS

w wf

w wbt wf wbt wiSwf

df 1 f 1

dS S S S S

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This provides another graphical significance enabling the average saturation to be determined.

w wf

w wbt wf wbt wiSwf

df 1 f 1

dS S S S S

The line of the tangent at breakthrough also cuts the fw=1 line at

Sw=Sw avg

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Oil Recovery CalculationsOil Recovery Calculations Analysis so far is for displacement process for a small Analysis so far is for displacement process for a small

core plug of oil by watercore plug of oil by water

In recovery calculations useful to express volumes in In recovery calculations useful to express volumes in pore volumes, PVpore volumes, PV

SwcSorA

L

PV A L

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Oil Recovery - Before BreakthroughOil Recovery - Before Breakthrough Before breakthrough only oil produced.Before breakthrough only oil produced.

At a rate the same as water injected because it is an At a rate the same as water injected because it is an incompressibile system.incompressibile system.

At breakthrough >At breakthrough > bt bt btpd id id bt w wiN W q t S S

bt

bt

pd

id

id

iid

bt

idbt

N pore volumes oil produced at breakthrough

W pore volumes water injected at breakthrough

q water injection rate in pore volumes

qq

A L

t is the time taken for water to breakthrough

Wt

t

bt

id

w

q

S average water saturation in reservoir at breakthough

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Oil Recovery - Before BreakthroughOil Recovery - Before Breakthrough At breakthrough when x=LAt breakthrough when x=L

In Buckley-Leverett eqn.In Buckley-Leverett eqn.

t bt w wiSw bt bt Sbt

Sw w wSw Sw

q t f fdx Wv t t x L

dt A S A S

iid

w

w Swbt

W 1W

LA dfdS

Oil recovery at breakthrough equal to inverse of the slope of breakthrough characteristic, the slope of line drawn from Swi to

tangent of fractional flow curve.

c ACTODD

Oil Recovery - Before BreakthroughOil Recovery - Before Breakthrough

t bt tpd id id bt w wibt

w

w Swbt

1N W q t S S

dfdS

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Oil Recovery - After BreakthroughOil Recovery - After Breakthrough After breakthrough the saturation at the exit of the After breakthrough the saturation at the exit of the

core increases until irreducible oil saturation reached.core increases until irreducible oil saturation reached.

From Welge equationw Swe

w we

w

w Swe

1 fS S

dfdS

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Oil Recovery - Oil Recovery - After BreakthroughAfter Breakthrough

In terms of pore volumes

w we w idSweS S 1 f W

Oil recovery

pd w wi we wi w idSweN S S S S 1 f W

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Oil Recovery - After BreakthroughOil Recovery - After BreakthroughThis equation pd w wi we wi w idSwe

N S S S S 1 f W

Also gives a useful construction to

determine average saturation

w Swew

w w weSwe

1 fdf

dS S S

w Swew

w w weSwe

1 fdf

dS S S

A line tangential to the FF curve at the exit water saturation intersects fw=1 at the S avg value.

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Oil Recovery Calculation procedureOil Recovery Calculation procedure 1. Generate a fractional flow vs. water saturation curve 1. Generate a fractional flow vs. water saturation curve

using appropriate rel.perm . data.using appropriate rel.perm . data.

2. Draw a tangent to the fractional flow curve from the 2. Draw a tangent to the fractional flow curve from the point Sw=Swi position at fw=0point Sw=Swi position at fw=0

(i) fw=fwbt, Sw=Swbt(i) fw=fwbt, Sw=Swbt

(ii)extrapolation of line to fw=1 gives average (ii)extrapolation of line to fw=1 gives average saturation value.saturation value.

(iii)(iii) t bt t

wpd id id bt w wibt

w Swbt

dfN W q t S S 1/

dS

(iii) Time for breakthrough is btidbt

id

Wt

q

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Oil Recovery Calculation procedureOil Recovery Calculation procedure After breakthrough.After breakthrough.

Select a saturation greater than breakthrough value and:Select a saturation greater than breakthrough value and:

(i) fw=fwe, Sw=Swe and extrapolation of tangential line to fw=1 (i) fw=fwe, Sw=Swe and extrapolation of tangential line to fw=1 gives average saturation value.gives average saturation value.

(ii) Oil recovery is (ii) Oil recovery is


iid

w

w Swe

W 1W

LA dfdS

(iii) Time for breakthrough is

e

e

idbt

id

Wt

q

This step is repeated for increasing values of Swe up to Sw=1-Sor.

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Oil RecoveryOil Recovery

Up to breakthrough the recovery is linear as the oil recovered is equal to water injected.

After breakthrough recovery determined by fractional flow curve.

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Impact of ViscosityImpact of Viscosity Viscosity has a very significant impact on the fractional flow curve

A - high oil/water viscosity ratio.

Dense, viscous oilunstable displacement.

Would require many pore

voumes of water to displace oil

M>>1

Improving recovery by reducing viscosity.

Thermal recovery EOR.Effect on oil greater than

water

c ACTODD


B - medium viscosity ratio.

More stable favourable displacement.

Shock front developed

M around 1

c ACTODD


C - very low oil:water viscosity ratio.

Light oil

Fractional flow curvature opposite to case A

Very stable displacement.

Piston like

M < 1

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Impact of ViscosityImpact of Viscosity

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Shape of the Fractional Flow for Most Common Shape of the Fractional Flow for Most Common Flooding Condition (M<1)Flooding Condition (M<1)

At point of chord contact

pd w wcN S S (PV)

pd or wcN 1 S S =1 (MOV)

Piston like displacement

c ACTODD

Shape of the Very Worst Type of Flooding Shape of the Very Worst Type of Flooding Condition (M>>1)Condition (M>>1)

Concave downward shape:implies that all saturations can be seen to have independent mobility

c ACTODD

Application to reservoir systemsApplication to reservoir systemsAnalysis so far is for one dimensional situations where a uniform saturation distribution can be assumed over the flow cross section. Diffuse flow.

For example a core plug.

Reservoirs generally cannot be handled with such simplicity.

The reservoirs have thickness and vertical permeability, giving rise to segregated flow.

The reservoirs may not be homogeneous but made up of layers which may or may not be in communication

c ACTODD

Two dimensional behaviour segregated flowTwo dimensional behaviour segregated flow In diffuse flow uniform saturation distributed over thickness.In diffuse flow uniform saturation distributed over thickness.

Need to scale up to 2 dimensions to give more realistic Need to scale up to 2 dimensions to give more realistic perspective.perspective.

Segregated flowSegregated flow

Leads to saturation distribution over the thickness.Leads to saturation distribution over the thickness.

DietzDietz

A sharp interface is assumed between oil bearing and A sharp interface is assumed between oil bearing and flooded part of formationflooded part of formation

c ACTODD

Segregated flow-Segregated flow-DietzDietz

Oil + immobile connatewater

Water + immobile residual oil

Core plug 1 dimensional

h

Assumptions:Homogeneous formation

Strong gravity segregationZero capillary pressure

c ACTODD

Segregated flowSegregated flow

Vertical equilibrium assumed to exist.

-large density contrast between

displacing fluid and oil.

High vertical permeability

Low oil viscosityLow fluid velocity

c ACTODD

Comparison of forcesComparison of forces

A useful indicator of dominating forces is the relation of viscous to capillary forces

ovc

uN

Cos

and viscous to gravity forces

o

vgo w o

uN

k g

Nvc and Nvg are the capillary and gravity numbers

In vertical equilibrium both of these are low.

c ACTODD

Segregated flowSegregated flowIn flooded part of the reservoir only water is

flowing.kw=kk’rw

In the unflooded zone only oil is flowing.ko=kk’ro

Distinct interface between oil and water

No capillary transition zone.

c ACTODD

Displacement Calculations for Displacement Calculations for Segregated FlowSegregated Flow

Consider a point in the displacement

path.We have a sharp interface over the

formation thicknessAt any point x, hw

represents fractional thickness below water

oil interface

Average saturation over this formation

thickness h is:

w w or w wcS h 1 S 1 h S

Solving for hw gives:w wc

wor wc

S Sh

1 S S

Since Sor & Swc are constant hw proportional to

Savg

c ACTODD


w

'rw w rwS

k h k

Average relative permeability Average relative permeability over the thickness, the over the thickness, the

thickness averaged relative thickness averaged relative permeabilitypermeability is for water; is for water;

w

rw w rw w rwSw 1 Sor Sw SwcSk h k 1 h k

'rw wc rw rw w ork 0 at S and k k at S 1 S

Therefore

w

'rw w rwS

k h k k’rw is the end point

relative permeability to water

c ACTODD


Average relative permeability Average relative permeability over the thickness, the over the thickness, the

thickness averaged relative thickness averaged relative permeabilitypermeability is for oil; is for oil;

w

ro w ro w roSw 1 Sor Sw SwcSk h k 1 h k

'rw ro wc ro w ork k at S and k 0 at S 1 S

k’ro is the end point

relative permeability to oil

Therefore w

'ro w rwS

k 1 h k

c ACTODD


If hww is now replaced in:

w wcw

or wc

S Sh

1 S S

w

'w wcrw rwS

or c

S Sk k

1 S S

we have

w

'w wcrw rwS

or wc

S Sk k

1 S S

and

w

'or wro roS

or wc

1 S Sk k

1 S S

Relative permeability linear functions of average water saturation

c ACTODD

Displacement Calculations for Segregated FlowDisplacement Calculations for Segregated Flow

Relative permeability linear functions of average water saturation

Diffuse flow curves

Linear thickness averaged relative permeability some times calledpseudo relative permeability

c ACTODD

Displacement Calculations for Segregated FlowDisplacement Calculations for Segregated Flow

ProcedureProcedure

Using core determined relative permeability curves Using core determined relative permeability curves generate linear thickness averaged relative permeability generate linear thickness averaged relative permeability lineslines

Only end point values required for thisOnly end point values required for this

Calculate fractional flow curve using the Calculate fractional flow curve using the pseudo( thickness) averaged relative permeabilities.pseudo( thickness) averaged relative permeabilities.

Apply one dimensional calculations to determine Apply one dimensional calculations to determine breakthrough and recovery.breakthrough and recovery.

c ACTODD

Reservoir HeterogeneityReservoir Heterogeneity Because of the depositional process reservoir formations are Because of the depositional process reservoir formations are

heterogeneous.heterogeneous.

This has considerable impact on displacement process.This has considerable impact on displacement process.

Two types of heterogeneityTwo types of heterogeneity

Vertical heterogeneityVertical heterogeneity

Horizontal heterogeneity.Horizontal heterogeneity.

The total recovery Np of the fluids in place N, is a combination The total recovery Np of the fluids in place N, is a combination of the recovery in the respective dimensionsof the recovery in the respective dimensions

pV A

NE E

N EA is the areal sweep efficiency,EV is

the vertical sweep efficiency

Formations swept by water leaves oil at irreducible saturation.Unswept oil at the original satuation

c ACTODD

Reservoir HeterogeneityReservoir Heterogeneity

c ACTODD

Vertical HeterogeneityVertical Heterogeneity The process of formation generates considerable variations in rock The process of formation generates considerable variations in rock

properties.properties.

Major impact on the permeability.Major impact on the permeability.

Permeability variation has considerable impact on water drive.Permeability variation has considerable impact on water drive.

Vital that this heterogeneity perspective is well characterised.Vital that this heterogeneity perspective is well characterised.

A layer in the formation of very high permeability can have a very A layer in the formation of very high permeability can have a very serious impact on the oil recovery process.serious impact on the oil recovery process.

Another aspect of importance is the connectivity between layersAnother aspect of importance is the connectivity between layers

Open hole pressure surveys very powerful after production has started Open hole pressure surveys very powerful after production has started in identifying layers in identifying layers

c ACTODD

Pressure Pressure profilesprofiles

c ACTODD

Examples of Examples of pressure pressure surveyssurveys

c ACTODD

Areal HeterogeneityAreal Heterogeneity

Areal heterogeneity also effects sweep Areal heterogeneity also effects sweep efficiencyefficiency

Other impacts from well spacing and Other impacts from well spacing and configuration.configuration.

More uncertainty in areal sweep than vertical.More uncertainty in areal sweep than vertical.

Dake has suggested vertical calculations are Dake has suggested vertical calculations are done in combination with production data to done in combination with production data to history match areal sweep efficiency.history match areal sweep efficiency.

c ACTODD

Vertical Sweep Displacement CalculationsVertical Sweep Displacement Calculations

Methods used for diffuse flow can be used for Methods used for diffuse flow can be used for layered systems.layered systems.

For layers with communication.For layers with communication.

Cross flowCross flow

and for layers with no communicationand for layers with no communication

No cross flowNo cross flow

The first step is to identify the layers and their The first step is to identify the layers and their individual characteristicsindividual characteristics

c ACTODD


hi

Thickness

ki

Permeability

i

Porosity

Swci

Connate water saturation

Sori

Irreducible oil saturation

k’rwi

End point relative permeability water

End point relative permeability oil

k’roi

i th layer

c ACTODD

Vertical Sweep Displacement CalculationsVertical Sweep Displacement CalculationsConsider vertical pressure communication between layers decide if there is cross flow or no cross flow.

Decide flooding order of layers

Generate thickness average saturations and relative permeabilties.

As each layer floods out

J is the ordering of the flooding

layers

j j

n

n N

j j or j j wcj 1 j n 1

w N

j jj 1

h 1 S h S

Sh

c ACTODD


j

nwn

n'

j j rwj 1

rw NS

j j1

h k k

kh k

and

j

nwn

N'

j j roj n 1

ro NS

j j1

h k k

kh k

These average saturations and thickness averaged relative

permeabilities are then used to generate fractional flow curve

The problem can now be solved using the I-D approach

c ACTODD

Vertical Sweep Displacement Calculations - Vertical Sweep Displacement Calculations - Ordering of the layersOrdering of the layers

The engineer has some control on the ordering of the layers

If there is very strong vertical equilibrium with strong segregation then the layers will flood out from the bottom layer.

c ACTODD

Vertical Sweep Displacement Calculations - Vertical Sweep Displacement Calculations - Ordering of the layers- No cross flowOrdering of the layers- No cross flow

When there is no communication between layers then the layers will flood out according to the velocity of displacement of each layer.

Velocity of each layer i, is given by

i

i i

'i rw

i

i or wc

k kv

1 S S

‘Stiles’ then flooded out the natural layers in order of decreasing velocity. A mobility ratio around 1 is assumed.

c ACTODD

Ordering of the layers- No cross flowOrdering of the layers- No cross flow

Natural order

Increasing velocity

c ACTODD

Ordering of the layers- No cross flowOrdering of the layers- No cross flow

Increasing velocity

Ordering of layers impacts on the pseudo relative permeabilities, which results in a subsequent fractional flow curve.When fractional flow curve generated breakthrough and recovery analysis as for 1-D case.

c ACTODD

Impact of Capillary Pressure in Homogeneous Impact of Capillary Pressure in Homogeneous systemssystems

So far neglected capillary pressure for diffuse flow where So far neglected capillary pressure for diffuse flow where saturation uniformly distributed over thickness.saturation uniformly distributed over thickness.

And segregated flow where we have an oil-water contact.And segregated flow where we have an oil-water contact.

If capillary pressure significant then the transition zone If capillary pressure significant then the transition zone cannot be ignored.cannot be ignored.

If reservoir thin relative to transition zone-diffuse flow If reservoir thin relative to transition zone-diffuse flow conditions.conditions.

If transition zone negligible displacement segregatedIf transition zone negligible displacement segregated

c ACTODD

Impact of Capillary Pressure in Homogeneous Impact of Capillary Pressure in Homogeneous systemssystems

If neither of these two extremes exist capillary If neither of these two extremes exist capillary pressure needs to be considered.pressure needs to be considered.

Dake gives a full account. Based on moving the Dake gives a full account. Based on moving the transition zone in stages and at each stage transition zone in stages and at each stage determining average saturations and average relative determining average saturations and average relative permeabilities. These form the basis of new fractional permeabilities. These form the basis of new fractional flow curve then treated as 1-D case.flow curve then treated as 1-D case.

Impact is between diffuse flow and segregated flow.Impact is between diffuse flow and segregated flow.

c ACTODD

Impact of Capillary Impact of Capillary Pressure in Pressure in Homogeneous systemsHomogeneous systems

Dake

Dake

c ACTODD

Impact of relative position of high permeability Impact of relative position of high permeability layerslayers Figures of pseudo rel. perms. below for cross flow cases of Figures of pseudo rel. perms. below for cross flow cases of

the impact of the high permeability layer being at the top as the impact of the high permeability layer being at the top as compared to the base.compared to the base.

An example from ‘Dake’ of a three layer system is illustrated.An example from ‘Dake’ of a three layer system is illustrated.

High perm at top High perm at base

Dake

c ACTODD

Impact of relative position of high permeability Impact of relative position of high permeability layerslayers Fractional flow curvesFractional flow curves

Very early breakthroughShock front developed

Dake

c ACTODD

Impact of Permeability Distribution on Impact of Permeability Distribution on WaterfloodingWaterflooding

Previous layered illustration demonstrated impact Previous layered illustration demonstrated impact of relative position of layers in a heterogeneous of relative position of layers in a heterogeneous system.system.

Absolute values of k not so significant more the Absolute values of k not so significant more the relative contrast in permeability.relative contrast in permeability.

c ACTODD

Impact of Permeability Distribution on Waterflooding - Impact of Permeability Distribution on Waterflooding - Unfavourable permeability distributionUnfavourable permeability distribution

c ACTODD

Impact of Permeability Distribution on Waterflooding - Impact of Permeability Distribution on Waterflooding - Favourable permeability distributionFavourable permeability distribution

Archer

c ACTODD

Impact of Impact of Permeability Permeability Distribution on Distribution on WaterfloodingWaterflooding

Dake

c ACTODD


Fractional flow curve determines quality of water Fractional flow curve determines quality of water flood.flood.

Demonstrated by saturation at breakthrough and Demonstrated by saturation at breakthrough and subsequent production.subsequent production.

Shape of FF curve influenced by various factors.Shape of FF curve influenced by various factors.

Relative viscositiesRelative viscosities

endpoint relative permeabilitiesendpoint relative permeabilities

permeability distributionpermeability distribution

c ACTODD


Impact of mobility ratio ( viscosity effect) and permeability Impact of mobility ratio ( viscosity effect) and permeability distribution illustrated belowdistribution illustrated below

Coarsening down Coarsening upward

Very early breakthrough

even with favourable M

Shock front even at high M

Piston like displacement

Dake

c ACTODD


Impact on water Impact on water production.production.

Early water Early water breakthrough at low breakthrough at low water cut requires a water cut requires a long time to displace long time to displace oil.oil.

Coarsening up with Coarsening up with favourable mobility favourable mobility ratio leads to delayed ratio leads to delayed breakthrough then breakthrough then rapid increaserapid increase

c ACTODD

Application to Field PerformanceApplication to Field Performance The displacement equations can be used to examine actual field The displacement equations can be used to examine actual field

waterflood performance of fields in production.waterflood performance of fields in production.

Then used to generate information to determine future waterflood Then used to generate information to determine future waterflood strategy.strategy.

Development of a reservoir fractional flow curve..Development of a reservoir fractional flow curve..

Production data provides;Production data provides;

Oil production, NpOil production, Np

Water injection, WiWater injection, Wi

Exit fractional flow, fwsExit fractional flow, fws

c ACTODD

Application to Field PerformanceApplication to Field Performance

Using recovery equation


The unknown Swe can be obtained, enabling a fw vs. Sw curve to be generated

This curve is much smoother than a typical eratic water cut vs. production curve

c ACTODD

Application to Field PerformanceApplication to Field PerformanceTypical water cut curve

Dake

Difficult to analyse

c ACTODD

Reservoir Fractional Flow CurveReservoir Fractional Flow Curve

Extrapolated Sw avgIncreased Sw avg due to increased water injection

Core data Sor

c ACTODD

Reservoir Fractional FlowReservoir Fractional Flow

Core data Sor

Allowing water cut to increase to high value would yield increased

recovery.However it would take a considerable time.

Another option is to increase injection rate if

facilties can be upgraded

. This would reduce time

to achieve recovery increase.

The option for the operator to progress round the reservoir FF curve is an

interesting approach

c ACTODD

Immiscible displacement in gas drive systemsImmiscible displacement in gas drive systems Method as generated for water-oil displacement can be Method as generated for water-oil displacement can be

used for gas-oil displacement.used for gas-oil displacement.

Mobility ratio for gas is very unfavourable because of large Mobility ratio for gas is very unfavourable because of large viscosity differenceviscosity difference

Gas drive is very unstable.Gas drive is very unstable. ' 'rg ro

g o

k k 0.5 1M / / 20

0.25 1

Fractional flow curve is also very unfavourable g

g ro

o rg

1f

k1

k

c ACTODD

gg ro

o rg

1f

k1

k

Immiscible displacement in gas drive systemsImmiscible displacement in gas drive systemsFractional flow curve is also very unfavourable

The full laboratory rel. perm. curves are not used since only the end point values used as gravity dominates leading to

segregated flow.

c ACTODD

Immiscible displacement in gas drive systemsImmiscible displacement in gas drive systems

Recovery calculations same as for water-oil.Recovery calculations same as for water-oil.

Can use the Buckley-Leverett and associated Can use the Buckley-Leverett and associated Welge analysis.Welge analysis.

Recovery of oil in gas drive equation is :Recovery of oil in gas drive equation is :

pd ge ge idN S 1 f G

Gid is the pore volumes of gas injected.

Equal to reciprocal of the slope of the FF curve at the exit gas saturation.

c ACTODD

Immiscible displacement in gas drive systems -Immiscible displacement in gas drive systems -Gravity segregationGravity segregation This is the dominant This is the dominant

perspective in gas driveperspective in gas drive

In gas cap drive the In gas cap drive the velocities are very low velocities are very low giving stable interface.giving stable interface.

When gas is injected When gas is injected along bedding.along bedding.

Unfavourable mobility plus Unfavourable mobility plus large gravity impact.large gravity impact.

Tendency for gas override.Tendency for gas override.

c ACTODD

Immiscible displacement in gas drive systems -Immiscible displacement in gas drive systems -Predicting performancePredicting performance

Carried out same way as for water-oil.Carried out same way as for water-oil.

Thickness average relative permeabilities generatedThickness average relative permeabilities generated

Layers flooded from the Layers flooded from the toptop..

Fractional flow curve generated.Fractional flow curve generated.

Calculations reduced to 1-D formCalculations reduced to 1-D form

j j

n

n

j j or wcj 1

g N

j jj 1

h 1 S S

Sh

j

n

n'

j j rgj 1

rg N

j j1

h k k

kh k

j

n

N'

j j roj n 1

ro N

j j1

h k k

kh k

c ACTODD

Immiscible displacement in gas drive systems -Immiscible displacement in gas drive systems -Gravity ImpactGravity Impact

gro g

rg o

1 Gf

k1

k

G is a positive gravity number 3 ro

o

kk A sinG 2.743 10

v

v is average Darcy velocity q/A at the injection front.

Actual velocity is: '

or wc

qv

A 1 S S

Oil recovery 'g g id

pdwc

S 1 f GN

1 S

'g gS and f are the averaged saturations and fractional flows

on the fractional flow curve after breakthrough

c ACTODD

Immiscible displacement in gas drive systems -Immiscible displacement in gas drive systems -

Impact of HeterogeneityImpact of Heterogeneity

Whereas for water-oil coarsening down gives rise to early water Whereas for water-oil coarsening down gives rise to early water breakthrough for gas injection it improves the flood profile.breakthrough for gas injection it improves the flood profile.

Higher permeability in lower layers counteracts density difference Higher permeability in lower layers counteracts density difference perspectiveperspective

Other parametersOther parameters

Stability of front affected by: k,q,Stability of front affected by: k,q,oo and and

Also affected by injection rateAlso affected by injection rate

c ACTODD

Immiscible Immiscible displacement in displacement in gas drive gas drive systemssystems

Injection rateInjection rate

A. Gravity is ignored.Unfavourable displacement

B. Impact of reducing velocity. Curve has inflection point.Therefore a breakthrough point.

C. Lower velocity gives a higher breakthrough condition

c ACTODD

Gas CyclingGas Cycling

Gas cycling is a possible Gas cycling is a possible option in the development option in the development of gas condensate of gas condensate reservoirs with a high reservoirs with a high condensate to gas ratiocondensate to gas ratio

In this process we have dry gas displacing wet gas.Very costly.Key issues are when does the dry gas arrive at the producer.?Although a miscible process can be analysed using the Buckley -Leverett approach.Usually modeled with a full compositional simulator

c ACTODD

Miscible Dry Gas-Wet gas DisplacementMiscible Dry Gas-Wet gas Displacement Core relative permeabilities are simple linear Core relative permeabilities are simple linear

functions since no residual gas saturations remains functions since no residual gas saturations remains after contact with dry gas.after contact with dry gas.

Mobilty ratio gives a reasonable displacement perspective

' 'rgd rgw

gd gw

k k 1 1M / / 1.5

0.02 0.03

c ACTODD

Miscible Dry Gas-Wet gas DisplacementMiscible Dry Gas-Wet gas Displacement Permeability variation has a Permeability variation has a

big impact.big impact.

High permeability at the top High permeability at the top favours early breakthrough.favours early breakthrough.

High permeability at the base High permeability at the base counterbalances slight counterbalances slight density impact.density impact.

High permeability layers High permeability layers within formation like water within formation like water flooding can give rise to early flooding can give rise to early breakthoughbreakthough

sc re chap18- immiscible displ

Documents

water production

water injection injectivity

cold water

modelling water drive

reinjection of produced

gas injection

injection pressures

respective injection