Dake L.P. Fundamentals of reservoir engineering

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SOME BASIC CONCEPTS IN RESERVOIR ENGINEERING 10

And, while it is easy to say, "simply by multiplying by the recovery factor", it is much

less easy to determine what the recovery factor should be for any given reservoir and,

indeed, it is the determination of this figure which is the most important single task of

the reservoir engineer.

For a start, one can clearly distinguish between two types of recovery factor. There is

one which is governed by current economic circumstances and, ever increasingly, by

environmental and ecological considerations, while the second can be classed as a

purely technical recovery factor depending on the physics of the reservoir-fluid system.

Regrettably, the former, although possibly the more interesting, is not a subject for this

book.

The two main categories of hydrocarbon recovery are called primary and

supplementary. Primary recovery is the volume of hydrocarbons which can be

produced by virtue of utilising the natural energy available in the reservoir and its

adjacent aquifer. In contrast, supplementary recovery is the oil obtained by adding

energy to the reservoir-fluid system. The most common type of supplementary

recovery is water flooding in which water is injected into the reservoir and displaces oil

towards the producing wells, thus increasing the natural energy of the system. The

mechanics of supplementary recovery will be described later, in Chapter 4, sec. 9 and

in Chapter 10; for the moment only primary recovery will be considered.

The entire mechanics of primary recovery relies on the expansion of fluids in the

reservoir and can best be appreciated by considering the definition of isothermal

compressibility.

∂

—

(1.11)

The isothermal compressibility is commonly applied in the majority of reservoir

engineering calculations because it is considered a reasonable approximation that as

fluids are produced, and so remove heat from the reservoir by convection, the cap and

base rock, which are assumed to act as heat sources of infinite extent, immediately

replace this heat by conduction so that the reservoir temperature remains constant.

Therefore, compressibility, when referred to in this text, should always be interpreted

as the isothermal compressibility.

The negative sign convention is required in equ. (1.11) because compressibility is

defined as a positive number, whereas the differential, ∂V/∂p, is negative, since fluids

expand when their confining pressure is decreased. When using the compressibility

definition in isolation, to describe reservoir depletion, it is more illustrative to express it

in the form

dV cV p=∆ (1.12)

where dV is an expansion and ∆p a pressure drop, both of which are positive. This is

the very basic equation underlying all forms of primary recovery mechanism. In the

reservoir, if ∆p is taken as the pressure drop from initial to some lower pressure,

SOME BASIC CONCEPTS IN RESERVOIR ENGINEERING 11

− p, then dV will be the corresponding fluid expansion, which manifests itself as

production.

The skill in engineering a high primary recovery factor, utilising the natural reservoir

energy, is to ensure that the dV, which is the production, is the most commercially

valuable fluid in the reservoir, namely, the oil. The way in which this can be done is

shown schematically in fig. 1.5.

aquifer gascap

oil

dV = oil production

tot

= dV + dV + dV

ow g

Fig. 1.5 Primary oil recovery resulting from oil, water and gas expansion

The diagram illustrates the fairly obvious fact that to produce an oil reservoir, wells

should be drilled into the oil zone. If the reservoir is in contact with a gascap and

aquifer, the oil production due to a uniform pressure drop, ∆p, in the entire system, will

have components due to the separate expansion of the oil gas and water, thus

TOT

= Oil Production = dV

+ dV

in which the balance is expressed in fluid volumes at reservoir conditions. Applying

equ. (1.12), this may be expressed as

TOT

= c

∆p + c

∆p

Considering the following figures as typical for the compressibilities of the three

components at a pressure of 2000 psia:

=15× 10

−6

/psi

=3× 10

−6

/psi

= 500 × 10

−6

/psi

; refer sec.1.5

æö

≈

ç÷

èø

it is evident that the contribution to dV

TOT

supplied by the oil and water expansion will

only be significant if both V

and V

, the initial volumes of oil and water, are large. In

contrast, because of its very high compressibility, even a relatively small volume of

gascap gas will contribute significantly to the oil production.

SOME BASIC CONCEPTS IN RESERVOIR ENGINEERING 12

Therefore, while it is obvious that one would not produce an aquifer, but rather, let the

water expand and displace the oil; so too, the gas in the gascap, although having

commercial value, is frequently kept in the reservoir and allowed to play its very

significant role in contributing to the primary recovery through its expansion. The

mechanics of primary oil recovery will be considered in greater detail in Chapter 3.

1.5 VOLUMETRIC GAS RESERVOIR ENGINEERING

Volumetric gas reservoir engineering is introduced at this early stage in the book

because of the relative simplicity of the subject. lt will therefore be used to illustrate

how a recovery factor can be determined and a time scale attached to the recovery.

The reason for the simplicity is because gas is one of the few substances whose state,

as defined by pressure, volume and temperature (PVT), can be described by a simple

relation involving all three parameters. One other such substance is saturated steam,

but for oil containing dissolved gas, for instance, no such relation exists and, as shown

in Chapter 2, PVT parameters must be empirically derived which serve the purpose of

defining the state of the mixture.

The equation of state for an ideal gas, that is, one in which the inter-molecular

attractions and the volume occupied by the molecules are both negligible, is

pV nRT= (1.13)

in which, for the conventional field units used in the industry

p = pressure (psia); V = volume (cu.ft)

T = absolute temperature − degrees Rankine (°R=460+°F)

n = the number of lb. moles, where one lb. mole is the molecular weight of the

gas expressed in pounds.

and R = the universal gas constant which, for the above units, has the value

10.732 psia.cu.ft/lb. mole.°R.

This equation results from the combined efforts of Boyle, Charles, Avogadro and Gay

Lussac, and is only applicable at pressures close to atmospheric, for which it was

experimentally derived, and at which gases do behave as ideal.

Numerous attempts have been made in the past to account for the deviations of a real

gas, from the ideal gas equation of state, under extreme conditions. One of the more

celebrated of these is the equation of van der Waals which, for one Ib.mole of a gas,

can be expressed as

()

(p ) V b RT

+−=

(1.14)

In using this equation it is argued that the pressure p, measured at the wall of a vessel

containing a real gas, is lower than it would be if the gas were ideal. This is because

the momentum of a gas molecule about to strike the wall is reduced by inter-molecular

attractions; and hence the pressure, which is proportional to the rate of change of

SOME BASIC CONCEPTS IN RESERVOIR ENGINEERING 13

momentum, is reduced. To correct for this the term a/V

must be added to the observed

pressure, where a is a constant depending on the nature of the gas. Similarly the

volume V, measured assuming the molecules occupy negligible space, must be

reduced for a real gas by the factor b which again is dependent on the nature of the

gas.

The principal drawback in attempting to use equ. (1.14) to describe the behaviour of

real gases encountered in reservoirs is that the maximum pressure for which the

equation is applicable is still far below the normal range of reservoir pressures.

More recent and more successful equations of state have been derived, - the Beattie-

Bridgeman and Benedict-Webb-Rubin equations, for instance (which have been

conveniently summarised in Chapter 3 of reference 18); but the equation most

commonly used in practice by the industry is

pV ZnRT= (1.15)

in which the units are the same as listed for equ. (1.13) and Z, which is dimensionless,

is called the Z−factor. By expressing the equation as

V=nRT

æö

ç÷

èø

the Z−factor can be interpreted as a term by which the pressure must be corrected to

account for the departure from the ideal gas equation.

The Z−factor is a function of both pressure and absolute temperature but, for reservoir

engineering purposes, the main interest lies in the determination of Z, as a function of

pressure, at constant reservoir temperature. The Z(p) relationship obtained is then

appropriate for the description of isothermal reservoir depletion. Three ways of

determining this relationship are described below.

a) Experimental determination

A quantity of n moles of gas are charged to a cylindrical container, the volume of which

can be altered by the movement of a piston. The container is maintained at the

reservoir temperature, T, throughout the experiment. If V

is the gas volume at

atmospheric pressure, then applying the real gas law, equ. (1.15),

14.7 V

= nRT

since Z=1 at atmospheric pressure. At any higher pressure p, for which the

corresponding volume of the gas is V, then

pV = ZnRT

and dividing these equations gives

SOME BASIC CONCEPTS IN RESERVOIR ENGINEERING 14

14.7V

By varying p and measuring V, the isothermal Z(p) function can be readily obtained.

This is the most satisfactory method of determining the function but in the majority of

cases the time and expense involved are not warranted since reliable methods of direct

calculation are available, as described below.

b) The Z-factor correlation of Standing and Katz

This correlation requires a knowledge of the composition of the gas or, at least, the gas

gravity. Naturally occurring hydrocarbons are composed primarily of members of the

paraffin series (C

2n+2

) with an admixture of non-hydrocarbon impurities such as

carbon dioxide, nitrogen and hydrogen sulphide. Natural gas differs from oil in that it

predominantly consists of the lighter members of the paraffin series, methane and

ethane, which usually comprise in excess of 90% of the volume. A typical gas

composition is listed in table 1.1.

In order to use the Standing-Katz correlation

it is first necessary, from a knowledge of

the gas composition, to determine the pseudo critical pressure and temperature of the

mixture as

pc i ci

pnp=

(1.16)

and

pc i ci

TnT=

(1.17)

where the summation is over all the components present in the gas. The parameters p

and T

are the critical pressure and temperature of the i

component, listed in table 1.1,

while the n

are the volume fractions or, for a gas, the mole fractions of each

component (Avogrado's law). The next step is to calculate the so-called pseudo

reduced pressure and temperature

(1.18)

and

(1.19)

where p and T are the pressure and temperature at which it is required to determine Z.

In the majority of reservoir engineering problems, which are isothermal, T

is constant

and p

variable.

With these two parameters the Standing-Katz correlation chart, fig. 1.6, which consists

of a set of isotherms giving Z as a function of the pseudo reduced pressure, can be

used to determine the Z−factor.

SOME BASIC CONCEPTS IN RESERVOIR ENGINEERING 15

For instance, for the gas composition listed in table 1.1, and at a pressure of 2000 psia

and temperature of 180° F, the reader can verify that

= 663.3 and T

= 374.1

giving

pr pr

2000 640

p3.02andT1.71

663.3 374.1

== ==

from which, using fig.1.6, the Z−factor can be obtained as 0.865.

Component Molecular

Weight

Critical

Pressure

(psia)

Constants

Temp.

(

Typical Composition

(volume or mole

fraction, n

)

Methane 16.04 668 343 .8470

Ethane 30.07 708 550 .0586

Propane 44.10 616 666 .0220

−C

Isobutane 58.12 529 735 .0035

−C

Normal butane 58.12 551 765 .0058

−C

Isopentane 72.15 490 829 .0027

−C

Normal pentane 72.15 489 845 .0025

−C

Normal hexane 86.18 437 913 .0028

−C

Normal heptane 100.20 397 972 .0028

n-C

Normal octane 114.23 361 1024 .0015

−C

Normal nonane 128.26 332 1070 .0018

−C

Normal decane 142.29 304 1112 .0015

Carbon dioxide 44.01 1071 548 .0130

S Hydrogen sulphide 34.08 1306 672 .0000

Nitrogen 28.01 493 227 .0345

TABLE 1.1

Physical constants of the common constituents of hydrocarbon

gases

, and a typical gas composition

Conventionally, the composition of natural gases is listed in terms of the individual

components as far as hexane, with the heptane and heavier components being

grouped together as

(heptanes-plus). In the laboratory analysis the molecular

weight and specific gravity of this group are measured, which permits the determination

of the pseudo critical pressure and temperature of the

from standard

SOME BASIC CONCEPTS IN RESERVOIR ENGINEERING 16

correlations

8,13

. This in turn facilitates the calculation of the Z−factor using the method

described above.

In calculating the Z−factor it has been assumed that the non-hydrocarbon components,

carbon dioxide, hydrogen sulphide and nitrogen, can be included in the summations,

equs. (1.16) and (1.17), to obtain the pseudo critical pressure and temperature.

This approach is only valid if the volume fractions of the non-hydrocarbon components

are small, say, less than 5% vol. For larger amounts, corrections to the above

calculation procedures are to be found in the text book of Amyx, Bass and Whiting

. If,

however, the volume fractions of the non-hydrocarbons are very large (the carbon

dioxide content of the Kapuni field, New Zealand, for instance, is 45% vol.) then it is

better to determine the Z−factor experimentally as described in a), above.

SOME BASIC CONCEPTS IN RESERVOIR ENGINEERING 17

1.3

1.2

1.3

1.2

1.05

1.1

1.0

0.9

0.8

0.7

0.6

Z -factor

Pseudo reduced pressure

0.5

Pseudo reduced temperature

0.4

0.3

0.2

0.1

012345678

Fig. 1.6 The Z−

−−

−factor correlation chart of Standing and Katz

(Reproduced by courtesy

of the SPE of the AIME)

If the gas composition is not available, the Standing-Katz correlation can still be used

provided the gas gravity, based on the scale air = 1, at atmospheric pressure and at

60°F, is known (refer sec. 1.6). In this case fig. 1.7, is used to obtain the pseudo critical

pressure and temperature; then equs. (1.18) and (1.19) can be applied to calculate the

pseudo reduced parameters required to obtain the Z−factor from fig. 1.6.

SOME BASIC CONCEPTS IN RESERVOIR ENGINEERING 18

MISCELLANEOUS GASES

CONDENSATE WELL FLUID

PSEUDO CRITICAL PRESSURE, psia

PSEUDO CRITICAL TE

PERATURE, degrees Rankine

GAS GRAVITY (Air = 1)

700

650

600

550

500

450

400

350

300

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

Fig. 1.7 Pseudo critical properties of miscellaneous natural gases and condensate

well fluids

c) Direct calculation of Z-factors

The Standing-Katz correlation is very reliable and has been used with confidence by

the industry for the past thirty-five years. With the advent of computers, however, there

arose the need to find some convenient technique for calculating Z−factors, for use in

gas reservoir engineering programs, rather than feeding in the entire correlation chart

from which Z−factors could be retrieved by table look-up. Takacs

has compared eight

different methods for calculating Z−factors which have been developed over the years.

These fall into two main categories: those which attempt to analytically curve-fit the

Standing-Katz isotherms and those which compute Z−factors using an equation of

state. Of the latter, the method of Hall-Yarborough

is worthy of mention because it is

SOME BASIC CONCEPTS IN RESERVOIR ENGINEERING 19

both extremely accurate and very simple to program, even for small desk calculators,

since it requires only five storage registers.

The Hall-Yarborough equations, developed using the Starling-Carnahan equation of

state, are

1.2(1 t)

0.06125p t e

−−

(1.20)

where p

= the pseudo reduced pressure

t = the reciprocal, pseudo reduced temperature (T

/T)

and y = the "reduced" density which can be obtained as the solution of the

equation.

234

232

yy y y

1.2(1 t)

0.06125 p t e (14.76t 9.76t 4.58t ) y

(1 y)

++−

−−

−+−−+

−

(2.18 2.82t)

(90.7t 242.2t 42.4t )y 0

+− + =

(1.21)

This non-linear equation can be conveniently solved for y using the simple Newton

Raphson iterative technique. The steps involved in applying this are:

1) make an initial estimate of y

, where k is an iteration counter (which in this case is

unity, e.g. y

= 0.001)

2) substitute this value in equ. (1.21); unless the correct value of y has been initially

selected, equ. (1.21) will have some small, non-zero value F

3) using the first order Taylor series expansion, a better estimate of y can be

determined as

k1 k k

yyF

=− (1.22)

where the general expression for dF/dk can be obtained as the derivative of

equ. (1.21), i.e.

234

dF 1 4y 4y 4y y

(29.52t 19.52t 9.16t )y

(1 y)

++ − +

=−−+

−

(1.18 2.82t)

(2.18 2.82t) (90.7t 242.2t 42.4t )y

++ − +

(1.23)

4) iterate, using equs. (1.21) and (1.22), until satisfactory convergence is obtained

≈ 0).

5) substitution of the correct value of y in equ. (1.20) will give the Z−factor.

(N.B. there appears to be a typographical error in the original Hall-Yarborough paper

in that the equations presented for F (equ. 8) and dF/dy (equ. 11), contain 1−y

and