Comparison of implicit-texture and population-balance foam models

Simulation models for foam enhanced oil recovery are of two types: those that treat foam texture or bubble size explicitly (population-balance models) and those that treat the effects of foam texture implicitly through a gas mobility-reduction factor. The implicit-texture models all implicitly assume local equilibrium (LE) between the processes of foam creation and destruction. In published studies most population-balance models predict rapid attainment of local-equilibrium as well, and some have been recast in LE versions. In this paper we compare population-balance and implicit-texture (IT) models in two ways. First, we show the equivalence of the two approaches by deriving explicitly the foam texture and foam-coalescence-rate function implicit in the IT models, and then show its similarity to that in population-balance models. Second, we compare the models based on their ability to represent a set of N 2 and CO 2 steady-state foam experiments and discuss the corresponding parameters of the different methods. Each of the IT models examined was equivalent to the LE formulation of a population-balance model with a lamella-destruction function that increases abruptly in the vicinity of the limiting capillary pressure P c* , as in current population-balance models. The relation between steady-state foam texture and water saturation or capillary pressure implicit in the IT models is essentially the same as that in the population-balance models. The IT and population-balance models match the experimental data presented equally well. The IT models examined allow for ﬂ exibility in making the abruptness of the coalescence rate near P c* an adjustable parameter. Some allow for coarse foam to survive at high capillary pressure, and allow for a range of power-law non-Newtonian behavior in the low-quality regime. Thus the IT models that incorporate an abrupt change in foam properties near a given water saturation can be recast as LE versions of corresponding population-balance models with a lamella-destruction function similar to those in current PB models. The trends in dimensionless foam texture implicit in the IT models is similar to that in the PB models. In other words, both types of model, at least in the LE approximation, equally honor the physics of foam behavior in porous media. © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).


Introduction
Enhanced oil recovery (EOR) techniques such as solvent, thermal, and chemical injection have the potential to increase oil production and oil recovery efficiency (Lake et al., 2014).With growth of global energy demand there is a significant interest by the oil industry in the development, optimization, and implementation of EOR methods.
Mobility control is essential to the effectiveness of EOR processes, such as surfactant/alkali flooding, miscible and immiscible gas flooding, and steam flooding.Gas-injection EOR projects often suffer from poor volumetric sweep efficiency due to low gas density and viscosity.The use of foam in gas-injection EOR applications has the potential to improve oil recovery by reducing gas mobility (Schramm, 1994;Kovscek and Radke, 1994;Rossen, 1996;Farajzadeh et al., 2011;Andrianov et al. 2012).
Foam in porous media can be defined as a dispersion of gas phase in liquid phase such that the liquid phase is connected and at least some gas flow paths are blocked by thin liquid films, called lamellae (Falls et al., 1988).It has been experimentally observed foam behavior in this regime.
Design of foam EOR processes for field application requires accurate simulation models.Foam simulation models come in two types: Population-balance (PB) models attempt to represent the dynamic processes of lamella creation and destruction as well as the effect of bubble size on gas mobility.These models can be set to assume local equilibrium (LE) between the processes of lamella creation and destruction.The second group of models reflects the effects of foam texture implicitly through a gas mobility-reduction factor that depends on saturations, superficial velocities and other factors.These models all assume LE.To avoid confusion with the LE version of PB models we refer to the second group here as implicittexture (IT) models.PB models are often assumed to be better because they are based on first principles.The dynamic version of the models are sometimes called "full physics" models (Chen et al., 2010;Ma et al., 2015).The IT models are often referred to as "empirical" (Rossen et al., 1999) or "semi-empirical" and lacking in essential physics (Kovscek et al., 1995;Chen et al., 2010;Skoreyko et al., 2012;Ma et al., 2015).Ma et al. (2015) review a wide range of foam models and their assumptions.All models, of course, incorporate only partial and imperfect physics.An essential test for models is their ability to fit available data.Skoreyko et al. (2012) represent foam generation, foam degradation and trapped foam by defining a set of first order, nonreversible reactions.They use Arrhenius-type equations to compute reaction rates.This model makes no reference to foam coarsening at a limiting capillary pressure, however, which distinguishes it from the models described here.Likewise, the population-balance models of Falls et al. (1988), Friedmann et al. (1991) and Zitha (2006) do not include foam coalescence at a limiting capillary pressure.Therefore, we do not address these models further in this paper.
In all the foam models discussed here, foam coalescence is related to P c coalescence as represented in the two types of models is essentially the same, at least under the conditions of the given experiments.
In addition, the experimental data for steady-state apparent foam viscosity (without oil present) versus foam quality are matched with the different foam models and the corresponding parameters in each model are discussed.The results confirm that the steady-state flow of foam in porous media can be adequately represented equally well by the simpler IT models.
Only population-balance models can represent the dynamics of foam creation and propagation at a shock front, the creation of foam at the entrance of the porous medium or near an abrupt change in permeability, and, possibly, foam dynamics in natural fractures.However, in published applications, PB models come to local equilibrium rapidly, suggesting that on the field scale LE applies, at least in relatively homogeneous formations (Rossen et al., 1999;Chen et al., 2010).The first step in fitting any foam model is to examine its ability to represent laboratory LE data, and this study focuses on that issue.

Foam models
Nearly all foam models alter the transport properties of gas only and assume that liquid mobility remains the same function of saturations as in the absence of foam, in accordance with laboratory observations (Bernard and Holm, 1964;Bernard et al., 1965;Sanchez et al., 1989;de Vries and Wit, 1990;Friedmann et al., 1991).In the presence of foam, gas trapped by stationary lamellae reduces mobile gas saturation, blocks gas flow and alters gas flow paths, and thus reduces gas relative permeability.The fraction of trapped gas is a function of pressure gradient, capillary pressure, aqueous-phase saturation, pore geometry and bubble size (Kovscek et al., 1995;Nguyen et al., 2007Nguyen et al., , 2009)).On the other hand, moving lamellae experience a drag force when they slide along the pore walls (Hirasaki and Lawson, 1985) that is complicated by capillary effects on the lamellae (Falls et al., 1989;Xu and Rossen, 2003).This effect is similar to an increase in gas viscosity.Because the viscosity of gas itself is not increased by foam, the effect of increased resistance to gas flow reflecting the presence of lamellae is termed "apparent (effective) gas viscosity."However, many models combine the effects of foam on gas relative permeability and apparent gas viscosity and reduce the gas mobility by a factor applied to either the gas viscosity or the gas relative permeability.In the following sections, IT and PB foam models are briefly discussed.Appendix A and Ma et al. (2015) provide a review of foam models.

Implicit-texture (IT) models
In this section we briefly discuss the UT (Cheng et al. (2000)), STARS (Computer Modeling Group, 2012) and Vassenden-Holt (1998) IT models.A summary of these models is provided in Table (A-1).

UT model
The UT model (Rossen et al., 1999;Cheng et al., 2000) was originally based on data of Persoff et al. (1991), which lies entirely in the high-quality regime.At fixed gas superficial velocity, this model gives a steep, linear increase in gas mobility as water saturation decreases through a narrow interval in the immediate vicinity of S w * , and a constant reduction in gas mobility for larger values of S w .The model allows for non-Newtonian behavior in the low-quality regime by making the mobility-reduction factor in the low-quality regime a power-law function of gas superficial velocity.This model is currently in use in compositional simulator UT-DOECO2 (Delshad et al., 2013;Naderi Beni et al., 2013) and chemical-flood simulator UTCHEM (Delshad, 2013).Because of the functional form chosen for the increase in gas mobility near S w * , the UT foam model cannot represent formation of strong foam during gas injection in a process of injection of alternating slugs of gas and surfactant solution (Dong, 2001;Shan and Rossen, 2004).

STARS model
In the STARS model (Computer Modeling Group (CMG), 2012), when foam is present, the gas relative permeability is multiplied by a factor FM, which is function of several factors that reflect the effects of different physical parameters, such as surfactant concentration, water saturation, oil saturation (and composition), salt concentration, and capillary number on foam behavior in porous media.In this paper, we focus on the dry-out function F 2 and shearthinning function F 5 , which are defined in  Vassenden and Holt (1998) proposed a foam simulation model in which the gas mobility reduction factor, F, is the sum of two exponential functions of water saturation.For water saturation slightly greater than S f (equivalent to S w * ), gas mobility decreases steeply because of the first exponential function; this corresponds to foam dryout and the high-quality regime.The second function decreases more gradually for higher water saturation and controls foam behavior in the low-quality regime.

Population-balance (PB) models
Foam mobility is influenced by its texture (Patzek, 1988;Falls et al., 1988).Foam texture is quantified as the number of lamellae per unit volume of gas.A foam with a fine texture has more lamellae in a given volume of gas and therefore induces more resistance to gas flow.Population-balance models incorporate foam texture explicitly to predict flow properties.A conservation equation for lamellae allows the simulator to track foam texture dynamically, i.e. without the local-equilibrium assumption.The rates of accumulation, convection, generation, and coalescence of foam bubbles are incorporated into the lamella balance, and, if desired, rates of trapping and mobilization as well, as they are for other molecular species in a reservoir simulator.
The transient population balance for the average flowing and trapped lamellae is written as (Chen et al., 2010): where S f and S t are flowing and trapped gas saturations, and n f and n t are number density of flowing and trapped foam lamellae, respectively.All current models assume n f ¼ n t are discussed below.Q b is a source/sink term, and q f is the net rate of generation of lamellae which can be defined as where r g and r c represent generation and coalescence rates, respectively.
The population-balance models can be simplified by assuming local equilibrium if desired (Ettinger and Radke, 1992;Myers and Radke, 2000;Kam and Rossen, 2003;Chen et al., 2010).In the LE version of the PB models, the rates of foam generation and coalescence are set equal to each other, which defines the LE value of foam texture n f at each location.Eq. ( 1) is eliminated from the set of governing equations.
In this section we briefly discuss the models of Chen et al. (2010), Kam et al. (2007), and Kam (2008).The models are summarized in Table (A-2).
The population-balance models examined here use the shearthinning expression for effective viscosity of Hirasaki and Lawson (1985), where m g is gas viscosity and m f is apparent (effective) gas viscosity in the presence of foam, n f is the foam lamella density (number of lamella per unit volume), v f is local gas velocity and a is a proportionality constant that depends on the surfactant formulation and permeability.
2.2.1.Kovscek et al. (1993) model, modified by Chen et al. ( 2010) Kovscek et al. (1993) considered Roof snap-off as the mechanism of lamella creation.The model employs a capillary-pressuredependent kinetic expression for lamella coalescence (to reflect the limiting capillary pressure) and also a term to represent the trapped fraction of foam.The gas relative permeability is then reduced according to the fraction of flowing gas to reflect the effect of gas trapping (Eq.(A-15)).The lamella-generation rate is taken as a power-law expression, proportional to the magnitude of the interstitial velocity of surfactant solution and 1/3 power of the interstitial gas velocity.Chen et al. (2010) introduced an upper limit for the concentration of lamellae that is related to pore size.The upper limit is achieved by the reducing generation rate as this limit is approached; they contended that this accounts for pre-existing gas bubbles that occupy foam-generation sites.They showed that the LE form of this model can predict both low-and high-quality regimes.
2.2.2.Kam et al. model (2007) Kam et al. ( 2007) presented a foam model in which lamella creation depends on pressure gradient and also on water saturation or capillary pressure, which governs the presence of lenses or lamellae available to be mobilized (Rossen and Gauglitz, 1990;Gauglitz et al., 2002).Specifically, lamella generation rate is proportional to water saturation and a power-law expression of pressure gradient.In this model, the generation rate monotonically increases with the pressure gradient.Lamella-coalescence rate is a power-law function of (S w -S w * ), with the exponent an adjustable parameter.This model can represent multiple (coarse and strong) foam states at the same superficial velocity and jumps between those states, as well as the low-and high-quality regimes for strong foam.

Kam model (2008)
In this extension of the model of Kam et al. (2007), for the lamella creation, the local pressure gradient must exceed the minimum pressure gradient required for lamellae mobilization and division.Kam (2008) proposed a new lamella-creation function, which reaches a plateau at larger pressure gradient (Eq.(A-22)).
Parameters R ref , fmmob and (1/F o ) represent reference (or maximum) mobility reduction factor that could be achieved by foam when all the conditions are favorable and are directly set in the IT models.Parameters n * and n max are upper limits for the concentration of foam bubbles in the Chen et al., Kam et al. and Kam models, respectively.They are related to pore size.More than one foam bubble per pore is not expected (Bertin et al. 1998;Kil et al., 2011).Parameters 3, epdry, and s 1 control the sharpness of the transition from high-quality to low-quality regimes in the IT models.The extent of the saturation range in the transition from high-quality to low-quality regimes is set to 2 3 in the UT model.In both STARS and the Vassenden-Holt models, for the large values of epdry and s 1 the transition is sharp and foam collapses within a narrow range of water saturation.In the Kam et al. and Kam models the coalescence exponent n controls the transition, with smaller n giving a sharper transition.The coalescence rate depends on nearness of capillary pressure to P c * with an exponent (À2) in the Chen et al. model (Eq.(A-12)).Parameters s, epcap and a account for shear-thinning behavior in the low-quality regime in UT, STARS, and modified Vassenden-Holt models, respectively; this would reflect both gas trapping and mobilization and the shear-thinning drag on individual moving bubbles.The population-balance models use the shear-thinning expression by Hirasaki and Lawson (1985) with an exponent of (À1/3) for the dependence of apparent gas viscosity on gas velocity.

Fitting foam models to experimental data
In this section we apply different foam models to match the steady-state CO 2 foam experimental data reported by Moradi-Araghi et al. (1997), and the N 2 foam experimental data of Alvarez et al. (2001).Then we discuss and compare the fits with different foam models.Moradi-Araghi et al. (1997) conducted experiments with CO 2 at 98 F and 2000 psi in a 551.5-md-permeability reservoir core from the South Cowden Unit in West Texas.The core plugs were 1 inch in diameter and 4.84 inches in length.Foam was made with 2000 ppm surfactant Chaser CD-1050 surfactant in synthetic South Cowden formation brine.The overall pressure drop in the foam experiments was measured and divided by that for single-phase water flow at the same total superficial velocity and reported as the reduction factor (RF).Therefore, to calculate the apparent foam viscosity as a function of gas fraction or foam quality (f g ), the reported reduction factor is multiplied by water viscosity (m w ) at the experimental conditions: where m f app is in cp.The apparent foam viscosity -m f app is defined in Eq. ( 4) in terms of the total mobility of foam treated as a single phase; the apparent gas viscosity m f is defined in the foam models above as the effective viscosity of the gas phase in foam.
Moradi-Araghi et al. did not report the injection rate in their experiments.Therefore, we assume a total injection rate of 5 ft/day in fitting their experimental data.In a separate study (Farajzadeh et al., 2015), it was found that the choice of flow rate affects only the MRF or fmmob values in the UT and STARS models and therefore does alter the ability of the models to fit data.Alvarez et al. (2001) conducted experiments with N 2 foam at room temperature and 600 psi outlet pressure in 530-md Berea sandstone core.The cores were 2 inches in diameter and 11 inches in length.Foam was made with 1 wt% Bio-Terge AS-40 surfactant in brine of 3 wt% NaCl and 0.01 wt% CaCl 2 .The pressure gradient at a fixed superficial velocity of 2.5 ft/day was measured and apparent foam viscosity is reported here by multiplication by absolute permeability and dividing by total velocity: Relative-permeability data are not reported for the porous media used in these experiments.Although the parameters depend somewhat on the choice of the relative-permeability parameters (Ma et al., 2014), the generality of our results will not be affected, e.g., with the choice of different relative permeability parameters the correlation between the basic foam properties such as P c * and rock properties like permeability will remain unaltered (Farajzadeh et al., 2015).For fitting the foam data we use a relative-permeability function fit to the data of Persoff et al. (1991) in Boise sandstone.We also use the water-gas relative permeability and capillary-pressure functions described in Appendix B. Table 1 summarizes all the rock and fluid properties used in this paper.
The objective function for the optimization of foam model parameters is defined as where x is the vector of foam-model parameters.For example in the STARS model, x ¼ [fmmob, epdary, fmdry, epcap].The reference capillary number, fmcap, is not an independent parameter and was set to a fixed value.In Eq. ( 6) m f ðexpÞ appi and m f appi are, respectively, the experimental data and values predicted by the foam model for the given set of foam-model parameters.A constrained least-squares algorithm in the optimization toolbox of MATLAB (lsqnonlin function) was used to solve our non-linear data-fitting problem.The constraints are set to physical limits for the foam parameters, e.g. S w * > S wc , S f > S wc or fmdry > S wc or, depending on the model; andP c@Swc > P * c .The final (best-fit) set of fitted parameters may depend on the initial guess.We discuss the issue of non-uniqueness in the various model fits below.In cases where the fit depends on the initial guess, we show here the best fit (i.e., the fit with the smallest value of f(x)).The model parameters are listed in Tables 2e7 Fig. 2    the different foam models.
The Vassenden and Holt model was not able to match the data in low-quality regime until we introduced an exponent a for the velocity term (see Eq. (A-8) and Fig. 4).The original model corresponds to a value of the exponent a in Eq. (A-8) of one.
The results show that the two types of models match the experimental data equally well.Some individual models do better in matching the data over some range and others better in other ranges.For instance, for the Moradi-Araghi et al. data the UT and STARS models do a little better than the others.
For the Chen et al. model we match the experimental data using three sets of foam parameters for both the data of Moradi-Araghi et al. and Alvarez et al. (Table 5).In set #1 we assume a maximum gas trapping saturation of 50% and we scale the gas relative permeability along with the gas viscosity to match the experimental results; the resulting fit is shown in Fig. 5. Since the value of trapped-gas saturation is uncertain, in set #2 we ignore gas trapping but we obtain the same quality match by adjusting n * and slightly adjusting P c regime.Dong (2001) examined this issue in the earlier model of Kovscek et al. (1995).Suppose, as posed in the Introduction, that foam is initially at LE in the high-quality regime, and then gas superficial velocity increases by a factor X. In this model, lamella-generation rate increases by X 1/3 (Eq.(A-9)).The coalescence rate increases by the factor X (Eq.(A-11)) because of the increase in gas velocity.This increase in coalescence rate does not depend upon nearness to P c * .Equilibrium foam texture n f therefore changes by a factor (X 1/3 / 3 ) ~(X À2/3 X À1/3 ) ¼ X À1 because of the dependence on texture and on gas velocity (Eq.( 3)).Through Darcy's law on the gas phase, the pressure gradient is proportional to (v g m f app ) ~(XX À1 ), i.e. is constant, with no change in capillary pressure.In this model, the pressure gradient is independent of gas superficial velocity in the high-quality regime because of the particular forms assumed for apparent gas viscosity (Eq.( 3)) and the lamella-generation and -coalescence rates, not the divergence of coalescence rate at the limiting capillary pressure.
In the model of Chen et al. (2010) the dependence of generation rate and gas mobility on gas superficial velocity is more complex.In our model fits, the water saturation and capillary pressure in the high-quality regime is the same for all three sets of parameters (#1 to #3) at S w ¼ 0. The water saturation in the high-quality regime can be derived from the slope of the pressure gradient versus f g in the experimental data, using Darcy's law for the water phase (Boeije and Rossen, 2013;Ma et al., 2013Ma et al., , 2014)).The water saturation in the high-quality regime is calculated to be 0.295 and 0.314 for the Moradi-Araghi and Alvarez data, respectively; the corresponding values of P c are 0.113 and 0.524 psi.

Coalescence function in implicit-texture models
In the IT models examined here foam experiences an abrupt change in its properties near the limiting water saturation.In this section we show that this abrupt change can be expressed by a lamella-destruction function similar to that in population-balance models with a limiting capillary pressure.
The coalescence rate r c in population-balance models (Eqs.(A-11), (A-20), and (A-23)) can be expressed as where c c is constant, n f is lamella density (foam texture), n D is dimensionless foam texture, n f * is foam texture in the low-quality regime (which we assume to be constant (Alvarez et al., 2001)), and f is a destruction function.In other words, lamellae have a probability of breaking proportional to f, where f can be expressed in terms of water saturation or capillary pressure.Kovscek et al. (1993) suggested that f f (P c * -P c ) À2 ; Bertin et al. (1998) proposed f f (P c * -P c ) À1 ; Kam et al. (2007) and Kam (2008) assumed f f (S w -S w * ) -n with n a fitted parameter.In all the models foam coalescence rate increases sharply at or near the limiting water saturation or the corresponding capillary pressure.Figs. 8 and 9 show the destruction function f used in the PB models in terms of water saturation and capillary pressure.However, there is no theoretical reason nor direct experimental evidence for the choice of one mathematical form of the lamella-destruction function over the others, as long as it increases greatly as P c approaches P c * .We show here that using the mobility-reduction function (specifically the dry-out function) in IT models implies a lamelladestruction function that is similar to the corresponding function in PB models.In other words, the physics of foam collapse near P c * is essentially the same in the IT models as in the PB models.
In the UT foam model the gas mobility-reduction factor corresponds to a dimensionless foam lamella density defined by where, for water saturations less than S w This function, related to the mobility-reduction factor F in Vassenden-Holt model, is inversely proportional to foam texture.Thus In the STARS model, gas mobility reduction is proportional to FM, with the maximum reduction when FM ¼ 1/(1 þ fmmob).Excluding the shear-thinning effect (F 5 function) from FM, the corresponding dimensionless foam texture is defined by Different population-balance models use different lamellageneration functions, and the best choice for this function remains controversial.Kam and Rossen (2003) show that different lamella-generation functions can give the same steady-state foam behavior.Moreover, behavior in the coalescence regime is dominated by an abrupt increase in coalescence rate, not a change in generation rate.Therefore, for simplicity, we assume here a constant bubble-generation rate, r g .Zitha (2006) assumes a constant generation rate in his model.More complicated generation functions could be used without changing our conclusions, which hinge on abrupt changes in foam behavior near P c * or S w * .Therefore, assuming local equilibrium, the destruction function is infinity at S w * -3.In the Vassenden-Holt model the destruction function increases sharply in the high-quality regime and it reaches its maximum value at S f .In this model the destruction function increases with larger slope for a larger value of s 1 .The STARS model implies a destruction function that increases sharply in the vicinity of fmdry but remains finite at all water saturations.In this model, the destruction function increases more abruptly as S w approaches fmdry for larger values of epdry.The lamella-destruction functions implied by these IT models are similar to those in the PB models.
There is no theoretical reason or experimental justification to prefer any of the functions in Figs. 9, 11 and 13, or 15 above the others.Thus, in terms of the most important mechanism in foam behavior without oil (Farajzadeh et al., 2012), i.e. foam collapse at the limiting capillary pressure, the IT models are as well-supported by theory and experiment as the population-balance models.Fig. 16 through 19 show dimensionless foam texture in the implicit-texture and population-balance models, using the parameter values fit to Moradi-Araghi et al. and Alvarez et al. experimental data.In the population-balance models, dimensionless foam texture is defined as the foam bubble density (n f ) divided by the maximum foam bubble density in the low-quality regime n max (n D ¼ n f /n max ).In all the models, dimensionless foam texture drops sharply close to S w * .Dimensionless foam textures implied in the IT models are similar to those in population-balance models.

Summary and conclusions
Implicit-texture (IT) models are often perceived as not reflecting the essential physics of foam in porous media.Although only population-balance models can represent the entrance region, dynamics at shock fronts, the process of foam generation, and regions in heterogeneous or fractured media, where abrupt heterogeneities mean that foam is not at local equilibrium, in this study we focus on the ability to fit steady-state data.IT models all assume local equilibrium (LE).Therefore, we compare IT and LE version of population-balance (PB) models.The main conclusions are as follows: An essential test of a model's usefulness is its ability to match available data and thereby give confidence in predicting new data.We show that both IT models and PB models at LE match the steady-state experimental data for CO 2 and N 2 foam presented here equally well.The corresponding parameters of the different foam models are presented and discussed.The original Vassenden-Holt model does not match the data in low-quality regime until we introduce an exponent a in the velocity term.The trapping parameter in the Chen et al. model does not play an essential part of the data-fitting procedure, for these data, at least.In the Vassenden-Holt and Chen et al. models, the high-quality or coalescence regime does not necessarily reflect a capillary pressure near the limiting capillary pressure (P c * ).Values of P c * fit to foam-mobility data with model of Chen et al. are sensitive to the value of the coalescence kinetic parameter k o À1 .We define dimensionless foam texture implicit in the IT models and derive the foam-coalescence-rate function implicit in these models.The results show that the IT models that incorporate an abrupt change in foam properties near a given water saturation can be recast as LE versions of corresponding populationbalance models with a lamella-destruction function similar to those in current PB models.The trends in dimensionless foam texture implicit in the IT models is similar to that in the PB models.In other words, both types of models, at least in the LE approximation and without oil, equally honor the physics of foam behavior in porous media.
where S wD is dimensionless water saturation and is defined as S wD ¼ S w À S wr 1 À S wr À S gr (B-3) The general form of gas-water capillary pressure represented by Li (2004) is P c ¼ P c@Swc ð1 À bS wD Þ À1=l (B-4) where, P c@Swc is the capillary pressure at the connate water saturation, S wc , when drainage capillary pressure is used.b and l are constants and are defined as a fmcap is a reference capillary pressure below which shear-thinning is assumed not to apply.The choice of this reference also affects the value of fmmob (Boeije and Rossen, 2013).
Fig.2.UT model fit to experimental data using the parameters in Table2

Fig. 3 .
Fig.3.STARS model fit to experimental data using the parameters in Table3
270e0.274 for Moradi-Araghi and S w ¼ 0.311 for Alvarez et al.Nevertheless, the fitted values of S w * and P c * vary among the parameter fits to the data: from S w * ¼ 0.265 to 0.263 to 0.219 (P c * between 0.19 and 0.80 psi) for parameter sets #1, 2 and 3, respectively for Moradi-Araghi and S w * ¼ 0.310 to 0.309 to 0.284 (P c * between 0.55 and 0.80 psi) for Alvarez et al.In the model of Chen et al., the high-quality regime water saturation need not be close to S w * , nor the capillary pressure close to P c * .

Fig. 6 .
Fig. 6.Kam et al. model fit to experimental data using the parameters in Table 6

Fig. 7 .
Fig.7.model fit to experimental data using the parameters in Table7

Fig. 9 .
Fig. 10 through 15 show the lamella-destruction functions implied by the UT, STARS, and Vassenden-Holt models, using the parameter values fit to the experimental data of Moradi-Araghi et al. and Alvarez et al. in Figs. 2, 3 and 4b.The UT model implies a destruction function that remains constant in the low-quality regime and increases sharply in the interval S w * ± 3, diverging to

Fig. 10 .
Fig. 10.Lamella-destruction function f implied by the UT foam model plotted as a function of water saturation.The vertical dotted lines represent (S w * -3 ).

Fig. 11 .
Fig. 11.function f implied by the UT foam model plotted as a function of capillary pressure.The vertical dotted lines represent the capillary pressure at (S w * -3 ).

Fig. 12 .
Fig. 12. Lamella-destruction function f implied by the STARS foam model plotted as a function of water saturation.The vertical dotted lines represent fmdry.

Fig. 13 .Fig. 14 .
Fig. 13.Lamella-destruction function f implied by the STARS foam model plotted as a function of capillary pressure.The vertical dotted lines represent capillary pressure at fmdry.

Fig. 15 .
Fig. 15.Lamella-destruction function f implied by the Vassenden-Holt foam model plotted as a function of capillary pressure.The vertical dotted lines represent the capillary pressure at S f .

Fig. 16 .
Fig. 16.Dimensionless foam texture obtained in the population-balance models to fit to Moradi-Araghi et al. data.

Fig. 18 .
Fig. 18.Dimensionless foam texture obtained in population-balance models to fit to Alvarez et al. data.

Fig. 19 .
Fig. 19.Dimensionless foam texture by implicit-texture models to fit to Alvarez et al. data.
Table (A-1).As in the UT model, gas mobility increases as S w decreases in the vicinity of S w Table 8 summarizes values of S w * or the corresponding parameters in other models (fmdry in STARS model and S f in Vassenden-Holt) used to fit the experimental data in different models.While in the UT, STARS, Kam et al., and Kam models the value of S w * was close to the water saturation in the high-quality regime, in the Vassenden-Holt and Chen et al. models water saturation in the high-quality regime is not close to S w * ; nor, is P c in the foam close to P c * .
Shell Global Solutions International for permission to publish this work.
(Coombe, 2012)cent version of STARS, the parameter fmdry is renamed sfdry, and epdry is renamed sfbet(Coombe, 2012).The water saturation around which foam collapses (fmdry) is no longer treated as a constant, but is a function of surfactant concentration, oil saturation, salt concentration, and capillary number.If one disables these other functionalities sfdry plays the same roll as fmdry as described here.

Table 1
Rock and fluid properties Parameters Moradi-Araghi et al.Alvarez et al.

Table 5
Chen et al. (2010)model parameters to fit experimental data

Table 6
Kam et al. (2007)model parameters to fit experimental data

Table 7
Kam (2008)model parameters to fit experimental data n w is the water Corey exponent and P ce is the capillary entry pressure.Parameters P c@Swc and P ce are assumed here to be 21.036 and 0.042 psi for the data of Alvarez et al.for foam made with N 2 .Assuming a similar rock type the capillary pressure for the Moradi-Araghi et al. (CO 2 ) data set can be scaled as CO2 (~5 dyne/cm) and s N2 (~30 dyne/cm) are CO 2 and N 2 interfacial tension with water, respectively.Figure(B-1) shows the capillary-pressure curves used in this study to model CO 2 and N 2 experiments.
where where s

Table 8
Values of limiting water saturation used in different models