Exercises Section6

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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.



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



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)



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?



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.



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


Water saturation




Relative permeability Depth 1552.5

Low rate Plug No. 15


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


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


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


Water saturation





Low rate Plug No. 19


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


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


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


Water saturation





Low

rate Plug No. 22


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


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


Water saturation





Low

rate Plug No. 25



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


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


Water saturation





Low

rate Plug No. 28



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


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


Water saturation





Low

rate Plug No. 32



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


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


Water saturation





Low

rate Plug No. 34



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


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


Water saturation



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?



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



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



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



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.



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



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



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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