SPE 124771

Unified Model of Drainage and Imbibition in 3D Fractionally Wet Porous MediaSiyavash Motealleh, SPE, Mandana Ashouripashaki, SPE, David DiCarlo, SPE, and Steven L. Bryant, SPE, The University of Texas at Austin

Copyright 2009, Society of Petroleum Engineers This paper was prepared for presentation at the 2009 SPE Annual Technical Conference and Exhibition held in New Orleans, Louisiana, USA, 4–7 October 2009. This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

Abstract We develop a grain based model for capillarity controlled displacement within 3D fractionally wet porous media. The model is based on a novel local calculation of the position of stable interfaces in contact with multiple grains. Each grain can have a different, arbitrary contact angle with the interface. The interface is assumed to be locally spherical for menisci separating the bulk non-wetting and wetting phases. The fluid/fluid interfaces between pairs of grains (surfaces of pendular rings) are assumed toroidal. Because the calculation of interface position is entirely local and grain-based, it provides a single, generalized, geometric basis for computing pore-filling events during drainage as well as imbibition. This generality is essential for modeling displacements in fractionally wet media. Pore filling occurs when an interface becomes unstable in a pore throat (analogous to Haines condition for drainage in a uniformly wet throat), when two or more interfaces come into contact and merge to form a single interface (analogous to the Melrose condition for imbibition in uniformly wet medium), or when a meniscus in a throat touches a nearby grain (a new stability criterion). The analytical solution for stable interface locations generalizes the Melrose and Haines criteria previously validated for pore-level imbibition and drainage events in uniformly wet media.

The concept of tracking the fluid/fluid interface on each grain means that a traditional pore network is not used in the model. The calculation of phase saturation or other quantities that are conveniently computed in a network can be done with any approach for defining pore bodies and throats (e.g. Delaunay tessellation, Voronoi tessellation, and medial axis methods). The fluid/fluid interfaces are mapped from the grain-based model to the network as needed. In addition, the model is robust as there is no difference in the model between drainage and imbibition, as all criteria are accounted for both increasing and decreasing capillary pressure.

To validate the model, we perform a series of drainage/imbibition experiments (oil/water) on fractionally wetted porous media prepared by mixing oil-wet grains with water-wet grains. In both experimental and simulation results, the drainage/imbibition curves shifts to lower capillary pressure with increasing fraction of oil-wet grains.

Using the model, we delineate which pore filling criteria occur as a function of initial wetting phase and wettability of grains. The shape and position of the pressure–saturation curve is shown to be a function of the pore filling types, and hysteresis arises naturally from the model.

Introduction A rock containing immiscible fluids is wetted by the fluid that has smaller surface energy of interaction with the rock (Dullien 1992). Consequently the wettability of reservoir rocks determines the distribution of fluids within the pore space of the rock (Anderson 1987a; Anderson 1987b). Fluid distribution at the pore scale is important because this affects the macroscopic rock/fluid properties such as capillary pressure curves and relative permeability curves.

Many rock minerals have a tendency to be wetted by water, and thus reservoir rocks are typically water-wet before they are filled with oil. However, chemical species within the oil that are charged can change the wettability of the reservoir rocks to oil-wet during geological time (Salathiel 1973). Reservoirs can be partly oil-wet and partly water-wet due to wettability alteration, which occurs on the part of the reservoir rock that is exposed to the crude oil (Salathiel 1973). This altered state is referred to in the literature variously as heterogeneous wettability (Laroche et al. 1999), fractionally wettability (Tsakiroglou and Fleury 1999) and mixed wettability (Van Dijke et al. 2000; van Dijke and Sorbie 2003; Al-Futaisi and Patzek 2004; Valvatne and Blunt 2004).

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Here, we focus on fractional wettability, defined as a state in which each grain in the porous medium is either water-wet or oil-wet. The water-wet and oil-wet grains are assumed to be distributed randomly within the porous medium. The contact angle on each grain is arbitrary, but we assume for convenience that all water-wet grains have the same contact angle, and all oil-wet grains have the same contact angle. This is depicted schematically in Fig. 1.

Fractionally wetted porous media occur naturally in certain soils, and they are the simplest heterogeneously wet media that can be reliably reproduced experimentally. In field soils, the fractional wettability may come about as the humic acids may coat grains of different sizes and mineralogies differently. In the laboratory, fractionally wetted porous media are easily prepared by mixing different fractions of oil-wet grains and water-wet grains. Because of this, fractionally wet media have been the focus of most experimental measurements (Brown and Fatt 1956; Sharma et al. 1991; Bradford and Leij 1995; Ustohal et al. 1998; Bauters et al. 2000; O'Carroll et al. 2005; Han et al. 2006).

Much modeling work has been done on heterogeneously wet porous media, usually approaching the problem with concepts developed originally for uniformly wet media. Most of these works consider variation of wettability within pores. McDougall and Sorbie (1993) modeled fractionally wet porous media by choosing certain fractions of pores to be water-wet or oil-wet, independent of the pore sizes. They used invasion percolation theory (Wilkinson and Willemsen 1983) to model the capillary displacement of water and oil phases. Other researchers (Heiba et al. 1983; Mohanty and Salter 1983; Blunt 1998; Ustohal et al. 1998; Piri and Blunt 2002) have studied heterogeneously wet porous medium using pore network modeling. Their pore network models are typically based on a lattice of pores and throats with different shapes (triangular, square, circular, star-shape, etc.) and randomly chosen sizes. The shapes are inspired by naturally occurring throats, which retain wetting phase at grain contacts after the throat has drained. The angular cross sections throats represent a useful advance over traditional cylindrical throats by retaining wetting phase in the corners of their throats (Singhal and Somerton 1970; Lenormand and Zarcone 1984). This enables direct implementation of heterogeneous wettability, e.g. the surface of a duct containing oil after drainage becomes oil-wet except in the corners where water phase remained at the drainage endpoint.

Grain-Based Modeling. A concept not explicitly considered in the above models is that "pores are where grains are not." In other words, pore shapes are not fundamental; rather, in nature the grain shapes are the geometric primitives which determine pore shapes. Moreover, wettability is a property of the grain surfaces, not of pores. A grain-based approach to this class of fluid displacements therefore seems natural.

The fundamental event in a grain-based approach to capillarity-controlled fluid displacement is the filling of an individual pore, just as in traditional pore network models. The difference from traditional network models emphasized in this work is that the criteria for such events are derived in terms of grain locations and contact angles, rather than in terms of the geometry of idealized pores and throats. The correct geometry and location of fluid/fluid interfaces (pendular rings at contacts between pairs of grains, menisci between three or more grains) are computed by a conceptually straightforward, deterministic algorithm in a grain-based model. Thus it is possible to study the movement, merger and splitting of interfaces in a mechanistic fashion. In contrast, the complementarity of pores and grains is not treated explicitly in traditional pore network models. As a result, ad hoc parameters and rules are often needed to describe the invasion of a pore, e.g. by the merger of two (or more) interfaces (Blunt and Scher 1995; Blunt 2001; Patzek 2001; Blunt et al. 2002; Piri and Blunt 2002).

By using the grain-based approach, we show by explicit calculation that pore-filling events usually associated with drainage in a water-wet medium (Haines jumps) and events usually associated with imbibition in a water-wet medium (merger of menisci and Melrose events) both occur during any cycle of drainage or imbibition in a fractionally wet medium. Accounting for this behavior in a traditional network model would require an additional level of complication in the "rulebook" for pore-filling events. It may be possible to develop such rules, but we believe the relative simplicity of a grain-based approach makes it preferable for predictive studies of displacements within fractionally wet media. .

Our grain-based model is founded on calculation of the stable positions of menisci on the grains. Thus the model requires knowledge of grain space geometry (shapes, spatial coordinates of individual grains, etc.), but it does not require a network representation of pore space. It proves convenient to use a network of pore bodies and pore throats to compute saturations or to track menisci motion, and for that purpose the subdivision of pore space can be accomplished by any approach (e.g. Delaunay tessellation (Bryant, S. L. et al. 1993; Bryant, S. et al. 1993), Voronoi tessellation (Jerauld et al. 1984a; Jerauld et al. 1984b), and medial axis (Lindquist et al. 1996; O'Rourke 1998) methods).

Terminology. To illustrate the grain-based model of menisci movement through 3D fractionally wet porous media, we use a dense disordered packing of equal spheres generated by a cooperative rearrangement algorithm (see ref. (Thane 2006)). The sphere pack is composed of 7000 equal size spheres. Each sphere can be made either water or oil-wet.

We use a Delaunay tessellation of the spheres centers to subdivide the pore space into pore bodies and pore throats. The tessellation yields tetrahedra (Fig. 2).

Pore. One pore (Delaunay cell) is constructed from four nearest-neighbor grains (spheres). In Fig. 2, the pore body is the void space inside tetrahedron O1O2O3O4. The centers of grains are the vertices of the tetrahedron. For this and subsequent figures grain numbers correspond to the subscripts on the labels for their centers. For instance, grain1 is centered at point O1 in Fig. 2.

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Throat. Three grains in a Delaunay cell define a pore throat. Thus each face of a Delaunay cell (pore) corresponds to a pore throat. A throat plane is a plane connecting the center of grains that define the throat. For instance, plane

1 2 3O O O in Fig. 2 is a throat plane.

Edge. Each edge of the tetrahedron (Fig. 2) corresponds to a grain-grain contact (if the spheres touch) or a gap between grains (if the spheres do not touch). Each edge connects the centers of two grains (spheres). For instance, line

1 3O O and 1 2O O

are two edges of a pore. The grain1 and grain3 are in a grain-grain contact, while there is a gap between grain1 and grain2. Nomenclature. We use a grain based nomenclature through the text for referring to pores, throats and edges. For instance,

pore1234 in Fig. 2 is defined by grain1, grain2, grain3 and grain4. In the same way throat123 is composed of grain1, grain2 and grain3.

Meniscus (Fluid/Fluid Interface Morphology Number 1). We model a fluid-fluid meniscus as a spherical cap. Figure 3 shows a throat between grains O1O2O3 and a meniscus located within the throat. The geometry of the meniscus is defined by the center of the meniscus (O), the radius of meniscus, and the intersections with the solid surface. The intersections of the meniscus with the grains are forced to be at the contact angle of each surface. In Fig. 3, the meniscus is the cap part (solid) of the yellow sphere (wiremesh). The radius and the center of the yellow sphere and the meniscus are the same. Importantly, the meniscus is restricted to the interior of the tetrahedron OO1O2O3, where the vertices of the tetrahedron are the centers of the meniscus and the three grains, respectively. At zero contact angles, the meniscus is tangent to each of the three grains.

Filling Angle of a Meniscus. In Fig. 4, the filling angle (ψ2) of the meniscus on the grain2 is the angle between the throat plane 1 2 3O O O and the line connecting the center of the grain2 (O2) to contact point (P2). Point P2 is the point where the meniscus touches the grain2.

Pendular ring/liquid bridge (Fluid/Fluid Interface Morphology Number 2). The edges of Delaunay tetrahedra within a sphere pack can hold pendular rings and liquid bridges of wetting phase, if and only if both grains associated with the edge have the same wettability. If the edge has a grain-grain contact, it holds the wetting phase as a pendular ring. An edge with a gap between its grains holds the wetting phase as liquid bridge. The morphologies of rings and bridges are similar, the only essential difference being that rings are unconditionally stable, while bridges can be ruptured at high capillary pressure. In this paper, for convenience we will refer to both as pendular rings. Figure 5 shows the schematic of a water-wet edge holding a pendular ring of the water phase. The contact angles between the water phase and grains are zero. Pendular rings between pairs of grains are assumed toroidal.

Filling Angle of Pendular Ring. The pendular ring filling angle ϕ is the angle between the line connecting the centers of the two grains defining the edge (line

1 2O O in Fig. 5) and the line connecting a grain center and any point at which the pendular ring touches the grain (line

1 1O P and 2 2O P in Fig. 5).

Capillary Pressure/Curvature. Because the solid surfaces in a fractionally wet porous medium have different wettability, the terms "wetting" and "non-wetting" phase are ambiguous. Hence we define capillary pressure to be the pressure of the non-aqueous phase (oil) minus pressure of the aqueous phase (water or brine).

= −c o wP P P (1)

This macroscopic capillary pressure is proportional to the curvature of the microscopic interfaces through the Young-Laplace equation,

1 2

1 1 2⎛ ⎞

= + =⎜ ⎟⎝ ⎠

c ow owP Cr r

σ σ (2)

where r1, r2 are radii of curvature and C is the mean curvature. The curvature is thus essentially a scaled capillary pressure. It is convenient in this study to make C dimensionless by multiplying by the radius of the grains in the porous medium. .When the pressure of oil phase exceeds the water phase pressure, the capillary pressure and curvature are positive and the interface curves toward the oil phase. When the capillary pressure and curvature are negative, the interface curves toward the water phase. These curved interfaces are referred to as menisci.

Drainage/Imbibition. Along these lines and following Morrow (Morrow 1990), we define drainage as a displacement for which the water saturation is decreasing, and imbibition to be when the water saturation is increasing.

Model Grain-Based Mechanistic Criteria for Menisci Movement. The essential input for a grain-based approach are (i) the locations of the grains and (ii) the wettability (contact angle) of each grain. The essential calculation is the location of a stable interface, given these data and a value of interface curvature. Geometric considerations (discussed in detail below) show that a fluid/fluid interface can become unstable if any of the criteria below is met.

1. Generalized Haines drainage: The curvature of meniscus exceeds the critical curvature for the throat holding the meniscus. The analogous condition in a uniformly wet medium is the Haines criterion for drainage.

2. Generalized Haines imbibition: Two menisci merge, forming a single meniscus. The analogous condition in a uniformly wet medium is the Haines criterion for imbibition.

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3. Melrose event: A meniscus touches a pendular ring. The analogous condition in a uniformly wet medium is the Melrose criterion for imbibition.

4. Coalescence: Two pendular rings touch. The analogous condition in a uniformly wet medium is the Melrose criterion for coalescence of pendular rings.

5. Meniscus-4th sphere: A meniscus associated with three grains touches a fourth grain located in front of the meniscus. This is a new criterion.

Generalized Haines Drainage Criterion. Haines (1925) defines the drainage critical curvature of a throat as the curvature of the biggest sphere that can pass through the opening of the throat. This definition is drawn from the physical situation in a fully water wet medium (i.e. contact angles of the grains equal to zero): when the radius of the meniscus becomes smaller than the critical value, the meniscus can no longer maintain contact with the grain surfaces, and it spontaneously jumps through the throat. However, when the contact angle between meniscus and grains which define the throat is nonzero, the above rule cannot be applied. We develop a general rule for drainage critical curvature of a throat with arbitrary contact angle on each grain defining the throat.

For a throat composed of three grains with arbitrary contact angle, the drainage critical curvature is given by the curvature of the meniscus when the center of the meniscus (cf. point O in Fig. 3) is located on the throat plane (cf. plane O1O2O3 in Figs. 2 and 3). As curvature increases, the center of the meniscus approaches the throat plane. When the center of the meniscus reaches the throat plane, the throat cannot hold the meniscus after any further increment in curvature. Thus the curvature at which the meniscus center reaches the throat plane is identified as the critical curvature for passing through the throat. This criterion reduces to Haines criterion for drainage in the case that all three grains make contact angle of zero with the meniscus. Thus, we refer to our criterion as a generalized Haines drainage criterion. We implement this criterion with the analytical solution for the position and stability of a meniscus in a throat with arbitrary contact angles on each grain (Motealleh 2009).

Figure 6a shows the meniscus (yellow cap) located on a water-wet throat. The curvature of the meniscus is smaller than the drainage critical curvature for that throat. As a result, the center of meniscus stands above the throat plane O1O2O3. In Fig. 6b, the curvature of the meniscus equals the drainage critical curvature of the throat. Thus, the center of the meniscus (brown cap, below gray throat plane) is positioned on the plane O1O2O3.

In Fig. 7 the throat which holds the meniscus is fractionally wet. In Fig. 7a, the curvature of the meniscus is smaller than the drainage critical curvature for that throat. As a result, the center of the meniscus stands above the throat plane O1O2O3, which connects the centers of three grains. The O1, O3 are the centers of water-wet grains and O2 is center of the oil-wet grain. The water-wet grains are tangent to the water side of the meniscus, and the oil-wet grain is tangent to the oil side. In Fig. 7b, the curvature of the meniscus equals the drainage critical curvature. Thus, the center of the meniscus is positioned on the plane O1O2O3.

Generalized Haines Imbibition Criterion. Haines (1925) defines imbibition critical curvature of a pore as the curvature of the biggest sphere that can be held inside a pore. In the case of fully water-wet medium (i.e. contact angle equal to zero), the imbibition critical curvature of the pore is thus equal to curvature of the sphere inscribed in the pore. However, when the contact angle between menisci and grains which define the pore is nonzero, the above rule cannot be applied. Here, we develop a general rule for the imbibition critical curvature of a pore with arbitrary contact angle.

We define the imbibition critical curvature of a pore as the curvature of a meniscus that results from merging two menisci, each located on adjacent throats in a pore. Merger has a specific meaning here: two menisci merge when their centers are located at the same point. This criterion reduces to Haines criterion for imbibition when all four grains of the pore make zero contact angles with the menisci. Thus, we call this a generalized Haines imbibition criterion.

Figure 8 shows a pore composed of four grains (1, 2, 3, and 4). Two menisci (blue and red) are located in two adjacent throats (throat123 and throat234). The red meniscus sits on the throat123 and the blue meniscus sits on the throat234. All grains make zero contact angles with two menisci. During imbibition, as the curvature decreases, the centers of these two menisci approach each other. Hence the angle between plane A and plane B get smaller. Plane A connects the center of the red meniscus and the centers of grain2 and grain3. Plane B connects the center of the blue meniscus and the centers of grain2 and grain3. In Fig. 8 the blue meniscus and red meniscus have different centers. As a result, the angle between plane A and plane B is non zero. We regard this as a stable configuration, analogous to Figs. 6a and 7a.

As the curvature decreases, the centers of these two menisci get closer together, until the centers of the two menisci are located at the same point. When this occurs, plane A and plane B are the same. In other words, the angle between plane A and plane B equals zero, Fig. 9 (this form of the criterion is convenient computationally). Hence two menisci become the part of a single, spherical surface. Since in this example the contact angles are zero, this sphere is the inscribed sphere of the pore. The moment the menisci touch, they become unstable and merge into one meniscus. As a result, a generalized Haines imbibition event occurs and the pore will be filled with the invading phase.

The main difference between the criterion developed above and Haines' original criterion is that the latter simply allows a pore to imbibe if the curvature is smaller than the curvature of the inscribed sphere. This criterion can be computed a priori from the pore and grain geometry. It is independent of the number of menisci within the pore during displacement. In contrast, our generalized Haines imbibition criterion requires the merger of two menisci centers. Thus if a pore has only one meniscus, it will not imbibe even if the curvature of the meniscus becomes smaller than the curvature of inscribed sphere of the pore. This is simply because a second meniscus is not present, and there is not another meniscus center with which to be

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co-located. Thus a pore must have at least two menisci located in its throats before it can be imbibed. This is consistent with the a priori simulations of meniscus movement during imbibition in uniformly wet media (Prodanovic and Bryant 2006).

During drainage or imbibition, the existence of two menisci on two adjacent throats belonging to a given pore is contingent on the particular sequence of local filling events. Consequently, the generalized Haines criterion for imbibition cannot be calculated a priori to the drainage/imbibition simulation. (Lenormand et al. 1983) and (Jerauld and Salter 1990) also suggest that it is easier to imbibe a pore when the number of menisci within the pore is greater than one. However, they use ad hoc parameters to take into account the effect of the number of menisci within the pore on critical curvature for imbibing the pore. For instance, Jerauld and Salter suggest that the critical curvature (Ccrit) for imbibing a water-wet pore equals 2 b nwr z where rb is the inscribed radius of the pore and znw is the number of pore throats holding menisci. Instead, here we calculate the critical curvature explicitly from the menisci positions.

Melrose Criterion for Imbibition. A Melrose event occurs when two or more separated interfaces come into contact and merge to form a single interface (Melrose 1965). As discussed in previous section, when two menisci merge to a single meniscus, we call the event generalized Haines imbibition. Note that two menisci in a 3D pore are connected, e.g. by an interface where wetting phase is held at a grain-grain contact. In the example of Fig. 8, the red and blue menisci are connected by the interface in the crevice between grains2 and grain3. The connecting interface has a different shape (toroidal) from the menisci (spherical), but the situation does not correspond to the notion of merging two previously disconnected interfaces. Consequently, merging two (or more) menisci in 3D cannot properly be called Melrose event. Here we operate under the understanding that the only interfaces separated from others are pendular rings. As a result, when a pendular ring touches a meniscus, or when two pendular rings touch each other (coalescence in a pore throat), then a Melrose event occurs.

A Pendular Ring and a Meniscus Touch (Melrose Event). Figure 10 shows a schematic of a pore that holds a meniscus in one of its pore throats (throat234). In addition it holds a pendular ring in one of its edges (edge13) which is not associated with throat234. In Fig. 10a, the pendular ring and meniscus do not touch. As a result, both interfaces (meniscus and pendular ring) are stable. A Melrose event occurs when the meniscus touches the pendular ring, Fig. 10b. As a result, a pore will be filled with the invading phase and the critical curvature for imbibition of the pore equals to the curvature of the meniscus. For simplicity, we call this event a Melrose event. The mathematical implementation of Melrose criterion is developed with details in Motealleh (2009).

Two Pendular Rings Touch (Coalescence Event). As curvature decreases, pendular rings grow inside pores and at some curvature two adjacent pendular rings can touch each other. When this happens, the two pendular rings coalesce. This results in closure of the pore throat to the nonwetting phase. Figure 11 shows this process. If the summation of two pendular rings filling angles (ϕ1 + ϕ2) is bigger than the angle between two edges holding pendular rings (angle α), then two pendular rings touch each other and as a result the coalescences occur. For simplicity, we call this event a coalescences event. In the uniformly wet medium, the coalescences event can result in disconnecting of the non-wetting phase (snap off). As a result, this event called snap off in several literatures(Mohanty et al. 1987; Blunt et al. 1992).

Unlike the three previous described mechanisms, coalescence is not a pore filling event. Upon occurrence of coalescence only a pore throat will be filled by invading fluid. The filling creates menisci in the throat, which may participate in other pore-filling events.

A Meniscus-4th Grain Contact. We develop a mechanistic criterion for the encounter of a meniscus with a fourth grain not involved in the throat holding the meniscus. To our knowledge this criterion has not been previously considered in pore-scale modeling of fluid displacement. A Meniscus-4th grain event is not a pore filling event per se, but it leads to several events that result in a pore-filling event. For better understanding of this event (a meniscus 4th grain contact), we use an example.

Figure 12 shows a schematic of two pores (pore1234 and pore1256). Throat123 and Throat125 hold menisci which are colored red and a dark blue, respectively. A light blue interface connects these two menisci. All grains are water-wet and they make contact angles of 30o with the menisci. During imbibition, the menisci advance farther into pore1234 and pore1256 as the curvature decreases. As a result of menisci movement, the dark blue meniscus in throat123 eventually touches the fourth grain located in front of the meniscus (grain4). This grain (grain4) is one of four grains which construct pore1234, but this grain does not belong to the throat123.

Figure 13 shows the evolution of the fluid configuration of Fig. 12 when the curvature is small enough for the meniscus (dark blue) to touch the 4th grain (grain4). When this happens, the water phase which is located below the meniscus becomes in contact with water-wet grain4. Consequently, the water phase rises along the surface of grain4. As a result, two pendular rings form on edge14 and edge24 (Fig. 14). To generalize, when the meniscus touches a fourth grain (e.g. grain4) near an edge joining two grains (e.g. grain1 and grain2), all three grains (e.g. grain1, grain2 and grain4) must have the same wettability in order for the invading phase to rise on the touched grain.

The pendular rings which form on edge14 and edge24 are touched by the dark blue meniscus (Fig. 14b). Consequently, Melrose events occur, and pore1234 will be filled with the water phase (Fig. 15).

In summary, when a meniscus touches a nearby grain that is not one of the three grains holding the meniscus (4th grain), but has the same wettability as at least two grains holding the meniscus, then the invading fluid rises on the touched grain and forms two pendular rings on the edges composed of the touched grain and two other grains. Subsequently, the formed pendular rings touch the original meniscus, and a Melrose event occurs, causing the pore to be filled with invading fluid. A

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grain can touch a meniscus during imbibition or drainage process. In the dense random packings of spheres studied here, this event occurs only when the absolute value of normalized curvature is small (approximately less than 1.5).

Simulation We simulate the menisci motion within a fractionally wet porous medium (composed of 7000 equal size grains) during drainage and imbibition process using the above criteria and an invasion percolation algorithm. The motion is actually a sequence of equilibrium configurations: We change curvature in small increments then compute the stable meniscus locations on all grains. It is convenient to organize the grains at the beginning of the simulation into groups that define pore throats and pores. We use Delaunay tessellation of grain centers for this purpose. At any given curvature (capillary pressure) all pores that are connected to the inlet (via invading fluid) will be members of a list of candidates for invasion. We check the candidate pores for each grain-based criterion of invasion defined above. If a candidate meets any of the criteria of invasion and it also connected to the outlet, we invade the candidate and add the neighboring pores to list of candidate. We continue until no more candidates remain in the list. Candidates that are not connected to the outlet are labeled “trapped” and removed from further consideration of events. Then we increase or decrease the curvature incrementally and repeat the procedure.

We emphasize that by checking generalized Haines and Melrose criteria developed in previous section for all grains, the algorithm generalizes the simulation of drainage and imbibition. Indeed, the only difference inside the code between drainage and imbibition is the sign of the increment in curvature. In contrast, traditional pore network models typically have algorithms specific to drainage and others specific to imbibition; only the machinery of invasion percolation is common to both.

Results In this section, we focus on the simulation results for drainage and imbibition for a fractionally wet porous medium. In the next section, we compare the simulation results for fractionally wet media with experimental data. Figure 16 shows the imbibition and drainage curves for a porous medium with 50% of its grains oil-wet.

Initially the porous medium was filled with the oil phase (Fig. 16, point P0), so the first process is primary imbibition where the water phase displaces oil until reaching residual oil phase saturation (imbibition end point P1). At the imbibition end point, the residual oil phase exists in two morphologies: the oil phase trapped inside the pores and the oil phase in the form of pendular rings at grain-grain contact of two oil-wet grains. The calculation for water saturation reported in Fig. 16 does not account for the volume of the oil phase held by oil-wet grains nor the volume of the water phase held by water-wet grains, as both are small. The drainage process starts from imbibition endpoint point P1) with increasing applied curvature (capillary pressure) and decreasing water saturation. Finally the secondary imbibition starts from drainage endpoint (point P2) and imbibe to porous medium to new value of residual oil phase saturation (point P3). The model allows seamless simulation of interface motion and filling events even through the zero curvature level. All menisci are tested for all pore filling events depending on their local circumstances. Thus the critical curvature for local events is dynamic and cannot be pre-calculated. This is unlike traditional invasion percolation models.

This gives the model an inherent robustness in the sense that all grains, throats, and pore bodies are considered equivalent, just with a different contact angle for each grain. This robustness can be seen by considering a 25% oil-wet medium, and the equivalent 75% oil-wet medium in which all of the oil-wet grains are made water-wet and vice-versa. Reversing the wettability of each grain should also reverse the capillary pressure curves. Figure 17a shows the 25% oil-wet capillary pressure curve obtained with the model, and Fig. 17b shows the 75% oil-wet capillary pressure curve obtained with the same model. The symmetry is evident, as the drainage in the 25% oil-wet medium is identical to the imbibition in the 75% water-wet medium. This symmetry is only seen if each grain’s wettability is reversed. If the positions of the grains are different, or the pattern of the oil-wet grains is different, the model will show slightly different capillary pressure curves.

We can now use the model to study how the drainage and imbibition curves vary with the change of fraction of oil-wet grains. Figure 18 shows the primary drainage curves for porous media with different fraction of oil wet grains. The oil-wet grains have a contact angle of 150° and the water-wet grains have a contact angle of 30°. For simplicity, we ignore the trapping of the water phase for the results presented in the Fig. 18. As a result, the oil phase is allowed to displace the water phase even though the water phase is not connected to exit pores. In addition, we disregard the volume of fluids in pendular rings for simplicity. Consequently, the curves in Fig. 18 show no residual saturation for the water phase.

The drainage curves move to lower capillary pressures (curvatures) with increasing fraction of oil-wet grains. Labeling the capillary pressure at which the saturation is equal to 50% as the percolation capillary pressure, we observe that the percolation capillary pressure is not a linear function of fraction of oil-wet grains. The largest change in the position of the drainage curve occurs from fully water-wet case to 25% oil-wet case causes a large change in the percolation capillary pressure, while going from 75% to 100% produces a much smaller change. All percolation capillary pressures for drainage are at positive capillary pressure except for the 100% oil-wet case.

Figure 19 shows the model results for the secondary imbibition curves for porous media with different fraction of oil-wet grains. The imbibition curves move to lower capillary pressures (curvatures) with increasing oil-wet fraction in the media. Again, the change in the percolation capillary pressures is not linear with oil-wet fraction, with the biggest changes now when the oil-wet fraction goes from 75% to 100%. Comparing Fig. 19 with Fig. 18 shows that there is hysteresis between

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imbibition curve and drainage curve of fractionally wet media. The imbibition curves always lie below the corresponding drainage curves. In the discussion section, we elaborate more on the reason for this observed hysteresis.

Experiments We performed laboratory experiments to obtain octane/water pressure-saturation curves for primary drainage and primary imbibition into fractionally wet sand packs. A water-wet sand (grain size d50 = 0.35 mm) was used as water-wet grains; the oil-wet grains were made by tumbling the same clean sand in a 5% OTS (octatrichlorosilane)-in-ethanol solution for 5 hours. After tumbling, the sand grains were rinsed in ethanol five or six times to remove the excess OTS, followed by air-drying the sand. Fractionally wet packs were made by mixing prescribed proportions of the water-wet and oil-wet grains.

The experiments were performed in 60 cm long columns (2.54 cm inner diameter), which were filled with the fractionally wet sand. The columns consisted of separate 3 cm and 1 cm long sections of polycarbonate tubing that were held together with shrink tubing. For drainage, the column was filled from below with water and flushed for 1 hour to remove entrapped air. Primary drainage was then performed by attaching a constant head tank to the outlet at the bottom of the column and octane was allowed to enter the top by attaching the top of the column to an oil tank of known height. Primary imbibition starts with an oil-filled column, and the attachment of the constant head tank at the bottom of the column. For imbibition, the height of the constant tank was chosen such that the water would rise roughly halfway through the column. Each column was allowed to equilibrate for 3 weeks. After this, the capillary pressure in each section was obtained assuming capillary gravity equilibrium. The water saturation was obtained by dissolving the contents of each section in ethanol and running the solution through a gas chromatograph.

Model Validation Drainage. Figure 20a shows the measured P-S curves for primary drainage (octane/water) for 5 different sands of varying oil-wet fraction. The measured drainage curves shift monotonically toward negative P with increasing oil-wet fraction. We simulate the primary drainage process for a media with the same fraction of oil wet grains as experimental data (e.g. 0, 25%, 50%, etc.). We multiply the normalized curvature by the known interfacial tension and divide by grains radii to obtain the capillary pressure in Pascal. Figure 20b shows the simulation results using contact angles for water-wet and oil-wet grains of θww= 0o and θοw= 120o, respectively.

The similarity of the trend with increasing oil-wet fraction between experimental data and simulation results is very good. The capillary pressure corresponding to the percolation threshold decreases monotonically in both figures (Figs. 20a and 20b). The percolation capillary pressure shows roughly the same non-linearity with the oil-wet fraction for both the simulation and experiment. In both experimental data and simulation result, the transition capillary pressure for all of drainage curves (except 100% oil-wet) are located above zero capillary pressure. In the experiment, part of the drainage curve for the fully oil-wet (100% oil-wet) medium lies below zero capillary pressure. However, in simulation results, only a small portion of the drainage curves for the fully oil-wet (100% oil-wet) medium is below the zero capillary pressure.

Imbibition. Figure 21a shows the measured P-S curves for primary imbibition (octane/water) for 5 different sands of varying oil-wet fraction. The imbibition curves move monotonically toward negative Pc with increasing oil-wet fraction. Figure 21b shows the results of simulating the imbibition process with the same fraction of oil-wet grains as experimental data, using the same normalization of curvature and the same contact angles as for drainage.

In the imbibition curves, Fig. 21a (experiment) and Fig. 21b (model), the percolation capillary pressure decreases monotonically with increasing oil-wet fraction in both figures. The percolation threshold changes nonlinearly with fraction of oil-wet grains in both experimental data and simulation results

Despite the overall match between experimental data and simulation results, several trends in the data are not captured by the simulation. For instance, in experimental data (Fig. 21a), the entire imbibition curve for 50% oil-wet fraction is positioned at negative capillary pressure. However, in simulation results (Fig. 21b), half of the imbibition curve for 50% oil-wet fraction located above zero capillary pressure line. In addition, in experimental data, the position of imbibition curve for fully oil-wet case is slightly lower than the position of imbibition curve for 75% oil-wet case. In contrast, in simulation results, the position of imbibition curve for fully oil-wet case is significantly lower compared to the position of imbibition curve for 75% oil-wet case.

Discussion A valuable feature of the mechanistic model is the opportunity to understand which pore filling processes dominate as a function of wettability and capillary pressure. In this section, we use the model to delineate the processes, and also study the effect of wettability on the fluid configuration during drainage and imbibition. In this discussion, we first analyze the effect of individual pore or throat filling event on drainage and imbibition of a fully water-wet medium. Later, we discuss the effect of individual pore filing event on drainage and imbibition of a fractionally-wet medium.

Effect of Each Event Type on Drainage/Imbibition Process Within a Water-Wet Medium. The Haines drainage event is the only pore filling event that occurs during the drainage process within a water-wet medium. However, several different events (i.e. Melrose, generalized Haines imbibition, coalescence, meniscus-4th grain) could occur during imbibition within a

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water-wet medium. In this section we study the effect of each event on the imbibition process of a fully water-wet medium by turning on and off each type of event in the simulation. For the following simulations, we disregard trapping and the volume of fluids in pendular rings for simplicity. However, we take into account the existence of pendular rings as irreducible water phase at the end of drainage process. The existence of pendular rings enables coalescence and Melrose events and thus affects the fluid displacement within porous media.

Effect of Melrose Events Versus Generalized Haines Imbibition Events in Water-Wet Media. We start the imbibition simulation with water existing only in pendular rings in a water-wet medium (secondary imbibition). First we turn on only the generalized Haines imbibition mechanism and simulate the imbibition process. The resulting P-S curve is the light blue line in Fig 22. Clearly the water phase does not percolate at positive capillary pressure, and thus little imbibition takes place from this initial state by means of generalized Haines imbibition events. This is a consequence of the criterion's requirement that (at least) two menisci must be present to merge. Since very few pores have two menisci initially, and since each imbibition event is likely to reduce the total number of menisci, the process is self-limiting.

Next we turn off the generalized Haines imbibition criterion and turn on the Melrose imbibition mechanism. The result is the red P-S curve in Fig. 22. Now the water phase does percolate through the water-wet porous medium at positive capillary pressures. Clearly Melrose events can be self-sustaining: menisci can "bootstrap" their way through the medium by touching successive pendular rings that are distributed throughout the medium.

Subsequently, we turn on both mechanisms. The percolation capillary pressure is even higher (green curve). The Melrose events increase the number of pores with at least two menisci located in their pore throats. Thus during imbibition in a water wet domain containing pendular rings, Melrose events are essential for enabling subsequent generalized Haines imbibition events. Finally, we also turn on the meniscus-4th grain events, and we observe that these events do not occur as they are superseded by the Melrose events, and thus do not change the P-S curve for secondary imbibition.

Figure 23 plots the number of total events and Melrose events at each step of curvature during imbibition. Both Melrose and generalized Haines imbibition are allowed to occur. The number of generalized Haines imbibition events is the difference between total events and Melrose events. The occurrence of the Melrose events is clearly necessary to initiate and sustain the chain of events which results in percolating the water phase through the porous medium. Nevertheless more than 20% of the events are generalized Haines imbibition events. When the greatest number of events occurs (Sw=0.4), more than 30% of them are generalized Haines imbibition events. Though the generalized Haines imbibition events do not dominate during imbibition of a water wet medium, their occurrence affects the imbibition process and thus cannot be ignored in any modeling attempt.

Effect of Melrose Events Versus Meniscus-4th Grain Events. Since the existence of pendular rings is necessary for the occurrence of Melrose event, the above results show that the existence (or absence) of pendular rings can greatly alter the imbibition process within a water-wet medium. However, at the beginning of the primary imbibition, the medium is completely filled with the oil (or gas) phase and no water phase pendular ring exists. In the absence of Melrose events, the possible events are generalized Haines imbibition and meniscus-4th grain events. Similar to the Melrose events, the menisci-4th grain events initiate a chain event results in percolating the water phase through the porous medium. However, the menisci-4th grain events do not occur until the applied curvature decreases to a small value (approximately less than 1.5). Figure 24 compares the P-S curve for primary imbibition (purple curve) and secondary imbibition (green curve). The primary imbibition curve percolates at a much lower curvature than the secondary imbibition curve. As seen earlier, the primary events occur in secondary imbibition are Melrose events. For primary imbibition, meniscus-4th grain events are necessary for percolation, but these occur at a lower capillary pressure than Melrose events. The difference between the primary and secondary imbibition curves in Fig. 24 show the importance of the pendular rings and its role during imbibition process.

Note that for the purpose of comparison between experimental data and simulation results for fractionally wet media, we assume that the pendular ring exist in front of the advancing menisci, even though the experimental data was measured for primary imbibition. The moisture within the porous media allows the formation of thin films. These thin films can carry small amount of water through roughness of the grains surfaces and build pendular rings in front of the advancing menisci. As a result, Melrose events can occur during primary imbibition experiments when the grains are rough.

Effect of Coalescence Events. Another type of event that can occur during imbibition process within a water-wet medium is the pendular ring coalescence. Motealleh (2009) reports that coalescence does not affect the position of P-S curve (i.e. percolation capillary pressure), but only causes a small increase in the amount of trapped nonwetting phase. The occurrence of pendular rings coalescence during drainage and imbibition within fractionally wet media is less when compared to the water-wet case. In fractionally wet media, not all edges can hold a pendular ring, because some edges join two grains with different wettability.

Effect of Meniscus-4th Grain Criterion on the Primary Drainage Curves of Fractionally Wet Media. The meniscus-4th grain criterion was not considered for fluid displacement within porous media by other researchers. Figure 25 compares the primary drainage simulation results without considering this criterion (Fig. 25a) with the primary drainage simulation results including this criterion (Fig. 25b). Figure 25a shows that primary drainage curves for the fully oil-wet medium, the medium with 75% oil-wet grains, and the medium with 50% oil-wet grains are positioned at the same curvature. In other word, the curvature corresponding to percolation threshold of their primary drainage curves are the same.

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During drainage of a fractionally wet medium, the oil phase invades to the porous medium and when a meniscus touches an oil-wet grain, the oil phase climb through the rough surface of the grain and creates pendular rings on the edges. This event results in pore-filling events. However, if the meniscus touches a water-wet grain, the oil phase does not climb the grain and does not create pendular rings. Consequently, no pore filling event occurs. As a result, the effect of meniscus-4th grain events on primary drainage curve decreases as the fraction of oil-wet grains decreases. We can observe this trend in Fig. 25. The position of primary drainage curve for the fully-water-wet medium, the medium with 25% oil-wet grains, and the medium with 50% oil-wet grains are the same in both Figs. 25a and 25b. The position of primary drainage curve for the fully oil-wet medium and the medium with 75% oil-wet grains are different in Figs. 25a and 25b.

During the primary drainage of fully oil-wet medium, the oil phase percolates at negative curvature if the meniscus-4th grain criterion is considered (Fig. 25b), while the oil phase does not percolate until the curvature increase to positive value when this criterion is not considered (Fig. 25a). The primary imbibition of the fully water-wet medium is mirror image of primary drainage of the fully oil-wet medium. Therefore, the meniscus-4th grain criterion is important to be considered for simulation of primary imbibition of the fully water-wet medium. Without its consideration, the water phase does not percolate through the medium at positive capillary pressure.

Compare the Occurrence of Each Event During Drainage/Imbibition Process Within Fractionally Wet Medium. Unlike in an uniformly wet medium, all pore filling events can occur during both drainage and imbibition of a fractionally wet medium. Here we study the fractions of events that occur as a function of saturation in fractionally-wet media. In these simulations we turn on all the possible events and record which events take place at which saturation.

Figure 26a shows the secondary drainage P-S curve for a medium with 25% oil-wet grains, and Fig 26b shows the fraction of each event during drainage. Following the drainage process from high to low saturation, we observe that at high saturations the majority of events occur during drainage of the medium with 25% oil-wet grains are generalized Haines drainage events (in a water-wet medium they are all Haines drainage events), with a smaller amount of generalized Haines imbibition and meniscus-4th grain events. This can be understood in terms of the wettability of the grains that combine to make the pores and throats. Here, the majority of pore throats are either fully water-wet (i.e. they have three water-wet grains) or have only one oil-wet grain. Since the generalized Haines drainage events occur on these pore throats at positive capillary pressure, hardly any generalized Haines drainage event occurs at negative capillary pressure. In addition, there are very few oil-wet edges that can hold the oil phase pendular rings. Consequently, no Melrose events occur at negative capillary pressure. As a result, the water phase saturation of the medium remains 100% at negative curvature. At small positive curvatures (less than 2), the generalized Haines drainage occurs on the fractionally wet throats (i.e. they have one or two oil-wet grains). These pore filling events establish enough pores with at least two menisci located in their pore throats for generalized Haines imbibition to occur subsequently. Occurrence of generalized Haines drainage and generalized Haines imbibition events results in percolating the oil phase through the porous medium at the curvature of 3. Above a curvature of 3 (and saturation of 0.4), primarily water-wet throats are left, which are only emptied by generalized Haines drainage events.

Figure 26c shows the drainage P-S curve for a medium with 75% oil-wet grains, and Fig 26d shows the fraction of each event during drainage. Again working downward in saturation, primarily Melrose events occur at high saturation (and negative capillary pressure), with a smaller amount of meniscus-4th grain and generalized Haines imbibition. At zero curvature, these events have caused the saturation of the water phase to be reduced to 0.6. At positive curvature, the Melrose events cease, and the main events are generalized Haines drainage and meniscus 4th-grain. In terms of the grain scale positioning, for a 75% oil-wet media, the majority of edges are oil-wet edges which can hold the oil phase pendular rings. Consequently, Melrose events can occur at negative capillary pressure. These pore filling events enable subsequent generalized Haines imbibition events. Occurrence of Melrose and generalized Haines imbibition events results in establish of enough pores with menisci located in their pore throats. At a small absolute value of curvature (|C|<1) some of these menisci touch 4th grains (meniscus-4th grain event) which results in more pore filling events. At curvature bigger than one, the main events are generalized Haines drainage events. These events occur on fractionally wet throats which were not invaded yet. In the medium with 75% oil-wet grains, there are not so many water-wet throats. Consequently, all throats invaded by the oil phase at the curvature smaller than 3. The curvature 3 corresponds to smallest critical curvature for occurrence of Haines drainage in a fully water-wet dense disordered packing of equal spheres (Mellor 1989; Motealleh 2009). The main events occur from Sw = 1 to Sw = 0.7 are Melrose and generalized Haines imbibition events. As a result, the shape of drainage curve from Sw = 1 to Sw = 0.7 are similar to shape of imbibition of a fully water-wet medium.

Figure 27 shows secondary imbibition into 25% and 75% fractionally-wet media. Here we observe basically a complete inverse from the secondary drainage into these media, with the 25% oil-wet switching with the 75% oil-wet. For the 25% oil-wet we see primarily Melrose events until negative curvature, where Haines drainage events take over. For the 75% oil-wet we see mainly generalized Haines events at negative curvatures. This is very reassuring and shows the robustness and the equal footing that all events are considered within the model.

The shape of imbibition curve for the medium with 25% oil-wet grains is similar to the shape of drainage curve for the medium with 75% oil-wet grains. The type of events occurs during these two different process is similar. In contrast, the shape of and quantitative values for the drainage curve and imbibition curve for the medium with 25% oil-wet grains are not similar. We argue that the occurrence of different type of pore filling events during drainage and during imbibition is the cause of the hysteresis between P-S curves. That is, the nature of a P-S curve is not a function of type of fluid displacement

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(imbibition or drainage), rather it is function of what type of pore filling events occur during the fluid displacement. For example, during the last half of drainage in the 25% oil-wet pack (Fig 26) only generalized Haines drainage events occur. During the last half of imbibition in the 75% oil-wet pack (Fig 27) only generalized Haines drainage events occur. Consequently, these portion of drainage and imbibition curves look like each other.

In our model drainage and imbibition of uniformly wet medium (fully water-wet or fully oil-wet) are special case of fluid displacement within fractionally wet media. Our model of fluid displacement allow all type of events occur during drainage/imbibition of a fully water-wet medium. However, the only event that occurs during drainage of a fully water-wet medium is the generalized Haines drainage event and the main events that occur during imbibition of a fully water-wet medium are the generalized Haines imbibition and Melrose events. The criteria for occurrence of these event were previously used by other researchers in order to model the drainage (Mason and Mellor 1995) and imbibition (Gladkikh and Bryant 2005)of a fully water-wet medium.

Effect of Wettability on Trapped Phase Saturation. In this section, we study the effect of wettability on residual phase saturation. The residual phase saturation exists in two morphologies: volumes of phase trapped inside pores and volumes trapped as isolated pendular rings. The saturation of the latter morphology (pendular rings) is small, and we neglect it here. Figure 28 shows the primary drainage curves for porous media with different fraction of oil wet grains. The oil-wet grains have a contact angle of 150° and the water-wet grains have a contact angle of 30o. The curves can be compared to those of Fig. 18, where trapping of the water phase was neglected.

Initially, the residual water saturation decreases when the fraction of oil-wet grains increases from 0% to 25%. Further increase in fraction of oil-wet grains (i.e. from 25% to 100%) increases the residual water saturation.

The value of the residual water phase saturation for a medium with specific fraction of oil-wet grains is a range, not an exact number. The primary drainage curves shown in Fig. 28 are corresponding to only one realization of each fractionally wet medium. The position of the oil-wet grains affects the residual saturation of water phase but not the position of corresponding drainage curves. The discussion regarding the effect of position of oil-wet grains on the position of P-S curves (value of percolation capillary pressure) is out of scope of this study.

Figure 29 compares the residual water phase saturation at the end of primary drainage for porous media with different fraction of oil-wet grains. The change in the value of residual water phase saturation with increase of fraction of oil-wet grains from 0 to 70% is small. This change is even smaller than the error bar associated with each point ‘O’ in Fig. 29. However, at high fraction of oil-wet grains (above 75%) the residual water saturation increase rapidly with increase of fraction of oil-wet grain. In addition, the residual water phase saturation shows significant sensitivity to the position of oil-wet grains at high fraction of oil-wet grains. As a result, the error bar associated with 75% and 82% oil-wet grains are bigger compare to other points ‘O’ in Fig. 29.

As we mention before, the imbibition of a fractionally wet medium is mirror image of drainage of another fractionally wet medium in which all of the oil-wet grains are made water-wet and vice-versa. Hence, we do not need to run the simulation for primary imbibition of fractionally wet medium in order to make prediction regarding the residual oil phase saturation at the end of primary imbibition. We can use the results presents in Fig. 29 to predict the value of residual oil phase saturation at the end of primary imbibition. For instance, the values of residual oil phase saturation at the end of primary imbibition of a fully water-wet medium equals to the value of residual water phase saturation at the end of primary drainage of a fully oil-wet medium (value of 0.26 in Fig. 29).

Conclusion We develop a grain-based, mechanistic model for oil/water displacement under capillary control in fractionally wet media. This is done by generalize the grain-based criteria previously developed for fully water-wet media (i.e. Haines and Melrose criteria) to be used for fractionally wet media. A novel criterion (i.e. meniscus-4th grain criterion) is discovered for instability of a meniscus when the meniscus touches a grain located in front of it. We implement and illustrate the model in dense disordered packings of spheres, solving analytically for stable configuration of the meniscus held between grains of arbitrary and unequal contact angle. Using the model, we carry out drainage and imbibition simulations that yield a priori predictions of the grain-scale configurations of water/oil interface (menisci) within the fractionally wet porous medium. The predictions of macroscopic properties (e.g. P-S curve for imbibition and drainage) agree semi-quantitatively with experimental data. The experimental data and simulation result show a non-linear behavior between the position of P-S curves and fraction of oil-wet grains.

The model allows an understanding and prediction of how the pore filling processes change as a function of the initial wetting state, the wettability of each grain and the fraction of each type of grain. We find for a water-wet medium, the difference between P-S curves of primary and secondary imbibition is due to existence of pendular rings during secondary imbibition process allows Melrose events to take place. In contrast, for primary imbibition, the meniscus-4th grain events are necessary to initiate imbibition, causing the imbibition to take place at a lower capillary pressure.

For fractional wettability, the grain based model of displacement has no criteria developed specifically for drainage or for imbibition. Instead we developed generalized grain-based criteria for five types of pore/throat filling events. As a result, our model unifies the simulation of drainage and imbibition processes. In this approach, the familiar hysteresis of P-S curves between drainage and imbibition processes in uniformly wetted media is due to difference in the type of pore/throat filling

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events that occur during each part of the cycle. The only event that occurs during drainage of a fully water-wet medium is the generalized Haines drainage event. The main events that occur during imbibition of a fully water-wet medium are the generalized Haines imbibition and Melrose events. In contrast, all types of pore/throat filling events can occur during drainage and imbibition of fractionally wet media. Consequently, the unified grain-based model is the only mechanistic way to simulate capillarity-controlled displacement (i.e. drainage and imbibition) in fractionally wet media.

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Morrow, N. R. (1990). "Wettability and Its Effect on Oil-Recovery." Journal of Petroleum Technology 42(12): 1476-1484. Motealleh, S. (2009). Mechanistic study of menisci motion within homogeneously and heterogeneously wet porous media. Petroleum

engineering, The university of Texas at Austin. PhD. O'Carroll, D. M., L. M. Abriola, C. A. Polityka, S. A. Bradford and A. H. Demond (2005). "Prediction of two-phase capillary pressure-

saturation relationships in fractional wettability systems." Journal of Contaminant Hydrology 77(4): 247-270. O'Rourke, J. (1998). Computational Geometry in C. Cambridge, Cambridge University Press. Patzek, T. W. (2001). "Verification of a complete pore network simulator of drainage and imbibition " Spe Journal 6(3): 251-251. Piri, M. and M. J. Blunt (2002). Pore-scale modeling of three-phase flow in mixed-wet systems. SPE Annual Technical Conference and

Exhibition, San Antonio, Texas. Prodanovic, M. and S. L. Bryant (2006). "A level set method for determining critical curvatures for drainage and imbibition." Journal of

Colloid and Interface Science 304(2): 442-458. Salathiel, R. A. (1973). "Oil Recovery by Surface Film Drainage in Mixed-Wettability Rocks." Journal of Petroleum Technology 155:

1216-1224. Sharma, M. M., A. Garough and H. F. Dunlap (1991). "Effects of Wettability, Pore Geometry, and Stress on Electrical Conduction in Fluid

Saturated Rocks." Log Analyst 32: 511-526. Singhal, A. K. and W. H. Somerton (1970). "Two-Phase Flow Through a Non-Circular Capillary at Low Reynolds Number." Journal of

Canadian Petroleum Technology July-Sept.: 197-205. Thane, C. (2006). Geometry and Topology of Model Sediments and Their Influence on Sediment Properties. Petroleum Engineering.

Austin, The University of Texas at Austin. MS. Tsakiroglou, C. D. and M. Fleury (1999). "Resistivity index of fractional wettability porous media." Journal of Petroleum Science and

Engineering 22(4): 253-274. Ustohal, P., F. Stauffer and T. Dracos (1998). "Measurement and modeling of hydraulic characteristics of unsaturated porous media with

mixed wettability." Journal of Contaminant Hydrology 33(1-2): 5-37. Valvatne, P. H. and M. J. Blunt (2004). "Predictive pore-scale modeling of two-phase flow in mixed wet media." Water Resources

Research 40(7). van Dijke, M. I. J. and K. S. Sorbie (2003). "Pore-scale modelling of three-phase flow in mixed-wet porous media: multiple displacement

chains." Journal of Petroleum Science and Engineering 39(3-4): 201-216. Van Dijke, M. I. J., K. S. Sorbie and S. R. McDougall (2000). A Process-Based Approach for Three-Phase Capillary Pressure and Relative

Permeability Relationships in Mixed-Wet Systems. SPE/DOE Improved Oil Recovery Symposium, Tulsa, OK. Wilkinson, D. and J. F. Willemsen (1983). "Invasion Percolation - A New Form of Percolation Theory." Journal of Physics a-Mathematical

and General 16(14): 3365-3376.

Water-wet surface

Oil-wet surface

Pore space Grain

Water-wet surface

Oil-wet surface

Pore space Grain

Water-wet surface

Oil-wet surface

Water-wet surface

Oil-wet surface

Pore space Grain

Fig. 1—Schematic of fractionally wetted porous medium viewed at the grain scale. Each grain is either entirely water--wet or entirely oil-wet.

SPE 124771 13

O2

O4

O3

O1

Fig. 2—Schematic of one pore (defined as one tetrahedral cell in the Delaunay tessellation of sphere centers) in a dense disordered packing of equal spheres.

O2

O3

O1

O

Fig. 3—The schematic of a meniscus (shiny triangular patch) located on three grains (spheres) with zero contact angles (fully water wet). O indicates the center of the meniscus and O1, O2, and O3 indicate centers of spheres constructing the throat. The meniscus is the indicated part of a cap of the yellow wiremesh sphere, restricted to the interior of the tetrahedron OO1O2O3.

O2

O3

O1

P2

ψ2

Fig. 4—A meniscus like that shown in Fig. 3 forms a filling angle (ψ2) on grain2 between line2 2O P and plane 1 2 3O O O . Point P2 is the

point where the meniscus touches the grain2.

14 SPE 124771

O2O1

ϕ1 ϕ2

P1P2

Fig. 5—Schematic of a contact between two water-wet grains (denoted a "water-wet edge") holding a pendular ring of the water phase. O1 and O2 indicate centers of grains constructing the edge. Points P1 and P2 are the points where the pendular ring touches the grain1 and grain2 respectively. The contact angle between the water phase and grains is zero. ϕ1 and ϕ 2 are the filling angles of the pendular ring on the grain1 and grain2 respectively.

O

O1 O2

O3

OO1 O2

O3

(a) (b)

Fig. 6—Illustration of generalized Haines drainage criterion for case of water-wet grains (zero contact angles). The position of a meniscus (yellow cap in (a), brown hemisphere in (b)) of prescribed curvature less than (a) and equal to (b) critical curvature is shown. (b) The criterion for drainage is that the center of the meniscus lie on (or below) the throat plane.

O

O1O2

O3 O

O1O2

O3

OO1

O2

O3

(a) (b)

Fig. 7—Illustration of generalized Haines drainage criterion for case of fractional wet throat where O1 and O3 is the center of water-wet grains (zero contact angle) and O2 is the center of oil-wet grain (contact angle=180o). The position of a meniscus (light brown cap in (a), brown hemisphere in (b)) of prescribed curvature less than (a) and equal to (b) critical curvature is shown. (b) The criterion for generalized Haines drainage is when the center of the menisci is on the throat plane even for arbitrary wettability.

SPE 124771 15

1

2

3

4

Plane A Plane B

1

2

3

4

1

2

3

4

Plane A Plane B

Fig. 8—Schematic of two menisci located in two adjacent throats. The red meniscus sits in throat123, the blue in throat234. All grains make zero contact angles with the two menisci. The menisci are connected by a wedge of wetting phase between grains 2 and 3. While clearly part of the same larger fluid/fluid interface in the pore, the menisci are not merged according to our definition.

Plane A Plane B=Plane A Plane B=Plane A Plane B=

1

23

4

Plane A Plane B=

1

23

4

1

23

4

Plane A Plane B=Plane A Plane B=

(a) (b)

Fig. 9—(a) Schematic of merging of two menisci located on two adjacent throats. The red meniscus sit on the throat123 and the blue meniscus sit on the throat234. Throat123 and throat234 share an edge composed of grain2 and grain3. b) When plane A and plane B are the same (the angle between plane A and plane B equals zero), we identify the generalized Haines imbibition criterion as the curvature of merged meniscus.

1 4

2

3

1 4

2

3

1 4

2

3

1 4

2

3

(a) (b)

Fig. 10—(a) A pendular ring (red) and a meniscus (blue) do not touch. Both interfaces are stable. b) When the pendular ring touches the meniscus, a Melrose event occurs. The curvature at which they touch is thus the critical curvature for this event.

16 SPE 124771

1

2

31

2

31

2

31

2

31

2

31

2

31

2

3

ϕ1αϕ2

1

2

31

2

31

2

31

2

3

ϕ1

αϕ2

1

2

31

2

31

2

31

2

3

(a) (b) (c)

Fig. 11—The stages of coalescence of pendular rings associated with a pore throat. a) Curvature of pendular rings is larger than the critical curvature for coalescence. Summation of pendular rings filling angles (ϕ1 and ϕ2) is smaller than the angle between the two edges holding pendular rings (angle α). b) Curvature of pendular rings equals the critical curvature for coalescence. Summation of pendular rings filling angles (ϕ1 and ϕ2) equals angle α. c) Coalescence has occurred, and the pore throat is filled by the wetting phase. This creates a meniscus in the throat.

1

2

3

4

5

6

Fig. 12—The fluid configuration within two pores (pore1234, and pore1256), before a meniscus–4th grain contact during imbibition. Throat123 and Throat125 hold menisci which are colored dark blue and red, respectively. A light blue interface connects the two menisci. All grains are water-wet and they make contact angles of 30o with the menisci. The water phase is located below the menisci, and the oil phase is located above. The 4th grain of pore1234 (grain4) which comes in contact with the meniscus in throat123 is colored green.

2

3

4

5

6

Fig. 13—Reducing the curvature in Fig. 12 leads to this fluid configuration (different view point than Fig. 12) in which the meniscus (dark blue) touches the green fourth grain (grain4). Grain1 has been removed expose the region of meniscus and fourth grain contact (the region is highlighted by red circle).

SPE 124771 17

(a) (b)

1

2

3

5

6

1

2

3

4

5

6

Fig. 14—The fluid configuration within two pores (pore1234, and pore1256), after the meniscus (dark blue) touches the 4th grain (grain4) (cf. Fig. 13). a) The water phase rises on the surface of grain4, and as a result two pendular rings are formed on the edge14 and edge24. b) Grain4 is removed in order to reveal rings (light blue) and meniscus (dark blue) contacts.

1

2

3

4

5

6

Fig. 15—The fluid configuration within two pores (pore1234, and pore1256), after the meniscus touches the 4th grain (grain4) (final stage). Three new menisci (dark blue) are formed on the throat correspond to the pore1234 (throat124, throat134, and throat243). The light blue interface is not a pendular ring, but a surface that connects adjacent menisci.

P0 P2

P1P3

Primary Imbibition

Secondary Imbibition

Drainage

Fig. 16—Model results for the primary imbibition and secondary drainage and imbibition curves for a porous medium with 50% of grains oil-wet. In all three curves, the curvature ranges from positive to negative values and thus passes through zero curvature. All three curves produced from the same code applying the same criteria for pore-filling events.

18 SPE 124771

PDPI

SD

D

I

SI

25 % oil-wet grains

75 % oil-wet grains

(a) (b)

Fig. 17—(a) The primary drainage (PD), imbibition (I) and secondary drainage (SD) curves for a porous medium with 25% oil wet grains. (b) The corresponding curves for a porous medium with 75% oil wet grains. The curves in Figs. 17a and 17b are symmetric, so if Fig. 17b rotates 180°, the curves will overlie the curves in Fig. 17a.

-12

-8

-4

0

4

8

12

0 0.2 0.4 0.6 0.8 1 Water Phase Saturation

Cur

vatu

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orm

aliz

ed b

y G

rain

Rad

ius

fully water-wet25% oil-wet50% oil-wet75% oil-wetfully oil-wet

Increase in fraction of oil-wet grains

Fig. 18—Comparison between primary drainage curves for porous media with different fraction of oil-wet (θΟw = 150o) grains distributed randomly among water-wet (θww= 30o) grains. The porous medium was originally filled with water. The oil phase pushes the water phase out as the capillary pressure (curvature) increases.

SPE 124771 19

-12

-8

-4

0

4

8

12

0 0.2 0.4 0.6 0.8 1Water Phase Saturation

Cur

vatu

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orm

aliz

ed b

y G

rain

Rad

ius

fully water-wet25% oil-wet50% oil-wet75% oil-wetfully oil-wetIncrease in fraction of oil-wet grains

Fig. 19—Comparison between secondary imbibition curves for porous media with different fraction of oil-wet grains; the contact angles on water-wet and oil-wet grains are θww= 30o and θΟw = 150o, respectively. The imbibition curves start from drainage endpoint of the curves in Fig. 18. The water phase pushes the oil phase out as the capillary pressure (curvature) decreases. As the fraction of oil-wet grains within porous medium increases the value for curvature which water phase percolate through porous medium decreases.

-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

5000

0 0.2 0.4 0.6 0.8 1Water Phase Saturation

P c [P

a]

fully water-wet25% oil-wet50% oil-wet75% oil-wetfully oil-wet

-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

5000

0 0.2 0.4 0.6 0.8 1Water Phase Saturation

P c [

Pa]

fully water-wet 25% oil-wet 50% oil-wet 75% oil-wetfully oil-wet

Increase in fraction of oil-wet grains

(a) (b)

Increase in fraction of oil-wet grains

Fig. 20—(a) Measured octane/water drainage curve for fractionally-wet media. The primary drainage curves move monotonically lower with increasing oil-wet fraction. (b) Simulation results for drainage of fractionally-wet media, neglecting trapping of water phase. The contact angles on water-wet and oil-wet grains are θww= 0o and θΟw = 120o, respectively.

20 SPE 124771

-4000

-3000

-2000

-1000

0

1000

2000

3000

0 0.2 0.4 0.6 0.8 1Water Phase Saturation

P c [P

a]

fully water-wet25% oil-wet50% oil-wet75% oil-wetfully oil-wet

-4000

-3000

-2000

-1000

0

1000

2000

3000

0 0.2 0.4 0.6 0.8 1Water Phase Saturation

P c [

Pa]

fully water-wet25% oil-wet50% oil-wet75% oil-wetfully oil-wet

Increase in fraction of oil-wet grains

(a) (b)

Increase in fraction of oil-wet grains

Fig. 21—(a) Measured octane/water imbibition curve for fractionally-wet media. The primary imbibition curves move monotonically lower with increasing oil-wet fraction. (b) Simulation results for imbibition of fractionally-wet media. The contact angles on water-wet and oil-wet grains are θww= 0o and θΟw = 120o, respectively.

0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1Water Phase Saturation

Cur

vatu

re N

orm

aliz

ed b

y G

rain

Rad

ius

DrainageOnly generalized Haines imbibition Only Melrose Generalized Haines imbibition and Melrose (secondary imbibition)

Fig. 22—Simulation of imbibition of a water-wet medium initially containing only pendular rings. Here we ignore the trapping of the oil phase. When only generalized Haines imbibition events are allowed to occur (light blue curve), the imbibition process is self-limiting. When only Melrose events (red curve) are allowed, the process is self-sustaining. When both types of events are allowed, the Melrose events enable generalized Haines imbibition events. The combination leads to imbibition at larger curvatures.

SPE 124771 21

0

1000

2000

3000

4000

5000

6000

7000

0 0.2 0.4 0.6 0.8 1

Water Phase Saturation

Even

ts O

ccur

at E

ach

Step

Total_eventsMelrose event

Fig. 23—Number of total events and Melrose events that occur during imbibition (green curve in Fig. 22) versus water phase saturation. The number of generalized Haines imbibition events equals to difference between total events and Melrose events.

0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1Water Phase Saturation

Cur

vatu

re N

orm

aliz

ed b

y G

rain

Rad

ius

DrainageGeneralized Haines imbibition and Melrose (secondary imbibition)Generalized Haines imbibition and meniscus_4th grain (primary imbibition)

Fig. 24—Compares the simulation results for primary imbibition of a water-wet medium (purple curve) with secondary imbibition of a water-wet medium (green curve). Haines imbibition event can occur in both primary and secondary imbibition. During secondary imbibition, pendular ring exist. Consequently Melrose events can occur. In contrast, in primary imbibition, no Melrose event occurs due to lack of existence of pendular rings. During primary imbibition, the meniscus-4th grain event occurs at small value of normalized curvature (approximately less than 1.5).

22 SPE 124771

-2

0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1 Water Phase Saturation

Cur

vatu

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y G

rain

Rad

ius

fully water-wet25% oil-wet50% oil-wet75% oil-wetfully oil-wet

without meniscus-4th grain criterion with meniscus-4th grain criterion

(a) (b)

-2

0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1 Water Phase Saturation

Cur

vatu

re N

orm

aliz

ed b

y G

rain

Rad

ius

fully water-wet25% oil-wet50% oil-wet75% oil-wetfully oil-wet

Fig. 25—(a) Primary drainage curves for porous media with different fraction of oil-wet (θOw = 150o) grains distributed randomly among water-wet (θww= 30o) grains. The meniscus-4th grain event is not considered. (b) Primary drainage curves for porous media with different fraction of oil-wet (θOw = 150o) grains distributed randomly among water-wet (θww= 30o) grains. The meniscus-4th grain event is considered.

75% oil-wet grains

-4

-2

0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1Water Phase Saturation

Cur

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25% oil-wet grains

-4

-2

0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1Water Phase Saturation

Cur

vatu

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y G

rain

Rad

ius

(a)

(b)

Drainage (c)

(d)

Drainage

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1Water Phase Saturation

Frac

tion

of T

otal

Eve

nts

Generalized Haines drainage Generalized Haines imbibitionMelrose Meniscus_4th grain

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1Water Phase Saturation

Frac

tion

of T

otal

Eve

nts

Generalized Haines drainage Generalized Haines imbibitionMelrose Meniscus_4th grain

Fig. 26—(a) The P-S curve for drainage of fractionally wet medium with 25 % oil-wet grains. The contact angles on water-wet and oil-wet grains are θww= 30o and θOw = 150o, respectively. (b) Fraction of each events during drainage simulation shown in (a). (c) The P-S curve for drainage of fractionally wet medium with 75 % oil-wet grains, same contact angles as in (a). (d) Fraction of each event during drainage simulation shown in (c).

SPE 124771 23

75% oil-wet grains

-12

-10

-8

-6

-4

-2

0

2

4

0 0.2 0.4 0.6 0.8 1Water Phase Saturation

Cur

vatu

re N

orm

aliz

ed b

y G

rain

Rad

ius

25% oil-wet grains

-12

-10

-8

-6

-4

-2

0

2

4

0 0.2 0.4 0.6 0.8 1Water Phase Saturation

Cur

vatu

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y G

rain

Rad

ius

(a)

(b)

Imbibition

(c)

(d)

Imbibition

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1Water Phase Saturation

Frac

tion

of T

otal

Eve

nts

Generalized Haines drainage Generalized Haines imbibitionMelrose Meniscus_4th grain

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1Water Phase Saturation

Frac

tion

of T

otal

Eve

nts

Generalized Haines drainage Generalized Haines imbibitionMelrose Meniscus_4th grain

Fig. 27—(a) The P-S curve for imbibition of fractionally wet medium with 25 % oil-wet grains. The contact angles on water-wet and oil-wet grains are θww= 30o and θOw = 150o, respectively. (b) Fraction of each events occur during imbibition of fractionally wet medium with 25 % oil-wet grains. (a) The P-S curve for imbibition of fractionally wet medium with 75 % oil-wet grains. The contact angles on water-wet and oil-wet grains are θww= 30o and θOw = 150o, respectively. (b) Fraction of each events occur during imbibition of fractionally wet medium with 75 % oil-wet grains.

24 SPE 124771

-2

0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1Water Phase Saturation

Cur

vatu

re N

orm

aliz

ed b

y gr

ain

radi

usfully water-wet 12% oil-wet 25% oil-wet37% oil-wet 50% oil-wet 62% oil-wet75% oil-wet 87% oil-wet fully oil-wet

Fig. 28—Comparison between primary drainage curves for porous media with different fraction of oil-wet (θOw = 150o) grains distributed randomly among water-wet (θww= 30o) grains. The water phase gets trapped when disconnected from the bulk phase.

Fig. 29—The residual water phase saturation at the end of primary drainage for porous media is similar for media with a wide range of fractions of oil-wet grains (0 to 70%). The oil-wet (θOw = 150o) grains are distributed randomly among water-wet (θww= 30o) grains. When the fraction of oil-wet grains exceeds 75%, the residual water saturation increases rapidly. This is consistent with a transition from trapping in irreducible wetting saturation morphologies to trapping in residual nonwetting morphologies (large blobs). The marker ' '○ and bar show the average, maximum and minimum value of residual water saturation for ten realizations of fractionally wet media.