Three-Phase Capillary Pressure and Relative Permeability Relationships in Mixed-Wet Systems

Transcription

1 Transport in Porous Media 44: 1 32, c 2001 Kluwer Academic Publishers. Printed in the Netherlands. 1 Three-Phase Capillary Pressure and Relative Permeability Relationships in Mixed-Wet Systems M. I. J. VAN DIJKE, S. R. MCDOUGALL and K. S. SORBIE Department of Petroleum Engineering, Heriot-Watt University, Edinburgh EH14 4AS, Scotland, U.K. (Received: 6 July 1999; in final form: 18 April 2000) Abstract. A simple process-based model of three-phase displacement cycles for both spreading and non-spreading oils in a mixed-wet capillary bundle model is presented. All possible pore filling sequences are determined analytically and it is found that the number of pore occupancies that are permitted on physical grounds is actually quite restricted. For typical non-spreading gas/oil/water systems, only two important cases need to be considered to see all types of allowed qualitative behaviour for non-spreading oils. These two cases correspond to whether water or gas is the intermediatewetting phase in oil-wet pores as determined by the corresponding contact angles, that is, cos θgw o > 0orcosθgw o < 0, respectively. Analysis of the derived pore occupancies leads to the establishment of a number of relationships showing the phase dependencies of three-phase capillary pressures and relative permeabilities in mixed-wet systems. It is shown that different relationships hold in different regions of the ternary diagram and the morphology of these regions is discussed in terms of various rock/fluid properties. Up to three distinct phase-dependency regions may appear for a non-spreading oil and this reduces to two for a spreading oil. In each region, we find that only one phase may be specified as being the intermediate-wetting phase and it is only the relative permeability of this phase and the capillary pressure between the two remaining phases that depend upon more than one saturation. Given the simplicity of the model, a remarkable variety of behaviour is predicted. Moreover, the emergent three-phase saturation-dependency regions developed in this paper should prove useful in: (a) guiding improved empirical approaches of how two-phase data should be combined to obtain the corresponding three-phase capillary pressures and relative permeabilities; and (b) determining particular displacement sequences that require additional investigation using a more complete process-based 3D pore-scale network model. Key words: three-phase flow, capillary pressure, relative permeability, capillary bundle, mixed-wet, spreading coefficient, saturation-dependency, pore occupancy, process-based model. 1. Introduction and Literature Review The study of three-phase flow in porous media is important in many subsurface processes such as those which occur in oil recovery and in aquifer remediation involving non-aqueous phase liquids (NAPLs). The principal multiphase flow parameters that appear in the governing transport equations are the three-phase capillary pressures and relative permeabilities. These quantities, especially the threephase relative permeabilities, have been the subject of much study over the past 40 years although much still remains to be understood about the behaviour of

2 2 M. I. J. VAN DIJKE ET AL. three-phase systems. The problem is compounded if we consider porous media of non-uniform wettability. In this paper, we focus on three-phase capillary controlled flow in simplified models of mixed-wet porous media. In the context of petroleum recovery, Leverett and Lewis (1941) and Corey et al. (1956) presented early discussions of the issues involved in three-phase flow. Since that time, a number of experimental studies of three-phase flow, principally measuring relative permeabilities, have appeared in the literature although such measurements are far from routine (Naar and Wygal, 1961; Saraf et al., 1982; Grader and O Meara, 1988; Oak, 1990, 1991; Oak et al., 1990; Kalaydjian et al., 1993; Nordtveldt et al., 1996). The difficult and time-consuming nature of these measurements has meant that a full parameteric study of all of the factors affecting three-phase flow in porous media has not been possible. Since so few factors are varied in any particular study and conditions change from one study to another, it is difficult to draw general conclusions from this body of experimental work. This is further complicated when petroleum reservoir rocks are considered, due to the difficulty in characterising the wettability states that such systems can adopt (Morrow, 1990; Cuiec, 1991; Jerauld, 1997; Jerauld and Rathmell, 1997). The extreme difficulty in measuring three-phase relative permeabilities (and capillary pressures), has led to a number of attempts to derive empirical expressions for the three-phase relative permeability isoperms by combining the two phase data in various ways (Stone, 1970, 1973; Dietrich and Bondor, 1976; Delshad and Pope, 1989; Fayers, 1984; Fayers and Mathews, 1984; Baker, 1988; Balbinski et al., 1997; Blunt, 1999). The early work of Stone (1970, 1973) has been both criticised and extended by later workers in the previous reference list. This approach is based partly on the fact that the three-phase flow parameters must limit appropriately to the various combinations of two phases which can occur. For example, in an oil/water/gas system, such as may be found in a petroleum reservoir, the three-phase relative permeability should limit correctly to the various two-phase oil/water, gas/oil and gas/water relative permeabilities. This fact has been used recently by both Hustad and Hansen (1996) and Blunt (1999) as the basis for a consistent three-phase relative permeability model. In addition to consistency, the correct phase trapping characteristics must be predicted by such empirical models and, for this purpose, early models due to Land (1968) and Carlson (1981) have been used. Recently, Skauge and Larsen (1994) have proposed more elaborate models for phase trapping in WAG (water-alternating-gas) processes in order to explain experimentally observed behaviour. The issue of residual oil determination in three-phase empirical models is also discussed by Balbinski et al. (1997) and Blunt (1999). In the groundwater/napl aquifer remediation literature, there have also been several attempts to predict three-phase flow parameters in various empirical ways (Lenhard, 1987, 1992; Parker et al., 1987; Parker and Lenhard, 1990). In empirical models of three-phase relative permeability, there is often little acknowledgement of the underlying pore-scale displacement physics. Therefore,

3 MIXED-WET SYSTEMS 3 basic questions cannot be answered from purely macroscopic considerations. For example, the precise phase-dependency of three-phase relative permeability in various regions of the saturation space cannot be determined aprioriand it is not clear if the isoperms should depend upon just one or on two of the phase saturations. Such issues are extremely important and, to our knowledge, no theoretical analysis has appeared which guides our choice of functional dependence of three-phase relative permeability in various regions of the saturation space. Indeed, no clear explanation of how the various fluid and rock properties (e.g. the interfacial tensions σ ow, σ go and σ gw, and rock wettability) in a three-phase system affect the potential phase-dependency regions has been presented. However, in recent years, a number of experimental studies have appeared where three-phase displacement processes have been observed directly in physical etched micromodels (Øren and Pinczewski, 1991, 1992, 1995; Soll and Celia, 1993; Mackay et al., 1997). Such studies have carefully identified many relevant pore-scale displacement mechanisms that occur in three-phase flow under various conditions. In addition, they have pointed out the importance of the oil spreading coefficient, C S,o = σ gw σ ow σ go,indetermining the fluid configurations and likely displacement mechanisms that occur for spreading (C S,o 0) and non-spreading (C S,o < 0) oils (Øren and Pinczewski, 1991, 1992, 1995; Kalaydjian et al., 1993; Vizika 1993; Vizika and Lombard, 1996; Fenwick, 1997; Zhou and Blunt, 1997; Mani and Mohanty, 1997). Strictly, threephase equilibrium systems which are spreading should have C S,o = 0 (Rowlinson and Widom, 1989), although experimental results for positive spreading systems are often reported which are probably quite far from equilibrium (e.g. Øren and Pinczewski, 1992, 1995; Skurdal et al., 1996). In spreading systems, the oil forms a film between the gas and water phases, whereas, in non-spreading systems, a three-phase contact line exists between the liquid phases. In this paper, we address both spreading and non-spreading cases. The important contribution of the direct visualisation of the pore-scale displacement physics in three-phase flow is that they can be built into a process-based model. By this, we mean a model where the emergent three-phase capillary pressures and relative permeabilities arise directly from the pore-scale physics of the process (plus some model assumptions). It is very important that the calculated behaviour be truly emergent, that is, that it genuinely arises in the calculations and is not already hidden in a very complex multi-parameter model. Previous process-based models of three-phase flow have been based on percolation theory approaches (Heiba et al., 1984; Soll and Celia, 1993). However, more accurate but much more complex process-based 3D network models of three-phase flow have recently been developed (Fenwick and Blunt, 1995, 1996, 1998a, 1998b; Fenwick, 1997; Øren et al., 1994; Mani and Mohanty, 1998; McDougall and Mackay, 1998). These models are very powerful and have given much insight into the physics of three-phase flow in strongly water-wet (or strongly oil-wet) systems. However, 3D network models are complex and are very time consuming to run. Also, to date, no systematic study has appeared which presents calculated three-

4 4 M. I. J. VAN DIJKE ET AL. phase flow results (capillary pressures and relative permeabilities) for mixed-wet systems. This paper takes a much simpler approach to modelling three-phase flow using a process-based model. The model porous medium considered here is a mixed-wet capillary bundle. Contact angles are assumed to have constant but different values in water-wet and oil-wet pores, reflecting strongly wetting conditions. Interfacial tensions are assumed to be constant throughout the calculations. We have as our central objectives: (i) to work through all of the possible pore filling sequences, thus determining all allowed pore occupancies; (ii) to determine all of the allowed regions of saturation dependence of capillary pressure and relative permeability; and (iii) to explain how these regions relate to the controlling factors in the model viz the interfacial tensions the parameters of the pore size distribution and the fraction of oil-wet pores. The key issue about our approach is that it yields the phase dependencies of capillary pressure and relative permeability for threephase flow in a mixed-wet system. Note that the base case mixed-wet system has small pores water-wet and large pores oil-wet (denoted MWL) although a brief discussion will be given of the case where the small pores are oilwet (denoted MWS). The three-phase relationships are expressed in terms of the two-phase curves (capillary pressures and relative permeabilities) obtained from the same systems. These predictions can be used to guide the application of a macroscopic empirical model such as those discussed above. In Section 2, we begin by describing the underlying capillary bundle model itself and derive a number of useful expressions relating contact angles and interfacial tensions for a non-spreading oil (C S,o < 0).InSection3,wegoontodiscuss a number of different two-phase displacements in a mixed-wet capillary bundle model. A clear understanding of these two-phase cases proves to be an important precursor to the more complex three-phase processes discussed later. The injection of a third fluid phase is considered in Section 4. After elucidating the different pore occupancy possibilities, we derive a number of relationships for three-phase capillary pressures and relative permeabilities in terms of their two-phase counterparts. This is then followed by a detailed analysis of the ternary diagram, which gives analytical boundaries in saturation space for the phase dependencies of the threephase parameters. In Section 5, we extend our results to the case of a spreading oil (C S,o 0) which we show to be a special case of the non-spreading system. Finally, a summary and conclusions are presented in Section Invasion in the Capillary Tube Model The porous medium is represented as a bundle of parallel cylindrical tubes with pore radii chosen from a uniform distribution varying from r = r min to r = r max.a

5 MIXED-WET SYSTEMS 5 given number fraction of pores x wet is strongly water-wet and the largest water-wet pore has radius r = r wet ; the remaining larger pores are strongly oil-wet. This situation is denoted as mixed-wet large (MWL) and these pore wetting assumptions are similar to those made in an earlier network modelling study of two-phase flow in mixed-wet systems (McDougall and Sorbie, 1995; Dixit et al., 1999). These earlier studies explicitly included intra-pore effects such as film drainage etc., but these are not considered here. In the bundle of tubes model, we consider flow of three phases, water w, oil o and gas g, which are assumed to be incompressible. Initially, all tubes are filled with one of the phases and connected at one side, say the outlet, to a vessel containing this phase which remains at a constant pressure at all times. At the inlet, the pores are connected to a vessel containing a different invading phase, whose pressure is varied to allow subsequent invasion of the pores. This sequence of events represents a two-phase displacement process. In addition, after partially flooding the tube model with the first invading phase, a second flood may be initiated with a third distinct phase. Hence, three-phase flow processes may also be considered. The condition under which the invading phase i invades a target pore occupied by phase j, is given by: P inv,i P entry,i, (1) where P inv,i denotes the pressure of the invading phase and P entry,i the entry pressure, which is given by P entry,i = P j + P c,ij, (2) for i = j, wherep c,ij denotes the capillary entry pressure of the target pore with respect to fluids i and j. The pore entry condition in (1), which is identical to the condition employed by Mani and Mohanty (1998), specifies the phase pressure of the invading phase that is necessary to enter a pore. The Young Laplace equation relates the three capillary pressures that arise in three-phase flow to the respective interfacial tensions σ contact angles θ and pore radii r according to the formulae: P c,ow = 2σ ow cos θ ow, (3a) r P c,gw = 2σ gw cos θ gw, (3b) r P c,go = 2σ go cos θ go. (3c) r Observe that relations (3) do not apply only in the present cylindrical pores, but in pores of any geometry where r is then taken as the inscribed radius (Bear, 1972). The θ ow and θ gw are measured through the water phase and the θ go is measured through the oil phase. We assume fixed values for the σ ij and the cos θ ij, although

6 6 M. I. J. VAN DIJKE ET AL. the latter differ for water-wet and oil-wet pores. We first work through the case of a non-spreading oil and later consider the case for a spreading oil in Section 5 below. For a non-spreading oil, the σ ij and the cos θ ij are related by cos θ w ow = cos θ o ow = cos θ o go = cos θ w gw = 1, (4a) cos θ w go = σ gw σ ow σ go, (4b) cos θ o gw = σ go σ ow σ gw, (4c) where the superscripts w and o denote water-wet and oil-wet pores, respectively. Relation (4a) reflect strongly wetting conditions. Relations (4b) and (4c) are based on a balance of interfacial tensions for flow of two phases in a pore with a wetting film of the third phase (see e.g. Øren and Pincewski, 1995). These relations impose certain restrictions on the values of the σ ij, because cos θ 1. One of the consequences of relation (4b) is that σ gw σ ow σ go or, equivalently, that the spreading coefficient for oil C S,o = σ gw σ ow σ go, (5) is non-positive, which is true for a non-spreading oil as C S,o < 0. Furthermore, the sign of the right-hand side of (4b) specifies whether gas is wetting to oil in waterwet pores (which is unlikely for realistic values of the σ ij ) and the sign of the righthand side of (4c) specifies whether gas is wetting to water (cos θgw o < 0) or water is wetting to gas (cos θgw o > 0) in oil-wet pores (which may occur for realistic values of the σ ij ). It transpires that the value of cos θgw o plays a very important role in determining the capillary pressure and relative permeability saturation-dependency regions. In Sections 3 and 4 below two cases of non-spreading three-phase systems are considered, indicated as Cases 1 and 2 in Table I, for which cos θgw o > 0 and cos θgw o < 0 respectively, being the only important generic cases for a nonspreading system. 3. Two-Phase Processes (Case 1 Parameters) To show the consequences of the entry condition (1), we simulate the possible pore occupancies for a specific combination of parameters, indicated as Case 1 in Table I. Observe that σ ow is smaller than σ go, hence water is wetting to gas in oil-wet pores. The boundaries of the radius distribution are r min = mandr max = m and the size of the largest water-wet pore r wet = = m. First we present the possible two-phase flow scenarios. The corresponding

7 MIXED-WET SYSTEMS 7 Table I. Number fraction of water-wet pores, values of the interfacial tensions, the resulting values of the oil spreading coefficient, C S,o (Equation 5) and cosines of the contact angles, cos θgo w and cos θgw o (Equations (4b) and (4c)) used in the capillary bundle model simulations Case x wet σ ow σ go σ gw C S,o cos θgo w cos θgw o (10 3 N/m) (10 3 N/m) (10 3 N/m) (10 3 N/m) Figure 1. Successive pore occupancies after oil-filled tubes are invaded by increasing amounts of gas, with parameters of Case 1. equations are valid for the specific examples, which relate to the most complex three-phase pore occupancies possible GAS-OIL DISPLACEMENT Starting with oil-filled pores, gas invasion leads to pore occupancies as shown in Figure 1. For the present choice of parameters, we obtain from Equations (1), (3c) and (4b) P entry,g (r wet ) = P o + 2(σ gw σ ow ) r wet <P o + 2σ go r max = P engry,g (r max ). (6) Hence, by increasing the gas pressure gas first invades the water-wet pores, filling them in decreasing order of size (Figure 1(a)). Because we have the ordering 2(σ gw σ ow ) r wet < 2σ go r max < 2σ go r wet < 2(σ gw σ ow ) r min, (7)

8 8 M. I. J. VAN DIJKE ET AL. conditions similar to (6) show that a further increase of the gas pressure results in gas filling oil-wet pores (starting at r = r max ), before all water-wet pores are invaded (Figure 1(b)). At this stage, gas fills water-wet and oil-wet pores alternately. The last inequality of (7) shows that gas invasion first reaches r = r wet from above (Figure 1(c)), before the smallest pores are filled WATER-OIL DISPLACEMENT For a water flood in oil-filled tubes we find the ordering 2σ ow r wet < 2σ ow r max < 2σ ow r wet < 2σ ow r min. (8) Hence, increasing the water pressure leads first to invasion of water-wet pores, filling them in increasing order of size. When all water-wet pores are invaded, water fills the oil-wet pores in decreasing order of size GAS WATER DISPLACEMENT For a gas flood in water-filled tubes the ordering 2(σ go σ ow ) < 2(σ go σ ow ) < 2σ gw r max r wet r wet < 2σ gw r min, (9) applies. Increasing the gas pressure leads to invasion of all oil-wet pores in decreasing order of size, after which the water-wet pores are also invaded in decreasing order of size. The above two-phase flow processes in a capillary bundle are non-hysteretic, implying that every possible two-phase occupancy can be reached both by invasion of phase one into phase two and by invasion of phase two into phase one. Furthermore, every two-phase occupancy leads to a unique saturation combination of the two phases involved. The saturation of phase j is defined by S j = r=r j V(r)f(r), (10) with r j indicating pores occupied by phase j,v (r) the pore volume function (relative to the total pore volume; e.g. see McDougall and Sorbie, 1995) and f(r) is the pore size distribution function. Hence, choosing V(r) r 2 and the uniform distribution, the two-phase relations between the capillary pressures and saturations can be written immediately. For the parameters of Case 1, these are presented in Figure 2. In Figure 2, we have introduced the following saturation values S wet = r=r wet r=r min V(r)f(r), (11a)

9 MIXED-WET SYSTEMS 9 Figure 2. Calculated two-phase capillary pressure-saturation relations: (a) P c,go (S o ), (b) P c,ow (S w ) and (c) P c,gw (S w ), with parameters of Case 1. S 1 = S 2 = r=r 1 r=r min V(r)f(r), r=r 2 r=r min V(r)f(r) + r=r max r=r wet V(r)f(r). (11b) (11c) Note that r 1 is the pore between r min and r wet that is invaded by gas precisely after all oil-wet pores are invaded. In other words, r 1 is the largest pore r g for which the occupancy of Figure 1(c) exists. r 2 is the last pore between r min and r wet that is invaded by gas before gas starts invading at r = r max. In other words, r 2 is the smallest pore r g for which the occupancy of Figure 1(a) exists. These definitions allow us to relate r 1 and r 2 to r wet and r max, respectively, as r 1 = r wet σ gw σ ow σ go, (12a) r 2 = r max r gw σ ow σ go. (12b)

10 10 M. I. J. VAN DIJKE ET AL. Figure 3. Three-phase pore occupancies after gas oil filled tubes of Figure 1(a) are invaded by different amounts of water with parameters of Case Three-Phase Displacements: The Non-spreading Cases 4.1. PORE OCCUPANCIES (CASE 1) Waterflood of Gas Oil System: Figure 1(a) To analyse three-phase flow processes, we first simulate water invasion in each of the three gas oil occupancies of Figure 1. In the situation of Figure 1(a), water invasion may start in three different pores: r = r min, r = r g or r = r max.the corresponding water entry pressures are P entry,w (r min ) = P 0 2σ ow r min, (13a) P entry,w (r g ) = P g 2σ gw = P o + 2(σ gw σ ow ) 2σ gw r g r g r g = P o 2σ ow r g, (13b) P entry,w (r max ) = P o + 2σ ow, (13c) r max where the third term of relation (13b) follows from the gas pressure at the end of the two-phase flow process of gas invasion into oil. Relations (13) show that r = r min is always invaded before the other two pores and that r = r g is invaded before r = r max (independent of the actual physical values chosen). Furthermore, generalisation of relation (13a) shows that all pores between r = r min and r = r g are invaded before r = r g. Hence, water invasion in the situation of Figure 1(a) by increasing the water pressure leads first to a pore occupancy as shown in Figure 3(a) and then to an occupancy as shown in Figure 3(b). Strictly speaking, water could now enter r = r max before it reaches r = r wet. However, the preceding two-phase flow process showed that gas went first in the water-wet pores, before entering r = r max, which implies P g <P o + 2σ go r max. (14)

11 MIXED-WET SYSTEMS 11 Figure 4. Three-phase pore occupancies after gas oil filled tubes of Figure 1(b) are invaded by different amounts of water; for parameters of Case 1. Hence, P entry,w (r wet ) = P g 2σ gw r wet <P o + 2σ go r max 2σ gw r wet <P o + 2(σ go σ gw ) (15) r max and with the restrictions on the right-hand side of (4c), we obtain P entry,w (r wet )<P o + 2σ ow = P entry,w (r max ). (16) r max Consequently, continued waterflooding displaces all of the oil from the water-wet pores before any of the oil-wet pores are invaded Waterflood of Gas Oil System: Figure 1(b) In the pore occupancy of Figure 1(b), water invasion starts again in r = r min, leading to a three-phase occupancy as shown in Figure 4(a). Using similar arguments as for the situation of Figure 1(a), we find that water first invades all water-wet oil-filled pores and then the water-wet gas-filled pores (Figure 4(b)). After all water-wet pores have been invaded, water next invades oilwet pores in the manner shown in Figure 4(c). This is due to the fact that, for the parameters of Case 1, water is the intermediate-wetting phase in oil-wet pores. Because the entry pressures for invasion into oil at r = r w,1 and for invasion into gas at r = r w,2 are of similar magnitude, water may next invade alternately

12 12 M. I. J. VAN DIJKE ET AL. oil-filled and gas-filled pores. The resulting two-phase situation has either water in the small and gas in the large oil-wet pores, or oil in the small and water in the large oil-wet pores. This depends upon the position of r = r g,2 in Figure 1(b), which relates to the preceding gas-flood. The precise condition for the occurrence of one of these two-phase occupancies follows by comparison of the water entry pressures in r = r max and r = r wet :if σ ow < σ go (σ go σ ow ), (17) r wet r g,2 r max all of the oil is recovered and some gas still remains within the system Waterflood of Gas Oil System: Figure 1(c) Water invasion in the situation of Figure 1(c) results in the three-phase pore occupancy plot shown in Figure 5. The situation here is more straightforward once water has invaded all oil-filled pores, the gas-filled pores are then invaded in increasing order of size Additional Three-Phase Displacements In Section 3, we described three different two-phase displacements: oil gas, oil water and water gas. We have shown that, for each of these occupancies, invasion of the remaining third phase leads to a wide range of three-phase pore occupancies. It can be demonstrated, however just as for two-phase pore occupancies that each saturation combination of the three phases as well as each pressure combination corresponds to a unique pore occupancy. As a consequence, gas invasion into an oil water system and oil injection into a water gas system, can only lead to pore occupancies that have already been found above for water invasion into the oil gas system the analysis is therefore greatly simplified. Only when one of the phases exactly occupies either all the water-wet pores or all the oil-wet pores at the end of the preceding two-phase flood, the pressure Figure 5. Three-phase pore occupancy after gas oil filled tubes of Figure 1(c) are invaded by water with parameters of Case 1.

13 MIXED-WET SYSTEMS 13 Figure 6. Ternary diagrams of saturation paths with parameters of Case 1. Flooding sequences are: (a) 100% oil-filled waterflood gasflood; (b) 100% oil-filled gasflood waterflood; and (c) 100% water-filled gasflood oilflood. differences are not exactly determined, which is reflected by the jumps in the P c,ow and P c,gw curves of Figures 2(b) and 2(c) at S w = S wet. This non-uniqueness is a result of the distinct difference in capillary entry pressures between the largest water-wet pore and the smallest oil-wet pore, which is expected to vanish in realistic porous media. Therefore, this rather special case will not be considered further in this paper THREE-PHASE CAPILLARY PRESSURE/SATURATION RELATIONSHIPS (CASE 1) To find the dependencies of the capillary pressures on saturations for three-phase flow, we consider the saturation paths of the flow processes described above. In Figure 6 the saturation paths for gas injection into oil water, water injection into oil gas and oil injection into water gas systems are presented for varying residuals after the first flood. All of the secondary floods are run to completion, that is, no residuals are considered. As for the pore occupancies, the three-phase saturation paths are independent of the two-phase flow process that has led to the saturations at the start of the second flood. Note, that the unique relation between pore occupancies and saturations guarantees that for a given combination of saturations, the subsequent injection of any one of the phases by increasing its phase pressure leads to a unique trajectory in saturation space.

14 14 M. I. J. VAN DIJKE ET AL. The ternary diagrams in Figure 6 show the different curves, identified by the number fraction of pores that are filled with the displaced phase after the first flood, for example, in (a) the residual indicates the number fraction of pores that are still filled with oil after waterflooding. These are of major importance in determining the capillary pressure/saturation relationships, because in each diagram one of the capillary pressures is constant along the saturation paths in the three-phase region. For example, in Figure 6(b) only the water pressure varies during the second flood and so each curve actually represents a gas oil iso-capillary-pressure ( isocap ). Similarly, Figure 6(a) provides the oil water and Figure 6(c) the gas water capillary pressure/saturation dependencies. As a result, straight saturation paths parallel to iso-saturation lines indicate dependence of the corresponding capillary pressure on only one saturation, whereas curved paths indicate dependence on more than one saturation. All diagrams of Figure 6 show that three distinct regions of phasedependency exist (denoted as regions I, II and III), as shown in Figure 7 for the saturation paths of Figure 6(c). The three-phase pore occupancies in Figures 3 5 correspond to one of the three regions of Figure 7. Figures 3(a), 4(a) and 5 belong to region I, Figure 4(b) belongs to region II and Figure 4(c) belongs to region III. The diagram of Figure 3(b) corresponds exactly to the lower part of the boundary between regions I and II. The precise morphology of regions I, II, and III is discussed more fully in Section 4.5. The saturation paths in the various regions, together with the pore occupancies, elucidate the dependencies of the capillary pressures on saturations completely. In region I, P c,ow depends on S w only (Figure 6(a)) and P c,go depends on S g only (Figure 6(b)). As a result, P c,gw depends on both S w and S g, since by definition P c,gw = P c,go + P c,ow (Figure 6(c)). This multiple dependency follows purely from the pore occupancy plot, as no boundary exists separating distributions of Figure 7. Regions in saturation space delineating different phase dependencies of the capillary pressure/saturation relationships. These are shown superimposed on the saturation paths of Figure 6(c).

15 MIXED-WET SYSTEMS 15 water-filled and gas-filled pores. In region II, P c,gw depends on S w only and P c,go depends on S o only. In region III, P c,gw depends on S g only and P c,ow depends on S o only. Hence, the phase-dependencies of the three three-phase capillary pressures are given as follows: in region I we obtain P 3 c,ow (S w) = P 2 c,ow (S w), (18a) Pc,go 3 (S g) = Pc,go 2 (1 S g), Pc,gw 3 (S w,s g )Pc,go 2 (1 S g) + Pc,ow 2 (S w); in region II Pc,gw 3 (S w) = Pc,gw 2 (S w), Pc,go 3 (S o) = Pc,go (S o) = Pc,go 2 (S o), Pc,ow 3 (S w,s o ) = Pc,gw 2 (S w) Pc,go 2 (S o); and in region III Pc,gw 3 (S g) = Pc,gw 2 (1 S g), Pc,ow 3 (S o) = Pc,ow 2 (1 S o), Pc,go 3 (S g,s o ) = Pc,gw 2 (1 S g) Pc,ow 2 (1 S o), (18b) (18c) (19a) (19b) (19c) (20a) (20b) (20c) where the superscripts 2 and 3 refer to the two and three-phase flow properties, respectively. The two-phase capillary pressure/saturation functions are those shown earlier in Figure 2. Note that in region II the three-phase gas oil capillary pressure is a function of S o only but this function is not the same as the two-phase capillary pressure since the three-phase gas occupancy (Figure 3(b)) is not the same as the corresponding two-phase occupancy for a gas oil system (Figure 1(b)). We refer to this as non-genuine phase-dependency and this is also seen in the corresponding relative permeability (see below). This issue is discussed in more detail elsewhere (van Dijke et al., 2000) THREE-PHASE RELATIVE PERMEABILITY/SATURATION RELATIONSHIPS (CASE 1) In addition to the relations between capillary pressure and saturation, we may also derive the dependencies of relative permeability on saturation. We define the phase j relative permeability as k r,j = r=r j g(r)f(r), (21)

16 16 M. I. J. VAN DIJKE ET AL. Figure 8. Ternary saturation diagrams of iso-relative permeability curves with parameters of Case 1. with g(r) the pore conductance function (relative to the total conductance). Choosing g(r) r 4 and still using the parameters of Case 1 (Table I), we derive the iso-relative permeability (isoperm) curves for the saturation paths shown in Figure 6. Just as each combination of S w, S o and S g corresponds to a unique pore occupancy, so too does each combination of the three-phase relative permeabilities k r,w, k r,o and k r,g. Indeed, we find that each set of three-phase flow processes produces exactly the same isoperm curves. Figure 8 shows isoperm plots for the three-phases. Three phase-dependency regions can again be identified in these plots. Clearly, k r,o depends on more than one saturation only in region I, whereas k r,w depends only on S w and k r,g depends only on S g. Similarly, in region II only k r,g and in region III only k r,w depends on more than one saturation. The precise relations between relative permeabilities and saturations follow from the pore occupancy plots, Figures 3 5. For example, Figure 4(c), corresponding to region III, shows that the distribution of oil-filled pores borders only water. Therefore, in region III the oil relative permeability is identical to the relative permeability in a two-phase oil water system. Similarly, gas only borders water and its relative permeability is identical to the gas relative permeability in a two-phase gas water system. The water relative permeability is now easily found, as in this capillary tube system the three relative permeabilities add up to one.

17 MIXED-WET SYSTEMS 17 Hence, we find in region I k 3 r,w (S w) = k 2,ow r,w (S w), (22a) k 3 r,g (S g) = K 2,go r,g (S g), (22b) kr,o 3 (S w,s g ) = 1 Kr,w 2,ow (S w) kr,g 2,go (S g); (22c) in region II k 3 r,w (S w) = k 2,gw r,w (S w), (23a) k 3 r,o (S o) = k r,o (S o) = k 2,go r,o (S o), (23b) kr,g 3 (S w,s o ) = 1 kr,w 2,gw (S w) kr,o 2,go (S o); (23c) and in region III k 3 r,o (S o) = k 2,ow r,o (S o), (24a) k 3 r,g (S g) = k 2,gw r,g (S g), (24b) kr,o 3 (S o,s g ) = 1 kr,o 2,ow (S o) kr,g 2,gw (S g), (24c) where the superscript 2, ij refers to the two-phase system occupied by phases i and j Similar to the gas oil capillary pressure in region II (see Equation (19b)) the three-phase gas oil relative permeability is not the same as the two-phase function, i.e. the phase-dependency of the oil relative permeability is non-genuine (van Dijke et al., 2000) GAS WETTING TO WATER IN OIL-WET PORES (CASE 2 PARAMETERS) To show that the phase-dependency regions depend strongly on the values of the interfacial tensions for a non-spreading oil, we consider a case for which gas is wetting to water in oil-wet pores; that is, cos θgw o < 0 (see Equation (4c)). The parameters used for this example are given as Case 2 in Table I. As a consequence of this choice of parameters, the ordering in Equation (7) becomes 2σ go < 2(σ gw σ ow ) < 2σ go < 2(σ gw σ ow ). (25) r max r wet r wet r min Hence, in a two-phase gas oil system, gas starts invading the oil-wet pores. This is in complete contrast to the earlier example depicted in Figure 1(a). The two-phase

18 18 M. I. J. VAN DIJKE ET AL. Figure 9. Ternary diagrams of saturation paths with parameters of Case 2. Flooding sequences are (a) 100% oil-filled waterflood gasflood; (b) 100% oil-filled gasflood waterflood; and (c) 100% water-filled gasflood oilflood. gas-oil capillary pressure curve changes accordingly and we define additionally S 3 and r 3 as S 3 = r=r 3 r=r min V(r)f(r), (26a) σ go r 3 = r wet, σ gw σ ow (26b) where r 3 is the smallest gas invaded pore between r wet and r max, for which gas is only present in oil-wet pores. Furthermore, instead of (9) we have 2(σ go σ ow ) < 2(σ go σ ow ) < 2σ gw r wet r max r wet < 2σ gw r min (27) hence, the two-phase P c,gw attains negative values for S w >S wet. In Figure 9, the saturation paths for gas injection into oil water, water injection into oil gas and oil injection into water gas systems are presented for varying residuals after the first flood. As indicated in Figure 9(b), only two phase-dependency regions can now be distinguished. Region III no longer appears. The pore occupancies corresponding to the various subregions are presented in Figure 10. They show that in region I oil borders both water and gas and in region

19 MIXED-WET SYSTEMS 19 Figure 10. Three-phase pore occupancies for parameters of Case 2 corresponding to the various subregions shown in Figure 9(b): (a) top subregion, (b) middle subregion and (c) bottom subregion of region I; (d) top subregion and (e) bottom subregion of region II. II gas borders both oil and water. Clearly, for S g < 1 S 3 region I is separated from region II by S o = 1 S wet.for1 S 3 <S g < 1 S 1, the boundary between regions I and II is again given by Equations (17), now with reference to Figure 10(b) when r = r w and r = g g,1 coincide. The capillary pressure/saturation and relative permeability/saturation relationships in regions I and II are found to be identical to those found earlier, when water was wetting to gas in oil-wet pores (Equations (18), (19), (22), and (23)) MORPHOLOGY OF THE PHASE-DEPENDENCY REGIONS Base Case with Large Pores Oil-Wet (MWL) From the examples presented so far, it has become clear that, in saturation space, at most three distinct regions exist that characterise the phase dependencies of the three-phase isocaps and isoperms. In each region, only one of the capillary

20 20 M. I. J. VAN DIJKE ET AL. pressures and only one relative permeability depends on more than one saturation, because only one phase borders the other two phases in the pore occupancy plot. In theory, a large number of three-phase pore occupancies may exist, but wettability rules exclude the majority of them. These wettability rules are based on the differences between pressures of the phases present in the porous medium. Because we investigate only systems at pressure equilibrium, the pressure within any given phase remains the same throughout the porous medium. We still assume that the mutual wettability of two phases in a pore having the wettability of the third phase is determined by Equations (4b) and (4c). Hence, the following rules apply (see Appendix A for derivation): (a) Within a cluster of either water-wet pores or oil-wet pores, the phase pressures are ordered according to their mutual wettability, with the wetting phase having the smallest and the non-wetting phase having the largest pressure. Consequently, in one cluster the wetting phase occupies smaller pores than the intermediate wetting phase, which occupies smaller pores than the non-wetting phase. (b) If the mutual wettability of two phases is different for water-wet pores and oil-wet pores, then these phases cannot coexist in both the water-wet and the oil-wet pores. This rule also applies if one of the phases is only present as a wetting film. An example of the latter is the situation in which only water and gas are present in oil wet pores. According to rule (b) the presence of oil-films in oil-wet pores specifies oil as wetting to water. This precludes oil from being present in water-wet pores, where oil is non-wetting to water. Conditions for the existence of the various regions and the change in boundaries between these regions can be derived for various parameter combinations. They follow from the possible orderings of the threshold values of the two-phase capillary pressure curves, of which (7) (9), (25) and (27) are examples. In general, the threshold values of the P c,go curve can have six different orderings, leading to the six possible morphologies shown schematically in Figure 11. For the P c,gw curve two possibilities exist, determining whether the saturation space for S w >S wet in Figure 11 belongs to either region II or region III. The P c,ow curve does not give rise to different morphologies. For each configuration of Figure 11, the corresponding constraints upon the P c,go values together with definitions for S 1, S 2, S 3, S 4 and S wet are given in Appendix B. In most cases, the regions shown in Figure 11 depend on the values of the interfacial tensions, the range of the radius distribution, that is, r min and r max,and the fraction of water-wet pores, indicated by the largest water-wet pore, r wet.from the conditions determining the division of saturation space, it follows that, for non-spreading systems, every division except the one shown in Figure 11(a) is possible in practice. If we take, for example, the values of the interfacial tensions of Case 2, we find that Figure 11(f), related to condition (B.1f), requires a nar-

21 MIXED-WET SYSTEMS 21 Figure 11. Possible positions of the boundary between regions I and II in saturation space, where the various regions reflect different dependencies of capillary pressure on saturation. The pore size distribution parameters and interfacial tensions satisfy the following (Appendix B): (a) condition (B.1a), (b) conditions (B.1b), (c) condition (B.1c), (d) condition (B.1d), (e) conditions (B.1e) and (f) condition (B.1f). row pore size distribution. Figure 11(e), related to conditions (B.1e), requires also a relatively narrow distribution, but has an extra restriction with respect to r wet. Figure 11(d), related to condition (B.1d), requires an almost completely water-wet system, whereas Figure 11(c), related to condition (B.1c), requires a mainly oilwet system. Finally, Figure 11(b), related to conditions (B.1b) requires a relatively wide pore size distribution. In each region, only one phase may be specified as being the intermediatewetting phase, as only one phase borders both other phases in the associated pore occupancy plot. Observe that this definition of intermediate-wetting is a generalisation of the usual definition for purely water-wet or purely oil-wet systems. In Figure 4(b), for example, gas is the non-wetting phase in both the water-wet and the oil-wet pores, but may be defined as the intermediate-wetting phase for the system as a whole. Furthermore, according to this definition, only the relative permeability of the intermediate-wetting phase depends on more than one saturation.

22 22 M. I. J. VAN DIJKE ET AL. The above analysis describes the purely water-wet and purely oil-wet systems as special cases of mixed-wet systems. By taking r wet = r max, it follows naturally that in purely water-wet systems the entire saturation space is covered by region I, which means that oil is the intermediate-wetting phase everywhere and is, therefore, the only phase whose relative permeability depends on more than one saturation. This is consistent with many observations in strongly water-wet systems (e.g. see the review in Honarpour et al., 1986). By taking r wet = r min it follows that in purely oil-wet systems the entire saturation space is covered by either region II or region III, depending upon whether gas (cos θgw o < 0) or water (cos θ gw o > 0) is the intermediate-wetting phase Mixed-Wet Case with Small Pores Oil-Wet (MWS) A straightforward but interesting extension of the above is the morphology of the phase-dependency regions for a mixed-wet system with the smaller pores oil-wet (denoted mixed-wet small, MWS). In this case the threshold values of both the P c,ow and the P c,go curves are uniquely ordered (for non-spreading oils). If gas is wetting to water in oil-wet pores (cos θgw o < 0), also the threshold values of the P c,gw curve are uniquely ordered and only one morphology exist, which is as shown in Figure 11(f) with the saturation space for S o < 1 S wet exclusively belonging to region II. If water is wetting to gas in oil-wet pores (cos θgw o > 0) the threshold values of the P c,gw curve may be ordered in six different ways. Hence, similar to Figure 11, six morphologies exist, with the saturation space for S w >S wet always belonging to region III and the space for S o > 1 S wet belonging to region I. The remaining part of the ternary diagram contains a non-linear boundary separating regions II and III, which varies with the orderings of the P c,gw values. Notice that if we assume, realistically, σ go σ ow <σ gw, one of these six morphologies will never occur. 5. Three-Phase Displacements: The Spreading Case Until now, the discussion has been restricted to negative spreading systems since all of the main principles can be shown for this (more general) case. In this section, results are extended to positive spreading systems (C S,o 0). In water-wet pores the spreading condition means that oil spreads as a layer between gas and water. Hence, a gas oil displacement occurs as if the pore were strongly oil-wet (cf. experiments by Kalaydjian (1992)) and a gas water displacement occurs as a double displacement, that is water displaces the oil layer, which in turn displaces gas, or vice versa. In oil-wet pores we find that water, gas and oil are unable to form a three-phase contact-line (Øren and Pinczewski, 1995), implying that water and gas are always separated by a thin oil-film. The latter is consistent with the spreading behaviour of oil in water-wet pores as gas and water are never in direct contact with each other. Hence, in an oil-wet pore a gas water displacement

23 MIXED-WET SYSTEMS 23 occurs also as a double displacement, similar to the gas water displacement in a water-wet pore. As a result, for positive spreading systems the relations between contact angles and interfacial tensions are given by cos θ w ow = cos θ o ow = cos θ o go = cos θ w go = 1, (28a) cos θ w gw = σ go + σ ow σ gw, (28b) cos θ o gw = σ go σ ow σ gw. (28c) Observe that the expressions for the gas water contact angles in oil-wet pores (relations (4c) and (28c)) are equal for spreading and non-spreading oils, both suggesting a double displacement process. Furthermore, under the spreading condition C S,o 0 relations (28b) and (28c), which follow naturally from the pore-scale physics, satisfy the restriction cos θ 1, similar to relations (4b) and (4c). With respect to the various capillary pressure-saturation and relative permeability saturation relations and the regions in which they apply, the only qualitative difference with a negative spreading system is that in an oil-filled medium gas will always invade the pores in decreasing order of size, because the entry conditions are ordered as follows 2σ go r max < 2σ go r wet < 2σ go r min. (29) Hence, for all radii distributions and fractions of water-wet pores, the phasedependency regions are always given by Figure 11(a). The two-phase gas oil capillary pressure curve has only one threshold value P c,go for S o = 1 S wet, but has no jump. Observe that this is consistent with condition (B.1a), related to Figure 11(a), which in turn is consistent with the spreading condition (C S,o 0). Consequently, the behaviour for a spreading oil can be considered as a limiting case of that for a non-spreading oil and the phase-dependencies of the capillary pressure and relative permeability relationships follow accordingly. It can be shown that for mixed-wet small (MWS) systems, the number of possible phase-dependency configurations is the same for positive and negative spreading systems.

24 24 M. I. J. VAN DIJKE ET AL. 6. Summary and Conclusions 6.1. SUMMARY In this paper, we have presented results from a simple process-based model of three-phase displacement cycles for non-spreading and spreading oils in a mixedwet capillary bundle model with the larger pores oil-wet. By studying the possible pore filling sequences, the three-phase flow relations between capillary pressure, relative permeability and saturation have been established for this case. Although it appears initially that many parameter combinations (e.g. of interfacial tensions etc.) may occur and that many pore occupancy situations may arise, it turns out that these are quite restricted. For typical non-spreading gas/oil/water systems found in oil reservoirs, only two important cases need to be considered to see all types of allowed qualitative behaviour for non-spreading oils; these depend upon whether water (cos θgw o > 0) or gas (cos θ gw o < 0) is the intermediate-wetting phase in the oil-wet pores. This leads to up to three phase-dependency regions for the capillary pressures and relative permeabilities. The behaviour of the spreading case turns out to be a special case of the non-spreading case and has only two phasedependency regions in practical cases. It is quite remarkable that such a simple mixed-wet capillary bundle model leads to a complex but constrained range of possible saturation-dependency regions for the capillary pressures and hence the relative permeabilities. A similar range of regions is found for a mixed-wet system with the smaller pores oil-wet. These results clearly indicate the important roles played by interfacial tension (σ ow, σ go and σ gw ), system wettability and pore size distribution during three-phase displacements. These may then be used within this simple process based model to determine the various possible regions, and hence the functional dependencies of the capillary pressures and relative permeabilities CONCLUSIONS The specific conclusions from this work for a mixed-wet system with the larger pores oil-wet (MWL) are as indicated below (the corresponding conclusions for a system with the smaller pores oil-wet (MWS) follow similarly): (i) Depending on the interfacial tensions, the range of the pore size distribution and the fraction of water-wet pores, up to three regions in saturation space can be identified and related to the phase dependencies of three-phase capillary pressures and relative permeabilities. Identification of the capillary pressure/saturation dependencies follows immediately from the presented three-phase floods, as the corresponding saturation paths are in fact isocapillary-pressure curves. The underlying two-phase capillary pressuresaturation relations are of major importance for determining the boundaries of these regions. (ii) In each region of saturation space, one phase turns out to be intermediatewetting for the mixed-wet system as a whole. In such a region, then (a)