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= 0.035

rog

= 0.175

Estimate the three-phase relative permeability at the existing satura-

tions by using:

a. Stone’s Model I

b. Stone’s Model II

Solution

a. Stone’s Model I

Step 1. Calculate S

by applying Equations 5-58 and 5-57, to give:

= (0.625) (0.15) + (1 - 0.625) (0.05) = 0.1125

Step 2. Calculate the normalized saturations by applying Equations 5-51

through 5-53.

Step 3. Estimate k

by using Equation 5-59.

b. Stone’s Model II

Apply Equation 5-60 to give:

0 3898

1 0 2034 1 0 4068

0 406 0 175

088

0 067

(. )(. )

1 0 15 0 1125

0 4068

030 015

1 0 15 0 1125

0 2034

0 4 0 1125

1 0 15 0 1125

0 3898

a= -

1 0 15 0 05

0 625

318 Reservoir Engineering Handbook

Reservoir Eng Hndbk Ch 05 2001-10-24 09:52 Page 318

PROBLEMS

1. Given:

• S

= 0.30 S

= 0.06 S

= 0.35

• unconsolidated-well sorted sand

Generate the drainage relative permeability data by using:

a. The Wyllie-Gardner correlation

b. Pirson’s correlation

c. Corey’s method

2. The capillary pressure data for an oil-water system are given below:

, psi

0.25 35

0.30 16

0.40 8.5

0.50 5

1.00 0

a. Generate the relative permeability data for this system.

b. Using the relative permeability ratio concept, plot k

versus S

on a semi-log scale and determine the coefficients of the following

expression:

= ae

3. Using the relative permeability data of Example 5-6, generate the rela-

tive permeability values for a layer in the reservoir that is character-

ized by the following critical saturations:

= 0.25 S

= 0.25 h = 1

4. Prepare a k

versus S

plot for the following laboratory data:

1.9 0.50

0.109 0.30

=088

0 406

088

003

0 175

088

0 035 0 03 0 035 0 044.

.(..).

Relative Permeability Concepts 319

Reservoir Eng Hndbk Ch 05 2001-10-24 09:52 Page 319

Find the coefficients of the following relationship:

= ae

b S

REFERENCES

1. Aziz, K., and Sattari, A., Petroleum Reservoir Simulation. London: Applied

Science Publishers Ltd., 1979.

2. Corey, A. T., and Rathjens, C. H., “Effect of Stratification on Relative Per-

meability,” Trans. AIME, 1956, pp. 207, 358.

3. Corey, A. T., “The Interrelation Between Gas and Oil Relative Permeabili-

ties,” Prod. Mon., 1954, pp. 19, 38, 1954.

4. Fayers, F., and Matthews, J. D., “Evaluation of Normalized Stone’s Method

for Estimating Three-Phase Relative Permeabilities,” SPEJ, April 1984, pp.

224–239.

5. Honarpour, M. M., Koederitz, L. F., and Harvey, A. H., “Empirical Equa-

tions for Estimating Two-Phase Relative Permeability in Consolidated

Rock,” Trans. AIME, 1982, pp. 273, 290.

6. Honarpour, M. M., Koederitz, L. F., and Harvey, A. H., Relative Permeabili-

ty of Petroleum Reservoirs. CRC Press, Inc., 1988.

7. Hustad, O. S., and Holt, T., “Gravity Stable Displacement of Oil by Hydro-

carbon Gas after Waterflooding,” SPE Paper 24116, EOR Symposium,

Tulsa, OK, 1992.

8. Pirson, S. J. (ed.), Oil Reservoir Engineering. New York: McGraw-Hill, 1958.

9. Rose, W. D., and Bruce, W. A., “Evaluation of Capillary Character in Petro-

leum Reservoir Rock,” Trans. AIME, 1949, pp. 127, 186.

10. Stone, H. L., “Estimation of Three-Phase Relative Permeability and Residual

Oil Data, “ J. of Can. Pet. Technol., 1973, pp. 12, 53.

11. Stone, H. L., “Estimation of Three-Phase Relative Permeability, JPT, 1970,

pp. 2, 214.

12. Torcaso, M. A., and Wyllie, M. R. J., “A Comparison of Calculated k

Ratios with Field Data,” JPT, 1958, pp. 6, 57.

13. Wyllie, M.R.J., and Gardner, G.H.F., “The Generalized Kozeny-Carmen

Equation—Its Application to Problems of Multi-Phase Flow in Porous

Media,” World Oil, 1958, pp.121, 146.

14. Wyllie, M.R.J., “Interrelationship Between Wetting and Nonwetting Phase

Relative Permeability,” Trans. AIME, 1961, pp. 83, 192.

320 Reservoir Engineering Handbook

Reservoir Eng Hndbk Ch 05 2001-10-24 09:52 Page 320

Flow in porous media is a very complex phenomenon and as such can-

not be described as explicitly as flow through pipes or conduits. It is

rather easy to measure the length and diameter of a pipe and compute its

flow capacity as a function of pressure; in porous media, however, flow

is different in that there are no clear-cut flow paths which lend them-

selves to measurement.

The analysis of fluid flow in porous media has evolved throughout the

years along two fronts—the experimental and the analytical. Physicists,

engineers, hydrologists, and the like have examined experimentally the

behavior of various fluids as they flow through porous media ranging

from sand packs to fused Pyrex glass. On the basis of their analyses, they

have attempted to formulate laws and correlations that can then be uti-

lized to make analytical predictions for similar systems.

The main objective of this chapter is to present the mathematical rela-

tionships that are designed to describe the flow behavior of the reservoir

fluids. The mathematical forms of these relationships will vary depend-

ing upon the characteristics of the reservoir. The primary reservoir char-

acteristics that must be considered include:

• Types of fluids in the reservoir

• Flow regimes

• Reservoir geometry

• Number of flowing fluids in the reservoir

321

CHAPTER 6

FUNDAMENTALS OF

RESERVOIR FLUID FLOW

Reservoir Eng Hndbk Ch 06a 2001-10-25 11:22 Page 321

TYPES OF FLUIDS

The isothermal compressibility coefficient is essentially the controlling

factor in identifying the type of the reservoir fluid. In general, reservoir

fluids are classified into three groups:

• Incompressible fluids

• Slightly compressible fluids

• Compressible fluids

As described in Chapter 2, the isothermal compressibility coefficient c

is described mathematically by the following two equivalent expressions:

• In terms of fluid volume:

• In terms of fluid density:

where V and ρ are the volume and density of the fluid, respectively.

Incompressible fluids

An incompressible fluid is defined as the fluid whose volume (or den-

sity) does not change with pressure, i.e.:

Incompressible fluids do not exist; this behavior, however, may be

assumed in some cases to simplify the derivation and the final form of

many flow equations.

∂

(6 - 2)

−∂

∂

(6 -1)

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Slightly compressible fluids

These “slightly” compressible fluids exhibit small changes in volume,

or density, with changes in pressure. Knowing the volume V

ref

of a

slightly compressible liquid at a reference (initial) pressure p

ref

, the

changes in the volumetric behavior of this fluid as a function of pressure

p can be mathematically described by integrating Equation 6-1 to give:

where p = pressure, psia

V = volume at pressure p, ft

ref

= initial (reference) pressure, psia

ref

= fluid volume at initial (reference) pressure, psia

The e

may be represented by a series expansion as:

Because the exponent x [which represents the term c (p

ref

−p)] is very

small, the e

term can be approximated by truncating Equation 6-4 to:

= 1 + x (6-5)

Combining Equation 6-5 with Equation 6-3 gives:

V = V

ref

[1 + c (p

ref

− p)] (6-6)

A similar derivation is applied to Equation 6-2 to give:

ρ=ρ

ref

[1 − c (p

ref

− p)] (6-7)

xx x

=++ + +⋅⋅⋅+1

!! !

(6 - 4)

VVe

ref

−

()

(6 - 3)

ref

()−

−=

∫∫

cdp

ref

Fundamentals of Reservoir Fluid Flow 323

Reservoir Eng Hndbk Ch 06a 2001-10-25 11:22 Page 323

where V = volume at pressure p

ρ=density at pressure p

ref

= volume at initial (reference) pressure p

ref

= density at initial (reference) pressure p

ref

It should be pointed out that crude oil and water systems fit into this

category.

Compressible Fluids

These are fluids that experience large changes in volume as a function

of pressure. All gases are considered compressible fluids. The truncation

of the series expansion, as given by Equation 6-5, is not valid in this cat-

egory and the complete expansion as given by Equation 6-4 is used. As

shown previously in Chapter 2 in Equation 2-45, the isothermal com-

pressibility of any compressible fluid is described by the following

expression:

Figures 6-1 and 6-2 show schematic illustrations of the volume and

density changes as a function of pressure for the three types of fluids.

FLOW REGIMES

There are basically three types of flow regimes that must be recog-

nized in order to describe the fluid flow behavior and reservoir pressure

distribution as a function of time. There are three flow regimes:

• Steady-state flow

• Unsteady-state flow

• Pseudosteady-state flow

Steady-State Flow

The flow regime is identified as a steady-state flow if the pressure at

every location in the reservoir remains constant, i.e., does not change

with time. Mathematically, this condition is expressed as:

=−

∂













(6 - 8)

324 Reservoir Engineering Handbook

Reservoir Eng Hndbk Ch 06a 2001-10-25 11:22 Page 324

Fundamentals of Reservoir Fluid Flow 325

Figure 6-1. Pressure-volume relationship.

Figure 6-2. Fluid density versus pressure for different fluid types.

Reservoir Eng Hndbk Ch 06a 2001-10-25 11:22 Page 325

The above equation states that the rate of change of pressure p with

respect to time t at any location i is zero. In reservoirs, the steady-state

flow condition can only occur when the reservoir is completely recharged

and supported by strong aquifer or pressure maintenance operations.

Unsteady-State Flow

The unsteady-state flow (frequently called transient flow) is defined as

the fluid flowing condition at which the rate of change of pressure with

respect to time at any position in the reservoir is not zero or constant.

This definition suggests that the pressure derivative with respect to time

is essentially a function of both position i and time t, thus

Pseudosteady-State Flow

When the pressure at different locations in the reservoir is declining

linearly as a function of time, i.e., at a constant declining rate, the flow-

ing condition is characterized as the pseudosteady-state flow. Mathemati-

cally, this definition states that the rate of change of pressure with respect

to time at every position is constant, or

It should be pointed out that the pseudosteady-state flow is commonly

referred to as semisteady-state flow and quasisteady-state flow.

Figure 6-3 shows a schematic comparison of the pressure declines as a

function of time of the three flow regimes.

RESERVOIR GEOMETRY

The shape of a reservoir has a significant effect on its flow behavior.

Most reservoirs have irregular boundaries and a rigorous mathematical

∂













cons t

tan (6 -11)

∂p

∂t













= f (i, t) (6 -10)

∂













0(6-9)

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description of geometry is often possible only with the use of numerical

simulators. For many engineering purposes, however, the actual flow

geometry may be represented by one of the following flow geometries:

• Radial flow

• Linear flow

• Spherical and hemispherical flow

Radial Flow

In the absence of severe reservoir heterogeneities, flow into or away

from a wellbore will follow radial flow lines from a substantial distance

from the wellbore. Because fluids move toward the well from all direc-

tions and coverage at the wellbore, the term radial flow is given to char-

acterize the flow of fluid into the wellbore. Figure 6-4 shows idealized

flow lines and iso-potential lines for a radial flow system.

Fundamentals of Reservoir Fluid Flow 327

Figure 6-3. Flow regimes.

Reservoir Eng Hndbk Ch 06a 2001-10-25 11:22 Page 327