2 Fractional-flow Waterflood
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Transcript of 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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Solution by method of characteristics
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
Solution by method of characteristics
Sw is constant along paths with slope dxD/dtD =dfw/dSw
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xD
0
1
inlet,inj.well
outlet,prod.well
tD0Sw = constant
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Solution by method of characteristics
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Sw is constant along paths with slope dxD/dtD =dfw/dSw
xD
0
1
tD0
IB
AI
J
Jfw
0
1
Sw0
J
I
A
B
Solution by method of characteristics
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 = 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)
From L W Lake, Enhanced Oil Recovery, Prentice Hall, 1989
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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)
From L W Lake, Enhanced Oil Recovery, Prentice Hall, 1989
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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