Module 4 - Solvent Flooding

8/12/2019 Module 4 - Solvent Flooding

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Solvent Flooding Copyright 1998-2001: Maria Barrufet

Class Notes for PETE 609Module 4 Page 1/105

Author: Dr. Maria Antonieta Barrufet - Fall, 2001

This modu le is protected by co pyr ight . No par t of any of these pages

may be reproduced in any form or by any means, electron ic or

otherwise, wi thout w r i t ten permiss ion from the copy r ight owner

Maria Barru fet 979-8450314 /[email protected]

Module 4 Solvent Flooding

Estimated Duration: 2 weeks

General overview of solvent methods.

Mechanisms of oil displacement.

Gas injection: Diffusion and Dispersion.

Hydrocarbon miscible displacement: First Contact Miscible process, Condensing-

Gas process, Vaporizing-Gas process.

Dissipation: Modeling Dispersion and Viscous Fingering

Minimum miscibility pressure (MMP). MMP estimation by correlations and by

equations of state.

Performance evaluation.

Suggested reading: MAB, R8, R18, S
mailto:[email protected]:[email protected]:[email protected]:[email protected]


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Learning Object ives

After completion of this module, you will be able to:

Identify various processes and solvent types

Isolate several solvent properties

Classify solvent floods

Isolate properties of immiscible displacements

Determine when viscous fingering may occur

Evaluate concentration profiles/histories in 1-D miscible displacements

Interpret slimtube experiments and evaluate Minimum Miscibility Pressure

Evaluate the performance of a solvent flood

The material for this module is mainly from Lake 1989, Stalkup 1992, and Orr 1994; as

well as MAB (your instructor).

In t rodu ct ion to Solvent Flooding

A solvent can be injected in the reservoir to displace oil. This injection can result in a

Miscible Displacement (1-phase), or in an Immiscible Displacement (2-phase).


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Figure 1 - Example of miscible and immiscible displacement.

InFigure 1,we can see that if solvent S1 is injected in the reservoir, the resulting

mixture M1 will be only liquid (1-phase, Miscible Displacement). However, if solvent S2

is used, the resulting mixture M2 will consist of a gas with composition G2 and a liquid

with composition L2 (2-phase, Immiscible Displacement). (Review Module 3.)

A solvent flood is any EOR process whose primary means for recovering oil is through

mass transfer or extraction. Though many chemicals can be used for this purpose, we

concentrate on gaseous solvents, primarily carbon dioxide. The process is frequently

called miscible flooding because many solvents either have developed or will develop

miscibility with the crude being displaced.

Solvent flooding is a more general term for what is commonly called miscible flooding.

It is also sometimes called gas flooding because most injected solvents in current use

are gases. In this module, you will learn the distinctions among types of solvent floods,

and the importance of common solvents (mainly carbon dioxide). We will analyze the

engineering aspects for designing a miscible flood and we will also see some simple

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scoping models to evaluate the performance of a miscible flood. You will also be able

to identify reasons for early solvent breakthrough.

The solvent can be introduced into the reservoir in one of four modes:

1. Continuous or pattern flooding,

2. Soak or stimulation,

3. Simultaneous or alternating injection with water,

4. In a saturated water solution.

Figure 2 illustrates the miscible gas injection process to be discussed further in this

module. Here, we will discuss primarily injection Modes 1 and 3.

Figure 2 - Illustration of miscible gas injection process.


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Potential Solvents

The number and type of potential solvents is very large. Among the gaseous solvents

are,

Gas: nitrogen, flue gas, methane, and carbon dioxide

Liquid solvents: liquefied petroleum gas (LPG), and any of several organic

alcohols

Carbon dioxide, LPG, nitrogen and combinations of these are in the most common

current use. Not all of these are particularly efficient displacing agents. However, all of

them recover oil through at least a limited amount of extraction.

CO2Solvent Properties

The oil extracting ability of many solvents can be related to their physical properties.

Carbon dioxide has a low critical pressure and a low critical temperature (1069.4 psia,

and 87.8oF respectively), this means that, for many applications, carbon dioxide, is a

supercritical fluid or even a liquid if the reservoir temperature is low. The near liquid

properties of carbon dioxide make it an excellent crude extractor.

CO2Density

Another consequence of the near-liquid character of CO2

is its high density. The CO2

density at reservoir conditions is very close to the oil density. Figure 3 shows CO2

density as a function of temperature and pressure. At typical reservoir pressures and

temperatures, the CO2density approaches that of a light crude. This means that gravity

segregation of CO2will occur with respect to water more readily than with crude,

particularly since there is some measure of miscibility with the latter.


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Figure 3 - Density of CO2as a function of pressure at various temperatures.

Note: Density expressed in [g/cm3] is very close to specific gravity.

CO2Viscosity

Figure 4 illustrates the viscosity of supercritical CO2. Note that pressure andtemperature are higher than the critical temperature and pressure for CO2, which are

Tc(CO2)=87.8F and Pc(CO2)=1069.4 psia.


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Figure 4 - Viscosity of CO2as a function of pressure at various temperatures.

Carbon dioxide viscosity, illustration inFigure 4,underscores other aspects. At typical

reservoir conditions, CO2viscosity is 0.06-0.08 cp. At the same conditions, methane

viscosity is 3-5 times less. This means that CO2-crude mobility ratios will not be nearly

as unfavorable as methane-crude mobility ratios.

Three-phase behavior in CO2floods is too complex to be modeled with currentsimulation capabilities. Three-phase behavior occurs in low temperature systems;

however, three-phase behavior disappears at T>107F.

The result of increasing the temperature is to shift the entire phase diagram to higher

pressures. This causes an expansion of the two-phase region, which suggests that it

will be more difficult for solvents to work at high temperature.


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In addition, many laboratory results indicate that a small amount of solid precipitate

(waxes, asphaltenes) can form at higher CO2concentrations. This means that four

hydrocarbon phases can form in at least a limited portion of the P-z plane at low

temperature. The precipitate is less prevalent at high temperature.

P-z Plots and Miscibility

The primary use of P-z diagrams is as a source of data to calibrate phase behavior

predictions using an Equation of State (EOS).

We will use them to illustrate an important insight into the issue of miscibility. A solvent

is said to be miscible with the crude if they mix in all proportions without forming an

interface. This means that CO2and crude are miscible at a temperature and pressure

outside the phase envelopes. They have limited miscibility elsewhere.

In a displacement prior to solvent breakthrough, fluid composition near an injector will

plot near the vertical axis on a P-z diagram. A point near a producer will plot on the

right vertical axis at some lower pressure. All other points in the reservoir are on some

line connecting these two points. Unless extraordinary pressure is used, this line must

necessarily cross a two-phase region. This means that CO2is rarely completely

miscible with crude at lower pressures.


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Figure 5 - Ideal composition profile.

Fortunately, such complete miscibility is not usually necessary because CO2extraction

can form a mixture, which will be miscible with both the crude and the pure solvent

(CO2). This is discussed next.

Ternary Plots and Miscibility

The compositional information not present on a P-z diagram is represented on a ternary

diagram (seen in Module 3).

We divide the non-aqueous components into "light", "intermediate", and "heavy". Then

we plot these on the top, lower right, and lower left apexes as inFigure 6.

InjectionWell

Production Well

InjectionWell

Production Well

Composition of CO2

presure


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Figure 6 - Ternary diagram illustrating single and two phase regions

Ternary diagrams may approximate phase behavior of multi-component mixtures by

grouping them into 3pseudocomponents. A frequent way of grouping different

components of a mixture based on similarities of critical and other physical properties is,

light (C1, CO2, N2- C1, CO2-C2, ...)

heavy (C7+)

intermediate (C2-C6)

Dilution lines

When representing phase behavior relations in a ternary diagram, the compositions of

ALL possible mixtures from mixing two fluids will fall in the straight line connecting the

points indicating the compositions of the two source fluids. For example, ALL mixtures

of n-C4 and bubble point fluid X in next figure are miscible in all proportions, while

mixtures of X with C1are miscible only at high concentrations of C1.


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C1

C10 n-C4

x

Figure 7 - Dilution lines example.

Exercise:

If 80 moles of oil with composition indicated with X inFigure 7 are mixed with 15 moles of n-

C4. What is the resulting composition of the mixture?

In practice, we must translate volumetric quantities to a molar basis.

The light and intermediate components are miscible, but the light and heavy

components are only partially miscible.

Exercise:

Indicate the "Heavy" and "Light" compositions at which mixtures of heavy and light

components are fully miscible. Consider that the amount of intermediate (n-C4) is zero.

Recall that the entire diagram is at fixed temperature and pressure.


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The two-phase region has tie lines, as discussed in Module 3, and three-phase regions

(if they occur) are smaller internal triangles within the 2-phase region.

Each of the apexes represents pseudocomponents in complex systems.


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Types of Displacement

Solvent displacements fall into two major categories, miscible and immiscible. Recall

the effect of pressure on miscibility by observing the two-phase behavior ofFigure 8.

Figure 8 - Effect of pressure on miscibility.


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To define the various types of solvent displacements, we analyze the data plotted on

ternary diagrams. A very important feature of the ternary representation is the critical tie

line. This is an extended tie line (in the single-phase region) which is tangent to the plait

point, as inFigure 9.The plait point is indicated by the critical composition where

bubble and dew curves converge at the pressure and temperature at which the ternary

phase diagram is calculated.

First Contact Miscible Recovery Processes (FCM)

The simplest and most direct method for achieving miscible displacement is to inject a

solvent that mixes completely with the reservoir oil in all proportions, such that all

mixtures are in single phase. Some examples are: intermediate molecular weight

hydrocarbon C3-C4or mixtures of LPG.

If mixing is solely by dispersion or diffusion, all possible compositions in the mixing zone

between the solvent (A) and the crude (O) plot on a straight line "dilution path" as in

Figure 9. If the dilution path entirely circumvents the two-phase region, the

displacement is first-contact miscible.


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Figure 9 - Example of a First Contact Miscible recovery process (FCM).

Reservoir oil with composition "O" could be diluted with methane up to concentration "A"

and still achieve FCM.

The highest methane concentration that would still achieve FCM is 30%.

Exercise:

If oil has a concentration of 0.8 C7+, 0.1 C1and 0.1 C2-C6, What would be the maximum C1

concentration in the injected solvent to achieve FCM?

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C1

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Figure 10 - Ternary diagram for FCM exercise.

For first contact miscibility to be achieved between solvent and oil, the displacement

pressure must be above the cricondenbar (CB) pressure of all possible combinations

between injected solvent and reservoir oil at the selected temperature. This ensures

that all solvent/oil mixtures above this pressure are single phase.

Problems associated with FCM:

Intermediate molecular weight hydrocarbon solvents for fixed contact FCM may

precipitate some of the asphaltenes from asphaltic crudes. Severe asphaltene

precipitation may reduce permeability and impair well injectivities and productivities. It

may also cause plugging in producing wells.

Pressure and temperature changes and/or the addition of intermediate molecular weight

hydrocarbons or CO2to some reservoir fluids may cause multiple phases to form.

Some of these phases are,

Solid precipitation of asphaltenes and/or waxes (supersaturation achieved due to P,

T, or composition changes).

Two or more liquid phases (i.e., Hydrocarbon-rich, CO2-rich)

Gas-liquid -solid-liquid phases.


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In the past, LPG solvents that are FCM have been too expensive to inject continuously.

Instead solvent was injected in a limited volume, or slug, and the slug was displaced

miscible with a less expensive fluid such as natural gas or flue gas.

SolventSlug

FlueGas

Oil

Figure 11 - Compositional grading.

Ideally with such a process scheme, solvent miscible displaces oil while drive gas

miscible displaces the solvent, propelling the small solvent slug through the reservoir.

Miscibility between solvent and driving gas normally determines the minimum pressure

required for miscible displacement in the FCM slug process with LPG solvents.

As solvent slug travels through the reservoir, mixes with oil at the leading edge and with

the drive gas at the trailing edge.

Quantitative Representation of Phase Compositions

Tie lines join equilibrium conditions of the gas and liquid at a given pressure and

temperature.

Dew point curve gives the gas composition.

Bubble point curve gives the liquid composition.

Hints: B.P. richer in the heavier component (oil).

D.P. richer in the lighter component (solvent).


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All mixtures whose overall composition (zi) is along a tie line have the SAME equilibrium

gas (yi) and liquid composition (xi), but the fractional amounts on a molar basis of gas

and liquid (fvand fl) change linearly (0vapor at B.P., 1liquid at B.P.).

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C10 n-C4

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Figure 12 - A ternary phase diagram illustrating the phase envelope and tie lines.

As the concentration of methane in the injection fluid increases (moving above point A

inFigure 10), the CB increases and will not have FCM. However, dynamic miscibility

can be achieved by multiple-contact-mechanisms (MCM). These are,

(1) condensing-gas drive

(2) or vaporizing gas drive

(3) condensing-vaporizing gas drive (most likely)


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Developed Miscibility

Suppose the reservoir oil and solvent (here taken to be the light component) are onopposite sides of the critical tie line as inFigure 13. The displacement is not first-

contact miscible because the dilution path passes through the two-phase region. The

solvent will develop miscibility with the crude, however, and this may be explained by

the following mixing cell arguments for vaporizing and condensing gas drives.

Figure 13 - 1-Vaporizing and 2-condensing gas drive processes.

Vaporizing Gas Drive

As shown inFigure 14,let the solvent mix with the crude to form mixture M 1which splits

into two phases G1and L1, provided by the tie line. The gas phase G1is less viscous

and it runs ahead of the liquid phase L1

to contact fresh crude. The result of this mixing

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is mixture M2which forms gas G2 and liquid L2. The gas again contacts fresh crude to

form mixture M3and so forth. See animation in the Powerpoint presentation.

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C 1

C2-C 6C 7+

O

M1

M2

M3

M4

G2

G3

G4

G1

Injection Gas

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M4

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G3

G4

G1

Injection Gas

Figure 14 - Vaporizing gas drive miscibility mechanism.

During these multiple contacts, the gas vaporizes intermediate components to such an

extent that it will form new mixtures at different locations (G1, G2, G3, ). These

mixtures approach the plait point which is first-contact miscible with the crude. The

solvent is not first-contact miscible with the crude, but it develops miscibility by

vaporization. This is the vaporizing gas drive process. The pressure required to bring

about such a process is far less than the pressure required for a first-contact miscible

process.

The mechanism for achieving dynamic miscible displacement in a vaporizing gas driveprocess relies on the in-situ vaporization of intermediate molecular weight hydrocarbons

from the reservoir oil into the injected gas to create a miscible transition zone.

Miscibility by this method uses N2, flue gas, or natural gas, provided that the miscibility

pressure is physically attainable in the reservoir.


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pushes the equilibrium gas G1, left after the first contact, further into the reservoir,

where it contacts fresh reservoir oil. Liquid L1, is left behind as a residual saturation.

As a result of this second contact, a new overall composition M2is reached with

equilibrium compositions L2-G2. Further injection causes gas G2to flow ahead andcontact fresh reservoir oil and the process is repeated.

In this manner, the composition of the gas at the displacing front is altered progressively

along the dew point line until it reaches the plait point composition. The critical point

fluid is fully miscible with the reservoir oil.

As long as the reservoir oil composition lies on, or to the right of, the

limiting tie line, miscibility with natural gas that has a composition lying to

the left of the limiting tie line can be achieved by the vaporizing gas drive

process.

Condensing Gas Drive

The condensing or rich gas drive process is the opposite of the vaporizing gas drive.

Now the solvent (rich-gas B inFigure 16)and crude are on opposite sides of the critical

tie line, but reversed from case 1 onFigure 13. The miscibility is again developed, but

through condensation of the intermediate components into the liquid phase. In the first

mixing-cell, the solvent contacts crude to form M1, the liquid L1will subsequently contact

fresh solvent. This multiple contacting will result in a mixture (again near the plait point

inFigure 16)which is first-contact miscible with the crude. The process is called rich

gas because of the intermediates (rich components) added to the solvent (gas).


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Figure 16 - Ternary diagram illustrating gas injection in a condensing gas drive process.

Injection gases with compositions between A & B (seeFigure 16)can still miscible

displace the reservoir oil even though they are not FCM with it. In this case, dynamic

miscibility results from in-situ transfer of intermediate molecular weight hydrocarbons

from the injected gas to the oil.

Assume a gas of composition B is injected to displace the oil inFigure 16. Oil and gas

B are not miscible in all proportions because most of their mixtures fall within the two-

phase region. Suppose mixture M1within the two-phase region results after the first

contact of reservoir oil by gas B (this will be dictated from material balance

computations, Lever rule). Point M1, which will be determined from the amount of gas

injected, will intersect a unique tie line. (Recall dilution lines!)

According to the tie line passing through M1, liquid L1and gas G1are in equilibrium at

this point in the reservoir, as shown inFigure 17. Further injections of gas B pushes the

mobile gas G1, ahead into the reservoir, leaving equilibrium liquid L1, for gas B to

contact.


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Figure 17 - Resulting liquid, L1, and gas, G1, from injecting gas B into the reservoir with

composition O.

By continuing injection of gas B, the composition of liquid at the wellbore is altered

progressively along the bubble curve until the plait point, or critical point, is reached.

The plait (or critical) point fluid is directly miscible with injection gas. By this multiple

contact mechanism, reservoir oil is enriched with intermediate molecular weight

hydrocarbons until it becomes miscible with the injected gas.

This mechanism is called condensing-gas drive process or enriched-gas drive process.

Sufficient gas/oil contacts must occur before the miscible transition zone is developed.

The multiple contacting mechanism creates a transition zoneof contiguously miscible

liquid compositions from reservoir oil composition through compositions L1-L2-L3...

Plait point (P). Likewise, we have gas/oil contacts along G1-G2-G3... plait point. In this

transition zone, two-phase gas liquid-gas flow can occur.


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For dynamic miscibility to be achieved by the condensing gas drive

mechanism with an oil whose composition lies on or to the left of the limiting

tie line (LTL) on a pseudo-ternary plot, the injected gas composition must lieon or to the right of the limiting tie line.

Miscibility Processes Summary

The vaporizing and condensing gas drives are both dynamically developed miscibility

processes. They are fundamentally different with respect to how they perform. For

example, the vaporizing process develops miscibility in the forward contacts or the front

mixing zone, and the condensing gas drive in the rear mixing zone. In practice, these

differences are rather subtle as, indeed, are the differences between first-contact and

developed miscible displacements. Because of this, it is frequently a good

approximation to treat developed miscibility displacements as though they were first-

contact. Such is not the case in an immiscible displacement.

Immiscible DisplacementEither vaporization or condensation can occur in an immiscible displacement though not

to the extent that it develops miscibility. Figure 18 shows an immiscible displacement

on a ternary diagram. Here both the crude and solvent compositions are on the same

side (the two-phase side) of the critical tie line. Now mixing between the crude and

solvent will result in two phases, a liquid phase which mixes with fresh solvent and a

vapor phase which mixes with fresh crude. Vaporization takes place at the forward

contacts at the expense of intermediates in the original crude. This vaporization is

limited, however, by a tie line whose extension passes through the crude composition.This must be true because any two-phase mixture which falls on this tie line will result in

phases with invariant composition when mixed with an arbitrary amount of crude.

Similarly, the intermediate extraction is limited by a limiting tie line in the reverse

contacts.


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Two variables can be adjusted to design a condensing-gas drive process to achieve

miscibility:

Reservoir pressure

Injected gas composition

For a given injection gas composition, there is a minimum pressure called the minimum

miscibility pressure(MMP) above which dynamic miscibility can be achieved.

As the pressure increases, it reduces the size of the two-phase region thus a lower

concentration of intermediate molecular weight hydrocarbons in the injection gas will

achieve miscibility as reservoir pressure increases. Condensing-gas drive miscibility

pressure is below both the CB and CP (or plait point) pressure on a PX diagram.

The requirement that the oil composition must lie to the right of the limiting tie line also

implies that only oils that are undersaturated with respect to methane (C1) can be

displaced miscible by methane or natural gas.

Unfortunately for a great many oils the miscibility pressure with methane/natural gas is

very high for reservoir flooding.

CO2is not FCM with reservoir oils at normally achievable reservoir pressures. But it is

possible to achieve MMP at much lower pressure than using flue gas, N2or a mixture of

LPG. This is a major advantage of the CO2miscible process because dynamicmiscibility can be attained at reasonable pressures on a broad spectrum of reservoirs.

References for this material: Miscible Displacement SPE/AIME - Monograph 8 by Fred

Stalkup (1992).


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Avoid ing Imm iscib le Displacement

An immiscible displacement is to be avoided, if possible. Two ways to do this are to

adjust the reservoir pressure, or to enrich the solvent.

Since the latter entails some expense, it is common to inject the solvent as a slug driven

by a second fluid. This circumstance is shown schematically in a ternary space in

Figure 19. The second or chase fluid (lean gas inFigure 19)is miscible with the solvent

but not with the crude. As the displacement progresses, the front and rear mixing zones

will overlay causing the maximum intermediate concentration to fall below that which

was injected. The curved dilution paths a-c inFigure 19 are showing compositions at

successively larger times (and successively larger dilutions). Clearly, it is desirable toinject just enough of the intermediates to avoid loss of miscibility.


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increased permeability, and

solution gas drive.

Under some circumstances, these immiscible effects can be highly effective.

Solvent Flood ing Exper iments

Slimtube Experiments

Several different types of experiments are common in solvent flooding, but one of the

most important is the slimtube experiment. In a slimtube experiment, a crude-saturatedporous medium is flooded with a solvent in a very long and thin tube. The porous

medium employed is usually unconsolidated and, in many cases, synthetic. This

causes the medium to have quite a large permeability, and the ensuing displacement to

take place at nearly uniform pressure. The packing material is clearly unlike any

realistic porous media. Therefore, the slimtube does not simulate displacement

efficiency. It is designed, through its long length and small diameter, to have an efficient

volumetric sweep efficiency. Most slimtube experiments are water-free. Figure 20

shows the schematic of a MMP experimental apparatus.

The standard technique for determination of minimum miscibility pressure (MMP) is the

slimtube displacement. It is an experiment that is intended to isolate the effects of

phase behavior on displacement efficiency. In the long tube of small diameter, effects

of viscous instability are minimized, and hence phase behavior dominates displacement

performance.

Different investigators have used experimental designs and conditions that vary widely,

and definitions of the MMP also differ. Thus, comparison of results from different

laboratories may include inevitable uncertainty that results from the differences in

technique and interpretation. The uncertainty is larger for heavy oils at highertemperatures.


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Slimtube MMP Apparatus

Figure 20 - MMP experiment apparatus.

If we perform a series of solvent displacements of a particular crude, each at a

successively higher pressure, we might see the behavior illustrated inFigure 21. This

figure plots oil recovery at 1.2 HCPV (hydrocarbon pore volumes) injected versus the

inlet pressure of the experiment. Recovery increases with pressure up to a certain

threshold beyond which further increases in pressure cause little increase in recovery.

The pressure where crude recovery levels out is called the minimum miscibility pressure(MMP).


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%O

ilR

eco

veryat1.2

PVofCO2in

jected

Test Pressure

MMP

%O

ilR

eco

veryat1.2

PVofCO2in

jected

Test Pressure

MMP

Figure 21 - Definition of MMP according to Yellig and Metcaf, 1980.

Minimum Miscibility Pressure

The ternary interpretation of the MMP is straightforward. For pressures lower than the

MMP the displacement is immiscible. As pressure increases, the displacement

becomes less immiscible, and attains completely developed miscibility at the MMP.

Above the MMP there is no character to the recovery-pressure curve because the

distinction between developed and first-contact miscibility is so subtle. Recalling from

Module 3 that the two-phase region shrinks with increasing pressure, the MMP

corresponds to the pressure at which the critical tie line in passes over the crude

composition as sketched inFigure 22.


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PETE 609 - Module 4




Figure 22 - Illustration of the MMP on a ternary diagram.

MMP Correlat ions

Because the minimum miscibility pressure at maximum recovery is an important design

variable, several correlations have been evolved for its prediction. One of the first was

the 1976 correlation for a CO2solvent by HoIm and Josendal shown inFigure 23. Thisfigure plots MMP versus temperature with the C5

+molecular weight of the crude as a

parameter. MMP increases with temperature and C5+molecular weight. The latter

trend is because the solvent has more difficulty vaporizing intermediates when they are

heavy.


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Figure 23 - MMP correlations for CO2 flooding, Holm and Josendal, 1976.

Next we review correlations that are commonly used to estimate MMP's for CO2/crude

oil displacements. Many correlations have been proposed. Those reviewed hereproduce reasonable estimates, but the correlations differ among themselves

substantially, particularly for heavy oils.

Effect of contaminants in injection fluid on MMP.

In many CO2projects, produced CO2is separated from produced oil and gas and

reinjected. Depending on the gas processing facilities used, the recycle CO2may

contain contaminants, and the question inevitably arises whether the contaminants

change the MMP. Some contaminants may increase the MMP (N2, C1), and others may

decrease the MMP (H2S). For more in-depth reading, see Stalkups Miscible

Displacement monograph.

MMP Correlation for CO2 flooding Holm and Josendal

Miscible Displacement SPE Monograph 1992, Stalkup


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Effect of dissolved gas on MMP.

We can also examine the effect of gas dissolved in the crude oil on the development of

miscibility. Calculations using the Peng-Robinson EOS to calculate phase behavior,

indicate that dissolved methane has small effect on development of miscibility because

methane partitions so strongly into the more mobile vapor phase that the methane flows

at the leading edge of the transition zone between injected fluid and original oil. The

injected CO2

then encounters oil that contains no C1.

Extrapolated vapor pressure of CO2as a MMP Correlation

Figure 24 sketches the vapor pressure of CO2and its extrapolated line which is used toestimate the MMP.

The extrapolated vapor pressure is given by

91.1015.273

2015exp7.14][

TpsiaEVP (1)

)32(9

5][ FCT (2)

For example, for T=200 F the CO2extrapolated vapor pressure is,

psiapsiaEVP 29.293,391.10

)32200(9

515.273

2015exp7.14][

(3)


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PETE 609 - Module 4




Nc

iiwKF

2

(4)

where

F = normalized partition coefficient

wi = weight fraction of carbon number i (normalized to remove methane)

i = carbon number

Nc = number of components in the mixture

2 - Calculate

761.00418.0)log( iKi (5)

3 - Calculate density of CO2at the MMP.

1.467F,189.1542.0 FMM P (6)

1.467F,42.0 MM P (7)

4 - Use EOS or density tables to find pressure at which MMPCO 2 at a given

temperature. (Note: for the homework you can use the CO2data fromFigure 3).

Glaso's MMP Correlation for Condensing Gas Drives

This correlation may provide quite different MMPs from Orr and Silva. We will see this

in an exercise. This correlation requires to evaluate the following parameters,


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PETE 609 - Module 4




588.6

846.0

7

622.2

C

y (8)

Variables:

MW = molecular weight of C2-C6

in the injection gas

z = mole percent methane in injection gas.

T = temperature F

Tey

zyy

zy)10127.1(

)185.0745.46(41.256329

34]MWfor[psigMMP

703.18.319258.512

(9)

Tey

zyy

zy)107.1(

)273.0913.80(238.195503

44]MWfor[psigMMP

058.1567.1373.39

(10)

Tey

zyy

zy

)1092.4(

)214.0515.73(703.257437

54]MWfor[psigMMP

109.1706.2152.514

(11)


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PETE 609 - Module 4




Estimation of Lean Gas or Nitrogen MMP

Stalkup provides several figures as a correlation for MMP's for condensing gas drives

seeFigure 25 toFigure 28.

Condensing gas drive miscibility pressure correlation

Miscible Displacement SPE Monograph 1992,Stalkup

Figure 25 - Condensing gas drive miscibility pressure correlation at T=100 F.


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Figure 26 - Condensing gas drive miscibility pressure correlation at T=150 F.

Condensing gas drive miscibility pressure correlation

(Miscible Displacement SPE Monograph 1992, Stalkup)


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Figure 29 - Location of natural CO2deposits in US. (Stalkup, 1992)

Firoozabadi and Aziz (SPERE, Nov. 1986, 575-582)

Firoozabadi and Aziz give an MMP correlation that can be used either for nitrogen or

lean gas.

X2-5= mole percent intermediates, ethane through pentane.

2

25.0523

25.0523

77

101430101889433

TM

x

TM

xMMP

CC

(12)

Location of Natural CO2Deposits (Stalkup, 1992)


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PETE 609 - Module 4




MC7+= molecular of heptane plus fraction. Firoozabadi and Aziz suggest that the

correlation is intended primarily for lean gas. It may be less accurate for nitrogen

systems.

MMP goes down with increasing temperature for oils with significant intermediate

fractions probably because the volatility of intermediates increases with the

temperature. Heavier oils (with higher nitrogen) MMP's show a slight increase with

temperature.

MMP goes down as the mole fraction of methane in the oil goes up. Methane behaves

like an intermediate component and is vaporized effectively by nitrogen.

For light oils at high temperature, MMP's for nitrogen, methane and carbon dioxide are

similar.

Solvent Displacement Mechanisms

Mixing

Solvent displacements are subject to mixing between the crude and the solvent. One

mixing is due to fractional flow effects, and another by dispersion, which is caused bymolecular diffusion and local fluctuations in the velocity field of a porous medium.

Dispersion, therefore, is large when these fluctuations are large (local heterogeneity is

large). Diffusion is negligible under typical conditions. The same is true of capillary

pressure except, perhaps, for an immiscible displacement far removed from the critical

point.

Segregation

Other effects exist which will cause the solvent to separate from the crude. One such

effect is gravity. A solvent injected into a reservoir will tend to be lighter than the crude

and much lighter than the connate water. If the gravity number and vertical

communication is good, this will cause the solvent to flow to the top of the reservoir.


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Figure 30 - Gravity effect when displacing fluid density is lower than displaced oil.

Such tonguing (Module 2) will leave the bottom portion of the reservoir unswept with a

corresponding loss of recovery. Gravity segregation can also occur when water is

injected simultaneously with the solvent. In this case, solvent flows to the top and water

to the bottom of the reservoir.

The importance of gravity varies from reservoir to reservoir. However, another type of

segregation phenomenonviscous fingeringis so pervasive that we devote a

separate section to it.


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Dissipation in Miscible Displacements

Viscous Fingering

You may be surprised about the discrepancy between the very high recoveries quoted

above in developed miscible floods and the about 15-20% recoveries actually observed

in field floods. Probably the single most important source of this discrepancy is viscous

fingering. We give only the barest details of this interesting phenomenon and a few

quantitative relations.

Actual fingering is quite chaotic as is shown inFigure 31,a reproduction of fingering

patterns in a scaled model experiment. Once the solvent breaks through to the

producers (this instant is shown in the right sketch ofFigure 31), large-scale bypassing

begins and successively more solvent cycling must be done to recover a given amount

of crude. It is clearly an effect to be avoided or at least anticipated.

Figure 31 - Viscous fingering.

Though the phenomenon is chaotic, it is not random and its most important features can

be illustrated with simple models.


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Flow regimes in miscible displacement depend upon mobility ratios and dimensionless

groups characterizing the ratio of viscous to gravity forces (Rv/g)

RatioMobilityM

o

sM

fluiddisplacedmobility

fluiddisplacingmobility (13)

oo

ss

k

kM

/

/ (14)

when mobile water is present,

o

w

Sw

w

o

o

Sw

w

s

s

kk

kk

M

(15)

The ratio of viscous to gravity forces is,

o

v/ g

o s

u LR

k ( ) h

(16)

In oil field units,


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o

v/ g

u [ B / D ft ] [ cp ] L [ ft ] R

k [ md ] [ g / cm ] h [ ft ]

2

3

2050

(17)

The sweepout efficiency at breakthrough has been correlated with the viscous gravity

force ration Rv/gand the mobility M, as indicated inFigure 32.

40

20

60

80

100

0

1 10 100 1,000 10,000 100,000

II III IV

I

III IV

II

I

M=1.35

M=6.5

M=27

Viscous Gravity Force Ratio Rv/gBreakthroughSweepoutEfficiency

,%

Figure 32 - Flow regimes in a 2-D, uniform linear system - Schematic. From Stalkup,1983.

Figure 32 illustrates conceptually the different flow regimes observed in a vertical cross-

sectional laboratory model packed with glass beads (Crane et al, 1963.) At very low


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PETE 609 - Module 4




values of Rv/g, the displacement is characterized by a single gravity tongue overriding

the oil. At higher values of Rv/g, vertical sweepout becomes independent of the

particular value of Rv/guntil a critical value is exceeded (seeFigure 33,Regions I and

II.) Beyond this critical value, a transition region is encountered (seeFigure 34,Region

III), where secondary fingers from beneath the main gravity tongue develop. Finally, at

even higher values of Rv/g, the displacement is entirely dominated by multiple fingering

in the cross section, and vertical sweepout again becomes independent of the particular

value of Rv/g, (seeFigure 35,Region IV.)

Figure 33 - Flow regimes for miscible displacements in a vertical cross section: Regions

I and II.


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Figure 34 - Flow regimes for miscible displacements in a vertical cross section: Region

III.

Figure 35 - Flow regimes for miscible displacements in a vertical cross section: Region

IV.

For a five-spot flow, Darcy's velocity (line driven flow) is calculated as


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hL

iu

251. (18)

where, iis the injection rate B/D per well

Examples

Problem

Calculate the viscous/gravity ratio for vaporizing-gas drive flooding in a reservoir that has

not been waterflooded previously. Assume these data: 40-acre five-spot pattern, i=2,000B/D

(gas injection at reservoir conditions), o=0.4 cp, k=75 md, h=35 ft, L=933 ft, =0.4

g/cm3, M=25, and kv/kh=1.

Solution

2/0766.0

)933(35

)000,2(25.1ftDBu

and

56)35)(4.0(75

)933)(4.0)(0766.0(050,2/ gvR

Therefore, fromFigure 32,flow is dominated by gravity tonguing.

Problem 2

Calculate the viscous/gravity ratio for CO2flooding in a reservoir that has not been

waterflooded previously. Assume these data: 40-acre five-spot pattern, i=500B/D (CO2


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injection at reservoir conditions), o=1.9 cp, k=4 md, h=25 ft, L=933 ft, =0.1 g/cm3,

M=25, and kv/kH=1.

Solution 2

2/0268.0

)933(25

)500(25.1ftDBu

and

734,9)25)(1.0(4

)933)(9.1)(0268.0(050,2/ gvR

Therefore, fromFigure 32,flow is dominated by viscous fingering.

Model ing Viscous Finger ing

The displacement of oil by FCM in homogeneous porous media is simple, when the

solvent oil M 1 and when gravity segregation does not influence the displacement by

segregating the 2 fluids. In those cases, the oil is displaced efficiently ahead of the

solvent, and the solvent does not penetrate into the oil except as dictated by dispersion.

The displacement front is stable, and a mixed zone develops and grows.

For M > 1, the solvent front becomes unstable, and numerous fingers of solvent develop

and penetrate into the oil in an irregular fashion. These cause a much inefficient oil

recovery. The problems that arise are: Earlier solvent breakthrough; and, poor oil

recovery.


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Xf+

Xf

L

Figure 37 - Simplified model of frontal instability (Collins, 1961).

Velocities of undisturbed front and disturbed front:

ffs

f

xxLM

Pk

dt

dx

)( (19)

where

s

oM

)()(

)(

ffs

f

xxLM

Pk

dt

xd (20)

It follows,


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Kovals Model for Finger Growth

Koval developed mathematical treatment analogous to Buckley-Leverett model for

immiscible displacement.

For linear flow,

S

StS

S dS

df

A

q

dt

dxS

S

(24)

Assume linear volume blending,

HS

Sf

oeff

Seff

S

SS

1)1(1

1

(25)

Eoeff

Seff

(26)

44/1

22.078.0

s

oE (27)


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PETE 609 - Module 4




ityheterogenerockH (28)

Characterization of Rock Heterogeneity H

Figure 38 - Characterization of rock heterogeneity

Solution of Equations(24) and(25) for pore volumes of solvent injected at solvent

breakthrough provide,

EHV BTPD

1: (29)

Characterization of Rock Heterogeneity H

50

60

70

80

90

100

1 2 3 4 5 6

H

%

RecoveryatoneVpinMatched

ViscosityFlood


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PD:BTV pore volumes solvent in jec ted

at solv ent breakthrough

The oil recovery after breakthrough,

/

PDi PDi

PV

EHV V N

EH

1 22 1

1

(30)

Where VPDiis the number of pore volumes injected, and NPVis the oil recovery as a

fraction of the pore volume.

The solvent fractional flow in the effluent is given by

/

PDi

Se

EHEH

Vf

EH

1 2

1

(31)

Total length of fingered region,

m

x

xK

Kl

m

1 (32)

EHK (33)

xm= mean displacement distance calculated as if the displacement had been piston

like.


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For radial flow,

KKrr m

1 (34)

0

20

40

60

80

100

120

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80

VPDi

%R

ecovery

Experimental

Predicted

Parameter = o/s

5

86

375

150

0

20

40

60

80

100

120

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80

VPDi

%R

ecovery

Experimental

Predicted

Parameter = o/s

5

86

375

150

Figure 39 - Comparison of Blackwell's experimental data with predictions based on K-

factor method; Stalkup, 1983.

Suggested additional reading: Chapter 3 of Miscible Displacement, monograph by

Stalkup, 1993.


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Simu lat ion of Miscib le Flood Perform ance

The design of a field-scale miscible flood is very likely to include some sort of simulation

effort as part of the assessment of project economics. In this section, we review the

types of reservoir simulators currently used to make such predictions.

Two types of simulators are widely used: miscible and compositional.

Miscible Simulators

These ignore the compositional behavior of the fluids. The implicit assumption in such

simulators is that the development of miscibility takes place over such a short fraction of

the reservoir length that the details of that development need not be represented.

Miscible simulators include an empirical representation of the effects of viscous

instability. The scheme used is related to Koval's model. Some versions of these

simulators include a limited representation of compositional behavior when the pressure

falls below the MMP.

Compositional Simulators

Compositional simulators use an EOS to calculate how individual components (C1,

C2) and pseudocomponents present in the solvent and the oil partition between

whatever phases are present. Fluid properties of phases are calculated from their

compositions, pressure and temperature in each grid block.

Compositional simulators use an equation of state in conjunction with a finite difference

solution of the flow equations. Thus, the details of component partitioning are

represented. Compositional simulators do not represent the effects of viscous fingering,

and numerical dispersion may cause problems. Numerical dispersion arises from

truncation error in the representation of the flow terms in the differential equations. It

can cause difficulties in compositional simulations because it alters the composition

path of the displacement. Since the composition path strongly influences displacement

performance in multicontact miscible floods, control of numerical dispersion may be

important.


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Simulators differ appreciably in the assumptions made about how the fluids mix within a

grid block and flow to adjacent blocks.

Miscib le Flood Simulators

Three-Component Todd-Longstaff Models

Treat the injected solvent as first-contact miscible with the oil. They use an empirical

model of the effects of viscous instability to determine the "effective" fluid properties.

Components are solvent, oil, and water (three-component version) or solvent, gas, oil,

and water (four-component version).

Solvent is treated as miscible with the oil. There is no representation of phase behavior.

(Miscibility develops over a length that is short compared to the displacement length.)

Calculate effective viscosities and densities for two pseudophases, oil and solvent.

The effective oil viscosity is,

mooe1

(35)

The effective solvent viscosity is,

msse

1

(36)

where the mixture viscosity is defined as,


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PETE 609 - Module 4




4

4141

11

oS

o

ooS

S

s

mSS

S

SS

S//

(37)

limits:

mixingNo0 sseooe , (38)

mixingComplete1 msemoe , (39)

These viscosities are then used in standard fractional flow expressions.

e

oS

S

oe

o

se

S

se

S

TLS

M

SS

S

SS

S

f

(40)

where

So = oil saturation

SS = solvent saturation

fsTL = Fractional flow of solvent as predicted by Todd-Longstaff

Me = Effective mobility ratio

Choosing the Value of the Mixing Parameter

Calculated results are quite sensitive to the value of . In one-dimensional flow,

breakthrough occurs at


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11

o

s

e

D

M

t (41)

and the displacement is complete at

1

s

oeD Mt

(42)

Hence, the length of the transition zone depends quite strongly on the value of .

The Todd-Longstaff model reproduces Koval's fractional flow expression if is chosen

to be

s

o

s

o

log

..log

/ 41

2207804

1 (43)

Both the Todd-Longstaff and Koval models predict that transition zones grow linearlywith displacement length. That prediction is roughly in accord with experimental

observations of fingering behavior.

Todd and Longstaff (JPT, 1972) recommended:


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PETE 609 - Module 4




32scaleLaboratory /

31scaleField /

Most simulations reported in recent years have used values between 1/2 and 2/3.

For field-scale simulations some investigators have attempted to determine an

appropriate value of by matching pilot performance.

Four-Component Todd-Longstaff Models

The components are solvent, gas, oil, and water.

Limited compositional capability. Pseudo K-values can be defined for pressures below

the minimum miscibility pressure for the solvent, gas, and oil components

Choose

Ks = Ks(P)

Ko = 0

Kg = Kg (P).

Using those K- values, the oil stays in the oil phase, the gas and solvent can dissolve in

undersaturated oil, and any excess gas phase is a mixture of initial equilibrium gas and

injected solvent.

SimpIe mixing rules can be used to determine oil phase properties:

3

1b ppwhen

i

o iio px (44)


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Todd-Longstaff Calculations for WAG Injection

Calculations for tertiary miscible displacement (no WAG) are typically performed with slightly less than 2/3.

WAG injection should partially stabilize the flood (depending on M, WAG ratio, gravity

effects, injection rates, )

It is not obvious how to select the appropriate value for 2D and 3D floods, however.

In one dimension, the optimum WAG ratio is of order 1. In terms of solvent utilization, it

is better to err on the side of more water injection. The selection of the optimum WAG

ratio is clearly a question to be investigated by simulation. Unfortunately, the fact that

the appropriate value of depends on WAG ratio makes the determination of the

optimum uncertain.

Gravity Segregation in Todd-Longstaff Models

Gravity segregation of the injected fluid is frequently quite important in field-scale

displacements.

The fractional flow equation for flow in the vertical direction is

o

oev

seoevTLs

verts S

q

gkff

)(1 (52)

when kvis relatively large or when qv is small, these fractional flow assumptions

generate considerable gravity segregation.


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Todd and Longstaff chose the values of oeand seto be consistent with the definitions

of the effective viscosities.

sooooooe SS )(1 (53)

sosoosse SS )(1 (54)

where the values of Sooand Sosare calculated from the quarter power viscosity blending

rule.

1

4/1

4/14/1

s

o

oe

o

s

o

ooS (55)

1

4/1

4/14/1

s

o

se

o

s

o

osS (56)

Todd and Longstaff attempted to model the effects of viscous fingering by modifying the

fractional flow expression.


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PETE 609 - Module 4




Their approach modifies the convective part of the differential equation. Hence, the

transition zone grows linearly with the distance traveled. The modifications simply affect

the rate of growth.

In 2D experiments, at least, given solvent concentrations are observed to move at

approximately constant velocity. Hence an appropriate fractional flow model is a

reasonable representation of the average flow behavior.

Other fractional flow models

Koval's model assumes that a solvent/oil mixture with fixed composition displaces pure

oil.

Fayers' model builds a fractional flow expression from a physical picture of a finger that

grows in width from its tip to its tail.

Koval's model gives reasonable agreement with Blackwell's 2D recovery curves, but it

does not produce physically realistic total fluid mobilities.

Fayers' model and the Todd-Longstaff formulation give reasonable mobility predictions.

The representation of gravity effects in Fayers' model has better physical justification

than does the Todd-Longstaff model.

Fayers Model of Viscous Fingering

Assume that solvent fractional flow can be related to a finger width function : Where

krse= , and kroe= 1 -

sbSa (57)


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PETE 609 - Module 4




where the growth exponent a is given by

40420 .. M (58)

The effective viscosity in the finger is taken to be

4

4141

1

//

o

s

s

sse

SS (59)

The fractional flow expression is then

o

se

ss

Sf

)(1

(60)

Effective densities are assumed to be

osssse

ooe

SS

)1( (61)

The resulting total mobility expression is

ose

t

1 (62)


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PETE 609 - Module 4




Formulation of the Multiphase Multicomponent

Reservoir Simulation Equations

Let us derive a generalized formulation for the reservoir simulation equations for

multicomponent multiphase systems as follows.

The more general case is to consider the pore space filled with gas (g), oil (o), and

water (w), and that a component j exists in these three phases.

Recall the following definitions

Oil Saturation

wgo

oo

VVV

VS

(63)

Gas Saturation

wgo

g

gVVV

VS

(64)

Water Saturation

wgo

ww

VVV

VS

(65)


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PETE 609 - Module 4




Pore space volume

wgof VVVV

(66)

An obvious constraint is

1wgo SSS (67)

Porosity is the ratio of pore volume and bulk volume

b

wgo

V

VVV

(68)

Darcys Law

x

Pkv

x

Pkv

x

Pkv

w

w

wwx

g

g

g

gx

o

o

oox

Water

Gas

Oil

(69)


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PETE 609 - Module 4




the velocity of phase o/w/galong xcoordinate is proportional to the pressure

gradient along the xcoordinate, the permeability to the phase k (o/w/g), and

inversely proportional to the viscosity of phase (o/w/g)

Mass Balance for Component j

Consider a control volume

zyxVb

(70)

jofcumulation AcofRateMassjofRateMass-jofRateMass OUTIN

Lets define the concentration of component j in phase i as a mass fraction

cN

j

i j

i j

i j , Njo,g,wim

m

C c 1and

1

(71)


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PETE 609 - Module 4




Note that mass fractions are not the same as a mole fraction.

Applying the conservation equation for component j in the control volume we have:

wjwwgjggojoo

xxwjwwxgjggxojoox

xwjwwxgjggxojoox

CVCVCVt

CvCvCv

CvCvCvzy

(72)

Dimensional Analysis:

timemasslength

lengthmass

timeCV

tRHS

time

mass

length

mass

time

lengthlengthlengthCyvxLHS

ojoo

ojoox

3

3

3

1

The oil, gas, and water volumes can be expressed using the following relations


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PETE 609 - Module 4




zyxSVSVSV obofoo (73)

zyxSVSVSV gbgfgg (74)

zyxSVSVSV wbwfww (75)

Replacing into the conservation equation

wjwwgjggojoo

xxwjwwxgjggxojoox

xwjwwxgjggxojoox

CSCSCS

t

zyx

CvCvCv

CvCvCvzy

(76)

Simplifying terms and letting 0x

wjwwgjggojoo

wjwwxgjggxojoox

CSCSCSt

CvCvCvx

(77)

Next substitute the velocities using Darcys expression


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PETE 609 - Module 4




wjwwgjggojoo

wjw

w

w

w

gjg

g

g

g

ojo

o

o

o

CSCSCSt

Cx

PkC

x

PkC

x

Pk

x

(78)

At this point we must determine the number of independent variables in the system. For

Nc components this is

Unknowns Number

Cij 3Nc

Pi 3

Si 3

i 3

i 3

ki 3

Total 3N+15

In order to solve this system uniquely we must have 3N+15 INDEPENDET

RELATIONSHIPS

Relationships can be algebraic or differential. These come from various sources,

1. Differential equations

2. Phase Equilibria Relations (EOS, correlations)

3. PVT Data/Correlations

4. Relative Permeability Data /Correlations)

5. Conservation Principles


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6. Capillary Pressure Data/Correlations

Letsdevelop the necessary correlations

#Eq Count Description Relationship

Nc Nc

One mass conservation

equation for every

componentConservation equation(78)

1 Nc+ 1 Fluid phase saturations must

always sum to one1 wgo SSS

3 Nc+ 4 Sum of mass fractions in

each phase must add to one c

N

j

ij N1,jgw,o,iCc

11

3 Nc+ 7 Phase densities can be

obtained from data, EOS, or

from correlations.cN1,jgw,o,ithwi

),,(

iijiii TCPf

3 Nc+ 10 Phase viscosities can be

obtained from data,

EOS+constitutive equations,

or from correlations.

cN1,jgw,o,ithwi

),,(

iijiii TCPg

3 Nc+ 13 Relative permeability can be

obtained from data or from

correlations.cN1,jgw,o,i with

),(

iiii TShk

2 Nc+ 15 Capillary pressure can be

obtained from data of from

correlationsgw,o,i)T,S(FPPP

gw,o,i)T,S(FPPP

iiiow,cwo

iiigo,cog

2Nc 3Nc+15 Phase equilibria relations

governing the distribution of

a component among phases

is obtained from correlations

or from EOS.

cj,gw

wj

gj

cj,go

oj

gj

N1,j)K(fC

C

N1,j)K(fC

C

2

1


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We therefore have 3Nc+15 independent relations and 3Nc+15 unknowns which can be

used to solve the system.

In practice several simplifying assumptions can be made to make the problem more

amenable to solution these are the following

Simplifications

1. Capillary pressure between oil and gas is generally neglected

2. The mass fraction of hydrocarbon components in water is usually very small

and can be neglected

3. Components are usually lumped together into hypothetical components which

must be PROPERLY CHARACTERIZED.

Sources & Sinks

The basic equation derived for the linear compositional model did not include sources or

sinks. These can be simply included as.

ojoo

ii

i ji

i

i ji

i

i

i CSt

xqCx

Pk

x

3

1

3

1

3

1

)( (79)

Where

iq Mass injection of phase i in suitable units

ij Mass fraction of component j in ith phase


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)x( Dirac delta function which is defined as

)x( 1Production or injection in cell at x

)x( 0No Production or injection in cell at x

The solution of the compositional reservoir system is by far the most difficult problem in

reservoir simulation.

Solution Technique

The equations shown are a large nonlinear set of equations that can be solved by

Newton-Raphson iterations the algorithm used in a sequential calculation is

1. Calculate new pressures from overall material balance equations using

compositions and saturations from the previous time step.

2. Calculate flow in and out of each grid block using new pressures and old

compositions and saturations. Obtain new overall composition in each grid

block.

3. Perform a flash calculation to obtain new phase compositions, saturations

densities and viscosities.

Some compositional simulators perform all three steps at once.


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Dissipat ion in Solvent Flood ing

The most important dissipation mechanisms in miscible displacements are by

dispersion

viscous fingering

In this section we will discuss the effects of dispersion on a miscible front in a one-

dimensional, homogeneous permeable medium. Dispersion is the mixing of two

miscible fluids caused by diffusion, local velocity gradients (as between a pore wall and

the center of the pore), and mechanical mixing in the pore bodies.

Recall the definition of dimensionless variables in Module 2.

D

xx

L

(80)

t t

D

po o

udt qdt t

L V

(81)

Where Vpis the pore volume

Modeling Dispersion

Assumptions

Isothermal miscible displacement

Incompressible rock and fluid

1-D

Homogeneous

Single phase


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Under these assumptions the convection-diffusion equation describes conservation of

displacingcomponent 'i' with mass concentration Ci.

i i iC C Cu -t x x

2

20

(82)

where K is the longitudinal dispersion coefficient.

Using the dimensionless Variables,

ti i I

Di D D

iJ iI

C C x udt C ; x ; t

C C L L

u = superf ic ia l veloci ty

u = in ters tit ial velo city

0=

(83)

Where the subscripts indicate

I= Initial concentration

J= Injection concentration

Equation(82) in dimensionless form becomes.

Di Di Di

D D e D

C C C-

t x Np x

2

2

10

(84)


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This is a second order partial differential equation (PDE) that requires one initial

condition (I.C.) and two boundary conditions (B.C.) to be solved.

I.C.Di D D

C ( x , ) x 0 0 0

(85)

B.C.1Di D D D C ( x ,t ) t 0 0 (86)

B.C.2Di D D D

C ( x ,t ) t * 1 0 (87)

*Original B.C. (XD= 0) changed as an artifact to build an analytical solution which is

approximate and valid for large tDor Npe.

The Peclet number is dimensionless and is defined as

euL convective transport NpK dispers ive transport

=

(88)

goal find ( , t ) Di D D

C x

The resulting analytical approximation valid for large tD or large Npclarge distances

from inlet boundary.

To solve Equation(84),use same techniques as for the heat transfer equation (Carslaw

& Jaeger, 1959).

Define a moving coordinate system


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PETE 609 - Module 4




LetD D D

x ' x t (moving coordinate)

The differential of Cican be expressed in terms of two different coordinate systems.

D D

Di Di

Di D D

D Dt x

C CdC dx dt

x t

(89)

or

D D

Di Di

Di D D

D Dt x '

C CdC dx ' dt

x ' t

(90)

Equation(89) = Equation(90) and after substituting

D D Ddx ' dx - dt (91)

D D

D D D

Di Di

D

D Dt t

Di Di D

D

D D Dx x ' t

C C- dx +

x ' x

C C C- + dt = 0

t t x '

(92)


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Since Equation(92) = 0 D Ddx , dt the brackets are = 0

This results in the following equalities

D D

Di Di

D Dt t

C C=

x x '

(93)

D D D

Di Di Di

D D Dx x ' t

C C C= -

t t x '

(94)

Substitute Equation(93) and Equation(94) into the Diffusion-Convection equation

D D D

Di Di Di

D D e Dx t t

C C Ct x Np x

2

21 0 (95)

D D D D

Di Di Di Di

D D D e D x ' t t t

C C C C

t x ' x ' Np x '

2

2

10 (96)

D D

Di Di

D e Dx ' t

C C

t Np x '

2

2

10 (97)


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PETE 609 - Module 4




Equation(97) looks like the heat conduction equation whose solution may be obtained

by the method of combination of variables. To do this define another dimensionless

variable

D

D

e

t= x '

Np

2 (98)

Redefine Equation(97) in terms of (exercise) and redefine B.C. and initial condition.

The goal was to transform the PDE into an ODE (ordinary differential equation). This

transformation is sometimes called the Boltzmanns transformation.

You should end up with the following equation

Di Di dC d C + = 0d d

2

22

(99)

Di

Di

C ( ) =C ( ) = 1

0 (collapsed B.C.)

Equation(99) is integrated twice to give

u

Di1 2C 1- e du 2

error funct ion

2

0

(100)


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PETE 609 - Module 4


Class Notes for PET

Documents

Module 4 - Solvent Flooding