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Modelling of Three-Phase Capillary Pressure for Mixed-Wet Reservoirs by

Johan Olav Helland

Thesis submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY (Dr. Ing.)

Faculty of Science and Technology Department of Petroleum Engineering University of Stavanger

July 2005

University of Stavanger N-4036 Stavanger Norway www.uis.no c 2005 Johan Olav Helland ISBN 82-7644-268-4 ISSN 1502-3877

Abstract Knowledge of the relationship between capillary pressure and saturation is required to describe the dynamics of two- and three-phase transition zones in mixed-wet reservoirs. When solving the equations governing fluid flow in reservoir simulation, this relationship is most conveniently formulated as a simple correlation with adjustable parameters that depend on the rock and fluid properties, the displacement process and the saturation history. In this work, I describe a pore model that simulates two- and three-phase, mixed-wet, capillary pressure curves for any sequence of gas, oil and water invasion processes after primary drainage. Three-phase capillary pressure correlations are proposed from the simulated results. The model assumes a bundle-of-triangular-tubes representation of the pore network. Although such a simple geometry does not include hysteresis and residual saturations caused by phase entrapment, we find that other characteristics of mixedwet capillary pressure curves can be reproduced. The triangular pore shape allows for representation of physical processes such as the establishment of mixed wettability within a single pore, and oil drainage through fluid layers in the corners. Contact angle hysteresis leads to a diversity of possible cross-sectional fluid configurations for different sequences of the invasion processes. Accurate expressions for the two- and three-phase capillary entry pressures are derived that truly account for mixed-wet conditions, contact angle hysteresis and the possibility of simultaneous displacement of the fluids occupying a cross-section. The model is employed to analyze the saturation dependencies of the three-phase capillary pressures for various conditions. In many cases we find that more than one capillary pressure depend strongly on two saturations during an invasion process. Based on these results, three-phase capillary pressure correlations are formulated as a sum of two terms, where one term is a function of a decreasing saturation and the other term is a function of an increasing saturation. Thus the correlations depend on the type of displacement process, i.e., the direction of saturation change. The two-saturation dependency, together with the inclusion of adjustable parameters, ensures that the correlations account for different rock and fluid properties, i

ii

saturation histories and displacement paths. The correlations are compatible with a smooth transition between two- and three-phase flow if one of the phases appears or disappears. In particular, if the gas saturation becomes zero, it is shown that the correlations are reduced to a previously published two-phase correlation validated for oil/water systems in mixed-wet rock. The correlations are fitted to capillary pressure curves computed by the pore model, and the match is excellent in all cases. The correlations are validated experimentally by centrifuge measurements performed on water-wet cores. The thesis is organized in two parts. Part I provides an overview of the fundamentals required for descriptions of three-phase pore-scale systems. A chapter is devoted to the description of the developed pore model, with emphasis on the three-phase capillary entry pressures, followed by a chapter where the proposed three-phase capillary pressure correlations are presented. Part I ends with a summary of the main developments and results from the four research papers that have been completed during this study. The research papers are listed below, and they are included in full-length versions in Part II. The papers demonstrate the applicability of the developed pore model to analyze relationships between key parameters in two- and three-phase mixed-wet capillary systems. • Paper A: Helland, J.O. and Skjæveland, S.M.:“Physically-based capillary pressure correlation for mixed-wet reservoirs from a bundle-of-tubes model,” paper SPE 89428 presented at the 2004 SPE/DOE Fourteenth Symposium on Improved Oil Recovery, Tulsa, OK, Apr. 17–21. Accepted for publication in SPE Journal. • Paper B: Helland, J.O. and Skjæveland, S.M.:“Three-phase mixed-wet capillary pressure curves from a bundle-of-triangular-tubes model,” paper presented at the 8th International Symposium on Reservoir Wettability, Houston, TX, May 16–18, 2004. Submitted for publication in J. Pet. Sci. Eng. • Paper C: Helland, J.O. and Skjæveland, S.M.:“Three-phase capillary pressure correlation for mixed-wet reservoirs,” paper SPE 92057 presented at the 2004 International Petroleum Conference in Mexico, Puebla, Mexico, Nov. 8–9. Submitted for publication in SPE Journal. • Paper D: Helland, J.O. and Skjæveland, S.M.: “The relationship between capillary pressure, saturation and interfacial area from a model of mixed-wet triangular tubes.” Unpublished report.

Acknowledgements The author is grateful to Professor Svein M. Skjæveland for helpful discussions and suggestions. Financial support was provided by Statoil through the VISTA program.

iii

Contents Abstract

i

Acknowledgements

iii

List of symbols

vii

I

1

1

Introduction 1.1 Scientific method . . . . . . . . . . . . . . . . . . . . . . . . . .

2

Principles of three-phase pore-scale systems 2.1 Spreading behaviour . . . . . . . . . . . . . . . . . . . . 2.2 Contact angles and wetting preferences . . . . . . . . . . 2.2.1 Wettability alteration and contact angle hysteresis . 2.3 Thermodynamic description . . . . . . . . . . . . . . . . 2.4 Displacement mechanisms . . . . . . . . . . . . . . . . . 2.4.1 Two-phase pore filling mechanisms . . . . . . . . 2.4.2 Three-phase mechanisms . . . . . . . . . . . . . .

3

The pore model 3.1 Pore geometry and network . . . . . . . . . 3.2 Simulation procedure . . . . . . . . . . . . 3.3 Primary drainage and wettability alteration . 3.4 Fluid configurations . . . . . . . . . . . . . 3.5 Three-phase capillary entry pressures . . . 3.5.1 Conditions for layer formation . . . 3.5.2 Layer collapse capillary pressures . 3.5.3 Gas invasion – general equations . . v

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21 21 21 22 23 29 29 30 33

vi

CONTENTS

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36 36 37 42 45

4

Correlations 4.1 Three-phase correlation design . . . . . . . . . . . . . . . . . . . 4.2 Applications and discussion . . . . . . . . . . . . . . . . . . . .

47 48 49

5

Summary of the papers 5.1 Paper A . . . . . . 5.2 Paper B . . . . . . 5.3 Paper C . . . . . . 5.4 Paper D . . . . . .

51 51 52 54 55

3.6

3.5.4 Water invasion – general equations . 3.5.5 Oil invasion – general equations . . 3.5.6 Capillary entry pressure algorithm . 3.5.7 Discussion . . . . . . . . . . . . . Applications and further work . . . . . . .

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Bibliography

II Paper A: Physically-based capillary pressure correlation for mixed-wet reservoirs from a bundle-of-tubes model Paper B: Three-phase mixed-wet capillary pressure curves from a bundle-oftriangular-tubes model Paper C: Three-phase capillary pressure correlation for mixed-wet reservoirs Paper D: The relationship between capillary pressure, saturation and interfacial area from a model of mixed-wet triangular tubes

57

List of symbols A Acij A(k) ij Al a bij(k) bpd c Cs d dx F Iij(k) Lf L (k) fij Ls L (k) sij Nij Pc X→Y Pcij

p Q r R S T V Vp W

Total cross-sectional area of a tube Total cross-sectional corner area bounded by ij AMs Cross-sectional bulk area bounded by ij AM k Cross-sectional layer area, see Eq. (3.42) Specific interfacial area Distance from the corner to the position of ij AM k Length of water-wet surface in a corner Parameter in capillary pressure correlation Spreading coefficient Parameter in capillary pressure correlation Virtual displacement Helmholtz free energy Indicator notation for ij AM k, see Eq. (3.23) Cross-sectional fluid-fluid length Cross-sectional fluid-fluid length of ij AM k Cross-sectional fluid-solid length Cross-sectional fluid-solid length of ij AM k Total number of ij AMs present in a corner Capillary pressure Capillary entry pressure between phases i and j for piston-like displacement from configuration X to Y Phase pressure Heat Radius of curvature Radius of the inscribed circle Phase saturation Temperature Phase volume Pore volume Work vii

viii

CONTENTS

x α

βij(k) η θ (k) θijh ξ σ l

Random number between 0 and 1 Corner half angle Angle defined from geometry of ij AM k, see Eq. (3.28) Interfacial area Parameter in the Weibull distribution Contact angle Hinging contact angle of ij AM k Entropy Geometry factor in expressions for layer collapse capillary pressures, see Eq. (3.12) Interfacial tension Effective perimeter of fluid layer, see Eq. (3.43)

Subscripts a c ch ext f g h max min o pd r s tot w

Advancing Capillary or corner Characteristic External Fluid Gas Hinging Maximum Minimum Oil Primary drainage Receding Solid Total Water

Superscripts col eq fin init (k) max

Collapse of fluid layer Equilibrium Final Initial AM number counted in order from corner towards center Maximum

CONTENTS X→Y

ix

Displacement from configuration X to Y

Abbreviations AM C D I ij AM k MS–P MTM WAG SWAG

Arc meniscus Constant saturation Decreasing saturation Increasing saturation The kth AM between phase i and j Mayer and Stowe – Princen Main terminal meniscus Water alternate gas Simultaneous water alternate gas

Part I

1

Chapter 1

Introduction A petroleum reservoir may have an oil zone located below a gas cap and above the water zone. Production from the oil zone causes an upward movement of the oil-water contact and a downward movement of the gas-oil contact. Water may displace oil and gas, and gas may displace water and oil. Water and/or gas may also be injected to increase oil recovery by pressure maintenance. The gas cap may expand if the pressure decreases below the dewpoint pressure, resulting in evaporation of the oil phase. Similarly, the oil zone may expand if the pressure increases above the bubblepoint pressure and gas dissolves in the oil phase. Other scenarios are possible as well. Modelling of the dynamics of gas-oil and oil-water transition zone movements in the reservoir requires three-phase capillary pressure curves. The transition zone dynamics is dominated by capillary-gravity forces, and the relationships between the capillary pressures and the fluid saturations are required in the entire saturation space. Capillary pressure is defined as the pressure difference across an interface between two fluids. Throughout this work, we consistently calculate capillary pressure through the lighter phase. For a system of gas, oil and water, this yields Pcow =

po − pw ,

(1.1a)

Pcgo =

pg − po ,

(1.1b)

Pcgw =

pg − pw .

(1.1c)

Eqs. (1.1a)–(1.1c) may be combined to obtain a relationship between the three capillary pressures, Pcgw = Pcgo + Pcow ,

(1.2)

implying that only two of the capillary pressures may vary independently. The 3

4

Introduction

fluid saturations are defined as Si =

Vi , Vp

i = g, o, w,

(1.3)

where V p is the total pore volume and Vi is the volume of phase i. By definition, the saturations satisfy (1.4) Sg + So + Sw = 1, implying that only two of the saturations may vary independently. Throughout this study, drainage is referred to as a process where a lighter fluid displaces a denser fluid. Likewise, imbibition is referred to as a process where a denser fluid displaces a lighter fluid. When solving the equations governing fluid flow in reservoir simulation, the capillary pressure vs. saturation relationship is most conveniently formulated as a simple correlation with adjustable parameters. In the reservoir, situations may occur where one of the phases appears or disappears, e.g., during phase transitions between gas and oil, or when a zero residual oil saturation is approached by drainage through continuous spreading layers in the crevices of the pore space [1]. To implement these scenarios in a numerical reservoir simulator without creating convergence problems, a smooth transition is required between two- and threephase flow. Among the empirical capillary pressure correlations reported in the literature, the Brooks-Corey [2] and the van Genuchten [3] expressions are some of the most frequently used because of the simple functional forms and the solid experimental validation. The correlations typically include two adjustable parameters, where one depends on the level of capillary pressure, and the other depends on the geometry of the pore network. However, most of the capillary pressure correlations have been developed for drainage of two-phase systems in water-wet media. Skjæveland et al. [4] extended the Brooks-Corey formula to also account for imbibition, secondary drainage and subsequent scanning loops for mixed-wet conditions. Huang et al. [5] derived analytical correlations for primary drainage and the bounding hysteresis loop for mixed-wet conditions assuming a bundle-of-tubes representation of the pore network. Three-phase capillary pressure curves have traditionally been predicted from corresponding two-phase measurements. This is based on the assumption made by Leverett [6] that the gas-oil capillary pressure in a three-phase system, Pcgo , is equal to the gas-oil capillary pressure in the corresponding two-phase gas-oil system, implying that Pcgo is a function of only Sg . It is also commonly assumed that the oil-water capillary pressure in a three-phase system, Pcow , is equal to the oil-water capillary pressure in a corresponding two-phase oil-water system, implying that Pcow is a function of only Sw [7]. These assumptions have been supported

1.1 Scientific method

5

by three-phase capillary pressure measurements in water-wet systems performed by Lenhard and Parker [8]. However, more recent measurements have shown that this practice may not be valid for other conditions [9]. Moreover, micromodel studies of three-phase flow have revealed that the fluid distribution and the displacement mechanisms at the pore scale may be more complex than for two-phase systems [10–12]. These findings emphasize the need for direct measurements of three-phase capillary pressure curves for various conditions. To our knowledge, measurements with three varying saturations have only been reported by Kalaydjian [13] and, more recently, Virnovsky et al. [14]. The capillary pressures were measured in water-wet sandstone core samples only. Bradford and Leij [9, 15, 16] measured three-phase capillary pressures in sandpacks for several wetting conditions achieved by mixing different fractions of water-wet and oil-wet sands. In these experiments one saturation was kept fixed. In three-phase flow there is an infinite number of possible displacement paths because of two independent saturations, by Eq. (1.4). Hence, it is impractical to perform time-consuming measurements of a vast amount of different processes for several rock and fluid properties. This points out the importance of developing physically-based pore-scale network models to compute the capillary pressure curves [17–22]. A pore-scale model, tuned to reproduce the measured data, may be employed to predict capillary pressure curves for other displacement paths not covered by the measurements. A plausible range of values of the parameters to be used in empirical capillary pressure correlations could then be estimated from the computed results for the specific rock and fluid properties under consideration.

1.1 Scientific method The primary objective of this work has been to develop a three-phase capillary pressure correlation for mixed-wet reservoirs that can be used to describe the dynamics of three-phase transition zones for various rock and fluid properties and displacement histories. The correlation should be based on sound physical principles yet sufficiently simple to be included in a reservoir simulator. Since an infinite number of possible displacement paths can occur in threephase flow, we have chosen to develop a simple pore model that computes the capillary pressure vs. saturation relationships for any displacement path in the three-phase saturation space. The model assumes a bundle-of-tubes representation of the pore network, the tubes having triangular cross-sections. This is a very simplistic approach for modelling of realistic reservoir rocks. Such a simple model does not incorporate the effect of interconnected pore networks, and hence phase entrapment is absent. However, the angular pore shapes allow for other impor-

6

Introduction

tant physical processes, such as the development of mixed wettability at the pore scale [23,24], and drainage through oil layers along the corners [25]. Moreover, the presence of the invading fluid in the corners of the pores may lead to a more complex capillary behavior [26]. Simple models are invaluable as they allow for careful interpretation of the simulated results, and hence the trends observed by varying a parameter may be analyzed completely and discerned from observed trends resulting from variations of other parameters. As already mentioned, simple models also offer the possibility of deriving analytical expressions between key parameters that describes the general trends observed in measured data. The developed bundle-of-triangular-tubes model is employed to simulate threephase capillary pressure curves for various rock and fluid properties, saturation histories and displacement paths. The simplicity of the model makes it possible to analyze the saturation dependencies of the capillary pressures in detail for different conditions. This has previously been done for a simpler bundle-of-cylindricaltubes model [27–30]. From the resulting saturation dependencies and analysis of the three-phase capillary pressure curves, we propose correlations that relate the capillary pressures to the appropriate saturations. The suggested correlations may be compared with capillary pressure measurements, from which physically realistic values of the correlation parameters are obtained. The model, tuned to reproduce the general trends observed in the measurements, may then be used to simulate other displacement paths not covered by the measured data. Reasonable values of the correlation parameters can eventually be obtained from the simulation results. The model may also prove useful to analyze how the correlation parameters generally relate to different rock and fluid properties, displacement paths and saturation histories. Such a systematic method anchored to a physically-based model to determine the correlation parameters will increase the reliability of the correlation for use in reservoir simulation. Part I of the thesis is organized as follows: Chapter 2 introduces parameters, concepts, theoretical approaches and mechanisms required to describe the physics of three-phase pore-scale systems. The three-phase mixed-wet bundle-oftriangular-tubes model is described in Chapter 3. Most of Chapter 3 is devoted to the derivation of the three-phase capillary entry pressures for the oil, water and gas invasion processes. General equations are written that accounts for all the assumed displacements between the different fluid configurations. The main steps in the capillary entry pressure algorithm is described next for cases of the oil, water and gas invasion processes. The proposed capillary pressure correlations are formulated in Chapter 4. Finally, a summary of each paper is given in Chapter 5.

Chapter 2

Principles of three-phase pore-scale systems The pore-scale distribution of three fluids in a porous medium is determined by knowledge of the interactions between the fluids (spreading behaviour), interactions between the fluids and the solid (wettability), and interactions between bulk fluids across curved interfaces (capillary pressure).

2.1 Spreading behaviour In a system of three fluids in contact with each other the force exerted at the contact line may be expressed in terms of the spreading coefficient. For systems of gas, oil and water, the initial spreading coefficient of oil is defined as [31] init init init − σgo − σow , Csinit = σgw

(2.1)

init init init , σgo and σow are fluid-fluid interfacial tensions evaluated for the three where σgw pairs of fluids before they are brought in contact with the third fluid. After oil has been introduced on a gas-water surface, and thermodynamic equilibrium is reached, the equilibrium spreading coefficient is nonpositive and given by [1, 32, 33] eq eq eq − σgo − σow . (2.2) Cseq = σgw eq

eq

eq

Thus, σgw , σgo and σow represent interfacial tensions measured in thermodynamic equilibrium of the three-phase system. The equilibrium and initial spreading coefficients are in general different because of the possibility of contamination of the gas-water interface and formation of thin molecular films when oil is added to the eq gas-water system. This may lead to a significantly lower value of σgw . According to Blunt et al. [1], three scenarios are possible: 7

8

Principles of three-phase pore-scale systems

eq s gw

g

eq s gw

eq s go

g

eq s go

o w

eq s ow

o w

eq s ow

s

s

(a)

(b)

Figure 2.1: Illustrations of three-phase systems in thermodynamic equilibrium. (a) eq eq Case (i) where Csinit < 0 and Cs < 0. (b) Case (ii) where Csinit > 0 and Cs < 0. The illustrations are reproduced from Blunt et al. [1].

eq

(i) If Csinit < 0 and Cs < 0, the contact line is stable and the drop of oil remains stationary on the water surface, as illustrated in Fig. 2.1(a). eq

(ii) If Csinit > 0 and Cs < 0, the three-phase contact line is unstable and the oil spreads on the water surface. Excess oil remains in a lens in equilibrium with the oil film as more oil is added to the system, see Fig. 2.1(b). eq

(iii) If Csinit > 0 and Cs = 0, the contact line is unstable and the oil spreads on the water surface. The film swells as more oil is added to the system. Throughout this work we only consider spreading coefficients and interfacial tensions evaluated at thermodynamic equilibrium, and thus the superscripts are omitted. Consequently, Cs ≤ 0. Cases where molecular oil films may exist between gas and water are modelled by assuming a reduced σgw .

2.2 Contact angles and wetting preferences The wetting preference of a solid surface in contact with two fluids is typically characterized by the contact angle. Assuming that the denser phase j is wetting relative to phase i, then cos θij ≥ 0, where the contact angle θij is measured through phase j . In equilibrium, the horizontal force balance at the fluid-fluid-solid contact line is given by Young’s equation [32]. For oil-water, gas-oil and gas-water

2.2 Contact angles and wetting preferences

s go

s ow w q ow s ws

o

s os

9

o q go

g

s gs

s gw g

s os

s gs

s

s

s

w q gw s ws

Figure 2.2: Force balance at the oil-water-solid contact line (to the left), gas-oilsolid contact line (in the middle) and gas-water-solid contact line (to the right).

interfaces located on a solid surface, the three relations are (see Fig. 2.2): σgs = σos + σgo cos θgo ,

(2.3a)

σos = σws + σow cos θow ,

(2.3b)

σgs = σws + σgw cos θgw .

(2.3c)

Elimination of the fluid-solid interfacial tensions σos , σws and σgs provides a constraint on the three-phase contact angles and the fluid-fluid interfacial tensions [33]: σgw cos θgw = σow cos θow + σgo cos θgo .

(2.4)

For strongly wetting conditions, the solid surface may be coated by a thick film of the wetting phase. If a thick water film is present and the deformation of the water phase at the three-phase gas-oil-water contact line is neglected, Eq. (2.3a) may be used with subscript s replaced by subscript w, resulting in the following relation [10, 31, 33, 34]: cos θgo =

σgw − σow Cs =1+ . σgo σgo

(2.5)

Thus, cos θow = cos θgw = 1 to satisfy Eq. (2.4). Generally, σgw > σow , and by Eq. (2.5), cos θgo > 0, implying that gas is non-wetting relative to oil. Correspondingly, if a solid surface is coated by an oil film, Eq. (2.3c) is employed with subscript s replaced by o, resulting in the expression cos θgw =

σgo − σow . σgw

(2.6)

Thus, by Eq. (2.4), the remaining contact angles satisfy cos θgo = 1 and cos θow = −1. Normally, σow > σgo , implying that cos θgw < 0, and hence gas is wetting relative to water in this case. Using similar reasoning, it is found that the wetting order in a three-phase fluid system of oil, water and gas may be divided into three categories [24, 35]:

10


(i) In water-wet media, water is wetting, oil intermediate-wetting, and gas nonwetting. The contact angles satisfy θow ≤ π2 , θgo ≤ π2 , and θgw ≤ π2 . (ii) In weakly oil-wet media, oil is wetting, water intermediate-wetting, and gas non-wetting (θow > π2 , θgo ≤ π2 , and θgw ≤ π2 ). (iii) In strongly oil-wet media, oil is wetting, gas intermediate-wetting, and water non-wetting (θow > π2 , θgo ≤ π2 , and θgw > π2 ). van Dijke and Sorbie [30] have proposed linear relationships of cos θgo and cos θgw as functions of cos θow accounting for the above wetting orders: 1 (Cso cos θow + Cso + 2σgo ), 2σgo

(2.7)

1 ((Cso + 2σow ) cos θow + Cso + 2σgo ). 2σgw

(2.8)

cos θgo = and cos θgw =

Thus, assuming that the underlying wettability is known in terms of the oil-water contact angles, calculations of θgo and θgw are possible by Eqs. (2.7), (2.8).

2.2.1 Wettability alteration and contact angle hysteresis According to Anderson [36], reservoirs range from weakly water-wet to oil-wet, even though most reservoir rocks originally are strongly water-wet. Laboratory experiments of crude oil/brine/rock systems indicate that the wettability of the rock surface changes by adsorption of crude oil components [36–38]. The degree of wettability alteration is affected by the mineralogy of the rock surface and the chemical composition of the oil and brine [37, 38]. This leads to development of heterogeneous forms of wettability, where some parts of the rock surface are altered oil-wet, while other parts remain water-wet. In the literature, it is often distinguished between fractional and mixed wettability [36]. Fractional wettability refers to the case when oil-wet surfaces or/and water-wet surfaces are discontinuous. Salathiel [39] introduced the term mixed wettability to describe cases where the oil-wet and water-wet surfaces form continuous pathways through the reservoir rock. He observed that mixed wettability may lead to little phase entrapment of oil. Kovscek et al. [23] developed a model to describe the development of mixed wettability at the pore level. They considered pores formed by the space between four cylinders. Initially, the pores are strongly water-wet and completely filled with water. During primary drainage, representing primary oil migration, oil invades the bulk of the pore space while thin water films protect the pore walls from

2.2 Contact angles and wetting preferences

11

being contacted by oil. The stability of these films depend on the prevailing capillary pressure and the shape of the disjoining pressure isotherm [31]. When a critical value of capillary pressure is reached, the water film collapses, oil contacts the pore walls, and the wettability changes. The corners where water is still present, remain water-wet. We adopt the somewhat simplified model for wettability change proposed by Hui and Blunt [24]. They did not consider stability and collapse of water films as this requires knowledge of the disjoining pressure isotherm and molecular properties. Instead they assumed that oil contacted the pore walls immediately after invasion, leaving the sides wettability-altered and the corners water-wet. To describe cases where the water films do not rupture, a small oil-water contact angle is assumed in subsequent displacement processes. This model has also been included in sophisticated pore-scale network models [21, 22]. The value of the contact angles is not only affected by the presence or absence of thin films. They also vary with the direction of the displacement. For example, if oil displaces water, the oil-water interface is receding with contact angle θowr , and if water displaces oil, the interface is advancing with a larger contact angle θowa . The difference between receding and advancing contact angles is referred to as contact angle hysteresis. This feature contributes to hysteresis between capillary pressure curves for drainage and imbibition [40]. The degree of contact angle hysteresis is affected by surface roughness [41], microscopic wettability heterogeneity [42], as well as fluid composition and adsorption of crude oil components [43, 44]. Experimentally measured oil-water receding and advancing contact angles often seem to differ by 60◦ , including cases where cos θowr > 0 while cos θowa < 0, see for example [42, 44]. This indicates oil-wet behaviour during imbibition and water-wet behaviour during secondary drainage. In this study we account for wettability alteration and contact angle hysteresis by specifying a small contact angle in primary drainage, θpd , and any advancing and receding oil-water contact angles, θowa and θowr , respectively, that satisfy θpd ≤ θowr ≤ θowa . We also take into account hysteresis in the gas-oil and gas-water contact angles. If gas displaces oil and water, the gas-oil and gas-water interfaces are receding with contact angles θgor and θgwr calculated from Eqs. (2.7), (2.8) with θow = θowr . Correspondingly, oil and water displace gas with advancing contact angles θgoa and θgwa calculated from Eqs. (2.7), (2.8), respectively, assuming θow = θowa . However, as noted by Piri and Blunt [22], there is an ambiguity by calculating receding and advancing contact angles from Eqs. (2.7), (2.8). To illustrate this, consider cases where Cs < 0 in Eq. (2.7), implying that cos θgo decreases with increasing cos θow (see also Fig. 2(a) in [30]). Thus, if θgor is calculated using θowr , and θgoa is calculated using θowa , we find that θgoa < θgor which is not physically realistic. To ensure that the advancing contact angles are larger than the receding contact angles, we first calculate the contact angles as described above, and then

12


we take the larger values as the advancing contact angles and the smaller values as the receding contact angles, as in the work by Piri and Blunt [22].

2.3 Thermodynamic description In this section we follow the approach by Morrow [45] and Bradford and Leij [46], and consider a closed system consisting of a porous medium saturated by gas, oil and water, see Fig. 2.3. The three fluids are assumed to be continuous and in contact with their respective reservoirs. The fluid saturations change if external work is done on the system, for instance by movement of frictionless pistons in contact with the reservoirs, according to pi d Si , (2.9) Wext = V p i=g,o,w

where V p is the total pore volume of the medium. The saturations and the fluidsolid interfacial areas satisfy d Sg + d So + d Sw = 0,

(2.10a)

dgs + dos + dws = 0.

(2.10b)

The total work carried out by the closed system consists of a term due to expansion or compression of the fluid volumes and a term representing the work done by the interfaces: pi d Vi − σij dij , (2.11) W = i=g,o,w

ij=go,ow,gw,gs,os,ws

where d Vg + d Vo + d Vw = 0, since the total volume is constant. By the first law of thermodynamics the change in internal energy is pi d Vi + σij dij . (2.12) dE = dQ − i=g,o,w


For a reversible change between energy states the heat flow into the system is d Q = T d, where T and are the temperature and entropy of the system, respectively. The change in Helmholtz free energy of the closed system is thus given by pi d Vi + σij dij . (2.13) d F = d E − d(T ) = −dT − i=g,o,w


2.3 Thermodynamic description

13 Piston

pg Gas reservoir Piston

po

Oil reservoir

Porous medium

Water reservoir

Piston

pw

Figure 2.3: Illustration of the system used in the thermodynamic description of three-phase capillary pressure. The decrease of Helmholtz free energy of the surroundings is due to the external work done on the system, Wext . Thus, for the porous medium and the surroundings the total Helmholtz free energy is given by pi d Si −dT − pi d Vi + σij dij . (2.14) d Ftot = −V p i=g,o,w

i=g,o,w


A system of constant total volume and temperature is at equilibrium with its surroundings when the total Helmholtz free energy is at a minimum [47], i.e., d Ftot = 0. If we assume that the system is isothermal (dT = 0) with incompressible fluids (d Vi = 0, i = g, o, w), then the change in Helmholtz free energy of the system is given by the change in surface free energy, i.e., σij dij , (2.15) dF = ij=go,ow,gw,gs,os,ws

and the condition for equilibrium is pi d Si + d Ftot = −V p i=g,o,w

σij dij = 0.

(2.16)


Eq. (2.16) states that the external work performed on the system is balanced by the increase of surface free energy of the system. Eq. (2.16) is revisited in Section 3.5 as it forms the basis of the derivation of capillary entry pressures in capillary tubes. For a two-phase system of oil and water, Eq. (2.16) reduces to V p Pcow d So = σow dow + (σos − σws )dos ,

(2.17)

by using the definition of capillary pressure and Eqs. (2.10a), (2.10b). Moreover, if we assume that Young’s equation, Eq. (2.3b), is valid at the porous media scale,

14


the fluid-solid interfacial tensions may be eliminated from Eq. (2.18), resulting in the expression [45, 46]: Pcow d So = σow (daow + cos θow daos ),

(2.18)

where the specific interfacial areas have been introduced by aij = Vijp . The use of Young’s equation in Eq. (2.18) has been questioned due to the mixing of scales [48]. As demonstrated in Section 2.2, Young’s equation is derived from a force balance at the three-phase contact line. Thus, Young’s equation is valid at the pore scale and may not be reformulated with a macroscopic contact angle to facilitate applications at the porous media scale. However, Eq. (2.18) is still valid for a single capillary tube. The derivation of Eqs. (2.17), (2.18) is based on reversible displacement processes. In drainage, d So > 0, and hence work is done on the system, resulting in a corresponding increase of surface free energy, d F > 0. In imbibition, d So < 0, and hence the system does work on its surroundings, resulting in a decrease of surface free energy, d F < 0. According to Morrow [45], hysteresis between drainage and imbibition results from spontaneous re-distribution of the fluids at constant saturations. In such displacements no work is done, while surface free energy is lost to heat. Since it is assumed that the heat capacity of the system is large, the accompanying increase of temperature is neglected. Eqs. (2.17), (2.18) relate capillary pressure to interfacial area and saturation. Hassanizadeh and Gray [48] suggested that such a formal relationship exists, and they hypothesized that the hysteresis in the Pc − S relationship was an artifact of projecting the a − Pc − S surface onto the Pc − S plane. Pore-scale network models may prove useful to investigate this conjecture [49, 50], while less sophisticated models are useful to develop analytical expressions to gain insight into the qualitative behavior of the relationship [51, 52].

2.4 Displacement mechanisms 2.4.1 Two-phase pore filling mechanisms In two-phase flow, pore filling can occur by piston-like invasion and snap-off [53, 54]. In piston-like invasion, the invading fluid enters the pore from one of the ends, displacing the fluid originally present. This displacement occurs during drainage when the capillary pressure is increased to the capillary entry pressure required for invasion of the interface. In imbibition, piston-like invasion occurs when the capillary pressure is decreased to the capillary entry pressure required for the displacement. The capillary entry pressures are calculated by the Laplace

2.4 Displacement mechanisms

15

equation [55], Pcij = σij

1 r1

+

1 , r2

(2.19)

where r1 and r2 are principal radii of curvature for the interface separating the phases i and j . Snap-off occurs as a result of increased content of wetting phase from the corners or by swelling of films. Eventually, the wetting phase may invade the entire cross-section and cut off the non-wetting phase, resulting in trapped compartments, bubbles or drops of non-wetting phase. According to Hui and Blunt [24] and Lenormand [53], snap-off does only occur if piston-like invasion is topologically impossible since the capillary pressure for piston-like invasion is favourable compared to the capillary pressure for snap-off. The Mayer–Stowe–Princen (MS–P) method A method to calculate capillary entry pressures for piston-like invasion into angular tubes is developed by Mayer and Stowe [56], who considered breakthrough pressure in the space between packed spheres, and Princen [57–59], who considered height of capillary rise in the space formed between cylinders. The method was referred to as the MS–P method by Mason and Morrow [60], and has later been extended to account for irregular pore shapes [18, 60, 61] and mixed-wet conditions [18, 26]. In this section the MS-P method is applied to derive the capillary entry pressures for piston-like invasion into uniformly wetted, regular, n-sided tubes. To explain the method we consider a tube with an equilateral, triangular, cross-section as the illustrating example. This cross-sectional shape is characterized by the radius R of the inscribed circle and the half-angle of a corner, α = π6 . The cross-section does not vary with respect to tube length. Because of symmetry, it suffices to consider the shaded area of the cross-section shown in Fig. 2.4(a). We consider invasion of oil into a water-wet tube completely filled with water. Contact angle hysteresis is neglected. Oil invasion may then result in a crosssectional fluid configuration in which oil has occupied the bulk area, while water is still residing in the corners as shown in Fig. 2.4(b). The invading interface moving in the direction along the tube, separating the bulk fluids, is referred to as the main terminal meniscus (MTM), and the interface located in the corners of the cross-section, separating bulk fluid from corner fluid, is referred to as the arc meniscus (AM). To calculate the entry pressure by the MS-P method, gravity effects are neglected, and thus the capillary pressure is uniform. Due to the constant cross-section of the tube, it is assumed that the MTM passes through the entire tube length at the capillary entry pressure. Therefore, the curvature of the AMs

16


AM

w o MTM

R

a

(b)

(a)

Figure 2.4: (a) Area of the cross-section employed in the derivation of the capillary entry pressure. (b) Illustration of piston-like oil invasion into a waterfilled tube.

is constant and equal to the entry curvature of the MTM during this displacement. Sufficiently far behind the MTM, the curvature of the AMs is represented by a cross-sectional circular arc. Thus, from Eq. (2.19) with r1 = row and r2 = ∞, the capillary entry pressure can be expressed as Pcow =

σow , row

(2.20)

where row is the entry radius of curvature measured through the oil phase. The MS-P method is founded on an energy balance equation which equates the virtual work with the associated change of surface free energy for a small displacement dx of the MTM in the direction along the tube. The energy balance equation then relates the entry radius of curvature expressed by Eq. (2.20) to the cross-sectional area exposed to change of fluid occupancy, Aow , the bounding cross-sectional fluidsolid and fluid-fluid lengths, Lsow and L fow , respectively, and the contact angle θow , as depicted in Fig. 2.5(b). For oil invasion into a uniformly wetted tube completely filled with water, there are generally two scenarios that must be considered separately depending on the contact angle: (i) If θow < π2 − α , formation of arc menisci occurs (see Fig. 2.4(b)), and the capillary entry pressure is positive. (ii) If θow ≥ π2 − α, AMs do not form, as there is no position inside the triangle where an AM can be located and at the same time have the same curvature and contact angle as the MTM. Thus, the entire cross-section is completely


17

AM

I

w

AM

II

o Aow

w

o

MTM dx

II I

b ow A ow w L fow row a q ow Lsow

R

(b)

(a)

Figure 2.5: Representation of the AM and MTM in a corner during oil invasion. (a) View in the direction along the tube length. (b) Cross-sectional view behind the MTM. The MS-P method relates the entry radius row to the parameters Aow , L sow , L fow and θow .

filled with oil after invasion. For the limiting case when θow = AM would touch the apex of the corner exactly.

π 2

− α, any

We consider case (i) first. The virtual work required to displace the MTM a distance dx is given by (2.21) Wext = Pcow Aow dx. The accompanying increase of surface free energy is given by d F = {(σos − σws )L sow + σow L fow }dx.

(2.22)

The energy balance equation, Wext = d F, then yields Pcow Aow = σow (L fow + cos θow L sow ),

(2.23)

where we have employed Eq. (2.3b) to eliminate the fluid-solid interfacial tensions. Notice that Eq. (2.23) is equivalent to Eq. (2.18) with Vp d So = Aow dx, V p daow = L fow dx and Vp daos = L sow dx. Substitution of Eq. (2.20) into Eq. (2.23) provides an equation that relates row to pure geometry: row =

Aow . L sow cos θow + L fow

(2.24)

18


From geometry considerations of Fig. 2.5(b) we find the following relations: βow =

π − α − θow , 2

(2.25)

L fow = row βow , L sow =

(2.26)

row sin βow R − , tan α sin α

(2.27)

2 r 2 sin βow sin(α + βow ) row R2 βow − ow + . (2.28) 2 tan α 2 sin α 2 Inserting Eqs. (2.25)–(2.28) into Eq. (2.24) then yields the following second order polynomial to be solved for row :

Aow =

−

cos(θow + α) cos θow π α θow 2 R cos θow R2 + − − row + row − = 0. 2 sin α 4 2 2 tan α 2 tan α

The solution must agree with a position of the AM located between the apex of the corner and the midpoint of the side. This condition may be formulated as 0 ≤ row ≤

R cos α . sin βow

(2.29)

The only solution that satisfies Eq. (2.29) is given by

row = cos θow +

R tan α (sin 2θow − 2θow − 2α − π ) 2

.

(2.30)

Finally, substituting Eq. (2.30) into Eq. (2.20) yields the expression for the capillary entry pressure. In case (ii), AMs do not form, and the cross-sectional bounding oil-water length, L fow , is zero. Furthermore, the area Aow is equal to the area of the entire cross-section, and the bounding cross-sectional fluid-solid length Lsow is equal to the total perimeter: L fow = 0,

Aow =

R2 , 2 tan α

L sow =

R . tan α

Inserting these expressions into the energy balance, Eq. (2.23), yields the simple Young-Laplace equation, 2σow cos θow , (2.31) Pcow = R which is also valid for cylindrical tubes.


19

2.4.2 Three-phase mechanisms When a third phase is introduced, other diplacement mechanisms can occur in addition to those mentioned above. As described in Section 2.1, oil spreads between gas and water if the initial spreading coefficient is positive. However, thick oil films or oil layers may also form in the corners of the pore space, separating water in the corners from gas in the bulk portion. Dong et al. [62] have shown that oil may spontaneously imbibe as a film over water, resulting in formation of oil layers in the corners. Their analysis were supported by experiments in micromodels. Fenwick and Blunt [63] and Firincioglu et al. [64] have formulated geometric conditions for the existence of these layers that depend on the contact angles, pore shape, the capillary pressures and the interfacial tensions. The oil layers are assumed to collapse when the interfaces surrounding the oil layers meet. Based on these geometric conditions, Fenwick and Blunt [63] and Keller et al. [11] showed that oil layers can be present even for large negative Cs , but as Cs decreases, oil layers are less likely to exist. This is in agreement with the micromodel experiments performed by Dong et al. [62]. Following the same approach, Piri and Blunt [22] derived expressions for the existence of gas, water, and oil layers surrounded by the two remaining fluids. Oil layers have been observed experimentally in glass tubes [25, 64], and they are believed to cause efficient oil recovery during gravity drainage [1]. In mixed-wet pores, oil layers surrounded by bulk water and water in the corners may also exist [18, 23, 24]. According to Kovscek et al. [23], such oil bridges have been observed in micromodels. The conditions for existence and stability of fluid layers are outlined in Section 3.5.2. The presence of more than one fluid in the cross-section of an angular pore leads to the possibility of simultaneous displacement of the three fluids during piston-like invasion. Recently, van Dijke et al. [65,66] have derived capillary entry pressures for such displacements when wettability is uniform. In these cases, the capillary entry pressure for displacement of the interface separating the bulk phases is also dependent on the pressure in the remaining phase. In Section 3.5, we extend their method to also account for mixed-wet conditions and contact angle hysteresis. The presence of three phases also make other types of displacement mechanisms possible. Mechanisms referred to as double displacements [10,11,17,63,67], are composed of two two-phase piston-like displacements: one fluid displaces another which again displaces a third. Double displacements can only occur if the first and the third fluid are continuous. Six different types of double displacements are possible [63]. As an example, if gas invades oil which again displaces water, the displacement is called double drainage [10]. Double drainage has been observed in micromodels and may result in coalescence and reconnection of discontinuous oil [10, 67]. Keller et al. [11] observed four of the six possible double

20


displacements in a water-wet micromodel. An extension of double displacements has been introduced by van Dijke and Sorbie [68]. They consider multiple displacement chains, i.e., when a continuous invading fluid triggers displacements of disconnected phase clusters by a chain of interface displacements throughout the pore network before a final displacement of a continuous fluid occurs. Such displacements have also been observed in micromodel experiments by van Dijke et al. [12]. The simple pore model developed in this work is not adapted for studies of double and multiple displacements as effects resulting from the interconnection of the pore network is neglected.

Chapter 3

The pore model 3.1 Pore geometry and network The pore network is represented as a bundle of parallel tubes, the tubes having equilateral, triangular cross-sections. As mentioned in Section 2.4.1, the geometry of an equilateral triangle is described by the half-angle of the corner, α = π6 , and the radius of the inscribed circle R. The pore-size frequency is described by a truncated two-parameter Weibull distribution. This is a flexible distribution that has often been employed frequently for this purpose [24, 63, 69]. The pore sizes R are selected from the cumulative distribution function in the following manner: Pick random numbers x ∈ [0, 1] and calculate the the inscribed radius from R − R η 1 max min ) + x] η + Rmin . (3.1) R = Rch − ln[(1 − x) exp(− Rch The cross-sectional area of a tube, A, is related to R by A=

3R 2 . tan α

(3.2)

3.2 Simulation procedure The model is programmed to simulate gas, oil and water invasion processes in any sequence starting with primary drainage of a waterfilled and water-wet medium. An invasion process is simulated by increasing or decreasing a capillary pressure stepwise until some maximum or minimum value is reached. At each step the cross-sectional fluid occupancies in the tubes are updated and the saturation is calculated. The saturations are calculated based on the fraction of the cross-sectional area that each phase occupies. Invasion of the oil phase is simulated by increasing 21

22

The pore model

Pcow at a constant Pcgw . At each pressure step Pcgo is calculated from Eq. (1.2). Oil invasion into a tube occurs if Pcow is larger than or equal to the oil-water capillary entry pressure of the tube. During water invasion Pcow is decreased at a constant Pcgo , and Pcgw is calculated by Eq. (1.2). Water invasion into a tube occurs if Pcow is smaller than or equal to the associated oil-water capillary entry pressure. During gas invasion, Pcgo is increased, and Pcgw is calculated from Eq. (1.2) assuming a constant Pcow . Gas invades a tube if Pcgo is larger than or equal to the gas-oil capillary entry pressure for that tube. This invasion algorithm has also been applied by van Dijke et al. [27, 28] in a bundle of cylindrical tubes. To simulate a predetermined sequence of several gas, oil and water invasion processes, a list of capillary pressures is specified where each value corresponds to the capillary pressure at which the specific process is terminated. The capillary pressures are expressed in terms of the radii of curvature by Eq. (2.20): σij , ij = go, ow, gw. (3.3) rij = Pcij A combination of Eqs. (1.2), (2.20) yields the useful relation σgo σow σgw = + . rgw rgo row

(3.4)

The contact angle hysteresis and the angular pore shape allows for formation of gas-oil, oil-water and gas-water AMs in the corners. The area in the three corners bounded by AMs is given by Acij (θij ) = 3rij2 {θij + α −

cos θ π ij + cos θij − sin θij }, 2 tan α

(3.5)

where ij = go, ow, gw. If bulk phase i is bounded by corner phase j , the corner area bounded by the ij AMs is calculated by Acij (θij ). If bulk phase j is bounded by corner phase i, the corner area bounded by the ij AMs is calculated by Acij (π − θij ). Thus, Eq. (3.5) with appropriate arguments θij , together with Eq. (3.2), constitute the equations employed in the saturation calculations accounting for all possible fluid configurations that may occur in the cross-sections.

3.3 Primary drainage and wettability alteration Initially all tubes are waterfilled and strongly water-wet, and hence the contact angle during primary drainage, θpd , is always small and satisfies θpd < π2 − α. The capillary entry pressure is therefore calculated from Eqs. (2.20), (2.30). It is assumed that oil always contacts the pore walls of the invaded tubes, and hence the

3.4 Fluid configurations

23

b pd

Figure 3.1: Final configuration of a tube after primary drainage. The bold lines along the sides represent the lengths of the pore wall where the wettability may have changed. The distances bpd in the corners remain water-wet. sides may experience a wettability alteration while the corners remain water-wet. The final configuration of a tube after primary drainage is shown in Fig. 3.1. The distance bpd of the solid surface that remains water-wet is given by bpd =

σow cos(θpd + α) , max sin α Pcow

(3.6)

max is the capillary pressure at the end of primary drainage. Irreducible where Pcow water saturations caused by phase entrapment do not occur in the model as we only consider a bundle of tubes. However, we may argue that a legitimate value of max is reached if the next pressure increase results in a saturation change smaller Pcow than some tolerance value. For subsequent invasion processes, receding and advancing oil-water contact angles are specified on the surface of potentially altered wettability. The receding and advancing gas-oil and gas-water contact angles are calculated as explained in Section 2.2.1.

3.4 Fluid configurations Contact angle hysteresis leads to a diversity of possible fluid configurations that can occur in simulations of different sequences of the invasion processes. The number of configurations are restricted by the following assumptions: (i) Only the three wetting orders described in Section 2.2 are allowed. (ii) A maximum number of two AMs are allowed to be present on the surface exposed to a potential wettability change. An additional AM may be located at position bpd .

24

The pore model

(iii) We do not study situations where the gas pressure is large enough for gas invasion into tubes, and corners of tubes, where oil has never been. With these constraints we find that the 17 configurations presented in Fig. 3.2 may occur during the simulations. Configuration A shows a tube that has always been waterfilled and water-wet. The configurations B–Q represent tubes that at some point have been invaded by oil and thus may have altered wettability. The curvatures of the gas-oil interfaces present in the configurations are always positive, whereas the gas-water and oil-water interfaces may have positive or negative curvatures to satisfy Eq. (3.4). In Table 3.1 we have specified the combinations of receding and advancing contact angles for which the different fluid configurations may occur. Contact angles are also specified to discriminate between the three wetting sequences whenever it is possible. Three-phase fluid configurations in mixedwet angular tubes have previously been analyzed by Piri and Blunt [21, 22, 70]. As opposed to us, they also consider cases where gas is wetting relative to oil. However, they have not accounted for our configurations F and K, which may occur when contact angle hysteresis is large. To our knowledge, only the configurations A–E, H, I, M and N have been observed by experiments in angular tubes or in micromodels [11, 23, 33, 62, 71, 72]. Assumption (i) implies that oil is always wetting relative to gas, i.e., θgo < π2 . Hence, bulk oil can not be bounded by gas layers in the cross-sections. Assumption (ii) is introduced to restrict the number of AMs in cases where contact angle hysteresis is large. If π π (3.7) θija > + α and θijr < − α, ij = ow, gw, 2 2 it is possible, in theory, that the number of AMs present in a cross-section could increase constantly as the number of saturation change reversals increases. For example, if Eq. (3.7) is satisfied for the oil-water contact angles, water invasion into configuration C may be a displacement to configuration E, while oil invasion into configuration E may be a displacement to configuration F. A subsequent water invasion into configuration F could then result in formation of a fourth AM in the corner, separating bulk water from a second oil layer. Even though Eq. (3.7) is satisfied, we do not allow formation of a new AM in the corner when two AMs are already present on the surface of altered wettability, by assumption (ii). We believe that this simplification is reasonable, since new AMs are likely to interfere with the AMs already present in most of these cases. Hence, water invasion into the bulk of configuration F is always assumed to be a displacement from configuration F to E. This sequence of oil-water displacements can only occur if the tube behaves as oil-wet during waterflooding and as water-wet during oil invasion. Similar configuration changes may occur during the gas-water displacements if Eq. (3.7) is


A

25

C

B

D

E

F

G

H

I

J

L

K

O

N

M

P

Q

Figure 3.2: Fluid configurations for any sequences of the invasion processes, with water in blue, oil in red, and gas in yellow. The bold lines along the sides represent the lengths with potentially altered wettability. Oil is always assumed to be wetting relative to gas.

26

The pore model

satisfied for both the gas-water and the oil-water contact angles, i.e., when the tube behaves as strongly oil-wet during waterflooding and as water-wet during gas invasion. Notice from Table 3.1 that the configurations F, G, K, L and Q can only occur if the wetting sequence of the three phases changes with the direction of the displacement. We allow for such capillary behavior since measurements indicate that contact angle hysteresis may be large if wettability alteration has occured [42, 44]. Furthermore, this effect may be more common when irregular geometries with different corner half angles α are assumed, since the contact angle hysteresis required to satisfy Eq. (3.7) are smaller in narrow corners. By assumption (iii), we do not allow for invasion of gas-water interfaces onto the water-wet surface where oil has never been, since we believe that the most realistic cases of three-phase flow in reservoirs can be studied without including this feature in the model. Nevertheless, such displacements could be accounted for by specifying gas-water and gas-oil contact angles on the water-wet surface as well, although this would increase the number of configurations. Assumption (iii) implies that any gas-water AMs located at position bpd are hinging with contact angles varying with Pcgw . Oil-water AMs located at this position are allowed to move on to the water-wet surface when the hinging contact angle has reached θpd . max , and a further increase of Pcow causes the length This happens when Pcow = Pcow of the water-wet surface, bpd , to decrease additionally. The AMs located on the surface of altered wettability may also hinge at fixed positions while the contact angles change with capillary pressure. The contact angles hinge according to  P b sin α cij ij   arccos − α if bulk phase i is bounded    σij    by corner phase j , (3.8) θijh = P b sin α cij ij   + α if bulk phase j is bounded arccos   σij     by corner phase i, where ij = go, ow, gw, and bij is the distance from the apex of the corner to the contact line. If the advancing or receding contact angle is reached, the AMs begin to move at constant contact angles during a further change of capillary pressure. The position bij is then changing according to  σij cos(θij + α)   if bulk phase i is bounded by corner phase j ,  P sin α (3.9) bij = σcij cos(θ − α) ij ij   if bulk phase j is bounded by corner phase i,  Pcij sin α where θij is equal to θijr or θija depending on the direction of the displacement.


Configuration A B C D E F G H I J K L M N O P Q

θowa n/a – – – > π2 + α > π2 + α > π2 + α – – > π2 > π2 > π2 – – > π2 + α > π2 > π2

27 θowr n/a < π2 − α – – – < π2 − α < π2 − α – – – – – π < 2 −α – – – < π2 − α

θgwa n/a – – – – – – – – > π2 + α > π2 + α > π2 + α – – – > π2 + α > π2 + α

θgwr n/a ≤ π2 – – – ≤ π2 ≤ π2 < π2 − α – – < π2 − α < π2 − α ≤ π2 – < π2 − α – ≤ π2

θgoa n/a ≤ π2 ≤ π2 ≤ π2 ≤ π2 ≤ π2 ≤ π2 ≤ π2 ≤ π2 ≤ π2 ≤ π2 ≤ π2 ≤ π2 ≤ π2 ≤ π2 ≤ π2 ≤ π2

θgor n/a ≤ π2 ≤ π2 ≤ π2 ≤ π2 ≤ π2 ≤ π2 ≤ π2 ≤ π2 ≤ π2 ≤ π2 ≤ π2 < π2 − α < π2 − α ≤ π2 < π2 − α ≤ π2

Table 3.1: Advancing and receding contact angles for which the cross-sectional fluid configurations are possible. It is always assumed that θpd < π2 − α. Contact angles are also specified to associate the configurations to the three wettability orderings, if possible. Empty spaces indicate that all values satisfying θija ≥ θijr , ij = go, ow, gw are allowed.

All the direct displacements implemented in the model are presented in Table 3.2, including piston-like invasion, collapse of fluid layers and change of the fluid areas in the corners. The configurations A–G may appear during the twophase oil-water displacements. Paper A provides a detailed description of the two-phase oil-water invasion processes, including expressions for the capillary entry pressures and the layer collapse capillary pressures. If the oil saturation has become zero, and only gas and water invasion processes are considered, the treatment of configuration H–L is analogous to the corresponding two-phase oil-water situation. If piston-like invasion occurs into a configuration containing multiple fluid layers in the corners, several displacements are possible, as shown in Table 3.2 for gas invasion into configuration F and water invasion into configuration N, for instance. Which particular displacement occurs is determined by the selected combination of contact angles and the capillary pressures. We only consider piston-like displace-

28

The pore model

Initial configuration A B C D E F G H I J K L M N O P Q

Final configuration Oil invasion Water invasion C – C D or G – D or E B or C – C or F D C E B D B or C D or L C D or J C or Q D C or Q J B or C D B or N D, G or H C D, E, I, M or P C, F or N E or H C or E E or J C B or J

Gas invasion – H, I or M I or N H or I I, N or O I, N or O H, I or M I – I or K I H H I N J or N I or K

Table 3.2: The programmed direct displacements during the oil, water and gas invasion processes.

ments where the invading fluid enters the bulk portion of the cross-section by a single MTM. This reduces the number of possible displacements in cross-sections where the invading fluid is already present as layers. As an example, piston-like water invasion into configuration O is always assumed to be a displacement to configuration E. If the oil layers collapse before an MTM invades into the bulk portion, the displacement O to H occurs, and MTM invasion is instead considered for configuration H. Thus, a configuration change from O to D is treated as two independent two-phase displacements that rarely occur simultaneously, since the capillary pressures associated with the two displacements then must satisfy Eq. (1.2). Similar reasoning applies to other three-phase configurations where the invading fluid already is present as layers. The algorithm used to determine the displacements and the associated expressions for the capillary entry pressures are described in the next section. A fluid layer is assumed to collapse when the bounding AMs meet at the contact lines or the midpoints. We employ the expressions derived by Piri and Blunt [22] for the collapse of fluid layers in a three-phase configuration. The displacement

3.5 Three-phase capillary entry pressures

29

resulting from water layer collapse in configuration Q is somewhat different from similar events in the other configurations since oil occupied in the bulk then becomes surrounded by gas, which is not allowed by assumption (i). However, oil is still wetting relative to gas, and thus a decreased Pcgo would cause spontaneous imbibition of oil into the corner and displace all of the gas phase immediately, resulting in a direct diplacement from configuration Q to C. This displacement has been identified in simulations using very large contact angle hysteresis satisfying the constraints for configuration Q in Table 3.1.

3.5 Three-phase capillary entry pressures An algorithm is formulated to determine the actual displacements occuring during piston-like invasion for all combinations of the contact angles and the capillary pressures. For each type of displacement the corresponding capillary entry pressures are calculated using the method proposed by van Dijke and Sorbie [65] who extended the MS–P method to also account for three phases. They calculate capillary entry pressures from an energy balance equation which equates the virtual work with the corresponding change in surface free energy for a small displacement of the MTM in the direction along the tube. The energy balance then relates the entry radius of curvature to the cross-sectional fluid occupancy, accounting for the possibility of simultaneous displacement of the fluids occupying the crosssection. We extend this method to account for mixed wettability and contact angle hysteresis following the approach by Ma et al. [26]. Thus, we incorporate the effect of hinging AMs stuck at fixed positions along the pore walls in the calculations. In Paper A we show for two phases that invasion does not necessarily proceed in the order of monotonic increasing or decreasing pore size when the AMs are hinging. Recently, Piri and Blunt [70] have used the same approach to study three-phase capillary entry pressure for mixed-wet conditions and contact angle hysteresis. They consider bulk gas-oil displacements that are affected by water present in the corners, e.g., displacements between the configurations C and I. We describe the more complicated cases occuring when a fluid phase invades configurations where two or three phases are arranged into several fluid layers with one of the phases occupied in the bulk.

3.5.1 Conditions for layer formation The conditions for layer formation are required to determine the correct displacement in each case. Formation of a new AM separating phase i from phase j at a position bij > bpd can only occur if the contact angles satisfy the following condi-

30

The pore model

tion: θij < θij >

π −α 2 π +α 2

if invading phase i is bounded by corner phase j , (3.10a) if invading phase j is bounded by corner phase i,(3.10b)

where ij = go, ow, gw, and θij is equal to θijr or θija depending on the direction of the displacement. However, if Eq. (3.10) is not satisfied, a new AM still forms at position bij = bpd since then the AM is assumed to hinge with contact angle θijh . If an AM is already present in the corner before invasion, a second condition required for layer formation is that the capillary pressure associated with the displacement must be favorable compared to the collapse capillary pressure calculated when the AMs surrounding the layer meet [65]. According to van Dijke et al. [66] and Piri and Blunt [70], these two geometric conditions are necessary but not sufficient for layer formation. Using free energy principles they argue that even if the above conditions are satisfied, one should also calculate the entry pressure for the displacement without layer formation and compare it with the entry pressure for the displacement with layer formation. The actual displacement occuring is the one associated with the most favorable capillary pressure. Thus, layers form if and only if the two geometric conditions are satisfied and the displacement is the most favorable. In Paper B and Paper D we follow [66, 70] and employ all the three conditions to determine if layers form. In Paper A and Paper C only the two geometric conditions are employed.

3.5.2 Layer collapse capillary pressures A fluid layer surrounded by the same phase on both sides collapses when the bounding AMs meet at their midpoints. In this case the AM separating the layer and the corner phase hinges at position bij , while the AM separating the layer from the bulk phase moves towards the corner. The AMs meet at a capillary pressure given by

col = Pcij

 σij (ξij2 − 1)   

    b cos α − 1 − ξij2 sin2 α ξ ij ij     σij (ξij2 − 1)   

  2  2  b cos α + 1 − ξ sin α ξ ij ij  ij   

if bulk phase i is bounded by layer phase j , if bulk phase j is bounded by layer phase i,

(3.11)


31

where  cos θijr   − 2, sin α ξij = cos θija   + 2, sin α

if bulk phase i is bounded by layer phase j , (3.12) if bulk phase j is bounded by layer phase i,

and ij = gw, ow. Fluid layers bounded by different corner and bulk fluids collapse when the bounding layers meet at the contact lines or at their midpoints depending on the combinations of contact angles, capillary pressures and the displacement history. The different collapse capillary pressures for gas, oil and water layers surrounded by the remaining two fluids are presented below. The expressions have also been derived by Piri and Blunt [22]. We consider collapse of oil layers bounded by different bulk and corner fluids first. By assumption (i), Section 3.4, such oil layers are surrounded by water on the corner side and by gas on the bulk side, as in configuration M, N and P. During gas invasion, Pcow is constant, and the gas-oil capillary pressure at which oil layer collapse occurs is given by

col Pcgo

 σgo cos θgoh − sin α   Pcow σow cos θowh − sin α = σ cos(θgoh + α)   Pcow go σow cos(θowh + α)

if θgoh < θowh , if θgoh ≥ θowh .

(3.13)

During water invasion, Pcgo is constant, and the oil layers collapse at an oil-water capillary pressure given by

col Pcow

 σow cos θowh − sin α    Pcgo σgo cos θgoh − sin α = σow cos(θowh + α)    Pcgo σgo cos(θgoh + α)

if θgoh < θowh , if θgoh ≥ θowh .

(3.14)

By assumption (i), Section 3.4, gas layers separated by different fluids are always bounded by oil on the corner side and by water on the bulk side, as in configuration P. During oil invasion, Pcgw is constant, and the gas layers collapse at a gas-oil capillary pressure given by

col Pcgo

 σgo cos θgoh − sin α    Pcgw σgw cos θgwh + sin α = σgo cos(θgoh + α)    Pcgw σgw cos(θgwh − α)

if θgoh > π − θgwh , if θgoh ≤ π − θowh .

(3.15)

32

The pore model

In water invasion, Pcgo is constant and the gas layers collapse at a gas-water capillary pressure given by  σgw cos θgwh + sin α   if θgoh > π − θgwh ,  Pcgo σgo cos θgoh − sin α col (3.16) Pcgw = σgw cos(θgwh − α)   if θgoh ≤ π − θgwh .  Pcgo σgo cos(θgoh + α) Yet more cases must be considered for water layers since they can be bounded by oil on the corner side and by gas on the bulk side, or vice versa. A water layer bounded by gas on the bulk side and oil on the corner side, as in configuration O, collapses during gas invasion when  σgw cos θgwh − sin α   Pcow if θgwh < π − θowh , col σow cos θowh + sin α = (3.17) Pcgw σ cos(θgwh + α)   Pcow gw if θgwh ≥ π − θowh , σow cos(θowh − α) and during oil invasion when  σow cos θowh + sin α    Pcgw σgw cos θgwh − sin α col = Pcow σow cos(θowh − α)    Pcgw σgw cos(θgwh + α)

if θgwh < π − θowh , if θgwh ≥ π − θowh .

(3.18)

A water layer surrounded by oil on the bulk side and by gas on the corner side, as in configuration Q, collapses during gas invasion at a gas-water capillary pressure given by  σgw cos θgwh + sin α   Pcow if θgwh < π − θowh , σow cos θowh − sin α col (3.19) = Pcgw σ cos(θgwh − α)   Pcow gw if θgwh ≥ π − θowh , σow cos(θowh + α) and during oil invasion at an oil-water capillary pressure given by  σow cos θowh − sin α   if θowh < π − θgwh ,  Pcgw σgw cos θgwh + sin α col = Pcow σow cos(θowh + α)   if θowh ≥ π − θgwh .  Pcgw σgw cos(θgwh − α)

(3.20)

The above expressions are written with hinging contact angles to accommodate for different displacement histories. However, in many cases at least one of the hinging contact angles in Eqs. (3.13)–(3.20) can be replaced by the appropriate receding or advancing contact angle depending on the invasion process. For example, if configuration N has formed by gas invasion into configuration C, the oil layer collapse is determined by Eq. (3.13) with θgoh = θgor .


I

o

w

L fow Lsow (a)

II I Aow

33

g

w

L fgw

R

II Agw

I

w R

Agw g

Aow

Lsgw

dx (b)

o

II

(c)

Figure 3.3: Representation of the cross-sectional parameters of the fluid-fluid and fluid-solid interfaces. (a) Configuration B. (b) Configuration H. (c) View in the direction along the tube length for the displacement from configuration B to H.

3.5.3 Gas invasion – general equations Consider the displacement from configuration B to H as an introductory example of piston-like gas invasion. Eq. (2.16) is reformulated to facilitate for the calculation of the capillary entry pressures. Using Eqs. (1.2), (2.3), (2.4), (2.10), we find that Eq. (2.16) may be written as Pcgw Vp d Sg + (Pcgw − Pcgo )Vp d So = σgw cos θgwr dgs + (σgw cos θgwr − σgo cos θgor )dos + σij dij .

(3.21)

ij=go,ow,gw

As illustrated in Section 2.4.1, it suffices to consider one sixth of the cross-sectional area of an equilateral triangle. For this part of the tube, and with reference to Fig. 3.3, we apply the relations Vp d Sg = Agw dx and Vp d So = − Aow dx, together with dgs = L sgw dx, dos = −L sow dx, dgw = L fgw dx, dow = −L fow dx and dgo = 0, to obtain Pcgw Agw + (Pcgw − Pcgo )(− Aow ) = σgw cos θgwr L sgw + σgw L fgw + (σgw cos θgwr − σgo cos θgor )(−L sow ) + σow (−L fow ).

(3.22)

Eq. (3.22) is employed to calculate the capillary entry pressure for a displacement from configuration B to H. The relation is a special case of the general equation derived by van Dijke and Sorbie [65] for uniform wettability with no contact angle hysteresis. To arrive at a general equation describing all piston-like displacements beetween the configurations shown in Fig. 3.2 during gas invasion, we first have to introduce some notation. AMs formed by the same pair of phases in a corner

34

The pore model

a

b ij(k )

j

bij(k ) (k ) q ijh

L(kfij)

rij

b ij(k )

i

a

Aij(k )

R

bij(k )

j

i

L(kfij) rij

) L(k sij

(k ) q ijh

Aij(k )

R

) L(k sij

(k)

(a) Iij

(k)

=1

(b) Iij

= −1

Figure 3.4: Representation of the cross-sectional parameters of ij AM k.

are numbered in order from the corner towards the center of the cross-section. The kth AM between phases i and j is referred to as ij AM k. We apply the indicator notation [66]   if ij AM k bounds bulk phase i and corner phase j , 1 (k) (3.23) Iij = −1 if ij AM k bounds bulk phase j and corner phase i,   0 otherwise, where ij = go, ow, gw. Furthermore, the total number of ij AMs present in a corner before displacement is denoted Nijinit , while the total number of ij AMs in the corner after displacement is denoted Nijfin . The cross-sectional bulk area bounded by ij AM (k) k is denoted A(k) ij . The bounding solid-fluid and fluid-fluid lengths are denoted Lsij and L (k) fij , respectively. With reference to Fig. 3.4, these parameters are defined as  rij bij(k) sin(α + βij(k) ) rij2 βij(k)  R2   − + if Iij(k) = 1, (k) 2 2 (3.24) Aij = 2 tan2 α  rij bij(k) sin(βij(k) − α) rij2 βij(k) R  (k)  − − if Iij = −1, 2 tan α 2 2 (k) L (k) fij = rij βij ,

and L (k) sij =

R − bij(k) , tan α

(3.25) (3.26)


where

and

 (k)   rij cos(θijh + α) if I (k) = 1,  ij sin(k) α bij(k) =  r cos(θijh − α)   ij if Iij(k) = −1, sin α π (k)  − α − θijh if Iij(k) = 1, (k) 2 βij = π  + α − θ (k) if I (k) = −1. ij ijh 2

35

(3.27)

(3.28)

(k) to account for conEqs. (3.24)–(3.28) are written with hinging contact angles θijh tact angle hysteresis and different displacement histories. However, in many cases, (k) θijh may be replaced by θijr or θija depending on the direction of the displacement. When the capillary entry pressure algorithms are described, we identify the dis(k) placements where Eqs. (3.24)–(3.28), written for the invading AM, apply with θijh replaced by θijr or θija . With the above notation, we find that the virtual external work required for gas invasion into the configurations presented in Fig. 3.2 may be written in generalized form as fin init Ngw Ngw (k) (k) (k) A(k) Wext ={Pcgw ( k=1 gw Igw − k=1 Agw Igw )+ Ngofin (k) (k) Ngoinit (k) (k) Ago Igo − k=1 Ago Igo )+ Pcgo ( k=1 fin init Now Now (k) (k) (k) A(k) (Pcgw − Pcgo )( k=1 ow Iow − k=1 Aow Iow )}dx.

(3.29)

The corresponding change in surface free energy may be written in generalized form as fin init Ngw Ngw (k) (k) (k) L (k) d F ={σgw cos θgwr ( k=1 sgw Igw − k=1 L sgw Igw )+ fin init Ngw Ngw (k) L (k) σgw ( k=1 k=1 L fgw )+ fgw − Ngofin (k) (k) Ngoinit (k) (k) L sgo Igo − k=1 L sgo Igo )+ σgo cos θgor ( k=1 fin Ngo (k) Ngoinit (k) L fgo − k=1 L fgo )+ σgo ( k=1 fin init Now Now (k) (k) (k) L (k) (σgw cos θgwr − σgo cos θgor )( k=1 sow Iow − k=1 L sow Iow )+ fin init Now Now (k) L (k) σow ( k=1 k=1 L fow )}dx. fow −

(3.30)

The energy balance equation, Wext = d F, which is equivalent to Eq. (2.16), is solved to obtain the capillary entry pressures for the various displacements occuring. For the fluid configurations considered in this work, Eqs. (3.29), (3.30)

36

The pore model

apply with some restrictions. By assumption (i), Section 3.4, gas is always wetting (k) = −1. By assumption (ii), ij=go,ow,gw Nijinit ≤ 3 and relative to oil, and thus Igo fin init fin ij=go,ow,gw Nij ≤ 3. Moreover, Ngo ≤ 1, and Ngo ≤ 1, since at most one gas-oil AM is present in a corner. Hence, the gas-oil parameters do not have to be specified with superscripts ‘(k)’. Eqs. (3.29), (3.30) are slightly reformulated with relevant capillary pressures and contact angles to facilitate for the calculation of the capillary entry pressures during the oil and water invasion processes.

3.5.4 Water invasion – general equations The virtual external work required for water invasion into any of the configurations in Fig. 3.2 may be written in generalized form as fin init Ngw Ngw (k) (k) (k) A(k) Wext ={Pcgw ( k=1 gw Igw − k=1 Agw Igw )+ fin init Now Now (k) (k) (k) A(k) Pcow ( k=1 ow Iow − k=1 Aow Iow )+ Ngofin (k) (k) Ngoinit (k) (k) Ago Igo − k=1 Ago Igo )}dx. (Pcgw − Pcow )( k=1

(3.31)

The corresponding change in surface free energy is given by fin init Ngw Ngw (k) (k) (k) L (k) d F ={σgw cos θgwa ( k=1 sgw Igw − k=1 L sgw Igw )+ fin init Ngw Ngw (k) L (k) σgw ( k=1 k=1 L fgw )+ fgw − fin init Now Now (k) (k) (k) L (k) σow cos θowa ( k=1 sow Iow − k=1 L sow Iow )+ fin init Now Now (k) L (k) σow ( k=1 fow − k=1 L fow )+ Ngofin (k) (k) Ngoinit (k) (k) L sgo Igo − k=1 L sgo Igo )+ (σgw cos θgwa − σow cos θowa )( k=1 Ngofin (k) Ngoinit (k) L fgo − k=1 L fgo )}dx. σgo ( k=1

(3.32)

3.5.5 Oil invasion – general equations The general expression for the virtual external work required for oil invasion into any of the configurations in Fig. 3.2 is given by fin init Now Now (k) (k) (k) A(k) Wext ={Pcow ( k=1 ow Iow − k=1 Aow Iow )+ Ngofin (k) (k) Ngoinit (k) (k) Ago Igo − k=1 Ago Igo )+ Pcgo ( k=1 fin init Ngw Ngw (k) (k) (k) A(k) (Pcow + Pcgo )( k=1 gw Igw − k=1 Agw Igw )}dx.

(3.33)


37

The corresponding change in surface free energy is given by the generalized expression fin init Now Now (k) (k) (k) L (k) d F ={σow cos θowr ( k=1 sow Iow − k=1 L sow Iow )+ fin init Now Now (k) L (k) σow ( k=1 k=1 L fow )+ fow − Ngofin (k) (k) Ngoinit (k) (k) L sgo Igo − k=1 L sgo Igo )+ σgo cos θgoa ( k=1 Ngofin (k) Ngoinit (k) L fgo − k=1 L fgo )+ σgo ( k=1 fin init Ngw Ngw (k) (k) (k) L (k) I − (σow cos θowr + σgo cos θgoa )( k=1 sgw gw k=1 L sgw Igw )+ fin init Ngw Ngw (k) σgw ( k=1 L (k) k=1 L fgw )}dx. fgw −

(3.34)

3.5.6 Capillary entry pressure algorithm Algorithms are formulated for each configuration to determine the actual displacements occuring and to calculate the associated capillary entry pressures from Wext = d F. Details are provided for cases of gas and water invasions in Paper B. Below we describe the general features of the algorithms for examples of gas, water and oil invasion. Gas invasion – configuration F The energy balance Wext = d F is obtained by equating Eq. (3.29) with Eq. (3.30). Since Pcow is constant, the energy balance is expressed in terms of Pcow in addition to one of the capillary entry pressures. The equation is solved, and finally the remaining capillary entry pressure is obtained from Eq. (1.2). For configuration F, init init init = Ngo = 0, and Now = 3. We consider first contact angles θgor and θgwr Ngw that satisfy Eq. (3.10a), implying that the allowed piston-like displacements from configuration F to I, N and O are all geometrically possible, as shown in Fig. 3.5. Since three AMs are already present in the corner, additional AMs are not allowed to form, by assumption (ii), Section 3.4. The following cases apply: fin fin = 0, Ngw = (i) Assume a displacement to configuration O. In this case Ngo fin 1 and Now = 2. The parameters of the invading gw AM 1 is given by (1) = θgwr . The energy balance Wext = d F is Eqs. (3.24)–(3.28) with θgwh F→O is obtained. formulated as a polynomial from which the entry pressure Pcgw (1) (3) ≤ bow , go to step (ii). (a) If the associated position satisfy bgw

38

The pore model

case (i)(a) case (ii)(b) case (ii)(a) case (iii)

case (i)(b)

o

I w (1) q gwh q gor (1) q goh q gwr

w

o

II

(1) q gwh

Figure 3.5: Cross-sectional view of configuration F. Oil-water AMs are represented by solid lines. Broken lines indicate possible locations of invading gas-water and gas-oil AMs that may form during gas invasion. The contact angles that the AMs make with the pore wall are also indicated for each scenario.

(1) (3) > bow , calculate the entry pressures again, assuming that gw AM (b) If bgw (1) (1) at the fixed position bgw = 1 hinges with an unknown contact angle θgwh (3) F→O bow . In this case the entry pressure Pcgw is calculated by solving Wext = d F iteratively together with Eqs. (3.24)–(3.28) written for gw (1) . AM 1 with θgwh fin fin = Ngo = 1 and (ii) Assume a displacement to configuration N. In this case Now fin Ngw = 0. The parameters of the invading go AM 1 is given by Eqs. (3.24)– (1) = θgor . In this case, Wext = d F is formulated as a polyno(3.28) with θgoh F→N is obtained. mial from which the entry pressure Pcgo (1) (2) ≤ bow , go to step (iii). (a) If the associated position satisfies bgo (1) (2) > bow , calculate the entry pressures again, assuming that go AM (b) If bgo (1) (1) at the fixed position bgo = 1 hinges with an unknown contact angle θgoh (2) F→N bow . In this case the entry pressure Pcgo is calculated by solving Wext = d F iteratively together with Eqs. (3.24)–(3.28) written for go (1) . AM 1 with θgoh fin fin = Ngo = 0 and (iii) Assume a displacement to configuration I. In this case, Now fin Ngw = 1. The capillary entry pressure is calculated by iterations since the


39

(1) invading gw AM 1 is assumed to hinge with an unknown contact angle θgwh (1) (1) at the fixed position bgw = bow . The energy balance is solved together with (1) F→I to obtain Pcgw . Eqs. (3.24)–(3.28) written for gw AM 1 with θgwh

(iv) Determine the most favorable displacement: (a) The displacement is from configuration F to O if and only if the capillary entry pressures satisfy F→O col < Pcgw , Pcgw

F→O F→N Pcgw < Pcgw

and

F→O F→I Pcgw < Pcgw ,

(3.35)

col is the gas-water capillary pressure at which the water layer where Pcgw in configuration O collapses, given by Eq. (3.17).

(b) The displacement is from configuration F to N if and only if Eq. (3.35) is not satisfied and the capillary entry pressures satisfy F→N col < Pcgo Pcgo

and

F→N F→I Pcgo < Pcgo ,

(3.36)

col is the gas-oil capillary pressure at which the oil layer in where Pcgo configuration N collapses, given by Eq. (3.13).

(c) The displacement is from configuration F to I if and only if the capillary entry pressures do not satisfy Eqs. (3.35), (3.36). If only θgwr satisfies Eq. (3.10a), formation of gas-oil AMs is not geometrically possible, and hence case (ii) does not apply. The actual displacement occuring is determined from the conditions stated in step (iv)(a), (c). If only θgor satisfies Eq. (3.10a), formation of gas-water AMs is not geometrically possible, and hence case (i) does not apply. The actual displacement is found from step (iv)(b), (c). If both θgwr and θgor do not satisfy Eq. (3.10a), only case (iii) applies, as the only geometrically possible displacement is from configuration F to I. Water invasion – configuration N In water invasion, Eqs. (3.31), (3.32) constitute the energy balance, Wext = d F. Since Pcgo is constant, the energy balance is expressed in terms of Pcgo in addition to one of the capillary entry pressures. The equation is solved, and finally the remaining capillary entry pressure is obtained from Eq. (1.2). In configuration N, init init init = Ngo = 1 and Ngw = 0. By assumption (ii), Section 3.4, additional AMs Now are allowed to form since only one AM is already present on the surface of altered wettability. We first consider the cases when both θowa and θgwa satisfy Eq. (3.10b) since then the allowed displacements from configuration N to P, E and D are all geometrically possible. The following cases apply:

40

The pore model

(i) Assume a displacement to configuration P. This is a two-phase displacement, N→P is calculated from the energy balance and the capillary entry pressure Pcgw (1) fin fin fin = θgwa in Eqs. (3.24)–(3.28). with Now = Ngo = Ngw = 1, and with θgwh fin fin = Ngw = 0 and (ii) Assume a displacement to configuration E. In this case Ngo fin Now = 2. The parameters of the invading ow AM 2 is given by Eqs. (3.24)– (2) = θowa . The energy balance, Wext = d F, is formulated as a (3.28) with θowh N→E is obtained. polynomial from which the entry pressure Pcow (2) (1) ≤ bgo , (a) If the associated position of the invading ow AM 2 satisfies bow go to step (iii). (2) (1) > bgo , calculate the entry pressures again, assuming that ow (b) If bow (2) at the fixed position AM 2 hinges with an unknown contact angle θowh (2) (1) N→E = bgo . In this case the entry pressure Pcow is calculated by bow solving Wext = d F iteratively together with Eqs. (3.24)–(3.28) written (2) . for ow AM 2 with θowh fin fin = Ngo = (iii) Assume a displacement to configuration D. In this case, Now fin F→I Ngw = 0. The capillary entry pressure Pcow is calculated by solving Wext = (1) at the fixed d F iteratively since ow AM 1 is assumed to hinge with θowh (1) position bow .

(iv) Determine the most favorable displacement: (a) The displacement is from configuration N to P if and only if the capillary entry pressures satisfy N→P col > Pcgw , Pcgw

N→P N→E Pcgw > Pcgw

and

N→P N→D Pcgw > Pcgw ,

(3.37)

col is the gas-water capillary pressure at which the gas layer in where Pcgw configuration P collapses, given by Eq. (3.16).

(b) The displacement is from configuration N to E if and only if Eq. (3.37) is not satisfied and the capillary entry pressures satisfy N→E col > Pcow Pcow

and

N→E N→C Pcow > Pcow ,

(3.38)

col is the oil-water capillary pressure at which the oil layer in where Pcow configuration E collapses, given by Eq. (3.11).

(c) The displacement is from configuration N to D if and only if the capillary entry pressures do not satisfy Eqs. (3.37), (3.38).


41

If only θgwa satisfies Eq. (3.10b), formation of new oil-water AMs is not geometrically possible, and hence case (ii) does not apply. The actual displacement occuring is determined from the conditions stated in step (iv)(a), (c). If only θowa satisfies Eq. (3.10b), formation of new gas-water AMs is not geometrically possible, and hence case (i) does not apply. The actual displacement is found from step (iv)(b), (c). If both θgwa and θowa do not satisfy Eq. (3.10b), only case (iii) applies, as the only geometrically possible displacement is from configuration N to D. Oil invasion – configuration J In oil invasion, Eqs. (3.33), (3.34) constitute the energy balance Wext = d F. Since Pcgw is constant, the energy balance is expressed in terms of Pcgw in addition to one of the capillary entry pressures. The equation is solved, and finally the remaining init capillary entry pressure is obtained from Eq. (1.2). In configuration J, Now = init init Ngo = 0 and Ngw = 2. Moreover, assumption (i), Section 3.4, implies that gas-oil fin = 0. Oil is always wetting AMs are not allowed to form in this case, and thus Ngo relative to gas, and thus Eq. (3.10b) is not met for θgoa . This restricts the number of possible displacements. By assumption (ii), Section 3.4, additional AMs are allowed to form since only one AM is present on the surface of altered wettability. We first consider the case when θowr satisfy Eq. (3.10a) since then the allowed displacements from configuration J to C and Q are all geometrically possible. The following cases apply: (i) Assume a displacement to configuration Q. This is a two-phase displaceJ→Q is calculated from the energy ment, and the capillary entry pressure Pcow (1) fin fin = θowr in Eqs. (3.24)– balance with Now = 1 and Ngw = 2, and with θowh (3.28). fin fin = 1 and Ngw = (ii) Assume a displacement to configuration C. In this case, Now (1) (1) 0. The invading ow AM 1 enters position bow = bgw with an unknown hing(1) J→C . Thus, the capillary entry pressure Pcow is calculated ing contact angle θowh by solving Wext = d F iteratively together with Eqs. (3.24)–(3.28) written (1) . for ow AM 1 with θowh

(iii) Determine the most favorable displacement: (a) The displacement is from configuration J to Q if and only if the capillary entry pressures satisfy J→Q col < Pcow , Pcow

and

J→Q J→C Pcow < Pcow ,

(3.39)

col is the oil-water capillary pressure at which the water layer where Pcow in configuration Q collapses, given by Eq. (3.20).

42

The pore model

(b) The displacement is from configuration J to C if and only if the capillary entry pressures do not satisfy Eq. (3.39). If θowr does not satisfy Eq. (3.10a), only case (ii) applies, as the only geometrically possible displacement is from configuration J to C.

3.5.7 Discussion The developed pore model employs accurate expressions for the capillary entry pressures to describe piston-like bulk invasion of an MTM that simultaneously may displace the different fluids that are occupied in the cross-section. The invading AMs may either hinge at fixed postions while the contact angle changes with capillary pressure by Eq. (3.8), or they may move with constant contact angles while the positions change with capillary pressure by Eq. (3.9). The method employed to derive the capillary entry pressures for the piston-like displacements implemented in the model makes it possible to analyze if other displacements can occur. A natural extension of the present model will be to include the following displacement types: (i) Invasion of fluid films on AMs formed by the other two phases, e.g., oil invasion as a spreading film on a gas-water surface. Such displacements have been analyzed using free energy principles [62, 66]. (ii) Piston-like displacement of fluid layers occupied in the cross-sections. To our knowledge, such displacements have not been analyzed before. The capillary pressures for the above displacements should be compared with the capillary pressures for the piston-like displacements and the layer collapse events presented in Table 3.2. The actual displacement occuring is the one associated with the most favorable capillary pressure. As a preliminary analysis of displacement type (ii), we consider water layer invasion following the displacement from configuration C to E in imbibition after primary drainage. Such a displacement is illustrated with the relevant parameters init fin init fin init fin = Ngw = Ngo = Ngo = 0, Now = 2 and Now = 0 in in Fig. 3.6. With Ngw Eqs. (3.31), (3.32), the energy balance equation Wext = d F becomes (1) (2) (2) (1) (2) Pcow ( A(1) ow − Aow ) = σow cos θowa (L sow − L sow ) + σow (L fow + L fow ).

(3.40)

Eq. (3.40) may be written in terms of row by Eq. (3.3). Moreover, when the expres(k) (k) sions for A(k) ow , L fow and L sow given by Eqs. (3.24)–(3.26) are introduced, Eq. (3.40) yields Al , (3.41) row = l


w

I 1) L(fow

43

w

o Al

2) L(fow

2) L(sow

1) L(sow

o

w R

I

w

II

dx

(1) Aow

( 2) w Aow

II (a)

(b)

Figure 3.6: Representation of the cross-sectional parameters employed in the calculation of the capillary layer entry pressure. (a) Cross-sectional view of configuration E. (b) View of the displacement in the direction along the tube length.

where (2) Al = A(1) ow − Aow

=− and

row (1) r2 (1) (2) (2) (1) (2) {bow sin(α + βow ) − bow sin(βow − α)} + ow (βow + βow ) 2 2 (2) (1) (1) (2) − bow ) cos θowa + row (βow + βow ). l = (bow

(3.42)

(3.43)

Evidently, the capillary layer entry pressure does not depend on pore size R. In (1) at the fixed configuration E, ow AM 1 hinges with an unknown contact angle θowh (1) position bow , while ow AM 2 moves towards the corner with constant contact an(2) (1) = θowa . Thus, the contact angle θowh , which is included in the expression gle θowh (1) (2) for βow , and the position bow must be updated in the calculations. Eqs. (3.41)– (3.43) are solved iteratively together with Eqs. (3.27), (3.28) in the following manC→E (1) (2) as the initial value. Calculate βow and bow from ner: choose row = σow /Pcow Eqs. (3.27), (3.28) and Al and l from Eqs. (3.42), (3.43), respectively. A new value of row is then obtained from Eq. (3.41). Finally, the capillary layer entry pressure is calculated from Eq. (3.3) using the converged value of row . In Fig. 3.7, the capillary layer entry pressure is compared with the collapse capillary pressure calculated by Eq. (3.11). The capillary entry pressures for the piston-like displacements from configuration C to D and E are also shown. Evidently, oil layer collapse does not occur, as the layer entry pressure is favorable

44

The pore model

0 −2

Pcow (kPa)

−4 −6 −8 −10 −12 0

P (C → E) cow P (E → D) cow Pcol (E → D) cow Pcow (C → D) 10

20

30

40

50

60

70

R (µm)

Figure 3.7: Capillary entry pressure as a function of pore size during imbibition following primary drainage. The oil layer collapse capillary pressure for configuration E, given by Eq. (3.11), is compared with the layer entry pressure for the max = 20 displacement from E to D. The following input parameters are used: Pcow ◦ ◦ kPa, θpd = 0 , θowa = 180 , σow = 0.045 N/m, Rmin = 1µm, Rmax = 100µm, Rch = 20µm and η = 2.

col , implying that water displaces the oil layer in a piston-like discompared to Pcow placement. Notice also that the layer entry pressure coincides with the minimum capillary level where displacements from configuration C to E occurs. Below this critical capillary level, configuration E is absent while water invasion proceeds by col is used instead as the minimum displacements from configuration C to D. If Pcow capillary pressure for oil layer existence, then configuration E may still exist while the displacement from configuration C to D occurs. However, if the pore space is represented by an interconnected pore network, the collapse capillary pressure may still be valid as a condition for layer existence in cases where layer invasion is topologically impossible because of reduced phase accessibility.

The capillary range for layer existence seems to be reduced if piston-like layer invasion is considered. Work is in progress to analyze capillary entry pressure for piston-like layer displacements when different fluids surround the layer on the bulk and corner side. The new expressions are compared with the collapse capillary pressures given by Eqs. (3.13)–(3.20). A paper is under preparation to provide a complete description of such layer invasion processes in mixed-wet two- and threephase systems.

3.6 Applications and further work

45

3.6 Applications and further work The developed pore model serves as a useful tool to efficiently generate capillary pressure curves that are expected to emulate general trends of measured data for various conditions and displacement histories. The model offers the possibility of carefully analyzing the computed data, in particular with respect to the pore-scale displacements occuring, the capillary levels and saturation dependencies, from which reliable capillary pressure correlations can be suggested. The model may be extended to also calculate three-phase relative permeability curves. This has been done with a similar model by Hui and Blunt [24]. However, they employed approximate expressions for the capillary entry pressures and considered a limited range of displacement paths. Using a similar scientific methodology as for capillary pressure, the model may prove useful in developing reliable three-phase relative permeability correlations for mixed-wet reservoirs. Although the model includes important aspects of pore-scale physics in threephase fluid systems, the bundle-of-tubes representation of the pore network is simplistic, as mechanisms leading to phase entrapment and residual saturations are absent. To overcome this, the pore geometry may be extended to three dimensions to also allow for converging–diverging pore shapes, i.e., tubes with cross-sections that vary in the direction along the tube lengths. Examples of such pore geometries include the space formed between packed spheres. Such models allow for formation of pendular rings and are known to display hysteresis also when contact angle hysteresis is absent [73]. Three-phase mechanisms such as double displacements may also be investigated by this model. Furthermore, by varying the converging–diverging shape as well as other input parameters, different trends in capillary behavior and pore-scale mechanisms may be detected, and the resulting effects on three-phase relative permeability and capillary pressure curves may be interpreted, leading to a more physically-based description of these quantities. This underlines the importance of having a simple, physically-based, model to compute relationships between key parameters.

Chapter 4

Correlations Among the two-phase capillary pressure correlations reported in the literature, the Brooks-Corey formula is one of the most frequently used because of its simplicity and solid experimental validation [2]. For primary drainage this correlation may be written as Pcow = cw Sw−dw ,

(4.1)

where cw is the entry pressure, 1/dw the pore-size distribution index, and Sw the normalized water saturation. Skjæveland et al. [4] extended the correlation to account for imbibition, secondary drainage and hysteretic scanning loops for mixed-wet conditions, by using the following reasoning: If Eq. (4.1) is regarded as valid for a complete water-wet system, then the same functional form should be equally valid for a complete oil-wet system when the subscript w for water is replaced by the subscript o for oil. For intermediate wetting states, both fluids contribute to the overall wettability, and hence the capillary pressure should be modelled as a sum of the water and oil terms, resulting in the correlation Pcow = cw Sw−dw + co So−do ,

(4.2)

where So is the normalized oil saturation. In general, Eq. (4.2) requires different sets of the parameters cw , co , dw and do for different drainage and imbibition capillary pressure curves. The constants dw , do and cw are positive, while co is negative. The Brooks-Corey formula is used as a basis for the design of the three-phase capillary pressure correlation because of its extensive validation for two phases [2, 4]. 47

48

Correlations

4.1 Three-phase correlation design In three-phase flow there is, for all starting positions in the saturation space, an infinite number of possible ways the three capillary pressures could relate to each other, corresponding to an infinite number of unique saturation trajectories. Thus we assume a process-based approach as a foundation for the correlation development, implying that the processes are known in terms of the saturation trajectories. The processes are sorted into classes of processes, defined by how the saturations change. We let the capital letters I, D and C denote increasing, decreasing and constant saturations, respectively. A class of processes is then denoted by a three-letter symbol where the first letter denotes the direction of water saturation change, the second letter denotes the oil saturation change, and the third letter denotes the gas saturation change. This is the same notation as employed by Oak [74, 75]. We propose to formulate three-phase correlations for each process class as a sum of two Brooks-Corey terms, with one term as a function of a decreasing saturation and the other term as a function of an increasing saturation. The two terms should then dominate in different regions of the saturation space. For all the classes of processes, the proposed correlations for the three capillary pressures, Pcij , ij = go, ow, gw, are formulated as follows: • The process classes XDI and XID where X = I, D or C: Pcij = cg (1 − Sg)−dg + co (1 − So)−do .

(4.3)

• The process classes IXD and DXI where X = I, D or C: Pcij = cg (1 − Sg)−dg + cw (1 − Sw )−dw .

(4.4)

• The process classes DIX and IDX where X = I, D or C: Pcij = co (1 − So)−do + cw (1 − Sw )−dw .

(4.5)

The parameters c and d have to be determined. By Eq. (1.2), correlations are needed only for two of the capillary pressures. This yields a total number of 8 parameters to be determined for each process. The reason for using the functional form (1 − Si ) instead of Si for phase i in the correlations is that the former allows for positive values of di without having capillary pressures starting from plus or minus infinity when Si = 0 initially. Notice also that the functional forms of the correlations are equal for all the three capillary pressures for a given process. The differences in the saturation dependencies and the capillary levels are included by using different values of the c’s and the d’s.

4.2 Applications and discussion

49

In the above formulation, Eqs. (4.3)–(4.5), it is stated that two different correlations could be used for processes where all the three saturations are changing. Since the capillary pressures are modelled as functions of two saturations, the actual choice of correlation is indifferent as long as the two saturation terms dominate in different regions of the saturation space. However, if only one of the phases is displaced by the invading phase initially, then the correlation with saturation terms of both the displaced and invading phase should be used.

4.2 Applications and discussion The proposed capillary pressure correlations with appropriate parameters may be employed to predict vertical fluid distributions as functions of the saturations for physically reasonable displacement processes in mixed-wet reservoirs. The correlations may also be implemented in reservoir simulators. Reliable correlation parameters are required for these applications. Knowledge of the wettability state, interfacial tensions, and geometry of the pore network may be transformed to input parameters for the developed pore model. From the simulated data, reasonable correlation parameters can be estimated by standard curvefitting procedures. The pore model, adjusted with the available data, may then be used to simulate three-phase capillary pressure curves for other displacement paths, from which a plausible range of values of the correlation parameters are determined for the specific rock and fluid properties in question. For implementation of the correlations in reservoir simulators, the capillary pressures are required in the entire saturation space. A capillary pressure surface may be generated by interpolating the capillary pressure curves described by the correlations and by the saturation paths from the pore model, which in turn describes the class of displacement processes relevant for the reservoir simulation. In Paper C, Eqs. (4.3)–(4.5) are successfully matched with a wide range of data computed by the pore model. If the correlations are compared with measured data, Eqs. (4.3)–(4.5) may in some cases require residual saturations as additional parameters to obtain a good agreement. As an example, trapping of the gas phase is likely to occur in WAG cycles [76, 77]. The pore model developed in this study is not adapted for estimation of residual saturations, and hence we have not yet included such features in the correlations. However, the correlations in their present forms are able to match capillary data from centrifuge measurements of gravity drainage [14], including experiments with a nonzero residual oil saturation. The two-saturation dependency of the correlations account for processes where the displacement path proceeds as a two-phase displacement after the residual saturation of the remaining phase is reached. Thus, the range of processes where residual

50

Correlations

saturations are required to obtain a good match with measured data may in fact be restricted. This is further emphasized by the possibility of treating different segments of a displacement path as individual displacement paths, each described by one of the correlations given by Eqs. (4.3)–(4.5). Obviously, such a procedure would increase the number of correlation parameters to be determined.

Chapter 5

Summary of the papers The scientific contributions of the thesis are presented in four papers, of which three have been published in conference proceedings. The last paper is an unpublished report. A summary of each paper is given below.

5.1 Paper A “Physically-based capillary pressure correlation for mixed-wet reservoirs from a bundle-of-tubes model,” Helland, J.O. and Skjæveland, S.M. This paper explains in detail how to calculate two-phase mixed-wet capillary pressure curves with hysteretic scanning loops from a bundle of tubes with equilateral triangular cross-sections. For the sequence of processes primary drainage, imbibition and secondary drainage, it is shown that six possible fluid configurations can occur in the cross-sections for all combinations of receding and advancing contact angles. One of these configurations (configuration F) has not been analyzed before and may only occur when the tube behaves as oil-wet during imbibition and water-wet during secondary drainage. Based on the work by Ma et al. [26], we have derived accurate expressions for the capillary entry pressures that account for hinging interfaces in the corners due to contact angle hysteresis. The effect of corner occupancy on the entry pressures is demonstrated, and it is shown that waterfilled corners bounded by hinging interfaces tend to increase the entry pressure, while oil layers bounded by hinging interfaces tend to decrease the entry pressure. Capillary pressure curves are computed for both uniform and randomly distributed contact angles assuming Weibull distributed pore sizes. In the uniform case, the curves may exhibit conspicuous steps in the transition between different 51

52

Summary of the papers

displacement types. In the distributed case, these transition steps are smoothed out as several displacement types can occur within the same range of saturations. Despite the simplicity of the model and the lack of residual saturations due to phase entrapment, we find that the simulated results display main characteristics of mixed-wet capillary pressure curves. This is demonstrated by curvefitting experimentally validated correlations [2,4] to the computed capillary pressure curves for primary drainage and the bounding hysteresis loop. The correlations match fairly well with the simulations. Using a pore-size distribution consistent with the Brooks-Corey correlation [2], we have derived a new analytical correlation for primary drainage accounting for the triangular geometry. The formula is reduced to the Brooks-Corey correlation when no water is residing in the corners, or when a model of cylindrical tubes is assumed.

5.2 Paper B “Three-phase mixed-wet capillary pressure curves from a bundle-oftriangular-tubes model,” Helland, J.O. and Skjæveland, S.M. This paper describes the extension of the bundle-of-triangular-tubes model to generate three-phase capillary pressure curves for mixed-wet conditions with contact angle hysteresis. The model allows for simulation of any sequences of the gas, oil and water invasion processes starting with primary drainage followed by a potential alteration of wettability. With certain restricting assumptions we find that 17 possible fluid configurations may occur in the cross-sections for all the allowed combinations of the contact angles. Based on the method proposed by van Dijke and Sorbie [65], we have derived accurate expressions for the three-phase capillary entry pressures that account for hinging interfaces in the corners and the possibility of simultaneous displacement of the fluids occupying the cross-sections. As a consequence, invasion does not necessarily proceed in the order of monotonic increasing or decreasing pore size. Several kinds of piston-like displacements are possible when a phase invades a configuration containing multiple fluid layers. Algorithms are formulated for each configuration to determine the actual displacement occuring in these cases. The algorithms are described in detail for the cases when gas invades configurations with oil layers bounded by bulk water and water in the corners (e.g., configuration E), and when water invades configurations with oil layers bounded by bulk gas and water in the corners (e.g., configuration N). The model is employed to simulate the sequence of processes primary drainage, imbibition, gas injection and waterflooding for a physically realistic set of interfacial tensions and three different combinations of advancing and receding contact angles representing oil-wet conditions with variable contact angle hysteresis. In

5.2 Paper B

53

these cases, gas invasion into configuration E may result in simultaneous displacement of bulk oil, the oil layers and some of the water in the corners. Similarly, water into configuration N may result in displacement of bulk gas and the oil layers. For the range of capillary pressures where such displacements occur, we have investigated the sensitivity of the three-phase capillary entry pressures to the capillary pressures at the end of the preceding invasion process and to the capillary max . The specific conclusions pressure where primary drainage was terminated, Pcow are as follows: (i) The gas-water capillary pressure, Pcgw , for gas invasion into configuration E, max . Furtheris more sensitive to variations of Pcow than to variations of Pcow π more, Pcgw increases with decreasing Pcow if θgwr < 2 , while Pcgw decreases according to Pcow if θgwr > π2 . (ii) The gas-water capillary pressure for water invasion into configuration N is max . less sensitive to variations of Pcgo than to variations of Pcow The three-phase capillary pressure vs. saturation relationships calculated for the gas and water invasion processes are compared with corresponding results from a bundle-of-cylindrical-tubes model, using the constraint that the capillary pressures in primary drainage and the pore volumes are identical for both geometries. The saturation dependencies of the three-phase capillary pressures are extracted from plots of capillary pressure iso-lines (“iso-caps”) and further analyzed by plots of capillary entry pressures and bulk pore occupancies [27, 28, 78]. The specific conclusions are summarized as follows: max , the two models may yield different saturation (i) For moderate levels of Pcow dependencies of three-phase capillary pressure. In the bundle of triangular tubes, two or even all three capillary pressures may depend strongly on two saturations in the same region of the saturation space, while the corresponding results from the bundle of cylindrical tubes often show that only one of the capillary pressures depend on more than one saturation in the same region.

(ii) The different saturation dependencies derived from the bundle of triangular tubes result from capillary entry pressures that are affected by hinging interfaces in the corners when contact angle hysteresis is assumed. In general, these entry pressures predict different bulk pore occupancies than the simple Young-Laplace equation which is valid for the cylindrical geometry. max small, (iii) The saturation dependencies derived from triangular tubes with Pcow agree with expected behavior for water-wet conditions, e.g., with Pcgo strongly

54

Summary of the papers max dependent on Sg [9, 15, 16], while for high Pcow , the results agree with expected behavior for oil-wet conditions, e.g., with Pcgo strongly dependent on So [9, 15, 16]. This is explained by a reduced area of water-wet surface when max is increased. Thus, triangular tubes induce a relationship between wetPcow tability and reversal point after primary drainage, as demonstrated in Paper max A and by Ma et al. [26]. This, in turn, leads to a relationship between Pcow and the three-phase saturation dependencies.

(iv) The level of Pcgw and Pcow is generally higher for the triangular tubes than for the cylindrical tubes during the gas and water injections. (v) The saturation dependencies, capillary levels and bulk pore occupancies calculated from triangular tubes approach the corresponding results calculated from cylindrical tubes when the capillary level at the end of primary drainage is increased. Despite its simplicity, the bundle-of-triangular-tubes model described in this paper is, to our knowledge, the first one capable of computing three-phase Pc − S relationships by employing true three-phase entry pressures for mixed-wet conditions. The results from this work indicate that three-phase capillary pressure correlations for mixed-wet reservoirs should be formulated as functions of two saturations.

5.3 Paper C “Three-phase capillary pressure correlation for mixed-wet reservoirs,” Helland, J.O. and Skjæveland, S.M. In this paper we propose new correlations for three-phase capillary pressure that could be used to model the dynamics of three-phase transition zones in mixedwet reservoirs. The Brooks-Corey formula has been chosen as the basis for the three-phase correlation due to its simple form and the solid experimental validation. The capillary pressures are modelled as a sum of two Brooks-Corey terms, with one term as a function of an increasing saturation and the other term as a function of a decreasing saturation. Thus the correlations depend on the direction of the displacement paths. The two-saturation dependency, together with the inclusion of adjustable parameters, ensure that the correlations account for different wettability states, saturation histories, and different relationships between the three capillary pressures. Using simple pore occupancy plots [27, 28], with invasion of the intermediate-wetting phase, we infer the saturation dependencies and the signs of the correlation parameters for the ideal cases of uniform wettabilities. We also

5.4 Paper D

55

demonstrate that the correlations are compatible with a smooth transition between two- and three-phase flow if one of the phases appears or disappears. In particular, if the gas saturation becomes zero, it is shown that the correlations are reduced to a previously published two-phase correlation validated for oil/water systems in mixed-wet rocks. The flexible pore model described in Paper B is employed to simulate capillary pressure curves for various wetting conditions, invasion processes and saturation histories. In particular, by assuming linear relationships between the gas-oil and oil-water capillary pressures we are able to simulate processes where two saturations increase, such as in simultaneous water-alternate-gas injections (SWAG). The three-phase capillary pressure correlations are fitted to the simulated data, and excellent agreement is obtained in all cases. Finally, the correlations are validated by recent centrifuge measurements of three-phase capillary pressure during gravity drainage in water-wet Berea sandstone cores.

5.4 Paper D “The relationship between capillary pressure, saturation and interfacial area from a model of mixed-wet triangular tubes,” Helland, J.O. and Skjæveland, S.M. In this paper we employ the bundle-of-triangular-tubes model to investigate the relationships between capillary pressure, interfacial area and saturation for two-phase mixed-wet systems. Since the model only accounts for cross-sectional configurations, incorporation of interfacial area between bulk phases is not possible. The contribution to interfacial area from possible thin films that may exist along the sides are also neglected. Thus, we only take into account the interfacial area from interfaces present in the corners of the cross-sections. Although this may underestimate the interfacial area derived from the model, we find that characteristics of interfacial area vs. saturation relationships measured in primary drainage and imbibition of real porous media can be reproduced when contact angle hysteresis is assumed [79]. Based on the model and the imposed assumptions, we have derived analytical correlations for primary drainage and suggested approximate correlations with adjustable parameters for imbibition and secondary drainage. The correlations are fitted to the simulated results, and a good match is obtained. Hassanizadeh and Gray [48] hypothesized that hysteresis in the capillary pressure vs. saturation (Pcow − Sw ) relationship is an artifact of projecting the Pcow − Sw − aow surface onto the Pcow − Sw plane. We investigate if the conjecture of Hassanizadeh and Gray [48] is compatible with our simple model by generating Pcow − Sw − aow surfaces based on imbibition and drainage scanning curves for

56

Summary of the papers

both mixed-wet and water-wet systems with contact angle hysteresis. We find that aij is not unique for constant Pcow and Sw . The differences between drainage and imbibition interfacial areas are significant for both water-wet and mixed-wet conditions, implying that hysteresis remains present in the relationship. However, if contact angle hysteresis is neglected, then hysteresis in the Pcow − Sw − aow relation is absent as well, and a more sophisticated model is required to investigate if the conjecture is also violated when hysteresis due to phase entrapment occurs.

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Part II

Paper A: Physically-based capillary pressure correlation for mixed-wet reservoirs from a bundle-of-tubes model

Paper B: Three-phase mixed-wet capillary pressure curves from a bundle-of-triangular-tubes model

Paper C: Three-phase capillary pressure correlation for mixed-wet reservoirs

Paper D: The relationship between capillary pressure, saturation and interfacial area from a model of mixed-wet triangular tubes