Orr, F. M. - Theory of Gas Injection Processes Episode 14 doc

It is a semishock that is faster than the composition velocities on the downstream side of the shock, the right state for this shock entropy conditions are frequently written in terms of

Bibliography 251

[90] Orr, F.M., Jr and Taber J.J Use of Carbon Dioxide in Enhanced Oil Recovery Science,

page 563, 11 May 1984

[91] Orr, F.M., Jr., Dindoruk, B., and Johns, R.T Theory of Multicomponent Gas/Oil

Displace-ments Ind Eng Chem Res., 34:2661–2669, 1995.

[92] Orr, F.M., Jr., Johns, R.T and Dindoruk, B Development of Miscibility in Four-Component

CO2 Floods Soc Pet Eng Res Eng., 8:135–142, 1993.

[93] Orr, F.M Jr., Silva, M.K., Lien, C.L and Pelletier, M.T Laboratory Experiments to Evaluate

Field Prospects for Carbon Dioxide Flooding J Pet Tech., pages 888–898, April 1982.

[94] Orr, F.M., Jr., Yu, A.D., and Lien, C.L Phase Behavior of CO2 and Crude Oil in

Low-Temperature Reservoirs Society of Petroleum Engineers Journal , pages 480–492, August

1981

[95] Pande, K.K Interaction of Phase Behaviour with Nonuniform Flow PhD thesis, Stanford

University, Stanford, CA, December 1988

[96] Peaceman, D.W Fundamentals of Numerical Reservoir Simulation Elsevier Scientific

Pub-lishing, New York, 1977

[97] Pedersen, K.S., Fjellerup, J.F., Fredenslund, A., and Thomassen, P Studies of Gas Injection into Oil Reservoirs by a Cell to Cell Simulation Model, SPE 13832, 1985

[98] Pedersen, K.S., Fredenslund, A., and Thomassen, P Properties of Oils and Natural Gases.

Gulf Publishing Company, Houston, Texas, 1989

[99] Peng, D.Y and Robinson, D.B A New Two-Constant Equation of State Ind Eng Chem Fund., 15:59–64, 1976.

[100] Perkins, T.K.and Johnston, O.C A Review of Diffusion and Dispersion in Porous Media

Soc Pet Eng J., pages 70–84, March 1963.

[101] Pope, G.A The Application of Fractional Flow Theory to Enhanced Oil Recovery Soc Pet Eng J., 20:191–205, June 1980.

[102] Pope, G.A., Lake, L.W and Helfferich, F.G Cation Exchange in Chemical Flooding: Part 1

- Basic Theory Without Dispersion Soc Pet Eng J., pages 418–434, December 1978.

[103] Ratchford, HH and Rice, J.D Procedure for Use of Electrical Digital Computers in

Cal-culating Flash Vaporization Hydrocarbon Equilibrium J Pet Tech., pages 19–20, October

1952

[104] Reamer, H.H and Sage, B.H Phase Equilibria in Hydrocarbon Systems Volumetric and Phase Behavior of the n-Decane–CO2 System J Chem Eng Data, 8(4):508–513, October

1963

[105] Reid, R.C., Prausnitz, J.M., and Sherwood, T.K The Properties of Gases and Liquids, 3rd

ed McGraw Hill, New York, NY, 1977.

252 Bibliography

[106] Rhee, H., Aris, R and Amundson, N.R First-Order Partial Differential Equations: Volume

I Prentice-Hall, Englewood Cliffs, NJ, 1986.

[107] Rhee, H., Aris, R and Amundson, N.R First-Order Partial Differential Equations: Volume

II Prentice-Hall, Englewood Cliffs, NJ, 1989.

[108] Rhee, H.K., Aris, R., and Amundson, N.R On the Theory of Multicomponent

Chromatog-raphy Phil Trans Roy Soc London, 267 A:419–455, 1970.

[109] Seto, C.J., Jessen, K., and Orr, F.M., Jr Compositional Streamline Simulation of Field Scale Condensate Vaporization by Gas Injection, SPE 79690, SPE Reservoir Simulation Sympo-sium, Houston, TX, February 3–5 2003

[110] Shaw, J and Bachu, S Screening, Evaluation , and Ranking of OIl Reservoirs Suitable for

CO2-Flood EOR and Carbon Dioxide Sequestration J Canadian Pet Tech., 41 (9):51–61,

2002

[111] Slattery, J.C Momentum, Energy, and Mass Transfer in Continua McGraw-Hill, New York,

NY, 1972

[112] Stalkup, F.I Miscible Displacement Society of Petroleum Engineers, Dallas, 1983.

[113] Stalkup, F.I Miscible Displacement Monograph 8, Soc Pet Eng of AIME, New York, 1983 [114] Stalkup, F.I Displacement Behavior of the Condensing/Vaporizing Gas Drive Process, SPE

16715, SPE Annual Technical Conference and Exhibition, Dallas, TX, September 1987 [115] Stalkup, F.I Effect of Gas Enrichment and Numerical Dispersion on Enriched-Gas-Drive

Predictions Soc Pet Eng Res Eng., pages 647–655, November 1990.

[116] Stein, M.H., Frey, D.D., Walker, R.D., and Pariani, G.J Slaughter Estate Unit CO2 Flood:

Comparison between Pilot and Field Scale Performance J Pet Tech., pages 1026–1032,

September 1992

[117] Taber, J.J., Martin, F.D., and Seright, R.D EOR Screening Critera Revisited–Part I:

In-troduction to Screening Criteria and Enhanced Recovery Field Projects Soc Pet Eng Res Eng., pages 189–198, August 1997.

[118] Tanner, C.S., Baxley, P.T., Crump, J.G., and Miller, W.C Production Performance of the Wasson Denver Unit CO2Flood, SPE 24156, SPE/DOE 8th Annual Symposium on Enhanced Oil Recovery, Tulsa, OK, April 22-24 1992

[119] Thiele, M.R Modeling Multiphase Flow in Heterogeneous Media Using Streamtubes PhD

thesis, Stanford University, Stanford, CA, December 1994

[120] Thiele, M.R., Batycky, R.P., and Blunt, M.J A Streamline-Based 3D Field-Scale Composi-tional Simulator, SPE 38889, SPE Annual Technical Conference and Exhibition, San Antonio,

TX, October 5-8 1997

[121] Thiele, M.R., Blunt, M.J., and Orr, F.M., Jr Modeling Flow in Heterogeneous Media Using

Streamlines–II Compositional Displacements In Situ, 19(4):367–391, 1995.

Bibliography 253

[122] van der Waals, J.D On the Continuity of the Gaseous and Liquid States In J.S Rowlinson,

editor, J.D van der Waals: On the Continuity of the Gaseous and Liquid States, pages 83–140.

North-Holland Physics Publishing, 1988

[123] Van Ness, H.C and Abbott, M.M Classical Thermodynamics of Nonelectrolyte Solutions With Applications to Phase Equilibrium McGraw-Hill, San Francisco, 1982.

[124] Varotsis, N., Stewart, G., Todd, A.C and Clancy, M Phase Behavior of Systems Comprising

North Sea Reservoir Fluids and Injection Gases J Pet Tech., pages 1221–1233, November

1986

[125] Wachmann, C The Mathematical Theory for the Displacement of Oil and Water by Alcohol

Society of Petroleum Engineers Journal, 231:250–266, September 1964.

[126] Walas, S.M Phase Equilibria in Chemical Engineering Butterworth Publishers, Stoneham,

MA, 1985

[127] Walsh, B.W and Orr, F.M Jr Prediction of Miscible Flood Performance: The Effect Of

Dispersion on Composition Paths in Ternary Systems IN SITU, 14(1):19–47, 1990.

[128] Wang, Y Analytical Calculation of Minimum Miscibility Pressure PhD thesis, Stanford

University, Stanford, CA, 1998

[129] Wang, Y and Orr, F.M., Jr Analytical Calculation of Minimum Miscibility Pressure Fluid Phase Equilibria, 139:101–124, 1997.

[130] Wang, Y and Orr, F.M., Jr Calculation of Minimum Miscibility Pressure J Petroleum Science and Engineering, 27:151–164, 2000.

[131] Wang, Y., and Peck, D.G Analytical Calculation of Minimum Miscibility Pressure: Com-prehensive Testing and Its Application in a Quantitative Analysis of the Effect of Numerical Dispersion for Different Miscibility Development Mechanisms, SPE 59738, SPE/DOE Im-proved Oil Recovery Symposium, Tulsa, OK, April 2000

[132] Watkins, R.W A Technique for the Laboratory Measurement of Carbon Dioxide Unit Dis-placement Efficiency in Reservoir Rock, SPE 7474, SPE Annual Technical Conference and Exhibition, Dallas, TX, Oct 1-3 1978

[133] Welge, H.J A Simplified Method for Computing Oil Recovery by Gas or Water Drive Trans., AIME, 195:91–98, 1952.

[134] Welge, H.J., Johnson, E.F., Ewing, S.P.,Jr., and Brinkman, F.H The Linear Displacement

of Oil from Porous Media by Enriched Gas J Pet Tech., pages 787–796, August 1961 [135] Whitson, C.H and Michelsen, M The Negative Flash Fluid Phase Equilibria, 53:51–71,

1989

[136] Wilson, G.M A Modified Redlich-Kwong Equation of State, Application to General Physical Data Calculation Paper 15C, presented at the 1969 AIChE 65th National Mtg., Cleveland,

OH, 1969

254 Bibliography

[137] Wingard, J.S Multicomponent, Multiphase Flow in Porous Media with Temperature Varia-tion PhD thesis, Stanford University, Stanford, CA, November 1988.

[138] Yellig, W.F., and Metcalfe, R.S Determination and Prediction of CO2 Minimum Miscibility

Pressures J Pet Tech., pages 160–168, January 1980.

[139] Zhu, J., Jessen, K., Kovscek, A.R., and Orr, F.M., Jr Analytical Theory of Coalbed Methane

Recovery by Gas Injection Soc Pet Eng J., pages 371–379, December 2003.

[140] Zick, A.A A Combined Condensing/Vaporizing Mechanism in the Displacement of Oil by Enriched Gas, SPE 15493, SPE Annual Technical Conference and Exhibition, New Orleans,

LA, October 1986

Appendix A 255

APPENDIX A: Entropy Conditions in Ternary Systems

In this appendix we consider the entropy condition for shocks in ternary systems The derivation

of the entropy condition for the shock between tie lines follows that of Wang [128], which is based,

in turn, on the approach used by Johansen and Winther [51] to study polymer displacements The derivation given here is for a specific system with constant K-values, but the patterns of behavior are the same for systems with variable K-values The use of the constant K-value example is an attempt to illustrate the abstract concept of an entropy condition in a concrete way

Entropy conditions are statements about the stability of a shock, written in terms of the relative magnitudes of eigenvalues of compositions on either side of the shock and the shock velocity If a shock is stable, it must be self-sharpening In other words, if a stable shock were to be smeared slightly by some physical mechanism, it must sharpen again into a shock in the limit as that physical mechanism is removed Dispersion is one physical mechanism that can create a continuously varying composition in place of a jump in composition In a binary displacement, the requirement of a stable shock can be translated easily into a statement about the eigenvalues on either side of the shock For example, the discussion in Section 4.2 states that the eigenvalue on the upstream side of a shock must be greater than the shock velocity, and the eigenvalue on the downstream side must be less than the shock velocity For a ternary displacement, however, there are two eigenvalues at each point in the composition space, so the statement of shock stability in terms of those eigenvalues is necessarily more complex In this appendix we consider the statement of an entropy condition for each of the shocks that can appear in the solution for a ternary displacement, leading, trailing, and intermediate, and we show that if there is an intermediate shock, it is a semishock

Leading Shock

To illustrate the statement of the entropy condition for the various shocks, we consider a specific

case: constant K-values, with K1 = 2.5, K2 = 0.5, K3 = 0.05, and M = 5 The solution for this

example is shown in Fig 5.16 The behavior of the leading shock, which connects a single-phase composition with a composition on the initial tie line, is exactly the same as that described for leading shock in a binary system (see Section 4.2) The leading shock is a shock that arises because

of the behavior of λ t, and it occurs along the extension of the initial tie line It is a semishock that

is faster than the composition velocities on the downstream side of the shock, the right state for

this shock (entropy conditions are frequently written in terms of left and right states, with the left state referring to upstream compositions and the right state to downstream compositions) Those velocities are are all one Fig A.1 shows the relationships between the shock velocity and the

eigenvalues λ t and λ nt for the leading shock The leading shock has a velocity, ΛLR equal to λ L t,

which is indicated by the fact that the line drawn from the right state composition, R, to the left state composition L is tangent to the fractional flow curve The tie line eigenvalue, λ t, is given by the slope of the fractional flow curve

The nontie-line eigenvalue is given by Eq 5.1.24 (see Section 5.1)

λ nt = F1+ p

C1+ p =

F1− C 1e

For constant K-values, the value of p, which is the negative of the volume fraction of component 1

on the envelope curve, C 1e, is given by Eq 5.1.49,

256 Appendix A

p = −C 1e = K1− K2

K2− 1

K1− K3

1− K3 x21 =

x2 1

with γ given by Eq 5.1.50,

γ = 1− K3

K1− K2

K2− 1

In the example considered here, an LVI vaporizing drive, K2 < 1, so γ is negative, as is p for any

tie line

The point labeled C L 1e is the composition at the point at which the extension of the initial tie line is tangent to the envelope curve (see Fig 5.12) Eq A.1 indicates that the slope of the line

drawn from the left state composition, L, to C L 1e is the nontie-line eigenvalue, λ L

nt Comparison of

the slopes for the leading shock and λ nt indicates that the leading shock velocity, ΛLR is greater

than λ L nt on the upstream side of the shock Hence, the relationships among the shock velocity and eigenvalues are

Thus, the leading shock is self-sharpening with respect to the tie line eigenvalue, but it is not with respect to the nontie-line eigenvalue This is another indication that the leading shock is a tie-line shock It must be self-sharpening for variations in the tie-line eigenvalue across the shock, but need not be self-sharpening for the nontie-line eigenvalue

Trailing Shock

In a ternary vaporizing gas drive, the trailing shock may or may not be a semishock Fig A.2 shows the shock constructions If the trailing shock is a semishock, then the shock velocity, ΛLR is

given by the slope of the line from the injection composition L to R t, the point at which the line

is tangent to the fractional flow curve If it is a genuine shock, as it would be for a shock from R g,

then the shock velocity is greater than λ R t, which is given by the slope of the fraction flow curve at

R g

The point labeled C R 1e is the point at which the extension of the injection tie line is tangent to

the envelope curve (see Fig 5.12) The value of λ R ntis given by the slope of the line drawn from R g

or R t to C R 1e It is clear from the slopes of the trailing shock lines and the lines corresponding to the nontie-line eigenvalue that the shock velocity is significantly lower than the nontie-line eigenvalue Here again, the shock is self-sharpening with respect to the tie line eigenvalue, but it is not with respect to the nontie-line eigenvalue

At a trailing semishock, then, the eigenvalue relationships are

λ R t

ΛLR < λ R t

and at a trailing genuine shock, they are

Appendix A 257

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Overall Volume Fraction of Component 1, C1 a

a

a

R

L

Figure A.1: Tangent construction for the leading shock The slope of the line from the initial (right

state) composition, R, to the left state composition, L, gives the velocity, Λ, of the leading shock The point labeled C L 1e is the point at which the extension of the initial tie line is tangent to the

envelope curve (see Fig 5.12) The slope of the line from L to C L 1e gives the value of λ nt at L.

0.8 1.0 1.2 1.4 1.6

Overall Volume Fraction of Component 1, C1

a

a a

a

L

Figure A.2: Tangent and genuine shock constructions for the trailing shock The slope of the line

from the injection (left state) composition, L, to one of the right state compositions, R g or R t, gives the velocity, ΛLR, of the trailing shock The slope of the line from R g or R t to C R 1e gives the

value of λ nt at R.

258 Appendix A

λ R g

ΛLR < λ R g

Both the leading and trailing shocks, therefore, are λ t shocks in the sense that they are

self-sharpening with respect to λ t That behavior is consistent with the discussion of shock stability given for binary displacements, which it must be if the ternary displacements are to reduce to the binary solution in the limit as one component disappears or when the initial and injection compositions lie on extensions of the same tie line There is no requirement, however, that these

shocks be self-sharpening with respect to λ nt The situation is reversed for nontie-line shocks that connect the injection and initial tie lines These are self-sharpening with respect to the nontie-line eigenvalue but not with respect to the tie line eigenvalue In the remainder of this appendix, we show why that must be true

Intermediate Shock The arguments given in Section 5.1.4 show that when variation along the nontie-line path is permitted by the velocity rule, the switch/indexpath switch from the tie line path to the nontie-line path must occur at the equal eigenvalue point If the line eigenvalue increases as the nontie-line path is traced upstream, however, then a shock replaces the variation along the nontie-nontie-line path Next we consider possible left and right states for that shock

Again, we consider the LVI vaporizing gas drive example shown in Fig A.3, which is the system shown also in Fig 5.16 Fig A.3 shows the compositions of potential left (L) and right (R) states, and it also shows the locations of the tie-line intersection point and the two envelope points on the

envelope curve that provide information about λ nt , C 1e L , and C 1e R, through Eq A.1 The tie-line

intersection point and the two envelope points do not change with changes in the left or right state compositions, so they are fixed for the purposes of the following discussion

We begin by considering possible landing points on the injection tie line, the left state Three

possible landing locations are shown in Fig A.4, left states L 1 , L 2 , and L 3 The intersection of

the line drawn from R to X with the fractional flow curve for the injection gas tie line (which

contains the left state compositions) gives possible landing compositions that satisfy the shock balance equations We consider each of those compositions in turn and show that only one satisfies all the requirements

The intermediate shock satisfies the shock balances, Eqs 5.2.23,

ΛLR = F

L

i − C X i

C i L − C X

i

= F

R

i − C X i

C i R − C X

i

Eq A.10 is represented in Fig A.4 by the line drawn from R to X For a given value of C R

i , the intersection of that line with the fractional flow curve for the injection gas tie line gives the value

of C i L that satisfies Eq A.10

At left state L 1, λ L t > Λ LR, because the slope of the fractional flow curve at L 1 is greater than

the slope of the shock line from R to X While a shock to L 1 does satisfy the shock balance, it can

be ruled out as a potential landing composition Any subsequent rarefaction along the injection gas tie line would violate the velocity rule, because the intermediate shock would be slower than the

Appendix A 259

CH4

C4

C10

a

a

a

a a

a a

a

a

R L

C1eL

C1eR X

Figure A.3: Composition path for a vaporizing gas drive with low volatility intermediate component

K1 = 2.5, K2 = 0.5, K3 = 0.05, and M = 5 (See Fig 5.16 and the accompanying discussion for a

description of the full solution and for the corresponding saturation profiles) Points L and R are the left and right states of the nontie-line shock Point X is the intersection point of the tie lines connected by the shock Points C L 1e and C R 1e are the tangent points on the envelope curve for the tie lines that contain the left and right states for the shock

260 Appendix A

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Overall Volume Fraction of Component 1, C1

a

a a

a

a

R

X

Figure A.4: Shock constructions for landing points of the intermediate shock on the injection gas tie line

compositions just upstream In addition, a direct shock from L 1 to the injection gas composition would violate the entropy condition for the trailing shock (see Section 4.2) Hence, a shock from

R to L 1 is prohibited

Point L 2 is the tangent point for a line drawn from the tie-line intersection point, X to the fractional flow curve for the injection tie line At L 2, λ L t = ΛLR This landing point can also be ruled out As the shape of the fractional flow curves in Fig A.4 show, the shock construction line

(X to L 2) does not intersect the fractional flow curve for the initial tie line (For the example of

this appendix, with constant mobility ratio, M , and x L1 > x R1, a tangent drawn to the fractional flow curve for the longer tie line does not intersect the fractional flow curve for the shorter tie line More care is required to show that a similar statement is true for more complex phase behavior

and mobility ratio that is not constant.) Thus, there is no solution for a shock that lands at L 2,

where λ L t = ΛLR, and satisfies the shock balance equations

Point L 3 is an acceptable landing point, however At L 3, the slope of the fractional flow curve

is lower than the slope of the shock line, and hence λ L t < Λ LR Variation along the injection gas tie line to a trailing semishock point would be consistent with the velocity rule, and an immediate genuine shock to the injection composition is also allowed Hence, we conclude that at the landing

point on the injection gas tie line, λ L t < Λ LR

Next we consider possible right states on the initial oil tie line Fig A.5 shows three possible

jump points on that tie line Point R 3 can be ruled out immediately At R 3, λ R

t < Λ LR, as

comparison of the slope of the fraction flow curve at R 3 and the slope of the shock line indicates

In other words, the intermediate shock moves faster than the compositions on the rarefaction along the initial tie line, a situation that would violate the velocity rule

Point R 1 is also not an acceptable right state, although more effort is required to show that it