2 fractional-flow waterflood

8/11/2019 2 Fractional-flow Waterflood

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Vermelding onderdeel organisatie

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Fractional Flow Method forModeling a Waterflood

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All models are false;

some models are useful

- George E P Box


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Assumptions of Fractional-Flow Method

1D flow through a homogenous permeable medium. No phasechanges.

Incompressible phases (and rock).

Uniform initial conditions.

Immediate attainment of local steady-state conditions, whichare a function of local phase saturations and compositions only.

Absence of all dispersive processes: diffusion, dispersion, heatconduction: governing equations are 1st-order p.d.e.s

Besides surfactant adsorption, fluids do not react with the rock.

Newtonian mobilities for all phases. For this lecture, only two phases present at any location.

Absence of gravity forces, fingering and dispersion

Almost all of these assumptions can be relaxed

Fractional-flow method applied towaterflood

Reference: document CHAP7 Multiphase Pore FluidDistribution, by G. J. Hirasaki; see this and othermaterials at

http://www.owlnet.rice.edu/~ceng571/

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Conservation equation

equation based on material balance:(flux in - flux out) = (change in saturation)

xd dimensionless position, defined as

x/L in linear flow

r2/Re2 in radial flow

td dimensionless time, pore volumes injected fw fractional flow of water

Sw water saturation

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Fractional-flow function (oil-water)

kri = relative permeability of phase i

i = viscosity of phase i fw is fraction of flowthat is water - not same as water

saturation (volume fraction in place), but isa functionof water saturation (and possibly other variables)

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Solving the equation

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Solution by method of characteristics

Sw is constant along paths with slope dxD/dtD =dfw/dSw xD dimensionless position, defined as volume back to

injection point:

x/L for linear flow

r2/re2 for radial flow

tD dimensionless time, pore volumes injected

Sw is function of xD, tD; plot Sw(xD,tD) on (xD,tD) plane

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Sw is constant along paths with slope dxD/dtD =dfw/dSw

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xD

0

1

inlet,inj.well

outlet,prod.well

tD

tD = 0:initial state of

formation or core

xD = 0:injection into

formation or core

0



10

xD

0

1

inlet,inj.well

outlet,prod.well

tD0Sw = constant


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xD

0

1

tD0

IB

AI

J

Jfw

0

1

Sw0

J

I

A

B



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xD

0

1

inlet,inj.well

outlet,prod.well

tD

tD = constant:state of formationor core at time tD

xD = constant:state at givenposition over time

0


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Exercise 2: Waterflood solution withlinear relative permeabilities

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Waterflood solution with linearrelative permeabilities: Lessons

Plot Sw(xD,tD) on (xD,tD) diagram

Constant Sw values propagate along straight lines(characteristics) on (xD,tD) plot

Slope of line = velocity = dxD/dtD = dfw/dSw for givenvalue of Sw

Slice through diagram at tD = constant gives Sw(xD):saturation profile

Slice through diagram at xD = constant (especially xD= 1, outlet) gives Sw(tD) or fw(tD): production history

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Real fractional-flowcurves have Sshape

Derives from nonlinearexponent in Corey relativepermeabilities

Physical cause: flowbecomes extremelyinefficient for phase justabove its residualsaturation.

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What if slopes dont increasemonotonically from J to I?

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Rules lead to multi-valued solutions

fw

0

1

Sw0

J

I

A

B

xD

0

1

tD0

IA

BJ

J

I


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How to eliminate multiple-valuedsolutions?

Allow for discontinuous solutions Sw(xD,tD); discontinuities areshocks

Solve for velocities of shocks by material balance at shocks17

How to eliminate multiple-valuedsolutions?

Allow for discontinuous solutions Sw(xD,tD); discontinuities are shocks

Solve for velocities of shocks by material balance at shocks (more onthis shortly)

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xD

0

1

tD0

I

J

constant stateor spreading

wave

constant stateor spreading

wave


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Fractional-Flow Method forWaterflood (water displacing oil)

Plot fw(Sw) function for oil and water

Locate initial condition I and injection condition J onthis plot

Find path from J to I along fw(Sw) with monotonicallyincreasing slope dfw/dSw

If such a path does not have monotonically increasingdfw/dSw, there is a jump, or shock

Slope of shock fw/Sw must fit into monotonically

increasing sequence of slopes from I to J Solution not allowed to cut fw(Sw) curve (entropy

condition)

See lecture notes, G. J. Hirasaki, ch. 7.

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Fractional-Flow Method forWaterflood

At each point along path, dimensionless velocity(dxD/dtD) is dfw/dSw; corresponding saturation, andphase properties, advance with this velocity

xD is fraction of pore volume back to injection point;defined for linear flow, radial flow, streamline flow

tD is pore volumes injected

Velocity of shock is fw/Sw Plot advance of saturations and shocks on xD-tD

diagram


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Waterflood Example

From L W Lake, Enhanced Oil Recovery, Prentice Hall, 1989

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Waterflood Example

Triple-valued Sw(xD)?

Resolve with adiscontinuity in Sw(xD)



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Material balance on water: (flux in-out)=accumulation

Material balance gives shock velocity

velocity of shock is slope of line on fw(Sw) representing shock

- --+ +

Is a shock really a discontinuity?

In reality, dispersive processes smear out a discontinuityinto a sharp, but continuous, transition

If there were a discontinuity in Sw, there would bediscontinuity in Pc, and infinite pressure gradient at shock;this pressure gradient smears out front

Within traveling wave at shock, self-sharpening effectsof fw(Sw) and dispersive effects of Pc result in sharptransition of fixed width as shock advances

As scale of displacement grows, this front looks more andmore like a discontinuity on scale of displacement; on labscale the front may be broad

Here we treat this front as discontinuity

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Analogy to interface between phases

On sufficientlysmall scale, thereis continuousvariation ofcomposition anddensity across aninterface

On larger scales,we treat interfaceas discontinuity

(figure from Marten

Buise, Shell GlobalSolutions BV)

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Waterflood Example

From xD-tD diagram can get Sw, mobility at anyposition, or flow rates at any position (including outlet,i.e. production history)


Other examples of shocks in nature

Automobile traffic on highways: an increase in traffic densityleads to a shock; a decrease to a spreading wave. (Whenyou enter a traffic jam, its sudden; when the traffic loosensup, it does so gradually)

Water flow in rivers and other channels: an increase in water

flow rate leads to a shock; a decrease spreads out. Floods hitsuddenly, but the afterwards the water level falls gradually.

Tsunami: the wave hits the shoreline suddenly, but thewater drains out gradually.

In all cases, the cause of the shock is the same; materialupstream is traveling faster than it is downstream, leading toan accumulation and a shock

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Review of Approach I Material balance on microscopic control

volume gives partial differential equation

Equation implies that Sw and all properties are constantalong characteristic paths with dxD/dtD = dfw/dSw

Plot initial condition I andfinal position J on fw(Sw)plot. Each saturation be-tween I and J moves withvelocity given by dfw/dSw

Plot these characteristics v. xD, tD; gives full solution If dfw/dSw does not monotonically increase from J to I,

method implies multivalued solutions not allowed

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Review of Approach I

In place of multivalued solutions,allow for traveling discontinuities(shocks) in solution

Material balance on shock gives graphical condition forshock: dxD/dtD=fw/Sw

Search for combination of continuoussolution and shock that gives mono=tonically increasing dfw/dSw from J toI on fw(Sw) curve

Plot these characteristics v. xD, tD;gives full solution

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Exercises 3 and 4: Solution forWaterflood with Shocks

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Waterflood with Shocks: Lessons

Have shocks where path from J to I along fw(Sw) doesnot have monotonically increasing slope

Even in 1D, with no sweep efficiency issues, last oil isproduced slowly because of shape of fractional-flowcurve (ultimately, because of shape of Corey relativepermeability at low oil saturation)

With increasing oil viscosity it takes even longer toproduce last oil

With increasing oil viscosity, viscous instability isexpected to worsen, BUT

Mobility is depends on rel perms, not just viscosities

Fingering is not represented directly in fractionalflow theory

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2 fractional-flow waterflood

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