Tarek Ahmed. Reservoir engineering handbook

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Calculate the oil formation volume factor at the bubble-point pressure

by using the six different correlations. Compare the results with the

experimental values and calculate the absolute average error (AAE).

Solution

Crude Exp. Method Method Method Method Method Method

Oil B

123456

1 1.528 1.506 1.474 1.473 1.516 1.552 1.525

2 1.474 1.487 1.450 1.459 1.477 1.508 1.470

3 1.529 1.495 1.451 1.461 1.511 1.556 1.542

4 1.619 1.618 1.542 1.589 1.575 1.632 1.623

5 1.570 1.571 1.546 1.541 1.554 1.584 1.599

6 1.385 1.461 1.389 1.438 1.414 1.433 1.387

%AAE — 1.7 2.8 2.8 1.3 1.8 0.6

where Method 1 = Standing’s correlation

Method 2 = Vasquez-Beggs correlation

Method 3 = Glaso’s correlation

Method 4 = Marhoun’s correlation

Method 5 = Petrosky-Farshad correlation

Method 6 = Material balance equation

Isothermal Compressibility Coefficient of Crude Oil

Isothermal compressibility coefficients are required in solving many

reservoir engineering problems, including transient fluid flow problems,

and they are also required in the determination of the physical properties

of the undersaturated crude oil.

By definition, the isothermal compressibility of a substance is defined

mathematically by the following expression:

For a crude oil system, the isothermal compressibility coefficient of

the oil phase c

is defined for pressures above the bubble-point by one of

the following equivalent expressions:

=-(1/V)(∂V/∂p)

(2-94)

∂

98 Reservoir Engineering Handbook

Reservoir Eng Hndbk Ch 02b 2001-10-24 09:26 Page 98

=-(1/B

)(∂B

/∂p)

(2-95)

= (1/r

)(∂r

/∂p)

(2-96)

where c

= isothermal compressibility, psi

-1

= oil density lb/ft

= oil formation volume factor, bbl/STB

At pressures below the bubble-point pressure, the oil compressibility

is defined as:

where B

= gas formation volume factor, bbl/scf

There are several correlations that are developed to estimate the oil com-

pressibility at pressures above the bubble-point pressure, i.e., undersaturat-

ed crude oil system. Three of these correlations are presented below:

• The Vasquez-Beggs correlation

• The Petrosky-Farshad correlation

• McCain’s correlation

The Vasquez-Beggs Correlation

From a total of 4,036 experimental data points used in a linear regres-

sion model, Vasquez and Beggs (1980) correlated the isothermal oil com-

pressibility coefficients with R

, T, °API, g

, and p. They proposed the

following expression:

where T = temperature, °R

p = pressure above the bubble-point pressure, psia

= gas solubility at the bubble-point pressure

= corrected gas gravity as defined by Equation 2-72

R T API

sb gs

-++ -

(

)

-+1 433 5 17 2 460 1 180 12 61

,.,.g

(2 - 98)

∂

(2 - 97)

Reservoir-Fluid Properties 99

Reservoir Eng Hndbk Ch 02b 2001-10-24 09:26 Page 99

The Petrosky-Farshad Correlation

Petrosky and Farshad (1993) proposed a relationship for determining

the oil compressibility for undersaturated hydrocarbon systems. The

equation has the following form:

where T = temperature, °R

= gas solubility at the bubble-point pressure, scf/STB

Example 2-30

Using the experimental data given in Example 2-29, estimate the

undersaturated oil compressibility coefficient by using the Vasquez-

Beggs and the Petrosky-Farshad correlations. Calculate the AAE.

Solution

Measured c

Vasquez-Beggs Petrosky-Farshad

Oil # Pressure 10

-6

psi 10

-6

psi 10

-6

psi

1 2689 22.14 22.88 22.24

2 2810 18.75 20.16 19.27

3 2526 22.60 23.78 22.92

4 2942 21.51 22.31 21.78

5 3273 24.16 20.16 20.39

6 4370 11.45 11.54 11.77

AAE 6.18% 4.05%

Below the bubble-point pressure, McCain and coauthors (1988) corre-

lated the oil compressibility with pressure r, the oil API gravity, gas sol-

ubility at the bubble-point R

, and the temperature T in °R. Their pro-

posed relationship has the following form:

= exp (A) (2-100)

where the correlating parameter A is given by the following expression:

A =-7.633 - 1.497 ln (p) + 1.115 ln(T) + 0.533 ln (API)

+ 0.184 ln(R

) (2-101)

c R API

osbg

=¥

1 705 10

460

7 0 69357 0 1885 0 3272

0 6729 0 5906

()

.. .

(2 - 99)

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Reservoir Eng Hndbk Ch 02b 2001-10-24 09:26 Page 100

The authors suggested that the accuracy of the Equation 2-100 can be

substantially improved if the bubble-point pressure is known. They

improved correlating parameter A by including the bubble-point pressure

as one of the parameters in the above equation, to give:

A =-7.573 - 1.45 ln (p) - 0.383 ln (P

) + 1.402 ln (T)

+ 0.256 ln (API) + 0.449 ln (R

) (2-102)

Analytically, Standing’s correlations for R

(Equation 2-70) and b

(Equation 2-85) can be differentiated with respect to the pressure p to give:

The above two expressions can be substituted into Equation 2-97 to

give the following relationship:

where p = pressure, psia

T = temperature, °R

= gas formation volume factor at pressure p, bbl/scf

= gas solubility at pressure p, scf/STB

= oil formation volume factor at p, bbl/STB

= specific gravity of the stock-tank oil

= specific gravity of the solution gas

RTB

¥+-

(. . )

..()

083 2175

0 00014 1 25 460

012

(2 -105)

∂

(

)

0 000144

083 2175

1 25 460

012

(2 -104)

∂

0.83 p 21.75

(2 - 103)

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Reservoir Eng Hndbk Ch 02b 2001-10-24 09:26 Page 101

Example 2-31

A crude oil system exists at 1650 psi and a temperature of 250°F. The

system has the following PVT properties:

API = 47.1 p

= 2377 g

= 0.851 g

= 0.873

= 751 scf/STB B

= 1.528 bbl/STB

The laboratory measured oil PVT data at 1650 psig are listed below:

= 1.393 bbl/STB R

= 515 scf/STB

= 0.001936 bbl/scf c

= 324.8 ¥ 15

-6

psi

-1

Estimate the oil compressibility by using:

a. McCain’s correlation

b. Equation 2-105

Solution

McCain’s Correlation:

• Calculate the correlating parameter A by applying Equation 2-102

A =-7.573 - 1.45 ln (1665) - 0.383 ln (2392) + 1.402 ln (710)

+ 0.256 ln (47.1) + 0.449 ln (451) =-8.1445

• Solve for c

by using Equation 2-100

= exp (-8.1445) = 290.3 ¥ 10

-6

psi

-1

Oil Compressibility Using Equation 2-105

c psi

¥+

=¥

515

1 393 0 83 1665 21 75

0 00014

0 851

0 792

515

851

792

1 25 250 0 001936

424 10

012

.[.( ) .]

.( ) .

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It should be pointed out that when it is necessary to establish PVT

relationships for the hydrocarbon system through correlations or by

extrapolation, care should be exercised to see that the PVT functions are

consistent.

This consistency is assured if the increase in oil volume with increas-

ing pressure is less than the decrease in volume associated with the gas

going into solution. Since the oil compressibility coefficient c

expressed by Equation 2-97 must be positive, that leads to the following

consistency criteria:

This consistency can easily be checked in the tabular form of PVT

data. The PVT consistency errors most frequently occur at higher pres-

sures where the gas formation volume factor, B

, assumes relatively

small values.

Oil Formation Volume Factor for Undersaturated Oils

With increasing pressures above the bubble-point pressure, the oil for-

mation volume factor decreases due to the compression of the oil, as

illustrated schematically in Figure 2-9.

To account for the effects of oil compression on B

, the oil formation

volume factor at the bubble-point pressure is first calculated by using any

of the methods previously described. The calculated B

is then adjusted

to account for the effect if increasing the pressure above the bubble-point

pressure. This adjustment step is accomplished by using the isothermal

compressibility coefficient as described below.

The isothermal compressibility coefficient (as expressed mathemati-

cally by Equation 2-94) can be equivalently written in terms of the oil

formation volume factor:

The above relationship can be rearranged and integrated to produce

ÚÚ

cdp

(2 -107)

∂

(2 -106)

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Evaluating c

at the arithmetic average pressure and concluding the

integration procedure to give:

= B

exp [-c

(p - p

)] (2-108)

where B

= oil formation volume factor at the pressure of interest,

bbl/STB

= oil formation volume factor at the bubble-point pressure,

bbl/STB

p = pressure of interest, psia

= bubble-point pressure, psia

Replacing with the Vasquez-Beggs’ c

expression, i.e., Equation 2-98,

and integrating the resulting equation gives:

BB A

oob

exp ln (2 -109)

104 Reservoir Engineering Handbook

Figure 2-9. Volume versus pressure relationship.

Reservoir Eng Hndbk Ch 02b 2001-10-24 09:26 Page 104

where

A = 10

-5

[-1,433 + 5 R

+ 17.2(T - 460) - 1,180 g

+ 12.61 API]

Replacing c

in Equation 2-107 with the Petrosky-Farshad expression

(i.e., Equation 2-99) and integrating gives:

with the correlating parameter A as defined by:

where T = temperature, °R

p = pressure, psia

= gas solubility at the bubble-point pressure

Example 2-32

Using the PVT data given in Example 2-31, calculate the oil formation

volume factor at 5000 psig by using:

a. Equation 2-109

b. Equation 2-110

The experimental measured B

is 1.457 bbl/STB.

Solution

Using Equation 2-109:

• Calculate the parameter A:

A = 10

-5

[-1433 + 5 (751) + 17.2 (250) - 1180 (0.873)

+ 12.61 (47.1)] = 0.061858

• Apply Equation 2-109:

B bbl STB

=1 528 0 061858

5015

2392

1 459. exp . ln . /

A R API T

sb g

4 1646 10 460

7 0 69357 0 1885 0 3272 0 6729

.() () ( )

.. . .

g (2 -111)

B B exp [ A (p p )] (2 - 110)

oob

0.4094

=--

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Using Equation 2-110:

• Calculate the correlating parameter A from Equation 2-111:

A = 4.1646 ¥ 10

-7

(751)

0.69357

(0.851)

0.1885

(47.1)

0.3272

¥ (250)

0.6729

= 0.005778

• Solve for B

by applying Equation 2-110:

= 1.528 exp [-0.005778 (5015

.4094

- 2392

.4096

)] = 1.453 bbl/STB

Crude Oil Density

The crude oil density is defined as the mass of a unit volume of the

crude at a specified pressure and temperature. It is usually expressed in

pounds per cubic foot. Several empirical correlations for calculating the

density of liquids of unknown compositional analysis have been pro-

posed. The correlations employ limited PVT data such as gas gravity, oil

gravity, and gas solubility as correlating parameters to estimate liquid

density at the prevailing reservoir pressure and temperature.

Equation 2-93 may be used to calculate the density of the oil at pres-

sure below or equal to the bubble-point pressure. Solving Equation 2-93

for the oil density gives:

where g

= specific gravity of the stock-tank oil

= gas solubility, scf/STB

= oil density, lb/ft

Standing (1981) proposed an empirical correlation for estimating the

oil formation volume factor as a function of the gas solubility R

, the spe-

cific gravity of stock-tank oil g

, the specific gravity of solution gas g

and the system temperature T. By coupling the mathematical definition

of the oil formation volume factor (as discussed in a later section) with

Standing’s correlation, the density of a crude oil at a specified pressure

and temperature can be calculated from the following expression:

osg

+62 4 0 0136..

(2 -112)

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where T = system temperature,

= specific gravity of the stock-tank oil

Example 2-33

Using the experimental PVT data given in Example 2-29 for the six

different crude oil systems, calculate the oil density by using Equations

2-112 and 2-113. Compare the results with the experimental values and

calculate the absolute average error (AAE).

Solution

Measured Oil

Crude Oil Density Equation 2-112 Equation 2-113

1 38.13 38.04 38.31

2 40.95 40.85 40.18

3 37.37 37.68 38.26

4 42.25 41.52 40.39

5 37.70 38.39 38.08

6 46.79 46.86 44.11

AAE 0.84% 2.65%

Density of the oil at pressures above the bubble-point pressure can be

calculated with:

exp [c

(p - p

)] (2-114)

where r

= density of the oil at pressure p, lb/ft

= density of the oil at the bubble-point pressure, lb/ft

= isothermal compressibility coefficient at average pressure, psi

-1

osg

(

)

62 4 0 0136

0 972 0 000147 1 25 460

1 175

.. .

(2 -113)

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