1

Section 2 Principles of Petroleum Production

System Analysis

• Introduction to well flow capacity calculationsØ Production from undersaturated oil reservoirØ Production from two-phase reservoirØ Production from natural gas reservoir

• Single phase flow of compressible vs. incompressible flow

• Introduction to two-phase flow systems

Introduction

Due to economy of a production system, the flow capacity of a well is a very important and therefore crucial to assess. It should be noted however that the flow capacity evolves over time and it tends to decrease as more fluid is produced.

The flow rate of the well depends on the reservoir driving force, i.e., the differential pressure between the reservoir and the back pressure exerted on the downhole (bottomhole pressure), and other parameters involving formation and fluid properties as it will be discussed in the next few slides.

2

Undersaturated Reservoirs – Darcy’s law application

• Darcy’s law in cylindrical coordinate – radial direction

• Transient flow equation: Production-Rate-Decline curve:drdpk

Aq

µ=

−+−=⇒

∂∂

=∂∂

+∂∂

2336162

1

22

2

.rc

klogtlog

khqB.

pptp

kc

rp

rrp

wtiwf

t

φµµφµ

−+

−=

23.3loglog6.162

)(

2wt

wfi

rck

tB

ppkhq

φµµ

(2.1)

(2.2)

(2.3)

Single Liquid (Oil) Flow – Udersaturated ReservoirsProduction-rate-decline curves are widely used throughout the producing side of oil industry in assessing individual well and field performance and in forecasting future behaviour. For an undersaturated reservoir (where no gas exist – the reservoir pressure is above bubble point), the analytical technique presented here through the use of radial diffusivity equation, provides a mathematical model by which the rate of decline of a producing well could be estimated. The parameters in the transient flow equation were all defined before. The only undefined parameter is the total system compressibility ct which determines how the volume of fluid system changes with pressure. This variable is known as the isothermal compressibility coefficient, determined by experiments, and is defined as follows:

In the transient flow equation in this slide, ct is the total system compressibility (change of fluid volume with pressure) in psi-1, k is in md, h and rw in ft, B in res bbl/STB, t in hours, µ in cp. Ø is porosity and dimensionless.As it is noticed the equation requires accurate reservoir parameters such as rock permeability and other parameters in addition to the initial (pi) and bottomhole (pwf) pressures which can be measured. Using the above equation production management could have an estimate of the life of the well if enhanced recovery techniques (gas injection, water flooding, etc.) are not employed to keep the reservoir pressure and production rate at a rather constant level.Note the above equation is more suitable to predict the decline of reservoir pressure when there is a constant production rate (q ) which should be close to the rate shortly after production. Therefore the equation provides an approximation of the decline rate as it can not be used to estimate for instance the production at time zero or production at the very early production period.

Tt P

VV

c

∂∂

−=1

3

• Rearranged Darcy’s equation:

• Well damage and skin effect:

Steady state well performance

wwf r

rhk

qpp

drdprhk

q ln2

2

πµ

µπ

=−⇒=

)(ln2

srr

hkq

ppw

wf +=−π

µ

)(ln2.141 srr

hkBqpp

w

ewfe +=− µ

(2.4)

(2.5)

(2.6)

If the reservoir pressure could be kept constant we could expect to have a rather steady flow of fluid through the production tubing. The Darcy’s equation can be used again to obtain the steady state flow rate for a constant differential pressure (driving force), between the reservoir and the bottomhole pressure or pe-pwf, also known technically as drawdown.

Equation 2.4 is logarithmic in nature therefore the near wellbore conditions (areas where ràrw) are extremely important in predicting well production this is where most of the pressure drop occurs. The oil which flows into the well during the afterproduction period comes from the intermediate surroundings of the wellbore. If the formation close to the wellbore is damaged in any way by drilling and formation of a filter cake of fluid mud or invasion of sand or if it is improved by acidizing or fracturing this damage or improvement will affect the production rate. The near wellbore condition is taken into account by introducing the concept of skin effect, presented by parameter s, which results in an additional steady state pressure drop which is added to the pressure drop in the reservoir (see equation 2.5). If s is negative, formation improvement has occurred, for s = 0 no improvement or damaged has occurred and for positive s formation damage has occurred.

If the reservoir exhibits a constant pressure outer boundary (at re), where the pressure can be assumed constant at pe and the well operates under steady state conditions therefore the radial flow equations becomes as shown by equation 2.6 in this slide.

The productivity index J can now be derived from this equation as the ratio of steady state production rate (q) over drawdown. The production engineer should maximize this parameter usually by reducing the skin effect by stimulation techniques such as acidizing and fracturing or reducing the drawdown. The bottomhole pressure can also be reduced (and therefore driving force increased ) by reducing the density of column of fluid above the bottomhole. For very heavy crude oils thermal recovery techniques such as steam or hot fluid injection or in-situ combustion are used to reduce viscosity and enhance the productivity index.

4

Inflow Performance Relationship - IPR

q (STB/d)

p wf(p

si)

S4

S3

S2 S

1

It is more useful to present the well performance rate as a function of the bottomhole pressure because if the bottomhole pressure is given, the production rate can be obtained readily. The bottomhole pressure however is a function of wellhead pressure and frictional pressure losses in the system and other parameters being dictated from the downstream operations. Therefore what the well can actually deliver is a function of the well potential (pressure availability in the reservoir) and hydraulics of the production system. The plot or relationship between the bottomhole pressure and production is called Inflow Performance Relationshipor IPR.

These plots can indicate for instance whether a certain series of remedial actions would lead to a targeted production rate or not and can help to identify the problems in smooth production of oil.

At a zero flowing bottomhole pressure the potential of a well to produce is determined; therefore the intersection of the plot with the q axis can be compared before and after treatment and decide whether the treatment has been successful or not. We expect to see an IPR curve with smaller absolute value for slope after a successful treatment.

5

Production from Horizontal Wells

• Permeability anisotropy:

• Skin effect:

eH

..

eH

.

V

Hani

LhI

aniw

ani

r.L/L

r..La

kkI

)I(rhI

/L

Laa:where

ani

902

for 2

250502

and , 1

, 2

4

50504

50

2

22

1

<

++=

=

+

=−+

=

ΓΓ

+

='eq

ani

H

sL

hIlnB.

phkq

212141 ΓΓµ

∆

( )21ln2.141)(

ΓΓ−

=µB

pphkq wfeH

(2.7)

(2.8)

Horizontal wells have become very popular as they provide a larger surface area by letting the well pass through the reservoir in a rather horizontal direction. This technique has proven very successful especially for reservoirs with small thicknesses and where the formation exhibit higher permeation rates in the vertical direction. A horizontal well with a length of L penetrating a reservoir with horizontal permeability kH and vertical permeability kV creates a drainage pattern that is different from a vertical well. The drainage shape is an ellipsoid with the larger half-axis length of a.

The equation presented in this slide (Equation 2.7) was presented for steady state well production calculations. In this equation Iani is the anisotropic factor which represents the effect of permeability anisotropy in horizontal and vertical directions and is defined as shown in the slide.

The effect of skin is considered by the introduction of the equivalent skin factor S’eq. The impact of the skin effect on the production rate can be very large.

6

Two-Phase Reservoir Production

• Fluid properties– bubble point– formation volume

factor– GOR

• Basic reservoir property estimation

• Standing Charts

C

Condensate reservoirOil reservoir

Gas reservoir

Pi, T

Pwf, Twf

C

C’

D

D’

E

E’

Temperature

Pre

ssu

reTwo-Phase Region

(Gas+Liquid)

Cri

con

den

ther

m

Ptf, Ttf

When the reservoir pressure falls below the bubble point two gas and liquid phases will form. This may happen both when the reservoir pressure is naturally below the fluids bubble point or when the pressure falls below the bubble point due to pressure losses as the fluid moves within the reservoir or natural loss of reservoir pressure due to production. The reservoir pressure and temperature, the bottomhole pressure and temperature as well as the surface facilities operating temperature and pressure are shown on the phase envelop in this slide.

Figure B-1 of Ref. 1 shows the typical variation of reservoir properties, including oil and gas formation volume factors Bo and Bg, and solution gas ratio Rs with pressure. Bo increases with increasing pressure because more gas is dissolved in the fluid and make it swell at higher pressures until the pressure reaches the bubble point above which we have only one single phase and the variation of volume with pressure is very small due to the low compressibility of liquids (large Bulk Modula). Bg declines with pressure as gases have significant change of volume change with pressure changes. The solution gas ratio Rs also increases with pressure (as discussed before more gas is dissolved in the liquid at higher pressures) and remains constant at bubble point.

The variation of Bo, Bg and Rs can be related through the definition of total formation volumefactor as follows:

Bt = Bo+(Rsb-Rs)Bg (2.9)

Where Rsb is the solution gas-oil-ratio at bubble point pressure. Note that when the above equation is used the unit for Bo and Bg should be consistent, i.e., if the unit of Bg is in res ft3/SCF it should be devided by 5.615 to convert to res bbl/SCF as the unit of Rs and Rsb are in SCF/STB.

Bo, Bg, and Rs and Rsb values are obtained through standard PVT laboratory tests. When no PVT data are available the Standing charts (Figures 3-2 and 3-3 of Ref. 1) can be used to estimate the reservoir bubble point and formation volume factor of bubble point fluids.

7

orS.G . of liquids/oil= density of liquids/density of pure water @ 60oF(15.5 oC)

oAPI = (141.5/ ) -131.5

Specific gravity (gases) = MW/29

Two-phase reservoir - Property estimation (cont.)• Liquid actual volumetric flow rate:

ql=Boqo (2.10)

• Gas actual volumetric rate: qg=Bg(GOR-Rs)qo (2.11)

• Solution gas -oil-ratio (Vasquez -Beggs correlations):

lγ

10.39311.172C3

0.842460.914328C2

56.1827.62C1

API>30API=30coefficient

×+=

+=

=⇔

=

−

−

).

plog(T)(.

andT

)(C

C

pRRC)(p

sslggs

l

aC/

gss

C

gs

sa

7114 1091251

460-a where

1010

5

3

1

11

22

γγγ

γ

γ

γ

gγ

lγ

The oil and gas formation volume factors - FVF (Bo and Bg see definition given before) and solution gas -oil ratio (Rs) determined in PVT labs can be used to relate the downhole flow rate to the production rate measured at the surface facility using the formula shown in this slide, where ql and qg are the actual liquid/oil and gas flow rates at some location in the well or reservoir, qo is volumetric flow rate of oil at measured at surface and Rs is solution oil-gas ratio or the amount of gas still remained dissolved in the liquid. GOR is the overall measured gas -oil ratio or the amount of gas released when the oil is completely degassed or technically speaking stabilized at atmospheric conditions. Bg is the gas formation volume factor. We shall discuss this parameter later when the gas reservoirs will be discussed.

The FVFs and Rs are measured in laboratory however empirical equations are also available to estimate these parameters if very accurate PVT lab reports were not available. The Standingequation were presented by charts (see Figures 3.2 and 3.3 discussed before). There are also Vasquez-Beggs equations which can be used to estimate these properties as shown in this slide. In Vasquez equation and are the specific gravities of oil and gas respectively. The specific gravity is a dimensionless property; in the field API degree is used to indicate the relative density of the crude with respect to water at a standard temperature (15.5oC or 60oF). Ts and ps are the surface separator temperature in oF and pressure in psia, respectively.

lγ gγ

8

FVF for pressures below bubble point: (2.13)Bo= 1.0 + 4.67x10-4 Rs + 0.1751x10-4F - 1.8106x10-8RsF ?l = 30o APIBo= 1.0 + 4.67x10-4 Rs + 0.11x10-4F + 0.1337x10-8RsF ?l > 30o API

Where:

For pressures above bubble point:

Two-phase reservoir - Property estimation (cont.)

−=

gs

lTFγγ

)60(

510

6112150121754331

×

+−++−=

p

..T.R.c lgss

o

γγ

( )[ ]boobop

T

o

oo ppcexpBB

pB

Bc −−=

∂∂

−= or 1

(2.14)

The Vasquez-Beggs equations for calculation of oil formation volume factor at pressures below bubble point and the way it could be translated to the values for pressures above bubble point using the bulk modulus of elasticity or the isothermal liquid compressibility coefficient co, are shown in this slide. co in the equation shown in this slide is the reciprocal of bulk modulus. Bob is the formation volume factor at the bubble point. GOR is the total amount of gas released when the reservoir fluid pressure is brought to atmospheric conditions. It is hence the total amount of gas that can be dissolved in the unit volume of crude at the reservoir bubble point pressure. GOR is measured in standard PVT tests by depressurizing a sample starting at its bubble point, therefore, GOR is equal to solution oil-gas ratio – Rs – at the bubble point,. The Vasquez-Beggs equation presented in the previous slide (Eq. 2.12} can be used to calculate the bubble point pressure and using this bubble point pressure the FVFequations given in this slide are used to calculate the FVF at bubble point Bob.

9

Two-phase reservoir - Property estimation (cont.): Density and viscosity• Fluid apparent density: (2.15)

• Fluid (oil) viscosity: (2.16)o

sgdl

o B

R.., γ

γρ013610

51318308 +

+= ( )[ ]boobo ppc −= expρρ

Beggs –Robinson equation for dead oil viscosity

µod= 10A - 1 and log A=3.0324-0.02023? l -1.163log T

Egbogah-Ng modified equation for A: log A=1.8653 – 0.025086( ? l) - 0.56441log T

Beggs-Robinson equation for live oil including dissolved gas: µob= CµodB

Where, C = 10.715 (Rs+100)-0.515 and B = 5.44 (Rs+150)-0.338

Vasquez-Beggs’ correction for pressure: µo= µob(p/pb)m

Where, m = 2.6p1.187exp (-11.513 - 8.98x10-5p)

Labedi’s pressure correction: µo= µob + (p/pb - 1)(10-2.488µod0.9036 pb

0.6151/ 10 )0.0197 lγ

The apparent density of the oil at pressures below bubble point ?ab, including the dissolved gas, at the standard conditions is determined by dividing the total mass by the apparent volume for one STB (standard barrel) of oil, as shown by the equation given in this slide. For pressures above the bubble point the concept of isothermal compressibility of liquids can be used here again. in the oil density equation can be estimated from the Katz gas density chart (Figure 3-4 - Ref.1)

The dead oil viscosity (oil with no dissolved gas) is estimated with the Beggs-Robinsonequation as shown in this slide. Parameter A in the Beggs -Robinson equation was modified by Egbogah and Ng later as shown in the slide. T is in oF and viscosity is calculated in centipoises in the above equations.

Beggs and Robinson proposed the shown equation in this slide to take into account the effect of dissolved gas on the oil viscosity. Vasquez and Beggs proposed the equation shown in this slide to correct for the effect of pressure. Later Labedi proposed the other equation shown in this slide for the pressure correction. In the above equations pressures are in psia, Rs in SCF/STB viscosity is calculated in centipoises and in oAPI.

Oil and water can make emulsions and depending on the water content of the emulsion viscosity can increase significantly. The Vand’s equation can be used to calculate the viscosity of emulsified oil as follows:

µeff=µc(1+2.5f+10 f 2)

Where f is the volume fraction of the discontinuous phase

lγ

gdγ

10

Fluid properties estimation - Accounting for water presence

ooow

oow

oow

wl

o

oowl

ool

BWORB

BWORWOR

BWORBWOR

)BWOR(qq

µρρ

ρµ

ρρρ

µ

ρρρ

+

+

+

=

++

=

+= (2.17)

(2.18)

(2.19)

The effect of water on the estimation of properties discussed before can be calculated using the volume fraction –weighted average equation presented in this slide. WOR is water oil ratio and can be measured experimentally or during surface operations when wells are tested.

11

Reservoir Saturation and Relative Permeabilities

• Relative permeability: ko= k kro , kw= k krw and kg=kkrg

• Inflow equations:

)(ln2.141

)( srr

hkkqB

ppw

e

ro

oooowfe +=−

µ

(2.20)

(2.21)

In a real reservoir water always exists. The permeability used in all equation discussed so far was for a fluid which consisted of only oil. Presence of water affects the permeability of the reservoir fluid. Therefore in all previous equations k should be considered effective permeability which is different from the individual permeabilities of oil and water. The permeability of pure oil and water can be measured in the laboratory. A relative permeability can also be measured which is the ratio of pure oil or water permeabilities and effective permeabilities (see the pertinent equations in this slide). Therefore the inflow equations should be modified as shown in the slide taking into account the effect of relative permeabilities in the flow of oil and water in the reservoir. The relative permeabilities depend on the saturation of water (fraction of formation volume rock occupied by water) in the reservoir and are normally less than the absolute permeabilities of water or oil.

12

IPR for two-phase reservoirs

• Vogel IPR equation:

• Fetkovich Approximation:

+

−

−

=

srr

B

pp

pp

hpk

q

w

eoo

wfwfo

o

ln2.254

8.02.012

µ

n

wfmax,oo p

pqq

−=

2

1

(2.22)

(2.23)

It is important to note to the effect of relative permeabilities on the well productivity of a reservoir. Vogel used the data from a number of various production systems and correlated the data using a second order equation as shown in this slide. Unlike the Inflow Performance Relationship (IPR) for a purely liquid system the relationship is not linear. The advantage of the Vogel equation is that only the properties of oil is used in a two-phase system to predict the well productivity.

In a similar attempt Fetkovich used a power law equation to relate the production rate and thebottomhole flowing pressure as shown in this slide. qo,max is the well absolute open flow potential (when there is no friction, hydrostatic head and resistance to flow within the production tubing and surface production facilities). Both qo,max and n are characteristic of a specific well and are determined through well testing operations by calculating the stabilized flow rates in various bottomhole pressures.

13

Natural Gas Production – Gas Properties and Phase Behaviour

• Behavior of ideal gas: PV=NRT• Behavior of a real (non-ideal) gas• Compressibility factor approach: PV=NZRT• Important equations of state – PVT Relationships üVirialüBenedict-Webb-Rubin (BWR)üCubic equationsüVan der WaalsüSaove-Redlich-Kwang (SRK)üPeng-Robinson (PR)üEtc.

In natural gas reservoirs the reservoir fluid exists in a gas phase only. It is important to know how the fluid behaves in gas reservoir. The prediction of gas properties and behaviour is important to study the production efficiency of the reservoir. All fluids follow physical laws that define their state under physical conditions. These laws are mathematically represented by equations of state(EOS) which essentially correlate pressure, temperature, and volume values. Many different empirical equations of state have been developed over the years, ideal gas equation of state being the simplest one used. An ideal gas is defined as a gas in which the molecules occupy negligible volumes and there is no interaction between the molecules, collisions between them are purely elastic – implying no energy loss on collision. At low pressures (<400 psi) most gases exhibit an almost ideal behavior. The ideal gas can be stated as follows: PV=nRT, where P and T are absolute pressure and temperature, n is the number of moles, and V is the volume occupied by the gas. R, the constant of proportionality in this equation is called universal gas constant. The value of R can be easily determined from the fact that 1 lbmole (pound mole) of any gas occupies 378.6 ft3 at standard conditions (14.73 psia and 60oF or 520o R). Similarly in CGS system of units 1 grmole of any gas occupies 22,400 cubic centimeters (cc or milliliters) in normal conditions (1 atmosphere and 0o C).R = PV/nT = (14.73 x 378.6)/(1 x 520) = 10.732 (psia)(ft3)/(lbmole)(oR)In SI units where P is in kPa, T is in K and V in m3, the value of R to use is 8.314 kPa-m3/kgmole.Kor kJ/kgmole.KIn general, gases do not exhibit ideal behavior. The reason is that non of the assumptions made above actually exist in reality. Molecules for even sparse systems occupy a finite volume; intermolecular forces such as electrostatic or Coulomb and attraction and repulsion forces exist even for a perfectly non-polar gas such as argon; and molecular collisions are never perfectly elastic. To correct for non-ideality, the simplest equation of state uses a correction factor known as compressibility factor, Z: PV=nZRT. Z, therefore, can be considered as the ratio of the volume of gas occupied by real gas to its volume under the same T and P if it is ideal. This is the most widely used real gas equation of state. Z is a function of temperature and pressure and usually can be estimated using Figures 23-3, 23-4, 23-5, 23-7, 23-8, and 23-9 of the GPSA data book.

14

Natural Gas Production – Gas Properties and Phase Behaviour: Physical Property Tables

15

Natural Gas Production – Gas Properties and Phase Behaviour: Physical Property Tables

16

§ Weight of a mole of any substance § Different units in Imperial, SI and CGS

systems§ Moles = Weight of a gas component divided by

its molecular weight§ Key’s mixing rule of molar averaging § Average molecular weight =

]).([ NN MWyMW ∑=

Natural Gas Production – Gas Properties and Phase Behaviour: Molecular Weight – Mole concept

(2.24)

Molar averaging is a technique used to find the average value for mixture properties when the values for pure components are known. For instance, for pure components, critical pressure and temperature data can be found from Figure 23-2 of the GPSA data book. For mixture, the Kay’s mixing rule can be used to find the effective critical properties as follows:

Ppc = S ynPcn and Tpc = S ynTcn

Average molecular weight for a mixture of natural gas is calculated using the same principles by the following formula:

yN = mole fraction of component N = moles of component N in gas phase divided by total moles in gas phase.

Moles = Weight of a gas component divided by its molecular weight

For instance moles for 32 lbs of methane is 32/16=2 lbmole. Molecular weight of individual compounds can be found in Figure 23-2 of GPSA data book. Since molecular weight is the weight of one mole of a compound it can have various weight units depending on the unit system used. Number of moles is also represented in different forms depending on the unit systems used; therefore for instance we may have 2.2 lbmoles of methane which equal to 1 kgmole of methane in SI system of units. Therefore, the number of moles is not the number of molecules rather an indication of weight of the compounds in molar basis. One grmole of each compound of course contain 6.02 x 1023 molecules (the Avogadro’s number), therefore one lbmole contains 6.02 x 1023/453.5 molecules of the same compound.

]).([ NN MWyMW ∑=

17

• Specific Gravity and Density – Density=mass of unit volume

(lb/ft3)– Specific gravity (liquids/oil)

• or S.G. = density of liquids/density of pure water @ 60oF

• oAPI=141.5/ S.G. -131.5– Specific gravity (gases)

• = MW/29

• Density: or

gγ

TZp

. gg

γρ 72=

TZp)MW(

.g 0930=ρ

Natural Gas Production – Gas Properties and Phase Behaviour: Properties estimation techniques

(2.25)

lγ

Fluid Properties estimation techniquesIn designing petroleum production, processing, transport, and handling systems, a complete knowledge of fluid properties is crucial. It is important to know and predict the amount, composition, and density of any phases present in any process situation. Specific Gravity and Density. Specific gravity (S.G.) of a liquid is the ratio of the density of the liquid at 60oF to the density of pure water. API gravity is related to specific gravity by the following equation:Density = mass of unit volume (lb/ft3 or kg/m3)

or S.G. = density of liquids/density of pure water @ 60oFoAPI=141.5/S.G. -131.5Specific gravity of petroleum fractions may be estimated using the chart of Figs. 23 -12 and 23-14 of the GPSAdata book.Specific gravity of a gas is defined as: density of the gas divided by density of air at standard conditions of temperature and pressure (i.e.,14.7 psi and 60oF);for gases: = MW/29

where , MW = molecular weight of gas = specific gravity (for air=1)

Density of the gas is given as:

Or:

where,= density of gas, lb/ft3= specific gravity of gas (air=1)

p= absolute pressure, psia (gauge pressure = atmospheric pressure; at sea level 14.7 psi )T = absolute temperature, oR (temperature in oF + 460)Z= gas compressibility factors (see charts given in Figures 23-7 to 23-9 from GPSA data book; will be discussed further in the following sections) MW= gas molecular weight

TZp

. gg

γρ 72=

TZp)MW(

.g 0930=ρ

lγ

gγ

gγ

gγ

ρ

18

• Critical temperature = the maximum temperature at which the component can exist as a liquid

• Critical pressure= vapour pressure of a substance at its critical temperature

• Beyond critical temperature and pressure there is no distinction between a liquid and a gas phase

• Principles of corresponding states and gas compressibility factor

ppc = S ynpcn and Tpc = S ynTcn (2.26)

pcn and Tcn from Figure 23-2 GPSA

Natural Gas Production – Gas Properties and Phase Behaviour: Properties estimation techniques, cont.

For pure compounds, critical pressure and temperature data can be found from Figure 23-2of the GPSA data book. For mixture, Kay’s mixing rule can be used to find the effective critical properties:ppc = S ynpcn and Tpc = S ynTCn

Where ppc and Tpc are the pseudocritical pressure and temperature, respectively, for the mixture and yN is the mole fraction of component N in the gas mixture. These are called pseudo because they are used a correlation basis rather than as a very precise representation of mixture critical properties. Gas compressibility factor Z. Several different correlations are available for this important parameter. The basic correlations use the corresponding states concept. According to the Van der Waals’ law of corresponding states, the physical characteristics of a substance are a function of its relative proximity to the critical point. This means that the deviation from ideal behavior of gases is the same if they are located at the same state relative to their critical state. This implies that all substances behave similarly at critical points, hence, should have equal critical point compressibility factor, Zc=pcVc/RTc. The real value of critical compressibility factor, however, is not the same for all compounds. The compressibility charts provide reliable estimation particularly for supercritical gases and at low pressure conditions. Therefore the relevant temperature and pressure values that express the departure of a real gas from ideal behavior are the reduced pressure, pr, and reduced temperature, Tr:Z = f (pr, Tr)Where Pr = p/pc and Tr= T/Tc. For gas mixtures, the reduced parameters are denoted as pseudo reduced temperature TPr (=T/Tpc), and pseudo reduced pressure, ppr (=p/ppc).

There are several ways to calculate compressibility factor that will be discussed later in the next sections of the course.

19

• Standing-Katz compressibility charts (Figures 23-3, 23-4, and 23-5 GPSA)

• Brown-Katz-Oberfell-Alden charts (Figures 23-7, 23-8, and 23-9 GPSA)

• Acid gas content consideration by Wichert-Aziz correction factors

e from Figure 23-10 GPSA• Compressibility from equations of state

)B(BT

TPPandTT

pc

'pcpc'

pcpcpc −+=−=

1εε

Natural Gas Production – Gas Properties and Phase Behaviour: Properties estimation techniques, cont.

(2.27)

There are several ways to calculate compressibility factor:

1- Standing-Katz charts: Figures 23-3, 23-4, and 23-5 of GPSA data book are used to estimate the compressibility using the reduced temperature and pressure values. These charts are generally reliable for sweet natural gas with minor amounts of non-hydrocarbons such as N2.

2- Brown-Katz-Oberfell-Alden charts: Figures 23-7, 23-8, and 23-9 of GPSA data book are used to predict the compressibility for low molecular weight natural gases. These figures cover a wide range of molecular weights (15.95 to 26.10), temperatures (-100o to 1000o F), and pressures (up to 5,000 psia). For gases whose molecular weights lie in between the molecular weights shown in Figs. 23-7 through 23-9, linear interpolation should be used to compute the compressibility. In general compressibilities for gases with less than 5% noncondensable components, such as nitrogen, carbon dioxide, and hydrogen sulfide, are predicted with less than 2% error. When molecular weight is above 20 and compressibility is below 0.6, errors as large as 10% may occur. Natural gases which contain H2S and CO2 exhibit different compressibility factor behavior than do sweet gases. The Wichert and Aziz equation should b e used to correct for the acid gas content. Figure 23-10 of the GPSA data book is used to find the correction factor and then correct the pseudocritical pressure and temperatures determined by Kay’s rule as follows:

where e is the correction factor and B is the mole fraction of H2S. These values of critical pressure and temperature are used to find the corrected Z factor for acid gas containing gases. Equations of state may also be used to calculate Z factor by replacing Z=pV/RT in EOS, a cubic equation is obtained which should be solved for Z values. Numerical techniques are used to solve these cubic equations.

)B(BTTp

pandTTPC

'PCPC'

PCPC'PC −+

=−=1ε

ε

20

Compressibility charts

Standing-Katz compressibility charts

Brown-Katz-Oberfell-Alden Z charts

When gas specific gravity is known, the Sutton correlations can also be used to estimate the pseudo critical properties as follows:

ppc and Tpc are in psia and oR, respectively.

The Thomas et al. equation takes a linear regression for the pseudo critical properties estimation and may also be used to obtain the pseudo critical properties from the gas specific gravity as follows:

Where is the specific gravity of gas. This equation is only accurate within the limits of up to 3% H2S, 5% N2, or a total impurity (non-hydrocarbon) content of 7% beyond which errors in critical pressure exceeds 6%. It should be noted that the gas gravity method is not very accurate. If the analysis of the gas is available, it must be used in accordance with the Kay’smixing rule.

(2.28)

(2.29)

2

2

756.8 131.0 3.6

169.2 349.5 374.0pc g g

pc g g

P

T

γ γ

γ γ

= − −

= + −

709.604 58.718

170.491 3097344pc g

pc g

p

T

γ

γ

= −

= +

gγ

21

• Carr et al. correlation (Fig. 23-32 and 23-33 GPSA)• Viscosity of gas mixture from single component data:

• Lee et al. for natural gas:

• GPSA charts – Figs. 23-30 through 23-38• Dean and Stiel method

∑

∑

=

== n

iii

n

iiigi

g

My

My

1

5.0

1

5.01

1

µµ

XyandMTXTM

TMKwhereXK

g

g

gygg

2.04.201.0/9865.319209

)02.04.9(10,)exp(

5.14

−=++=++

+==

−

ρµ

[ ]9/8

Pr5

Pr

9/5

Pr5

Pr3/22/1

6/1

)10(0.34,5.1

,0932.01338.0)10(8.166,5.1;)(

4402.5

TTforand

TTforPMy

T

g

gPCii

PC

−

−

=≤

−=>=∑

ξµ

ξµξ

Empirical Correlations – Natural Gas: Viscosity

(2.30)

(2.31)

(2.32)

The most accurate method to determine viscosity is, obviously, to measure it for a given fluid under the desired conditions. This, however, is not generally possible, given the various complex mixtures of hydrocarbon and non-hydrocarbon compounds present in gas mixtures. Some common methods of predicting gas viscosity are given below:

1. The Carr et al. correlation for natural gas: This method only needs gas gravity or average molecular weight to calculate viscosity. The correlations were presented in graphical forms as given in Figures 23-32 and 23-33 of the GPSA data book. Figure 23-32 is used first to calculate the viscosity at one atmosphere pressure and any given temperature. Corrections for non-hydrocarbons such as CO2, H2S, and nitrogen are provided Then, Figure 23-33 can be used to correct for pressure based on the corresponding state principle.

2. Viscosity from single-component data: If the analysis of gas is known, it is possible to calculate the viscosity of the gas mixture from the component viscosities. First viscosity is determined at one atmosphere (or any low pressure) and the given temperature using the Herning and Zippere mixing rule as:

where, yi= mole fraction of component i in the gas mixture; and µ1gN =pure component viscosity at 1 atmosphere pressure and the desired temperature. The pressure correction is made using the Carr et al.’scharts. Viscosity of some natural gas constituents can be found from Figure 3-16 the Kumar’s Gas Production Engineering book .

3. The Lee et al. Correlation for natural gas: Lee et al. provide an analytical expression which can be used for programming purposes. where, T is oR, ?g is in g/cm3,, and µg is in cp. This equation however does not correct for impurities such as CO2, H2S, and N2.

4. The Dean and Stiel method:

∑

∑

=

== n

iii

n

iiigi

g

My

My

1

5.0

1

5.01

1

µµ

XyandMTXTM

TMKwhereXK

g

g

gygg

2.04.201.0/9865.319209

)02.04.9(10,)exp(

5.14

−=++=++

+==

−

ρµ

[ ]9/8

Pr5

Pr

9/5Pr

5Pr3/22/1

6/1

)10(0.34,5.1

,0932.01338.0)10(8.166,5.1;)(

4402.5

TTforand

TTforPMy

T

g

gPCiN

PC

−

−

=≤

−=>=∑

ξµ

ξµξ

22

Viscosity Charts

23

• Gas formation volume factor FVF

)unitsField(PT

Z.B

)unitsSI(PT

Z.B

PP

TTZ

VVB

g

g

SC

SCSC

Rg

=

×=

==

−

02830

10473 4

SC= Standard Conditions:

P = 14.696 psia = 101.325 kPa,abs

T = 60 oF (520 oR) = 15.5 oC (288.7 K)

TTg P

ZZPP

VVulusmodbulk

C

∂∂−=

∂∂−== 1111

• Gas isothermal compressibility coefficient:

Natural Gas Production – Gas Properties and Phase Behaviour: FVF

(2.33)

(2.34)

Dry gas PVT tests are rather simple and straight forward. No phase change is expected for a dry gas sample and therefore tests only involve density and compressibility factor (Z) measurement routines.

Procedure:

• A known volume of gas is loaded in the PVT cell at the reservoir temperature

• Weight of the known gas volume is determined

• Molecular weight is determined using the gas chromatography technique

• Gas formation volume factor is determined based on the following definition:

Using the real gas relation PV=nZRT the definition is converted to the following mathematical expressions:

The gas isothermal compressibility coefficient, Cg, can also be calculated using the variation of Z with pressure. This parameter should not be confused with gas compressibility factor Z. For ideal gases cg is the reciprocal of absolute pressure as the bulk modulus equals absolute pressure.

conditions standardat gas of Volumeconditions reservoir at gas of Volume

Bg =

)unitsField(pT

Z.B

)unitsSI(p

TZ.B

pp

TT

ZVV

B

g

g

SC

SCSC

Rg

=

×=

==

−

02830

10473 4

TT

g pZ

ZPpV

Vulusmodbulkc

∂∂−=

∂∂−== 1111

24

•For Darcy flow:

•high gas flow rates, turbulent (non-Darcy flow):

Natural Gas Production – Well Deliverability

+=−⇒

+= s

rr

khTZq

pppp

TZB

w

ewfe

wfeg ln

1424

2/)(0283.0 22 µ

)pp(Cq wf22 −=

10.5 re whe22 <<−= n)pp(Cq nwf

(2.35)

(2.36)

(2.37)

The relationship developed for the Darcy incompressible flow can also be used for the compressible flow if the average compressibility and formation volume factors are used. The equation shown in this slide is used to calculate the deliverability for a gas well where q is the gas standard volumetric flow rate in 1000 St’dft3/day (MSCFD), the values for viscosity µ and compressibility factor Z should be calculated in average pressures.

The equation can be shown in a simplified form as shown in this slide for reasonably small flow rates (e.g., ~1,000-5,000 MSCFD or 1-5 MMSCFD). For larger flow rates a power law form of equation can be used as shown in this slide. A plot of q vs the difference between the pressure squares in a log-log coordinate should result in a straight line with slope of n and intercept of C.

25

• Aronofsky-Jenkins equation: (2.38)

• Determining the non-Darcy coefficient - D:

Natural Gas Production – Well Deliverability: Non-Darcy Flow

2

22 000260 where51 and

1424 wtDD

w

d

w

d

wf

rckt.tt.

rr

DqsrrlnTZ

)pp(khq

φµµ

==

++

−=

2

105

22222

106

:using estimated becan ely alternativ

or

perfw

.s

wfwf

hrhk

D

D

bqaq

ppbqaqpp

µγ −−×=

+=−

+=− (2.39)

(2.40)

The well deliverability can be more precisely estimated using the Aronofsky-Jenkins equation presented in this slide in which D is the non-Darcy coefficient and rd the time dependent Aronofsky effective radius. rd varies with time until it reaches a minimum value of rd=0.472re, otherwise it is calculated using the equations shown in this slide. The term Dq is called turbulent skin effect.

The Anorofsky equation can be rearranged and shown in a quadratic (2nd order) equation. The parameter a can be defined by comparing the actual Anorofsky equation and is very similar to equation discussed before for Darcy gas flow equation (Eq. 2.35). The q2 multiplier b accounts for non-Darcy effects and can be used to obtain the parameter D through actual tests. In the absence of field measurements and experimental data the empirical equation shown in this slide can be used to approximate the value of D, where ? is gas gravity, k s is the near wellborepermeability in md, h and hperf the net and perforated thickness, both in ft and µ is the gas viscosity in cp evaluated at the bottomhole pressure.

26

Natural Gas Production: Production-Rate-Decline curve or Transient Flow

t)p(m

kc

r)p(m

rr)p(m

dpZp

m(p)t

pkc

rp

rrp

t

p

p

t

o

∂∂=

∂∂+

∂∂

=∂

∂=

∂∂

+∂

∂∫

φµ

µφµ

1

2 and 1

2

2

22

2

22

[ ]

−+

−=

2336381 2 .r)c(

klogtlogT,

)p(m)p(mkhq

wit

wfi

µφ

Combing Darcy’s and conservation of mass laws:

Similar solution to the oil diffusion equation results:

(2.41)

(2.42)

Using the Darcy’s law and writing the relevant equations for a cylindrical element in the reservoir one can obtain the partial differential equation (PDE) similar to what we had for oil reservoirs. Only pressure terms will be replaced with square pressure terms. The same solution that we had for oil reservoirs can be obtained here, the only problem is the variation of viscosity and gas density/fluid compressibility with pressure which should be taken into account when a solution is developed. The use of the concept of real gas pseudo pressure function m(p) was used to obtain the solution to the above partial differential equation. Change of variable using this concept results in a PDE very similar to oil flow equation and therefore a similar solution, only pressure terms will be replaced with pseudo pressure functionterms as shown in this slide. In this equation q is in 1000 standard cubic ft per day (MSCFD) and the product µct should be calculated at the original reservoir pressure. T is the absolute temperature (oR) and t is time in hours.

27

Natural Gas Production: Horizontal Wells

eH

..

eH

.

V

Hani

LhI

aniw

ani

r.L

/Lr

..L

a

kk

I)I(r

hI/L

Laa

:where

ani

902

for 2

250502

and , 1

, 2

4

50504

50

2

22

1

<

++=

=

+

=−+

=

ΓΓ

+

−=

21

22

1424 ΓΓµ lnDqL

hIZT

)pp(hkq

ani

wfeH

(2.43)

For horizontal wells similar equations to those of the oil reservoirs can be developed. Turbulence effects ( Dq ) in horizontal wells can be neglected as they are multiplied by the scaled aspect ratio Ianih/L term and since L is usually large the term will be very small. The skin effect can also be included in the first term (the non logarithmic term) in the denominator of the flow equation. All parameters are defined as they were for the oil system.