zyxw zy zyxw zyxwvu Environmental and Energy Engineering A Pore-Level Scenario for the Development of Mixed Wettability in Oil Reservoirs zyxw A. R. Kovscek, H. Wong, and C. J. Radke Earth Sciences Div. of Lawrence Berkeley Laboratory and Dept. of Chemical Engineering, University of California, Berkeley, CA 94720 Understandingthe role of thinfilms in porous media is vital to elucidate wettability at the pore level. The type and thickness of films coating pore walls determine reservoir wettability and whether or not reservoir rock can be alteredfrom its initial state of wettability. Pore shape, especial& pore wall curvature, is important in determining wetting-film thicknesses. Yet, pore shape and physics of thin wetting films are generally neglected in flow models in porous rocks. Thin-film forces incorporated into a collection of star-shaped capillary tubes model describe the geological development of mixed wettability in reservoir rock. Here, mixed wettability refers to continuous and distinct oil and water-wetting surfaces coexisting in the porous medium. This model emphasizes the remarkable role of thin films. New pore-levelfluid configurations arise that are quite unexpected. For example, efficient water displacement of oil (low residual oil saturation) characteristic of mixed-wettability porous media is ascribed to interconnected oil lenses or rivulets which bridge the walls adjacent to pore corners. Predicted residual oil saturations are approximately 35% less in mixed-wet rock than in completely water-wet rock. Calculated capillary pressure curves mimic those of mixed-wetporous media in the primary drainage of water, imbibition of water, and secondary drainage modes. Amott-Harvey indices range from - 0.18 to 0.36 also in good agreement with experimental values (Morrow et al., 1986; Jadhunandan and Morrow, 1991). zyxwvut zyxwv introduction The wettability of reservoir rock is a critical factor in determining the displacement effectiveness and ultimate oil recovery by drive fluids, such as water. Since the most wetting fluid tends to occupy the smallest and, hence, most hydrodynamically resistive pore channels while the least wetting fluid distributes to the largest and least resistive pore channels, wettability is a prime factor controlling multiphase flow and phase trapping. Therefore, understanding how wettability is established at the pore level is crucial if predictive flow models are to be developed. Wettability in porous media is generally classified as either homogeneous or heterogeneous. For the homogeneous case the entire rock surface has a uniform molecular affinity for either water or oil. Conversely, heterogeneous wettability in- Correspondence concerning this article should be addressed to C. J . Radke 1072 dicates distinct surface regions that exhibit different affinities for oil or water. Three broad classifications of homogeneous wetting exist: strongly water-wet, strongly oil-wet, and intermediate-wet. If smooth representative rock surfaces can be prepared, then contact angles for water-wet surfaces, measured through the water phase, are near zero. Whereas for oil-wet surfaces they are near 180". In the case of intermediate-wetting the rock has neither a strong affinity for water nor oil and contact angles range roughly from 45" to 135" (Craig, 1971). Two types of heterogeneous wettability are generally recognized. Mixed wettability refers to distinct and separate waterwet and oil-wet surfaces which coexist and span a porous medium. Dalmatian, also speckled or spotted, wettability refers to continuous water-wet surfaces enclosing regions of discontinuous oil-wet surfaces or uice uersa (Cuiec, 1991). For many years, it was common petroleum-engineering prac- June 1993 Vol. 39, No. 6 zyxw z AIChE Journal zyxwvuts zyxwvutsrqpo tice to assume that oil reservoirs are strongly water-wet. Because most reservoir rock is highly siliceous and oil reservoirs evolve by oil migrating into initially brine-occupied pore space, it was thought that the rock surface maintains a strong affinity for water even in the presence of oil (Morrow, 1991). In 1973, however, Salathiel established that reservoirs with mixed wettability display low residual oil (that is, oil trapped as isolated globules) and consequently high displacement efficiency, as gauged by the ratio of oil recovered after waterflooding to the original oil in place. Heiba et al. (1983) and Mohanty and Salter (1983) used network models to study the distribution of oil under mixed-wettability conditions. These methods relied on ad hoc rules for the creation of the oil-wet surfaces of mixed-wettability porous media. More recently, Fassi-Fihri et al. (1991) use cryoscanning electron microscopy to confirm that wettability is heterogeneous on the pore scale in actual reservoir media and that pore geometry plays an important role in determining the wettability of pore surfaces. The purpose of this article is to develop a pore-level picture of how mixed wettability might form and evolve in reservoir rock initially filled with brine. Before embarking on this task we briefly review the characteristics of mixed wettability as observed by Salathiel (1973). Figure 1. Mechanisms of oil surface drainage in mixedwettability systems (after Salathiel, 1973). pore to pore, allow oil to be swept along the rock surface almost indefinitely by the viscous traction of flooding water. Oil trapping is thereby obviated, but production rates are small. As wettability is molecular in origin, a succinct description of mixed wettability follows from considering the effects of thin films which coat and adhere to solid surfaces. Thin-film forces, in fact, control wettability. Before discussing the geological evolution of mixed wettability, it is necessary to explore the relationship between thin films and wettability. zyxwvutsr Mixed Wet tability Following standard protocol, Salathiel oilflooded cores that were initially fired at 400°C and filled with brine until little or no water was produced. This simulates the original migration of oil into a water-wet oil-bearing reservoir rock. The water saturation following oil migration is commonly referred to as connate saturation. Next, the core was waterflooded to residual oil saturation where little or no more oil elutes. Prior to achieving residual oil saturation, oil that may still be displaced is called remaining oil. When using an asphaltic East-Texas crude oil with either actual reservoir core or the fired sandstones, Salathiel found residual oil saturations of less than 10%. Small amounts of oil continued to be produced even after flowing thousands of waterflood pore volumes. This result is quite opposite to that usually encountered. Indeed, when n-heptane or a viscous white oil replaced the crude oil, Salathiel established residual oil saturations near the commonly observed values of 25 to 40%, within several waterflood pore volumes. Fascinatingly, when a small volume of the asphaltic crude was diluted into the n-heptane, the waterflood performance was essentially the same as that of the original reservoir crude oil. Further, for the asphaltene-laden oils, lower initial connate water saturation correlated with lower residual oil content. Oilflooding a previously waterflooded core led to irreducible water saturations much higher than the initial connate value. Salathiel’s explanation of his experimental results is shown in Figure 1, which depicts proposed pore-level configurations of oil and water after extensive waterflooding of an asphaltic oil. Water flows into the plane of the drawing. Near the cusps of the grain contacts, the rock is protected from the effects of the asphaltic oil and remains water-wet. On the central portion of each grain surface away from the cusps, an oil-wet surface is created due to the deposition of asphaltic components of the crude oil. The continuous oil-wet paths, which span from zyxwv Thin Films and Wettability Figure 2 displays an apparatus used to quantify thin-film forces in liquid films adjacent to a solid surface (Derjaguin et al., 1978). An annular porous disk, which allows flow of wetting fluid, is sealed against a solid substrate. A nonwetting fluid meniscus enters from below the solid substrate disk assembly. Far away from the solid substrate, the meniscus is hemispherical. When the central portion of the meniscus is pressed against the wall, a thin film forms. As the pressure of the nonwetting fluid rises compared to that in the wetting fluid, the film thins and the meniscus or Plateau border recedes into the corner. An experiment proceeds by applying a capillary pressure difference ( P ,= PnW- P,) across the Plateau-border meniscus and measuring the resulting thickness of the film by interferometry or ellipsometry. At equilibrium, the pressure of the wetting fluid in the thin film and the Plateau border are identical. The film is flat, yet the capillary pressure is nonzero. Consequently, the standard Young-Laplace equation of capillarity does not describe the thin-film portion of Figure 2. Rather, thin-film forces must be incorporated into the Young-Laplace equation by adding a term, l l ( h ) , referred to as disjoining pressure (Derjaguin zyx zyxwvuts AIChE Journal zyxwvu Figure 2. Apparatus for measuring thin-film forces in solidlliquid systems. June 1993 Vol. 39, No. 6 1073 zyxwvutsrq zyxwvutsrqp zyxwvutsrq zyxwvutsr zyxwvut zyxwvu zyxwvuts and Obukhov, 1936; Derjaguin and Kussakov, 1939a,b; Derjaguin et al., 1987): Pc=u (il-+-3 +rI(h). In the augmented Young-Laplace equation, u refers to the bulk interfacial tension between the wetting and nonwetting phases for an interface with principal radii of curvature rl and r,. Disjoining pressure is a function of film thickness, h. If disjoining pressure is positive, the two interfaces repel each other, whereas if disjoining pressure is negative, the two interfaces attract. When films are thin (for example, 100 nm), the disjoining pressure is significant compared to the other terms in Eq. 1 . For the flat film in Figure 2 the curvature is zero so that at equilibrium P,=II(h). Equation 1 is cast in dimensionless form when both sides of the equation are multiplied by the ratio of a characteristic pore dimension, a,, to a characteristic surface tension, u. Representative values u and a, in this work are 50 mN/m and 150 pm, respectively. With this scaling the magnitude of the first term on the right side of Eq. 1 is unity, whereas a disjoining pressure of 500 Pa makes the second term of order 1 . A schematic disjoining pressure isotherm is drawn in Figure 3. For certain values of disjoining pressure, for example, n,, there are three possible equilibrium film thicknesses. The outermost films have thicknesses of order 10 to 100 nm. The central portion of the ll vs. h curve (where d n / d h is positive) is an unstable region (Vrij, 1966; Chambers and Radke, 1991). A film in this thickness region spontaneously thickens or thins to a new stable configuration. The innermost films are quite thin, possibly on the order of one or several monolayers of solvent molecules. Many factors contribute to the shape of a disjoining pressure isotherm. Three major force components are electrostatic interactions, van der Waals interactions, and hydration forces (c.f., Melrose, 1982; Chambers and Radke, 1991). The first, electrostatic forces, originate from the overlap of ionic clouds n I I li z Figure 3. Disjoining pressure isotherm for wetting films on solids. 1074 present at each interface known as electrical double layers. For symmetric charged films, these are repulsive stabilizing forces (Callaghan and Baldry, 1978). Melrose (1982) and Hall et al. (1983) give thorough discussions of double-layer repulsive forces and their effects on film stability. Dispersive van der Waals forces are usually attractive and destabilize thin-aqueous films (Melrose, 1982). Lastly, strong repulsive hydration forces are important at film thicknesses approaching molecular dimensions (Hirasaki, 1991a). Clearly, the mineral content of pore walls has an effect on thin-film interaction forces. For example, when a drop of crude oil is contacted with a brine-covered surface, crude oil adheres more readily (that is, thick aqueous films break more easily) on calcite than quartz (Morrow, 1990). Because calcite exhibits a smaller negative surface charge than quartz, under identical conditions there is less stabilizing repulsive action from the electrostatic overlap forces. Oil-drop adhesion tests provide a simple approach to characterize the wetting behavior of crude oil against solid surfaces (Morrow et al., 1986; Buckley et al., 1989; Buckley and Morrow, 1990). When adhesion occurs, it is found that the solid/ oil/water boundary is fixed or “pinned” at the three-phase contact line (Morrow, 1990). Accordingly, the contact line of a retracting oil drop does not recede along the solid substrate. In subsequent sections, the importance of contact-angle pinning in mixed-wettability systems will be evident. Thin-film forces determine the contact angle, the most widely used measure of wettability. Direct integration of the augmented Young-Laplace equation yields the following relationship between equilibrium contact angle, 0, and disjoining pressure (Derjaguin, 1940): In Eq. 2 h is the equilibrium film thickness of interest. Integration over the positive purely repulsive portions of Figure 3 (the thick outer films) results in zero contact angle. Conversely, nonzero contact angles up to 90”are possible when Eq. 2 is integrated over the attractive negative portions of Figure 3 (Wong et al., 1992). Equation 2 is derived strictly for a meniscus attached to a flat solid surface. Nevertheless, it still applies for solid curved surfaces as long as the film thickness is much smaller than the radius of curvature of the surface (Wong et al., 1992). Churaev (1988) and Hirasaki (1991b,c) provide thorough reviews on thin films and contact angles. Consider now the case of films on curved surfaces. Figure 4 displays one such surface: two spherical solid beads with diameters dl and d2covered with a layer of wetting fluid. Hall et al. (1983) considered a similar situation for Athabasca oil sands. The curvature of the films is nonzero, and the curvature term on the right side of Eq. 1 now contributes. If the wetting films are convex, as in Figure 4, the curvature of the interface is deemed negative. Assume now that the films in Figure 4 are under an imposed capillary pressure and that the films are thin enough so that the curvature of the film far away from the bead contact is roughly that of the bead (hi/di<<l). In this situation, the smaller the bead diameter, the more negative is the film curvature. Rearranging Eq. l and invoking the geometry of Figure 4 we write that: June 1993 Vol. 39, No. 6 AIChE Journal 5 0 . zyxw . , . . . , . . . , . . . , . . . zyxwvu zyxwvutsr zyxwvuts zyxwvuts zyxwvutsrq zyxwvuts Figure 4. Wetting films covering sand grains. 40 rI(hj)= P , + - . di (3) inage 5 cn P Thus, to maintain equilibrium with a fixed capillary pressure, the film disjoining pressure must rise as the diameter, d,, of a bead decreases. Accordingly, the thin film coating the smaller bead is thinner than that coating the larger bead. The second term on the right side of Eq. 3 contributes 2 kPa (0.3 psi) when d, is 100 pm and u is SO mN/m. According to Figure 3, film thickness decreases monotonically with increased capillary pressure until the local maximum in disjoining pressure, XImax, is reached. This corresponds to a critical capillary pressure, Pr, for thick-film rupture. When the capillary pressure exceeds Pr, the thick outer film sheets aways and only a molecularly adsorbed film resides next to the surface. Because the disjoining pressure is the largest in the film coating the smallest bead, that film becomes unstable first and consequently exhibits the smallest Pr. Inserting XImax into Eq. 3 gives Pr for any bead diameter. Equation 2 teaches that exceeding the critical capillary pressure also signals a transition in the contact angle. Our discussion so far is quite general and applies to oil films between a solid and brine as well as to aqueous films between a solid and oil. When a thick aqueous wetting film collapses into a molecularly thin one, surface-active components from asphaltic oil can, in some instances, adsorb irreversibly onto the surface. Asphaltenes (high-molecular-weight aggregates, insoluble in light normal alkanes but soluble in benzene or pyridine) occur in relatively large quantities in many crude oils. It is currently believed that asphaltenes are colloidal polydispersions comprising flat, disklike aggregates (Dubey and Waxman, 1991). They are usually coated with lower-molecular-weight resins (Chung et al., 1991). Resins adsorbed to the asphaltene surfaces apparently stabilize the asphaltene colloidal dispersions. Clementz (1976, 1982) demonstrated that the heavy ends fraction of crude oil (asphaltenes) adsorbs strongly to clay materials at low water content. Also, Dubey and Waxman (1991) showed that asphaltenes adsorb quite readily to clay materials in the absence of both brine and resins. The asphaltene aggregates apparently adsorb with their disk faces against the planar faces of the clay. Asphaltene adsorption AIChE Journal Y no 0 0.2 0.6 0.4 0.8 1 SW (b) Figure 5. Typical experimental capillary pressure vs. aqueous-phase saturation curves. (a) Water-wet 48-60 mesh glass beads (data of Morrow and Harris, 1965); (b) weakly mixed-wet sandstone (data of Mohanty and Miller, 1991). appears irreversible for all practical purposes (Morrow et al., 1986; Hirasaki et al., 1990; Dubey and Waxman, 1991). Strong solvents and extensive cleaning procedures are necessary to desorb asphaltenes and resins. The key factor in asphaltene adsorption appears to be direct access of the asphaltic components of the oil to the rock surface without an intervening thick layer of water in which they are highly insoluble. Thus, when the critical capillary pressure is exceeded, asphaltenes may adsorb because only a molecular aqueous film protects the solid surface. In practice, surfaces within porous reservoir rock are curved and rough (Dullien, 1979). Hence, it is difficult to measure contact angles and to determine directly the extent of connectivity of water- and oil-wetting surfaces. Fortunately, capillary pressure as a function of water saturation exhibits characteristic shapes for different types of wettability in porous media. Morrow (1990) and Cuiec (1991) review how wettability may be assessed from experimental capillary-pressure information. Figure 5 contrasts typical experimental capillary-pressure curves for water-wet (Figure Sa, Morrow and Harris, 1965) and mixed-wet rock (Figure Sb, Mohanty and Miller, 1991). June 1993 Vol. 39, No. 6 1075 Measurement of capillary pressure usually begins with the porous rock sample saturated with the wetting phase. Under the action of, for example, a centrifuge (Slobod, 1951) nonwetting fluid is driven into the sample causing an increase in capillary pressure and primary drainage of the wetting phase. Large capillary pressures leading to low wetting-phase saturations are obtained at high rotational speeds. When the applied capillary pressure is then lessened incrementally and provided that a capillary connection to the wetting phase exists, the porous sample spontaneously imbibes the wetting phase thereby displacing the nonwetting fluid. Equilibrium capillary-pressure measurements are made possible in this fashion. Spontaneous imbibition continues until the interfacial curvature reaches zero. For the mixed-wetttability rock, additional wetting phase is imbibed under applied (centrifugal) pressure. Hence, the wetting-phase pressure exceeds that of the nonwetting phase and negative capillary pressures occur, as illustrated in Figure 5b. Strongly water-wetting rock does not display negative capillary pressures under forced imbibition. Finally, secondary drainage of water follows imbibition. Hysteresis is evident in the curves shown in Figure 5. The wetting phase drains from the largest pores first causing relatively large changes in saturation with moderate changes in capillary pressure. Conversely, wetting phase is imbibed into the smallest pore spaces first because of capillary suction. This causes large changes in capillary pressure with only moderate changes in saturation. The extent of imbibition and drainage processes defines empirical wettability indices (Morrow, 1990; Cuiec, 1991). One such index is the Amott-Harvey index, Iwo.This index ranges from + 1 for strongly water-wetting rock to - 1 for strongly oil-wetting rock. A value of zero indicates neither strong water nor strong oil wettability. Thus, the I , for the water-wet rock in Figure 5a is 1, while the weakly mixed-wet rock of Figure 5b has an Iwoof - 0.05. Further discussion of wettability indices and their significance is deferred until later. Geological evolution of mixed wettability is described by following pore-level events through the cycle of primary drainage, spontaneous imbibition, forced imbibition, and secondary drainage. Asphaltene adsorption onto solid surfaces is allowed if thick protective water films break. Immediately following the outline of pore-level events, theoretical capillary-pressure curves are calculated to gauge the extent of oil-wettability via Amott-Harvey indices. zyxwvu zyxwvutsr zyxwvutsrqp zyxwvutsr Figure 6. Cross section of a translationally invariantstarshaped pore. thin films in controlling wettability. Later, we discuss the consequences of interconnecting the pores. Because reservoir-scale flow rates and pore-scale gravitational forces are small, all continuous, immiscible phases are separated by constant curvature surfaces in capillary equilibrium. End effects are not considered. Capillary pressure is consistently defined as the oil-phase pressure minus the aqueous-phase pressure. Single-pore events We begin with a single, brine-filled, star-shaped pore of size a that is initially strongly water-wet and consider pore-level events as a function of changing capillary pressure. Later, we incorporate a pore-size distribution and generate capillary pressure curves. Oil cannot invade into the single pore until the capillary entry pressure, PF, is exceeded. This initial drainage of brine is called pristine drainage. For a star-shaped, translationally invariant pore with zero contact angle, the capillary entry pressure is (Mayer and Stowe, 1965): Theory zyxwv aP,F Initially, reservoir rock is saturated with brine (the terms brine and water are used interchangeably) and strongly waterwetting. Reservoir rock unaltered from its original state is referred to as pristine. Actual pores in reservoir media are noncircular and cornered (Duilien, 1979). Accordingly, the porous medium is modeled as a collection of star-shaped translationally invariant pores, as illustrated in Figure 6. The star shape is chosen because it is nonaxisymmetric, cornered, and resembles the open area between four rods or four sand grains in cross section when they are in contact. The radius of the largest circle which can be inscribed in a pore is denoted as a. Actual porous media display a much greater degree of pore connectivity. However, our simple model of a porous medium specifically emphasizes the fascinating and important role of -= U 1.86. (4) For P,? P:, oil enters the tube with a constant meniscus curvature displacing brine from the central portion of the tube leaving brine only in the corners, as illustrated in Figure 7 . Light shading represents brine, while dark shading represents oil. Thick aqueous films lie between the solid pore walls and the oil. Their thickness is determined by the capillary pressure and the form of the disjoining pressure isotherm (c.f., Figure 3). The configuration of the brine-filled corners away from the front end of the invading meniscus is that of constant curvature circular arcs. The curvature of the so-called arc menisci (Morrow and Mason, 1991) does not change until the invading meniscus zyxwvu 1076 June 1993 Vol. 39, No. 6 AIChE Journal In accordance with Eq. 2, the ultrathin molecular films, produced for Pc>PT, do not exhibit the same contact angle as that for thick wetting films. Disjoining pressure isotherms have not been measured for solid/water/oil systems of interest. Therefore, we approximate contact angles for the molecular aqueous films as 20" rather than calculate specific contact angles for each film thickness. The contact angle is 0" for the thick films (P,"""<P:),since they lie on the purely repulsive portion of the disjoining pressure curve. When the P y imposed is quite large, the brine content of the pore in Figure 7 is very low. It is important to note that P y is not a fundamental physical property of the rock or the rock surface. Oil reservoirs remain at low water saturations for geological periods of time. We assert that asphaltenes adsorb along the walls of the oil-occupied pores where thick protective water films are broken (those pores for which P- > P: ) . The location along pore walls, where asphaltenes adsorb, is displayed in Figure 8 as a white region recessed from the pore wall. No asphaltene adsorption occurs within larger pores for which P: > P,""",because thick protective water films coat pore surfaces. We designate this type of pore as water-wet. Because asphaltenes are insoluble in water, it is apparently not possible for them to dissolve in and transport across the thick-water films to contact the pore walls. The resin-coated asphaltene aggregates, however, are much larger than the molecular dimensions of water. Hence, they can, over time, span the molecular layers of water coating smaller oil-occupied pores in which P: < P y causing some water molecules to desorb, and, thus, directly contact the rock surface to adsorb irreversibly. In this model, the presence of asphaltenes in the oil phase is indispensable for the generation of oil-wet surfaces. The underlying mineralogy of the rock surface, although important in the asphaltene adsorption process, is not the prime source of wettability alteration in our scenario. Salathiel (1973) observed such asphaltene adsorption on a variety of sandstones and glass beadpacks within hours. It is not clear whether the resins coating the asphaltene colloids or the asphaltene aggregates themselves actually bind to the surface in the presence of molecular water films. Figure 8 also shows that there are two distinct regions of wettability in pores where asphaltenes adsorb. A step change in wettability exists at the contact line between the asphaltenecoated portion of the pore wall and the water-filled corner. This contact line is pinned at the position it established at P y , The corners of the pore retain bulk water preventing asphaltene adsorption and, thus, preserving their water wettability. Away from pore corners, the highly hydrophobic adsorbed asphaltenes make that portion of the pore wall oil-wet. We designate this type of pore as mixed-oil-wet. In mixed-oil-wet pores the three-phase contact line is pinned and does not move in response to variation in capillary pressure. Instead, resulting curvature changes at the oil/water interface are accommodated by changes in the angle of contact. This angle of contact at the pinned contact line is free to assume values here between 20" and 180". Previous network model descriptions (Heiba et al., 1983; Mohanty and Salter, 1983) of mixed wettability use randomly distributed oil wetness or make some fraction of the total oil-occupied pore space oil-wet. A single pore displays only one type of wettability. Following adsorption of asphaltenes, the course of pore- zyxwvut zyxwvutsrq zyxwvutsr zyxwvut zyxwvutsr zyxw zyxwvut Figure 7. Water-wet pore just after oil entry. empties the entire middle portion of the tube. After which, the arc menisci increase their curvature in response to higher imposed capillary pressures and advance toward the cusp. Thus, as more oil enters the tube, the capillary pressure climbs, the aqueous films coating pore walls thin, and the disjoining pressure climbs. At the critical capillary pressure, P:, for this sized pore, the thick films become unstable and spontaneously thin to molecularly adsorbed films. Since h/a << 1 , the thick films coating the walls of star-shaped pores have approximately the same curvature as the pore walls. Consequently, the critical capillary pressure for thick wetting-film collapse is: P: = nman - a(JZ - l ) / a . (5) The second term on the right incorporates the curvature of the film adhering to the pore wall. It is negative (because the pore wall is convex) causing P: to be less than Pax, the primary maximum in disjoining pressure (c.f., Figure 3). For smaller pore sizes, P: is smaller and vice versa. Hence, P: may not be attained in all pores. The drainage and film-breakage processes continue as the capillary pressure is increased to a maximum, P?. h I pinned contact oil-wet (adsorbed asphaltenes) ' pinnbd contact angles Figure 8. Mixed-oil-wet pore showing the location of asphaltene deposition and coexistence of oilwet and water-wet regions within a single pore. AlChE Journal June 1993 Vol. 39, No. 6 1077 z z zyxwvu zyxwvutsrq zyxw zyxw Figure 10. Imbibition processes in a mixed-oil-wetpore. (a) Spontaneous imbibition: (b) just after brine entry during forced imbibition. Figure 9. Imbibition processes in a water-wet pore. level events through imbibition and secondary-drainage processes differs radically between a mixed-oil-wet and a waterwet pore, as discussed below. Water-wet pore Mixed-oil- wet Dore Upon imbibition, the water-wet pore immediately refills with water (Provided that a capillary connection to a water SUPPlY exists). The menisci here are not pinned and thus f o l h the exact reverse course O f pristine drainage in response to a decreasing applied capillary pressure. As the capillary pressure falls, the menisci move away from the pore corners at zero equilibrium contact angle. This continues until the menisci touch to form an inscribed circle (see Figure 9). However, the inscribed circle configuration is unstable (Ransohoff et al., 1987) in that any infinitesimal disturbance in the streamwise direction must grow. As first described by Roof (1970), the oil snaps off to achieve a minimum energy configuration. Morrow and Mason (1991) argue that for perfectly wetting triangular pores, the resulting disconnected cylinders of oil created by snap-off exhibit curvatures corresponding to the entry curvature. We adopt the same reasoning here and demand that the curvature of the snapped-off oil drop be that given by Eq. 4. Snapped-off oil is distributed along the length of the water-wet pores as isolated cylinders of oil resembling sausages with hemispherical-like end caps and curvatures corresponding to the appropriate entry curvature. The length of the isolated oil cylinders is determined by long wavelength disturbances which cause breakup of the oil column, roughly 27ra (Chambers and Radke, 1991). After snap-off, the isolated oil sausages are disconnected from the continuous oil fraction and are no longer under the influence of the applied capillary pressure. In our equilibrium scenario, there is no facility to displace this discontinuous oil. Even if small pressure gradients are imposed, the disconnected 1078 oil eventually encounters a pore constriction and traps. Therefore, snapped-off oil is termed here residual oil. The fluid configuration in the water-wet pore remains unchanged until secondary drainage of water occurs. At the appropriate entry pressure for this sized pore, oil enters the waterwet pore, as during pristine drainage, and reconnects with the residual oil already present. Once the central portion of the pore refills with oil (c.f., Figure 7), the arc menisci again move toward the pore corner as the capillary pressure climbs. Upon completion of secondary drainage, the water-wet pore contains continuous oil in the center of the pore and continuous brine in the corners connected to thick continuous water films along the pore walls in an exact replica of pristine drainage. Figures 10 and 11 display pertinent events in a mixed-oilwet pore (one in which p- > p,' SO that adsorbed asphaltene films coat the pore walls away from the corners), which proceed through imbibition and secondary-drainage processes. Recall that the three-phase Contact line in a mixed-oil-wet pore is zyxwv Figure 11. Secondary drainage in mixed-oil-wet pores. June 1 9 3 Vol. 39, No. 6 (a) Case I , unstable lenses and unstable O/W emulsion films; (b) case I . unstable lenses and stable O/W emulsion films; (c) case 2, stable W/O emulsion films AIChE Journal zyxwvutsr pinned at the location it attained at PT". Upon gradual lowering of the imposed capillary pressure, water imbibes spontaneously due to capillary suction until the curvature of the pinned water-oil interfaces falls to zero (Figure 10a). At zero imposed capillary pressure the mixed-oil-wet pore contains continuous oil in the center and continuous brine in the corners with an interfacial curvature of zero. Further imbibition into the mixed-oil-wet pore occurs when water is forced into the rock. Water-phase pressure now exceeds oil-phase pressure. Hence, negative capillary pressures arise, and this process is referred to as forced imbibition. Because the fraction of water-wet surfaces in a mixed-oil-wet pore is quite small, the pore appears completely oil-wet (a contact angle of 180" measured through the water phase) to the water-invasion process. Thus, forced water invasion is directly analogous to forced oil entry into completely water-wet pores (a contact angle of zero). Accordingly, the entry curvature of the eightfold symmetric star-shaped pore is given by: 0 emulsion films are stable within the range of applied capillary pressures. If the W/O emulsion films are not stable over the range of capillary pressures applied, case 1 is recovered. Regardless of which picture is adopted, the mixed-oil-wet pore becomes completely filled with water except for a small oil fraction that exists as either thin oil films along the solid surface of the pore or as W/O emulsion films which span each pore corner and which are connected to the thin oil films lying along the solid pore surface. Pore-level events differ significantly for cases 1 and 2 during secondary drainage of water. In the case of unstable lenses, beginning at the minimum negative imposed capillary pressure, the oil films lining the walls of the mixed-oil-wet pore thicken following Eq. 1. The oil resides as thin films until the capillary pressure is positive and exceeds the magnitude of the wall curvature [ ~ a ,r ( 4-I)/~I. Upon increasing the positive capillary pressure further, oil invades along the pore walls, as shown in Figure 1 la. Pinning is still demanded at the step change in wettability on the pore surface. Invasion proceeds until the expanding oil/water interfaces touch. If sufficient stabilizing forces are present, a thin oil-in-water (O/W) emulsion film forms in between the oil growing from adjacent pore walls. Figure l l b illustrates that increasing P, forces the fraction of water contained in the center of the pore to decrease and the extent of the thin films to grow. If the O/W emulsion films are stable at all applied capillary pressures, no water trapping is possible. However, irreducible water saturation is clearly observed in mixed-wet porous media (Salathiel, 1973). Consequently, we presume that the emulsion films in Figure 1 l b are unstable. Thus, once the expanding oil interfaces touch in Figure 1la, an unstable fluid configuration arises. The brine in the very center of the pore rearranges into cylindrical droplets with rounded end caps which span the pore. In cross-section, these cylindrical droplets are very similar to those displayed in Figure 10b for forced imbibition except that the outermost pinned oil/water interfaces bow toward the corners. Oil maintains continuity along the length of the pore, but the brine in the center of the pore is no longer continuous. Individual discontinuous brine droplets are separated axially by thick oil bridges perpendicular to pore walls and connected to the oil lenses running the length of the pore. This discontinuous brine is the direct analogue to discontinuous trapped oil and is referred to as irreducible water. Mixed-oil-wet pores are requisite to producing irreducible trapped water. In strongly water-wet porous media there is no irreducible water. Rather, connate water in the pore corners is continuous, and the connate saturation simply decreases as the imposed capillary pressure rises (c.f., Figure 7). In the case of stable W/O emulsion films (case 2), oil refills the thin oil films bridging the corners of the pore beginning at P p , as illustrated in Figure 1 lc. The free oil meniscus (the inner most one which is unpinned) grows at 180" contact angle, while the outermost water/oil interface remains pinned at the contact line for the step change in wettability. Once the unstable inscribed circle configuration is attained (see Figure 1 lc), snapoff ensues, and the water disconnects giving rise to irreducible trapped water. The shape of the isolated cylinders of irreducible water is quite identical to that for case 1 except that the isolated water cylinders are separated axially by smaller oil bridges. At the completion of secondary drainage for either case 1 or 2, the mixed-oil-wet pore contains continuous brine in the zyxwvuts zyxwvuts zyxwvutsrq aP,'_ - - 1.86, U where the negative sign arises because of the 180" contact angle (c.f., Eq. 4). Figure 10b displays the oil and water configurations in the mixed-oil-wet pore just after water invasion. A thick lens or rivulet of oil bridges the corner of the pore separating brine in the pore center from that in the pore corner. This bridging, oil-lens configuration is readily argued to be stable (Ransohoff et al., 1987). An oil film coats the central portion of the pore wall. The original three-phase contact line nearest the corner remains pinned, while the newly created meniscus establishes an equilibrium contact angle against the pore wall. Bridging oil lenses have been observed in etched-glass micromodels containing asphaltic oil and brine (Buckley, 1992). Since asphaltene adsorption is virtually irreversible (Morrow et al., 1986; Hirasaki et al., 1990; Dubey and Waxman, 1991), we postulate that all thicknesses of oil films deposited on asphaltene-coated surfaces are stable. It follows that the oil-film disjoining pressure is purely repulsive. Thus, the solid surface is completely wetted by oil, and the newly created meniscus establishes a contact angle of 180" measured through the water phase. If measured through the oil phase, the contact angle is zero reflecting a monotonic, repulsive disjoining pressure isotherm and Eq. 2. More structure to the oil-film disjoining pressure isotherm is possible (Hirasaki, 1991a), but leads to no significant differences in overall pore-occupancy configurations. Oil contained in the lenses of Figure 10b is translationally continuous. Consequently, all of the oil drains, although slowly, as the capillary pressure becomes more negative. When the lenses are nearly empty of oil, the two water/oil interfaces composing each lens meet. Two possibilities exist for the behavior of the oil lenses when the waterloil interfaces touch. In case 1, the lens is unstable and breaks. The small amount of oil contained in the lens when it breaks, flows via the continuous oil-wet surfaces to surrounding pores. In case 2, a stable, thin water-in-oil (W/ 0)emulsion film is created which bridges the pore corners. A repulsive, stabilizing disjoining pressure in the thin water/oil/ water film is required for case 2. We assume here that any W/ AIChE Journal z zyxwv June 1993 Vol. 39, No. 6 1079 corners of the pore, continuous oil lenses, and disconnected irreducible brine in the center of the pores. If W/O emulsion films do not form (unstable lenses), irreducible brine is formed while the capillary pressure is positive during secondary drainage. If W/O emulsion films are stable within the range of applied capillary pressure, then irreducible brine is formed while the capillary pressure is negative. 100 I zyxwvuts zyxwvutsrqp zyxwvu zyxwvuts zyxwv 1 80 r 0 C al -E! & Results 5 The above series of pore-level fluid configurations leads directly to calculation of capillary pressure vs. aqueous-phase saturation curves, once a distribution of pore sizes is specified. Figure 12 shows on a semilogarithmic scale the discrete size distribution of star-shaped pores employed here. The distribution is nondimensionalized by the mean of the distribution, a,, chosen as 150 Fm. This particular distribution approximates roughly that found by Wardlaw et al. (1987) for pores in a Berea sandstone. In addition to pore bodies, the adopted distribution also contains a significant number of pores which are exceptionally small. These represent pore throats and micropores. We make no distinction between pore throats or bodies. The frequency scale indicates the actual number of pores of a particular size used in the calculations to follow. Capillary pressures are imposed by incrementing interfacial curvature, pore-level reconfiguration of fluids is assessed, and the saturation is calculated from a knowledge of interfacial curvature and fluid configuration. The known phase configurations lead directly to the aqueous-phase saturation, S,. = A ,,./ A,, where A, is the cross-sectional area of the pores occupied by water and A, is the total cross-sectional area of the pores. Once generated, the theoretical P,. vs. S,,,curves may be compared to typical experimental data. Numerical and geometrical details of the calculation are available elsewhere (Kovscek, 1993). Capillary-pressure curves Figure I3 depicts capillary pressure vs. aqueous-phase saturation for case 1, unstable lenses. On Figure 13, a nondimensional capillary pressure, a,,J',./a, of 20 corresponds to approximately 1 .O psi (6.8 kPa). Pristine drainage occurs between the points labeled A through C. No oil enters the porous medium until point A where the capillary entry pressure is exceeded for the largest-sized pore. Each particular pore size has a unique entry pressure determined by Eq. 4 with the fluid configuration after entry shown in Figure 7. As the capillary pressure climbs oil flows into the porous medium, more capillaries are entered and the water saturation declines. The smallest-sized pores are not entered by oil, because their capillary entry pressure is not exceeded. Consequently, they remain brinefilled. The curve displayed in Figure 13 is not smooth, because a discrete rather than smooth distribution of pore sizes is employed (c.f., Figure 12). Following Eq. 5 , each sized pore in the size distribution has a unique critical capillary pressure for thick film stability. Films are thinnest along pore walls with the most negative curvature (those pore walls with the smallest radii of curvature). Thick protective films correspondingly break first in the smallest oiloccupied pores. The area near point B on Figure 13 denotes the region where thick protective water films coating walls of intermediate-sized 1080 a,,,= 150 pm 60 40 C 20 0 0.01 0.05 0.15 0.1 1 3 a am Figure 12. Discrete size distribution of star-shaped pores. pores break. Larger pores do not exceed their particular P: and thus remain water-wet. The drainage and thick film-breakage processes continue to P,""",the highest imposed capillary pressure (point C on Figure 13). Water saturation at this point is connate saturation, S,. Figure 7 reveals that large capillary pressures, such as P,""",correspond to low connate water saturation. This is consistent with Salathiel's (1973) original observations. It is possible to reduce water saturation further by exceeding P,""", because both phases maintain continuity throughout the pore structure. That is, no water is trapped at connate saturation in Figure 13. Asphaltene adsorption also occurs at point C in those intermediate-sized pores which lack thick water films. After asphaltene adsorption, the smallest pores that were never entered by oil are completely brine-filled. The intermediate-sized pores, that exceeded their critical capillary pressure for thick-film stability, P:, have asphaltenes adsorbed on the exposed rock surface and, hence, are mixed-oil-wet with a pinned contact line between the oil- and water-wet regions. The largest pores, -15 zyxwvut J , , ,: zyxw June 1993 Vol. 39, No. 6 - I I I . :G,alnPcmln I . n , , , AIChE Journal zyxwvutsr zyxwvuts zyxwvutsrqp which did not exceed their corresponding Pr , remain waterwet. Upon a controlled decrease in capillary pressure, spontaneous imbibition commences between the points labeled C through E, and all pores refill with water (as illustrated by Figures 9 and 10a). The water-wet and mixed-oil-wet pores follow individual imbibition paths. Near the point labeled D, oil snaps off and traps in the water-wet pores. As the capillary pressure approaches zero, the large water-wet pores are filled with residual oil and no longer change saturation in response to capillary pressure changes. Further examination of Figure 10a teaches that small changes in the brine saturation of mixedoil-wet pores cause large changes in interfacial curvature, and accordingly, large changes in capillary pressure. This explains the near step change in P, in the vicinity of point E on Figure 13. Forced imbibition at negative capillary pressures occurs along points E through G of Figure 13. Saturation changes occur only in the mixed-oil-wet pores because, as noted above, the smallest pores are completely brine-filled and the largest pores are filled with trapped oil. Water imbibes into progressively smaller mixed-oil-wet pores as the capillary pressure becomes more negative (such as point F on Figure 13), and oil slowly drains from the bridging lenses or rivulets. As capillary pressure is made increasingly negative and decreases to P,"'" (point G of Figure 13), the oil saturation of the porous medium decreases to So,, the residual oil saturation. Secondary drainage for case 1, or unstable lenses, is denoted by points G through K. Beginning at point G, the oil films coating pore walls thicken, but the capillary pressure must be positive and exceed the magnitude of the pore-wall curvature before significant amounts of oil can reinvade the mixed-oilwet pores. Correspondingly, the capillary pressure rises sharply from point G to point H with little alteration in saturation. Oil then reoccupies the mixed-oil-wet pores, as illustrated in Figure 1 la. Water traps near point I and leads to irreducible water saturation. With P, now positive, oil reenters the waterwet pores at the appropriate P,' (point J), as it did during pristine drainage. Oil filling continues until P y is reattained at point K with saturation S,,,,. When another cycle of imbibition and drainage beginning from P y is made, a scanning or hysteresis loop is found between points K and G. A dashed line on Figure 13 illustrates how the scanning loop connects. The pore-level events which distinguish case 1 from case 2 happen strictly during secondary drainage. Hence, the capillary pressure curve for case 2 differs from case 1 only during secondary drainage. On Figure 14 secondary drainage for case 2 is marked by the points G through P. Point G of Figure 14, the minimum imposed capillary pressure, is identical with point G of Figure 13. Large volumes of oil, relative to case 1, immediately reinvade the stable W/O films at P,"'" during secondary drainage for case 2. Irreducible water is created at negative capillary pressures as marked by region M in Figure 14, consistent with the secondary-drainage mechanism illustrated in Figure 1lc. Once snap-off and trapping are completed in the mixed-oil-wet pores, only the pinned interface can respond to changes in capillary pressure (c.f., Figure llc). Due to the sensitivity of the pinned interface, the applied capillary pressure rises quite rapidly with little change in saturation in the region around point N on Figure 14. Similar behavior was observed at point . . ( 02 O swc . . . , . . . , I . . , . zyx . 08 06 04 SW,,, . (1% 1 ) sw Figure 14. Capillary pressure vs. aqueous-phase saturation, case 2 stable WIO emulsion films. E on Figure 13 during spontaneous imbibition. With capillary pressures now positive, oil enters the water-wet pores, for instance, at point 0 on Figure 14. Again the capillary pressure increases to P y , point P on Figure 14. Similar to case 1, a scanning or hysteresis loop is found, now between points P and G, when the imbibition and drainage processes are repeated. Now that the theoretical curves have been established, they are evaluated to gauge the wettability of the porous medium. Amott-Harvey indices are the requisite tool. zyxwvutsrqp AIChE Journal Porous-medium wettability Amott-Harvey indices are based on the saturation changes associated with spontaneous imbibition, ASws, spontaneous secondary drainage, ASos, and the overall saturation change during imbibition, AS,, (Amott, 1959; Morrow, 1990). These quantities are defined on Figure 5b. First, the wettability index to water, Z,=AS,,JAS,f, is established as the ratio of spontaneous and overall saturation changes during imbibition. Next, the wettability index to oil, I, = AS,,/AS,,, is found from the ratio of saturation change during secondary drainage to the overall saturation change during imbibition. The difference between water and oil wettability indices, I,, = I , - I,, is the Amott-Harvey index. Amott-Harvey indices and residual oil saturations are presented in Table 1 for both mixed-wet cases on Figures 13 and 14 and for a water-wet case. Case 3, the water-wet example, has a nondimensional disjoining pressure maximum (umIImax/ a), for aqueous films coating solid surfaces, that is 10 times larger than the mixed-wettability cases. This ensures that none zy Table 1. Amott-Harvey Indices and Residual Oil Saturations for Water-Wet and Mixed-Wet Porous Media Case 1. Unstable lenses 2. Stable W/O emulsion films 3. Water-wet June 1993 Vol. 39, No. 6 amnmaX u I,, 18 0.36 S", 0.35 18 -0.18 0.35 180 1 .O 0.54 1081 zyxwvutsrq zyxwvuts zyxwvutsrqp zyxwvu zyxwvuts of the thick water films coating pore walls rupture to molecularly adsorbed films within the range of applied capillary pressures. Asphaltene adsorption is thereby prohibited. Consequently, this case exhibits an I,, of 1, as expected for strongly water-wetting rock. The Amott-Harvey index for W/O emulsion films (case 2) is less than that for unstable lenses (case 1). Since W/O emulsion films refill immediately with oil, while capillary pressure is negative during secondary drainage, AS,, and Z, are significant for case 2. Accordingly, Iwois smaller. The unstable lenses and the stable W/O emulsion films exhibit identical residual oil saturations, because in each case an identical amount of oil is contained in lenses, such as those shown in Figure lob. Recovery of oil from these lenses is eventually complete. Pore-level events that change pore occupancy differ only after P,"'" and So,are attained (point G of Figures 13 and 14). Comparison with experiment The theoretical capillary-pressure curves and Amott-Harvey indices compare favorably to experimental data for mixed-wet porous media. As illustrated in Table 1, residual oil saturation decreases as Zwo tends toward mixed-wet conditions. For the mixed-wet cases, So, is 20 saturation units lower than the waterwet case. This is consistent with experimental observations (Salathiel, 1973; Jadhunandan and Morrow, 1991). However, the absolute magnitude of So,obtained experimentally is much lower than that predicted here. Our scenario overestimates residual oil saturation, primarily because the pores are not explicitly interconnected. In actual porous media, the connectivity or topology of the pore space strongly influences the displacement and entrapment of oil (Mohanty et al., 1987). In water-wet pore space that lies adjacent to and is connected to oil-wet pore space, oil cannot snap off and trap (Mohanty and Salter, 1983). Also, it is well known that a network of completely water-wet pores with multiple connections allows more escape routes for oil to remain continuous (Heiba et al., 1992). Salathiel (1973) originally attributed extended oil recovery from mixed-wet rocks to thin continuous oil films along the rock surface which allow almost indefinite oil displacement. In the scenario proposed here, we predict that most remaining oil occurs as lenses or rivulets which span the corners of mixedoil-wet pores (c.f., Figure lob). The oil films coating the oilwet fraction of the pore walls allow surface drainage of oil, but the oil volume they contribute is much less than that associated with lens drainage. Further, the general shapes of the capillary-pressure curves and corresponding hysteresis loops in Figures 13 and 14 reflect those measured experimentally (compare to Figure 5b or the measurements of Sharma and Wunderlich, 1987, or Hirasaki et al., 1990). In particular, mixed-wet rocks exhibit both large positive and negative capillary pressures over a significant range of saturation. Also, the area enclosed by the forced-imbibition portion of the capillary-pressure curve and the line Pc=O is large indicative of mixed-wettability. Figures 13 and 14 show that consistent with Salathiel's observations, connate and irreducible water saturations are not equal. This is a result of the wettability alteration that occurs at P y during primary drainage. Oil migrates into a reservoir when it is water-wet. Thus, no brine is trapped, and the connate or initial water saturation is much less than the irreducible 1082 water attained after oilflooding a previously waterflooded reservoir. The pore-level events which occur along the path to P y during secondary drainage now reflect a mixed-wet porous medium. Although the same capillary pressures are imposed, the endpoint saturations at P y (points K on Figure 13 and P on Figure 14) are quite different. Figures 1l a and 1l c display the water/oil configurations prior to water snap-off which determine S,,,,. It is evident that Figure 1l a (unstable lenses) leads to less irreducible water than Figure 1 l c (W/O emulsion films). As with residual oil saturation, our model predicts irreducible water saturations larger than those found experimentally, again due to a lack of multiple interconnectedness among pores. Discussion Our proposed model of mixed-wettability has five important parameters which can be altered: maximum imposed capillary pressure, minimum imposed capillary pressure, disjoining pressure maximum, pore shape, and pore-size distribution. We consider briefly each in turn. As the maximum imposed capillary pressure is increased, more thick protective water films spontaneously break to molecular films during pristine drainage. A larger fraction of the rock surface is then coated with asphaltenes and becomes oilwet. This effect is consistent with Salathiel's findings. As he decreased connate water saturation by increasing the imposed capillary pressure, his core samples displayed increased oil-wet behavior. More recently, Jadhunandan and Morrow (1991) show experimentally that Berea sandstone changes from fairly strongly water-wet to mixed-wet in the presence of crude oil and brine when connate or initial water saturation is reduced by 10%. If all thick protective water films lining pore walls are broken under case 1 (unstable lenses), the porous medium exhibits an Amott-Harvey wetting index of approximately zero. Since there are no longer large water-wet pores, the capillary pressure drops precipitously from P y to zero with little change in saturation upon imbibition. Upon secondary drainage, the capillary pressure still rises rapidly from P y to zero with little change in saturation. These two results balance to give near neutral wettability. However, because the pore corners remain water-wet, the porous medium retains mixed-wettability character. Under case 2 (stable W/O emulsion films), strongly oilwet behavior ( I w o near - 1) results, because the stable W/O emulsion films span all pores except the very smallest ones. These films all refill with oil, while P, is negative giving greater oil-wet character. Again, mixed-wettability is retained because the pore corners remain water-wet. When P y becomes infinite, S,, approaches zero. All of the brine residing in pore corners is eliminated. In this instance, all solid surfaces of the porous medium become oil-wet, and an I,, of - 1 results. In this situation oil recovery by waterflooding is potentially complete. The minimum imposed capillary pressure is set to - P,"" for all calculations. A less negative P,"'" does not force all of the water/oil interfaces of the bridging-oil lenses to touch. A distribution of lenses in which the interfaces do and do not touch arises in the mixed-oil-wet pores. In case 1, not all of the oil lenses reach the unstable configuration. A mixture of broken and unbroken lenses at P y gives capillary pressure curves which June 1993 Vol. 39, No. 6 zyxw AIChE Journal zyxwvutsrqpo zyxwv zyxwvutsr zy zyxwvu zyxwvutsrqp are a mixture of cases 1 and 2. The unbroken oil lenses refill in a manner analogous to the filling of the W/O emulsion films in case 2. In case 2, not all of the pores develop W/O emulsion films as P,""' is made less negative. However, the capillary pressure curve is unaffected. A more negative P p also has no effect, because there is very little additional oil to be recovered from the mixed-oil-wet pores in cases 1 and 2. Increasing the maximum in the solid/water/oil disjoining pressure curve (c.f., Figure 3) adds stability to the thick brine films coating pore walls. Thus, the critical capillary pressure, F(*,required to rupture thick films is increased following Eq. 5. As ITax is increased, the porous medium portrays more water-wetting character. In the limit of a very large disjoining pressure maximum (such as case 3), a completely water-wet porous medium emerges. Thus, our picture explains mechanistically how surfaces may remain water-wet even when highly asphaltic oil is present. Water wettability is also maintained when asphaltenes or other strongly adsorbing polar organics are absent from the The absence of oil phase, regardless of the value of Fax. asphaltenes prohibits wettability reversal even when thick brine films rupture to molecularly adsorbed ones. Although molecularly adsorbed and thick brine films reflect different contact angles (c.f., Eq. 2), such shifts in the contact angle do not dramatically alter imbibition and drainage capillary-pressure curves. Pore shape has a significant effect on which pores become mixed-oil-wet. For instance, in a concave pore, such as that with an eye-shaped cross section in Figure 15a, the thinnest brine films line the walls of the largest pores, whereas a convex pore shape (c.f., Figure 6) places the thinnest brine films in the smallest pores. Hence, with eye-shaped pores, the largest oil-filled pores become mixed-oil-wet rather than the smallest oil-occupied pores. Remaining oil is still present as lenses or rivulets which bridge the corners of the pores. Since oil traps only in the smaller volume water-wet pores, residual oil saturation is now lower. A triangular pore (Figure 15b) allows all of the thick water films coating pore walls to break at the same capillary pressure. Results are then similar to those when the imposed capillary pressure is very large. In this instance, Pf;corresponds exactly to Pax, because the walls of triangular pores are of zero curvature. Residual oil saturation is drastically lower than with either star or eye-shaped pores, because all pores become mixedoil-wet and undergo efficient drainage. As with both the star and eye-shaped pores, bridging-oil lenses are the source of remaining oil saturation. If bridging-oil lenses are unstable as in case 1, Zwo is approximately zero. For stable W/O emulsion near - 1) results, films in case 2, strongly oil-wet behavior (Iwo but there pores retain some mixed-wettability character, because the corners of the pore remain water-wet. Morrow and Mason (1991) find that in perfectly wetting triangular capillaries the amount of trapped oil is reduced as the triangle becomes more irregular. Recall that the area of the largest circle that can be inscribed in a pore (displayed in all pore shapes of Figure 15) determines the amount of trapped oil. The more irregular a triangle, the smaller the possible inscribed circle relative to the total triangle area. Asymmetry in both convex (Figure 15c) or concave (Figure 15a) pore shapes also leads to less oil trapping and greater oil recovery. The last pore shape considered is shown in Figure 15d. The walls of this star-shaped pore do not meet at a point. This shape arises when minerals deposited between grains (Salathiel, 1973). The corners are rounded and do not retain significant amounts of brine at large imposed capillary pressures. Consequently, the entire pore wall becomes continuously oil-wet. If the imposed capillary pressure is again large, all pores are made oil-wet in this manner, the Amott-Harvey wetting index equals - 1 indicative of strongly oil-wet conditions. The fifth model parameter is the pore-size distribution. By decreasing the ratio of large-sized pores to small-sized pores, the amount of residual oil is reduced. Likewise, increasing the fraction of intermediate-sized mixed-oil-wet pores raises the amount of recoverable oil. Both of these changes shift Iw, toward more oil-wet conditions. When the smallest pores (representing micropores and pore throats) are trimmed from the distribution of pore sizes in Figure 12, small completely brinefilled pores no longer exist and s,, falls to zero. The shapes of the capillary-pressure curves (Figures 13 and 14) are not changed. They merely shift toward the left. Wettability indices are unaffected. The pore surfaces considered in this work are smooth, whereas roughness is the norm in reservoir media. When the protrusion size of roughness asperities is much smaller than the average film thickness, the film completely overlays and levels the roughness. A smooth water/oil interface emerges. When leveling occurs, the average pore-wall curvature is the relevant curvature in Eq. 5. Conversely, when the protrusion length scale of asperities is much larger than the film thickness, wetting films contour or hug the solid surface. In the hugging regime, it is the local roughness curvature which determines the capillary pressure necessary to rupture thick films to molecularly thin ones; wetting-film behavior does not likely correlate with pore shape. Moreover, wettability reversal occurs on the exposed portions of the asperities. Between the leveling and hugging regimes, wetting films can make rough surfaces appear dramatically smoother if the distance separating the asperities is less than the average film thickness (Robbins et al., 1991). This prediction is borne out by the experiments of Garoff et al. (1989) who measured the gadliquid interfacial roughness of a wetting film on a rough substrate. Thus, for our star-shaped pore model, the behavior of films on pores zyxwvutsrqp Figure 15. Alternative cross-sectional pore shapes. (a) Eye-shaped; (b) irregular-triangle;(c) irregular-star;(d) starshaped with mineral deposits in pore corners. AIChE Journal June 1993 Vol. 39, NO. 6 1083 with rough walls mimics that of smooth pores whenever the asperities are dull and/or closely spaced. The converse situation warrants investigation. The collection of nonaxisymmetric capillaries model allows exploration of not only mixed-wettability behavior, but a wide variety of wetting behavior. Intermediate-wettability can be accommodated by altering the oil/water/solid disjoining pressure isotherm (c.f., Figure 3) to represent surfaces with finite contact angles. In fact, a surface with virtually any type of wettability can be modeled by adopting the appropriate disjoining pressure isotherm. We also are not limited to pore surfaces of uniform or symmetric wettability. One portion of the pore surface may be represented by a disjoining pressure isotherm for water-wetting minerals and another for oil-wetting minerals. The case of multiple occupancy of a single cornered pores by three phases (such as, gas, oil, and water) appears even more intriguing and intricate. Conclusions to pore walls. Oil drainage is slow due to large hydrodynamic resistances. With star-shaped pores these lenses or rivulets reside in pores of intermediate size. The notion of contact anglepinning allows development of the bridging oil lenses. Because the largest pores remain water-wet, oil snaps off and is trapped in a manner identical to the creation of residual oil in completely water-wetting porous media. Irreducible snapped-off water is also formed during secondary drainage, because brine is trapped in mixed-oil-wet pores in a fashion similar to the creation of residual oil in water-wet porous media. In contrast to water-wet systems, connate and irreducible water saturation are shown to be quite different in mixedwettability systems. Since wettability states undergo change at or near S,,., there is a corresponding change in pore-level occupancy events for mixed-wettability systems. Irreducible water, consistent with experimental observations, is predicted to be greater for mixed-wettability systems than for waterwetting systems due to the formation of discontinuous trapped water. zyxwv zyxwvuts Currently most models of flow in porous media are based on interconnected networks of pores whose typical shapes are either cylinders or unspecified. Although networks models are very successful in explaining concepts such as relative permeability and trapping, they d o not address the important issues of wettability and wettability alteration. At a fundamental level, nonaxisymmetric pore shapes and thin films are necessary to understand wettability. We explore the stability of both brine and oil films wetting pore walls. The roles of both thin water-in-oil and oil-in-water emulsion films are also examined. In addition to these thin-film phenomena, our scenario incorporates not only cornered pores, but pore convexity, contact angle pinning, and explicit pore-size distributions. We find that a delicate interplay between pore shape and thin-film chemistry and physics can predict mixed wettability in porous rocks. The presence of asphaltenes in the oil phase is indispensable for the generation of mixed wettability. In fact, our scenario suggests that asphaltenes play a greater role in determining the evolution of mixed wettability than the underlying mineralogy of the rock. In a distribution of starshaped pores, the largest pores are prevented from becoming mixed-oil-wet, because their pore walls are protected by thick continuous water films. These films prohibit asphaltene adsorption and subsequent alteration in the wettability state of the pore surface. Conversely, ultrathin, molecular films form on the walls of intermediate-sized star-shaped pores during pristine drainage. These films permit irreversible asphaltene adsorption, and subsequently these pores become mixed-oilwet. The smallest pores which are never entered by oil remain water-wet. Our proposed suite of pore-level events correctly describes the essential observations associated with mixed-wettability reservoir rocks. Capillary-pressure curves mimic those generated experimentally in the laboratory. Foremost, a significant reduction, compared t o water-wet rock, in residual oil saturation is predicted. The correct trends of wettability index are found in that lower initial brine saturations lead to greater oil Acknowledgment This work was supported by the U.S. Department of Energy under Contract No. DC03-76F00098 to the Lawrence Berkeley Laboratory of the University of California. Parts of this work were presented as SPE 24880 at the 1992 SPE Technical Conference and Exhibition in Washington, DC. zyxwv zyxwv zyxwvu zyxw Notation a = radius of largest circle that can be inscribed in pore, m A, = cross sectional area, mz d, = bead diameter, m h = equilibrium film thickness, m h, = equilibrium film thickness overlaying a bead of diameter d,, m I, = wettability index I," = Amott-Harvey wettability index P,. = capillary pressure, difference in oil and water phase pressure, N/m' P", = nonwetting phase pressure, N/m' P , = wetting phase pressure, N/m2 r , , r2 = principal radii of curvature, m s, = phase saturation, ratio of the area occupied by a phase to total area of the system zyxwvutsrq wettability. zyxwv * -- critical value for thick film rupture e = entry for a particular size pore max = maximum m = mean o = oil or = residual oil t = total w = water wc = connate water wirr = irreducible water Greek letters zyxwvut Remaining oil saturation, the long period of oil production, and high-efficiency waterfloods of mixed-wettability porous media are attributed to the formation of oil lenses or rivulets which span the corners of pores in addition to oil films adhering 1084 Subscripts and superscripts n disjoining pressure, Pa interfacial tension, N/m 0 = equilibrium contact angle measured through the aqueous phase ASoT = saturation change during secondary drainage ASwc = saturation change during spontaneous imbibition AS",, = overall saturation change during imbibition € = dummy variable of integration, m = u = June 1993 Vol. 39. 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