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Core Petrophysics
Section Six: Advanced Interpretation Methods forEngineers
Exercises
Question 6.1: Interpreting relative permeability measurementsThe objectives of this question are to
1. Understand the physics behind a steady-state experiment
2. Determine relative permeability curves from a steady state experiment
Assume that the capillary number for this experiment is small; this is equivalent to
assuming that P c = 0. A small capillary number would be observed in low permeability
plugs flooded at high rate.
BackgroundTo show some of the specific features of a typical steady-state experiment, an example
experiment will be interpreted in detail. Before doing this, answer the following 'general'
questions about the technique. The answers will be of use in the remainder of the exercise.
6.1.1. Starting with Darcy’s Law for multiphase flow, write down an expression for the ratio
of oil/water flow rates (qo/qw) as a function of k ro, k rw, µ o and µ w. The ratio (qo/qw) is
known as the ‘fractional flow’.
6.1.2. Is the average oil saturation at steady-state during a 100% water flood the same as the
residual oil saturation (S or )?
The experimentThe rest of this exercise involves a typical steady-state imbibition experiment (S w increasing).
A core plug has been brought to an initial water saturation of S wirr = 0.2 by capillary
desaturation using the porous-plate method. Assume that this saturation is homogeneously
distributed over the core. The experimental results of the steady-state experiment, consisting
of 9 different fractional flow rates (step 1-9) are reported in table 6.1.1. This table resembles
the data sheet which a third party laboratory might use. Be aware, however, that these sheets
are usually not provided if not specifically asked for. The tabulated data have been circulated
in Excel format.
Analysis
6.1.3. Complete the table to report the relative permeability values for each flow rate.
Relative permeability here is reported relative to the oil permeability at the irreducible
saturation (K oil(S wirr )).
6.1.4 Plot the resulting relative permeability curves and label them.
6.1.5. What is the residual oil saturation in this experiment?
Table 6.1.1 (overleaf). Data from a steady-state relative permeability experiment.
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Plug data Fluid data at experimental temperature
orientation: horizontal oil: decane brine: simul. formation brine
length: 8.5 cm density: 730 kg/m3 density: 995 kg/m
3
diameter: 2.54 cm viscosity: 1.0E-03 Pa.s viscosity: 1.0E-03 Pa.s
area 5.07 cm2 IFT to brine: 3.5E-02 N/m
Swi 0.2
Test data
step 1 2 3 4 5 6 7 8 9
o:w ratio 100:0 90:10 70:30 50:50 30:70 10:90 5:95 1:99 0:100
flow rate(cm
3/hr)
50 50 50 50 50 50 50 50 50
qw (cm3/hr) 0 5 15 25 35 45 47.5 49.5 50
qo (cm3/hr) 50 45 35 25 15 5 2.5 0.5 0
delta P (Pa) 15580 79940 109400 117930 111700 84700 69940 46855 32535
Kw,eff (mD)
Ko,eff (mD)
k rw (to Koil@Swi)
kro
(to Koil@Swi
)
Sw (avg.) 0.2 0.435 0.49 0.525 0.56 0.61 0.635 0.685 0.72
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Question 6.2: Handling relative permeability dataThe aim of this exercise is to give you some experience of handling relative permeability
data. Figures 6.2.1-6.2.8 show a suite of relative permeability curves obtained from a vertical
well in an oil-bearing sandstone reservoir. The data are presented in the form they are
typically reported by the measuring laboratory. The tabulated data have been circulated in
Excel format.
Section 1: Re-scaling reported relative permeability data for analysis and application
Measured permeability data for a given phase p (in this case oil and water) during a relative
permeability experiment are reported relative to the oil permeability at irreducible oil
saturation (denoted ‘Oil permeability at SWI’ in the tables)
( ) ( )
( )wirr
measured
o
w
measured
p
w
reported
rpS k
S k S k = (6.2.1)
However, as given in notes, the correct definition of relative permeability for a given phase is
( ) ( )
abs
w
measured
p
wrpk
S k S k = (6.2.2)
Consequently, prior to analysis/interpretation, the reported relative permeability data must be
re-scaled so that it is relative to the absolute permeability of the plug, rather than the
permeability to oil at irreducible saturation.
6.2.1: Derive an expression to convert the reported data to the appropriate formUse equations (6.2.1) and (6.2.2) above to deduce a new equation which will allow you to
express the relative permeability of a given phase p (rpk ) in the correct form, in terms of the
reported relative permeability ( reported
rpk ), the absolute permeability of the plug (
absk ), and the
permeability to oil at irreducible saturation ( ( )wirr
measured
o S k ).
6.2.2: Apply your expression and plot the correctly re-scaled data
Using your expression, re-scale the reported relative permeability curves for all plugs and
plot them on the graphs provided. Plot the original (reported) data on the same axis for
comparison. How much do the relative permeability curves changed when they are properly
scaled?
6.2.3: Identify the end-point values
Using your properly scaled curves, identify the values of the following parameters:
1. End-point relative permeability to oil (k roe)
2. End-point relative permeability to water (k rwe)
3. Irreducible water saturation (S wirr )
4. Residual oil saturation to waterflooding (S orw)
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Section 2: Variation in relative permeability with rock quality
Handling relative permeability data is challenging. A large number of curves may be
reported, and it can be difficult to identify a sensible way of analyzing and distributing them
for reservoir characterization.
One approach which is commonly used is to break down the curves into their constituentcomponents of end-points and shape, and to investigate how these components vary with rock
quality. The rock quality is often expressed quantitatively in terms of the rock quality index
(RQI)
φ
k RQI = (6.2.3)
6.2 4. Investigate variations in irreducible water saturation with RQI
Plot a graph of S wirr versus RQI for each plug in the dataset. Can you identify a trend? Can
you explain this trend in terms of the pore-scale distribution of oil and water?
6.2.5. Investigate variations in other end-point properties with RQI
Repeat task 6.2.4 for the other end-point values (k rwe, k roe, S orw).
Can you identify any trends? Can you explain these trends in terms of the pore-scale
distribution of oil and water?
6.2.6 Building a relative permeability model
A typical model for relative permeability uses power-law functions of the normalised water
saturation, given by
or wirr
wirr w
wnS S
S S S
−−
−
=
1 (1.6.3)
with the relative permeability to water and oil given by
( ) p
wn
e
rwwrw S k S k = (1.6.4)
( ) ( )qwn
e
rowro S k S k −= 1 (1.6.5)
The relative permeability values can also be normalised by dividing by the end-point value,
yielding
( ) ( ) e
rwwrw
p
wnwwn k S k S S k == (6.2.4)
( ) ( ) ( ) e
rowro
q
wnwon k S k S S k =−= 1 (6.2.5)
For each plug, plot the normalised relative permeability curves (so they scale between one
and zero on each axis). Can you fit a single normalised curve through all the data (i.e. can
you identify a single value of p and q for all plugs?
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You have developed a simple relative permeability model here which allows curves to be
developed for any interval of rock for which the permeability and porosity values are known.
The regressions you have identified allow the end-points to be predicted, and the shape of the
curve is given by the values of p and q.
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Water/oil Well Test 1
Relative permeability Depth 1550
Low
rate Plug No. 10
Mildly cleaned sample
Oil permeability at
SWI 170.966 mD
Sw kro krw
0.32 1 0
0.369 0.6561 0.000466
0.418 0.4096 0.003726
0.467 0.2401 0.012575
0.516 0.1296 0.029808
0.565 0.0625 0.058219
0.614 0.0256 0.100603
0.663 0.0081 0.159753
0.712 0.0016 0.238466
0.761 1E-04 0.339534
0.81 0 0.465753
Figure 6.2.1. Relative permeability data reported by contractor for plug 10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
R e l a t i v e p e r m e a b i l i t y
Water saturation
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
R e l a t i v e p e r m e a b i l i t y
Water saturation
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Water/oil Well Test 1
Relative permeability Depth 1552.5
Low rate Plug No. 15
Mildly cleaned sample
Oil
permeability
at SWI 442.68 mD
Sw kro krw
0.25 1 0
0.302 0.69159 0.000893
0.354 0.457947 0.005051
0.406 0.286974 0.013919
0.458 0.167313 0.028572
0.51 0.088388 0.049913
0.562 0.040477 0.078735
0.614 0.014789 0.115754
0.666 0.003578 0.161628
0.718 0.000316 0.216969
0.77 0 0.282353
Figure 6.2.2. Relative permeability data reported by contractor for plug 15
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
R e l a t i v e p e r m e a b i l i t y
Water saturation
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
R e l a t i v e p e r m e a b i l i t y
Water saturation
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Water/oil Well Test 1
Relative permeability Depth 1554.5
Low rate Plug No. 19
Mildly cleaned sample
Oil
permeability
at SWI 232.8 mD
Sw kro krw
0.26 1 0
0.312 0.713799 0.000655
0.364 0.489652 0.004562
0.416 0.319384 0.014198
0.468 0.195022 0.031774
0.52 0.108819 0.059349
0.572 0.053283 0.098884
0.624 0.021222 0.152256
0.676 0.005798 0.221285
0.728 0.000631 0.307737
0.78 0 0.413333
Figure 6.2.3. Relative permeability data reported by contractor for plug 19
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
R e l a t i v e p e r m e a b i l i t y
Water saturation
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
R e l a t i v e p e r m e a b i l i t y
Water saturation
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Water/oil Well Test 1
Relative permeability Depth 1556
Low
rate Plug No. 22
Mildly cleaned sample
Oil permeability at SWI 183.057 mD
Sw kro krw
0.22 1 0
0.268 0.784798 0.002533
0.316 0.598559 0.010858
0.364 0.440276 0.025441
0.412 0.30885 0.046548
0.46 0.203063 0.074372
0.508 0.121545 0.109066
0.556 0.062716 0.150758
0.604 0.024681 0.199555
0.652 0.005012 0.255554
0.7 0 0.318841
Figure 6.2.4. Relative permeability data reported by contractor for plug 22
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
R e l a t i v e p e r m e a b i l i t y
Water saturation
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Water/oil Well Test 1
Relative permeability Depth 1557.5
Low
rate Plug No. 25
Mildly cleaned sample
Oil permeability at SWI 62.452 mD
Sw kro krw
0.43 1 0
0.463 0.663049 0.012191
0.496 0.418843 0.042453
0.529 0.248818 0.08808
0.562 0.136392 0.147831
0.595 0.066986 0.220904
0.628 0.028057 0.306711
0.661 0.009136 0.404793
0.694 0.001879 0.514776
0.727 0.000126 0.636346
0.76 0 0.769231
Figure 6.2.5. Relative permeability data reported by contractor for plug 25
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
R e l a t i v e p e r m e a b i l i t y
Water saturation
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Water/oil Well Test 1
Relative permeability Depth 1559
Low
rate Plug No. 28
Mildly cleaned sample
Oil permeability at SWI 372.69 mD
Sw kro krw
0.19 1 0
0.248 0.69159 0.001234
0.306 0.457947 0.006981
0.364 0.286974 0.019237
0.422 0.167313 0.03949
0.48 0.088388 0.068986
0.538 0.040477 0.108821
0.596 0.014789 0.159986
0.654 0.003578 0.223389
0.712 0.000316 0.299876
0.77 0 0.390244
Figure 6.2.6. Relative permeability data reported by contractor for plug 28
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
R e l a t i v e p e r m e a b i l i t y
Water saturation
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Water/oil Well Test 1
Relative permeability Depth 1561
Low
rate Plug No. 32
Mildly cleaned sample
Oil permeability at SWI 260.224 mD
Sw kro krw
0.21 1 0
0.257 0.801511 0.00381
0.304 0.625877 0.014219
0.351 0.472831 0.030722
0.398 0.342072 0.053067
0.445 0.233258 0.081088
0.492 0.145991 0.114657
0.539 0.079791 0.153674
0.586 0.034054 0.198055
0.633 0.007943 0.247728
0.68 0 0.302632
Figure 6.2.7. Relative permeability data reported by contractor for plug 32
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
R e l a t i v e p e r m e a b i l i t y
Water saturation
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Water/oil Well Test 1
Relative permeability Depth 1562
Low
rate Plug No. 34
Mildly cleaned sample
Oil permeability at SWI 92.125 mD
Sw kro krw
0.38 1 0
0.417 0.635686 0.001882
0.454 0.383078 0.009933
0.491 0.215735 0.026285
0.528 0.111186 0.052427
0.565 0.050766 0.089565
0.602 0.019447 0.138731
0.639 0.005644 0.200838
0.676 0.000987 0.276711
0.713 5.01E-05 0.367107
0.75 0 0.472727
Figure 6.2.8. Relative permeability data reported by contractor for plug 34
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
R e l a t i v e p e r m e a b i l i t y
Water saturation
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Question 6.3 Integration of permeability from core and pressure transient test
The aim of this exercise is to compare permeability estimates from core and from an
interpretation of a pressure transient test. Data integration at this basic level is essential to
ensure consistency between different data sources and interpretation models.
Background
Table 6.3.1 reports values of porosity and horizontal permeability against depth. The data
were obtained from a vertical well in an oil bearing sandstone reservoir which contains some
shale. The gamma ray log is also shown. Figure 6.3.2 shows a relative permeability curve
measured over the same interval. A pressure transient test has also been obtained over the
interval 1551 – 1561m.
6.3.1: Calculate the average horizontal permeability of the tested interval
Use the expression for flow parallel to layering you derived yesterday. Be careful to note the
irregular data spacing. What limit of effective permeability does your average represent?
6.3.2: Calculate the average horizontal permeability of the tested interval to oil
To compare permeability estimates from core and test, it is important to remember that the
test measures the permeability of the reservoir to the flowing phase. In this case, the test
flowed oil in the presence of connate water.
Use the data shown in Figure 6.3.2 to convert your average reservoir permeability to an
average permeability to oil, so that it can be compared with the estimate from test.
6.3.3: Compare your average horizontal permeability value with that obtained from the
pressure-transient test interpretation.
Interpretation of a pressure-transient test over the same interval yield an estimate of
horizontal permeability of 363.4mD (you will learn more about interpretation methods next
week). How does this compare with your calculated average value? Can you explain why
they might be different?
6.3.4: Calculate the average vertical permeability of the tested interval
Use the expression for flow perpendicular to layering you derived yesterday. Be careful to
note the irregular data spacing. What limit of effective permeability does your average
represent? What assumption have you made about permeability at the plug scale? Is this
reasonable? What value of k v/k h ratio do you obtain?
6.3.5: Calculate the average vertical permeability of the tested interval to oil
6.3.6: Compare your average vertical permeability value (and k v/k h ratio) with that obtained
from the pressure-transient test interpretation.
Interpretation of a pressure-transient test over the same interval yields an estimate of vertical
permeability of 2.4mD. How does this compare with your calculated average value? What
value of k v/k h ratio does this yield, and how does this compare with your calculated value?
Can you explain why they might be different?
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6.3.7: Investigate the impact of varying the end-point permeability to oil
So far, you have assumed that the end-point relative permeability to oil is constant and
independent of rock quality. However, yesterday you found that relative permeability end-
points can correlate to rock quality.
Use the information provided in Figure 6.3.3 (which shows end-point oil relative permeability correlated to rock quality index) and Table 6.3.1 (also available in Excel) to
calculate an appropriate value of permeability to oil at each depth. You will need to calculate
the rock quality index (RQI) at each depth, and use Figure 6.3.3 to calculate the appropriate
end-point relative permeability to oil.
Repeat tasks 1-6 using these new values of oil-phase permeability at each depth. How does
your match to the test data change?
Plug Depth (m) Poro (%) kh (mD)
10 1550 18.1 234.2
11 1550.5 21.3 843.2
12 1551 18.1 310.2
15 1552.5 23.2 520.8
16 1553 19.2 89.5
17 1553.5 18.7 201.4
18 1554 21.3 345.3
19 1554.5 20.2 310.4
20 1555 21.4 523.2
21 1555.5 24.3 1564.5
22 1556 19.3 256.3
24 1557 17.4 45.7
25 1557.5 15.8 120.1
26 1558 22.3 289.5
27 1558.5 21.3 313.4
28 1559 23.2 454.5
29 1559.5 20.2 245.3
32 1561 19.4 342.4
33 1561.5 21.3 876.5
34 1562 17.8 167.5
Table 6.3.1 Porosity and permeability data
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Figure 6.3.2. Gamma ray and porosity as a function of depth
Figure 6.3.3. Relative permeability curves, plotted with respect to K abs (not K o). These are
the same curves you de-normalized in the previous question.
1548
1550
1552
1554
1556
1558
1560
1562
1564
0 50 100 150
D e p t h ( m )
GR (GAPI) and Porosity (%)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
R e l a
t i v e p e r m e a b i l i t y
Water saturation
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Figure 6.3.4. k roe versus RQI. The regression yields k roe = 1.7245RQI + 0.0531. These are
the same data that you plotted in the previous exercise.
y = 1.7245x + 0.0531
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.2 0.25 0.3 0.35 0.4 0.45 0.5
E n d p o i n t s
RQI
Kroe
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Question 6.4One approach to analysing multiple capillary pressure curves from a given rock type is to
convert them to a J-Function and fit a single curve through the dimensionless data. This
curve can then be rescaled to local values of k and φ within the rock type.6.4.1 Convert the gas-brine capillary pressure data in table 6.4.1, obtained at laboratory
conditions, to a J-Function form, assuming the plug from which these data werecollected had a porosity of 0.21 and a permeability of 245 mD, and using appropriate
values of contact angle and IFT. It is recommended that you convert all parameters to
SI units before calculating the J-Function; this will make it easier to apply in later
analysis.
6.4.2 Plot the J-function data as a function of water saturation and try to fit a curve to the
data. Common curve fits include
( )C wirr w S S B A J −+= (6.4.1)
E S S D J wirr w +−= )log()log( (6.4.2)
where A, B, C , D, E and S wirr are adjustable parameters to fit the data. It is
recommended for this exercise to use equation 6.4.2. Try a match by eye; if you have
done this before, try calculating the R2 fit of your curve to the data. This latter step is
not obligatory.
6.4.3 Table 6.4.2 shows permeability and porosity data as a function of depth within the
reservoir. Use you J-Function curve, along with the fluid properties from question
4.4, and assuming capillary-gravity equilibrium, to predict and plot water saturation as
a function of height above the FWL, accounting for the variations in k and φ . This is
a common application of capillary pressure data. Note that capillary-gravityequilibrium yields
( ) ( )φ θ σ
ρ ρ
φ θ σ
k ghk S PcS J oww
wcoscos
)( −
== (6.4.3)
The approach is to calculate J (S w) for each height h using 6.4.3 and the corresponding
values of k and φ (Table 6.4.2), and then calculate S w for this value of J by re-
arranging your chosen curve fit.
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Sw Pc (Pa)
0.23 13131
0.27 5848
0.31 2968
0.35 2062
0.38 1625
0.42 1369
0.46 1202
0.50 1085
0.54 998
0.58 932
0.62 879
0.65 836
0.69 801
0.73 772
0.77 746
0.81 725
0.85 706
0.88 690
0.92 675
0.96 662
1.00 650
1.00 0
Table 6.4.1. Air-brine capillary pressure data measured during drainage at laboratory
conditions using the porous-plate method
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Depth (mTVDSS) φ k (mD)
1393.1 0.21 234
1393.4 0.19 58
1393.7 0.15 45
1394 0.16 56
1394.3 0.23 890
1394.6 0.21 670
1394.9 0.18 210
1395.2 0.17 105
1395.5 0.19 98
1395.8 0.21 234
1396.1 0.23 670
1396.4 0.22 125
1396.7 0.21 216
1397 0.24 703
1397.3 0.19 324
1397.6 0.18 126
1397.9 0.2 453
1398.2 0.13 34
1398.5 0.08 21
1398.8 0.09 11
1399.1 0.03 34
1399.4 0.18 21
1399.7 0.21 345
1400 0.12 321
Table 6.4.2 Porosity and permeability as a function of depth
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Question 6.5: Integration of data to calculate STOIIP
The equation below is used to calculate the Stock Tank Oil Initially In Place. Next to each
parameter, write down all data sources you can think of that contribute to calculating that
parameter. Then, think of as many links (i.e. possible integration routes) as you can between
each data source.
S T OI I P
= GR V
n t g a v
φ a v
( 1 - S w ) a v
(
1 / B o ) a v
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