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Unstable miscible liquid—liquid displacement in porous media: A new model for predicting displacement performance in homogeneous beds
103
The Chemxal Engmeerzng Journal, 18 (1979) 103 - 116 0 Elsevler Sequoia S A , Lausanne - Prmted m the Netherlands
Unstable Miscible Liquid-Liquid Displacement in Porous Media: a New Model for Predicting Displacement Performance in Homogeneous Beds H H NGUYEN Commonwealth
Industrzal
Gases, Surry Hrlls, Sydney
2010 (Australza)
D F BAGSTER Department
of Chemzcal Engzneerzng,
The Unzverszty of Sydney,
Sydney
2006 (Australza)
(Received 6 December 1977, m final form 27 October 1978)
Abstract A theory of fmgerrng was developed to predtct the volume of the mrxang zone which 1s likely to result during an unstable mtscable dtsplacement m a homogeneous porous bed This theory 1s based on the apphcatton of Darcy ‘s Law to a sample model representing the flow sttuatton, and the calculated energy dtsstpatton wtthm the model Phystcal factors believed to govern the growth of fingers are considered vlscosaty ratio, dens1 ty difference, flow rate, medium porost ty and permea bdt ty. The theory only constders flows m homogeneous beds Avaalable experimental data, those of this present work and of Slobod and Howlett, confirm the tmportance of the first three factors mentioned The expenmental study also found that the model as superror to a previous one of Koval m predactmg mixing zone volumes of displacement processes rnuolvmg both vtscous fmgenng and gravity segregatton However, a method to extend the model to heterogeneous systems was not achieved
INTRODUCTION
The displacement of a liquid by a second miscible liquid m a porous bed is of importance m many industrial processes. Examples are found m the secondary recovery of petroleum by solvent-floodmgs. The idea of using miscible phase displacements to increase recovery of crude oil has been under active study for more than 25 years [l, 21. Other kinds of rmscible liquid-liquid displacement
can be found m the washing stages m the operation of an ion-exchange column, or m sugar refineries where sugar-liquor displaces, or is displaced by, pure water from beds of solid adsorbent. The particular phenomenon being mve,tigated is that of “fingering” due to density and viscosity differences m the liquids concerned [3 - 51. In the present work a new model has been developed for predictmg miscible displacement performance. Expenmental results are produced for comparison with the model. The attempts of Koval [6] and Dougherty [ 71 to predict displacement performance are also taken into account. Koval was successful m obtaining very good agreement between his model and his own experimental data, which involve only viscous fmgermg. However, the method, as has been pointed out, is considered unsatisfactory because Koval had to employ rather arbitrary composition figures (78% oil, 22% solvent) for the so-called effective displacmg agent which he believed must exist to account for the mixing effect within the medium. Dougherty’s model and his trial and error numerical computation found that B, the empirical parameter accountmg for transverse mixing, is velocity and viscosity dependent. The information obtamed from Dougherty’s work, however, is still mconelusive. In this work, different from Koval’s, the approach used is to consider every probable physical factor which is thought to control the behaviour of fingers as they grow m the medium. Another feature of the model is that it also takes into account the gravity segregation effects, as well as viscous finger-
104
mg. The model is described m four stages: (1) description of the model, (2) method of determinmg the finger-zone cross-sectional area Af, (3) effect of transverse mixmg; (4) effect of medium heterogeneity.
Condz tzon 2 hydrodynamzc condz tzon The pressure gradient Inside the fmger zone must be equal to the pressure gradient m the surroundmg fluld. (3)
(grad p)f = (grad PI,
where THE MATHEMATICAL
MODEL
Uf ---ml)
(grad p)f = --F&I The mathematical model, which IS twodimensional, IS based on known facts about finger behaviour m porous beds The first step m the model development IS to replace the actual picture of fmgermg shown m Fig. l(a) by a much simpler one (Fig. l(b)). 0
0'
I
(4)
(grad P), = -(Kc~(zu~ -@z)
(5)
according to Darcy’s Law. Within Zone II, it IS considered that the fluid velocity IS uniform and equal to Uf within the finger zone A’BCD and U, m the surroundmg zone ABCD’. The displacement can then be assessed by determmmg the growth m length h of this finger zone. This can be done by considermg the motion of the finger-zone “front” CD and the finger-zone “base” AB. Putting
and (a)
(b)
Fig 1 Fmgermg m two-dlmenslonal (a) actual, (b) present model
(7)
porous beds
The total volume of fingers IS represented by a finger zone, denoted by the area A’BCD, m Zone II. This finger zone has a crosssectional area equal to a fraction At of the total bed cross-sectional area A. Fluid velocities m Zones I and III are considered uniform and equal to the average displacement rate U, whereas the fmgermg phenomenon takes place only m Zone II. This phenomenon is considered to be governed by the followmg conditions (cf Scheidegger’s concept [ 31). Condztzon 1 con tznuz ty condz tzon Referring to Zone II, one can write
a(1 --Ad + qi4
=Q
(1)
where qf IS the flow rate of liquid Inside the fingers, q, IS the flow rate of the surroundmg liquid, and Q IS the total flow rate. In terms of the mterstitial velocities, (1 -A,)U,
+A,U,
= U
where I, and lf are the distances from the origm 00’ to the base AB and the front CD, respectively (Fig. l), one can rewrite eqns. (2) and (3), which become A,%+(1-A,)s=U
dlf dt
KE,A~
-gp,
= Kw2
2
-@2
(9)
Solvmg the two simultaneous eqns. (8) and (9)
dls -= dt
U - CIAf 1 + (R - l)Af
dlf -=Rz+& dt
(11)
where Cl
= dP1 Kw
P2)
I
and
R =!? E.cl
dlf/dt and dZ,/dt can be calculated numerically with given values of the displacement
105
rate U, the physical properties of the hquids and the medium, and the value A, which is to be determined However, as long as At remains constant, one can write U--&At
h(t) =
l+(R-l)Af
1 t
-Af) l*(t) = lzU+ &(l l+(R--l)Af
I
(13)
t
1 Af =R+1
and U(R -1)
h(t) =
+ cr
l+(R-l)Af
1
(14)
t
This finger-zone length h(t) can only be observed inside the porous bed itself As mixing zone volume V,, 1s defined as the volume of the efflux collected between the moment the first trace of the displacing liquid appears, to, and the moment the last trace of the displaced fluid is detected, tlm, one can write u vMZ
= t
(hO0
(15)
-to)
where
(WW = L[l + CR- 1)&l U - CIAf and t
L loo = (dl,/dt) =
Method of determmmg Af In the model, Scheidegger’s mmimum energy concept [3] was adopted. The energy dlssipated m the displacement process was calculated (see Appendix 1). It was taken that the cross-sectional area Af of the finger zone will have the value correspondmg to the stationary point of the energy dissipation function. This value was proven in the energy calculation (Appendix 1) to be
L[l + (R - l)A,]
(17)
U(R-l)+C1
Hence, the mlxmg zone volume V,,
is
1 + (R - l)A, U - CIAf
1 (18)
This value VMz has been used as the parameter measurmg the efficiency of the displacement process [3, 51. To predict V,, one needs to determine At.
(19)
Effect of transverse maxmg Handy [8] studied the effect of transverse diffusion m miscible displacement, using a double-tracer method m which two soluble tracers were added to the displacmg liquid and one tracer has a diffusion coefficient much greater than the other. If transverse diffusion is important, then substances with high molecular diffusion coefficients will have sharper concentration profiles than those with comparatively low molecular diffusion coefficients. Results of Handy’s experiment with a 30 m. high column show an absence of any appreciable effect of displacement rate or of diffusional separation of the tracers on the observed displacement curves. This led Handy to the conclusion that transverse diffusion has no important effect on the finger growth and shape; as Handy argued, the distances between fingers are large compared with diffusional distances. Another study of transverse diffusion was made by Slobod and Thomas [9]. Experiments were carried out with a two-dimensional porous bed of 18 m. height. Fast displacement rate (29 ftjday) and slow displacement rate (1.6 ft/day) were used to determine the effect of residence time on the geometry of the fingers The shape of the fingers was observed directly with an X-ray technique. These authors reported that at the fast rate numerous narrow fingers were observed, but at the slow rate “a simple somewhat bulgmg finger” was produced. This observation was also supported by photographs. It 1sthus obvious that the amount of material moving transversely by diffusion was sufficient to modify the finger geometry m the slow-rate run because of the long rewdence time. This is in contradiction to Handy’s conclusion about finger shape. On
106
the other hand, Slobod and Thomas also reported that the displacement curves (or breakthrough curves) observed m both cases of slow and fast rates are very much the same. The results obtamed m Slobod and Thomas’s study tend to support those of Handy when only the analysis of the effluent IS considered. Transverse mlxmg is thus not important m determining the breakthrough curves, and hence the mixmg zone. This, however, seems to contradict Slobod and Thomas’s earher observation on finger shape, which was supported strongly by the X-ray evidence Nevertheless, an explanation was proposed by those authors They argued that the material which flows from the core IS produced as a result of the longitudmal motion of fluid m the core. Thus the composition averaged over the entire crosssection of the flowmg stream is not changed by transverse nuxmg of the components withm the core, so long as the longitudmal flow pattern is unchanged. The above facts obtained from the studies of Handy and of Slobod and Thomas thus tend to discount Koval’s consideration that the transverse mlxmg effect is important, and also his concept of effective vlscosrty ratio. However, without this effective viscosity ratio and its fixed composition figures (78/22), Koval’s model is no longer successful m predlctmg his expenmental data. Effect of medmm heterogeneaty Concemmg the medium heterogeneity, Koval based his model on the H-factor concept m chummg that the model is applicable to the heterogeneous porous system. In this model, he claimed that H should be independent of the viscosity ratio and purely reflect the heterogeneity of the medium. A closer look mto Koval’s method of predlctmg the Hfactor, however, will reveal that the magmtudes of those H-factor values determined by Koval are, in fact, dependent on the values of E, the effective viscosity ratio, as H was calculated from K/E, where K was determined by matchmg the expenmental result with Koval’s model. Since the use of KovaI’s composition figures (78/22) is questionable, the validity of Koval’s H-factor values is also m doubt. This cnticlsm ISfurther supported by the fact that, using Koval’s experimental results, the values
of the H-factor determined by Dougherty’s model are much higher than Koval’s values. Added to this, it was found that a plot of Dougherty’s H-factors versus the corresponding values of (Vpl), does not result m a straight line like Koval’s data. This either undermines the validity of Dougherty’s model, as far as heterogeneous beds are concerned, or there is a possibihty that the straight lme of H uersus (Vpl)n established by Koval is Just the result of comcidence The model developed up to this point is applicable to homogeneous beds that have (Vpi), > 99%.
DISCUSSION
OF THE MODEL
The model developed thus provides a method to predict the mixing zone volume V,, resulting from an unstable miscible displacement process m homogeneous beds The expression of the finger-zone length h(t), eqn. (14), indicates a linear relationship of h with time, that is, with the mean distance travelled 1 (= Ut). This is m agreement with the conclusion of Perkms et al [lo] Furthermore, one can write eqn. (14) as h(t) = Nt
(20)
where
(21) Since Af < 1 and the denommator is always positive, the sign of the coefficient N is dependent on the numerator. Thus (a)KeU(cc, -3~~) -i&2 --Ed> We N> the finger zone will grow steadily with time UNSTABLE
0,
(b) KeWs --ccl) -g(ps -PJ< 0, i.e N< 0, any initial disturbance h (at t = 0) will be suppressed* STABLE The above conditions for stability and mstabllity denved from the model are identical with Hill’s [4] cntena for overall stability. In our model, the velocity distnbutlon of fluids inside a fingenng system (Fig. 1) was based on the assumption that cross-flow, which is known to exist around the finger tips
107
and ends [lo] , IS negligible. This assumption is considered reasonable as the volume which is mvolved m this lateral motion of liquid due to cross-flow is much smaller than the volume of hquids m which the actual longitudmal motion takes place In the development of the model, Af was taken to be a smgle value throughout the displacement process. Saffman and Taylor’s observation [ 51 on an immiscible system mdicated a tendency for the fingers to space themselves such that the relative finger width approached 0.50. This value, however, is not applicable to three-dimensional beds [ 111 smce, even if the relative finger width is kept at 0.50, one can see from Fig. 6 that the finger-zone area Af m such beds is not equal to, but much less than, 0.5 of the total flow area. Scheidegger’s mmimum energy concept was thus adopted m the determmation of Af . The energy calculation made m this study, however, has found that the system can finger not only at the mmlmum energy dissipated in certam situations but also at maxunum energy dissipation m another case (see Appendix 1). A proven example of the maximum energy dissipation case has been pomted out by H. Brenner (private commumcation, 1974) and by Harper and Chang [13] m the problem of the motion of a dumb-bell-shaped particle m a slow viscous shear flow. The particle, accordmg to the result of these studies, is shown to have a preferred periodic orbit which corresponds to maxmum dissipation of energy. The model as completed can be used to predict the displacement performance m homogeneous beds only. Koval’s H-factor method is a sound idea to extend the model to heterogeneous systems. However, since heterogeneity of a porous bed also includes rmcroscoplc dispersion, it should be a function of the physical properties of the liquids concerned and the displacement rate [3]. As a result, (Vpi)n and hence the H-(Vpi), relationship are also dependent on those variables. The problem of predictmg the displacement performance m mhomogeneous beds is thus far more complex.
EXPERIMENTAL
DETAILS
Apparatus The apparatus and the flow system are similar to those used m a study of dispersion reported earlier [ 111. A 7 cm 1.D column of 31 cm height was used. It was randomly packed with -35 + 48 mesh glass beads. The bed porosity and permeabihty were found to be 38 5% and 82 3 Darcy, respectively. The packed bed is considered homogeneous as an average value (Vpl), of 98.6% was obtamed. Procedure The same experimental procedures as described m ref. 11 were followed. In the process, the resident fluid, a sucrose solution, was displaced vertically downward by the mvadmg fluid, water. This arrangement is thus identical to the “sweetenmg-off” stage m the sugar refmmg process Flmd properties Four sucrose solutions of different concentrations were used. They were prepared from 67 “Bnx liquid sugar supplied by the CSR Co Ltd. (Australia). Viscosities and densities of the solutions were determined by B.S. U-tube viscometers and Griffin and Tatlock (London) S.G. hydrometers, respectively Results of the measurements are shown m Tables 1 and 2. Data from Slobod and Howlett’s [ 131 expenments are shown m Tables 3 and 4 Experiments were carried out with the entire flow system immersed in a constanttemperature bath maintamed at 25 f 0.1 “C. Sampling of the effluent stream was done manually with measunng cylinders. TABLE 1 Fluid propertles - expemments m present work Fluid
Sucrose solution Sucrose solution Sucrose solution Sucrose solution Dlstllled water
I II III IV
“BrlX
vlscoslty @ 25°C (W
Den&y @ 25°C (CP)
45 54 60 63 -
8 20 41 14 0
1 1 1 1 0
80 40 50 20
30 20 30 30 894
205 252 284 304 997
108 TABLE 2 ExperImental condltlons - present work (@ 25 “C) Experzment set No
A B C D
vlscosl ty ratio R
Flwd Dlsplacmg
Displaced Sucrose Sucrose Sucrose Sucrose
solution solution solution solution
I II III IV
Dlstdled Dlstdled Dlstllled Dlstdled
water water water water
9 22 53 83
30 60 00 00
Dens1 ty difference k/cm31 -0 -0 -0 -0
208 255 292 307
TABLE 3 Fluld propertles - Slobod and Howlett’s experiments (from Table 1 of ref 13) Fhds
Den&y @ 30 “C k/cm31
vlscoslty @ 30 “C (CP)
Refractive rndex @ 30 “C n
cc14 @)
Ethanol Naphtha Soltrol 170
0 7819 0 7172 0 7660
1030 0 448 2 220
1 3592 14023 14298
_ _ -
Ethanol + Ccl* (1) Ethanol + Ccl4 (2) Ethanol + Ccl4 (3) Ethanol + Ccl4 (4) Ethanol + Ccl4 (5) Soltrol 170 + Ccl4 Naphtha + Ccl4
0 0 0 0 0 0 0
1 020 1 020 1035 1033 0 830 2 160 0 450
1 1 1 1 1 1 1
8 13 16 26 6 11 14 26 22 39 196 7 48
8466 9113 8307 8954 9601 7819 7819
3668 3752 3649 3732 3809 4302 4060
TABLE 4 Experimental condltlons - Slobod and Howlett’s experiments (from Table 3 of ref 13) Bed permeablhty = 18 Darcy, bed porosity = 37% Experiment set No
9
10 11 12 13 14 15 16
Flurds Displaced
Dlsplacmg
vlscoslty ratio R @ 30°C
Ethanol Ethanol Ethanol + CCL (1) Ethanol + Ccl4 (2) Soltrol 170 + CC14 Soltrol 170 Soltrol 170 Soltrol 170
Naphtha + CCL Naphtha Naphtha Naphtha Ethanol Ethanol + CC14 (3) Ethanol + CC14 (4) Ethanol + CC14 (5)
2 2 2 2 2 2 2 2
290 295 275 275 100 145 150 675
Dens1 ty difference @ 30 “C 0
-0 -0 -0
065 129 194 0 0 065 0 129 0 194
Vlscoslty ratio R = vlscoslty of displaced fluld/vlscoslty of dlsplacmg fluid (P~//JI) Den&y difference = den&y of dlsplacmg fluld - den&y of displaced fluld (~1 - pz)
The sucrose concentration of each sample was determmed by means of Its refractive index. EXPERIMENTAL
RESULTS
Experimental results obtained m this work are tabulated m Table 5 and shown m Fig. 2,
together wrth predlctrons by the models Slmllarly, Slobod and Hewlett’s data [13] are shown in Table 6 and Fig. 3 Expenmental data from the present work are also plotted m Fig. 4 for comparrson, showmg the mstabihty charactenstic of the “sweetening-off” process as the strength of the hqurd sugar increases.
109 TABLE 5 Experlmental results -this Set No
A
Run No
work Dzsplacement (cm/s)
rate
Mlxmg zone volume (pore volume) Observed
Predrcted
by model of
Koval
Thus work
4 2 3 4 5 6
0 0 0 0 0 0
008208 012818 018103 022488 026705 031437
0 380 0 736 0 910 1410 1078 1 150
0 830 0 980 1 604 1 105 1133 1 156
0 697 1106 1172 1 248 1 300 1 343
7 2 3 4 5
0 0 0 0 0
004295 007028 008096 013380 017990
0 422 0 721 0.906 1 240 1 368
1424 1 655 1 708 1857 1 920
1 007 1 280 1 340 1509 1583
1 2 3 4 5 6 7
0 0 0 0 0 0 0
002811 002811 005487 006746 012087 016866 017990
0 827 0 850 1 234 1375 1 590 2 035 2 143
2 2 2 2 2 3 3
311 311 712 804 997 072 084
1 290 1 290 1 549 1.609 1737 1786 1795
1 2 3 4 5 6 7
0 0 0 0 0 0 0
002619 002664 006117 007713 010120 016866 017428
0 861 0 870 1 520 1926 2 164 2 435 2.474
3 3 3 3 3 3 3
067 081 600 697 791 917 923
1 459 1465 1708 1 754 1798 1857 1 859
Fig 2 Comparison of present expenmental results with predIctIons of Koval and present work
110 TABLE 6 Experimental results - Slobod and Howlett’s experiments (from Table 3 of ref 13) Set No
Displacement rate
(ftldw)
Mccmg zone volume (pore volume)
(cm/s)
0 008818 0 017773 0 035277
Observed
Predicted by model of Koual
Thus work
0 667 0 751 0 775
0.397 0 397 0 397
0 784 0 784 0 784
9
25 50 100
10
25 50 100
0 407 0 536 0 696
0.234 0.316 0 357
0 338 0 501 0 673
11
25 50 100
0 155 0 366 0 525
Stable 0 227 0 310
Stable 0 326 0 551
12
25 50 100
0 072 0 238 0 455
Stable 0 146 0 268
Stable 0 084 0 440
13
25 50 100
0 704 0 693 0 710
0 352 0 352 0 352
0 708 0 708 0 708
14
25 50 100
0 902 0 820 0 772
0 443 0 403 0 383
0 951 0 836 0 781
15
25 50 100
1 105 0 869 0 825
0 525 0 443 0 403
1 210 0 952 0 838
16
25 50 100
1454 1.088 0 836
0 744 0 607 0 543
1800 1.288 1 090
IXSCUSSION
The first observation is that both models, Koval’s and that of the present work, predict the correct trends whereby the mixing zone volume should vary with displacement rate. These trends reflect the important phenomenon due to the difference in fluid densities, called gravity segregation in vertical flow. They show that the liquid residence time, which is inversely proportional to the displacement rate, IS the maJor factor controlling the degree of gravity segregation effect in the displacement process. The second observation is that the trend whereby mixing zone volume varies mth viscosity ratio is also predicted by both models. An increase m viscosity ratio should
favour more fingering and hence a larger muring zone volume, as shown expenmentally in Fig. 4. An mteresting observation was made here concernmg the physical characteristic of sucrose solution as it is displaced by water through a porous medium (“sweetening-off” process). Table 1 indicates that as sucrose solution gets more concentrated, the magnitudes of its viscosrty and density also steadrly increase. As is known, an increase in viscosity ratio (cc[sucrose] /p [water] ) destabrlises the process whereas, on the contrary, an increase m density difference (p [sucrose] - p [water] ) tends to stabihse it. The trend observed in Fig. 4, showing a steady increase m mixing zone volume mth sucrose concentration, thus implies that the viscous fingering effect is dominant over the
111
4
Fig 3 Comparison
of experlmental results of Slobod and Howlett wltb predlctlons of Koval and present work
opposing effect of gravity, m this case. This observation suggests that, if the strength of the sucrose solution mvolved m the process may be reduced, the process can be conducted at a faster rate and still mamtam the same efficiency (Fig. 4).
The comparrson between the two models and Slobod and Howlett’s experimental results mdicates very good agreement between those experimental results and the model of this work. Figure 3 shows that the present model closely predicts Slobod and Howlett’s
112
SUCROSE
SOLUTION
i
(45 8"Br)
SUCROSE
SOLUTION
i-i
(54 4OBr)
A m
SUCROSE
SOLUTION
?Ti-
(60 50°Br)
0
SUCROSE
SOLUTION
F
(63 ZO'Br)
0 0
10 INTERSTITIAL
VELOCITY
(CM SEC-')
Fig 4 Instablllty characteristic of sucrose solution m the “sweetenmg-off”
data [ 131, whereas the values predicted by Koval’s model are much lower. For unstable displacement cases mvolvmg a higher viscosity ratio (experiments in this work, Fig. 2), comparison with the experimental results of ref. 13 also indicates that the model of this work gives better predictions of displacement performance than does Koval’s theory. In general, Koval’s model 1snot successful in predicting the experimental results of Slobod and Howlett and those of the present work, although it does predict very well Koval’s own experimental data. This is further evidence to show that the arbitrary composition figures used m Koval’s model for the effective dlsplacmg agent are not universally applicable. The model developed in the present work employs the actual values of the physical properties of the liquids concerned. This approach IS Justified by the experimental evidence of Handy [8] and Slobod and Thomas [9] and also, by the good agreement obtamed by the present model with the experimental results, especially with those of Slobod and Howlett 1131. However, it was observed that the model developed seems to diverge sigmficantly from the actual experimental results for viscosity ratios over 20. Within this high viscosity ratio range (20 - 80) the model predicts much lower values of mixing zone volume than those obtained experimentally-
process
CONCLUSIONS
The theoretical model developed can predict the volume of the mixing zone which is likely to result during an unstable displacement process m a homogeneous porous bed. Available experimental data, those of this present work and of Slobod and Howlett, confirm the predicted trends whereby the three important factors, viscosity ratio, density difference and the flow rate, affect the size of the mixing zone. The experimental study also found that the model is superior to Koval’s m predicting mixing zone volumes of displacement processes involving both viscous fingering and gravity segregation. However, a method to extend the theory to heterogeneous porous beds was not achieved.
APPENDIX 1
Rate of energy dassipation in the mucable displacement process In this calculation, it is assumed that the displacement process has already taken place for a period of time At. The rate of energy dissipation inside the system is now calculated at the tune At’ for the stable and unstable cases (Fig. 1). The stable and unstable displacements at At are described in Figs. 5(a) and 5(b), in which particular interest is drawn to Zone
113
(a) The rate of energy dissipation is calculated at At’. (b) Equation (Al) is apphcable throughout the energy calculation. (c) The calculation is based on the displacement configurations described m Fig. 5 for the stable and unstable cases. These configurations are considered unchanged during the time between At and At + dt m which this calculation is made. (d) The calculation is only made for the section of the bed between ZIZ; and ZzZa that is Zone II, for each case. For the same reason as in (c) above, these boundaries are assumed stationary durmg d t. (e) The calculation is based on one umt of bed cross-sectional area.
(4
(b) Fig 5 Stable and unstable displacements m present model
ZIZz (Zone II in Fig. 1). In the unstable case, fluid 1 inside the finger zone A’BCD moves with velocity dZt/dt and fluid 2 in the surroundmg zone ABCD’ moves with velocity dZ,/dt. Fluid outside Zone II moves with the mean displacement rate U In the stable case (Fig. 5(a)) there exists a smgle interface II only and the whole liquid system moves with the mean displacement rate U Hence it can be seen that the difference between the two cases has arisen from the flow distribution in Zone II. Consequently, to compare the rates of energy dissipation in the two cases, one needs to consider only the energy dissipation within Zone II in each case. General energy equataon As an incompressible fluid flows in a packed column with a specific rate q, the total rate of energy dissipation dE/dt (per umt crosssectional area) between I1 and Zz can be expressed as dE
‘2
= (-gradp)q s dt 11
Case A Stable dtiplacement (Fig. 5(a)) In the stable case, Zone II includes a section ZIZ; -11’ of fluid 1 and a sectionII’-ZzZ; of fluid 2. The rate of energy dissipation within zr zl-II’ 1s
El, =J (-grad l=UAt
P)~ Uc dl
=
(KeplU-gpl)( U-z)UeAt
(A3)
Similarly, the rate of energy dissipation within II’-ZzZa is dE
[
1
dtz = (KwaU-gpd
(z-U)UeAt
(A4)
Hence, the total rate of energy dissipation withm Zone II in the stable case will be (A51 = (KepcU-gp,)
dl
W)
I= U,At
(U-$)UeAt
+
(AlI
where E denotes the energy dissipated and (grad p) is the pressure gradient between I1 and Zz. Basis of the energy calculation The following pomts should be noted m the calculation
+(Kep2U-gp2)($-U)W
(A6) Case B. Unstable displacement (Fig. 5(b)) In the unstable case, Zone II is composed of a portion A’BCD of fluid 1 having a cross-
114
sectional area Af and a portion ABCD’ of fluid 2 having a cross-sectional area 1 - Af For the hydrodynamic eqmhbnum condltion to be satisfied,
In the unstable displacement, the front CD always travels faster than AB; hence
dls ,
(grad P)I = (grad ~)2
(A7)
or
dlf
-
(KEpl = -gpl (Aa)
The total rate of energy dissipation wlthm Zone II can be calculated as
where (grad p) can be either (grad P)~ or (grad ~3)~.Hence
(AlO) = Kq.t2
$gp2)(
(
f
-2)
&At
(All) After some rearrangement, one can prove that
[fl,.,,,.
(A15)
UL
Consequently, the expression for A [dE/dt] indicates that the sign of A [dE/dt] 1s only dependent on the viscosity difference Ap = p1 - p2, regardless of the density difference, if any As a result the followmg conclusions may be drawn (1) p1 < p2, then A [dE/dt] < 0- the rate of energy dlsslpatlon m the stable case 1s greater than that m the unstable case. (u) p1 > p2, then A [dE/dt] > 0 the rate of energy dlsslpatlon m the stable case 1sless than that m the unstable case One can see that whereas Scheldegger’s minimum energy dlsslpatlon theory does fit case (I), there could also exist cases of unstable displacement m which the energy dissipated 1s greater than that m the stable case (11).Those cases should involve fmgermg due to gravity segregation only. cc1 > p2 but Pl
>
P2.
Returning to the mathematics developed, as it 1staken that the likely situation, m either case (1) or (u), corresponds to the stationary point m the energy dlsslpatlon function, the value Af can be determined by dlfferentlatmg A [dE/dt] (eqn. (A14)) with respect to Af and putting
=[(Kwl~-gPl)(tJ-~)+
+ Kw2
2 -m,
It can be proved that
)if-U)]&At
(A121 To compare the rates of energy dlsslpatlon m the stable and unstable cases, one needs to evaluate A [dE/dt] , where
Gl
= iil.&_
I
I
--IfI,,,,. (Al31
It can be proved that
(A14)
Fig 6 Cross-sectlon of fmgers with relative finger width 0 5 (shaded regions are fingers)
115
(Al@ where R IS the actual viscosity the liquids.
APPENDIX
ratio F&.I~ of
2
Modafzcatzon of Koval’s model to anclude the effect of gravity segregation in the miscible dwplacement process Koval considered viscous fmgermg only m his model. In order that Koval’s theory can be used for vertical systems, the effect of gravity must be mcluded. Koval applied the theory of simultaneous, imrmscible two-phase flow m porous media, and then the Buckley-Leverett equation to the rmsclble displacement process. Hence, eqn (A17) is an appropriate starting point m this derivation +ApgcosZZ’)
X
Koval’s simple form of the Buckley-Leverett equation was written as 1 -= VP1
-df, d% t
( )
(AW
where Vpi is the number of pore volumes of the displacmg fluid inJected (= qt). It can now be seen that, with the three equations (A20), (A21) and (A22), the breakthrough curve fi versus Vpi can be predicted, with any given set of experimental conditions by assignmg S1 a value between 0 and 1, fi 1s evaluated by eqn. (A20) and the corresponding value of Vpi is calculated from eqns. (A21) and (A22) The above approach has thus included the effect of the density difference Ap m Koval’s model of predlctmg the breakthrough curve m miscible displacement. Also, it should be noted that m order to apply Koval’s model accurately the term cl2 m eqns (A20) and (A21) must have the value pe, the viscosity of his 78/22 effective displacmg agent.
-1
6417) Since the liquids are miscible, pC = 0, and K1 = S1
(Al8)
K2=SP=1-S1
(Al3
where subscripts 1 and 2 now denote the displacmg and displaced fluids, respectively. Also, for vertical porous beds cos ZZ’ = 1. Equation (A17) then becomes fi =
Kwz + (1 --s,)Am?
(A201
Kq(Erl/Sl +1-(2---c(1) Equation (A20) expresses the relationship between the saturation S1 of the displacmg fluid mside the porous medium and the volume fraction fi of the displacmg fluid m the efflux, at any one time t. By dlfferentiatmg fi with respect to S1, one would obtam afl as, t =
( 1
NOMENCLATURE
4 Cl
Df E E f g h H k kw, k, K K L 1 n P PC z!
relative cross-sectional area of the finger zone = g(p 1 - P 2YKw1 spacing between fingers energy effective viscosity ratio, Koval’s model efflux composition, Koval’s symbol acceleration due to gravity length of the finger zone Koval’s heterogeneity factor permeability relative permeabllities with respect to the wetting and non-wetting fluids, respectively flow resistivity of the packed bed (l/permeability) Koval’s K-factor (= HE) bed length length from the ongm 00’ refractive index pressure capillary pressure flow rate per unit area total flow rate
116
R s19 s2
t u
V MZ (VdD
X Wf
viscosity ratio saturation of dlsplacmg fluid and displaced fluid, respectively, m the packed bed time mterstitial velocity volume of mrxmg zone (pore volume) per cent recovery of the residence fluid at one pore volume dlsplacing fluid inJected in a matched fluid flood (dlsperslon only) length finger width
P AP
porosity absolute vlscoslty density density difference, p1
4 5 6 7 8 9 10 11
Greek symbols e P
REFERENCES
12 13 -
p2
W R Schowalter, Am Inst Chem Eng J, 11 (1965) 99 R L Perrme, Sot Pet Eng J, I (1961) 9 A E Scheldegger (ed ), The Physzcs of Flow Through Porous Media, Umv Toronto Press, Toronto, 1974 S Hdl, Chem Eng Scz , 1 (1952) 247 P G Saffman and G I Taylor, Proc R SOC London, Ser A, 245 (1958) 312 E J Koval, Sot Pet Eng J, 3 (1963) 145 E L Dougherty, Sot Pet Eng J, 3 (1963) 155 L L Handy, Trans A I M E , 216 (1959) 382 R L Slobod and R A Thomas, Sot Pet Eng J, 3 (1963) 9 T K Perkms, R N Hoffmanand 0 C Johnstone, Sot Pet Eng J, 3 (1963) 70 H H Nguyen, A Study of Axral Mlxzng zn Lzqutd-Laquzd Dzsplacement Through Porous Beds, Ph D Theses, Umverslty of Sydney, 1975 E Y Harper and I D Chang, J Flurd Mech , 33 (1968) 209 R L Slobod and W E Howlett, Sot Pet Eng J, 4 (1964) 1
The Chemxal Engmeerzng Journal, 18 (1979) 103 - 116 0 Elsevler Sequoia S A , Lausanne - Prmted m the Netherlands
Unstable Miscible Liquid-Liquid Displacement in Porous Media: a New Model for Predicting Displacement Performance in Homogeneous Beds H H NGUYEN Commonwealth
Industrzal
Gases, Surry Hrlls, Sydney
2010 (Australza)
D F BAGSTER Department
of Chemzcal Engzneerzng,
The Unzverszty of Sydney,
Sydney
2006 (Australza)
(Received 6 December 1977, m final form 27 October 1978)
Abstract A theory of fmgerrng was developed to predtct the volume of the mrxang zone which 1s likely to result during an unstable mtscable dtsplacement m a homogeneous porous bed This theory 1s based on the apphcatton of Darcy ‘s Law to a sample model representing the flow sttuatton, and the calculated energy dtsstpatton wtthm the model Phystcal factors believed to govern the growth of fingers are considered vlscosaty ratio, dens1 ty difference, flow rate, medium porost ty and permea bdt ty. The theory only constders flows m homogeneous beds Avaalable experimental data, those of this present work and of Slobod and Howlett, confirm the tmportance of the first three factors mentioned The expenmental study also found that the model as superror to a previous one of Koval m predactmg mixing zone volumes of displacement processes rnuolvmg both vtscous fmgenng and gravity segregatton However, a method to extend the model to heterogeneous systems was not achieved
INTRODUCTION
The displacement of a liquid by a second miscible liquid m a porous bed is of importance m many industrial processes. Examples are found m the secondary recovery of petroleum by solvent-floodmgs. The idea of using miscible phase displacements to increase recovery of crude oil has been under active study for more than 25 years [l, 21. Other kinds of rmscible liquid-liquid displacement
can be found m the washing stages m the operation of an ion-exchange column, or m sugar refineries where sugar-liquor displaces, or is displaced by, pure water from beds of solid adsorbent. The particular phenomenon being mve,tigated is that of “fingering” due to density and viscosity differences m the liquids concerned [3 - 51. In the present work a new model has been developed for predictmg miscible displacement performance. Expenmental results are produced for comparison with the model. The attempts of Koval [6] and Dougherty [ 71 to predict displacement performance are also taken into account. Koval was successful m obtaining very good agreement between his model and his own experimental data, which involve only viscous fmgermg. However, the method, as has been pointed out, is considered unsatisfactory because Koval had to employ rather arbitrary composition figures (78% oil, 22% solvent) for the so-called effective displacmg agent which he believed must exist to account for the mixing effect within the medium. Dougherty’s model and his trial and error numerical computation found that B, the empirical parameter accountmg for transverse mixing, is velocity and viscosity dependent. The information obtamed from Dougherty’s work, however, is still mconelusive. In this work, different from Koval’s, the approach used is to consider every probable physical factor which is thought to control the behaviour of fingers as they grow m the medium. Another feature of the model is that it also takes into account the gravity segregation effects, as well as viscous finger-
104
mg. The model is described m four stages: (1) description of the model, (2) method of determinmg the finger-zone cross-sectional area Af, (3) effect of transverse mixmg; (4) effect of medium heterogeneity.
Condz tzon 2 hydrodynamzc condz tzon The pressure gradient Inside the fmger zone must be equal to the pressure gradient m the surroundmg fluld. (3)
(grad p)f = (grad PI,
where THE MATHEMATICAL
MODEL
Uf ---ml)
(grad p)f = --F&I The mathematical model, which IS twodimensional, IS based on known facts about finger behaviour m porous beds The first step m the model development IS to replace the actual picture of fmgermg shown m Fig. l(a) by a much simpler one (Fig. l(b)). 0
0'
I
(4)
(grad P), = -(Kc~(zu~ -@z)
(5)
according to Darcy’s Law. Within Zone II, it IS considered that the fluid velocity IS uniform and equal to Uf within the finger zone A’BCD and U, m the surroundmg zone ABCD’. The displacement can then be assessed by determmmg the growth m length h of this finger zone. This can be done by considermg the motion of the finger-zone “front” CD and the finger-zone “base” AB. Putting
and (a)
(b)
Fig 1 Fmgermg m two-dlmenslonal (a) actual, (b) present model
(7)
porous beds
The total volume of fingers IS represented by a finger zone, denoted by the area A’BCD, m Zone II. This finger zone has a crosssectional area equal to a fraction At of the total bed cross-sectional area A. Fluid velocities m Zones I and III are considered uniform and equal to the average displacement rate U, whereas the fmgermg phenomenon takes place only m Zone II. This phenomenon is considered to be governed by the followmg conditions (cf Scheidegger’s concept [ 31). Condztzon 1 con tznuz ty condz tzon Referring to Zone II, one can write
a(1 --Ad + qi4
=Q
(1)
where qf IS the flow rate of liquid Inside the fingers, q, IS the flow rate of the surroundmg liquid, and Q IS the total flow rate. In terms of the mterstitial velocities, (1 -A,)U,
+A,U,
= U
where I, and lf are the distances from the origm 00’ to the base AB and the front CD, respectively (Fig. l), one can rewrite eqns. (2) and (3), which become A,%+(1-A,)s=U
dlf dt
KE,A~
-gp,
= Kw2
2
-@2
(9)
Solvmg the two simultaneous eqns. (8) and (9)
dls -= dt
U - CIAf 1 + (R - l)Af
dlf -=Rz+& dt
(11)
where Cl
= dP1 Kw
P2)
I
and
R =!? E.cl
dlf/dt and dZ,/dt can be calculated numerically with given values of the displacement
105
rate U, the physical properties of the hquids and the medium, and the value A, which is to be determined However, as long as At remains constant, one can write U--&At
h(t) =
l+(R-l)Af
1 t
-Af) l*(t) = lzU+ &(l l+(R--l)Af
I
(13)
t
1 Af =R+1
and U(R -1)
h(t) =
+ cr
l+(R-l)Af
1
(14)
t
This finger-zone length h(t) can only be observed inside the porous bed itself As mixing zone volume V,, 1s defined as the volume of the efflux collected between the moment the first trace of the displacing liquid appears, to, and the moment the last trace of the displaced fluid is detected, tlm, one can write u vMZ
= t
(hO0
(15)
-to)
where
(WW = L[l + CR- 1)&l U - CIAf and t
L loo = (dl,/dt) =
Method of determmmg Af In the model, Scheidegger’s mmimum energy concept [3] was adopted. The energy dlssipated m the displacement process was calculated (see Appendix 1). It was taken that the cross-sectional area Af of the finger zone will have the value correspondmg to the stationary point of the energy dissipation function. This value was proven in the energy calculation (Appendix 1) to be
L[l + (R - l)A,]
(17)
U(R-l)+C1
Hence, the mlxmg zone volume V,,
is
1 + (R - l)A, U - CIAf
1 (18)
This value VMz has been used as the parameter measurmg the efficiency of the displacement process [3, 51. To predict V,, one needs to determine At.
(19)
Effect of transverse maxmg Handy [8] studied the effect of transverse diffusion m miscible displacement, using a double-tracer method m which two soluble tracers were added to the displacmg liquid and one tracer has a diffusion coefficient much greater than the other. If transverse diffusion is important, then substances with high molecular diffusion coefficients will have sharper concentration profiles than those with comparatively low molecular diffusion coefficients. Results of Handy’s experiment with a 30 m. high column show an absence of any appreciable effect of displacement rate or of diffusional separation of the tracers on the observed displacement curves. This led Handy to the conclusion that transverse diffusion has no important effect on the finger growth and shape; as Handy argued, the distances between fingers are large compared with diffusional distances. Another study of transverse diffusion was made by Slobod and Thomas [9]. Experiments were carried out with a two-dimensional porous bed of 18 m. height. Fast displacement rate (29 ftjday) and slow displacement rate (1.6 ft/day) were used to determine the effect of residence time on the geometry of the fingers The shape of the fingers was observed directly with an X-ray technique. These authors reported that at the fast rate numerous narrow fingers were observed, but at the slow rate “a simple somewhat bulgmg finger” was produced. This observation was also supported by photographs. It 1sthus obvious that the amount of material moving transversely by diffusion was sufficient to modify the finger geometry m the slow-rate run because of the long rewdence time. This is in contradiction to Handy’s conclusion about finger shape. On
106
the other hand, Slobod and Thomas also reported that the displacement curves (or breakthrough curves) observed m both cases of slow and fast rates are very much the same. The results obtamed m Slobod and Thomas’s study tend to support those of Handy when only the analysis of the effluent IS considered. Transverse mlxmg is thus not important m determining the breakthrough curves, and hence the mixmg zone. This, however, seems to contradict Slobod and Thomas’s earher observation on finger shape, which was supported strongly by the X-ray evidence Nevertheless, an explanation was proposed by those authors They argued that the material which flows from the core IS produced as a result of the longitudmal motion of fluid m the core. Thus the composition averaged over the entire crosssection of the flowmg stream is not changed by transverse nuxmg of the components withm the core, so long as the longitudmal flow pattern is unchanged. The above facts obtained from the studies of Handy and of Slobod and Thomas thus tend to discount Koval’s consideration that the transverse mlxmg effect is important, and also his concept of effective vlscosrty ratio. However, without this effective viscosity ratio and its fixed composition figures (78/22), Koval’s model is no longer successful m predlctmg his expenmental data. Effect of medmm heterogeneaty Concemmg the medium heterogeneity, Koval based his model on the H-factor concept m chummg that the model is applicable to the heterogeneous porous system. In this model, he claimed that H should be independent of the viscosity ratio and purely reflect the heterogeneity of the medium. A closer look mto Koval’s method of predlctmg the Hfactor, however, will reveal that the magmtudes of those H-factor values determined by Koval are, in fact, dependent on the values of E, the effective viscosity ratio, as H was calculated from K/E, where K was determined by matchmg the expenmental result with Koval’s model. Since the use of KovaI’s composition figures (78/22) is questionable, the validity of Koval’s H-factor values is also m doubt. This cnticlsm ISfurther supported by the fact that, using Koval’s experimental results, the values
of the H-factor determined by Dougherty’s model are much higher than Koval’s values. Added to this, it was found that a plot of Dougherty’s H-factors versus the corresponding values of (Vpl), does not result m a straight line like Koval’s data. This either undermines the validity of Dougherty’s model, as far as heterogeneous beds are concerned, or there is a possibihty that the straight lme of H uersus (Vpl)n established by Koval is Just the result of comcidence The model developed up to this point is applicable to homogeneous beds that have (Vpi), > 99%.
DISCUSSION
OF THE MODEL
The model developed thus provides a method to predict the mixing zone volume V,, resulting from an unstable miscible displacement process m homogeneous beds The expression of the finger-zone length h(t), eqn. (14), indicates a linear relationship of h with time, that is, with the mean distance travelled 1 (= Ut). This is m agreement with the conclusion of Perkms et al [lo] Furthermore, one can write eqn. (14) as h(t) = Nt
(20)
where
(21) Since Af < 1 and the denommator is always positive, the sign of the coefficient N is dependent on the numerator. Thus (a)KeU(cc, -3~~) -i&2 --Ed> We N> the finger zone will grow steadily with time UNSTABLE
0,
(b) KeWs --ccl) -g(ps -PJ< 0, i.e N< 0, any initial disturbance h (at t = 0) will be suppressed* STABLE The above conditions for stability and mstabllity denved from the model are identical with Hill’s [4] cntena for overall stability. In our model, the velocity distnbutlon of fluids inside a fingenng system (Fig. 1) was based on the assumption that cross-flow, which is known to exist around the finger tips
107
and ends [lo] , IS negligible. This assumption is considered reasonable as the volume which is mvolved m this lateral motion of liquid due to cross-flow is much smaller than the volume of hquids m which the actual longitudmal motion takes place In the development of the model, Af was taken to be a smgle value throughout the displacement process. Saffman and Taylor’s observation [ 51 on an immiscible system mdicated a tendency for the fingers to space themselves such that the relative finger width approached 0.50. This value, however, is not applicable to three-dimensional beds [ 111 smce, even if the relative finger width is kept at 0.50, one can see from Fig. 6 that the finger-zone area Af m such beds is not equal to, but much less than, 0.5 of the total flow area. Scheidegger’s mmimum energy concept was thus adopted m the determmation of Af . The energy calculation made m this study, however, has found that the system can finger not only at the mmlmum energy dissipated in certam situations but also at maxunum energy dissipation m another case (see Appendix 1). A proven example of the maximum energy dissipation case has been pomted out by H. Brenner (private commumcation, 1974) and by Harper and Chang [13] m the problem of the motion of a dumb-bell-shaped particle m a slow viscous shear flow. The particle, accordmg to the result of these studies, is shown to have a preferred periodic orbit which corresponds to maxmum dissipation of energy. The model as completed can be used to predict the displacement performance m homogeneous beds only. Koval’s H-factor method is a sound idea to extend the model to heterogeneous systems. However, since heterogeneity of a porous bed also includes rmcroscoplc dispersion, it should be a function of the physical properties of the liquids concerned and the displacement rate [3]. As a result, (Vpi)n and hence the H-(Vpi), relationship are also dependent on those variables. The problem of predictmg the displacement performance m mhomogeneous beds is thus far more complex.
EXPERIMENTAL
DETAILS
Apparatus The apparatus and the flow system are similar to those used m a study of dispersion reported earlier [ 111. A 7 cm 1.D column of 31 cm height was used. It was randomly packed with -35 + 48 mesh glass beads. The bed porosity and permeabihty were found to be 38 5% and 82 3 Darcy, respectively. The packed bed is considered homogeneous as an average value (Vpl), of 98.6% was obtamed. Procedure The same experimental procedures as described m ref. 11 were followed. In the process, the resident fluid, a sucrose solution, was displaced vertically downward by the mvadmg fluid, water. This arrangement is thus identical to the “sweetenmg-off” stage m the sugar refmmg process Flmd properties Four sucrose solutions of different concentrations were used. They were prepared from 67 “Bnx liquid sugar supplied by the CSR Co Ltd. (Australia). Viscosities and densities of the solutions were determined by B.S. U-tube viscometers and Griffin and Tatlock (London) S.G. hydrometers, respectively Results of the measurements are shown m Tables 1 and 2. Data from Slobod and Howlett’s [ 131 expenments are shown m Tables 3 and 4 Experiments were carried out with the entire flow system immersed in a constanttemperature bath maintamed at 25 f 0.1 “C. Sampling of the effluent stream was done manually with measunng cylinders. TABLE 1 Fluid propertles - expemments m present work Fluid
Sucrose solution Sucrose solution Sucrose solution Sucrose solution Dlstllled water
I II III IV
“BrlX
vlscoslty @ 25°C (W
Den&y @ 25°C (CP)
45 54 60 63 -
8 20 41 14 0
1 1 1 1 0
80 40 50 20
30 20 30 30 894
205 252 284 304 997
108 TABLE 2 ExperImental condltlons - present work (@ 25 “C) Experzment set No
A B C D
vlscosl ty ratio R
Flwd Dlsplacmg
Displaced Sucrose Sucrose Sucrose Sucrose
solution solution solution solution
I II III IV
Dlstdled Dlstdled Dlstllled Dlstdled
water water water water
9 22 53 83
30 60 00 00
Dens1 ty difference k/cm31 -0 -0 -0 -0
208 255 292 307
TABLE 3 Fluld propertles - Slobod and Howlett’s experiments (from Table 1 of ref 13) Fhds
Den&y @ 30 “C k/cm31
vlscoslty @ 30 “C (CP)
Refractive rndex @ 30 “C n
cc14 @)
Ethanol Naphtha Soltrol 170
0 7819 0 7172 0 7660
1030 0 448 2 220
1 3592 14023 14298
_ _ -
Ethanol + Ccl* (1) Ethanol + Ccl4 (2) Ethanol + Ccl4 (3) Ethanol + Ccl4 (4) Ethanol + Ccl4 (5) Soltrol 170 + Ccl4 Naphtha + Ccl4
0 0 0 0 0 0 0
1 020 1 020 1035 1033 0 830 2 160 0 450
1 1 1 1 1 1 1
8 13 16 26 6 11 14 26 22 39 196 7 48
8466 9113 8307 8954 9601 7819 7819
3668 3752 3649 3732 3809 4302 4060
TABLE 4 Experimental condltlons - Slobod and Howlett’s experiments (from Table 3 of ref 13) Bed permeablhty = 18 Darcy, bed porosity = 37% Experiment set No
9
10 11 12 13 14 15 16
Flurds Displaced
Dlsplacmg
vlscoslty ratio R @ 30°C
Ethanol Ethanol Ethanol + CCL (1) Ethanol + Ccl4 (2) Soltrol 170 + CC14 Soltrol 170 Soltrol 170 Soltrol 170
Naphtha + CCL Naphtha Naphtha Naphtha Ethanol Ethanol + CC14 (3) Ethanol + CC14 (4) Ethanol + CC14 (5)
2 2 2 2 2 2 2 2
290 295 275 275 100 145 150 675
Dens1 ty difference @ 30 “C 0
-0 -0 -0
065 129 194 0 0 065 0 129 0 194
Vlscoslty ratio R = vlscoslty of displaced fluld/vlscoslty of dlsplacmg fluid (P~//JI) Den&y difference = den&y of dlsplacmg fluld - den&y of displaced fluld (~1 - pz)
The sucrose concentration of each sample was determmed by means of Its refractive index. EXPERIMENTAL
RESULTS
Experimental results obtained m this work are tabulated m Table 5 and shown m Fig. 2,
together wrth predlctrons by the models Slmllarly, Slobod and Hewlett’s data [13] are shown in Table 6 and Fig. 3 Expenmental data from the present work are also plotted m Fig. 4 for comparrson, showmg the mstabihty charactenstic of the “sweetening-off” process as the strength of the hqurd sugar increases.
109 TABLE 5 Experlmental results -this Set No
A
Run No
work Dzsplacement (cm/s)
rate
Mlxmg zone volume (pore volume) Observed
Predrcted
by model of
Koval
Thus work
4 2 3 4 5 6
0 0 0 0 0 0
008208 012818 018103 022488 026705 031437
0 380 0 736 0 910 1410 1078 1 150
0 830 0 980 1 604 1 105 1133 1 156
0 697 1106 1172 1 248 1 300 1 343
7 2 3 4 5
0 0 0 0 0
004295 007028 008096 013380 017990
0 422 0 721 0.906 1 240 1 368
1424 1 655 1 708 1857 1 920
1 007 1 280 1 340 1509 1583
1 2 3 4 5 6 7
0 0 0 0 0 0 0
002811 002811 005487 006746 012087 016866 017990
0 827 0 850 1 234 1375 1 590 2 035 2 143
2 2 2 2 2 3 3
311 311 712 804 997 072 084
1 290 1 290 1 549 1.609 1737 1786 1795
1 2 3 4 5 6 7
0 0 0 0 0 0 0
002619 002664 006117 007713 010120 016866 017428
0 861 0 870 1 520 1926 2 164 2 435 2.474
3 3 3 3 3 3 3
067 081 600 697 791 917 923
1 459 1465 1708 1 754 1798 1857 1 859
Fig 2 Comparison of present expenmental results with predIctIons of Koval and present work
110 TABLE 6 Experimental results - Slobod and Howlett’s experiments (from Table 3 of ref 13) Set No
Displacement rate
(ftldw)
Mccmg zone volume (pore volume)
(cm/s)
0 008818 0 017773 0 035277
Observed
Predicted by model of Koual
Thus work
0 667 0 751 0 775
0.397 0 397 0 397
0 784 0 784 0 784
9
25 50 100
10
25 50 100
0 407 0 536 0 696
0.234 0.316 0 357
0 338 0 501 0 673
11
25 50 100
0 155 0 366 0 525
Stable 0 227 0 310
Stable 0 326 0 551
12
25 50 100
0 072 0 238 0 455
Stable 0 146 0 268
Stable 0 084 0 440
13
25 50 100
0 704 0 693 0 710
0 352 0 352 0 352
0 708 0 708 0 708
14
25 50 100
0 902 0 820 0 772
0 443 0 403 0 383
0 951 0 836 0 781
15
25 50 100
1 105 0 869 0 825
0 525 0 443 0 403
1 210 0 952 0 838
16
25 50 100
1454 1.088 0 836
0 744 0 607 0 543
1800 1.288 1 090
IXSCUSSION
The first observation is that both models, Koval’s and that of the present work, predict the correct trends whereby the mixing zone volume should vary with displacement rate. These trends reflect the important phenomenon due to the difference in fluid densities, called gravity segregation in vertical flow. They show that the liquid residence time, which is inversely proportional to the displacement rate, IS the maJor factor controlling the degree of gravity segregation effect in the displacement process. The second observation is that the trend whereby mixing zone volume varies mth viscosity ratio is also predicted by both models. An increase m viscosity ratio should
favour more fingering and hence a larger muring zone volume, as shown expenmentally in Fig. 4. An mteresting observation was made here concernmg the physical characteristic of sucrose solution as it is displaced by water through a porous medium (“sweetening-off” process). Table 1 indicates that as sucrose solution gets more concentrated, the magnitudes of its viscosrty and density also steadrly increase. As is known, an increase in viscosity ratio (cc[sucrose] /p [water] ) destabrlises the process whereas, on the contrary, an increase m density difference (p [sucrose] - p [water] ) tends to stabihse it. The trend observed in Fig. 4, showing a steady increase m mixing zone volume mth sucrose concentration, thus implies that the viscous fingering effect is dominant over the
111
4
Fig 3 Comparison
of experlmental results of Slobod and Howlett wltb predlctlons of Koval and present work
opposing effect of gravity, m this case. This observation suggests that, if the strength of the sucrose solution mvolved m the process may be reduced, the process can be conducted at a faster rate and still mamtam the same efficiency (Fig. 4).
The comparrson between the two models and Slobod and Howlett’s experimental results mdicates very good agreement between those experimental results and the model of this work. Figure 3 shows that the present model closely predicts Slobod and Howlett’s
112
SUCROSE
SOLUTION
i
(45 8"Br)
SUCROSE
SOLUTION
i-i
(54 4OBr)
A m
SUCROSE
SOLUTION
?Ti-
(60 50°Br)
0
SUCROSE
SOLUTION
F
(63 ZO'Br)
0 0
10 INTERSTITIAL
VELOCITY
(CM SEC-')
Fig 4 Instablllty characteristic of sucrose solution m the “sweetenmg-off”
data [ 131, whereas the values predicted by Koval’s model are much lower. For unstable displacement cases mvolvmg a higher viscosity ratio (experiments in this work, Fig. 2), comparison with the experimental results of ref. 13 also indicates that the model of this work gives better predictions of displacement performance than does Koval’s theory. In general, Koval’s model 1snot successful in predicting the experimental results of Slobod and Howlett and those of the present work, although it does predict very well Koval’s own experimental data. This is further evidence to show that the arbitrary composition figures used m Koval’s model for the effective dlsplacmg agent are not universally applicable. The model developed in the present work employs the actual values of the physical properties of the liquids concerned. This approach IS Justified by the experimental evidence of Handy [8] and Slobod and Thomas [9] and also, by the good agreement obtamed by the present model with the experimental results, especially with those of Slobod and Howlett 1131. However, it was observed that the model developed seems to diverge sigmficantly from the actual experimental results for viscosity ratios over 20. Within this high viscosity ratio range (20 - 80) the model predicts much lower values of mixing zone volume than those obtained experimentally-
process
CONCLUSIONS
The theoretical model developed can predict the volume of the mixing zone which is likely to result during an unstable displacement process m a homogeneous porous bed. Available experimental data, those of this present work and of Slobod and Howlett, confirm the predicted trends whereby the three important factors, viscosity ratio, density difference and the flow rate, affect the size of the mixing zone. The experimental study also found that the model is superior to Koval’s m predicting mixing zone volumes of displacement processes involving both viscous fingering and gravity segregation. However, a method to extend the theory to heterogeneous porous beds was not achieved.
APPENDIX 1
Rate of energy dassipation in the mucable displacement process In this calculation, it is assumed that the displacement process has already taken place for a period of time At. The rate of energy dissipation inside the system is now calculated at the tune At’ for the stable and unstable cases (Fig. 1). The stable and unstable displacements at At are described in Figs. 5(a) and 5(b), in which particular interest is drawn to Zone
113
(a) The rate of energy dissipation is calculated at At’. (b) Equation (Al) is apphcable throughout the energy calculation. (c) The calculation is based on the displacement configurations described m Fig. 5 for the stable and unstable cases. These configurations are considered unchanged during the time between At and At + dt m which this calculation is made. (d) The calculation is only made for the section of the bed between ZIZ; and ZzZa that is Zone II, for each case. For the same reason as in (c) above, these boundaries are assumed stationary durmg d t. (e) The calculation is based on one umt of bed cross-sectional area.
(4
(b) Fig 5 Stable and unstable displacements m present model
ZIZz (Zone II in Fig. 1). In the unstable case, fluid 1 inside the finger zone A’BCD moves with velocity dZt/dt and fluid 2 in the surroundmg zone ABCD’ moves with velocity dZ,/dt. Fluid outside Zone II moves with the mean displacement rate U In the stable case (Fig. 5(a)) there exists a smgle interface II only and the whole liquid system moves with the mean displacement rate U Hence it can be seen that the difference between the two cases has arisen from the flow distribution in Zone II. Consequently, to compare the rates of energy dissipation in the two cases, one needs to consider only the energy dissipation within Zone II in each case. General energy equataon As an incompressible fluid flows in a packed column with a specific rate q, the total rate of energy dissipation dE/dt (per umt crosssectional area) between I1 and Zz can be expressed as dE
‘2
= (-gradp)q s dt 11
Case A Stable dtiplacement (Fig. 5(a)) In the stable case, Zone II includes a section ZIZ; -11’ of fluid 1 and a sectionII’-ZzZ; of fluid 2. The rate of energy dissipation within zr zl-II’ 1s
El, =J (-grad l=UAt
P)~ Uc dl
=
(KeplU-gpl)( U-z)UeAt
(A3)
Similarly, the rate of energy dissipation within II’-ZzZa is dE
[
1
dtz = (KwaU-gpd
(z-U)UeAt
(A4)
Hence, the total rate of energy dissipation withm Zone II in the stable case will be (A51 = (KepcU-gp,)
dl
W)
I= U,At
(U-$)UeAt
+
(AlI
where E denotes the energy dissipated and (grad p) is the pressure gradient between I1 and Zz. Basis of the energy calculation The following pomts should be noted m the calculation
+(Kep2U-gp2)($-U)W
(A6) Case B. Unstable displacement (Fig. 5(b)) In the unstable case, Zone II is composed of a portion A’BCD of fluid 1 having a cross-
114
sectional area Af and a portion ABCD’ of fluid 2 having a cross-sectional area 1 - Af For the hydrodynamic eqmhbnum condltion to be satisfied,
In the unstable displacement, the front CD always travels faster than AB; hence
dls ,
(grad P)I = (grad ~)2
(A7)
or
dlf
-
(KEpl = -gpl (Aa)
The total rate of energy dissipation wlthm Zone II can be calculated as
where (grad p) can be either (grad P)~ or (grad ~3)~.Hence
(AlO) = Kq.t2
$gp2)(
(
f
-2)
&At
(All) After some rearrangement, one can prove that
[fl,.,,,.
(A15)
UL
Consequently, the expression for A [dE/dt] indicates that the sign of A [dE/dt] 1s only dependent on the viscosity difference Ap = p1 - p2, regardless of the density difference, if any As a result the followmg conclusions may be drawn (1) p1 < p2, then A [dE/dt] < 0- the rate of energy dlsslpatlon m the stable case 1s greater than that m the unstable case. (u) p1 > p2, then A [dE/dt] > 0 the rate of energy dlsslpatlon m the stable case 1sless than that m the unstable case One can see that whereas Scheldegger’s minimum energy dlsslpatlon theory does fit case (I), there could also exist cases of unstable displacement m which the energy dissipated 1s greater than that m the stable case (11).Those cases should involve fmgermg due to gravity segregation only. cc1 > p2 but Pl
>
P2.
Returning to the mathematics developed, as it 1staken that the likely situation, m either case (1) or (u), corresponds to the stationary point m the energy dlsslpatlon function, the value Af can be determined by dlfferentlatmg A [dE/dt] (eqn. (A14)) with respect to Af and putting
=[(Kwl~-gPl)(tJ-~)+
+ Kw2
2 -m,
It can be proved that
)if-U)]&At
(A121 To compare the rates of energy dlsslpatlon m the stable and unstable cases, one needs to evaluate A [dE/dt] , where
Gl
= iil.&_
I
I
--IfI,,,,. (Al31
It can be proved that
(A14)
Fig 6 Cross-sectlon of fmgers with relative finger width 0 5 (shaded regions are fingers)
115
(Al@ where R IS the actual viscosity the liquids.
APPENDIX
ratio F&.I~ of
2
Modafzcatzon of Koval’s model to anclude the effect of gravity segregation in the miscible dwplacement process Koval considered viscous fmgermg only m his model. In order that Koval’s theory can be used for vertical systems, the effect of gravity must be mcluded. Koval applied the theory of simultaneous, imrmscible two-phase flow m porous media, and then the Buckley-Leverett equation to the rmsclble displacement process. Hence, eqn (A17) is an appropriate starting point m this derivation +ApgcosZZ’)
X
Koval’s simple form of the Buckley-Leverett equation was written as 1 -= VP1
-df, d% t
( )
(AW
where Vpi is the number of pore volumes of the displacmg fluid inJected (= qt). It can now be seen that, with the three equations (A20), (A21) and (A22), the breakthrough curve fi versus Vpi can be predicted, with any given set of experimental conditions by assignmg S1 a value between 0 and 1, fi 1s evaluated by eqn. (A20) and the corresponding value of Vpi is calculated from eqns. (A21) and (A22) The above approach has thus included the effect of the density difference Ap m Koval’s model of predlctmg the breakthrough curve m miscible displacement. Also, it should be noted that m order to apply Koval’s model accurately the term cl2 m eqns (A20) and (A21) must have the value pe, the viscosity of his 78/22 effective displacmg agent.
-1
6417) Since the liquids are miscible, pC = 0, and K1 = S1
(Al8)
K2=SP=1-S1
(Al3
where subscripts 1 and 2 now denote the displacmg and displaced fluids, respectively. Also, for vertical porous beds cos ZZ’ = 1. Equation (A17) then becomes fi =
Kwz + (1 --s,)Am?
(A201
Kq(Erl/Sl +1-(2---c(1) Equation (A20) expresses the relationship between the saturation S1 of the displacmg fluid mside the porous medium and the volume fraction fi of the displacmg fluid m the efflux, at any one time t. By dlfferentiatmg fi with respect to S1, one would obtam afl as, t =
( 1
NOMENCLATURE
4 Cl
Df E E f g h H k kw, k, K K L 1 n P PC z!
relative cross-sectional area of the finger zone = g(p 1 - P 2YKw1 spacing between fingers energy effective viscosity ratio, Koval’s model efflux composition, Koval’s symbol acceleration due to gravity length of the finger zone Koval’s heterogeneity factor permeability relative permeabllities with respect to the wetting and non-wetting fluids, respectively flow resistivity of the packed bed (l/permeability) Koval’s K-factor (= HE) bed length length from the ongm 00’ refractive index pressure capillary pressure flow rate per unit area total flow rate
116
R s19 s2
t u
V MZ (VdD
X Wf
viscosity ratio saturation of dlsplacmg fluid and displaced fluid, respectively, m the packed bed time mterstitial velocity volume of mrxmg zone (pore volume) per cent recovery of the residence fluid at one pore volume dlsplacing fluid inJected in a matched fluid flood (dlsperslon only) length finger width
P AP
porosity absolute vlscoslty density density difference, p1
4 5 6 7 8 9 10 11
Greek symbols e P
REFERENCES
12 13 -
p2
W R Schowalter, Am Inst Chem Eng J, 11 (1965) 99 R L Perrme, Sot Pet Eng J, I (1961) 9 A E Scheldegger (ed ), The Physzcs of Flow Through Porous Media, Umv Toronto Press, Toronto, 1974 S Hdl, Chem Eng Scz , 1 (1952) 247 P G Saffman and G I Taylor, Proc R SOC London, Ser A, 245 (1958) 312 E J Koval, Sot Pet Eng J, 3 (1963) 145 E L Dougherty, Sot Pet Eng J, 3 (1963) 155 L L Handy, Trans A I M E , 216 (1959) 382 R L Slobod and R A Thomas, Sot Pet Eng J, 3 (1963) 9 T K Perkms, R N Hoffmanand 0 C Johnstone, Sot Pet Eng J, 3 (1963) 70 H H Nguyen, A Study of Axral Mlxzng zn Lzqutd-Laquzd Dzsplacement Through Porous Beds, Ph D Theses, Umverslty of Sydney, 1975 E Y Harper and I D Chang, J Flurd Mech , 33 (1968) 209 R L Slobod and W E Howlett, Sot Pet Eng J, 4 (1964) 1