1 opening lecture

8/11/2019 1 Opening Lecture

1/15

1

Vermelding onderdeel organisatie

1

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?

2


2/15

2

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

3

Solution

Even though the change in flow rate upstream wasabrupt, the decrease in river height and flow rate isgradual 100 km downstream.

4

time

riverheight


3/15

3

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?

5

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)

6

time

riverheight


4/15

4

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

7

Lesson

In convection problems, some changes propagategradually downstream, and others abruptly (as a

shock)

8


5/15

5


9

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

10


6/15

6

11

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

12


7/15

7

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

13


14

Background and Definitions


8/15

8

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

15

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

16


9/15

9

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

17

Typical relative-permeabilityfunctions

Plot assumeswater-wet rock

krw stronglynonlinear

kro less nonlinear

18

Sw00

1

1

Swr

krw

kro

kri

(1-Sor)


10/15

10

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:

20


11/15

11

Water fractional-flow function

Water superficial velocity:

Oil superficial velocity:

Ignore capillary pressuregradients: pw = po

Water fractional flow : fw = [uw/(uw+uo)]

21

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

22


12/15

12

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

23

Mixed-wet sandstone

Water viscosity 1 cp

Oil viscosity 5 cp

24

totalrelativemobility

waterfractional

flow

relativepermeabilities


13/15

13

Mixed-wet sandstone


Oil viscosity 5 cp

25

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


Oil viscosity 50 cp

26

fractional-flowcurve moves up

and to left

Minimummobility is

lower


14/15

14

Water more viscous


Oil viscosity 5 cp

27

fractional-flowcurve movesdown and to

right

Minimum

mobility is atintermediate

saturation

Water-wet sandstone

Water Corey exponent nw= 4, oil exponent no = 2


Oil viscosity 5 cp

28

water is lessmobile at given

saturation

water is lessmobile at given

saturation


15/15

Gas flood (no oil)


Gas viscosity 0.02 cp

Strongly water-wetrelative permeabilities

29

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

30

* idealized low-tension flood

1 opening lecture

Documents