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The onset of blob motion in a random sphere packing caused by flow of the surrounding liquid
ChemicalEngineetingScienceVol. 38. No. 9, pp. 1451-1465.19R3 Prmkd in Great Britain.
THE ONSET OF BLOB MOTION IN A RANDOM SPHERE PACKING CAUSED BY FLOW OF THE SURROUNDING LIQUID G. D. YADAVt
and G.
MASON*
Department of Chemical Engineering, Loughborough University of Technology, Loughborough, Leicestershire. England (Received 26 October 1981; in revised form 20 December 1982; accepted
2 February 1983)
Abstract-Entrapped liquid blobs in the pore space of a packing of equal spheres were made visible by matching the refractive index of the spheres with that of the bulk fluid filling the pore space. By pumpingbulk fluid through the surrounding pore space an experimental determination of the curvature difference between the ends of the blobs at the onset of mobilisation was made. This was repeated for blobs with special histories. The probability of blob movement was calculated. Good agreement between the theory and experiments was found.
INTRODUCTION
During the commercial extraction of oil a significant amount of oil is entrapped in the porous reservoir rocks as small blobs or ganglia, typically a few pore diameters long Displacement of such residual oil might be achieved by reduction of the oil/water capillary forces together with the application of a pressure gradient such as that produced by pumping water through the adjacent pores in the rock. The published work [ l-71 indicates that the capillary forces dominate over the viscous forces and that for blob movement the interfacial tensions have to be reduced to very low levels indeed. The exact mechanism of blob motion is of particular interest because it seems Iikely that the ease of mobilisation is directly related to the length of the blob, and so increasing the blob length as well as reducing the interfacial tensions might make mobilisation commercially practicable. Ng et a/.[31 have presented a theory for the mobilisation of a blob in the pore space of a random packing of equal spheres. The blob should move when the hydrodynamic pressure difference between its ends is greater than the capillary pressure difference. They tested this theory with an experiment in which the movement of blobs could be directly observed. Their blob visuahsation experiment consisted of 3.2mm Perspex spheres in a square column of 20 mm x 20 mm x 150mm: used glycerol-water mixture with matching refractive index as continuous phase; and Meriam manometer oil of similar density for the blob phase. Blobs were observed to move in quantum-like jumps. During their experiments, the history of each blob was always the same: the blob was stationary when the bulk fluid flow was zero. Their measurement was of the flow required to advance an isolated blob of desired orientation by one sphere diameter and this involved several jumps. The theoretical analysis of Ng et al. used auerage values of the meniscus
*Author to whom correspondence should be addressed. tLeverhulme Visiting Fellow from Department of Chemical Technology, University of Bombay, Matunga Road, Bombay409019. India.
curvatures and so could only predict the auerage conditions for blob movement and not specific probabilities. The distributions of meniscus curvature for both advancing and receding menisci in the pore space of a random packing of equal spheres have been given by Mason[8]. Subsequently these were used to calculate the probability of blob motion for two simplified mechanisms-namely, branching and no branching, and indicated that the past history of the blob was especially important in determining the probability of the initial quantum-like jump [9]. The object of the present paper is to report a blob visualisation technique similar to that of Ng et al. [3], but with experiments better designed to test Mason’s theory[9]. The basic method was as follows. An isolated blob of known length and orientation was given a specific history, that of being stationary in a relatively high superficial velocity of the bulk liquid. The blob was observed while the flow of the liquid was gradually and continuously reduced to zero and then reversed and increased. The flow at which the blob made the first jump in the opposite direction to the initial history flow was noted. The cumulative probability so obtained is important in calculating the subsequent probabilities of blobs advancing larger distances. THEORY OF BLOB MOTION
The theory of Ng et al.[3] for blob motion requires that the capillary pressure between the ends of the blob be less than the hydrodynamic pressure produced by the viscous fluid flowing in the pore space alongside the blob. In a packing of spheres the menisci at the advancing end (head) of the blob are forced through constrictions in the pore space and the menisci at the retreating end (foot) of the blob move through larger cavities. In general the head men&i are more highly curved than the foot menisci and so a pressure is required to move the blob. The pore space in a random packing of equal spheres can be modelled by assembling tetrahedra whose six edges are chosen according to an edge length distribution function (similar to the radial distribution function)[8] and centering spheres at each vertex. These tetrahedra 1461
1462
G. D. YADAVand G. MASON
winhave an internal cavity and four oonstrictions.or dows leading in and out, one for each face (see Fig. 1). Using the Haines approximation[lO] that a capillary meniscus in such complex geometry is approximated by a spherical surface it is possible to calculate the window (throat) radius distribution, f(r), and the cavity (body) radius distribution, g(r). There is a single cavity radius associated with each tetrahedral pore, but there are four window radii. Although there is some correlation between the sizes of adjacent windows (they each have a tetrahedron edge in common), this is not strong and the approximation that they are independent is not unreasonable [ 1I]. For an individual tetrahedral pore there is a strong correlation between its largest window radius and the cavity radius [12], but for blobs more than one pore long the cavity radius at the foot of the blob is unrelated to the window radii several pores away at the head of the blob. The maximum capillary pressure difference, AP,,, along a blob is given by:
The probability that the trailing foot has a radius between rb and rb f dr,, is g(rb)drb. Combination of these two probabilities gives: 3[c
fWdr]2f(r,l
dw(d
drb
(3)
this being the probability that the largest radius of the head lies between r, and r, + dr, and the radius of the foot lies between rb and rb + dra. Let the total curvature difference between the ends of the blobs produced by the flow of the bulk fluid be 2X. Let the probability of half of the capillary curvature difference (l/r, - l/rtJ being greater than X be H(X). H(X) will be given by the integral of (3) over all r, and r, for which &$>X
or, rearranging
rb
>
(1
_rir~)
H(X)=6[~~f(r)dr]lf(r.)~~~(r)drdr,
(4)
(5)
~~[~f(r)dr]lf(r.)10_R(r)dldlr’ O*rb
where rr is the interfacial tension, r, and rb are the radii of the window (throat) and cavity (body) respectively, and /3, the parameter used by Ng et al.[3], is (I - (dra). In the analysis of Ng et al., /3 was assigned an average value. As the blob moves so its head enters a tetrahedral pore through one face and leaves through the largest of the three remaining faces. The probability that any two faces out of three have radii less than r,, the radius of the head end of the blob, and the third face has a radius between r, and rt + dr, is:
3[~f(r)dr]2f(ddTr-
(2)
H(X) represents the probability that a blob with no special past history does not move at a curvature difference X, this being half of the curvature difference produced between its ends by the bulk fluid flow. The upper bound of the first integral is limited to 1/X because rb becomes infinite at that value. The difference in curvature X is a function of blob length, fluid density difference, interfacial tension, and the Darcy pressure gradient in the pore space of the sphere packing. Mason[S] gives distributions for the functions g(r) and f(r) and so H(X) can be evaluated[9]. Every experimental blob has a special past history and so H(X) is not directly applicable. If a blob is formed and is stationary at zero flow (X = 0) then the probability of it mooing by at least a single step when the curvature between its ends produced by the fluid flow is increased to 2X is
HUN-H(X) H(O)
(6)
because H(O) is the probability of being stationary at 0 and H(X) of being stationary at X; the difference represents the probability of moving. H(0) in the denominator normalises the probability so that it becomes unity when X = m (and hence H(X) = 0). If the special past history of the blob is that it was stationary at curvature Y then the probability (Qu) of moving by a single step at some subsequent increase in curvature X is: WYi;yT(X)
Fig. 1. View of a single irregular tetrahedral pore. Centred at each vertex are portions of the spheres of the packing. The pore space is a central cavity with constrictions leading out.
= QY.
(7)
It is possible to establish experimentally the probability Qy by observing the behaviour of many isolated blobs when subjected to different bulk flows (varying X and Y) and so the theoretical function H(X) can be compared with the experimentally established one.
The onset of blob motion in a random sphere packing -AL
The experimental
determination of Qy can be achieved by directly observing the mobilisation of individual blobs in a randomly packed column as the flow of the bulk fluid is changed. Observation of the blob demands that the refractive indices of the packing and the bulk fluid be quite accurately matched. Also a very steady controllable flow of bulk fluid is essential as any flow pulsations vary the pressure gradient along the blob and cause it to mobilise. A 35 mm i.d. glass column was packed with 3.125 mm dia. precision ground glass spheres. The effective length of the column was 150mm and this section had a flat glass window. The refractive index of the glass spheres was 1.524. The bulk phase (wetting) was a suitable mixture of benzyl acetate and benzyl benzoate which had a refractive index very close to that of the glass balls thus rendering them virtually invisible. The mixture had a density of 1.0665 g/cm3 and viscosity of 2.18 CP at 25°C. The blobs were water containing ammonium purpurate dye. The column was provided with pressure tappings and had injection ports at either end to facilitate the introduction of blobs into the packing. Figure 2 shows a general outline of the experimental set-up. The production of a controllable flow of the organic phase proved to be quite difficult and the system used in Fig. 2 went through several modifications before a satisfactory solution was found. There were two thermostated reservoirs (6 and 7 on Fig. 2), each of 21. capacity, containing the organic phase and connected to the inlet and outlet of the packed column (1) through an inverted U-tube flowmeter (3) and two 3-way cocks (4). A steady flow of the organic liquid was created by pressuring the inlet reservoir (6) with a small air-pump of a type used for aerating fish-tanks. The flow rate of the liquid could be maintained by controlling the air flow rate into or out of the air-space above the liquid in the reservoir by needle valves (5). The downstream reservoir (7) was open to atmosphere and was fitted with a level detector (11) and controller (12) which varied the speed
Fig. 2. Schematic diagram of the experimental set-up: (1) packed column containing 3.175 mm dia. glass spheres (top and bottom injection tubes extended into the effective zone of the column); (2) inverted [I-tubedifferentialmanometer with wateraslight Ruid;(3) inverted U-tube capillary flow-meter; (4) three-way plug cock; (5) needle valve; (6) continuous phase (organic) reservoir; (7) receiver for continuous phase with level indicator; (8) peristaltic pump; (9) pressure surge vessel; (10) air-pump; (11) lever mechanism for controller; (12) recorder; (13) digital volt meter; (14) polyethylene float; (15) mercury manometer.
14.63
of a peristaltic pump (8) and this pump returned the organic phase to the pressurised upstream reservoir. The flow of organic phase was thus independent of the pressure pulses produced by the pump. A precalibrated inverted U-tube capillary flowmeter (3) was used to measure the liquid flow rate. The two 3-way plug cocks (4) in the line as shown in Fig. 2, made it possible to reverse the flow of liquid through the column. The pressure drop per unit length of the packed bed was measured directly with a coloured-water/organic phase manometer (2). Coloured-water blobs of known volume were introduced by a hypodermic syringe through the injection tube at the bottom of the column into the packing. The blob lengths were measured with a cathetometer. The interfacial tension between the two phases was measured by a ring tensiometer and was found to be 12dynelcm at 25°C. The densities of the liquids were measured with a specific gravity bottle and the viscosities with an Ostwald viscometer. Procedure The packed column was first filled with the organic phase and all air bubbles were removed from the packing and the connecting lines. This was quite awkward but once flow was established air bubbles could be cleared by simply increasing the fluid Row. To avoid refilling the tubes a special provision was made for isolation of the packed zone and for its emptying and refilling and this was the method for removing spent blobs from the column. Blobs were of water coloured with ammonium purpurate, a dye which was insoluble in the organic phase. At zero flow rate of the bulk organic phase, a blob of known volume was introduced into the packing, and an upward flow used to move the blob to the middle of the packing. An upward (or sometimes downward) flow of liquid was arranged to give a value of Y, the half curvature between the blob ends, of about 2.5 thus giving the blob a special history. The blob would move some distance before stopping and if its length changed, the Bow had also to be changed to bring the imposed history half curvature back to 2.5. The blob length was then measured and also its orientation noted. Then the flow rate of liquid was slowly reduced whilst the blob was observed. When the first quantum movement of the blob in the direction opposite to the impoged history flow was observed, the pressure drop across the packing was recorded. Often the flow would reach zero with no observed step. Then the flow would be reversed and increased until the stepwise movement was observed. The flow of liquid was increased or decreased by controlling the flow of air into the reservoir. In calculating the value of the curvature at which the blob moved, correction had to be made for the hydrostatic pressure difference caused by the small density difference (0.00665 g/cm3) of the two phases (see for example Ref. [ 131). It was experimentally quite difficult to perform experiments with Y = 2.5 and so in addition experiments were also performed where the blob was static in zero flow (and had the curvature history produced entirely by buoyancy), this making Y approximately zero. These
G. D. YADAVand G. MASON
H(X) = 1 - QY.
H(X) Fmbobillfy
-‘2
-I Normelised
Curvalure
Difference
RX
Fig. 3. Histogram of the normalised half curvature difference at rhe onset of blob motion for blobs which were stationary at RX = - 2.5. There is about a 50% probability (i.e. W(0)) of a blob moving in the opposite direction to the flow producing the historical curvature as that flow is reduced from the historical value to zero.
(8)
The values of X at which 35 blobs first moved were measured and the distribution of Qu used to find an experimental distribution of H(X) vs X. The results are shown in Fig. 3 together with the theoretical function of H(X) (eqn 5) obtained from j(r) and g(r). The agreement is quite good and provides a better test of the derivation of H(X) than the use of Ng et al.% results[3] by Mason[9]. The structure of H(X) was also investigated using blobs which were stationary at zero flow because, as explained above these were experimentally much easier to obtain. At zero flow the history curvature is approximately zero, but because of the buoyancy forces produced by the density difference between the bulk organic phase and the water blobs, these blobs actually had a history curvature, dependent upon their length, of between + 1 and - 1. These results, shown in Fig. 4, cannot give an absolute value for H(O), but they do confirm the general shape of H(X) where X >O. This region is important for the continual progressive movement of blobs in the flow field. CONCLUSIONS
Curvature
,
RX
0
I
I m
indicates initial
II
< 2
the curvature
The experimental measurements of the onset of movement of blobs with different histories confum the overall model of blob motion used by Ng et al.[3] and also the modifications put forward by Mason[9]. Bearing in mind the approximations involved in obtaining f(r) and g(r) from the model of a sphere packing, the confirmation is surprisingly good. It may be that there is a degree of self compensation in approximating the meniscus profiles to spherical surfaces because the same approximation is made for both the head of the blob and for the foot. Acknowledgmcnl-G. D. Yadav acknowledges the receipt of a Leverhulme Visiting Fellowship for support during the course of this work. NOTATION
0
Curvature
RX
I-
I
2
Fig. 4. Histogram for the onset of blob motion for blobs which
f(r) g(r) H H(O)
were stationary at .zero flow. The small density difference between the bulk fluid and blob fluid makes the historical initial half curvature differ from zero and dependent upon blob length. These histograms represent a portion of H(X) of Fig. 3.
H(X)
experiments duplicated those of Ng et al. [3] except that the first step was observed and not the flow required to advance the blob by one sphere diameter.
Q(Y)
EXPERIMENTAL
RESULTS
For blobs stationary at a curvature of Y = -2.5 (the minus sign is used here to indicate that the curvature difference is in the opposite direction to the blob motion) the value of H( -2.5) is virtually 1. So QY becomes 1-H(X)or
WY)
window (throat) radius distribution function cavity (body) radius distribution function probability probability of a blob being stationary at a curvature difference X = 0 probability of a blob with no special past history being stationary at X probability of a blob with a special past history being stationary at a curvature difference 2Y where Y
radius of curvature of blob, L radius of body (cavity) of blob (retreating meniscus), L radius of throat (window) of blob (advancing meniscus). L radius of spheres in packing, L
The onset of blob motion in a random sphere packing X
half curvature difference between the advancing and retreating menisci of blob,
Y Greek
, L-’
special half curvature difference, L-’ symbols
P (r
a factor defined as (I - (r,/r& interfacial tension, MT-*
dimensionless
REFERENCES [ll Melrose J. C. and Brandncr C. F., J. Can. Pet. Technol. 1974 13 54. [21 Shah D. 0. and Schechter R. S., Improved Oil Recovery, Academic Press, New York 1977.
1465
I31 Ng K. hf., Davis H. T. and Scriveo L. E., Chem. Engng Sci. 197833 1009. I41 Ng K. M. and Payatakes A. C., A.LCh.E.L 198026 419. [51 Payatakes A. C., N8 K. M. aad Flumerfelt R. W., A.I.Ch.E..J. 198026 430. [61 Larson R. G., Striven L. E. and Davis H. T., Chem. Ettgng Sci. 198136 57. 171Larson R. G., Davis H. T. and &riven L. E., Chem. Engny: Sci. 1981 36 75. I81 Mason G., I. Colloid Interface Sci. 197135 279. 191Mason G., Chem. Engng Sci. 198338 1455. [IO] Haines W. B., J. Agric. Sci. 192717 264. [I 11 Mason G., I. Colfoid Interface Sci 197241 208. [I21 Mason G., I. Cofloid Interface Sci. 198288 36. 1131Morrow N. R., J. Can. Pet Technol. 197918 35.
THE ONSET OF BLOB MOTION IN A RANDOM SPHERE PACKING CAUSED BY FLOW OF THE SURROUNDING LIQUID G. D. YADAVt
and G.
MASON*
Department of Chemical Engineering, Loughborough University of Technology, Loughborough, Leicestershire. England (Received 26 October 1981; in revised form 20 December 1982; accepted
2 February 1983)
Abstract-Entrapped liquid blobs in the pore space of a packing of equal spheres were made visible by matching the refractive index of the spheres with that of the bulk fluid filling the pore space. By pumpingbulk fluid through the surrounding pore space an experimental determination of the curvature difference between the ends of the blobs at the onset of mobilisation was made. This was repeated for blobs with special histories. The probability of blob movement was calculated. Good agreement between the theory and experiments was found.
INTRODUCTION
During the commercial extraction of oil a significant amount of oil is entrapped in the porous reservoir rocks as small blobs or ganglia, typically a few pore diameters long Displacement of such residual oil might be achieved by reduction of the oil/water capillary forces together with the application of a pressure gradient such as that produced by pumping water through the adjacent pores in the rock. The published work [ l-71 indicates that the capillary forces dominate over the viscous forces and that for blob movement the interfacial tensions have to be reduced to very low levels indeed. The exact mechanism of blob motion is of particular interest because it seems Iikely that the ease of mobilisation is directly related to the length of the blob, and so increasing the blob length as well as reducing the interfacial tensions might make mobilisation commercially practicable. Ng et a/.[31 have presented a theory for the mobilisation of a blob in the pore space of a random packing of equal spheres. The blob should move when the hydrodynamic pressure difference between its ends is greater than the capillary pressure difference. They tested this theory with an experiment in which the movement of blobs could be directly observed. Their blob visuahsation experiment consisted of 3.2mm Perspex spheres in a square column of 20 mm x 20 mm x 150mm: used glycerol-water mixture with matching refractive index as continuous phase; and Meriam manometer oil of similar density for the blob phase. Blobs were observed to move in quantum-like jumps. During their experiments, the history of each blob was always the same: the blob was stationary when the bulk fluid flow was zero. Their measurement was of the flow required to advance an isolated blob of desired orientation by one sphere diameter and this involved several jumps. The theoretical analysis of Ng et al. used auerage values of the meniscus
*Author to whom correspondence should be addressed. tLeverhulme Visiting Fellow from Department of Chemical Technology, University of Bombay, Matunga Road, Bombay409019. India.
curvatures and so could only predict the auerage conditions for blob movement and not specific probabilities. The distributions of meniscus curvature for both advancing and receding menisci in the pore space of a random packing of equal spheres have been given by Mason[8]. Subsequently these were used to calculate the probability of blob motion for two simplified mechanisms-namely, branching and no branching, and indicated that the past history of the blob was especially important in determining the probability of the initial quantum-like jump [9]. The object of the present paper is to report a blob visualisation technique similar to that of Ng et al. [3], but with experiments better designed to test Mason’s theory[9]. The basic method was as follows. An isolated blob of known length and orientation was given a specific history, that of being stationary in a relatively high superficial velocity of the bulk liquid. The blob was observed while the flow of the liquid was gradually and continuously reduced to zero and then reversed and increased. The flow at which the blob made the first jump in the opposite direction to the initial history flow was noted. The cumulative probability so obtained is important in calculating the subsequent probabilities of blobs advancing larger distances. THEORY OF BLOB MOTION
The theory of Ng et al.[3] for blob motion requires that the capillary pressure between the ends of the blob be less than the hydrodynamic pressure produced by the viscous fluid flowing in the pore space alongside the blob. In a packing of spheres the menisci at the advancing end (head) of the blob are forced through constrictions in the pore space and the menisci at the retreating end (foot) of the blob move through larger cavities. In general the head men&i are more highly curved than the foot menisci and so a pressure is required to move the blob. The pore space in a random packing of equal spheres can be modelled by assembling tetrahedra whose six edges are chosen according to an edge length distribution function (similar to the radial distribution function)[8] and centering spheres at each vertex. These tetrahedra 1461
1462
G. D. YADAVand G. MASON
winhave an internal cavity and four oonstrictions.or dows leading in and out, one for each face (see Fig. 1). Using the Haines approximation[lO] that a capillary meniscus in such complex geometry is approximated by a spherical surface it is possible to calculate the window (throat) radius distribution, f(r), and the cavity (body) radius distribution, g(r). There is a single cavity radius associated with each tetrahedral pore, but there are four window radii. Although there is some correlation between the sizes of adjacent windows (they each have a tetrahedron edge in common), this is not strong and the approximation that they are independent is not unreasonable [ 1I]. For an individual tetrahedral pore there is a strong correlation between its largest window radius and the cavity radius [12], but for blobs more than one pore long the cavity radius at the foot of the blob is unrelated to the window radii several pores away at the head of the blob. The maximum capillary pressure difference, AP,,, along a blob is given by:
The probability that the trailing foot has a radius between rb and rb f dr,, is g(rb)drb. Combination of these two probabilities gives: 3[c
fWdr]2f(r,l
dw(d
drb
(3)
this being the probability that the largest radius of the head lies between r, and r, + dr, and the radius of the foot lies between rb and rb + dra. Let the total curvature difference between the ends of the blobs produced by the flow of the bulk fluid be 2X. Let the probability of half of the capillary curvature difference (l/r, - l/rtJ being greater than X be H(X). H(X) will be given by the integral of (3) over all r, and r, for which &$>X
or, rearranging
rb
>
(1
_rir~)
H(X)=6[~~f(r)dr]lf(r.)~~~(r)drdr,
(4)
(5)
~~[~f(r)dr]lf(r.)10_R(r)dldlr’ O*rb
where rr is the interfacial tension, r, and rb are the radii of the window (throat) and cavity (body) respectively, and /3, the parameter used by Ng et al.[3], is (I - (dra). In the analysis of Ng et al., /3 was assigned an average value. As the blob moves so its head enters a tetrahedral pore through one face and leaves through the largest of the three remaining faces. The probability that any two faces out of three have radii less than r,, the radius of the head end of the blob, and the third face has a radius between r, and rt + dr, is:
3[~f(r)dr]2f(ddTr-
(2)
H(X) represents the probability that a blob with no special past history does not move at a curvature difference X, this being half of the curvature difference produced between its ends by the bulk fluid flow. The upper bound of the first integral is limited to 1/X because rb becomes infinite at that value. The difference in curvature X is a function of blob length, fluid density difference, interfacial tension, and the Darcy pressure gradient in the pore space of the sphere packing. Mason[S] gives distributions for the functions g(r) and f(r) and so H(X) can be evaluated[9]. Every experimental blob has a special past history and so H(X) is not directly applicable. If a blob is formed and is stationary at zero flow (X = 0) then the probability of it mooing by at least a single step when the curvature between its ends produced by the fluid flow is increased to 2X is
HUN-H(X) H(O)
(6)
because H(O) is the probability of being stationary at 0 and H(X) of being stationary at X; the difference represents the probability of moving. H(0) in the denominator normalises the probability so that it becomes unity when X = m (and hence H(X) = 0). If the special past history of the blob is that it was stationary at curvature Y then the probability (Qu) of moving by a single step at some subsequent increase in curvature X is: WYi;yT(X)
Fig. 1. View of a single irregular tetrahedral pore. Centred at each vertex are portions of the spheres of the packing. The pore space is a central cavity with constrictions leading out.
= QY.
(7)
It is possible to establish experimentally the probability Qy by observing the behaviour of many isolated blobs when subjected to different bulk flows (varying X and Y) and so the theoretical function H(X) can be compared with the experimentally established one.
The onset of blob motion in a random sphere packing -AL
The experimental
determination of Qy can be achieved by directly observing the mobilisation of individual blobs in a randomly packed column as the flow of the bulk fluid is changed. Observation of the blob demands that the refractive indices of the packing and the bulk fluid be quite accurately matched. Also a very steady controllable flow of bulk fluid is essential as any flow pulsations vary the pressure gradient along the blob and cause it to mobilise. A 35 mm i.d. glass column was packed with 3.125 mm dia. precision ground glass spheres. The effective length of the column was 150mm and this section had a flat glass window. The refractive index of the glass spheres was 1.524. The bulk phase (wetting) was a suitable mixture of benzyl acetate and benzyl benzoate which had a refractive index very close to that of the glass balls thus rendering them virtually invisible. The mixture had a density of 1.0665 g/cm3 and viscosity of 2.18 CP at 25°C. The blobs were water containing ammonium purpurate dye. The column was provided with pressure tappings and had injection ports at either end to facilitate the introduction of blobs into the packing. Figure 2 shows a general outline of the experimental set-up. The production of a controllable flow of the organic phase proved to be quite difficult and the system used in Fig. 2 went through several modifications before a satisfactory solution was found. There were two thermostated reservoirs (6 and 7 on Fig. 2), each of 21. capacity, containing the organic phase and connected to the inlet and outlet of the packed column (1) through an inverted U-tube flowmeter (3) and two 3-way cocks (4). A steady flow of the organic liquid was created by pressuring the inlet reservoir (6) with a small air-pump of a type used for aerating fish-tanks. The flow rate of the liquid could be maintained by controlling the air flow rate into or out of the air-space above the liquid in the reservoir by needle valves (5). The downstream reservoir (7) was open to atmosphere and was fitted with a level detector (11) and controller (12) which varied the speed
Fig. 2. Schematic diagram of the experimental set-up: (1) packed column containing 3.175 mm dia. glass spheres (top and bottom injection tubes extended into the effective zone of the column); (2) inverted [I-tubedifferentialmanometer with wateraslight Ruid;(3) inverted U-tube capillary flow-meter; (4) three-way plug cock; (5) needle valve; (6) continuous phase (organic) reservoir; (7) receiver for continuous phase with level indicator; (8) peristaltic pump; (9) pressure surge vessel; (10) air-pump; (11) lever mechanism for controller; (12) recorder; (13) digital volt meter; (14) polyethylene float; (15) mercury manometer.
14.63
of a peristaltic pump (8) and this pump returned the organic phase to the pressurised upstream reservoir. The flow of organic phase was thus independent of the pressure pulses produced by the pump. A precalibrated inverted U-tube capillary flowmeter (3) was used to measure the liquid flow rate. The two 3-way plug cocks (4) in the line as shown in Fig. 2, made it possible to reverse the flow of liquid through the column. The pressure drop per unit length of the packed bed was measured directly with a coloured-water/organic phase manometer (2). Coloured-water blobs of known volume were introduced by a hypodermic syringe through the injection tube at the bottom of the column into the packing. The blob lengths were measured with a cathetometer. The interfacial tension between the two phases was measured by a ring tensiometer and was found to be 12dynelcm at 25°C. The densities of the liquids were measured with a specific gravity bottle and the viscosities with an Ostwald viscometer. Procedure The packed column was first filled with the organic phase and all air bubbles were removed from the packing and the connecting lines. This was quite awkward but once flow was established air bubbles could be cleared by simply increasing the fluid Row. To avoid refilling the tubes a special provision was made for isolation of the packed zone and for its emptying and refilling and this was the method for removing spent blobs from the column. Blobs were of water coloured with ammonium purpurate, a dye which was insoluble in the organic phase. At zero flow rate of the bulk organic phase, a blob of known volume was introduced into the packing, and an upward flow used to move the blob to the middle of the packing. An upward (or sometimes downward) flow of liquid was arranged to give a value of Y, the half curvature between the blob ends, of about 2.5 thus giving the blob a special history. The blob would move some distance before stopping and if its length changed, the Bow had also to be changed to bring the imposed history half curvature back to 2.5. The blob length was then measured and also its orientation noted. Then the flow rate of liquid was slowly reduced whilst the blob was observed. When the first quantum movement of the blob in the direction opposite to the impoged history flow was observed, the pressure drop across the packing was recorded. Often the flow would reach zero with no observed step. Then the flow would be reversed and increased until the stepwise movement was observed. The flow of liquid was increased or decreased by controlling the flow of air into the reservoir. In calculating the value of the curvature at which the blob moved, correction had to be made for the hydrostatic pressure difference caused by the small density difference (0.00665 g/cm3) of the two phases (see for example Ref. [ 131). It was experimentally quite difficult to perform experiments with Y = 2.5 and so in addition experiments were also performed where the blob was static in zero flow (and had the curvature history produced entirely by buoyancy), this making Y approximately zero. These
G. D. YADAVand G. MASON
H(X) = 1 - QY.
H(X) Fmbobillfy
-‘2
-I Normelised
Curvalure
Difference
RX
Fig. 3. Histogram of the normalised half curvature difference at rhe onset of blob motion for blobs which were stationary at RX = - 2.5. There is about a 50% probability (i.e. W(0)) of a blob moving in the opposite direction to the flow producing the historical curvature as that flow is reduced from the historical value to zero.
(8)
The values of X at which 35 blobs first moved were measured and the distribution of Qu used to find an experimental distribution of H(X) vs X. The results are shown in Fig. 3 together with the theoretical function of H(X) (eqn 5) obtained from j(r) and g(r). The agreement is quite good and provides a better test of the derivation of H(X) than the use of Ng et al.% results[3] by Mason[9]. The structure of H(X) was also investigated using blobs which were stationary at zero flow because, as explained above these were experimentally much easier to obtain. At zero flow the history curvature is approximately zero, but because of the buoyancy forces produced by the density difference between the bulk organic phase and the water blobs, these blobs actually had a history curvature, dependent upon their length, of between + 1 and - 1. These results, shown in Fig. 4, cannot give an absolute value for H(O), but they do confirm the general shape of H(X) where X >O. This region is important for the continual progressive movement of blobs in the flow field. CONCLUSIONS
Curvature
,
RX
0
I
I m
indicates initial
II
< 2
the curvature
The experimental measurements of the onset of movement of blobs with different histories confum the overall model of blob motion used by Ng et al.[3] and also the modifications put forward by Mason[9]. Bearing in mind the approximations involved in obtaining f(r) and g(r) from the model of a sphere packing, the confirmation is surprisingly good. It may be that there is a degree of self compensation in approximating the meniscus profiles to spherical surfaces because the same approximation is made for both the head of the blob and for the foot. Acknowledgmcnl-G. D. Yadav acknowledges the receipt of a Leverhulme Visiting Fellowship for support during the course of this work. NOTATION
0
Curvature
RX
I-
I
2
Fig. 4. Histogram for the onset of blob motion for blobs which
f(r) g(r) H H(O)
were stationary at .zero flow. The small density difference between the bulk fluid and blob fluid makes the historical initial half curvature differ from zero and dependent upon blob length. These histograms represent a portion of H(X) of Fig. 3.
H(X)
experiments duplicated those of Ng et al. [3] except that the first step was observed and not the flow required to advance the blob by one sphere diameter.
Q(Y)
EXPERIMENTAL
RESULTS
For blobs stationary at a curvature of Y = -2.5 (the minus sign is used here to indicate that the curvature difference is in the opposite direction to the blob motion) the value of H( -2.5) is virtually 1. So QY becomes 1-H(X)or
WY)
window (throat) radius distribution function cavity (body) radius distribution function probability probability of a blob being stationary at a curvature difference X = 0 probability of a blob with no special past history being stationary at X probability of a blob with a special past history being stationary at a curvature difference 2Y where Y
radius of curvature of blob, L radius of body (cavity) of blob (retreating meniscus), L radius of throat (window) of blob (advancing meniscus). L radius of spheres in packing, L
The onset of blob motion in a random sphere packing X
half curvature difference between the advancing and retreating menisci of blob,
Y Greek
, L-’
special half curvature difference, L-’ symbols
P (r
a factor defined as (I - (r,/r& interfacial tension, MT-*
dimensionless
REFERENCES [ll Melrose J. C. and Brandncr C. F., J. Can. Pet. Technol. 1974 13 54. [21 Shah D. 0. and Schechter R. S., Improved Oil Recovery, Academic Press, New York 1977.
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I31 Ng K. hf., Davis H. T. and Scriveo L. E., Chem. Engng Sci. 197833 1009. I41 Ng K. M. and Payatakes A. C., A.LCh.E.L 198026 419. [51 Payatakes A. C., N8 K. M. aad Flumerfelt R. W., A.I.Ch.E..J. 198026 430. [61 Larson R. G., Striven L. E. and Davis H. T., Chem. Ettgng Sci. 198136 57. 171Larson R. G., Davis H. T. and &riven L. E., Chem. Engny: Sci. 1981 36 75. I81 Mason G., I. Colloid Interface Sci. 197135 279. 191Mason G., Chem. Engng Sci. 198338 1455. [IO] Haines W. B., J. Agric. Sci. 192717 264. [I 11 Mason G., I. Colfoid Interface Sci 197241 208. [I21 Mason G., I. Cofloid Interface Sci. 198288 36. 1131Morrow N. R., J. Can. Pet Technol. 197918 35.