. \

The Effects of Dissolved Methane on Composition Paths

in Quaternary CO2 - Hydrocarbon Systems -

A Report

Submitted to the Department of Petroleum Engineering

of Stanford University

In Partial Fulfillment of the Requirements

for the Degree of

Master of Science

by

William Wesley Monroe

August, 1986

X r

s:

s:

s:

ACKNOWLEDGMENTS

I would like to thank my advisor Dr Franklin M. Orr, Jr., for his enthusiasm and guidance in the pursuit

of this rqx)rL His trust and expertise were greatly appreciated Also I would lilpe thank Stanford

University and the Stanford Research Institute under Department of Energy Contract DE AC03

81SF11564, and SUPRI-C for the financial support which made tfiis work possible. Finally I would like

thank Glenn Kioeger» for contributing flie GKS gr^hic primatives which made the clear presentation of

four component phase diagrams possible.

-u-

ABSTRACT

This report presents several analytical results concerning composition patlu for quaternary

systems. The behavior of displacements in the CO2 - Ci - C4 - C\q system is investigated to

answer questions pertaining to tiie effects of the presence of meAane on displacement

efficiency in CO^crude oil systems. An interactive computer program was developed to

calculate composition paths in a step-wise manner. The solutions generated explain the

experimental observation that addition of methane to a dead oil has little effect on the

measured minimum miscibili^ pressure. They also indicate that high efficiency displacements

are possible even when the initial fluid forms two phases.

-ui-

TABLE OF CONTENTS

Page

Acknowledgments ii

Abstract iii

List of Figures v

List of Tables vii

1. Introduction 1

2. Mathematical Model..i/2.I. Calculation of Component Velocities 7

2.2. Calculation of Shock Velocities..... 12

2.3. Determination of Composition Grid Paths 14

3. CompositionPaths in a Ternary Systemi 19

Path Description 22

4. Composition Paths in a Four Component System..J 28

/4i

Quaternary Grid Topology 30

Path Description 32

4.1. Effect of Oil Composition on Development of Miscibility 43I

Comparisonof Live and Dead Oil Displacements 43

Development of Miscibility 45

Effert of the Amount of Methane Dissolved in the Original Oil 50

.2. Discussion 54

5. Summary and Conclusions 57

Nomenclature

References - 59

Figure 1.1

Figure 2.1

Figure 2.2

Figure 3.1

Figure 3.2

Figure 3.3

Figure 3.4

Figure 3.5

Figure 3.6

Figure 3.7

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4

Figure 4.5

Figure 4.6

Figure 4.7

Figure 4.8

Figure 4.9

Figure 4.10

Figure 4.11

Figure 4.12

UST OF nCURES

Phase diagram for Cj - C4 - Cjosystem at 3200 psia and 160 ®F.

Shock position at time t and t +At.

Composition paths for Ci - C4- Cio system at 1600psia and 160 ®F.

Composition paths for CO2 "€4- Cjo system at 1600psia and 160 ®F.

Solution path for the example ternary problem.

Saturation profile for example ternary problem at //> = 0.5.

Composition profile for example ternary problem at tp = 0.5.

Variation of tie-line and nontie-line eigenvalues along the initial tie line.

Variation of tie-line and nontie-line eigenvalues along the injection tie line.

Recovery of decai» and butane for a ternary displacement by COi-

Composition paths for CO2 - Cj —Cio system at 1600 psia and 160 ®F.

Composition paths for CO2 ~ Ci —C4 —Ciosystem at 1600 psia and 160 ®F.

Variation of composition velocities along a tie line.

Solutionpath for quaternary example problem.

Variation of tie-line and nontie-line eigenvalues along die initial tie line.

Variation of tie-line and nontie-line eigenvalues along the cross-over tie line.

Variation of tie-line and nontie-line eigenvalues along the injection tie line.

Saturation profile for example quaternary problem at fp = 0.5.

Composition profile for example quaternary problem at tp = 0.5.

. Composition paths which do not meet the criteria for path construction.

- Methane, butane and decane recovery curves.

- Composition paths for systems 1» 2, 3, aix! 4.

-V-

Figure 4.13 - Change in injection tie line for oils with varying amounts of methane.

Figure4.14 - Tie line surface associated with a vertical path.

-VI-

UST OF TABLES

Table 3.1 - Inidal and injected fluid compositions and phase densities for examine temaiyproblem.

Table 3.2 - Compositions, saturations and velocities for a ternary displacement path.

Table 4.1 - Tm'tial and injected fluid compositions and phase densities for examplequaternary problem.

Table 4.2- Compositions, saturations and velocities for a quaternary displacement path.

Table 4.3- Shock compositions and velocities for system 1, an "oil" with low butaneconcentration (p = 0.3553 Ib-moHf?).

Table 4.4- Shock compositions and velocities for system 2, an"oil" with intermediatebutane concentration (p = 0.4005 Ib-moUf?).

Table 4.5- Shock compositions and velocities for system 3, an"oil" with intermediate butaneconcentration (p = 0.4206 Ib-mollf^).

Table 4.6- Shock compositions and velocities for system 4, an "oil" with high butaneconcentration (p = 0.4367 Ib-moHf?).

Table 4.7- Effect of the addition of methane to the displaced oil on methane bank behavior.

-vii-

Section 1: Introduction

The use of high pressure flue gases, light hydrocarbon gases and CO2 as injection fluids

for the displacement of residual oil has been studied by many investigators; Many factors

contribute to the enhanced recovery in these multi-contact miscible type displacements/

Displacement efficiency during miscible type EOR processes depends on the mass transfer of

components between the phases that are present and on the resulting variations in physical

properties such as density and viscosity. These properties vary with the phase composition

which, in turn, varies as components partition between phases. Thus, quantitative prediction of

the performance of a miscible-type displacement requires a detailed understanding of the

impact of phase behavior on the displacementprocess.

Several authors have shown tiiat for a ternary or pseudo ternary system in which the

initial and injection fluid compositions are constant, the ternary diagram can be divided into

regions of immiscibility and miscibility; the dividing line between the two regions is the

extension of the critical tie-line, the tie line tangent to the binodal curve at the plait point.

Slobod and Koch were the first to publish such a description in 1953. In their paper they

divided the temaiy diagram into two sections as shown in Fig. 1. They called the two zones

the *'!" zone, where phases in equilibrium at the displacement front are immiscible, and the

"M" zone, where the injected fluid could be enriched through multiple contacts with the

reservoir oil to a point where the fluids at the displacement front were miscible, resulting in the

elimination of the capillary entrapment of oil by displacing gas. In this case the oil was said to

be essentially "washed out", meaning 100% recovery at one pore volume (PV) injected. They

indicated that this result was due to the removal of capillary pressure and relative permeability

cffects which are present when more than one phase is present. Thus, Slobod and Koch were

the first to divide the ternary diagram based on the region of tie line extensions. They argued

that displacement of oil by mediane is "immiscible", when the initial oil composition lies

' "miscible type" as used throughout this paper refers to displacements in which component tiMsferplays an important role in the displacement Iwhavior. This could include immicible, miscible or mulU-contact miscible displacements.

Mo

Figure 1.1 - Phase Diagram for Cj - C4- Cjo system.at 3200 psia and 160 ®F.

within the region of tie line extensions. Stone and Crump (1955) presented results which

showed that if the composiuon of the injected fluid were such that it had high solubility in the

reservoir oil, miscibility could develop resulting in very high recovery, the first quantitative

description of the condensing or enriched gas drive mechanism. A more thorough description

of die i^se transitions that accompany the development of miscibility was given by

Hutchinson and Braun (1964). In their paper they described in detail the phase relations and

mass transfer mechanisms of the high pressure gas, enriched gas and miscible slug processes.

The authors used temaiy diagrams to provide a clear conceptual analysis of the phase

transitions that occur in the three miscible processes. They also outlined the effects of variables

such as temperature, pressure and composition on performance of a given process.

Rathmellet el. (1974) applied the vaporizing gas drive ideas developed by Slobod and

Koch and Hutchinson and Braun to C02-crude oil systems and noted the ability of the CO2 to

volatilize intermediate and heavy hydrocarbons much better than methane or flue gases. They

Tielines

-2-

Limiting tieline

-3-

also gave the first description of the composition path for a four component system. That

description is discussed in more detail in §4. They also noted the presence of a methane bank

which preceded the CO2 breakthrough when an immicible displacement was thought to occur.

They suggested that the presence of such a bank might be used as the signal of an immiscible^

displacement, l^etcalfe and Yarborough (1979) also noted the presence of methane banks

during immiscible processes. They observed that the banks did not disappear until the the

process was first contact miscible.

Welgeetal (1960) were the first to quantify the effects of phase behavior on

displacement behavior for an enriched (condensing) gas drive system. In that work the authors

showed how the compositions of the gas and oil present in the transition zone of an immiscible

displacement* are those of the bubble point and dew point loci between the limiting tielines.

They also outlined calculations which could be performed to predict the composition paths for

a three component system based on component material balances. The equations derived were

ordinary algebraic equations when applied to shocks (step changes between the single and two

phase region), and partial differential equations if they are applied to component velocities in

the two phase region. Along tie lines where phase compositions are fixed and only phase

saturations change the Buckley-Leverett formulation of the flow equations was shown to apply.

In that work, effectsof volume change on mixing were included.

Helfferich (1981) generalized the mathematical approach of Welge et al. and presented

rigorous arguments that confirmed the qualitative arguments of Slobod and Koch and

Hutchinson and Braun, but restricted his analysis to the case where volume change on mixing

is negligible. The mathematical approach was developed to study composition paths for a

system with an arbitrary number of components and phases. A general solution to the

convection equations was presented in which no assumptions were made about the dependence

of phase equilibria, relative permeabilities, and viscosity on phase compositions. The solution

* Immiscible as refened to here means not miscible. It does not mean that components do not transferbetween the phases, only that the two phases present do not develop miscibility after infinite contacts.

-4-

could be applied for art)itraiy initial and injection compositions. Hirasaki (1981) applied the

theory to study the effects of phase behavior on surfactant flooding using ternary

representations of surfactant-oil-brine phase behavior. Dumore' et al. (1984) took the analysis

a step further by including the effects of volume change on mixing. They reported analyses of

composition paths in vaporizing and condensing gas drives in ternary systems. Because effects

of volume change on mixing are included, the approach used by Dumore et al. allows the use

of an equation of state (EOS) to model the phase behavior and jAase densities. The

mathematical approach developed by Heljferich (1981) and Dumore' et al. (1984) can be used

to examine a long standing question concerning the effect of the presence of methane on

miscible displacement in C^?2"Crude oil systems.

Experimental investigations of the effect of phase behavior on displacement performance

have relied on measurements of the minimum miscibility pressure (MMP). Experimentalists

have presented a variety of definitions for MMP. Each definition attempts to specify

quantitatively the lowest pressure at which two fluids can dynamically develop miscibility

during a displacement Displacement experiments are carried out in long slim tubes at a

variety of pressures until a practical ultimate recovery is discerned at some minimum pore

volumes (PV) of CO2 injected, typically 1.2 PV. The displacements are performed in slim

tubes because mechanisms such as dispersion, viscous instability and gravity segregation are

nearly eliminated, and hence phase behavior plays die key role. Because of the importance of

phase behavior, die composition of the displaced oil and the displacing fluid have been

examined to determine its effect on the MMP.

Several authors include a factor in their MMP correlations which accounts for the effects

of variations in oil composition. Holm and Josendal (1974) developed a correlation for

prediction of CO2 MMP that was based on the reservoir temperature and molecular weight of

the Cj fraction. They neglected the amount of Cj - C4 present, speculating that these lighter

hydrocarbon components fingered out ahead of the region where miscibility was developing

and thus should not effect the predicted MMP. The presence of the light components was

-5 -

thought to be important only if there was enough Cj present to raise the bubble point pressure

(BPP) of the dead oil above the predicted MMP. In such case the bubble jwint pressure is

taken to be the MMP. Yellig and Metcalfe (1980), in their attempts to develop a more accurate

CO2 MMP correlation, concluded diat the recombined oil composition had little or no effect on

die MMP at lower temperatures and only slight effects at higher temperatures. Holm and

Josendal (1982) attempted to improve the correlation which they published in 1974 by

including a factor that could account for the ability of CO2 to solublize intermediate

hydrocarbon components. They argued that CO2 density was an appropriate correlating

parameter. They noted that the gaseous CO2 appeared to extract only the lightest components,

while more dense CO2 could vaporize a much wider range of molecular weight components,

thus aiding in the development of miscibility. The MMP correlation developed by Orr and

Silva (1985) was based on the molecular weight distribution within the displaced oil. Each of

these correlations included the empirical rule that if the BPP of the displaced oil were above

the predicted MMP, then the BPP was to be taken as the MMP.

This correction _is_ apparently-inconsistent- with-the-definition-oLmultirCDniaci..m.iscibility

based on the analysis of coupled ^vecrion and componenjLP.artitiom>^ The analyses ofSlobod and Koch (1953), and Hutchinson and Braun (1964) indicate that miscibility develops

only if the oil composition lies outside the region of the line extensions. In fact, Helfferich

showed that in the absence of mechanisms such as dispersion and capillary pressure, the

composition path of adisplacement can enter the two phase region on a ternary diagram only

via a shock (step change) along a tie-line extending through the single phase composition, a

rigorous statement of the quantitative arguments of Slobod and Koch and Hutchinson and

Braun. The use of the BPP as the MMP for oils with BPP^s jgreaterjhAnjhe_,empj^^^

predicted MMP contradicts the developed miscibility argument, since the bubble point of an. oil

is one end ofa tie-line and hence is inevitably within the region oftie line extensions.

This report makes use of an analysis of composition paths for the CO2 - Cj - C4 - Cio

system to resolve the inconsistency. That analysis is used to answer the following questions:

-6-

1) How does the presence of methane in an oil affect the recovery inmiscible ty^ displacements?

2) Is the presence of light hydrocarbon components such as methanean important factor to be considered in some cases and not inothers?

3) What does the composition grid path look like for a quaternarysystem and what are its pertinent features?

4) Is the presence of a methane bank before CO-i breakthrougha good criterion for judging whether miscibiliiy has developed ornot?

In §2 the mathematical model is developed, and the method for solution construction is

outlined. §3 illustrates the calculations necessary for determination of the composition path for

a COi —C4 —Cio ternary system in which pure CO2 is injectcd into an "oil" composed of

C4 and CjQ. In §4, the composition paths for the COi - Cj —C4—Cjo quartemary system are

studied in detail to answer the questions raised above. Detailed examples of calculations for

four-component systems with a variety of compositions are described, and a comparison of

model predictions with available experimental evidence are reported in §4. Finally, conclusions

and results are summarized in §5.

-7 -

Section 2: Mathematical Model

The equations governing the flow of multiphase, multicomponent reSfervoir fluids, in

which the motion of components is due to convection, dispersion and component partitioning,

are quite complex and are generally solved numerically. Analytical solutions are possible,

however, after several key assumptions are made. These solutions are imjx)rtant because they

can be used to validate the results from large numerical simulators, and because they isolate

the effects ofphase behavior, making clear the dependance ofsolutions on key variables such

as temperature, pressure and composition. These solutions also usually require less

computation than is needed to produce numerical solutions. The analytical model proposed by

Dumore etal is used in this study. Composition patiis are calculated for a four component

system with methane (Ci), normal butane (C4), decane (Cio). and carbon dioxide (CO2). This

system was chosen because of the availability of experimental phase compositions anddensities. For this system, the Peng-Robinson equation of sute was tuned and found to

represent the phase behavior accurately {Orr and Taber 1984). In addition, the interaction ofthe three hydrocarbon components (light, intermediate and heavy) with the CO2 could beinvestigated to answer the questions raised, and to supplement some of the qualitative

arguments proposed by authors about the mechanisms of CO2 enhanced oil recovery. Themathematical formulation is presented here to clarify nomenclature and review the terms and

concepts necessary for understanding development and interpretation of the solutions.

Section 2.1*; Calculation of Component Velocities

Consider the flow of components that form up to phases. Ifdispersion is absent, a

material balance on tiie i"* component gives.

-8-

ot >1 >.1(2.1)

where:

rip is the number of phases,tie is the number of components,Xij is the mole fraction of component i in phase j,Pj is the molar density (kg-mole/m^) ofphase y,V,- is the phase velocity vector (m/day) andSj is the saturation of phase y, and^ is the porosity.

If the flow is one dimensional, the porosity is constant and the velocity of a phase is given by

(.2.2)

r

where v is the total velocity ^ vy= ^M, and fj is the fractional flow of phase j, Eq. (2.1)

becomes,

Eq. (2.3) can be written more simply by defining.

=S R/ ~T ^ '/=i ^

(2.3)

(2.4)

From the above definitions it can be seen that G,- represents an overall composition, and F,-,

represents the overall flux of component i in all phases. Substitution of the definitions for G/

and Fi into (2.3) yields

aCi n . 1(2.6)

It should be noted that the functions G,- and Fi depend on the local overall component mole

fractions, Q =L*,- + (1 - L) y,-, and f,- also depends on the local flow velocity, v,

G,= C(C.) i=l.n,-l (2.7)

F,. = f(C„v) i=l,n.-l. (2.8)

-9-

Therefore, the derivatives of G; andF,- can be written

and

dCt . ,at IjdCt at

dFi dCt , dFi av . .= -:r-+-7- j=l, «CBx dCj^ dt ' Jv

Now suppose that in the plane of the dependent variables Q and v (this plane is referred to by

Dumore' et al. as the "hodograph space") the solution can be represented as,

Q = C,<t1) and v = v(ti) (2.11,2.12)

This lumps the two^ependent variables, xand / into the single^ependent variable Ti, and thederivatives of overall composition C,- and v with respect tox and t in equations (2.9) and (2.10)

can be written in terms of T|,

and

dG;_"':^dGi dCtdn hdCt A\ '

V'dP; dCtfci 'iCt dt\ dv dr\

Using the above definitions, equation (2.6) becomes

^i!L +£ii!i =o «=in.dt] dt dr\ dx ' '

Now T] is a fimction of both x and t, so expanding in a one term Taylor series gives

dt dx

(2.9)

(2.10)

(2.13)

(2.14)

(2.15)

(2.16)

Now we look for solutions along characteristics, that is along lines of constant T[. Thus, along

the characteristics.

<ic|!l+A^=0dt

(2.17)

. 10-

Equations (2.15) and (2.17) form aset of linear algebraic equations with and ^ as theunknowns. This system ofequations has a nontrivial solution if and only if the determinant of

the coefficients is zero.

'dFi dCi

0 = dr\ dr\

dx dt(2.18)

Expanding the determinant in eq. (2.18) gives

dt\'^''dt =—dx (219)

Uponrearranging (2.19) becomes,

£i^ =vc.= -^ (2-20)dt dGj

dn

Note that there is one equation that is the same form as (2.20) for each of the components,

and each equation has an unknown composition velocity, =vq. The solution sought is one

in which the compositions move together. Such solutions are called coherent sets of

compositions such that vc, =vc,= =vc. (Helfferich 1981). Coherence is the key to

development of solutions to this problem. §2.^ gives a detailed definiuon of coherence. For

now it is important to understand that finding sets of compositions that move together is the

idea which leads into the formulation of an eigenvalue problem. With this in mind equation

(2.20) has the following matrix form.

-11 -

3Fi 3F, 3F, 3Fi dCi

3Ci dCi 3v dn

dFi 3^2 3F2 3F2 dC2

3Ci 3C2 "

• » • •

dC^l

P • $ •

3v d([

3f«

• f • '• • # •

3F,^i 3F^ dv

3C, 3C2 " dv dr\

30, 3G,

3C, SCi

3G2 3G23Ci 3C2

3G„ 3G«

3C, 3C2

3G,

3C«.,

3G2

3Cnc-1

3G,

3C«C-1

dCx

d\\

dC2

dn

dv

dn

(2,21)

dF: dG:where the definitions for and given in (2.13) and (2.14) have been substituted into

dt\ dc\

eq. (2.20).

Eq. (2.21) is a general eigenvalue problem for which a solution exists if and only if

det F-31G = 0 . For a three component system F and G are 3 x 3 matrices defined in

(2.21). The eigenvalues correspond to /ig - 1 characteristic rates^o^omposition velocities and

fee last eigenygliift For a four component system the matrices are 4 x 4. In both

cases, the eigenvectors correspond to characteristic directions in the hodograph space. Given

the complexity of the equation-of-state representation of phase compositions that appear in G,

and F/, finite difference representations are used to evaluate the derivatives necessary to

assemble the matrix problem presented in (2.21). The exact form of finite difference used to

calculate the derivatives is dependent on the overall composition, but either a two point

forward or backwards difference is used. In a ternary system where the dependent variables are

two independent overall component mole fractions and the total velocity,

dFi F,<Ci + ACi. C2, u) - F.<Ci. C2. u)

dCt AC,i = 1, . (2.22)

and.

- 12-

dFj _ fi(Ci. Ci ->• AC2. «) - ^2> j- 2 ^<iC2 " ^^2

<ff-; F.<Ci. C2. u + Am) - fi(Ci. Cj- ")du ~ An

(2.23)

(2.24)

The solution presented above is valid in those regions where the overall compositionvaries continuously. The composition velocity of adiscontinuity resulting from astep changebetween a single phase and two phase region or across a self-sharpemng wave can becalculated by a component material balance across the shock. The derivation of the materialbalance equations is given in the next section.

Section 12i Shock Solution between Single Phase and Two Phase Region.

the motion of such a discontinuity during an incremental penod of time At.

During this incremental time, the shock travels adistance Ax as illustrated in Figure 2.1.

Position of Shock atTime, t

Position of Shock att + A t

Ax

Figure 2.1: Shock position at time ^and f+ Af.

Let 11 and / denote compositions on the upstream and downsp-eamjide of the sho^At lime t the amount of mass of component / in all phases within the differential

clement A* is given by

Ax

and at time t + Af,

Ax

PyIri

= AxG{

£*!>• P>Iri

//

= AxG?.

.(2.25)

(2.26)

^r r

/ c

' 13 -

The change in mass over the time increment At is given by

Accumulation = Ax (G{' - G{) ^ (2.27)

Now the net inflow of mass to the differential element is given by the difference ^tween what

came into the element during Ar

VI

Ir^ow = Af

and what went out during that same period

Outflow = Af

Now to conserve mass, accumulation must equal the net inflow, thus

Ax(G;'-G{) = A/(f<'-f<).

or.

Ax F'i-F'iAf

And in the limit as the volume element becomes infinitesimally small,

dt c'!-g['

This is the material balance which must be satisfied oy eacH of the components across a shock.

(2.28)

(2.29)

(2.30)

(2.31)

(2.32)

It is important to note that there are two different types of shocks. A shock may be the

limit of a continuous variation along a tie line, in which case the calculated component material

balance velocity must equal the Buckley-Leverett (1942) velocity.

IliIL =v (2.33)Gf-G? dS

Because the above equation is equivalent to performing a tangent construction QVelge, 1952),

such shocks throughout this paper will be called "tangent" shocks. In some cases, the

composition path will reach,the entry or exit tie line at a point from which a continuous

variation is not possible. In such cases, an immediate jump to the initial or injection

compsoition occurs. In tfiis case, the overall samrations G,- and the overall fractional flows F/

- 14-

are known and the material balance calculation using Eq. 2.32 can be performed to determine

the velocity of the shock. A similar situation occurs when a jump occurs as the result of a

self-sharpening wave. To distinguish between die two types of shocks, such shocks will be

referred to as "nontangent shocks".

Section 23: Determination of Composition Grid Paths

Sections 2.2 and 2.3 outlined the mathematics nccessaiy for composition grid path

construction, but did not detail the steps which are taken to develop the composition grid or the

criterion which must be met for the determination of the composition route for a given set of

boundary conditions. That is the objective of the present section.

To trace out the composition grid path one must choose a starting point somewhere

within the two phase region. At this point a flash calculation is performed to determine the

phase compositions. In this study the Peng-Robinson equation of state was used to calculate

the phase compositions and densities. The subroutine used to calculate fluid properties was

developed by Nutakki et al (1985). The routine uses the Lohrenz, Bray, Clark (1964) version

of the JossU Stiel and Thodos (1962) correlations to calculate the phase viscosities, which are

used to calculate the fluid mobilities. Once the phase compositions are calculated, the overall

composition is perturbed and more flash calculations are performed at various other overall

compositions to generate the overall compositions (G,) and overall fractional flows (F,) of each

of the components. Those values are then used to obtain finite difference derivatives (Eq.

2.22-2.24) for the matrix problem Eq. 2.21. Next the eigenvalue problem is solved using the

IMSL routine EIGZC (1982). For an system with components this results in the generation

of rtc independent eigenvalues and associated eigenvectors*. For athree-component system, two

of the eigenvalues represent composition velocities. The third eigenvalue is infinite.^ An infiniterate just means that changes in the wave velocities that result from volume change on mixing

* It will be 'seen that there are two ringular points associaied with each tie line, atwhich two oftheeigenvalues are equal. At those points the eigenvalues are not independent

^ This a results of the fact that one of the columns of the G matrix is all zeros.

- 15-

are propagated instantaneously throughout the system. In aquaternary system, there are threefinite eigenvalues that give composition velocities and three associated eigenvectors that givecomposition directions. The fourth eigenvalue in this case is also infinite.

At this point it is important to distinguish between "wave velocities" and 'particle

velocities". Particle velocity implies the rate of advance of material. The phase velocity vy is a

particle velocity. Wave velocity on the other hand refers to the rate of propagation of agivenphysical property such as composition. Consider the motion of a given component thai ispresent in more than phase as it passes a given point. The wave velocity represents thevelocity of that component in all of the phases, as such it has been referred to as a "blind

dC- AFi

mans" variable (Helfferidi, 1980). The composition velocities, X=-^ and A=—. which

are calculated from the eigenvalue problem and material balance equations respecnvely. are

wave velocities.

For a system with n, components Aere will be n,- 1 eigenvalues which represent

velocities ofcoherent sets of compositions. Helfferich (1980) defined coherence as follows.

"Coherence is a shon-hand expression for what in the present context could becalled propagational subiUty. An arbiu-ary composiuon vanation - «.g.. existinginitially at some distance from the injection point, or generated at that pomt byiniection of a fluid different from that present imaally - in general ispropagationally unsuble; that is. it cannot be propagated with intepi^ as one singlewave Rather, it separates into several propagationally stable ( coherent ) waveswhich travel at different speeds and between which new zones of different, umformcompositions arise.Coherence requires alt dependent variables at any given point in space and time tohave the same wave velocity."

The eigenvectors associated with die velocity eigenvalues represem composition variations thatare coherent Thus, the eigenvectors represent the overall compositions that can occur ahead ofand behind a given overall composition within a transition zone at agiven time and distancealong the displacemem path. On a ternary diagram, the eigenvectors poim to overallcompositions that represent those neighboring overall compositions. As mentioned previously,one of the eigenvalues of the matrix problem is infinite. The eigenvector which is associated

- 16-

wiA the_infini^eigenvalue represents the variation in the phase velocities (again the phase

velocity is a particle velocity) which occurs as the result of tfie volume change that

accompanies a composition variation.

Once the eigenvalues and eigenvectors are determined at one point within the two phase

region, a small step is taken in the direction indicated by one of the two eigenvectors. Again

the flash calculations are performed and the matrix problem assembled. This procedure is

continued until enough paths have been integrated to give a sufficient indication of the general

grid topology.

The grid topology for a C, - C4 - Cio ternary system is shown in Fig. 2.2. Remember

that for a ternary system there are three eigenvalues and eigenvectors associated with each

overall composition within the two phase region. Two of the eigenvalues represent wave

velocities, and their eigenvectors represent changes in compositions which satisfy the coherence

condition, that is, compositions that can exist ahead of and behind a given overall composiDon.

Ci

Equivelocity curve

Fast paths

Slow paths

Figure 2.2 - Composition paths for Cj - C4 - C,o system at1600 psia and 160 ®F.

eCC

- 17-

The grid is composed of two distinct sets of paths. These are referred to as tie-line and

nontie-line paths. The tie-line paths represent composition variations along a tie line. Along tie

line paths it can be shown tfiat the Buckley-Leverett solution for immiscibte displacements

applies (y^elge et a/., 1960). Each path is "slow" in certain sections and "fast" in others.

Because there are two velocities associated with each overall composition within the two phase

region one velocity (eigenvalue) is slow, the other velocity is fast, and their eigenvectors

represent variations along slow and fast paths respectively. The distinction between slow and

fast paths is depicted on Figure 2.2 by drawing the slow paths as bold lines.

It should also be noted that the binodal curve is a path {Heljferich, 1980). One other

subregion of the grid topology which should be mentioned is the "equivelocity space". The

cquivelocity space is the set of overall compositions within the two phase region which have

associated phase compositions which travel at the same velocity (particle velocity).

Composition variations represented by the equivelocity space are coherent and thus are paths

(Helfferich, 1980). For a three component system die equivelocity space is a curve. For a four

component system it is a surface. The equivelocity curve is also shown in Figure 2.2.

After a general layout of the composition gnd path has been established, the next step is

to construct a solution for a given set of initial and injection compositions. The rules for the

route construction are quite simple.

1) The solution path must vary exclusively along paths. Only along the paths are

composition variations propagated with stability.

2) The solution traverses the hodograph space in a sequence of increasing wave velocities.

(This just means that the compositions which exist furthest down stream travel fastest)

Thus when tracing out the composition route from the initial to the injection composition,

the only allowable path switch is from a fast path to a slow path.

It has been proved by Dumore' et al (1984) diat the solution path must enter the two

phase region via a shock (step change) along the tie line that, when extended, passes through

the single phase composition. Therefore to construct a solution for a system which is not first

- 18-

contact or multiple conuct miscible. the first step is to calculate an entry or exit tie linevelocity. In the discussion that follows, the "initial" tie line refers to the tie line that passestoough the initial composition when it Uextended, and the "exit" or "injection^ tie line is thatwhich passes tiirough the injected composition. From the entry point within the two pregion the eigenvalue solution outlined in §2.1 is used to trace out the composition path. Ifupon landing in the two phase region .he path switches immediately from one path to another,forming abank, flien the actual landing point is indetermment. and the composition must betraced from the opposite end. The goal of the stepwise procedure is to trace out apath from.he landing poim on the initial tie line to athe injection tie line which satisfies the velocityrules outlined above.

mre is one final poim that requires explanation. While tracing out the compositionroute from the initial to the injection composition, along agiven path the wave velocities mayeither increase, decrease, or remain constant; the composition variations they represent arereferred to as "self-sharpening", "non-sharpening", or "indifferent" respectively. Self-sharpening waves are represented as shoclcs on asaturation profile. Non-sharpening waves arerepresented as continuous variations.

- 19 -

Section 3: Composition Paths in a Ternary System.

In this section the CO2 - C4 - Cjo system is studied to illustrate how to develop a

soludon path for a given set of boundary conditions. The displaced fluid in this problem is a

single phase, dead "oil" composed of butane and decane. The displacing gas is pure CO2.

Once most of the important features of this example are illustrated, the quartemary system can

be studied for a recombined oil that is composed of methane, butane and decane to answer

questions about to the effects of the presence of methane on displacement efficiency.

The grid topology for the CO2 - C4 - C|o ternary system is shown in Fig. 3.1.

^10

Figure 3.1 - Composition paths for CO2 - Q - Cjo system at1600 psia and 160 °F.

The general grid topology for the CO2 - C4 - Ciq ternary system is similar to that of the

Cj - C4 - Cio system shown in Figure 2.2. Both grids have slow, fast, tie-line and nontie-line

paths. The primary difference between the two systems is the presence ofa plait point in the

CO2 - C4 - Cio system. Not all of the nontie-line paths are tangent to a tie line in the

Ci - C4 - Cio system where there is no plait point.

Equivelodty curvePlait point

< C

-20-

The composition data for this example are given in Table 3.1.

Table 3.1 - Initial and injected fluid compositions for example ternary problem.

Composition (mole %)CO? Butane Decane

Molar

Densitv(gms/cc)

Injection Gas(mole %) 1.00 0.00 0.00 0.4133

Initial (Dil(mole %) 0.0 0.4176 0.5824 0.3735

To calculate the relative permeabilities of the fluid phases that are used to calculate the overall

fractional flows F,-, the following p^ametric equations were used.

and,

^ro ~

1.-5^,-

1. - .Sj, - S„1. —S^l —Sgr

The following constants were used in the above relative permeability equations.

Kgm-Kom- 0-8 = 3

5^= 0.05 0.00

To calculate the mobility of the fluid phases, the the following fractional flow relation for

horizontal flow was used was used.

Kj

j-o, gfj = ^ro ^

(3.1)

(3.2)

(3.3)

At time zero pure gaseous CO2 is injected at a constant velocity of 1.0 m/d into a core

tfiat contains single-phase oil. The porosity of the core is 20%. The solution path is plotted

on a ternary diagram in Figure 3.2 and the resulting composition, saturation and velocity data

are given in Table 3.2.

-21 -

Displacement of Oil (C4 + Ciq) by CO2at 160® F and 1600 psia

Comment Composition (mole %)Total

VelocityGasSaturation

Wave

Velocity

CO2 C4 C]o u (m/d) S V (m/d)

InjectionComposition

1.000 0.0 0.0 1.0 1.0

Slow Shock 0.958 0.0 0.042 0.9537 0.9194 0.2787

Zone ofConstant

State

0.958 0.0 0.042 0.9537 0.9194 0.76

SelfSharpeningWave

0.85 0.0986 0.0514 0.9587 0.65 0.76

ContinuousVariation

0.8480.846

0.09960.1005

0.05240.0535

0.95870.9587

0.6497

0.631

0.7974

0.942

Fast Shock 0.0 0.4176 0.5824 0.8774 0.0 0.942

Initial OilComposition

0.0 0.4176 0.5824 0.8774 0.0

Table 3.2 Composition, saturations and velocities for a ternary displacement path.

Trailing edge shock v

d

Leading edge shock

Figure 3.2 - Solution path for the example temaiy problem.

-22-

Path Description

As was in §2.4, the solution path can enter or exit the two-phase region only viashocks along tie lines. Thus, the first step toward path construction is to perform amaterialbalance calculation to detemine the landing points within the two phase region. Thecalculations done by Dumore' « al. indicated that the leading edge shock for vaporizing gasdrive systems was atangent shock. Direct calculations for the COi - C4 - C,o temaiy systemyielded the same results. Therefore to constnia asolution, the path is traced out from theinitial oil composition to the injection gas composition. As was discussed in §2.2, the exactlanding point can be determined by matching ihe shock velocities calculated from thecomponent material balances and the tie-line path characteristic velocity (Eq. 2.23). To findthe exact point, composition and shock velocities are calculated for a sequence of overallcompositions along the initial tie line to find where tiie ^and tie-line velocities are equal,•nie overall composition associated with this landing poim represents acomposition that movescoherently with the single phase oil.

For the leading shock in this problem, the step change is from point ato bin Figure 3.2.The composition and sawration profiles for this example are shown in Figures 3.3 and 3.4.The points that are labeled in Figure 3.2 correspond to tfiose labeled in Figures 3.3 and 3.4.At the leading edge, the composition jumps from that of initial oil, which contains no CO,, toatwo phase composition where all three of the components are present. Note the large jumpin saniration at this shock. This occurs because COj extracts the intermediate hydrocarbonsefficientiy. 'From the shock composition the composition path varies continuously along the

de line from poim bto c. Along atie line path the phase compositions remain constant,

^.he phase saturations are changing. In this region the solution behaves as an immiscibledisplacement and the Buckley-Leverett equations for two phase flow apply (Wetge «1960). In this region, the composition velocities vary quite rapidly. Figures 3.5 and 3.6 showhow the eigenvalues vary along the initial and injection tie lines. Nori« that the high slope ofthe tie-line path variation from bto cin Figure 3ii. Recall tiiat the exponents used in the

-23-

trailing edge shockT—I y'' I I I I—I I i I I ' ^1.0

0.8

^ 0.6%^ 0.4

t/3ed

o0.2

0.0

„ c zone ofconstant state

§

j I—I—L -i_i I—I

self-sharpening wave

b

a

leading edgeshock

I I I 1 j I L

0.0 0.25 0.5

xd

0.75 1.0

C 1.0o

o

2 0-8tu

S"S 0.4a

I"^ 0.0

Figure 3.3: Saturation profile for|xample ternary problem

r—I r—JT 1 [ 1 1 Jh 1 1 ' ' ' ' 1 ' ^ -' C02

11111111111111111InjectionGas Composition(e)11

1 -:i

Original jComposition (a) —

1 • 1 » ' 1 ' 1—1—t-J

0.0

Xd

DfllFigure 3.4: Composition profile for example ternary problemattr»=0.5.

oo 8o

>

a.

4

12

-24-

till -T-r 1 1 —T"! I 1

(r _

1 1 1 1

-

-

- JLb-

, L.VlU_ 1 1 1 1 till

0.0 0.25 0.5 0.75 1.0

Gas SaturationFigure 3.5: Variation of tie-line and nontie-line eigenvalues

along the intial tie line.

12

oo 8

:>

—T-n 1 1 1 i 1 —n 1 1 —TT I 1

i-

/

\lI1..

r 1 1-1- 1 1 1 1 1 1 1

0.0 0.25 03 0.75 1.0

Gas SaturationFigure 3.6; Variation oftie-line and nontie-line eigenvalues

along theinjection tie line.

-25-

lelative permeability equations, rig and were both set to three. Thus, small changes in

saturation cause large changes in relative permeabilities, which are reflected as a large change

in fractional flow, and hence the tie-line velocity - which is proportional to - also vanes

rapidly. Consequently in the region of continuous variation, a large change in velocity isaccompanied by only a small change in saniration and overall composition. This fact is seen

clearly in Figures 3.3 and 3.4.

At point c the path switches from a tie-line path to the nontie-line path that is tangent_to

the tie line at point c. The composition path continues along the nontie-line path from pomt c

to d. Notice that between points c in Figure 3.5 and point d in figure 3.6, the velocity

(eigenvalue) increases. It must be remembered that the only allowable path switch is from afast path to a slow oath when tracing the composition path_from the-initial_CQropQSition_to_the

injection oil composi^on, but the velocities along apath may increase or decrease, reflectingthe behavior of the composition variation as it is propagated through the system. In this case

the wave between points cand d ^Therefore the composition path is drawn

as adashed line on die ternary diagram in Figure 3.2 to reflect the fact that there is ajump, not

a continuous variation between the two points. Figures 3.3 and 3.4 show the step change in

. both the overall composition and saturation between c and d. Because the variation is not

continuous, amaterial balance must be calculated to determine the velocity of the discontinuity

(Eq. 2.32). Also note that the overall mole fraction of butane on the upstream side of the self-sharpening wave i^M^, All of the butane moves in abank near the leading edge shock wheremiscibility is developing.

Point d lies on the tie line that passes through the injection gas composition. Thus the

final in the composition path is between points dand e. If the nontie-line path between

points cand dhad intersected the injection tie line at asaturation less than that of the smgularpoint, there would have been acontinuous variation followed by a tangent shock. This is

'tWsbehavior is mresult of the f«ct that the sIop« of the tie lines increase as they move away fromtheCOj- C,o binaiy face.

-26-

obviously not the case heie. TTie final jump is a nontangent shock to flie injection gase. Calculation of component material balances aaoss the shock (Eq.2.32) results

in a wave velociqr that is less than the nontie-line velocity at point d, Aus^there is a stepchange in velocity. TWs fact is best illustrated in Figure 3.6 which shows the step change invelocity that occurs at point d. Because there are two different velocities associated with thesame overall composition d. feis path switch results in the formation of a zone of constant state

or fluid bank. This region is labeled inFig 3.3.

Figure 3.7 shows recovery curves for butane and decane.

^ 60

I I J I 1 1/ I I

— —— Butane

• Dccane

1 2

PV Injected

Figure 3.7: Recovery of decane and butane for aternary displacement by COj.Tlus slope of the recovery curves up to one PV injected is not unity because of the volumechange that occurs as the components partition between phases. Tbt pore volume scale inFigure 3.7 is based on Ae densiqr of die pure injected fluid. Once fliat COi dissolves in theoil. however, it occupies less volume, and hence, the volumetric production rate is lower than

-27-

tfie injection me. The points where Ae recovery curves change slope conespond to thebreakthrouBh of the leading shock, flie self-sharpening wave and the trailing edge shock. Notethat all of the butane has been recovered when the self-sharpening wave breaks through, butthe df?"" is not completely recovered until the traUing shock breaks tteough. This is as"expected because butane is more soluble than decane in the COj. The recovety curves indicatelhat avery efficient displacement occurs even though the composition path passes through thetwo-phase region. In fact, this displacement would be "miscible" if the definition of miscibilityis 95%recovery at 1.2PV injected.

With most of the concepts and feawres of asolution path understood, the next task is to

develop the solution path for a quartemary system. The same two rules apply for paththat is, the solution must vary from the initial to the injection com^smon along

paths, and the only allowable path switch is from i^pto an iptwrnS^. to acjlow^g!,

-28-

Section 4: Composition Paths in a Four Component System.

The discussion that follows makes use of quatcmaiy phase diagrams. A quaternary

diagram is a four component phase diagram that is plotted within a tetrahedron. Each comer

of the tetrahedron represents 100 mole percent of a given component and the base opposite a

given comer represents 0% of that component. Planes which cut through the tetrahedron

parallel to the base corresponds to increasing mole fractions of the component which is

represented at the apex. A point within the quaternary diagram represents an. overall

composition for a four component system just as a point in a temary diagram represents an

overall composition for a three component system. The faces of the quaternary diagram are

temary diagrams because they are phase diagrams in which one of the component mole

fractions is zero.

Figure 4.1 is a plot showing composition paths for the CO^ — —̂ lo system.

CO2

Slow paths

Fast paths

Equivelocity curve

'10

Figure 4.1 - Composition paths for COi ~ Cj - C\o system. I at 1600 psia and 160 **F.

This figure coupled widi Figures 2.2 and 3.1 show three of the faces of the

I C

-29-

COj - Ci - C4 - C)o quaternary diagram examined in this section. Understandmg these threesimplifies the interpretation of the quaternary diagram. Figure 4.2 shows selected

composition paths traced on the CO2- C, - C4- C,o quaternary phase diagram (1600 psia

and 160 ®F).

Horizontal paths

Vertical paths

Locus of

plait points

Slow

Intermediate

Fast

COz- Cw C«f

Ci to

Figure 4.2 - Composition paths for CO2 - Ci - C4 - Ciq system* at 1600 psia and 160 ®F.

For afour component system, the matrix problem Eq. 2.21 yields four independent eigenvaluesand associated eigenvectors for each point within the two-phase region. Three of theeigenvectors represent composition variations and founh corresponds to the change in totalvelocity resulting from volume change on mixing.

-30-

Qualernary Grid Topology

Several features of flie grid topology are worth noting. Because there are threeeigenvectors associated with each overall composition, there are three paths thM pass throughM.y point within the two-phase region. Following HetfferiMs terminology the paths arelefetred to as the "slow", "intermediate", and "fast" paths. As in the ternary case, tie lines are

and the characteristic velocity associated with atie line is proportional to Figure 4.3

shows aplot of the eigenvalues' (composition velocities) vs gas saturaUon along atypical tieline.

Oo

>

12

-T-1 1 1 •• -T-i 1 1 ••

rie-line pat]

-r-n I •

1 . -

- 11 -

/ \ Nontie-l\ ^

ine paths'

V ^

^ / 5f/\ 1 1

lingular poj 1 IJ-L.

ints v.

1 1 1 1 t

0.25 0.5 0.75

Gas Saturation

Figure 4.3 -Variation of composition velocity along alie line.

Notice that the tie-line path is slow at either end of the tie line, fast in the central region andiKermediate in between. TOs fact is also illustrated in Figure 4.2 by showing the velocities of

—: ; « Keures 43 «nd 4.5 - 4.7. include a multiplicative factor ii/^ . To normalize •-c in Tab.es 4.2 - 4.7. <Uv«c by U.. «»-

pies discussed here, «/$ =5.

-31 -

the paths with different line widths.

In addition to the tie-line path, there are also two nontie-line paths associated with a

given composition (see Figures 4.2 and 4.3). As shown in Figure 4.2 die irontie-line paths

move both "horizontally", as in the Ci - C4 - Cio face and the CO2 -Ca" Cjo face, and

"vertically" as in the CO2 - Cj - Cio face. The terms "vertical" and "horizontal" are often used

to describe these paths. Figure 4.2 and 4.3 shows that there are four points along a tie line

where the nontie-line paths are ungent to the tie-line path. Two of the singular points are at

the intersection of a tie-line path and horizontal nontie-line paths, and the other two singular

points are the intersection of tie-lines line path and vertical nontie-line paths path. Two of thesingular points, one horizontal and one vertical path intersection point, are on either side of the

equivelocity surface.

The equivelocity surface is an intermediate path which can be traversed either venicallyor horizontally. Finally, the two-phase surface is also apath {Dumore et al, 1984).

The displacement conditions for this problem are similar to those used in the ternary

example in §3. At time zero, pure CO2 is injected into aporous media containing single-phaseoil whose composition is constant throughout. The injection velocity is 1.0 m/d and the

*porosity is 20%. In the example reported in this section, asmall amount of methane is addedto the "oil" of §3 to study how the dissolved gas affects composition paths and displacement

efficiency. The initial and injection fluid compositions and densities for this example are givenin Table 4.1. The same relative permeability data that were used in §3 are used in this section.

0u

Composition (mole %)Methane Butane Decane

Molar

Densitv(gm/cc)

Injection Gas 1.00 • 0.0 0.0 0.0 0.4133

Initial Oil 0.0 0.010 0.416 0.574 .3766

Table 4.1 - Compositions and Molar Densities of the Initial and Injected fluids.

^ c

CO2

Trailing edge shock--

-32-

Leading edge shock

Figure 4.4 - Solution path for the example quaternaiy problem.Path Description

The solution path is plotted on a quaternary diagram in Figure 4.4 and the composiuon,

saturation and velocity data are given in Table 4.2. The path enters the two phase region, as .

before, via a shock to a fast path along a tie-line from the initial oil composition, pomt a, to y

point b. Across Ag-^ock-there-is-a-step-in-methane-concentration^with no Atpoint b the path switches immediately from a fast path to an intermediate path. The stepchange from the initial composition to the two-phase region is not a limit of a continuousvariation. Hierefore. it is a nontangent shock. Figure 4.5 illustrates how the composition

velocities (eigenvalues) vary along the initial tie line. The path switch which occurs at point bin Figure 4.4 is shown in Figure 4.5. Because there are two velocities associated with thesame overall composition, a fluid bank is formed. This is illustrated clearly in the saturationand composition profiles shown in Figures 4.8 and 4.9. Along the vertical, intermediate

-33 -

Displacement of oil(Ci. C4 .Cjo) by CO2 in a Four Component SystemAt IWO psia and 160 ®F.

Comment Composition (mole %)Total

VelocityGas -Saturation

wave

Velocitv

CO2 C, C4 Cio u (m/d) S V (m/d)

InjectionComposition

1.0 0,0 0.0 0.0 1.0 1.0

Slow Shock 0.9580 0.0 0.0 0.0420 0.9537 0.919 0.2787

Zone ofConstant

State

0.9580 0.0 0.0 0.0420 0.9537 0.919 0.76

SelfSharpeningWave

0.85 0.0 0.098 0.0515 0.9588 0.66 0.76

Continuous

Variation

0.8469

0.84370.83950.46830.1773

0.0

0.0

0.00.00100.2027

0.3847

0.4989

0.09950.10080.10200.16710.20910.2332

0.0535

0.05550.05750.16200.2290

0.2678

0.9588

0.95880.95900.9888

1.00101.0210

0.636

0.6110.5900.375

0.3230.298

0.9691

0.9826

0.9827

0.9910

1.00101.0071

Zone ofConstant

State

0.0 0.4989 0.2332 0.2678 1.0210 0.298 1.0607

Fast Shock 0.0 0.4989 0.2332 0.2678 1.0210 0.00 1.0607

Initial OilComposition

0.0 0.0100 0.4160 0.574 0.9489 0.0

Table 4.2 Composition, saturation and velocities for a quatemaiy displacement path.

*oo

%>

12

-34-

-T—r-i 1 jiP~\ « 1 -I—I 1 \

_1.1—L_

/

-

> I —

f»n

cr

-

/ 1 1 j

/

--1 f t 1

Gas Saturation

Figure 4.5: Variation of tie-line and nontie-line eigenvaluesalong initial tie line.

Gas Saturation

Figure 4.6: Variation of tie-line and nontie-line eigenvaluesalong thecross-over tie line.

12

O 8V

>

-35-

-Ill 1

A-T 1 1 1""" 1 1 -1" 1

-

-

/

j/\ 1 I 1 1-1—L-

V1 1 l-J-

^1.

'-^1 ] \

0 0.25 0.5 0.75 1

Gas Saturation

Figure 4.7: Variation of tie-line and nontie-line eigenvaluesalong Injection tie line.

§^ 0.6i

0.4%/icd

o0.2

trailin? edee shock

zone ofconstant state

-36-

,.'.0 =.CD\ •

C"*'

L f a

§ 1

p2 0.8

PUa>

•o 0-62"g 0.4

Ie- 0.2B

d n

leading t6%tf nshock ^ ' / -

I « » I I—I—I—I—L« • I t i J I 1 L1

0.25 0.5

Xd

0.75

Figure 4.8: Saturation profile for example quartcmaiy problem

T—r

—s

p J

• «s: o

-.1- f

&t t^ =0.5.

|L J_l—!-i-I'e ' I ' ' • •rr.

' 1 -• — Decane• - • Butane ~~

C02—— Methase

jL—I t—I—0.25

I

OS

original

imposition (a)

A

0.75

Rgure 4.9: Composition profile for example quartcmaiy problemat « 0.5.

-37-

velodty path between points b and c the concentration of the COi is increasing while theconcentration of flie other fliree components is decreasing. The path moves along the vertical

path tip to point^c^hich lies in the COi ~̂ 4 ~̂ lo After point cthe path is exactly asitwas in the ternary case. From point c to d there is acontinuous variation along atie line. At

point d the solution switches from atie-line path to anontie-line path which is tangent to thetie line at that point. Between points d and e the wave velocities increase, and as in the

ternary case, a self-sharpening wave is formed. Point e is on the tie Une which extendsthrough the injection gas composition. Finally there is ajump from e to f. the injecuon gascomposition. The velocity of the nontangent shock is calculated using Eq. (2.32). The velocitycalculated is greater than the composition velocity associated with the injection ne line,therefore the jump is not the limit of acontinuous variation and atraiUng bank is fonned.

Figures 4.5, 4.6 and 4.7 indicate that the route described does not include path switchesthat violate the rule that wave velocities must deaease from the initial to the injecuoncomposition. The first switch is at the leading edge shock point b. Fig. 4J shows that thispath switch is from afast to an intermediate path. Between points band e, the compositionpath varies continuously along the intermediate path. Path switches occur at points cand dwhere eigenvalues are equal (Figure 4.6). The only other path switch occurs at the trailingedge shock where the path switches from the intermediate path to aslow path at point e. Thecomposition route described above travels along paths. Therefore the description of thecomposition route given above is valid, but it has not yet been shown conclusively that it is theonly legitimate path. It is necessary, therefore, to investigate other possible paths to determinewhether or not Aere are other routes which meet the criterion for path construction.

When considering die possible composition paths, it is necessary to remember that thepath must enter the two-phase region via ashock along atie line. The tie line which passesthrough the injection gas composition is on the CO2 - C,o bimuy edge, and for the oil, (whichcontains no COj), the initial tie line must lie in the C, - C4 - C,o ternary face. IHus, it ispossible fliat the path may either move vertically to the COj-€4-0,0 face and then

-38-

horizontally to tfie injection tie line, or it may move horizontally to the CO2 - Cj - Cjo face

and then vertically to die injection tie line.

Calculation ofmaterial balance velocities for shocks to compositions on^^the lower side

of the cquivelocity space result in composition velocities which are slower than the average

interstitial velocity —. and hence there could be no fast shock, which, of course, is impossible.

Therefore in order for there to be a leading edge shock, the step change from initial oil

composition to tfie two-phase region must cross the cquivelocity surface. This fact aloneeliminates many possible solution paths.

Suppose diat the leading edge shock lands on afast portion of atie-line path with atwo-phase composition which is between die equivelociQ^ surface and the one intermediate verticalpath which is also tangent to atie line in the COj- C4- Cio face. Figure 4.10 show whatseveral different these paths starting from this point would look like. If that shock were a

tangem shock, two composition variations from the landing point are possible. TTiecomposition path could either switch immediately to the intermediate vertical path whichpasses through the landing point (a) ' or there could be acontinuous variation along the tieline. If the immediate switch occurred (Figure 4.10 (a)), the path would then travel vertically tothe COi - C4 - C,o face, and arrive on afast tie-line path (point b). From this point, the onlypath switch which could occur would be an immediate switch to aslow horizontal path. Eventhough the slow path reaches the entry tie-line, the injection shock velocity calculated from Eq.(2.32) is faster than the nontie-line path velocity, apath switch that is not allowed.

Suppose that instead of the immediate switch at point a. there was acontinuous vanauonalong the tie line path. The next path switches to be considered are at overall composmonsto are beyond the vertical path which is tangent to Ae COi - C4 - Cm face (for examplepoint c). Now there are again die two possibilities, an immediate switch to the intermediatevertical path or acontinued continuous variation. If the former took place, it can be seen in

• Thetennl^ is used 10 describe the lide of the equivdocity curve where the gas ««uiralion is low.' This is what would happen ifthe teiding shock were «nonlangcDt shock.

-39-

Equivelocity curve

Initial composition

Figure 4.10 (a-b) -Hypothetical paths which do not meet the requirements forpath construction.

-40-

Figure 4.10 (c) -Hypothetical paths which do not meet the requirements forpathconstruction.

c

r C

-41 -

Figure 4.10 (b) that the path does i>ot reach the CO^-C,- C,o face. TTie path could continueto the singular point d. at which point the interniediate path changes from being avertical pathto being ahorizontal tie-line path. The intermediate path switches from being atie-line path tobeing anontie-line pad> at the next singular point e. Continuation along that path would carryAe overall composition to the COj - C, - C,o face (point f). Now the only possible path toAe injection tie line which passes through fis avertical path. The vertical path through pointf is afast path, therefore aswitch to that path from the intermediate nontie-line path is notallowable. If there had been acontinuous variation after point c, it can be seen that beyond thenext singular point (point g) the only vertical paths are fast paths, and switches to those pathsare not feasible.

•me last path which should be discussed is one which upon landing in the two-phaseregion continues to the singular point g. At that point the intermediate horizontal tie-line pad,switches to anontie-line pad» (Figure 4.10 (c)). But continued variation along this path resultsin acomposition on the CO, - C. - C.o face at point h. m tie-line path associated wimpoint gis aslow path and switching to diat path is adead end. as there is no vertical pathwhich can cany the composition variation to the entry tie line. THe only vertical path is afastpath and switching to that path is not allowed. Similar arguments can be used to eliminate allpaths except that shown in Figure 4.4.

It is important to note here that neither the leading nor the traUing edge shocks aretangent shocks, and the overall fractional flows and samrations on the two-phase side of theshocks canL.t be deu^rmined without knowledge of which nontie-line composition path wastraversed to arrive at the given bounding tie line. How then is the path to be determined if wecamiot start at one boundary and work to the other as was done in the ternary example?

•me key to finding the solution path for the quaternary system is the detennination of the•cross-over" tie-Une. R.ints dand ein Figure 4.4 are on this tie line. Tliis tie-line is keybecause it has vertical and horizontal nontie-line paths that are tangent to it which also intersect.he entrance and exit tie-lines. This idea is illustrated in Figure 4.3. TWs idea is important

c

-42-

because it implys that for each tie-linc in the CO, -C,- C,o face there is an associated tie-line on the C, - C4 - C,o face. This observation will prove useful when analyzing the results.Thus, construction of asolution for given initial and injection composition requires that theCTOSS-Over tie line be located. In general, atrial and error solution is required. The followingsteps can be used to find that path for asystem with no CO, present initially (refer to Figure4.4).

1) Guess point e on the injection tie line.

2) Follow the horizontal nontie-line path to the wss-over tie ^Wint d. follow die crossK.ver tie line path to the singular point c. and follow thevertical tie line to the C, - C4 - C,o base at point b.

3^ Determine whether the tie line through point bpasses through the^ ^mSn If not adjust point eand repeat steps (2) unul the landing point

in the C, - C4 - Cio base is on the exit tie Une.

If .he amount of C4 along the base tie line extension is too low. point eshould be movedto ahigher mole fraction of CO,. Otherwise move point eto lower CO, concentrations. Asimilar procedure can be used if the initial mixture lies within the interior of the tetrahedron,but the test of whemer the final tie line passes through .he initial composition becomes morecomplex.

-43-

Section 4.1: Effect ofOil Composition on Development ofMiscibility

Based on the discussion in §4 titiat shows that the path description given is/u^^e^^dsatisfies the criterion for path construction, solutions for several different "live" oils can be

generated to study quantitatively the effects of tiie presence of dissolved methane on

displacement efficiency. First, the composition path for an oil containing methane is compared

with the ternary composition path from §3 to see how recovery is effected in this vaporizing

gas drive system. Next, the development of miscibility is investigated. Miscibilitydevelopment can be simulated in one of two ways. The composition of the displaced oil can

be enriched with intermediates (C4). or the system pressure can be increased to reduce the size

of the two phase envelope. The idea behind both of these techniques is to move the overallcomposition of the displaced oil outside the region of tie-line extensions. Theoretically, only

oirs whose composition lies outside the region of tie-line extensions can develop miscibility

through multiple contacts. The last question to be investigated is how varying the amount ofmethane present in the initial oil effects the displacement behavior.

Comparison ofLive and Dead Oil Displacements

Comr""""" and saturation profiles for the problem outlined in §4 are given in Figures4.8 and 4.9. Comparison of these figures with those for the methane free oil illuminate thefiuidamental differences in displacement behavior resulting from the addition of methane to the

dead oil. Several facts are worth noting. The saniration profile shows that there are two zones

of constant state at points b and e. Between the zones of constant state, there is a self-shaipening wave from dto eand acontinuous variation from d to b. One of the primarydifferences between the profiles in Figures and 3.3 isjie.position.of tHeJeading edge ,

jihock. Overlaying the two figures also indicates that flie volume of the gas sawrated zonetwhinH the displacement front is significantly greater in the system where methane is present.

All of the resulting volume increase is between the oil composition aand the leading shock forthe three component system (poim con Figure 4.?). Hie composition profile indicates that the

within that region is mostly methane. Behind point c there is no methane, an

-44-

indication that all of the methane moves in a bank ahead of the COi front .The increase in gas

volume is reflected by the change in the total velocity + u^. Remember that the porosity is

constant and the flow is one dimensional, thus an increase in total velocity is equivalent to

increasing the flow rate, which means that there will be a larger gas saturated zone in the live

oil system. When methane is present, there is less shrinkage volume resulting from the mass

transfer of components between phases. When CO2 dissolves in the live oil, methane also

transfers to the vapor, replacing much of th volume of the CO2 that dissolved. When CO2

dissolves ina dead oil, it occupies much less volume, and hence the total velocity must decline

to compensate for the volume change on mixing. At tp =0.5, the position of the leading

shock in Figure 3.3 is at =0.47, and at the same time the shock in the four component

system has advanced to =0.58. Most of the volume change occurs between b and c.

Between points c and b, there is acontinuous variation and throughout this region the overall

composition is becoming enriched with methane and the COi concentration goes to zero.

Behind the methane bank in the zone where the phase compositions are fixed but the

saturations are varying, tiie butane travels in a bank. This is exactiy what was seen in the

ternary case. Thus the lightest component methane, has apparently volatilized into the vapor,phase and has traveled out ahead of the CO2 front. The butane is the next component to besolubilized by the CO2. Just as in the ternary case, all of the butane travels as abank between

the COi front and the self-sharpening wave. Decane, the heaviest component is the lastcomponent to be extracted completely at the solubilization shock.

Figure 4.11 is a plot of butane, decane and methane recovery, which shows how theeffluent composition varies as the displacement is carried out Up until breakthrough therecovery of each of the components is the same because the oil recovered up to that pomt hasnot been elfected by the presence of After that point, there are several noticeable breakover points which correspond to the breakthrough of CO2 . the self-sharpening wave, and thetrailing edge shock. Again, nearly all of the methane is recovered at CO2 breakthrough, andall ofthe butane is recovered after the self-sharpening wave breaks through. When companng

80

E:*^ 60o

40

20

-45 -

• • I T—1—r-j 1^ 1 1 1 1 1 -L—u_

—

/ . - - - . Methane—

- j Butane —

/ TVr.ane

-

r 1 1 1 1 i 1 1 1 1 i 1 1—L 1

0 12 3

PV Injected

Figure 4.11 - Methane, butane and decane recovery curves.

the results of the displacements in §3 and §4 it is apparent that the breakthrough of the leading

edge shock occurs earlier in the four-component case, (again this occurs because of the

reduced volume change of mixing caused by the presence of methane), but the ultimate

recovery occurs at the same time for both systems. This second fact results because there is

litde difference in the saturation and composition profiles behind the self-sharpening wave.

Development of MiscibilityI

In this section four oils with differem amounts of butane are studied. Addition of butane

moves the overall composition of the oil towards the CO2 - Ci - C4 face of the quaternary

diagram. By analyzing the four composition paths in this section, it is possible to investigatehow miscibility development is effected by the presence of methane. The amount of methane

in each of the four oils studied is held constant in order to see if the presence of methane

might aid in the development of miscibility for aheavy or volatile oil. Also, the question ofwhether the presence of a methane bank is indicative of an immiscible displacement can be

^ C

fls [

-46-

investigated.

The four oils and aportion of their composition path data are tabulated in Tables 4.3, 4.4.and 4.6 and will be referred to as systems 1. 2. 3and 4respectively. Tlieir paths are all

shown on a quaternary diagram inFigure 4.12.

CO2

Systems

Figure 4.12 -Composition paths for systems 1. 2. 3, and 4.

Hie portion of the data which has been eliminated is between the trailing edge shock and theback edge of the leading zone of constam state. In the discussion that follows we examine how^S^"o"irmd"^locities"^Ae shocks jto see what happens as miscibility develops, for these

"STme key parameters and zones which determine ulUmate recovery and breakthrough times.Several of the important changes that occur as d.e overall composition moves towards the

COi-Ci-C, face should be highlighted. First, the velocity of the trailing solubilizationshock inaeases from 0.147 mid to 0.83 mid (the velodty of pure injected fluid is 1.0 mid).•nius the addition of butane causes the trailing shock to move more rapidly. When asystemhas truly developed miscibility. in the absence of other transport mechanisms such as

-47-

Displacement ofOU (Cj. C4, Cjo) by CO^ in a Four Componem systemat 1600 psia and 160 ®F.

Q

Comment Comoosition (mole %)

Total

VelocityGasSaturation

Wave

Velocity

CO2 C, C4 Cio u (m/d) S V (m/d)

InjectionComposition

1.0 0.0 0.0 0.0 1.0 1.0

Slow Shock 0.92 0.0 0.0 0.08 0.9453 .147

Methane

Bank

0.0

0.0

0.51200.5120Aft ^ A

0.12670.1267A 10A*7

0.36130.3613

1.03171.0317

1 0317

0.30100.3010

0.0

1.02351.1466

1.1466Fast Shock

Initial OilComposition

0.0

0.0

0.5120

0.1000

U.1/0/

0.2085

u.ooi j

0.6916 0.9314 0.0

Table 4.3 Shock composiuons and velocities for system 1,an "oil" with low butane concentration (p = 0.3553 H^-moUfr)

Displacemem of Oil (C, A C,o) by COj in aFour Component Systemat 1600 psia and 160 ®F.

Comment Cbmpositioln (mole %)C,n

TotalVelocity_u (m/d)

Uas

Saturation

S

Wave

VelocityV (m/d)

InjectionComposition

CO2

1.0

Ci

0.0

/\ f\

C4

0.0

n A

v^lO

0.0

n 0490

1.0

0.9537

1.0

0.919 0.2787Slow ShockMethane

Bank

0.9580

0.00.0

0.0

0.49890.4989

Af\OC\

U.U

0.23320.2332A

0.26780.26780 2678

1.02101.0210

1.0210

0.2980.298

0.00

1.00711.0607

1.0607Fast ShoQkInitial OilComposition

0.0

0.0

0.4989

0.1000

U.Z33^

0.3823 0.5177* 0.9404 0.0

Table 4.4 Shock compositions and velocities for ^"em^an "oil" with intermediate buune concentranon (p =0.4005 Ib-moUfn

-48-

Displacement ofOil (C,, C4. Cjo) by COj in a Four Component systemat 1600 psia and 160 ®F.

Comment Composition fmole %)

TotalVelocity

GasSaturation

Wave

Velocity

CO2 Ci C4 Cio u (m/d) Saturation V (m/d)

InjectionComposition

1.0 0.0 0.0 0.0 1.0 1.0

Slow Shock 0.975 0.0 0.0 0.25 0.9681 0.9645 0.5029

Methane

Bank

0.00.0

0.492

0.492

0.2745

0.2745

0.2332

0.2332

1.01051.0105

0.28990.2899

1.0042

1.0468

Fast Shock 0.0 0.492 0.2745 0.2332 1.0105 0.0 1.0468

Initial OilComposition

0.0 0.1 0.4484 0.4516 0.9739 0.0

Table 4.5 Shock compositions and velocities for system 3,an"oil" with intermediate butane concentration (p = 0.4206 Ib-moUfr)

Displacement of Oil (Cj, C4, Cjo) by CO2 in iat 1600 psia and 160 *

I Four Component systemT.

L

Comment Composition (mole %)Total

VelocityGasSaturation

Wave

Velocity

CO2 Ci C4 Cio u (m/d) Saturation V (m/d)

InjectionComposition

1.0 0.0 0.0 0.0 1.0 1.0

Slow Shock 0.9560 0.0 0.0 0.015 0.9890 c0..9^Zj 0.8280

Methane

Bank

0.00.0

0.4900.490

0.3040.304

0.206

0.206

1.0205

1.0205

0.2892

0.2892

1.0156

1.0462

Fast Shock 0.0 0.490 0.304 0.206 1.0205 0.0 1.0462

Initial OilComposition

0.0 0.1 0.4968 0.4032 ^.9904-i 0.0

Table 4.6 Shock compositions and velocities for systeman "oil" with high butane concentration (p = 0.4367 Ib-moVfr)

-49-

dispersion, the two phases behave as one and the .resulting slug moves at the average interstmaJvelocity, which in this case is constant and equal to the injection rate. In this system theaddition of butane clearly improves displacement efficiency. The gas saturation just inside thetwo phase region on the downstream side of the trailing shock increases from 0.8 in theheavier oU (system 1) to 0.99 in the more volatile oil (system 4). Note mFigure 4.12 that theportion of the composition path which corresponds to the trailing edge shock is approachingthe two phase envelope as miscibiliQ? develops, reflecting the changes in saturation which areseen. When the system is miscible, the composition path follows "niusas in system 4. most of the hydrocarbons have been solubilized fimher up in the transitionzone, and the system is nearly miscible.

•me second set of importam changes occur towards the leading edge. Figure 4.8 showsAzt most all of the methane moves directly behind the displacement from in abank. Itinteresting, therefore, to note what happens as the system approaches miscibility. Does thebank disappear? The ubulated results show clearly thatjtdoKjSQL From system 1to system2. the velocity of both the leading and trailing edge of the methane bank do decrease, butbetween systems 2and system 4. the size and velocity of the methane bank is not greadyeffected by the increased concentration of the intermediate hydrocarbon component, C4. Also,the gas samration of the bank is unaffected, as it remains at about 30% in all of the systems.TWs occurs because it is impossible to have-an initial composition outside the region of tie-lineextensions on the C, - C4 - C,o base, because the two-phase region spans the the diagramfrom the CO. - C.0 binary edge to the CO, - Qedge. Hius no "oil" composition in ti,e basecan meet the strict definition of miscibiliiy. and avoid_^o-phase regionjomp^y. Itshould be emphasized, however, that the recovery efficiency improves considerably as the«nount of butane is increased. Tlie displacement of system 4would clearly be judged to bemiscible by the criteria of Holm and Josendal (1982) or Ycllig and Metcplfe (1980).

THe explanation for the improvement in recovety is based on the observation that theaoss-over tie lines for the four oUs approach the plait poiM in the CO, - C4 - C,o face as the

-50-

amount of butane is increased from system 1to 4. Observe that for the oils of systems 3and4. the cross-overover tie Unes are verj; near the plait point on the COi - C4 - C,o face eventhough the composition paths fall well into the two-phase region in ihe C, - C4 - C,o face.That behavior is another indication that the presence of methane has Utde effect on thedisplacement efficiency even though it does affect the composition path. The final portion ofthe composition path passes so close to the plait poim and the dew poim portion of the bmodalcurve that recovery is very high despite the fact fliat the displacement passes through the two-phase region.

Effect of the Amount ofMethane in the Original Oil

The next idea to investigate is how the composition path is affected by varying theamount of methane in the original oil. To simplify the calculations necessary for thecomputation of the shock velocities, flie oils smdied in this section were chosen so they alllie alone the same tie-line extension. Hence, the composition path and the saturation inside thetwo phase region behind the leading zone of constam state remain constant for all of the oilcompositions studied. TTierefore. only the shock velocities vary. The compositions of the oilsand their phase densities are given in Table 4.7. Also tabulated are the leading edge shockvelocities, and the ratio of the velocities on either side of the shock for each of the oils.

Comp<3sition (mc>le %)7S

Mo

Dens

0 •

liar

iities0

Shock

Velocity

V,

VelocityRatio

Ci0.01000.10000.20000.30000.39000.4500

C40.41600.38230.34500.30760.07390.2515

Clo0.57400.51770.45500.39240.33610.2985

P/

0.37660.40050.43010.46300.49470.4969

Hv

0.2875

5.3035.4135.6136.0137.002

13.5798

1.07641.07091.06141.0436

1.00411.0

Table 4.7 -Effect of the addition of methane on leading bank behavior.

Tht addition of methane to the oil increases the size of the leading zone of constant state, andas expected, increasing the amoum of methane in the dead oil increases the velocity of the

-51 -

edge shock. In addition, as the methane concentration increases, the amount of volumedecreases, and hence the velocity on the two-phase side of the shock approaches that

on the single-phase side.

It should be noted that studying the leading shock velocities for oU compositions that arealong an extension of one tie Une is not strictly equivalent to smdying the effect of addition ofmethane to adead oil. In theory, the compositions that should be examined fall on the linethat comiects the dead oil composition and the pure methane composition. Figure 4.13 showsthat addition of methane in tfiis manner causes the composition of the oil to move slightly offthe initial dead oil composition tie line.

' c 10

. y

/Initial tie lines

// with ' .

/ dissolved/ methane ^

/ * w * ^

'Dead" oil composition

Figure 4.13 -Change in injection tie line for oils with varying amounts of methane.However for the oil smdied here the difference between tie Unes through the bubble point liveoil composition and the dead oil composition is small. Also, remember that tie lines in theC,-C4-C,o face are associated with tie lines in tfie COj-Q-Qopreviously, small differences between phase compositions on the C, - C4 - C,o result meven

differences between the cross-over tie lines on flie COj - C4 - C,o face. Agam. most

r

. \

-52-

of what determines how oil is recovered in the quaternary system is a function of thecomposition path behind the methane bank. Finally, it was pointed out in the previous sectionthat the saturation of the zone of constant state where the methane is flowinris not affectedgreatly by the addition of butane, which is equivalent to moving off one tie line extension ontoanother. These facts aU indicate that the computations presented above give a goodquantitative feeling for the d>ange in the displacement behavior when there is alarge amountof methane present in the initial oil.

The results shown in Table 4.7 suggest one additional important pomt. Ininalcompositions inside the two-phase region on the C, - C4- C.o face can show recoveries ashigh as diose for compositions the lie outside the two-phase region. Thus, it is possible tohave avery efficient displacement even if enough methane has been added that the recombined-oil" is below its bubble point pressure. As methane is added, the leading shock velocityincreases, but the composition path behind the leading shock remains unchanged. Forexample, the velocity of the leading shock for the highest methane concentration in Table 4.7.acomposition which gives an initial gas sanitation of 0.2884 gives aleading shock velocitylarger than any of thise obtained for methane concentrations in the single phase region. Hence,use of the BPP as the MMP for systems which have BBP's that are above the predicu:d MM?Bay be overly conservative. Instead, it is apparently possible, from aphase behavior point ofview at least, to have anearly miscible displacement in areservoir tiiat is below its bubblepoint pressure.

•n«i effect of addition of CO2 to the original oU can also be seen from the compositionpath shown in Figure 4.4. The vertical path intersects asequence of tie lines in the interior of•he quaternary diag«m. as Ulustrated in Figure 4.14. THus. the cross-over tie line in theCOj - C. - C,o face is associated with asurface made up of tie lines that intersect the verticalpatii that is tangent to the cross-over tie line. Any initial oil composition that lies on anextension of a tie line in that surface follows exactly the same composition path in theCO2 - C4 - C,o face. THUS, fte only effect of changing oil composition within that surfSce of

k' c-53-

Figure 4.14 - Tie line surface associated with a vertical path.

tie line extensions is to change the velocity and composition at the leading shock. As in the

case of "oils" with no CO2 present initially, it is location of the cross-over tie line that has the

largest effect on recovery efficiency.

-54-

Section 4^: Discussion

The path descriptions and results for a quartemary system presented in §4.2 explain some

of the inconsistency between experimental observations and the analytic^ description of

developed miscibility. Consider for example an oil composition which lies ofthe extension of

the critical tie-line in the CO2 - C4 - Cio face. Pure CO2 would be able to develop miscibility

with such an oil after sufficient multiple contacts enriched the injection gas towards the critical

point In other words the displacement pressure would be the MMP exactly. If a small

amount of methane were added to the oil the composition route would no longer remain

exclusively in die face. Instead, the path would be similar to the one

described in §4.2. Thus if there is any methane present the displacement does not avoid the

two phase region, and hence is not miscible it\the strict sense.

Oil recovery, however, remains very high. As was noted earlier, each tie-line on the

CO2 - C4 - Cio face is associated with atie-line on die Ci - C4 - Cjo face. For an oil that isoutside the region of tie-line extensions for the CO2 - C4 - Cjo system but inside the region of

tie-line extensions for the C, - C4-C,o system, the cross-over tie line was very near the

critical point on the CO2 - C4 - Cio face. Hence the saturation of the oil-rich phase at thesolubilization shock is small. Thus, while the displacement is not strictly miscible. it recovers

nearly all of the oil anyway. Hence, it is also not surprising that the expenmental results,which are frequently interpreted based on recovery efficiency, show little change in MMP

when methane is added to the oil.

Several of the authors who have studied the effects ofcomposition on the MMP have

noted that there was amethane bank which preceded the CO2 breakthrough when the displaced

oil contained dissolved methane. RathmcU et al. (1971) were the first to repon such an

observation. The four component composition path computed in fliis report analyncally

predicts similar behavior. In dieir paper, Rathmell et al. suggested that if alight componentsudi as methane virere present in tfie oil, a quaternary diagram would be necessary to display

the r<'T"i"«rinn path. TTiey presented results of displacement experiments and discussed the

< f -55-

lesulting composition path qualitatively, using aCO2 - C, - quaternaiy phase, f diagram. In their composition path description, they argue that overall compositions resulting

from displacemem of acrude oU containing dissolved C. by CO, would pass from the initialoil composition to apoint near the plait pomt on the CO2 - C2-6 - C7. fee®.

C »mt be some C. in the composition at the leading edge of the transition zone. Though theirdescription is slightly different from that presented in this report, the key idea that the mostimportant portion of the composition path must lie along the dew point portion of the phase

C envelop in the CO, -C-^-Cr face is exactly what was found here.Rmhrnell et al. also suggested that the presence of amethane bank might be used to

signal an immiscible process. TTus idea was funher investigated by Mccalfe and Yarborough• (1979). In their work they studied the displacement of oil systems under immiscible, MCM

and fim contact miscible conditions. They concluded that they were unable to determine if thepresence of amethane bank was asufficiently accurate criterion for judging miscibility. m

; C factor contributing to their indecisiveness was the fa« that during their displacementexperiments the methane bank did not completely disappear until the process was first contactmiscible. Because of the wide range of oil compositions which do exhibit high recovery even

; C W the process is not strictiy miscible. this criterion is obviously too conservative to beused as an indication of an immiscible displacement.

On.ot""

, —„ ^ te c, - c. - a. oil«. dumt. I" P""- I" "I"™ " •"

miscibili^ developed, just as noted in results presented here.

-56-

in the recovery curves in §3 and §4. Yellig and Metcalf (1980) stated that two phases were

visible during many oftheir displacements though tiieir presence was not necessarily indicative

of an immicible process. Both of these observation are consistent with the results of this

study. Thus, the qualitative predictions of the one-dimensional theory presented in this report

are consistent with and provide an explaination for experimental observations in similar

systems.