A Brief Review of Capillary Number and its Use in Capillary Desaturation Curves

Abstract

Capillary number, understood as the ratio of viscous force to capillary force, is one of the most important parameters in enhanced oil recovery (EOR). It continues to attract the interest of scientists and engineers, because the nature and quantification of macroscopic capillary forces remain controversial. At least 41 different capillary numbers have been collected here from the literature. The ratio of viscous and capillary force enters crucially into capillary desaturation experiments. Although the ratio is length scale dependent, not all definitions of capillary number depend on length scale, indicating potential inconsistencies between various applications and publications. Recently, new numbers have appeared and the subject continues to be actively discussed. Therefore, a short review seems appropriate and pertinent.

1 Introduction

A primary objective of enhanced oil recovery (EOR) methods is to extract original oil in place (OOIP) efficiently and economically from oil reservoirs (Lake 1989; Green and Willhite 2018). Despite many years of study, the best way to achieve this objective has remained one of the hottest research topics for petroleum engineers, geologists, chemical engineers and hydrologists.

Most EOR-methods attempt to increase the overall recovery efficiency (defined in Eq. (23)) of the amount of oil or gas. As stated on the back cover of the textbook by Lake et al. (2014), the main theme in such attempts is the frequent reference to two fundamental principles: lowering the mobility ratio and increasing the capillary number. In this paper, the capillary number, defined as the quotient

$$\begin{aligned} \mathrm {Ca}=\text {(capillary number)}= \frac{\text {(viscous forces})}{\text {(capillary forces)}} , \end{aligned}$$

(1)

of viscous to capillary forces, will be the focus of interest. One of the most widely accepted expressions for capillary number is the ratio of viscosity $\mu$ times velocity to interfacial tension . To mobilize the residual oil, this ratio must be increased by several orders of magnitude from the value $10^{-6}$ it normally has in a waterflood, and the only realistic way to achieve this is by drastically lowering the interfacial tension . This is the most important reason for the popularity of surfactant EOR.

Table 1 Overview of correlating groups and capillary numbers. The numbers are denoted as ${{}^{n}\mathrm {Ca}}$ where n is the row number in this table. The third column indicates whether all quantities entering ${{}^{n}\mathrm {Ca}}$ can be measured independently in experiment. The fourth column, labelled Pure, indicates whether ${{}^{n}\mathrm {Ca}}$ is formulated purely with or purely with quantities. Its entries are Yes, No or Yo. An entry “Yes” is coloured or according to whether the number is purely or . An entry “Yo” indicates that the number is “almost pure”

Dullien (1992) and Hilfer and Øren (1996) noticed serious limitations of the capillary number . Experiments indicate that the force balance and hence capillary number should depend on length scale, which is not the case for . In addition, the experimentally observed values of in capillary desaturation experiments are much too small, and Dullien (1992, p. 450) concludes “it is certainly peculiar that, when the viscous and capillary forces acting on a blob are equal, the capillary number, that is supposed to be equal to the ratio of viscous-to-capillary forces, is equal to $2.2\times 10^{-3}$”.

Given that remains widely used, the main objective of this article is to review the various definitions of capillary numbers with respect to their importance for capillary desaturation curves. Let us recall that capillary desaturation curves play a crucial role in reservoir simulation and the planning of EOR-strategies. Our objective is therefore to review capillary numbers mainly with a view towards their use in capillary desaturation curves and their application in chemical EOR. Our definitions of recovery efficiencies, residual, remaining oil saturations and other important symbols are summarized in Appendix A. Definitions of capillary desaturation curves and capillary number correlations are given in Appendix B.

2 Capillary Numbers

It is now well accepted that the capillary number is a dimensionless group that quantifies the ratio of viscous force to capillary forces, as shown in Eq. (1). The problem of finding suitable definitions for the capillary number is identical to the problem of finding precise and, if possible, concise characterizations for the viscous and the capillary forces in Eq. (1). Lots of studies were devoted to this problem. Larson et al. (1981) summarized 14 correlating groups for the period from 1927 to 1979 and Taber (1980) summarized 17 correlating groups reported before 1979. According to our present survey, at least 41 different capillary number definitions or correlating groups have been reported. They are listed in Table 1. The first correlating group is reported to have been established in 1927 (Larson et al. 1981, Table 1). An early discussion of dimensionless groups can be found in (Leverett’s work 1939; 1941).

There are 41 numbers in this table. Most of them are mixed capillary numbers in the sense of Sect. 2.3. Many of them are closely related with each other or identical, such as ${{}^{8}\mathrm {Ca}}={{}^{23}\mathrm {Ca}}$. To appreciate the relation between various numbers it is necessary to consult the original references or the brief definition of symbols given in Tables 2 and 3. Note that for ${{}^{8}\mathrm {Ca}}$ the symbol K in Dombrowski and Brownell (1954) has dimensions of volumetric flow rate, not area. Many numbers contain quantities that appear to lack a clear definition or are otherwise difficult to access experimentally, such as the length of effective continuity behind front, the linear extent of trapped clusters, the mean length of an oil ganglion et cetera (see Table 3). Such numbers will be discussed in more detail in Sects. 2.3, 2.4, 2.5, 2.6 and 2.7. Other groups contain quantities that are not parameters of the problem, but elements of its solution, such as local microscopic or macroscopic pressure gradients. Although all of them have been used as capillary number correlations, they may not always be interpreted or intended as dimensionless groups expressing a force balance.

Some entries in Table 1 and numerous references contain or discuss gravity effects. For brevity, we limit the scope to viscous and capillary forces and do not discuss gravitational or centrifugal forces in this review. Similarly, we exclude drying (Yiotis et al. 2004) and diffusion–reaction problems (Kechagia et al. 2002) from the purview.

2.1 The Problem of Scale

While it is well accepted that the capillary number quantifies the ratio of viscous forces to capillary forces, it is rarely appreciated that this ratio depends on length scale. It should be emphasized that the length scales over which the two forces act need to be carefully checked in applications. Viscous and capillary forces on the pore scale differ from those on the laboratory scale and, again, from those on the field scale.

The upscaling problem from pore scale to laboratory scale is mathematically complicated by the emergence of measure valued functions (Hilfer 2018). This scientific fact is often overlooked by engineers in the field. Instead, it is generally assumed that the plateau region in conceptual pictures such as Hubbert (1957, Fig. 5, p. 35), Whitaker (1969, Fig. 3, p. 17), Bear (1972, Fig 1.3.1, p. 17), or Lake et al. (2014, Fig. 2.2, p. 21) exist. For convenience Figure 5 from Hubbert (1957, p. 35) is reproduced as Fig. 1. This assumption amounts to assuming that Eq. (57c) in Hilfer (2018) holds true. Precision measurements in Hilfer and Zauner (2011) and Hilfer and Lemmer (2015) tested such assumptions for natural and synthetic porous media. Figure 2 from Hilfer and Lemmer (2015) demonstrates that fluctuations are larger than commonly thought.

In this review a strict distinction between physical quantities on the pore scale and on the laboratory scale will be maintained. Pore scale quantities such as the interstitial phase velocity , the interfacial tension or the contact angle are denoted in serif font and coloured blue. Laboratory scale quantities such as porosity , permeability or capillary pressure ${P_\mathrm {c}}$ are denoted in a boldface sans-serif font and coloured magenta. The only exceptions to this convention will be densities and viscosities. For densities and viscosities it is assumed that

(2a)

(2b)

holds, where is the pore scale viscosity resp. density, and is the lab scale viscosity resp. density.

Fig. 1

Conceptual figure 5 from Hubbert (1957, p.35) depicting the commonly expected behaviour of porosity as a smooth function of the volume of the averaging window

Fig. 2

Four high precision measurements of porosity $\phi (\ell )$ as a function of side length $\ell$ obtained from four cubic averaging windows coloured black, blue, red and green (Hilfer and Lemmer 2015, Fig. 3). The sample is a cube of sidelength 15mm containing a pore space resembling that of Fontainebleau sandstone. The pore space is known with floating point precision. The porosity of the full sample is 0.134310101335984. The four averaging windows are disjoint for $\ell <7.5$mm. They overlap for $\ell >7.5$mm so that the decrease in fluctuations at large $\ell$ is not significant. These data do not support the conceptual Fig. 1. For $\ell <7.5$mm the fluctuations are much larger than commonly thought

2.2 Microscopic Capillary Numbers

The most widely used capillary number is the microscopic capillary number. It is defined generically as

(3)

where is the interstitial velocity and the pore scale viscosity of the displacing fluid. The capillary forces in this group are represented by the interfacial tension . The microscopic capillary number is identical or closely related as

(4)

to the numbers ${{}^{1}\mathrm {Ca}}$, ${{}^{10}\mathrm {Ca}}$, ${{}^{12}\mathrm {Ca}}$, ${{}^{15}\mathrm {Ca}}$, ${{}^{16}\mathrm {Ca}}$, and as

(5)

to the numbers ${{}^{21}\mathrm {Ca}}$, ${{}^{24}\mathrm {Ca}}$ from Table 1.

2.3 Mixed Capillary Numbers

Mixed capillary numbers arise from microscopic capillary numbers by replacing in eq. (3) with using Darcy’s law. The result is the number

(6)

discussed already in Ojeda et al. (1953) (see also ${{}^{8}\mathrm {Ca}}$ in Dombrowski and Brownell (1954) or ${{}^{20}\mathrm {Ca}}$ in Stegemeier (1974)). The popular variant

(7)

discussed in Melrose and Brandner (1974) arises when the generalized Darcy law for two phase flow is used instead of Darcy’s law for single phase flow. While the replacement is convenient when or $k^r$ is known, its drawback is that the replacement mingles pore scale with Darcy scale. Inconsistencies can arise from this. As shown in Hilfer (1996, Eq. (6.51)) the replacement amounts to normalizing the macroscopic pressure using Darcy’s law instead of the capillary pressure function. This is tantamount to the hidden assumption that

$$\begin{aligned} (\text {macroscopic viscous forces}) = (\text {macroscopic capillary forces}) \end{aligned}$$

(8)

as discussed in Hilfer (1996, Sec.IV.C.2). While this assumption may hold for some processes and conditions it may fail for others (Hilfer 2020). For instance, in Fig. 8 the assumption holds for the leftmost points, and in Fig. 9 for points close to the vertical dashed line. For more examples see Hilfer (2020, Fig. 5). It follows that mixed capillary numbers do not give the correct force balance in EOR applications.

The great majority of capillary numbers in Table 1 are of mixed type. In column 4 of Table 1 the mixed numbers are listed with “No”.

An interesting type of mixed number was obtained in Armstrong et al. (2014, Eq. (7)). The development starts from the macroscopic numbers

(9)

and introduces an average cluster length defined in Armstrong et al. (2014, Eq. (4)). As discussed in Hilfer et al. (2015, Eq. (32)) the cluster length, the computed effective permeability and the computed capillary pressure are not experimental quantities, but computational pore scale quantities that depend on numerous parameters of the underlying computer algorithms.

2.4 Macroscopic Capillary Numbers

The macroscopic (or Darcy scale) capillary number is defined in Hilfer (1996, Eq. (6.54)) or Hilfer and Øren (1996, Eq. (50))

(10a)

where is the Darcy velocity of the displacing fluid , $\mu _i$ is its effective viscosity on the REV-scale, is the permeability, and

(10b)

is the midpoint capillary pressure (see Anton and Hilfer (1999, Eq. (18)) for details). The midpoint capillary pressure $P_\mathrm {b}$ has been expressed in Hilfer (1996, Eq. (6.70)) and Hilfer and Øren (1996, Eq. (66)) as

(10c)

in terms of interfacial tension and the midpoint value of the empirical Leverett $J$-function correlation. Note that Eq. (10c) is not an “explanation” or “prediction” of $P_\mathrm {b}$ based on interfacial tension. It is merely another way of writing Eq. (10b) by defining . Both, $P_\mathrm {b}$ as well as $J_{\mathrm {b}}$, depend on as or . But these dependencies remain unknown.

The midpoint capillary pressure $P_\mathrm {b}$ is extracted from the capillary pressure function ${P_\mathrm {c}}(S)$ for the process of interest. The function ${P_\mathrm {c}}(S)$ can be measured directly in the laboratory or estimated for larger scales from saturation profiles in hydrostatic gravitational equilibrium. Consider a porous column experiment. Let x denote position along the column and let $x_0$ be a reference location at which the macroscopic pressure is the reference pressure . Solving the equations of motion for two immiscible phases with $\varrho _{\mathbbm {w}}>\varrho _{\mathbbm {o}}$ when all velocities and time derivatives vanish yields the well known result (Bear 1972)

(11a)

(11b)

for pressure and saturation, where $g$ is the acceleration of gravity. In this way ${P_\mathrm {c}}(S)$ and hence $P_\mathrm {b}$ are obtained experimentally from measuring $S(x)$ and inverting Eq. (11b).

The macroscopic capillary number is identical with ${{}^{32}\mathrm {Ca}}$ and related as

(12)

to the numbers ${{}^{7}\mathrm {Ca}}$ and ${{}^{36}\mathrm {Ca}}$. It was recently rediscovered in Andersen et al. (2017). The main difference between ${{}^{32}\mathrm {Ca}}$ and ${{}^{7}\mathrm {Ca}}$, apart from being inverted, is that $P_\mathrm {b}$ is defined for all processes, while $P_\mathrm {T}$ is only defined for primary drainage. The number

(13)

depends on scale, saturation and velocity. Quite different from previous expressions its definition involves relative permeability and capillary pressure. It may be called “localized macroscopic capillary number”, because it gives the local force balance between viscous and capillary forces within a macroscopically small but microscopically large region with saturation $S$.

Another purely macroscopic number in Table 1 is

(14)

introduced in Yortsos and Fokas (1983) and discussed in Dong et al. (1998). Note that the symbol S in Yortsos and Fokas (1983) equals $1-S$ in this paper. More seriously, the quantity $(d{P_\mathrm {c}}/d(1\!-\!S))_{\mathrm {ch}}$ can vanish in circumstances where the capillary forces do not vanish. This can occur e.g. for non-monotone ${P_\mathrm {c}}(S)$ functions such as those shown in Hilfer et al. (2012, Fig.1 and 2). Therefore, $(d{P_\mathrm {c}}/d(1\!-\!S))_{\mathrm {ch}}$ does not represent macroscopic capillary forces. This problem with ${{}^{27}\mathrm {Ca}}$ cannot be remedied easily.

2.5 Numbers Containing Contact Angle

A total of 13  capillary numbers in Table 1, namely ${{}^{4}\mathrm {Ca}}$, ${{}^{8}\mathrm {Ca}}$, ${{}^{9}\mathrm {Ca}}$, ${{}^{10}\mathrm {Ca}}$, ${{}^{11}\mathrm {Ca}}$, ${{}^{23}\mathrm {Ca}}$, ${{}^{17}\mathrm {Ca}}$, ${{}^{24}\mathrm {Ca}}$, ${{}^{30}\mathrm {Ca}}$, ${{}^{29}\mathrm {Ca}}$, ${{}^{31}\mathrm {Ca}}$, ${{}^{39}\mathrm {Ca}}$ and ${{}^{40}\mathrm {Ca}}$, contain a factor , where is some contact angle on the pore scale. All of these numbers diverge or vanish for neutral wettability, i.e. when .

The divergence of these 13  capillary numbers implies by virtue of the general relation (1), that the capillary forces vanish for . Indeed, some text books (Dullien 1992, p.456) teach erroneously “that for neutral wettability (i.e. ) Eq. (5.3.83) predicts $\mathrm {CA}_{\mathrm {imb}}=\infty$, because with a flat meniscus present the capillary forces are equal to zero—they don’t exist”. In this sentence the number $\mathrm {CA}_{\mathrm {imb}}$ is ${{}^{31}\mathrm {Ca}}$ from Table 1. The sentence is erroneous, because the contact angle obeys Young’s equation (Rowlinson and Widom 1982, Eq. (1.21))

(15)

where are the surface tension of water, resp. oil, with the matrix and . If and , then vanishes. Thus, the capillary rise in a circular capillary of radius r

(16)

vanishes. But that does not imply that “capillary forces are equal to zero”. Indeed only means that the capillary forces pulling the meniscus upward are equal to the capillary forces pushing it downward.

For this reason, it was emphasized in Hilfer (1996, p. 400), that the microscopic “capillary number is a measure of velocity in units of , a characteristic velocity at which the coherence of the oil-water interface is destroyed by viscous forces”. The estimate m/s was given in Hilfer (1996, Table V) for the critical velocity.

In summary, capillary forces exist also for , i.e. for neutral wetting and . They vanish only for . The balance of viscous and capillary forces on the pore scale is correctly quantified by . The capillary numbers ${{}^{4}\mathrm {Ca}}$, ${{}^{8}\mathrm {Ca}}$, ${{}^{10}\mathrm {Ca}}$, ${{}^{11}\mathrm {Ca}}$, ${{}^{17}\mathrm {Ca}}$, ${{}^{24}\mathrm {Ca}}$, ${{}^{30}\mathrm {Ca}}$, ${{}^{29}\mathrm {Ca}}$, ${{}^{31}\mathrm {Ca}}$,${{}^{39}\mathrm {Ca}}$ and ${{}^{40}\mathrm {Ca}}$ do not give the correct balance of forces. In EOR applications such as Lashgari et al. (2016) or Alvarado and Manrique (2010, p.13) they should be used with special care.

2.6 Numbers Containing Viscosity Ratio

Among the definitions of $\mathrm {Ca}$ in Table 1, the one proposed by Abrams (1975) is quite different from others because it incorporates the viscosity ratio. Abrams observed that multiplying the viscosity ratio as

(17)

seemed to decrease the scatter in the data points (Abrams 1975, Fig.3,4). The viscosity ratio is a key factor affecting the sweep efficiency ${E_\mathsf {V}}$.

Doorwar and Mohanty (2017) combined capillary number with other dimensionless scaling groups to predict recovery for unstable immiscible flows. Their so-called instability number reads

(18)

where is the absolute permeability. The length $D$ is called ”diameter” and seems to mean the diameter of the core, i.e. $D\approx 2.5$cm. It is unclear what $D$ is for general EOR processes. Note also, that the exponent of the viscosity ratio is negative in ${{}^{38}\mathrm {Ca}}_{\mathbbm {w}}$ while it is positive in ${{}^{21}\mathrm {Ca}}_{\mathbbm {w}}$. If ${{}^{38}\mathrm {Ca}}$ can be regarded as a capillary number, then, at fixed and constant $D$ and , Eqs. (18) and (17) are in conflict. We remind readers, however, that ${{}^{38}\mathrm {Ca}}$ was introduced not as a capillary number, but as an “instability number” in Doorwar and Mohanty (2017). More generally, any dependence of $\mathrm {Ca}$ on viscosity ratio raises the question whether viscosity ratio influences not only sweep efficiency ${E_\mathsf {V}}$, where this influence is well established, but also displacement efficiency ${E_\mathsf {D}}$.

2.7 Dimensional Groups

Not all groups listed in Table 1 are dimensionless. The group ${{}^{9}\mathrm {Ca}}$ has dimensions of length and the group ${{}^{14}\mathrm {Ca}}$ has dimensions of (length)$^{-2}$. These groups are listed in Table 1 because of their historical importance.

The ”interplay” between viscous and capillary forces “at the flood front” was discussed in great detail by Moore and Slobod (1955) in terms of the group ${{}^{9}\mathrm {Ca}}$, which led them to introduce the dimensionless group ${{}^{10}\mathrm {Ca}}$. They noticed important differences between drainage and imbibition based on their “Viscap concept” which they describe as a “capillary pressure-viscous force competition theory”. Because the pore size in laboratory core floods is the same as in reservoir floods they observe that “we generally cannot scale everything down in size”. They propose the dimensional group

(19)

as the correlating group for scaling. Here, is the Darcy velocity of the displacing phase, is the pore scale viscosity of the displacing phase, is the interfacial tension (IFT) between wetting and non-wetting phase, and is the contact angle. The length ${L_{\mathrm {ec}}}$ in Moore and Slobod (1955) is not the “length of the flow path” but the “length of effective continuity of the oil phase behind the flood front”, or “ the distance over which pressure difference between the oil and the water phases is developed as a result of the phases flowing”.

Moore and Slobod argue that ${L_{\mathrm {ec}}}$ is large for waterflooding of oil-wet systems, but small for water-wet systems, because in the latter case oil is discontinuous behind the front. Next, they claim that “$L$ may be considered as an invariant, geometric property of the rock”. They then drop $L$ “for simplicity”. In their words : “For simplicity, then, for water-wet systems $L$ may be eliminated, and we then have the dimensionless group

(20)

as the scaling term for water-wet systems” (Moore and Slobod 1955, p. 5). If is omitted from Eq. (20) and the Darcy velocity is replaced with the interstitial velocity , then Eq. (20) becomes the microscopic capillary number from Eq. (3).

The Taber group is not a dimensionless group, but has dimensions of m$^{-2}$. It has been influential in EOR, because Taber demonstrated the existence of a critical pressure gradient m$^{-2}$ for the mobilization of discontinuous oil in Berea sandstone Taber (1969). The critical gradient is then sometimes inserted into the condition ${{}^{6}\mathrm {Ca}}\approx 1$ to estimate the interfacial tension below which mobilization occurs from for a sandstone of permeability . It should be clear that this can be misleading, because the mingles the pore scale and the Darcy scale.

2.8 Summary of Capillary Numbers

• (1) At least 41 different expressions for capillary number have been put forward in the literature.

• (2) The 41 capillary numbers fall into three categories: 5  macroscopic ’s, 12  microscopic ’s and 24  mixed $\mathrm {Ca}$’s. Macroscopic ’s contain only macroscopic quantities, microscopic ’s contain only microscopic quantities and mixed $\mathrm {Ca}$’s contain both kinds.

• (3) Most $\mathrm {Ca}$’s are proposed for pore-matrix systems, and two for fracture-matrix systems.

• (4) Only two Ca’s incorporate the viscosity ratio, but in quite opposite dependencies.

• (5) Several definitions for $\mathrm {Ca}$ were proposed only recently, indicating that $\mathrm {Ca}$-theory still needs to be developed, clarified and improved further.

3 Capillary Desaturation Curves

Capillary desaturation curves (CDCs), also known as capillary number correlations (CNCs), were defined in Eq. (28a). They correlate residual oil saturation $ROS$ with capillary numbers. Accordingly, there are as many types of CDCs as there are capillary numbers, namely

(21)

where $n=1,\dots ,41$ and the dots indicate other parameters. CDCs are among the most important curves in EOR (Lake 1989; Green and Willhite 2018), because the key issue in EOR is to reduce residual oil saturation $ROS$ significantly and economically by various EOR methods and processes. Capillary desaturation curves enter as important input parameters into reservoir simulations (Youssef et al. 2013). In a numerical simulation the applicable capillary numbers will ensue automatically from correctly non-dimensionalizing the quantities in the mathematical model being simulated. A pore scale simulation will have pore scale numbers, while Darcy scale simulations will automatically exhibit the macroscopic capillary numbers. Mixed numbers will usually not arise.

3.1 Classic Microscopic and Mixed CDCs

To the best of our knowledge, one of the first CDCs was $CDC_{10}\left( {{}^{10}\mathrm {Ca}}\right)$ in Moore and Slobod (1955, Fig.5a). It was replotted horizontally in Abrams (1975, Fig. 2) and the latter plot is reproduced here as Fig. 3. The curve is monotone decreasing. The viscosity ratio between displacing and displaced fluid is close to one. An immediate problem with Fig. 3 is that the capillary number ${{}^{10}\mathrm {Ca}}$ on the abscissa diverges for . Note also, that the ordinate is oil saturation at breakthrough. Therefore, the saturations in Fig. 3 are expected to decrease further when more pore volumes are flooded through the sample. This expectation is confirmed in Abrams (1975, Fig. 5) (reproduced as Figure 10  in  Sect. 3.3) showing the mixed $CDC_{21}\left( {{}^{21}\mathrm {Ca}}\right)$ for seven reservoir rocks. The dash-dotted curve in Fig. 10 is for Berea sandstone.

Fig. 3

$CDC_{10}\left( {{}^{10}\mathrm {Ca}}\right)$ for Berea sandstone from Abrams (1975, Fig. 2). The ordinate is oil saturation at breakthrough. The abscissa becomes ill defined for contact angle

Fig. 4

$CDC_{28}\left( {{}^{28}\mathrm {Ca}}\right)$ for Fontainebleau sandstone from Chatzis and Morrow (1984), Fig. 13

The influence of the continuity of the initial oil configuration on CDCs was investigated in Chatzis and Morrow (1984, Fig.13). Figure 4 shows $CDC_{28}\left( {{}^{28}\mathrm {Ca}}\right)$, for Fontainebleau sandstone from Chatzis and Morrow (1984, Fig. 5). The solid lines in Fig. 4 represent floods with an initial oil configuration that is continuous throughout the sample. This is achieved by refilling the sample with oil after each flood. The dash-dotted line for initially discontinuous oil is obtained by increasing the flow rate or pressure drop without refilling the sample. It is seen that displacement of initially continuous oil is much easier than displacement of initially discontinuous oil. To achieve a given target $ROS$ smaller capillary numbers are necessary for continuous oil than for discontinuous oil displacement.

Fig. 5

Compilation of $CDC_{20}\left( {{}^{20}\mathrm {Ca}}\right)$ for various experiments from Stegemeier (1977, Fig. 13) illustrating influence of wettability

Fig. 6

Conceptual from Lake (1989, Fig. 3–17, p. 70) to illustrate wettability effects

Figure 5 from Stegemeier (1977, Fig. 13) collects the curves $CDC_{8}\left( {{}^{8}\mathrm {Ca}}\right)$,  $CDC_{10}\left( {{}^{10}\mathrm {Ca}}\right)$, $CDC_{15}\left( {{}^{15}\mathrm {Ca}}\right)$, $CDC_{21}\left( {{}^{21}\mathrm {Ca}}\right)$, $CDC_{16}\left( {{}^{16}\mathrm {Ca}}\right)$ and others into a single plot. Of course, such a comparison is problematic, because the rock samples, the fluids and the EOR-processes are different in the various studies. Nevertheless, the summary shows common features. In many cases a plateau region, where $ROS$ is nearly independent of $\mathrm {Ca}$ is followed by a more or less gradual decrease. The two dashed curves show non-wetting phase residual oil saturation, while the solid curves show residuals for the wetting phase.

The influence of wettability is depicted schematically in Fig. 6 from Lake (1989, Fig. 3-17) using standard microscopic . The CDC labelled “Wetting phase” is a drainage curve, the curve labelled “Non-wetting phase” is an imbibition curve. As a rule, the plateau saturation for wetting CDCs is believed to fall below that for non-wetting CDCs Lake (1989). Experimental data that seem to confirm these EOR concepts are shown in Fig. 7 for various cores of uniform and mixed wettability. “WF” in the legend of Fig. 7 stands for waterflood, “SF” stands for surfactant flood. The cores are Berea outcrop sandstone. Compare the blue circles and the blue diamonds for uniformly water wet samples with the yellow triangles and magenta squares for two mixed wet cores. For uniformly water wet cores $ROS$ remains constant, while for the mixed wet systems it decreases.

The CDCs discussed in this section are regarded as classic, because they are based on the widely used microscopic and mixed capillary numbers such as ${{}^{6}\mathrm {Ca}}$ and ${{}^{12}\mathrm {Ca}}$. Classic CDCs are usually measured on high permeability cores. Berea sandstones have values of between 40 and 2500 mD (Chatzis and Morrow 1984). Examples for low permeability media are Bandera sandstones with 10mD in Moore and Slobod (1955), 32 mD Abrams (1975) and a limestone with 26 mD in Abrams (1975).

Fig. 7

Experimental from Abeysinghe et al. (2012, Fig. 9) for Berea outcrop sandstone

The common features of classical CDCs exhibited above are :

• (1) The larger the lower is $ROS$. In other words classical microscopic and mixed CDCs $ROS=CDC(\mathrm {Ca})$ are monotone decreasing functions of $\mathrm {Ca}$. Maximum oil recovery requires maximum $\mathrm {Ca}$.

• (2) At the highest values of to achieved to date in the laboratory residual oil often approaches zero. Classical CDCs are expected to approach zero, i.e. , for .

• (3) There exists a breakpoint or critical capillary number $\mathrm {Ca}_c$ such that $ROS=CDC(\mathrm {Ca})$ is approximately constant for all $\mathrm {Ca}<\mathrm {Ca}_c$. For many media the critical capillary number $\mathrm {Ca}_c$ corresponds to the beginning of mobilization. For many porous media, such as sandstones, $\mathrm {Ca}_c\approx 10^{-6}\dots 10^{-5}$.

• (4) Initially, continuous oil is displaced more easily than initially discontinuous oil.

• (5) CDCs for water-wet and oil-wet rocks are different. The breakpoint $\mathrm {Ca}_c$ for the wetting phase is typically one or two decades above that for the non-wetting phase.

• (6) CDCs for mixed wet rocks seem not to show plateau where oil- and water-wet rocks exhibit their plateau saturation.

Fig. 8

Macroscopic for bead packs and Berea sandstone from Anton and Hilfer (1999, Fig.2)

3.2 Macroscopic CDCs

The macroscopic $CDC_{32}\left( {{}^{32}\mathrm {Ca}}\right)$ based on ${{}^{32}\mathrm {Ca}}$ was computed for the first time in Anton and Hilfer (1999, Fig. 2) and it is reproduced here as Fig. 8. Desaturation data for bead packs from Morrow et al. (1988) are shown as symbols connected by a solid line (continuous oil) and as a dashed line (discontinuous oil). Desaturation data for Berea sandstone from Abrams (1975) are shown as stars in Fig. 8. Figure 8 shows lower values for $ROS$ than Fig. 3 at breakthrough. Figure 8 also confirms the results for continuous versus discontinuous mode displacement seen in Fig. 4. The main result of Fig. 8 is the values of on the abscissa. These are five decades larger than for classic CDCs. The breakpoint obeys . This confirms that is an accurate measure of viscous-to-capillary force balance during two phase flow in porous media, while is not.

Figure 9 shows a comparison of three capillary desaturation curves, the microscopic (as tringles), the macroscopic (as circles), and the local macroscopic $CDC_{36}\left( {{}^{36}\mathrm {Ca}}\right)$ (as pentagons), for the desaturation experiments on Dalton sandstone (sample 798) in Abrams (1975). The vertical dashed line at $\mathrm {Ca}=1$ marks equality of viscous and capillary forces. Figure 9 directly visualizes the enormous shift by 7-8 decades. It also demonstrates that the global macroscopic CDC based on can differ substantially from the local macroscopic CDC based on ${{}^{36}\mathrm {Ca}}$. This may have important consequences for EOR screening patterns.

Fig. 9

Comparison of three CDCs for Dalton sandstone (sample 798) from Abrams (1975). The microscopic is shown as tringles, the macroscopic $CDC_{32}\left( {{}^{32}\mathrm {Ca}}\right)$ as circles, and the local macroscopic $CDC_{36}\left( {{}^{36}\mathrm {Ca}}\right)$ as pentagons. The vertical dashed line at $\mathrm {Ca}=1$ marks equality of viscous and capillary forces

The importance of macroscopic CDCs arises from :

• (1) Macroscopic CDCs are based purely on macroscopic quantities appearing in the generalized Darcy law and the capillary pressure saturation relation.

• (2) Macroscopic CDCs represent the balance of macroscopic viscous to macroscopic capillary forces correctly. Their breakpoint occurs at resp. ${{}^{36}\mathrm {Ca}}_c\approx 1$ while the breakpoint of microscopic and mixed CDCs occurs at $\mathrm {Ca}\ll 1$.

• (3) Macroscopic CDCs depend on petrophysical properties of the reservoir rock while microscopic CDCs do not.

• (4) Macroscopic CDCs depend only on quantities that are measurable experimentally by standard methods and do not depend on mesoscopic cluster sizes, computational modelling or 3d image analysis of displacements.

• (5) Macroscopic CDCs suggest new and hitherto unexplored EOR methods and EOR processes.

Fig. 10

$CDC_{21}\left( {{}^{21}\mathrm {Ca}}\right)$ for various sandstones and one limestone from Abrams (1975, Fig. 5) illustrating the influence of viscosity contrast

Fig. 11

$CNC_{38}\left( {{}^{38}\mathrm {Ca}}\right)$ for various cores and floods summarized in Doorwar and Mohanty (2017, Fig. 10). Note the scale on the abscissa

3.3 Influence of Viscosity Ratio on CDCs

A thorough study of the influence of viscosity ratio on CDCs for six sandstone cores and one limestone was carried out in Abrams (1975) using 21 fluid pairs with different viscosities and interfacial tensions. A schematic summary of the highly fluctuating results is given in Abrams (1975, Fig. 5) and reproduced in Fig. 10. The factor was introduced because it decreases the scatter in the data, albeit only very slightly. No theoretical justification was given for either the factor or the exponent 0.4. Recently, the factor in Abrams (1975) was contradicted in Doorwar and Mohanty (2017). These authors multiply a factor $(\mu _{\mathbbm {w}}/\mu _{\mathbbm {o}})^{-2}$ and propose a theoretical basis for it. In Doorwar and Mohanty (2017) eight corefloods were conducted on a single Boise sandstone core. The brine permeability of the core was measured to be approximately 6 darcies, and the porosity was 29%. Tests showed good obvious trend of cumulative recovery with ${{}^{38}\mathrm {Ca}}$, as shown in Fig. 11 from Doorwar and Mohanty (2017). However, it is interesting that ${{}^{38}\mathrm {Ca}}$ in Fig. 11 is as large as ${{}^{38}\mathrm {Ca}}\approx 10^{10}$.

3.4 CDCs for Polymer Floods

In some classic microscopic $CDC$ tests, polymers were used to increase the viscosity of the displacing phase. It is well accepted that polymers without surfactants cannot increase the microscopic capillary number to reach the critical value $\mathrm {Ca}_c$. Indeed, this would reduce $ROS$ to a very low value, which is not observed. This is important for EOR, because polymers can increase the brine viscosity by at most two orders of magnitude due to the reservoir pressure gradient constraint. Therefore, it was believed for a long time that polymers cannot reduce $ROS$ (Needham and Doe 1987; Lake 1989; Lake et al. 2014; Green and Willhite 2018).

Fig. 12

(lower group of six decreasing curves) from polymer floods affected by viscoelasticity. from Wu et al. (2007)

However, some recent polymer flooding experiments produced CDCs that are different from previously observed CDCs. Figure 12 shows such a $CDC$ polymer flooding reported in (Wu et al. 2007). In these experiments surfactants and alkali were used. The cores were flooded horizontally. In Figure 12, the upper group of curves shows the Oil Recovery Factor while the lower group shows $ROS$. Figure 12 shows that the viscoelasticity of a HPAM polymer might play an important role in reducing $ROS$ (Wu et al. 2007). Other studies also showed that the viscoelasticity of polymers seems to contribute to the reduction in $ROS$ (Wang et al. 2007; Azad and Trivedi 2017; Wang et al. 2004, 2000).

Qi et al. (2017) reported a quite different polymer flooding CDC, as is shown in Figure 13. In these experiments, the cores were displaced vertically. Figure 13 shows that polymers without the addition of surfactants can reduce $ROS$ to almost zero. They also attributed this to the viscoelasticity effect of HPAM to the ROS (Erincik et al. 2018). However, the extremely low ROS in their tests were also possibly caused by the gravity-stabilization effect (Adebayo et al. 2017; Hilfer 1996; Cinar et al. 2006; Schechter et al. 1994, 1991).

In another vertically placed flood test (Lu and Pope 2017), gravity-stable surfactant flooding without addition of the polymer (to control the mobility) achieved a very high recovery, which was not reported for horizontally placed core floods or field tests. In addition, this CDC (Figure 13) was obtained in a secondary recovery model at high initial oil saturation where oil was more continuously distributed. However, some cores were oriented vertically while others were horizontally oriented. This great difference should be given special attention.

A non-monotonic $CNC$ was reported by Qi et al. (2014). It is shown in Fig. 14. In these experiments $CNC_{12}\left( {{}^{12}\mathrm {Ca}}\right)$ was used. However, the three-layer long cores (4cm*4cm*30cm) used to get this CNC were quite different from typical short cores that are used in classic CDC tests. These core flooding tests were horizontally placed. The same physical model used was well shown in Hou et al. (2005). Note that in Fig. 14 the data points are not from a single core, but from different cores. It is interesting that the ROS is not further reduced when Ca goes higher than 10${}^{-2}$. In addition, the minimum ROS does not go to zero. The Ca values in their tests went high to 10${}^{-1}$, which were rarely reported. However, the ordinate in Figure 14 is not the actual residual oil saturation $ROS$ but remaining oil saturation due to insufficient sweep efficiency.

Fig. 13

Polymer flooding with sharper decay from Qi et al. (2017)

Fig. 14

Polymer flooding from Qi et al. (2014)

Fig. 15

Fracture $CDC_{39}\left( {{}^{39}\mathrm {Ca}}\right)$ from AlQuaimi and Rossen (2018, Fig. 10)

3.5 Non-monotonic CDCs

The CDCs mentioned in Sect. 3.1 through 3.4 are generally based on core flooding tests. Numerous studies use other physical or numerical models to obtain CDC. These CDCs help to understand the flow in porous media at different scale and provide insights into displacement efficiency and sweep efficiency which are important for EOR.

Mai et al. (2009) and Mai and Kantzas (2009) found that waterflooding after primary production yields an improved viscous oil recovery at slower injection rates. This means lower improved oil recovery at higher . Thus, a non-monotonic $CNC$ can be obtained. Doorwar and Mohanty (2017) also reported lower breakthrough viscous oil recovery at higher Ca as defined in Eq. (3). See Fig. 7 on p. 25 in Doorwar and Mohanty (2017).

AlQuaimi and Rossen (2018) proposed the capillary number definition ${{}^{39}\mathrm {Ca}}$ for fractures that incorporated a geometrical characterization of the fracture. It is called fracture capillary number in this paper and the $CDC_{39}\left( {{}^{39}\mathrm {Ca}}\right)$ is called fracture CDC. It seems that the CDCs based on ${{}^{39}\mathrm {Ca}}$ scatter less than the curves obtained for the conventional definition of the Ca, which can be seen in Fig. 15 from AlQuaimi and Rossen (2018). In addition, the values of ${{}^{39}\mathrm {Ca}}$ are much higher than values of the conventional Ca and their largest Ca values approach unity. It is worth mentioning that the models used for their $CDC_{39}\left( {{}^{39}\mathrm {Ca}}\right)$ are different from the cores used for a typical CDC.

Chang et al. (2019) used the “complete” capillary number ${{}^{40}\mathrm {Ca}}$ which includes pore-scale characteristics of micromodels such as pore-throat diameter and micromodel depth. Their study concerns unstable drainage of brine by supercritical CO${}_{2}$. Their most unusual observation is the non-monotonic increase of the CO${}_{2}$ saturation for increasing $\mathrm {Ca}$. This CNC (Fig. 6, p. 19) in Chang et al. (2019) is the opposite process of the CDC. Although transition from capillary fingering to viscous fingering during the unstable drainage process may help to account for their test results, the CNC is quite inconsistent with previous studies (Lenormand and Zarcone 1988).

A non-monotonic CNC for a 2-D network model was reported in Rodriguez de Castro et al. (2015). Their study showed that no plateau of ROS is observed when the Ca decreases. Figure 6 (p. 8525) and Figure 7 (p. 8526) in Rodriguez de Castro et al. (2015) show that the ROS was not only affected by the capillary number, but also by the viscosity ratio. It seems that a monotonic CDC is only possible when the displacing phase viscosity is larger than the displaced phase (viscosity ratio is larger than unity). In another test using the Hele-Shaw cell model, non-monotonic CNCs were obtained as well in Khosravian et al. (2015) (see Figure 4 (p. 1388) in Khosravian et al. (2015)). Interestingly, the non-monotonicity is very obvious when M is 0.22 ($= 1 / 4.5$). However, some extent of non-monotonicity is also seen when viscosity ratio M is 1 (Figure 4, in Khosravian et al. (2015)). The non-monotonicity of CNC was also reported by other researchers. Zhao et al. (2016) studied impact of wettability on viscously unfavourable fluid-fluid displacement in disordered media by means of high-resolution imaging in microfluidic flow cells. Their study showed that even when the Ca (defined as ) was the same, ${E_\mathsf {D}}$ and ${E_\mathsf {V}}$ can vary strongly regarding the Ca. The displacement efficiency can be seen in Fig. 3 (B) on p. 10253 from Zhao et al. (2016). Effect of Ca on ${E_\mathsf {V}}$ can be seen in Fig. 2 in this reference. Fig. 2 and Fig. 3 show obvious non-monotonicity of CNC. It is interesting that ${{}^{10}\mathrm {Ca}}$ was not used in Zhao et al. (2016), although the contact angles in their tests can be more easily fixed and obtained than in other tests.

Rabbani et al. (2018) also reported microfluidic tests that yield non-monotonic CNCs. These showed a gradual and monotonic variation of the pore sizes along the front path suppresses viscous fingering (Rabbani et al. 2018). The non-monotonicity was well seen in Fig. 2 and Fig. 3 in (Rabbani et al. 2018). With the same Ca (defined as , the displacement efficiency (Fig. 3E in Rabbani et al. (2018)) or sweep efficiency (Fig. 2 in Rabbani et al. (2018)) varied. Their study showed that CNC is not only affected by Ca, but also by pore-size gradient. Since the viscosity ratio in their tests was kept constant, it is interesting that the flow does not follow a monotonic pattern, as would be expected. Since the models used in these tests can be regarded as homogeneous, the non-monotonicity of CNCs deserves further study. Note that all these non-monotonic CNCs were based on microscopic .

It is interesting that most CDCs from core flooding tests are monotonic (see e.g. Figs.5 and 6), while CDCs from micromodels can be monotonic (Zhang et al. 2011, Fig. 5) or non-monotonic (An et al. 2020, Fig. 4). It remains to be further investigated what major factors cause nonmonotonicity. Possible candidates are the two-dimensionality of micromodels or possibly insufficient scale separation (see e.g. Karadimitriou et al. 2013, Fig. 9). Other factors for the nonmonotonicity in micromodels may be associated with the flow regime crossover from capillary fingering to viscous fingering (An et al. 2020; Khosravian et al. 2015; Chen et al. 2017; Wang et al. 2013; Ferer et al. 2004), model inhomogeneity (Wang et al. 2013), solid surface wettability (Zhao et al. 2016), flow history (Krummel et al. 2013; Khosravian et al. 2015), and rock surface roughness (Glass et al. 2003; Chen et al. 2017).

4 The Use of Capillary Numbers in EOR

To illustrate the importance of $\mathrm {Ca}$ and $CDC$ for EOR, consider the mobilization of trapped oil. If for fixed and given initial oil saturation the functions $CDC_n$ are decreasing with ${{}^{n}\mathrm {Ca}}$, then different CDCs lead to very different EOR strategies. EOR strategies based on the traditional microscopic would tend to increase the velocity or viscosity of the displacing fluid and to decrease the interfacial tension, but would predict recovery to be independent of length scale. EOR strategies based on the mixed would predict recovery to be independent of viscosity and to decrease with length scale because the pressure drops and hence  the  velocity decreases.  EOR strategies based on the macroscopic , however, would predict recovery to increase with increasing length scale and viscosity, but to decrease with increasing permeability.

Standard EOR techniques are based on the classical microscopic number . Among different EOR techniques, chemical flooding is more closely related to the capillary number (Green and Willhite 2018) than other methods. Surfactants in chemical EOR serve to reduce the oil/water interfacial tension from typically around mN/m to 10$^{-3}$ mN/m to reach sufficiently high capillary numbers (Guo et al. 2018, 2019). This is well accepted by many, if not all, petroleum engineers.

Microscopic CDCs also suggest that reduction in interfacial tension has the same effect on ROS as increasing the viscosity $\mu$ of the displacing phase by an equal percentage. It should be kept in mind, however, that $\mu$ affects sweep efficiency as well as displacement efficiency, while this is not the case for . Laboratory tests (Zhang et al. 2010; Hou et al. 2005; Shen et al. 2009; Wang et al. 2010) and current chemical flooding practice (Guo et al. 2017, 2019) suggest that increasing viscosity has a higher effect on ROS, because it improves not only capillary number but also viscosity ratio. This seems especially true, when heterogeneity is taken into consideration. Heterogeneity effects have been well investigated in (Zhou et al. 1997) and (Hilfer and Helmig 2004).

5 Summary

• 1. At least 41 different expressions for capillary number have been identified in the literature. The 41 capillary numbers fall into three categories: 5  macroscopic ’s, 12  microscopic ’s and 24  mixed $\mathrm {Ca}$’s. Macroscopic numbers contain only Darcy or field scale quantities, microscopic numbers contain only pore scale parameters, mixed numbers contain both.

• 2. For macroscopic capillary numbers the accurate characterization of viscous and capillary forces is length scale dependent.

• 3. Mixed capillary numbers containing contact angle have difficulty in describing neutral wettability.

• 4. The most widely used classic definitions of $\mathrm {Ca}$ in EOR are shown to be microscopic or mixed. They often contain the hidden assumption . This fact has been overlooked for a long time.

• 5. Different from microscopic and mixed capillary number, which are directly based on interfacial tension as a crucial parameter, macroscopic capillary numbers are based on macroscopic capillary pressure which is affected not only by fluid-fluid interfacial tension , but also by numerous other geometrical and physical parameters of the displacement.

• 6. Classic capillary desaturation curves are mainly based on microscopic and mixed $\mathrm {Ca}$’s. Classic CDC’s are monotonic, and hence small residual oil saturation $ROS$ requires large values of implying the use of surfactants in EOR to reduce interfacial tension to ultra-low values and/or polymers to increase viscosity. This doctrine to reduce to ultra-low values is very widespread in chemical EOR.

• 7. Different from classic based on microscopic or mixed numbers, for macroscopic the transition from capillary force dominated flow to viscous force dominated flow happens at macroscopic , which seems to be more reasonable.

• 8. Non-monotonic $CNC$’s from many tests may reflect the importance of viscosity ratio and capillary number on . A notable attempt to correlate them was made with ${{}^{38}\mathrm {Ca}}$. However, it remains to be investigated whether and how to use macroscopic capillary numbers to account for these non-monotonic $CNC$’s.

• 9. Macroscopic functions may call for alternative screening EOR strategies and new experiments.

Appendices

Definition of Symbols

Recovery efficiencies are generally introduced by writing

$$\begin{aligned} \text {(amount initially present)}&= \text {(amount produced)} +\text {(amount remaining)} \end{aligned}$$

(22a)

$$\begin{aligned} \text {(amount initially present)}&= \text {(amount contacted)} +\text {(amount missed)} . \end{aligned}$$

(22b)

The total recovery efficiency is then defined as

(23a)

the (“microscopic”) displacement efficiency as

(23b)

and the (“macroscopic”) volumetric displacement (or sweep) efficiency as

(23c)

In this paper, we do not use the words microscopic and macroscopic in this way. Instead we use microscopic to mean “pore scale” and macroscopic to mean “Darcy scale” or “field scale” as discussed in subsection 2.1 of Sect. 2. The second equality in Eq. (23) identifies “amount” with “volume occupied”. And , , denote remaining, missing and initial oil saturation. Efficiencies are pure numbers obeying $0\le {E_\mathsf {R}},{E_\mathsf {D}},{E_\mathsf {V}}\le 1$, and the relation

$$\begin{aligned} {E_\mathsf {R}}= {E_\mathsf {D}}\,{E_\mathsf {V}}\end{aligned}$$

(24)

holds by definition. Because the amount contacted cannot be measured as easily as the amount produced, the sweep efficiency remains somewhat ill defined.

Displacement efficiency ${E_\mathsf {D}}$ depends on many factors such as pore structure, pore distribution, microscopic heterogeneity, wettability and interfacial tension. It is generally believed that the sweep efficiency ${E_\mathsf {V}}$ is significantly affected, among other factors, by the mobility ratio between the displacing phase and the displaced phase (Lake 1989).

General EOR desaturation processes decrease the oil saturation starting from an initial oil saturation . If , then a process dependent residual oil saturation $ROS$ is defined as

(25)

Equivalently, one has

(26)

where Eq. (23) has been used. In the literature the symbol is often used for residual oil saturation instead of our symbol $ROS$. Here, we prefer to define the symbol as the solution of the equation

(27)

where is the relative permeability function of oil, i.e. is a zero of .

Let us now recall that the most significant parameter to affect or influence $ROS$ (or ), and hence ${E_\mathsf {D}}$, is capillary number

Capillary forces are responsible, in part, for the inefficiency of the displacement of oil from petroleum reservoirs during water flooding. In this paper, the convention is used to call a function that relates and other parameters of the EOR-process with residual oil saturation $ROS$

(28a)

a capillary desaturation curve (CDC), while the function

(28b)

is called a capillary number correlation (CNC). For one finds $CDC=CNC$. More generally, for and the $CDC$-functions and $CNC$-functions are related as

(28c)

by virtue of Eq. (26). More precise definitions are given in Appendix B.

Definition of Capillary Desaturation Curves

An EOR process starting from an initial oil saturation may depend on many parameters and conditions, but it always ends at some residual oil saturation $ROS$. To define precisely the capillary desaturation function $CDC$ in Eq. (28a) some standard mathematical notation is used, that may be unfamiliar to the reader. The notation

Table 2 List of commonly used symbols

Table 3 List of special symbols specific to certain authors

$$\begin{aligned} f:A\times B&\rightarrow C\nonumber \\ (a,b)&\mapsto c=f(a,b) \end{aligned}$$

(29)

is used to denote a mapping f that maps the domain $A\times B$ to a set C (called range of f), where $(a,b)\mapsto c=f(a,b)$ means that the element $(a,b)\in A\times B$ is mapped to the element $c=f(a,b)\in C$. The symbol $\times$ is the Cartesian product of sets. To define $CDC$ one first introduces a mapping

(30)

that maps initial saturation and other parameters of the EOR process, the fluids, the rock, etc., to $ROS$ called desaturation correlation. In Eq. (30) the set {EOR} stands for the set of parameters of the EOR desaturation process, such as number of pore volumes flooded, {fluids} stands for fluid parameters such as viscosities, interfacial tension, initial fluid configuration, etc., and {rocks} stands for reservoir parameters such as rock types, topological pore structure, pore geometry, surface roughness, wetting properties, etc. If the capillary number can be used to reduce the number of independent parameters in the desaturation correlation Eq. (30), then a simplified function

(31)

can be defined that has fewer arguments than $DC$. It is called capillary number correlation (CNC) or capillary desaturation curve (CDC).