## Phase Behavior

#### Phase Diagram of a pure substance

*Vapor-pressure line*This line separates the pressure-temperature conditions for which the substance is a liquid from gas form.*Critical Point*the upper limit of the vapor-pressure line is the critical point, indicated by point C. The temperature and pressure represented by this point are called the*critical temperature*, $T_{c}$, and the*critical pressure*, $P_{c}$. for a pure substance, the critical temperature may be defined as the temperature above which the gas cannot be liquefied, regardless of the pressure applied. similarly, the critical pressure is defined as the pressure above which liquid and gas cannot coexist, regardless of the temperature. these definitions of critical properties are invalid for systems with more than one component.*Triple point*, this point represents the pressure and temperature at which solid, liquid, and gas coexist under equilibrium conditions.

###### The Clausius-Clapeyron equation

The Clausius-Clapeyron equation expresses the relationship between vapor pressure and temperature.

$dTdP_{v}β=T(V_{Mg}βV_{ML})L_{v}β$

- $L_{v}$ is the heat of vaporization of one mole of liquid
- $V_{Mg}βV_{ML}$ represents the change in volume of one mole as it goes from liquid to gas weβd get the following equation after integrating:

$lnP_{v}=βRL_{v}β(T1β)+C$

$ln(P_{v1}P_{v2}β)=βRL_{v}β(T_{1}1ββT_{2}1β)$

if we plot $lnP_{v}$ against $T1β$ it would be a straight with the slope $βRL_{v}β$ and intercept $C$.

###### Pressure-Volume Diagram

Phase behavior of a pure component can also be portrayed on a pressureβvolume (PV) diagram at various isotherms, such as the one shown in this figure.

- if the temperature of the system is less than the critical temperature, with a huge amount of decrease in pressure the volume of liquid would change slightly, because liquid is slightly uncompressible.
- once it reaches the vapor pressure it would start to change phase and at the same pressure it would start to change in volume, while liquid is turning into vapor.
- once it all turned into vapor the change in volume with pressure would be with a lower slope.
- the isothermal line of $T_{c}$ goes through the critical point.

###### Density-Temperature diagram

The curves basically represent the densities of the vapor and the liquid phases that coexist in the two-phase region, that is, along the vapor pressure curve.

- it can be seen that at the critical point, the density of liquid and vapor is equal
- also it shows the average densities of the vapor and the liquid phases as a function of temperature, indicating a straight line that passes through the critical point. this particular property is known as the
.**law of rectilinear diameters**

###### Phase diagrams of two-component

Similar to the single-component systems, the phase behavior of a binary system is also described by a phase diagram. However, the most significant difference between the phase behavior of a pure component and a binary system is the difference in the characteristics of the phase diagram itself. For a pure component, a single vapor pressure curve represents the two-phase vaporβliquid equilibrium, whereas for a binary system, there is a broad region in which the two phases coexist in equilibrium.

- unlike the single-component system, in a binary system, vapor and liquid can coexist in equilibrium at pressures and temperatures even above the critical point.
- the left hand side of the critical point above the phase envelope is single phase liquid like
- the right hand side is vapor like.
and**Cricondenbar**are defined as the highest pressure and highest temperature, respectively, on the phase envelope.**cricondentherm**- this figure shows that phase envelope of the two component system lies between vapor-pressure lines of each component.
- the critical temperature of the mixture lies between the critical temperature of the two pure components.
- the critical pressure of the mixture is above both if the components.
- when the percentages of each component are almost the same, the phase envelope is at itβs widest Some general points about phase diagrams:
- the heavier the components are the farther right it would be shifted, and the lighter they are it would be shifter to left.
- the higher the difference in mass between components the wider the phase envelope gets.
- phase envelope for gas reservoirs is tighter and more on the left while for oil reservoirs is wider and on the right.
- the higher the percentage of heavier components the critical point would converge to the peak of the phase envelope.
- reservoirs with their critical point on the right side of the phase envelope are rare and it would mean that they have low amount of average sized components and nitrogen dissolved in it.
- the shape of the phase envelope is a function of composition of each of component.

###### Density-Temperature diagram for two-components

the only difference between a binary system and a single component system is that while phase change between the bubble point and dew point is occurring, pressure would drop because each component has a different vapor pressure value.

###### Pressure-composition diagram for two-components

the horizontal line that connects bubble-point line to dew-point line is called a * Equilibrium tie line*.

- the length of line $12$ divided by the length of the tie line is the ratio of moles of gas to total moles of mixture.
- the length of line $13$ divided by the length of the tie line is the ratio of moles of liquid to total moles of mixture.

###### Three-component phase diagrams

itβs used to show the phase of a 3 component system.

- the lightest component is on the top and the heaviest is on the bottom left.
- both pressure and temperature are constant and the only changing part is composition.

###### Multicomponent mixtures

As the number and complexity of the molecules in a mixture increase, the separation between the bubble-point line and dew-point lines on the phase diagram becomes greater.

## The Five Reservoir Fluids

#### Black oil

Black oils consist of a wide Variety of chemical components including large, heavy, nonvolatile molecules. The phase diagram predictably covers a wide temperature range. The critical point is well up the slope of the phase envelope.

- When reservoir pressure lies anywhere along line $12$, the oil is said to be
. the word**undersaturated**is used in this sense to indicate that the oil could dissolve more gas if more gas were present.**undersaturated** - if the reservoir pressure is at point 2 and below, the oils is at its bubble point and is to be saturated.

##### characteristics of Black oils:

- initial GOR of 2000 scf/STB
- Producing GOR will increase production when reservoir pressure falls below the bubble point pressure.
- the stock-tank oil usually will have a gravity below 45 API
- stock-tank oil gravity will slightly decrease with time until late in the life of reservoir when it will increase.
- The stock-tank oil is very dark, often black, sometimes with greenish cast or brown.
- initial $B_{o}=2.0Βbbl/STB$ or less
- $C_{7}+>30%$

#### Volatile oil

volatile oils contain relatively fewer heavy molecules and more intermediates (defines as ethane through hexanes) than black oils.

- The temperature range covered by the phase envelope is somewhat smaller than black oil.
- The critical temperature is much lower than for a black oil and, in fact, is close to reservoir temperature.
- The iso-vols are not evenly spaced but are shifter upwards toward the bubble-point line.

##### characteristics of Volatile oils:

- initial GOR between 2000 and 3300 scf/STB
- Producing GOR will increase production when reservoir pressure falls below the bubble point pressure.
- the stock-tank oil gravity of 40 API or higher
- stock-tank oil gravity will increase during production as reservoir pressure falls below the bubble point.
- The stock-tank oil is colored, usually brown, orange or sometimes green.
- initial $B_{o}>2.0Βbbl/STB$
- $12.5%<C_{7}+<30%$
- the dividing line between volatile oils and retrograde gases of 12.5 mole percent heptanes plus is fairly definite.

#### Retrograde gas

- the phase diagram of retrograde gas is somewhat smaller than that for oils, and the critical point is further down the left side of the envelope.
- These changes are a result of retrograde gases containing fewer of the heavy hydrocarbons than do the oils.
- it has a critical temperature less than reservoir temperature and a cricondentherm greater than reservoir temperature.

##### characteristics of Retrograde gases:

- 3300 < initial GOR < 50,000
- GOR constant above dew-point pressure and below that it would increase.
- stock-tank liquid gravity between 40 and 60 API
- API constant above dew-point pressure and below that it would increase.
- the liquid can be lightly colored, brown, orange, greenish, or water-white.
- $C_{7}+<12.5%$

#### Wet gas

itβs composed of entirely smaller molecules and the entire phase diagram will lie below reservoir temperature.

- a wet gas exists solely as a gas in the reservoir throughout the reduction in reservoir pressure.
- no liquid is formed in the reservoir
- however, separator conditions lie within the phase envelope, causing some liquid to be formed at the surface.

##### characteristics of Wet gases:

- initial GOR higher than 50,000 scf/STB
- stock-tank gravities same as retrograde gases, 40-60 API
- both would stay constant during production.
- the liquid is water-white.
- $C_{7}+<12.5%$

#### Dry gas

Dry gas is primarily methane with some intermediates.

- hydrocarbon mixture is solely gas in the reservoir
- no liquid formed at the surface either.

## Equations of state

#### ideal gas equation

$PV=nRT$

where: $P$ absolute pressure, Psia $V$ volume, $ft_{3}$ $n$ moles of gas, lbmole $R$ gas constant, for field units 10.73 $T$ temperature, $Β°R$

$n=Mmβ$

where: $m$ mass of the gas, $lb_{m}$ M molecular mass of gas

$PV_{m}=RT$

$V_{m}$ molecular volume (the volume of a mole of gas) $V_{m}$ for standard conditions is $380.7ΒSCF$

#### Mixtures of ideal gases

the following laws only apply to ideal gasses

###### Daltonβs law of partial pressures

$P=P_{A}+P_{B}+P_{C}$

$P_{i}=y_{i}P$

###### Amagatβs law of patrial volumes

$V=V_{A}+V_{B}+V_{C}$

$V_{i}=y_{i}V$

###### Apparent molecular weight

$M_{a}=βy_{i}M_{i}$

#### Specific gravity of gas

$Ξ³_{g}=Ο_{air}Ο_{g}β$

$Ξ³_{g}=M_{air}Mgβ=29M_{g}β$

#### Behavior of real gases

###### Compressibility equation of state

$PV=znRT$

$Ο_{g}=zRTPMβ$

where the correction factor, $z$ is known as the * compressibility factor* and the equation is known as the compressibility equation of state.
The z-factor is the ratio of volume actually occupied by a gas at given pressure and temperature to the volume the gas would occupy at the same pressure and temperature if it behaved like and ideal gas.

$z=V_{Ideal}V_{Real}β$

- at low pressures, molecules are relativity far apart and they behave like an ideal gas.
- at moderate pressures, the molecules are close enough to exert some attraction between them to reduce the volume thus z is less than 1.0.
- at higher pressures, the molecules are forced close together, repulsive forces come into play, the actual volume is greater than ideal volume, thus z is greater than 1.0.

###### The law of corresponding states

$T_{r}=T_{c}Tβ$

$P_{r}=P_{c}Pβ$

and extended to mixtures:

$T_{pc}=βy_{i}T_{ci}$

$P_{pc}=βy_{i}P_{ci}$

### other equations of state

TBD

## Properties of Dry gases

###### Gas formation volume factor

$B_{g}=V_{sc}V_{res}β$

since$V_{res}=PznRTβ$ and $V_{sc}=P_{sc}z_{sc}nRTscβ$ and $T_{sc}=520ΒΒ°R$, $P_{sc}=14.65Βpsia$, and for all practical purposes $z_{sc}=1$, then

$B_{g}=0.0282PzTβ(SCFcuftβ)$

$B_{g}=0.00502PzTβ(SCFresbblβ)$

###### The coefficient of isothermal compressibility of gas

$c_{g}=βV1β(βPβVβ)_{T}$

for an ideal gas:

$V=PnRTβ$

$(βPβVβ)_{T}=βP_{2}nRTβ$

$c_{g}=βV1β(βP_{2}nRTβ)$

$c_{g}=(βnRTPβ)(βP_{2}nRTβ)=P1β$

for real gas:

$c_{g}=P1ββz1β(βPβzβ)_{T}$

###### viscosity of gas

$Ξ½=Ο_{g}ΞΌβ$

$Ξ½$ kinematic viscosity, centipoise $g/cm.sΓ10_{β2}$ $ΞΌ$ dynamic viscosity, centistokes $cm_{2}/sΓ10_{β}2$ $Ο_{g}$ density, $g/cm_{3}$

###### Heating Value

The * Heating Value* of a gas is the quantity of heat produced when the gas is burned completely to carbon dioxide and water. Heating value usually is expressed as $BTU/scf$.

- wet and dry refer to the condition of the gas prior to combustion.
means that the gas is saturated with water vapor, about 1.75 volume percent.**Wet**means that the gas contains no water vapor.**dry**- gross and net refer to the condition of the water of combustion after burning takes place.
is the heat produced in complete combustion under constant pressure with the combustion products cooled to standard conditions and the water in the combustion products condensed to the liquid state. This quantity also is called**Gross heating value**.**total heating value**is defined similarly, except the water of combustion remains vapor at standard conditions.**Net heating value**- the difference between net and gross heating values is the heat of vaporization of the water of combustion.

$GHV=NHV+heatΒ ofΒ waterΒ vapor$

- in industry the dry gross heating value is used.

$NHV_{ideal}=βy_{i}NHV_{i}$

$GHV_{ideal}=βy_{i}GHV_{i}$

$NHV_{real}=ZNHV_{ideal}β$

$GHV_{real}=ZGHV_{ideal}β$

$Z=1β(βy_{i}1βZ_{i}β)_{2}$

the important equations are:

$(NHV)_{wet}=(1β0.0175)(NHV)_{dry}$

$(GHV)_{wet}=(1β0.0175)(GHV)_{dry}+0.9$

where $0.9ΒscfBTUβ$ is the heat released from the condensation of water.

###### Joule-Thomson effect

Temperature changes as pressure is reduced when a flowing stream of gas passes through a throttle, i.e., a valve, choke, or perforations in casing. this is called the * Joule-Thomson effect*.

$ΞT=C_{p}ZVTβ(βTβZβ)_{P}βΓΞP$

- since $V,T,ZΒandΒC_{P}$ are always positive, the direction of the change of temperature is dependent on $ΞPΒandΒ(βTβZβ)_{P}$ .
- and since in petroleum engineering conditions pressure drops and $ΞP$ is negative, the change of temperature is only dependent on $(βTβZβ)_{P}$ .
- at pressures above $7000Βpsi$ with the decrease in pressure, temperature increases.
- at pressures below $5000Βpsi$ with the decrease in pressure, temperature drops as well, and the most of the drop happens between $1500β2000Βpsi$.

## Properties of Wet gases

The key to analysis of the properties of a wet gas is that the properties of the surface gas are not the same as the properties of the reservoir gas. At the surface, the well stream is separated into stock-tank liquid (condensate), separator gas, and stock-tank gas. All three of these fluids must be included in the recombination calculation.

#### Recombination of surface

###### Separator Gas and stock-tank vent gas properties known

in this method a barrel of stock-tank liquid is the basis, and with using the ratio between gas and liquid we can calculate the produced gas from the reservoir.

$R=R_{P}=βR_{sep.i}+R_{ST}$

where: $R_{P}$ is the ratio of all the produced gas to the stock tank liquid. $(STBSCFβ)$ $R_{sep.i}$ is the ratio of gas produced in each separator to the stock tank liquid. $(STBSCFβ)$ $R_{ST}$ is the ratio of stock gas to stock tank liquid. $(STBSCFβ)$

- since the lighter components stay in vapor phase at higher pressures, the gas leaving the first separator is the lightest, and the gas leaving the stock tank is the heaviest.

$Ξ³_{gsep1}<Ξ³_{gR},ΒΞ³_{gsep2},...,Ξ³_{gsepΒn}<Ξ³_{gST}$

$Ξ³_{gs}=βR_{sep.i}+R_{ST}βR_{sep.i}ΓΞ³_{gsep.i}+R_{ST}ΓΞ³_{gST}β$

where: $Ξ³_{gsep.i}$ is the specific gravity of each separator gas. $Ξ³_{gST}$ is the specific gravity of the stock tank gas. $Ξ³_{gs}$ is the specific gravity of surface gas.(the recombination of all gases produced at the surface) $Ξ³_{gR}$ is the specific gravity of the reservoir gas.

$Ξ³_{gR}$ can be calculated in the following way:

$Ξ³_{gR}=28.97MW_{gR}β$

$MW_{gR}=molesΒ ofΒ stockΒ tankΒ liquid(n_{l})+molesΒ ofΒ producedΒ gas(n_{g})theΒ massΒ ofΒ stockΒ tankΒ liquid(m_{l})+theΒ massΒ ofΒ gasΒ producedΒ atΒ theΒ surface(m_{g})β$

$Ξ³_{gR}=R_{p}+133,300MWΞ³βR_{p}ΓΞ³_{gs}+4600Ξ³_{o}β$

$GE=133,300MW_{o}Ξ³_{o}β$

where: $GE$ is the gas equivalent volume.

###### Properties of stock-tank gas is unknown

* Vapor Equivalent* and

*are defined as the following:*

**Additional gas produced**, VEQ, is the volume of second-separator gas and stock-tank gas in $scf/STB$ plus the volume in scf that would be occupied by a barrel of stock-tank liquid if it were gas.**Vapor equivalent volume**

$VEQ=β_{i=2}R_{sep.i}+R_{ST}+GE$

- The
, AGP, is related to the mass of gas produced from the second separator and the stock tank.**additional gas produced**

$AGP=β_{i=2}R_{sep.i}ΓΞ³_{gsep.i}+R_{ST}ΓΞ³_{gST}$

$Ξ³_{gR}=R_{sepΒ1}+VEQR_{sepΒ1}ΓΞ³_{gΒsepΒ1}+AGP+4600Ξ³_{o}β$

in a two stage system:

$AGP=R_{ST}ΓΞ³_{gST}$

$VEQ=R_{ST}+GE$

in a three stage system:

$AGP=R_{sepΒ2}ΓΞ³_{gΒsepΒ2}+R_{ST}ΓΞ³_{gST}$ $VEQ=R_{sepΒ2}+R_{ST}+GE$

#### Formation volume factor of wet gas

The * formation volume factor of a wet gas* is defined as the volume of reservoir gas required to produce one stock-tank barrel of liquid at the surface. by definition:

$B_{wg}=volumeΒ ofΒ stock-tankΒ liquidΒ atΒ standardΒ conditionsvolumeΒ ofΒ reserviorΒ gasΒ atΒ reserviorΒ pressureΒ andΒ temperatureβ=(V_{o})_{P,ΒT}(V_{g})_{P,ΒT}β$

where: $B_{wg}$ is the formation volume factor of a wet gas. $(STBcuftβ)$ $(V_{g})_{P_{R},ΒT_{R}}$ is the volume of gas at reservoir pressure and temperature. $(cuft)$ $(V_{g})_{P_{R},ΒT_{R}}$ is the volume of stock-tank liquid at standard conditions. $(STB)$

$volumeΒ ofΒ reserviorΒ wetΒ gas=R_{sepΒ1}+VEQΒΒΒΒΒΒΒ(STBSCFβ)$

$B_{wg}=0.0282PZTβ(R_{sepΒ1}+VEQ)ΒΒΒΒΒ(STBcuftβ)$

$B_{wg}=0.00502PZTβ(R_{sepΒ1}+VEQ)ΒΒΒΒΒ(STBbblβ)$

#### Plant products

Frequently, processing surface gas to remove and liquefy the intermediate hydrocarbons is economically feasible. these liquids often are called * plant products*. the quantities of liquid products which can be obtained usually are determined in gallons of liquid per thousand standard cubic feet of gas processed, gal/Mscf, or GPM.

$GPM_{i}=(y_{i}lbΒmoleΒgaslbΒmoleΒiβ)(380.7ΒscflbΒmoleΒgasβ)(Mscf1000Βscfβ)(MW_{i}lbΒmoleΒilbΒiβ)(Ο_{oΒi}ΒlbΒicuftΒliqβ)(cuftΒliq7.481Βgalβ)=19.65Ο_{oΒi}y_{i}MW_{i}βMscfgalβ$

$GPM_{i}=0.3151Ξ³_{oΒi}y_{i}MW_{i}βMscfgalβ$

## Properties of Retrograde gases

The discussion above for wet gasses applies to retrograde gases as long as the reservoir pressure is above the dew-point pressure of the retrograde gas. At reservoir pressures below the dew point, none of the recombination calculations given in this section are valid for retrograded gases. Special laboratory analyses are required for engineering of retrograde gas reservoirs.

## Properties of Black Oils

###### Specific gravity of oil

$Ξ³_{o}=Ο_{w}Ο_{o}β$

$Β°API=Ξ³_{o}141.5ββ131.5$

where $Ξ³_{o}$ is the specific gravity at $60Β°/60Β°$

###### Formation volume factor of oil

The volume of oil that enters the stock tank at the surface is less than the volume of oil which flows into the wellbore from the reservoir. This change in volume is due to **three factors**:

- the most important factor is the evolution of gas from the oil as pressure is decreased from reservoir pressure to surface pressure. this causes a rather large decrease in volume of the oil when there is a significant amount of dissolved gas.
- The reduction in pressure also causes the remaining oil to expand slightly, but this is somewhat offset by the contraction of oil due to the reduction of temperature.
The change in oil volume due to these three factors is expressed in terms of the
. itβs defined as the volume of reservoir oil required to produce one barrel of oil in the stock tank. since the reservoir oil includes dissolved gas:**formation volume factor of oil**

$B_{o}=volumeΒ ofΒ oilΒ enteringΒ stockΒ tankΒ atΒ standardΒ conditionsvolumeΒ ofΒ oilΒ inΒ reservoirΒ conditionsβ$

- This figure shows the initial reservoir pressure to be above the bubble point pressure of the oil. As reservoir pressure is decreased from initial pressure to bubble-point pressure, the formation volume factor increases slightly because of the expansion of the liquid in the reservoir.
- A reduction in reservoir pressure below bubble-point pressure results in the evolution of gas in the pore spaces of the reservoir. The liquid remaining in the reservoir has less gas in solution and, consequently, a smaller formation volume factor.
- If the reservoir pressure could be reduced to atmospheric, the value of the formation volume factor would nearly equal 1.0 $resΒbbl/STB$. A reduction in temperature to $60Β°F$ is necessary to bring the formation volume factor to exactly 1.0 $resΒbbl/STB$.

###### Shrinkage factor

The reciprocal of the formation volume factor is called the * shrinkage factor*.
$b_{o}=B_{o}1β$

###### Solution Gas-Oil Ratio

The quantity of gas dissolved in an oil at reservoir conditions is called * solution gas-oil ratio*. Solution gas-oil ratio is the amount of gas that evolves from the oil as the oil is transported from the reservoir to surface conditions. This ratio is defined in terms of the quantities of gas and oil which appear at the surface during production.

$R_{s}=VolumeΒ ofΒ oilΒ enteringΒ stockΒ tankΒ atΒ standardΒ conditionsVolumeΒ ofΒ gasΒ producedΒ atΒ surfaceΒ atΒ standardΒ conditionsβ$

- This figure shows the way the solution gas-oil ratio of a typical black oil changes as reservoir pressure is reduced at constant temperature.
- The line is horizontal at pressures above the bubble-point pressure because at these pressures no gas is evolved in the pore space and the entire liquid mixture is produced into the wellbore. When reservoir pressure is reduced below bubble-point pressure, gas evolves in the reservoir, leaving less gas dissolved in the liquid.

###### Total Formation volume factor

This figure shows the volume of occupied by one barrel of stock-tank oil plus its dissolved gas at bubble-point pressure. The figure also shows the volume occupied by the same mass of material after an increase in cell volume has caused a reduction in pressure. The volume of oil has decreased; however, the total volume has increased.
The volume of oil at the lower pressure is $B_{o}$. the quantity of gas evolved is the quantity of gas evolved is the quantity in solution at the bubble point, $R_{sb}$, ,minus the quantity remaining in solution at the lower pressure, $R_{s}$. The evolved gas is called * free gas*. It is converted to reservoir conditions by multiplying by the formation volume factor of gas, $B_{g}$.
This total volume is the

*. $B_{t}=B_{o}+B_{g}(R_{sb}βR_{s})$ The gas formation volume factor must be expressed in units of $resΒbbl/scf$, and total formation volume factor has units of $resΒbbl/STB$.*

**total formation volume factor**- This figure gibes a comparison of total formation volume factor with the formation volume factor of oil. The two formation volume factors are identical at pressures above the bubble-point pressure since no gas is released into the reservoir at these pressures.
- The difference between the two factors at pressures below the bubble-point pressure represents the volume of gas released in the reservoir. The volume of this gas is $B_{g}(R_{sb}βR_{s})$.

###### The coefficient of isothermal compressibility of Oil - Pressures above the bubble point pressure.

At pressures above the bubble point, * the coefficient of isothermal compressibility* of oil is defined exactly as the coefficient of isothermal compressibility of gas. At pressures below the bubble point an additional term must be added to the definition to account for the volume of gas which evolves.

$c_{o}=βV1β(βPβVβ)_{T}$

$c_{o}=βB_{o}1β(βPβB_{o}β)_{T}$

$c_{o}=Ο_{o}1β(βPβΟ_{o}β)_{T}$

- this figure shows that black oil compressibility is virtually constant except at pressures near the bubble point.
- Values rarely exceed $35Γ10_{β6}Βpsi_{β1}$ with integrating the above equations we get:

$V_{2}=V_{1}exp[c_{o}(P_{1}βP_{2})]$

$B_{o2}=B_{o1}exp[c_{o}(P_{1}βP_{2})]$

$Ο_{o2}=Ο_{o1}exp[c_{o}(P_{2}βP_{1})]$

###### The coefficient of isothermal compressibility of Oil - Pressures below the bubble point pressure

When reservoir pressure is below bubble-point pressure, the situation is much different. As this figure shows, the volume of reservoir liquid decreases as pressure is reduced. However, the reservoir volume occupied by the mass that was originally liquid increases due to the evolution of gas. The change in liquid volume may be represented by:

$(βPβB_{o}β)_{T}$

the change in amount of dissolved gas is:

$(βPβR_{s}β)_{T}$

and so, the change in volume of free gas is:

$β(βPβR_{s}β)_{T}$

Thus, at reservoir pressures below the bubble point, the total change in volume is the sum of the change in liquid volume and the change in free gas volume.

$[(βPβB_{o}β)_{T}βB_{g}(βPβR_{s}β)_{T}]$

Where $B_{g}$ is inserted to convert the volume of evolved gas to reservoir conditions. Consequently, the fractional change in volume as pressure changes is:

$c_{o}=βB_{o}1β[(βPβB_{o}β)_{T}βB_{g}(βPβR_{s}β)_{T}]$

- the evolution of the first bubble gas causes a large shift in the value of compressibility.

###### Coefficient of viscosity of oil

The * coefficient of viscosity* is a measure of the resistance to flow exerted by a fluid.

*appears as a coefficient in any equation which describes fluid flow.*

**Viscosity**- Viscosity, like other physical properties of liquids, is affected by both pressure and temperature.
- An increase in temperature causes a decrease in viscosity.
- A decrease in pressure causes a decrease in viscosity, provided that the only effect of pressure is to compress the liquid.
- in addition, in the case of reservoir liquids, there is a third parameter which affects viscosity. A decrease in the amount of gas in solution in the liquid causes an increase in viscosity, and, of course, the amount of gas in solution is a direct function of pressure.
- At pressures above bubble point, the viscosity of the oil in a reservoir decreases almost linearly as pressure decreases. At lower pressures the molecules are further apart and therefore move past each other more easily.
- However, as reservoir pressure decreases below the bubble point, the liquid changes composition. The gas that evolves takes the smaller molecules from the liquid, leaving the remaining reservoir liquid with relatively more molecules with large complex shapes.

###### Volatile Oils

Formation volume factors and solution gas-oil ratios normally are not measured for volatile oils. These quantities are used primarily in material balance calculations which do not apply to volatile oils. if these quantities were measured for volatile oils, they would have the shapes indicated in these figures.

- The large decrease in both curves at pressures immediately below bubble point are due to the evolution of large quantities of gas in the reservoir at pressures just below the bubble point.

## Reservoir Fluid Studies

###### Flash Vaporization

A sample of the reservoir liquid is placed in a laboratory cell. Pressure is adjusted to a value equal to or greater than initial reservoir pressure, Temperature is set at reservoir temperature. Pressure is reduced by increasing the volume in increments. The procedure is illustrated in this figure.

- The cell is agitated regularly to ensure that the contents are at equilibrium. No reservoir gas or liquid is removed from the cell.
- At each step, pressure and volume of the reservoir fluid are measured.
- The volume is termed total volume, $V_{t}$, since at pressures below the bubble point the volume includes both gas and liquid.
- Pressure is plotted against total volume in this figure.
- The pressure at which the slope changes is the bubble-point pressure of the mixture. The volume at this point is the volume of the bubble-point liquid. Often it is given the symbol $V_{sat}$.
- The volume of the bubble-point liquid can be divided by the mass of reservoir fluid in the cell to obtain a value of specific volume at the bubble point.

###### Differential Vaporization

The sample of reservoir liquid in the laboratory cell is brought to bubble-point pressure, and temperature is set at reservoir temperature.
Pressure is reduced by increasing cell volume, and the cell is agitated to ensure equilibrium between the gas and liquid.
Then, All the gas is expelled from the cell while pressure in the cell is held constant by reducing cell volume.
The gas is collected, and its quantity and specific gravity are measured. The volume of liquid remaining in the cell, $V_{o}$ , is measured. This process is shown in this figure.
The process is repeated in steps until atmospheric pressure is reached. then temperature is reduced to $60Β°F$ , and the volume of remaining liquid is measured. This is called * residual oil from differential vaporization*.
Each of values of volume of cell liquid, $V_{o}$ , is divided by the volume of the residual oil. The result is called

*and is given the symbol $B_{oD}$. The volume of gas removed during each step is measured both at cell conditions and at standard conditions. The z-factor is calculated as:*

**relative oil volume**$z=V_{sc}P_{sc}T_{R}V_{R}P_{R}T_{sc}β$

Where the subscript R refers to conditions in the cell. Formation volume factors of the gas removed are calculated with these z-factors using this equation:

$B_{g}=0.0282PzTβscfcuΒftβ$

- The total volume of gas removed during the entire process is the amount of gas in solution at the bubble point.
- This total volume is divided by the volume of residual oil, and the units are converted to standard cubic feet per barrel of residual oil. The symbol $R_{sDb}$ represents standard cubic feet of gas removed per barrel of residual oil.
- The gas remaining in solution at any lower pressure is calculated by subtracting the sum of the gas removed down to and including the pressure of interest from the total volume of gas removed. The result is divided by the volume of residual oil, converted to $scf/residualΒbbl$, and reported as $R_{sD}$. Relative total volume at any pressure is calculated as:

$B_{tD}=B_{oD}+B_{g}(R_{sDb}βR_{sD})$

###### Separator Tests

A sample of reservoir liquid is placed in the laboratory cell and brough to reservoir temperature and bubble-point pressure. Then the liquid is expelled from the cell through two stages of separation. See this figure.

- The vessel representing the stock tank is a stage of separation if it has lower pressure than the separator.
- Pressure in the cell is held constant at bubble point by reducing cell volume as the liquid is expelled.
- The temperatures of the laboratory separator and stock tank usually are set to represent average conditions in the field.
- The stock tank is always at atmospheric pressure.
The
is calculated as:**formation volume factor of oil**

$B_{oSb}=volumeΒ ofΒ liquidΒ arrivingΒ inΒ theΒ stockΒ tankvolumeΒ ofΒ liquidΒ expelledΒ fromΒ theΒ cellβ$

The subscript $S$ indicates that this is a result of a separator test, and the subscript $b$ indicates bubble-point conditions in the reservoir.

- The volume of liquid expelled from the cell is measured at bubble-point conditions.
- The volume of stock-tank liquid is measured at standard conditions.
The
is calculated as:**solution gas-oil ratio**

$R_{sSb}=volumeΒ ofΒ liquidΒ inΒ theΒ stockΒ tankvolumeΒ ofΒ separatorΒ gasΒ+ΒvolumeΒ ofΒ stock-tankΒ gasβ$

with all volumes adjusted to standard conditions. The subscripts $S$ and $b$ have the same meanings as discussed above.

###### Selection of Separator Conditions

The separator pressure which produces the maximum amount of stock-tank liquid is known as * optimum separator pressure*.
It is identified from the separator test as the separator pressure which results in:

- a minimum of total gas-oil ratio
- a minimum of formation volume factor of oil (at bubble point)
- and maximum in stock-tank oil gravity Most black oils have optimum separator pressures of 100 to 120 psig at normal temperatures.

###### Formation volume factor of oil

At pressures * above* bubble-point pressure, oil formation volume factors are calculated from a combination of

*data and*

**flash vaporization***.*

**separator test**$B_{o}=(V_{b}V_{t}β)_{F}B_{oSb}Β atΒPβ₯P_{b}$

$B_{o}=(resΒ bblΒ ofΒ oilΒ atΒP_{b}resΒ bblΒ ofΒ oilΒ atΒPβ)(STBresΒ bblΒ ofΒ oilΒ atΒP_{b}β)=STBresΒ bblΒ ofΒ oilΒ atΒPβ$

At pressures * below* the bubble-point pressure, oil formation volume factors are calculated from a combination of

*data and*

**differential vaporization***data.*

**separator test**$B_{o}=B_{oD}B_{oDb}B_{oSb}βΒ atΒP<P_{b}$

$B_{o}=(residualΒ bblΒ byΒ diffΒ vapresΒ bblΒ ofΒ oilΒ atΒPβ)(residualΒ bblΒ byΒ diffΒ vapresΒ bllΒ ofΒ oilΒ atΒPβ)(STBresΒ bblΒ ofΒ oilΒ atΒPβ)β=STBresΒ bblΒ atΒPβ$

###### Solution Gas-Oil Ratio

solution gas-oil ratio at pressures * above* bubble-point pressure is a constant equal to the solution gas-oil ratio at the bubble point.

$R_{s}=R_{sSb}Β atΒPβ₯P_{b}$

solution gas-oil ratios ate pressures * below* bubble-point pressure are calculated from a combination of differential vaporization data and separator test data.

$R_{s}=R_{sSb}β(R_{sDb}βR_{sD})B_{oDb}B_{oSb}βΒ atΒP<P_{b}$

###### Formation volume factor of gas

Gas formation volume factors are calculated with z-factors measured with gases removed from the cell at each pressure step during differential vaporization.

$B_{g}=0.0282PzTβscfresΒcuΒftβ$

###### Total formation volume factor

Total formation volume factors may be calculated as

$B_{t}=B_{o}+B_{g}(R_{sb}βR_{s})$

using the fluid properties calculated from the reservoir fluid study. if relative total volumes, $B_{tD}$ , are reported as a part of the results of the differential vaporization, total formation factors can be calculated as:

$B_{t}=B_{tD}B_{oDb}B_{oSb}β$

$B_{t}=(residualΒ bblΒ byΒ diffΒ vapresΒ bblΒ oilΒ +Β gasβ)(residualΒ bblΒ byΒ diffΒ vapresΒ bllΒ ofΒ oilΒ atΒPβ)(STBresΒ bblΒ ofΒ oilΒ atΒPβ)β=STBresΒ bblΒ oilΒ +Β gasβ$

###### Lab tests of gas condensate reservoir

TBD

## Properties of Oilfield Waters

TBD

## Gas-Liquid Equilibria

The area bounded by the bubble point and dew point curves on the phase diagram of a multicomponent mixture defines the conditions for gas and liquid to exist in equilibrium.

#### Ideal solutions

An * Ideal liquid solution* is a solution for which:

- Mutual solubility results when the components are mixed, no chemical interaction occurs upon mixing, the molecular diameters of the components are the same. and
- The intermolecular forces of attraction and repulsion are the same between unlike as between like molecules.

###### Raoultβs Equation

* Raoultβs equation* states that the partial pressure of a component in the gas is equal to the mole fraction of that component in the liquid multiplied by the vapor pressure of the pure component. Raoultβs equation is valid only if both the gas and liquid mixtures are ideal solution. The mathematical statement is

$P_{i}=x_{i}P_{v_{i}}$

Where $P_{i}$ represents the partial pressure of component $i$ in the gas in equilibrium with a liquid of composition $x_{i}$. The quantity $P_{vi}$ represents the vapor pressure that pure component $i$ exerts at the temperature of interest.

###### Daltonβs Equation

We saw that * Daltonβs equation* can be used to calculate the partial pressure exerted by a component of an ideal gas mixture:

$P_{i}=y_{i}P$

###### Compositions and quantities of the Equilibrium Gas and Liquid Phases of an Ideal Solution

Raoultβs and Daltonβs equations both represent the partial pressure of a component in a gas mixture. In the case of Raoultβs equation, the gas must be in equilibrium with a liquid. These equations relates the compositions of the gas and liquid phases in equilibrium to the pressure and temperature at which the gas-liquid equilibrium exists.

$k_{i}=x_{i}y_{i}β$

$y_{i}P=x_{i}P_{v_{i}}$

$k_{i}=x_{i}y_{i}β=PP_{v}β$

The above equation shows that for an ideal solution, $k_{i}$ is a function of system pressure $(P)$ and temperature $(T)$ (since $P_{v_{i}}$ is a function of temperature. )

A material valance of the i-th component is:

$z_{i}n=x_{i}n_{l}+y_{i}n_{g}$

combining with the previous equation gives:

$z_{i}n=x_{i}n_{l}+x_{i}PP_{v}βn_{g}$

or

$x_{i}=n_{l}+PPβn_{g}z_{i}nβ$

since $n=n_{l}+n_{g}=1$

$βx_{i}=β1+n_{g}(k_{i}β1)z_{i}β=1$

$βy_{i}=β1+n_{l}(k1ββ1)z_{i}β=1$

###### Calculation of the Bubble-Point Pressure of an Ideal Liquid Solution

The bubble point is the point at which the first bubble of gas is formed. For all practical purposes, the quantity of gas is negligible. Thus, we can take $n_{g}$ to be equal to zero and $n_{L}$ to equal the total moles of the mixture. Substitution of $n_{g}=0$, $n_{L}=0$, and $P=P_{b}$ into the equation results in:

$βP_{b}/P_{v}z_{i}β=1$

or

$P_{b}=βz_{i}P_{v_{i}}$

Therefore, the bubble-point pressure of an ideal liquid solution at a given temperature is simply the summation of the products of mole fraction times vapor pressure for each component.

###### Calculation of the Dew-Point Pressure of an Ideal Gas Solution

At the dew point, the quantity of liquid is essentially is negligible, so for $P=P_{d}$, we can substitute $n_{L}=0$ and $n_{g}=0$ into the equation, this results in:

$βP_{v}/P_{d}z_{i}β=1$

or

$P_{d}=βz_{i}/P_{v}1β$

The dew point of an ideal gas mixture at a given temperature is simply the reciprocal of the summation of the mole fraction divided by vapor pressure for each component.

#### Non-Ideal solutions

TBD