Tarek Ahmed. Reservoir engineering handbook

Подождите немного. Документ загружается.

Solution

Step 1. Calculate the normalized water saturation for each core sample by

using Equation 5-36.

Core Sample #1 Core Sample #2 Core Sample #3

0.20 — — 0.000

0.25 0.000 — 0.111

0.30 0.125 0.000 0.222

0.40 0.375 0.238 0.444

0.50 0.625 0.476 0.667

0.60 0.875 0.714 0.889

0.65 1.000 0.833 1.000

0.72 — 1.000 —

Step 2. Determine relative permeability values at critical saturation for

each core sample.

Core 1 Core 2 Core 3

0.850 0.800 1.000

0.400 0.500 0.35

Step 3. Calculate (k

–

)

and (k

–

)

by applying Equations 5-39 and

5-40 to give:

–

)

= 0.906

–

)

= 0.402

Step 4. Calculate the normalized k

and k

for all core samples:

Core 1 Core 2 Core 3

0.20 — — — — — — 0.000 1.000 0

0.25 0.000 1.000 0 — — — 0.111 0.872 0.023

0.30 0.125 0.887 0.045 0.000 1.000 0 0.222 0.839 0.077

0.40 0.375 0.655 0.230 0.238 0.741 0.154 0.444 0.663 0.251

0.50 0.625 0.414 0.495 0.476 0.491 0.382 0.667 0.463 0.503

0.60 0.875 0.154 0.818 0.714 0.252 0.646 0.889 0.215 0.817

0.65 1.000 0.000 1.000 0.833 0.139 0.788 1.000 0.000 1.000

0.72 — — — 1.000 0.000 1.000 — — —

308 Reservoir Engineering Handbook

Reservoir Eng Hndbk Ch 05 2001-10-24 09:52 Page 308

Step 5. Plot the normalized values of k

and k

versus S

for each core

on a regular graph paper as shown in Figure 5-9.

Step 6. Select arbitrary values of S

and calculate the average k

and k

by applying Equations 5-37 and 5-38.

(k*

)

Avg

(k*

)

avg

Core Core Core Core Core Core

123 123

0.1 0.91 0.88 0.93 0.912 0.035 0.075 0.020 0.038

0.2 0.81 0.78 0.85 0.821 0.100 0.148 0.066 0.096

0.3 0.72 0.67 0.78 0.735 0.170 0.230 0.134 0.168

0.4 0.63 0.51 0.70 0.633 0.255 0.315 0.215 0.251

0.5 0.54 0.46 0.61 0.552 0.360 0.405 0.310 0.348

0.6 0.44 0.37 0.52 0.459 0.415 0.515 0.420 0.442

0.7 0.33 0.27 0.42 0.356 0.585 0.650 0.550 0.585

0.8 0.23 0.17 0.32 0.256 0.700 0.745 0.680 0.702

0.9 0.12 0.07 0.18 0.135 0.840 0.870 0.825 0.833

Relative Permeability Concepts 309

Figure 5-9. Averaging relative permeability data.

Reservoir Eng Hndbk Ch 05 2001-10-24 09:52 Page 309

Step 7. Using the desired formation S

and S

(i.e., S

= 0.30, S

0.27), de-normalize the data to generate the required relative per-

meability data as shown below:

(k*

)

avg

(k*

)

avg

= S*

(1 - S

- S

= 0.906 k

= 0.402

+ S

(k*

)

avg

(k*

)

avg

0.1 0.912 0.038 0.313 0.826 0.015

0.2 0.821 0.096 0.356 0.744 0.039

0.3 0.735 0.168 0.399 0.666 0.068

0.4 0.633 0.251 0.442 0.573 0.101

0.5 0.552 0.368 0.485 0.473 0.140

0.6 0.459 0.442 0.528 0.416 0.178

0.7 0.356 0.585 0.571 0.323 0.235

0.8 0.256 0.702 0.614 0.232 0.282

0.9 0.135 0.833 0.657 0.122 0.335

It should be noted that the proposed normalization procedure for

water-oil systems as outlined above could be extended to other systems,

i.e., gas-oil or gas-water.

THREE-PHASE RELATIVE PERMEABILITY

The relative permeability to a fluid is defined as the ratio of effective

permeability at a given saturation of that fluid to the absolute permeabili-

ty at 100% saturation. Each porous system has unique relative permeabil-

ity characteristics, which must be measured experimentally. Direct exper-

imental determination of three-phase relative permeability properties is

extremely difficult and involves rather complex techniques to determine

the fluid saturation distribution along the length of the core. For this rea-

son, the more easily measured two-phase relative permeability character-

istics are experimentally determined.

In a three-phase system of this type it is found that the relative perme-

ability to water depends only upon the water saturation. Since the water

can flow only through the smallest interconnect pores that are present in

the rock and able to accommodate its volume, it is hardly surprising that

the flow of water does not depend upon the nature of the fluids occupy-

ing the other pores. Similarly, the gas relative permeability depends only

upon the gas saturation. This fluid, like water, is restricted to a particular

range of pore sizes and its flow is not influenced by the nature of the

fluid or fluids that fill the remaining pores.

310 Reservoir Engineering Handbook

Reservoir Eng Hndbk Ch 05 2001-10-24 09:52 Page 310

The pores available for flow of oil are those that, in size, are larger

than pores passing only water, and smaller than pores passing only gas.

The number of pores occupied by oil depends upon the particular size

distribution of the pores in the rock in which the three phases coexist and

upon the oil saturation itself.

In general, the relative permeability of each phase, i.e., water, gas, and

oil, in a three-phase system is essentially related to the existing saturation

by the following functions:

= f (S

) (5-41)

= f (S

) (5-42)

= f (S

, S

) (5-43)

Function 5-43 is rarely known and, therefore, several practical

approaches are proposed and based on estimating the three-phase relative

permeability from two sets of two-phase data:

Set 1: Oil-Water System

row

= f (S

)

= f (S

)

Set 2: Oil-Gas System

rog

= f (S

)

= f (S

)

where k

row

and k

rog

are defined as the relative permeability to oil in the

water-oil two-phase system and similarly k

rog

is the relative permeability

of oil in the gas-oil system. The symbol k

is reserved for the oil relative

permeability in the three-phase system.

The triangular graph paper is commonly used to illustrate the changes in

the relative permeability values when three phases are flowing simultane-

ously, as illustrated in Figures 5-10 and 5-11. The relative permeability

data are plotted as lines of constant percentage relative permeability (oil,

water, and gas isoperms). Figures 5-10 and 5-11 show that the relative per-

meability data, expressed as isoperms, are dependent on the saturation val-

ues for all three phases in the rock.

Relative Permeability Concepts 311

Reservoir Eng Hndbk Ch 05 2001-10-24 09:52 Page 311

Three-Phase Relative Permeability Correlations

Honarpour, Keoderitz, and Harvey (1988) provided a comprehensive

treatment of the two- and three-phase relative permeabilities. The authors

listed numerous correlations for estimating relative permeabilities. The

simplest approach to predict the relative permeability to the oil phase in a

three-phase system is defined as:

= k

row

rog

(5-44)

There are several practical and more accurate correlations that have

developed over the years, including:

• Wyllie’s Correlations

• Stone’s Model I

• Stone’s Model II

• The Hustad-Holt Correlation

312 Reservoir Engineering Handbook

Figure 5-10. Three-plate relative permeability imbibition. (After Honarpour et al., 1988.)

Reservoir Eng Hndbk Ch 05 2001-10-24 09:52 Page 312

Wyllie’s Correlations

Wyllie (1961) proposed the following equations for three-phase rela-

tive permeabilities in a water-wet system:

In a cemented sandstone, Vugular rock, or oolitic limestone:

SS S S

owo wc

()

(5 - 46)

SS SSS

gwcwowc

--+-

22 2

[( ) ( ) ]

()

(5 - 45)

Relative Permeability Concepts 313

Figure 5-11. Three-phase drainage. (After Honarpour et al., 1988.)

Reservoir Eng Hndbk Ch 05 2001-10-24 09:52 Page 313

In unconsolidated, well-sorted sand:

Stone’s Model I

Stone (1970) developed a probability model to estimate three-phase

relative permeability data from the laboratory-measured two-phase data.

The model combines the channel flow theory in porous media with prob-

ability concepts to obtain a simple result for determining the relative per-

meability to oil in the presence of water and gas flow. The model

accounts for hysteresis effects when water and gas saturations are chang-

ing in the same direction of the two sets of data.

The use of the channel flow theory implies that water-relative perme-

ability and water-oil capillary pressure in the three-phase system are func-

tions of water saturation alone, irrespective of the relative saturations of

oil and gas. Moreover, they are the same function in the three-phase sys-

tem as in the two-phase water-oil system. Similarly, the gas-phase relative

permeability and gas-oil capillary pressure are the same functions of gas

saturation in the three-phase system as in the two-phase gas-oil system.

Stone suggested that a nonzero residual oil saturation, called minimum

oil saturation, S

exists when oil is displaced simultaneously by water

and gas. It should be noted that this minimum oil saturation S

is differ-

ent than the critical oil saturation in the oil-water system (i.e., S

orw

) and

the residual oil saturation in the gas-oil system, i.e., S

org

. Stone intro-

duced the following normalized saturations:

SSSS

owowc

()( )

()

(5 - 50)

()

(5 - 49)

wwc

(5 - 48)

wwc

(5 - 47)

314 Reservoir Engineering Handbook

Reservoir Eng Hndbk Ch 05 2001-10-24 09:52 Page 314

The oil-relative permeability in a three-phase system is then defined as:

= S

(5-54)

The two multipliers b

and b

are determined from:

where S

= minimum oil saturation

row

= oil relative permeability as determined from the oil-water

two-phase relative permeability at S

rog

= oil relative permeability as determined from the gas-oil

two-phase relative permeability at S

The difficulty in using Stone’s first model is selecting the minimum oil

saturation S

. Fayers and Mathews (1984) suggested an expression for

determining S

=aS

orw

+ (1 -a) S

org

(5 - 57)

with

a=1 -

1 -S

- S

org

(5 - 58)

rog

-1

(5 - 56)

row

-1

(5 - 55)

wc om

()

--1

(5 - 53)

for S S

wwc

wc om

wwc

()

≥

(5 - 52)

for S S

oom

wc om

oom

()

≥

(5 - 51)

Relative Permeability Concepts 315

Reservoir Eng Hndbk Ch 05 2001-10-24 09:52 Page 315

where S

orw

= residual oil saturation in the oil-water relative

permeability system

org

= residual oil saturation in the gas-oil relative permeability

system

Aziz and Sattari (1979) pointed out that Stone’s correlation could give

values greater than unity. The authors suggested the following nor-

malized form of Stone’s model:

where (k

)

Swc

is the value of the relative permeability of the oil at the

connate water saturation as determined from the oil-water relative per-

meability system. It should be noted that it is usually assumed that k

and krog curves are measured in the presence of connate water.

Stone’s Model II

It was the difficulties in choosing S

that led to the development of

Stone’s Model II. Stone (1973) proposed the following normalized

expression:

This model gives a reasonable approximation to the three-phase rela-

tive permeability.

The Hustad-Holt Correlation

Hustad and Holt (1992) modified Stone’s Model I by introducing an

exponent term n to the normalized saturations to give:

kkk

ro ro

row

rog

rg rw rg

(

)

(

)

(

)

(

)

(5 - 60)

row rog

ro S

()()

()

(5 - 59)

316 Reservoir Engineering Handbook

Reservoir Eng Hndbk Ch 05 2001-10-24 09:52 Page 316

where

The b term may be interpreted as a variable that varies between zero

and one for low- and high-oil saturations, respectively. If the exponent n

is one, the correlation is identical to Stone’s first model. Increasing n

above unity causes the oil isoperms at low oil saturations to spread from

one another. n values below unity have the opposite effect.

Example 5-5

Two-phase relative permeability tests were conducted on core sample

to generate the permeability data for oil-water and oil-gas systems. The

following information is obtained from the test:

= 0.10 S

= 0.15

orw

= 0.15 S

org

= 0.05

= 0.88

At the existing saturation values of S

= 40%, S

= 30%, and S

= 30%

the two-phase relative permeabilities are listed below:

row

= 0.403

= 0.030

SSS

wwc

wc om gc

---1

(5 - 65)

SSS

ggc

wc om gc

-- -1

(5 - 64)

SSS

oom

wc om gc

---1

(5 - 63)

()()11

(5 - 62)

row rog

ro S

()

()b (5 - 61)

Relative Permeability Concepts 317

Reservoir Eng Hndbk Ch 05 2001-10-24 09:52 Page 317