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 S w is usually be expressed as a fraction) S w , fraction k ro k rw k ro /k rw 0.100 0 inf. <-- extrapolated endpoint for k rw 0.216 0.84 0.03 25.33 <-- data for sample calculation 0.280 0.59 0.08 7.32 0.321 0.48 0.11 4.41 0.375 0.33 0.15 2.19 0.410 0.23 0.18 1.31 0.451 0.16 0.21 0.73 0.494 0.10 0.26 0.38 0.524 0.07 0.30 0.24 0.548 0.05 0.33 0.17 0.565 0.04 0.35 0.12 0.586 0.03 0.38 0.07 0.627 no value 0.44 0.64 0.00 0.00 <-- extrapolated endpoint for k ro See Chart 1 for a graph. Finally, the endpoints, S w and 1-S or were obtained by inspecting the graph and 1. extrapolating the k rw line to the x axis (defining S wirr ) THIS IS THE MOST IMPORTANT DATA POINT 2. extrapolating the k ro line to the x axis (defining 1- S or ) This is quite difficult in this case because the original data was given on a semi-log scale; some latitude should be given to students because 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
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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)
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.
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 )
PET E 471 Homework #1 Solutions 2/7
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:
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.
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
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 Homework #1 Solutions 7/7
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
Water Saturation, Sw
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
Water Saturation, Sw
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