Waterflooding Solutions

Homework #1 Solutions

The Taber Waterflood

Part 1. Obtaining relative permeability data from chart.The relative perm curves are given in the paper by Shaw and Stright.Unfortunatey, the semi-log scale makes it somewhat difficult to obtain the data.Some software is available to do this (e.g. GraphClick for Mac OS X), or students may have simply measured the data points directly.

A table of values should look something like this:(Note that Sw is usually be expressed as a fraction)

Sw, fraction kro krw kro/krw

0.100 0 inf. <-- extrapolated endpoint for krw

0.216 0.84 0.03 25.33 <-- data for sample calculation0.280 0.59 0.08 7.320.321 0.48 0.11 4.410.375 0.33 0.15 2.190.410 0.23 0.18 1.310.451 0.16 0.21 0.730.494 0.10 0.26 0.380.524 0.07 0.30 0.240.548 0.05 0.33 0.170.565 0.04 0.35 0.120.586 0.03 0.38 0.070.627 no value 0.44

0.64 0.00 0.00 <-- extrapolated endpoint for kro

See Chart 1 for a graph.Finally, the endpoints, Sw and 1-Sor were obtained by inspecting the graph and1. extrapolating the krw line to the x axis (defining Swirr) THIS IS THE MOST IMPORTANT DATA POINT 2. extrapolating the kro line to the x axis (defining 1- Sor)This is quite difficult in this case because the original data was givenon a semi-log scale; some latitude should be given to studentsbecause the rest of the homework is anchored to this data

Sample calculation (not necessary for this part):

PET E 471 Homework #1 Solutions 1/7

Part 2. Fitting the permeability ratio to an exponential equation

A graph of the perm ratio, kro/krw vs. Sw should reveal a linearrelationship on a semi-log plot. From this, values of the coefficientsa and b may be calculated by selecting reasonable endpoints of the line.The selection of points from the original data is somewhat subjective.See class handout, "PET E 471 Dispacement 1 Handouts.pdf" for details.

Points selected to fit to exponential:

Sw, fraction kro/krw Coefficients0.216 25.33 a= 678.020.524 0.24 b= 15.19

Results will vary slightly depending on the two points chosen to evaluate a and b.Sample hand calculation (note slight roundoff error compared with spreadsheet):

Using the newly obtained coefficients a and b, the relative permeabilitycan be evaluated for any value of Sw.For comparison, the results are tablulated below (also see Chart 2)Note that the equation does not do well at the end points of the curve;there the values should be inserted manually in future calculations.

!

b = lnko1kw2

kw1ko2

"

# $

%

& ' /(Sw2 ( Sw1)

a =ko1

kw1

eb(Sw1 )


Sw

kro/krw from graph

kro/krw from eq'n

0.1 inf. 148.51 <-- this is Swirr, so krw is really zero by definition0.216 25.33 25.330.280 7.32 9.59 <-- data for sample calculation0.321 4.41 5.190.375 2.19 2.280.410 1.31 1.340.451 0.73 0.720.494 0.38 0.370.524 0.24 0.240.548 0.17 0.160.565 0.12 0.130.586 0.07 0.090.627 no value 0.050.640 0.00 0.04 <-- this is 1-Sor, so kro is zero by definition

Sample hand calculation:

Part 3. Fractional Flow CurveThe analytical expressions derived in class notes for fw and dfw/dSw may be used.There are actually two variants of the equations, one which includes the term kro/krw

Keep in mind that the equation may not give the proper endpoints, which are:

fw = 0 at Swirr

fw = 1 at 1-Sor

Dataµo 58 cpµw 5 cp (polymer)

µo/µw 11.6a 678.02 from Part 2b 15.19 from Part 2

!

fw

=1

1+µw

µo

ae"bSw

#

$ %

&

' (

!

"fw"Sw

=

bµw

µo

ae#bSw

1+µw

µo

ae#bSw

$

% &

'

( )

2


Sw

ko/kw from equation

fw from equation

dfw/dSw from equation

0.10 148.51 0.00 1.02 <-- Swirr

0.15 69.50 0.14 1.860.20 32.53 0.26 2.94 <-- data for sample calculation0.25 15.22 0.43 3.730.30 7.12 0.62 3.580.35 3.33 0.78 2.630.40 1.56 0.88 1.590.45 0.73 0.94 0.850.50 0.34 0.97 0.420.55 0.16 0.99 0.200.60 0.07 0.99 0.100.64 0.04 1.00 <-- 1-Sor

See Chart 3 for fw and dfw plots

Sample calculation:


Part 4. Water Saturations at flood front, breakthrough

Saturations Sw' and Swbt may be obtained either from a graphical solutionor analyticallly by iteration. See Chart 4

Sw' by Goal Seek:a= 678.02b= 15.19uo/uw 11.60Swirr 0.1

Change this in Goal Seek Goal Seek to zero

Initial Guess Sw'

Goal Seek Sw'

ko/kw from equation fw fw'

fw'-fw/(Sw'-Swirr)=0

0.2 0.327 4.75 0.710 3.129 0.0000.5 0.327 4.75 0.710 3.130 0.000

Sometimes two different guesses are required if the roots are double-valued.Got lucky this time.

Summary of Results:

MethodFlood Front Saturation, Sw'

Breakthrough Saturation, Swbt

Graphic (tangent line) 0.33 0.42

Analytic (Goal Seek) 0.327

TOO MESSY, NOT DONE

No sample calculations required, but a graph showing tangent line is necessary.

Part 5 Saturation Front Distance TravelledAccording to the Welge refinement to the Buckley-Leverett method, the waterflood fronthas a water saturation of Sw', therefore only values above that need to be evaluated.

Reservoir Parameters

L 460 mW 100 mH 10 mphi 0.27

Injection Dataqi 187 m3/d

!

L =Wi

A"

dfw(Sw)

dSw


time, days50 100 150 200 216.8

Wi= 9350 m3

Wi= 18700

m3

Wi= 28050

m3

Wi= 37400

m3

Wi= 40544

m3

Sw ko/kw fw dfw/dSw

Front dist, L,

m

Front dist, L,

m

Front dist, L,

m

Front dist, L,

m

Front dist, L,

m0.10 148.51 0.072 1.0210.15 69.50 0.143 1.8610.20 32.53 0.263 2.9430.25 15.22 0.432 3.7270.30 7.12 0.620 3.5790.33 4.52 0.720 3.063 106.1 212.2 318.2 424.3 460.00.35 3.33 0.777 2.633 91.2 182.4 273.6 364.8 395.40.40 1.56 0.881 1.587 55.0 109.9 164.9 219.8 238.30.45 0.73 0.941 0.846 29.3 58.6 87.9 117.2 127.10.50 0.34 0.971 0.422 14.6 29.2 43.9 58.5 63.40.55 0.16 0.986 0.204 7.1 14.1 21.2 28.2 30.60.60 0.07 0.994 0.097 3.4 6.7 10.1 13.4 14.50.64 0.04 1.000 0.053 1.8 3.7 5.5 7.3 8.0

Sample calcluation (using Sw=0.33, t=50d):

The time to breakthrough was found to be 216.8 days.This can be determined in at least two ways:1. Using the spreadsheet, simply insert values of t and find when L=460 by trial and error.2. Set up a goal seek in Excel. An answer +/- say 5 days is ok.

Part 6. Post-Breakthrough PerformanceWelge gives a method to calculate post-breakthrough performance of a waterflood.After breakthrough, however, it's a game of diminishing returns andin the real world it plays out only as long as the economics are favourableResults are presented graphically in Chart 6

Swirr= 0.1 EXTRA CALCULATIONSNOT REQUIRED

Exit Water Sat, Sw2

Exit Flowing Water Fraction

fw2

Slope of fw(Sw)

PV Water Injected, Qi

Avg Water Sat, S

PV Oil Recover

ed Qo

PV Water Prod WOR

% Oil Rec

0.33 0.72 3.063 0.326 0.421 0.321 0.00 0.00 0.360.35 0.78 2.633 0.380 0.435 0.335 0.04 0.13 0.370.40 0.88 1.587 0.630 0.475 0.375 0.26 0.68 0.420.45 0.94 0.846 1.182 0.520 0.420 0.76 1.81 0.470.50 0.97 0.422 2.369 0.568 0.468 1.90 4.06 0.520.55 0.99 0.204 4.908 0.617 0.517 4.39 8.50 0.570.60 0.99 0.097 10.336 0.666 0.566 9.77 17.25 0.630.65 1.00 0.053 18.863 0.707 0.607 18.26 30.09 0.67


Sample calculation:For post-breakthrough calculations, the procedure is:

1. Examine the last part of the handout, "Displacement 3" because you haven't read it yet!

2. Start with the water front saturation obtained in Part 4: Sw'=0.33 At breakthrough, this is the water saturation at the outlet end, Sw2

3. Use the equation for fw (Part 3) to obtain flowing water fraction at outlet: fw2=0.72 A Visual Basic function macro would be handy about now, n'est-ce pas?

4. Use the equation for fw' to obtain slope of the line: fw'=3.063

5. From the Welge soluton, PV water injected is simply the inverse of fw' Check your handouts to be sure.

5. Calculate average water saturation in the reservoir by rearranging this equation: For the first calculation, this should match the value of Swbt you determined in Part 4.

6. Then by material balance, the quantity of oil recovered is simply the same as the increase in AVERAGE water saturation so far.

7. ASSUME an increase in outlet water saturation, Sw2 and repeat the calculations starting at step 3. It is best to use small increments, such as .05 to start See if you can work out the values in the EXTRA CALCULATIONS columns.

!

Sw " Sw2 =Qi fo2

!

Qi =1/ f2

" =LA#

Wi

=1

dfw

dSw

$

% &

'

( ) Sw 2

!

Qo

= S " Swirr


PET E 471 Fall Term 2006, Homework #1Part 1. Relative Permeability Data

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700

Water Saturation, Sw

Rel

ativ

e Pe

rmea

bilit

y, k

r

krwkro

PET E 471 Fall Term 2006 Homework #1Part 2. Relative Permeability Ratios

0.01

0.10

1.00

10.00

100.00

0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700


Rel

ativ

e P

erm

eabi

lity

Rat

io, k

ro/k

rw

kro/krw from original graph datakro/krw from equation1st point used

2nd point used

PET E 471 Fall Term 2006, Homework #1Part 3. Fractional Flow Curve and Derivative

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.20 0.40 0.60 0.80 1.00


fw

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

dfw/d

Sw

fw from equationdfw/dSw from equation

The equation doesn't work well at the endpoint Swirr, so fw was forced to zero. This is important because it is the anchor point of your tangent

PET E 471 Fall Term 2006, Homework #1Part 4. Water Saturation at Flood Front and Breakthrough

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00


fw

fw from equation

Sw'=0.33 at fw=0.71

Swbt=0.42 at fw=1

Remember to start the tangent line here at Swirr

PET E 471 Fall Term 2006, Homework #1Part 5. Waterflood Front Saturation vs. Time

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 50 100 150 200 250 300 350 400 450

Distance, L [m]

Wat

er S

atur

atio

n, S

w

50100150200216.8

Swirr = 0.1

Time, days

Breakthrough!

unrecoverable oil

unrecovered oil

PET E 471 Fall Term 2006, Homework #1Part 6. Oil Recovered vs. Water Injected

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8 10 12 14 16 18 20

Water Injected, PV

Oil

Rec

over

ed, P

V

Oil recovery at breakthrough = 32.1%

Waterflooding Solutions

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