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8/11/2019 1 Opening Lecture
1/15
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Vermelding onderdeel organisatie
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Fractional Flow Methods forModeling Enhanced OilRecovery
W. R. RossenProfessor of Reservoir EngineeringDepartment of Geoscience and EngineeringDelft University of Technology
Thought experiment 1
A river, 100 km long, is contained in a concretechannel. Suddenly the flow of water into the riverfrom upstream stops. What is the height of the water100 km downstream as a function of time?
In particular, does it decline gradually with time ateach point, or does it decline abruptly to zero?
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Details
River is in a rectangular channel, open on the top, 10m wide. The river declines 19 m over its length.
Assume friction factor f = 0.005. g = 9.81 m/s2
Assume at start height of river is 2 m
Q = (2 H3 g (0.00019) W2/ f )1/2
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Solution
Even though the change in flow rate upstream wasabrupt, the decrease in river height and flow rate isgradual 100 km downstream.
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time
riverheight
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Thought experiment 2
The same river starts with the same flow rate as inthought experiment 1. Suddenly the flow rate into theriver doubles.
What is the height of the river 100 km downstream asa function of time?
In particular, does the river rise abruptly in height at agiven time, or does it rise gradually over a long time?
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Solution
The change in height at the inlet propagates down-stream as an abrupt rise. (Not a discontinuous rise,because of small-scale effects, but an abrupt rise)
It would be abrupt even if the change at the inlet werenot abrupt (say, changing over a few hours)
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time
riverheight
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Its not fair. When the river is rising, itrises about an inch an hour. When it isfalling, it falls only about an inch a day
- citizen of Missouri, USA, quoted in news
broadcast during period of flooding ofMississippi river, Aug. 3, 1993
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Lesson
In convection problems, some changes propagategradually downstream, and others abruptly (as a
shock)
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Vermelding onderdeel organisatie
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Fractional Flow Methods forModeling Enhanced OilRecovery
Goals of Course
Learn the tool of fractional-flow analysis applied toboth waterflood and for Enhanced Oil Recoveryprocesses
Practice the approach with examples: miscible EORflood, polymer flooding, foam EOR
Work with your neighbors
If you are already familiar with some of theconcepts, help your neighbors!
Understand the limitations of fractional-flow analysis
Appreciate the advantages of fractional-flow analysis
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All models are false but
some models are useful.
- George E P Box
Philosophical Foundation
Assumed student background
Some familiarity with oil recoveryand enhanced oil recovery
Knowledge of Darcys law for singleand two-phase flow
Some familiarity and comfort withcalculus
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References
Course notes of Prof. George Hirasaki, Rice University,available athttp://www.owlnet.rice.edu/~ceng571/
Larry W Lake, Enhanced Oil Recovery; 2nd Editionavailable Aug. 2014 from Society of PetroleumEngineers
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Vermelding onderdeel organisatie
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Background and Definitions
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Some definitions
Porosity : fluid-filled void space in rock, as fraction oftotal volume
Darcys law for horizontalflow of single phase:
u = superficial velocity, Q/A, i.e. (volumetric flowrate/cross-sectional area)
k = permeability (property of rock)
= viscosity (property of fluid)
p = pressure
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Darcys law for multiphase flow
ui = superficial velocity of phase i; (Qi/A)
k = permeability (property of rock)
kri = relativepermeability of phase i; for given fluidsand rock, a function of the saturation Si of phase i
Si = saturation of phase i, volume fraction of phasei among fluid phases in pore space
= viscosity (property of phase i)
pi = pressure in phase i
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Relative permeability function kri
For given rock and fluids, ability of fluid to flowdepends on its saturation Si
At and below some minimum residualsaturation Sir,fluid i stops flowing completely
For saturations greater than this, relative permeabilityis a nonlinear function of phase saturation
Popular representation:Corey relative permeability
Exponent ni is larger (kri morenonlinear) for wetting phase than nonwetting phase
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Typical relative-permeabilityfunctions
Plot assumeswater-wet rock
krw stronglynonlinear
kro less nonlinear
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Sw00
1
1
Swr
krw
kro
kri
(1-Sor)
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1D flow: Material balance on water in1D incompressible flow
Material balance on small volume of thickness x,cross-sectional area A, porosity
Constant total superficial velocity ut = Q/A
Define fractional flow of water fw uw/ut Water flow in: (ut fwA t)x
Water flow out: (ut fwA t)x+x Accumulation of water inside control volume:
(A x Sw)19
x x+x
(ut fwA t)x (ut fwA t)x+x
Material balance on water in 1Dincompressible flow
(ut fwA t)x - (ut fwA t)x+x = (A x Sw)
Let x, t shrink to zero:
Change in water saturation with time dependson fractional-flow function fw
Define dimensionless position xd = x/L; dimensionlesstime td = L/(ut/), i.e.
pore volumes injected:
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Water fractional-flow function
Water superficial velocity:
Oil superficial velocity:
Ignore capillary pressuregradients: pw = po
Water fractional flow : fw = [uw/(uw+uo)]
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Total relative mobility
Total superficial velocity is the sum of the phasesuperficial velocities
Total relative mobility rt is a measure of the ability ofthe phases to flow; the inverse of resistance to flow,i.e. the inverse of apparent viscosity of two-phaseflow
Total relative mobility governs viscous instability, thatharms sweep efficiency in enhanced oil recovery
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Exercise: water fractional-flowfunction, effect of parameters
Recall Corey representation of relative permeabilities:
Consider effects of residual saturations, relative-
permeability parameters, and viscosities on relativepermeabilities, fractional-flow function, and totalrelative mobility
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Mixed-wet sandstone
Water viscosity 1 cp
Oil viscosity 5 cp
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totalrelativemobility
waterfractional
flow
relativepermeabilities
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Mixed-wet sandstone
Water viscosity 1 cp
Oil viscosity 5 cp
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Minimum
mobility atintermediate
saturation
S shape tofractional-flow
curve;fw = 0 at Swr;
fw = 1 at (1-Sor)
Residualwater
saturation
Swr
Residualoil
saturation
Sor
Oil more viscous
Water viscosity 1 cp
Oil viscosity 50 cp
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fractional-flowcurve moves up
and to left
Minimummobility is
lower
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Water more viscous
Water viscosity 10 cp
Oil viscosity 5 cp
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fractional-flowcurve movesdown and to
right
Minimum
mobility is atintermediate
saturation
Water-wet sandstone
Water Corey exponent nw= 4, oil exponent no = 2
Water viscosity 1 cp
Oil viscosity 5 cp
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water is lessmobile at given
saturation
water is lessmobile at given
saturation
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Gas flood (no oil)
Water viscosity 1 cp
Gas viscosity 0.02 cp
Strongly water-wetrelative permeabilities
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fractional-flowcurve moves far
down and toright
mobility ofgas is muchgreater thanwater (or oil)
Ultra-low-interfacial-tension (surfactant) flood*
Residual saturations = 0
Nearly linear rel perms
Water viscosity 5 cp (addpolymer)
Oil viscosity 5 cp
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* idealized low-tension flood