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Core Petrophysics Section Six: Advanced Interpretation Methods for Engineers Exercises Question 6.1: Interpreting relative permeability measurements The 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. Background To 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 (q o /q w ) as a function of k ro , k rw , µ o and µ w . The ratio (q o /q w ) 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 experiment The 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: length: diameter: area

horizontal 8.5 cm 2.54 cm 2 5.07 cm

S wi

0.2

oil: density: viscosity: IFT to brine:

decane 3 730 kg/m 1.0E-03 Pa.s 3.5E-02 N/m

brine: density: viscosity:

simul. formation brine 3 995 kg/m 1.0E-03 Pa.s

Test data step

1

2

3

4

5

6

7

8

9

100:0

90:10

70:30

50:50

30:70

10:90

5:95

1:99

0:100

50

50

50

50

50

50

50

50

50

3

0

5

15

25

35

45

47.5

49.5

50

3

q o (cm /hr)

50

45

35

25

15

5

2.5

0.5

0

delta P (Pa)

15580

79940

109400

117930

111700

84700

69940

46855

32535

0.2

0.435

0.49

0.525

0.56

0.61

0.635

0.685

0.72

o:w ratio flow rate 3 (cm /hr) q w (cm /hr)

K w,eff (mD) K o,eff (mD) k rw (to K oil@Swi ) k ro (to K oil@Swi ) S w (avg.)

Question 6.2: Handling relative permeability data The 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) k

reported rp

(S w ) =

k pmeasured (S w )

k omeasured (S wirr )

(6.2.1)

However, as given in notes, the correct definition of relative permeability for a given phase is k rp (S w ) =

k pmeasured (S w ) k abs

(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 form Use 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 ( k rp ) in the correct form, in terms of the reported relative permeability ( k rpreported ), the absolute permeability of the plug ( k abs ), and the

permeability to oil at irreducible saturation ( k omeasured (S wirr ) ).

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 constituent components 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) RQI =

k

(6.2.3)

φ

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

S wn =

S w − S wirr 1 − S wirr − S or

(1.6.3)

with the relative permeability to water and oil given by e p k rw (S w ) = k rw S wn

(1.6.4)

k ro (S w ) = k roe (1 − S wn )

q

(1.6.5)

The relative permeability values can also be normalised by dividing by the end-point value, yielding p e k wn (S w ) = S wn = k rw (S w ) k rw

(6.2.4)

k on (S w ) = (1 − S wn ) = k ro (S w ) k roe

(6.2.5)

q

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 Relative permeability Low rate Mildly cleaned sample Oil permeability at SWI

170.966

Well Depth

Test 1 1550

Plug No.

10

mD

Sw kro 0.32 1 0.369 0.6561 0.418 0.4096 0.467 0.2401 0.516 0.1296 0.565 0.0625 0.614 0.0256 0.663 0.0081 0.712 0.0016 0.761 1E-04 0.81 0

krw 0 0.000466 0.003726 0.012575 0.029808 0.058219 0.100603 0.159753 0.238466 0.339534 0.465753

Relative Relative permeability permeability

1 0.9 1 0.8 0.9 0.7 0.8 0.6 0.7 0.5 0.6 0.4 0.5 0.3 0.4 0.2 0.3 0.1 0.2 0 0.1 0

0.2

0.4

0.6

0.8

1

0.8

1

Water saturation

0 0

0.2

0.4

0.6

Water saturation

Figure 6.2.1. Relative permeability data reported by contractor for plug 10

Water/oil Relative permeability Low rate Mildly cleaned sample Oil permeability at SWI 442.68

Well Depth Plug No.

Test 1 1552.5 15

mD 1

0 0.000893 0.005051 0.013919 0.028572 0.049913 0.078735 0.115754 0.161628 0.216969 0.282353

kro krw

0.9

1

0.8

Relative permeability

0.406 0.458 0.51 0.562 0.614 0.666 0.718 0.77

1 0.69159 0.457947 0.286974 0.167313 0.088388 0.040477 0.014789 0.003578 0.000316 0

0.9 0.8

Relative permeability

Sw 0.25 0.302 0.354

0.7 0.6 0.5

0.7 0.6 0.5 0.4 0.3

0.4

0.2

0.3

0.1

0.2

0 0

0.1

0.2

0.4

0.6

0.8

Water saturation

0 0

0.2

0.4

0.6

Water saturation

Figure 6.2.2. Relative permeability data reported by contractor for plug 15

0.8

1

1

Water/oil Relative permeability Low rate Mildly cleaned sample Oil permeability at SWI 232.8

Well Depth Plug No.

Test 1 1554.5 19

mD 1

0 0.000655 0.004562 0.014198 0.031774 0.059349 0.098884 0.152256 0.221285 0.307737 0.413333

kro krw

0.9

1

0.8

Relative permeability

0.416 0.468 0.52 0.572 0.624 0.676 0.728 0.78

1 0.713799 0.489652 0.319384 0.195022 0.108819 0.053283 0.021222 0.005798 0.000631 0

0.9 0.8

Relative permeability

Sw 0.26 0.312 0.364

0.7 0.6 0.5

0.7 0.6 0.5 0.4 0.3

0.4

0.2

0.3

0.1

0.2

0 0

0.1

0.2

0.4

0.6

0.8

Water saturation

0 0

0.2

0.4

0.6

Water saturation

Figure 6.2.3. Relative permeability data reported by contractor for plug 19

0.8

1

1

Water/oil Relative permeability Low rate Mildly cleaned sample

Well Depth

Test 1 1556

Oil permeability at SWI

183.057 mD

Plug No.

22

1

0.22 0.268 0.316 0.364 0.412 0.46 0.508 0.556 0.604 0.652 0.7

kro

krw

1 0.784798 0.598559 0.440276 0.30885 0.203063 0.121545 0.062716 0.024681 0.005012 0

0 0.002533 0.010858 0.025441 0.046548 0.074372 0.109066 0.150758 0.199555 0.255554 0.318841

0.9 0.8

Relative permeability

Sw

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.2

0.4

0.6

Water saturation

Figure 6.2.4. Relative permeability data reported by contractor for plug 22

0.8

1

Water/oil Relative permeability Low rate Mildly cleaned sample

Well Depth

Test 1 1557.5

Plug No.

Oil permeability at SWI

25

62.452 mD 1

0.43 0.463 0.496 0.529 0.562 0.595 0.628 0.661 0.694 0.727 0.76

kro

krw

1 0.663049 0.418843 0.248818 0.136392 0.066986 0.028057 0.009136 0.001879 0.000126 0

0 0.012191 0.042453 0.08808 0.147831 0.220904 0.306711 0.404793 0.514776 0.636346 0.769231

0.9 0.8

Relative permeability

Sw

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.2

0.4

0.6

Water saturation

Figure 6.2.5. Relative permeability data reported by contractor for plug 25

0.8

1

Water/oil Relative permeability Low rate Mildly cleaned sample

Well Depth

Test 1 1559

Plug No.

Oil permeability at SWI

28

372.69 mD 1

0.19 0.248 0.306 0.364 0.422 0.48 0.538 0.596 0.654 0.712 0.77

kro

krw

1 0.69159 0.457947 0.286974 0.167313 0.088388 0.040477 0.014789 0.003578 0.000316 0

0 0.001234 0.006981 0.019237 0.03949 0.068986 0.108821 0.159986 0.223389 0.299876 0.390244

0.9 0.8

Relative permeability

Sw

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.2

0.4

0.6

Water saturation

Figure 6.2.6. Relative permeability data reported by contractor for plug 28

0.8

1

Water/oil Relative permeability Low rate Mildly cleaned sample

Well Depth

Test 1 1561

Oil permeability at SWI

260.224 mD

Plug No.

32

1

0.21 0.257 0.304 0.351 0.398 0.445 0.492 0.539 0.586 0.633 0.68

kro

krw

1 0.801511 0.625877 0.472831 0.342072 0.233258 0.145991 0.079791 0.034054 0.007943 0

0 0.00381 0.014219 0.030722 0.053067 0.081088 0.114657 0.153674 0.198055 0.247728 0.302632

0.9 0.8

Relative permeability

Sw

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.2

0.4

0.6

Water saturation

Figure 6.2.7. Relative permeability data reported by contractor for plug 32

0.8

1

Water/oil Relative permeability Low rate Mildly cleaned sample

Well Depth

Test 1 1562

Plug No.

Oil permeability at SWI

34

92.125 mD 1

0.38 0.417 0.454 0.491 0.528 0.565 0.602 0.639 0.676 0.713 0.75

kro

krw

1 0.635686 0.383078 0.215735 0.111186 0.050766 0.019447 0.005644 0.000987 5.01E-05 0

0 0.001882 0.009933 0.026285 0.052427 0.089565 0.138731 0.200838 0.276711 0.367107 0.472727

0.9 0.8

Relative permeability

Sw

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.2

0.4

0.6

Water saturation

Figure 6.2.8. Relative permeability data reported by contractor for plug 34

0.8

1

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 endpoints 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 10 11 12 15 16 17 18 19 20 21 22 24 25 26 27 28 29 32 33 34

Depth (m) 1550 1550.5 1551 1552.5 1553 1553.5 1554 1554.5 1555 1555.5 1556 1557 1557.5 1558 1558.5 1559 1559.5 1561 1561.5 1562

Poro (%) 18.1 21.3 18.1 23.2 19.2 18.7 21.3 20.2 21.4 24.3 19.3 17.4 15.8 22.3 21.3 23.2 20.2 19.4 21.3 17.8

kh (mD) 234.2 843.2 310.2 520.8 89.5 201.4 345.3 310.4 523.2 1564.5 256.3 45.7 120.1 289.5 313.4 454.5 245.3 342.4 876.5 167.5

Table 6.3.1 Porosity and permeability data

GR (GAPI) and Porosity (%) 0

50

100

150

1548 1550

Depth (m)

1552 1554 1556 1558 1560 1562 1564

Figure 6.3.2. Gamma ray and porosity as a function of depth

1 0.9

Relative permeability

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.2

0.4

0.6

0.8

1

Water saturation

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.

1 0.9 0.8

Endpoints

0.7

Kroe y = 1.7245x + 0.0531

0.6 0.5 0.4 0.3 0.2 0.1 0 0.2

0.25

0.3

0.35

0.4

0.45

0.5

RQI

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.

Question 6.4 One 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 were collected 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

J = A + B(S w − S wirr )

(6.4.1)

log( J ) = D log(S w − S wirr ) + E

(6.4.2)

C

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-gravity equilibrium yields

J (S w ) =

Pc(S w ) k (ρ w − ρ o )gh k = σ coσ θ φ σ coσ θ φ

(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 rearranging your chosen curve fit.

Sw 0.23 0.27 0.31 0.35 0.38 0.42 0.46 0.50 0.54 0.58 0.62 0.65 0.69 0.73 0.77 0.81 0.85 0.88 0.92 0.96 1.00 1.00

P c (Pa) 13131 5848 2968 2062 1625 1369 1202 1085 998 932 879 836 801 772 746 725 706 690 675 662 650 0

Table 6.4.1. Air-brine capillary pressure data measured during drainage at laboratory conditions using the porous-plate method

Depth (mTVDSS) 1393.1 1393.4 1393.7 1394 1394.3 1394.6 1394.9 1395.2 1395.5 1395.8 1396.1 1396.4 1396.7 1397 1397.3 1397.6 1397.9 1398.2 1398.5 1398.8 1399.1 1399.4 1399.7 1400

φ 0.21 0.19 0.15 0.16 0.23 0.21 0.18 0.17 0.19 0.21 0.23 0.22 0.21 0.24 0.19 0.18 0.2 0.13 0.08 0.09 0.03 0.18 0.21 0.12

k (mD) 234 58 45 56 890 670 210 105 98 234 670 125 216 703 324 126 453 34 21 11 34 21 345 321

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.

STOIIP = GRV ntgav φav (1-Sw)av (1/Bo)av