A Novel Model of Foam Flooding Considering Multi-Factors for Enhancing Oil Recovery

Foam flooding is a promising technique for achieving mobility control and diverting fluid into low-permeability strata in post-water-flooding reservoirs. However, foam flow is very complicated and is influenced by many factors which have not been studied and explored very rigorously (i.e. permeability, surfactant concentration, foam quality, reservoir temperature, oil saturation, water saturation and seepage velocity). Based on core flooding experiments of foam flowing and blocking rules using two kinds of foaming agents, a novel model of foam flooding considering the influences of the above factors is established and solved using a reservoir simulator which is formulated using the IMPES method in conjunction with a Runge-Kutta method. Then, the validation is performed by core flooding experiments in both the absence and presence of oil. Finally, the simulator is used to investigate the effects of the permeability max-min ratio, ratio of vertical to horizontal permeability, gas-liquid ratio, depositional sequence, foaming agent concentration, and reservoir temperature.


INTRODUCTION
Foam flooding is an EOR (Enhance Oil Recovery) technique taking foam as the displacing fluid.Foam not only effectively decreases the mobility ratio of gas to oil but also weakens the phenomena of gas channeling and steam override [1,2].Moreover, due to the particular mechanisms of foam generation and collapse, foam presents the selective blocking features of "big block and small not block" and "water block and oil not block" when it flows in porous media [3], which is very favorable for enhancing oil recovery.In the past few years, foam flooding has been carried out in many pilots in China, and most of them have shown a good performance [4].
In a study of foam triggering mechanisms, Kovscek et al. [5] attributed triggering mechanisms of foam to increasing the viscosity of the driving phase; Rossen et al. [6] attributed the triggering mechanism to declining gas phase mobility; Kovscek and Bertin [7] reported that both increasing gas phase viscosity and decreasing the gas phase relative permeability take effect simultaneously, and hence they improved the models for foam viscosity and the gas phase relative permeability.However, it is controversial that relative permeability and viscosity are considered separately [8].As can be seen from these studies, it is recognized that the triggering mechanism of a foam system is decreasing the gas mobility, but many different approaches, e.g.modifying the gas phase viscosity or gas phase relative permeability or modifying both the gas phase viscosity and relative permeability are considered to achieve this mechanism.Nevertheless, foam flow in porous media is a very complicated process, since the lamellae in the flowing state will increase the viscosity of the driving phase and the lamellae adsorbed on the pore surface will reduce gas relative permeability.Unfortunately, because of the surrounding environment, especially the properties of the rock and fluid, both mechanisms are difficult to distinguish and test by experiments.Therefore, the above approaches for representing foam mechanisms are partial and difficult to quantify.In this paper, the mobility of the gas phase is considered as a whole to test and be measured by the resistance factor.This method includes increasing the viscosity of the displacing phase as well as decreasing the gas relative permeability, avoids repeated tests, and takes into account the effects of these two mechanisms.Numerical simulation is a technique to study the flow characteristics described by mathematical models [9].It has features of low cost, high efficiency and good repeatability, and can demonstrate the entire complex flow process.Because foam is a very complex system where gas is the dispersed phase and liquid is the continuous phase, accurate prediction is difficult.There are several approaches to deal with its complexity.The first approach is called "population balance", which quantifies the relation between foam mobility and foam texture and all the mechanisms of creation and destruction of liquid films, or lamellae [10][11][12][13].It uses a differential equation to represent foam texture in situ, but the basic concept is consistent with the local-equilibrium version, in which foam texture is an algebraic function of local conditions [6].In all population balance models, the rate of creation of lamellae depends on gas velocity through pore throats where snap-off takes place [10][11][12][13].The second approach represents foam mobility as an empirical function of many influence factors [14][15][16][17].The third approach is the "fixed-P c * model", which relies on the relation between capillary pressure, foam texture and foam mobility [18].In essence, it is also a localequilibrium version of the population balance for strong foams under conditions where capillary pressure dominates foam texture and gas mobility [6].The first approach is more complex, and there is no way to separate the effects of lamella generation and destruction in history-matching coreflood results [6]; the second approach is simple, but it disregards the relation between foam mobility and texture; the third approach is between the other two, but it disregards the foam strength influenced by many key factors.Based on the advantages and disadvantages of the above three models, a synthesized multi-factor foam model is presented.In the novel model, the foam generation depends on the critical gas velocity, which is a function of the aqueous phase mobility fraction [12].The foam collapse depends on limiting capillary pressure, at which foam collapses abruptly [19].Since S w is related to P c through the capillary-pressure function P c (S w ), this means that foam remains at a given water saturation S w * S w (P c *). Foam strength, which is influenced by permeability, surfactant concentration, foam quality, reservoir temperature and oil saturation, affects the foam mobility, which is evaluated by the foam resistance factor.Therefore, a foam resistance factor model derived from our coreflood experiments using two kinds of foam agents is used to represent foam strength features [20].The advantages of this approach reflect the competing goals of simplicity and completeness; however, the model is not able to describe slow generation or destruction processes thoroughly, considering the influence of flow velocity on foam mobility.
Finally, the model validation was performed by core flooding experiments; the factors which affect the EOR effects of foam flooding were investigated using the novel model: the permeability max-min ratio, the ratio of vertical to horizontal permeability, gas-water ratio, foaming agent concentration, temperature and depositional sequence.

GOVERNING EQUATION 1.Mass Conservation Equations
The assumptions imposed on the flow equations include: -four components: oil, water, gas and surfactant; -gas, water and surfactant generate foam which reduces the gas mobility; -the whole process is isothermal, and the energy transfer is neglected; -fluid flow obeys the generalized Darcy's law; -phase equilibrium is reached instantaneously; -foam only affects mobility (the ratio of gas relative permeability to viscosity).Mass balance equation for the oil phase: Mass balance equation for the aqueous phase: Mass balance equation for the gas phase: Mass balance equation for the surfactant (foaming agent): In which k is the absolute permeability, lm 2 ; k rl is the relative permeability of phase l; l l is the effective viscosity of phase l, mPaÁs; B l is the formation volume factor of phase l, m 3 /m 3 ; 5 is the divergence operator; q l is the source and sink for phase l, m 3 /(dayÁm 3 ); R g is the gas resistance factor; U l is the potential of phase l, MPa; S l is the saturation of phase l; u is porosity; c s is the surfactant concentration, kg/m 3 ; d s is the diffusion coefficient of the surfactant, m 2 /s; q r is the rock density, kg/m 3 ; ĉs is the adsorption concentration of the surfactant, kg/kg.

Auxiliary Equations
Let S w , S o and S g be water, oil and gas saturation, respectively.Then the saturation constraint equation is: Also, p w , p o and p g are water, oil and gas phase pressures, respectively; the capillary pressure equations are: In which p cow is the capillary pressure between oil and water, Pa; and p cgo is the capillary pressure between gas and oil, Pa.

Process Mechanism Models of Foam Flooding 1.3.1 Foam Resistance Factor
The effects of different factors on foam block characteristics are obtained by experiments using two kinds of surfactants.One is sodium dodecyl sulfate (#1) and another is an industrial foaming agent used in the Henan oilfield (#2).Then, a range of variograms and their correction models are applied to fit the relationship linking the foam resistance factor and other factors.The undetermined coefficients can be obtained using DATAFIT software [21] based on experimental results.Because of the consistency in physical meaning among variograms, the foam generation and collapse mechanisms, this method evades the defect from the conventional mathematical regression method that can merely be applied to a local scale but not to the extended area.

Resistance Factor vs Foaming Agent Concentration
A logistic model is a straightforward model used to forecast the population growth and presents an "S" tendency which can be divided into elementary, acceleration, transfer, deceleration and saturation periods.The adsorbance of active molecules on the gas/liquid interface has a similar trend, which determines the variation of foam blocking ability.As a result, the blocking capacity increases as the surfactant concentration increases, although there is a break in the curve at about 0.5% by weight.Based on experimental results (Fig. 1) and the Critical Micelle Concentration (CMC) introduced as a characteristic parameter, a logistic function was used to model the relationship between the resistance factor and foaming agent concentration: Using DATAFIT and the above model, the undetermined coefficients were obtained.For surfactant #1, a 1 =100 and r 1 =19.5; and for surfactant #2, a 2 =230 and r 2 =16.
Resistance Factor vs Foam Quality Khatib et al. [19] showed that at a given gas flow rate, the foam resistance factor increases slightly with increasing foam quality ranging from 50% to 98%.De Vries and Wit [22] reported that at an imposed total flow rate, the foam resistance factor increases as foam quality increases until the break point; beyond that it decreases.Chang and Grigg [23] found that the foam resistance factor increases with increasing foam quality ranging from 33.3% to 80%.We found that the resistance factor first increases and then decreases as the foam quality increases, and the break point is at about 80%.However, it is challenging for the foam quality to reach above 80% in foam flooding pilots.Hence, the proportional region between the resistance factor and foam quality is of concern from the practical application viewpoint.Derived from the data of this region (Fig. 2), the variation resembles the exponential variogram model.After deliberating the endpoint feature, the relationship between the resistance factor and foam quality is: Using DATAFIT and the above model, the undetermined coefficients were obtained.For surfactant #1, b 1 = 60; and for surfactant #2, b 2 = 45.

Resistance Factor vs Permeability
One merit of the foam used for EOR is "big block and not small block", meaning foam has a better blocking ability in high-permeability regions or layers and the blocking ability nearly disappears in low-permeability regions or layers.This indicates shear thinning of the foam, which has been experimentally confirmed by many researchers [24][25][26][27][28].In order to characterize the relation between the resistance factor and permeability (Fig. 3), dimensionless permeability (k D ) is used.The relation between the resistance factor and dimensionless permeability is: Using DATAFIT and the above model, the undetermined coefficients were obtained.For surfactant #1, m 1 =3 and n 1 =2; and for surfactant #2, m 2 =8.5 and n 2 =1.05.

Resistance Factor vs Temperature
Influencing mechanisms of temperature on foam resistance are diverse [29].First of all, the dissolving capacity increases as the temperature increases, so the adsorbance of active molecules on the gas/liquid interface decreases, which results in a decrease in the strength of the lamellae.In addition, an increase in temperature will accelerate the drainage rate of liquid from lamellae to the liquid phase and promotes the collapse and coalescence of the bubbles.Moreover, an increase in temperature will also speed up the degradation of the foaming agent.Friedmann et al. [12] and Maini and Ma [30] reported that the foam resistance ability largely decreases as the temperature increases in porous media.Based on experimental results and the break point denoted as T r (Fig. 4), the relation between temperature and foam quality is modeled as: Using DATAFIT and the above model, the undetermined coefficients were obtained.For surfactant #1, a 1 =140 and b 1 =30; for surfactant #2, a 2 =70 and b 2 =38.

Resistance Factor vs Oil Saturation
The mechanisms by which oil affects the foam stability are diverse.Some studies have confirmed that the foam can be generated in the presence of oil by selecting appropriate foaming agents [31][32][33][34].Some studies have stated that foam can be generated when the oil saturation is below a critical value [35][36][37], but some other studies have shown that foam is generated at relatively high oil saturation [34,38,39].The characteristics of foaming agents and oil samples used in the experiments may cause these differences.In general, it is commonly agreed that the presence of oil is detrimental to the foam stability [35,36,38,40], and our experimental results also support this view.Therefore, oil saturation is more likely to be regarded as an influencing factor of the resistance factor than a critical condition of foam collapse.Based on experimental results (Fig. 5), the foam resistance factor exponentially decreases as the oil saturation increases, and different values of c will be achieved for different foaming agents and oil samples:   Using DATAFIT and the above model, the undetermined coefficients were obtained.For surfactant #1, c 1 =9.0; for surfactant #2, c 2 =9.2.

Seepage Velocity -The Critical Generation Condition of Foam
The generation mechanisms of foam in porous media include lamellae snap-off, lamellae lag and lamellae trapping.Many experiments indicated that the lamellae snap-off or lamellae flow will not occur until the pressure or velocity overpasses the critical value.In fact, it is a triggering mechanism.Once foam is in place, velocity can be reduced, but the foam remains.Fridmann et al. [12] found that the critical velocity increases as the liquid volume fraction decreases according to the following: where v gc is the critical velocity of foam generation, m/day, and V L is the mobility fraction of the aqueous phase.

Water Saturation -The Critical Collapse Condition of Foam
Based on the DLVO theory [41,42], the collapse of foam is related to critical capillary force.When the practical capillary force exceeds the critical value, foam will collapse.However, the capillary force is a function of water saturation; the higher the water saturation, the smaller the capillary force will be.Therefore, once the water saturation is lower than the critical value corresponding to the critical capillary force, the foam will collapse.Khatib et al. [19] found that the critical capillary force logarithmically decreases as the permeability and gas velocity increase, which also reflects the shear-thinning characteristics of the foam system.Based on the definition of the Leverett J-function and the empirical relation of the Leverett J-function [43,44], we have: Khatib et al. [19] also found that the critical capillary pressure logarithmically decreases with increasing permeability (may reflect the effect of the pore structure) and gas velocity, which also reflects the shear-thinning characteristic of foam: Substituting Equation ( 16) into Equation ( 15), the limiting water saturation was obtained to be: In Equations (15)(16)(17), J(S w *) is the J function; P c * is the critical capillary force, MPa; v g is the gas velocity, cm/s; r gw is the gas-water interfacial tension, mN/m; h is the wetting angle; a 1 , b 1 , a J and b J are undetermined coefficients; S w * is the critical water saturation.

Multi-factor Gas Mobility Model
Based on a previous study [6], the resistance factor of the gas phase in porous media with the presence of foam is: where e is a special parameter; e = 0.001 was used in this paper.It will not affect the simulation of the displacement effect.It is only used to avoid the jump of the calculated gas mobility due to the actual water saturation fluctuation above and below the critical value.The influencing factors of foam blocking ability in the process of foam flooding are numerous, and play a role simultaneously, hence it is very valuable to quantitatively characterize the foam resistance factor under multiple scenarios.Also, it is beneficial to the advancement of numerical simulation technology of foam flooding.Therefore, a novel foam resistance factor model that comprehensively considers the influences of the above factors is achieved: The validation of the multi-factor gas mobility model can be verified by the experimental data, which is shown in Figure 6.The values of the undetermined coefficients are shown in Table 1.Equation ( 19) not only comprehensively considers the influences of the rock properties, fluid properties and surroundings on foam block characteristics, but also avoids the discontinuous behavior of gas mobility caused by the critical foaming agent concentration and critical oil saturation, which will greatly improve the stability of the numerical simulation process.

Correction of the Resistance Factor for Weak Foam
It was reported that flow resistance is small and foam textures are coarse, consistent with the injection of unfoamed gas or weak foam in the inlet region, and there exists a strong foam piston-like front moving through the porous media [6,44].This phenomenon has also been verified by many experiments [45,46].However, most of the existing foam flow models cannot simulate this process [6].In this model, the above resistance factor model is based on strong foam, so the following correction function is introduced to represent this effect: where R kf 0 is the resistance factor of strong foam; L is the flow distance of weak foam or unfoamed gas and foaming agent solution, m; AA and RR are model coefficients.

Adsorption Model for the Surfactant
The adsorption of the foaming agent will occur when it flows in the porous media and obeys the typical Langmuir law: where b s is the Langmuir parameter, (kg/m 3 ) À1 ; ĉsmax is the maximum adsorbed concentration, kg/kg.

Capillary Desaturation and Interfacial Tension Model
The foaming agent can also reduce the oil-water interfacial tension by its surfactant characteristics and decrease the residual oil saturation [47]: where N co is the capillary number; r ow is the oil-water interfacial tension, mN/m; S or min is the limit of residual oil saturation under a high capillary number; S or max is the residual oil saturation under a low capillary number; a r , b r and A N are undetermined coefficients.

Relative Permeability Model
The Stone II model is used to calculate the three-phase relative permeability [48]: where k rocw is the oil phase relative permeability corresponding to connate water; k rw and k rg are the water and gas phase relative permeability, respectively; k row and k rog are the oil phase relative permeability in the oil-water system and oil-gas system, respectively; n w , n ow , n w , n og and n g are input parameters; S wc , S orw , S gc and S org are the end-points in oil-water and oil-gas systems' relative permeability functions.

Physical Properties of the Gas Phase [49]
The formation volume factor is: The expansion factor is: where Z is the compressibility factor; t is the reservoir temperature, °C.

Capillary Pressure Model
Capillary pressure is a function of saturation: where A 1 and A 2 are input parameters; B 1 and B 2 are capillary pressure exponents.

SOLUTION AND VALIDATION OF THE MATHEMATICAL MODEL 2.1 Solution Method
In this model, gas flow velocity is used as the triggering mechanism; water saturation obtained based on DLVO theory is used for the bubble collapse mechanism.If foam exists in the porous media, the foaming agent concentration, foam quality, permeability, temperature and oil saturation come into play for calculating the resistance factor in the grid.This method avoids solving the foam population balance equation.A method for solving this mathematical model is as follows: -the IMPES (Implicite Pressure Explicite Saturation) method is used to solve the pressure and saturation equations to obtain the pressure and saturation distributions for each phase; -the classical four-order Runge-Kutta method is applied to solve the concentration equation of the foaming agent; -the local gas flow velocity, critical gas flow velocity and critical water saturation based on the grid pressure and saturation are solved;  -the critical generation condition and critical collapse condition are used to determine whether there is foam in the grid.If foam exists in the grid, using permeability, the foaming agent concentration, gas-liquid ratio, temperature and oil saturation are used to calculate the resistance factor; otherwise, the resistance factor is set to one; -then the resistance factor will be substituted into the gas flow equation; -repeat steps (1-5) until the end of the simulation time.

Validation of the Novel Model In the Absence of Oil
The model includes all mechanisms that are: -crucial to the process; -known with reasonable accuracy.
With the purpose of validating the model, the foam flow data obtained from Kovscek [44] is used.Core and fluid data are based on reference [44], and other model parameters are listed in Table 2.The fitting result of the multi-factor model shown in Figure 7 is a slightly better fit than the population balance model although the curve for 0.23 PV is not a close fit due to the lack of strong foam and gas channeling.

In the Presence of Oil
Because the impact of oil presence is considered, one group of foam flooding experimental data is also used.In this experiment, foaming agent solution (SDS: surfactant #1) with the concentration of 0.4 wt% was injected continuously into linear unconsolidated media of length 0.3 m at a rate of 2 mL/min, while gas was also injected continuously to give a foam quality of 80% at the sandpack entrance.The permeability is 4.5 lm 2 and porosity is 0.33.The initial oil saturation is 0.4 and oil viscosity is about 2.0 mPaÁs.At 120 min, continued water flooding was performed at a rate of 5mL/min.Other model parameters are listed in Table 3.The comparison between computed and experimental data is shown in Figure 8 and the agreement is excellent.

RESULTS AND DISCUSSION
In order to study thoroughly the differences between the foam flooding with the gas-water flooding and the influences of many parameters on the EOR effect, a synthetic geological model is built.The reservoir scale is 10 9 10 9 2 with the dimensions of dx = dy = 10 m and dz = 5 m, as shown in Figure 9.The time step is 1 d.The reservoir temperature is 30°C and pressure is 20 MPa.The viscosity of oil is 20 mPaÁs and initial oil saturation is 0.8.The porosity is 0.3, and the permeability is 1 lm 2 and 4 lm 2 for the top and bottom layers, respectively.The ratio of vertical permeability to horizontal permeability is 0.1.The production rate is 20 m 3 /day, with an injection-production ratio of 1:1.The foam is injected when the water-cut reaches 90% (t = 2 000 d) and the slug size is 0.4 PV.The foaming agent concentration is 0.5 wt% and the injection gaswater ratio is 1:1 in formation conditions.The gas viscosity is 0.03 mPaÁs and the water viscosity is 1.0 mPaÁs.The difference between gas-water flooding and foam flooding is that the foaming agent concentration equals zero or not.Other model parameters are listed in Table 4.
Comparison between the computed value using the multifactor model, Kovscek's coreflood data, and the computed value using a population balance model

Comparing Foam Flooding with Water + Gas Flooding
Gas has a much smaller viscosity and lower density; therefore, if gas is used as the displacing fluid, gas channeling and override will occur.The major advantage of foam flooding is that the lamella of the foam can cut off the gas phase, effectively decrease the mobility ratio of gas to oil, and weaken the phenomena of gas channeling and override.Figure 10 shows the distribution of gas saturation at different times by foam flooding and water + gas flooding (the foam slug was injected between 2 000 d and 2 600 d).It is clear that, when t = 2 500 d, the gas phase area of L 1 is larger than L 2 for both water + gas flooding and foam flooding; the gas saturation of water + gas flooding is lower than foam flooding, but the gas phase area is larger for both L 1 and L 2 ; the amount of gas in L 1 is more than L 2 for gas-water flooding, which is the reverse for foam flooding.One explanation is that in transgressive depositional sequence reservoirs [50], the fractional flow of bottom strata is larger; but when the gas is injected into the reservoir, the gas is inclined to enter the top strata because of gravitational differentiation.At the same time, the gas viscosity is much lower than both water and oil; thus, the gas flows fast in both lateral and vertical directions.However, if the gas exists in the form of foam, the gas mobility will greatly decrease.Thus, the gas flows slowly and the gas saturation is larger for foam flooding.With lower permeability and   higher oil saturation in L 1 , the resistance factor is lower, which results in the gas phase area being larger in L 1 .When t = 2 750 d, the gas has already channeled to the producer along the L 1 , and the gas saturation is close to zero in the L 2 for water + gas flooding.Conversely, gas channeling does not appear and the gas saturation is still about 0.3 and exists in the form of foam in L 2 for foam flooding.The foam located in L 2 makes water divert into the lowerpermeability strata (L 1 ) and displaces more additional remaining oil than water + gas flooding.The development effects of different methods are shown in Figure 11.It is clear that the ultimate recovery order is foam flooding (48.8%) > water + gas flooding (46.7%) > water flooding (43.7%) and the flood-response time of water + gas flooding is earlier and shorter than foam flooding because of the densities of water, oil and gas.Due to gravitational differentiation, both top and bottom layers are well swept by gas and water, so the ultimate recovery of water + gas flooding is larger than water flooding.In the foam flooding process, some of the unfoamed gas flows in the top layer to sweep the remaining oil, and some of the gas in the form of foam flows in the bottom layer.As previously analyzed, the blocking ability of foam in the high-permeability layer is stronger than in the lowpermeability layer.As a result, the gas flows in the highpermeability layer the same as in the low-permeability layer, and more water enters the lower-permeability layer and consequently displaces more oil.Hence, foam can both effectively inhibit the gas override and establish a large flow resistance to realize the fluid diversion, which is favorable for enhancing oil recovery in heterogeneous reservoirs.

Sensitivity Analysis of Foam Flooding Enhancing
Oil Recovery In a lab study, foam flooding can enhance oil recovery by more than 20% in suitable conditions.Actually, most of the fields in China have carried out foam flooding and obtained satisfactory results [4], but the incremental oil recoveries are less than lab results due to their adverse conditions.Hence, studying the effects of reservoir and foam properties for choosing suitable reservoirs to perform foam flooding is very important.The permeability max-min ratio, ratio of vertical to horizontal permeability, depositional sequence, reservoir temperature, foam  quality and foaming agent concentration are very practical, common and important influencing factors in actual oilfields.Therefore, we performed a sensitivity analysis of these factors using the above heterogeneous reservoir and parameters.The parameters of the simulated projects are shown in Table 5.

Permeability Max-Min Ratio
The permeability max-min ratio is defined as the ratio of the highest permeability to the lowest permeability.
Because foam has the feature of "big block and not small block", if the reservoir has a suitable permeability maxmin ratio, the EOR effect will be superior.Figure 12 shows the ultimate and incremental oil recoveries for different permeability max-min ratios.It is shown that when the permeability max-min ratio is small, the incremental oil recovery increases sharply as the ratio increases; when it reaches 10, the oil recovery increment reaches a plateau.The reason for this is the lower the permeability max-min ratio will result in the closer the permeability of each stratum.From Equation (10), the foam resistance factor is a function of permeability.For a smaller permeability max-min ratio reservoir, after foam flooding, the water diverting into the low-permeability layers is lowered; for a larger permeability max-min ratio reservoir, the foam resistance factor is larger in the high-permeability layer and more water diverts into the low-permeability layer to sweep the remaining oil.Due to the limited blocking capacity of foam, the foam strength decreases if the permeability is too large.Therefore, the ultimate recovery is not very good for a reservoir with an extremely large permeability maxmin ratio.Consequently, foam flooding is suitable for reservoirs with a moderate permeability max-min ratio.

Ratio of Vertical to Horizontal Permeability
Override is a problem in gas flooding.In gas flooding, the ratio of vertical to horizontal permeability should be considered.Figure 13 shows the ultimate and incremental oil recoveries for different ratios of vertical to horizontal permeability.It is shown that as the ratio of vertical to horizontal permeability increases, the incremental oil recovery sharply decreases.When the ratio is bigger than 0.2, the incremental oil recovery decreases slowly and is only about 2%.The reason is that if the ratio of vertical to horizontal permeability is large, the gas gravitational variations will be more pronounced.As a result, all the injected gas overrides into the top layer, and no foam is generated in the bottom layer.The continued water diversion would not take place.Therefore, foam flooding is suitable for reservoirs with a lower ratio of vertical to horizontal permeability.
The depositional sequence is defined as the regulation by which the sand particles deposit in sequence to form rock strata.A transgressive depositional sequence is composed of several sand layers where the sand diameter declines from the bottom to the top, i.e. the permeability is reduced in the vertical profile from the bottom to the top; a regressive depositional sequence is just the reverse.The depositional  Oil recovery increment (%) Ultimate recovery and the increment vs ratio of vertical to horizontal permeability.
sequence has a significant influence on the gas override to affect the ultimate recovery.Figure 14 shows the development effects with different displacing modes and depositional sequences.Apparently, the oil recovery in a regressive depositional sequence reservoir is higher than that in a transgressive depositional sequence reservoir for all the three displacing modes.The water + gas flooding oil recovery is higher than water flooding.However, the incremental oil recovery in regressive depositional sequence reservoirs is lower than transgressive depositional sequence reservoirs.The foam flooding oil recovery is higher than water + gas flooding.However, the incremental oil recovery in a regressive depositional sequence reservoir is higher than in a transgressive depositional sequence reservoir.
For the rationale, in transgressive depositional sequence reservoirs, water is inclined to enter the high-permeability layer due to gravitational variations and lower flow resistance, which results in some remaining oil being confined within the lowpermeability layer; continued gas is inclined to override into the low-permeability layer under the gravitational variations and displaces the remaining oil.In the regressive depositional sequence reservoirs, due to gravitational variations and lower flow resistance, both the high-permeability and low-permeability layers are well swept by water, and the remaining oil is lowered in the high-permeability layer (top layer); the gas is still inclined to enter the top layer due to gravitational variations and lower flow resistance, but the EOR effect is not obvious.Comparing foam flooding with water + gas flooding, in the transgressive depositional sequence reservoirs, more gas flows in the low-permeability layers, and the amount of foam in high-permeability layers is less than regressive depositional sequence reservoirs, which results in less water diverting into lower-permeability layers after foam flooding.Conversely, in the regressive depositional sequence reservoirs, since more gas flows in the high-permeability layers, consequently, this results in more flow of foam in highpermeability layers as well.As a result, more water diverts into lower-permeability layers after foam flooding.

Reservoir Temperature
Figure 15 shows the variation of ultimate oil recovery with reservoir temperature.The ultimate oil recovery decreases as the temperature increases.When the temperature is lower than 100°C, the ultimate oil recovery varies very little; when the temperature is between 100 and 200°C, the ultimate oil recovery decreases sharply; when the temperature exceeds 200°C, the ultimate oil recovery of foam flooding is close to that of water + gas flooding.Therefore, the reservoir temperature should not be high, or temperature-tolerant foaming agents should be used for high-temperature reservoirs.

Foam Quality
Foam quality is an important factor determining the blocking capacity of the foam texture.Figure 16 shows the ultimate oil recovery of different foam qualities.Oil recovery (%) Comparison of the developments effects with different methods and depositional sequences.Temperature ( °C) Ultimate recovery vs reservoir temperature.
Evidently, as the foam quality increases, the ultimate oil recovery first increases and then decreases with an optimal value of 50%.This is due to the fact that a lower foam quality means a thicker lamella and a smaller bubble; under these conditions, the Jamin effect [51] is very small; conversely, a higher foam quality leads to a thinner lamella, a faster gas velocity and a bigger bubble; under these conditions, the bubble is more likely to collapse by shearing when it flows through the porous media.Therefore, a more stable bubble with a better Jamin effect can obtain a better EOR effect.Based on the relation between the foam quality and gas-water ratio, the optimal injection gas-water ratio should be chosen at the volume ratio of 1:1 under the reservoir conditions.

Foaming Agent Concentration
Figure 17 shows the variation of ultimate oil recovery with foaming agent concentration.Noticeably, the ultimate oil recovery reaches a plateau at 0.6 wt% after increasing along with the foaming agent concentration.
Based on the parameters used in this case, the CMC is about 0.5 wt%.Optimal concentration is a little higher than the CMC since some of the foaming agent is adsorbed on the rock.So, the foaming agent concentration of foam flooding should be a little higher than the CMC of the surfactant.

CONCLUSIONS
A novel model of multi-factor foam flooding is established.
The IMPES method in conjunction with a fourth-order Runge-Kutta method were used to solve the governing equation.The validation of this model was confirmed by matching the experimental data.Finally, the effects of several parameters including the permeability max-min ratio, ratio of vertical to horizontal permeability, depositional sequence, foam quality, foaming agent concentration, and reservoir temperature on oil recovery were studied.
The following are the main conclusions: 1) A foam resistance factor model considering the influences of permeability, foaming agent concentration, foam quality, temperature and oil saturation is obtained.Then, taking the gas velocity as the foam generation condition and water saturation as the foam collapse condition, the gas mobility model was achieved; 2) Foam can effectively weaken the phenomena of gas override in the vertical direction due to gravitational variations and gas channeling in a horizontal direction caused by large gas mobility; 3) The mechanism of water + gas flooding enhanced oil recovery is that due to the gravitational variations, gas increases the swept area of the injected fluid; and the mechanisms of foam flooding are both the gravitational variations and the water diversion; 4) With a smaller permeability max-min ratio, the incremental oil recovery increases sharply and reaches a plateau at the ratio of 10.Foam flooding is suitable for reservoirs with a moderate permeability max-min ratio (k r = 5-10); 5) As the ratio of vertical to horizontal permeability increases, the incremental oil recovery sharply decreases.When the ratio is bigger than 0.2, the incremental oil recovery decreases slowly, and is only about 2%.Foam flooding is appropriate for reservoirs whose ratio of vertical to horizontal permeability is lower than 0.2; Ultimate recovery vs foam quality.
6) The oil recovery in regressive depositional sequence reservoirs is higher than that of transgressive depositional sequence reservoirs for water flooding, gaswater flooding and foam flooding.The gas-water flooding incremental oil recovery for water flooding in regressive depositional sequence reservoirs is less than that in transgressive depositional sequence reservoirs, but the foam flooding increment for water + gas flooding in regressive depositional sequence reservoirs is more than that in transgressive depositional sequence reservoirs; 7) The ultimate recovery decreases as the temperature increases.The reservoir temperature should not be high for foam flooding; otherwise, temperature-tolerant foaming agents should be used for high-temperature reservoirs; 8) As the foam quality increases, the ultimate oil recovery first increases and then decreases, and the optimal value is 50%.The optimal injection for the gas-water ratio should be chosen at the volume ratio of 1:1 under the reservoir conditions; 9) Due to adsorption of the surfactant, the foaming agent concentration during foam flooding should be a little higher than the CMC of the surfactant.

Figure 1 Figure 2
Figure 1Comparison between experiments and the model of the resistance factor for different surfactant concentrations.

Figure 4
Figure 4 Comparison of experiments and the model of the resistance factor for different temperatures.

Figure 5 ComparisonFigure 3
Figure 5Comparison of experiments and the model of the resistance factor for different oil saturation ratios.

Figure 6
Figure 6 Validation of the multi-factor gas mobility model using experimental data.a) Surfactant #1, b) Surfactant #2.

Figure 8 ComparisonFigure 10 Distribution
Figure 8Comparison between computed and experimental pressure drop for foam flooding in oil-bearing conditions.

Figure 11 Development
Figure 11 Development effects of different methods.

Figure 12 Ultimate
Figure 12Ultimate recovery and increment vs permeability max-min ratio.

Figure 17 Ultimate
Figure 17Ultimate recovery vs foaming agent concentration.

TABLE 3
Parameter values for simulations of the foam flooding experiment

TABLE 5
Parameters of simulated projects for sensitivity analysis