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A filtration model for the flow of dilute, stable emulsions in porous media—I. Theory
Chemicd Jh&eerimg Science. Printed in Great Britain.
Vol.
41, NO. 2. pp.
263~272,
1986.
0009-2509/86 S3.00+0.00 Pergamon Press Ltd.
A FILTRATION MODEL FOR THE FLOW OF DILUTE, STABLE EMULSIONS IN POROUS MEDIA-I. THEORY H. SOOT and C. J. RADKE* Chemical Engineering Department, University of California, Berkeley, CA 94720, U.S.A. (Received
25 June
1984)
Abstract-Flow of dilute, stable emulsions in porous media is important in several oil recovery processes. Because underground media have relatively low permeabilities, the emulsion drop sizes may overlap the pore sizes. Hence, strong interaction occurs between the emulsion droplets and pore constrictions, and local flow redistribution occurs within the porous medium. To predict quantitatively how emulsions are transported in underground media, a theoretical model is required which correctly accounts for the interactions between the flowing droplets and the pore walls. In Part I of this work, we present a simplified filtration model describing the flow of stable, dilute emulsions in unconsolidated porous media. In the model, emulsion drops are captured in pores by straining and interception and, thus, reduce the overall permeability. Transient flow behaviour is characterized by three parameters: a filter coefficient, a flow-redistribution parameter and a flow-restriction parameter. The filter coefficient controls the sharpness of the emulsion front, the flow-redistribution parameter dictates the steadystate retention, as well as the flow redistribution phenomenon, and the flow-restriction parameter describes the effectiveness of retained drops in reducing permeability. Critical comparison is made between the new filtration theory and the current continuum-viscous and retardation models for emulsion flow in porous media. Only the filtration picture is able to explain all the experimental observations. Quantitative comparison between the filtration flow theory and experiment is presented in Part II.
INTRODUCTION Emulsion flow in porous media can occur either by accident or by design in enhanced oil recovery, including several chemical (Jennings et al., 1974, Willhite et al., 1980) and steam-flooding (Doscher, 1967) processes. The emulsions, which may be either of the oilin-water or of the water-in-oil type, have drop sizes of a few micrometres, and they must transport in reservoir rock having permeabilities ranging from a tenth to several Darcys. In order to evaluate quantitatively the effects of such emulsions on oil recovery, it is essential to develop a mathematical model which is based on the actual physical mechanisms of droplet percolation. For the case of surfactant-stabilized, oil-in-water emulsions flowing in both consolidated and unconsolidated media, experimental findings of several researchers (McAuliffe, 1973; Devereux, 1974b; Alvarado and Marsden, 1979; Soo and Radke, 1984a) reveal the following features: (a) the permeability of the porous medium decreases monotonically with time, (b) the particles do not appear in the effluent directly after a throughput of one pore volume of emulsion injected, (c) the rate and the amount of permeability decrease and the droplet breakthrough time increase with increasing drop-size to pore-size ratios, (d) in some cases the rate and the amount of permeability reduction increase slightly with decreasing flow rates (McAuliffe, 1973), (e) eventually a steady state is reached, and (f) when emulsion injection is followed by many pore volumes of water flow, droplets elute for +Present address: Union Carbide Corporation, South Charleston,WV 25303, U.S.A. *To whom correspondence should be addressed.
about the first pore volume but the original permeability decrease caused by the emulsion remains (Soo and Radke, 1984a). Currently, two theories are available to describe this emulsion transport behaviour: the bulk viscosity model of Alvarado and Marsden (1979) and the droplet retardation model of Devereux (1974a). Both treatments are limited. In the viscosity model the emulsion is viewed as a continuum, single-phase fluid. No interaction between drops and pore walls is allowed, even though drop sizes and pore sizes overlap. The mathematical description of the viscosity model is based on porous-medium scaling laws identical to those used for non-Newtonian polymer solutions (Savins, 1969). Hence, emulsion flow is predicted to differ from Darcy’s law only when the bulk emulsion viscosity varies with shear rate. No permeability reduction, transient or steady state, is predicted, and the emulsion particles must elute exactly at one pore volume of fluid injection. When compared to the previously summarized experimental results, the viscosity model is seen to have validity only in the case of high-concentration emulsions, which approach a steady state quickly and which have small drop-size to pore-size ratios. The retardation model, originally delineated by McAuliffe (1973), does consider transient permeability reduction. In this model, the emulsion drops are retarded during their passage along the tortuous paths of the medium. When a droplet encounters a pore of throat size smaller than its own diameter, it deforms and squeezes through. In the constricted pore it experiences a capillary resistance force (Jamin, 1860, Gardescu, 1930) and, therefore, moves at a slower
H. Soo and C. J.
264
speed than the continuous phase, thereby causing an overall permeability reduction. A steady state is reached when the emulsion breaks through the porous medium. To describe this physical picture mathematically, Devereux (1974a) modifies the classical BuckleyLeverett theory for two-phase flow in porous media (Buckley and Leverett, 1942) by including a retardation factor in the pressure driving force of the dispersed oil phase. He presents a solution for the case of constant-pressure flow. The case of constantvelocity flow follows directly [a derivation is outlined by Soo (1983)]. The resulting expression describing the transient permeability reduction, K/Ko, for constant-velocity flow is as follows: -.--I K/K,
=
(1 -zZ2,)+
pdRf
9 (1)
Cl&, L
-1
where Z2,is the dimensionless position of the emulsion front, or f, =
Cir/Sd
for ciT/Sd < 1
1
for cir/Sd 2 1.
In eqs (1) and (2), pd and cc, denote the bulk viscosity of the droplet phase and of the continuous water phase, respectively; u is the superficial velocity, Sd is the constant fractional pore space occupied by oil droplets (i.e. the oil saturation) behind the emulsion front, r is the pore volume of the injected emulsion, and ci is the inlet volume concentration of the emulsion. Three unknown parameters appear in eq. (1) which must be evaluated from the experimental flow results. They are y, the total capillary retarding force per unit volume of the porous medium, and k, and k,, which are the relative permeabilities of the oil droplets in the emulsion and of the continuous water phase. Both k,, and k, are taken as known functions of the oil saturation S, (Buckley and Leverett, 1942; Devereux, 1974a). Equations (1) and (2) correctly reflect several features of the experimental emulsion flow results cited above. The permeability of the porous medium decreases as emulsion is injected (i.e. as ff increases), and a steady state is reached. Further, if the emulsion droplet saturation S, is not a function of flow rate, and if S, and y increase with drop-size to pore-size ratio, then eq. (1) also correctly implies that the permeability is more reduced with decreasing flow rate (u) and with increasing drop-size to pore-size ratio. Devereux’s theory, however, possesses two characteristics which do not match the experimental results: (1) the emulsion breaks through as a step change, and (2) the permeability of the porous medium ultimately rises back to its initial value when emulsion injection is followed by water. Thus, although the retardation model properly describes the transient, permeabilityreduction flow behaviour, it must misrepresent the underlying physical mechanisms. The recent experimental study of Soo and Radke
RADKE
(1984a) reveals that emulsion droplets are not simply retarded when they flow through porous media; they are actually captured. As demonstrated in Fig. 1, when drops flow in the porous medium, they clog pore constrictions of sizes smaller than their own by lodging between sand grains. This mode of capture is denoted as straining. Also, they are trapped in recirculation eddies, wedge in crevices, or sometimes attach to the pore walls due to van der Waals, electrical, gravitational and hydrodynamic forces. Interception is then the designated capture mode. Drop interception near pore constrictions may reduce the effective pore diameter and thus permit straining of other drops. As drop retention builds, flow diverts to the larger pores where capture probability is lower because of the increased pore sizes. This physical picture is very similar to that of a deepbed filtration process. However, it differs from traditional deep-bed filtration in the following respects. In a traditional filtration process, the particle-size to pore-size ratio is small. Therefore, the interception capture mechanism dominates, and the particles are captured mostly on the surfaces of the media grains. Flow distribution in the porous medium is not altered much by the presence of the captured particles. In this case, the whole process can be modelled by considering only the interaction between a single particle and a single collector (e.g. a sand grain). Permeability reduction caused by the retained particles is not significant and is sometimes overlooked. For the type of emulsion flow considered in Fig. 1, however, the drop size is of the order of the pore size, and drops are captured both by the straining and the interception mechanisms, with the possibility of straining being the dominant mechanism. Since the pores are either totally blocked or greatly restricted by the retained drops, flow redistribution takes place among neighbouring pores. To model emulsion flow, one must consider this flow-redistribution phenomenon and, consequently, treat the porous medium as a whole. As a corollary, the permeability reduction caused by the retained drops is large, and knowledge of this permeability reduction is important.
IN-l
Fig. 1. The mechanism of emulsion flow in porous media.
Filtration model for the flow of dilute, stable emulsions in porous media-1 Additionally, liquid drops are deformable. Hence, strained drops may reentrain into the flowing stream, once the local pressure drop increases enough to overcome capillary restraining forces (Jamin, 1860; Gardescu, 1930; Oh and Slattery, 1979). Under this condition, the amount of drops retained locally couples strongly to the local pressure drop. Finally, the shape of the captured drops and, hence, their influence on local permeability depend on the contact angle. Thus, for many reasons, conventional deep-bed filtration theory developed for undeformable solid suspensions is not strictly applicable for dispersed liquid droplet flow. Part I of this work presents a genera1 theory, based on deep-bed filtration concepts, to describe dilute, stable emulsion flow in porous media, where drop sizes and pore sizes overlap. The flow-redistribution phenomenon, as well as the possibility of large permeability reductions; is included in the model. The treatment is restricted to monodisperse emulsions of such high interfacial tensions that capillary reentrainment is not possible. In Part II, which follows, the theory is compared quantitatively with data appropriate for these restrictions, and experimental values of the parameters are obtained. Extensions of the theory for the cases of low tensions and drop-size distributions are handled separately (Soo, 1983; Soo and Radke, 1984b). Three parameters with clear physical interpretation arise in the theory. One indicates the probability of drop capture, the second reflects local flow redistribution, and the third measures the effectiveness of captured drops in restricting flow. Upon comparison with the viscosity and retardation models, only the filtration-based flow theory is able to explain both transient pressure-drop behaviour and effluent drop concentrations. GENERAL THEORY
As in deep-bed filtration theory (Herzig et al., 1970; Tien and Payatakes, 1979), the procedure is to predict the local disperse-phase retention profiles in time and distance. Then, by relating disperse-phase retention to local permeability, the overall transient flow rate-pressure drop behaviour may be established. Compared to classical colmatage theory, however, dilute emulsion flow of relatively large drops in finely grained porous media is complicated by the strong interaction of drop retention and local flow redistribution. One simplification does arise for highly stable emulsions, in that droplet coalescence does not occur. Local retention We begin by writing the unsteady, disperse-phase continuity equation in one-dimension with constant flow rate and negligible longitudinal dispersion (Herzig et al., 1970): r Pm -I
at
aPdDP 1 1 Jo-
a +c+
qeoc+ u] =
it
= -u$
(3)
265
where c(t. x) is the volume concentration of oil drops (still suspended in the flowing phase) at time t and axial position x, u is the constant superficial velocity, .Q,is the clean-bed porosity, 0 is the volume of retained drops per bed volume (thus removed from the flowing phase), and a*dD, is the volume of the drops retained per bed volume in pores of throat diameters between D, and D, + dD, (i.e. 10”a,dD, = u). Inherent in eq. (3) is the approximation that the clogged-bed porosity E is not much different than the initial bed porosity, .Q,. According to Herzig et al. (1970), this approximation is appropriate for dilute suspensions for which EC < u. To proceed, an expression is needed for the capture rate of drops in pores of throat diameter D,. We postulate that the capture kinetics are first order in the convected flux of drops, UC:
where 0,~ is the local fraction of emulsion fiow to pores of throat diameter Dp, and 1, is the filter coefficient of drops in that size of pore. The filter coefficient has the dimension of inverse length, and the inverse of the filter coefficient has the physical meaning of the distance a drop travels before being captured. Hence, the pore filter coefficient is related to the efficiency of drop capture in given sized pores by 3 I,$,, where 1, is the pore length. A linear relation ‘IP between capture efficiency and I, is demanded in this work. The reasons are as follows. When D, 2 D, (i.e. straining capture), the capture efficiency is near unity. We anticipate that the length the drop travels before being captured is the pore length. Accordingly, 1, equals l/l,. Conversely when D, Q D, (i.e. interception capture), 1, is close to zero, and again, as others have shown, A,, = qp/l, (Tien and Payatakes, 1979). Finally, we caution that when drop capture efficiencies are near unity, the continuum suspension balance in eq. (3) no longer holds. Its use here implies that the fraction of pores which have large capture efficiencies is small so that the flowing drop concentration can be treated in a continuum sense. The pore capture kinetics in eq. (4) imply that decolmatage does not occur. For the case of significant electrical double-layer repulsion between drops and between drops and sand grains, high drop tensions, and relatively low flow rates, the experimental results of Soo and Radke (1984a) prove that captured drops do not re-entrain into the flow stream. Other surface chemistry and flow conditions may require an additional term in the drop-capture kinetic expression. Sincere-entrainment is not permitted in eq. (4), steadystate flow can occur only in those pores for which I, eventually approaches zero. Knowledge of I, and 8, is required in eq. (4). With dilute emulsions, where no simultaneous multipledrop capture occurs, the filter coefficient is independent of drop-volume concentration. However, it is a complicated function of surface chemistry, drop size, pore size, interstitial velocity, and the amount and distribution of oil retained in pores of a given size.
H. Soo and C. J. RADKE
266
Likewise, 13~is a complicated function of the amount and distribution of oil retained locally in all pores. Some progress can be made by adopting a model for the porous medium under study. We utilize the unitbed-element treatment of Payatakes et al. (1973). In their model the porous medium is comprised of a series of statistically identical unit bed elements. Each unit bed element in turn consists of a number of parallel, constricted tubes with a throat-size distribution corresponding to the measured pore-size distribution of the porous medium. Payatakes et al. (1973) assume that for each constricted tube, the length of the tube is proportional to the throat size of the tube based on the following argument. In the porDus medium, on the average, large throat sizes form among grains which are large, and/or are loosely packed. Therefore, large diameter pores are expected to be longer. The average pore length is defined as the unit-bed-element thickness. Since the pressure drop across each constricted tube is assumed to be the same in a unit bed element and since tube lengths and throat sizes are proportional, it follows that the local volume flow through each tube is directly proportional to the volume of that tube. Hence, in a clogged unit bed element, 0, is written approximately as: 8, = (ar - e&o0
-cb
(5)
where a,, is the clean porous-medium porosity contribution of pores of throat diameter D, (i.e. the variation of &p/&o with D, is the traditional pore-volume frequency distribution). After multiplication of each side by the superficial velocity u, eq. (5) states that as smaller pores are blocked, flow diverts proportionally to the larger-volume pores. The expression for Op is simplistic in that it ignores how the retained drops are distributed in a particular pore. Nevertheless, it is exact for the cases in which the drop size is small compared to the pore size, the flow through the pore is not greatly disturbed by the retained drops, or the pore is totally blocked. Combination of eqs (4) and (5), followed by integration over all pore sizes, yields the overall dropcapture rate:
au _-
cu
at- (Eo:@ o 4%
- 3) d%
05)
Equation (6) directly reflects the flow-redistribution phenomenon. When drop capture commences, the smaller pores in the bed element clog quickly, since they have a higher capture efficiency (i.e. a larger Q_ Flow shifts to the larger pores where the capture efficiency is lower. When all the small pores (i.e. those pores in which mainly straining capture occurs) are blocked, the drops flow through the porous medium in a network of large pores. The process reduces to conventional deep-bed filtration, in which the particle sizes are smaller than the pore sizes. Equations (3)-(6) provide the drop retention profiles in each size of pores, and the drop volume concentration profiles in time and distance, once 1, is specified as a function of
drop size, pore size, and the amount and distribution of drops retained in the pore. Local permeability To calculate the overall pressure drop-flow rate behaviour, a relation is required between local permeability, K,, at position x and local retention cr. Unfortunately, a rigorous identity is difficult to construct. In addition to the amount of oil retained in the various pores, the location and configuration of that oil along the pore also influence the local permeability. Within the context of the unit-bed-element model we write: K, -_=I-’ KO
T” BpQD,,
&o I 0
(7)
where fl, is a parameter measuring the effectiveness of the retained drops in pores of throat diameter D, in restricting flow. For straining capture, /I, is simply the ratio of the volume of the pore to the volume of the retained drops. Naturally, /I, is a function of drop size, pore size, up and the distribution of captured drops within the pore. Combination of eqs (3), (6) and (7) and Darcy’s law (i.e. u = (K,/pc) ( - dp/dx), where ps is the bulk emulsion viscosity) gives a complete description of the flow behaviour including the desired transient permeability. However, the necessary detailed information on the quantities A, and 8, is not available. An exact theoretical treatment of this problem is not undertaken here. Rather, we employ a phenomenological approach in which average filtration parameters are defined and evaluated from experiments. PHENOMENOLOGICAL
MODEL
Local retention We classify the pores of the medium into two types: non-pluggable, in which the flow behaves like traditional deep-bed filtration and no straining occurs; and pluggable pores, which are eventually completely blocked by the drops. We stress that the pluggable pores do not necessarily have pore sizes smaller than the drop size. They may be two or three times larger. When drops flow in these pores, they may first capture on the surface of the pore and reduce the throat diameter. The following drop is then strained and completely blocks the pore. This multi-drop straining mechanism differs from the formation and bridging of dendrites reported in the past for solid suspensions (Sakthivadivel, 1966, Payatakes and Tien, 1976, Payatakes, 1977, Payatakes and Gradon, 1980, Soo and Radke, 1986). The transition pore-throat diameter between pluggable and non-pluggable constricted tubes is denoted as D,. The drop capture efficiency of the pluggable pores is either unity or close to unity initially and increases to unity very quickly. Consequently, it is taken as unity here so that the filter coefficient in these pores is just l/l,. In the non-pluggable pores, which eventually support the steady flow, the drop capture efficiency must
Filtration model for the flow of dilute, stable emulsions in porous media-1
decrease with increasing retention and eventually drop to zero when all capture sites are filled. The exact dependence of the capture efficiency on retention is complicated. However, in the case of small retention, we follow Ives (1967) in writing J-, = A,,(1
--Q+~,~)
for D, > D,,
(8)
where the subscript 0 denotes the initial clean-bed condition, and the subscript m indicates a maximum. The physical meaning of eq. (8) is that drops retained on the surface of the non-pluggable pores do not provide sites for further droplet capture. This is because capture must cease when yp reaches upVm. Herzig et al. (1970) and the companion Part II give further insight into eq. (8). Combination of eqs (6) and (8) and the condition of unit capture efficiency for the pluggable pores gives the overall capture rate:
au -=
267
straining capture in the small size pores. As retention builds, more pores are plugged. Local flow redistribution is no longer significant, because flow is mostly in the larger pores where capture is slight. Then a shifts its role to reflect steady-state retention. From eq. (12), l/a is established as the steady saturation of captured oil (i.e. when &r/at = 0, then l/a = u, /so). If straining capture dominates, then the occupied pores should not have a wide range of pore-throat diameters. In this case, eq. (11) shows that a has the value of so/j$epdDpt and is therefore a constant. Local permeability To simplify the expression of the local permeability reduction, we introduce an average flow-restriction parameter, j?, as follows: (13) Use of eq. (13) transforms eq. (7) to the desired result:
at
K -r=l-!?_ Ko
(9) We now introduce two phenomenological parameters. The clean-bed average filter coefficient, including both straining and interception, is defined by
where the meaning of L,, is generalized to include its value of l/l,, for pores of throat diameters less than D,. Equation (10) indicates that the average filter coefficient is obtained from the individual pore clean-bed filter coefficients by averaging over the initial porevolume distribution_ Similarly, based on examination of eq. (9), a flow-redistribution parameter, a, is defined as: LYE-
1 4u
O”(A P,o%%&,m)dDp. s 0
(11)
Here again the meaning of crp,,, is generalized to include its value of sp for pores of diameter less than D, (i.e. aside from a geometric factor of order unity b,_ = sp for D, -C D,). Physically, a accounts for the flowdiversion effect discussed above. Additional insight into the meaning of a is provided in Part II, where eq. (11) is utilized to calculate a directly [see eqs (5) and (11) of Part II]. For small retentions (i.e. u c so and up < Ed when D, > DJ, Appendix A demonstrates that eqs (9), (10) and (11) permit a phenomenological expression for the droplet capture kinetics:
au y$=
ASI 1 -dLQ (
% >
UC.
The parameter a describes the role of flow redistribution in the capture kinetics, due to drops retained in the pores. In general, a is a function of u. At the start of colmatage, significant flow redistribution occurs due to
(14)
so
This phenomenological expression has been proposed previously by Stein (1940) and Camp (1964). The flowrestriction parameter, 8, measures how effective retained drops are in reducing fluid permeability, per unit volume of drops retained. The effectiveness of a single large drop in reducing the local permeability is different from that of several small drops having the same total volume; j? reflects this. Clearly, j? also is a function of retention, u. At the beginning of the filtration process, most drops retain in the small pores; the value of j3 is close to the value of fi for straining capture. As cohnatage progresses, the value of /3 increases to its steady-state value. If straining capture dominates, p is again a constant at a value of about unity. Parameter retention dependence The dual-pore filtration model expressed in eqs (12) and (14) is most useful when the dependence of the three phenomenological parameters d,,, a, and fl on total retention is weak. For a given drop size and poresize distribution, 1,, is a constant and does not vary with retention. However, a and #I are constants only if straining is the dominant capture mechanism. When t When straining capture dominates, up,m = ep and eq. (11) becomes
Since for D, -c D, the pore-throat diameters (thus also the pore lengths) do not vary much, I,, is a weak function of D, and with cq. (10) we have
n, %dDP I 0 Dt t7 %dDP I 0
Eo a=
= Eg /I
4 cl
ep dD,.
H. so0 and C. J.
268
both straining and interception control the capture process, a and /3vary with retention_ To gain some insight into this retention dependence, we adopt a twopore model of the porous medium, as suggested by the work of Gruesbeck and Collins (1982). In the two-pore model, the porous medium consists only of two sizes of pores: small pores of throat diameter D, and porosity contribution sS, in which only straining occurs, and large pores of throat diameter D, and porosity contribution aI (i.e. E~+E~ = E,,), in which only interception occurs_ Without completely solving the two-pore model, it is possible to estimate the retention variations of a and 8. The mathematical calculation of a and p with the two-pore model is summarized in Appendix B. It demonstrates that these two parameters are functions of the ratio of the retention in each size of pores (i.e. of ~,/a~). The initial retention ratio, according to Appendix B, is written in the limit of zero time as
( >=ll,E,. 4-3
-QI
us
0
Equation (15) indicates that at the beginning of colmatage, the retention ratio is determined by the ratio of the fraction of flow in each of the two sizes of pores [i.e. according to the constricted-tube porous medium model adopted here, the fractional flow in each size pore is proportional to its void fraction (Payatakes et al_, 1973)]. It is also proportional to the ratio ofcapture probabilities in each sized pore. As the filtration process continues, the retention ratio varies from the initial ratio [eq. (15)] to the steady state of (a&,), = a&es, where tri mis the maximum retention in the interception pores. ’ Figure 2 displays how the flow-redistribution parameter a is predicted by eq. (B5) to vary from its initial value, aO, to its steady-state value, a,, as a function of the retention ratio eI /us based on the two-pore model. Although the function a/a_ in eq. (B5) ranges mathematically over the interval 0 c el/cs -Z 00, Fig. 2 shows only that portion of the range that is physically meaningful (i.e. those a,/~, ratios corresponding to a variation of a from a0 to a,). In Fig. 2, a, is given the typical value of 15 (i.e. see the experimental results of
RADKE
Part II) so that the steady retained oil saturation is the clean-bed porosity is l/a, = 0.067. Likewise, e. = 0.34, and the ratio of individual-pore filter coefficients is 1,/,X, = 100. Porosities of the strainingdiameter pores, sS, range from 2.5 x lo-4 to 1.9 x lo-‘; arrows indicate the time direction of the process. In the early stages of filtration, a/a, has its largest value, because a initially reflects the flow-redistribution phenomenon. During the later stages of the filtration process, a reflects the steady-state retention, and its value falls monotonically to a,. For very small or very large straining-pore porosities, es, only one capture mechanism dominates, either interception or straining. At intermediate values of es, where both straining and interception control filtration behaviour, the variation of a is maximal (about a factor of 2 in Fig. 2). With the two-pore model, calculated changes in the range of a increase when the amount of drop capture increases, or, equivalently, when a, is lowered. In actual porous media, however, a, and sS are interrelated so that es cannot be independently varied over large ranges as in Fig. 2. That is, at low values of a,, straining is more important and +, accordingly, is larger. Accordingly, variation in a with retention is not of consequence. Our experimental results (Soo and Radke, 1984a, Part II) show that a, is usually larger than 5 (i.e. a steady retained oil saturation less than l/z- - 0.2), in which case the maximum change in a is much less than two-fold_ The maximum change in a also depends on the choice of 1,/1,. However, this dependence is weak when &/A, is large. Similar behaviour is found for the flow-restriction parameter 8, except that the variations in fi are weaker than those for a. Therefore, based on the simplified results of the two-pore model, we treat a and #Jhere as constants, independent of total retention. Filtration model behaviour Once the filtration parameters are taken as constants, eqs (3) and (12) provide an analytical solution for the concentration and retention profiles. For a step increase injection of emulsion into an initially clean bed, this solution is well known (Herzig et al., 1970; Aris and Amundson, 1973):
u(T, 2) -= so
1 - exp (aAs,ci T) a[1 -exp(As,X)-exp(ahs,ciT)]’
(16)
and
Fig. 2. Retention dependence of the flow-redistribution parameter from the two-pore model.
c(T, 2) =- a(Z a) (17) a,(T) ’ ci where f = x/L is a reduced axial distance, T = T - 3 is a shifted time variable with r = ut/EOL defining the volume of injected emulsion to time t in units of pore volumes. The filter coefficient appears in reduced form with A,, = &tL, and the subscript i denotes the condition at the inlet to the packed bed. Equations (16) and (17) display exponential-like decaying concentration and retention profiles with a shock front at T = 0.
Filtration model for the flow of dilute, stable emulsions in porous media-I
r
Note that the effluent drop concentration cL = c( T, 1) a function of the variable ci(r - l), where ciz corresponds to the injected pore volumes of oil. The overall transient permeability K readily follows from eq. (14) and Dar&s law for the whole bed (i.e. under constant flow-rate conditions. K - ’
I
I
I
269 I
I
I 04
,
I
is
KO -=
K(r)
sl
[I -fiBa/so]-lda.
(18)
0
Equation (18) can be numerically integrated in general, using a/~~ from eq. (16), and in certain special cases analytical integration can be performed. For example, when the retention at the front of the core ci is close to the steady-state retention, the analytical result? is:
0
0
0.2 -Cc,.
x (1 + aci)-’
In
( >
L! l--
=L so
QL l-6’ so > (
,
(19)
wherein all the r-dependence resides within aL(7). In this expression the subscript L gives the condition at the end of the core of length L, so that aL = a(7,l). The emulsion viscosity pc does not appear in the transient permeability as long as the drop concentrations are low enough that the suspension viscosity is essentially that of the continuous fluid. Any hydrodynamic resistances due to flow of drops in narrow passages are neglected in this treatment. At steady state (ai = aL = 0, ), the retention approaches so/a and the steady permeability becomes K, /K, = 1 -B/a_ The role that the filtration parameters A,, and a and the flowrestriction parameter p play in determining the transient overall permeability and the ef3uent concentration of emulsion is now briefly discussed. Sample calculations are based on eqs (16), (17) and (19) with the base parameter values chosen for illustrative behaviour. The calculations are restricted to injected drop concentrations less than about 1 %. Figure 3 shows the influence of the reduced filter coefficient on permeability-reduction and emulsioneflluentconcentration histories for fixed values of a and J?.Solid lines correspond to transient permeabil-
VOLUME
0.6
OIL
Fig_ 3. Phenomenological
model calculations: the effect of the reduced filter coefficient A, on breakthrough-concentration and permeability-reduction histories.
ities and dashed concentrations. a;
PORE
lines correspond
to effluent emulsion
The em&ion breakthrough times [inflection points of C,(T)] and the steady-state permeabilities are the same in each case, since these quantities are controlled by a and fi. We see that A, determines the spread of the effluent-concentration histories and the time necessary for the permeability to reach its steady-state value. Thus, for small filter coeffifzients (i.e. low capture efficiencies) or for small bed lengths, drop retention profiles are diffuse, but as the filter coefficient increases or as the bed is lengthened they approach a shock front. Because dispersion is not included in the model, the effluentconcentration history exhibits a step-change at one pore volume of emulsion injected or r = 1. This is seen most clearly when A,, = 2. An important finding from Fig. 3 is that once the reduced filter coefficient exceeds a value of about 50, the transient permeability history is independent of the filter coefficient. Figure4 demonstrates the influence of the flowredistribution parameter a for a fixed and large reduced filter coefficient (A,, > 50) and for fl equal to unity. As a diminishes, the steady-state retention increases. Hence, with more net droplet capture a longer time is required to reach steady state, and also a larger permeability reduction is evidenced_ Figure 5 depicts the very important influence of the permeability-reduction parameter /I. Because A,, and a
t At the instant t, a relation between dZ and da may be derivedfrom an integralmass balance (Herzig et aI., 1970). The resulting equation is recorded as:
or, using eq. (17), 0
02 TC,.
With this relation, eq. (18) now may be analytically integrated to yield eq. (19) if oi is taken as a constant.
0.4 FORE
0.6 VOLUME OIL
0.6
Fig. 4. Phenomenological model calculations: the effect of the flow-redistribution parameter (x on breakthroughconcentration and permeability-reduction histories.
H.. Soo and C. J.
270 r-------
-
0
----
a=3
i
I
I
0
0.2
1
1
I
0.4 -Cc, . PORE
VOLUME
0.6 OIL
Fig. 5. Phenomenological model calculations: the effect of the flow-restriction parameter fl on breakthroughconcentration and permeability-reduction histories.
are fixed in Fig. 5, there is only one emulsion effluent concentration curve. For the identical retention profiles in the porous medium, slight increases in p cause large increases in the amount of permeability reduction. The filtration flow model, as depicted in Figs 3-5, well represents all the behaviour of dilute emulsion flow in porous media outlined in the Introduction. A more critical comparison to the viscosity and retardation models is outlined below. Quantitative evaluation of the parameters AsI, a and fl from experimental data is relegated to Part II. Comparison between models Figure 6 compares the calculated results of the viscosity model, the retardation model and the proposed filtration model for a dilute emulsion. In Fig. 6 the reduced effluent emulsion concentration is plotted against the pore volumes of emulsion injected, z, as dashed lines. Lines denoted by (1) represent the calculations from the viscosity model, which predicts emulsion breakthrough at r = 1 and no permeability reduction. This clearly contradicts experimental results which show large permeability reductions (McAuliffe, 1973; Soo and Radke, 1984a). Lines denoted by (2) represent the calculations from the retardation model, according to eqs (1) and (2). The
RADKE
parameters used are K. = 0.606 pm’, u = 0.05 cm/s, S, = 0.685, ci = 0.02, k,/p, = 1.26 x 10-2cms/g, k,/pw = 4_8cms/g and y = 1.692 x lo6 Pa/m, as reported by Devereux (1974b). Lines denoted by (3) represent the results of the filtration model, with the parameters (l/a = S, = 0.685 and p = 1.436) chosen to represent the same breakthrough time and steadystate permeability as in the calculated retardation model. The difference between the two models is small, both for emulsion elution and transient permeability. A flow experiment corresponding to Fig. 6 could not distinguish between the filtration and retardation models. The major difference between the retardation model and the filtration model is shown in Fig. 7. After 21 pore volumes of emulsion injection, the system is switched back to water injection. Arrows in the dashed lines show the time evolution of the efRuent drop concentration. The retardation model predicts that all drops in the porous medium eventually elute and that permeability rises back to its initial value. Conversely, the filtration model implies that only the non-captured drops are produced and the permeability remains the same no matter how many pore volumes of water are injected. Figure 7 also demonstrates that if water injection is not carried out long enough, permeability data alone are not sufficient to discriminate between the models. The effluent concentration data provide the discerning evidence. Figure 7 of Soo and Radke (1984a), which reports results for this suggested critical experiment, demonstrates that the filtration model is correct. CONCLUSIONS
A filtration model based on the flow-redistribution phenomenon due to drops being captured in pores is presented to describe the flow of dilute, stable emulsions in unconsoiidated porous media. Emulsion percolation is characterized by three parameters: a filter coefficient, an inter-pore flow-redistribution factor and a local flow-restriction factor. The filter coefficient controls the sharpness of the emulsion front, the flow-redistribution parameter dictates the flowredistribution phenomenon as well as the steady-state
I
I
I
1
,
2 3-
RETARDATlON FlLTRATlON
1 - VISCOSITY Z-RETARDATION 3 -FILTRATION
I : 0 OJ
0
, 30
I
I
20
IO T.
PORE
VOLUME
0
1 40
INJECTED
Fig. 6. Comparison between the viscosity, retardation and filtration models for continuous emulsion injection.
L 10
T.
PORE
I JWATER
III 20 0 VOLUME
,NJECTlON I 10
INJECTED
Fig. 7. Comparison between the retardation and filtration models for a 21 pore-volume pulse of emulsion followed by continuous water injection.
Filtration model for the flow of dilute, stable emulsions in porous media-1 retention, and the flow-restriction parameter describes the effectiveness of retained drops in reducing permeability. The dependence of these parameters on retention is shown to be weak, and therefore they may be treated as a function of drop size and pore-size distribution only. Part II addresses this treatment quantitatively. Comparisons among the filtration mode1 and previously developed emulsion flow models show that pressure drop-flow rate data alone are not sufficient to discriminate between various models. Effluent concentration data provide the discerning evidence. Available experimental results are consistent only with the new filtration theory.
Subscripts
d ; i L m 0 P S SI t W X al
Acknowledgements-This research was supported by the U.S. Department of Energy under Grant W-7405-ENG-48 to the
Lawrence Berkeley Laboratory. H.S. acknowledges financial assistance from the Chevron Oil Field Research Company. NOTATION
c
Dd
DP Dt K
Kx k rd kv L 4 P ‘d t
T IJ x R
ff Greek
volume concentration of oil drops in emulsion, volume of drops/flowing volume drop diameter, m pore throat diameter, m transition pore diameter, m overall permeability, m2 local permeability, mz relative permeability of the emulsion droplets relative permeability of the continuous water core length, m length of a pore with throat diameter D,, m pressure, N/m” oil saturation time, s reduced shifted time variable, 7 -X velocity, m/s distance, m reduced axial distance, x/L reduced emulsion front position letters
flow-redistribution parameter flow-restriction parameter flow-restriction parameter of retained drops in pores of throat diameter D, retardation force, N/m’ bed porosity, void volume/bed volume porosity contribution of pores of throat diameter D,, m- ’ drop capture efficiency in pores of throat diameter D, fractional porosity open for flow in tubes of throat diameter D,, m-l filter coefficient, m-l reduced dimensionless filter coefficient, ilL viscosity, mPa s local oil retention, oil-drop volume/bed volume local retention of drops in pores of throat diameter D,, m-I emulsion pore volumes injected, ut/eOL
271
droplet emulsion interception inlet exit maximum initial pore straining straining and interception transition water phase denotes local value steady state REFERENCES
Alvarado, D. A. and Marsden, S. S., 1979, Flow of oil-in-water emulsions through tubes and porous media. Sot. Petrol. Engng J. 19, 369-377. Aris. R. and Amundson, N. R., 1973, Mathematical Methods in Chemical Engineering, Vol. 2. Prentice-Hall, Englewood Cliffs, NJ. Buckley, W. E. and Leverett, M. C., 1942, Mechanism of fluid displacement in sands. Trans. AIME 146, 1077116. Camo. T. R.. 1964.Theorv of water filtration. Proc. Am. Sot. C&l Engig 9O(SA4) paper 3990, l-30. Devereux. 0. F., 1974a. Emulsion flow in porous solids-I. A flow model. Chem. Engng J. 7, 121-128. Devereux, 0. F., 1974b, Emulsion flow in porous solids--II. Exueriments with crude oil-in-water emulsion. Chem. E&g J. 7, 129-136. Doscher, T. M., 1967, Technical problems in in-situ methods for recovery of bitumen from tar sands. Proc. 7th World Petroleum Congress 3, 628. Gardescu, I. I., 1930, Behavior of gas bubbles in capillary spaces. Trans. AIME 86. 351-370. Gruesbeck, C. and Collins, R. E., 1982, Entrainment and deposition of fine particles in porous media. Sot. Petrol. Engng J. 22, 847-856. Herzig, J. P., Leclerc, D. M. and LeGoff, P., 1970, Flow of suspensions through porous media-applications to deep bed filtration. Ind. them. Engng 62, 8-35. Ives, K. J., 1967, Deep filters. Filtr. Separ. March/April, 125135. Jamin, J., 1860, On the equilibrium and motion of liquids in porous media. Phil. Mag. 4th Ser. 19, 204-207. Jennings, H. Y.. Jr.,Johnson, E. E., Jr. and McAuliffe, C. D., 1974, A caustic waterflooding process for heavy oils. J. Petrol. Technol. 26. 1344-l 352. McAuliffe, C. D., 1971, Oil-in-water emulsions and their flow properties in porous media. J. Petrol. Technol. 25727-733. 06, S-G. and Sl&ttery, J. C., 1979, Interfacial tension required for significant displacement of residual oil. Sot. Petrol. Engng J. 19, 83-96. Payatakes, A. C., 1977, Model of transient aerosol particle deposition in fibrous media with dendritic pattern. A.I.Ch.E. J. 23, 192-202. Payatakes, A. C. and Gradon, L., 1980, Dendritic deposition of aerosol particles in fibrous media by inertial impaction and interception. Chem. Engng Sci. 35; 1083-1096: Payatakes, A. C. and Tien, C., 1976, Particle deposition in fibrous media with dendrite-like pattern: a preliminary model. J. Aerosol Sci. 7, 85100. Payatakes, A. C., Tien, C. and Turian, R. M., 1973. A new model for granular porous media. A.I.Ch.E. J. 19, 58-66. Sakthivadivel, R., 1966, Theory and mechanism of filtration of non-colloidal fines through a porous medium. Tech. Rep. HEL _ ..> _15-5. . . Hydraulic Engng Lab., University of Laurornla, aerrerey.
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272
Savins, J. G., 1969, Non-Newtonian flow through porous media. Znd. Engng Chem. 61, 1847. Soo, H., 1983, Flow of dilute, stable emulsions in porous media. Ph.D. Thesis, University of California, Berkeley. Soo, H. and Radke. C. J., 1984a, The flow mechanism of dilute, stable emulsions in porous media. Znd. Engng Chem. Fundam. 23, 342-347. Soo, H. and Radke, C. J., 1984b, Velocity effects in emulsion flow through porous media. J. Coil. Znt. Sci. 102,462474. Soo, H. and Radke, C. J.. 1986, Flow of dilute, stable liquid and solid dispersions in underground porous media. A.Z.Ch.E. J. (in press). Stein, P. C., 1940, A study of the rapid filtration of water through sand. D.Sc. Dissertation, MIT, Cambridge, MA. Tien, C. and Payatakes, A. C., 1979, Advances in deep bed filtration. A.Z.Ch.E. J. 25, 737-759. Willhite, G. P., Green, D. W., Okoye. D. M. and Looney, M. D., 1980, A study of oil displacement by microemulsion systems-mechanisms and phase behavior. Sot. Petrol. Engng J. 20, 459472.
APPENDIX A. DERIVATION OF THE PHENOMENOLOGICAL CAPTURE RATE [EQ. (12)] For a small overall retention, (r < rr,, and for up c sp when D, > D,, eq. (9) reduces to (remember that ep,m = a,, when D, c LB,): ~=%f~l~,,sp(l-~~)dD~,
(Al)
where, as before, Rp,e has been generalized. Substitution of eqs (10) and (11) mto eq. (Al) gives the desired result in eq. (12). APPENDIX B. TWO-PORE MODEL In this appendix the filtration model is presented for a porous medium comprised of only two pore sixes. In the large size pores, only interception capture occurs, and in the small size pores, only straining capture exists. The drop capture kinetics of this system follow directly from eq. (9): a~
aus
-=dr+ar=&“C
right-hand side of eq. (Bl) (i.e. corresponding to &a/at) describes the drop capture rate in the small pores and the second term describes the dron canture rate in the large - _wres _ (i.e. corresponding to do,/&): In the mathematical limit of o < es (small overall capture) and et -c et (small interception capture), eq. (Bl) reduces to:
This result may be rewritten in the form of eq. (12) using eqs (10) and (11). The parameters Aal and a in the two-pore model are then (B3) and
&us + h&l
m=E” u
[
*I
=I QI
6&s + &,A1
.
WI
In the two-pore model, at steady state, es- approaches aaand cr __ approaches ~1,~. Thus, either from eq. (B4) or from thd steady-state form of eq. (12) we find that a, = se/(&S + err,,), and that -______ (l ‘=
+?#s+Y.q ($+z)(l+$y
(B5)
Likewise, the flow-restriction parameter for the two-pore model follows from its definition in eq. (13): 8=
Bscs + Z&c1 0 .
(86)
Equations (B5) and (B6) permit estimates of the retention dependence of a/a, (and fi) with the two-pore model, as presented in Fig. 2. Equation (15) of the text follows from the ratio of the capture rates in each pore as specified in eq. (Bl) or
au1
at
++c(l
RADKE
au, iat -= -2)
(z),
.
(B71
(Bl)
where the subscript S denotes straining, the subscript I denotes interception and se = es + et. The first term on the far
In the limit of zero time or equivalently in the limit of zero retention (u = (TS= ~1 = 0), eq. (B7) reduces to eq. (15).
Vol.
41, NO. 2. pp.
263~272,
1986.
0009-2509/86 S3.00+0.00 Pergamon Press Ltd.
A FILTRATION MODEL FOR THE FLOW OF DILUTE, STABLE EMULSIONS IN POROUS MEDIA-I. THEORY H. SOOT and C. J. RADKE* Chemical Engineering Department, University of California, Berkeley, CA 94720, U.S.A. (Received
25 June
1984)
Abstract-Flow of dilute, stable emulsions in porous media is important in several oil recovery processes. Because underground media have relatively low permeabilities, the emulsion drop sizes may overlap the pore sizes. Hence, strong interaction occurs between the emulsion droplets and pore constrictions, and local flow redistribution occurs within the porous medium. To predict quantitatively how emulsions are transported in underground media, a theoretical model is required which correctly accounts for the interactions between the flowing droplets and the pore walls. In Part I of this work, we present a simplified filtration model describing the flow of stable, dilute emulsions in unconsolidated porous media. In the model, emulsion drops are captured in pores by straining and interception and, thus, reduce the overall permeability. Transient flow behaviour is characterized by three parameters: a filter coefficient, a flow-redistribution parameter and a flow-restriction parameter. The filter coefficient controls the sharpness of the emulsion front, the flow-redistribution parameter dictates the steadystate retention, as well as the flow redistribution phenomenon, and the flow-restriction parameter describes the effectiveness of retained drops in reducing permeability. Critical comparison is made between the new filtration theory and the current continuum-viscous and retardation models for emulsion flow in porous media. Only the filtration picture is able to explain all the experimental observations. Quantitative comparison between the filtration flow theory and experiment is presented in Part II.
INTRODUCTION Emulsion flow in porous media can occur either by accident or by design in enhanced oil recovery, including several chemical (Jennings et al., 1974, Willhite et al., 1980) and steam-flooding (Doscher, 1967) processes. The emulsions, which may be either of the oilin-water or of the water-in-oil type, have drop sizes of a few micrometres, and they must transport in reservoir rock having permeabilities ranging from a tenth to several Darcys. In order to evaluate quantitatively the effects of such emulsions on oil recovery, it is essential to develop a mathematical model which is based on the actual physical mechanisms of droplet percolation. For the case of surfactant-stabilized, oil-in-water emulsions flowing in both consolidated and unconsolidated media, experimental findings of several researchers (McAuliffe, 1973; Devereux, 1974b; Alvarado and Marsden, 1979; Soo and Radke, 1984a) reveal the following features: (a) the permeability of the porous medium decreases monotonically with time, (b) the particles do not appear in the effluent directly after a throughput of one pore volume of emulsion injected, (c) the rate and the amount of permeability decrease and the droplet breakthrough time increase with increasing drop-size to pore-size ratios, (d) in some cases the rate and the amount of permeability reduction increase slightly with decreasing flow rates (McAuliffe, 1973), (e) eventually a steady state is reached, and (f) when emulsion injection is followed by many pore volumes of water flow, droplets elute for +Present address: Union Carbide Corporation, South Charleston,WV 25303, U.S.A. *To whom correspondence should be addressed.
about the first pore volume but the original permeability decrease caused by the emulsion remains (Soo and Radke, 1984a). Currently, two theories are available to describe this emulsion transport behaviour: the bulk viscosity model of Alvarado and Marsden (1979) and the droplet retardation model of Devereux (1974a). Both treatments are limited. In the viscosity model the emulsion is viewed as a continuum, single-phase fluid. No interaction between drops and pore walls is allowed, even though drop sizes and pore sizes overlap. The mathematical description of the viscosity model is based on porous-medium scaling laws identical to those used for non-Newtonian polymer solutions (Savins, 1969). Hence, emulsion flow is predicted to differ from Darcy’s law only when the bulk emulsion viscosity varies with shear rate. No permeability reduction, transient or steady state, is predicted, and the emulsion particles must elute exactly at one pore volume of fluid injection. When compared to the previously summarized experimental results, the viscosity model is seen to have validity only in the case of high-concentration emulsions, which approach a steady state quickly and which have small drop-size to pore-size ratios. The retardation model, originally delineated by McAuliffe (1973), does consider transient permeability reduction. In this model, the emulsion drops are retarded during their passage along the tortuous paths of the medium. When a droplet encounters a pore of throat size smaller than its own diameter, it deforms and squeezes through. In the constricted pore it experiences a capillary resistance force (Jamin, 1860, Gardescu, 1930) and, therefore, moves at a slower
H. Soo and C. J.
264
speed than the continuous phase, thereby causing an overall permeability reduction. A steady state is reached when the emulsion breaks through the porous medium. To describe this physical picture mathematically, Devereux (1974a) modifies the classical BuckleyLeverett theory for two-phase flow in porous media (Buckley and Leverett, 1942) by including a retardation factor in the pressure driving force of the dispersed oil phase. He presents a solution for the case of constant-pressure flow. The case of constantvelocity flow follows directly [a derivation is outlined by Soo (1983)]. The resulting expression describing the transient permeability reduction, K/Ko, for constant-velocity flow is as follows: -.--I K/K,
=
(1 -zZ2,)+
pdRf
9 (1)
Cl&, L
-1
where Z2,is the dimensionless position of the emulsion front, or f, =
Cir/Sd
for ciT/Sd < 1
1
for cir/Sd 2 1.
In eqs (1) and (2), pd and cc, denote the bulk viscosity of the droplet phase and of the continuous water phase, respectively; u is the superficial velocity, Sd is the constant fractional pore space occupied by oil droplets (i.e. the oil saturation) behind the emulsion front, r is the pore volume of the injected emulsion, and ci is the inlet volume concentration of the emulsion. Three unknown parameters appear in eq. (1) which must be evaluated from the experimental flow results. They are y, the total capillary retarding force per unit volume of the porous medium, and k, and k,, which are the relative permeabilities of the oil droplets in the emulsion and of the continuous water phase. Both k,, and k, are taken as known functions of the oil saturation S, (Buckley and Leverett, 1942; Devereux, 1974a). Equations (1) and (2) correctly reflect several features of the experimental emulsion flow results cited above. The permeability of the porous medium decreases as emulsion is injected (i.e. as ff increases), and a steady state is reached. Further, if the emulsion droplet saturation S, is not a function of flow rate, and if S, and y increase with drop-size to pore-size ratio, then eq. (1) also correctly implies that the permeability is more reduced with decreasing flow rate (u) and with increasing drop-size to pore-size ratio. Devereux’s theory, however, possesses two characteristics which do not match the experimental results: (1) the emulsion breaks through as a step change, and (2) the permeability of the porous medium ultimately rises back to its initial value when emulsion injection is followed by water. Thus, although the retardation model properly describes the transient, permeabilityreduction flow behaviour, it must misrepresent the underlying physical mechanisms. The recent experimental study of Soo and Radke
RADKE
(1984a) reveals that emulsion droplets are not simply retarded when they flow through porous media; they are actually captured. As demonstrated in Fig. 1, when drops flow in the porous medium, they clog pore constrictions of sizes smaller than their own by lodging between sand grains. This mode of capture is denoted as straining. Also, they are trapped in recirculation eddies, wedge in crevices, or sometimes attach to the pore walls due to van der Waals, electrical, gravitational and hydrodynamic forces. Interception is then the designated capture mode. Drop interception near pore constrictions may reduce the effective pore diameter and thus permit straining of other drops. As drop retention builds, flow diverts to the larger pores where capture probability is lower because of the increased pore sizes. This physical picture is very similar to that of a deepbed filtration process. However, it differs from traditional deep-bed filtration in the following respects. In a traditional filtration process, the particle-size to pore-size ratio is small. Therefore, the interception capture mechanism dominates, and the particles are captured mostly on the surfaces of the media grains. Flow distribution in the porous medium is not altered much by the presence of the captured particles. In this case, the whole process can be modelled by considering only the interaction between a single particle and a single collector (e.g. a sand grain). Permeability reduction caused by the retained particles is not significant and is sometimes overlooked. For the type of emulsion flow considered in Fig. 1, however, the drop size is of the order of the pore size, and drops are captured both by the straining and the interception mechanisms, with the possibility of straining being the dominant mechanism. Since the pores are either totally blocked or greatly restricted by the retained drops, flow redistribution takes place among neighbouring pores. To model emulsion flow, one must consider this flow-redistribution phenomenon and, consequently, treat the porous medium as a whole. As a corollary, the permeability reduction caused by the retained drops is large, and knowledge of this permeability reduction is important.
IN-l
Fig. 1. The mechanism of emulsion flow in porous media.
Filtration model for the flow of dilute, stable emulsions in porous media-1 Additionally, liquid drops are deformable. Hence, strained drops may reentrain into the flowing stream, once the local pressure drop increases enough to overcome capillary restraining forces (Jamin, 1860; Gardescu, 1930; Oh and Slattery, 1979). Under this condition, the amount of drops retained locally couples strongly to the local pressure drop. Finally, the shape of the captured drops and, hence, their influence on local permeability depend on the contact angle. Thus, for many reasons, conventional deep-bed filtration theory developed for undeformable solid suspensions is not strictly applicable for dispersed liquid droplet flow. Part I of this work presents a genera1 theory, based on deep-bed filtration concepts, to describe dilute, stable emulsion flow in porous media, where drop sizes and pore sizes overlap. The flow-redistribution phenomenon, as well as the possibility of large permeability reductions; is included in the model. The treatment is restricted to monodisperse emulsions of such high interfacial tensions that capillary reentrainment is not possible. In Part II, which follows, the theory is compared quantitatively with data appropriate for these restrictions, and experimental values of the parameters are obtained. Extensions of the theory for the cases of low tensions and drop-size distributions are handled separately (Soo, 1983; Soo and Radke, 1984b). Three parameters with clear physical interpretation arise in the theory. One indicates the probability of drop capture, the second reflects local flow redistribution, and the third measures the effectiveness of captured drops in restricting flow. Upon comparison with the viscosity and retardation models, only the filtration-based flow theory is able to explain both transient pressure-drop behaviour and effluent drop concentrations. GENERAL THEORY
As in deep-bed filtration theory (Herzig et al., 1970; Tien and Payatakes, 1979), the procedure is to predict the local disperse-phase retention profiles in time and distance. Then, by relating disperse-phase retention to local permeability, the overall transient flow rate-pressure drop behaviour may be established. Compared to classical colmatage theory, however, dilute emulsion flow of relatively large drops in finely grained porous media is complicated by the strong interaction of drop retention and local flow redistribution. One simplification does arise for highly stable emulsions, in that droplet coalescence does not occur. Local retention We begin by writing the unsteady, disperse-phase continuity equation in one-dimension with constant flow rate and negligible longitudinal dispersion (Herzig et al., 1970): r Pm -I
at
aPdDP 1 1 Jo-
a +c+
qeoc+ u] =
it
= -u$
(3)
265
where c(t. x) is the volume concentration of oil drops (still suspended in the flowing phase) at time t and axial position x, u is the constant superficial velocity, .Q,is the clean-bed porosity, 0 is the volume of retained drops per bed volume (thus removed from the flowing phase), and a*dD, is the volume of the drops retained per bed volume in pores of throat diameters between D, and D, + dD, (i.e. 10”a,dD, = u). Inherent in eq. (3) is the approximation that the clogged-bed porosity E is not much different than the initial bed porosity, .Q,. According to Herzig et al. (1970), this approximation is appropriate for dilute suspensions for which EC < u. To proceed, an expression is needed for the capture rate of drops in pores of throat diameter D,. We postulate that the capture kinetics are first order in the convected flux of drops, UC:
where 0,~ is the local fraction of emulsion fiow to pores of throat diameter Dp, and 1, is the filter coefficient of drops in that size of pore. The filter coefficient has the dimension of inverse length, and the inverse of the filter coefficient has the physical meaning of the distance a drop travels before being captured. Hence, the pore filter coefficient is related to the efficiency of drop capture in given sized pores by 3 I,$,, where 1, is the pore length. A linear relation ‘IP between capture efficiency and I, is demanded in this work. The reasons are as follows. When D, 2 D, (i.e. straining capture), the capture efficiency is near unity. We anticipate that the length the drop travels before being captured is the pore length. Accordingly, 1, equals l/l,. Conversely when D, Q D, (i.e. interception capture), 1, is close to zero, and again, as others have shown, A,, = qp/l, (Tien and Payatakes, 1979). Finally, we caution that when drop capture efficiencies are near unity, the continuum suspension balance in eq. (3) no longer holds. Its use here implies that the fraction of pores which have large capture efficiencies is small so that the flowing drop concentration can be treated in a continuum sense. The pore capture kinetics in eq. (4) imply that decolmatage does not occur. For the case of significant electrical double-layer repulsion between drops and between drops and sand grains, high drop tensions, and relatively low flow rates, the experimental results of Soo and Radke (1984a) prove that captured drops do not re-entrain into the flow stream. Other surface chemistry and flow conditions may require an additional term in the drop-capture kinetic expression. Sincere-entrainment is not permitted in eq. (4), steadystate flow can occur only in those pores for which I, eventually approaches zero. Knowledge of I, and 8, is required in eq. (4). With dilute emulsions, where no simultaneous multipledrop capture occurs, the filter coefficient is independent of drop-volume concentration. However, it is a complicated function of surface chemistry, drop size, pore size, interstitial velocity, and the amount and distribution of oil retained in pores of a given size.
H. Soo and C. J. RADKE
266
Likewise, 13~is a complicated function of the amount and distribution of oil retained locally in all pores. Some progress can be made by adopting a model for the porous medium under study. We utilize the unitbed-element treatment of Payatakes et al. (1973). In their model the porous medium is comprised of a series of statistically identical unit bed elements. Each unit bed element in turn consists of a number of parallel, constricted tubes with a throat-size distribution corresponding to the measured pore-size distribution of the porous medium. Payatakes et al. (1973) assume that for each constricted tube, the length of the tube is proportional to the throat size of the tube based on the following argument. In the porDus medium, on the average, large throat sizes form among grains which are large, and/or are loosely packed. Therefore, large diameter pores are expected to be longer. The average pore length is defined as the unit-bed-element thickness. Since the pressure drop across each constricted tube is assumed to be the same in a unit bed element and since tube lengths and throat sizes are proportional, it follows that the local volume flow through each tube is directly proportional to the volume of that tube. Hence, in a clogged unit bed element, 0, is written approximately as: 8, = (ar - e&o0
-cb
(5)
where a,, is the clean porous-medium porosity contribution of pores of throat diameter D, (i.e. the variation of &p/&o with D, is the traditional pore-volume frequency distribution). After multiplication of each side by the superficial velocity u, eq. (5) states that as smaller pores are blocked, flow diverts proportionally to the larger-volume pores. The expression for Op is simplistic in that it ignores how the retained drops are distributed in a particular pore. Nevertheless, it is exact for the cases in which the drop size is small compared to the pore size, the flow through the pore is not greatly disturbed by the retained drops, or the pore is totally blocked. Combination of eqs (4) and (5), followed by integration over all pore sizes, yields the overall dropcapture rate:
au _-
cu
at- (Eo:@ o 4%
- 3) d%
05)
Equation (6) directly reflects the flow-redistribution phenomenon. When drop capture commences, the smaller pores in the bed element clog quickly, since they have a higher capture efficiency (i.e. a larger Q_ Flow shifts to the larger pores where the capture efficiency is lower. When all the small pores (i.e. those pores in which mainly straining capture occurs) are blocked, the drops flow through the porous medium in a network of large pores. The process reduces to conventional deep-bed filtration, in which the particle sizes are smaller than the pore sizes. Equations (3)-(6) provide the drop retention profiles in each size of pores, and the drop volume concentration profiles in time and distance, once 1, is specified as a function of
drop size, pore size, and the amount and distribution of drops retained in the pore. Local permeability To calculate the overall pressure drop-flow rate behaviour, a relation is required between local permeability, K,, at position x and local retention cr. Unfortunately, a rigorous identity is difficult to construct. In addition to the amount of oil retained in the various pores, the location and configuration of that oil along the pore also influence the local permeability. Within the context of the unit-bed-element model we write: K, -_=I-’ KO
T” BpQD,,
&o I 0
(7)
where fl, is a parameter measuring the effectiveness of the retained drops in pores of throat diameter D, in restricting flow. For straining capture, /I, is simply the ratio of the volume of the pore to the volume of the retained drops. Naturally, /I, is a function of drop size, pore size, up and the distribution of captured drops within the pore. Combination of eqs (3), (6) and (7) and Darcy’s law (i.e. u = (K,/pc) ( - dp/dx), where ps is the bulk emulsion viscosity) gives a complete description of the flow behaviour including the desired transient permeability. However, the necessary detailed information on the quantities A, and 8, is not available. An exact theoretical treatment of this problem is not undertaken here. Rather, we employ a phenomenological approach in which average filtration parameters are defined and evaluated from experiments. PHENOMENOLOGICAL
MODEL
Local retention We classify the pores of the medium into two types: non-pluggable, in which the flow behaves like traditional deep-bed filtration and no straining occurs; and pluggable pores, which are eventually completely blocked by the drops. We stress that the pluggable pores do not necessarily have pore sizes smaller than the drop size. They may be two or three times larger. When drops flow in these pores, they may first capture on the surface of the pore and reduce the throat diameter. The following drop is then strained and completely blocks the pore. This multi-drop straining mechanism differs from the formation and bridging of dendrites reported in the past for solid suspensions (Sakthivadivel, 1966, Payatakes and Tien, 1976, Payatakes, 1977, Payatakes and Gradon, 1980, Soo and Radke, 1986). The transition pore-throat diameter between pluggable and non-pluggable constricted tubes is denoted as D,. The drop capture efficiency of the pluggable pores is either unity or close to unity initially and increases to unity very quickly. Consequently, it is taken as unity here so that the filter coefficient in these pores is just l/l,. In the non-pluggable pores, which eventually support the steady flow, the drop capture efficiency must
Filtration model for the flow of dilute, stable emulsions in porous media-1
decrease with increasing retention and eventually drop to zero when all capture sites are filled. The exact dependence of the capture efficiency on retention is complicated. However, in the case of small retention, we follow Ives (1967) in writing J-, = A,,(1
--Q+~,~)
for D, > D,,
(8)
where the subscript 0 denotes the initial clean-bed condition, and the subscript m indicates a maximum. The physical meaning of eq. (8) is that drops retained on the surface of the non-pluggable pores do not provide sites for further droplet capture. This is because capture must cease when yp reaches upVm. Herzig et al. (1970) and the companion Part II give further insight into eq. (8). Combination of eqs (6) and (8) and the condition of unit capture efficiency for the pluggable pores gives the overall capture rate:
au -=
267
straining capture in the small size pores. As retention builds, more pores are plugged. Local flow redistribution is no longer significant, because flow is mostly in the larger pores where capture is slight. Then a shifts its role to reflect steady-state retention. From eq. (12), l/a is established as the steady saturation of captured oil (i.e. when &r/at = 0, then l/a = u, /so). If straining capture dominates, then the occupied pores should not have a wide range of pore-throat diameters. In this case, eq. (11) shows that a has the value of so/j$epdDpt and is therefore a constant. Local permeability To simplify the expression of the local permeability reduction, we introduce an average flow-restriction parameter, j?, as follows: (13) Use of eq. (13) transforms eq. (7) to the desired result:
at
K -r=l-!?_ Ko
(9) We now introduce two phenomenological parameters. The clean-bed average filter coefficient, including both straining and interception, is defined by
where the meaning of L,, is generalized to include its value of l/l,, for pores of throat diameters less than D,. Equation (10) indicates that the average filter coefficient is obtained from the individual pore clean-bed filter coefficients by averaging over the initial porevolume distribution_ Similarly, based on examination of eq. (9), a flow-redistribution parameter, a, is defined as: LYE-
1 4u
O”(A P,o%%&,m)dDp. s 0
(11)
Here again the meaning of crp,,, is generalized to include its value of sp for pores of diameter less than D, (i.e. aside from a geometric factor of order unity b,_ = sp for D, -C D,). Physically, a accounts for the flowdiversion effect discussed above. Additional insight into the meaning of a is provided in Part II, where eq. (11) is utilized to calculate a directly [see eqs (5) and (11) of Part II]. For small retentions (i.e. u c so and up < Ed when D, > DJ, Appendix A demonstrates that eqs (9), (10) and (11) permit a phenomenological expression for the droplet capture kinetics:
au y$=
ASI 1 -dLQ (
% >
UC.
The parameter a describes the role of flow redistribution in the capture kinetics, due to drops retained in the pores. In general, a is a function of u. At the start of colmatage, significant flow redistribution occurs due to
(14)
so
This phenomenological expression has been proposed previously by Stein (1940) and Camp (1964). The flowrestriction parameter, 8, measures how effective retained drops are in reducing fluid permeability, per unit volume of drops retained. The effectiveness of a single large drop in reducing the local permeability is different from that of several small drops having the same total volume; j? reflects this. Clearly, j? also is a function of retention, u. At the beginning of the filtration process, most drops retain in the small pores; the value of j3 is close to the value of fi for straining capture. As cohnatage progresses, the value of /3 increases to its steady-state value. If straining capture dominates, p is again a constant at a value of about unity. Parameter retention dependence The dual-pore filtration model expressed in eqs (12) and (14) is most useful when the dependence of the three phenomenological parameters d,,, a, and fl on total retention is weak. For a given drop size and poresize distribution, 1,, is a constant and does not vary with retention. However, a and #I are constants only if straining is the dominant capture mechanism. When t When straining capture dominates, up,m = ep and eq. (11) becomes
Since for D, -c D, the pore-throat diameters (thus also the pore lengths) do not vary much, I,, is a weak function of D, and with cq. (10) we have
n, %dDP I 0 Dt t7 %dDP I 0
Eo a=
= Eg /I
4 cl
ep dD,.
H. so0 and C. J.
268
both straining and interception control the capture process, a and /3vary with retention_ To gain some insight into this retention dependence, we adopt a twopore model of the porous medium, as suggested by the work of Gruesbeck and Collins (1982). In the two-pore model, the porous medium consists only of two sizes of pores: small pores of throat diameter D, and porosity contribution sS, in which only straining occurs, and large pores of throat diameter D, and porosity contribution aI (i.e. E~+E~ = E,,), in which only interception occurs_ Without completely solving the two-pore model, it is possible to estimate the retention variations of a and 8. The mathematical calculation of a and p with the two-pore model is summarized in Appendix B. It demonstrates that these two parameters are functions of the ratio of the retention in each size of pores (i.e. of ~,/a~). The initial retention ratio, according to Appendix B, is written in the limit of zero time as
( >=ll,E,. 4-3
-QI
us
0
Equation (15) indicates that at the beginning of colmatage, the retention ratio is determined by the ratio of the fraction of flow in each of the two sizes of pores [i.e. according to the constricted-tube porous medium model adopted here, the fractional flow in each size pore is proportional to its void fraction (Payatakes et al_, 1973)]. It is also proportional to the ratio ofcapture probabilities in each sized pore. As the filtration process continues, the retention ratio varies from the initial ratio [eq. (15)] to the steady state of (a&,), = a&es, where tri mis the maximum retention in the interception pores. ’ Figure 2 displays how the flow-redistribution parameter a is predicted by eq. (B5) to vary from its initial value, aO, to its steady-state value, a,, as a function of the retention ratio eI /us based on the two-pore model. Although the function a/a_ in eq. (B5) ranges mathematically over the interval 0 c el/cs -Z 00, Fig. 2 shows only that portion of the range that is physically meaningful (i.e. those a,/~, ratios corresponding to a variation of a from a0 to a,). In Fig. 2, a, is given the typical value of 15 (i.e. see the experimental results of
RADKE
Part II) so that the steady retained oil saturation is the clean-bed porosity is l/a, = 0.067. Likewise, e. = 0.34, and the ratio of individual-pore filter coefficients is 1,/,X, = 100. Porosities of the strainingdiameter pores, sS, range from 2.5 x lo-4 to 1.9 x lo-‘; arrows indicate the time direction of the process. In the early stages of filtration, a/a, has its largest value, because a initially reflects the flow-redistribution phenomenon. During the later stages of the filtration process, a reflects the steady-state retention, and its value falls monotonically to a,. For very small or very large straining-pore porosities, es, only one capture mechanism dominates, either interception or straining. At intermediate values of es, where both straining and interception control filtration behaviour, the variation of a is maximal (about a factor of 2 in Fig. 2). With the two-pore model, calculated changes in the range of a increase when the amount of drop capture increases, or, equivalently, when a, is lowered. In actual porous media, however, a, and sS are interrelated so that es cannot be independently varied over large ranges as in Fig. 2. That is, at low values of a,, straining is more important and +, accordingly, is larger. Accordingly, variation in a with retention is not of consequence. Our experimental results (Soo and Radke, 1984a, Part II) show that a, is usually larger than 5 (i.e. a steady retained oil saturation less than l/z- - 0.2), in which case the maximum change in a is much less than two-fold_ The maximum change in a also depends on the choice of 1,/1,. However, this dependence is weak when &/A, is large. Similar behaviour is found for the flow-restriction parameter 8, except that the variations in fi are weaker than those for a. Therefore, based on the simplified results of the two-pore model, we treat a and #Jhere as constants, independent of total retention. Filtration model behaviour Once the filtration parameters are taken as constants, eqs (3) and (12) provide an analytical solution for the concentration and retention profiles. For a step increase injection of emulsion into an initially clean bed, this solution is well known (Herzig et al., 1970; Aris and Amundson, 1973):
u(T, 2) -= so
1 - exp (aAs,ci T) a[1 -exp(As,X)-exp(ahs,ciT)]’
(16)
and
Fig. 2. Retention dependence of the flow-redistribution parameter from the two-pore model.
c(T, 2) =- a(Z a) (17) a,(T) ’ ci where f = x/L is a reduced axial distance, T = T - 3 is a shifted time variable with r = ut/EOL defining the volume of injected emulsion to time t in units of pore volumes. The filter coefficient appears in reduced form with A,, = &tL, and the subscript i denotes the condition at the inlet to the packed bed. Equations (16) and (17) display exponential-like decaying concentration and retention profiles with a shock front at T = 0.
Filtration model for the flow of dilute, stable emulsions in porous media-I
r
Note that the effluent drop concentration cL = c( T, 1) a function of the variable ci(r - l), where ciz corresponds to the injected pore volumes of oil. The overall transient permeability K readily follows from eq. (14) and Dar&s law for the whole bed (i.e. under constant flow-rate conditions. K - ’
I
I
I
269 I
I
I 04
,
I
is
KO -=
K(r)
sl
[I -fiBa/so]-lda.
(18)
0
Equation (18) can be numerically integrated in general, using a/~~ from eq. (16), and in certain special cases analytical integration can be performed. For example, when the retention at the front of the core ci is close to the steady-state retention, the analytical result? is:
0
0
0.2 -Cc,.
x (1 + aci)-’
In
( >
L! l--
=L so
QL l-6’ so > (
,
(19)
wherein all the r-dependence resides within aL(7). In this expression the subscript L gives the condition at the end of the core of length L, so that aL = a(7,l). The emulsion viscosity pc does not appear in the transient permeability as long as the drop concentrations are low enough that the suspension viscosity is essentially that of the continuous fluid. Any hydrodynamic resistances due to flow of drops in narrow passages are neglected in this treatment. At steady state (ai = aL = 0, ), the retention approaches so/a and the steady permeability becomes K, /K, = 1 -B/a_ The role that the filtration parameters A,, and a and the flowrestriction parameter p play in determining the transient overall permeability and the ef3uent concentration of emulsion is now briefly discussed. Sample calculations are based on eqs (16), (17) and (19) with the base parameter values chosen for illustrative behaviour. The calculations are restricted to injected drop concentrations less than about 1 %. Figure 3 shows the influence of the reduced filter coefficient on permeability-reduction and emulsioneflluentconcentration histories for fixed values of a and J?.Solid lines correspond to transient permeabil-
VOLUME
0.6
OIL
Fig_ 3. Phenomenological
model calculations: the effect of the reduced filter coefficient A, on breakthrough-concentration and permeability-reduction histories.
ities and dashed concentrations. a;
PORE
lines correspond
to effluent emulsion
The em&ion breakthrough times [inflection points of C,(T)] and the steady-state permeabilities are the same in each case, since these quantities are controlled by a and fi. We see that A, determines the spread of the effluent-concentration histories and the time necessary for the permeability to reach its steady-state value. Thus, for small filter coeffifzients (i.e. low capture efficiencies) or for small bed lengths, drop retention profiles are diffuse, but as the filter coefficient increases or as the bed is lengthened they approach a shock front. Because dispersion is not included in the model, the effluentconcentration history exhibits a step-change at one pore volume of emulsion injected or r = 1. This is seen most clearly when A,, = 2. An important finding from Fig. 3 is that once the reduced filter coefficient exceeds a value of about 50, the transient permeability history is independent of the filter coefficient. Figure4 demonstrates the influence of the flowredistribution parameter a for a fixed and large reduced filter coefficient (A,, > 50) and for fl equal to unity. As a diminishes, the steady-state retention increases. Hence, with more net droplet capture a longer time is required to reach steady state, and also a larger permeability reduction is evidenced_ Figure 5 depicts the very important influence of the permeability-reduction parameter /I. Because A,, and a
t At the instant t, a relation between dZ and da may be derivedfrom an integralmass balance (Herzig et aI., 1970). The resulting equation is recorded as:
or, using eq. (17), 0
02 TC,.
With this relation, eq. (18) now may be analytically integrated to yield eq. (19) if oi is taken as a constant.
0.4 FORE
0.6 VOLUME OIL
0.6
Fig. 4. Phenomenological model calculations: the effect of the flow-redistribution parameter (x on breakthroughconcentration and permeability-reduction histories.
H.. Soo and C. J.
270 r-------
-
0
----
a=3
i
I
I
0
0.2
1
1
I
0.4 -Cc, . PORE
VOLUME
0.6 OIL
Fig. 5. Phenomenological model calculations: the effect of the flow-restriction parameter fl on breakthroughconcentration and permeability-reduction histories.
are fixed in Fig. 5, there is only one emulsion effluent concentration curve. For the identical retention profiles in the porous medium, slight increases in p cause large increases in the amount of permeability reduction. The filtration flow model, as depicted in Figs 3-5, well represents all the behaviour of dilute emulsion flow in porous media outlined in the Introduction. A more critical comparison to the viscosity and retardation models is outlined below. Quantitative evaluation of the parameters AsI, a and fl from experimental data is relegated to Part II. Comparison between models Figure 6 compares the calculated results of the viscosity model, the retardation model and the proposed filtration model for a dilute emulsion. In Fig. 6 the reduced effluent emulsion concentration is plotted against the pore volumes of emulsion injected, z, as dashed lines. Lines denoted by (1) represent the calculations from the viscosity model, which predicts emulsion breakthrough at r = 1 and no permeability reduction. This clearly contradicts experimental results which show large permeability reductions (McAuliffe, 1973; Soo and Radke, 1984a). Lines denoted by (2) represent the calculations from the retardation model, according to eqs (1) and (2). The
RADKE
parameters used are K. = 0.606 pm’, u = 0.05 cm/s, S, = 0.685, ci = 0.02, k,/p, = 1.26 x 10-2cms/g, k,/pw = 4_8cms/g and y = 1.692 x lo6 Pa/m, as reported by Devereux (1974b). Lines denoted by (3) represent the results of the filtration model, with the parameters (l/a = S, = 0.685 and p = 1.436) chosen to represent the same breakthrough time and steadystate permeability as in the calculated retardation model. The difference between the two models is small, both for emulsion elution and transient permeability. A flow experiment corresponding to Fig. 6 could not distinguish between the filtration and retardation models. The major difference between the retardation model and the filtration model is shown in Fig. 7. After 21 pore volumes of emulsion injection, the system is switched back to water injection. Arrows in the dashed lines show the time evolution of the efRuent drop concentration. The retardation model predicts that all drops in the porous medium eventually elute and that permeability rises back to its initial value. Conversely, the filtration model implies that only the non-captured drops are produced and the permeability remains the same no matter how many pore volumes of water are injected. Figure 7 also demonstrates that if water injection is not carried out long enough, permeability data alone are not sufficient to discriminate between the models. The effluent concentration data provide the discerning evidence. Figure 7 of Soo and Radke (1984a), which reports results for this suggested critical experiment, demonstrates that the filtration model is correct. CONCLUSIONS
A filtration model based on the flow-redistribution phenomenon due to drops being captured in pores is presented to describe the flow of dilute, stable emulsions in unconsoiidated porous media. Emulsion percolation is characterized by three parameters: a filter coefficient, an inter-pore flow-redistribution factor and a local flow-restriction factor. The filter coefficient controls the sharpness of the emulsion front, the flow-redistribution parameter dictates the flowredistribution phenomenon as well as the steady-state
I
I
I
1
,
2 3-
RETARDATlON FlLTRATlON
1 - VISCOSITY Z-RETARDATION 3 -FILTRATION
I : 0 OJ
0
, 30
I
I
20
IO T.
PORE
VOLUME
0
1 40
INJECTED
Fig. 6. Comparison between the viscosity, retardation and filtration models for continuous emulsion injection.
L 10
T.
PORE
I JWATER
III 20 0 VOLUME
,NJECTlON I 10
INJECTED
Fig. 7. Comparison between the retardation and filtration models for a 21 pore-volume pulse of emulsion followed by continuous water injection.
Filtration model for the flow of dilute, stable emulsions in porous media-1 retention, and the flow-restriction parameter describes the effectiveness of retained drops in reducing permeability. The dependence of these parameters on retention is shown to be weak, and therefore they may be treated as a function of drop size and pore-size distribution only. Part II addresses this treatment quantitatively. Comparisons among the filtration mode1 and previously developed emulsion flow models show that pressure drop-flow rate data alone are not sufficient to discriminate between various models. Effluent concentration data provide the discerning evidence. Available experimental results are consistent only with the new filtration theory.
Subscripts
d ; i L m 0 P S SI t W X al
Acknowledgements-This research was supported by the U.S. Department of Energy under Grant W-7405-ENG-48 to the
Lawrence Berkeley Laboratory. H.S. acknowledges financial assistance from the Chevron Oil Field Research Company. NOTATION
c
Dd
DP Dt K
Kx k rd kv L 4 P ‘d t
T IJ x R
ff Greek
volume concentration of oil drops in emulsion, volume of drops/flowing volume drop diameter, m pore throat diameter, m transition pore diameter, m overall permeability, m2 local permeability, mz relative permeability of the emulsion droplets relative permeability of the continuous water core length, m length of a pore with throat diameter D,, m pressure, N/m” oil saturation time, s reduced shifted time variable, 7 -X velocity, m/s distance, m reduced axial distance, x/L reduced emulsion front position letters
flow-redistribution parameter flow-restriction parameter flow-restriction parameter of retained drops in pores of throat diameter D, retardation force, N/m’ bed porosity, void volume/bed volume porosity contribution of pores of throat diameter D,, m- ’ drop capture efficiency in pores of throat diameter D, fractional porosity open for flow in tubes of throat diameter D,, m-l filter coefficient, m-l reduced dimensionless filter coefficient, ilL viscosity, mPa s local oil retention, oil-drop volume/bed volume local retention of drops in pores of throat diameter D,, m-I emulsion pore volumes injected, ut/eOL
271
droplet emulsion interception inlet exit maximum initial pore straining straining and interception transition water phase denotes local value steady state REFERENCES
Alvarado, D. A. and Marsden, S. S., 1979, Flow of oil-in-water emulsions through tubes and porous media. Sot. Petrol. Engng J. 19, 369-377. Aris. R. and Amundson, N. R., 1973, Mathematical Methods in Chemical Engineering, Vol. 2. Prentice-Hall, Englewood Cliffs, NJ. Buckley, W. E. and Leverett, M. C., 1942, Mechanism of fluid displacement in sands. Trans. AIME 146, 1077116. Camo. T. R.. 1964.Theorv of water filtration. Proc. Am. Sot. C&l Engig 9O(SA4) paper 3990, l-30. Devereux. 0. F., 1974a. Emulsion flow in porous solids-I. A flow model. Chem. Engng J. 7, 121-128. Devereux, 0. F., 1974b, Emulsion flow in porous solids--II. Exueriments with crude oil-in-water emulsion. Chem. E&g J. 7, 129-136. Doscher, T. M., 1967, Technical problems in in-situ methods for recovery of bitumen from tar sands. Proc. 7th World Petroleum Congress 3, 628. Gardescu, I. I., 1930, Behavior of gas bubbles in capillary spaces. Trans. AIME 86. 351-370. Gruesbeck, C. and Collins, R. E., 1982, Entrainment and deposition of fine particles in porous media. Sot. Petrol. Engng J. 22, 847-856. Herzig, J. P., Leclerc, D. M. and LeGoff, P., 1970, Flow of suspensions through porous media-applications to deep bed filtration. Ind. them. Engng 62, 8-35. Ives, K. J., 1967, Deep filters. Filtr. Separ. March/April, 125135. Jamin, J., 1860, On the equilibrium and motion of liquids in porous media. Phil. Mag. 4th Ser. 19, 204-207. Jennings, H. Y.. Jr.,Johnson, E. E., Jr. and McAuliffe, C. D., 1974, A caustic waterflooding process for heavy oils. J. Petrol. Technol. 26. 1344-l 352. McAuliffe, C. D., 1971, Oil-in-water emulsions and their flow properties in porous media. J. Petrol. Technol. 25727-733. 06, S-G. and Sl&ttery, J. C., 1979, Interfacial tension required for significant displacement of residual oil. Sot. Petrol. Engng J. 19, 83-96. Payatakes, A. C., 1977, Model of transient aerosol particle deposition in fibrous media with dendritic pattern. A.I.Ch.E. J. 23, 192-202. Payatakes, A. C. and Gradon, L., 1980, Dendritic deposition of aerosol particles in fibrous media by inertial impaction and interception. Chem. Engng Sci. 35; 1083-1096: Payatakes, A. C. and Tien, C., 1976, Particle deposition in fibrous media with dendrite-like pattern: a preliminary model. J. Aerosol Sci. 7, 85100. Payatakes, A. C., Tien, C. and Turian, R. M., 1973. A new model for granular porous media. A.I.Ch.E. J. 19, 58-66. Sakthivadivel, R., 1966, Theory and mechanism of filtration of non-colloidal fines through a porous medium. Tech. Rep. HEL _ ..> _15-5. . . Hydraulic Engng Lab., University of Laurornla, aerrerey.
H. Soo and C. J.
272
Savins, J. G., 1969, Non-Newtonian flow through porous media. Znd. Engng Chem. 61, 1847. Soo, H., 1983, Flow of dilute, stable emulsions in porous media. Ph.D. Thesis, University of California, Berkeley. Soo, H. and Radke. C. J., 1984a, The flow mechanism of dilute, stable emulsions in porous media. Znd. Engng Chem. Fundam. 23, 342-347. Soo, H. and Radke, C. J., 1984b, Velocity effects in emulsion flow through porous media. J. Coil. Znt. Sci. 102,462474. Soo, H. and Radke, C. J.. 1986, Flow of dilute, stable liquid and solid dispersions in underground porous media. A.Z.Ch.E. J. (in press). Stein, P. C., 1940, A study of the rapid filtration of water through sand. D.Sc. Dissertation, MIT, Cambridge, MA. Tien, C. and Payatakes, A. C., 1979, Advances in deep bed filtration. A.Z.Ch.E. J. 25, 737-759. Willhite, G. P., Green, D. W., Okoye. D. M. and Looney, M. D., 1980, A study of oil displacement by microemulsion systems-mechanisms and phase behavior. Sot. Petrol. Engng J. 20, 459472.
APPENDIX A. DERIVATION OF THE PHENOMENOLOGICAL CAPTURE RATE [EQ. (12)] For a small overall retention, (r < rr,, and for up c sp when D, > D,, eq. (9) reduces to (remember that ep,m = a,, when D, c LB,): ~=%f~l~,,sp(l-~~)dD~,
(Al)
where, as before, Rp,e has been generalized. Substitution of eqs (10) and (11) mto eq. (Al) gives the desired result in eq. (12). APPENDIX B. TWO-PORE MODEL In this appendix the filtration model is presented for a porous medium comprised of only two pore sixes. In the large size pores, only interception capture occurs, and in the small size pores, only straining capture exists. The drop capture kinetics of this system follow directly from eq. (9): a~
aus
-=dr+ar=&“C
right-hand side of eq. (Bl) (i.e. corresponding to &a/at) describes the drop capture rate in the small pores and the second term describes the dron canture rate in the large - _wres _ (i.e. corresponding to do,/&): In the mathematical limit of o < es (small overall capture) and et -c et (small interception capture), eq. (Bl) reduces to:
This result may be rewritten in the form of eq. (12) using eqs (10) and (11). The parameters Aal and a in the two-pore model are then (B3) and
&us + h&l
m=E” u
[
*I
=I QI
6&s + &,A1
.
WI
In the two-pore model, at steady state, es- approaches aaand cr __ approaches ~1,~. Thus, either from eq. (B4) or from thd steady-state form of eq. (12) we find that a, = se/(&S + err,,), and that -______ (l ‘=
+?#s+Y.q ($+z)(l+$y
(B5)
Likewise, the flow-restriction parameter for the two-pore model follows from its definition in eq. (13): 8=
Bscs + Z&c1 0 .
(86)
Equations (B5) and (B6) permit estimates of the retention dependence of a/a, (and fi) with the two-pore model, as presented in Fig. 2. Equation (15) of the text follows from the ratio of the capture rates in each pore as specified in eq. (Bl) or
au1
at
++c(l
RADKE
au, iat -= -2)
(z),
.
(B71
(Bl)
where the subscript S denotes straining, the subscript I denotes interception and se = es + et. The first term on the far
In the limit of zero time or equivalently in the limit of zero retention (u = (TS= ~1 = 0), eq. (B7) reduces to eq. (15).