,Effects of Dispersion and Dead-End Pore Volumein Miscible Flooding

L. E, BAKERMEMBER SPE-AIME

ABSTRACT

The design of the solvent slug size /or a miscibleflood process can be improved with data on holdup(or capacitance effects) and dispersion of thesolvent slug in the reservoir. A modified version ofthe Coats-Smith dispersion-capacitance mode I andan improved solution method for the model wereused to study dispersion avd capacitance effectsin cores. The velocity dependence of the modelparameters is sbotvn. A correlation is developed forestimating effective dispersion coefficients forfield application.

The method described provides a means forcharacterizing the properties of dispersive mixingand micro heterogeneity of reservoir cores and aidsin the design of the volume o~ solvent for misciblefloods.

INTRODUCTION

The amount of solvent that must be injected is acritical factor in the success of a miscible flood.Because of the cost of miscible solvents such ascarbon dioxide or rich gas, slug processes generallyare used. If the solvent slug used is larger thannecessary, the solvent cost will be increasedwithout compensatory increases in oil recovery. Iftoo small a slug is used, some of the oil that couldhave been moved will be left behind. The slug sizerequired is affected by many variables, includingreservoir geometry, interwell spacing, gravityeffects, mobility ratios, etc. Slug degradation iscaused by mixing (by dispersion) of solvent withoil at the leading edge of the solvent bank and withchase fluid (for example, dry gas) at the trailingedge. Trapping of oil and solvent in microscopicheterogeneities (regions of dead-end pore volume orrelatively stagnant flow) also contributes to themixing-zone growth. This trapping, known ascapacitance, may be caused by rock heterogene-ities 1 or by shielding of oil and solvent by waterfilms. 293

Original manuscript received in Society of Petroleum Engineersoffice July 23, 1975. Paper accepted for publication Nov. 21,197S. Revised manuscript received Jan, 24, 1977. Paper (SPE5632) was first presented at the SPf+A2ME SOth Annual FallTechnical Conference and Exhibition, held in DaI1as, Sept. 28-Oct. 1, 1975. (@ Copyright 1977 American Institute of Mining,Metallurgical, and Petroleum Engineers, Inc.

2%1spaper will be included In the 1977 ?%msactfone volume.

JUNE.1977

I AMOCO PF?ODUCT/ON CO.TULSA-This paper is concerned with predicting solvent

slug requ~rements in an idealized linear systemwhere gravity, mobility ratio, and areal sweepeffects are unimportant, but where longitudinaldispersion (mixing at the leading and trailing edgesof the bank) and capacitance effects are significant.An example might be a miscible displacement Inthe pinnacle reef formations of Alberta. A predictionof the effects of dispersion and capacitance wasneeded for the design of a miscible flood of thistype. The oil-column height was about 350 ft, andthe flood advance rate was to be downward at0.0384 ft/D. The oil/solvent viscosity ratio of 10was unfavorable; however, it was expected that theunfavorable mobility effects would be largelycompensated for by the stability effects of gravityat the low flow rate.

Publish ed data4-6 relating to similar reservoirsindicated that stagnant volume that could causetrapping and degradation of the solvent slug mjghtbe as much as 10 percent of the reservoir volume,Based Oi] these data, preliminary calculations weremade using the Coats-Smithl dispersion-capacitancemodel to predict the mixing-zone profiles. Theresults indicated that this level of stagnant volumemight cause the solvent requirement to be increasedby 30 to 90 percent over the amount predicted by asimple dispersion model without capacitanceeffects if the peak solvent concentration in theenriched gas bank did not drop below 99 percentthroughout the life of the flood.

Coats and Smithl indicated that tests in shortcores would show extended mixing zones ifcapacitance effects were present, but that if themagnitude of the traq,$fer group MD = ML/u waslarge (as it woulck be i; a field situation, where Lmay be very large), the influence of capacitancewou Id be minimized. The prediction of a 30- to90-percent increase iii solvent requirements for thecase described above prompted a review of methodsfor measuring capacitance effects and a search fora more convenient method for predicting the severityof capacitance effects in field application. Animproved method for modeling data from short coretests was developed, and experimental work wasperformed to investigate the factors influencing thecapacitance-model parameters.

219

DISCUSSION

CAPACITANCE MODEL

Dispersion during flow in porous media has beenstudied from both experimental and theoreticalviewpoints. Perkins and Johnston7 and Greenkornand Kessler8 have reviewed much of the literatureon this topic. Of particular interest are thepapersl~2~6~9- 13 dealing with the effects ofheterogeneity and capacitance on the developmentof mixing zones in porous media.

Capacitance or stagnant pore-volume effects aremost apparent when a concentration step-changemiscible-displacement test is performed. This is atest in which a continuous solvent bank is injected,beginning at time t = O, to displace the in-situ fluidfrom a core. The simple dispersion model charac-terized by Eq. 1,

JC ac=ac-ax%

.(1)aX2

when solved using the appropriate initial andboundary conditions,b yields a solution for theeffluent concentration:

()C = 0.5 erfc wm ()+ 0.5 e K) erfc * .(2)The concentration profile predicted by Eq, 2 isnearly symmetrical around the PV = 1 point, asshown in Fig. 1. Also shown in Fig. 1 IS a typicaleffluent concentration profile exhibiting capacitanceeffects. The asymmetric profile is attributed toholdup of in-situ fluid in regions of stagnant orvery SIOW flow, with subsequent bleeding out ofthe in-situ fluid as the mixing zone passes.

Several models incorporating dispersion andcapacitance have been proposed. Each modelincludes dispersion and convection in a flowingregion of the core and fluid transfer between flowingand stagnant regions. The model described byCoats and Smith 1 was chosen for the presentanalysis. Experimental studies performed by Coatsand Smith 1 and by Goddard 12 indicated the transfercoefficient, M, of the model was velocity-dependent,

1.00

IL,0

o.w /,/

/

0.60 OISPIRSION--WITH CAPACITANCE

/

0.40/

If

0, xl /

/

o. m0.0 0.2 0.4 0.6 0.8 1.0 i.? 1.4 1.6 1.8 2.0

HYDROCARBON PORE VOLUMES OF FLUIO INJECTIO

FIG. 1 EXAMPLE EFFLUENT CONCENTRATIONPROFILES.

The models proposed by Turnerg and Gottschlic~10assumed a velocity-independent transfer coefficient,while the Coats-Smith model does not impose thislimitation.

Brigham6 showed that Eq. 1 could describe eitherthe flowing (effluent) or the in-situ concentration,depending on the boundary conditions chosen forsolution of the equation. Appendix A shows that asimilar choice of boundary conditions determineswhether the in-situ or the effluent concentrationprofile is obtained from the Coats-Smith capacitancemodel. The method described in Appendix A ismuch simpler than the method proposed by Brigham6to determine the effluent concentration for thecapacitance model, and results in the followingequations:

K*IU acf%

+ (1-f) g+ ,ax2 x

. . . . . . . . . . . . . . . . . (3a)

and

~~-f) ac+~= M(C-C+), . . . ..(3b)

to be solved with the initial and boundary conditions

c+ (xJo) = o @x>o. o.. .. (4a)

c (X,o) = o @x>o . . . . ..(4b)

c (Ott) q 1 @t>() . . . . ..(4c)

c(oo, t)=o Ant . . . . ..(4cI)These equations are derived from the equationsobtained by Brigham by means of a simple trans-formation of variables.

Eqs, 3a and 3b may be solved in the s (complexfrequency) domain by taking the Laplace transformsof the equations and combining the resultingordinaty differential equations. The solution withinitial and boundary conditions of Eqs, 4a through4d is

. . . . . . . . . . . . . . . . . . . . .(5)Eq. 5 may be inverted to the time domain by the

use of the Cauchy Integral Theorem 14:

c (x)t)=+ ~jw ~(c), stds . . . . . .(6)-P

However, the numerical integration involved isquite time-consuming, especially when it must bedone many times during the iterative process offitting the ~odel to the experimental data.

2m SOCIETY OF PETROLEUM ENGINEERS JOURNAL

The form of Eq. 5 suggests that, rather thantransforming the model solution to the time domain,the data could be transformed into the frequencydomain, An algebraic curve-fitting procedure thencould be used, reducing the computing time andinaccuracies introduced by the numerical integrationof Eq. 6. A method for transforming time-domaindata to the frequency domain and for fitting themodel is outlined in Appendix B. This method wasfound to be quite satisfactory. An additionaladvantage of this procedure is that the frequencyresponse of the flow system can be obtained directlyfrom step, impulse, sinusoidal, or many otherconcentration signals.

EXPERIMENTAL PROCEDURE

Step-change tests of miscible displacement incores were made with benzene (p = 0.604 cp, p =0.8734 gin/cc) and meta-xylene (p = 0.584 cp, p =0,8601 gin/cc). The flow system is shown in Fig.2. A constant-rate positive-displacement pumpmanifolded to two fluid reservoirs controlled flowrates at 1.45 x 10-5 to 9.88 x 10-3 cm/sec (0.041to 28 ft/D). A three-way valve at the core inletdetermined which fluid was injected into the core.The pump maintained equal pressure on both fluidreservoirs to minimize pressure surges when flowwas switched from one fluid to the other, Volumesof the core end-caps and inlet and outlet tubingwere less than 1 percent of the core volume.

A Berea sandstone core (chosen for its homoge-neity) and a vugular limestone core (whichpreliminary tests indicated was heterogeneous)were used. The cores were sealed with epoxy resinand reinforced with Fiberglas tape. The sandstonecore (Core 1) was 22,86 cm long and 5.o8 cm indiameter, with a permeability of 175 md and aporosity of 21.4 percent. The limestone core (Core2) was 9.4 cm long and 7.60 cm in diameter, andhad a permeability of 5,4 md and a porosity of 11.9percent. No brine saturation was present in thecores. The cores were evacuated before beingsaturated with liquid.

The procedure for each test was to saturate thecore with one of the miscible fluids and establisha stable flow rate. The injection valve was thenswitched to the other fluid, and injection continuedat the same rate as samples of the effluent fluidwere taken for analysis. Concentration profileswere monitored by gas chromatographic analysis of

DPUMP {BENZENE

3- WAY VALVE

n

!YLENE

WATER CORE

L I J II

SAMPLECOLLECTIONl!!!?

FIG, 2 FLOW SYSTEM.

the effluent samples. The precision of the sampleanalysis were ~ 1 percent.

For the initial runs, samples were collectedmanually. For later tests, an in-line gas chromato-graphy with an automatic sample valve was used. Inthese later tests, a backpressure of JO psig wasmaintained at the sample-valve outlet. Repeat testsshowed no effect of the change in backpressure orsampling procedure.

The run variables are listed in Table 1, andexample effluent profiles are plotted in Figs. 3an; 4.

DISCUSSION OF RESULTS

The capacitance model was fitted to the data ofeach run as described in Appendix B, Fig. 3 showsthe experimental data and the best-fit calculatedpoints for Run 2. The data of Runs 1 through 3 (inthe 23-cm Berea core) were fitted very well by thesimple dispersion model (to which the capacitancemodel reduces when / = 1.0). This is illustrated bythe straight-line fit of the data of Run 2 on theprobability plot 1s of Fig. 4.

Runs 4 through 10 were made v.ith a vugularlimestone core at flow velocities of 0.04 to 19.9ft/D (1.45 x 10-5 to 7.02 x 10-3 cm/see). Anexample concentration profile and best-fit calcu-lated curve (for Run IO) are shown in Fig. 3. Theconcentration profile is also shown on the probabilityplot of Fig. 4. The curvature of the data as shownon this plot indicates that some type of holdup(capacitance) occurred.

Fig. 5 shows the dispersion coefficients forRuns 1 through 10. The dispersion coefficients

TABLE 1 RUN VARIABLES

Model ParametersDisplacing Displaced Flow Rate

Run tire Fluid Fluid.

(CIII/S@C X 105) (ft/D) (m Cn&c x 105) f (s@c-lX 106)11 Mets-xylene Benzene 104 2,95 19.321 Benzene

1.000 -Mets-xyl ene 490 13.9 157

31 Benzene1.000 -

Mets-xylene 988 28.0 301 1,000 -4 2 Mets-xylene Benzene 1.45 0,047 1,36 0.36352 Benzene

1,80Mets-xylene 14.7 0,416 28.6

620.670 5,46

Mets-xylene Benzene 30.6 0.668 85.3 0.404 13,872 Eerrzene Mete-xylene 31.9 0.803 46.482

0.562 20,0Benzene Met&xylene 109 3.08 225 0.691 25.5

9 2 Mets-xylene Benzene 123 3.49 174 0.579 53.610 2 Met&xylene Benzene 702 19,9 1,590 0.666 222

J!.NE, 1977 221

increase with velocity; the line shown connectingthe points for the limestone core has a slope of 1.0and that for the sandstone core has a slope of 1.25.This indicates that the correlations suggested forhomogeneous cores (Eq. 7) is also valid for coresexhibiting capacitance effects:

~ , ~l. oto 2,0 . . . . . . . . . .(7)

Fig. 5 also shows the transfer coefficients, M,computed for Runs 4 through 10. The data indicatea dependence of M on velocity to the 0.84 power,Coats and Smith* also noted a possib!e velocityvariation of M, as did Goddard. 12 If fluid transferwas dependent on transverse diffusion alone, Mshould be independent of velocity. The velocitydependence of M confirms that the transfermechanism involves mixing other than diffusion,

The flowing fractions (/) for Runs 4 through 10are shown in Fig. 6. A large amount of scatter inthe data is evident; / values range from 0.383 to0.691, with no apparent velocity correlation. If thetwo lowest values are disregarded, the other datapoints are fit fairly well by an average value of / =0.64. Some variation of / with velocity might beexpected; however, the data of Fig. 6 do not justifyany conclusion except that the flowing fractionappears to be constant in these runs for velocitiesgreater than 10-4 cm/sec (O. 28 ft/D).

l.al ~ -r-@--w+-! -& , ..- - -

F

JJ~ io.IxJ 1. .L ..-. ..1. l_ d0.0 0.5 1.0 1.5 20 25 3.0 3.5 4.0 4.5 5.0

FOR[ vOLUMIS lNJftl[O, I

FIG. 3 EFFLUENT CONCENTRATION PROFILES.

Zm

[

mRW2 .

1.54 A RLN 10 b.l

Lal l. I

0.50 - .

/flu

I-1 o,~ . *Sn

~a.aemmm~emeeao l*

+.30 ~ . l*l*

-Lm - . !.b-1,54-.

-zoo -

-250, , , ,0.010 0.100 O.m 0.6UI awl o. Wo

EFFUKNTCONCLNTR41IW

APPLICATION

The number of data points that can be used inmatching the capacitance model to experimentaldata may be limited by the repetitive and time-consuming numerical integration of the inverseLaplace transform (Eq. 6). The frequency-domaincurve-fitting procedure outlined in Appendix B doesnot require the inversion of the Laplace transform;thus, for a given amount of computing time, thefrequency-domain fit allows the use of many moredata points. The result is a more precise fit of themodel to the data.

Application of the capacitance model to predictionof solvent concentration profiles in field applicationalso can be simplified. Stalkup2 and Brigham G haveshown that for reservoir-scale systems, the concen-tration curves predicted by the capacitance model(Eqs. 3a and 3b) approach the symmetrical profilepredicted by the simple dispersion model (Eq. 1).The apparent or effective dispersion coefficient(Ke), however, is greater than the true dispersioncoefficient (K). To confirm this, mixing-zone lengths(the distance between the IO- and 90-percent solventconcentration points for a step-change test) werecalculated (using the capacitance model) forseveral values of K and M at 100 and 1,000 ft offlood advance. For each set of parameters, the

m 0)S3 1 DISPERSION

,0-2 X 03SS 2 OISPIRSION

l CORE 2 MASS TRANSFER

1~

m@ x

*

~o-i

.&10+

a

10+ ]~+ @ 10-3 ,0-2..

VELOCITY, WSEC

FIG. 5 VELOCITY DEPENDENCE OF THE DIS.PERSION COEFFICIENT AND MASS -TRANSFER

COEFFICIENT.

:r-

0.al I~y 10-2

FIG, 6VELOCITY DEPENDENCE OF THE FLOWINGFRACTION.FIG, 4 EFFLUENT CONCENTRATION PROFILES.

222 SOCIETY OF PETROLEUM ENGINEERS JOURNAL

mixing-zone length increased as the square root ofthe distance that was traveled. The simpledispersion model also predicts growth of the mixingzone with the square root of distance traveled;thus, it appeared that it should be possible to usethe simple dispersion model (with an effectivedispersion coefficient) to predict the mixing-zonegrowth for field applications.

To determine the effects of M and / on theeffective dispersion coefficient, mixing-zone lengths(10- to 90-percent solvent concentration) were cal-culated for a range of values of the dimensionlessparameters NBO, MD, and /. The effective dispersioncoefficient, K@, was defined by 15

LMZ 3.62S W........(8)For constant values of K, M, /, and u, the

effective dispersion coefficient increased asymp-totically to a limiting value, Klim, as NBO increased(increasing L), The limiting values, expressed asK1im/K, were correlated as a function of f and thedimensionless group

~= MDINBO O o-;(9)

as shown in Fig. 7. The curves shown in thisfigure are for values of NBO greater than 1,000, andwill overestimate Ke for lower values of NBO.

It follows from Eq. 8 that the ratio Kc/K is alsothe square of the ratio of the mixing-zone lengthswith and without capacitance effects:

Ke ,()

LMZ2 . . . . . . . . . .. (10)T LMZr

Fig. 7 shows that, for a given dispersioncoefficient and velocity, the mixing-zone length isless for a larger value of / (less stagnant volume)or for a larger value of the transfer coefficient. Therange of KMln2 values shown includes thosemeasured in the present work and those mentionedin earlier papers. 1~2*4-G~12

The correlation of Fig. 7 can be derived fromEqs. ja and 3b. If Eq. 3b is differentiated with

t\\\x

0.001 a01 0.10 1.0Y

FIG. 7 DEPENDENCE OF THE EFFECTIVE DIS-PERSION COEFFICIENT ON THE MASS-TRANSFER

COEFFICIENT AND FLOWING FRACTION.

respect to t and the resulting expression for r3C+/dtis substituted into Eq. 3a, the dispersion-capacitance model becomes

[1#~+,,+g.~$-ug=%

~ ax;..........(11)

This is analogous to the simple dispersion model(Eq. 1), with an effective dispersion coefficient,Ke, defined by

&&

-( )

.l+~2 K ,.. (12)K &_

2 3X2

If we assume that d2C/dx2 and d2&/dx 2 areequal (at a large rravel distance, L), the values ofKc/K given by Eq. 12 agree within a few percentwith the values shown in Fig. 8 for Khf/u2 valuesgreater than 0.001. For smaller values of KM/u2,the approximation d% /dx2 = &C+/dx2 is appar-ently invalid; as M approaches zero (no fluidtransfer between flowing and stagnant regions), theeffective dispersion coefficient should againapproach the true dispersion coefficient.

Deansll used a different approach to derive arelation similar to Eq. 12, but limited in applicationto small values of the group KM/u2. His correlationpredicts thar Kc/K will approach zero for largevalues of KM/u2 or for values of / approachingunity.

This correlation shows that the solvent concen-tration is a function of the dimensionless groupsNBO, KM/u2, and f, and that it is the group KM1u2rdther than MD that determines whether capacitanceeffects will be negligible in a reservoir application.Thus, capacitance effects do not disappear with~~creasing system length, as implied by Coats and~mith I and Stalkup. 2

-w1?

-i10-111 t , , ,1q 1 , , I , I o I

104 10-3 10-2

Ke, CM21SEC

FIG. 8 ERROR IN MODEL FIT AS A FUNCTION OFEFFECTIVE DISPERSION COEFFICIENT.

JWE, 1977 223

For the field example cited in the Introduction,the dispersion group NBO was about 12,250. Thetransfer group MD was estimated to range from 46(Case 1) to 132 (Case 2). Concentration profilespredicted by the simple dispersion and thecapacitance models for these values, and with aflowing fraction of 0.9 (10 percent {stagnantvolume), are shown in Fig. 9. The 10- to 90-percentmixing zone is 11.2 ft long for the simple dispersioncase, increasing to 15.45 ft (Case 1) or to 21.7 ft(Case 2) with the inclusion of capacitance effects.The same prediction can be obtained from Fig. 7;at KM/u2 = 1,07 x 10-2 (Case 1), the Kc/K ratio is1.96, corresponding to a mixing-zone length ratio ofw= 1.4. At KM/u2 = 3.74 x 10-3 (CaSe 2), theKc/K ratio is 3.61, corresponding to a 90-percentincrease in mixing-zone length.

.

EFFECT OF PARAMETER VALUES ON MODEL FITThe correlation of Eq. 12 suggests that it may not

be possible to specify unique K, M, and / values tofit the data of any given run. If one of the valuesis specified arbitrarily, the others can be chosento obtain any desired value of Ke (within certainlimits). To investigate this possibility, the error infitting the model to the experimental data wascalculated for a range of values of the parameters(K and M values ranging from 0.1 to 10.0 times thebest-fit values found by the fitting proceduredescribed in Appendix B, and values of / rangingf 0.2 from the best-fit values). The results areshown in Fig. 8 for a representative case as a plotof E (sum of squared errors) vs K@. Nea: thebest-fit region (Ke = 1.2 x 103 sq cm/see), thereare several combinations of K, M, and / that giverelatively good fits. However, there are mar,y morecombinations giving the same valuee of Ke that donot fit the data well, Examination of the error in fitas a function of each of the variables K, M, and /shows that there is a relatively narrow region foreach parameter within which values can be chosento produce acceptable fits.

SUMMARY

A rapid and efficient method for matching adispersion-capacitance model to experimental data

L m

%am -

~~

o.al -

a mm I IMl 350 360 370

DI STANCE, FE~

FIG. 9 PREDICTED MIXING ZONE AT 350 FTi

224

was developed and used to model unit-viscosityratio, laboratory miscible-displacement tests over awide rc.nge of velocities. The method allowsfrequency response data to be obtained withnonsinusoidal input signals.

The dispersion coefficients, K, and fluid transfercoefficients, M, for the dispersion-capacitancemodel were found to be velocity-dependent, whilethe flowing fractions, /, were relatively independentof velocity.

A correlation was developed for estimating theeffective dispersion coefficient for field applicationfrom the values of the dispersion coefficient,transfer coefficient, and flowing fraction derivedfrom short core tests. It was shown that / and thedimensionless group KM/u2 determine the value ofthe effective dispersion coefficient.

Application of the capacitance model to anexample reservoir shows that the exaggeratedtailing noted in laboratory tests using short coresis not as pronounced in longer systems. However,mixing-zone lengths and solvent require~n~s maybe increased significantly by capacitance effectsin reservoirs.

c.Cj =c.C* .

c+ .

h-=

/=

FA .

9=/=j=

K=K= =

Klim =

L=f?(c) =

M=MD =

LMZ =

LMZ, =

n=

NBO &

%=s=

NOMENCLATURE

{in-situ concentration of injected fluidconcentration of injected flttid at core inletflowing concentration of injected fluidconcentration of in jetted fluid in stag-

nant regiondefined by C+ = C* - (K/u) (dC*/dx)sum of squared errors in model fitfraction of pore volume readily available

for flow; (1 -/) is the stagnant fractionampIitude ratio (see Appendix B)imaginary part of Laplace transformpore volumes of fluid injectedW-dispersion coefficient, sq cm/seceffective dispersion coefficient (defined by

Eq. 8), sq cm/seclimiting value of Ke (at large values of

uL/K)core length, cmLaplace transform of Cfluid transfer coefficient, seeldimensionless transfer group, ML/umixing-zone length ( 10- to 90-percent sol-

vent concentration), cmmixing-zone length without capacitance

effects, cm (reference length)integral number of cycles of phase lag (see

Appendix B)dimensionless dispersion group (uL/K);

Bodenstein numberreal part of Laplace transformLaplace transform variable, @

SOCIETY OF PETROLEUM ENGINEERS JOURNAL

.1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

1s.

16.

t = time, sectn = time at which the output concentration

returns to its initial valuetl = time at which the output concentration C

reaches steady-state valuetq = time at which the input concentration Ci

17.

18.

19.

reaches a steady-state valueu = velocity, cm/secx = distance relative to core inleta = proportionality constant, defined by K = au~ = phase angle (see Appendix B)p = viscosity, cpp = density, gin/ccu = frequency, radians/see

REFERENCES

Coats, K. f-f. and Smith, B. D.: tDead-End PoreVolume and Dispersion in Porous Media,$ Sot. Pet.Eng. J. (March 1964) 73-84 Trans., AIME, Vol. 231.Stalkup, F. I.: Displacement of Oil by Solvent atHigh Water Saturation, jr SO=, Pet. Errg. J. (Dec. 1970)337-348.

Shelton, J. L, and Schneider, F. N.: The Effects ofWater Injection on Miscible Flooding Methods UsingHydrocarbons and Carbon Dioxide, ~ Sot, Pet. Eng.J. (June 1975) 217-226.AERCB Application 4404, Supplementary Information,Texaco, Inc., Wizard Lake field (Oct. 1969, 197 1).AERCB Application 5065, Hudsons Bay Oil andGas Co., 2ama-Virgo field (Msrch 1970).Brigham, W. E.: {tMixingEquationa in Short Labora-tory Cores, !) SO=. Pet. Etz& ]. ( Feb. 1974) 91-99;Trans., AIME, Vol. 257.Perkins, T. K., Jr,, and Johnston, O. C.: A Reviewof Diffusion and Dispersion in Porous Media, Sot.Pet. Eng. ], (March 1963) 70-84; Trans., AIME, Vol.228.Greenkom, R A, and Kessler, D. P.: Dispersionin Heterogeneous Non-Uniform Porous Media, k!& f?ng. Cbem. (Sept. 1969) 14-32.Turner, G. A,: The Flow Structure in Packed Beds, Cbetn. Ens Sci. (1958) 156-165.Gottschlich, C. F.: /lAxial Disperaiofl in a PackedBed, AICbE Jour. (1963) 88-92.Deana, H. A,: A Mathematical Model for Dispersionin the Direction of Flow in Porous Media, Sot. Pet,Eng. ]. (March 1963) 49-52; Tratis., AIME, Vol, 228,Goddard, R. R.: Fluid Diapersion and Distributionin Porous Media Using the Frequency ResponseMethod With a Radioactive Tracer, Sot. Pet. Eng.~. (June 1966) 143-152; Trans., AIME, Vol. 237.Goaa, M. J.: Determination of Diapersion andDiffusion of Miscible Liquids in Porous Media by aFrequency Response Method, paper SPE 3525presented at the SPE-AIME 46th Annual Fall Meet-ing, New Orleans, Oct. 3-6, 1971.

Churchill, R. V.: Complex Variables and Applica-tions, McGraw-Hill Book Co., Inc., New York (1960)118.

Brigham, W. IL, Reed, P. W., and Dew, J. N.:~~ExPeriments on Mixtig During Miscible Displacementin Porous Media, Sot. Pet. Ens J. (March 1961) 1-8; Trans., AIME, Vol. 222.Clements, W. C., Jr., and SchnelIe, K. B., Jr.: rPulseTesting for Dynamic Analysis, S ln~ & t%g. Chem.

Proc, Design and Deu, (1963) 94-102.

JUNS, 1S77

Filon, L. N. G.: Proc., Royal Society of Edinburgh( 1928) vol. 49, 38-47.Hougen, J. 0.: Experiments . and Experiences inProcess Dynamics, *~ Cbem. Eng. Prog,, MonographSeries (1960) Vol. 4, 33-34.

Hays, J. R., Clements, W. C,, and Harris, T. R.:~~~e Frequency Domain Evaluation of Mathem$tk&l

Modela for Dynamic Systems, AIChE Jour. (1967)374-378.

APPENDIX A

DETERMINING CONCENTRATIONPROFILE OBTAINED

Brighamb pointed out that the dispersion equation

ac=ac*C ._K% .. ., . . ..(A-l)b% at

which describes the in-situ concentration, C, of afluid flowing through a porous medium, has a formidentical to that of the equation

~ 32C _aJ , acax TiT ....

.(A-2)~

This equation describes the flowing concentration,C, which is equivalent to the effluent or cup-mixingconcentration. The two concentrations are relatedby

K acc=c-~~.... .(A-3)

Whether the in-situ or the flowing concentrationis obtained upon solution of Eq. I or Eq. 2 dependson the initial and boundary conditions chosen. Thein-situ concentration is obtained when the followingconditions are used:

C(X, O)= O, X= O . . . . . . ..(A-~a)

C(m, t)=o, t=o . . . . . . ..(i%dq

C(o,t) = I +: ~ , t 2?0 . l (A-4C)

The flowing concentration (which is the oneusually measured in laboratory core tests) isobtained when the conditions

C(X, O)=O, x20, . . . . .( A-5a)

C(w, t)= (l, t~(j , . . . . .( A-5b)

C(O, t)=l, t= O , . . . . .( A-5c)

are used. The differences in the two solutions areminor except when the dimensionless dispersiongroup NBO . ux/K is small (for example, 20 or

22s

.less). However, this is precisely the range of NBOthat is most likely to be encountered In testsperformed in reservoir core samples.

The same problem appears in the application ofthe Coats-Smith 1 three-parameter capacitance modelto short core data. The model may be written as

. . . . . . . . . . . . . . . .. (A-6a)

a co(1 -f) ~~

= M(C -C). , . . .( A-6b)

or, with a transformation of variables, as

~ a*c m .f ac + (l-f) -# . . (A-is)-uax~ax2

(l-f) ~ =M(C -C+}. . . . . (A-7b)

Again, C and C are related by

C,.CX3C ,.. . . . . . . (A-8)u ax

and C+ and C * are related by

C+=c+$....... (A-9)Substitution of Eqs. A-8 and A-9 into Eqs. A-6a

and A-6b leads directly to Eqs. A-7a and A-7b.The proper boundary conditions for solving Eqs.

A-7a and A-7b to obtain C are Eqs. A-5a throughA-~c.

APPENDIX B

FREQUENCY-DOMAIN FITTING OFEXPERIMENTAL DATA

The solution of the capacitance model in thetime domain is given by a complex numericalintegral:

c(x,t)= z% ~jmX(C) et ds . . . . .(B-l)-jce

where .$?(C) is given by

X((v.. .

J3c,) e,w,+/~j

. . . . . . . . . . . . . . . . . .(B-2)

The computing time and round-off errors involvedin evaluating this integral are undesirable, espe-cially for an iterative curve-fitting process thatrequires the integration to be performed many times.

A more satisfactory approach is the transformationof the experimental data to the frequency domainand fitting of the model in the frequency domain.This can be accomplished easily using the followingmethod, as described by Clements. 16

The Laplace transform of a function C(x, t) isdefined by

where s = jm. The frequency u may be chosenarbitrarily, within certain limits. The trmtsforrn maybe written as

co

X(C) = { C(x, t) [COS (@t) -j sin (Wt)] dt. . . . . . . . . . . . . . .

.(B-4)

Consider the response of a flow system to aconcentration input pulse that begins at time t = 0,rises to a finite value, and decreases again to zero.The output concentration also will rise from zeroto a finite value, then will drop again to zero at afinite time tn. Thus, the integral of Eq. B-4 needbe evaluated only for the finite period for whichC(X, t) + o:

Z(C) = {C(L t) [COS (Wt) - j sk (tit)] dt. . . . . . . . . . . . . . . . . . . (B-5)

The result is a complex function,

ac) = t?(x,td)+ jd(x, ~) . . . . . (B-6)with real and imaginary parts 9 and 9, respectively.The numerical integration of Eq. !3-5 can beaccomplished quickly and accurately using Filonsmethod. 17

For an input signal that does not return to itsinitial value but does reach and hold a steady-statevalue within a finite time (such as a step-change),the method of Walsh and Wiesner18 m~y be used totransform the data. This method is equiva!.ent totransforming the derivatives of the input and outputsignaIs, and is performed as follows:

( mC fx,t)e StdtX(c) .m f= Ci (O, t) e t dt

o

ymacg p e -5t dt.>, t~c. (O,t)e -Stdt (B-7)

a

The equivalence of these ratios is based on thefact that C(Y., O) = Ci(O, O) = O. Since the input andoutput functions reach steady-state values at finitetimes t2 and tl, respectively, then

226 SOCIETY OF PETROLEUM ENGINEtiRS JOURNAL

ac (x t)~=ot?t=t l...... (B-8~)at

and

2Ci (O, t)at

O@tt2 . . . . . ..(B-8b)

Thus, Eq. B-7 can be integrated by parts to give

tl

C (x, tl)e -Stl + ~x(c) J C (xjt) e tit:~(Ci) 0 t2

Ci (O,t 2) e-Stz +5

J Cl (O, t) e-St ~

o

. . . . . . . . . . . . . . . . . .(B-9)Both these integrals are finite, and can be evaluatedfrom the original experimental data without furthermanipulation.

It should be noted that the input signal Ci maybe a step, pulse, or any other wave form withsufficient frequency content to be Laplace-transformed. It is not limited to a step change asused in this paper.

Fitting of the model to the data then proceeds bythe following steps:

1. The experimental data are transformed to thecomplex (frequency) domain using Eq. B-9 for aseries of values O ~ 0< timax (~max is chosen asthe value of ~ at which the absolute value of thetransfer function Y (C)/l? (C~) is equal to 0.3).16

Values of o are chosen by trial and error,beginning at u = O, such that for each succeedingvalue of w the absolute value of the transferfunction !? (C)/!? (Ci) is abour 93 to 95 percent ofthe transfer-function magnitude for the precedingvalue of ~. Other techniques used to choose thefrequencies (within the range specified) resulted indifferent frequency sets, but the same best-fitvalues of K, f, and M.

2. A material balance is made by integrating thedimensionless effluent concentration profile withrespect to time. (In most cases, the integral wasfound to be within 2 percent of its theoreticalvalue: core pore voIume divided by injection rate.)The model fit, in some cases, may be very sensitiveto errors in the assumed velocity; therefore, an

initial velocity estimate, Ul, is made by dividingthe concentration vs time integral into the corelength.

3. Estimates of Kl, /1, and Ml are made.4. Eq. B-2 is evaluated for each value of s = j~

and the sum of squared errors

2

E= IX(c) .&u2!-Zza,c x(Ci) ~XPtl

s

. . . . . . . . . . . . . . . . . . (B-IO)is calculated.

5. A Taylor-series expansion of Eq. B-2 aroundthe parameter estimates of Ki, /i, Mi, and ui isis made to determine the dmection of steepestdescent of the sum of squared errors.

6, New parameter estimates of Ki+l} /i+ 1, Mi+ 1,and ui+ ~ are made, and Steps 4 through 6 arerepeated until a minimum sum of squared errors isreached.

In all cases, the final (best-fit) veIocity differedfrom the estimated (material-balance) velocity onlyby a few percent.

Hays et al. 19 have shown that the method ofleast-squares fitting in the frequency domain isequivalent to least-squares fitting in the timedomain. In this study, few problems with convergenceresulted when reasonable initial estimates wereused, and the substitution of different initialestimates led to essentially the same final parametervalues.

The transformed data are equivalent to the dataobtained from frequency-response testing using asinusoidal input signal for exam le, the methodsused by Goss 13 and by Goddard. $1 The amplituderatio of the concentration signal at any frequencyu is given by

and the phase lag is given by

$ = tan-l (d?/~)+Zn~ . . . .( B-12)where n is the integral number of cycles of lag.

*x~*