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Miscible Flooding : Introduction

Displacement of crude oil with drive agents such as water or natural gas leaves behind an immiscibly-trapped oilsaturation, which is a large fraction of the initial oil saturation. This is the case even when the amount of drive agentis equal to a very large number of pore volumes. It would seem advisable, therefore, to use a displacing agent that ismiscible with the crude oil. By "miscible," we mean that there is only a single non-aqueous phase present when any

proportions of the crude oil and displacing agent are mixed together and allowed to come to equilibrium. The phrase"miscible in all proportions" is often used to describe this condition.

While the above definition of "miscible" is generally true, there is a distinction between a displacing agent that isimmediately miscible with the crude oil and one that develops miscibility after a series of equilibrium contact stages.In the first case, the agent isfirst-contact miscible with the crude oil. In the second case, the agent is multiple-contact, or conditionally miscible. Another phrase used to describe this second case is developed miscibility.

Miscible Flooding Processes

There should clearly be an advantage in using a displacing agent that is either first-contact miscible or multiple-contact miscible with the crude oil. Under secondary recovery conditions (i.e., at the original or connate watersaturation), it should be possible to displace all of the crude oil, since there is no trapped or residual oil saturation.

(This is almost, but not completely, true). However the pore volume that was occupied by crude oil will then beoccupied by the displacing agent. For such a displacement to be economical, it is necessary either to recover thisdisplacing agent or to use an agent that is much less valuable than crude oil.

This constraint on the displacement presents a problem. Few displacing agents are both miscible with and worth lessthan crude oil. Natural gas, flue gas, and even nitrogen separated from air in an air liquefaction plant are likely to becheaper than crude oil on an "equal reservoir volume" basis. (If air itself is injected, in situ combustion occurs ratherthan miscible displacement.) Until about 1970, LPG (propane and butane) was also much cheaper than crude oil, butit has since become worth almost as much as the crude oils that it might be used to displace.

Carbon dioxide is also cheaper than crude oil when it is recovered from natural deposits or from gas mixtures richerin carbon dioxide than flue gas. These carbon dioxide-rich gas mixtures are found, for instance, with natural gas(e.g., in the gas fields of the Delaware/Val Verde Basin in west Texas), with hydrogen in the effluents from steamreforming of natural gas and from the combined partial oxidation and steam reforming of natural gas to produceammonia synthesis gas. These carbon dioxide-rich gas mixtures are usually processed with amine-type solvents,which dissolve the carbon dioxide under pressure in an absorption tower, leaving the valuable natural gas, hydrogenor ammonia synthesis gas in the exiting gas phase. The carbon dioxide is removed from the solvent in a secondtower by heating at a lower, usually near-atmospheric, pressure. Most of the cost of the removed carbon dioxide isfor dehydration and compression from near atmospheric pressure to a pressure sufficient to inject in a reservoir, or totransport it to a reservoir and then inject it.

Carbon dioxide is also found in large natural deposits (like natural gas), often containing small amounts of methane.Levorsen (1954) believes that these deposits result from an intrusion of hot magmatic rock into sedimentarycarbonate rocks, such that the temperature of the carbonate rock rises sufficiently to calcine it-that is, to convert it tocalcium oxide, which dissolves in the magmatic rock-and free carbon dioxide. The carbon dioxide then rises until itencounters a trap containing pore space. Such carbon dioxide deposits exist in northeastern New Mexico, in southcentral Colorado, in southwestern Colorado, in eastern Utah, in southwestern Wyoming and in north centralMississippi. Carbon dioxide at injection pressures can be obtained more cheaply from these natural sources thanfrom the carbon dioxide scrubbing processes discussed above, because much less compression cost is involved.

A large fraction of an injected miscible agent can be recovered from a reservoir by waterflooding after the crude oilhas been displaced. However, since we could have recovered about the same fraction of the original oil in place by awaterflood, the economic gain by using the miscible drive agent is equal to the difference in value between thewaterflood residual oil and the same residual saturation of the cheaper displacing agent. It is common practice inmost miscible floods, therefore, to follow the miscible drive by a water drive, and to evaluate the results in terms of


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the value of the extra oil recovered (beyond the waterflood-recoverable oil) minus the cost of performing themiscible flood prior to the waterflood. This cost must include the cost of the maximum amount of the displacingagent which must be purchased, minus any credit for the displacing agent that is produced back from the reservoir.

In most cases, a waterflood will already be in progress by the time a miscible flood is considered. In fact, thereservoir may be approaching the economic limit of water cut (currently considered to be about 98%). When the

waterflood has progressed beyond the point where water has broken through at production wells, a subsequentmiscible flood is considered to be a tertiary recovery process. In this case, the miscible drive agent must immisciblydisplace water in order to contact the crude oil. Much of this oil will be trapped and immovable, extending throughseveral pores but cut off at the pore throats by water films bridging the pore throats. Therefore, the amount ofdisplacing agent required is considerably greater than the volume of crude oil remaining in the pores.

As a rough guide, it is generally necessary to displace at least half as much water as the residual crude oil saturationto remobilize all of the oil in the water-swept regions of the reservoir. It is then possible, with a final waterflood, todrive out the excess miscible displacing agent, down to the nonaqueous phase residual saturation. Again, the processmust be debited with the maximum amount of the miscible agent that must be used, and credited with the value ofthe excess miscible agent driven out by the final waterflood.

Both secondary and tertiary miscible drive processes are therefore judged on the same basis: the value of the extra

oil recovered over that which could be obtained by waterflooding, and the cost in terms of the gross miscible agentused less credit for excess miscible agent produced. Of course, the miscible process also entails extra operatingcosts.

Furthermore, all known miscible displacing agents that are cheaper than crude oil are also much less viscous andusually less dense than crude oil. This means that these agents will have a mobility ratio much greater than one, andviscous fingering of the miscible agent through the oil can be expected, as well as gravity tonguing (override) due tothe lower density. The same is true during the immiscible displacement of water in excess of the connate watersaturation during a tertiary miscible flood. The sweep efficiency at any given throughput will therefore be less thanthat of a waterflood. Also, miscible drive agents are generally more expensive than water, so is not possible toovercome the poor sweep efficiency just by injecting large number of pore volumes (as can usually be done in awaterflood when the sweep efficiency is poor because of unfavorable mobility ratio). Generally, the amount ofmiscible agent will be limited to considerably less than a pore volume, and must itself be driven through the

reservoir by a following waterflood. Despite these drawbacks, miscible floods have been found to be economicallyadvantageous. There are two main variations of miscible drives: horizontal drives and gravity-stabilized vertical(downward) drives. The process efficiency is strongly affected by whether or not the drive is assisted by gravity.The results of these projects, in terms of the percentage of original oil-in-place (OOIP) recovered, range from 3-15%greater than water-flooding for horizontal drives to 15-25% for gravity-assisted drives.

There are a variety of miscible recovery processes, which are characterized by the type of miscible agent used, andalso, in the case of multiple-contact miscibility, by the procedure through which miscibility is attained. The mainmechanismsworking in a miscible process are extraction, solubilization, vaporization, condensation and dissolution.These processes trigger other mechanisms such as oil swelling, viscosity reduction and solution gas drive.


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Miscible Flooding with LPG

Natural gas is first-contact miscible with LPG, and LPG is first-contact miscible with most crude oils under mostreservoir temperatures at pressures of 600 to 1,700 psia. This suggests a process in which a slug of LPG is injectedand then natural gas is injected to displace as much crude oil and LPG as possible before the producing gas/oil ratiorises to an economic limit.

We can expect recovery of both oil and LPG to be incomplete, since the LPG fingers through crude oil and thenatural gas fingers through the LPG and then through the crude oil. In a homogeneous, single-layer reservoir, asimple estimation using the five-spot correlation I developed in 1972 gives 52% oil recovery for an 18% (basishydrocarbon-filled pore volume) LPG slug followed by dry gas. In a heterogeneous, multi-layer reservoir, thisrecovery would be reduced by a factor of 1/8 to 1/4, depending on the degree of heterogeneity, because of gravitysegregation in each layer and varying slug sizes in different layers. Some further reduction would result from areasoutside of well patterns, near the edges of the reservoir, not being significantly swept.

Miscible Flooding with Natural Gas

At sufficiently high pressure, lean natural gas (natural gas containing less than 10% of components other thanmethane), flue gas, or pure nitrogen becomes multiple-contact miscible with most crude oils. The pressure required

depends upon the temperature and the nature of the crude oil. At temperatures in the 100-130 F range, and withcrude oils of 35-100 API gravity, the miscibility pressure is about 4,000-5,000 psia. At elevated temperatures (200-

250 F) the pressure is about 7,000-8,000 psia. For lower API gravity crude oils, the pressure rises by about 500 psiafor each 5 degrees lower API gravity, and vice versa.

The cost of natural gas has historically been lower in the U.S., relative to its fuel value, than the price of crude oil,and still lower based on the cost of a barrel of gas at reservoir conditions compared to the value of the oil that itdisplaces. Regulation by the Federal Power Commission of the price of natural gas which was transported ininterstate commerce began in 1954. Due to the initial decision which placed interstate gas under federal control, andsubsequent FPC rulings (particularly the "Phillips Case" in 1960, and the "Permian Basin" ruling in 1965), producersof natural gas have until recently only been able to charge a price for natural gas which was based on its generallylow production cost, not on its competitive fuel value in terms of heat of combustion per standard cubic foot relative

to the cost of the same amount of heat when derived from crude oil. The latter ratio would require about the sameprice for 5.5 thousand standard cubic feet of natural gas as for a barrel of crude oil. When crude oil was $3 perbarrel, this would have called for a price of 3.00/5.5 or about 55 cents per thousand standard cubic feet (mscf).Instead, the price was set by law at levels ranging from 15 to 20 cents at the wellhead.

The volume of the gas at reservoir conditions can be estimated roughly by the ratio of the pressure to atmosphericpressure. Thus, at 4,500 psi relative to 15 psi (atmospheric pressure) the volume is about 300 times smaller, so that1,000 cubic feet of gas at atmospheric pressure would occupy about 3 1/2 cubic feet at reservoir pressure. In mostcases, about 1.5 to 2.0 mscf would equal the volume of one barrel of crude oil.

It is evident, therefore, that natural gas qualifies as a miscible drive agent which is cheaper than crude oil and cantherefore be left in place of crude oil in a reservoir.

It is possible to obtain multiple-contact miscibility at a pressure that is lower than that required for first-contactmiscibility. The maximum pressure that can be applied in the process is generally limited by the fracturing pressureof the reservoir. This fracture gradient is seldom more than 0.7 psi per foot of depth and is sometimes as low as 0.5psi/ft. For example, the maximum injection pressure at 5,000 ft vertical depth will be from 2,500 to 3,500 psia.

The pressure required for miscibility depends on the reservoir temperature. The temperature of the reservoir depends

on the natural rate of rise of temperature with depth from the surface temperature, which averages about 70 F. Ifthis geothermal gradient (as this rate of rise is called) is as low as 0.6 F per 100 feet of depth, as it is in some partsof west Texas, then the reservoir temperature at 5,000 feet depth is only 100 F. Such a relatively low temperature


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(for the reservoir pressure which can be employed) is advantageous; it is easier to obtain miscibility with any given

crude oil. Typical geothermal gradients are 1.02.5 F per 100 feet (1.84.60 C per 100 m).

The small amounts of ethane and propane normally present in natural gas that has been processed to remove LPGmake the miscibility pressure slightly lower than that for pure methane; high pressure miscible gas drives do, in fact,use processed ("dry" or "lean") natural gas rather than pure methane, which would be hard to obtain.

The pressure not far from an injection well is considerably lower than the injection well bottom-hole pressure, whichis limited by the fracturing pressure. Therefore, it is usually necessary to take advantage of the lower pressurerequirement of the conditionally-miscible or multiple-contact miscible process. In this process, the equilibriumcontact of the lean gas with the first crude oil it encounters does not lead immediately to miscibility; but it doesproduce a gas containing an increased proportion of the low molecular weight hydrocarbons normally present in thecrude oil. This addition of ethane, propane, butane and pentanes to the methane brings the gas closer to miscibilitywith the crude oil. As this enriched gas moves away from the injection well, it acquires, by a series of such contacts,a sufficiently high content of these intermediate hydrocarbons as to be immediately miscible with the next crude oilwhich it encounters. This progressive enrichment of the gas as it moves away from the well is considered avaporization process, and so this high pressure gas miscible drive process is called a vaporizing miscible drive.

Both nitrogen (available from the atmosphere) and flue gas (which can be obtained by burning a hydrocarbon fuel to

obtain about ten times as much flue gas volume as the hydrocarbon fuel in a gaseous state) can be used instead oflean natural gas, often at an economic or logistic advantage.

Miscible Flooding with Enriched Gas

It is possible to achieve multiple-contact miscible displacement at much lower pressures than that required for thevaporizing gas drive process by using natural gas to which a moderately high proportion of intermediatehydrocarbons (LPG and natural gasoline) has been added. These intermediates are transferred from the injected gasto the crude oil that it first encounters, and by a series of such transfers (the stripped gas moving on out into thereservoir), the crude oil near the well is so enriched in the intermediates that it is immediately miscible with the nextincrement of injected gas. Since the injected gas must be externally enriched in intermediate hydrocarbon content,this is often called an enriched gas drive process. Since the process of transfer of intermediates from gas to oil maybe considered a condensation process, it is also called a condensing gas drive.

Despite the apparently clear distinction between the vaporizing and the condensing gas drive processes, it waslearned that, in quite a few cases, the miscible displacement mechanism is not just one or the other of theseprocesses; rather it is a combination of both. Zick (1986) first described this process, while Stalkup (1987) andNovosad and Costain (1989) further developed the subject. In a triangular diagram, where a pseudo-critical point orplait point is normally shown on a two-phase envelope having vapor phase on one side of the plait point and liquidphase on the other side, and which is often used to explain either vaporizing or condensing gas drive mechanisms,instead of the plait point, a neck appears and the two-phase envelope then widens into a new region in which bothphases are dense super-critical fluids.


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Miscible Flooding with Carbon Dioxide

Carbon dioxide flooding is currently a popular form of multiple-contact miscible flooding. It has somecharacteristics which are significantly different from those of the hydrocarbon miscible solvents, and others whichare generally similar. The principal differences are:

1. It is significantly soluble in reservoir brine or injected flood water, whereas hydrocarbons are not (however,it is not miscible with water under any conditions where water is a liquid phase).

2. At typical reservoir conditions, it has a density near that of reservoir crude oils, while the density of thelight hydrocarbon miscible drive agents is considerably lower. In some cases, carbon dioxide is slightlymore dense than the crude oil; however, it is always considerably less dense than liquid water. Hence, thereis much less gravity segregation of carbon dioxide than there is of hydrocarbon solvents from crude oil,though it still occurs relative to a mobile water phase.

3. At any given pressure, more carbon dioxide than methane will dissolve in crude oil. The swelling of crudeoil due to dissolved carbon dioxide is about the same as the swelling due to the same ratio of methane,hence, since more dissolves at a given pressure, it swells crude oil more at a given pressure than doesmethane.

4. At any given pressure, it reduces the viscosity of crude oil more than does the amount of methane whichwill dissolve at the same pressure. A consequence of its solubility in water is that mobile carbon dioxidecan transfer through water phase to dissolve in and swell waterflood-trapped oil ganglia.


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Phase Behavior in Multiple-Contact Miscible Processes

Phase behaviorrefers to the existence of one or more phases at given conditions of temperature, pressure andcomposition, and the relationships between the phase compositions.

Equilibrium Phase Compositions

It is common engineering practice to deal with processes where components present in two or more phases (solid,liquid or gas) change from one equilibrium set of phase compositions through a series of other such compositionsuntil they attain a desired degree of transfer from one phase to another. Usually, these phase composition changesresult from changes in temperature, pressure, or the total composition of all of the phases present in a volume regionsmall enough that the phase compositions throughout the region, as well as the total composition, are constant. Thisuniformity in composition must be brought about by some mixing process.

In chemical plants and refineries, this uniformity can be attained by powerful mixing devices in pipelines or vessels.Even more common is the use of an elongated, horizontal or vertical, vessel containing mixing devices such asdistillation trays or zones filled with loose packing material. Different phases, such as liquid and gas, or two onlypartially miscible liquids, are often injected into these vessels at different locations so that the phases move inopposite directions through the vessel. The desired result is that the exiting phase compositions are very different

from the entering phase compositions.

While the exchange of components between the phases as they pass through the system may be continuous(especially in the case of the packed vessel or tower), the exchange process can still be accurately described as beingequivalent to some number of stages of equilibrium component exchange, or alternatively as a larger series of stepsthat partially approach equilibrium compositions (e.g., in each plate space of a distillation tower). In the componentexchange which may take place between an injected fluid and in-place fluids in porous, permeable rock surroundingan injection well in an oil reservoir, the mixing effects are due to molecular diffusion and, to a much larger extent, tothe variations in flow path lengths and velocities of the fluid traveling through interconnected pores and to thedividing and rejoining of these flow paths. The latter mixing effects are proportional to the velocity of flow and arestronger in the direction of flow than in directions transverse to the flow direction (Perkins and Johnston, 1963).This mixing is similar to molecular diffusion but is distinguished from it by calling it dispersion instead of diffusion.

Because of these mixing effects, exchange of components between an injected stream and the in-place fluids takesplace, and these exchanges can be interpreted as comprising a series of equilibrium steps. In such a series, after agiven equilibrium is reached, either one or both of the separate phases move into other regions, where they comeinto contact (and mix) with another, different portion of the other phase. After a series of such sequential contacts,one of the phases may approach a composition which is miscible with subsequent portions of the other phase.

This desired result can be studied in terms of the series of equilibrium steps that can bring it about. It is helpful toconsider how equilibrium compositions are determined, based on the pressure, temperature and overall compositionin a given mixed region, and how the desired result of miscibility can be brought about for crude oil and possiblesolvents.

Gibbs Phase Rule

In general, phase behavior is governed by Gibbs Phase Rule (see Findlay, 1951), which states that the number ofdegrees of freedom (i.e., the number of system conditions such as temperature, pressure and composition which canbe altered independently of each other) is given by:

F=C-P+2 (2.1)where

F=the number of degrees of freedom


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C=the number of components (different chemical compounds or uncombined chemical elements)

P=the number of phases present in the system (a phase being defined only relative to another phase, suchthat between the phases there exists a surface which completely separates one phase from the other with asurface tension force consequently existing in that surface).

The system is defined as a region in which the conditions of temperature, pressure and composition are uniform, andequilibrium is defined as the condition in which the properties of the system and the phase distribution ofcomponents within the system do not change as time progresses, and will return to the same state if an independentvariable (such as temperature) is altered slightly and then returned to the initial value.

This rule is very helpful when the system has only a few components. For example, in the case of steam inequilibrium with liquid water, there are two phases but only one component, so there is only one degree of freedom.Thus, if the pressure is changed, the temperature adjusts automatically, and vice versa. The tables of saturated steamproperties thus have only one independent variable.

However, whenever crude oil is involved in a phase equilibrium, there are so many different chemical compoundspresent in the crude oil that the degrees of freedom may seem to be almost infinite. Nevertheless, even in this case,the distribution of each chemical component between such phases still follows definite physical and chemical rules.

In general, the behavior can be specified in terms of the ratio of the concentration (mole fraction) in a given phaserelative to that in another phase; these values are called K values.

Flash Calculations From K Values

If K values are given for the components of a mixture, then the composition of the liquid and vapor phases can berigorously determined using an equilibrium flash calculation (named for the rapid or "flash" vaporization of gasfrom a liquid phase when pressure is lowered). For more than two components, though, this procedure requires atrial-and-error calculation involving the variation of the liquid or the vapor mole fraction until the correct value isfound.

The behavior of these vapor/liquid K values as temperature and pressure vary has been studied extensively forcompounds present in crude oils. The results are presented in the Gas Processors Suppliers Association (GPSA)Engineering Data Book(1972) as a series of K-value charts for each of the hydrocarbons from methane through iso-and normal pentane, and for the mixed hexanes through decanes. This publication is presently the most authoritativecompilation of these data.

Calculations From Equations of State

As an alternative, the equilibrium K values can be calculated by means of an equation of state, plus constantsappropriate to each of the chemical components present in any given mixture (see Reid, Prausenitz and Poling,1987).

The simplest of such equations is the ideal gas law, PV = nRT, where n is the number of moles present (thus, PV =RT when there is only one mole present). In that law, there is only one constant (R), in addition to the state variables

of system volume (V), system temperature (T), and system pressure (P). R is called the universal gas constant, andvalues are given for different sets of units in handbooks such as theCRC Handbook of Chemistry and Physics.

Unfortunately, real compounds and elements do not obey the ideal gas law at temperatures and pressures very far

from typical ambient temperatures (e.g., 15-38 C) and atmospheric pressure. It is therefore necessary to makesuitable corrections.

For a single component, it is necessary only to add an empirical adjustment factor Z which varies with temperatureand pressure, so that the non-ideal gas law PV= ZRT gives an accurate description of the behavior (per mole) of the


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given component as the pressure and temperature vary, with one set of Z values applying to the component in thegaseous state and another set of Z values for the liquid state. This need for an empirical correction factor Z has madeit evident that a more elaborate gas law is needed, and efforts have been made to supply more correct gas laws.

The first moderately successful elaboration of the ideal gas law was made in 1873 by van der Waals. More recent,more successful ones are those of Redlich and Kwong (especially in the version as modified by Soave), and Peng

and Robinson. These are described in detail in physical chemistry texts and in Reid, Prausnitz and Poling (1987).The original articles in which these improved equations of state were presented by their authors are reprinted in SPEReprint Series Book No. 15, Phase Behavior (1981).

In most cases, the engineer will use an equation-of-state computer program to calculate phase equilibria, or will usea flash equilibrium computer program which requires K values to be supplied (e.g., by taking them from the GPSAEngineering Data Book) In either case, it is helpful if the engineer knows what is taking place. For this reason, somesimple examples of flash calculations and of equation-of-state calculations are given in the Appendix.


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Phase Equilibrium Diagrams

Equilibrium compositions may be calculated for a given mixture at a variety of pressures and temperatures, or, if asingle temperature is more appropriate (such as a fixed reservoir temperature), for a variety of pressures. This willgive direct calculations of bubble-point pressure, and of phase proportions and compositions at pressures below thebubble point. It is of interest in some cases to calculate the behavior (versus pressure at a reservoir temperature) of a

series of mixtures of a crude oil with solvent at increasing mole fractions (x) of solvent. This is called a p-x diagram.

If, at a given reservoir temperature and pressure, the phase behavior of various mixtures of solvent with a givencrude oil is calculated in a region of mixtures in which a vapor and a liquid phase result, then these pairs of vaporand liquid compositions may be plotted on a triangular equilibrium phase diagram, and the liquid and vaporcompositions trace out a curve which contains all of the two-phase compositions in that diagram. Such a diagram ishelpful in explaining the behavior of the multiple-contact miscible flooding processes mentioned previously.

An equilateral triangle is usually used to construct the diagram, though it is also possible-and sometimesadvantageous-to use a right triangle. The discussion below assumes an equilateral triangle.

The key assumption of triangular equilibrium diagrams is that a given mixture of components is composed of onlythree pure components, or that the mixture behaves as if it were composed of three components. (The latter may be

called pseudocomponents.) It is, in fact, often possible for hydrocarbon systems containing a light (low viscosity)crude oil, LPG and natural gas to be adequately characterized as composed of just three pseudocomponents. Thefirst component is methane, the second component is LPG, and the third component is C7+(i.e., heptanes and allhigher mole weight components lumped together). The natural gas is considered to be composed of methane plus aslight amount of LPG, and the crude oil is considered to be composed of C7+, LPG and methane. The mixturereferred to as "LPG" in this case is actually a mixture of C2-to-C6hydrocarbons; this is a somewhat broader carbonnumber range than we usually include in the term LPG. The C2-to-C6 range is commonly termed "intermediatehydrocarbons," but we shall use the term "LPG" to designate this wider range.

It is normal practice in the ternary representation to place C 1at the top vertex, C7+ at the lower left vertex, and C2-C6(or LPG) at the lower right vertex, as shown inFigure 1.

Figure 1
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Each vertex represents the point of 1.0 mole fraction, or 100 mole percent, of the component named at that vertex,and the opposite side is 0.0 mole fraction, or zero mole percent, of that component.

Any point within the triangle represents a mixture of the three components, and the amount of each component is

determined by the distance of the point from each of the three sides. The sum of these distances is equal to thedistance of any vertex from its opposite side.

That is, the sum of the three mole fractions always equals 1.0 (the sum of the mole percentages equals 100).Furthermore, with regard to any three points in a straight line within the diagram (including a point or points on theboundaries), the intermediate point represents a mixture of the compositions represented by the two points at theextremities. Such a set of points represents a material balance, and the amounts present are determined by the lever-arm principle. That is, if the intermediate point is the fulcrum, the amount at one end times its distance from thefulcrum is equal to the same product for the other end.

The diagram usually contains at least one two-phase region, and may contain more than one such region and/or athree-phase region. The two-phase region is bounded by a "two-phase envelope," and contains tie lines that end onthe envelope at each end. These two endpoints represent the compositions of the two phases which co-exist at

equilibrium as a result of de-mixing of any mixture on the tie-line (always a straight line) joining the two ends. Theproportions of the two phases which will result from a given mixture point on the tie-line are determined by thelever arm principle stated above.

There are an infinite number of tie-lines present in the two-phase region, just as there are an infinite number ofpoints on the two-phase envelope. Where the twophase envelope terminates at one side of the diagram, a tie-lineexists which lies in that side of the diagram. As the tie-lines diverge from this limiting tie-line on the side of thediagram, their slope usually changes, and they usually get shorter when the two-phase envelope is convex toward theopposite vertex as shown. The tie-lines may eventually get so short that they converge into a point, which is calledthe plait point or pseudocritical point. The tangent to the curve at the plait point is called the critical tie-line, and isan important determinant with regard to attainment of multiple-contact miscibility, as will be shown.

The area of the two-phase region varies with temperature and pressure. In reservoir engineering applications, the

temperature is normally determined by the depth and the geothermal gradient, such that there is a reservoirtemperature which does not vary much from the top to the bottom of oil-bearing formations of moderate thickness.In this case, the two-phase area is primarily determined by pressure, which does vary as a consequence ofproduction operations in the reservoir. The two-phase area decreases with increasing pressure. It is often possible, byraising pressure, to attain miscibility when it is impossible at a lower pressure. The area lying outside the two-phaseenvelope is a region where only a single phase exists. If a displacing fluid comprised of methane and LPG (andtherefore represented by a point on the right side of the triangle) can be connected with a point representing thecomposition of reservoir crude oil by a straight line which does not cross the two-phase region, then the displacingfluid is first-contact miscible with the crude oil. This is illustrated byFigure 2andFigure 3.
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Figure 3

Figure 2


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The variation in size of the two-phase region with change in pressure is illustrated by the sequence inFigure 4,

Figure 4

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Figure 5

andFigure 6,

Figure 6

with the pressure decreasing fromFigure 4 toFigure 6.

As can be seen, the relative size of the two-phase region grows as the pressure falls. Eventually the two-phase regionbulges out of the right side of the triangle, and the result is seen inFigure 6. If the pressure were loweredconsiderably more, the two-phase region would also bulge out of the bottom of the triangle. Then there would be leftthree one-phase regions close to the apexes of the triangle. This last occurrence is at such a low pressure (nearatmospheric) that we are not concerned about this case in miscible drive processes. There is a case where a neckdevelops instead of a plait point; this is discussed later. It is a case of incomplete miscibility, but high oil recovery ispossible.

If the reservoir pressure were high enough that the two-phase region disappeared out of the left side of the triangle,then all of the components shown would be miscible in all proportions, and any injected fluid, including pure

methane, would be first-contact miscible even with a crude oil entirely lacking in C2-C6intermediate hydrocarbons,i.e., just C7+. This situation is not a real case, however. The usual case is one similar toFigure 5.
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Phase Behavior in Hydrocarbon Miscible Floods

High Pressure or Lean Gas (Vaporizing) Drive

In the past the relative value of the components has generally been: methane lowest, C2-C6next, and C7+ highest. It istherefore desirable to use a drive agent containing as little LPG and as much methane as possible to attain

miscibility. The case of almost pure methane (natural gas from which as much LPG has been removed as iseconomically possible) is that which we call the high pressure gas drive, or lean gas (vaporizing) drive. But thepossibility of using this lowest cost drive agent depends on the composition of the crude oil (as a point in thetriangular diagram) relative to the envelope of the two-phase region. If the line connecting the crude oil compositionand the drive gas composition lies outside the two-phase region, then we have a first-contact miscible high pressuregas drive situation (highly desirable). Note that for this to happen, the crude oil must contain a rather high content ofboth methane and LPG, not very much C7+.While this does happen, it is not very common. A more common case isshown inFigure 1, where the line between the crude oil point and the drive gas (nearly pure methane) passesthrough the two-phase envelope.

Figure 1

InFigure 1, however, note that the crude oil composition still lies to the right of the critical tie line. This is mostimportant, and if it is not the case, as shown inFigure 2, then multiple-contact miscibility is not attainable.
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Figure 2

It is of course possible to move the critical tie line by changing the pressure level in the reservoir, but it is notpossible to change the location of the crude oil composition (other than to reduce its methane content by pressure

depletion, which is usually unfavorable from this standpoint). If the pressure can be raised sufficiently, either by gasinjection or by water injection or both, a high pressure miscible flood might become possible even if it had not beenat the time when miscible flooding was first contemplated. Otherwise, consideration must be given to the possibilityof enriched gas as the miscible displacing agent.

If nitrogen or flue gas is the high-pressure miscible displacing agent, either of these would take the place of methaneas the lowest molecular weight component. A complication arises when methane is present as a major component inthe crude oil. In this case, a three-dimensional phase diagram may be needed (see Stalkup 1984) with nitrogen (orflue gas) at one apex of a tetrahedron, methane at a second apex, and LPG and C 7+ at the other two apexes. The two-phase region becomes a mound within the tetrahedron, and every tie-line has ends at specific points, which togethercomprise a curve on the surface of the mound. There is still a critical tie line which is tangent to the surface of themound at the pseudo-critical point. To achieve multiple-contact miscibility, it is still necessary that a line from thecrude oil composition be able to reach a gas-phase composition, which can be achieved by multiple contacts of the

injected gas with the crude oil. Such a gas phase composition will therefore be a point on the tie-line curve on thesurface of the mound, and will lie on the gas-phase side of the pseudocritical point. If a straight line connecting thecrude oil composition with such a point is not possible, then a high pressure miscible gas drive is not possible withthe given system, but if it is possible, then such a drive is possible.

The reason for this difference is as follows: InFigure 1, a mixture of drive gas and crude oil falling in the two-phaseregion (for example at the point marked "x" on the tie-line nearest to the pseudocritical point Pc) will separate into aliquid phase lying at the lower end of the tie line and a gas phase with the composition of the upper end of the tieline. In this case, note that the dotted line connecting the crude oil composition with this gas phase composition lies


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entirely in the single-phase region, i.e., the first (immiscible) contact of the crude oil and drive gas forms a gas phasewhich is much richer in LPG (and incidentally in C7+ ) so that this gas is miscible with crude oil on ensuing contacts.This illustrative case, in which only one immiscible contact was sufficient to produce a gas phase miscible withcrude oil, is not really representative. It normally requires quite a few stages of immiscible contact before the linebetween the richest gas so produced and the crude oil composition grazes the two-phase envelope. Thus, the laststage required is where the gas phase is at the point where a tangent line from the crude oil composition touches thetwo-phase envelope. This point must lie above the pseudocritical point, since all points above are gas phases and allpoints below are liquid phases. The tangent to the two-phase envelope at a point above the pseudocritical point musthave a slope such that it crosses the critical tie-line at a point above the pseudocritical point. Therefore, all possiblecompositions for crude oil lying on the tangent line must lie to the right of the critical tie line extended downward.

We might ask, why can the crude oil composition not lie to the left of the critical tie line if it is also above thepseudocritical point? The answer is that, in that case, we would call it a gas instead of a crude oil. We may define agas in the single-phase region to be a composition which lies above the pseudocritical point and to the left of thecritical tie line. The region to the right of the critical tie line is ambiguous. At most of the pressures of interest, twophases may exist here as dense supercritical fluids, but this occurs only when there are more than three componentspresent, and when some of them are of relatively high molecular weight.

Enriched Gas (Condensing) Drive

Now, what can be done if the crude oil composition lies to the left of the critical tie line ( Figure 2)? In this case wemust simply use a drive fluid composition that lies on the other side of the critical tie line. This is shown in Figures 3and 4.

InFigure 3, the gas has not been sufficiently enriched in LPG to meet this criterion, and so all mixtures of the drivegas and crude oil remain immiscible.

Figure 3


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The first contact cannot produce a liquid richer in LPG than the lower end of the tie line, where the line connectingcrude oil with the drive gas meets the upper part of the two-phase envelope. The contact of drive gas with thatenriched liquid cannot produce any mixtures lying to the right of the tie line which, when extended, passes throughthe drive gas composition (the second dotted line in the diagram).

InFigure 4,

Figure 4

however, the first contact of the enriched gas with crude oil, forming the mixture marked "x" on the first tie line tothe left of the pseudocritical point, separates into a gas (at the top end of the tie line) that contains less LPG than thedrive gas and a liquid (at the lower end of the line) which contains more LPG and methane than the crude oil. Notethat the tangent to the two-phase envelope from the enriched gas drive composition touches the envelope at thelower end of this tie line. The enriched drive gas is miscible with that liquid phase, since the dotted line betweenthem lies entirely in the single-phase region.

In most cases, many more stages of contact would be required to reach a liquid composition that would be misciblewith the drive gas (for example, suppose that the enriched drive gas were only slightly to the right of the point wherethe critical tie line intersects the right side of the triangle). Nevertheless, so long as the drive gas and crude oilcompositions lie on opposite sides of the critical tie line, multiple-contact miscibility can be achieved.

Notice that we have only two degrees of freedom in adjusting conditions to make this possible: pressure and drivegas composition. We cannot readily (or economically) change the reservoir temperature from its initial value just forthe purpose of attaining miscibility; hence, we may consider the temperature a constant. The crude oil composition


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is likewise fixed at whatever composition exists when the miscible flood is being considered. Note, however, that ifthe reservoir has undergone pressure depletion, the pressure may vary over different regions of the reservoir, and sothe dissolved gas content and thus the composition may vary (in the terms we have been considering) on thetriangular diagram. This introduces complications which we shall not consider further here; they are best taken careof by reservoir simulation, using a compositional simulator in which the initial composition may be varied asrequired over the area and depth of the reservoir.

The LPG Slug Process

There is, of course, one more case which has been mentioned-that of using an LPG slug to drive crude oil and thenusing dry gas (nearly pure methane) to

drive the LPG. This case is shown inFigure 5, where the LPG is shown as containing a slight amount of methaneand C7+ in order to move the point from the apex into the single-phase region close to the apex.

Figure 5

This process allows the lowest pressure of all of the processes discussed, while still maintaining miscibility.

The minimum pressure that may be allowed in the LPG slug process is that at which the two-phase region begins tobulge out of the right side of the triangle. This means that the dry drive gas is then immiscible with the LPG. Thishappens at a higher pressure than that at which LPG and crude oil become immiscible. Hence, the critical conditionfor attaining miscibility in this method is to maintain a pressure high enough for the drive gas and LPG to remainmiscible.


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In Chapter 5 of Miscible Displacement (SPE Monograph No. 8), Stalkup (1984a) gives plots from which thispressure may be determined. Stalkup also describes circumstances where the two-phase region bulges out of thebottom of the triangle before it bulges out of the right side; this occurs primarily when the reservoir temperature isnear or above the critical temperature of the LPG. The principal hazard involved in the LPG slug process (in regardto miscibility) occurs during the process rather than at the beginning. Diffusion and dispersion cause mixing at thefront of the LPG slug between the crude oil and the LPG; on the triangular diagram ( Figure 5), these mixtures areon the line joining the LPG and the crude oil. At the back of the slug, diffusion and dispersion create mixtures alongthe line joining the LPG and the drive gas compositions. When these dispersion zones overlap, mixtures are createdthat lie between these two lines-and that is, in the two-phase region. When the peak LPG concentration falls belowthe peak of the two-phase envelope, miscibility will be lost.

Viscous fingering contributes strongly to this last-mentioned effect. LPG fingers through the crude oil, and the drygas follows the LPG fingers, usually occupying the middle. The result is that crude oil, LPG and gas are flowing inparallel rather than in sequence. An LPG slug size calculated on the basis of dispersion rates to be adequate to avoidloss of immiscibility when the fluids travel in sequence may be inadequate when they travel in parallel. Thisdepends on the magnitude of the transverse dispersion rate relative to the longitudinal dispersion rate used in thesequential calculation, and also on the thickness of the sheath of LPG between the gas fingers and the crude oil.

In areal terms, as determined by study of such viscous fingering in a Hele-Shaw (parallel-plate) model, it appears

that the band of "LPG" surrounding fingers of "gas" is about one-fifth as thick as the annular band would be iffingers did not occur. If dispersion is in the low range where molecular diffusion is the controlling factor in bothlongitudinal and transverse mixing, then the slug would need to be approximately five times as large as would berequired if fingering did not occur. In three dimensions, this translates to 5 3/2, or about eleven times as large.However, if both longitudinal and transverse dispersion are in the velocity-controlled region, where transversedispersion is about one-tenth as great as longitudinal dispersion, then the hazard is not loss of miscibility in thetransverse direction but in the longitudinal direction. In this case, the slug is thinned by fingering just as much (if notmore) at the tips of the fingers as at the sides. If the displacement becomes immiscible at the tips of the fingers, thenpenetration of crude oil by immiscible fingers of gas occurs. This immiscible displacement of oil is much lessefficient (on the microscopic scale) than the miscible displacement, and it may be expected that much more oilwould be left behind in the paths of the fingers than would be the case if immiscible gas fingering did not occur

The Combined Vaporizing/Condensing Process

At the 1986 SPE Annual Meeting, Aaron Zick of ARCO presented a paper (SPE 15493; see the References) which

showed that in experimental contacts of moderately heavy crude oils (at temperatures from 160-205 F andpressures from 3,100-3,600 psia), with solvent compositions representing enriched gas such that a condensingmultiple-contact miscibility mechanism was expected (with pressures 500 psi or more above the multiple-contactmiscibility pressure as found by slim-tube tests), miscibility did not actually occur. The gas and liquid phasesproduced by multiple contact approached each other in composition, but then held constant or diverged; the Kvalues approached 1.0, but the values for the lightest components did not get as low as 1.0, and those of the heaviercomponents rose close to, but not as high as, 1.0. The densities of the two phases became close but not equal.Simulations with the Peng-Robinson equation of state using only one C7+ pseudocomponent did not show thisbehavior but rather showed the typical condensing mechanism with miscibility attainment. On the other hand, moreextended analyses of the crude oils were available, and when three or more pseudocomponents were used with thehighest one being a C30+ pseudocomponent, the P-R equation of state was able to match the experimental behavior.

The triangular diagram calculated in the latter case is compared with that of a true three-component (C1, C4, C10)condensing multiple-contact miscibility system inFigure 6andFigure 7(sketches similar to figures in Zicks paper).
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Figure 6

Figure 7


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Later papers (Stalkup, 1987; Novosad and Costain, 1989) verify Zicks conclusions, both with respect to the

experimental behavior and to the need for several pseudocomponents, ranging up to a relatively high molecularweight pseudo-component, in order for an equation of state to match the experimental behavior.

Stalkup and other authors have pointed out that the simple three-component representation, which is so convenientfor describing the miscibility mechanisms as discussed above, is not an adequate representation for crude oils. Only

in the case of a light crude oil practically a condensate-is this representation valid. For real crude oils, the nearapproach to miscibility permits a high oil recovery, butin essentially every case, a small amount of high molecular weight oil is left behind by the displacement process. Ineffect, this is always shown by the slim-tube tests commonly used to determine "miscibility" pressure. The oilrecovery is never 100%, either at breakthrough or at a throughput of 1.0 or 1.2 pore volumes of solvent (thesethroughputs are recommended by different authors). The oil left behind in the tube is cleaned out using an aromaticsolvent such as benzene or toluene; these are good asphalt solvents. If the solvent is evaporated, the remaining oil isfound to be of much higher molecular weight, density and viscosity than the crude oil, and it contains more asphalticcomponents. This kind of heavy oil residue is predicted by the vaporizing/condensing process. Even propane solventwill leave such a heavy oil precipitate. The amount of this heavy oil residue is somewhat greater with CO2(3%-8%)than it is with propane or LPG-enriched natural gas (2%-4%). This is in accord with what is known about therelative precipitation behavior of these solvents from their behavior in lubricating oil refining processes and catcracker feed preparation by solvent precipitation of heavy ends. This does not preclude high oil recoveries bysolvent displacements in oil reservoirs, but it means that a small residual saturation of heavy, asphaltic oil willremain in almost every case. We still use the phrase "miscible flooding" for this process, as a matter of convenience.

In all of the processes discussed, if a waterflood has taken place before the miscible flood is carried out, then animmiscible displacement of water by the miscible drive agent must occur in order to mobilize all of the oil, at leastpart of which will exist in a water-trapped saturation.


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Phase Behavior in Carbon Dioxide Miscible Floods

The solubility of CO2in water is determined by the local CO2partial pressure at any given temperature and by thewater salinity (solubility decreasing with increasing salinity and with increasing temperature, but increasing withincreasing pressure). The CO2partial pressure at equilibrium is equal to the mole fraction of carbon dioxide in thenon-aqueous phase times the pressure in the that phase. If a phase consisting of 100 mole percent carbon dioxide is

introduced, at a given pressure, into a system containing rock, brine and a waterflood-trapped oil phase, then beforethe CO2phase becomes diluted by dissolved oil or hydrocarbon gas, that pressure is the CO 2 partial pressure, and anaccording amount of carbon dioxide dissolves in the water phase to arrive at equilibrium with the pure carbondioxide. Any other non-aqueous phase which is present (though not in direct contact with the CO2) must arrive atequilibrium with this water phase, and can do so only by arriving at 100% carbon dioxide content.

Of course, this means that if oil is anywhere present, it will continue to acquire carbon dioxide from the water phaseuntil its mole fraction of carbon dioxide matches that of any other non-aqueous phase within mass-transfer distance.The swelling effect would make the trapped oil phase become at least partly mobilized, and thus would dilute themobile carbon dioxide phase with oil. Equilibrium could only be achieved when the non-aqueous phase has thesame content of carbon dioxide and of oil, and the CO2content in the water is the appropriate solubility for thepressure and carbon dioxide mole fraction in the non-aqueous phase.

The phase behavior of pure carbon dioxide resembles that of ethane. Its equilibrium K value in mixtures withhydrocarbons, from methane to the higher molecular weight components of crude oil, is intermediate between the Kvalues of methane and ethane; the GSPAEngineering Data Bookrecommends using the square root of the productof the values for methane and ethane as the K value for carbon dioxide.

In mixtures with crude oil at temperatures up to about 130O F; CO2swells the oil while raising the bubble-pointpressure until a pseudocritical point is reached. The results are often plotted on a p-x (pressure/mole fraction ofsolvent) diagram. If further carbon dioxide is added beyond the pseudocritical point, the system then exhibits a smallamount (by volume) of hydrocarbon liquid dew phase. Both phases contain a high mole fraction of carbon dioxide atthis point, but the upper phase has a higher mole fraction of carbon dioxide than the lower dew phase. The molecularweight of the hydrocarbons in the dew phase is considerably higher than that of the hydrocarbons in the upper phase.If the pressure on the system is lowered, a bubble of gas phase appears, which is lighter than the upper phasepreviously present (so it was also a liquid phase). Further reduction in pressure leads to division of the middle phase

between the gas phase and the heavier liquid phase until the middle phase disappears, leaving only a gas phase and aliquid phase. In addition to the three phases mentioned, a tiny amount of asphaltene particles may be precipitatedfrom solution as a semi-solid material. If the experiment described is performed in a gauge glass, the asphalteneparticles make black specks on the glass.

At temperatures above 130 F, when a dew phase appears, the upper phase is a vapor phase rather than a secondliquid phase, and on lowering pressure below the two-phase boundary the amount of liquid (dew) phase firstincreases, then at considerably lower pressure decreases again (similar to retrograde condensation).

Due either to the liquid dew phase, or to the asphaltenes, or to both in varying degrees, the mobility of carbondioxide in an oil reservoir is much lower than would be expected from its viscosity. This has been observed in bothsecondary miscible floods (Pontious and Tham, 1978) and in tertiary miscible floods (Hansen, 1977; Youngren andCharlson, 1980).

Gardner et al. (1981) indicate that a lower relative permeability curve, which applies in a partly oil -wet rock in thepresence of both crude oil and carbon dioxide (as well as brine), is responsible for this lower mobility of the carbondioxide. Brannan and Whittington (1977) attribute lowered injectivity for water following enriched gas injection atthe Levelland field to similar relative permeability effects (in their case, to three-phase relative permeability effects).

According to Stalkup (Chapter 8 of his monograph, 1984) carbon dioxide achieves multiple-contact miscibility viathe vaporization of hydrocarbons. It appears that carbon dioxide is somewhat unusual in that it primarily takes up


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hydrocarbons extending up through the gasoline, kerosene and gas oil molecular weight range, rather than the lighthydrocarbons ethane through pentane, as in the case of natural gas vaporizing gas drives.

In common with propane and butane, CO2tends to precipitate high molecular weight fractions such as asphalt fromcrude oils when the crude oil is in small proportions-for instance, in 2:1 to 5:1 ratio of the solvent to crude oil. Mostof the crude oil dissolves in the solvent, and in the second phase containing the asphaltic components, a considerable

proportion of solvent is present (although less than in the less dense phase).

Thus, according to the available evidence, CO2is a poorer solvent for crude oil than either propane or butane.Miscibility pressures are therefore generally higher than for LPG, though far less than the miscibility pressure fordry natural gas (95% methane).

The different behavior of CO2observed between its critical temperature of 88 F and 130 F as compared to thatobserved above 130 F is shown by the p-x diagrams ofFigure 1andFigure 2.

Figure 1

The triangular phase diagrams for these systems contain, in the case of the systems between 88 F and 130 F, atriangular three-phase region, in addition to two or three two-phase regions.


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Figure 2


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Miscible Flood Phase Behavior: Combined Effects of Fingering and Heterogeneity

In the gas-driven LPG slug process, the fingering of LPG through crude oil is accompanied by fingering of gasthrough the LPG fingers. When the gas fingers break through the forward ends of the LPG fingers, an immisciblecontact of gas with crude oil occurs. When we consider the transition in composition through the sides of the fingers,it is also possible that the maximum LPG concentration will fall below the value needed to maintain miscibility.

This could result from the thinning of this transition region as the fingers lengthen, and from the accompanyingtransverse dispersion, which causes crude oil to move inward and the miscible agent(s) to move outward.

These phenomena indicate the need for a much larger slug size than would be needed in the absence of fingering.Even a larger slug, however, may not entirely prevent immiscible regions from developing, because the fingerthicknesses are determined by formation heterogeneity as well as by the mobility ratio and the velocity. For anygiven combination of these variables, there is a natural spectrum of finger size in a homogeneous formation, butthese finger sizes are even more strongly affected by rock heterogeneity.

Formation heterogeneity has many forms, which may be generally distinguished between layered and non-layeredrock formations. Non-layered rock heterogeneity is mostly of weathered reef origin-the weathering creates irregularvertical channels in coral or stromatolite reefs, which do not have strong layering before the weathering. There arealso fractured shales and breccias which have relatively little layer character. Most rock formations, however, have

been formed by deposition of coarse-to-fine particles of rock (either of sandstone or calcium carbonate shellfragments) and these are called clastic formations. They typically form layers, because long periods of time wereinvolved in laying down successive deposits of significant thickness (several inches to a foot or more), and theconditions of transport and deposit of the particles changed over such time periods. The size range of the particles insuccessive layers varies and this results in layers of different porosity and permeability. The variation of layerpermeabilities is often described by measures such as the Dykstra-Parsons coefficient of permeability variation.

Changes in formation properties subsequent to the original deposition are also important. The processes causingthese change are called diagenetic (rebirth) processes. They include dissolution and precipitation of various rockminerals, and transformations such as that of limestone (calcium carbonate) to dolomite and transformations such asthat of limestone (calcium carbonate) to dolomite (calcium/magnesium carbonate). Between periods when fairlycoarse sand or limestone particles were being deposited, very fine particle beds, which were mud-like in texture,were often deposited; after compaction these turned into shales in sandstones or their equivalent in limestones and

dolomites. These shale beds, and the permeable beds between them, have a variety of thicknesses, ranging from theorder of a centimeter to multiple-meter scale In the course of the climate and weather changes during deposition ofsequences of layers, the edges of rivers, lakes and ocean beaches on which much of the layer deposition occurredshifted from side to side and back and forth; often, the water level changed significantly over time. This resulted inboth permeable and impermeable layers ending over some finite distances, so each layer may not continue overtypical well distances.

From the changes in weight of formations above them (including alternate deposition and erosion), and regionaltectonic movements, many rock beds have undergone faulting, in which a broad slab of rock many tens or hundredsof meters thick moves up or down relative to rock on each side. The friction as the rock faces slide past each othercauses local grinding and melting so that the faults are often impenetrable to fluid movements. In some cases, therock cracks in many more places, creating fractured rock formations. These heterogeneities control fluidmovements within the permeable formations to such a degree that fluids can be transported into contact with each

other (for example in adjacent layers) that are not in phase equilibrium. The processes which lead only gradually bymultiple contacts to a miscible state are not able to prevail in such three-dimensional heterogeneous circumstances,rid contacts between immiscible phases can occur in many places during a displacement that would be miscible in aone-dimensional system. This leads to incomplete displacement of some of the crude oil.Even in a homogeneousformation or layer, besides well-pattern sweep efficiency, there are fluid-related mechanical effects-viscousfingering and gravity tonguing-which limit the volume fraction of the layer being swept by the miscible fluids,resulting in incomplete displacement of the crude oil. Even when individual layers are relatively homogeneous, thevariation of properties among the different layers often results in some layers being incompletely swept, while inother, more permeable layers, high flow rates result in the economic limit of gas/oil ratio or of water cut beingreached at the production wells.


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Physical Dispersion of Miscible Fluids

The basic literature on this subject is an article by Perkins and Johnston (1963), which is also reprinted in SPEReprint Book No. 8, Miscible Processes (1965). Other significant articles are in the same reprint book, namely thoseby Blackwell (1962) and van der Poel (1962). The articles by Pozzi and Blackwell (1963) and by Blackwell, Rayneand Terry (1959) are also recommended.

In SPE Reprint Book No. 18,Miscible Processes II(1985)) see the two articles by Koonce and Blackwell (1965), aswell as those by Warren and Skiba (1964), Coats and Smith (1964) and Baker (1977). The articles by Coats andSmith and by Baker deal with the contribution of dead-end pores to the overall dispersion behavior. Fatt (1959) hadsuggested that such dead-end pores would affect pressure transient behavior. Coats and Smith, and later Stalkup(1970), thought that the effect of dead-end pores on dispersion would be significant on the laboratory scale but noton the field scale; the article by Baker concludes that this is not the case; it is also important at field scale, althoughmuch less so than in the laboratory.

The articles by Warren and Skiba and by Blackwell and Koonce introduce the effects of heterogeneity on dispersion.Warren and Skiba deal with random three-dimensional heterogeneity, and conclude that its effects on dispersion canbe described adequately by the standard treatment of dispersion in one dimension (with a much larger dispersioncoefficient). Blackwell and Koonce also deal with the broadening of a dispersion zone by transverse dispersion

between adjacent layers. This is taken up in more detail by Lake and Hirasaki (1981), who give criteria for decidingwhen adjacent layers will have sufficient overlap in dispersion zones to be considered a single layer with a greaterlongitudinal dispersion coefficient.

Mahaffey et al. (1966) show that using laboratory sand packs or slabs of natural rock for laboratory miscible floodsresults in relatively few viscous fingers, because the dispersion rate is high compared to the convection rate, whileflooding on a field scale results in a relatively high degree of viscous fingering, because the transverse dispersionrate in the field is almost at the molecular diffusion level, and is much lower than the convection rate. Theyrecommend the use of Hele-Shaw (parallel plate) models for experimental study of miscible areal sweep efficiencybecause, using an analysis similar to that of Taylor (1953), they found that the dispersion coefficient is proportionalto the square of the plate spacing, enabling models with very close spacing to simulate field behavior more closelythan sand-packs or rock slabs.

Perkins and Johnstons (1963) equation for the longitudinal dispersion coefficient KL for clastic rocks (such assandstone) is:

KL/D0 = 1/Fc + 0.5(u dp/D0) (3.1)The similar equation for the transverse dispersion coefficient KT is:

KT/D0 = 1/F + 0.0157(u dp/D0) (3.2)where

D0= molecular diffusion coefficient for the fluid pair involved

F = formation electrical resistivity factor (ratio)

= porosity (1/F = 1/ , = tortuosity), fraction or ratio

u = superficial interstitial velocity = q/A+, cm/s

= pore shape factor based on average particle size of grains

dp = average grain size of the clastic rock, cm


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Table 1 of their paper gives values of ad for several outcrop sandstone samples; the average value is 0.36. Below (u

dp/0) values of about 1.0 for longitudinal dispersion, or about 10 for transverse dispersion, the second term on the

right is not to be counted. The ratio K/D0is just equal to 1/F = 1/ below that point. In oil reservoirs, the value of

the second term is often low enough to place the dispersion value in that range where it is equal to 1/ , while inlaboratory test equipment it is usually in a much higher range of the second (velocity) term.

The quantity is the average tortuosity of the flow path through the pores; it usually has a value on the order of 1.5to 2.5 (length of the crooked path divided by the straight distance).

Most miscible flood calculations involve numerical methods utilizing computer simulations. In such computations,numerical dispersion can actually overshadow the physical dispersion effects discussed here.

Numerical Dispersion in Computer Simulations

What is called "truncation error" by mathematicians causes a dispersion of fronts in reservoir simulations done withfinite-difference computer displacement simulation programs, which is similar to physical dispersion in some waysand different in others. Lantz (1971) gives formulas for the equivalent dispersion due to different forms ofdiscretization of space and time and of calculation of transmissibility between blocks.

For the case of a simple backward difference and the IMPES (Implicit Pressure, Explicit Saturation) method ofsolving the simultaneous flow equations, the diffusion coefficient is:

D(total) = D(physical) + (uL) * [_x/L - ut/L]/2,Or, dimensionlessly,

[KLeff./D0] = [KL/D0] + (uL/D0) * [1 - Ni/(PV)n]/2n (3.3)where

u = the linear velocityL = the length in the general coordinate direction xn = the number of grid blocks in the coordinate direction xNi = the fraction of a mobile pore volume injected per time step t(PV)n = the mobile pore volume per grid block.

In general, an attempt to reduce the quantity [1 - Ni/(PV)n] below about 0.90 to 0.95 (for the IMPES method ofsolution) results in a significant increase in error of the mass balance with increasing number of time steps. For agiven L and u, the only effective way to cut dispersion is to increase the number of grid blocks (n). Note that for agiven flow velocity (u), the number of blocks (n) must be increased along with an increase in size (L) of the totalsystem, just to keep dispersion constant.

In the case of immiscible displacements, physical dispersion can be introduced into the computer simulationequations either by means of capillary pressure or by adding a second-order term containing KL to the first orderterms of the Buckley-Leverett differential mass-balance equation. Adding the second order term is the only way toaccount for physical dispersion in miscible displacement simulations. Only in unusual circumstances would it benecessary to add this type of term, because the numerical dispersion due to the second term in Equation 3.3 isusually so large compared to physical dispersion that it is difficult to make it small enough to equal the physicaldispersion expected in a field process.

For example, for a flow velocity u of one centimeter per hour (0.000278 cm/s), an L of 10,000 cm, and a D 0 of0.0003 cm2/s, and n = 10 grid blocks (10 meters length per grid block), we get about 400 for the second term ofEquation 3.3. That is, the effective dispersion is about 400 times the ratio KL/D0. This is why it is difficult incomputer simulation to match the sharpness of the fluid fronts that actually occur in oil reservoirs. Furthermore, it isimpossible to display features such as viscous fingers with a limited number of grid blocks, such as the ten just usedin the example.


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Numerical dispersion in a finite-difference simulator is similar to the dispersion due to discretization of a linearreaction system, as expressed by a series of ideally-mixed reaction vessels or "CSTRs" in the field of reactionengineering. This was discussed by Kramers and Alberda (1953). In a single ideally-mixed vessel, dispersion is sogreat that an injected tracer emerges immediately from the exit, while as the number of vessels approaches infinity,the displacement behavior of an injected tracer approaches plug flow (i.e., zero dispersion). The dimensionlessnumber uL/D0 is approximated by (1 + n)/2, where n is the number of vessels in series.


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Miscible Flood Sweep Efficiency

Injected fluids not only disperse into fluids already present in an oil reservoir, but also fail, to some extent, to contactthe fluids in place and drive out the desired oil. There are a variety of reasons why this sweep-out of crude oil failsto take place.

When water is the displacing fluid, capillary forces cause it to travel more rapidly than oil, through some of thesmaller rock pores. Water thereby reaches pore throats ahead of the retreating oil phase and forms tiny films acrossthe mouths of these pore throats, preventing any further flow of oil through not only the small pores, but through thelarger ones as well. Oil is trapped in isolated multi-pore "ganglia" several millimeters in length and less in width.The enormous multitude of these fine ganglia adds up to a significant fraction of the crude oil encountered by thewater-from one third to one half-not being produced at the production wells. Thus, while waterflooding is relativelyinexpensive and sweeps most of the area of well patterns, the efficiency of displacement of oil on a microscopicscale is disappointing.

On the other hand, when miscible flooding agents are used, the efficiency with which they sweep out well patterns isnot very good either. This is partly because of gravity segregation and viscous fingering, but also because the wellpatterns for reservoir flooding processes have relatively stagnant regions, where oil can be driven out only by usingmany volumes of injected fluid relative to the volume of the reservoir. Water is low enough in cost so that many

volumes can be used, but all of the other flooding agents, including those miscible with the crude oil, are relativelyexpensive; when using these agents, we can afford only a fraction of the reservoir volume. The sweep efficiency ofthese well patterns thus becomes a matter of major concern.

Sweep efficiency varies not only with the amount of agent injected, but also with its mobility, or relative ease offlow through a resisting porous medium. For a single fluid, mobility is proportional to the reciprocal of the fluidviscosity, which measures the resistance to flow of the fluid. When one miscible fluid displaces another, we refer tothe mobility ratio (M), which is the ratio of the mobility of the displacing fluid to that of the displaced fluid; themobility ratio is the reciprocal of the viscosity ratio. For the miscible drive fluids currently available, this alwaysturns out to be a number considerably greater than one, and these mobility ratios give relatively poor well-patternsweep efficiency ( Figure 1).


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Figure 1

Note: The equations referenced below may be found in the section titled "Well Pattern Sweep Efficiency."

Muskats Equations

Muskat (1937, 1949) derived equations for the sweep efficiency of ideal well patterns for unit mobility ratio and in asingle homogeneous layer, with no gravity segregation effects.

Deppes Approximation Method

Deppe (1961) gives important extensions of Muskats work. (There is a closely related article by Prats et al. (1959)that is also recommended reading.) Deppe gives the equations referred to above, but in slightly different form,expressing permeability in millidarcys instead of in darcys: hence the numerical constant at the front of the righthand side is a thousand times smaller. He also uses base 10 logarithms rather than natural base logarithms, so his

constants are also reduced by the factor of the natural logarithm of 10. His constant of 0.0011538, when multipliedby 1,000 and then by 2.303, gives the value 3.54 used in the section titled "Well Pattern Sweep Efficiency."

The constant in the denominator as Deppe gives it is 0.2688; when this is multiplied by 2.303 we get the value of0.619 as given by Muskat. Among other information, Deppe gives equations for patterns or parts of patterns on theedges of fields, which were not previously available. He shows that both regular well patterns and irregular wellpatterns can be considered to be composed of injection circles around injection wells and production circles aroundproduction wells, with the sum of the areas equal to the total area of the well pattern.

Claridges Equations

Claridge has developed a set of equations (not published elsewhere) for the influence of mobility ratio and arealheterogeneity on areal sweep efficiency (E

a) at injected fluid breakthrough in ideal well patterns.

Applying Claridges Equations to Deppes Method

Claridges breakthrough sweep efficiency equations, mentioned above, may be used to modify Deppes Method.

Deppe states that at breakthrough of injected fluid, the flow changes from being radially inward over the entireperimeter of the circle representing the interface between the injected and displaced fluid to being radial flow inwardwithin a sector of fixed radius but increasing central angle, while the displaced fluid flows inward in the remainingsector of the circle. Deppe places this circle generally at a radius less than the radius of his production circle, andbases this lesser radius on the breakthrough sweep efficiency.

Claridge proposes the modification of making the production circle radius just equal to this radius at which the flowchanges from being concentric to parallel flow in sectors. This implies that the area of the injection well circle

divided by the total pattern area should be made equal to the areal sweep efficiency Ea. The production circle areawould then be proportional to (1 - Ea), and flow inside it would always be sectorwise. When the injected fluidreaches the periphery of the injection circle by radial flow, it then begins to flow into the production well through asector which begins with a zero central angle and increases in central angle while the d isplaced fluid continues to beproduced from the production circle in a sector which is correspondingly narrowing.


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Irregular Well Patterns in Miscible Flooding

The shapes of actual well patterns often do not closely correspond to those of ideal well patterns. They may bedistorted versions of ideal patterns, or they may have no obvious resemblance. Furthermore, in multiple wellpatterns, injection and/or production wells generally do not have the same injectivity or productivity for all wells ofeach type, as required by ideal pattern relationships. The reasons for irregular pattern shapes are (1) it is difficult to

drill wells to a series of exact predetermined bottom-hole locations, and (2) the number of wells that can be drilled tocover an irregular reservoir is limited. Differences in injectivity and productivity are caused by variations in thethicknesses and permeabilities of the layers penetrated by each well.

When planning oil recovery projects under such circumstances, it may be helpful to get some indication of the flowbehavior as determined by actual well locations and flow rates. In the cases discussed, we need to assume that onlyone fluid is flowing (or that the mobility of displacing fluid and displaced fluid are the same), that the density isconstant, and that the formation thickness, porosity and permeability are constant over the area of interest.

It is possible to make a set of points track the streamlines; the curves connecting these points at any given timeoutline the flood front. They also show, by assigning a given volume of flow to be represented by each point, howmuch of the flow from a given injection well arrives at different producers.

Finally, it is possible to define the space between adjacent streamlines in a given well pattern as "stream tubes," andto calculate the flow through these stream tubes by Darcys Law and the Buckley-Leverett method. It is necessary toallocate the injected fluid to the different stream tubes, and to calculate the rate of flow based on the total pressuredrop and the length and average cross-sectional area of each stream tube. With this method, it is further possible tostack one layer on top of another, and (by computer) carry out the accounting of flows of the components such asoil, water and gas from each stream tube and in each layer, to obtain a total o

eor miscible flooding

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