Foam flow in porous media: Concepts, models and challenges

Journal of Natural Gas Science and Engineering 53 (2018) 163–180

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Review Article

Foam flow in porous media: Concepts, models and challenges a,∗

b

b

Hamed Hematpur , Syed Mohammad Mahmood , Negar Hadian Nasr , Khaled Abdalla Elraies a b

T b

Reservoir Studies Department, Research Institute of Petroleum Industry (R.I.P.I), Iran Petroleum Department, Universiti Teknologi Petronas, Malaysia

A R T I C LE I N FO

A B S T R A C T

Keywords: Foam flooding Mobility control Foam flow modeling Foam model parameters

This paper aims to elaborate foam concepts and foam flow modeling approaches in porous media. Furthermore, this review summarizes and compares all existing foam models approaches including Mechanistic, SemiEmpirical and Empirical. Finally, it discusses foam models in different reservoir simulators in detail and presents different approaches for obtaining models’ parameters in simulators. The comparison results showed that Emprical models are more suitable for simulation study due to less required paramters and faster calculation; however, these models might not be a appropriate in transient foam flow. Moreover, the challenges about he results of this review provide an valuble insight about foam behaiviour.

1. Introduction Recently, gas flooding became one of the most accepted and widely used methods for enhanced oil recovey (EOR) (Franklin and Orr, 2007). Fig. 1 illustrates the 38.4% and 68.4% contribution of gas flooding methods in the worldwide enhanced oil recovery. The gas injection phases are nonhydrocarbon gases such as flue gas, nitrogen, carbon dioxide, and even hydrogen sulfide and hydrocarbon gases such as methane and mixture of methane to propane (Mohamed El Gohary, 2012). There are two different schemes for gas injection process; miscible gas flooding and immiscible gas flooding. In the first scheme, the governing mechanisms for oil production are swelling the oil phase as well as reducing the oil viscosity. This mechanism leads to an increase in the microscopic efficiency compared to water flooding (Lake, 1989). Generally, the hydrocarbon gases and carbon dioxide are utilized as the gas phase for miscible gas injection. In the second scheme, only a small portion of the gas phase is dissolved in the oil and the main purpose of this method is to increase the reservoir energy (Green and Willhite, 1998). This mechanism increases the macroscopic efficiency which is usually is less than water flooding. The nitrogen gas is a good candidate for the immiscible process because it is hard to achieve the miscibility point at common reservoir pressures. The adverse mobility ratio of gas during gas displacement is considered as the crucial problem of gas flooding (Hanssen et al., 1994; Liu et al., 2011; Farzaneh and Sohrabi, 2013; Rossen et al., 2014; Farajzadeh et al., 2016). Mobility ratio is the mobility of the displacing fluid divided by that of the displaced fluid (Mdisplacing / Mdisplaced ). The

∗

favaroble mobility ratio is one or less than one to make a piston like displacment; however, the mobility ratio for gas flooding is between 10 and 100 which is considered as unfavorable mobility ratio (Displacement Efficiency of Immiscible Gas Injection, 2013). This poor mobility ratio arises from the significant difference between the viscosity of gas and oil compared to the viscosity of water and oil. This difference in viscosity results in higher mobility of gas to oil, consequently, leads to an unfavorable mobility ratio. This unfavorable mobility ratio results in viscous fingering phenomena and premature breakthrough of the gas phase, eventually, poor sweep efficiency (Boeije and Rossen, 2015). The poor sweep efficiency increases the cost of gas injection and recycling process (Krause et al., 1992). This situation becomes worse in heterogeneous reservoir since gas channeling phenomena occurs through the higher permeable layers and the gas bypasses the oil (Chang et al., 1990). Furthermore, normally the gas density is less than one-third of oil density at reservoir condition (Jamshidnezhad et al., 2008), this large difference brings another disadvantage. It causes phase segregation and gas overriding to the top of the reservoir (Rossen et al., 2014). The segregation phenomenon also decreases the sweep efficiency of gas flooding, consequently, the huge amount of oil phase will be bypassed by the gas phase. Fig. 2 illustrates these gas flooding mobility issues. One of the common methods to reduce the gas mobility as well as increase the sweep efficiency is water-alternating-gas (WAG) injection. The WAG was introduced by Caudle et al., in 1957 to mitigate the gas mobility problem (Dyes et al., 1954). The microscopic displacement of the oil by water is less that by gas due to higher interfacial tension between water and oil. On the other hand, the macroscopic

Corresponding author. Reservoir Studies Department, Research Institute of Petroleum Industry (R.I.P.I), Iran. E-mail address: [email protected] (H. Hematpur).

https://doi.org/10.1016/j.jngse.2018.02.017 Received 2 November 2016; Received in revised form 19 February 2018; Accepted 19 February 2018 Available online 24 February 2018 1875-5100/ © 2018 Elsevier B.V. All rights reserved.

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Fig. 1. The contribution of different EOR methods in the world for onshore and offshore fields in 2016. (a) Onshore fields; (b) Offshore fields (Kang et al., 2016).

Fig. 2. Poor sweep efficiency of gas flooding issues listed as; (1) viscous fingering, (2) gas channeling, and (3) gas overriding (Hanssen et al., 1994).

There are various examples of foam's field application for EOR such as Kern River and Midway Sunset fields in the US (Hirasaki, 1989; Patzek and Kolnls, 1990; Friedmann et al., 1994), Snorre field in Norway (Tore et al., 2002; Aarra et al., 2002), Prudhoe Bay field in the US (Krause et al., 1992), San Andres field in the US (Prieditis and Paulett, 1992), and Oseburg field in Norway (Aarra and Skauge, 1994; Hoefner et al., 1995). These implementations, results in significant increase of recovery factor, for instance in San Andres using foam–assisted process improve an oil production by 10%–30% (Prieditis and Paulett, 1992). The mode of foam application would depend on the nature and source of the problem (Turta and Singhal, 2002). In the field application, foam is injected in different ways which are more diverse compared to that in the laboratory (Turta and Singhal, 2002). There are fives types of foam injection methods in the field and laboratory applications;

displacement of the oil by water is better than by gas due to lower mobility of the water. Therefore, the combination of these two methods improves the recovery due to the reduction of gas mobility (Christensen et al., 1998). However, the pilot test for WAG application and simulation studies shows only a modest reduction in gas mobility; the segregation and viscous fingering reoccur (Surguchev et al., 1995; Righi et al., 2004; Sohrabi et al., 2000; Christensen et al., 1998). The foam assisted process is a potential solution to tackle all the mentioned problems in gas flooding as well as WAG (Bond and Holbrook, 1958; Kovscek et al., 1995; Farajzadeh et al., 2008; Wang and Li, 2016). Foam is able to control the gas mobility properly, by ceasing a large amount of gas phase through the porous media and increase the apparent viscosity of gas phase (Bernard et al., 1980; Hirasaki and Lawson, 1985). Wang and Li also showed that mobility reduction by SAG was much higher than by WAG Method while flowing surfactant solution and propane alternately through glass beadpack. They also observed that higher SAG ratio resulted in greater mobility reduction. (Wang and Li, 2016). Adebayo, Kamal, & Barri conducted a series of laboratory experiment on rock samples in both vertical and horizontal flow directions to compare the effects of water alternating gas (WAG) and surfactant alternating gas (SAG) as mobility control methods versus continuous gas injection method with respect to pressure drop and trapped gas saturation. Their results showed that the SAG method significantly increased trapped gas saturation for both horizontal and vertical flows while WAG method showed opposite behavior (Rasheed et al., 2017). Foam also reduces the relative permeability of gas drastically. Bond and Holbrook (1958) defined the concept of foam application for gas mobility reduction for the first time. This reduction is shown by mobility reduction factor (MRF) which is the mobility ratio of foam divided by gas mobility. The MRF can be calculated by dividing the pressure drop of foam flooding by pressure drop of gas flooding (MRF = ΔPfoam/ ΔPgas ) (Nguyen et al., 2000).

⁃ Pre-formed foam injection: In this method, foam is generated outside the porous media. Foam can be generated either at the surface via a foam generator or through the tubing during downward flow (Turta and Singhal, 2002). The potential of controlling the foam injection quality and foam's strength are the feature of this method. ⁃ Co-injection foam: In this method, foam is formed inside the formation few meter from the injector well by injecting both phases of gas and surfactant solution. This method is also called “in-situ foam” generation. Two tubing strings are required for this method, one for gas phase and second for surfactant solution (Turta and Singhal, 2002). ⁃ Surfactant-Alternating-Gas (SAG) foam injection: Foam is formed inside the porous media by consecutive injection of gas and surfactant solution in this method. During SAG foam process, a surfactant solution is drained by the gas phase, therefore, this method is also called “drainage foam” injection. The foam under this method is not limited to entry zone but wherever the gas has the contact with invaded surfactant solution, the foam can be generated (Rossen and Boeije, 2013). ⁃ Dissolved surfactant foam injection: according to several studies, some surfactants are able to dissolve in carbon dioxide under the supercritical condition (Le et al., 2008; Ashoori et al., 2010). In this method, only one phase is injected into the reservoir and the foam is generated once it meets the formation water. ⁃ Simultaneous different layers foam injection where the gas phase and surfactant solution phase are injected simultaneously but in different sections of the well. This method can be performed in both vertical and horizontal wells. The gas is injected from the lower section/lower horizontal well and the surfactant is injected from upper section/upper horizontal well. Because of gravity segregation phenomena, the gas and surfactant phase can meet each other and generate the foam inside the formation (Stone, 2004; Rossen et al., 2010).

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The foam applications are not limited to EOR, several studies were cconducted on foam applicaton in matrix acidization treatments (Rossen and Wang, 1999; Alvarez et al., 2000; Kam et al., 2007a,b; Li et al., 2008), gas leakage prevention (Bernard and Holm, 1970; Smith and Jikich, 1993) and foam contaminated aquifer remediation (Hirasaki et al., 1997; Mulligan, 2009). The succesful foam application requires a predective mathematical model to be a represntative of the foam flow behavior. Even though various works have been carried out in this area, the lack of comprehensive and comparative study on foam modeling is the motivation of this study. Therefore, the objective of this study is to review the previous studies on foam flow modeling and its challenges. This literature review focuses on the history of foam flow models in porous media and the methods to derive the model's parameters for simulation studies. In the first section of this review paper the basic concepts of foam are explained briefly. Then, the backgrounds of different foam model approaches including the mechanistic,semi-empirical and empirical are elaborated. After that, the models which are utilized in commercial simulators are explained. methods for determining the parameters of simulator models are presented. Lastly, the summary section highlightes the key points of this study.

fg =

Vg VL + Vg

(1)

Where Vg and VL are volume of gas and liquid, respectively. The bubble size of foam is another parameter which can be determined as an effective parameter on foam rheology. The average value of foam bubble size is considered as bubble size which is related to foam quality. David and Marsden, 1969) performed several tests with a commercial foamer known as O.K. which was a mixture of an anionic and one nonionic surfactant. They used uniform crushed pyrex bead (150–250 mesh) and generated the foam using compressed air at 100 psi. Fig. 4 shows their results for typical ranges of bubble size at different foam quality. This figure shows that higher foam qualities have wider ranges of bubble sizes compared to lower foam qualities. The high frequencey of bubble sizes for lower foam quality is around 0.20 mm; however, it is around 0.35 mm for highest foam quality (93%). 2.2. Foam in porous media Foam in porous media is defined as a dispersion of gas phase inside a liquid phase in the way that the liquid phase remains as a continuous phase with the lamella creating a discontinuity of the gas phase (Hirasaki, 1989). When foam's lamella is generated in the porous media, the flow of gas in some parts is ceased. Therefore, the subsequently injected gas can sweep the oil in the pores which would not have been reached in the absence of foam (Rossen, 1999). Fig. 5 shows the three different states that may exist for gas in porous media during foam generation; continuous gas, flowing discontinuous gas and trapped discontinuous gas. Also, it shows that, unlike the bulk foam, the foam behavior in porous media is mainly dependent on size, connectivity, and the shape of pores. Foam inside the porous media, same as bulk foam, is categorized into two strong foam and weak foam. Weak foam with coarse texture shows a moderate increase in differential pressure, However, strong foam with fine texture shows a large increase in differential pressure and decreases the gas mobility. The dynamic nature of lamella creation and coalescence inside the porous media lead to changes in the foam texture during the flow. Therefore, all parameters (such as surfactant type and concentration, rock mineralogy and wettability, pore structures) which affect the coalescence and creation of lamella in porous media, influence the foam texture. Foam liquid film inside the porous media can be created by three mechanisms (Afsharpoor et al., 2010): leave behind, snap-off, and lamella mobilization and division. In leave behind mechanism, during the drainage process (gas invasion into the water-saturated zone), the lamella is left behind the gas. In the snap-off mechanism, the gas (nonwetting phase) start snapping off when the capillary pressure fluctuates. In the lamella mobilization and division, pre-existing lamella starts to flow due to local pressure gradient and when it hits the pore junctions it can be divided into several lamellae. Fig. 6 shows schematics of these three mechanisms. Wang and Li also showed that foam stability increased with increasing surfactant concentration up to critical micelle concentration (CMC), beyond which stability did not change. Higher temperature reduced stability and increased effective CMC. Increase in salinity decreased stability only slightly (Wang and Li, 2016). On the other hand, the instability of foam liquid films leads to coalescence of lamella due to the reduction of interfacial area between two immiscible phases (Kovscek and Radke, 1994; Rossen, 1996). Basically, the foam structure at high capillary pressure environment is unstable and resulted in sudden rupture of the lamella. Khatib et al. (1988), showed that there is a threshold for capillary pressure in foam generation which above that the foam can not survive, they refer this capillary pressure as “ limiting capillary pressure (Pc∗)” and the corresponding saturation as “ limiting water saturation (Sw∗ )”. Moreover, many other factors including

2. Foam fundamentals 2.1. Foam definition Foam defined as a structured two phases in such a way to create a systematic hexagonal structure as illustrated in Fig. 3. As this figure shows, foam consists of water, gas and foamer agent which create the liquid films (lamellas) and plateau borders. The lamella is the thin liquid film which divides gas bubbles and requires the sufficient disjoining pressure to sustain. The lamella is strengthened via increasing the surfactants amount at the liquid-gas interface (Ma et al., 2014). The meeting point of three lamellae is named as “plateau border” which has an angle of 120° (Fig. 3). However, in 3D, four plateau borders join together with an angle of 109°. In thermodynamically point o view, foam is considered as unstable and this stability is directly affected by lamella stability. The number of lamellae per unit volume is defined as foam texture which usually is shown by “nf ”. Although this parameter is essential to determine the foam rheology, there is no reliable technique to measure it directly. Considering the amount of lamellae in the foam texture, the foam can be categorized into two groups: weak foam which is coarsetextured leading to moderate reduction in gas mobility and strong foam which is fine-textured foam that generates a drastic reduction in gas mobility. In the laboratory to infer foam texture, usually the pressure gradient is measured (Afsharpoor et al., 2010). The foam quality is another effective parameter, which has a significant effect on the foam rheology. The foam quality ( fg ) is defined as the percentage or fraction of gas volume to total volume of foam (Equation (1)):

Fig. 3. Structure of foam (Schramm, 1994).

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Fig. 4. Frequency of bubble size at different foam quality (David and Marsden, 1969).

Fig. 5. Foam in porous media (Afsharpoor et al., 2010).

evaporation of liquid, diffiusion of the gas phase, the presence of another phases influence liquid film stability (Aronson et al., 1994; Rossen, 1996; Dholkawala et al., 2007). Several studies have been carried out on foam rheology based on the data from laboratory experiments (David and Marsden, 1969; Weaire, 2008; Vries and Wit, 1990; Osterloh and Jante, 1992; Alvarez et al., 2001). Most of the results showed two distinct flow regimes in porous media: strong foam regime and weak foam regime. The pressure gradient is independent of the velocity of the liquid phase in the weak foam regime (low-quality regime), on the contrary, the pressure gradient is independent of the velocity of the gas phase in the strong foam regime (high-quality regime). The weak foam regime is governed by bubble trapping and mobilization mechanism, on the other hand, the strong foam regime is governed by bubble coalescence because the water saturation is in the vicinity of liming value (Afsharpoor et al., 2010). Fig. 7 (a) and (b) show these two regimes in each steady state pressure gradient for sharp and moderate transition zone, respectively.

Fig. 6. Foam generation mechanisms in porous media, (a) leave behind, (b) snap-off and (c) lamella mobilization and division (Afsharpoor et al., 2010).

instabilities should be considered during SAG process design calculations of optimum slug size, well injectivity and foam strength (Farajzadeh et al., 2016).

2.3. Foam instability in porous media 3. Foam flow modeling in porous media The heterogeneities in porous media, even at small-scale, can perturb the front between displacing and displaced fluid. Such perturbations can cause fingering and bypassing of fluids if not countered by stabilizing forces. An implicit-texture foam model simulation study showed that the instabilities within the foam bank may modestly distort the foam front, even when the mobility ratio at the front is favorable. Therefore,

A number of models have been developed and employed to simulate the foam behavior in porous media, during the past three decades. According to the literature, modeling approaches were categorized into three different groups: mechanistic approaches, Semi-empirical approaches, and empirical approaches (Hematpour et al., 2016). These three approaches are described in the following sections. 166

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Fig. 7. Two different regime in steady state pressure gradient in psi/ft, (a) shows the sharp transition between foam regimes, (b) moderate transition between foam regimes. Dotted lines indicate the foam quality at fixed superficial velocity (15 ft/day) (Farajzadeh et al., 2015).

Where the gas relative permeability is reduced by multiplying by the fraction of flowing gas ( Xf ), and the increase in gas viscosity is derived by considering pore constrictions (GσnL r / vg ) where σ , nL , vg , r and G are interfacial tension, number of lamella per unit length, interstitial velocity, equivalent capillary radius and geometrical factor, respectively. Lp is characteristic length of pore constriction and t form and tconv are the time to form the lamella by snap-off and the time to convect the existing lamella, respectively. The apparent gas viscosity in smooth, uniform pores shows by μs in the presence of foam. The foam model modifies gas mobility by changing both gas relative permeability and gas viscosity. They performed their experiments in glass bead packs with a step change in permeability. After Falls, Chang and his co-worker (Chang et al., 1990), continued Falls's work in similar way to modify the gas relative permeability for flowing foam. As the first step of his model, it has to be proven whether or not foam exists. If foam exists, the gas mobility is modified according to Equation (3).

3.1. Mechanistic modeling approches Generally, the aim of mechanistic models is to track changes in foam texture. These changes result from dynamic mechanisms of in-situ lamella creation and coalescence, which thusly, results in alternating gas mobility and pressure gradients. Population balance models developed base on the concept of tracking the bubble population are common models in mechanistic approaches which have been used in many methods for foam simulation (Lotfollahi et al., 2016). The pioneering study on utilizing bubble population balance models for foam flow through porous media dates back to the 1980s (Falls et al., 1988). Although the rates of lamella creation and coalescence are the main parameters to determine the foam texture, in some studies the two are considered having equal values. Therefore population balance models categorized in two major groups: general and local equilibrium population balance models. The difference between population balance models and local-equilibrium models is that general population balance models use a mass conservation equation to dynamically track the changes in foam texture (nf ) while local-equilibrium models either utilize an algebraic equation to derive foam texture or do not explicitly derive foam texture (Ma et al., 2014). However, the difficulty in determining some of the parameters for the general version with core flood and field data could be a disadvantage of these models. Additionally, solving the bubble population balance equation numerically adds computational cost to reservoir simulators and leads to additional numerical challenges. In order to tackle these drawbacks, some researchers proposed local-equilibrium methods or semi-empirical approaches to calculate the foam texture as an algebraic function of variables that can be obtained through Darcy's law and mass conservation equations (Hematpour et al., 2016). The details of these two modeling approaches were discussed in the following section.

f f 1.5 c ⎧ μg = μg k3 nf k (Uref 3/ ug ) 3, if μg > μg ⎪ μgf = μg , if μgf ≤ μg ⎨ ⎪ krgf = krg × Xf ⎩

Where k3 is geometric factor and ug is gas superficial velocity. Chang figured out that to generate foam, the conditions of Sg > 0 , So < Solim and Cs > Cslim should be satisfied. Chang considered another relative permeability modification for leave-behind lamellae:

krgf =

⎨ Rg − R c = ⎪ ⎪ ⎩

krgf = krg × Xf μgf nn

( )

if Pc < Pc∗ and t form < tconv

Rg − R c = 0 otherwise

(4)

e

−1

⎧ Rg = k1 ⎡1 + nf 1⎤ nc ⎪ ⎣ ⎦ ⎪ Rg = 0, ⎨ ⎪ R c = k2 nf ⎪ ⎩

= μ w + GσnL r / vg

a + Lp / vg Pc∗ − Pc

krg 1 + bSg

The Equation (4) shows that the gas relative permeability is altered as a function of gas saturation in leave behind mechanism, b is a fitting parameter which can be derived by experiment. They also explicitly presented the foam generation function (Rg ) and the foam coalescence function (R c ) to obtain bubble population balance. Both Rg and R c functions depend on foam texture and velocity of the gas. Regarding the minimum pressure gradient needed to generate foam, there is critical gas velocity (ugc ) below which the Rg function is zero. Moreover, the R c function also relies on surfactant concentration in view of the assumption that the higher the surfactant concentration, the more stable the bubbles.

3.1.1. General population balance models These approaches take the lamella creation and coalescence rates into account to calculate the foam texture. Falls et al. (1988), developed the population balance model based on one-dimensional displacement and considered capillary snap-off as the main mechanism for foam generation. They didn't define separate functions for generation and coalescence rates as shown in Equation (2).

⎧ ⎪ ⎪

(3)

c1

( ), ug

Uref 1

if ug / ugc ≥ 1

otherwise e2

( )( )

(2) 167

Csref Cs

ug

Uref 2

c2

(5)

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model for viscosity and lump some of the parameters into a coefficient α (Equation (11)).

Chang and his co-worker (Chang et al., 1990) constructed their own core flooding system and utilized CO2 as injected gas to generate the foam. They performed (1) simultaneous injection of CO2 and brine and (2) simultaneous injection of CO2 and surfactant solution in a brine saturated Berea sandstone. They matched their model with experimental data for pressure drop and sweep efficiency. Friedmann and his co-workers (Friedmann et al., 1994) presented a new model which was similar to Chang's model and modified the gas relative permeability to reduce gas mobility in the presence of foam.

krgf = krg × Xf ⎧ ⎪ f μ = μg Fg k 3/2nf (ug / Uref )n − 1, if μgf > μg ⎨ g ⎪ μgf = μg , if μgf ≤ μg ⎩

(

( )

(6)

−1

krgf = krg0

(

Xf Sg 1 − Swc

)

( )

(7)

(

1 − Sw ⎞ k−1 = k −01 ⎜⎛ ∗⎟ ⎝ 1 − Sw ⎠

(8)

Where is critical water saturation corresponding limiting capillary pressure and k −01 is a reference coalescence rate coefficient. They investigated gas saturation profiles utilizing Gamma-ray technique and pressure drop history in both experimental data and simulation throughout a transient foam process. They injected N2 to surfactantsolution-saturated silica sand pack. Myer and Radke (Myers and Radke, 2000) presented their model to investigate the transient foam displacement in the presence of residual oil. They revealed the same function as those in Kovscek's model to determine the gas relative permeability reduction factor (Equation (14)). The only difference is that in viscosity equation they considered the constant value for the exponent of foam velocity for all cases.

(9)

(

βnt 1 + βnt

)

X S

)

f g ⎧ krgf = krg0 1 − Swc ⎪ αnf ⎨ μgf = μg + 1 ⎪ uf3 ⎩

Xf = 1 − Xt

(

(13)

Sw*

Kovscek and co-workers (Kovscek and Radke, 1994) proposed a model in which the flowing gas saturation was taken into account to obtain gas relative permeability. This saturation depends on the amount of trapped foam ( Xt ), as a function of pressure gradient, capillary pressure, water saturation, and pore geometry (Equation (10)).

⎨ Xt = Xt , max ⎩

(12)

Where ω is a constant determining the shape of inverse proportionality of foam-generation sites to pre-existing gas bubbles and n∗ is an upper limit for the concentration of foam bubbles that is related to pore size (Chen et al., 2010). This model was applied on co-injection experimental data in steady state condition to match transient foam experiments. Fergui and co-workers (Fergui et al., 1998), modified the Kovscek's model slightly. They defined foam coalescence rate coefficient k−1 as a function of water saturation instead of capillary pressure.

where A is the scaling constant. In order to model foam generation function Rg and coalescence function R c , they assumed surfactant concentration, liquid velocity, and saturation are constant. In these cases, Rg only relies on gas velocity and R c relies on gas velocity as well as foam texture (Equation (9)). Although the coefficient k1 and k (−1) are considered as constant in this model, they brought up that k1 shifted with liquid saturation and k (−1) is reliant strongly on the type and concentration of surfactant as well as liquid saturation.

⎧

)

g

⎨ μ f = μ + An d 3 [σ /(μ u )]1/3 μ f b g w g w ⎩ g

m ⎧ Rg = k1 ug n ⎨ R c = k−1 ug nf ⎩

(11)

Rg = k1 u w u1/3 ⎧ f ⎪ nf ω ⎪ k1 = k10 ⎡1 − n∗ ⎤ ⎣ ⎦ ⎨ R c = k−1 (Cs, Pc ) uf nf ⎪ 2 Pc ⎪ k−1 = k 0 −1 Pc∗ − Pc ⎩

if vg / vc ≥ 1

They applied their model on core flooding data in Berea sandstone using nitrogen foam. This model predicts pressure drop and gas saturation matched data for both coinjection and SAG core flooding, appropriately. Ettinger and Radke (1992), proposed a new model to modify mobility, in which the relative permeability function is modified by using the flowing gas saturation ( Xf Sg ) in the Corey-type relative permeability curve. Also, the gas viscosity function is modified by including a function of pore-body (db is a characteristic pore-body diameter), aqueous viscosity ( μ w ), foam texture (nf ), gas velocity (ug ) and the foam-free gas viscosity ( μg ).

⎧

g

They incorporated the effect of liquid velocity in the foam generation function (Rg ) which has not been considered in the previous model. The coefficient k1 in the Rg function is taken as a constant initially similar to Ettinger's model, however, Chen et al. (2010), incorporated the effect of increasing bubble texture on foam-generation sites on the change in k1. The foam coalescence function R c relies on foam velocity and foam texture, and the coefficient k (−1) is a function of surfactant concentration and capillary pressure. They also realized that surfactant concentration alters the limiting capillary pressure Pc∗ in foam coalescence, therefore k (−1) can be written as a function of capillary pressure (Pc ) as well as limiting capillary pressure (Pc∗).

They proposed Rg function which incorporated foam texture and gas velocity with a somewhat different mathematical formulation compared with Chang's model. The R c function in this model takes both foam texture and surfactant concentration (Cs ) into account.

⎧ R = k ⎡ 10(ug / uc − 1) ⎤ × ⎡1 + nf 5⎤ , 1 1 + 10(u / u − 1) nc g c ⎪ g ⎣ ⎦ ⎣ ⎦ Rg = 0 , otherwise ⎨ ⎪ R c = k2 nf Cs−1.4 ⎩

)

X S

f g ⎧ krgf = krg0 ⎪ 1 − Swc ⎨ μ f = μ + αnf g ⎪ g ucf ⎩

g

(14)

Where g is called as Corey exponent. The novelty of their model is that they considered the presence of oil on foam coalescence function R c , which is not considered in previous population-balanced models. Notwithstanding the dependence of their model on foam velocity and foam texture, they proposed the new R c as a function of oil saturation and the capillary pressure on a pseudo-emulsion film Pcpf in light of a pinch-off rupture mechanism. They additionally proposed the concept of the rupture capillary pressure on a pseudo-emulsion film Pc∗pf near which pseudo-emulsion films are unstable and rupture.

(10)

Where Xt , max is the maximum fraction of trapped foam, β is trapping parameter and nt is trapped foam texture. This equation shows the correlation between trapped gas fractions with trapped foam texture, which is assumed to be equal to flowing foam texture (local equilibrium assumption). They presented their model for relative permeability of gas similar to Ettinger's model, however, they simplified Ettinger's 168

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⎧ ⎪ ⎪

based on a simpler but realistic foam rheology and stochastic bubble generation ideas. This model focuses on the changes of gas viscosity and modifies the mobility based on gas viscosity. They presented the particular yield stress τy , below which foam does not shear and above which foam flows with a power-law behavior (Eqaution (21)).

Rg = k1 ufa u wb 2

P

cgw R c = k−1 ⎛ P∗ − P ⎞ vf nf ⎝ cgw cgw ⎠

⎨ εPcpf / Pc∗pf ⎪ ⎪ R c = k−2 So ⎛ 1 + εPc / Pc∗ ⎞ vf nf pf pf ⎠ ⎝ ⎩ ⎜

⎟

in absence of oil in presence of oil (15)

⎧

All parameters in the absence of oil are same as Kovscek's model, the only difference is in the presence of oil when the capillary pressure on a pseudo-emulsion film Pcpf plays the main role k−2 is a constant and ε determines the initial slope at which pseudo-emulsion function (analogous to a Langmuir adsorption isotherm) rises. Myer and Radke (Myers and Radke, 2000) conducted foam experiments using N2 in the absence and presence of residual oil in 60-cm long Berea sandstone cores and reached to a good agreement between experiments and simulations of transient foam flow in porous media of different permeabilities. Kam and Rossen (2003), proposed a new mechanistic model in which the relative permeability and viscosity of foam are mostly the same as Myer's model.

(

⎧ krgf = krg0 Xf 1 − Sw − Sgr ⎪ 1 − Swr − Sgr αnf ⎨ μgf = μg + 1/3 ⎪ uf ⎩

)

τ

⎨ μgf = μg + K1 ugm − 1 + K2 y ug ⎩

⎧ K = c0 m − 1 c1 ⎪ 1 ⎨ c K2 = c 1 ⎪ 0 ⎩

Later, Roostapour (Roostapour and Kam, 2013), modified the function of foam generation utilizing the error function to calculate Rg . Implementation of error function expressed the behavior of Rg when it comes to plateau at high-pressure gradient. He claimed his model to match previous experimental data (Kam et al., 2007a,b). ⎜

⎟

(19)

Beyond that, they followed the previous concept of the catastrophic behavior of foam near the limiting capillary pressure Pc∗ . Therefore they proposed a model for foam coalescence in light of limiting capillary pressure but replaced the Pc∗ by an equivalent value of limiting water saturation Sw∗ .

⎧ R c = Cc nf ⎪ ⎨ ⎪ R c = Cc nf ⎩

( (

1 ∗ Sw − Sw Sw ∗ Sw − Sw

n

) )

n

(23)

krgf = krg × Xf ⎧ ⎪ am ⎪ f σ μg = μg + Anb∞ DP3 μ u μ w if ∇p > ∇p g w ⎨ crit ⎪ Kg a n nb∞ = K ug ⎪ d ⎩

( )

if Sw < Sw∗ if Sw > Sw∗

(22)

3.1.2. Local equilibrium population balance models In these population balance models, the concept of local equilibrium (LE) is employed. The local equilibrium concept has been developed based on the assumption of equivalency of bubble generation and bubble coalescence rates. Therefore, the numerical method to solve a partial differential equation to compute foam texture is not required. In solving the partial differential equation, the algebraic functions are applied to determine the foam texture; however, these algebraic functions have been developed based on population balance equations. Hatzlavramldls and co-workers (Hatzlavramldls et al., 1995) presented the local equilibrium model relied on Ettinger's model (Ettinger and Radke, 1992). Although they followed the equations of Ettinger's model to alter the krgf and μgf in every respect, they showed that the critical pressure gradient ( ∇p crit ) plays an important role below which the foam would not have been generated. Therefore they applied this restriction on their model to calculate the model's parameters. Based on the concept of LE they assumed that the rate of bubble generation (Rg ) and bubble coalescence (R c ) are equal in Ettinger's model, consequently they expressed a method to calculate the local equilibrium bubble density nb∞. This expression illustrates that LE bubble density is directly related to gas velocity.

(18)

⎟

1 2

In order to validate the new stochastic model, Zitha and Du, 2009) compared the analytical results of water saturation and bubble density with Kovscek's model. Due et al. (Du et al., 2011) simulated their model and compared it with the results obtained from CT scan images of surfactant solution displacement using foam in a Bentheim core, which demonstrated a good consistency in transient liquid saturation profiles.

Since increasing the water saturation increases the foam generation incidence (Kam et al., 2007a,b), in the later development of this model, water saturation was also taken into account to calculate the Rg function (Dholkawala et al., 2007). This model for foam generation depicted an appropriate match with the steady state foam experimental data.

⎜

k ϕ

⎧ Rg = K g (n∞ − n) ⎨ Rc = Kd n ⎩

(17)

Cg ⎡ ∇p − ∇p0 ⎞ −∇p0 ⎞ ⎤ erf ⎛ − erf ⎛ 2 ⎢ 2 ⎠ ⎝ 2 ⎠⎥ ⎣ ⎝ ⎦

1

Where ϕ and k are porosity and permeability respectively and c0 , c1 and m are fitting parameters. Moreover, this model simplified the bubble generation (Rg ) and bubble coalescence (R c ) remarkably, in contrast with previous population-balance models. As Equation (23) shows both Rg and R c functions are only reliant on bubble density (n), where K g and K d are the bubble generation and coalescence rate coefficient, respectively. A maximum bubble density is obtained when K d → 0 and t → ∞.

On the other hand, Kam proposed the foam generation function based on the pressure gradient which is the primarily driving force for lamella mobilization and division to occur. At the outset, a simple relationship is utilized (Equation (17)), where Cg and m are two modeling parameters (Kam and Rossen, 2003). This model matched the experimental data of Gauglitz and co-workers (Gauglitz et al., 2002) perfectly and showed the three foam regimes (the coarse foam regime, the strong foam regime, and the unstable transient regime). Additionally, their model revealed the behavior of a strong foam in both high and lowquality regimes.

Rg =

(21)

ϕ 2 (m − 1) k

( ) () ()

(16)

Rg = Cg Sw (∇P )m

if τ > τy

With

g

Rg = Cg (∇P )m

μgf = 0 if τ ≤ τy

(20)

Where Cc is the fitting parameter (constant value) for foam coalescence. Zitha and Du (Zitha and Du, 2009), changed the way to face population balance models and they proposed an improved foam model

(24)

Where all parameters are same as Ettinger's model which are mentioned in the previous section. Hatzlavramldls et al. (Hatzlavramldls et al., 1995) simulated the steam flooding using this model, consequently the results showed good agreement with pilot field data in term of injection 169

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permeability of gas in the presence of foam can be calculated by Equation (30).

pressure as well as oil rate production. Kovscek and Bertin (2003a) illustrated a model to modify the foam mobility in porous media and relied on the population balance Kovscek's model (Kovscek et al., 1995). They altered the α parameter in viscosity function in Kovscek's model and took other parameters into accounts, such as capillary pressure (Pc ), permeability (k ) and interfacial tension (σ ) (Equation (25)).

β 3/2k 3/2Pc nf 2σ v1/3 f

μgf ≅

1 − Sw − Sgr ⎞ krgf = krg0 ⎛⎜Xf ⎟ ⎝ 1 − Swc − Sgr ⎠

μgf

= Ψ1

(25)

k1 − 2aηkrg ∇p aη v1/3 f Pc n1f + aη

(26)

Where a , η and vf are a fitting parameter, percolation exponent and interstitial velocity of foam, respectively. Also, Ψ1 is a constant of proportionality and a constrain on the right recovers the gas mobility (in the absence of foam) when nf approaches zero. In order to determine the foam texture, the LE concept has been applied on Kovscek's model by putting both bubble generation and coalescence as equal. Accordingly, they derived Equation (27) to compute foam texture using their model.

⎧ nf ≅ ⎨ ⎩

(

3/2

ϕ3 (1 − ϕ)2k

) (

Pc∗ − Pc 2 v w Pc v 2/3 f

)

(

(27)

nfω +

Cf nf [ug / ( ϕ Sg Xf )]1/3

(28)

Kam (2008), considered the local equilibrium concept in his model and put foam generation function (Rg ) and foam coalescence function (R c ) as equal, therefore he derived the expression to compute the LE foam texture value (Equation (29)).

nf =

Cg ⎛ Sw − Sw∗ ⎞n ⎡ ∇p − ∇p0 ⎞ −∇p0 ⎞ ⎤ erf ⎛ − erf ⎛ if nf < 2Cc ⎝ Sw ⎠ ⎢ 2 2 ⎠⎥ ⎝ ⎠ ⎝ ⎣ ⎦ ⎜

nf

max

⎟

⎜

⎟

⎜

g

(31)

n∗ωk−1 vf k10 vw

2 3

nf − n∗ω = 0

(32)

Where ω is a constant to determine the shape of inverse proportionality of foam-generation sites to pre-existing gas bubbles and n∗ is an upper limit for the concentration of foam bubbles that is related to pore size. For ω = 3, this equation becomes a cubic equation that is easy to solve for nf at the given liquid velocity, gas velocity, and capillary pressure using standard analytical formulas. This equation leads to one single real root otherwise the condition for foam existence should be checked again. The foam texture becomes zero in the absence of a surfactant. This model was validated by performing N2 foam coreflood experiments (using X-ray to measure saturation). Chen illustrated the comparison between both steady-state and transient experiments with their model generated terms such as aqueous saturation, foam texture, and pressure drop. Moreover, the comparative study was conducted between their local equilibrium model and Kovscek's general population balance model. They inferred that the primary distinction in the prediction of these two models was in entrance area since a certain distance is required for the foam to reach local equilibrium condition in the experiments.

no changes in krgf

⎨ μgf = μg + ⎩

)

As mentioned before, they modified the kovscek's model for bubble generation function to some extent, they employed two parameters, ω and n∗ to compute k 1 in foam generation function. Meanwhile, they tried to simplify this model by the implementation of local equilibrium concept. They considered both bubble generation and coalescence functions as equal value to compute foam texture directly. Eventually, the equation was obtained to calculate the foam texture (nf ) value rather than solving the partial differential equation.

Where ϕ and k are porosity and permeability respectively. Also vw and vf are the interstitial velocity of water and foam. This model showed appropriate match with experimental data of Kovseck et al. (Kovscek et al., 1995) and Persoff et al. (1991). Kam (2008), proposed a modified model for his own general population balance model (Kam and Rossen, 2003), in which the local equilibrium has been applied. In order to modify the mobility of gas in the present of foam, he assumed that it is enough to alter the viscosity of gas only, therefore he utilized the same expression to alter gas viscosity in the presence of foam as shown in Equation (28).

⎧

X S

f g ⎧ krgf = krg0 ⎪ 1 − Swc ⎨ μ f = μ + αnf g ⎪ g vf 1/3 ⎩

if Pc < Pc∗

nf = 0 if Pc ≥ Pc∗

(30)

Where g is considered Corey's exponent to model the relative permeability. Roostapour (Roostapour and Kam, 2013), also applied this model on his simulation study to find out the behavior of fractional flow curve under different foam flooding regimes. He conjectured that this model is able to explain the three different regions in fractional flow curve for three foam regimes (weak, intermediate and strong foam). Chen and co-workers (Chen et al., 2010), proposed the new LE model based on their general population balance model. Bearing in mind, not only gas relative permeability was changed in the presence of foam in this model but also gas viscosity. As shown in Equation (31) the flowing gas term ( Xf ) was incorporated to alter the gas relative permeability of this model.

Where β is constant value. In this model, the relative permeability of the gas has been changed by applying percolation analysis of the flowing foam. Eventually, the expression for calculating the foam mobility has been derived as shown in Equation (26).

krgf

g

⎟

(29)

3.1.3. Percolation models These models have been developed based on the pore network theory which is called the percolation theory. These models are considered as mechanistic models since they employ the foam texture in their calculations. Percolation models try to model the porous medium using a network of pores which are connected by capillary tubes as pore throats (Hunt, 2001). The capillary tube models also can be utilized to model the porous medium. These capillary tubes may have different radius sizes to illustrate the foam trapping and mobilization mechanism of fixed-size bubbles (Rossen and Wang, 1999). These models have been developed in 2-D and 3-D network models to simulate the foam invasion process inside the porous media (Kharabaf and Yortsos, 1998).

Where Cg and Cc are foam model parameters. This model is consistent with foam catastrophic theory as well as it reduces the instability of numerical method for general population balance model. Later, Afsharpoor (Afsharpoor et al., 2010), conducted the study on surfactant alternating gas (SAG) process using Kam's LE model and compared these results with fractional flow analysis results. He concluded that analysis of foam fractional flow requires a new way to solve the problem analytically rather than the typical method of constructing a tangent line from the initial condition. Although he utilized the same expression for foam texture and foam viscosity as Kam (2008) proposed, he modified the relative permeability term. The flowing gas term ( Xf ) was considered in Afsharpoor's model, hence the relative 170

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In order to calculate the foam texture (nf ), three different threshold values for gas saturation were defined in this model; reference gas saturation (Sgm ), Sg* − ε and Sg* + ε . Accordingly, Li and co-workers (Li et al., 2006), determined four different regions during foam flooding process:

These models are able to quantify the heterogeneity by changing the connections of volumes, area or line segments. This is one of the advantages of percolation models. Besides that, in foam modeling applications, the foam generation, and coalescence mechanisms can be studied with spatial distribution in pore-level. For instance, Rossen and Gauglitz (Rossen and Gauglitza, 1990), found out the role of the minimum pressure gradient to mobilize the foam lamella, using the percolation models. Chen et al. (2005) carried out a study on the interactions between three pore-level foam generation mechanisms using the pore network models. Furthermore, some required parameters for population balance models which are difficult to be obtained may be derived from the pore network models (Kharabaf and Yortsos, 1998; Kovscek and Bertin, 2003b). However, application of these models in the large scale cases due to its high computational time and cost is limited.

1. Lamella density accumulating region: In this region the lamella density persistently aggregating, more lamellae are created inside the porous medium, and the lamella coalescence rate increases. This occurs when 0 ≤ Sg ≤ Sgm . 2. Lamella density steady state region: In this region, the rate of foam generation and coalescence are equal to each other and occurs when Sgm < Sg ≤ (Sg* − ε ). 3. Foam breaking transient region: This region is located between steady state region and no foam region ((Sg* − ε ) < Sg ≤ (Sg* + ε ) ). In this region the capillary pressure exceeds limiting capillary pressure therefore the foam coalescence is a dominant phenomenon in this region. 4. No foam region: In this region, the gas saturation exceeds the limiting saturation (Sg > (Sg* + ε ) ) therefore foam cannot be generated.

3.2. Semi-empirical approaches These types of models are categorized neither as population balance nor empirical. Since they use a direct expression to calculate the foam texture rather than solving the partial differential equation, therefore, they could not be categorized as population balance models. Although these models utilized the algebraic function to derive foam texture, these function was not derived by equating the foam generation rate to the foam coalescence rate. As an alternative to that, these models compute the foam texture based on correlation. Bertin and co-workers (Bertin et al., 1998) developed a model in which Equation (33) was utilized to alter gas viscosity. They assumed that in order to modify the mobility of gas in the presence of foam only the viscosity function change is sufficient.

μgf = μg +

According to these regions, Li and co-workers proposed different equation for computing foam texture which was illustrated in the following equation (Li et al., 2006). m

( )

Sg ⎧ nf = Cnf S if 0 ≤ Sg ≤ Sgm; gm ⎪ ⎪ nf = Cnf if Sgm < Sg ≤ (Sg∗ − ε ) ⎪ Sg∗ + ε − Sg ⎨ ∗ ∗ ⎪ nf = Cnf ⎛ 2ε ⎞ if (Sg − ε ) < Sg ≤ (Sg + ε ) ⎝ ⎠ ⎪ ⎪ nf = 0 if Sg > (Sg∗ + ε ) ⎩

αnf 1 vg3

They validated their model by experimental results for both 1D and 3D sand pack tank foam flooding.

(33)

They presented their model to calculate foam texture based on the correlation. The function of foam texture was proposed which was dependent on total number of generation site in the pore volume (nf ), gas saturation (Sg ), flowing gas fraction ( Xf ) and function of capillary pressure including the limiting capillary pressure (Pc∗). They defined nf , as a function of absolute permeability (k ) and porosity (ϕ ). Equation (34) shows the expression to compute the foam texture value. −3/2

180 × (1 − ϕ)2 ⎞ k⎟ nf = ⎜⎛ ϕ3 ⎠ ⎝

× Sg Xf ⎛ ⎝ ⎜

Pc∗ − Pc (Sw ) ⎞ Pc (Sw ) ⎠

3.3. Empirical approaches Generally speaking, the empirical models do not endeavor to directly illustrate the relationship between the mobility of gas and foam texture. However, they adapt gas relative permeability and/or gas viscosity in the light of experimental or field data. A majority of mathematical formulations in this classification turn out without much consideration of the theories of foam lamellae transport. The significant advantage of these models is that there are fewer parameters required compared with full-physics versions of population balance models. These models are used in reservoir simulators, make them free to solve foam texture equations. Consequently, the reservoir simulators need less time by utilizing these models in their simulation. Although these models do not go through the first principal concepts of foam generation and destruction, they attempt to take all effective parameters into account. Numerous studies have been conducted to develop and improve these models. In the following section, a brief literature review about empirical model is explained. As it has been mentioned above in empirical models some studies try to modify foam mobility by altering the gas viscosity or gas relative permeability or both of them. Marfoe et al. (1987), considered gas viscosity as the only parameter needing modification to derive foam mobility. Regarding this assumption, they derived Equation (38).

⎟

(34)

In order to validate this model, Bertin and co-workers conducted a simulation study to fit experimental data. The simulation results show a good agreement with unconsolidated sand pack flooding experimental data (Liu et al., 1992) as well as consolidated sandstone core flooding experimental data (Persoff et al., 1991). Li and co-workers (Li et al., 2006), continued Bertin's idea for the correlation-based model and elaborated the effect of gas saturation in foam texture equation. They utilized the viscosity Equation (35) which is the modified version of Friedmann's model (Friedmann et al., 1994). f ⎧ ⎪ μg = μg Fg Cnf nf ⎨ μgf = μg ⎪ ⎩

n−1

( ) ug

uref

if μgf > μg

if μgf ≤ μg

(35)

Where μg is gas viscosity without foam, Fg is a geometric factor, Cnf is a constant coefficient and uref is a reference superficial velocity beyond which the shear thinning effect occurs. They also simplified the Fall's model for relative permeability (Falls et al., 1988) using Sgrf instead of gas flowing fraction ( Xf ) as shown in Equation (36). f ⎛ Sg − Sgr ⎞ krgf = krg0 ⎜ f ⎟ ⎝ 1 − Sgr ⎠

(37)

μgf = μg [1 + RCs (Sw − Swc ) f (ug )]

(38)

Where f (ug ) is a function of gas superficial velocity and R is the constant parameter which are 1 and 0.01, respectively. The above model considers the effect of surfactant concentration, saturation of water and gas velocity on the mobility of foam. However, these parameters can be determined by fitting experimental data. This model and related parameters have been drived from experimental results of sandpack foam

g

(36) 171

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flooding conducted by Nutt et al. (1979). Moreover, Marfoe and coworkers (Marfoe et al., 1987) made a comparison simulation study between foam and gas injection in water-gas system and oil-water system. Their simulation resulted in an appropriate match with experimental data using water saturation and cumulative production. Marfoe, concluded that this formulation might be adjusted to a wide variety of applications including nitrogen foam, carbon dioxide foam and, foam steam flooding. Islam and Farouq Ali (Islam and Farouq Ali, 1988) continued Marfoe's study and modified his model as shown in Equation (39). Their model considered water saturation and surfactant concentration function ( fc ) like Marfoe's model. However, they involved the effect of formation permeability ( fk ) and oil saturation in the viscosity function of gas when foam exists. Also, they utilized pressure gradient function instead of velocity function ( fp ). The parameters D and E are two constants in this model. Islam and Farouq Ali performed the simulation study using the experimental data which had been conducted in five sets for N2-foam core flooding in the presence of a bottom water zone. Although Islam et al. (Islam and Farouq Ali, 1988) observed that increasing in pressure gradient raises the foam mobility, in their simulation study consider fp (∇p) as zero. The results of their simulation showed acceptable consistency with experimental data in cumulative oil production.

μgf =

modify the gas viscosity function or the gas relative permeability function to show the foam effect. For foam remained at pc∗ over various water and gas flow rates, the pressure gradient (∇P ) and the gas phase mobility can be determined by Darcy's law at the critical water saturation. Regarding Equation (42) for foam at Sw∗ , the value of ∇P can be determined without knowing foam texture if the water relative permeability at Sw∗ is known. They showed that in some data pc∗ is independent of both gas and liquid rates (fixed- pc∗ model), therefore gas mobility (λ g ) is a function of water fractional flow ( fw ) which becomes the concept of foam mobility in the high-quality regime controlled by Sw∗ mentioned in a later work. Note that this model describes foam at steady state condition, it may not be applicable to the period of foam generation or at the leading edge of an advancing foam front. μ

∇p = u w kkw∗ = u w λ w∗ where ⎧ ⎪ rw ug 1−f ⎨ λ λ w∗ u = λ w∗ f w = g ⎪ w w ⎩

( )

(39)

Chang and co-workers (Chang et al., 1990) developed a model to modify the mobility of gas when the foam is presented as depicted in Equation (40). They developed a relationship between gas mobility and interstitial gas velocity (vg ), the fractional flow of gas ( fg ), surfactant concentration (Cs ) and phase saturations (oil and water). The parameters k1, k2 , k3 , k 4 , and n can be derived experimentally. The authors derived the parameters k1 , k2 and n from their own experimental data which has been conducted by CO2 foam with different gas fractional flow and interstitial gas velocities. However, they did not mention how to derive k3 and k 4 in their report.

kkrgf μgf

= (k1 fg + k2 vgn )[1 + k3 Cs (Sw − Swr )][1 + k 4 (So − Sor )]

krgf μgf

(42)

u w + ug k − rw ⎤ = max ⎡0, − ⎢ ∇ k p μw ⎥ ⎣ ⎦

(43)

They applied their model in their finite difference simulator to modify the mobility of gas in the presence of foam at each time step. Surfactant transport and adsorption and gas solubility have also been considered in the simulator. Robert and Mack utilized a conventional dynamic first-order model (Chung, 1991) to model surfactant adsorption. They contrasted simulation results and their own experimental data of N2-foam core floods with both single-core and two-core flooding. In the single-core simulation during the surfactant post flush, they found that the effect of nitrogen solubility in the aqueous phase should be considered during the second imbibition stage to match the experiments with an acceptable range of precision. For this reason, Modified Henry's Law (Prausnitz et al., 1986) is used to describe the process of nitrogen dissolution. Rossen and co-workers (Rossen et al., 1999) extended the limiting capillary pressure model as shown in Equation (44), they proposed that no foam is created if the water saturation Sw is less than (Sw∗ − ε ) or the surfactant concentration (Cs ) is below a threshold surfactant concentration for foam formation (Cs0 ); the reduction of gas relative permeability rises linearly with Sw when (Sw∗ − ε ) ≤ Sw ≤ (Sw∗ + ε ) and Cs = Cs0 ; full-strength permeability reduction with a permeability reduction factor R is achieved when Sw > (Sw∗ + ε ) and Cs ≥ Cs0 . Both ε and R are model parameters. They showed that if ε approaches zero and R is large, this model turns to the fixed- Pc* model.

(40)

Mohammadi et al. (Mohammadi et al. (1995), for the first time, proposed an empirical model focusing only on relative permeability of gas in the presence of foam as illustrated in Equation (41). This model was utilized as first foam model by Computer Modeling Group (CMG) in STARS software. They considered the influence of surfactant concentration (Cs ) and pressure gradient ∇pfoam ) on mobility reduction factor (FM), consequently, the relative permeability of foam (krgf ). The pressure gradient and surfactant concentration are normalized with a pressure gradient in the absence of foam (∇pnofoam ) and maximum surfactant concentration (Csmax ) respectively. The fitting parameter in this model is es which is an exponent of surfactant concentration function. They applied this model in their simulation usnig nitrogen foam injection for field case study in order to match the history production data from the field. ∇ pfoam Cs ⎧ ⎪ FM = ⎡1 + ∇ p max nofoam Cs ⎣ ⎨ ⎪ krgf = krg × FM ⎩

at Sw = Sw∗

Robert and Mack (1997), developed a model to show the effect of foam on the mobility of gas. They explained foam behavior based on a set of experimental data collected in cores at relevant to matrix acidizing conditions. They found that foam behaves as two-phase flow, with a minimum yield stress needed to displace gas phase. At high foam injection rate, they found that foam was highly shear-thinning and the pressure gradient through the foam bank was independent of the rate of foam. In their model (Equation (43)), the pressure gradient ∇P is identified by the foam-injection conditions, which depends on surfactant concentration, foam velocity and quality, temperature, and rock permeability. However, the mathematical details of these relationships are not addressed.

μg [1 + Dfc (Cs )(Sw − Swc ) fk (k ) + fp (∇p)] 1 + ESo2

∗ krw = krw (Sw ) Sw = Sw∗

es −1

( ) ⎤⎦

∗ 0 f ⎧ if Sw < (Sw − ε ) or Cs < Cs then krg = krg ⎪ if (Sw∗ − ε ) ≤ Sw ≤ (Sw∗ + ε ) and Cs = Cs0 then krgf ⎪ ⎪ krg = ⎨ ∗ + ε) (R − 1)(Sw − Sw + 1 ⎪ 2ε ⎪ ⎪ if Sw > (Sw∗ + ε ) and Cs ≥ Cs0 then krgf = krg / R ⎩

(41)

Zhou and Rossen (1995) developed a new model based on the concept of critical capillary pressure as shown in Equation (42). This model, also called as “fixed pc∗- model” which characterize a critical capillary pressure ( pc∗) as the maximum value of stable foam regime. Since capillary pressure is a function of water saturation then critical water saturation (Sw∗ ) can be characterized corresponding to critical capillary pressure. This model is different from earlier models that

(44)

In this model R considered as constant value and rheology is Newtonian in the low-quality regime. Cheng et al. (2000) modified the 172

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parameter of R to incorprate the shear thinning effect according to gas flow rate as shown in Equation (45).

ug ⎞ R = Rref ⎛⎜ ⎟ u ⎝ gref ⎠

k

rg ⎧ krgf = 1 + fmmob × F × F × F w o s ⎪ arctan [epdry (Sw − fmdry )] ⎪ w = = F F ⎧ w ⎪ w π ⎪⎪ arctan[epdry (Swr − fmdry )] ⎪ − if Swr ≤ Sw ≤ (1 − Sor ) ⎪ π ⎪⎪ 0 if Swr ≤ Sw ≤ fmdry ⎪⎨ ⎧ ⎪ Sw − fmdry epdry ⎪⎪ k if fmdry < Sw < fudry ⎪⎪ Fw = Fw = ⎨⎡ fudry − fmdry ⎤ ⎣ ⎦ ⎪⎪ ⎪ ⎪ 1 if fudry ≤ Sw ≤ (1 − Sor ) ⎩ ⎩ ⎨ ⎪ ⎧ Fo = Fow = 0.5 + arctan[epoil (fmoil − So)] if Sor ≤ So π ⎪ ⎪ ⎪ ⎪ ≤ (1 − Swr ) ⎪ ⎪ 1 if So < floil ⎪ ⎧ ⎪ ⎨ ⎪ fmoil − So epoil k ⎪ ⎪ Fo = Fo = ⎡ if floil < So < fmoil ⎤ ⎨⎣ fmoil − floil ⎦ ⎪ ⎪ ⎪ ⎪ ⎪ 0 if fmoil ≤ So ≤ (1 − Swr ) ⎩ ⎪ ⎩ eps ⎪ F s = tanh [RCs ] ⎩

σ−1

(45)

Where ug is the superficial velocity of the gas, Rref is the value of R at a reference gas superficial velocity (ugref ) and σ is the power-law exponent. The value of exponent for Newtonian behavior and shear thinning behavior are equal to 1 and less than 1, respectively. Vassenden and Holt (2000) proposed a model (Equation (46)) based on relative permeability concept which was well suited for simulation of foam processes. This model considered the effect of water saturation and gas velocity on foam mobility. They avoided narrowing the range of Sw to (Sw∗ − ε ) ≤ Sw ≤ (Sw∗ + ε ) . Instead, they defined a function which includes two exponential functions to show the dependency of foam mobility reduction on Sw over a range of water saturations (Sw ≥ Sf ). Sf is the critical water saturation which above that the foam can be survived and it is equal to Sw∗ .

⎧ ⎪ ⎨ ⎪ ⎩

krgf

(47)

Where epdry , fmdry and fudry are fitting parameters for dry out function, epoil and eps are exponents for oil and surfactant function respectively, fmoil is the critical oil saturation above which foam is completely destroyed, floil is the lowest oil saturation to destabilize foam. In this model a new function Fww is presented to ensure that Fw = 0 when Sw is equal to Swr . They also studied a case in which foam is killed at low water saturations (Swr ≤ Sw ≤ fmdry ) using the function Fwk . The oil saturation-dependent function Fok is similar to the function F2 in STARS model, except this model specifies the upper bound 1 (when So < floil ) and the lower bound 0 (when fmoil ≤ So ≤ (1 − Swr ) )) for this function. In addition to the function Fok , it was also proposed a function Fow to characterize the weakening effect of oil on foam. In this model, a hyperbolic tangent function is proposed to describe the surfactant concentration effect, which helps reduce numerical dispersion in foam simulations (Zanganeh et al., 2011). However, a comparison with laboratory experiments and field tests using this model is needed to understand how these changes the STARS model affects the simulation performance.

= krg × F

ug

F = exp[(Sf − Sw ) s1] + ⎛ 0 ⎞ F0 exp[(Sf − Sw ) s2] when Sw ≥ Sf ⎝ ug ⎠ F = 1 when Sw < Sf (46)

Regarding this model as shown in Fig. 8, there is no foam effect when water saturation is below Sf . For higher saturations, the gas mobility was decreased exponentially with the slope s1 within the limiting capillary pressure region, until it reached the limiting pressure gradient region, where the gas mobility was reduced by a factor F0 (ug / ug0) . The reference mobility factor F0 is the mobility factor at the reference gas velocity ug0 . In the limiting pressure gradient region, the mobility factor changed by the slope of s2 . Using this model, Vassenden and Holt fitted their experimentally measured gas relative permeability data with foam in bead pack and Snorre's reservoir sandstone, as well as reported laboratory data in the sand pack (Osterloh and Jante, 1992) and bead pack (Khatib et al., 1988). Namdar Zanganeh and her co-worker (Zanganeh et al., 2011) figured out that foam is not totally destroyed at the residual water saturation (Swr ) using previous models, which is an artifact as foam should collapse as the water saturation approaches Swr and Pc approaches infinity. Therefore, they proposed a new model for foam as shown in Equation (47).

3.4. Comparison between foam modeling approaches To the best of the author's knowledge, there is no single approach which claims that is able to fit all various foam flooding into porous media at different condition. Mechanistic approaches consider the foam texture and the rate of generation and coalescence in their approaches. Most of the mechanistic models utilize continuous mathematical functions; however, some of them employ discontinues functions to model the catastrophic phenomena in the vicinity of limiting capillary pressure, as well as in the vicinity of critical surfactant concentration. The main issue of mechanistic models is the number of required fitting parameters and the number of laboratory experiments needed to obtain fitting parameters. Besides that, mechanistic models require many computations to derive the foam generation and coalescence rate as well as the foam texture. On the other hand, the empirical models calculate the mobility reduction factor without calculating the foam texture and foam coalescence and generation because the local equilibrium concept (equal values for foam generation and coalescence) is considered in this approach. Lotfollahi et al. dispelled the notion that Implicit-Texture (IT) models are not adequate in reflecting the essential physics of foam in porous media. Their steady-state experimental data showed that both IT models and PB models at local equilibrium equally honor the physics of foam behavior in porous media. However, population-balance models are better in representing abrupt heterogeneities that distort the local equilibrium, shock fronts, and the entrance region (Lotfollahi et al., 2016). In semi-empirical models, the method for foam texture is different from population balance, therefore they cannot be categorized as the mechanistic approach.

Fig. 8. Characteristic foam relative permeability (Vassenden and Holt, 2000).

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Table 1 Comparison between foam models approaches.

Empirical Approach Semi-Empirical Approach Mechanistic Approach

9 Number of model parameters to be determined

10 Difficulty of method for determining model parameters

11 Accuracy to fit experimental data

12 Time consumption for simulation

13 Commercial 14 simulator 15 software

A large number of model's parameter should be determined a large number of model's parameter should be determined few number of model's parameter should be determined

easy

good fit for steady state experiment but weak for unsteady state good fit for steady state experiment but weak for unsteady state good fit for both steady and unsteady state experiment

short time

widely used

short time

not used

long time

rarely used

difficult difficult

All three approaches aim to model the foam flooding by reducing the gas mobility by reducing the gas relative permeability or increasing the gas viscosity. These approaches showed the acceptable results for steady-state experiments; however, the mechanistic approach showed better results for unsteady state experiments. This comparison is listed in Table 1.

with minimum-size bubbles, in the absence of factors increasing bubble size. The normal range for fmmob may vary between 0 and 100000. Krgf is the gas relative permeability in the presence of foam and Krg0 is the gas relative permeability in the absence of foam. F1 represents the effects of surfactant concentration; surfactant mole fraction in the aqueous phase is divided by parameter fmsurf (the allowed range is 0–1 and typical value is 0.00001) and then raised to power of epsurf . This exponent control the sharpness of transition zone and the allowed range is −4 to 4. F2 describes the effect of oil saturation. fmoil is the critical oil saturation (the allowed range is 0–1 and typical value is 0.2), epoil is exponent for oil saturation and the allowed range is 0–5 but usually is considered as 1 and floil is the lower oil saturation value. The allowed range is 0–1. Foam is completely destroyed at oil saturations greater than fmoil (the allowed range is 0–1) and there is no effect of oil on foam for oil saturations less than floil (the allowed range is 0–1). F3 and F4 represent the shear-thinning effect of pressure gradient on gas mobility in the low-quality regime. fmcap is a reference capillary number value (the allowed range is 0–1) and epcap is the exponent for the effect of capillary the number on gas mobility which has a value between −10 and 10. F4 shows critical generation capillary number effect on gas mobility. fmgcp is the critical generation capillary number value used in dimensionless foam interpolation calculation (the allowed value is 0–1) and epgcp is exponent for the generation capillary number (the allowed range is −10 to 10). F5 illustrates the effect of critical oil mole fraction on gas mobility in which fmomf is the critical oil mole fraction for component numx which has a value is 0–1 and epomf is its exponent (the allowed range is 0–5). F6 represents the salt effect on gas mobility in which flsalt and fmsalt are lower and critical salt mole fraction values, respectively. The allowed range for these two parameters are 0–1. epsalt is exponent for salt contribution to the dimensionless foam interpolation calculation and the allowed range is 0–5. One of the main functions which plays a critical role in foam behavior is a dry-out function, also called water saturation function fdry . This function is able to apply the water saturation effect on mobility reduction factor. Also, it should be able to model the limiting water saturation (Sw∗ ) below which the foam coalescence occurs. sfbet is a parameter to control the sharpness of transition zone between two different foam regimes. The minimum value for this parameter is 0 and it can have higher values of up to 100000. SF in previous version of CMG (Computer Modelling Group, 2006) called fmdry and considered as critical water saturation. In the latest version of CMG, SF is a function of sfdry (equivalent to fmdry ) and other fitting parameters (Computer Modelling Group, 2015). This modification have applied to overcome the malfunctioning of the previous dry-out function to derive the foam mobility back to gas mobility when the water saturation becomes lower than the limiting water saturation. However, this modification increases the number fitting parameters and there is no robust justification for them. In enormous related studies, the last three functions F4, F5 and F6 were considered as one and their effects can be neglected (Ma et al.,

4. Foam models in reservoir simulators Nowadays, various numbers of reservoir simulators have been developed. However, most of them do not employ complex EOR models in their backgrounds. The complex EOR mechanisms such as low saline flooding, wettability alteration by chemical flooding, foam flooding, and different thermal processes. Among all reservoir simulator, three of them which have been commercialized are a pioneer in chemical flooding and foam flooding: ECLIPSE from Schlumberger company, STARS from CMG and UTCHEM from University of Texas at Austin. the foam models which are utilized in these three simulators are described in detail as following. 4.1. STARS model STARS is one of the widely used commercial foam simulators. It fits foam behavior in both the high- and low-quality foam-flow regimes reasonably well STARS includes a simple mechanistic model in which it considers the lamella as a separate phase (Skoreyko et al., 2012). This model was developed based on the reaction concept between phases, all reactions are irreversible types using the Arrhenius equation for reaction rates (Skoreyko et al., 2012). CMG also includes an empirical model which represents the effects of bubble size implicitly in factors for reducing gas mobility as a function of phase saturations, compositions, and other properties. In STARS, the foam model modifies the gas relative permeability to represent the effect of foam by multiplication with a factor FM which is inversely proportional to several other factors as in Equation (48). 0

f

Krg = Krg (Sw ) FM ⎫ ⎧ ⎪ ⎪ 1 FM = ⎪ 1 + fmmob . F 1.F 2.F 3.F 4.F 5.F 6. fdry ⎪ ⎪ ⎪ epsurf ⎪ ⎪ F 1 = mole fraction (ICPREL) fmsurf ⎪ ⎪ ⎪ ⎪ (fmoil − So) epoil ⎪ ⎪ F 2 = (fmoil − floil ) ⎪ ⎪ epcap ⎪ ⎪ fmcap F 3 = capillary number ⎨ ⎬ ⎪ F 4 = (capillary number − fmgcp) epgcp ⎪ fmgcp ⎪ ⎪ ⎪ ⎪ epomf − fmomf X ( ) numx ⎪ ⎪ F5 = fmomf ⎪ ⎪ ⎪ (mole fraction − flsalt ) epsalt ⎪ ⎪ F6 = ⎪ (fmsalt − flsalt ) ⎪ ⎪ ⎪ fdry = 0.5 + arctan(sfbet (Sw − SF ) ⎪ π ⎭ ⎩

(

)

(

(

) )

(

)

(

(

)

)

(48)

Where fmmob describes the normalized resistance to flow of the foam 174

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sharp increase of gas mobility when water saturation reduces in the vicinity of limiting water saturation (Sw∗ ). The details of this model was described in the emprical model section.

Table 2 The equivalent model's parameters in the three different simulators. STARS, CMG (2015)

ECLIPSE (2015)

UTCHEM (2000)

fmmob

Mr

Rref ε

sfbet

fw

sfdry

Swl

Sw∗

fmcap epcap

Ncr ec

ug σ

∗ f 0 ⎧ if Sw < (Sw − ε ) or Cs < Cs then krg = krg ⎫ ⎪ ∗ ∗ f⎪ 0 ⎪ if (Sw − ε ) ≤ Sw ≤ (Sw + ε ) and Cs = Cs then krg ⎪ ⎪ ⎪ krg = ⎪ ⎪ (R − 1)(Sw − S ∗ + ε )

⎨ ⎪ ⎪ ⎪ ⎪ ⎩

2013a; Rossen and Boeije, 2013; Rossen and Boeije, 2015).

1+

w

2ε

if Sw > (Sw∗ + ε ) and R = Rref

Cs ≥ Cs0

( ) ug

then

(σ − 1)

ugref

krgf =

krg R

⎬ ⎪ ⎪ ⎪ ⎪ ⎭

(50)

4.2. ECLIPSE model 4.4. Comparison between simulator models

ECLIPSE is one of the popular reservoir simulators which have been developed by Schlumberger. This simulator employs the empirical model to simulate the foam flooding process. Although this empirical model is same as CMG's model in several aspects, the influences of some parameters are ignored in ECLIPSE's model. In order to model the gas mobility reduction factor, two different approaches can be followed in ECLIPSE, firstly, using a set of tables to specify the mobility reduction factor based on concentration, pressure and shear stress, secondly, defining a function of mobility reduction factor (Empirical model). The mobility reduction function which is used in ECLIPSE (Schlumberger, 2015) is depicted in Equation (49).

Three different simulator models were explained in the previous section. This explanation shows that the nature of the empirical model and affecting parameters for all three simulators are the same. However, ECLIPSE's and CMG's models can be considered as part of UTCHEM's model because of the similarity of these two models. All these three models focus on the mobility reduction factor of gas and the affecting parameters in the presenece of foam. The common foam model's parameters for three different simulators models are summarized in Table 2. This table shows that all three simulators employ different terminologies to model the foam behavior. Some of these, provide the same concept but some of them not. For instance, fmmob , Mr and Rref are considerd as maximum or references mobility reduction factor but in different terminology. Nevertheless, to apply the shear thinning effect, the capillary number value is considered in both ECLIPSE and CMG's models but the gas superfacial velocity is considered in UTCHEM's model. The sharpness of transition zone between two foam regimes are presneted with the same concept but by three different terminologies with opposite directions, increasing the sfbet and fw equal to decreasing ε . Moreover, some relations between these parameters have been found, e.g. Cheng et al. (2002) revealed that the power-law exponent (σ ) in UTCHEM's model is related to epcap and ec by the Equation (51).

krgf = krg × Mrf ⎫ ⎧ ⎪ ⎪ 1 = M ⎪ rf 1 + (Mr × Fs × Fw × Fo × Fc ) ⎪ ⎪ ⎪ C es ⎪ ⎪ Fs = Csr s ⎪ ⎪

( )

l )) atan(fw (Sw − Sw

⎨ Fw = 0.5 + π ⎪ S m − S eo ⎪ Fo = o S m o ⎪ o ⎪ Ncr ec ⎪ Fc = N c ⎩

(

)

( )

⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(49)

Where in the surfactant function (Fs ), Mr is the reference value for mobility reduction factor, Cs and Csr are effective and reference surfactant concentration values, respectively, es is an exponent to control the steepness of the transition. In the water function (Fw ), Swl refers to the limiting water saturation and fw is the weighting factor to control the sharpness of regimes changing zone. The effect of oil saturation is presented in oil function (Fo ) which is considered as zero when (So > Som ). Som refers to maximum oil saturation above which foam coalescence occurs and eo is an exponent to control the steepness of the transition. The shear thinning effect is included in capillary function (Fc ), where Nc and Ncr are capillary number and reference capillary number, respectively and ec is an exponent to control the steepness of the transition. The above equation shows that the ECLIPSE's model, unlike the CMG's model, only consider the effect of water saturation, oil saturation, surfactant concentration and capillary number.

σ=

1 (1 + epcap)

(51)

The main difference between UTCHEM and ECLIPSE (or CMG) model is the nature of mathematical function utilized in these models. The ECLIPSE and CMG models employ a continuous function; however, the UTCHEM utilizes a discontinues function to simulate the limiting water saturation phenomenon in foam flow through the porous media. The ECLIPSE and CMG models are not able to show fully dry-out phenomenon at water saturation below the limiting water saturation. In other words, they are not able to make the mobility reduction factor to 1 when the water saturation reaches limiting water saturation (Leeftink et al., 2015).

4.3. UTCHEM model

5. Obtaining models parameters for simulator models

Researchers at the University of Texas at Austin have developed an open source three-dimensional multiphase multicomponent chemical compositional reservoir simulator and named it as “UTCHEM”. This simulator is capable of simulating very complex chemical flooding process (Delshad et al., 1996; Korrani et al., 2014; Lashgari et al., 2015). This simulator is equipped with both population balance and empirical models for foam flooding process (The University of Texas at Austin, 2000). In population balance simulator, the local equilibrium model which has been developed by Chen et al. (2010) is utilized. The Rossen's model (Rossen et al., 1999) is utilized as an empirical model for UTCHEM foam simulator (Equation (50)). This model leads to a

According to the literature, most of them have not mentioned a specific method to obtain model's parameters for mechanistic models because there are several parameters which cannot be derived directly from experimental data. They assumed some values as fitting parameters or performed the history matching via numerical simulator to find models parameters (Gauglitz et al., 2002; Afsharpoor et al., 2010; Bertin et al., 1998; Zitha and Du, 2009). However, various studies have been conducted to present a method to obtain model's parameters for empirical models which are utilized in reservoir simulators (Ma et al., 2013, 2013a; Rossen and Boeije, 2015; Rossen and Boeije, 2013). Most of these methods only focus on parameters of dry-out (water saturation) 175

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Fig. 9. Dataset which is required to obtain model's parameters (Rossen and Boeije, 2015). Fig. 10. Fitting the foam model on steady-state data of Ma et al. (Boeije and Rossen, 2015).

function in mobility reduction factor and neglecting oil, surfactant, and salinity functions. These methods may be categorized into three groups: characteristic method, graphical method and numerical method. The details of these methods are described in the following.

−1

krg0 (Sw∗ ) μw ⎞ ⎛ f g∗ = 1 − ⎜1 + fmmob krw (Sw∗ ) μg ⎟ ⎝ ⎠

5.1. Characteristic method

Finally, based on simulation results, sfbet in STARS ( fw in ECLIPSE) is selected in such a way as to be the largest value consistent with acceptable simulator stability and runtime. The larger value of sfbet leads to the vertical trend of ∇P in the high-quality regime. Boeije et al.(Boeije and Rossen, 2015), extended the same procedure and applied it on other steady state data. They applied their study of pressure gradient (∇P ) versus foam quality ( fg ) data from Ma et al. (2013b) experiments (Fig. 10), instead of flow rates data. This graph shows that the pressure gradient reaches its maximum value at a certain foam quality (50%) which is called transition foam quality. Boeije et al. (Boeije and Rossen, 2013), also applied their method on the apparent viscosity data from steady state experiments of Moradi-Araghi et al. (Moradi-Araghi et al. (1997). Fig. 11 shows the fitting results which is acceptable fitting because the model met the maximum apparent viscosity and transition foam quality accurately.

This method was presented by Cheng et al. (2000). Their method requires the data which exhibits at least two flow regimes as a function of gas and water velocities as shown in Fig. 9. Each dot represents a pressure gradient for a specific steady state experiment. The pressure gradient is independent of gas flow rate when the foam quality is high. This flow regime is controlled by bubble coalescence at the “limiting capillary pressure”. On the other hand, in the low-quality regime, this gradient is independent of liquid flow rate. While the water saturation is increased by the flow rate, the bubble size remains fixed in this regime. The low-quality regime exhibits shear-thinning as a function of overall flow rate. The transition between the regimes occurs at a foam quality f g∗. To fit models parameters to core flood data, the free foam relative permeability function for both gas and water phases are required. In order to obtain models parameters, initially, the steady state data of changing rate of both gas and water for each pressure gradient should be plotted (Fig. 9). If data are not already smoothed, it may be necessary to smooth them. As it was mentioned above, the model assumes vertical line for high quality and a horizontal line for the low-quality regime. This method assumes sfdry in STARS and Swl in ECLIPSE as limiting water saturation (Sw∗ ) and identify it as follows. Select one pressure gradient contour in high quality regime as representative. The value of Sw∗ is obtained from Equation (52).

∇P = u w

μw Kkrw (Sw∗ )

(54)

5.2. Graphical method In this method, presented by Ma et al. (2013b), the apparent viscosity is employed instead of pressure gradient (∇P ). They estimated fmmob (Mr in ECLIPSE) and sfdry (Swl in ECLIPSE) by matching the t transition foam apparent viscosity ( μfoam , app ) at the transition foam quality ( f gt ). Also, sfbet ( fw in ECLIPSE) is estimated by fitting the curve

(52)

Where μ w and u w are the water viscosity and water velocity, respectively. Once the Sw∗ is determined for specific pressure gradient contour, that contour should be followed to find out the transition between high quality and low quality regime (the corner from vertical to horizontal slope). From the ratio of flow rates at the corner, the transition foam quality can be calculated from Equation (53).

ug ⎞ f g∗ = ⎜⎛ ⎟ u ⎝ g + uw ⎠

(53)

fmmob in STARS (Mr in ECLIPSE) is another parameter which is determined after f g∗ is determined. This parameter is fixed along the pressure gradient contour and transition zone. The reference contour (used in previous step) is utilized to determine fmmob which is derived by fractional flow equation at Sw = Sw∗ as shown in Equation (54).

Fig. 11. Fitting the foam model on steady-state data of Moradi-Ashraghi (Boeije and Rossen, 2015).

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to the rest of the steady-state data and the transient experiment with continuous gas injection. In order to estimate fmmob and sfdry via this method, a system of equations should be created with three unknowns (Equation (55)). t ⎧ μfoam, app (Sw, fmmob, sfdry ) = maxμfoam, app (Sw, fmmob, sfdry ) ⎫ ⎪ ⎪ 1 t μfoam ⎪ ⎪ f t , app = t) , fmmob, sfdry ) krg (Sw krw (Sw ⎪ ⎪ + μw μg ⎨ ⎬ 1 ⎪ ⎪ f gt = t μ k (S ) g ⎪ ⎪ 1 + rw w f t μ ⎪ ⎪ w krg (Sw, fmmob, sfdry ) ⎩ ⎭

(55) Two different plots are utilized in this method to solve the system of equations. The first plot is a contour plot of f gt as a function of fmmob and sfdry . To find Swt , the golden section search method is employed for each pair of fmmob and sfdry based on Equation (55). The second plot is a contour plot of μfoam, app as a function of fmmob and sfdry based on Equation (56).

μfoam, app =

Fig. 13. Tuning sfbet (epdry) in the foam model using continues gas injection data (Ma et al., 2013a).

5.3. Numerical method The concept and procedure of this method, presented by Ma et al. (2013) are as same as the graphical method. However, it hires the iteration method instead of using superposition of contour plot. In this method, the value of fmmob and sfdry are estimated utilizing two objective functions (Fun1 and Fun2) as shown in Equation (58).

1 krw (Sw ) μw

f

+

krg (Sw, fmmob, sfdry )

(56)

μg

The Sw of above equation is solved by zero-finding function using the experimentally measured transition foam quality ( f gt , measured ) for each pair of fmmob and sfdry , then according to Sw , fmmob , and sfdry t the value of μfoam , app is calculated. By performing superposition of both contour plots, the intersection point shows values of fmmob and sfdry t t where f gt = f gt , measured in the first plot and μfoam , app = μfoam, measured in the second plot. Moreover, the residual sum of squares deviation (Equation (57)) is employed to determine the best value of sfbet to fit all experimental data. The minimum value of RSS provides the best sfbet value to fit the experimental data.

⎜

⎜

∑ ⎜⎛

μi, calculated − μi, measured ⎞

⎝

⎠

i=1

μi, measured

2

⎟

⎟

⎜

⎜

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

2

n

RSS =

2

⎛ ⎞ t t ⎧ ⎛ μfoam ⎛ f gt − f gt ⎛measured⎞ ⎞ , app − μfoam, app measured ⎞ ⎪ ⎝ ⎠ ⎝ ⎠ ⎟ +⎜ ⎟ ⎪ Fun1(fmmob, sfdry ) = ⎜ ⎛measured⎞ t μfoam f gt ⎛measured⎞ ⎟ ⎜ ⎟ ⎜ , app ⎨ ⎝ ⎠ ⎝ ⎠ ⎠ ⎝ ⎠ ⎝ ⎪ ⎪ Fun2(Sw ) = −μfoam, app (Sw ) ⎩

⎟

(57)

⎟

⎟

(58)

f gt

t μfoam , app

and are defined as Equation (55). The minimum Where values for both functions are determined by Simplex method in the outer loop and Golden section method for the inner loop. The inner loop and outer loops solution, determine the values for Swt and both fmmob and sfdry , respectively. Ma et al. (2013) applied this method on their foam displacement experimental data (sand pack foam flooding using nitrogen and IOS (2 wt%)) to determine the fitting model parameters. The result of this fitting procedure is shown in Fig. 14.

Ma et al. (2013a) conducted a set of steady state foam displacement tests on the sand pack at ambient condition, using IOS (0.2 wt%) as surfactant and nitrogen as the injection gas. Then they applied the graphical method on the experimental data to find the fitting parameters for ECLIPSE model. The results of fitting the ECLIPSE model on the experimental data (apparent viscosity versus injected gas fraction) is shown in Fig. 12. This graph shows the acceptable fit for ECLIPSE model because the fitting model properly captured the maximum apparent viscosity (transition apparent viscosity) and transition foam quality. Ma and co-workers (Ma et al., 2013a), also conducted the continuous gas injection (one cycle of SAG) on the same sand pack system to tune the sfbet (epdry) value rather using Equation (57). The result of this tuning is shown in Fig. 13. Moreover, this figure shows the general behavior of continues gas injection to generate the foam.

5.4. Comparison between different methods of obtaining model parameters Three different methods for obtaining model parameters (in reservoir simulators) were introduced in this study. In the characteristic method, the fitting procedure is divided into two parts; fitting the highquality and fitting the low-quality regimes considering pressure

Fig. 12. Fitting results using graphical method (Ma et al., 2013a).

Fig. 14. Fitting results using numerical method (Ma et al., 2013).

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gradient. It also assumes the same values for residual water saturation and limiting water saturation. On the other hand, both graphical and numerical methods derive the limiting water saturation which is different residual water saturation. Moreover, both numerical and graphical methods fit the whole data (high-quality and low-quality regimes considering apparent viscosity) at the same time. The numerical method uses the mathematical equation to fit the foam model on the experimental data; however, the graphical method employs different plots (in the entire ranges of fitting parameters) to find the model parameters. 6. Summary In this study, we introduced the foam concepts and its applications in EOR. We also reported the foam modeling approaches in porous media. The foam models utilized in reservoir simulators were discussed and compared. The methods for obtaining the model parameters were reviewed and compared. Based on the conducted survey, a summary of the takeaway points can be made as 1. The foam application in EOR was introduced after WAG introduction to reduce the gas mobility problems. 2. Three different mechanisms are dominated in foam generation in porous media including snap-off, leave behind and lamella division. However, the only mechanism for foam coalescence is dominated by limiting capillary pressure. 3. Three different approaches were developed for foam flow modeling through porous media including mechanistic, empirical and semiempirical models. Among these model, the mechanistic approach is the most complicated approach; however, it is the most accurate approach because it is able to model the dynamic phenomena of foam generation and foam coalescence. 4. Three commercial reservoir simulators equipped with foam model including ECLIPSE, CMG, and UTCHEM. The UTCHEM utilizes a discontinuous function but ECLIPSE and CMG employ a continues function which is not able to show the fully foam dry-out phenomena. 5. Three different methods to obtain the model parameters (used in the simulators) were developed including characteristic, graphical, and numerical. The concept of the graphical and numerical method are same, they fit the entire data (apparent viscosity) at the same time. The characteristic method uses two different line to fit both highquality and low-quality regimes considering pressure gradient. Acknowledgements This research has been done at Universiti Teknologi PETRONAS under the SHELL-PETRONAS project. The authors are grateful to EOR center and Petroleum department of Universiti Teknologi PETRONAS for their support. References Aarra, M.G., Skauge, A., 1994. A foam pilot in a North sea oil reservoir: preparation for a production well treatment. In: SPE 69th Annual Technical Conference and Exhibition, New Orleans, Louisiana, 25–28 September. Aarra, M.G., Skauge, A., Martinsen, H.A., 2002. FAWAG: a breakthrough for EOR in the North sea. In: Proceedings - SPE Annual Technical Conference and Exhibition, San Antonio, Texas, 29 September-2 October. Afsharpoor, A., Lee, G.S., Kam, S.I., 2010. Mechanistic simulation of continuous gas injection period during surfactant-alternating-gas (SAG) processes using foam catastrophe theory. Chem. Eng. Sci. 65 (11), 3615–3631. Alvarez, J.M., Rivas, H., Navarro, G., 2000. An optimal foam quality for diversion in matrix-acidizing projects. In: SPE International Symposium on Formation Damage Control Held in Lafayette, Louisiana, 23–24 February. Alvarez, J.M., Rivas, H.J., Rossen, W.R., 2001. Unified model for steady-state foam behavior at high and low foam qualities. SPE J. 6 (3), 325–333. Aronson, A.S., Bergeron, V., Fagan, M.E., Radke, C.J., 1994. The influence of disjoining

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