ENHANCED RECOVERY
Role of Capillary Forces in Determining Microscopic Displacement Efficiency for Oil Recovery by Waterflooding
J. C. Melrose and C. F. Brandner, When displacement is carried out by flooding with Mobil Research and Development Corporation, an immiscible fluid such as water or gas, the surface Dallas, Texas forces (or, more accurately, the capillary forces) are responsible for trapping a large portion of the oil Abstract phase within the interstices or pores of the rock. For typical, homogeneous sandstone rocks, under usual waterflooding conditions, trapping will occur within The factors controlling the distribution of two immiscible fluids, SUJch as oil and wa,ter, within the interstices of a an appreciable fraction of the total number of pores porous solid are reviewed. It is shown that an improved constituting the pore space of the rock. The fraction understanding of these factors forms the basis of much of the pore volume which is involved may range from Tecent work aimed at the development of new methods fOT 10 to 50 per cent. Thus, capillary forces playa major reducing capillary forces and, consequently, eliminating residual oil saturations. role in limiting the recovery efficiency of such dis placement processes. Capillary forces will clearly be eliminated if the Introduction drive fluid is preceded by a slug or bank of fluid which is miscible both with the reservoir oil and with the THE PETROLEUM INDUSTRY has for many years devoted displacing phase(4l. However, the costs of the solvent much research effort to the development of new pro materials required in processes based on this principle cesses for achieving improved oil recovery efficien may be prohibitive. Alternatively, water-soluble sur cies. The current status of the most promising of these factants, such as petroleum sulphonates, can be em new processes has been reviewed by Elkins(l) and by ployed to eliminate the tendency of the oil phase to ArnoldC'). For the most part, the research leading to remain trapped in the pores of the rock. Details of these processes has developed along lines pointed out several processes based on the use of such surfactants by Muskat(3) in a review published about 20 years ago. have been recently reported<5-9J. In addition to thermal processes, which are used In this paper, the nature of the surface or capillary primarily to overcome adverse crude oil viscosity forces encountered in petroleum reservoirs, together characteristiCs, a variety of processes are designed with the microscopic features of the associated trap to eliminate or reduce the so-called "surface forces" ping mechanism, will first be reviewed. Some fea within the crude-oil-displacing fluid-reservoir rock tures of a new recovery process based on the use of system. surfactants, i.e., the low-tension waterflooding process(7-9J, will then be described. This discussion will emphasize one of the principal microscopic mechan James C. Melrose is a senior research isms on which the process is based. An analysis of this associate with the gxploration and mechanism indicates that ultra-low values of the oil Production Research Division of Mobil water interfacial tension are required in order to Research and Development Corpora tion in Dallas, Texas. He holds an achieve improved recovery. S.B. degree from Harvard University and a Ph.D. degree from Stanford University in physical chemistry. He joined Mobil in 1947 and has worked Hydrostatics, Capillarity and in the areas of drilling fluid rheology, clay chemistry, reservoir fluid prop Residual Saturations erties and the application of surface chemistry to improved oil recovery processes. HYDROSTATIC PRINCIPLES IN MULTIPHASE FLUID SYSTEMS Carl F. Brandner is a research as sociate with the Production Research Two or more fluid phases confined within a porous Division of Mobil Research and Devel opment Corporation in Dallas, Texas. solid phase of very small average pore size will gen He holds a B.S. from Iowa State Uni erally be microscopically commingled. The fluid-fluid versity and a Masters degree from interfacial areas associated with fluid distributions University of Minnesota in physics. of this type are large and characterized by high cur He joined the Field Research Labora tory in 1949 and has worked in the vatures. The precise configurations of the individual rheology of drilling fluids, formation segments of the total fluid-fluid interfacial area in evaluation, core analysis, reservoir such a system are described by two equations derivable description, and the physics and chem- from the fundamental principle of fluid hydro istry of fluid flow in porous media, especially as applied to improved oil recovery processes. statics 54 The Journal of Canadian Petroleum fluids, e.g., oil and water, will be explicitly discussed. The extension to cases involving three fluids presents / / / / / / / / // / '// / some complexities(ll), but, in principle, can be carried ///// /SOLI D PHASE / ~ /// out without recourse to any additional fundamental / / ~ / / // / / // / / / / considerations. ~ The first of the two hydrostatic equations is the Yos Laplace equation; Yow WATER OIL = Yow Po -P w Jow...... (1) PHASE PHASE Here, P denotes pressure in a fluid phase, ')I the in Pw Po terfacial tension and J the curvature of the fluid radius of fluid interface (c.f. Fig. 1). The subscripts 0 and w cu rvatu re refer to oil and water, and the double subscript ow refers to the corresponding interface. The thermo dynamic derivation of Equation (1) provides a con text within which the effects of temperature, pressure and interfacial curvature on the interfacial tension can be interrelatedOo,121, These considerations also apply, of course, to interfaces for which one of the FIGURE 1 - Hydrostatic equilibrium two fluid phases in contact with a solid phase. contiguous fluids is a gas (denoted by the subscript g). In this case, the quantities ')IgO and ')Igw are referred to as surface tensions. within porous solids presents difficulties not readily The second hydrostatic equation is the Young apparent from the usual textbook treatments of the equation, problem. Because the average pore size is assumed to be small, the pressures in the two fluid phases may Yos = Yws + Yow cos 60ws · ...... (2) be taken as constant over distances comparable to the Here, (Jows represents the angle of contact, as measured dimensions of an average pore. Also, the interfacial through the water phase, which the oil-water interface tension may be taken as independent of the magnitude OO (fluid meniscus) makes with the solid surface, denoted of the curvature ). According to Equation (1), then, by the subscript s (c.f. Fig. 1). The contact angle, the interfacial configurations will correspond to sur hereafter denoted simply as 0, is clearly a property faces of constant curvature. Even under this simplifi characterizing the three-phase line of contact. Meas ~ation, the algebraic expression for the curvature, J, urements of this property typically display hysteresis IS not a simple one. The expression includes terms effects(12). It should also be pointed out that a more which involve the various first and second partial rigorous version of Equation (2) will include a term derivatives of the function representing the shape of accounting for the line-tension in the three-phase line the interface. In fact, the expression corresponds to of contact°l). This "line property" is the analogue of a second-order non-linear differential equation for the interfacial tension for surfaces. which general solutions in terms of known functions are not available. The problem is generally circum The contact angle provides the only direct and un vented by the use of highly simplified geometrical ambiguous specification of the so-called "wettability" models to represent the shape of the actual pores property characteristic of a given oil-water-reservoir involved. The necessary boundary conditions are then rock system. If the angle (J is small, say less than 40 provided by the assumed configuration of the pore degrees, the system is usually said to be water-wet. walls and by the contact angle, 0, defined by Equation Similarly, an oil-wet system is characterized by an (2) . angle 0 greater than 140 degrees. For angles of inter mediate magnitude the wettability is best described as For example, in the case of a porous solid formed intermediate. by a random packing of uniform spherical particles, the pore openings may be approximated by the inner The measurement of the interfacial properties de surface of a torus(20, 21). In such a model, the radius of fined by Equations (1) and (2) under the tempera the torus ring corresponds to the radius of the ture and pressure conditions typical of petroleum re particles forming the packing. The equatorial distance servoirs presents a number of difficulties. Even for from the axis of symmetry to the ring surface is systems of relatively simple composition, such data then adjusted to provide pore openings of the ap are only gradually being reported in the literature. propriate sizes. Recent studies of this type which should be noted are those of Jennings(13, 14) and McCafferyOS-l7). An earlier A further problem arises in connection with the (8 stability of the fluid-fluid interfacial configuration study by Hocott ) reported surface and interfacial tension data for several field systems. Measurements which results from the solution of the differential of contact angles for crude oil - reservoir water sys equation represented by Equation (1). For typical tems on flat, polished surfaces of quartz and calcite pore models, such as packings of spherical particles have heen carried out by Treiber, Archer and of uniform size, it is easily shown(22,23) that a limited (9 range of such configurations are stable (c.f. Fig. 2). Owens ). This work indicates that many petroleum reservoirs may be intermediate in wettability type, a This fact accounts for the hysteresis in the relation conclusion which is in disagreement with the com ship between the capillary pressure, Pc = Po - Pw, monly held belief that water-wet conditions prevail in and the saturation of the aqueous phase, expressed as nearly all cases. Sw = Vw/(Vo + Vw), where V is the volume of a particular fluid in a test sample. (Under the implicit assumption that 0 = 0, the quantity Sw is often re APPLICATION TO POROUS SOLIDS ferred to as the wetting-phase saturation.) The oil The application of Equations (1) and (2) to deter phase saturation, So, is similarly defined, so that mine the configurations of the fluid-fluid interfaces Sw + S, = 1. Technology, October-December, 1974, Montreal 55 pIe, if the model is a random packing of uniform spherical particles, R may be taken to be the particle radius. WETIING NON - WEnI NG PHASE PHASE Experimental systems approximating this model indicate that H dr == 2.70 and H;mb == 1.75. The model calculations(21) also show that, for drainage conditions, UNSTABLE STABLE ~INTERFACE INTERFACE (6) 1 if 6 (53) CONFI GURATIONS CONFIGURATIONS Zdr > > o...... and for imbibition conditions, Zimb (6) < 1 if 6> 0 (5b) Under the definitions adopted for H dr and H;mb, it is clear that both Zdr (In and Zimb (e) approach unity as the contact angle, e, vanishes. FIGURE 2 - Stable and unstable interface configurations. CAPILLARY PRESSURE VERSUS FLUID SATURATION RELATIONSHIPS If the limits of stability are exceeded for a given Consider a sample of reservoir rock or similar pore, the system will respond by a sudden and local porous solid which is fully saturated with an aqueous ized flow in which one fluid is displaced from the pore phase, and assume that the condition e = 0 holds. by the other fluid. Such microscopic flow processes The quasi-static displacement of this phase by a non were first discussed by Haines(24) and are usually re wetting phase is then defined by the set of in ferred to in the field of soil physics as "Haines creasing values of the capillary pressure, Pc, and a jumps"(""). The alternative designation of "rheon" corresponding set of decreasing values of the wetting has been suggested(12J, and this term is used by Mor phase saturation, Sw (c.f. Fig. 3a). The relationship row'26\ although in a more restricted sense than defined by such data is called the drainage capillary originally proposed. Rheons can be easily observed pressure curve. The pressure-versus-saturation rela visually by allowing air to displace a liquid phase (or tionship for the reverse process, in which the wetting vice versa) from the pore space formed by packing phase displaces the non-wetting phase, is known as small glass beads in a glass capillary tube. A study of the imbibition capillary pressure curve. Because, for a the pressure oscillations associated with a displace typical pore configuration, the imbibition capillary ment experiment of this type has been reported by pressure is of the order of half of the drainage pres C27l Crawford and Hoover • sure, the curves will display considerable hys The stability limits applicable to interfacial con teresis'24, 33). figurations with axial symmetry have recently been Excellent examples of such curves, with well-defined investigated by Dyson and co-workers,28-3.). This work interior scanning loops, have been reported by Morrow has shown that the conjugate-point criterion of the and Harris'34l, Topp and Miller'3.) and Bomba(36). A calculus of variations leads to explicit solutions for the review emphasizing the fundamental physical prin limiting interfacial configurations which are observed ciples underlying such hysteresis behaviour has been (26 experimentally. A thermodynamic approach to this given by Morrow ). An earlier, but illuminating, problem, which is applicable to more general inter account of the manner in which interface stability facial shapes, has been initiated by Everett and conditions give rise to rheons or Haines jumps, on the Haynes'31,32). An intuitive treatment(l2), also thermo microscopic or pore-size level, and to hysteresis ef dynamic in nature, suggests that the appropriate con fects on the macroscopic level is due to Miller and (25 dition for configurational stability can be expressed Miller ) • as As rheons are associated with both branches of the 0 capillary pressure curve, it may be convenient to dis dJow/dVw< 0, for 6 < 40 ••••••••• .. .. (3a) 0 tinguish between the two resulting types of rheons. dJow/dVo < 0, for 6 > 140 ••••••••. •••.•••••••• •• (3b) On the drainage branch, the rheons correspond to a As mentioned previously, solutions to Equation (1) process in which the wetting-phase saturation is can only be obtained for highly simplified models of decreasing; hence such rheons may be referred to as the pore geometry thought to be characteristic of xerons. Similarly, rheons on the imbibition branch cor porous solids C2·,21). Regardless of the particular geo respond to an increasing wetting-phase saturation metrical forms which may be adopted as the basis and may be referred to as hygrons. of such a model, the meniscus configurations corres A characteristic feature of the drainage capillary ponding to the limiting conditions for stability will pressure curve for two nearly incompressible fluids, be given by Equations (3). The maximum and mini such as oil and water, is the minimum value of the mum curvatures so obtained are referred to as drain wetting-phase saturation. It appears that this re age and imbibition curvatures, J dr and J;mb, respect sidual saturation, SWi, is reached asymptotically as the ively. The effect of varying contact angle can then be capillary pressure is increased indefinitely. In the expressed(21) in terms of a cos e factor and normalized case of a typical unconsolidated pack of closely sized function, Z (e), as follows: sand particles, the wetting-phase fluid corresponding Jdr = 2 Hdr Zdr (6) cos 6/R. (4a) to the minimum saturation is largely in configurations know as pendular rings. These are individually isolated J;mb = 2 H;mb Zimb (6) cos 6/R...... (4b) segments of fluid surrounding the particle-to-particle Here, H dr and H;mb are the normalized values of the contact points. As the particle size distribution is made two limiting curvatures for the case of zero contact broader, or possibly bimodal in character, it is possible angle. The normalization factor is 2/R, where R is to develop local, small-scale packing heterogeneities. a characteristic distance for the pore model. For exam- If this occurs, the magnitude of the residual wetting- 56 The Journal of Canadian Petroleum NARROW PORE WIOE PORE SIZE 01 ST'N SIZE OIST'N t i 1 u u 0- c... u..J' u..J- a: :::J :::J Vl ""Vl Vl Vl u..J L.W c... "'-"" ORA INAGE "" >- >- "" "" DRAINAGE :s-' :s-' 0- c... « « '-' u IMBIBITION IMBIBITION 0.0 1.0 0.0 1.0 WETTING PHASE SATURATION, Sw WE;TiNG PHASE SATURATION, Sw FIGURE 3a - Capillary pressure versus wetting FIGURE 3b - Capillary pressure versus wetting phase saturation - narrow pore size distribution. phase saturation - wide pore size distribution. phase saturation for a drainage process increases, as saturation. For typical sandstone rocks, this residual Morrow(31,38) has recently shown (c.f. Fig. 3b). The saturation is of the order of 30 to 40 per cent. Because additional fluid is now trapped in such a way that such rocks are usually considered to be water-wet, small clusters of pores remain entirely filled. The this residual saturation, denoted as Sor, corresponds relationship between the residual wetting-phase satu to residual oil. It also defines what is termed the ration, as measured in the laboratory, and the water microscopic displacement efficiency, saturation in a petroleum reservoir at the time of discovery (the connate water saturation) is discussed E _ 1 - Sor - Swi ...... (6) m- I-S i by Morrow(39). w In a manner somewhat analogous to the drainage If, as may well be the case, many oil-water-reservoir case, the imbibition capillary pressure curve is charac rock systems are not completely water-wet, it may be terized by a minimum value of the non-wetting-phase possible to alter the wettability in such a way as to saturation. The residual non-wetting fluid is, however, reduce the value of Sor and hence recover additional trapped in configurations which are quite different oil. In this connection, Morrow and co-workers(44, 45) are from those of the residual wetting fluid under currently studying the dependence of both the drain drainage conditions. In an imbibition process, the non age and imbibition capillary pressure versus saturation wetting fluid is trapped in such a way as to nearly relationships on the wettability of the system, as de fill individual pores, or small clusters of neighbouring fined by the contact angle. The model pore studies(2l), pores. This trapping occurs because any given pore mentioned above, suggest that if the angle is larger will have only a limited number of openings leading than 45 degrees, a wetting phase, which has been dis to other pores. There is then a finite probability that placed through the application of a positive capillary any given pore will be by-passed"·). Evidence has been pressure, will not be able to reimbibe. This prediction reported(4l) which indicates that this probability in has been borne out, at least qualitatively, by the re creases as the larger pores of the pore-size dis cently reported experimental results"4,45), as well as tribution are subjected to displacement; i.e., as the by unpublished work by Sutula and Wilson"6). capillary pressure decreases. As was pointed out some time ago by Frisch and Low Interfacial Tensions Hammersley(42\ a rather sophisticated theory for For Improved Recovery treating the statistical aspects of the trapping process has been developed. This theory, known as the theory SIMULTANEOUS FLOW OF IMMISCIBLE FLUIDS of percolation processes, is closely related to the well When oil is displaced from a homogeneous porous known problem of treating a random, self-avoiding solid by a waterflood process, both fluid phases un "walk" on an idealized lattice system. It is apparent dergo hydrodynamic transport through the pore space. from a recent review(43) that a direct application of Only when, in a given region of the porous solid, the this theory to the prediction of residual non-wetting oil-phase saturation is reduced to its residual value phase saturations for porous solids has not yet been does the oil phase cease to flow. Under simultaneous developed. two-phase flow conditions, the relationships between Experimental results for various porous systems the normalized coefficients of permeability for the clearly show that the broader the pore-size distribu two phases and the water saturation are known as tion, the larger is the residual non-wetting-phase relative permeability curves(47). Technology, October-December, 1974, Montreal 57 Many fundamental studies of these relationships to the magnitude of the capillary forces'6o, 61). For have been reported in the literature. It is found that ordinary waterflooding conditions, Nca is of the order 6 for so-called steady-state conditions (saturations not of 10- (8). changing with time), such curves are closely related Laboratory studies of the relationship between the to the corresponding capillary pressure curves. Thus, capillary number and the microscopic displacement hysteresis behaviour, as defined by differing curves efficiency have usually been carried out by means for drainage and imbibition conditions, is ob of "tertiary" waterfloods. Such an experiment is pre served'4S,49). Also, the final saturations reached for ceded by a conventional waterflood. The resulting both drainage (oil displacing water) and imbibition value of the residual oil saturation is that which cor (water displacing oil) are those which are defined responds to secondary recovery under field conditions. by the capillary pressure curves. In particular, it is The tertiary flood then involves greatly increasing found that the relative permeability relationships are the flow rate, Uw, greatly decreasing the interfacial independent of the flow rates employed and of the tension, yow, or both. An increased wetting-phase vi.3 viscosities of the two fluids'50, 5l). This evidence suf cosity, p.,w, may also be used. As the capillary number fices to prove that, on a microscopic level, the com is substantially increased, eventually some additional mingled fluids are substantially in mutual equilibrium oil is recovered. This increased recovery is attributed as far as capillary pressure is concerned'52,53). to the tertiary flood. Studies of actual displacement processes, under non For packs of very coarse sand, Leverett(47) reported steady-state conditions, are also revealing"O). Per data which showed indications of slightly improved kins(54) has shown that when unconsolidated sand packs recovery efficiency for capillary numbers as low as are waterflooded, the residual oil saturation is inde 10-5. These measurements were not extended to higher pendent of flood rate. Loomis and Crowell(55) found values of Nca. Studies on several sandstone samples that for gas-oil systems under drainage conditions were carried out by Moore and Slobod"9), who found in homogeneous rocks, the relative permeability rela the critical value of Nca for the initial reduction in tionships are in good agreement with the relationships residual oil saturation to be about an order of magni measured by steady-state methods. In the case of tude higher; i.e., about 10-4. Results reported by water-oil systems, however, these workers found that Taber'62,63) suggest that this lower critical value of steady-state and non-steady-state methods were not Nca is of the order of 5 x 10-" whereas values reported in agreement, unless oil-wet conditions were main by Lefebre du Prey"6, 57) and by Foster'S) are again tained. Because for water displacing oil an oil-wet about 10-4. Variations in the lower critical value of condition gives rise to a drainage process, it is seen Nca are to be expected, as various reservoir rocks will that only for the non-steady-state imbibition process differ widely with respect to pore structure and pore are departures from local capillary equilibrium signif size distribution. Also, it is by no means certain that icant. This strongly suggests the possibility that this all of the reported investigations were carried out type of behaviour is due to dynamic contact-angle under the same contact-angle conditions. effects. However, it is not known at present how in Evidence for the magnitude of the upper critical formation on the flow-rate dependence of the contact value of N Ca, which corresponds to complete oil angle is actually to be applied in this context. recovery, has not been so extensively reported in the literature. It appears that in order to reduce the value of the residual oil saturation by a factor of about CRITICAL RANGE OF THE CAPILLARY NUMBER one-half, it is necessary to increase the capillary num ber by a factor of 10'62, 63) to 100 (56,57,59). Work re- It has been stated in the previous section that for ported by Foster'S) indicates that increasing N Ca by a completely water-wet system the residual oil satura tion is independent of the flood rate. This implies a factor of 200-300 will result in microscopic displace ment efficiencies which approach 100 per cent (c.f. that, under ordinary flooding conditions, capillary Fig. 4). Thus, the upper critical value of the capillary forces dominate the macroscopic displacement process number is of the order of 10-2to 10-'. In a recent study and that the microscopic distribution of the oil and by Dullien and co-workers(64), the upper critical value water phases is determined by the conditions for hy of Nca was not established. Although water-wet con drostatic equilibrium. Clearly, however, if the flood ditions apparently were maintained in these experi rate is made sufficiently high, this situation will no ments, the techniques used to establish that such longer be maintained. Viscous forces will begin to conditions existed are not specified. have an effect on the magnitude of the residual oil; Le., on the microscopic displacement efficiency as de fined by Equation (6). 1.0 ..------:77",.....-----, In order to assess the transition between a displace ment process dominated by capillary forces and one dominated by viscous forces, it is convenient to con ~E PORE SIZE DI ST'N sider the dependence of the microscopic displacement NARROW ~ efficiency, Em, on a suitable dimensionless parameter, AVERAGE 0.5 such as the "capillary number", defined as -'CL V> WIDE C cC N Ca = ...... (7) u ~ WATER, WET SYSTEMS Here, J1-w and Uw are the water phase viscosity and flow rate per unit cross-sectional area, respectively, 0.0 10'4 10,2 and qJ is the porosity of the solid. This number, or its equivalent, has been used by a number of CAP ILlARY NUMBER. NCA authors'S, 47, 56-63). Physically, it represents the non-di FIGURE 4 - Correlation of microscopic displacement effi mensional ratio of the magnitude of the viscous forces ciency with capillary number. 58 The Journal of Canadian Petroleum MOBILIZATION OF RESIDUAL OIL DIRECTION OF ~ WATER FLOW > The correlation between the displacement efficiency and the capillary number, as obtained from tertiary waterflood experiments, strongly suggests that the process of mobilizing residual oil depends on a com petition between viscous and capillary forces. A description of the mechanism by which this compe tition is brought into play would obviously be in structive. Although various authors have attempted to develop such mechanistic descriptions, these efforts are widely scattered in the literature and have not been subjected to critical examination. Clearly, a test of any such description is whether it is consistent with the observed critical range of the capillary num ber, N ca. As pointed out by Taber(62l, the classical "Jamin effect", discovered over 100 years ago, provides the basic concept required for the development of a "'L mechanistic interpretation of the process of mobilizing FIGURE 5 - Configuration of trapped oil ganglion. residual oil. An early attempt in this direction is due to Gardescu(65). This work is concerned, however, with two distinct effects, both referred to as the Jamin denotes the radius of the uniform spherical particles effect. As pointed out by Smith and Crane(66), these forming the porous solid, and N is the ratio of the two effects are associated with (a) contact-angle hys length of the mobilized ganglion, 6 L, to the particle teresis, as observed in cylindrical capillary tubes, and diameter. The quantity N thus measures the minimum (b) capillary-pressure hysteresis, as observed in non number of pores comprising a particular ganglion; cylindrical tubes, i.e. tubes in which a series of con i.e., the number of pores encountered if the ganglion strictions are introduced. is traversed in the direction of the surrounding water phase flow. The resulting confusion in terminology seems to have been compounded by an insufficient apprecia In order to relate the critical pressure gradient given tion of the actual basis of capillary pressure hys by Equation (8) to the corresponding capillary num teresis effects(59J. As discussed above, this hysteresis ber, (NCa)cd" it is only necessary to introduce Darcy's is a consequence of the stability conditions applicable law. Equation (7) can then be written as to the configuration of any fluid-fluid interface which is bounded by contact with a solid phase(21.23J. These (Nca)"it = k,w K(6"PI:) ...... (9) 'PlOW L.:::.. edt stability conditions are given by Equations (3). Thus, the model calculations summarized in Equations (4) where K is the value of the single-phase permeability and (5) should provide a more satisfactory basis for and k rw is the relative permeability to water. As some developing the desired mechanistic interpretation of authors report the results of tertiary waterflood the critical range of N Ca than does the approach experiments in terms of the quantity (6Pw/6L)critlyow, followed by Gardescu(65J. Equation (9) can be used to compute the correspond ing values of the capillary number. In evaluating the Figure 5 is a schematic representation of an isolated range of capillary numbers used in the work reported segment of non-wetting-phase fluid trapped in the by Taber(62,63), for instance, it is necessary to employ interstices of a porous solid by an invading wetting this relationship. phase. The residual oil saturation (water-wet condi tion) is the sum of such trapped volumes of oil, divided The permeability, K, may also be related to the by the total pore volume. Microscopic studies indicate capillary pressure for the case of zero contact angle that these oil "ganglia" usually extend over no more through a non-dimensional quantity, the Leverett than about 10 neighbouring pores. If now the sur number(33), defined as rounding water phase is caused to flow at an extremely N Lc = Jdr vK/ cp (6 = 0)...... (10) high rate, such an isolated segment will also be caused to flow. However, this flow will be accompanied by at If now Equations (9) and (10) are combined with least two individual displacement events, or rheons. Equations (4) and (8), the following expression is One of these will be of the drainage type, in which obtained for the case of () = 0, oil moves out of the original volume occupied by the (N )._ k nv (NLY G (11) ga~glion. The other will be of the imbibition type, in Ca en' - 4 N H dr WhICh water moves into the original volume. The where drainage rheon (xeron) occurs at the downstream or G = 1 - (Himb/Hdr) · · (12) low-pressure side of the ganglion; the imbibition rheon (hygron) occurs at the upstream or high-pressure side. In order to test Equation (11), the following aver This picture of the oil mobilization process thus age values of the various parameters may be used: requires that a critical pressure gradient be exceeded k rw = 0.3, N Le = 1/\1'5, H dr = 2.70, G = 0.35 and in the surrounding water phase. For the simplified N = 2. These values are typical for unconsolidated pore-shape model discussed above(21J, this critical packs of uniform glass beads or nearly spherical sand gradient is given by grains. Using these values, the computed value of (Nca)cd' is approximately 10-3. Because this value falls 6 P /6 L). = Yow Ode - Jimb) within the critical range of capillary numbers observed ( w en' 2 N R .. (8) experimentally, the mechanism for oil mobilization Here, Jdr and J imb are the drainage and imbibition cur which has been described is reasonable and can be vatures given by Equations (4a) and (4b), Ragain accepted as a working hypothesis. Technology, October-December, 1974, Montreal 59 Further consideration of Equation (11) also sug altering the value of the interfacial tension, yow, may gests that the observed critical range of capillary be approached with added confidence as to the 2 numbers (10-4 to 3 X 10- ) is associated with the ex magnitude of the required change. Thus, if Nca is to pected ranges for the parameters appearing in this be increased by a factor which is of the order of 10~, equation. In particular, the lower critical value of the interfacial tension must be decreased by a similar N ca is seen to be related to those ganglia, formed factor. In other words, the interfacial tension must be during the original trapping processes under ordinary reduced to what may be called an ultra-low value. waterflood (imbibition) conditions, which have the Such a value is of the order of 10-3 to 10-4 dynes/em. largest values of the ratio N/(NLe )2. In a similar Earlier estimates'6,,69) of the required level suggested manner, the upper critical value of N Ca will correspond that 0.05 dynes/em was an appropriate target for to the mobilization of oil trapped in single pores the purposes of tertiary oil recovery by interfacial (N = 1) of the smallest sizes within the pore-size tension alteration. These estimates, however, were distribution. Thus, the proposed mechanism easily based on limited evidence concerning the relationship accounts for a rather broad critical range of capillary between the capillary number and displacement ef numbers. ficiency. It should be noted that, according to Equation (11), neither the size of the spherical particles forming SURFACE CHEMISTRY OF the porous solid nor the permeability of the solid LOW-TENSION SYSTEMS directly influences the critical value of the capillary number. Therefore, even though reservoir rocks will The achievement of ultra-low values of the inter differ widely with respect to these parameters, it is facial tension between oil and water phases represents not expected that such variables will contribute a challenge of considerable magnitude. The literature significantly to the observed range of critical capillary dealing with the fundamental aspects of the adsorption numbers. On the other hand, variations in the width of surface-active chemical species at the oil-water of the pore-size distribution and in the shapes of the interface offers limited guidance in this respect. The early work of Harkins and Zollman'70), who re pores could, as already indicated, contribute to varia 3 tions in the critical capillary number. ported a single instance of a tension as low as 2 x 10- dynes/em, appears never to have been followed up by It is clear that the picture of the oil mobilization any systematic study of the composition variables process which leads to Equation (11) is based on the suggested by this observation. Presumably, it has same concepts which are used to explain the occurrence been believed that any such system will inevitably un of hysteresis in the capillary pressure versus fluid dergo spontaneous or self-emulsification. If this type saturation relationship. Thus, the pressure difference of behaviour were of universal occurrence, the study in the wetting phase which appears in Equation (8) of such interfacial states by conventional techniques is directly related to the difference between the drain based on thermodynamics would, of course, be impos age and imbibition interfacial curvatures discussed in sible. a previous section. It should be remarked in this con nection that Dullien(67) has suggested a somewhat In a pioneering study of the interfacial behaviour of similar expression. However, the relationship which crude oil-water systems, Reisberg and Doscher(7l) is proposed appears to be derived solely by considering attributed the observation of an ultra-low value for the magnitudes of the equivalent radii of a void and of yow by Harkins and Zollman to the in-situ formation its pore entry. Thus, the mechanism of the oil mobil of a highly surface-active component. Using a sample ization process envisaged by Dullien does not explicitly, of crude oil from Ventura, California, and the pendant at least, involve the condition for the configurational drop technique for measuring interfacial tension, Reisberg and Doscher found that a tension as low as stability of fluid interfaces. This condition, as has 3 been pointed out above, is responsible for the occur 10- dyne/em could be observed when the concentration rence of hysteresis in the capillary pressure versus of NaOH in the aqueous phase was about 0.1 N. The saturation relationship. tensions for such systems, however, were rather un stable, so that equilibrium values of yow were not The model calculations(2l) on which Equation (11) obtained. Jennings(72) also used the pendant-drop is based can also be used to predict the effect of method and reported an ultra-low value of yow for a contact angle. If a normalized function expressing this crude oil-surfactant solution system. effect is defined by the relationship More recently, studies carried out with aqueous W(6) = {Nca (6)/(Nca)6=O}e,il . .. (13) solutions of the petroleum sulphonate class of sur C7 9 factants - ,73) have also demonstrated that ultra-low then the model leading to Equations (4) will yield the values of yow can be achieved. It has been found that following result, the average surfactant molecular weight, the nature W(6) = {Zd,(6) - (1 - G)Zimb(6) I (cos 6/G) ...... (14) of the surfactant molecular weight distribution and the electrolyte concentration of the aqueous phase If now Equations (5) are taken into account, it is are significant variables. In fact, a rather close seen that the effect of increasing 0 on Zd' and Zimb specification of each of these parameters is essential. may be such that W(O) will actually increase. This is The interfacial compositions of such systems will be, in contrast to the effect predicted by taking W (0) to of course, of considerable complexity. It appears, be identical with cos O. It is found, in fact, that W (0) however, that the tendency for spontaneous emul may be as large as 2.2 when 0 is about 45 degrees. sification is greatly reduced, if not eliminated, in Thus, the optimum wettability condition for tertiary these cases. The conditions under which instabilities waterflooding is predicted to be one of complete of this type could arise are also complex. These con water-wetness (0 = 0). ditions are the subject of an analysis recently reported Having provided a satisfactory mechanistic basis by Miller and Scriven'74'. for the observed range of critical capillary numbers, Although it seems quite likely that the basic me the problem of achieving improved oil recovery by chanism involved in low-tension waterflooding for ter- 60 The Journal of Canadian Petroleum tiary oil recovery is as described above, the design kr relative permeability of a practical field process must take into account a K single-phase permeability L distance in direction of macroscopic flow number of other significant factors. The various re N number of pores in trapped oil ganglion C75 quirements involved have been surveyed by Ahearn J, N Ca capillary numher, non-dimensional who includes references to the patent literature. Ex N Le Leverett number, non-dimensional cessive adsorption of the surfactant components on P fluid phase pressure Pc capillary pressure the reservoir rock surfaces must be avoided, as was R grain or particle radius emphasized a decade ago by Wyllie'7.) and even earlier S saturation, fraction of void volume occupied by a fluid by Muskat(3). The latter author also foresaw the need U fluid phase flow rate per unit cross-stlctional area fluid phase volume for using a polymeric material as a thickener in the V W(6) function expressing effect of contact angle on NCa aqueous drive fluid. By this means, the viscosity of Z(6) function expressing effect of contact angle on ratio, the·displacing fluid can be made larger than that of J/cos6 the displaced fluid, and the necessary conditions for Greek Letters hydrodynamic stability can be satisfied. Practical techniques for meeting these and other critical re "'( interfacial or surface tension quirements are discussed in recently reported work'7.". 6 contact angle IJ. fluid phase viscosity '!' porosity, fractional void volume Summary and Conclusions Subscripts 1. The distribution of commingled immiscible fluid = oil phase = water phase phases, e.g., oil and water, within the interstices of a go = gas-oil interface homogeneous porous solid, e.g., reservoir rock, is de gw = gas-water interface termined by capillary forces. When the conditions for = oil-water interface the configurational stability of a particular fluid-fluid = oil-solid interface w. water-solid interface interface within an individual pore are not satisfied, oil-water-solid line of contact a sudden and rapid burst of flow occurs. The ensemble dr = drainage branch of hysteresis curve of these microscopic flow events, or rheons, constitutes imb = imbibition branch of hysteresis curve the macroscopic displacement process. wi = residual wetting phase, water-wet condition Or' residual non-wetting phase, water-wet condition 2. This mechanism accounts for hysteresis in the edt = critical value of the capillary number capillary pressure and relative permeability versus saturation relationships; i.e., the differences between the drainage and imbibition processes. It also ac Literature Cited counts for the existence of residual saturations under both types of displacement process; Le., the minimum (1) Elkins, L. E., World Oil, June, 1971, p. 69. (2) Arnold, C. W., AIChE Symp. Ser. No. 127, 69, 25 wetting-phase saturation for drainage and the mini (1973). mum non-wetting-phase saturation for imbibition. (3) Muskat, M., Ind. Eng. Chem., 45, 1401 (1953). 3. If a lower critical value of a non-dimensional quan (4) Craig, D. R, and Bray, J. A., Eighth World Petro Congr. Proc., 3, 275 (1971). tity, the capillary number, is exceeded, the residual (5) Gogarty, W. B., and Tosch, W. C., Trans. AIME, 243, non-wetting-phase saturation can be reduced. When 1407 (1968). the upper critical value of the capillary number is (6) Davis, J. A., and Jones, S. C., Trans. AIME, 243, reached, the residual non-wetting-phase saturation 1415 (1968). (7) Hill, H. J., Reisberg, J., and Stegemeier, G. L., J. is eliminated. This range of critical capillary numbers Petro Tech., February, 1973, p. 18'6. can be predicted from simplified pore-shape models (8) Foster, W. R, J. Petro Tech., February, 1973, p. 205. and a consideration of the conditions for the con (9) Gale, W. W., and Sandvik, E. I., Soc. Petro Eng. J., 13,191 (1973). figurational stability of fluid interfaces. (10) Melrose, J. C., Ind. Eng. Chem., 60, No.3, 53 (1968). 4. In order to achieve a value of the capillary number (11) Pujado, P. R, and Scriven, L. E., J. Colloid Inter sufficiently high that the residual oil saturation can fa,ce Sci., 40, 82 (1972). (12) Melrose, J. C., Can. J. Chem. Eng., 48,638 (1970). be eliminated, an ultra-low value of the oil-water in (13) Jennings, H. Y., Jr., J. Colloid Interface Sci., 24, 323 terfacial tension is required. Such a value is of the (1967). order of 10-3 to 10-4 dynes/cm, and it has been recently (14) Jennings, H. Y., Jr., and Newman, G. H., Soc. Petro shown that aqueous solutions of petroleum sulphonates Eng. J., 11,171 (1971). (15) McCaffery, F. G., and Mungan, N., J. Canadian Petro can provide tensions in this range. This finding forms Te,ch., July-Sept., 1970, p. 185. the basis of several new processes for tertiary water (16) McCaffery, F. G., J. Canadian Petro Tech., July flooding. Model calculations indicate that a condition Sept., 1972, p. 26. of complete water-wetness should provide the optimum (17) McCaffery, F. G., and Cram, P. J., Paper No. 56, Div. ColI. and Surf. 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